UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Mechanical and Thermal Properties of Ceramics

Proceedings of a Symposium

Held at Gaithersburg, Maryland

April 1-2, 1968

Edited by

J. B. Wachtman, Jr. Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

Sponsored by

The American Ceramic Society

The American Society for Testing and Materials

and

The National Bureau of Standards

,^ , National Bureau of Standards, Special Publication 303

Issued May 1969

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $4. 25
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Abstract

The proceedings of the Symposium on Mechanical and Thermal Properties of Ceramics,

held at the National Bureau of Standards in Gaithersburg, Maryland, on April 1-2, 1968, are presented. The symposium was cosponsored by the American Ceramic Society, the

American Society for Testing and Materials, and the National Bureau of Standards. Attention was concentrated upon properties primarily of structural and/or high temperature use

including melting points, thermal expansion, thermal conductivity, thermal radiation

properties, elasticity, viscoelasticity, inelastic deformation, and fracture. The dependence

of these properties upon composition and microstructure was surveyed and the importance

of controlling these features of character to insure reliability of ceramics was emphasized.

The procedures by which American ceramists formulate standards and specifications to

assist in insuring reliability were summarized.

Key Words: Ceramics; elasticity; fracture; inelastic deformation; melting point;

standards; thermal conductivity; thermal expansion; thermal radiation;

viscoelasticity.

Library of Congress Catalog Card Number: 68-61960

ii
Foreword

A major responsibility of the NBS Institute for Materials Research is to insure the

availability of those basic data on the properties of materials which are urgently needed to

meet the requirements of the Nation's scientists and engineers. To obtain reliable data of

this kind not only must the experimental procedures be tailored to permit the required measurement accuracies, and the results correctly analyzed, but also the sample materials must be

sufficiently uniform and well characterized to permit measurements amenable to meaningful

interpretation and practical use. The reliability of data needed for structural or high-temperature use of ceramics is an

important example. Sensitivity of strength data to test conditions and complications introduced

by high measurement temperatures plague the measurements and make experimental procedures

a prime concern. Also, several important properties, including thermal conductivity, inelastic

deformation, and strength, are sensitive to small variations in microstructure and composition.

To provide a basis for understanding the dependence of ceramic properties on various characterizing features, a Symposium on Mechanical and Thermal Properties of Ceramics was held

at the National Bureau of Standards laboratories in Gaithersburg, Md., in April 1968. The

meeting was jointly sponsored by the American Ceramic Society, the American Society for

Testing and Materials, and NBS.

The National Bureau of Standards is pleased to have the opportunity of making the

papers presented at the Symposuim available in this publication. In this way we hope to assist

those attending the conference in optimizing the use of ceramics, to make the conference

results available to a wider audience, and to show appreciation for the essential contributions

of the participants and the cosponsor organizations.

A. V. Astin, Director

m
Introduction

This is a collection of the proceedings of the symposium, Mechanical and Thermal

Properties of Ceramics, held at the National Bureau of Standards laboratories at Gaithersburg,

Maryland, on April 1-2, 1968. Approximately 210 scientists and engineers participated.

The proper selection and effective use of materials is one of the fundamental factors upon

which advances in technology depend. Ceramic materials have a long history of effective use in

many applications for which their hardness, stiffness, strength, resistance to corrosion, thermal

and electrical insulating qualities, high melting points, optical properties, semiconducting

properties, dielectric properties, magnetic properties, piezoelectric properties, nuclear properties, or other special qualities are appropriate. The ceramic industry in the United States

continues to grow and is approaching the ten billion dollar a year level. Every type of material,

including ceramics, is challenged, however, both by competition from other materials in its

traditional areas of use and by the requirements for improved properties needed in new

applications.

The development of new materials and the improvement of properties depend partly

upon an empirical approach in which the composition and manufacturing procedure are systematically varied and correlation with resulting properties is attempted. This empirical

method will probably continue indefinitely to be an important part of materials development

and need not conflict with the use of materials science. The latter is helpful in predicting areas where improvement appears possible, choosing the parameters to be varied, designing an

efficient processing—property study, and interpreting the results.

Ceramics are frequently manufactured by a heating process which appears deceptively

simple but which usually involves a number of competing reactions and transport processes

occurring at different rates. Optimizing a particular property requires a starting composition

and heat treatment which produces the final composition and microstructure (usually denoted

by the word "character") needed for that property. In principle one should begin with the

properties needed, determine the character required from an understanding of the propertycharacter relationship, and design the processing from an understanding of the chemistry and

mass transport which are involved. In practice the understanding of both areas is limited so that empirical procedures, guided by some degree of fundamental understanding, are generally

used.

The current status of ceramic processing was surveyed by the Materials Advisory Board

ad hoc Committee on Ceramics Processing, chaired by Professor J. A. Pask, and their conclusions are available in Ceramic Processing, National Academy of Sciences Publication 1576

(1968). The importance of an improved understanding and use of the property-character

relationship by users of ceramics as well as producers is emphasized. In particular, the user should understand that many properties depend on other aspects of character in addition to

gross chemical composition. The engineer who specifies that a part be made of "alumina" is leaving the strength very poorly specified; commercial alumina ceramics range in strength

from values below 2000 psi for certain refractories to values approaching 100,000 psi for certain

tool bits and other special purpose ceramics. The present symposium is aimed primarily at

the users of ceramics but it is hoped that producers will also find a systematic review of property-character relationships useful.

Properties of ceramics in general form too broad and diverse a subject to be effectively

treated in this symposium; the properties selected for treatment are those related to mechanical and thermal behavior. Ceramics are frequently used to exploit their thermal properties

and also as structural members in which case their deformation and ultimate fracture under

load are of obvious importance. In many applications ceramics are used to exploit other properties, but mechanical and thermal properties are usually still important in determining the

size, weight, and cost of the parts involved. Ceramics are, of course, particularly well suited

to high temperature applications but the conditions of high temperature use frequently create

thermal gradients with accompanying thermal stress. The symposium accordingly includes

IV
thermal factors not only because of their direct engineering importance, but also because of

their role in limiting load-bearing capacity at high temperatures; i.e., the melting points,

thermal expansion, thermal conductivity, and thermal radiation properties are surveyed.

The practical utilization of high-performance ceramics requires more than their successful

production on a laboratory scale and favorable economic factors. Such ceramics must be reliable

in production, that is, their properties must be uniform within each piece and reproducible

from piece to piece. Small lots of special ceramics frequently show significant variation in

structure-sensitive properties including strength. This variation has raised questions con- cerning the permissible stress levels in structural applications and is currently inspiring research

aimed at determining to what extent this variability in strength results from variability in

processing, variability in testing, or from the intrinsic nature of brittle materials. This research

aims at the ultimate establishment of a degree of processing control, testing accuracy, and

design understanding to permit ceramics to be used routinely and reliably at high stress levels.

Such use depends on acceptance of testing standards by the producers and users of ceramics.

This symposium accordingly opens with a survey of the activities of the American Ceramic

Society and the American Society for Testing and Materials, two of the principal institutions

through which American ceramists communicate and formulate standards. The needs for

high-performance ceramics are next discussed and the importance of reliability is stressed.

The thermal properties are then treated, being taken first because of their role in determining

stresses resulting from heat flow. Finally, the mechanical properties which determine response

to stress, whether of thermal or mechanical origin, are discussed. Throughout the discussion

of properties the emphasis is placed on the property-character relationship; i.e., on the features

of composition and microstructure upon which the properties depend. Some property tabulations are given to illustrate property ranges, but exhaustive tabulation of numerical values

has not been attempted. References are given to many existing tabulations which, excellent as some are, need to be used carefully with an understanding of how variation in character of

the ceramic under consideration can cause properties to vary from the handbook values.

Each of the following papers was written as a separate entity and can be read without

reference to the others. A few comments on some of the highlights of the papers and on the

interrelation of the various properties should, however, give a helpful perspective on the general subject of the symposium. An important point concerning high-performance ceramics is made in the paper on the American Ceramic Society and Ceramic Materials Technology.

The example of ceramic insulators for spark plugs is quoted to make the point that when large

volume requirements exist ceramics can be and are routinely produced at modest cost to

reliably meet high-performance specifications. This example raises the question of how well

expertise developed in producing a specialized ceramic can be transferred to the production

of other ceramics to meet somewhat different high-performance requirements. A purely empirical production process developed around a specific raw material and product may not be a good guide in developing a new product; understanding of the property-character relationship

and of the process of character development during processing are needed. Standard test

methods for both character determination and property measurement are therefore important

to aid in developing this understanding in relation to commercial manufacturing of ceramics.

The process by which United States Standards are developed and the role of the American

Ceramic Society and the American Society for Testing and Materials in the development of

standards for ceramics is accordingly described.

Commercial development of new ceramics will occur only if justified by the demand; the

paper on "Properties of Ceramics for Structural and/or High Temperature Use; Need for

Control, Measurement and Compilation" discusses new markets open to ceramics and emphasizes the requirement of reliability and the consequent need for understanding the characterproperty relationship and for good testing procedures.

The discussion of thermal properties begins with a paper on melting points. This might

incorrectly be thought to be a simple subject but the engineer unfamiliar with phase equilibria

who attempts to use ceramics at high temperatures may be in for a surprise. Thus, for example,

Ti02 , melting at about 1840 °C, and CaO, melting at about 2570 °C, might appear to be safe

for use at 1500 °C. Anyone who has tried using CaO and Ti02 in contact with each other at

1500 °C or many similar combinations of ceramics above their eutectic temperature is referred

to the paper on melting points for an explanation of what happened to his ceramic parts.

v
One of the factors limiting use of ceramics at high temperature, as mentioned previously,

is the development of stress accompanying thermal gradients. Thermal shock parameters

involving elastic moduli and thermal expansion have been used as a rough guide to the "thermal

shock resistance" of materials, but it has long been recognized that a complete analysis of

the heat flow and temperature distribution for each particular device is needed. High-speed

computers make this feasible so that modern designs are increasing based on an analysis which

requires the thermal radiation properties (emissivity, etc.) and thermal conductivity to calculate the temperature distribution and then requires the thermal expansion and elastic moduli

(plus equations giving plastic response if it is involved) in order to calculate the stress distribution.

Fracture of polycrystalline ceramics as a result of stress of thermal or mechanical origin

usually occurs suddenly and completely, at least at temperatures below the range of appreciable

plastic deformation. The need to understand the characteristics of brittle fracture and to

design for the use of ceramics with this in mind is apparent. The fracture paper in this symposium deals with this subject from the point of view of fracture mechanics as a branch of

continuum mechanics as well as from the point of view of the effects of defects and environment. Finally, the subject of mechanical testing is treated in a separate paper because of its

difficulty and importance.

J. B. Wachtman, Jr.

Acknowledgment

The editor wishes to thank the following for their assistance in organizing and holding

the symposium:

R. T. Cook, National Bureau of Standards

P. M. Corbett, Glidden-Durkee Division, SCM Corp.

J. J. Diamond, National Bureau of Standards

G. T. Leighty, National Bureau of Standards

J. L. Pentecost, W. Pv. Grace & Co., Washington Research Center

D. W. Robertson, General Refractories Company

VI
Contents

Page

Foreword Ill

Introduction IV

The American Ceramic Society and Ceramic Materials Technology 1

J. S. Owens

ASTM: Its Function and Philosophy 5

F. J. Mardulier

ASTM Activities in Ceramics 9

J. C. Richmond

Properties of Ceramics for Structural and/or High Temperature Use: Need for

Control, Measurement, and Compilation 13

C. S. Bersch

Melting Points of Ceramics 19

S. J. Schneider

Thermal Expansion of Ceramics 41

R. K. Kirby

Thermal Conductivity of Ceramics 63

D. R. Flynn

Thermal Radiation Properties of Ceramic Materials 125

J. C. Richmond

Elastic Deformation of Ceramics and Other Refractory Materials 139

J. B. Wachtman, Jr.

Viscoelasticity of Glasses 169

P. B. de Macedo

Inelastic Deformation of Oxide Ceramics 189

R. M. Spriggs

Plastic Deformation of Carbides 201

G. E. Hollox

Fracture of Ceramics 217

S. M. Wiederhorn

Strength Testing of Ceramics—A Survey 243

L. Mordfin and M. J. Kerper

Subject Index 263

vn

The American Ceramic Society and Ceramic Materials Technology

James S. Owens

President, The American Ceramic Society

Champion Spark Plug Company, Ceramic Division, Detroit, Michigan 48211

The ceramic industry in the United States, through the continued growth of the use of

traditional ceramics and the development of many new products and applications, has a

sales volume approaching ten billion dollars a year. The American Ceramic Society, which was founded 69 years ago and represents all segments of the industry, has grown into the

world's largest ceramic technical organization and is still growing at the rate of 50 percent

per decade. It has diversified into ten Divisions to serve the economic, technological and

scientific interests associated with ceramics. High performance ceramic materials with

unique properties inherently dependent upon microstructure have been developed for

specific applications that require either limited quantities or high volume mass production.

Careful production control to insure a high level of performance is receiving close attention.

Key Words: Ceramics; ceramic industry; American Ceramic Society; microstructure;

properties; special ceramics.

It is an honor to be invited to take part with

you in this significant Symposium on Mechanical

and Thermal Properties of Ceramics, and it is a

pleasure to do so. Since its founding, The American Ceramic Society has been extremely interested in encouraging and supporting the continuation of the technical education of the members of

the ceramic industry and of those associated with

it. This meeting is an excellent example of such

continuing education. Our Society is happy and

proud to sponsor this Symposium through its

Baltimore-Washington Section in cooperation with

the Middle Atlantic District of the American

Society for Testing and Materials and the

National Bureau of Standards.

It is particularly fitting that this program

should be held here at the National Bureau of

Standards since the Bureau has done outstanding

research in ceramics and this Symposium was

organized by J. B. Wachtman, Jr., who is Chief

of the Physical Properties Section of the Bureau

and is an officer of the Baltimore-Washington

Section of The American Ceramic Society.

The ceramic industry is a large one with

estimated sales this year of about ten billion

dollars. Ceramics include the inorganic nonmetallic solid materials. They have great importance

for a wide variety of industrial, commercial and

consumer uses, and to our Government for both

national defense and civilian purposes. Their

broad range of uses includes, for example, refractory ceramics for application at very high temperatures, in nuclear and aerospace technology, and

in industrial furnaces and kilns; electronic ceramics in miniaturized microelectronic circuitry that

helps to make possible new generations of computers and other complex, yet compact, electronic

equipment; spark plug and jet igniter insulators;

ceramic protective coatings for metals; brick,

glass, and tile structural and decorative ceramics;

glass containers; and dinnerware.

Since the success of many applications of

ceramic materials depends on their mechanical

and thermal properties, this Symposium is an

important and timely one. The American Ceramic Society had its inception

seventy years ago in February 1898 at the 12th

Annual Convention of the National Brick Manufacturers' Association in Pittsburgh where the

first scientific paper of its kind was presented

before that Association. As a result, there developed in the next two days the idea for a separate society for the presentation of technical

papers on ceramics that culminated a year later on

February 6, 1899 in the official chartering of The

American Ceramic Society in Columbus, Ohio,

with 15 charter members. These included General

Edward Orton, Jr., who was Secretary of the

Society until 1917, and A. V. Bleininger who was

associated with the National Bureau of Standards

for several years.

Our Society has grown from those very small

beginnings into the largest ceramic technical

organization in the world with just under 10,000

members and subscribers. It is dynamic and progressive and has developed to its present significant position because it was needed to aid in the

development and the improvement of the ceramic

profession and industry, and it has made a continuing, conscientious, successful effort to fill this

1
need. As a result, the Society has grown tremendously and is still growing in membership at the

rate of over 50 percent per decade. However,

many of the activities of our members are increasing twice that rapidly. Thus, the number of

technical papers published annually, the number

of papers presented at the Annual Meeting and

the fall Divisional Meetings, the attendance at

the fall meetings, and the number of abstracts

published yearly has doubled in the past decade.

The primary services of the Society are the

holding of meetings for the sharing of technical

ceramic information by the presentation and

discussion of papers and the publication of these

papers for the wider dissemination of the information. These are extremely important services,

but, in addition, the Society provides other

significant services for its members and for the

whole ceramic industry and the public generally.

These include authoritative, specialized publications on selected, important subjects such as Phase Diagrams for Ceramists, Engineering Properties of Selected Ceramic Materials, Large Scale

Phase Diagrams of Oxide Systems, and Symposium

on Nucleation and Crystallization in Glasses and

Melts; the encouragement and sponsorship of

continuing education programs such as this

Symposium and the programs carried out under

the auspices of other Sections of the Society;

investigation of the needs of the ceramic industry

for technical information services and its possible

sequel for the determination of optimum methods

to provide the services found to be needed; liaison

through the National Institute of Ceramic Engineers with other engineering disciplines; cooperation with related scientific and technical societies

in this country and with ceramic organizations

abroad on matters of common interest; and

cooperation with governmental agencies and

public organizations such as the National Research Council through members of the Society

serving on advisory panels, boards and committees

of these organizations in the public interest. I

might also mention that many members of our

Society serve on appropriate committees of

ASTM to help advance the work of that distinguished organization for the public welfare.

The American Ceramic Society is international

in scope with the staff housed in a functional

headquarters building in Columbus, Ohio. The

Society holds an annual meeting each spring at

which technical programs are presented in the

fields of interest of all of its Divisions and programs are also presented by its Classes. The

Society is organized functionally into two Classes

and ten Divisions. The Classes are the National

Institute of Ceramic Engineers which provides

coordination between our Society and other engineering fields and the Ceramic Educational

Council which is concerned primarily with educational policies, college curricula, and other similar

educational aspects of ceramics. The Divisions

represent ten different categories of interest in

ceramics of the members of the Society namely

Basic Science, Ceramic-Metal Systems, Design,

Electronics, Glass, Materials and Equipment,

Nuclear, Refractories, Structural Clay Products,

and White Wares. Each Division, with one ex- ception, holds a fall meeting for the presentation

of technical papers of interest to its members.

The Society is also organized regionally by means

of thirty Sections located throughout the United

States which carry out its activities on a local or

regional basis. The Sections hold evening meetings at regular intervals; some also hold meetings

of one to three days duration, from once to three

times a year, at which papers are presented on

different technical ceramic subjects. Some Sections

carry on continuing education programs, of

which this Symposium is a good example.

We in The American Ceramic Society are proud

to be able to contribute in a measurable way to the

technical advancement of our country. We invite

you who may be concerned with ceramic materials,

but who are not members of the Society, to join

with us in this progress.

I should like now to discuss briefly some general

aspects of the microstrueture of ceramic materials

in view of its influence on their properties. Polycrystalline ceramic systems are quite complex,

usually consisting of two or more phases, pores of

varying size and size distribution and of varying

shape and shape distribution, grain boundaries,

grains of varying size and size distribution that

may range from quite broad to very narrow,

distribution of impurities within grains and in

grain boundaries, partitioning of impurities between phases, and a crystallographic orientation

that may range from a random one to an ex- tremely well defined one. The arrangement of

such internal features on a microscopic scale

constitutes the microstructure. Sub-grain features,

such as dislocations, complicate the microstructure

even further. While shape, size, surface condition,

and other physical factors of a polycrystalline

ceramic article of a given composition may have

definite effects, the microstructure will, to a considerable extent, influence its physical, mechanical, thermal, and electrical properties.

An understanding of the effect of specific

microstructural characteristics on a property or

properties of a ceramic is often an inducement to

endeavor to change these characteristics. Thus

Coble [1] achieved translucency and enhanced

strength in a high alumina ceramic, later called

Lucalox, by the elimination of porosity and the

control of grain growth by a special sintering

process. Spriggs [2] has reported that a transverse

bend strength of up to 130,000 psi has been

obtained for dense, submicron grain size alumina

prepared by vacuum pressure sintering to maintain

minimum grain size and porosity. These developments were made possible by the application of

the knowledge that porosity and large grain size

weaken brittle materials and that porosity in- creases the reflection and refraction of light. These
microstructural characteristics could readily be

observed, but there are other characteristics that

are not so easily measurable. Insley and Barczak

[3] demonstrated that polycrystalline alumina

ceramics could be strengthened by heat treatment

that causes a re-solution of phases present at

grain boundaries that have precipitated from

solution in the grains. This thermal conditioning

illustrates the dependence of the strength of a ceramic on a rather subtle change of its microstructure caused by heat treatment. Jorgensen and

Westbrook [4] believe that solute segregation at

grain boundaries is a significant mechanism for

control of grain growth and porosity; they have

used the results of such refined techniques as autoradiography and microhardness testing in

support of their hypothesis.

The announcement of this Symposium pointed

out that the control and reproducibility of many

desirable physical properties of a given ceramic

material depend upon the characterization and

control of its microstructure and composition. It

has been demonstrated, for example, that the

mechanical strength of a particular type of ceramic

will depend markedly on both grain size and

porosity and that the thermal conductivity can be strongly affected by composition and also by

compound formation. Riddle [5] reported experimental results which show that an intermediate

compound in a binary system will have a lower

conductivity than either end member.

Any type of ceramic with specifically required

properties will require careful control of its production to assure its optimum behavior in the

environment to which it is to be subjected. The

specifications for a ceramic do not usually include

a specification of the microstructure although such

properties as nominal chemical composition, specific gravity, flexural strength, dielectric constant,

and thermal conductivity, for example, may be

specified. However, almost all of such properties are dependent to some extent on the microstructure, and the specification of microstructure

might be helpful for the improvement of the

quality and uniformity of ceramic materials. In

this regard, some more effort might be directed

toward more rapid and efficient methods of preparing sections for microstructural examination

and determination. We are able to determine the

composition of materials by instrumental techniques in a matter of minutes. However, the

composition alone will not adequately characterize a material for all purposes and a rapid method of

microstructural analysis would undoubtedly be

of great value for process control.

Some of you may be concerned with ceramic

materials that have rather unique properties for

highly specialized applications and which may be

required in only relatively small quantities. The

development of such materials may be very time

consuming and costly, and their subsequent

small scale production may likewise be expensive.

It should, however, be realized that there are other ceramic materials that have the excellent

properties needed to meet equally stringent re- quirements but which are used in much larger

volume and have much lower production costs. One example would be the ceramic insulators for

spark plugs. These are necessarily made of highly

refined, carefully developed and manufactured

ceramics that must withstand extremely stringent conditions in use. These ceramic insulators

must have excellent electrical resistance over a broad temperature range, high dielectric strength,

excellent resistance to thermal shock, high mechanical strength, good thermal conductivity, great

hardness and resistance to abrasion, and excellent

resistance to chemical attack by combustion

products. The development of a ceramic with a

suitable balance of these electrical, thermal,

mechanical, and chemical properties and the

development of well controlled processes to manufacture insulators from it require a great amount

of intensive research and engineering work. However, the development costs may be distributed over a large volume of product so that the special

ceramic insulators with carefully specified and

controlled properties can be produced at moderate

cost in the large volume required.

The technical papers that will be presented in

the program that has been arranged for this

Symposium should be of great benefit to all of us who have the responsibility of fundamentally

studying, developing, endeavoring to improve,

producing, testing, specifying, or using ceramic

materials.

References

[1] Coble, R. L., Sintering crystalline solids: I, Intermediate and final state diffusion models, J. Appl. Phys.,

32, 787 (1961).

Coble, R. L., Sintering crystalline solids: II, Experimental test of diffusion models in powder compacts,

ibid., 32, 793 (1961).

[2] Spriggs, R. M., Chapter 7, Microstructure and its in- fluence on strength, in Strengthening Mechanisms,

Metals and Ceramics (Syracuse University Press,

Syracuse, New York, 1966).

[3] Insley, R. H., and Barczak, V. J., Thermal conditioning

of polycrystalline alumina ceramics, J. Am. Ceram.

Soc, 47, 1 (1964).

[4] Jorgensen, P. J., and Westbrook, J. H., Role of solute

segregation at grain boundaries during the final- stage sintering of alumina, J. Am. Ceram. Soc. 47,

332 (1964).

[5] Riddle, F. H., Ceramic spark-plug insulators, J. Am.

Ceram. Soc. 32, 333 (1949).

3

ASTM—Its Function and Philosophy

Francis J. Mardulier, President, ASTM

Manager, Rock Products Chemicals, Dewey and Almy Chemical Division, W. R. Grace and Co.,

Cambridge, Mass. 02140

The American Society for Testing and Materials is a nonprofit privately operated

interdisciplinary materials-oriented society which specializes in standards rather than a

discipline-oriented society. ASTM provides the meeting ground for the technically competent engineers and scientists who comprise the technical committees which develop the

realistic compromises, known as standards, which govern our economic life. Its membership

comes from and is supported by industrial companies, government bureaus, educational

institutions, testing laboratories, etc., all of whom are vitally interested in the voluntary

standardization process.

Key Words: Compromises; consensus; materials; standards; technology; voluntary

standardization process.

It is a privilege to be here to tell you about

ASTM and the part it plays in voluntary standardization and its relation to the other organizations represented here in this forum. I welcome

particularly this opportunity to discuss ASTM's

principles of operation. We are grateful indeed

to the National Bureau of Standards for providing

this second opportunity to join with them and

the American Ceramic Society in a technical

conference.

Founded in 1898 as the American Section of

the old International Association for Testing and

Materials, ASTM's original charter, dated 1902,

stated its purpose to be "the promotion of the

knowledge of materials of engineering, and the

standardization of specifications and the methods

of testing." Inherent in this statement of purpose

is ASTM's fundamental interest in research and

education as well as its standardization work.

ASTM's specialty, therefore, is materials research

and standards, although it has formulated nonmaterials standards on request when no other

organization was available to fill an existing need.

As a result of its research and standards development activities, ASTM has achieved worldwide

recognition in the field of standard methods of

testing, whether applied to materials, products, or

assemblies.

ASTM, like the American Ceramic Society, is not a discipline-oriented professional society, like

ASCE, AIA, AIME, ASME, etc. Instead, because

it is materials oriented and its specialization is standards, it must be considered an interdisciplinary society in which the standardization

interests of its members predominate. Its membership comes from and is supported by industrial

companies, governmental bureaus, educational

institutions, testing laboratories, etc., all of whom

are vitally interested in the voluntary standardization process. ASTM provides the meeting

ground for the technically competent engineers

and scientists who comprise the technical committees which develop the realistic compromises,

known as standards, which govern our economic

life.Standards are necessarily compromises—compromises among the many conflicting producer and

nonproducer interests involved. Each participant

in the standardizing process wants a standard

that helps him but the self-interests of these

of these participating individuals or organizational representatives are necessarily diverse. The

standardization process thus must involve concessions by all the interested parties to the extent

needed to achieve an acceptable compromise which

protects all and seriously injures no one. Such compromises—not of the technical truths or principles involved but rather of the quantitative restrictions needed for the maintenance of

reasonable quality levels—represents a realistic

evaluation of the relationship between quality and

cost. Highest quality and lowest cost to the con- sumer are not compatible. However, with an agreed

upon quality level acceptable to the producer,

user, and general interests involved, goods may be

mass produced utilizing the most modern technologies and processes to provide the consumer

with standard quality at minimum cost.

I can assure you, ladies and gentlemen, that

the ASTM procedures involved in the development of standards by our technical committees

do guarantee a consensus of the parties involved,

and, at the same time, protect the interests of

minorities. In addition, these procedures require

that the representatives of the various interests

involved be technically competent and support

5
actively the interests they represent to the end

that the resulting standard be an effective,

realistic compromise of all the interests involved.

The procedural details whereby this is accomplished will be given to you by Mr. Richmond, who

follows me.

Reference to the word "consensus" brings to

light one of the probably unique features of our

technical committee procedures. Webster's

Seventh New Collegiate Dictionary defines

"consensus" as "general agreement—unariimity."

To me there is a gap of about 49 percentage

points between these two definitions, since the

first can be taken to mean majority agreement,

while the second definitely calls for 100 percent

agreement. In ASTM balloting on proposed or

revised standards, a consensus of the magnitude of

95 percent or more is usually achieved, and even

then it is required that negative votes be carefully considered and resolved if possible. A large

majority of ASTM standards are, as a result,

unanimously approved and accepted.

However, consensus of the committee membership would be meaningless if any one interest

particularly a producer interest—predominated.

For this reason, each ASTM technical committee

is required to maintain a balance of interest

between the producer and nonproducer interests

so that the producer voting membership never

exceeds 50 percent. It may be less, but too much

less is not desired either. Unfortunately, for a variety of reasons—mostly financial—it is much

more difficult to get technically competent nonproducers to assist in the preparation of ASTM

standards. We are particularly grateful, therefore,

to the General Services Administration, the

National Bureau of Standards, the U.S. Bureau

of Reclamation, the U.S. Corps of Engineers

and many other governmental agencies for their

active participation in our standardization work

as representatives of consumer interests. I might

mention in passing that the National Bureau of

Standards whose primary interest is the standardization of units of measurement has one or more

representatives on a majority of ASTM's technical

committees and contributes significantly to the

technical soundness of ASTM standards.

We would similarly welcome the greater participation by consumer and general interest representatives from other areas of our free-enterprise

economy so that its present and future needs will

be made known and provided for in terms of new

standards or revisions of old standards. Technological change is impossible without the development of appropriate standards. Consumers and

industry use ASTM standards extensively by re- ferring directly or indirectly to many ASTM

standards in purchasing, design, and construction

specifications. Yet consumers and even industry

in some cases participate in ASTM standards work

in far less proportion than established by their

standards usage and needs. To maintain ASTM's

balance of interest in its technical committees, we

need the active participation of all interested

parties, but more particularly that of consumers. ASTM's scrupulous observance of its democratic principles in its standardizing procedure

have caused ASTM standards not only to receive

national and international acceptance, but has

legal implications of tremendous importance to

industry and our free enterprise system. It also

guarantees the continuation of industry's direct

participation in standards development and of

government's participation as a representative of

the consuming public. Strict adherence to ASTM

operating procedures make it possible for technical

committee members to articulate their organization's special interests without fear of antitrust

violation, provided, of course, there is adequate

representation of the parties at interest on the

committee. This emphasizes the balance-of-interest

requirement in the membership of ASTM committees.

As an interesting side effect of its requirement

that there be balance of interest in committee

membership, ASTM's standards when submitted

through USASI for acceptance as an ISO standard

are generally well received by the other members

of the ISO committees having jurisdiction over

the materials area in question. Thus, ASTM

standards proposed for such acceptance are frequently converted into ISO standards with little or no revision of technical content and requirements. For example, the bulk of the ISO standards

on plastics are essentially ASTM standards as

originally formulated by ASTM Committee D-20.

The significance of this fact to the U.S. plastics

industry which is interested in export business is inescapable.

It seems appropriate at this point to digress in

order to describe the relationship between ASTM

and the USA Standards Institute (USASI). USASI

is a private organization, of which the former

American Standards Association was the prototype, whose function is that of a USA standards

coordinator and clearing house of information on

standards. It is the official USA member of those

international organizations which establish international standards. Except for the unusual circumstances when existing standards organizations

are unable or unwilling to provide needed standards, USASI does not itself actively participate

in standards development and then only through

the medium of a USA Standards Committee

sponsored or administered by some outside organization or a USASI member body.

ASTM is such a member body, competent and

willing to develop standards where needed. ASTM

has formulated more standards than any other

organization, private or public. It has in fact

provided approximately 40 percent of the standards which to date have been processed through

USASI and published as USA Standards. Its

only requirement for such standards development

activity is that there be a clearly defined standards

need and to this end ASTM has issued an invita6
tion to all organizations, be they companies,

governmental bureaus, educational institutions,

etc., to bring to our attention standards needs

where they exist, and join with us in formulating

the appropriate standards.

I hope that in this brief presentation I have

given you a clear picture of the basic principles

governing ASTM procedures, and the importance

of meticulous observance of its democratic procedures to the continuation of voluntary standardization in the U.S.A. I hope I have made clear

ASTM's relationship to USASI as one of the

latter's most important member bodies and overseas representatives. I would be remiss indeed if I did not emphasize that USASI is also dedicated

to the same principles in the development of

standards and as U.S. coordinator of standards is in a position to see that all USA standards

similarly conform.

In closing, I should like to call to your attention

a final and most important requirement of a good

standard—it should be realistic and legally defendable. A unilaterally developed standard may be

technically correct but legally and politically

unacceptable if it does not recognize the needs

and rights and represent the participative effort

of all the parties at interest. This is a fact of life which becomes increasingly clear as the needs and

rights of the consumer are daily accorded growing

recognition.

Two years ago at a similar symposium I told

you of a decision which had been made in a

restraint of trade case which ASTM had instigated

to clear itself of any stigma which might exist as a result of its involvement in a prior criminal suit

brought against three USA manufacturers of

asbestos-cement pipe for alleged antitrust violation. One of Judge Van Deusen's findings of fact

in the U. S. District Court for Eastern Pennsylvania sums up far better than I can the basis on

which ASTM won its case. And so, in closing, I take the liberty of quoting these findings once more:

"The technically qualified balanced membership of ASTM committees, and the democratic procedure governing their operation,

make it likely that the results reached by them

will be scientifically sound and will represent

the general interest".

"Because of the heavy reliance of federal,

state and municipal governments on ASTM

for specifications, the society may be regarded

as an essential arm or branch of the government and its acts may be entitled to the

immunity from antitrust laws accorded governmental acts".

7

ASTM Activities in Ceramics

Joseph G. Richmond

Institute for Basic Standards,

National Bureau of Standards, Washington, D.C. 20234

The organization of the American Society for Testing and Materials is described, the

procedure followed in the promulgation of ASTM Standards is briefly outlined, and the

scope of ASTM activities in the field of ceramics is indicated.

Key words: ASTM; committees; definitions; promulgation; standards; tentatives.

The American Society for Testing and Materials

is the largest voluntary standardizing organization

in the world. Its membership is made up of over

11,000 individual engineers and scientists, 2700

companies, associations, and research institutes,

1300 educational institutions, governmental agencies (federal, state, and municipal) and professional societies, and 900 engineering and science

students. It functions by the committee system,

with some 106 main technical committees that are primarily responsible for the development of

standards.

The large membership of highly qualified

individuals with widely diverse technical backgrounds is particularly advantageous for the

development of standards. A technically sound

standard is likely to be approved by such technically qualified members and the standards that

are approved will receive wide distribution and

acceptance.

The membership of each technical committee

is made up of individuals knowledgeable in the

specific field covered by the committee, each of

whom, it is assumed, represents the interests of

his employer, or himself, if self-employed. Membership on a committee is attained by election to

the committee after application by an individual,

stating his qualifications for membership. In

general only those considered capable of contributing to the work of the committee are elected

to membership. Membership is lost automatically

when a member changes employment, or ceases

to participate in the work of the committee.

The primary objective of the society is to

promulgate the standards prepared by the technical committees. The generic term "standard"

includes documents that differ both in the degree

of acceptance and in content. With respect to

content, they generally fall into one of the following classes: a) definitions of terms, b) recommended

practices, c) methods of test, d) classifications,

and e) specifications. With respect to the degree

of acceptance they are published as: a) Proposed

Standards, b) Tentatives, or c) Standards. As

a matter of Society policy its Standards are submitted for recognition as USA Standards, and

currently more than 40 percent of all USA

Standards are ASTM Standards.

Many technical terms are not adequately

defined in standard dictionaries or if they are

defined, the technical meaning is different from

that which is technically accepted. Adequate

definitions of terms are thus the first prerequisite

for meaningful communication, not only between

individuals in the same area of technology, but

particularly between individuals in different areas

of technology.

Engineers and scientists in many fields have

found that in many cases certain ways of doing

things are likely to lead to trouble or yield erro- neous results, while other procedures will generally

avoid such trouble or errors. Information of this

kind that is not properly a part of a test procedure

or specification is termed a recommended practice.

The detailed description of test apparatus and

techniques to be used in evaluating the properties

of materials and equipment form the basis for a method of test. If the techniques are intended for

verifying the calibration of testing machines or equipment, such a document is another form of an

ASTM method of test. A specification may include

one or more test procedures, but its primary

purpose is to set limiting values for the parameters

of a material or equipment for a specific application; it may classify a material into different grades

on the basis of its physical or chemical properties.

Documents may be published for information

only, particularly when in a new or rapidly ex- panding area of technology. This type of publication implies no acceptance or approval by the

Society. In the case of new areas of technology,

or controversial subjects, a document may be

published as a proposed method or specification in

order to bring it to the attention of interested

members of the Society, so that if they wish they

may express their opinion regarding the merits of

323-655 O - 69 - 2
the proposal before it is accepted as a Tentative.

Publication as a Tentative implies that the document is new and requires more testing data. Such

a document has the approval of the sponsoring

committee and the Administrative Committee on

Standards representing the Society. Publication as a Standard implies full approval by the sponsoring

committee and the Society as a whole. Acceptance

of an ASTM Standard by the U.S.A. Standards

Institute transforms it into a U.S.A. Standard.

Not all U.S.A. Standards originate in ASTM, but

far more come from ASTM than from any other

source. The term "Tentative" has an unfortunate connotation, and the Society is actively searching for a better term to express the thought that such a document does indeed represent the best available

technical knowledge on the subject, but cannot be

granted the status of a Standard because of lack

of knowledge of the accuracy and precision of the

test methods involved, or the correlation of test

and service data, or some similar lack.

The procedures followed in the promulgation of

an ASTM Standard are quite involved. This involvement results in part from the elaborate safeguards that have been instituted (1) to ensure that

every interested individual member of the Society

will have an opportunity to express his opinion

and be heard, and (2) to prevent domination

of any group within ASTM by special interest

representatives.

ASTM Standards are developed by the democratic process implied in the term "consensus."

Each Standard is subjected to letter ballot at least

three times before adoption, twice in the technical

committee and once by the Society. A key provision in the adoption process is that any negative

vote in a letter ballot must be accompanied by

valid technical reasons, and must be considered and

reconciled if possible before adoption of the

Standard.

The first draft of a standard is generally prepared by a small task group appointed for that

purpose, made up of technically qualified individuals who represent as many diverse interests as is practical. Development of a suitable test procedure, for instance, may involve research work

in the laboratories of the members of the task

group, followed by round-robin tests to prove that

the method is precise and reliable. After the group

has agreed upon the technical details, a draft is prepared and presented to the subcommittee,

where it is discussed, revised if necessary in the

light of the discussion, and after approval by

letter ballot, is presented to the main committee,

where it is again submitted to letter ballot. Although only a two-thirds favorable vote is re- quired in subcommittee and committee letter

ballots, the ballot is usually unanimous in the

subcommittee and nearly unanimous in the main

committee. When approved by the committee it

is submitted to the Society as a whole, either at

the Annual Meeting or through the Administrative

Committee on Standards, where it is accepted as a Tentative. After one year as a Tentat ve, the

sponsoring committee may recommend that it be

advanced to a Standard, with or without revision,

which must again be confirmed by a two-thirds

favorable letter ballot. After such a recommendation, it is submitted to letter ballot of the Society as a -whole. After approval by two-thirds of these

voting, it is formally accepted and published as an

ASTM Standard. If immediate adoption as a Standard is recommended by sponsoring committee, approval by nine-tenths of those voting is required in the Society letter ballot.

Any technically sound proposal that accurately

represents the best knowledge available, that is capable of the desired precision and accuracy,

and that is arrived at by the concensus process

mentioned above, may and should be submitted

for immediate adoption as a Standard, without

prior adoption as a Tentative.

ASTM Standards are subject to periodic re- vision and reapproval by the sponsoring committee at least once in each six year period. They do

not remain in force unless the committee makes

such a recommendation, confirmed by letter

ballot.

The Regulations Governing ASTM Technical

Committees provide that the members of each

Committee shall be classified as either producers,

consumers, or general interest members, on the

basis of their interest in the material or product

involved in a document. The regulations also re- quire that in no case shall the producers dominate

any committee. No producer shall serve as chairman of a committee that prepares or recommends adoption of a specification. As a further

check on domination by producer interests, the

regulations require that each member indicate on

each letter ballot his classification with respect to

the material or product involved, and a two-third

vote requires that two-thirds of the producers and

two-thirds of the nonproducers separately approve

the recommendation. This insistance on participation by consumers and general-interest members

in the development and approval of ASTM documents has contributed greatly to their wide ac- ceptance by industry, government, and consumers

generally, and has kept the Society free from

involvement in antitrust actions.

As a further restriction, while each member has

an individual vote in the subcommittee, the representatives of each organization have only one vote

collectively in the letter ballots of the committee

and of the Society.

There are 106 technical committees. These are

classified as follows: 7 "A" Committees on Ferrous

Metals, 9 "B" Committees on Nonferrous Metals,

24 "C" Committees on Cementitious, Ceramic,

Concrete, and Masonry Materials, 30 "D" Committees on Miscellaneous Materials, 27 "E"

Committees on Miscellaneous Subjects, 6 "F"

Committees on Materials for Specific Applica10
tions, and 3 "G" Committees on Corrosion, Deterioration, and Degradation of Materials.

Most of the "C" Committees are concerned

with ceramic materials. Time does not permit a

detailed account of the activities of these Committees, but a list of the major "C" Committees

will indicate something of the breadth of their

scope: C-l on Cement, C-3 on Chemical Resistant

Nonmetallic Materials, C-4 on Clay Pipe, C-7 on

Lime, C-8 on Refractories, C-ll on Gypsum,

C-14 on Glass and Glass Products, C-15 on

Manufactured Masonry Units, C-l6 on Thermal

Insulating Materials, C-17 on Asbestos-cement

Products, C-20 on Acoustical Materials, C-21 on

Ceramic Whitewares, C-22 on Porcelain Enamel

and Related Ceramic-Metal Systems, C-25 on

Ceramics for Electronics. In addition, several other

committees also deal with ceramic materials. D-9

on Electrical Insulating Materials has a subcommittee on Ceramic Products. E-12 on Appearance

of Materials covers test methods for measuring

reflectance, gloss, and color of ceramic as well as other materials. E-20 on Temperature Measurement covers methods of calibrating thermometers,

thermocouples, and pyrometers. F-l on Materials

for Electron Devices covers ceramic-metal seals.

The standardization of methods of testing and

specifications is the primary objective of ASTM.

In 1967, 2565 Standards and 1403 Tentatives were

published in 32 volumes, with over 25,000 pages.

The complete Book of Standards is republished

annually, in order to incorporate all revisions as well as new Tentatives and Standards. Standards

dealing with ceramic materials are included primarily in Part 13, and comprise 127 Standards

(including 34 U.S.A. Standards) and 31 Tentatives.

A recent bulletin from ASTM Headquarters

states that about 30 percent of the contents of the

ASTM Book of Standards is obsolete after one

year, and 65 percent after two years. This statement indicates the effort put forth by ASTM

technical committees to keep their standards up

to date.

In addition to its primary work of promulgating

standards, ASTM sponsors and publishes the proceedings of a wide variety of symposia, generally

in the fields of properties of materials and materials research. The fact that our meeting here is jointly sponsored by ASTM is an example of such

sponsorship. ASTM also sponsors research, either

directly or through the technical committees. The

Cement Reference Laboratory and the project on

x-ray powder diffraction patterns here at NBS are good examples.

ASTM has a number of other publications.

Materials Research and Standards is a monthly

magazine which includes announcements of all ASTM sponsored meetings, general information

on Society activities, and technical articles of

general interest. The quarterly Journal of Materials contains technical papers on materials

research. The annual Proceedings contains the

annual reports of all technical committees. Many

special publications containing technical papers

on a single general topic are published each year.

This is an abbreviated account of some of the

activities of the American Society for Testing and

Materials, but indicates something of their scope,

particularly in the field of ceramics. We expect

that these activities will continue and even

expand in years to come.

11

Properties of Ceramics for Structural and/or High Temperature Use:

Need for Control, Measurement, and Compilation

Charles F. Bersch

Naval Air Systems Command,

Code AIR-52032A, Washington, D.C. 20360

Extensive new markets are available to ceramics as soon as weight-efficient applications

involving primary and secondary structural requirements can be satisfied. The major obstacle

and its causes are discussed and long and short range approaches for dealing with those

causes are suggested.

Key Words: Applications of ceramics; brittleness; brittle design; ceramics; strength;

structural ceramics; variability; weight-efficiency.

1. New Markets Open to Ceramics

The opportunities for new or improved materials with the properties to perform new functions, or to operate in more extreme environments, or to fulfill their functions more efficiently are commonplace in the aerospace industry and

are increasingly evident in the hydrospace and

automotive industries.

Ceramics, by virtue of the spectrum of functional and structural properties they offer, are an

especially attractive source for such materials.

Prominent among their functional properties are extreme resistance to heat, oxidation, corrosion

and abrasion; outstanding dimensional stability

(in thermal, corrosive, abrasive or mechanical

environments) ; and unique nuclear, optical, magnetic, electronic and energy-conversion capabilities. Unparalleled stiffness and specific stiffness are available for structural purposes, and achievements of substantial portions of the predicted

theoretical strength (and the associated specific

strength) are increasingly frequent and sometimes commercially successful [4, 8, 9].

Functional applications of ceramics are widespread and common. They often include lowlevel structural requirements (secondary structural

requirements), as in sanitary ware, electrical insulators, windows, and corrosion-resistant processing equipment, but functional considerations are predominant. Ceramics have also been used

routinely for many years in a wide variety of

thermal and structural applications, oftentimes

in systems incorporating both modes simultaneously, such as kiln construction and furnace

linings.

However, virtually all current applications of

ceramics in which there have been appreciable

mechanical stresses due to thermal or mechanical

loading have been successful, or at least adequate,

because one or more of three conditions prevailed:

(1) The loads were accommodated by gross

overdesign

.

(2) All appreciable stresses were successfully

confined to the compression mode.

(3) Failure, if it occurred, was not catastrophic

either in mode or else in consequences.

In almost no instance could one say that the

system or part was weight-efficient, that is, that

it fulfilled its purpose at the lowest possible

weight. Yet weight-efficiency is almost a sine qua

non in industries where each pound saved in

functional or structural weight during the conceptual stage can be cascaded on the drawing board,

via savings in structural, powerplant, and fuel

requirements, into savings of tens and hundreds

of pounds, and, in the case of missiles, thousands

of pounds in total systems weights.

2. The Obstacle to Weight- Efficient

Uses of Ceramics

One persistent obstacle prevents a burgeoning

use of ceramics in applications where primary or secondary structural properties are needed. That

obstacle has many descriptors. Most commonly

they are "brittleness" or an "inability to design

with brittle materials." But one can live with the

brittle failure mode when one is certain that

failure will not occur or when one can predict and

control when it will occur. And, though the

techniques may be primitive because of insufficient

usage and lack of accrued experience there is some ability to design using brittle materials.

A more accurate descriptor is "lack of reliability"

or, as some designers say, the "variability" of

13
ceramics. But even controlled or bounded variability could be endured if its limits could be predicted. "Poor predictability" is probably the best

descriptor for the major obstacle.

An even finer focus on the real obstacle to more

widespread applications of ceramics is that

"variability" and "unpredictability" apply primarily to the strength properties of ceramics.

Though thermal, electrical, elastic, and other

properties are influenced by compositional and

microstructural variations, they are usually much

less variable than the strength properties. That

lesser variability is usually attributed to a lower

sensitivity, but it may also suggest that the first

two of the following suggested causes of variability

in strength properties are the major ones, since

they should have no effect on the nonstrength

properties and they are also the most readily

identifiable causes of variability in strength.

3. The Causes of Variability

Possibly the major contributor to variability

in strength data is the testing situation. Just as

special design techniques are needed for brittle

materials, so also special test procedures performed by skilled and careful operators are needed

to evaluate brittle materials. A ductile material

will ignore or forgive parasitic stresses, due to

misalignment and mistreatment, that are catastrophic to a ceramic material. The paper by

Mordfin and Kerper in this proceedings discusses

this subject in greater detail.

The surface of a ceramic specimen is also of

major importance, especially when the part is loaded in tension or flexure. But the variability

of a surface finish is likely to be a greater cause of

variability in strength data than is the surface

finish itself. A uniformly poor surface finish will cause low strength values but will not necessarily cause a scatter in data. However, as the average

surface finish improves, the effects of surface

finish variations will be more prominent. An

added complication becoming increasingly evident

is environmental sensitivity, which is also experienced via the surface and which is also more prominent as the surface finish improves. This is discussed further in the paper by S. M. Wiederhorn.

The character of the ceramic specimen, the

resultant of the raw materials and the processing

as evidenced by the composition and microstructure, has a prominent but mixed role in

ceramic property variability. Pores, microcracks

and cracks will have some effect on variability,

but the size, shape, location, and distribution of

defects are probably more critical than their

frequency. Grain size variability, grain shape, and

orientation can also be factors. But, except for

cracks, it is probable that body character is important to variability only after test and surface

deficiencies are under control.

4. Steps to a Brighter Future for

Ceramics

4.1. Design Techniques Based on

Predictability

From the foregoing it would appear that there

are several steps that must be taken for ceramics

to fulfill their potential in future hardware. One is a substantial advance in ability to design for

brittle materials. This subject is beyond the scope

of this discussion; however, it must be obvious

that the availability of ceramic materials having

predictable performance would provide both an

impetus to advances in design techniques and the

means or substance for physical substantiation of

theoretical advances.

One contribution to predictability might be a

total confrontation with brittle failure—an intensive study and analysis of the failure mechanism

to determine where and how and why failures

begin and how and why they proceed to completion. Modest efforts of this nature have been very

fruitful. Griffith flaws have provided a useful

concept for theoretical analysis. Their physical

reality has been demonstrated in a variety of

forms ranging from mechanical damage [1-3]

through thermal shock damage [4] to stress cor- rosion sites [5] and possibly phase separation

boundaries that occur at the surface. Once the

flaw and its genesis have been identified, it is possible to conceive of prevention, elimination, or

at least control. Current techniques for providing

compressive stresses in the surface of a body [6, 7] are essentially efforts to prevent Griffith flaws from

acting as stress concentrators and failure initiation

sites. Efforts along this line are encouraging;

bending strengths above 200,000 psi after abrasion

have been achieved with glass ceramics [8] and the

techniques have been shown to be feasible with

conventionally processed ceramics [4, 7]. More

recently a procedure for coating glass that appears

to prevent Griffith flaws due to either mechanical

or stress corrosion effects has allowed bending

strengths of seven percent of the elastic modulus

to be realized in glass rods of five different

compositions [9]. Such studies, some aimed at understanding

brittle failure and some aimed at enhancing useful

strength, tread a common ground and are mutually

supporting, suggesting that predictability and

great strength may be achieved simultaneously.

What an impetus to advances in design techniques

that would be!

4.2. More Useful Tests and Data

A second essential step toward weight-efficient

structural applications of ceramics is an improvement in data, primarily strength data. Several

facets are involved: improvements of tests to

minimize variability due to the tests; sufficient

identification of any test specimen or part so

14
it can be duplicated; and a concept of strength

and measurements of strength that extends from

properties to behavior.

An improvement in test procedures and techniques with a resultant reduction in variability

of strength data and a concomitant increase in

predictability is necessary but not sufficient to

advance the use of ceramics. It is necessary to

provide sufficiently sensitive comparative data to

facilitate process or product improvement studies

and it is necessary to provide useful numbers to

evaluate probable material performance in various

stress states. Part of the insufficiency in the

case of development studies is obvious, since the

development activity itself is also essential. And

part of the insufficiency in design analysis is so fundamental that its neglect is possibly the

greatest paradox in the ceramic industry: the

failure to identify or characterize the test specimen

or the piece of hardware sufficiently so it can be

reproduced so that the test data or the service

experience that have been acquired achieve

meaning via future utility. This deficiency is by

no means confined to strength data though it

may be more critical for strength data. The distressing fact is that much of the data in the

literature including some of the most carefully

acquired data is useful only to suggest trends in

behavior; it cannot be used with exactitude

because the ceramic material from which it was

taken can never intentionally be reproduced. The

incredible truth is that there are a few instances

where there was no further identification than

the word "alumina."

The accumulation of carefully controlled and

reliable data on reproducible ceramic materials is necessary but not yet quite sufficient; the data

must be meaningful and useful. Strength is not

an invariable condition; it is rather a numerical

measure of a process that is influenced by many

external features. It is sensitive to the rate of

loading; to the duration over which the load is applied; to the temperature of the stressed

material; to chemical influences such as those

which cause corrosion or stress-corrosion; to the

kind of stress, its origin and uniformity; to the

simultaneous presence of other stresses, macro

and micro; to surface energy effects which, in

boundary cases, may include size and geometry

effects. In fact, in many cases test data do not

so much describe a property as a behavior in the

presence of a variety of conditions. Thermal

shock behavior is a good example. A less clearcut

but equally important example is the ability to sustain long enduring loads during immersion in

seawater or some other uncommon medium.

It should be evident that test data obtained at

random, that is, without a specific use in mind

for that data with that use determining the

conditions under which the data are obtained,

will be of value only to suggest trends in behavior.

It will not suffice for any specific decisions.

A supplier of metals to the aerospace industry

reports that tensile data on his material was all that the designer desired 35 years ago. Today he

must obtain data from 63 different test specimen

configurations to obtain the variety of data

demanded by aerospace designers. The ceramic

industry, with a product that is presently much

less predictable, will need more sophistication

than an advance from three to four point loading.

4.3. Improved Ceramic Processes and Parts

Several volumes might be written about the

general and specific actions related to processing

that coidd be taken to improve the predictability

of ceramic materials. Each individual composition

and each separate process has its merits and

shortcomings, its areas especially susceptible to

improvement. But what is improvement? It may

be higher or more uniform properties achieved via

smaller average grain size, or smaller grain size

range, or more or less orientation of anisotropic

grains, or fewer, or rounder, or smaller pores.

An almost endless list can be made and it would

be rather fruitless to complete it or to attempt to

act on it. What seems more appropriate is to apply to the materials presently in use, and to a few model materials, the best principles available

at the time, such as uniform small grain size,

minimum porosity, etc., as the material and

process combination allow, in order that uniformity within a part, reproducibility from part to

part, and reliability in service may be improved.

Usable strength levels would undoubtedly rise as a consequence.

One area that merits special attention is the

surface, and one process that requires special

attention, since it is used on most test specimens

and parts, for shaping, dimensioning, or application of a specified finish, is abrasive machining or

grinding. Another paradox of the ceramic industry

is the evident recognition of the contribution of

the surface of a ceramic part to the strength and

reliability of the part set off against the failure

to study what grinding, which is a rather brutal

treatment, does to that surface. A knowledge of

the nature and the extent of the influence of

grinding on a ceramic surface is a prerequisite to

any efforts to prevent, control, or cure such

influence.

5. Characterization

The character of a ceramic has been defined

as the sum of the compositional and structural

descriptions which identify a specimen uniquely,

may be correlated with its properties, and suffice

for the reproduction of the material _ [10]. It is apparent that the need for characterization has

been a consistent thread throughout the previous

discussion. In particular, characterization was emphasized as essential to provide meaning to test

data by providing the opportunity to use that

15
knowledge again in further production of the

same material. Less emphasized, but equally

important, is the need for characterization during

process and product improvement. This is particularly so as technology becomes more sophisticated and subtle and the earlier direct relationships

between processing and properties become insufficient. Character then becomes, in addition to

all above, the link joining processing and

properties.

Complete or total characterization (CT) consists of the sum of all character features on the

atomic scale (CA ), on the microstructural scale

(C(x) and on the macrostructural scale (CM) and

is often expressed in the equation:

CT = CA -\-Cfi-\- CMAtomic character includes chemical composition and uniformity. Microstructural character

includes the size, shape and orientation of grains;

the nature of interfaces and grain boundaries;

the presence and disposition of additional phases;

the distribution of dislocation and intergranular

stresses. Macrostructural character includes the

size and shape of the part, its uniformity, density,

gross chemical composition and surface topography, and the major stresses present.

Complete characterization is an ideal state;

it may never be realized in entirety for any

body and it is certainly too extensive, complex,

and expensive to do routinely. Accordingly, a second concept is available to provide, simultaneously, a means to approach the ideal state

and a more practical exercise, so benefits can

be realized at once. The concept is labeled "description'' (D) and it is the sum of the total

characterization available (CT) plus data on the

properties (P) of the body plus the sample history (H), including raw material background and

processing data. This too has an equation:

D=CT +P+H.

In the hope that the current partial reliance

on properties and sample history may eventually

be discarded, the equation is expressed as

where as X approaches 1, description will equal

total characterization.

The provision of the second and third equations

recognizes the need for a practical approach to

characterization but it still does not suggest what

a happy medium might be or should be. In trying

to make specific recommendations one is torn

between the desirability of having the maximum

amount of information available versus the

practicality of a minimum expenditure of time

and effort. The most practical suggestion is not,

of course, to do no characterization; that is

shortsighted and self destructive; it is, rather, to

do as much as is known to be important, to do as much as provides a useful immediate return in

fostering growth. Wachtman's [11] recommendations concerning conventional, augmented conventional and more intensive characterization, as shown in table I, seem to be the most reasonable.

Thus, in the most routine cases one might accept

only chemical composition, average grain size,

and density. This is not in any way adequate, but

if confined to production materials it has the

added benefits of accumulated experience in the

typical properties and histories of the materials.

Table 1. Parameters needed to specify the mechanical

properties of a ceramic

Conventional Characterization: chemical composition (1 percent level) average grain size density

Augmented Conventional Characterization

crystalline and amorphous phases

grain shape

grain size distribution degree of preferred orientation

distribution of porosity

surface roughness

Fine Microstructure and Other Characteristics

trace impurities, amount and state of aggregation

dislocation concentration twin concentration point defect concentration

residual stresses ambient atmosphere

radiation field state of temper (glass)

In any effort other than routine production,

for example in product or process development,

the augmented conventional characterization outline should be followed. This, again, is minimally

adequate, but assumes prior knowledge about the

process or product that helps complete the

knowledge and understanding of the material.

Lastly, in any research study on ceramics, the

last grouping of character features must be considered in addition to the ones already discussed.

Not all of them will be needed in all circumstances. In fact, some may be quite difficult to

determine. But the effort must be made to foster an optimum practical growth of the state of the

art and the range of applications of ceramics.

6. Summary

Broad opportunities exist for much wider and

more extensive applications of ceramics, in primary structural applications ranging to the hulls

of deep submergence vehicles and beyond, and in

secondary structural applications from rod end

bearings in helicopter blade control rods to gas

bearings, fuel cell components, and yet unconceived applications, if: • poor predictability is overcome by confronting

brittle behavior and learning to prevent, over- come or control it; • poor predictability is eliminated by the skillful use of well-conceived test procedures;

• poor predictability is overcome through

achievement of understanding and control of

16
ceramic processing, in all stages from the

preparation of raw materials through surface

finishing and assembly.

7. References

[1] Griffith, A. A., Phenomena of rupture and flow in

solids, Phil. Trans. Roy. Soc. (London) 221A,

163-98 (1920).

[2] Ernsberger, F. M., Detection of strength-impairing

surface flaws in glass, Proc. Roy. Soc. (London)

A257, 213-23 (1960).

[3] , A study of the origin and frequency of occurrence of Griffith microcracks on glass surfaces, pp 511-24 in Advances in Glass Technology

(Plenum Press, New York, 1962).

[4] Kirchner, H. P., Gruver, R. M., Platts, D. R.,

Rishel, P. A., and Walker, R. E., Chemical

Strengthening of Ceramic Materials, Summary Report for Naval Air Systems Command Contract

N00019-67-C-0489 (6 April 1968).

[5] Schmitz, G. K., Gulden, M. E., and Metcalfe, A. G.,

A Study of Stress Corrosion of Glass Fibers, Final

Report for Naval Air Systems Command Contract N00019-67-C-0409 (June 1968).

[6] Stookey, S. D., Olcott, J. S., Garfinkel, H. M., and

Rothermel, D. L., Ultra-high strength glasses by

ion exchange and surface crystallization, Advances

in Glass Technology (Plenum Press, New York,

1962).

[7] Kirchner, H. P., and Gruver, R. M., Chemical

strengthening of polycrystalline ceramics, J. Am.

Ceram. Soc. 49, 330-3 (1966).

[8] Duke, D. A., MacDowell, J. F., and Karstetter, B. R.,

The crystallization and chemical strengthening of

nepheline glass-ceramics, J. Am. Ceram. Soc. 50,

67-74 (1967).

[9] Hallman, R. W., and Anderson, J. S., Ultra-High

Strength Bulk Glass, Final Report for Naval Air

Systems Command Contract N00019-67-C-0435

(15 May 1968).

[10] Ceramic Processing, Materials Advisory Board, National Academy of Sciences, Publication MAB-233,

February 1968. Also National Academy of Sciences

Publication 1576.

[11] Wachtman, J. B., Jr., Mechanical properties of ce- ramics: An introductory survey, Bull. Am. Ceram.

Soc. 46, 756-72 (1967).

17

Melting Points of Ceramics

S. J. Schneider

Institute for Materials Research,

National Bureau of Standards, Washington, D.C. 20234

The melting behavior of a ceramic, as for any material, can be described by the phase

rule and its graphical expression, the phase equilibrium diagram. The solid to liquid transition (melting point) is discussed in terms of a one-component system and extended to more

complex systems utilizing unary principles as basic building blocks. Phase equilibrium

diagrams of hypothetical and of real ceramic systems are presented to illustrate the relationship between melting point and the pertinent intensive variables—pressure, temperature

and composition. The effect of metastable equilibrium upon the melting point is also emphasized. Experimental methods for melting point determinations are outlined and include

the thermal analysis and quenching techniques. Relevant details of temperature measure- ment are given.

Key Words: Ceramics; fusion curve; melting point; melting point methods; metastability; phase rule.

1. Melting Point and the Phase Rule

Ceramics as a general class are materials with

rather high melting points, characteristically

having a somewhat viscous liquid phase. Although

ceramics possess distinguishing features which

impart experimental difficulties in their study, the

general principles governing the fusion and crystallization process are the same as that inherent in

all materials. Conceptually, the melting point of a ceramic cannot be viewed any differently from

that of any other material and is the temperature

at which solid and liquid of the same chemical

composition are in equilibrium for a given confining pressure. Normally, the pressure is inferred

to be equal to 1 atm but most certainly is not

limited to that value. An equally valid but less

widely used definition would be the pressure at

which solid and liquid are in equilibrium for a given temperature.

The significant thoughts embodied by both

definitions are related by the familiar Gibbs phase

rule for heterogeneous systems:1

P+^=(7+2 (1)

where

P=the number of homogeneous phases present

at equilibrium. Phases may consist of either

solid, liquid, or vapor and must be mechanically separable one from another.

.F=the number of intensive variables of a system such as pressure, temperature, and

composition which can be arbitrarily changed

1 Numerous publications are available describing the theory and utilization

of the phase rule. Some of these are tabulated in references [1] through [14]. (Figures in brackets indicate the literature references at the end of this paper.)

without disturbing the equihbrium. These

variables are commonly termed degrees of

freedom or variance of the system.

C=the least number of independent components

from which P phases can be made.

Equation (1) applies to all noncondensed systems.

Inasmuch as many ceramic materials have low

vapor pressures, the vapor phase is often neglected,

thus reducing eq (1) to

{P-\)+F= (7+2-1

or (2) p+F=C+\

where F and C have the same meaning as previously stipulated and P refers only to solid and

liquid phases. The condensed phase rule, however,

is not strictly applicable to true one-component

systems.

The phase rule says nothing about rates of

reactions between the components, nor does it

infer the amount of each phase present at equilibrium. It does, however, stipulate the number

of phases allowable for a given set of conditions.2 At times the utility of the phase rule may appear

to be purely academic, but a real insight into

the behavior of materials can be gained through

its use. Of course, the entire field of phase equilibria is merely the experimental extension of the

phase rule. When applied to the melting phenomenon the aforementioned definitions of melting

point become more clear. For example, in the

familiar water system, ice and water are in equilibrium at the melting point. Here the number of

2 For noncondensed and condensed systems the maximum number of phases equals C+2 and C+l, respectively.

19
components is one (H20), the number of phases,

two (solid and liquid), and the variance, one,

(F=C+2-P=l+2-2=l). Thus, if one degree of

freedom is utilized by holding the system at

constant pressure, the temperature of melting is automatically determined. Conversely, when the

temperature is held constant the pressure is then

specified.

At the melting point the number of degrees of

freedom (F) always equals unity for a noncondensed single-component system. Moreover, ex- pansion to a multicomponent system does not

permit an increase in the variance of the solidliquid equilibrium at the melting point of a compound. Since the composition is fixed by specifying

the compound, the only remaining variables are

either pressure or temperature. Hence F—l, the

same as for a one-component system. In essence,

regardless of the number of elemental constituents

involved in the reaction, proper application of

the phase rule requires that only the smallest

number of components necessary to describe the

equilibria need be considered. By way of illustration, the compound CaO-Al2 03 -2Si02 melts to all liquid at 1553 °C [15]. Treating the system as containing three components (CaO , A1203 , Si02 ) would yield at the melting point of the compound

a variance of three, an obviously false conclusion.

The degree of freedom, of course, equals unity

as for a one-component system.

Furthermore, the number of components under

consideration may be altered within a system

depending upon the conditions of experiment. The

compound 3CaO-Al203 , for instance, melts at

1535 °C and atmospheric pressure to CaO and

liquid [16]. At temperatures below melting and

at fixed pressure, the compound can be thought

of as one component possessing one degree of freedom

for the condensed system. At 1535 °C three

phases, CaO, 3CaO-Al203 , and liquid are in equilibrium with a variance of zero. The number of

components then must equal two. Basically, the

system appears to change from one to two components (3CaO-Al203 —>CaO and A1203) all within

the framework of the CaO-Al2 03 binary.

The phase rule is basically a statement of the

conditions which exist when a system is at thermodynamic equilibrium. Theoretically, equilibrium

conditions require that a system cannot change

with time without a gain or loss in energy. Here,

the Gibbs energy of the system is at a minimum

and the entropy at a maximum. The attainment

of equilibrium may be difficult and often presents

a dilemma to the investigator. In certain systems,

particularly those involving ceramics, reactions

and transformation may proceed at such a slow

rate that a quasi state of equilibrium will be obtained. Such a situation, termed metastable equilibrium [2, 3], is often consistent with the phase

rule. A good case in point is supercooled water

which is in metastable equilibrium with its vapor

at temperatures below its freezing point. More

pertinent to this discussion is the metastable equilibrium exhibited by fused silica when supercooled

to room temperature [17]. The liquid is metastable

with respect to quartz, the stable room temperature modification. The system strictly obeys the

phase rule and yet is not at thermodynamic equilibrium. It is axiomatic that compliance with the

phase rule is not a test for equilibrium but only

a guide. However, disobedience of the phase rule

furnishes positive evidence that equilibrium conditions are absent. The importance of metastable

equilibrium cannot be stressed too strongly. Many

significant processes depend upon it for the manufacture of common, everyday products such as

glass.

2. One-Component Systems

2.1. Stable Equilibria

For practical use the phase rule must be combined with experimental information. The graphical representation of the data encompassing the

principles set forth by the phase rule is the phase

equilibrium diagram. Figure 1A gives a hypothetical phase diagram for a single-component system

using pressure and temperature as the independent

variables. Real systems can be infinitely more complicated in detail but the figure illustrates the es- sential features inherent in all single-component

systems. The diagram consists of three individual

curves, intersecting at point b, which separate

solid, liquid, and vapor phases. The variance of

each area and curve is indicated by an F value,

the maximum being two (divariant) for singlephase areas and the minimum, zero (invariant) for

point b. Univariant curves ab and be are the vapor

pressure curves for the solid and liquid phases,

respectively. At point b, the triple point, three

phases, solid, liquid, and vapor, are in equilibrium.

(A) (B)

TEMPERATURE

Figure 1. Schematic phase diagrams for one-component

systems. F equals number of degrees of freedom.

Systems A and B

ab—univariant sublimation curve, be—univariant liquid-vapor curve (boiling point curve), bd—univariant melting point curve, b—invariant triple point (solid, liquid, and vapor).

System B only: ad—univariant solid I-solid II transition curve, de—univariant melting point curve of solid I, d—invariant triple point (solid I, solid II, and liquid),

20
Curve bd represents the melting curve and indicates the change in the solid-liquid equilibria

(melting points) with pressure and temperature.

The curve indicates quite clearly that any definition of melting point must include statements

regarding both variables. It is insufficient to quote

a melting temperature without specifying the

pressure.

Theoretically, the melting curve is involved

only with solid-liquid equilibria and only at the

triple point does the vapor phase enter into

the reaction. Take for instance, the compound

AX 0V which vaporizes congruently to a number

of gaseous species. For simplicity the reaction can be written as Ax Ov (s, l)^>Ax {g) -\-Ov {g). The

reaction equation describes the stoichiometry of

vaporization and will be reflected in the one- component phase diagram by the location of curves ab and be, figure 1. Even here the twovapor species, A<t(g) and 0v (g), still only constitute a single-vapor phase and consequently obey the

phase rule. In effect, the vaporization curves

for ^.^(s, I) are dissociation curves. The mechanism of vaporization will have no definitive effect

upon the fusion curve in a one-component system,

for in theory, there is no vapor phase to contend

with at melting and the equilibrium of interest

is: Ax Ov (s) —>AX 0V {1). In this hypothetical situation

the pressure confining the system must be visualized as that exerted by a frictionless piston. In

actuality, however, all real experiments must be

carried out in some type of environment, say air or inert gas. Now the system becomes binary

because a second component, the environment

(air), has been added and the melting equilibrium

is describable as: Ax Ov (s) +environment(air) -*AxOy

(l) +environment( ai r >

•

Both the solid and liquid then are also in equilibrium with a vapor phase and vapor pressures

may be influential. In terms of the phase rule

along the melting curve of AX 0V : [P-\-F=C+2

]

= [3(s, I, v)+F=2+2] or F=l, the same as for a true one-component system. If the vapor

pressure of the compound is very low, as is the

case for many ceramics, the condensed phase

rule can be applied: [P+F=C+l] = [2(s, l)+F

=2+1] or F=l, again the same F value as previously obtained when considering the vapor

phase. It should be emphasized that neglecting

the vapor as a possible phase does not remove

it as one of the components {Ax Oy

-\- environment).

The condensed phase rule, eq 2, only accounts

for a vapor phase deletion and the system, in

essence, is not truly a one-component system.

The assumption that the presence of an environment will not affect a melting curve is normally

valid, provided the vapor pressures of the con- densed components are low and provided that

the environment is nonreactive. For example, if the component Ax Oy dissociated to Ax Ov - z (s)

+Oz (g) or Ax (s)-\-Oy(g), the determined melting curve then would be representative of the new

system, AxOy - z (s) or ^lx(s), not the original system

under investigation. In this context, the dissociation (or oxidation) pressures become extremely

important to the melting point determination.

Figure IB indicates another hypothetical diagram for a one-component system in which a polymorphic change occurs in the solid state.

Curve ad represents the transition curve for

solid phases I and II. Point a is another triple

point where the two solids and vapor are in

equilibrium. The transition curve, ad, is drawn to intersect the melting curve, bd, at the triple

point, d, where solids I and II and liquid exist.

With increased temperature and pressure solid I melts directly to liquid, thus exhibiting its own

melting curve, de. This situation, where at high

pressures the low form melts at a higher temperature than the maximum fusion temperature of

the high form, is not at all uncommon. In ceramic

systems there is a general scarcity of data showing

such phenomena because of the extreme experimental conditions involved. Silica, the mainstay

of all ceramics, does exhibit this type of behavior

when subjected to high pressures. Figure 2

gives a composite phase diagram for Si02 drawn

from the data presented by several investigators

[6, 18, 19, 20]. High quartz is seen to melt at

higher temperatures than cristobalite, the high

STISHOVITE /

/

/

COESITE

/ LIQUID

/ HIGH — / QUARTZ /

LOW / X \S

QUARTZ / / \s\

/ /\/ CRISTOBALITE

/ TRIDYMITE

/ / VAPOR

TEMPERATURE —•>

Figure 2. Schematic phase diagram for the one- component system, S1O2. Portions of diagram

after Levin et al. [6], Ostrovsky [18], Boyd and

England [19], and Fenner [20].

21
temperature-atmospheric pressure form. Coesite,

the high pressure-low temperature phase, melts

at still higher temperatures and although data

are not available, stishovite conceivably would

fuse at temperatures successively higher.

In the silica diagram the majority of the

melting and transition curves all have positive

slopes, that is, an increase in pressure will cause an increase in temperature for the transition.

Negative slopes, although somewhat rare, do

occur in a number of materials including water,

bismuth, and gallium [2, 211. Normally the overall

change in melting point with pressure is rather

small and generally is disregarded. However

slight, these changes can be approximated from

the Clapeyron equation provided values of the

molar volumes of the solid and liquid phases as

well as the heat of fusion are available:

dT TUVt-Vs) Tm AV

dP~ AH, ~ AH, w

where

dT= change in melting temperature from Tm, dP= change in confining pressure,

Tm=melting temperature at given pressure,

Vi=mo\tibr volume of liquid,

Va =molar volume of solid,

AH,=molar heat of fusion.

A semi-empirical equation describing the melting curve has been derived by Simon [21, 22, 25]

for the case where the coordinates of the triple

point are known:

These small pressure effects are characteristic of

ceramic systems involving only solid-liquid

equilibria.

The differential form of the Clapeyron equation

is seldom applicable over very wide pressure

ranges, for both AH, and AV are functions of

pressure and temperature. The compressibility of

the liquid phase generally is greater than that of

the solid [3] so at very high pressures Vi will

become less than VB. At some point Vi will equal

d T VB and ^p- will then be zero, resulting in a maximum or reversal point on the melting curve. Few

ceramics have been studied at sufficiently high

pressures and temperatures to determine whether

or not they exhibit this type of behavior. However, some data are available and figures 3 and 4

illustrate such maxima found for the carbon [18,

24] and Li2 Cr04 [25] systems. The reversal phenomenon, of course, is an experimental observation and a detailed explanation of why it occurs

for some materials involves careful analysis of

the structures of both the liquid and solid phases

at fusion temperatures.

A maximum in the fusion curve dictates that

AV is zero. If, coincident with AV=0, the change

in entropy were also zero, the system then would

cease to be heterogeneous and the solid and

liquid phases would be indistinguishable, analogous to the critical phenomena which exist for

liquid-vapor transitions. The fusion curve at this

P-Po

a

(4)

where

P0 =pressure of the triple point,

T0 =temperature of the triple point,

P and T=pressure and temperature of some

point on melting curve, a and c= constants characteristic of the material.

At the triple point the Simon equation is related

to the Clapeyron equation by [21]:

dT

dP c{a+P) (5)

where P»P0 The Simon equation is in essence a curve-fitting

equation requiring extensive and accurate ex- perimental data to be of any great utility. For

ceramic systems the Clapeyron relationship will

suffice for most purposes. As a practical example

the slope of the melting curve for A1203 is 0.02

deg per atm assuming Afl/=28.3 kcal/mol,

y,=0.338 cm»/g and F8 =0.251 cm3/g. For a change in melting point of only 2 deg, a pressure

of at least 100 atm would be required—indeed a very negligible pressure effect. Similar calculations

applied to the cristobalite-liquid transformation

yield a 0.07 deg/atm slope, still a slight effect.

CARBON

/

m J

LIQUID CARBON

DIAMOND

120 kbors r

ge

1 ^"^^ >v 1200 °C J SYNTHESIS^- I 2400 °C \

1 ^^Z—.1 1 REVERSAL ^*~-*$>Q kbor T POINT

/ 4000-4700°C

/ 60-70 kbars

GRAPHITE

/ VAPOR

TEMPERATURE -

Figure 3. Schematic phase diagram for the one-component system carbon. Diagram re- drawn from that given by Bundy [24~\.

22
400' 1 1 1 1 1 1 1 I 10 20 30 40

PRESSURE -KILOBARS

Figure 4. Phase diagram for the one component system,

Li2 Cr04. After Pistorius [25].

critical temperature and pressure point would

terminate in a manner similar to the upper limit

found for liquid vaporization curves. Critical

melting has been postulated but not verified for

several materials [21]. If such melting occurs in

ceramics the pressures involved would be exceedingly high.

2.2. Metastable Equilibria

Metastable equilibria can be depicted graphically in a manner similar to that illustrated for

stable systems. However, a certain definite relationship always exists between the two types of

equilibria which dictates the behavior of metastable phases with pressure and temperature.

Figure 5 gives diagrams for three one-component

(A) (B) (C)

i c 9 i c h i d c

SOLID

liquid/ liquid/

\

1 LIQUID/

e

-4

f /

/ / //

is

j

n |

e " A/

'is

a —

)

1 I HI

1

1

/ g' / b

0 VAPOR VAPOR

a

VAPOR

TEMPERATURE —

Figure 5. Schematic phase diagrams for one-component

systems showing stable (solid lines) and metastable (dashed

lines) univariant curves.

eb—supercooling, bf—superheating (system A)

,

fg and gi—metastable melting point curves.

systems showing the effect of metastability upon

the fusion curve. In the general case, (A), the

metastable univariant curves (dashed lines) are extensions through the triple point, b, of the

stable curves (solid fines) with bf representing

superfusion, and eb, supercooling. Superfusion,

that is, the heating of a solid above the melting

point without the appearance of the liquid phase,

has seldom, if ever, been found experimentally

[3]. Supercooling on the other hand, is quite

common and as mentioned previously, ordinary

glass offers a perfect example.

Both metastable extensions, curves bf and eb,

must be drawn between the stable univariant curves so that the stable phases will be always

illustrated as having the lowest vapor pressures.

This requirement is necessary because a phase is always stable with respect to another if it possesses a lower Gibbs energy or, in practical terms, lower

vapor pressure [2, 3].

In case (B), figure 5, the system is seen to

contain a phase II which has no stability field

whatsoever. The vapor pressure curve of II, ef,

intersects the extension of the liquid-vapor curve

at f, a metastable triple point. The fusion curve,

fg, indicates metastable melting at temperatures

lower than those for stable melting, curve bd.

Example (C) of the figure illustrates metastable

melting, curve gi, as influenced by the presence

of a stable polymorph, I. Here solid I melts

rather than transforming to solid II as required

by the stable equilibrium relationships. It is interesting to note that both situations, (B) and

(C), show how it is possible for certain materials

upon heating, to melt, then solidify, and finally

remelt again.

The question sometimes arises whether or not

a metastable phase can melt at temperatures

higher than its stable counterpart. The metastability represented by (B) and (C), fig. 5, indicate

lower melting. In these examples the metastable

liquid phase is always unstable with respect to a

true equilibrium solid. For metastable melting to occur at higher temperatures, a liquid phase would

23
have to be formed which is unstable with respect

to an equilibrium liquid. Figure 6 (A) and (B)

gives hypothetical examples of higher melting in

which liquid II would be unstable. In essence two

immiscible liquids would have to be postulated

for a one-component system. This, of course,

represents an improbable situation and conse- quently higher melting cannot occur. It should

be pointed out that example (A), figure 6, represents nothing more than superfusion previously

illustrated in figure 5A. The unlikelihood of liquid

immiscibility in a one-component system accounts

for the fact that superfusion has never been demonstrated experimentally.

Instances of metastable equilibria occurring in

ceramic materials are numerous. The Ge02 system [26], figure 7, perhaps offers a perfect

example of the types of metastability given in

figure 5. It is evident from the figure (curve jc)

that liquid Ge02 can be supercooled from above

1116° C, the true melting point, to room temperature. The rutile form of Ge02 , the stable low

temperature form, is seen also to melt metastably

at 1086 °C instead of transorming to the stable /3 quartz polymorph at 1049 °C. Furthermore, the

/3 quartz form on cooling, can transform metastably

at 1000 °C to a quartz, a form that presumably

has no stability field whatsoever.

Other examples of the prominent role played by

metastable equilibria in the behavior of ceramics

are given by the Si02 [20] and Bi203 [27] systems,

figures 8 and 9. Here again metastable melting

is a significant feature of both systems.

3. Binary Systems

Although few commercially important ceramics

involve only single-component systems, thenbasic behavior can be developed rather nicely from

one-component principles. Consider first a one- component system, solid A, to which small

increments of an impurity, solid B, have been

(A) (B)

SOLID SOLID

VAPOR VAPOR

TEMPERATURE-*

Figure 6. Phase diagrams for one-component systems showing

two situations of superheating.

Solid lines—stable univariant curves, Dashed lines—metastable univariant curves,

I—stable liquid, II—metastable liquid.

VAPOR

O 01 ID O I" 00 ID O O O -

TEMPERATURE °C

Figure 7. Phase diagram for the one-com- ponent system, GeC>2. Essentially after Sarver and Hummel [26\.

Dashed lines—metastable univariant curves,

Solid lines—stable univariant curves,

fe—metastable melting point curve of rutile form, de—stable melting point curve of 0-quartz form,

hi—metastable transition curve of metastable form

a-quartz and stable form a-quartz.

SiOa

{3 Quartz

/

I

^13 Tridymite

/

-a. Tridymite

a. Quartz

-/? Quartz

1^. ro a> -Hi o

> Temp. ( °C )

Figure 8. Stability relations of the silica minerals. After

C. N. Fenner [20].

24
0 9 —i

0 —r ""i 0 o 0 o o If) ^> ro o f\J <D a CO

i CD >\ !

1

|; 1 C. c ^^^^

i § 0 200 400 600 800 TEMPERATURE . °C

Figuee 9. Postulated stability diagram of Bi203.

After Levin and Roth [27\.

Polymorphs: Mon—monoclinic, C—cubic, tet—tetragonal, b.c.c—body-centered cubic, Liq—liquid. Metastable melting

point ~800 °C.

added. Further presuppose that no reaction occurs between solids, but that the impurity does react

to form a liquid solution containing both constituents. If each binary combination is viewed as an individual one component, then a series of

typical solid-liquid-vapor curves can be established for every compositional increment as given

in figure 10A. An orthogonal projection of the

pseudo one-component curves on the 100 percent

composition plane of A produces a two-dimensional

representation, figure 10B, which clearly indicates

the effect caused by the addition of the second

component. The impurity, B, reacting in the liquid

only, causes a sequential lowering of the liquidvapor pressure curves as more and more of the

second component goes into solution. Because of

nonreaction between solids, the solid-vapor pressure curve remains unchanged. Intersection of the

liquid-vapor curves with the unaltered solidvapor univariant forms a series of triple points

and consequently a number of melting point curves (1 through 5, fig. 10B). The fusion curves

are seen to occur at progressively lower temperatures, indicative of the amount of impurity added

to the pure component. The area defined by

curves 1 and 5 and the solid-vapor pressure curves

represent, in three dimensions, a volume containing both solid, A, and liquid phases.

For experimental convenience, it is customary,

and sometimes necessary, to investigate the

behavior of binary and other multicomponent

systems at a fixed pressure. The constant pressure

plane of importance to ceramics normally would

be at pressures greater than those of the binary

triple points. Graphically, figure 11 would represent an isobaric study of a binary system in

which the pressure plane intersects only the

melting and liquid-vapor curves. The melting

point of the pure component, A, at the fixed

pressure is indicated by Tx while T2 , T3 , Tiy and

T5 are melting points of the binary increments.

If the various melting points (Ti—»T5) are con- nected as shown in figure 12A, the resulting curve,

TEMPERATURE

Figure 10. Schematic representation showing the effect on the univariant curves of a one-component system by

the addition of a second component.

(A) Three-dimensional model illustrating progressive change of one-component univariant curves. (B) Projection of binary univariant curves on pressure-temperature plane of pure component A. Curve 1—melting point curve of

solid A; Curves 2, 3, 4, and 5—melting point curves of binary

mixtures.

Figure 11. Isobaric cut through melting point and boiling

point curves of binary mixtures and pure component, A.

Tt, Ti, T%, Tt , and TSl —melting points at constant pressure for pure component A and binary mixtures.

COMPOSITION 100%—

Figure 12. Schematic derivation of liquidus curve for portion

of binary system at constant pressure.

(A) Portion of three-dimensional model given in figure 11. (B) Conventional two-dimensional phase diagram for a portion of a two

component system. Ti, Tt, T3, Ti, and Ts—liquidus temperatures

at constant pressure.

25

323-655 O - 69 - 3
given on the two dimensional temperature-composition plot, figure 12B, becomes the familiar

liquidus curve found for so many phase diagrams

of binary systems.

The same general treatment can be applied to

the other end member, B, with component A now

acting as the impurity. Lower melting temperatures for B would occur with the end result being

a simple eutectic system as shown in figure 13.

The intersection of the two depressed liquidus curves defines the eutectic temperature and

composition for a given pressure.

The extent of the melting point depression of

each end member can be approximated from the

Raoult-Van't Hoff equation for dilute solutions

[2, 3]

RT2 AT=2^(tfBL) (6)

where

A T=change in melting point,

T=melting point of pure component A,

A^BL=niole fraction of impurity B in liquid

solution,

i?=gas constant,

AHf =molar heat of fusion.

Quite obviously eq (6) is based on ideal solutions

and therefore is of limited use for real systems.

General trends, however, can be anticipated from

the relationship and when applied to A1203 , for

instance, it is found that the melting point would

be depressed approximately 3° for every mole

percent contaminant. For Si02 a similar calculation indicates a change in melting point of about

80° for each mole percent of added impurity.

Thus, these data suggest that the slope of any

liquidus curve involving A1203 as the pure component would be less than that for Si02

. This fact

is borne out by experimental results on the A12 03- Si02 system [28], figure 14, which indicates a

greater depression for the melting point of Si02 than for A1203

. Systems exhibiting only liquid solution are the

ultimate in simplicity and seldom are found in

real situations. Most systems contain one or

20 40

Wt %

60 80

COMPOSITION

Figure 13. Schematic derivation of liquidus curves for binary

system A-B at constant pressure.

i—i—i

1 1 1 1 1 1

Li(l uid

i00[- Mullite

Mullite +|_iq . \ ss /Xor.

^

Liq~"X 1/ +Liq.

]B50^^/ ,940°

S 1

0

2

+ L i q

.

1800

1600

1400

1595°

Si0

2+ Mullite

Cor. + Mullite ss

V-Mullit e ss

i i SiC

20 40 60

Mol. %

80

Al

203

Figure 14. Phase diagram for the Si02-Al203

system. After Aramaki and Roy [28]. Figure

reproduced from Levin et al. [6].

more intermediate compounds, and more often

than not, some solid solubility between end

members. Compounds which melt to a liquid and

solid of different composition are termed incongruent and their presence will be reflected by

an inflection or break in the liquidus curves

indicative of the melting temperature (peritectic

point) . Congruently melting compounds are those

which fuse to a liquid of the same composition.

They can be treated in entirety as a one-component system with the melting point always

appearing as a maximum on the liquidus curve of

the full binary.

Approximations are difficult to establish concerning the effect of solid solution upon the melting

point of a compound or pure component. Solid

solution can either raise or lower the melting

point depending upon the solubility of the solute

in both the liquid and solid phase. To show the

influence of solid solution the Raoult-Van't Hoff

relationship, eq (6), can be modified to [3]:

RT2 ri-^i (7)

where, AT, R, T, and AHf have the same meaning

as described for eq (6) and

ATBL =mole' fraction of impurity B in liquid

solution,

AfB8 =mole fraction of impurity B in solid

solution.

It is readily apparent that if iVBs is zero then eqs

(6) and (7) become equivalent and a simple

melting point depression will occur. On the other

hand, if NBS is greater than NBU then the melting

point will be elevated. Figure 15 graphically

illustrates this principle by indicating an effect on

the vapor pressure curves caused by both liquid

and solid solutions. Example A shows greater

vapor pressure change for liquid solution

(A^bl>A^bs) and thus a lower melting curve. For case B the solid vapor pressure curve is

26
(A) (B)

SOLID SOLUTION^ PURE / s

J

SOLID LIQUID / /

VAPOR

SOLID

SOLID

SOLUTION

VAPOR

TEMPERATURE —

Figure 15. Schematic phase diagrams for one-component

systems showing effect of solid solution on the melting point

curve.

(A) Melting point curve lowered by solid solution. (B) Melting point curve raised by solid solution.

affected to a greater degree (NBIj <iNBS) and a

raised melting point results.

4. Pseudo One-Component Systems

Many ceramic materials seemingly behave as

single components in that they exhibit all the

overt characteristics of one-component systems,

melting at defined pressures and temperatures.

Some compounds, for example, will melt at a given temperature in one atmosphere oxygen and

simply by changing the environment to one

atmosphere of air, a different melting point will

be observed. If the course of each different heat

treatment is followed with, say, simultaneous

x-ray examination of the samples, it will become

evident that compositional changes have taken

place. Because of the environment the resulting

equilibrium cannot be described in terms of the

starting material. Titanium dioxide for instance

is known to lose oxygen upon heating in a supposedly oxidizing environment such as air. Consequently the determined melting point probably

will not be representative of stoichiometric Ti02 and the composition of interest would lie in the

Ti-02 binary system. Examination of the Ti-Ti02 diagram [29], figure 16, clearly indicates that an

Ti-0 2000

1800-

1600

1400

1200

1000

800

Ti 305 ss + Liq.

TiO, ss

200-

low - TiO s s +

high -Ti 203

s s

low -TiO s s +

low - Ti 2 03 s s

high -Ti 2 03

s s a>

in + o

Ti02

s s DUO (anatase form )

low -Ti 20 3 s s + Ti02

s s (anotose )

i i 0.2 0.4 0.6 0.8 1.0 1.2

Atomic Rotio (O/Ti)

1.4 1.6 TiO,

Figure 16. Phase diagram for the Ti-Ti02 system. After DeVries and Roy [29]. Figure reproduced from Levin et al. [6].

27
I

1

1 I I I 1 I I

1600 -

0 1i I . 1 1 I I I i -80 -70 -60 -50 -40 -30 -20 -10 0 10 LOG. Po2

Figure 17. Phase diagram for the iron-iron oxide system as a function of oxygen partial pressure

in atm. After Muan [30] and after Darken and Gurry [31, 32].

Dot-dash-lines of equal Ph2 o/Ph2 ratio of gas phase at 1 atm.

infinite number of "melting points" are possible,

all dependent upon the extent of reduction and

the final composition at fusion. Unfortunately,

the type of diagram illustrated lacks one important

feature; it gives no information concerning the

amount of reduction or change in the condensed

phases which could occur for each environment.

Knowledge of the phase relations as a function

of both temperature and oxygen partial pressures

is therefore of great importance, particularly in

those systems involving the transition metal

oxides where oxidation and reduction are common

occurrences. As an illustration, figure 17 gives

information of this type for the Fe-0 binary

system [30, 31, 32]. In this plot the stability

fields of the condensed phases, iron and iron oxides

in various oxidation states, are given as a function

of temperature and oxygen partial pressure.

Overall, the diagram has the appearance of the

typical one-component system with each boundary curve representing a univariant situation.

Although not shown, each phase is assumed to be

in equilibrium with a gas phase. If the oxygen

partial pressure were held constant say at log

Po2 ~ — 0.7 (one atmosphere air) hematite would

upon heating transform through reduction to

magnetite before melting. On the other hand, if the hematite composition were fixed, the oxygen

partial pressure would continually change during

the course of the heat treatment. It is interesting

to note that for stoichiometric Fe^ (hematite)

to melt, the oxygen pressure would have to be

much greater than 1 atm. As far as is known,

"Fe2 03 " as such has never been melted.

The phases listed in fig. 17 have been designated

as magnetite, hematite, etc. for good reason. The

psuedo one-component diagram provides no information on the exact compositions (Fe/O ratios) of

the condensed phases. Wiistite, for example, can vary in composition depending upon the temperature and oxygen partial pressure. The same

is true for hematite and magnetite. To alleviate

this problem the conventional binary plot can be

28
1600

CI

1400

P.

UJ

or

h—<

orUJ

CL

UJ

1200

1000

800

600

-LIQUID (METAL)

LIQUID^METALh LI0UID

2 (OXIDE)

H .

MAGNETITE

+ LIQUID

2 io°\ HEMATITE

J-IRON + LIQUID (METAL)

<f- IRON + LIQUID (OXIDE)

r-lRON+WUSTITE

«-lR0N+WUSTITE - I0" 20 -

I 1 10' 30

400 -

200

<- 1 RON + MAGNETITE

10 40

Fe

20 22 ^FeO

WT.% OXYGEN

24 26

— io" 4- H

MAGNETITE

+ HEMATITE

10" -

10 20.

io" 30 H

10,-0 -

^28-

FeOFe

2

0

3

HEMATITE

FeO

2 3

Figure 18. Phase diagram of the Fe-Fe203 system. After Darken and Gurry [31, 32] and after Muan [30]. Heavy solid lines are boundary curves separating labeled phase areas. Dash-dot lines are 02 isobars in atm.

combined with the one-component representation

in a manner as shown in figure 18 [30, 31, 32].

Here, various oxygen isobars are plotted across compositional lines and indicate the pressure of

the gas in equilibrium with the condensed phases.

Each two-phase region is equivalent to the

univariant boundary curves given in figure 17.

Single-phase areas in addition to being defined

by temperature and oxygen pressure, are also

located with respect to composition.

Some exceedingly useful information can be

gained from proper utilization of this type of

diagram. Certain rules, however, must be adhered

to in its use. If the oxygen partial pressure is held

constant, the path or course of events for a

particular composition would follow that given

isobar. For instance, Fe304 (FeO-Fe203), upon

heating from room temperature in a gas of 10-4

atm oxygen partial pressure and one atmosphere

total pressure, would oxidize at low temperatures

until Fe2 03 is formed. This is necessary because

the IO-4 atm isobar does not intersect the Fe304 composition at low temperatures, thus making the

phase metastable with respect to Fe203

. Further

29
increases in temperature would cause a subsequent

reformation of magnetite ("Fe3 04 ") but of a

different composition than the starting material.

The magnetite phase would continue to change in

composition with temperature until melting oc- curred. At the melting point the solid (magnetite)

would be in equilibrium with a liquid only at one

temperature for the 10-4 atm isobar, thus giving

the appearance of congruently melting. If the

oxygen partial pressure were altered, a different

"congruent" melting temperature would be

observed. In these cases, congruent melting is a misnomer since the solid and liquid in equilibrium

do not have the same composition.

5. Multicomponent Systems

To the novice, the phase diagram of a multicomponent system may appear as a quagmire of

points, curves, areas, and polyvariant situations.

Multicomponent systems present a problem not

only in their investigation but also in the mere

graphic representation of their equilibrium relations. As stated previously, the independent

variables of any system are pressure, temperature,

and composition (of n components), each requiring

an axis in space for pictorial representation. It is apparent that a major problem exists in representing all variables in three-dimensional space.

A plane, two-dimeDsioual figure is suitable for a one-component system with an axis each for

pressure and temperature. Likewise, a binary

system can be depicted in two dimensions if one

of the variables is fixed; or in three dimensions, if both the pressure and temperature variables are

utilized. Ternary systems can be illustrated fully

by a three-dimensional model only by fixing the

pressure (or temperature) and setting up a co- ordinate system of temperature and the concentration of the three components. For systems

containing four components graphic representation is impracticable because both pressure and

temperature then must be held constant. Thus,

for a system of 3-i-n components a complete

graphic model cannot be formulated since 3+n

dimensional space would be required. Fortunately,

phase equilibrium relationships between materials

are such that the multicomponent system can be

viewed as groups of interrelated limiting boundary

systems consisting of a lesser number of components. The four-component system, A-B-C-D,

for example, can be thought of as the four boundary ternary systems; [A-B-C], [A-C-D], [A-B-D],

and [B-C-D]. These ternary systems in turn can be broken down into other boundary subgroups

which are more conducive to investigation.

For apparent reasons, the ternary system plays

a prime role in the understanding of more complex

systems. Similar to binary mixtures, the ternary

system also can be developed using one-component

systems as basic building blocks. Consider first,

three one-component systems A, B, and C, which

react to form only liquid solution. The individual

Figure 19. Schematic derivation of the three- component system, A-B-C, from individual

one-component systems.

X—Isobaric lines.

systems can be arranged about a common temperature axis 120 deg apart in the manner illustrated

by figure 19. Lines of equal pressure (X) drawn to

intersect the univariant curves, in essence fix the

pressure variable at the same value for each one- component system. By connecting the isobaric

lines with planes of constant pressure a threedimensional model (figure 20) is obtained which

relates the three components with temperature

at constant pressure. Points TA, TB, and Tc indicate the melting temperatures of components

A, B, and C for that given pressure. The resulting

Figure 20. Schematic derivation of the three- component system, A-B-C, showing the three-dimensional relationships between

components at constant pressure.

Ta, Tb, and Tc—melting points of components A, B, and C, respectively, at constant pressure.

30
model, more clearly shown in figure 21 A, consists

of three planes, each representing a limiting

boundary binary system at constant pressure,

which intersect at 60 deg. Temperatures are indicated by the vertical axis. Concentration of

each end member can be located by the coordinate

system shown by the equilateral triangle projected

below the solid figure. The apex and corresponding

base of the triangle represent 100 percent and

0 percent, respectively, of the designated component. Point 4, for example, would have an overall

composition of 40 percent A, 40 percent B, and 20

percent C.

As described before, the melting point of each

component in binary combination will be depressed

to form a eutectic. Figure 2lB indicates the

liquidus curves for the three binary systems with

points 1, 2, and 3 locating the binary eutectic

compositions and temperatures. The addition of

the third component to a binary combination will cause further depression of melting points, thus

forming a series of curved liquidus surfaces meeting at a common ternary eutectic point. The

complete ternary model is given in figure 21C.

Projection of the various important points to the

base of the solid model results in the two dimensional plane triangle given below the triangular

prism. Point 4 is the ternary eutectic and as before

points 1, 2, and 3, the binary eutectics. Arrows

indicate the direction of falling temperatures.

Although not shown, temperatures on the liquidus

surface can be indicated by contour lines similar

to those designating elevations on topographic

maps. The entire liquidus surface can be seen as a group of individual surfaces, each indicating a primary phase field. Upon cooling from a melt the

primary phase would be the first solid phase to

(A) (B) (O

B B B

B B B

Figure 21. Schematic representation of the three-component system, A-B-C, in various stages of develop- ment. Upper figures indicate three-dimensional models;

lower, the projection of important points on base.

T\, TB,and Tc—melting points of components A, B, and C. Numerals (1, 2, 3, and 4) —eutectic temperatures and compositions. Curves

1-4, 3-4, and 2-4—boundary curves separating primary phase areas. Arrows—indicate direction of falling temperatures.

crystallize. The intersection of the individual

surfaces form a series of boundary curves 1-4,

2-4, and 3-4. The projection shows clearly which

primary phase is in equilibrium with the melt

along the liquidus surface. Phase A, for illustration, will be in equilibrium with liquid for those

compositions in area A-l-4-3. Along univariant curve 1-4, phases A, B, and liquid will coexist.

Likewise, curves 3-4 and 2-4 separate primary

phases A and C, and B and C, respectively. At

point 4 (invariant) all three components plus

liquid will comprise the equilibrium assemblage.

As a consequence of Raoult's law, congruently

melting compounds will always melt at temperatures higher than those of adjacent compositions

(excluding some solid solutions) . Thus, the liquidus

surface will appear as a dome or mountain with

the peak occurring at the composition of the

compound. On the other hand an incongruently

melting compound will produce a terrace effect in

the liquidus surface. A general guide to the

behavior of ternary liquidus surfaces is given by

the Alkemade theorem [6] which in essence states

that the direction of falling temperatures on a boundary curve is always away from the straight

line (Alkemade line) connecting the composition

points of adjacent primary phase fields. The point

of intersection of an Alkemade line and a boundary

curve or its extension will be a maximum on that

boundary curve. In figure 21C (lower triangle)

line AB (or AC and BC) can be thought of as an

Alkemade line. The intersection of AB with

boundary curve 1-4 at point 1 is seen to be at a

higher temperature than any ether point along

1-4. Similarly points 2 and 3 must represent the

highest temperatures on boundary lines 2-4 and

3-4, respectively.

Figure 22 (upper) gives six hypothetical systems

which demonstrate typical situations encountered

in ternary mixtures. The middle set of binary

systems indicate the phase relations between the

two base end members of the ternary system.

The lower diagrams illustrate binary representations of segments or joins (dashed lines) through

the ternary systems. With the exception of system

A, the join (dashed line) is coincident with an Alkemade line. Temperatures are indicated on

the three sets of diagrams by numerals.

The ternary system A, is a simple eutectic

system similar to that developed for figure 21.

The binary system, A-B, exhibits elementary

eutectic behavior while the join, X-C, does not.

Line X-C fails to meet the requirement of an

Alkemade fine in that composition X has no primary phase field. Consequently point 5 will not

be a maximum along boundary curve 1-4 or along

the extension of curve 2-4.

The remainder of the ternary systems, B

through F, are somewhat more complicated with

each containing an intermediate binary or ternary

compound. System B indicates a congruently

melting compound AB and two ternary eutectics.

In this case line AB-C (Alkemade line) represents

31
9; A BAAB BAAB BAAB BAAB B A B

X ABC C AB C AB C AB C AB C AB ABC

D E F

Figure 22. Typical three-component systems of various complexity.

Upper—After Levin et al. [6]. Arrows indicate direction of falling temperatures. Middle—Phase diagram of one binary system included in respective ternary system (upper). Numerals indicate temperatures of points corresponding to those given in respective ternary systems. Lower—Binary representations of joins through ternary systems (dashed lines). Systems B and F (lower) are true binary systems.

a true binary subsystem and triangles A-AB-C

and AB-C-B true subternaries. Each subsystem

may be considered individually, independent of

the overall larger system. The distinction here

between true and pseudo system becomes important. A true system is one in which all primary

phases formed in that system can be expressed in

terms of the end member components. As illustrated by the binary diagram, AB-C (figure 22B,

lower), the primary phases are either AB or C

and thus AB-C is a true system.

Conversely, system C, figure 22, indicates a

situation in which compound AB does not form a

true subsystem with component C. The diagram

of the AB-C join (Alkemade line) shows this

clearly. Phase A which occurs throughout cannot

be formed from any combination of AB and C,

thereby relegating system AB-C to a pseudo

binary status. Furthermore the AB-C pseudo

system yields three phase equilibrium at regions

other than invariant points, a situation which is not allowed by the phase rule for two components.

It should be noted that points 7 and 8 will not

be temperature maxima of curves 1-5 and 4-5,

respectively. This is due to the fact that the

Alkemade theorem is applicable only to boundary

curves of adjacent primary fields. The intersection

of line AB-C with the extension of the C and AB

boundary curve, line 5-6 will be at a maximum.

Thus, point 5 in the ternary system C represents

a ternary peritectic and point 6 a ternary eutectic.

System D, figure 22, illustrates an incongruently

melting compound, AB, with its primary phase

field indicated by area 3-4-5-6. Again, point 5 is a peritectic and point 6 a eutectic. As before in

system C, the subsystem AB-C is not a true

binary.

System E indicates a situation not at all uncommon for ceramic systems. The binary compound, AB, decomposes before melting to solids

A and B. In the ternary system, however, AB is seen to be the primary phase in equilibrium with

liquid in field 4-5-6. Both points 4 and 6 are

peritectics while point 5 is a ternary eutectic. It

is evident that the appearance of AB as primary

phase in ternary mixtures offers interesting

possibilities for single crystal growth not possible

in the binary system, A-B, alone.

System F, figure 22, contains two intermediate

congruently melting compounds, ABC (ternary)

and AB (binary). The larger system, A-B-C, can be subdivided into four true ternary systems,

A-ABC-AB, A-ABC-C, ABC-C-B, and ABCB-AB. Each sub-binary in turn is a true system.

System AB-ABC (lower, figure 22), typical of

each subsystem, illustrates only eutectic configuration.

Often it is difficult to visualize the distinct

relationships which exist between various subregions of multicomponent systems. Levin et al.

32
[6] have formulated a useful graph, reproduced

in figure 23, which aids immensely in clarifying

matters. For a complete discussion of the figure

reference must be made to the original article.

In essence the figure basically relates the variance

and its geometrical representation to the number

of solid phases (squares) in equilibrium with

liquid for a given number of components in a condensed system. As a simple example, the figure

indicates that at a eutectic point in a condensed

ternary system, three solid phases and liquid will

be in equilibrium. Similarly the liquidus surface

of a ternary is a divariant situation in which one

solid phase and liquid are in equilibrium. The

vertical arrows and numbers in parentheses

indicate the interrelationships between regions of

any given system. For illustration, in a ternary

system two divariant surfaces will intersect to

form an univariant curve. Three univariant curves

in turn will intersect to form an invariant point

(eutectic) at which three solid phases are in equilibrium with liquid. Finally, the diagonal arrows

relate the origin or terminus of regions at boundary systems. For instance, a univariant curve in a ternary system will originate (or terminate) at an invariant point of a binary boundary system.

In essence the region loses one degree of freedom

because the boundary system always has a lesser

number of components.

Degrees

of Freedom

0

Invariant

I Univariant

2

Divariant

3 Invariant

Tetravariant

Pentavariant

F + P = C + I =

Geometrical

Representa -

tion

Point

Curve

Surface

Volume

None

(4 dimensional )

None

( 5 dimensional )

NUMBER OF SOLID PHASES IN EQUILIBRIUM WITH LIQUID IN SYSTEMS:

Binary Ternary

Hi

Quater- nary

Quinary

4 5

(4) \ 15)

3 4

(3) _N(4)

n-i

n-2

ln-2)

n-3

n- 4

n-5

n + i

Figube 23. The interrelationships of subregions of multicomponent condensed systems at constant pressure. After

Levin et al. [6].

procedures in establishing the melting or freezing

point of a material. All first order transitions

such as the solid to liquid and solid I to solid II

transformations, produce an energy change which

is reflected by a corresponding change in the heat

content of the system. Heat will be absorbed from

the surrounding areas upon heating, and conversely, will be evolved during cooling. At the

transformation point, simultaneously with the

heat change, there will also be a discontinuous

change in a number of physical properties, as for

example, thermal expansion or electrical resistivity. Basically the thermal analysis method

utilizes these changes to detect the occurrence of

the transition. Once detected, the problem re- solves itself to one of the accurate measurement

of the temperature at which these overt changes

indicative of melting take place.

The thermal analysis method most commonly

employed makes use of the change in heat content

inherent in the solid to liquid transition. Generally, the experimental arrangement simply consists

of a crucible containing the sample in which a temperature measuring device has been centrally

located. A continuous record during controlled

heat treatment of the specimen will result in a time-temperature plot commonly referred to as a heating or cooling curve (direct method). If no

transition takes place during cooling the curve

will be smooth and continuous similar to that

shown by A, figure 24. At the freezing point heat

will be evolved thus causing an arrest or halt in

the cooling process. For ideal situations the

temperature of the specimen will remain constant

until the last trace of liquid has solidified, at which

point further cooling takes place. The freezing

point will be reflected by that portion of the

cooling curve which is flat and parallel to the

time axis (B, figure 24). Usually, the solidification

rate of most materials is slow enough to cause some supercooling, the end result being a cooling curve illustrated by C, figure 24. If the rate of

cooling exceeds the rate of evolution of heat the

true freezing point will not be indicated by the

F.P.

TIME

6. Methods

6.1. Dynamic Methods

Tradit: onally, the thermal analysis method has

been widely accepted as one of the more desirable

Figure 24. Typical thermal analyses cooling

curves. Essentially after Weber [36]. A—no transition, B—theoretical cooling curve with freezing arrest, C—cooling curve illustrating supercooling,

D and E—imperfect cooling curves which give false indication of freezing point.

33
cooling curve. Curves D and E, figure 24 exemplify

this type of behavior with both curves yielding a freezing point less than the true value.

Freezing and melting points are numerically

equal, the terminology being merely a reflection

of the experimental approach to the solid-liquid

transition. Heating curves generally are less well

defined than the corresponding cooling curves. A

cooling liquid is more likely to possess a uniform

temperature because of greater thermal conductivity throughout the melt. Furthermore on heating, small variations in power supplied to the

furnace will cause abnormalities in the heating curve not related to melting. Supercooling, especially for ceramics, can be troublesome in utilizing

cooling curves. On the other hand, superheating

does not occur, at least readily, thus making the

heating curve method more desirable in this re- spect. The choice between a heating or cooling curve as a means of establishing the transition

point is strictly material dependent. Figures 25

[33] and 26 [34] give examples of experimental

determinations of the freezing points of gold and

several ceramics. Figure 27 [35] shows both the

heating and cooling curve obtained for alumina

(A12 03). Here supercooling is so pronounced that

the observed freezing point of A1203 is some 200

degrees less than the observed melting point.

The thermal analysis method, of course, is not

limited to the study of pure materials. It is equally

adaptable to binary and other multicomponent

systems. Figure 28A, curves I and II, give illustrations of cooling curves for a pure component and

a mixture. Line A-B, curve I, represents the freezing temperatures for the pure component while

the inflections C and D indicate the liquidus and

solidus temperatures, respectively, for the mixture. The temperature interval between C and D

indicates the region in which both the solid pure

component and liquid containing the mixture are

in equilibrium. Obviously, it is possible to construct a phase diagram from thermal analysis data

105.00

o

a 104.60 S

<

_i

104.20

1

\

1

1 1 I

1

1 1 1 1 1 I ! 1 1

jo.5°C

0

O ^—

0

0 >^

1 1 1 > 1 1 1 1 1 1 1 1 0 10 time, min

Figure 25. Cooling curve of gold. After Kostkowski and Lee

[33].

3000

2800

Tempera -urt C

V

2600

\ CaO

2500

2400

\ SrO

2300

2200

2100

BaO

1900

0 2 6 \

Temps

CsecJ

8

Figure 26. Cooling curves of several ceramics. After Foex [34].

1

- 0

0- O000 —I P-U.J

-- O

_£N -

- «—

ix

e

°c0a

>-

>

n U—

1

r 0 -1

- if)

-—

1 ^ c

: E

in

Figure 27. Heating and cooling curves of A12 03 . After

Gitlesen and Motzfeld [35]. Heating curve begins at right axis; cooling curve begins at maximum

temperature. Sharp vertical breaks in curve are due to pyrometer

measurements.

34
A TIME

1 2* 4 S 6789

B TIME

Figure 28. Typical cooling curves as applied to a binary mixture. After McNamara [42]. A—Curve I—pure component; Curve II—a mixture,

B—relationship between cooling curves and a binary

phase diagram. Curves 1 and 9—pure components; Curves 2 through 8—mixtures. Time axis and composition axis are superimposed.

on various compositional increments. The relationship between these data and the phase diagram

for a simple eutectic system is given by figure 28B.

Curves 1 and 9 indicate the cooling behavior for

the end members while curves 2 through 8 represent similar data for intermediate compositions.

Normally the heat of fusion (or crystallization)

is not large in comparison with the total heat

content of the system; consequently, the melting

or freezing arrest usually is not very pronounced.

To alleviate this problem, various alternate

methods have been devised to enhance the inflection points. In all, there are four principal methods

which are used to obtain thermal data [36, 37]

1. Direct (heating and cooling curves),

2. Inverse rate,

3. Differential (D.T.A.),

4. Derived differential.

The direct and inverse rate procedures differ only

in the method which is used to record the data.

The same experimental apparatus is appropriate

for both. The inverse rate method consists of

determining the time required for the specimen

to change a definite temperature interval, which

in effect, amounts to differentiating the temperature-time curve (direct method) with respect

to temperature. Methods 3 and 4 require the same

experimental setup and, as before for methods 1 and 2, differ only in the way the data is collected.

The differential method records the difference in

temperature between the specimen and an inert

body which undergoes no transformation. These

differences are plotted against either the temperature of the specimen or that of the furnace cavity.

The derived-differential method (4) consists of

differentiating the differential curve with respect

to temperature. Typical examples [36] of the four

types of plots are given in figure 29. Each indicates

the relative sensitivities in estabhshing the same

transition point.

Various other dynamic methods have been

devised for melting point determinations [37, 38,

39, 40, 41]. Although these methods differ somewhat in detail, conceptually, they all involve

visual observation of a specimen during heat

treatment. The temperature at which a specimen

is seen to fuse is taken as the melting point. Here,

the measurement of temperature becomes the

major obstacle in obtaining an accurate value.

The direct thermal analysis method is generally

more conducive to accurate temperature measurement in that a blackbody can be built into the

experimental setup, thereby permitting a realistic

assessment of true temperatures. The other

dynamic methods depend on the ability of an

observer to distinguish a specimen from surrounding areas to detect melting. This requirement is

not compatible with the achievement of the

1 b /c [d

At / ( AT

\ aT 1 aT \ T

Figure 29. Four methods

of thermal analysis.

After Weber [36].

a—direct b—inverse rate c—differential (D.T.A.)

d—derived differential T=temperature

i=time

35
blackbody conditions as a specimen located in a blackbody enclosure for all practical purposes

becomes invisible. The obvious recourse has been

to allow the necessary deviation from perfect

blackness to see the specimen and to incorporate

a temperature correction factor obtained from

emissivity data or from in situ calibration.

Published emissivity data as well as some calibration procedures, however, are not always reliable

and are generally not applicable to specific

experiments.

6.2. Static Methods

The quenching method [6, 42, 43, 44] is the one

procedure specifically developed for study of

ceramic cystems. The method relies upon the

fact that many reactions in ceramics are sluggish

and may be arrested or "frozen in" by rapid

cooling or quenching. In general, a small homogeneous charge is placed in a container which in

turn is heated at a constant temperature until

equilibrium is attained. The specimen is then

quenched by dropping the container into a cooled

liquid or alternatively, by rapidly cooling the

furnace. Examination of the cooled specimen by

various means reveals the degree of melting.

If the specimen shows no signs of melting, additional heat treatments are performed at successively higher temperatures. After a series of

heatings the melting point can be established as that temperature midway between the maximum

temperature at which no melting occurred and

the minimum temperature where fusion was

complete. Quite obviously the success of the

method depends upon the ability of the examiner

to establish the degree of melting the specimen

had undergone. For glass-forming materials evidence of fusion can be detected by microscopic

techniques which will identify the presence of

liquid by the appearance of isotropic material

(glass). For nonglass formers the problem of

liquid detection becomes more formidable.

Physical appearance of the cooled specimen is a

useful guide. A sample which had lost its original

shape and flowed to surrounding areas can, of

course, be considered completely melted. Unfortunately, many ceramics when melted tend

to form viscous liquids and consequently little

slumping or flow occurs. In these cases the

problem is minimized if single crystals are used

as test specimens [45] . Unless melted, the single

crystal will retain its identity, that is, remain

single. If fusion occurs the cooled specimen will

appear as a polycrystalline mass which is readily

identifiable.

7. Temperature Measurement

7.1. Temperature Scales

Perhaps the greatest difficulty inherent in the

determination of a melting point is the accurate

measurement of temperature. The thermodynamic

temperature scale [46] is the fundamental scale,

to which all temperatures should be referred.

The scale is based on the second law of thermodynamics which allows complete definition by

fixing one thermodynamic temperature. At present

this temperature is designated as exactly 273.16

K, the triple point of water [46]. Because of

experimental problems associated with the measurement of temperatures on the thermodynamic

scale (gas thermometry), a practical temperature

scale was established in 1927 by international

agreement [47]. A major revision of the 1927

scale was made in 1948 [48] with subsequent

text revision in 1960 [46]. Further changes in

this scale are anticipated to make it conform

more closely to the thermodynamic scale. The

International Practical Temperature Scale of

1948 (IPTS) [46], the scale now in use,

3 in essence

represents a realistic experimental approach to

the ideal thermodynamic scale.

Currently six points define IPTS between the

lowest assigned value, the oxygen point (—182.97

°C) and the highest, the gold point (1063 °C).

Certain thermometric instruments and formulas

are specified as a means to realize these fixed

points and intermediate temperatures. Above

the gold point, temperatures are based upon the

Planck relationship

where,

JAu and Jt =radiant energies per unit wavelength emitted by unit area of a blackbody at temperature t and

the gold point Au (1063 °C),

\=wavelength,

C2 =second radiation constant (1.438

cm deg),

jT0 =273.15 deg.

Although no instrument of measurement is designated by IPTS for temperatures above 1063 °C,

the optical pyrometer has been traditionally

utilized for this purpose. In addition to the six

primary points IPTS recommends certain other

points which may be used for calibration purposes.

Table 1 summarizes important features of the

practical scale.

7.2. Instruments

Since IPTS specifies certain instruments it is generally advisable to limit their use to the designated temperature ranges. This is not always

possible, however, and frequently the investigator

must extend the range of the instrument beyond

that stated by IPTS. Of prime importance to the

study of ceramics are the (1) thermoelectric, and

(2) radiation pyrometers. The theory, use, and

calibration of these temperature measuring dea Recently a new scale (IPTS 1968) was adopted, official text of which will appear in Metrologia, April 1969.

36
Table 1. Fixed and secondary points on the International

Practical Temperature Scale of 1948 1 [46].

Material

Oxygen _ Carbon dioxide Mercury

Water _._ Water Phenoxybenzene

Sodium sulphate decahydrate _ Water.. Benzoic acid— _ Indium

Naphthalene

Tin

Benzophenone

Cadmium

Lead. Mercury _ Zinc... _. . Sulfur Antimony.

Aluminum

Silver _ Gold.. Copper

Nickel Cobalt Palladium Platinum _ Rhodium

Iridium Tungsten

Property 2

B.P.

S.P. F.P.

F.P. T.P.

T.P.

Tr.P. B.P.

T.P.

F.P. B.P.

F.P. B.P.

F.P.

F.P. B.P.

F.P. B.P.

F.P.

F.P.

F.P.

F.P. F.P.

F.P.

F.P.

F.P.

F.P.

F.P.

F.P. M.P.

Temperature, "C.

Primary

point

-182. 97

+0. 01

"166

444.

6

960.8

1063

Secondary

point

-78.5

-38. 87 0.000

26.88

32.38

122.36

156.61 218.0

231. 91 305.9

321. 03 327.3

356.58

419. 505

630.5

660.1

1083

1453 1492 1552

1769

1960 2443 3380

1 Certain instruments of measurement are specified as a means to realize the scale between different fixed points. These are: —182.97 to 630.5 °C, platinum resistance thermometer; 630.5 to 1063 "C, platinum versus platinum10% rhodium thermocouple; 1063 °C and above, no designated instrument

although an optical pyrometer is generally utilized.

2 The symbols B.P., S.P., T.P., F.P., M.P., and Tr.P. refer to boiling

point, sublimation point, triple point, freezing point, melting point, and

transition point, respectively. All points except the triple points applicable

when determined under one standard atmosphere. The copper point re- quires a reducing environment.

vices have been well documented [33, 36, 49,

50, 51, 52, 53, 54].

Thermoelectric pyrometers, thermocouples, are, of course, the mainstay of the ceramic industry.

Quite simply, thermocouples consist of two dissimilar electrical conductors joined together at

their ends. A temperature difference between the

two junctions will produce an emf which is characteristic of the type of material and of the

magnitude of the temperature gradient. Calibration of a particular thermocouple at several

fixed and secondary reference points will yield an

emf versus temperature plot generally in the

form of

E=a+bM+c(M)2 where,

E=emf,

At= temperature of hot junction minus

temperature of cold junction,

a, b and c= constants.

The types of thermocouples available are many

and varied [50, 52, 53]. Generally it is advisable

for the thermocouple to have as large a thermoelectric power /dE\ as possible. Resistance to de-

\dtj

terioration by oxidation or reduction is also of

great importance. The melting point of the lowest

melting member will set the maximum temperature

of use. However, other factors generally limit its

utility to lower values.

Radiation pyrometry can be divided into two

categories: (a) total, and (b) spectral. Total

radiation pyrometry employs radiation from a large part of the spectrum according to the StefanBoltzmann relationship

J=eToT (10)

where,

J= total radiant flux per unit area per second,

T— temperature (K),

eT =total emissivity,

cr=Stefan-Boltzmann constant.

The radiation pyrometer in effect collects the

energy emitted by a definite area of a hot body

withm a limited solid angle and converts that

energy to some measurable quantity such as emf.

The emf or other deflection, D, of the indicating

instrument is related to temperature by:

D=c(T*-T0*) (11)

or D~cTn (when T»T0) (12)

where,

D=deflection of indicating instrument,

7

1 =blackbody temperature (K),

T0 =pyrometer reference temperature (K),

c= empirical constant,

n=exponent, approximately equal to 4.

For a nonblackbody the total radiation pyrometer

will indicate temperatures lower than the actual

target temperatures. Observed temperatures are

related to the true temperature by:

T0 =T,fc (13)

where,

T0 =observed temperature (K),

T,=true temperature (K),

e t = total emissivity.

Spectral pyrometry uses only a narrow region

of the spectrum, traditionally about 0.65 (im.

The spectral pyrometer compares the intensity of

filtered radiation emitted from a target with that

of a comparison source of known but variable

temperature. Since its inception, this type of

pyrometer has utilized the human eye as the

means to compare the two intensities and thus

has become known as an optical pryrometer. Of

late, however, a photoelectric tube has been used

for this purpose [45, 51]. The comparison source

is generally a lamp filament contained within the

pyrometer. By varying the current of the lamp

filament its intensity can be adjusted until a match in brightness is obtained with the target.

37
Calibration of the pyrometer against targets of

known temperatures gives an equation:

T=a+bI+cP+dP (14)

where,

T=blackbody temperature (°C),

7=pyrometer lampcurrent,

a, b, c and d= empirical constants.

In spectral radiation pyrometry the measuring

instrument determines a brightness temperature,

not the true temperature of an incandescent body.

Using Wien's law (an approximation) the relationship between these two temperatures is given by:

where,

T^true temperature (K),

T0 =observed (brightness) temperature (K),

\e = effective wavelength,

Ct =second radiation constant,

ex =spectral emissivity.

7.3. Blackbodies

Radiation pyrometry plays a prominent role in

the determination of the melting point of ceramics

primarily because of the high temperatures involved. The theory behind this type of pyrometry

is well established and easy to utilize. Equations

13 and 15 of the previous section relate the true

temperature (Tt ) of a body with the observed

temperature (T0) as monitored by a total and a

spectral pyrometer, respectively. Quite obviously,

for a given wavelength band or effective wavelength, the difference between Tt and T0 may be

rather large, depending upon the emissivity. For

example, if the true temperature of a material

having a total or a spectral emissivity of 0.5 is

2000 °C, an optical pyrometer would indicate

about 1880 °C or 120 °C too low. Similarly, for

the same situation, a total radiation pyrometer

would read a temperature of about 1640 °C or be

in error about 360 °C.

Emissivity is known but for a few materials;

thus, a real problem exists in the accurate measurement of temperatures by radiation methods. The

only practical solution available is to incorporate a blackbody into the temperature measuring system

.

By definition, a blackbody is a substance or object

which absorbs all incident radiation, and reflects or transmits none. A blackbody is also a perfect

radiator, emitting the maximum possible energy

as the result of temperature alone. Consequently,

the emissivity of a perfect blackbody is equal to

unity. It is evident, then that if a radiation

pyrometer is sighted on a blackbody the observed

temperature would necessarily be equal to the true

temperature.

Fortunately, experimental blackbodies can be

constructed so that their effective emittances are

sufficiently close to unity as to make errors in

temperature measurement small. An experimental

blackbody generally consists of a uniformly

heated opaque enclosure containing a small aperture. The enclosure may be any shape but usually

is in the form of a sphere, cylinder, or closed end

tube. Providing isothermal conditions exist, radiation within the enclosure, after numerous reflections, emerges through the hole as nearly black.

All interior details of the enclosure are indistinguishable. The smaller the area of the aperture

in comparison with the total internal area of the

enclosure, the greater the emittance. The omittance of any enclosure supposedly built as a blackbody is extremely dependent upon the nature

of the internal reflections. Even enclosures in

which the geometric design is correct, can have low

emittances if the internal reflections are specular

rather than diffuse. A roughened internal surface

aids in this respect.

A perfect blackbody, like a perfect gas, represents an ideal situation unobtainable in practice.

Therefore, an estimate must be made of the quality

of the experimental blackbody. Excellent papers

by several authors [55, 56, 57, 58] give methods of

evaluating the effective emittances of blackbodies

having different configurations. It should be emphasized that these estimates can be valid only for

enclosures which are uniformly heated. If thermal

gradients exist along any enclosure there is little

possibility that its effective emittance can approach

the theoretical limit of unity.

8. Melting Point Standards

Numerous ceramics have been utilized throughout the years as high temperature melting point

standards. Materials such as A12 03 , Cr2 03 , MgO,

NiO, CaMgSi203 (diopside), 15%CaO : 85%Si02 , and CaSi03 [59] all, at one time or another, have

been used, often as fixed points on practical

temperature scales [37]. These materials, however,

only can be regarded as pseudostandards for there

is no common agreement as to their exact melting

point. There is such a wide divergence between

published melting points for the same material [59]

that the user more often than not selects a value

that best conforms with his own experimental data.

In reality, the only materials acceptable for use as high temperature standards are those having internationally recognized melting points such as given

by the International Practical Temperature Scale.

Unfortunately, the materials specified by IPTS

which melt above the gold point are too few in

number and generally not adaptable to a wide

variety of experimental conditions. Furthermore,

many of the melting points were established many

years ago [60, 61, 62, 63] and thus a reevaluation

might be in order. The need for additional secondary reference points is all too apparent. The metal

oxides as well as other ceramics, indeed, warrant

consideration for this purpose.

38
Many thanks are due to Dr. T. Negas and other

members of the phase equilibria group, High

Temperature Chemistry Section, National Bureau

of Standards, for their helpful discussions and

suggestions which aided greatly in the preparation

of the text. 9. References

[1] Bowden, S. T., Phase Rule and Phase Reactions

(Macmillan and Co., London, 1938).

[2] Findlay, A., Campbell, A. N., and Smith, N. D., The

Phase Rule and Its Application, 9th ed., 494 pp. (Dover Pubis., Inc., New York, 1951).

[3] Darken, L. S., and Gurry, R. W., Physical Chemistry

of Metals, 535 pp. (McGraw-Hill Book Co., New

York, 1953).

14] Gibbs, J. W., Trans. Conn. Acad. Sci. 3, 108-248,

343-525 (1874-78); also Gibbs, J. W., The Collected Works of J. Willard Gibbs 1, 54-371 (Longmans, Green and Co., New York, 1928).

[5] Knapp, W. J., J. Am. Ceram. Soc. 36, 43-47 (1953).

[6] Levin, E. M., Robbins, C. R., and McMurdie, H. F., Phase Diagrams for Ceramists, 601 pp. (Am.

Ceram. Soc, Columbus, Ohio, 1964).

[7] Marsh, J. S., Principles of Phase Diagrams, 1st ed.,

193 pp. (McGraw-Hill Book Co., New York, 1935).

[8] Morey, G. W., The Glass Industry 12, No. 4, 69-80

(1931).

[9] Newton, R. F., Science 79, 321-322 (1934).

[10] Ricci, J. E., The Phase Rule and Heterogeneous

Equilibrium, 505 pp. (D. Van Nostrand Co., Inc., New York (1951).

[11] Slater, J. C, Introduction to Chemical Physics, 3-32,

166-182, 256-307 (McGraw-Hill Book Co., New

York, 1939).

[12] Turnbull, D., Phase Changes, Solid State Physics,

Advances in Research and Application, 225-306

(Academic Press Inc., New York, 1955).

[13] Wetmore, F. E. W., and Leroy, D. J.. Principles of Phase Equilibria, 1st ed., 200 pp. (McGraw-Hill

Book Co., Inc., New York, 1951).

[14] Zernike, J., Chemical Phase Theory, N. V. UitceversMaatschappij, 494 pp. (Ae. E. Kluwer DeventerAntwerp-Djakarta, 1955).

[15] Osborn, E. F., and Muan, A., Phase Equilibrium Diagrams of Oxide Systems, Plate 1, Large Scale

diagrams (Am. Ceram. Soc. and Edward Orton

Ceram. Foundation, 1960).

[16] Lea, F. M., and Desch, C. H., The Chemistry of

Cement and Concrete, 2nd ed., 52 (Edward

Arnold and Co., London, 1956).

[17] Sosman, R. B., The Phases of Silica, 388 pp. (Rutgers

University Press, New Brunswick, New Jersey,

1965).

[18] Ostrovsky, I. A., Geol. J. 5, No. 2, 321-328 (1967).

[19] Boyd, F. R., and England, J. L., J. Geophys. Res.

65, No. 2, 752 (1960).

[20] Fenner, C. N., Am. J. Sci., 4th Ser. 36, 338 (1913);

and Greig, J. W., Am. J. Sci., 5th Ser. 13, 1-44

(1927).

[21] Ubbelohde, A. R., Melting and Crystal Structure

(Clarendon Press, Oxford, 1965).

[22] Simon, F. E., and Glatzel, G., Z. Anorg. u. Allgem.

Chem. 178, 309 (1929).

[23] Babb, Jr., S. E., Rev. Mod. Phys. 35, 400-413 (1963).

[24] Bundy, F. P., Science 137, 1057-1058 (1962).

[25] Pistorius, C. W. F. T., J. Phys. Chem. Solids 28,

Pergamon Press, Great Britain, 1811-1819 (1967).

[26] Sarver, J. F., and Hummel. F. A., J. Am. Ceram.

Soc. 43, 366 (1960).

[27

[28

[29

[30

[31

[32

[33

[34

[35

[36

[37

[38

[39

[40

[41

[42

[43

44

45

[46

[47

[48

[49

[50

[51

[52

[53

[54

[55

[56

[57

[58

[59

[60

[61

[62

[63

Levin E. M., and Roth, R. S., J. Res. NBS 68A

(Phys. and Chem.) No. 2, 189-195 (1964).

Aramaki, S., and Roy, R., J. Am. Ceram. Soc. 42, 644

(1959).

DeVries, R. C, and Roy, R., Am. Ceram. Soc. Bull. 33, 370-372 (1954).

Muan, A., Am. J. Sci. 256, 171-207 (1958).

Darken, L. S., and Gurry, R. W., J. Am. Chem.

Soc. 68, 798-816 (1946).

Darken, L. S., and Gurry, R. W., J. Am. Chem.

Soc. 67, 1398-1412 (1945).

Kostkowski, H. J., and Lee, R. D., NBS Mono. 41

(1962).

Foex, M., Solar Energy 9, 61-67 (1965).

Gitlesen, G., and Motzfeldt, K., Rev. Hautes Temper.

et Refract. 3, 343-350 (1966).

Weber, R. L., Heat and Temperature Measurement

(Prentice-Hall, Inc., New York, 1950).

Foote, P. D., Fairchild, C. O., and Harrison, T. R.,

Tech. Pap. BS, 14, 1-326 (1920-1921) T170.

Gutt, W., J. Sci. Instrum. 41, 393-394 (1964).

Hill, V. G., Roy, R., and Osborn, E. F., J. Am.

Ceram. Soc. 35, 135-142 (1952).

Riley, B., Rev. Hautes Temp6r. et Refract. 3,

327-336 (1966).

Statton, W. D., J. Chem. Phys. 19, 33-40 (1951).

McNamara, E. P., Ceramics II, 350 pp. (Penn.

State College, State College, Penn., 1949).

Rankin, G. A., and Wright, F. E., Am. J. Sci. 39,

4th Ser., 1-79 (1915).

Schneider, S. J., and Waring, J. L., J. Res. NBS

67A, (Phys. and Chem.) No. 1, 19-25 (1963).

Schneider, S. J., and McDaniel, C. L., J. Res. NBS

71A, (Phys. and Chem.) No. 4, 317-333 (1967).

Stimson, H. F., J. Res. NBS 65A, (Phys. and Chem.)

No. 3, 139-145 (1961).

Burgess, G. K, BS J. Research 1, 635-640 (1928)

RP22.

Stimson, H. F., J. Res. NBS 42, 209-217 (1949)

RP1962.

Olsen, L. O., and Halpern, C, Bibliography of Temperature Measurement, NBS Mono. 27, Supplement 2 (1967).

Margrave, J. L., Temperature Measurement, Physicochemical Measurements at High Temperatures,

Edited by Bockris, J. O'M., White, J. L., Mackenzie, J. D., 6-46 (Butterworth Scientific Publ.,

1959).

Lee, R. D., Metrologia 2, No. 4, 150-162 (1966).

Roeser, W. R., and Lonberger, S. T., NBS Circ. 590

(1958).

Caldwell, F. R., NBS Mono. 40 (1962).

Temperature, Its Measurement and Control in Science and Industry 3, Parts 1 and 2, Edited by

C. M. Herzfeld (Reinhold Publ. Corp., New York

1962).

Gouffe, A., Rev. Opt. 24, 1-10 (1945).

DeVoss, J. C, Physica 20, 669-689 (1954).

Quinn, T. J., Brit. J. Appl. Phys. 18, 1105-113 (1967).

Sparrow, E. M., Albers, L. U., and Eckert, E. R. G.,

J. Heat Transfer, Trans. Amer. Soc. Mech. Engs.

Series C, 84, 73-81 (1962).

Schneider, S. J., NBS Mono. 68, 31 pp. (1963).

Wensel T., and Roeser, W. F., BS J. Research 5,

1309-1318 (1930) RP258.

Roeser, W. F., and Wensel, H. T., BS J. Research 12,

519-526 (1934) RP676.

Roeser, W. F., Caldwell, F. R., and Wensel, H. T., BS J. Research 6, 1119-1129 (1931) RP326.

Henning, F., and Wensel, H. T., BS J. Research 10,

809-827 (1933) RP568.

39

Thermal Expansion of Ceramics

Richard K. Kirby

Institute for Basic Standards,

National Bureau of Standards, Washington, D.C. 20234

Various types of coefficients of thermal expansion are defined and their application

described. The theory of thermal expansion is also reviewed and several applications to ceramics are given. Variations of four general test methods are described with special emphasis

placed on techniques of obtaining high accuracy. Included are three variations of the vitreous

silica dilatometer, two types of interferometers, an optical comparator, and two types of

x-ray diffractometers. The effect of several characterization parameters on the expansion

of ceramics is discussed. These include crystalline and amorphous phases, Active temperature

of glassy phase, quantity of phases, grain size, orientation of anisotropic grains, microcracks,

and stress.

Key Words: Anisotropic grains; ceramics; composites; glass-crystal system; interferometer; microcracks; optical comparator; stress; thermal expansion;

vitreous-silica dilatometer; x-ray diffractometer.

1. Introduction

During the last twenty years over 200 papers

have been published in the Journal of the American

Ceramic Society that have contained information

on thermal expansion [1-216] 1

. All but a few of

these papers have presented experimental data

obtained by the author(s), two have been review

papers on thermal expansion [17 and 35], and

several have described equipment for measuring

thermal expansion [16, 37, 81, 85, 130, 138, 139,

150, 156, 168, 169, and 180]. This represents about

12 percent of all papers published by the Journal

during this period. Of these papers about 30 percent have given data on crystalline oxides, 20

percent on glasses, 15 percent on multiphase systems, 15 percent on porcelains, enamels, etc., and

about 6 percent on carbides, nitrides, and borides.

Very little of this body of information, however,

has been obtained on materials sufficiently characterized to be of usefulness beyond knowing the

expansion of some particular specimens.

In general, thermal expansion is not greatly

dependent upon Che history and character of

materials, the variation of coefficient of linear

thermal expansion being about the same order as the coefficients themselves. This is in contrast with

the behavior of transport properties and some

mechanical properties where the variations reach

several orders of magnitude. History, however, is

of relative importance in its effect on thermal

expansion of ceramics.

1 Figures in brackets indicate the literature references at the end of this paper.

History includes the method of preparation

and thermal treatment and to a great extent

determines the character of the material. Of the

many parameters of character which affect the

properties of ceramics, the following will sometimes have a significant effect on thermal expansion:

Chemical composition (1 percent level)

Crystalline and amorphous phases [32, 141,

185, 197]

Quantity and size of phases [92, 125, 184,

201]

Orientation of anisotropic grains (87, 165,

183]

Residual stress [55, 93, 198]

Fictive temperature of glassy phase [6, 83,

94, 155]

Formation of cracks [113]

Formation of point defects

Chemistry of surface [205, 206, 207]

Those parameters which usually do not have a

significant effect on thermal expansion include:

Density [83]

Grain size [65, 75, 165]

Porosity [59, 82, 165]

Stoichiometry of grains [196, 212]

Impurities (<1 percent level)

Dislocations

Grain boundaries

Topography of surface [60]

There are always exceptions to the above rule,

however. For instance, in graphitic materials the

grain size and porosity have very significant

41

323-655 O - 69 - 4
effects on the thermal expansion through the

orientation of the anisotropic grains [217].

The general range of values of the coefficient

of linear thermal expansion of ceramics is indicated in table 1. Compilations of thermal expansion can be found in five readily available sources

[218-222]. Literature values for the expansion

of ceramic materials vary on the average by

about 10 percent but variations as large as 40

percent also appear. It is difficult to judge how

much of this variation is due to experimental

procedures and how much is due to history and

character differences. An interesting interlaboratory study has been made on magnesium oxide

[223], however, in which it was found that an

average variation of about 5 percent was due to

experimental procedures.

Table 1. Coefficients of linear thermal expansion of some

ceramic materials at 300 K

Material Coefficient

NaCl [224] 40 X 10-«/K PbS [224] _ 18 MgO [224].. ._ 11 Th0 2 [224] _ 7 Al2 03 [224] 5 InSb [224] 5 SiC [218] _ 4 Borosilicate glass [218] _ 3 Vitreous silica [224] 0.4

(3-Eucryptite (c axis) [113] -17

2. Definitions

The thermodynamic coefficient of thermal ex- pansion is defined as

V\dTjP

0=

where the volume, V, is a function of temperature, T, at constant pressure, P. The coefficient

of linear thermal expansion is defined as 1/dZA

a=L\j5T)P

where the length, L, is a function of temperature

at constant pressure. If the material is isotropic

then

/3=3a.

Single crystals having cubic symmetry, annealed

polycrystalline solids having randomly oriented

grains, and annealed glasses or amorphous materials are isotropic.

The coefficients of linear thermal expansion

needed to describe the expansion of crystals of

various symmetries are as follows [225]

Cubic Oil Oil Oil 0 0 0

Hexagonal Oil Oil Oi3 0 0 0

Tetragonal Oil Oil «3 0 0 0

Trigonal Oil Oil Oi3 0 0 0

Orthorhombic Oil OL2 <*3 0 0 0

Monoclinic Oil Oi2 Oi3 0 Oi5 0

Triclinic Oil a2 «3 Oi5 <*6

The principal coefficients (ai, a2) and a3) describe

the volume change of the crystal so that

&=ot\-\-ct2 -\-a3

.

These coefficients are not necessarily in the direction of the crystallographic translation vectors

but are in the direction of the principal axes

[225]. The remaining coefficients (<x4 , a5 , and a6 ) describe the change in shape (but not the symmetry) of the crystal. A monoclinic or triclinic

crystal will change shape during a temperature

change even if it is cut in the crystallographic

reference system. For example, the lattice parameters of a monoclinic crystal are given as a,

b, c, 90°, /3 90° (see fig. 1). The special relations

a=7=90° cannot be changed by temperature

change. While a2 and a3 are the coefficients along

the crystallographic translation vectors b and c,

cci is the coefficient in the direction of the principal

axis Xi which is orthogonal to the X2 (b) and

X3 (c) axes. The remaining coefficient, a5 , represents

the change in the angle /3 and is given by

Oi5

'- J_ dp

V/2 df

Except for monoclinic and triclinic symmetries,

the expansivity of a single crystal in a direction

different from the crystallographic axes is given

by

aw =2ai COS2aj j

where coj is the angle between the direction of at and the direction of interest. For hexagonal,

tetragonal and trigonal symmetries this relationship reduces to

a w = ai + (a3 — ai) cos

2o>3

.

Monoclinic a*b*cCoefficients

a, along X, a 2 along X 2 (b) a 3 along X 3 (c) a 5 the change in /3

Figure 1. Coefficients of thermal expansion for a

monoclinic crystal.

The direction X\ (orthogonal to X2 and X, in the xz plane) varies with temperature and a5 is a measure of the change in the angle (I.

42
If the values of au are plotted against the corre- sponding values of cos

2 w3 the result is very close

to a straight line. Deviations from the straight

line can result from residual stresses. Data

obtained by the author on several crystals of

Ti02 having different orientations are illustrated

in figure 2. The thermodynamic coefficient of thermal

expansion can be calculated from an equation

that represents the expansion of the material

or by graphically determining the slope of the

expansion data at a particular temperature and

dividing by the volume at that temperature. The

difficulty with the first procedure is that while

equations (usually polynominals) can be obtained

in limited temperature ranges that appear to fit the data, a closed equation will not in fact exactly

represent the thermal expansion. This is especially

true in the lower temperature range as can be

seen in figures 3, 4, and 5. Equations derived from

Griineisen's theory of the solid state are probably

the best to use for representing thermal expansion.

Derivation of these equations will be given in

the next section.

Under certain conditions the instantaneous

coefficient of thermal expansion will satisfactorily

0 0

Figure 2. Coefficients of thermal expansion of

single-crystal rutile in various directions from

the c axis.

200 300 400

TEMPERATURE, K

500 600

Figure 4. Expansivity of single-crystal rutile as function of

temperature.

150 200 250 TEMPERATURE, K

Figure 5. Expansivity of indium antimony as a function of

temperature.

0 200 400 600 800 1000 TEMPERATURE, K

Figure 3. Expansivity of magnesium oxide as a function of temperature.

represent the thermodynamic coefficient. The

instantaneous coefficient is defined as

R _l_dV_ limit V2-V1

at

2

where V0 is the volume at some reference temperature, usually room temperature. The instantaneous

coefficient of linear thermal expansion is given by

limit L2—h\

If the expansion over a limited temperature range can be approximated by a quadratic,

43
then

and

L=L0 +aT+bT2

a+2bT

dietL0

a-\-2bT

L0 +aT+bT2

For the case of a quadratic, the limit of Tl -^T2 does not need to be applied. For low-expansion

materials such as vitreous silica, cti equals a for

temperatures as high as 1200 K. If the expansion

curve takes the shape of higher polynominals the

limit of T\—>T2 must be applied if it is desired

that cti —>a. For practical applications the mean coefficient

of linear thermal expansion between temperatures

Tx and T2 ,

<Xm= L2 L\

is very useful. Values of the mean and instantane- ous coefficients for quartz are shown in figure 6. It is quite apparent that the mean coefficient is relatively insensitive to changes in expansion. The

mean coefficient of volume thermal expansion is not equal to three times the mean coefficient of

linear thermal expansion but can be approximated

/Jm~3«m +3(T2 -TiVm .

reliable information about the anharmonic terms

in the potential energy function. The inelastic

scattering of neutrons by a lattice has been

studied as a function of volume by application

of external force [227]. The frequency shifts of

the lattice vibrations that are thus produced

reflect the deviation of the crystal potential from

the harmonic approximation. These measurements

have resulted in calculated values for thermal

expansion and confirm what has been approximated by simple approaches.

From thermodynamics we have the following

set of definitions and relationships:

and

£=

KT

1 fdV\ a

1 fbV\

(1)

where k t is the isothermal compressibility, S is the entropy, and F is the free energy. Applying

statistical mechanics to the quasi-harmonic

approximation (the interaction constants of the

harmonic theory are allowed to be volume

dependent) for a set of harmonic oscillators it can be shown that

d

2 F I 3N

V i=i where

For example, if <xm =10X 10- 6/K and T3 — 2V=1000

K then /3m =30.3X 10~6/K.

3. Theory of Thermal Expansion

It has been known for years [226] that thermal

expansion originates in the anharmonic vibrations

of atoms in a solid. It is only recently, however,

that real progress has been made in obtaining

l.8|— —50"

TEMPERATURE, K

Figure 6. Comparison of the mean and instan- taneous coefficients of quartz.

and

C^kihoiJkT) 2 exp (hco ( /kT)/[exp (hUi/kT)-l]2

. Ct is the contribution of each vibrational mode

to the heat capacity, k is Boltzmann's constant,

h=h/2T where h is Planck's constant, and is the

frequency of the ith mode. If y is the weighted

average of the individual y^'s,

3N 3N

8=1 i=l

then

&F _y_Cy

bVdT V

' K

'

Gruneisen's relation is obtained from eqs (1) and

(2),

0=7^' (3)

where y, the Griineisen parameter, is a measure

of the average anharmonic interaction.

At temperatures higher than the Debye characteristic temperature, 6D, the value of y is nearly

constant since the whole spectrum of vibrational

44
frequencies is excited and ym is merely the arithmetic average of the 7/s,

I 3N

7co= 3iV§ 7< "

In the low-temperature limit as T—>0 K, the

Debye continuum is a valid model and again 7

will approach a constant value,

Vdd0

where 6 0 is the limiting value of 6D at low temperatures and is directly related to the elastic constants. At intermediate temperatures the behavior

of 7 may be investigated through Gruneisen's

relation,

The variation of 7 for A12 03 is shown in figure 7. Since the free energy is an additive function of

state, eq (3) can be expressed as a sum of the

contributions from the lattice vibrations (both

the acoustical and optical), the conduction electrons in metals, magnetic interactions, etc.,

0~!p(7acoCa co+7optC,opt +7eCrc +7mCm +. . .). (4)

In the case of nonmagnetic metals, the heat

capacity of the conduction electrons and the

lattice can be approximated by

Ce ^aT and C,~bT3

at very low temperatures.

Equation (4) can therefore be written as

l3=^(yeaT+yi bT3 ).

If the terms are rearranged,

0V

=ye a+yibT2

,

and the values of are plotted against (T2)

a straight fine should result. Such a plot for copper

is shown in figure 8. The separation of the free energy terms for

the acoustical and optical vibrations in a polyatomic crystal can also be obtained [228]. The

heat capacity of the optical modes is determined

from Einstein's function using the value of the

infrared resonance frequency at the absorption

maximum,

Copt —/ (j^) where QE =he

The heat capacity of the acoustical modes is taken

as the difference between observed values of Cv and calculated values of Copt. If values of

(/3T7Cac0 KT) are plotted against the values of

(Copt/Caco), a straight line should result. Such a

plot for Ti02 is shown in figure 9. Equations can also be derived to represent the

anisotropic expansion of any class of crystal [226].

Using matrix notation, eq (3) can be expressed

as a sum of the expansion in the crystallographic

directions,

Cv 6 "1=^7 2jS«7i, i=l, 2, 3,

where s%j are the elastic compliances for the lattice.

This equation gives a linear relation between

components of the expansion tensor and the heat

capacity. The volume coefficient is the trace of

the expansion tensor so that

Figure 7. Gruneisen's parameter for alumina as a function

of temperature.

100 (TEMPERATURE)f K

z 200

Figure 8. Contributions to the thermal expansion of

copper of the conduction electrons and the lattice vibrations.

45
Figure 9. Contributions to the thermal expansion of rulile of the acoustical and optical vibrational modes.

C 3 6

V i=l 7 =1

When Sy 0=4, 5, 6) =0, as in cubic, hexagonal,

orthorhombic, and some tetragonal crystals, a

single value of y can be determined by

3 i3

».i=i / i,;'=i

For the case of a hexagonal crystal,

Cv

ai=-y [(Sli+Si2)7l+Sl373]j

Q!3=^7 [2 S1371 +S33Y3],

|3=^ [2(sn+si2+s13)7i+ (2^13+533)73],

and

7=

2(gii +g1 2 +513)71 +2(g1 3 +S33)73

2sii+s33+2s12 +4si3

Observed and calculated values of ax and a3 for

A1203 are shown in figure 10.

The use of the equations derived so far re- quires knowledge of the compressibility or the

elastic compliances as a function of temperature.

A useful variation of Griineisen's relation can be

obtained that circumvents a possible lack of

this knowledge [226]. The Debye equation of

state [229] is given by

(5)

200 400 600 800

TEMPERATURE, K

1000

Figure 10. Comparison between observed (solid lines)

and calculated values (points) of the expansivity of alumina in the crystallographic directions.

where U0 is the internal energy at 0 K and UD is

the contribution of the lattice vibrations to the

internal energy. On setting P equal to zero eq (5) becomes

V^=G(V)=yUD. (6)

On expanding G(V) in a Taylor series, eq (6) beJG(Vo) , (V-V0) 2 comes G(Vo) + (V-V0y- dV0

1

d

2 G(V0) dV0

2 -...=yUD (7)

where V0 is the volume at 0 K. Assuming a reasonable model for the potential energy of the

lattice, it can be shown that

G(y0)=0

dG(V0) 1 dV0

d

2 G(V0) 2A

dV02 V0K0

where k0 is the compressibility at 0 K. Substituting these terms in eq (7) one obtains

(8)

If the approximation

Y^Jh where q=Yi V0 Q «o7

46
is used in eq (8), a variation of Griineisen's relation is obtained:

V-Vo= UD

V0 Q—AUd

This equation is best used to represent expansion

data by determining the values of Q and A

through a least squares calculation. This has been

done for A1203 and Th02 with excellent results over a temperature range of 100 to 1100 K [139].

4. Methods of Measuring Thermal

Expansion

While there are many different methods and

variations of methods of measuring thermal

expansion, for reasons of space, only four general

types will be described: (1) The vitreous-silica

dilatometer is very useful in the range 100 to

1200 K. The measurements of thermal expansion

("IT") a dilatometer are usually accurate 2

to 25 ppm but the precision can be as good as

0.1 ppm. (2) The interJerometric method is useful

in the range 20 to 1200 K. Smaller specimens are used, which increases the accuracy of the temperature measurements. The measurements of expansion with this method are usually accurate to

4 ppm but can be accurate to 1 ppm. (3) The

optical comparator method is most useful in the

range 1000 to 3500 K. The measurements of

expansion can be accurate to 50 ppm. (4) The

Debye-Scherrer x-ray diffractometer can be used

in the range 20 to 2500 K. Only crystalline materials can be measured but very small amounts

are sufficient. The measurements of expansion can be accurate to 50 ppm.

4.1. Vitreous Silica Dilatometers

Three general types of dilatometers are shown

in figure 11. Two of these types are described in

an ASTM method of test [230]. The tube type,

shown on the right, is the best known and the

most widely used. The single-rod type shown on

the left is the easiest to use and is more accurate

than the tube type. The double-rod type shown

in the center is in principle the most accurate but

has been used very little. In all three types, the

thermal expansion of the test material is given by

where (AZ) 0 is the observed length change, A is a calibration constant for the system, and B is the net expansion of the dilatometer.

There are three ways of using and calibrating a

dilatometer. These are, (1) Very careful calibra2 The following values for the accuracy of the thermal expansion measure- ments are based on a temperature interval of 100 K and a material having

an expansivity of 10 X 10-6/K.

Figure 11. Variations of the vitreous-silica dilatometer.

tion of the extensometer and thermometer so that

A=l and assume a value for the expansion of

the vitreous silica. (2) Same as (1) but a reference

material of known expansion is used to determine

the expansion of the vitreous silica. (3) Two

reference materials are used to determine the

values of A and B (this is not a substitute for the

calibration of the extensometer and thermometer

but provides additional information on the behavior of the system). It can be seen in table 2

that the method of using two reference materials

is in general somewhat better than the other

methods except that a more accurate calibration can be obtained with just one reference material

if the expansivity is within 2 percent of the test

material.

The vitreous silica parts of a dilatometer

should be fine annealed 3 after they have been

fabricated and should thereafter not be heated

above 1200 K. Dial indicators and linear variable

differential transformers (LVDT) [81] are used

the most for measuring the length change but

many other types of extensometers are also used.

These include optical levers [231], interferometers

[232], strain gages [16], and optical gratings [37].

One of the most common sources of error in

using dilatometers is the measurement of temperature. All too often, the temperature that is measured is not the temperature of the specimen.

This is especially true if measurements are made

while the temperature is changing. If a thermocouple is used it is recommended that the junction

and at least 3 cm of the leads should be at the same

temperature (preferably by being in contact with

the specimen). This procedure will prevent heat

from being transported away from or to the

junction.

3 Heat at 100 K/hr to 1475 K, hold at that temperature for 2 hr, cool at 60 K/hr to 1275 K, at 120 K/hr to 900 K, and at 200 K/hr to room temperature. Care should be taken to minimize viscous distortion during the annealing.

47
Table 2. Accuracy that can be obtained with a vitreous silica dilatometer on measurements of coefficients of thermal expansion

(a one percent linear error is assumed in either the length or temperature measurements), X10~6/K.

Method of using dilatometer

1* 2 2 3 3

1st reference material.. . Accuracy ....

None 0. 50 10. 00 0. 50 0. 50

used ±0.02 ±0.04 ±0.02 ±0.02

2nd reference material

Accuracy . None None None 4. 00 10. 00

used used used ± 0. 03 ± 0. 04

Nominal expansivity of 0. 0 ±0. 10

test material and pre- 0. 5 ±0. 10

cision of measurement. 4. 0 ±0. 10

10. 0 ±0. 12

±0.17 ±0.15 ±0.19 ±0.16 ±0.15

±0. 17 ±0. 15 ±0. 19 ±0. 14 ±0. 14

±0. 17 ±0. 15 ±0. 18 ±0. 15 ±0. 13

±0.20 ±0.18 ±0.17 ±0.36 ±0.18

•For these calculations it was assumed that the value used for the expansivity of the dilatometer was accurate to ±0.12X10_e/KDilatometers may be used in either the vertical or horizontal position. For use at temperatures

above 1200 K, a dilatometer can be made from

sapphire [233], SiC [234], or other refractory

materials [150].

4. 2. Interferometers

Three general types of interferometers that

produce fringes of equal thickness are also described in an ASTM method of test [235]. These

are the Fizeau, Abbe-Pulfrich, and Priest interferometers. Because of its limited use, the Priest

type will not be described here. The AbbePulfrich type (see fig. 12), like the dilatometer,

measures the thermal expansion of a specimen

relative to the expansion of the material from

which it is constructed. The Fizeau type, on the

L~l. L1

Fizeau

1

*\ mm

it

< i 1

im A

V mi 1 Mil V

II!

Figure 12. The Fizeau and AbbePulfrich types of interferometer.

other hand, measures the "absolute" expansion

of the specimen.

When using a Fizeau interferometer the ex- pansion of the specimen from TVto T2 is given by

AL NXV nx —n2

where N is the number of fringes, including

fractional parts, that have moved past the

reference point, X v is the vacuum wavelength of

the monochromatic light used to produce the

fringes, and «i and n2 are the indices of refraction

of the gas between the optical flats at pressure P

and temperatures Ti and T2 respectively.

Experimental values of the index are not available over extended temperature ranges and therefore it is recommended that the Fizeau interferometer be placed in a reduced pressure atmosphere. When helium is used, a pressure of 5 torr or less will usually not produce a significant fringe

shift over a range of 50 to 1200 K. Air should not

be used in any case at temperatures below 250 K.

Vacuum is not recommended, however, except in

specially designed equipment because of the

difficulty of temperature control and measurement. The index of refraction at a fixed wavelength

can be estimated by

n=l +(nT —1) 288P/760T

where nT is the index at a reference temperature

of 288 K and a pressure of 760 torr. For air, nr =

1.0002771 (yellow light) or nr = 1.0002779 (green

light). For helium, nT =1.000036.

In the case of the Abbe-Pulfrich interferometer,

the effect of the index change is made insignificant

by reducing the length of the optical path to a

48
sufficiently small value. Under this condition the

expansion of the specimen is given by

AL N\v

| (AZ) r L0 2L0 ni L0

where (AL) r is the length change of the material

from which the interferometer is made.

With the Fizeau interferometer reliable results can be obtained with pin specimens, see figure 13,

but special care must be taken. One technique is to place the three pins so that one of them carries

jy J± A,

A

vv—w

Figure 13. Variations of specimen shape for use in inter- ferometers.

most of the weight of the top optical flat [236].

That pin is taken as the reference point for the

fringe system so that if one or both of the other

pins tip it will not effect the measurement of

expansion. A "ring" or "T" shaped specimen,

however, is the easiest to work with because it cannot tip. The specimen used in the AbbePulfrich interferometer may be the "ring" etalon or the center post. The ends of the post are polished optically flat and parallel and it is rung

or contacted to the lower optical flat. A Pulfrich viewer, shown in figure 14, is usually

used to observe the fringe system. The position of

the fringes with respect to a reference point, see

figure 15, can be estimated with the unaided eye

to 0.1 of a fringe separation (~0.03 nm) and

easily measured with a micrometer eyepiece to

0.03. The number of fringes that move past the

reference point can be counted by eye or by

photographic [237, 238] or photoelectric [239]

techniques. It is also possible to avoid the counting problem by determining the correct order of

the fringes at each temperature with two or more

different wavelengths of light [240].

4.3. Optical Comparators

The optical comparator method is most useful

for measuring thermal expansion at high temperatures. There are many variations [85, 156, 168,

169, 241] of this type of apparatus but the best

results are obtained when two general techniques

are followed, see figure 16. The first of these is that two microscopes should be used. They should

be rigidly mounted and the length change measured

with filar micrometer eyepieces. The second technique is that the fiducial marks observed with the

POINT

Figure 14. Pulfrich viewer.

49
b — —

s 4 fj o I

1

Fractional Part of Fringe Displacement is —

b

Figure 15. Measurement of fringe fraction.

(a)

vertically in the furnace, see figure 17. The gage

length of the specimen, between the fiducial

marks, is 10 cm. Since the windows may cause a

significant error in the measurement of absolute

length, a light source is built into the furnace to

illuminate the specimen for a room-temperature

measurement. The windows are flat to one wavelength and parallel to one second. Since the maximum temperature of the furnace is 1900 K, the

measurement of temperature can be made with

thermocouples. Both W-Re and Pt-Rh 30-6

thermocouples are used. These thermocouples are

calibrated in the furnace by a probe thermocouple

inserted in place of the specimen.

This probe is also used to measure the temperature profile in the furnace. As seen in figure 18,

the microscopes are mounted on an invar bar and

as a unit can be rotated and moved vertically.

The microscopes have a magnification of 50 X

and a working distance of 16 cm. During each

measurement, the length of the specimen is compared to the length of a reference scale kept at

room temperature. This procedure compensates

for any apparent change in distance between the

microscopes that might be caused by the expansion

of the invar bar or by the tilting of a microscope.

When using the optical comparator method, a vacuum can be used but an argon atmosphere is sometimes necessary to suppress vaporization. When

an atmosphere is used, care must be taken in the

measurement of length to avoid refraction effects

in optical paths that travel through temperature

gradients [110, 241]. An optical pyrometer should

be used to measure temperature in the range

above 2000 K. When a pyrometer is used, care must be taken to provide blackbody conditions

and to keep the furnace window clean. The pyrometer and the window should be calibrated together.

Figure 16. (a) Schematic drawing of optical comparator.

Microscopes mounted on Invar bar can be either vertical or

horizontal.

Figure 16. (b) Two types of fiducial marks that can

be cut into the specimen.

Parallel (=) hairs in the microscope eyepiece should be used to sight on the cut or nick type of fiducial mark and cross (X)

hairs should be used to sight on the step or knife edge type.

the microscopes should be machined into the

specimen in such a way that they will be in sharp

focus. If a cut or nick is used then it should be

symmetrical and of a size that when viewed with

the microscope it appears to be comparable to the

separation between the parallel hairs in the eyepiece. If a knife edge is used then crossed hairs

(X) should be used in the eyepiece. Small, clean

holes drilled through the specimen are also satisfactory for fiducial marks.

In the optical comparator used at the National

Bureau of Standards [242] the specimen is hung

4.4. X-Ray Techniques

The thermal expansion of crystalline materials can be measured with x-ray cameras or with x-ray

diffractometers. This method is important because

it provides accurate data under conditions that

preclude the use of any other method, for instance,

when only a small amount of material is available.

The method is unique in that it can provide data

on expansion in different directions in anisotropic

crystals and permits direct observation of phase

changes, measurement of the expansion of each

phase when two phases coexist [243], and measurement of the change in specific volume resulting

from the phase change. There is a further advantage in that the expansion measured is that of the

crystal lattice and does not include effects that are observed in the macroscopic expansion [244].

The use of the x-ray method depends upon the

Bragg equation

d=n\/2 sin 8

where d is the distance between crystal planes, n

is the order of reflection, A is the wavelength of

50
Figure 17. Schematic drawing of controlled-gradient vacuum furnace and probe thermocouple for measurement of expansion.

A. Specimen support B. Bellows

C. Position adjusting screw, one of three D. Thermocouple feedthrough E. Electrical feedthrough, one of ten

F. Tungsten wire loop G. Water cooling coil H. Resistance bleating wire

I. Thermocouple protection tube

J. Radiation heat shields

K. Alumina Core

L. Specimen M. Window

N. Shutter O. Vacuum port

P. Vycor tube

Q. Vacuum

R. Thermocouple protection tube

S. Pyrometer viewing hole T. Tantalum tube

51
Figure 18. Photograph showing relative position of comparator, furnace, and reference scale.

the x-ray beam, and 6 is the angle of incidence of

the beam with the crystal plane. From the Bragg

equation it is easy to show that the change of the

spacing between crystal planes is given by

Ad sin di— sin 02 d sin 62

Except for a small correction due to refraction of

the x-rays [223] this measurement of expansion is independent of wavelength.

The distance between planes in crystals of

various symmetries is given by

[1 —cos

2 a—cos

2 j3— cos2 7+2 cos a cos B cos 7]

2kl . q v 2hl

,

—t— (cos a—cos 8 cos 7) (cos 8

be ac

2hk 1/2 —cos a cos 7) — (cos 7—cos a cos 8) >

where h, k, and I are the Miller indices. Using this

equation it can be shown that for cubic crystals

(a=b =c and a=0=7=90°) the expansion

Ad_Aa

d a

can be measured by using the reflection from any

crystal plane. In the case of hexagonal crystals

(a=b^c and a=./3=7=90°)

Ad

d

/[*~(h2 +hk+k2 )+l2~]

the expansion in the a and c directions must be

determined from simultaneous equations except

for special planes such as (100) and (010) where

and (001) where

Ad_Aa

d a

Ad_Ac

d~~ c

To obtain accurate data with an x-ray camera it

is necessary to measure the film shrinkage so that

the angles can be determined. The position of the

diffraction lines should be determined with a

travelling microscope or a microdensitometer. To

obtain accurate data with an x-ray diffracto meter

the furnace must be mounted on a precision wideangle goniometer. In both techniques it must be

possible to aline and maintain the sample on the

focusing circle, see figure 19. Any tilt of the sample

surface about the axis of the goniometer or the

plate holder results in broadened x-ray lines, as part of the surface will be above or below the

focusing circle. A displacement of the sample sur- face in the X direction from the focusing circle

by an amount S will cause an error in the

measurement of d by as much as

Ad

d''

s cos

2 e R sine

where R is the radius of the circle. Determinations

of d, therefore, are very sensitive to sample disFocus of X-Ray Beam

Focussing

Circle

Film or Counter Tube

Figure 19. Focusing conditions for an x-ray diffraclometer

.

52
placement in the front reflection region with sensitivity rapidly decreasing at higher angles.

Furnaces that have precision sample movements

[245, 246] provide results that are equivalent

in resolution and line profile to conventional x-ray

techniques. There is a 10 to 25 percent loss of

intensity due to absorption in the furnace windows

and a reduction of from 5 to 40°, 20, in useful

angular range due to limiting windows, radiation

shields, etc. Because of this reduction in angular

range some back-reflection cameras and diffractometers are not able to measure the more complex

crystal systems such as the orthorhombic, monoclinic, and triclinic.

Because of the small size of x-ray furnaces and

the relatively large window area, thermal gradients

exist in the sample which makes accurate temperature measurements difficult to obtain. Temperature

gradients across the sample also results in broadened x-ray lines. At temperatures around 1300 K

there can be thermal gradients of 20 to 30 K/cm

across the sample and 100 to 600 K/cm through

the sample holder. Since the penetration depth of

the x-rays is only about 0.01 cm, the temperature

of the surface must be measured. If a thermocouple is used it should, therefore, be attached to

the front surface of the sample with the junction

in the x-ray beam and the leads also in contact

with the surface. If possible, providing there is no

alloying or reactions, an internal standard of

known expansion should be used to measure the

temperature.

4.5. Reference Materials

Two reference materials, chemically pure platinum and copper, are recommended by ASTM

standards [230, 235] for calibrating thermal

expansion apparatus in the range 100 to 1200 K.

Platinum, silver, gold, iron, tungsten, and MgO

have also been used with success in high-temperature x-ray measurements. In using these materials

it is hoped that their thermal expansion, even when

purchased from different suppliers, will be close to

the tabulated literature values. Since there is some

doubt that these materials will behave as hoped,

the National Bureau of Standards, through its

Office of Standard Reference Materials, will shortly

offer a group of materials for sale that will have

certified values of thermal expansion. These refer- ence materials will include vitreous silica, borosilicate glass, single-crystal sapphire, copper, and

tungsten and will be certified within the range 20

to 1900 K.

5. Effect of Some Characterization

Parameters on Thermal Expansion

As indicated in the introduction, the thermal

expansion behavior of a ceramic is in general not

greatly influenced by its history or character. A

glass, for example, should have about the same

expansivity as a crystal from the same melt even

though it does not have the long-range structure

of the crystal. Porosities of up to 50 percent do not

alter the expansivity but do have a drastic effect on mechanical properties. Shifts from stoichiometry, a slight loss of oxygen in Ti02 , for instance, can cause a striking change in electrical conductivity but little or no change in expansivity. Some

parameters of character that do affect thermal ex- pat sion will be discussed.

5.1. Amorphous and Crystalline Phases

Materials that form glasses usually have a

crystalline form characterized by an open structure

and because of the high viscosity of their melts a random structure characteristic of the melt can be

frozen in by rapid cooling. Glasses are therefore

said to be in metastable equilibrium relative to

the crystalline form. An idealized diagram of a

glass-crystal system is shown in figure 20. In the

range between the melting temperature and the

glass transition temperature, Tg , the atomic configuration possesses a degree of randomness which

is temperature dependent. Because the viscosity

is increasing very rapidly in this range, it is possible

to freeze in different configurations by varying the

rate of cooling or by quenching from different

temperatures within the range. This results in

glasses having different specific volumes at room

temperature.

Figure 21 indicates the sort of results that are obtained with borosilicate glass [247]. On fast

cooling from above Tg a high volume structure is obtained and on very slow cooling a low volume

structure is obtained. On slow heating of the

quenched glass it will expand normally to a temperature near Tg but then its volume will decrease

toward that of the fine annealed glass before

V

g 'm

Figure 20. Volume-temperature diagram for an

ideal glass former.

53
V

T

Figure 21. Volume-temperalure diagram for a

borosilicate glass.

again expanding. On the other hand, the rapid

heating of the fine annealed glass will carry its

low volume structure to temperatures above Tg

. It is expected that a glass characterized by

different densities would have different expansivities. Eitland [83] has found this to be the case for

borosilicate glasses, as indicated in figure 21, and

Brueckner [248] has found the same behavior in

vitreous silica. Ritland also pointed out that there

is a difference between the expansivity of a glass

that has been quenched and that of the same

glass that has been slowly cooled, even though

the same specific volume had been obtained.

The volume dependence of several glass-crystal

systems is shown in figures 22, 23, and 24. In the

B203 system the relationship between the crystalline and glassy states is more or less normal

with the glass having the largest expansivity. The

situation for Si02 is not so clear, however, because

of the 22 phases that can form. It is of interest,

however, that all of the stable high-temperature

100 200 300 400 500 TEMPERATURE, °C

0.46

0.45

044

0 43

e 0 42

UJ5 041

O 040 >

0.39

0.38

0.37

VITREOUS SILICA TRIOYMITE

SiO,

CRISTOBALITE

J

400 800 1200 1600 TEMPERATURE, °C

2000

Figure 23. Volume-temperature diagram for S1O2.

UJ2

o

>

CRYSTAL

BETA-EUCRYPTITE

200 400 600 800

TEMPERATURE, °C

1000

Figure 22. Volume-temperature diagram for B2O3.

Figure 24. Volume-temperature diagram for eucryptile

(Li2 0 • AI2O3 • 2 Si02).

crystalline phases have near-zero expansivity and

that when silica glass is heated to a high enough

temperature it will crystallize in the cristobalite

phase. Silica is therefore normal in that the glassy

and crystalline states of the same configuration

have about the same expansivity. It is, of course,

the near-zero expansivity of vitreous silica that

makes it so useful.

A metastable crystalline phase of Li2OAl2 03 •2Si02 , /3-eucryptite, also has a near-zero ex- pansivity at room temperature. This glass-crystal

system does not seem to be normal in that the

glassy phase has a higher density and expansivity

than the crystalline phase. The structure of

/3-eucryptite is similar to that of silica, however,

and is described as a stuffed derivative of the

/3-quartz structure where Al3+ ions are substituted

for Si4+ ions in the network and Li+ ions are positioned in the interstitial openings to preserve

electrical balance [207]. In these materials the

near-zero expansion is presumably associated

54
with transverse optical modes, the transverse

vibrations of the oxygen atoms in the open

network structure.

Figure 25 indicates other compositions in the

Li20-Al203-Si02 system which also have the

property of near-zero expansion. Compositions

in this system are used in the glass ceramic

materials that are finding wide application.

The expansivities of compositions along the

section between Si02 and Li2 OAl2 03 are shown

in figure 26. The expansivity of the crystalline

phases stays near zero from high Si02 content to

eucryptite. The expansivity of the glassy phase,

however, increases linearly from Si02 to spodumene. This type of behavior is expected when two

network formers are involved. The significantly

lower expansivity of the glass corresponding to

the eucryptite composition is probably caused

by the interstitial Li+ ions.

5.2. Quantity of Phases

The thermal expansion of two-phase composite

bodies can be predicted with some but not

SiO„

Li 2 0 AI.O,

Figure 25. Ternary diagram of the Li2 O-Al2 03-Si02

system showing the compositions that have nearzero expansivity.

10 —

—Li 2 0-Al2 0, WEIGHT PERCENT

Figure 26. Expansivities of the crystalline and glassy phases

of the Si02-Li2 0 A12 03 system.

complete success. An equation derived by Turner

[249] seems to be very useful in spite of its

simplicity. In his derivation the average coefficient of the composite is given by

aj-E^Pj 012E2P2

di do

E1P1 , E2P2

d1 d2

where and a2> Ex and E2 , dx and d2 , and Pi

and P2 are the coefficients, Young's modulus,

density, and weight fractions of the two phases,

respectively. Kerner [250] has developed an

equation based on a model that takes shear

stresses into consideration. In his derivation

the average coefficient is given by

_(4+s) aiPi <*2P2

L

4 |

+3 4 I+3

where G and K are the shear and bulk moduli

of the composite body and Ki and K2 are the

bulk moduli of the two phases. Kerner has also

developed equations for predicting the values

of the shear and bulk moduli in composites but

they are very difficult to use. Fortunately, a paper by Hunter and Brownell

[201] provides all of the data that is needed to

compare Turner's and Kerner's equations. Figure

27 shows such a comparison for composites

formed from spinel and alumina. It can easily be

seen that the values of the coefficient calculated

with Turner's equation are in good agreement

with the experimentally determined values. Figure

28 shows the same comparison for composites

formed from a borosilicate glass and alumina. In

this case neither equation represents the experimental values.

5.3. Grain Size and Formation of Cracks

Grain size, as such, does not affect the thermal

expansion of ceramics. In anisotropic crystals,

however, two effects of grain size have been

2§ 8

Li_Z ,

Li- < ' bJ QL O X

OUJ 6

KERNER

0

1

0 0 TURNER

1

1

1 MgO-AI2 0, 0.2 0.4 0.6 0.8 WEIGHT PERCENT

AI.O,

Figure 27. Comparison of the Turner and Kerner

equations for calculating expansivity.

Points indicate experimental values for the spinel-alumina

composite bodies.

55
§ GLASS 0.2 0.4 0.6 0.1 WEIGHT PERCENT

Al,0,

Figure 28. Comparison of the Turner and Kerner

equations for calculating expansivity

.

Points indicate experimental values for the alumina-borosilicate glass composite bodies.

observed. Fryxell and Chandler [165] found that

increasing grain size in BeO specimens was

accompanied by a preferred orientation which

caused a change in expansivity. Charvat and

Kingery [90], on the other hand, found that

large-grained rutile specimens formed microcracks

because of grain boundary stresses. This effect

caused a hysteresis loop on heating and cooling

between 300 and 1400 K. Gillery and Bush [108]

observed the same behavior in polycrystalline

/3-eucryptite, see figure 29. Because of the extreme

anisotropy of (3-eucryptite, high stresses arise

between grains during cooling from the firing

temperature and fracture occurs, causing a decrease in density. During reheating the fractures

heal and a hysteresis loop is formed.

5.4. Stress

From thermodynamic relationships it can be

shown that the effect of an elastic stress, a (it

does not make any difference whether the stress

is applied externally or arises internally from

0.8

1— 0.4

z.UJ

RC 0.0

Q_ -0.4 Z*o

(/) -0.8 z

<

Cl -1.2 X

LU

-1.6

/3-EUCRYPTITE

200 400 600

TEMPERATURE, °C

800

STRESS, x 10" N/m*

UJO

o

Figure 29. Thermal expansion of 0-eucryptite.

Figure 30. Effect of stress on the expansivity of alumina.

mechanical or thermal causes), on the expansivity

of a solid is given by

Wr E2 \dTj;

Rosenfield and Averbach [251] have demonstrated

this effect in metals, but there is no indication that

anyone has done as much with ceramics. Using a

literature value for the temperature coefficient of

Young's modulus, a calculation of this effect in

alumina was made: the results are shown in

figure 30.

6. Summary

The definition of terms used in the field of

thermal expansion and a brief review of the theory

involved have been given. Four general methods

of measuring this thermophysical property have

been described, with emphasis placed on the

techniques needed to obtain high accuracy. The

effects of some characterization parameters on

thermal expansion have been discussed. All of the

papers published in the Journal of the American

Ceramic Society from 1948 through 1967 that

contain information on thermal expansion have

been referenced. Other survey papers on the

expansion of ceramic materials are also referenced

[17, 35, 252].

7. References

Note: References 1—216 were published in J. Am. Ceram.

Soc.

[1] Staff Owens-Illinois Glass Co., Properties of sodastrontium oxide-alumina-silica glasses, 31, 1 (1948).

[2] Staff Owens-Illinois Glass Co., Effect of boric oxide on properties of soda-dolomitic lime-silica glass,

31,8 (1948).

[3] Bryant, E. E. and Amnion, M. G., Determination

of compressive stress present in porcelain enamel

on sheet iron, 31, 28 (1948).

[4] Haystek, H., and Campbell, R. M., Factors involved

in production of thin wall tile, 31, 59 (1948).

[5] Plankenhorn, W. J., Refractory ceramic base coats

for metal, 31, 145 (1948).

56
[6] Tool, A. Q., Effect of heat-treatment on the density

and constitution of high-silica glasses of the

borosilicate type, 31, 177 (1948).

[7] Shelton, G. R., Creamer, A. S., and Bunting, E. N.,

Properties of barium-magnesium titanate dielectrics, 31,205 (1948).

[8] Lamar, R. S., Development of cordierite bodies with

sierralite, a new ceramic material, 32, 65 (1949).

[9] Hursh, R. K., Development of a procelain vacuum

tube, 32, 75 (1949).

[10] Beck, W. R., Crystallographic inversions of the

aluminum orthophosphate polymorphs and their

relation to those of silica, 32, 147 (1949).

[11] Gebler, K. A., and Wisely, H. R., Dense cordierite

bodies, 32, 163 (1949).

[12] Hummel, F. A., Properties of some substances iso- structural with silica, 32, 320 (1949).

[13] Riddle, F. H., Ceramic spark-plug insulators, 32,

333 (1949).

[14] Shevlin, T. S., and Blackburn, A. R., Fundamental

study and equipment for sintering and testing of cermet bodies: IV, Development and testing

equipment, 32, 363 (1949).

[15] Cook, R. L., and Brunner, C. D., Correlation of glaze-body stresses with thermal properties of whiteware bodies, 32, 401 (1949).

[16] Turnbull, J. C, Recording dilatometer for measuring

thermal expansion of solids, 33, 54 (1950).

[17] Hummel, F. A., Observations on the thermal ex- pansion of crystalline and glassy substances, 33,

102 (1950).

[18] Johnson, A. G., Scholes, S. R., and Simpson, H. E., Volume expansion of glass at high temperatures,

33, 144 (1950).

[19] Smoke, E. J., Thermally resistant whiteware, 33,

174 (1950).

[20] Staff Owens-Illinois Glass Co., Effect of K2 0 and

Li2 0 on properties of soda-dolomite lime-silica

glasses, 33, 181 (1950).

[21] Duwez, P., and Odell, F., Phase relationships in the

system zirconia-ceria, 33, 274 (1950).

[22] Gangler, J. J., Some physical properties of eight

refractory oxides and carbides, 33, 367 (1950).

[23] Van Gordon, D. V., High-fire opaque glazes for zircon bodies, 34, 33 (1951).

[24] Smoke, E. J., Ceramic compositions having negative

linear thermal expansion, 34, 87 (1951).

[25] Brackbill, C. E., McKinstry, H. A., and Hummel,

F. A., Thermal expansion of some glasses in the

system Li2 0-Al2 03-Si02 , 34, 107 (1951).

[26] Hummel, F. A., Thermal expansion properties of some synthetic lithia minerals, 34, 235 (1951).

[27] Ricker, R. W., and Hummel, F. A., Reactions in the

system Ti02 -Si02 revision of the phase diagram,

34, 271 (1951).

[28] Yoshiki, B., and Matsumoto, K., High-temperature

modification of barium feldspar, 34, 283 (1951).

[29] Hummel, F. A., and Reid, H. W., Thermal expansion

of some glasses in the system MgO-Al2 03-Si02 , 34,

319 (1951).

[30] Blackburn, A. R., and Sherlin, T. S., Fundamental

study and equipment for sintering and testing of

cermet bodies: V Fabrication, testing, and properties of 30 chromium-70 alumina cermets, 34, 327

(1951).

[31] Beals, R. J., and Cook, R. L., Low-expansion

cordierite porcelains, 35, 53 (1952).

[32] Hummel, F. A., Significant aspects of certain ternary

compounds and solid solutions, 35, 64 (1952).

[33] Yoshiki, B., and Yoshida, R., Composition of lowalkali glass, 35, 166 (1952).

[34] Karkhanavala, M. D., and Hummel, F. A., Thermal

expansion of some simple glasses, 35, 215 (1952).

[35] Austin, J. B., Thermal expansion of nonmetallic

crystals, 35, 243 (1952).

[36] Abe, T., Borosilicate glasses, 35, 284 (1952).

[37] Lieberman, A., and Crandall, W. B., Design and

construction of a self-calibrating dilatometer for high-temperature use, 35, 304 (1952).

[38

[39

[40

[41

[42;

[43

[44

[45

[46

[47

[48

[49

[50

[51

[52

[53

[54

[55

[56

[57

[58

[59

[60

[61

[62

[63

[64

[65

Schwartz, B., Theimal stress failure of pure refrac-

^ tory oxides, 35, 325 (1952).

Shartsis, L., Capps, W., and Spinner, S., Density

and expansivity of alkali borates and density

characteristics of some other binary glasses, 36,

35 (1953).

Curtis, C. E., and Sowman, H. G., Investigation of

the thermal dissociation, reassociation, and synthesis of zircon, 36, 190 (1953).

Comeforo, J. E., Hatch, R. A., Humphrey, R. A.,

and Eitel, W., Synthetic mica investigation: I, A hot-pressed machinable ceramic dielectric, 36,

286 (1953).

Walton, J. D., Jr., and Sweo, B. J., Determination

of strains between enamel and iron by means of

split rings, 36, 335 (1953).

Skinner, K. G., Cook, W. H., Potter, R. A., and

Palmour, H., Ill, Effect of Ti02 , Fe2 03 , and alkali on mineralogical and physical properties of mullite-type and mullite-forming Al2 03-Si02 mixtures:

I., 36, 349 (1953).

Goodman, G., Ferroelectric properties of lead

metaniobate, 36, 368 (1953).

Karkhanavala, M. D., and Hummel, F. A., The

polymorphism of cordierite, 36, 389 (1953).

Quirk, J., and Harman, C. G., Properties of a tin

oxide-base ceramic body, 37, 24 (1954).

Walton, J. D., Jr., Study of strains between enamel

and iron as related to physical properties of each,

37, 153 (1954).

Mitoff, S. P., and Pask, J. A., Thermal strains

during firing of a ceramic body, 37, 168 (1954).

Harrison, D. E., McKinstry, H. A., and Hummel,

F. A., High-temperature zirconium phosphates,

37, 277 (1954).

Plankenhorn, W. J., Effect of ceramic coatings on

fatigue strength of metal, 37, 281 (1954).

Navias, L., Advances in ceramics related to elec- tronic tube developments, 37, 329 (1954).

Curtis, C. E., Doney, L. M., and Johnson, J. R.,

Some properties of hafnium oxide, hafnium sili- cate, calcium hafnate, and hafnium carbide, 37,

458 (1954).

Shartsis, L., and Shermer, H. F., Surface tension,

density, viscosity, and electrical resistivity of

molten binary alkaline-earth borates, 37, 544

(1954) . Lamar, R. S., and Warner, M. F., Reaction and

fired-property studies of cordierite compositions,

37, 602 (1954).

Kingery, W. D., Factors affecting thermal stress resistance of ceramic materials, 38, 3 (1955).

Buessem, W. R., Thermal shock testing, 38, 15

(1955) . Manson, S. S., and Smith, R. W., Theory of thermal

shock resistance of brittle materials based on

Weibull's statistical theory of strength, 38, 18

(1955).

Buessem, W. R., and Bush, E. A., Thermal fracture

of ceramic materials under quasi-static thermal

stresses (ring test), 38, 27 (1955).

Coble, R. L., and Kingery, W. D., Effect of porosity on thermal stress fracture, 38, 33 (1955).

Baroody, E. M., Simons, E. M., and Duckworth,

W. H., Effect of shape on thermal fracture, 38, 38

(1955).

Crandall, W. B., and Ging, J., Thermal shock

analysis of spherical shapes, 38, 44 (1955).

Brown, F. H., Jr., and Duwez, P., The systems

zirconia-lanthana and zirconia-neodymia, 38, 95

(1955).

Smyth, H. T., Thermal expansion of vitreous

silica, 38, 140 (1955).

Nadler, M. R., and Fitzsimmons, E. S., Preparation

and properties of calcium zirconate, 38, 214 (1955).

Koenig, C. J. Influence of particle-size distribution on

the properties of nepheline syenite, 38, 231 (1955).

323-655 O - 69 - 5 57
King, B. W., and Suber, L. L., Some properties of the oxides of vanadium and their compounds, 38,

306 (1955).

Knudsen, F. P., Moreland, R. E., and Geller, R. F., Physical characteristics of titanium carbide type

cermets at elevated temperatures, 38, 312 (1955).

Van Gordon, D. V., and Spangenberg, W. C, Adjustment of thermal expansion of cone 8 glazes,

38, 331 (1955).

Wuellner, J. F., and Sweo, B. J., Determination of strains in enameled cast iron, 38, 404 (1955).

Sherlin, T. S., and Hauck, C. A., Fundamental

study and equipment for sintering and testing of cermet bodies: VII, Fabrication, testing, and

properties of 34 Al2 03 -66 Cr-Mo Cermets, 38, 450

(1955) . Sutton, W. H., and Matson, F. R., Factors affecting

strength of clays in the temperature range 110 to 800° C, 39, 25 (1956).

Lachman, I. M., and Everhart, J. 0., Development

of safe cooling schedules for structural clay

products, 39, 30 (1956).

Russell, R., Jr., Valencia, C, and Emrich, H. W.,

Barite in ceramic whitewares, 39, 73 (1956).

Dalton, R. H., Solder glass sealing, 39, 109 (1956).

Burdick, M. D., and Parker, H. S., Effect of particle

size on bulk density and strength properties of

uranium dioxide specimens, 39, 181 (1956).

Hedden, W. A., and King, B. W., Antimony oxide

glasses, 39, 218 (1956).

Godejahn, G. C., Jr., and Cook, R. L., Effect of

zircon and zirconia on thermal expansion of lead

borosilicate glazes, 39, 256 (1956).

Lauchner, J. H., Cook, R. L., and Andrews, A. 1., Fundamental thermal deflection analysis of

enamel-metal systems, 39, 281 (1956).

Vukovich, M., Jr., Effect of wollastonite substitutions in a semivitreous dinnerware body, 39, 323

^ (1956).

Shafer, E. C, and Roy, R, Studies of silica-structure

phases: I, GaP04 , GaAs04 , and GaSb04 , 39, 330

(1956) . Kelly, H. J., and Harris, H. M., An automatically

recording thermal-expansion apparatus, 39, 344

(1956) . Coble, R. L., and Kingery, W. D., Effect of porosity on physical properties of sintered alumina, 39,

377 (1956).

Ritland, H. N., Limitations of the Active temperature concept, 39, 403 (1956).

Lang, S. M., and Kundsen, F. P., Some physical

properties of high-density thorium dioxide, 39,

415 (1956).

Whittemore, O. J., Jr., and Ault, N. N., Thermal

expansion of various ceramic materials to 1500 °C,

39, 443 (1956).

Curtis, C. E., Properties of yttrium oxide ceramics,

40, 274 (1957).

Belas, R. J., and Cook, R. L. Directional dilatation of crystal lattices at elevated temperatures,

40, 279 (1957).

Juggins, M. L., and Abe, T., Structure of borate

glasses, 40, 287 (1957).

Mattyasovszky-Zsoluoy, L., Mechanical strength of porcelain, 40, 299 (1957).

Charvat, F. R., and Kingery, W. D., Thermal conductivity: XIII, Effect of microstructure on conductivity of single-phase ceramics, 40, 306 (1957).

Duwez, P., and Loh, E., Phase relationships in the

system zirconia-thoria, 40, 321 (1957).

Kingery, W. D., Note on thermal expansion and

microstresses in two-phase compositions, 40, 351

(1957) . Lauchner, J. H., Cook, R. L., andAn drews, A. I., Effect of firing schedules on stress-temperature

relations in enamel-metal systems, 40, 410 (1957).

Hagy, H. E., and Ritland, H. N., Effect of thermal

history on glass expansion characteristics, 40,

436 (1957).

[95

[96

[97

[98

[99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

Sastry, B. S. R., and Hummel, F. A., Studies in lithium oxide systems: I, Li2 OB2 03 -B2 03 , 41, 7

(1958).

Newkirk, H. W., Jr., and Sisler, H. H., Determination of residual stresses in titanium carbide-base

cermets by high-temperature x-ray diffraction,

41, 93 (1958).

Simon, I., Note on thermal expansion of neutronirradiated silica, 41, 116 (1958).

Bush, E. A., and Hummel, F. A., High-temperature

mechanical properties of ceramic materials: I, Magnesium dititanate, 41, 189 (1958).

Strimple, J. H., and Giess, E. A., Glass formation

and properties of glasses in the system Na2 0-B2 03- Si02-Ti02 , 41, 231 (1958).

Hafner, H. C, Kreidl, N. J., and Weidel, R. H.,

Optical and physical properties of some calcium

aluminate glasses, 41, 315 (1958).

Goughanour, L. W., Shartsis, L., and Shermer, H. F.

Viscosity, density, and electrical resistivity of

molten alkaline-earth borate glasses with 3 mole %

of potassium oxide, 41, 324 (1958);

King, B. W., and Kelly, G. D., Infrared-transmitting

glasses in the system K2 0-Sb2 03-Sb2 S 3 , 41, 367

(1958).

Ploetz, G. L., Krystyniak, C. W., and Dumas, H. E.,

Sintering characteristics of rare-earth oxides, 41,

551 (1958).

Long, G., and Foster, L. M., Aluminum nitride, a

refractory for aluminum to 2000 °C, 42, 53 (1959).

Ranch, H. W., Commons, C. H., Jr., and Blau, H. H.,

Exploration of lithia glassforming systems: I, Li2 0-CaO-Al2 03-Si02 at different levels from 0 to 30 weight % CaO, 42, 113 (1959).

Lauchner, J. H., and Bennett, D. G., Thermal

fracture resistance of ceramic coatings applied to metals I, Elastic deformation, 42, 146 (1959).

Curtis, C. E., and Tharp, A. G., Ceramic properties

of europium oxide, 42, 151 (1959).

Gillery, F. H., and Bush, E. A., Thermal contraction

of /3-eucryptite (Li2 0 • A12 03

• 2 Si02 ) by x-ray and

dilatometer methods, 42, 175 (1959).

Schneider, S. J., Effect of heat-treatment on the

constitution and mechanical properties of some

hydrated aluminous cements, 42, 184 (1959).

Engberg, C. J., and Zehms, E. H., Thermal expansion

of A12 03 , BeO, MgO, B4 C, SiC and TiC above

1000 °C, 42, 300 (1959).

Mackenzie, J. D., Density and expansivity of vitre- ous germania, 42, 310 (1959).

Carter, R. E., Thermal expansion of MgFe2 04 , FeO and MgO-2 FeO, 42, 324 (1959).

Bush, E. A., and Hummel, F. A., High-temperature

mechanical properties of ceramic materials: II, /3-eucryptite, 42, 388 (1959).

Harrison, D. E., and Hummel, F. A., Reactions in

the system Ti02-P2 05 , 42, 487 (1959).

Ramsey, T. H., Jr., Summary of some properties of

yttrium iron garnet and study of dislocations

observed in an yttrium iron garnet crystal, 42,

645 (1959).

Tilton, L. W., Structural and thermal expansions

in alkali silicate binary glasses, 43, 9 (1960).

Sastry, B. S. R., and Hummel, F. A., Studies in lithium oxide systems: VII, Li2 0-B2 03-Si02 , 43,

23 (1960).

Wilson, W. B., High-temperature x-ray diffraction

investigation of the uranium-carbon system, 43,

77 (1960).

Goodman, G., Ferroelectric behavior in barium

zirconium metaniobate, 43, 105 (1960).

Subbarao, E. C, Studies on lead titanate ceramics

containing niobium or tantalum, 43, 119 (1960).

Ploetz, G. L., Muccigrosso, A. T., Osika, L. M., and

Jacoby, W. R., Dysprosium oxide ceramics, 43,

154 (1960).

Flaschen, S. S., Pearson, A. D., and Northover,

W. R., Formation and properties of low-melting

58
glasses in the ternary systems As-Tl-S, As-Tl-Se

and As-Se-S, 43, 274 (1960).

[123] Subbarao, E. C, X-ray study of phase transitions in

ferroelectric PbNb2 06 and related materials, 43,

439 (1960).

[124] Argyle, J. F., and Hummel, F. A., Dialatometric

and x-ray data for lead compounds: I, 43, 452

(1960) . [125] Baskin, Y., Harada, Y., and Handwerk, J. H., Some

physical properties of thoria reinforced by metal

fibers, 43, 489 (1960).

[126] Grisaffe, S. J., Thermal expansion of hafnium

carbide, 43, 494 (1960).

[127] Subbarao, E. C., Reactions in the system PbO-Ta2 05 , 44,92 (1961).

[128] Fenstermacher, J. E., and Hummel, F. A., Hightemperature mechanical properties of ceramic

materials: IV, Sintered mullite bodies, 44, 284

(1961) . [129] Dormer, G. J., Norton, G. R., and Payne, B. S., Glassing characteristics of low-alloy steels, 44,

375 (1961).

[130] Warshaw, I., and Roy, R., Thermal-expansion

measurements from nonindexed high-temperature

x-ray powder patterns, 44, 421 (1961).

[131] Pappis, J., and Blum, S. L., Properties of pyrolytic

graphite, 44, 592 (1961).

[132] Hirayama, C, Properties of aluminobarate glasses

of group II metal oxides: I, Glass formation and

thermal expansion, 44, 602 (1961).

[133] Taylor, R. E., Thermal conductivity and expansion

of beryllia at high temperatures, 45, 74 (1962).

[134] Pei-Ching, L., Ghose, A. C., and Gong-Jen, S., High

temperature density determination of boron

oxide and binary rubidium and cesium borates,

45, 89 (1962).

[135] Sarver, J. F., and Hummel, F. A., Stability relations

of magnesium metasilicate polymorphs, 45, 152

(1962) . [136] Studt, P. L., and Fulrath, R. M., Mechanical

properties and chemical reactivity in mullite, 45,

182 (1962).

[137] Thornton, H. R., Bond strength and elastic properties of ceramic adhesives, 45, 201 (1962).

[138] Burk, M., Thermal expansion of ceramic materials

at -200 to 0°C, 45, 305 (1962).

[139] Wachtman, J. B., Jr., Scuderi, T. G., and Cleek,

G. W., Linear thermal expansion of aluminum

oxide and thorium oxide from 100 to 1100 °K,

45, 319 (1962).

[140] Kim, K. H., and Hummel, F. A., Studies in lithium

oxide systems: XII, Li2 0-B2 03-Al2 0 3 , 45, 487

(1962).

[141] Merz, K. M., Brown, W. R., and Kirchner, H. P., Thermal-expansion anisotropy of oxide solid

solutions, 45, 531 (1962).

[142] Subbarao, E. C., Polymorphism and dielectric

properties of Bi2 Ti4 0„, 45, 564 (1962).

[143] Murthy, M. K., Thermal-expansion properties of

BaTiGe3 09, 45, 566 (1962).

[144] Tien, T. Y., and Carlson, W. G., Effect of additives on properties of lead titanate, 45, 567 (1962).

[145] Pask, J. A., and Fulrath, R. M., Fundamentals of

glass-to-metal bonding: VIII, nature of wetting

and adherence, 45, 592 (1962).

[146] Chessin, H., and Turkdogan, E. T., A crystallographic investigation of calcium diferrite, 45, 597

(1962).

[147] Murthy, M. K., Thermal-expansion properties of

vitreous and crystalline germania, 45, 616 (1962).

[148] Argyle, J. F., and Hummel, F. A., Dialtometric and

x-ray data for lead compounds, II, 46, 10 (1963).

[149] Eppler, R. A., Glass formation and recrystallization

in the lithium metasilicate region of the system

Li2 0-Al2 03-Si02 , 46, 97 (1963).

[150] Shaffer, P. T. B., and Mark, S. D., Inert-atmosphere

dilatometer for use to 2000° C, 46, 104 (1963).

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

Rao, Bh. V. J., Influence of Ti02 on properties of

glasses in the system K2 0-PbO-Si02-Ti02 and

its relation to structure, 46, 107 (1963).

Kirchner, H. P., Merz, K. M., and Brown, W. R.,

Thermal expansion of uranium pyrophosphate and

and of ceramic bodies in the system U02-UP2 07,

46, 137 (1963).

Shalek, P. D., Preparation and properties of uranium

and thorium monosulfides, 46, 155 (1963).

Sarver, J. F., Thermal-expansion data for rutile- type Ge02 , 46, 195 (1963).

Takamori, T., Effects of thermomechanical history on properties of glasses, 46, 366 (1963).

Nielsen, T. H., and Leipold, M. H., Thermal ex- pansion in air of ceramic oxides to 2200° C, 46,

381(1963).

Carroll, D. F., Thermal expansion of plutonium

nitride, 46, 406 (1963).

Mistier, R. E., Ploetz, G. L., and Smith, J. A.,

Thermal expansion of polycrystalline lanthanum

and neodymium sesquioxides, 46, 561 (1963).

Gillam, E., and Holden, J. P., Structure of nickel

oxide containing alumina, 46, 601 (1963).

Christensen, J. A., Thermal expansion and changes

in volume of uranium dioxide on melting, 46, 607

(1963) . Mendez-Penalosa, R., and Talyor, R. E., Thermal

expansion of uranium monocarbide, 47, 101 (1964).

Herczog, A., Microcrystalline BaTi03 by crystallization from glass, 47, 107 (1964).

Nielsen, T. H., and Leipold, M. H., Thermal expansion of yttria-stabilized zirconia, 47, 155 (1964).

Nielsen, T. H., and Leipold, M. H., Thermal expansion of yttrium oxide and of magnesium oxide with

yttrium oxide, 47, 256 (1964).

Fryxell, R. E., and Chandler, B. H., Creep, strength,

expansion, and elastic moduli of sintered BeO as a

function of grain size, porosity, and grain orientation, 47, 283 (1964).

Houska, C. R., Thermal expansion of certain group

IV and group V carbides at high temperatures,

47, 310 (1964).

Murthy, M. K., and Ip, J., Studies in germanium

oxide systems: I, Phase equilibria in the system

Li2 0-Ge02 , 47, 328 (1964).

Miccioli, B. R., and Shaffer, P. T. B., High-temperature thermal expansion behavior of refractory

materials: I, Selected monocarbides and binary

carbides, 47, 351 (1964) (Correction to paper by

B. R. Miccioli and P. T. B. Shaffer, 47, 536 (1964).

Ohnysty, B., and Rose, F. K., Thermal expansion

measurements on thoria and hafnia to 4500 °F, 47,

398 (1964).

Mendez-Penalosa, R., and Taylor, R. E., Thermal

expansion of uranium sesqui carbide, 47, 416 (1964)

.

Murthy, M. K., and Agnayo, J., Studies in germanium oxide systems: II, Phase equilibria in the

system Na2 0-Ge02 , 47, 444 (1964).

Soga, N., Thermal expansion properties of lithium

orthogermanate and orthosilicate, 47, 469 (1964).

Takamori, T., and Tomozawa, M., Thermal expansion of a cubic sodium tungsten bronze, 47, 472

(1964) . Takamori, T., Thermal expansion characteristics of polycrystalline tungsten oxides, 47, 534 (1964).

Floyd, J. R., Effect of secondary crystalline phases on dielectric losses in high-alumina bodies, 47, 539

(1964).

Tien, T. Y., and Hummel, F. A., Studies in lithium

oxide systems: XIII, Li20 • A12 03

• 2 Si02-Li20 A12 03

• 2Ge02 , 47, 582 (1964).

Lang, S. M., Axial thermal expansion of tetragonal

Zr02 between 1150 and 1700 °C, 47, 641 (1964).

Fehrenbacher, L. L., and Jacobson, L. A., Metallographic observation of monoclinic-tetragonal

phase transformation in Zr02 , 48, 157 (1965).

Nielsen, T. H., and Leipold, M. H., Thermal expansion of nickel oxide, 48, 164 (1965).

59
Merryman, R. G., and Kempter, C. P., Precise

temperature measurement in Debye-Scherrer

specimens at elevated temperatures, 48, 202

(1965).

Takamori, T., Low-temperature thermal expansion

characteristics of a cubic sodium tungsten bronze,

48, 277 (1965).

Krochmal, J. J., Comment on paper "High-temperature thermal expansion behavior of refractory

materials, I", 48, 328 (1965).

Corvin, I., and Cartz, L., Anisotropic thermal ex- pansion of V2 05 , 48, 328 (1965).

Takamori, T., and Tomozawa, M., Behavior of

interlayers of glass-to-tungsten seals, 48, 405

^ (1965).

Stansfield, O. M., Thermal expansion of polycrystalline Hf02 -Zr02 solid solutions, 48, 436 (1965).

Branson, D. L., Thermal expansion coefficients of

zirconate ceramics, 48, 441 (1965).

Richardson, J. H., Thermal expansion of three

group IV A carbides to 2700°C, 48, 497 (1965).

Tennery, V. J., High-temperature phase transitions

in NaNbOa, 48, 537 (1965).

Napolitano, A., Macedo, P. B., and Hawkins, E. G.,

Viscosity and density of boron trioxide, 48, 613

(1965) . Schreiber, E., and Anderson, O. L., Pressure derivatives of the sound velocities of polycrystalline

alumina, 49, 184 (1966).

Ulrich, D. R., and Smoke, E. J., Devitrified barium

titanate dielectrics, 49, 210 (1966).

Allen, W. C., Temperature dependence of properties

of magnesium ferrite, 49, 257 (1966).

Arias, A., Thermal shock resistance of zirconia with

15 mole % titanium, 49, 334 (1966).

Soga, N., and Anderson, 0. L., High-temperature

elastic properties of polycrystalline MgO and

A12 03 , 49, 355 (1966).

Anderson, 0. L., and Andreatch, P., Jr., Pressure

derivatives of elastic constants of single-crystal

MgO at 23 and -195.8 °C, 49, 404 (1966).

Aronson, S., Cisney, E., and Auskern, A. B., Thermal

expansion of nonstoichiometric zirconium carbides,

49, 456 (1966).

Tennery, V. J., High-temperature phase transitions

in PbZr03 , 49, 483 (1966).

Foster, P. K., and Hughes, I. R., Internal stress in ceramics containing cristobalite, 49, 515 (1966).

Baskin, Y., Properties of uranium monophosphide,

49, 541 (1966).

Kruger, O. L., and Moser, J. B., Sintering characteristics and properties of PuS and PuP, 49, 661

(1966) . Hunter, O., and Brownell, W. E., Thermal expansion

and elastic properties of two-phase ceramic bodies,

50, 19 (1967).

Riebling, E. F., Volume relations in Na2 0-B2 03 and

Na2 OSi02 -B2 03 melts at 1300 °C, 50, 46 (1967).

Vergano, P. J., Hill, D. C., and Uhlmann, D. R.,

Thermal expansion of feldspar glasses, 50, 59

(1967) . Keihn, F. G., and Keplin, E. J., High-temperature

thermal expansion of certain group IV and group

V diborides, 50, 81 (1967).

Karstetter, B. R., and Voss, R. O. , Chemical strengthening of glass-ceramics in the system Li2 0-Al2 03-

SiO-2, 50, 133 (1967).

Kirchner, H. P., Gruver, R. M., and Walker, R. E.,

Chemically strengthened, leached alumina and

spinel, 50, 169 (1967).

Beall, G. H., Karstetter, B. R., and Rittler, H. L.,

Crystallization and chemical strengthening of

stuffed /3-quartz glass-ceramic, 50, 181 (1967).

Soga, N., and Anderson, O. L., High-temperature

elasticity and expansivity of forsterite and

steatite, 50, 239 (1967).

Aronson, S., Cisney, E., and Gingerich, K. A., Thermal expansion of some cubic refractory

compounds of thorium, 50, 248 (1967).

[210

[211

[212

[213

[214

[215

[216

[217

[218

[219

[220

[221

[222

[223

[224

[225

[226

[227

[228

[229

[230

[231

[232

[233

[234

Felty, E. J., and Myers, M. B., Thermal expansion

of arsenic-selenium glasses, 50, 335 (1967).

Warshaw, S. I., and Seider, R., Comparison of strength of triaxial porcelains containing alumina

and silica, 50, 337 (1967).

Fries, R. J., and Wahman, L. A., Effect of stoi- chiometry on the thermal expansion of Tad, 50,

475 (1967).

Provance, J. D., and Wood, D. C, Molybdenum

phosphate glasses containing Ag2 0 or K2 0, 50,

516 (1967).

Mazdiyasni, K. S., Lynch, C. T., and Smith, J. S., II, Cubic phase stabilization of translucent

yttria-zirconia at very low temperatures, 50, 532

(1967).

Riebling, E. F., Blaszyk, P. E., and Smith, D. W.,

Structure of glasses and melts in the Na2 0 Ge02• B2 03 System, 50, 641 (1967).

Weirauch, D. F., and Tennery, V. J., Electrical, x-ray, and thermal expansion studies in the

system KNb03-AgNb03 , 50, 671 (1967).

Hove and Riley, Ceramics for Advanced Technologies,

John Wiley & Sons, Inc., New York, 1965.

Goldsmith, Waterman, and Hirschhorn, Handbook

of Thermophysical Properties of Solid Materials,

MacMillan, New York, 1961.

Touloukian, Thermophysical Properties of HighTemperature Solid Materials, Macmillan, New

York, 1967.

Lynch, Ruderer, and Duckworth, Engineering

Properties of Selected Ceramic Materials, The Am.

Ceram. Soc, Inc., Columbus, 1966.

Thielke and Henry, Refractory Materials for Use in

High-Temperature Areas of Aircraft, The Pennsylvania State College, AF Technical Report No.

6080, ASTIA, ATI-80745, 1950.

Stutzman, Salvaggi, and Kirchner, Summary Report

on an Investigation of the Theoretical and

Practical Aspects of the Thermal Expansion of

Ceramic Materials, Volume One—Literature Survey, Cornell Aeronautical Laboratory, Inc., Report PI-1273-M-4, ASTIA, AD 220685, 1959.

Campbell, W. J., Thermal Expansion of Magnesium

Oxide: An Irterlaboratory Study, Bureau of

Mines Report of Investigation 6115, 1962.

Gray, American Institute of Physics Handbook,

Second Edition, McGraw-Hill, New York, 1963.

Nye, J. F., Physical Properties of Crystals—Their

Representation by Tensors and Matrices, Oxford

University Press, New York, 1957.

Griineisen, E., U.S. NASA Publ. No. RE2-18-59W

(translation of Handb. Physik, 10, 1 (1926)).

Lechner, R., and Quittner, G., Pressure-induced

phonon frequency shifts in lead measured by

inelastic neutron scattering, Phys. Rev. Letters

17, 1259 (1966).

Kirby, R. K., Thermal expansion of rutile from 100

to 700°K, J. Res. NBS 71A, 363 (1967).

Kittel, C, Introduction to Solid State Physics,

John Wiley & Sons, Inc , New York 1956.

ASTM Method of Test, E228-66aT, for Linear

Thermal Expansion of Rigid Solids with a Vitreous

Silica Dilatometer, 1967 Book of ASTM Standards, Part 31. Douglas, R. W., and Isard, J. O., Apparatus for the

measurement of small differential expansions and

its use for the study of fused silica, J. Sci. Instr. 29, 13 (1952).

Meyerhoff, R. W., and Smith, J. F., Anisotropic

thermal expansion of single crystals of thallium,

yttrium, beryllium, and zinc at low temperatures,

J. Appl. Phys. 33, 219 (1962).

Hyde, G. R., Domingues, L. P., and Furlong, L. R.,

Improved dilatometer, Rev. Sci. Instr. 36, 204

(1965).

Mark, S. D., and Emanuelson, R. C, Thermal

expansion apparatus with a silicon carbide

60
dilatometer for temperatures, to 1500°C, Am.

Ceram. Soc. Bull. 37, 193 (1958).

[235] ASTM Method of Test, E289-65T, for Linear

Thermal Expansion of Rigid Solids with Inter- ferometry, 1967 Book of ASTM Standards, Part

31.

[236] Saunders, J. B., Improved interferometric procedure

with application to expansion measurements, J. Res. NBS 23, 179 (1939).

[237] Nix, F. C, and MacNair, D., An interferometricdilatometer with photographic recording, Rev.

Sci. Instr. 12, 66 (1941).

[238] Saunders, J. B., An apparatus for photographing

interference phenomena, J. Res. NBS 35, 157

(1945).

[239] Ersner, R. L., Reversible photoelectric fringe

counting, Rev. Sci. Instr. 29, 521 (1958).

[240] Willey, L. A., and Fink, W. L., An Interferometer

Type of Dialtometer, and Some Typical Results,

Am. Inst. Mining Met. Engrs, Technical Publication No. 1859 (1944).

[241] Conway, J. B., and Losekamp, A. C, Thermalexpansion characteristics of several refractory

metals to 2500° C, Trans. Metall. Soc. AIME

236, 702 (1966).

[242] Rothrock, B. D., and Kirby, R. K., An apparatus

for measuring thermal expansion at elevated

temperatures, J. Res. NBS 71C, 85 (1967).

[243] Peiser, Rooksby, and Wilson, X-ray Diffraction

by Polycrystalline Materials, Reinhold, New

York, 1960.

[244] Fouchaux, R. D., and Simmons, R. O., Measurements of thermal expansion and thermal equilibrium defects in silver chloride, Phys. Rev.

136, A1664 (1964).

[245] Mauer, F. A., and Bolz, L. H., Measurement of Thermal Expansion of Cermet Components by

High Temperature X-Ray Diffraction, WADA

Technical Report 55-473, Supplement 1, ASTIC

Doc. No. AD 155555 (1957).

[246] Campbell, W. J., Stecura, S., and Grain, C, Hightemperature Furnaces for X-ray Diffractometers,

Bureau of Mines Report of Investigation 5738,

1961.

[247] Tool, A. Q., Lloyd, D. B., and Merritt, G. E.,

Dimensional changes caused in glass by heating

cycles, J. Res. NBS 5, 627 (1930).

[248] Brueckner, R., Characteristic physical properties of

the chief oxide glass formers and their relation to the structure of glasses: II, Mechanical and

optical properties as a function of thermal history,

Glastech. Ber. 37, 459 (1964).

[249] Turner, P. S., Thermal-expansion stresses in reinforced plastics, J. Res. NBS 37, 239 (1946).

[250] Kerner, E. H., The elastic and thermo-elastic

properties of composite media, Proc. Phys. Soc. (London) B69, 808 (1956).

[251] Rosenfield, A. R., and Averbach, B. L., Effect of

stress on the expansion coefficient, J. Appl. Phys.

27, 154 (1956).

[252] Kirchner, H. P., Thermal Expansion of Ceramic

Crystals, Progress in Solid State Chemistry, Vol.

2, H. Reiss, Editor, Macmillan, 1964.

61

Thermal Conductivity of Ceramics

D. R. Flynn

Institute for Applied Technology,

National Bureau of Standards, Washington, D.C. 20234

A description is given of the underlying phenomena governing thermal conduction in

ceramic materials. Heat transfer via lattice vibrations, electronic conduction, and thermal

radiation are discussed in some detail. The effects of environmental variables (temperature,

atmosphere, electromagnetic fields, pressure and stress, heat flow) on the thermal conductivity of ceramics are discussed. The effects of composition (molecular structure, impurities, stoichiometry, alloying) and structure (crystallinity, imperfections, additional phases,

microstructure) on thermal conductivity are discussed.

It is pointed out that many thermal conductivity data reported in the literature are

unreliable, either due to experimental errors or to inadequate sample characterization. A

number of methods of measuring thermal conductivity are described, with attention being

given to precautions to assist in avoidance of measurement errors. The use of thermal

conductivity reference standards is discussed and a list is given of materials which may be

used, on an interim basis, as reference standards.

Key Words: Ceramics; conductivity; glass; iieat conduction; heat transfer; thermal

conductivity.

CONTENTS

Page

1. Introduction and scope 63

2. Definition and phenomenological theory 64

2.1. Definition 64

2.2. Phenomenological theory 64

a. Lattice conduction (phonons) 65

b. Electronic conduction 70

c. Radiation "conduction" 73

d. Gas conduction 77

e. Other transport mechanisms 78

3. Effects of environment 78

3.1. Temperature 78

3.2. Atmosphere 81

3.3. Electromagnetic fields 81

3.4. Pressure and stress 81

3.5. Heat flow 82

4. Directional effects 83

5. Effects of composition and structure 83

5.1. Composition 83.

a. Molecular structure - 83

b. Impurities 84

c. Stoichiometry 85

1. Introduction and Scope

The thermal conductivity of ceramics and other

nonmetalhc solids is of considerable practical

and theoretical importance. Many of the practical

applications which require accurate knowledge of

thermal conductivity are concerned with devices

for conversion of heat into other forms of energy.

Furnace and boiler design requires knowledge of

the thermal conductivity of insulating bricks and

refractories from ambient to high temperatures.

In a nuclear reactor, the maximum temperature

of the fuel elements is directly related to the

thermal conductivity of these fuel elements. The

Page

d. Alloying 85

5.2. Structure 85

a. Crystallinity 85

b. Imperfections 86

c. Additional phases 87

d. Microstructure 93

6. Reliability of existing data 95

7. Measurement methods 96

7.1. Longitudinal heat flow 96

a. Mathematical development 96

b. Experimental techniques 97

7.2. Radial heat flow 107

a. Mathematical development 107

b. Experimental techniques 108

8. Reference standards 108

8.1. Current status 109

a. Metals 109

b. Nonmetals 110

8.2. Ongoing activities 111

9. Conclusions 112

10. References 113

efficiency of a thermoelectric power generator is a function of the thermal conductivity of the thermoelement.

The extreme temperatures encountered in aero- space applications, ranging from containment of

cryogenic fuels to ablative heating during re- entry, have created yet another area where

accurate thermal conductivity data are required.

In many applications of solid-state electronics,

materials used as heat sinks or as potting compounds must have as high a thermal conductivity

as possible in order to dissipate the heat generated

by the device.

63
At very low temperatures (i.e., below the boiling

point of nitrogen), the thermal conductivity of

solids is extremely sensitive to imperfections.

Thus thermal conductivity measurement provides

a valuable tool for the solid-state physicist.

In this paper we define thermal conductivity

and present a brief phenomenological description

of the energy transport mechanisms involved. We

then discuss the influence of environmental

effects, such as temperature, pressure, and stress,

on the thermal conductivity of solids. This is followed by a discussion of the effects of composition and structure on the thermal conductivity

of ceramics and other nonmetallic solids. Some

sources are cited for thermal conductivity data

and a few remarks are made regarding the reliability of thermal conductivity data. We describe some of the techniques which are used to

measure thermal conductivity and comment on

the availability and use of thermal conductivity

reference standards.

2. Definition and Phenomenological

Theory

2.1. Definition

The thermal conductivity, X, of a substance is related to the rate of heat flow per unit area,

q, and the temperature gradient, vT, by Fourier's

law, which for an isotropic medium can be written

as q=-XV T. (1)

For one-dimensional heat flow, eq (1) is simply

5T

2*= —

X

bz (2)

Equation (1) may be regarded as the definition

of thermal conductivity.

If one considers an elemental volume in an

isotropic medium which is opaque to thermal radiation, the following second-order differential equation is easily derived

V • (\VT)+W=C dT

(3)

where C is the heat capacity per unit volume, t is time, and, if heat is produced in the solid, Wis the

rate of heat production per unit time and unit

volume. In general the thermal properties, X and

C, can be functions of position and of temperature.

For a homogeneous material which has no

internal heat production and in which thermal

conductivity is independent of temperature, eq

(3) reduces to the form

a Ot

(4)

independent of temperature and position over the

temperature range of interest but C may still be

temperature-dependent.

Many engineering applications involving timedependent temperatures require a knowledge of

the thermal diffusivity rather than, or in addition

to, the thermal conductivity. In this paper we

shall confine our attention to thermal conductivity

and will not discuss thermal diffusivity. If needed,

the thermal diffusivity of a material can be

measured directly or can be computed from the

thermal conductivity and the volumetric heat

capacity. Since heat capacity is relatively insensitive to variations of composition and structure,

remarks we make concerning the effects of these

on thermal conductivity are applicable to thermal

diffusivity as well.

Many applications which require a knowledge

of thermal conductivity or thermal diffusivity

involve analytical or numerical solutions of some

variation of eq (3). The reader who is interested

in mathematical solutions of the heat conduction

equation should consult, for example, the books

by Carslaw and Jaeger [l] 1

, Schneider [2], and

Arpaci [3].

2.2. Phenomenological Theory

The theory of thermal conduction is the subject

of many research papers and review papers and

forms a major portion of the material in quite a few books. Thus we can only briefly touch on

the theory of thermal conduction and will limit

ourselves to a phenomenological discussion of the

processes and principles involved. For detailed

discussions of this subject the reader can consult

the various books and review papers cited [4-15].

It is important to realize that the form of eq (1)

implies that heat conduction is a diffusion process.

The thermal energy diffuses through the material.

In conduction heat transfer the flow of heat is proportional to the temperature gradient. This

may be contrasted with radiative heat transfer

through a perfectly transparent medium (e.g., in

vacuo) where the thermal flux is related to the

temperatures of the surfaces which are exchanging

heat. In many cases the heat conducted through

a material is directly proportional to the temperature difference—however, these are special cases which arise, for example, because the

thermal conductivity is essentially independent

of temperature over the temperature range of concern. A well-known result of the kinetic theory of

gases is that the thermal conductivity of a gas

is given by

(5)

where a=\/C is the thermal diffusivity. Note

that in order for eq (4) to be valid, X must be

where C is the heat capacity per unit volume,

v is the average particle velocity (group velocity),

1 Figures in brackets indicate the literature references at the end of this paper.

64
and A is the mean free path of a gas particle

between collisions with another gas particle. It

is extremely useful to carry eq (5) over to the

case of heat conduction in a solid. Before discussing this it is instructive to consider one- dimensional heat flow in an elemental length,

Az, of material supporting a temperature gradient

AT/Az. From (2) and (5) the rate of heat flow in

the 2-direction is

*-=-xtf=-*(CAr>(^> oo

The quantity CAT is the excess thermal energy

per unit volume at one end of the elemental length

with respect to the other end. This excess energy

is propagated down the specimen with an effective

transport velocity which is just the particle

velocity reduced by the ratio of the mean free

path to the length of the element.

In a solid material, heat can be transported by

a variety of carriers or mobile excitations. For

this "gas" of heat carriers we can write, by analogy

to eq (5),

X=*SC>«A„ (7)

i

where the summation is over all carriers. Equation

(7) simply states that the heat transported by

different carriers may be added to obtain the

total heat transported. In a solid where thermal

transport is via several types of carriers, the total

thermal conductivity is given by

\=\i+\e +K+. • (8)

where the different subscripts refer to different

types of carriers. One definitely should not infer

from eqs (7) and (8), however, that the different

mechanisms of heat conduction do not interact.

The different components of the total thermal

conductivity are additive but they are not

independent.

Of the three quantities on the right hand side

of eqs (5) and (7), the carrier heat capacity and

the carrier velocity normally have a relatively

well-known temperature dependence and are

relatively insensitive to material imperfections

and impurities. The carrier mean free path is the

troublemaker. It may have a rather complicated

and elusive temperature dependence and may be

very sensitive to material imperfections and

impurities. The mean free path of carriers is determined by the interactions ("collisions") between

the carriers and by the interactions between the

carriers and material imperfections and impurities.

In the remainder of this section we will discuss

the theory of thermal conductivity as it applies to

the several mechanisms of heat transfer which

take place in ceramic materials.

a. Lattice Conduction (Phonons)

A solid material may be thought of as a threedimensional array of masses (atoms) held together

by springs (interatomic forces) . Each atom vibrates

with an amplitude which is dependent on temperature; atoms which are at higher temperatures

vibrate more vigorously than those at lower temperatures. This thermal motion of the atoms can be resolved into traveling elastic waves which

transport thermal energy through the solid. The

energy carried by these waves may also be thought

of as being carried as a number of quanta of energy,

called phonons, which can be treated as particles.

This is analogous to the better known practice

of treating light waves (or traveling electromagnetic waves) as particles called photons.

In order to better discuss the properties of

phonons, it is helpful to consider certain aspects of

the vibrational spectrum of a discrete lattice. For

this purpose it will suffice to consider a one- dimensional lattice with two kinds of atoms

equally spaced as shown in figure 1 . If we consider

each atom to behave as a classical harmonic

oscillator which interacts only with its nearest

neighbors, it can be shown that the only waves

which can be propagated along this chain of

atoms are those having an angular frequency

(rad s-1

) given by

2 of 1

, 1 V alY 1, 1 V 4 sin2 ka j

2

where m and M are the masses of the two kinds of

atoms, a is the separation between adjacent atoms,

j8 is the force constant acting to restore each atom

to its equilibrium position, and k is the wave

vector, which has a sign corresponding to the

direction in which the wave travels and a magnitude equal to 27r/(wavelength). The two branches

of eq (9) are shown plotted in figure 2. The limiting

values of eq (9) are shown in figure 3. The lower

branch is called the acoustical branch and the

upper branch is called the optical branch.

Phonons transport energy with a velocity

(known as the group velocity), v =du/dk. For

small values of k (long wavelengths), the group

velocity in the acoustical branch is

v=dk=lm^nl a=

k

(10)

For wavelengths long in comparison to the interatomic spacing, adjacent atoms vibrate in phase

in the acoustic branch and our one-dimensional

M m M m M

Figure 1. Schematic representation of a

diatomic linear lattice.

65
CO

-3TT/2a -TT/a -TT/2a 0 TT/2a TT/a 3TT/2a

_^ Brillouin zone

Figure 2. Plot of angular frequency, versus wavenumber,

k, for a diatomic linear lattice.

00

(2/?/ M) 1

(Zp/m)

2a

Figure 3. Positive half of the first Brillouin zone for a diatomic linear lattice.

lattice behaves like a continuous elastic string

down which phonons propagate with a constant

velocity given by eq (10) 2

. In the higher frequency, or optical branch, the

phonons have a constant frequency for long wavelengths. Adjacent atoms vibrate 180° out of phase

for the long-wavelength optical phonons. Since

dw/dk=0 for these phonons, their group velocity

is zero and they can transport no energy.

From figure 3, we see that there is a forbidden

band, _2/3/m<co2 <2/3/M, in which no phonons can be excited.

The periodic structure of eq (9), as shown in

figure 2, has a very important consequence.

Looking at figure 2 and recalling that the energy

of a phonon is propagated with the group velocity,

v=dw/dk, we see that a phonon having a wave

vector between w/2a and ir/a has a group velocity

opposite in sign to k. Although the wave is traveling to the right, the energy is traveling to the left. As an example consider a phonon of wavelength

8a/3 traveling in the positive direction (i.e., to

2 The "density" (mass per unit length) of our one-dimensional lattice is d=(M+m)l2a. The "Young's modulus" is E=£a. Thus eq (10) is simply the usual expression, »= -^E/d, for the velocity of sound in an elastic continuum.

the right) in our one-dimensional lattice; this

phonon has a wave vector +37r/4a. Consider also a phonon of wavelength 8a traveling in the negative direction; this phonon has a wave vector — x/4a. Since 3ir/4a and — ir/4a are completely

equivalent in eq (9) , we see that these two phonons,

of quite different wavelengths, are completely

equivalent both as to the frequency of vibration

and the direction of energy propagation. In view

of this periodicity, we need only be concerned with

phonons having wave vectors whose magnitude

is less than ir/2a (wavelength greater than 4a).

The region —7r/2a<&<7r/2a, indicated by the

vertical dashed lines in figure 2, is known as the

Brillouin zone. Wave vectors falling outside the

Brillouin zone can be brought back inside by

adding or subtracting the appropriate integral

multiple of ir/a.

The equivalence of two phonons having wave

vectors differing by nw/a arises because of constructive or destructive interference effects due

to reflections of the phonons from the discrete

lattice. Phonons of wave vector equal to nw/a

cannot propagate at all because their reflections are exactly 180° out of phase. For phonons of

wave vector equal to («—};>) 7r/a, the reflections are in phase and the phonons at the edge of the

Brillouin zone correspond to standing waves for

which the group velocity vanishes.

It should be noted that all of the aspects discussed so far in this section are classical. We have

used the term phonon to represent quanta of

energy traveling through the lattice—however,

this was done for convenience and not from

necessity. Thus, to this point in our discussion, it has not been necessary to invoke any quantum

theory.

The extension of the above discussion to a threedimensional lattice of harmonic oscillators presents

no particular difficulty. The wave vector k indicates the direction of propagation and has the

magnitude 2x/(wavelength). For a given k, there

are three independent directions in which the

atoms can vibrate. For a not-too-complicated

lattice, there is a single longitudinal wave having

particle displacements parallel to k and two

transverse waves having displacements perpendicular to k. Depending on the symmetry of the

crystal, the longitudinal phonon and the two

transverse phonons for a given k can have

different frequencies. For a diatomic crystal

there may be three acoustical branches and

three optical branches for a given direction of

propagation.

References [4-15] all contain some discussion

of phonons. Other useful references to the theory

of lattice vibrations include [16-28].

Having introduced the concept of phonons, we

can think of an ideally perfect crystal as an

otherwise-empty box containing a "gas" of

phonons which collide with one another and with

the walls of the box (i.e., the surface of the

crystal). These phonons will have different wave66
vectors and frequencies as discussed above.

Bringing in some quantum theory, each phonon

transports energy in the amount hu, where

ft= 1.05450X10-^ Js. 3 In our discussion above, we assumed that the

atoms behaved as purely harmonic oscillators. If

this were strictly so, the phonons would not

interact with one another and the thermal conductivity of a perfect crystal would be infinite.

In fact, there are anharmonic terms which allow

the phonons to interact and cause thermal resist- ance. The simplest type of interaction which

can occur is when two phonons combine to form

a third phonon as shown in figure 4.

Conservation of energy requires that

hwi-{-ha)2=fow3', (11)

while conservation of the wave vectors (analogous

to conservation of momentum) requires that

k1 +k2 =k3 . (12)

Phonon interactions such as described by eqs (11)

and (12) are known as normal processes or re- processes. Both energy and quasi-momentum (a

name sometimes given to hk) are conserved and

there is no attendant thermal resistance.

Consider, however, an interaction in which kx and k2 aie roughly in the same direction and

their magnitudes are sufficiently large that k3 lies outside the Brillouin. zone and, therefore,

corresponds to energy transport in a different

direction. For a square, two-dimensional lattice,

such an interaction is illustrated in figure 5. The resultant vector, k3 , can be replaced by

k4 =k3 — G, where G is a vector perpendicular to

the boundary of the Brillouin zone and equal

in magnitude to the width of the zone. For such an interaction, eq (12) can be replaced by

k,+k2 =k4 =k3 -G. (13)

k 2

, co 2

k,, u>,

Figure 4. A normal- or Nprocess in a square, two- dimensional lattice. A phonon of wavevector ki (wi) combines with a phonon of wavevector k2 (102) to form a new phonon

of wavevector ks (wi) Both energy and quasi-momentum are con- served.

ft=A/2jr, where A is Planck's constant.

—k 2

» co 2 0<

k 4 ,co4/^ \

Hi*

*k, , co,

Figure 5. An umklapp- or U-process in a square, two-dimensional lattice. A phonon of wavevector ki (on) combines with

a phonon of wavevectork2 (w). Since the result- ant vector k3 (0)3) crosses the boundary of the

first Brillouin zone, the direction of energy

flow is changed so that the newly-formed phonon

has the wavevector k* (ut), where wi =oi3. In a

U-process, energy is conserved but quasimomentum is not conserved.

In such an interaction, known as an umklapp

process, or U-process, the direction of energy

flow is changed and there is an attendant thermal

resistance. The change in direction of energy

flow which accompanies a U-process can be

thought of as an interference effect in which the

phonons are reflected by the lattice; this is analogous to Bragg reflection of x-rays.

According to quantum theory, the energy of an

oscillator can assume only the discrete values

E(\t)=fu*{N+%), (14)

where N is a positive integer.4 The energy corres- ponding to a given k thus consists of a zero-point

energy, hu/2, plus N phonons, each carrying an

energy hu. For a crystal in thermodynamic equilibrium, the average number of phonons in a mode

of frequency w is

r) =;xP (Hi D-i (16)

where ^ =1.38054 X10-23 JK"1 is the Boltzmann

constant, and T is the absolute temperature.

Classically, the energy corresponding to a given

temperature would be simply k

T. The number of

units of energy of size fiw in a classical oscillator

is thus T/6, where 0 is a vibrational temperature

defined by d=fiu/ Jt

. In figure 6 we have plotted the

quantity ON/T, N being given by eq (15). This is simply the ratio of the number of phonons in a mode of frequency co to the number of phononsized units of energy that would exist if there were

no quantum effects. We see that at temperatures

which are low in comparison to 0, there are very

few phonons available.

The paucity of phonons at low temperature

affects the lattice thermal conductivity in two

* We can let it be understood that k specifies the polarization as well as the magnitude and direction of the wave vector.

67
ways: (1) the specific heat of the solid decreases

with decreasing temperatures thus tending to

decrease the thermal conductivity; (2) the number

of U-processes decreases with decreasing temperature thus increasing the phonon mean free path

and tending to increase the thermal conductivity.

It is customary to discuss the heat capacity of a solid in terms of the Debye model; in this model

the low temperature heat capacity is proportional

to (T/6) 3

, where the Debye temperature, 6, is defined by / 6=/icomax , where comax is the highest

frequency of vibration in the lattice. From our

discussion above, it is seen that in order for

U-processes to occur it is necessary that the

interacting phonons have wave vectors extending

about halfway to the boundary of the Brillouin zone. This would correspond to phonon energies of

about /Lgj=^ 9/2. The number of such phonons is given by eq (15); for T<<G this is exp(— 6/2 T).

The rate at which U-processes occur, then, should

be approximately proportional to this factor.

Since the mean free path of phonons in an infinitely

large ideal crystal is limited only by U-processes,

the mean free path should be approximately

inversely proportional to this factor. Assuming

that the phonon velocity is essentially independent

of temperature, the low temperature thermal conductivity of an ideal crystalline lattice is given by

x < =a (£f exp(^r)' when T«e > < 16 )

where a depends on the structure of the crystal,

n has a value of 3 or less, and b is of the order of 2.

At temperatures larger than the Debye temperature, the specific heat of a dielectric crystal

becomes essentially constant. At such temperatures, most phonons will have wave vectors of

sufficient magnitude to permit U-processes to occur. The number of U-processes will therefore

be approximately proportional to the number of

phonons; from figure (6) we see that this is proportional to the temperature. Again assuming

that the phonon velocity is essentially independent

of temperature, the high temperature thermal

conductivity of an ideal crystalline dielectric is given 5 by

\ t =Aj> when 7»9> (17)

At low temperatures, the phonon mean free

path would increase exponentially with decreasing

temperature in a sufficiently large ideally perfect

crystal. In a crystal of finite size, the mean free

path of a phonon is limited to a value of the order

of the crystal diameter. This was first predicted

by Pieirls [30], observed experimentally by de

5 Equation (17) is not necessarily valid for T'=e.Leibfried and Sehlomann

[ 29] obtain x.=4Mf)'+. • }>

where the coefficient of (6/T) 2 is positive and has a magnitude depending on

the frequency spectrum of the lattice.

1.0

1 r

3 T/e

Figure 6. The number of phonons in a

given mode, as predicted by quantum

mechanics, divided by the number of phonon-sized units of energy in a classical

oscillator at the same temperature.

Haas and Biermasz [31], and calculated by

Casimir [32]. The reflections of phonons from the

specimen boundaries is analogous to the flow of a

rarified gas in a tube as first discussed by Knudsen

[33] and Smoluchowski [34]. Ziman [7, ch. XI]

and Klemens [11] discuss boundary scattering of

phonons in some detail. Other references include

[10, 35-43]; Guyer and Krumhansl [42] and

Thacher [39] give still more references to theoretical and experimental work on boundary

scattering of phonons.

At temperatures low enough that the specific

heat varies as T3, the thermal conductivity of a

finite crystal varies approximately as

(18)

where d is the effective diameter of the crystal;

the proportionality constant depends on the

nature of the crystal surface—a roughened surface

which scatters phonons diffusely will result in a lower thermal conductivity than, for example, a cleaved surface which scatters specularly.

Figure 7 shows the data obtained by Thacher

[39] for single crystals of isotopically pure LiF

of different sizes. Note that this is a log-log plot

and the differences among the four curves shown

are large.

Qualitatively, at least, we now have discussed

the factors which affect the thermal conductivity

of a dielectric single crystal which is free from

defects. In figure 8 the thermal conductivity of a sample of single-crystal artificial sapphire (AZ2 03 )

is shown plotted versus temperature. At higher

temperatures, the thermal conductivity is controlled by Umklapp processes and goes approximately as T' 1 (eq (17)). As the temperature

decreases, Umklapp processes are frozen out,

the mean free path increases exponentially, and

the thermal conductivity varies with temperature

68
o

o

1 III 1

// /* » fl A A \

A J A A \

lb

- lit

ill

( r \

I H

» MEAN CRYSTAL Hit WIDTHS ftt I

0 7 25 mm

• 400 mm i

A 214 mm it

A 1.06 mm III 1 2 5 10 20 50 100

TEMPERATURE, K

Figure 7. Thermal conductivity of different sized crystals of lithium fluoride, illustrat- ing the effect of boundary scattering of

phonons (after Thacher [39]).

according to eq (16). At still lower temperatures,

the mean free path becomes constant at a value

of the order of the effective crystal diameter and

the thermal conductivity varies approximately as T3 (eq (18)); in this temperature range it is no

longer meaningful to speak of the thermal conductivity of a given material—one can only

speak of the thermal conductivity of a given

specimen.

In a real crystal, the height of the peak in the

thermal conductivity will be reduced, due to

scattering of phonons by impurities and imperfections, to a value less than that which would

be extant in a perfect crystal. We shall now

briefly discuss theoretical predictions of the influ- ence of impurities and imperfections on the

thermal conductivity of a dielectric crystal. The

very large amount of theoretical work in the

area and the complexity of the calculations

involved make it impossible for me to do much

more than state the results. The interested

reader should consult one of the review papers by,

for example, Klemens [9-11] or the book by

Ziman [7]. Other references include [4-6, 8, 12-15,

29-30, 42, 44-61].

Consider an imperfection (this could be an

impurity, vacancy, dislocation, etc.) in an otherwise perfect crystalline lattice. In the vicinity

of this imperfection the elastic properties of the

lattice will differ from those in the perfect lattice.

Consequently, a lattice wave or phonon will be

deflected or scattered when it encounters an

imperfection. If one thinks of a perfect dielectric

0 25 50 75 100 TEMPERATURE, K

Figure 8. Idealized temperature-dependence

of thermal conductivity for a single-crystal

(AI3O3) in which the only mode of heat conduction is via lattice vibrations (after

Ziman [7, p. 292}).

crystal as an otherwise empty box containing

a gas of phonons, then an imperfect crystal can

be thought of as a box containing a number of

stationary obstacles with which the phonons

interact or collide. The effective scattering cross- section of an imperfection is dependent on the

extent of the surrounding strain field. If a disordered region is small (point defects) compared

to the wavelength of the interacting phonon,

the details of the strain field are not important.

If a disordered regions is comparable to, or larger

than, the wavelength of the interacting phonon,

interference effects will enter in and the detailed

shape of the strain field must be considered. If a number of impurities or point defects are arranged

in some regular array, interference effects again

must be considered. In the case of a substitutional

impurity, the mass difference between the impurity

atom and a normal atom will lead to scattering

of phonons, irregardless of the associated strain

field.

Quantum-mechanically, one usually considers

the Hamiltonian of a system of phonons in the

absence of impurities and then applies timedependent perturbation therory to find the effect

caused by impurities or imperfections. The

references cited above may be consulted for details.

The total lattice thermal resistivity (Wi=l/Xi)

can be expressed as the sum of the thermal

resistivities due to various scattering mechanisms

Wl =Wlu +Wl »+Wlp + . . ., (19)

where Wtu is the thermal resistivity due to

U-processes, Ww is that due to boundary scattering, Wip is that due to point defects, etc. In

table 1 we present the temperature dependence

69
Table 1. Temperature variation of Thermal Resistivity due

to Various Interaction Mechanisms (after Klemens [10])

Temperature

Scattering mechanism dependence of thermal

resistivity

Low temperatures (T<6):

Umklapp processes-. T-> e -e/tT Point defects... Ti Long cylinders yo

Dislocations (core) yo

Dislocations (strain field) Conduction electrons in metals y-2

Stacking faults... T-i Thin sheets embedded in crystal of continuous orientation T-i Grain boundaries. _ T-3

External boundaries y-3

High temperatures (T>6):

Umklapp processes Ti

All imperfections __

of the component of thermal resistivity due to

these various scattering mechanisms. If the relative strengths of the different scattering mechanisms are known, eq (19) can be used to obtain

the temperature dependence of the total lattice

thermal resistivity.

At temperatures well above the Debye temperature, we see that the thermal resistivity due to all imperfections should be independent of temperature and the total thermal resistivity should be

simply

W,=l/\t=A+BT, (20)

where A and B are constants which can be obtained

empirically. For fairly pure dielectric crystals,

experimental data conform reasonably well to

eq (20) in many cases. In section 5 we shall discuss experimentally

obtained thermal conductivity values with respect

to the influence of composition and structure.

b. Electronic Conduction

In many traditional ceramic materials such as metallic oxides, the electrical resistivity is sufficiently high that the electronic component of

thermal conductivity is negligible. However, in

metallic carbides, graphite, and many semiconductors, for example, the electronic portion of

thermal conductivity may be comparable with or

larger than the lattice portion. Cermets, or mixtures of metals and ceramics, are also of interest

to the ceramic industry; in such two-phase materials, the thermal conductivity of the metallic

phase may be almost entirely due to electronic

conduction. It is therefore quite appropriate that

we briefly discuss the theory of electronic thermal

conductivity.

The electrons which conduct heat and electricity

may be thought of as a gas of essentially free

electrons wandering aimlessly around in the periodic potential field of the crystalline lattice. In

the presence of an electric field or temperature

gradient, the electrons are motivated in a particular direction and there is a net transport of charge

or thermal energy.

These wandering electrons have energies that

fall within certain "bands". Between these bands

there are gaps, or energy regions which may not

be occupied. The two bands which are of interest

to us are the valence band and the conduction

band. The valence band is analogous to the outermost shell of electrons in a free atom—the electrons are bound to the atom but can be set free if they

acquire sufficient kinetic energy. Electrons in the

conduction band have sufficient kinetic energy

to permit them to travel through the crystal.

In a metal the conduction band is partially filled

with electrons even at 0 degrees kelvin. The aver- age number of electrons in a state of energy E is given by the Fermi function:

f(E) =

1 exp l(E-t)UT\+l (21)

where the Fermi energy, f, is determined by the

condition that 2/(2?) over all quantum states

must equal the total number of electrons in the

conduction band.6 The behavior of the Fermi

function is shown in figure 9. At a temperature of

absolute zero, all states having energies less than

f are filled and none of the higher energy states are filled. As temperature is increased, j{E) no

longer is a step function but rather decreases

from j{E)=\ tof(E) =0 over an energy interval

of width approximately equal to ^ T centered at

E=£. Because of this behavior of the Fermi

function, only those electrons having energies near f are affected by temperature change. An

electron gas is considered to be "degenerate" when

T<^<^TF, where TF =£/Jl is known as the Fermi

temperature.7 For a degenerate electron gas, the

specific heat of the electrons is proportional to the

absolute temperature.

At very low temperatures, electrons could

travel essentially unimpeded through a perfect

metallic crystal and the mean free path of the

electrons would increase very rapidly as the

temperature is decreased toward absolute zero. The electronic thermal conductivity of a

perfect metallic crystal would thus be limited at

low temperatures by boundary scattering as was

discussed for the phonon thermal conductivity.

However, in a real crystal, as opposed to a perfect

UJ

Figure 9. The Fermi function, eq{2i).

6 Electrons obey the Pauli exclusion principle so that only two electrons

(of opposite spin) may occupy a given energy level.

7 Typically, V» K< 2X105 K for metals.

70
one, the limiting factor for electronic thermal

conductivity is scattering of electrons by impurities and perfections. In the case of very thin

films or fine wires, the electronic component of

thermal conductivity will be limited by boundary

scattering.

At high temperatures, electrons are scattered

very effectively by phonons and the mean free

path of the electrons is approximately inversely

proportional to the number of phonons and thus

inversely proportional to temperature. With a heat capacity which is proportional to temperature, an electron velocity (corresponding to f)

which is essentially constant, and a mean free

path which is inversely proportional to temperature, we see that eq (7) predicts an essentially

constant electronic thermal conductivity for a pure metal at temperatures which are high compared to, say, the Debye temperature, 0, but

low compared to the Fermi temperature, TF . If

impurities are added to a metal, the electronic

heat capacity will not be seriously affected and the

mean free path will be decreased more at low

temperatures than at high. For a completely

disordered alloy, the mean free path would be

essentially constant and the electronic thermal

conductivity would be proportional to the electronic specific heat and hence to temperature.

In discussing electronic thermal conductivity,

it is customary and useful to discuss it in terms of

the Lorenz function

L =\> (22)

0~±

where Xe is the electronic component of thermal

conductivity, o~ is electrical conductivity, and T

is absolute temperature. An expression analogous to (19) may be written for the electronic

thermal resistivity:

We =l=We0 +Wel + . . .

, (23)

where We0 is the thermal resistivity due to

scattering of electrons by static imperfections

and Wel is the thermal resistivity due to scattering

of electrons by phonons [7, 60-61]. Similarly, the

electrical resistivity may be expressed as

P=-=Po+P*+ ... .

, (24)

0~

which is exactly analogous to eq (23). We now

assume that each component of electronic thermal

resistivity is related to the corresponding electrical

resistivity by a relation such as eq (22) but with

a different characteristic "Lorenz function" for

each scattering mechanism. The electronic thermal

resistivity may then be written as w-=h=T[£+£+ ]•

TM

These "Lorenz numbers" are not, in general,

constant but may vary with temperature. To

a good approximation, the structural electronic

thermal resistivity (i.e., that due to scattering

of electrons by impurities and imperfections) is related to the structural electrical resistivity at

all temperatures by a Lorenz relation where L0 has the constant theoretical value Z0 =2.443X 10-8

V2K

-2

. 8 (In the present paper, the nomenclature

L0 will always be used when this theoretical value

is meant.) The coefficient, Lh associated with the

electronic thermal resistivity due to lattice vibrations, is not a constant. At low temperatures,

electrons are scattered inelastically and nonisotropically by electron-phonon interactions. Thus

the effective mean free path for thermal conduction is smaller than for electrical conduction.

Therefore Lh as here defined, rises from a value

of zero at 0 degrees kelvin and asymptotically

approaches the theoretical value, L0 , at temperatures well above the Debye characteristic temperature, 0.

If there are no additional scattering mechanisms,

so that We =We0 +Wel and p = Po -\- Pl, then the

effective Lorenz function is, from eqs (22) and

(25),

l=l{

i

--X

i -t)T- (26)

For a very pure metal where po/p<C<11, L~Lt ; for a highly disordered metal where p = p0 , L=L0

. In figure 10, the theoretical temperature dependence [62-64] of L/L0 is shown for various

values of p0

. For any real specimen, p0 is finite so that at sufficiently low temperatures p=p0 and L=L0

. At low temperatures, the electronic thermal

conductivity of a fairly pure metal or dilute

alloy is given closely by

±=aT*+P T«Q> (27)

TEMPERATURE

Figure 10. The temperature variation of the Lorenz function as a function of purity and

temperature.

La is the limiting value of L at low and high temperatures. The top curve relates to the most impure specimen (after Rosenberg [8, p. 120]).

8 £o=y(j)

2 where k is Boltzmann's constant and e is the charge on an

electron.

71
where n>2, 0~po/Lo , and a is an empirical

constant which is characteristic of the particular

metal. At higher temperatures, electrons are scattered elastically and isotropically by phonons,

and Li~L0 ; therefore

-s(?+i

>

T>6 ' <28>

since p~p0 -\-bT at higher temperatures. For

highly pure metals, the electronic thermal con- ductivity approaches the constant value \e =L0/b

at high temperatures. For extremely disordered

alloys, the structural resistivity, p0 , dominates

and the electronic thermal conductivity increases

linearly with temperature.

In figure 11 we see the idealized behavior of the

thermal conductivity of copper as derived by

Cezairliyan and Touloukian [65-67]. At the lowest

temperatures, the thermal conductivities increase

linearly with increasing temperature. The thermal

conductivity values pass through a maximum

value, Xm , at a different temperature, Tm , for each

sample and then decrease again at higher temperatures to an approximately constant value. These

recommended values of Cezairliyan and Touloukian are in only fair agreement with experimental

data obtained on copper [68-72].

The electronic component of thermal conductivity for ferromagnetic metals, transition metals,

and semi-metals (e.g., bismuth) is less well

understood than for more ideal metals and the

above is of somewhat restricted applicability.

Additional references to the theory of the thermal

conductivity of metals include [7-9, 11, 73-79].

10 io2 TEMPERATURE ,K

Figure 11. Thermal conductivity of samples of

copper of different purity as a function of

temperature.

The parameter /3, expressed in units of mK2 W->,

(see eq (27)) is approximately equal to the residual

electrical resistivity of the sample divided by the

theoretical value of the Lorenz function (after Cezairliyan and Touloukian [65-67]).

In an "intrinsic" semiconductor at 0 degrees

kelvin, the valence band is completely filled and

the conduction band is completely empty; thus

the electrical conductivity and the electronic

thermal conductivity of an intrinsic semiconductor

are zero at absolute zero. As the temperature

increases, some electrons attain sufficient thermal

energy to get into the conduction band. This

results in "holes" being left in the valence band.

It is possible for energy and charge to move

through the crystal by means of electrons moving

from one hole in the valence band to another hole.

The semiconductor acts as if there were particles

of positive charge moving through it. Thus it has

become common to talk about the motion of the

holes, which under the influence of a temperature

gradient or electric field will result in a net transfer

of thermal energy or electric charge.

In an intrinsic semiconductor, the number of

electrons in the conduction band and the number

of holes in the valence band increase very

rapidly with increasing temperature. This results

in a rapid increase in electrical conductivity with

temperature. The relation between electronic

thermal conductivity and electrical conductivity

for an intrinsic semiconductor is rather complex

[7, 12, 14, 80-81]. At higher temperatures, the

Lorenz function is given, if certain assumptions

are met, by

HH(0I1+1?(4+iI)1 (29>

where o-=ae

-\- ah , ae and ah being the portions of

the electrical conductivity due to electrons and

holes, respectively, and Ee is the width of the forbidden gap between the valence and conduction

bands. Equation (29) has its maximum value

when ah =ae , L then being greater than 6(^/e) 2 by an amount which depends on Eg

. An "extrinsic" semiconductor, which may in

its purest state be an electrical insulator, contains

impurities which have either more or fewer

valence electrons per atom than the normal atom.

This results in the creation of either electrons in

the conduction band («.-type semiconductor) or holes in the valence band (p-type semiconductor).

Either results in an increase in the electrical

conductivity. 9 In a compound semiconductor, a

stoichiometric deficiency of one constituent will

act as an impurity. 10

If the conduction band of an extrinsic semiconductor is only slightly filled, the Fermi energy,

and hence the Fermi temperature, will be low so

that, at sufficiently high temperatures, T^>y>TF

and the electron gas is nondegenerate. Similarly,

if there is a small number of holes in the valence

band, the hole gas is nondegenerate. Under such

conditions, the Lorenz function is simply L—

2 U leY, which is what eq (29) reduces to if either

9 For example, 10 ppm of boron in pure silicon will increase the room temperature electrical conductivity by a factor of 103

. 10 For example, CU2O and ZnO.

72
ue or 0-

ft vanish. If many electrons (or holes) are present so that T<C<CTF, the carrier gas becomes

degenerate and L=L^=)'%-k2 (11 /e)

2

, as for a metal.

The curve in figure 12 shows how the electronic

thermal conductivity of a simple extrinsic semiconductor varies with the electrical conductivity.

c. Radiation "Conduction"

Planck's law [82-84] states that the energy

emission per unit time and per unit area (Wm~2) from a blackbody in a range of angular frequency

du (rad s_1

) is given by

eba (T)d<j}=

47r

2 c2 [exp (fa/AT)—l]

(30)

where fi and k have previously been identified,

c is the speed of light in vacuum, and n is the

index of refraction (at u>) of the medium which

bounds the blackbody. Expressing eq (30) in

terms of frequency rather than wavelength is preferable for our present purpose since a does

not change when radiation travels from one

medium to another medium which has a different

index of refraction (i.e., speed of light). Equation

(30) can be rearranged as

eba _15h / %w \

3 1 <TM2 T3 ~irH \kTJ [exp (ha/tT)-!]

(31)

where a-- «

2 k -- 5.6697 X 10- 8 Wm-zK~i is known

60h3c

2

~

as the Stefan-Boltzmann constant. In this form

the right hand side of eq (31) is a universal function of / T/hco and is so plotted in figure 13.

We note that

kT=

kT^nlT

j % o) hv Ci (32)

where v—2tw is frequency, I is wavelength (in the

medium having n as its index of refraction), and

c2 =27rftc/^ =hcjk= 1-43879 X10-2m K is known as the second radiation constant. 11 The total energy radiated is obtained by integrating eq (30) over all frequencies:

Jo 4*VJ0 [exp (hoof kT)-l]

(33)

If we assume that the index of refraction of the

bounding medium is independent of frequency, eq

(33) can be integrated to yield

eb {T)=n2 aT\ (34)

Eq (34) is usually seen without the factor n

2

, since

7i=l in a vacuum.

The monochromatic energy emitted from a

real (non-black) surface is

(35)

where e a is the hemispherical emittance of the

surface for radiation of angular frequency w (i.e.,

2.0

rueT, m K

0 0.01 0.02

Figure 13. Spectral emissive

power of a blackbody.

Figure 12. Plot of electronic thermal

conductivity versus electrical conductivity for a simple extrinsic semiconductor at constant temperature.

Region A is the nondegenerate region in which

the carriers obey classical statistics and the Lorenz function is given by 1=2 We) 2

. In the

degenerate region, C, the carriers obey FermiDirac statistics and L=l/3ir2 (k/e) 2

. Region B is intermediate between these two limiting cases.

11 Usually one sees en, the energy emission in a range of wavelength dl, rather than etu, which we have discussed here and plotted in figure 13. The

quantity which is usually plotted is some multiple of

n3 T« (nl T) » [exp (cj/«tf T)—l]'

where c\=2ivh c

2 =3.7405X10"16 Wm2 is known as the first radiation constant. This is most frequently seen without the n appearing explicitly, n=l being assumed. Even if n is included, this expression of Planck's law in terms of wavelength rather than frequency is not generally valid since it depends on

the assumption that n is independent of wavelength.

323-655 O - 69 - 6 73
photons of energy hco). The total energy emitted

from a surface is

/» 00 /» 00 2= I e adto= I e aebada

Jo Jo

(36)

The total hemispherical emittance is defined as the ratio of the emissive power, e, of a given

surface to that of a black surface at the same

temperature, i.e.,

(37)

Note that e is a property of the surface and its

temperature.

The monochromatic energy per unit time and

area absorbed by a real surface is

where Ha is the intensity of the incident radiation

and au is the monochromatic hemispherical

absorptance of the surface. The total hemispherical absorptance is

a

(38)

ta

Note that a is not a property of the surface alone

but is a function of the nature of the incident

radiation.

Kirchhoff's law states that

au = t a . (39)

One frequently sees the assumption being made

that Kirchhoff's law holds for total absorptance

and emittance. This assumption is valid in some

cases [84, pp. 41-42] but in general

a 7^ €. (40)

We now wish to discuss radiative transfer of

energy within a diathermanous material; that is,

one which is partially transparent to infrared

radiation. Radiation (photons) traveling through

a medium can suffer attenuation due to absorption

and/or scattering. Consider a monochromatic

beam of radiation of intensity Ia (energy per

unit time per unit solid angle) as shown in figure 14.

The intensity will be attenuated as

la, =I0ae~ u »+y «> s = Ioa

e-f>»s (41)

where I0a) is the intensity at s=0, the apex of the

beam, ku is the monochromatic absorption coefficient for photons of energy frw, ya is the monochromatic scattering coefficient, and (8 u =Ka,+7u

is the monochromatic extinction coefficient.

+ dl<

-s = 0

Figure 14. Attenuation of radiation

due to absorption and scattering

{after Sparrow and Cess [84, p. 17]).

There is a fundamental difference between absorption and scattering of photons which should

be kept in mind. When a photon is absorbed it is out of the picture, so to speak, and we need no

longer be concerned about it. When a photon is

scattered, on the other hand, it is still active as an

energy carrier and hence we must consider where

it goes after being scattered. In isotropic scattering, which is often assumed (and probably rarely

justified), a photon has no memory of which way

it was going before being scattered and hence has

equal probability of traveling in any direction.

In small-angle scattering the photon is deflected

only slightly by interaction with a scattering

center so that this type of scattering has much

less effect on energy transfer.

From an extension of Kirchoff's law, it is

apparent that any medium which absorbs radiation must also emit radiation. For an isotropic

medium, the monochromatic emission per unit

time and volume is

4«ue

The total emission is

where

4/Cp6&a),

/» CO

. Jo

u

(42)

(43)

(44)

Total absorption and scattering coefficients

may be defined as

I k u \ IudQ,do)

Jo J 4jt K= ~ n IadQ.dw

JO J 4ir

(45)

and

f"T.f

Jo Jin

IadSldv

f f Undo

Jo Ji-r

(46)

74
where dti is the element of solid angle and the 4tt signifies that the integration is to be oyer all directions. However, since will depend on conditions throughout the medium and at the sur- rounding surfaces, a and y are not generally

properties of the medium and hence are of rather

limited usefulness.

We now have set forth the basic equations

which are necessary in order to calculate the heat

flux through a diathermanous material. Let us

consider one-dimensional heat flow in a slab of

thickness L. It is convenient to define a monochromatic optical thickness:

(47)

where 2 is a position coordinate measured from

the lower surface of the slab. Sparrow and Cess

[84, Ch. 7], who cite earlier work, show that the

intensity of radiation at any point in the slab can

be divided into two components as shown in

figure 15. The monochromatic radiation having a component in the positive 2-direction is designated

as Iu+

(tu,h) ; that which has a component in the

negative 2-direction, by Ia~(T 0,n), where m=cos 0

designates the direction in which the photons are traveling (0 is measured relative to the normal to

the slab). The component of radiation in the

positive direction is given by

It{r a, n)=H(0, n)e-T J»

+

where

- r^l^eM+^G^y^'^ (48)

=2

-X'

(49)

is the incident energy per unit area inside the

slab. Both e ba (t) and Ga (t) which appear in eq

(48) are unknown functions of position. The first

term on the right hand side of eq (48) is the

energy that enters at the 2=0 surface of the slab

and is attenuated in accordance with eq (41).

The first term in the integral in eq (48) represents

energy which is emitted by the medium in the

region below t„; the second term in the integral

represents energy scattered in the 0 direction in

the region below r u . An equation analogous to

eq (48) can be written for Iu

~, the component in

surface 2

SURFACE I

Figure 15. Coordinate system for one- dimensional radiative transfer (after

Sparrow and Cess [84, p. 196]).

the negative 2-direction. The projection of

I o=l\> +Jrl\>~ in the 2-direction gives the total

monochromatic radiation flux normal to the slab

at any position. Integration over all values of co gives the total radiation heat transfer, gr

. If both radiative heat transfer and conductive

heat transfer are present, conservation of energy

requires that, at steady state,

V.(\VT)=V-q T , (50)

where A does not include any radiative contribution to the thermal conductivity. Equation (50)

implies that there is an interaction between heat

transfer by conduction and that by radiation.

One cannot, in general, compute the heat flow

and temperature distribution due to conduction

and, separately, those due to radiation and then

add them. As we shall see, this can be done under

certain rather restrictive circumstances. In general,

the system of equations describing energy transfer

in a diathermanous medium consists of two

rather formidable simultaneous, nonlinear, integrodifferential equations that must be integrated over all photon frequencies. Solution of these

equations under various conditions is the subject

of a great deal of recent research [85-117]. In this

paper we shall only briefly discuss some limiting

cases. Consider first the optically thin limit Avhere

the optical thickness of the slab is small, i.e., t m (z=£X'0> for all co. In this case, the mean

free path of a photon is much larger than the

thickness of the slab so that absorption, scattering,

and volume emission within the slab are not important. We are only concerned with radiation

between the two surfaces. The monochromatic

radiation flux between the two surfaces, of

emittances e l01 and e2u , is [e^{Tx )—

e

ba (T2 )] . (l/ei J+(l/e2 J-l (51)

where Tx and T2 are the temperatures of the

surface. In general, eq (51) would have to be

integrated over all frequencies. For the special case in which the bounding surfaces of the slab are gray,

12 diffuse 13 emitters (and absorbers) of

energy, the total radiation flux is

where

1 E=

(52)

(53)

ex and e2 being the total hemispherical emittances

of the two surfaces. Let us further assume that

\T1 -T2 \«T=(Tl +T2 )/2; then

q T =4*miET3 {Tl -Ti ). (54)

> 2 That is, e, and t 2u are the same for all u.

> 3 That is, the intensity of the radiation leaving either surface is the same in

all angular directions.

75

\
In the optically thin limit it can be shown [84]

that the total heat flux is the sum of the radiative

flux and the conducted flux; thus for a medium of

constant thermal conductivity,

2=2c+2r=(X+W#f3Z) ^=^2

> (55)

where L is the total thickness of the slab. The

apparent total thermal conductivity is

Xapp =\+W#r3Z. (56)

Note that in the optically thin limit, the apparent,

or effective, radiative component of thermal conductivity is proportional to thickness and hence

is not an intrinsic property of the material.

Consider next the optically thick limit where the

optical thickness of the slab is large, i.e., r u (2=i)]>>l, for all co. In this case the mean

free path of a photon is much less than the thick- ness of the slab so that conditions at the surface

become unimportant and radiative heat transfer

is controlled by absorption, scattering, and volume

emission. In the optically thick limit, the monochromatic radiant energy content per unit volume

in a frequency interval dw is

so that the monochromatic contribution to the

volumetric heat capacity is

Ca —

4nde

c dT fCO, (57)

the velocity of a photon is v=c/n; the mean free

path of a photon is A.=l/pu . With these quantities

substituted into eq (5) or eq (7), the monochromatic contribution to the thermal conductivity is

4 debaj

Xra~dpa ~dTau'

The total radiative thermal conductivity is

4 deb 16riVT3

3PR dT 3/3,

(58)

(59)

where pR is the Rosseland mean extinction coefficient defined by

1= f°°JL

PB oj j8„

deb01 deb da- (60)

Thus under conditions where the optically thick

approximation holds, it is meaningful to define a radiative component of thermal conductivity.

In many materials, the extinction coefficient,

pa, varies strongly with co. The factor de ba/de b in

the integrand of eq (60) gives greater weight to

the values of p u in the frequency region where

there is the most energy. Thus even if p a were

independent of temperature for all co, the effective

extinction coefficient, pR, could be a very strong

function of temperature. For certain materials

\ r could, therefore, increase as Tn

, where n might

be as large as 10.

In figure 16 we show the qualitative behavior

of the apparent thermal conductivity of a diathermanous material as a function of thickness. The

increase in the apparent thermal conductivity

for small thicknesses will vary as EL, where E (see

eq (54)) is the effective emittance of the bounding

surfaces and L is thickness. As the thickness is increased, the increase in the apparent thermal

conductivity will asymptotically approach XT as defined in eq (59).

In summary, the radiative contribution to

thermal conductivity can be considered as a unique property of a material only when the

characteristic dimensions of the sample in question

are large compared to the average mean free path

for photons. 14 This is analogous to the situation

with the lattice component of thermal conductivity at low temperatures, where we noted that

it was necessary to speak of the thermal conductivity of a particular size of specimen in the region

where the lattice thermal conductivity is limited

by boundary scattering of phonons.

Values for the monochromatic extinction coefficient and index of refraction of a number of

materials are given in [118]. Most of these corres- pond to room temperature. Gardon [119] presented

values for the spectral absorption coefficient of

window glass in the solid and liquid state. Inspection of these values, which are illustrated in figure

17, reveals that below 600 °C the monochromatic

absorption coefficients are practically independent

THICKNESS

Figure 16. Apparent thermal conductivity of a diathermanous material as a function of thickness and the effective emittance of the bounding

surfaces.

» The average mean free path for photons is simply I/Pr, as defined in eq (60).

76
of temperature so that one could use room temperature values [118, 120-121] of monochromatic

absorption coefficients to estimate, using eqs (60)

and (61), the radiative component of thermal

conductivity for glasses. At temperatures sufficiently high that photons are excited in the wavelength region 15 shorter than about 2.8 /mi, the

weighting factor, deiw/deb, in eq (60) will cause the

Rosseland mean extinction coefficient to decrease

very rapidly with increasing temperatures and

thus cause a rapid increase in the radiative thermal

conductivity. There do not appear to be many

high temperature data on the extinction coefficients of ceramics. Folweiler and his group [122-

124] have carried out high temperature absorption

coefficient measurements on single crystal A12 03 , MgO, SrTi03 , and CaF2 and on fused Si02 glass.

Chapin and Howe [125] have made measurements

on radiant energy transport in BeO.

In general, the radiative thermal conductivity

will be reduced below the value in, say, a single

crystal by grain boundaries, pores, and any impurities which absorb or scatter photons. In other

sections of this paper we will see some experimental

results which illustrate certain aspects of radiant

energy transfer in ceramics.

Since there are no comprehensive review papers

(to my knowledge) which adequately cover the

subject of the radiative component of thermal conductivity, I have included a rather large number

of references [126-223] to individual papers and

books which have bearing on this subject. These

references, plus those previously cited [82-125],

while not exhaustive, cover a good portion of the

pertinent literature.

d. Gas Conduction

In porous ceramics, it is necessary to be con- cerned with the thermal conductivity of the gas

which fills the pores. For a classical gas, the heat

capacity per atom is/^/2, where/ is the effective

o

K

Q.

2 3 4 WAVELENGTH, pm

Figube 17. Monochromatic absorption coeffi- cients of window glass at different temperatures

(after Gordon [119]).

The original reference also shows data at 1000 and 1300 °C.

number of degrees of freedom (translational-h

vibrational +rotational) of the gas molecule.

For an ideal classical gas, then, the heat capacity

per unit volume at cons'tant volume is simply

C= 2T (61)

where p is the pressure of the gas. The mean free

path of a gas molecule for a not-too-dense gas is

Ps

(62)

where S is the effective cross sectional area of a gas molecule. This relation follows easily from

the reasonable assumption that the mean free

path is inversely proportional to the density of

the gas. The average translational velocity of

an ideal classical gas is

3RT W (63)

where m is the weight of the gas molecule, M is

the molecular weight of the gas, and i?=8.3143

JK_1 mol-1

is the gas constant.

Combining eqs (5, 61, 62, and 63), the thermal

conductivity of an ideal classical gas is given by

6SV

3RT

M

(64)

s Wavelengths measured in air, not in the glass.

The implications of this equation are as follows

1. Thermal conductivity is independent of

pressure.

2. Thermal conductivity increases as the square

root of absolute temperature.

3. Thermal conductivity is proportional to the

number of degrees of freedom of the gas

molecule.

4. Thermal conductivity decreases with increasing molecular weight. The scattering

cross section, S, in general increases with

molecular weight, so that thermal conductivity decreases more rapidly than M-1/2

. For gases with the same number of degrees

of freedom, thermal conductivity is very

roughly inversely proportional to molecular

weight.

Perhaps the most surprising thing about the

thermal conductivity of a gas is that it is independent of pressure. This happens because the

specific heat is proportional to pressure, the mean

free path is inversely proportional to pressure,

and the average velocity of a molecule is independent of pressure; thus the pressure dependences

cancel out. At low pressures, the mean free path

becomes large and may be limited by the size of

the pores in which it is contained, thus resulting

in a decrease in the effective thermal conductivity

of the gas. At high pressures, the mean free path

77
approaches the size of the molecules and the

thermal conductivity increases with pressure.

Real gases depart somewhat from the simplified

analysis given above. At low temperatures, particularly in the case of helium, quantum effects

enter in. At higher temperatures, additional

degrees of freedom may become excited. In general,

however, eq (64) provides a fair quantitative

description of the thermal conductivity of a gas.

For an extended treatment of all the factors which

affect the thermal conductivity of gases, the

interested reader may consult the treatise by

Hirschfelder, Curtis, and Bird [224].

e. Other Transport Mechanisms

There are some mechanisms, in addition to

those discussed above, by which energy may be

transported in a ceramic material. In general,

these are much less important than phonons,

electrons, photons, and gas molecules so we will

accordingly only mention them briefly.

In section 2.2b we briefly discussed the independent motion of electrons and holes in a semiconductor and the associated thermal conductivity

(see eq (29) and accompanying text). It is also

possible for an electron and a hole to form a bound state known as an exciton in which an

electron and a hole are bound together (in a manner analogous to the formation of a hydrogen

atom by a proton and an electron) and can travel

through a solid as a single entity. Since an exciton

is electrically neutral, there is no electrical current

associated with its motion.

There has been some speculation in the literature [225-228] regarding the existence of exciton

heat transfer in ceramics at high temperatures.

To date, however, there is insufficient evidence to

prove that this mode of heat conduction is of any

practical importance. Recent literature concerning

the theory of excitons includes [229-231]; these

references cite some of the earlier work.

Ziman [7, p. 319] points out that in some solids

the molecules have a significant rotational specific

heat which presumably can be transferred between neighboring molecules, thus contributing

to the thermal conductivity.

Ziman [7, p. 319] also mentions the possibility

of heat conduction by spin waves in which energy

is transferred by interaction of spin magnetic

moments.

We have not mentioned thermal conductivity

in superconductors. In brief, the thermal conductivity of a material will be lower in the superconducting state than in the normal state since

the superconducting electrons do not contribute

to heat transport. At sufficiently low temperatures, there is no electronic contribution to the

thermal conductivity of a superconducting material. Rosenberg [8] discusses the thermal conductivity of superconductors.

If there is mass transport of any kind in a

material, there will be an associated energy of

transport which can augment heat conduction.

The most obvious mechanism that we have not

discussed is gaseous convection. In a fibrous or granular material having a continuous gaseous

phase, there can be convection within the material

which under certain circumstances can transfer

very large quantities of heat. Even in a material

having closed pores so that there is no net transfer

of gas through the material, convection within

the pores can cause increased heat transfer across the pores. As a rule-of-thumb, convection within

closed pores is not important unless the pores

are larger than, say, 5 mm.

A more subtle type of mass transfer is the

diffusion of individual molecules or imperfections

under the influence of a potential field. For

example, in electromigration, molecules or impurities can migrate under the influence of an applied

electric field. In the Soret effect, molecules or impurities can migrate under the influence of the

temperature gradient. Since the energy transport

is proportional to the temperature gradient, the

Soret effect is a legitimate augmentation (or

depletion, depending on the sign of the heat of

transport) of the thermal conductivity. However,

since the Soret effect can result in a redistribution

of impurities, vacancies, etc., the thermal conductivity could change with time. Both electromigration and the Soret effect become significant

only at temperatures approaching the melting

point of the solid. I do not know of any cases where the effect of either mechanism on thermal

conductivity has been observed.

If the vapor pressure of a material or one of its

constituents becomes significant, a large energy

transfer can occur through porous materials due

to sublimation (or evaporation) and condensation.

Even in nonporous materials, the heat of vaporization could make it very difficult to make an

accurate thermal conductivity measurement.

3. Effects of Environment

Before discussing the effects of composition and

structure on the thermal conductivity of ceramics,

it is important to briefly discuss the influences of

environment.

3.1. Temperature

As we have seen in section 2.2, thermal conductivity is influenced by temperature for all of

the heat conduction mechanisms which were

discussed. In later sections, we shall see examples

of how the dependence of thermal conductivity

on composition or structure varies with temperature. In this section we present several figures

showing thermal conductivity versus temperature

for a number of ceramic materials and then show

a few examples to illustrate the relative importance

of different mechanisms of heat conduction at

different temperatures.

In figure 18 we compare on a log-log plot the

low temperature thermal conductivity of several

78
TO 20 50 1013 200 TEMPERATURE,

K

Figure 18. Thermal conductivity

of some non-metallic crystals compared with very pure copper

(after Berman [232]).

nonmetallic single crystals with the thermal

conductivity of a very pure copper sample. Heat

conduction in these dielectric crystals is entirely

due to phonons at low temperatures while the

thermal conductivity of copper is chiefly due to

electrons. At higher temperatures the thermal

conductivity of copper decreases slowly with

increasing temperatures, falling from 400 Wm_1K

_1

at room temperature to 340 Wm^E"1 at 1200 K.

Conversely the thermal conductivity of the dielectric materials continues to decrease fairly rapidly,

going roughly as T

_1

. In figure 19 we compare the high temperature

thermal conductivity of several dense ceramic

materials. In figure 20 the thermal conductivities

of a number of porous ceramics are shown as functions of mean temperature. We shall discuss

the effect of porosity in section 5.2.

>

\ 1 1

""Vs.Be0

1 1 1 1

100

SiC — Graphite \MgO^~--^\.

Spinel ^^S.

10 —Nmo^C^^.

Mullrte

1

Fused S0? -^^

1 1

Zr02

1 1 1 1 300 600 900 1200 1500 1800 2100 TEMPERATURE, K

Figure 19. High temperature thermal

conductivity of several dense ceramic

materials (after Kingery, Klein, and

McQuarrie [238])

.

400 800 1200

MEAN TEMPERATURE, °C

Figure 20. Thermal conductivity of several refractory brick (after Ruh and McDowell

[234]).

The thermal conductivity of titanium carbide

is shown in figure 21. The electronic component

of thermal conductivity, computed from eq (22)

using L=L0 for the Lorenz function, is seen to be

about one-half of the total thermal conductivity,

the remainder being due to lattice conduction

(phonons). This behavior can be contrasted with

that of oxide ceramics which generally have a

negligible electronic contribution to the thermal

conductivity. This separation of thermal conduction into an electronic and a lattice component

should not be taken too seriously since L may

not be equal to L0 for titanium carbide.

In figure 22, the thermal resistivity data of

Taylor [236] for beryllium oxide are shown as a

function of temperature. The samples were hotpressed to 99 percent of theoretical density. From

1000 to 1850 K the thermal resistivity increases

T

Ql 1 1

'

1 0 500 1000 1500 2000 TEMPERATURE, °C

Figure 21. High temperature thei-mal conductivity of titanium carbide.

The upper curve represents the total measured

thermal conductivity, X = X«+Xi. The lower curve

represents the electronic contribution, \«, computed

using the theoretical Sommerfeld value for the Lorenz function (after Williams [235]).

79
1x1 X

o I 1 I I I

i I i 800 1200 1600 2000 2400

TEMPERATURE, K

Figure 22. High temperature thermal resistivity of two samples of high density

beryllium oxide (after Taylor [236]).

linearly with temperature, as predicted by eq

(20). At higher temperatures, the thermal resistivity departs from this linear relation, passes

through a maximum near 2100 K and then decreases with increasing temperature. The departure of the high temperature data from the

extrapolation of the line through the data below

1850 K increases as approximately T

17

. There

are several possible reasons for this departure.

At these very high temperatures, the mean free

path of a phonon is becoming very short—of

the order of interatomic distances. Thus the

mean free path of a phonon may become essentially constant at sufficiently high temperatures.

Since specific heat and phonon velocity are also

essentially constant at very high temperatures,

this would result in a temperature-independent

phonon contribution to the thermal conductivity. Such behavior of the mean free path would

explain the departure from a linear relation

but would not explain the decrease in thermal

resistivity (increase in thermal conductivity) at

the highest temperatures. To explain this it is necessary to invoke an additional mechanism of

heat conduction. The most plausible candidate is radiative heat transfer, or photon conduction.

Equation (59) predicts a T3 temperature dependence for the total radiative thermal conductivity

if the index of refraction and the extinction

coefficient are independent of temperature. Chapin

and Howe [125] found that beryllia became more

opaque to thermal radiation at very high temperatures; if this is so the radiative component of

thermal conductivity would increase less rapidly

than T3

. However, the combination of a constant

mean free path and a radiative contribution might

still lead to the behavior observed in figure 22.

There is also the possibility of some more exotic

mechanism such as exciton conduction. At such

high temperatures, experimental errors could be

quite significant and might account for part of

the turn-down in figure 22.

In the case of glasses, the importance of radiative heat transfer is quite clear. Referring back to

figure 19, the thermal conductivity of fused silica

is seen to increase slowly with temperature near room temperature and then to increase much

more rapidly at higher temperatures. Since fused

silica is an amorphous material, the phonon mean

free path above room temperature is essentially

constant at a value approximating the size of a

silicon dioxide tetrahedron [237]. The increase

with temperature in the thermal conductivity of

fused silica near room temperature is due to the

increase in the heat capacity. The spectral absorption coefficient for fused silica is similar to that for

window glass as shown in figure 17. Fused silica

is essentially opaque to wavelengths (measured in

air) longer than 2.8 jum. At temperatures below,

say, 400 K, there is negligible thermal radiation

(cf. figure 13) at wavelengths below 2.8 so that

the radiation contribution to thermal conductivity

is completely negligible. Above approximately

600 K, there are sufficient short-wavelength

photons to transfer significant amounts of energy

and the radiative component of thermal conductivity rises very rapidly (perhaps as Ti to T6

) with increasing temperature.

Figure 23 shows some values for the apparent

radiative contribution to thermal conductivity in a sample of fused silica and in single crystals of

three oxide ceramics. These samples were probably

not large enough, in comparison to the photon

mean free paths, for these measured values to be

true properties of the materials.

The temperature dependence of the thermal

conductivity of a number of oxide ceramics can be

observed from the recommended values given by

2

1 1 1 /n 1

Si02 jj

n /

Ti °z ^

J

MgO

1

1

I 1 250 500 1000 2000

TEMPERATURE, K

Figure 23. Effective radiative component of thermal conductivity in 2.5 cm samples of silica glass and three single crystal oxide ceramics (after

Kingery, Klein, and McQuarrie

[233]: the original data are due to Charvat and Kingery [439] and

Lee and Kingery [189]).

80
Powell, Ho, and Liley [238]. Recommended values

for the thermal conductivity of several types of

graphite are given by Ho, Powell, and Liley [239].

If one wishes to study the temperature dependence

of the thermal conductivity of other ceramic materials, the best place to find the appropriate ref- erences is from one of the various data compilations issued by the Thermophysical Properties

Research Center [240]. A good review paper,

which includes discussion and examples of the

temperature dependence of thermal conductivity,

has been written by Kingery [241].

3.2. Atmosphere

The thermal conductivity of a porous material

is dependent on the type of atmosphere pervading

the material. In the case of materials having no

continuous solid phase, the difference between

the thermal conductivity in vacuo and in the

presence of, say, atmospheric pressure gas may be

several orders of magnitude. Many of the refer- ences cited in Section 5.2b are concerned with

multiphase materials having no continuous solid

phase.

In the case of porous ceramic materials, the

thermal conductivity will vary in different atmospheres provided the atmosphere can pervade the

pores. In figure 24 we show the thermal conductivity of a commercial insulating fire brick in

helium, air, and argon. As is apparent from eq

(64) , the low-molecular-weight gas has the highest

thermal conductivity. Reducing the gas pressure

would not significantly affect the thermal conductivity values shown in figure 24 until the mean

free path of the gas molecules approached the

pore size. A reasonably good vacuum would result

in thermal conductivity values below those shown

for argon. We will return to porous materials in

Section 5.2b.

G.2-

z>Q

X

H ol I I I 300 500 700 900

MEAN TEMPERATURE, K

Figure 24. Thermal conductivity of a commercial insulating firebrick in

various gases (after Young, Hartwig,

and Norton [242])

.

3.3. Electromagnetic Fields

The effect of an electromagnetic field on thermal

conductivity is much too complex a subject to

discuss in any detail here. Consequently, I shall

only make a few brief remarks.

In general, thermal conductivity will not be

significantly affected by an electric field of the

magnitude normally encountered in engineering

applications. In the case of a metal, platinum,

Flynn and O'Hagan [243] found no measurable

effects on thermal conductivity due to electric

current densities of 108 Am-2

; it should be noted

that this corresponded to electric fields of less than

400 Vm_1

. In semiconductors, or insulators, larger

electric fields can be sustained and effects on

thermal conductivity might conceivably become

significant.

A magnetic field of rather small magnitude

exerts a much greater force on an electron than

can be exerted by any attainable electric field. In

brief, a magnetic field causes an electron (or hole)

to follow a curved path so that the effective mean

free path in the direction of the temperature

gradient is reduced. This can result in a very

significant reduction in the electronic component

of thermal conductivity. The application of a magnetic field can cause the thermal conductivity

to be anisotropic. Ziman [7, pp. 483-523] may be

consulted for details. In a dielectric solid, where

there is no significant electronic contribution to

the thermal conductivity, the application of a magnetic field would normally not affect thermal

conductivity.

We have previously discussed the transfer of

heat by photons through partially transparent

materials. In that discussion, it was tacitly as- sumed that the spectral distribution of radiation

impinging on a volume element of the material

was, at least roughly, the same as the spectral

distribution of energy emitted by the volume

element. If a material is subjected to radiation

with an electromagnetic field (photons) of quite

different spectral characteristics from those of

the field which would be extant in an infinitely

large, approximately isothermal, body of the same

material (at the same temperature), the radiative

heat transfer through the material may be very

dependent on theinttnsity and spectral distribution

of this applied electromagnetic field. For example,

at room temperature there is a negligible radiative

contribution to the thermal conductivity of

window glass. However, high frequency photons

due to solar radiation will pass readily through

window glass with a large attendant transfer

of heat. Under such conditions, it is meaningless

to attempt to associate a thermal conductivity

with this heat transfer.

3.4. Pressure and Stress

In sections 3. Id and 3.2 we mentioned briefly

the effect of pressure on the thermal conductivity

of gases. Near atmospheric pressure, the thermal

81
conductivity of a gas is almost independent of

pressure.

There has been very little experimental work

on the effect of hydrostatic pressure on the

thermal conductivity of solids. Bridgman [244,

245] measured the thermal conductivity of a number of rocks and minerals as a function of

pressure. Hughes and Sawin [246] have recently

measured the pressure dependence of thermal

conductivity of several dielectric solids at pres- sures up to about 1.9X109 Nm"2 (~1.9X103 atm)

and temperatures to 450 °C. They found the

(lattice thermal conductivity to increase by 50

to 100 percent over the range 0 to 2X108 Nm-2

(~2X103 atm) and then to increase less rapidly

at higher pressures. This corresponds to less than

0.05 percent for a pressure change of 1 atm.

In a dielectric solid, the increase in thermal

conductivity with pressure can be associated

with the increase in the Debye temperature

(see eqs (16) and (17)). Drickamer et al. [247]

review the effect of pressure on the lattice parameters of solids and cite a number of pertinent

references. The effect of pressure on the thermal

conductivity of a dielectric solid is much greater

in the case of solid helium [248-250] where the

Debye temperature can be changed very drastically

by applied pressure.

At high pressures, of the order of 1010 Nm-2

, many insulators and semiconductors transform

to metallic behavior so that the electrical conductivity, and presumably the electronic component

of thermal conductivity, increase by several

orders of magnitude. Drickamer [251] has surveyed

the literature on this increase in the electrical

conductivity; I do not know of any experimental

thermal conductivity work in this pressure region.

Bridgman [252] found that the Lorenz function,

X e/a-T, of metals could either increase or decrease

with increasing pressure. Starr [253], however,

found the ratio of thermal to electrical conductivity at constant temperature to remain essentially

constant at pressures up to about 109 Nm-2

. If

this is so, one can predict the change in electronic

thermal conductivity from the change in electrical

conductivity. Review papers on the effect of

pressure on the electrical properties of metals or semiconductors include [245, 251, 254-258]. In

brief, the electrical conductivity of most metals

increases by an amount less than 0.01 percent for

each atmosphere of pressure increase. Most semiconductors, and some metals, exhibit a decrease

in electrical conductivity with increasing pressure.

In the vast majority of engineering applications,

the pressure dependence of thermal conductivity

for solid materials can be neglected.

There is also very little information on the effect

of elastic strain, due to nonhydrostatic compression,

to tension, or to torsion, on thermal conductivity.

Any strain, other than hydrostatic, will distort

the lattice and thus make it more difficult for

energy carriers (e.g., electrons, phonons) to travel

through the lattice. This will decrease the thermal

conductivity. Bridgman [259] has measured

the effect of tensile stress on the thermal conductivity of a number of metals near room temperature and found the decrease in thermal conductivity to be proportional to the stress. As an

example, he found that a tensile stress of about

108 Nm-2 (~1.5X104 lb kr2

) decreased the

thermal conductivity of copper by about 0.2

percent. The apparent thermal conductivity,

however, computed using the unstressed dimensions of the specimen rather than the stressed

dimensions, would have increased by about 0.8

percent under this stress. For metals in torsion,

the thermal conductivity decrease is approximately proportional to the square of the twist

[260-261]. In the case of copper, a twist of 1 rad m_1 of length reduced the thermal conductivity by about 0.2 percent.

At low temperatures (say, <100 K), the thermal

conductivity of relatively pure metals and dielectric crystalline solids might be significantly affected by elastic deformation. Above room temperature, any such effects are probably of no

practical importance in most engineering applications. An exception to this would occur if stresses

caused microcracks to appear in the material.

If a solid undergoes plastic deformation, its

thermal conductivity will be significantly decreased due to the large number of dislocations

formed in the lattice. Cold-working of a metal

could reduce its thermal conductivity by an order

of magnitude. Such a large effect would be much

less likely in most ceramic materials due to their

brittle nature. We will return to the subject of

dislocations and other lattice imperfections in

section 5.2.

3.5. Heat Flow

Fourier's law, eq (1), defines thermal conductivity as the negative ratio of the heat flux to th e temperature gradient. In practically all applications of thermal conductivity, it is tacitly assum ed

that the thermal conductivity of a material does

not depend on the heat flux (or the temperature

gradient) within the material. In principle, this

assumption need not be valid. In practice, no one

has yet conclusively proven it is not.

Austin [262] reviewed some of the early attempts

to find a variation of thermal conductivity with

heat flow and concluded that the existence of such

an effect had not been established. Recently,

Patrassi [263] reported a very large dependence of

thermal conductivity on temperature gradient (or

heat flow) for uranium dioxide. His experimental

procedure and conclusions have been severely

criticized by Fulkerson, McElroy, and Moore

[264], who made independent measurements of the

thermal conductivity of uranium dioxide and found

no measureable effect due to temperature gradient.

Other laboratories, as a result of Patrassi's article,

have attempted to find an effect such as he describes. None of this other work has yet been

82
published but as far as I know, there has been

no confirmation of Patrassi's results.

There are several possible effects, other than

experimental error, which could result in measured thermal conductivity values being dependent

on heat flow:

1. Further extension of the formal theory of

heat conduction might reveal the necessity

of including some higher order terms which

would indicate that heat flux is not strictly

proportional to temperature gradient.

2. Steep temperature gradients may induce sufficient strain in the solid to scatter energy

carriers and lower the thermal conductivity

(see section 3.4). Alternatively, the induced

strain may open microcracks in the sample.

3. The differential equation for heat flow in a

solid may not be valid in the presence of

very steep temperature gradients. Examination of the derivation of the diffusion equation reveals that an implicit assumption is made which may not be valid at very high

temperature gradients. Without going into

detail, it is necessary that

!$X§XA«1.

Essentially, we are saying that if the diffusion equation is to be valid, it is necessary

that there be no significant change in the

thermal conductivity over a distance of the

order of the mean free path of the energy

carriers.

It appears that some further theoretical and

experimental work will be required before the issue

of the possible dependence of thermal conductivity

on heat flux is resolved.

4. Directional Effects

Many materials are anisotropic as regards

thermal conductivity. All single crystals of other

than cubic symmetry may be expected to possess

thermal conductivities which are directionaldependent. For single-crystal quartz, the ratio

of the thermal conductivity parallel to the c-axis

to that perpendicular to the c-axis drops from 2.0

at 40 K to 1.4 at 700 K. The thermal conductivity

of pyrolytic graphite is extremely anisotropic,

typically by a factor of as much as 100 to 200,

depending on temperature and details of

fabrication.

Polycrystalline materials will generally be more

isotropic than single crystals of the some material.

Hydrostatically pressed and sintered materials

are usually reasonably isotropic. Extruded materials normally retain a significant portion of the

anisotropicity characteristic of the single-crystal

material.

Carslaw and Jaeger [1] discuss the mathematical theory of heat flow in anisotropic materials. The thermal conductivity of an anisotropic

material is best expressed as a tensor rather than

a scalar. Drabble and Goldsmid [14, p. 67] discuss

the phenomenological theory of heat conduction

in anisotropic solids.

In highly-anisotropic materials, it is questionable whether or not the intrinsic thermal conductivity in the lowest-conductivity direction has

ever been measured. Kaspar and Zehms [265],

who cite a few earlier references, point out that

the observed thermal conductivity value for

pyrolytic graphite in the c-direction is often simply

a component of the conductivity perpendicular

to the c-axis which arises due to departures from

ideal crystalline orientation.

In an application where anisotropicity of thermal

conductivity may be important, experimental

checks are in order unless previous measurements

have been made on the same material produced

by the same process.

5. Effects of Composition and

Structure

It would be possible to write a fairly large book

solely concerned with the effects of composition

and structure on the thermal conductivity of

solids. In this paper I can only attempt to touch

on most of the factors of importance and give

examples of a number of them.

The influence of compositional and structural

variations at low temperatures is covered in some

depth, either directly or by reference, in the

reviews by Ziman [7], Rosenberg [8], and Klemens

[9-11]. At high temperatures, the somewhat more

pragmatical reviews by Austin [78, 262] and

Kingery [233, 241] are very helpful.

5.1. Composition

In this section we shall be concerned with the

effect of the chemical composition of a solid

material on thermal conductivity.

a. Molecular Structure

The electronic thermal conductivity of a material is related to the number of "free" con- duction electrons in the material. As discussed

in 2.2b, the electronic component of thermal

conductivity is approximately proportional to the

electrical conductivity. Thus copper and silver

have very high electronic thermal conductivities

as a result of their high electrical conductivities.

Many metallic carbides, borides, silicides, nitrides,

and sulfides have sufficiently high electrical conductivities to possess significant electronic components of thermal conductivity. The magnitude

of the electronic component of thermal conductivity of electrically conductive ceramics can be

estimated using the Lorenz relation (eq (22)) with

the theoretical Lorenz number, L0 =2.443 X

10- 8V2K-2

. The lattice component of thermal conductivity

in a pure, dense, crystalline material is strongly

83
correlated with the complexity of the molecular

structure. For example, referring back to figure

19, the thermal conductivity of alumina is greater

than that of spinel which in turn is greater than

that of mullite. A rule of thumb is that the simpler

structure will have the higher lattice thermal

conductivity. For similar molecular structures,

the material with the lowest molecular weight

will generally have the highest intrinsic thermal

conductivity. This trend can be observed for the

nonmetallic elements by looking at the data

compilation due to Ho and Powell [266]. In the

case of metallic oxides, this trend is seen in figure

25, where thermal conductivity is plotted versus

the atomic weight of the cation.

b. Impurities

Introduction of impurities into a pure crystalline

material normally reduces the thermal conductivity. This is true for metals where the foreign

atoms scatter electrons, thus reducing the electronic thermal conductivity. It is true for dielectric materials where the foreign atoms scatter

phonons and reduce the lattice thermal conductivity. The thermal resistivities due to impurities

are additive for each conduction mechanism (see

eqs (19) and (23) for phonons and electrons,

respectively)

.

The additional electronic thermal resistance

due to impurities is related to the corresponding

increase in electrical resistivity by the Lorenz

relation (see eq (25)). Thus we can confine our

attention to the effect of impurities on electrical

resistivity. The increase in electrical resistivity,

due to x atomic percent of a given impurity, is

essentially independent of temperature and is_

given remarkably well by Nordheim's rule [267]

general, the constant A is eq (65) will be smallest

if the impurity atom has the same valency as the

parent material. In the case of metals, Linde's

rule [268], although not very accurately obeyed,

serves as a rule of thumb for the effect of valency

Ap=a-\-bZ2

, (66)

where Z is the valency difference and a and b are constants for a given solvent for each row in the

periodic table from which the solutes are taken.

There is no simple relation, analogous to the

Lorenz relation, which permits one to use a more

easily measured property to compute the change

in the lattice thermal conductivity due to introduction of impurities. In general, the effect of a given atomic percent of impurity on the lattice

thermal conductivity will be greatest when the

atomic mass difference between solvent and solute

is greatest. Except at very low temperatures^ the

additional thermal resistivity due to impurities

is essentially independent of temperature. Since

the thermal resistivity due to phonon-phonon

interactions (Umklapp-processes) increases approximately linearly with temperature, the relative

effect of impurities decreases approximately linearly with temperature in a dielectric crystal. In

figure 26 we see the thermal conductivity of a number of specimens of potassium chloride doped

with different amounts of potassium nitrite.

Increasing the concentration of impurities lowers

the peak in the thermal conductivity curve and

shifts it to a higher temperature. Different types

of impurities will affect the thermal conductivity

by different amounts. The curves shown in figure

26 are typical, however, in that the effect is very

Ap(x) =Ax(\ — x) (65) iowhere A is a constant which depends upon the

parent material (solvent) and the impurity

(solute). For small concentrations, the increase

in electrical resistivity is proportional to the

atomic concentration of impurity atoms. In

5 10 30 100 300

ATOMIC WEIGHT OF CATION

Figure 25. Thermal conductivity of

several dense ceramic oxides (after

Kingery, Klein and McQuarrie

[233]).

E

/d y /

Y

/ / /

KCI + KN02 A - "PURE"

B - 9xl016 cm" C- 4xl017 D- 5xl017 E- 1.6 xlO,B F— 4xl018 12 5 10 20 50 100 TEMPERATURE, K

Figure 26. Thermal conductivity

of -potassium chloride crystals

doped with N02 ~ ions (after Pohl

[286]).

The curves are identified according to the density of NO2-

ions in each sample.

84
t 1 r

TEMPERATURE , K

Figure 27. Thermal conductivity of

single crystals of LiF with various 6 Li

contents (after Berman and Brock

[269]):

Symbol Percent «LiF Symbol Percent «LiF A 0.02 D 90.4 B 0.01 E 25.0 C 4.6 F 50.1

pronounced at low temperatures and then decreases rapidly with increasing temperature.

At low temperatures the presence of different

isotopes of the same element can cause large

differences in thermal conductivity. An example

of this is shown in figure 27.

The different survey papers cited [7-15] may

be consulted for the many references to the effect

of impurities on thermal conductivity. Some refer- ences which are too recent to be covered in existing

reviews include [39, 270-280].

c. Stoichiometry

Departures from stoichiometry in dielectric

compounds will decrease the lattice thermal conductivity since the interstitials or vacancies

present act as impurities. In some cases, departures

from stoichiometry will increase the electrical conductivity by a sufficient amount to significantly

contribute to the electronic thermal conductivity.

The resultant thermal conductivity, arising from

the sum of the lattice and electronic components,

could, in principle, be either increased or decreased

by departures from stoichiometry. Pertinent refer- ences to the effect of nonstoichiometry on thermal

conductivity include [281-285].

d. Alloying

Alloying is simply the addition of controlled

amounts of "impurities", perhaps to the extent

that the solvent and solute lose their identities.

In the case of two metals which form a con- tinuous series of solid-solution disordered alloys,

the electronic component of thermal conductivity

is related, by the Lorenz function, to the electrical

conductivity. In the absence of ordering (e.g.,

intermetallic "compounds"), Nordheim's rule, eq

(65), holds reasonably well over the entire composition range. Austin [78] shows several examples

of the thermal conductivity of binary solid solutions of two metals.

In the case of solid solutions of dielectric materials, the thermal conductivity usually varies

with composition in a manner similar to that

shown in figure 28 for the KCl-KBr system and

in figure 29 for the MgO-NiO system. The chief

characteristic feature of these plots in that the

thermal conductivity values always fall below the

straight line connecting the end points, the departures from this line being roughly parabolic in

shape. As shown in figure 29, the relative effect of

alloying decreases with increasing temperature.

Abeles [289] has presented a phenomenological

theory describing the variation of lattice thermal

conductivity with alloy composition and temperature. Other pertinent references include [290-

303].

5.2. Structure

a. Crystallinity

In a highly crystalline material, the mean free

path of an energy carrier is much longer, and thus

the thermal conductivity much higher, than in a

disordered or amorphous solid. This effect is most

pronounced at lower temperatures. In figure 30,

the thermal conductivity of fused quartz is contrasted with that of crystalline quartz. The curves shown differ by almost three orders of magnitude

near 40 K and then approach a common value at

higher temperatures. As mentioned in section 3.1,

the phonon mean free path in fused quartz above

room temperature is approximately constant at a value approximating the size of the silica tetrahedron. At some sufficiently high temperature,

the increased number of Umklapp processes will

u

20 40 60 80 IOO KCI MOLE PERCENT KBr

Figure 28. Thermal conductivity of

solid solutions of potassium chloride

and potassium bromide (after Eucken

and Kuhn [287]).

85
3U

25 -

O 20

15

/

0 20 40 60 80 100 VOLUME % NiO

Figure 29. Thermal conductivity of solid solutions of

magnesium oxide and nickel

oxide (after Kingery [£88])

.

(The left-hand axis represents pure MgO).

in much the same manner as we saw for impurities.

Table 1, in section 2.2a, lists the temperature

dependence of the thermal resistivity due to

different types of imperfections. The various

review papers cited [7-15] discuss the effects of

imperfections on the thermal conductivity of

solids and cite numerous pertinent references.

Recent additional references include [304-315].

One class of imperfections that is of particular

importance, because of the extensive use of ceramic

materials in nuclear applications, is that of the

imperfections caused by irradiation. In figure 31,

the effect of fast neutron irradiation on the lowtemperature thermal conductivity of single-crystal

sapphire is shown. The radiation damage lowers

the thermal conductivity, especially near the peak.

Part of the damage due to irradiation can be

annealed out at high temperatures. Figure 32

shows the thermal conductivity of beryllium oxide

at 50 °C as a function of irradiation dose and

irradiation temperature. The effect of a given dose

100

10

E

01

I -

FUSED QUARTZ

200 400

TEMPERATURE, K

600

6.0x10 Sapphire

Neutron Irradiation

20 40 60

TEMPERATURE, K

80 100

Figure 31. Thermal conductivity of synthetic sapphire,

A12 03 , before and after the indicated reactor irradiations.

The doses were 1.5, 8.9, 20.2, and 50.2X1021 fast neutrons m"2

, respectively

(after Berman, Foster, and Rosenberg [316]).

Figure 30. Thermal conductivity of crystalline and fused quartz.

The uppermost curve represents the thermal conductivity parallel to the c-axis while the curve just below it represents the conductivity perpendicular

to the c-axis. These are the recommended values from Powell, Ho, and Liley [238].

decrease the mean free path in crystalline quartz

until it reaches the size of the silica tetrahedron.

Above this temperature the phonon mean free

path will be essentially the same in crystalline

and amorphous quartz.

We will see the effect of less drastic departures

from crystallinity in the following section.

b. Imperfections

Dislocations and other imperfections perturb

the perfect periodic structure of a crystalline

lattice and hence lower the thermal conductivity

2.4x10'

E

>

t-o

Z>Q

-z. o

o

_J<

<r

UJ 0 I ^ I 1—^ 1 0 5 10 15 20 25x10'"* IRRADIATION DOSE, neutrons m"2

Figure 32. The effect of neutron irradiation

dose and irradiation temperature on the thermal

conductivity of BeO at 50 °C (after Moore et al.

[317]).

86
is less if the irradiation occurs at an elevated

temperature.

It is very interesting to compare the effects of

neutron irradiation on fused and crystalline quartz.

Berman and his co-workers [320-321] found that

neutron irradiation lowered the thermal conductivity of crystalline quartz. Crawford and Cohen

[322-323] found that irradiation significantly increased the low-temperature thermal conductivity

of fused quartz.

References on the effect of neutron and other

types of irradiation on thermal conductivity include [324-341]. A few general references on the

effects of radiation damage in solids are [342-345].

c. Additional Phases

All of our previous discussion has been concerned

with homogeneous materials. Except for single

crystals, most ceramic materials are heterogeneous

at least to the extent of having grain boundaries

or small porosities. Many ceramic materials have

very significant amounts of a second phase present.

In this section we will discuss the effects of additional phases on thermal conductivity.

In section 2.1, thermal conductivity was defined

as X=- VT

For a heterogeneous material, we extend this definition to

q . ^effective —

(VT) (67)

where (VT) is the average value of the temperature gradient over a region large in comparison

with the size of the inhomogeneities. Unless the

sample is large in comparison with the inhomogeneities, it is scarcely meaningful to attempt to

define an effective thermal conductivity.

In a multiphase material the effective thermal

conductivity will depend upon:

1. The thermal conductivity of each phase.

2. The proportions of each phase.

3. The manner in which the phases are distributed; in particular

a. whether or not the phase is continuous in

the direction of heat flow,

b. whether the phase distribution is ordered

or random,

c. the size, shape, and orientation of each

segment of each phase.

4. The nature of the contacts between the

different phases.

5. The emissive and absorptive properties of the

phases if there is significant radiative heat

transfer through one or more of the phases.

The problem of computing the effective thermal

conductivity of a mixture from the thermal conductivities of the components is mathematically

the same as the problem of computing the electrical conductivity, dielectric constant, or magnetic permeability of a heterogeneous mixture.

There exists a large body of pertinent literature,

much of which is covered in the review articles by

Powers [346] and by Meredith and Tobias [347].

Additional references, not given by either [346] or [347], include [78, 140, 149, 162-164, 175, 178,

188, 207, 213, 219, 221, 223, 233-234, 241, 242,

262, 281, 286, 288, 348-434]. These references

include most of the mathematical analyses of

multiphase media and also contain a more limited

amount of experimental data on multiphase

media. The papers by Everest, et al. [149],

Gorring and Churchill [376], Nahas, et al. [400],

and Krupiczka [391] contain other reviews of

various analytical expressions for predicting conductivity of multiphase media.

We shall now discuss a few of the available

mathematical relations for correlating the effective

thermal conductivity of a mixture with the

thermal conductivities of the individual components. We shall limit our discussion to two-phase

systems.

The simplest model for purposes of analysis is that in which the two phases are arrayed in

alternative parallel layers as shown in figure 33.

If the heat flow is parallel to the layers, the

effective thermal conductivity is given by

X—/1X1+/2X2, (68)

where fi and f2 are the volume fractions of the

phases having thermal conductivities X, and X2 respectively. If the heat flow is perpendicular to

the phase layers

/2X1 +/1X2

Equations (68) and (69) represent the extreme

limits of the thermal conductivity of a two-phase

mixture. These limits are shown in figure 34 for

the case X1 = 10X2 . Although both eq (68) and (69)

predict thermal conductivity values intermediate

between the conductivities of the individual components we see that the conductivity obtained

is very different for the two cases. Thus these

limits are of relatively little use except for laminated materials.

HEAT FLOW

PARALLEL

TO LAYERS

HEAT FLOW

PERPENDICULAR

TO LAYERS

Figure 33. Two-phase material with

phases distributed as parallel slabs.

87
1.0

.8 X

_

6

.4

.2

0

s

1 1 1 1 1

\ PARALLEL

\

SERIES

_i i i i_

.2 .4 .6

f,

.8 1.0

Figure 34. Effective thermal

conductivity of a laminated

material with heat flow

parallel or perpendicular to laminations.

It is possible to obtain tighter limits for the

thermal conductivity of a two-phase mixture

by calculating the apparent effective conductivity

by each of two simple methods:

Series Slabs

The material is divided into thin slabs perpendicular to the direction of principal heat flow.

The effective conductivity of each slab is computed by assuming that the two phases act as conductors in parallel. The effective conductivity of the mixture is then computed by assuming that the slabs act as conductors in series.

Parallel Tubes

The material is divided into thin tubes or rods

parallel to the direction of principal heat flow.

The effective conductivity of each tube is computed by assuming that the two phases act

as conductors in series. The effective conductivity of the mixture is then computed by assuming that the tubes act as conductors in parallel.

Jackson and Coriell [435] have very recently

shown that these two methods provide upper and

lower bounds for the true effective conductivity

of a mixture. In order to calculate these bounds it

is necessary to assume some sort of model representing the manner in which the phases are deployed. Several investigators have represented a

disperse second phase by a cubic array of cubes

as shown in figure 35. The two methods described

above can then be used to calculate limits for the

effective conductivity of the mixture. Since there is some confusion in the literature on these calculations, it is worthwhile to spell out the steps and

assumptions involved.

Series Slabs

1. As shown in figure 35, the mixture is divided

into slabs (A) containing no disperse second

phase and into slabs (B) containing both con- tinuous and disperse phases.

2. The effective conductivity of the B-slabs is computed, using eq (68), by assuming the

PARALLEL TUBES

SERIES SLABS

p.x.

(l-Py) rA/WWVS

Pi

(i-P„> K

Figure 35. Cross-section of the model in which a disperse second phase is considered to be a cubic array of cubes.

88
disperse and continuous phases act as conductors in parallel

XB =PaXd +(l-Pa )\c ,

where Pa is the fraction of the total area which

contains the disperse phase of conductivity

Xd and (1—

P

a) is the fraction of the area which

contains the continuous phase of conductivity

Xe

. 3. The effective conductivity of the mixture is computed, using eq (69), by taking the A-slabs

and the B-slabs in series:

X= Xaab AX

P,Xa+(1-Pi)Xb

where XA =XC , XB was given above, and Pz is the fraction of the total length containing the

disperse phase. Evaluation of this equation

yields

(l-Pa)+PaX,/Xc Xc {l-Pa +PaPl)+Pa{l-Pl)\al\ (70)

Parallel Tubes

1. In this approximation, the mixture is divided

into parallel "tubes" (A) containing no disperse

second phase and "tubes" (B) containing both

phases.

2. The effective conductivity of the B-tubes is computed, using eq (69) , by assuming the disperse and continuous phases to be in series

XR = X„X

pl xc +{l-pl )\d

3. The effective conductivity of the mixture is

computed, using eq (68), by assuming the

A-tubes and B-tubes to be in parallel:

X=PaXB +(l-Pa )XA.

Evaluation of this yields

\_Pl (l-Pa) +(l-Pl +PaPl)Xd/\c Xc Pl +(l-Pl)K*c (71)

In the above derivations, we stated that our model

was a cubic array of cubes. In fact, it is not neces- sary to be so restrictive. For eq (70) to be valid,

it is only necessary that the model can be divided

into two types of slabs (perpendicular to the flow

of heat) —one containing no disperse phase and

one having a fraction, Pa, of disperse phase which

can be distributed in any manner. For eq (71)

to be valid, it is only necessary that the model

can be divided into the two types of "tubes"

(parallel to the flow of heat). Thus eqs (70) and

(71) are, in principle, also applicable to dispersions

of, for example, fibers or platelets oriented parallel or perpendicular to the flow of heat. However,

these equations will not necessarily bound the true

effective conductivity unless both are based on

the same phase deployment.

For a disperse phase in the form of cubes, or in

which cubes may be used to approximate an

isometric disperse phase, equations (70) and (71)

may be recast in terms of the volumetric fraction

of disperse phase, which we shall designate as /.

For the model used, it is easily seen that Pi=f1/3 and Pi=J2/3

; with these substitutions, we obtain

the forms usually seen

16

Series Slabs

(i-/»/3)+/2/yxe xc (i-/2/3 +/)+(/2/3 -jr )x,Ac

Parallel Tubes

X l/

1/3 -/)+(i-/1/3 +/)xg/xc k /

i/3 +(i-/i/3 )x,/xc

(72)

(73)

Although one would appear to be, on the face of

things, considering a fairly complicated model in

deriving eqs (70) to (73), a little thought reveals

that the model reduces to the two simple electrical

networks shown in figure 35. In the series-slabs

model, one effectively assumes that the continuous

phase has an infinite thermal conductivity normal

to the principal flow of heat; thus this approach,

resulting in eqs (70) and (72), always overestimates the effective thermal conductivity. In the

parallel-tubes model, one effectively assumes that

the continuous phase has zero thermal conductivity normal to the principal flow of heat; thus

this approach, resulting in eqs (71) and (73),

always underestimates the effective thermal conductivity.

Maxwell [346, 347] derived an expression for the

conductivity of a two-phase dispersion of spherical

particles of conductivity Aa- imbedded in a medium

of conductivity Xc- This expression is rigorously

valid for dilute dispersions where the average

distance between dispersed particles is much

larger than the particle size. Maxwell's relation can be written in the form:

X =

2-2y+(l+2/)\t /Xe Xe 2-h/+(l-/)Xd/Xc (74)

The behavior of eq (74) for small / is more easily seen by expanding it in the form:

X 3/(l-X,/Xc ) 3/2 (l-Xg/Xc ) 2

\c 2+X./X, (2+Xd/Xc) 2

(75)

In figure 36, we compare the predictions of the

series-slabs expression (eq (72)), the paralleltubes expression (eq (73)), and the Maxwell

dilute dispersion expression (eq (74)) for the case y=0.1. For values of Xd/Xc near 1, all three expres-

> 6 Equation (4) of Powers [346] and Equation (5) of Nahas, et al. [400]

,

which should agree with our eq (73), are obviously erroneous in that they

do not reduce to X=X e when Xd=X e.

323-bDO O - 69 - 7 89
.7 I I I | I .01 .1 I 10 100

Xd / Xc

Figure 36. Computed effective conductivity of a dispersion

of 0.1 volume fraction of a material of conductivity Xd in a continuous matrix of material of conductivity \c . S Series-slabs expression, eq (72); P Parallel-tubes expression, eq (73); M Maxwell dilute dispersion expression, eq (74).

sions agree. However, if the thermal conductivities

of the two phases differ significantly, the seriesslabs and parallel-tubes expressions disagree with

the Maxwell expression, which should be very

accurate for /< 0.1.

To predict the thermal conductivity of a dispersion having less than about 0.1 volume fraction

of isometric dispersed phase imbedded in an

isotropic continuous phase, one should use the

Maxwell dilute dispersion expression, eq (74) or

(75). The expressions obtained from the cubic

array of cubes, eqs (70) to (73), should not be

used for dilute dispersions in media having an

isotropic continuous phase. For anisotropic media,

the Maxwell expression would have to be modified.

For highly anisotropic media, such as pyrolytic

graphite, the implicit assumptions of either

infinite or zero lateral thermal conductivity of

the continuous phase might be more nearly met so that one of the series-slabs or the parallel-tubes

expressions (eqs (70) to (73)) might be more

accurate than in isotropic materials. For heat

conduction in the poorest conducting direction,

the series-slabs expressions, eqs (70) and (72),

should be more accurate while for heat conduction

in the best-conducting direction, the paralleltubes expressions, eqs (71) and (73) should be

more accurate.

A special case of interest for ceramics is the

effect of porosity on thermal conductivity. If the

thermal conductivity of the continuous ceramic

material is much greater than the effective conductivity of the pores, we can set Xtf/Xc =0 and

our expressions reduce to:

Series-Slabs

x l —

j~l=fk+f **«V (76)

which for small / reduces to

~=l-J-f^+. Xd «Xc- (77)

Ac

Parallel -Tubes

r=l-/^ Atf «Xc (78)

Maxwell Dilute Dispersion

l=2W' X' <<V (?9)

which for small / reduces to

~1-|/+|/2 -. • X,«XC . (80)

In figure 37 the predictions of eqs (76) , (78) , and

(79) are shown for void volume fractions up to 0.1.

Over this porosity range the Maxwell equation

should be rather accurate if the porosity is in

the form of dispersed, disconnected, isometric

pores. In many ceramics this will not be the case and eq (79) should then serve only as an upper

limit for the effective thermal conductivity of a porous ceramic material.

For dispersions which are sufficiently dilute

for eq (74) to be valid, neither the size distribution

of the disperse particles nor the manner in which

Figure 37. The effect of porosity on thermal conductivity as computed by:

S Series-slabs expression, eq (76); P Parallel-tubes expression, eq

(78); M Maxwell dilute dispersion ex- pression, eq (79).

90
they are deployed are of consequence. However,

these factors must be considered if the concentration of the dispersed phase is increased. Lord

Rayleigh treated the case of uniform spheres

arrayed in a cubic lattice distribution. Meredith

and Tobias extended Rayleigh's derivation by

an additional term and obtained [347]

which reduces to

A=1

K 2+Xd/Xc

3/

-/-

1.315(l-Xd/Xc )/

10/3 4/3+Xd/\c +0.409(l+Xtf /Xc )/

7/3

(81)

for the conductivity normal to a side of the cube.

If the term involving f

10 '3 in the denominator of

the right hand side is dropped, this expression

reduces to the Maxwell dilute dispersion expression, eq (74). Equation (81) should be more

accurate than eq (74) for values of / up to x/6 =0.524, which is the maximum possible value for a cubic array of spheres.

A rigorous solution for the effective conductivity

of a concentrated random array of particles of

varying sizes has not been achieved. Several

approximations have been developed which are

useful in many cases. Bruggeman (see [346-347])

developed an expression,

X-Xd GO" =1-/' (82)

which has proved rather effective in predicting

the conductivity of a dispersion containing a wide

range of particle sizes. For Xtf —>0, eq (82) reduces to _=(1_/)3/2

j X,«X„ (83)

while for \d—>»

,

Ac (1-/) i' K»K (84)

For a concentrated dispersion containing only a narrow range of particle sizes, the Bruggeman

variable dispersion expression (eq (82)) tends to. overestimate the effect of the disperse phase while

the Maxwell expression (eq (74)) tends to underestimate. Meridith and Tobias [347] suggested an

alternative semiempirical expression that predicts

conductivity values intermediate between the

Bruggeman variable dispersion equation and the

Maxwell dilute dispersion equation:

X _r2(X,/Xe -r-2)+2(XgAc- !)./

"!

X, L.2(Xtf/Xc +2)-(X,/Xc -l)/J

X

r(2-/) (x,/xe +2) +2(xtf/xc -1) n

L (2-/)(X,Ac+2)-(Xd/Xc -l)/

J

(85)

and

X 8(2—/)(!—/)

Xc (4+/) (4-/)

X =

(l+/)(2+/)

Xc (l-/)(2-/)'

(86)

(87)

Meredith and Tobias [347] contrast the predictions of the various expressions we have given

above. For a given two-phase system, all of these

expressions predict , ]two different conductivities,

dependent on which phase is assumed to be

disperse. Bruggeman (see [346]) derived another

approximate expression that should be applicable

to mixtures where neither phase is necessarily

continuous

-/l X1 +2X+-/2X2 +2X~° (88)

In figure 38, the predictions of the Bruggeman

mixture equations are compared with those from

the Maxwell dilute dispersion equation. We see that for small values of f2 the Bruggeman mixture

equation is in agreement with a dilute dispersion

of particles of conductivity X2 in a matrix of conductivity Xi while for values of j2 approaching

unity the mixture equation predicts a conductivity

due to particles of conductivity Xi dispersed in a matrix of conductivity X2 . There have not been enough accurate measurements of the thermal conductivity of

well-characterized ceramics to provide adequate

experimental confirmation of any of the above

equations. For well-defined systems (such as spheres dispersed in a continuous matrix) which

are in good correspondence to the models used in

deriving these equations, measurements of electrical conductivity or dielectric constant have shown

Figure 38. Commuted effective thermal conductivity of a

mixture.

B Bruggeman mixture expression, eq (88); H Maxwell dilute dispersion ex- pression, eq (74), high-conductivity phase continuous; L Maxwell dilute dispersion expression, eq (74), low-conductivity phase continuous.

91
good agreement with the theoretical predictions of

these equations. Two-phase ceramics, in general,

cannot be readily described in terms of a simple

model such as spheres dispersed in a uniform medium, and hence in many cases the expressions we

have cited will indicate qualitatively the effect of

an additional phase but should not be relied upon

for accurate quantitative predictions.

In the remainder of this section we shall cite a few examples from the literature to illustrate the

effect of an additional phase on the thermal conductivity of ceramics. The next section, on microstructure, is also pertinent here.

In figure 39 we show the data of Kingery [288]

on six samples of the two-phase system magnesiaforsterite (MgO-Mg2 Si04 ) which is reported to

form essentially no solid solution. The solid curve shown represents the Bruggeman mixture expression (eq (88)) fitted to the end points. The deviations of the data from the smooth curve could be

due to a combination of several factors:

1. The Bruggeman mixture equation is only

approximately valid.

2. The data may contain some experimental

error.

3. There may be some formation of a solid solution phase which would have a lower con- ductivity than would a mixture.

4. Errors may have been introduced by the

procedure Kingery used to correct his data

to correspond to 100 percent dense material.

5. The thermal conductivity of these samples

may have been affected by their (perhaps

differing) microstructures.

Kingery corrected his data to correspond to theoretically dense material by dividing the measured

0 .2 .4 .6 .8 1.0 VOLUME FRACTION

Mg 2 Si04

Figure 39. Thermal conductivity of mixtures of magnesia

and forsterite as measured by

Kingery [288].

The solid curve represents the Bruggeman mixture expression, eq

(88).

values by (1—/), where / is the volume fraction

porosity. As we have seen (e.g., eqs (76 to 80) and

figure 37), porosity would be expected to have a greater effect than was assumed by Kingery. For

small values of /, it would seem preferable to

divide measured conductivity values by (1 —3//2)

to obtain an estimate of the conductivity of the

dense material.

As an example of the effect of porosity on the

thermal conductivity of magnesia, we show the

data of Moore et al. [317] and of Koenig [436, 437]

in figure 40. The data shown for zero porosity were

obtained on single crystals; the other data points

correspond to polycrystalline specimens of the

porosity shown. Moore, et al. wrote a semiempirical expression to relate conductivity and

porosity:

The three solid curves in figure 40 correspond to

different values of 0 in eq (89). For 0=0, eq (89)

reduces to the simple expression \=(1—f)\c used

by Kingery, among others, for adjusting thermal

conductivity data to correspond to theoretical

density. For 0=0.5, eq (89) is the same as the

Maxwell dilute dispersion expression, eq (79), for

the case where the disperse phase has zero conductivity. Moore, et al. found that an empirical value

of 0=1 fit the data reasonably well for values of/

up to 0.08 but that at higher porosities the conductivity decreased more rapidly.

In section 2.2c we saw that the apparent contribution to thermal conductivity due to radiation

through a transparent slab was proportional to

the thickness of the slab in the optically thin

limit where there is negligible absorption or emission of radiation within the slab. By analogy to

0 MOORE, el al. [317]

X KOENIG [437, 438]

\

\

I I I I

I I I 1 0 .04 .08 .12

Figure 40. Effect of porosity on the thermal conductivity of MgO at 90 °C (after Moore, et al. [317].

92
eq (56), the radiative component to the termal

conductivity of a pore is

\ r =4an2ET3 -gL, (90)

where a is the Stefan-Boltzmann constant, n is the index of refraction of the material filling the

pore, E is the effective emittance of the boundary

of the pore, T is mean temperature, L is the pore

size in the direction of heat flow, and g is a geometrical factor dependent on the pore shape.

Thus the effective thermal conductivity of a pore

is proportional to the size of the pore, all other

factors being equal. This effect is shown in figure

41 where the calculated effect of two different pore

sizes is shown. For pores less than about 0.01 cm

in diameter, the effect of radiation is slight. At

larger pore sizes, however, radiation across the

pores can make a very significant contribution to

the overall effective thermal conductivity of a porous ceramic, particularly if the emittance is high.

One other effect of porosity, not previously

discussed, is to decrease the radiative contribution

to heat transfer through diathermanous materials.

Even quite small amounts of porosity will act as

scattering centers thus reducing the photon mean

free path so that the high temperature radiative

component of thermal conductivity is less than

in single crystals or glasses.

In this section we have specifically discussed

only those cases of two-phase media in which

the disperse phase was isometric, e.g., spheres or

cubes. Powers [346] and Meredith and Tobias

[347] also discuss cases in which the disperse

phase is anisometric, e.g., ellipsoids or cylinders.

E=0.9

SOLID

0 400 800 1200 1600 2000 TEMPERATURE, °C

Figure 41. Thermal conductivity of zirconia sample with 20 -percent

porosity, assuming various emit- tances (after Kingery [438].

d. Microstructure

The microstructure, or internal structure on a

microscopic scale, of a ceramic material can have

a significant effect on the thermal conductivity

of the material. It is helpful to think of a polycrystalline ceramic material as a concentrated

dispersion of grains of conductivity Xg in a continuous matrix (the grain boundary region) of

average conductivity X6 . In this way, a polycrystalline ceramic may be treated as a two-phase

system and the thermal conductivity estimated

using the expressions cited in the previous

section.

We have seen that the two expressions (eqs

(72) and (73)) derived from the model of a cubic

array of cubes disagree seriously with one another

in the limiting case of small volume fractions of

disperse phase. However, for volume fractions of

the disperse phase approaching unity, eqs (72)

and (73) converge to a common expression which,

quite interestingly, is identical to the Maxwell

dilute dispersion expression, eq (74) . The Maxwell

equation can be put in the form

X =

3-2(l-X6/Xg)/»

\g 3+(VV-l)/» (91)

where jb is the volume fraction of the grain

boundary material. If the average grain size is g

and the average boundary thickness is b, the

volume fraction of grain boundary material is given by

fb =j' b«9- (92)

With this substitution, eq (91) becomes

X =

l-2(l-X6Ag)%

Xg

l + (Xg/X6 -l)% (93)

If b and g are measured from a photomicrograph

of the ceramic, the approximate effect of grain

boundary size on thermal conductivity may be

calculated using eq (93), if a reasonable estimate can be made of the conductivity of the grain

boundary region. In figure 42 we show a family of curves computed from eq (93). As an example of

the use of these curves, if the average grain size

is about 200 times the average boundary thickness

and the grain boundary conductivity is estimated

to be about one-fifth of the conductivity of the

grains, then the effective thermal conductivity

of the polycrystalline material would be about

97 percent of the conductivity of the grain material.

The catch here, of course, is to come up with a

reasonable value for the average conductivity

of the grain boundary material.

As figure 42 illustrates, thermal conductivity can be very sensitive to both the relative thickness

of the grain boundary material and to the effective

thermal conductivity of the boundary region. If

93
1.00

1000

Figure 42. Calculated effect of grain size on the effective thermal conductivity, X, of polycrystalline materials.

The grains, of average size g in the direction of heat flow, have a thermal

conductivity X, while the grain boundary region, of average thickness 6

,

has an average conductivity Xi,.

there are no pores or microcracks in the grain

boundaries, the thermal conductivity of the

boundary region should be at least as high as that of glasses since short-range order persists in

the boundary region. The distribution of impurities, pores, and microcracks in a ceramic can be

very significant. If such imperfections occur in

the grain boundaries they can both lower the

conductivity and increase the width of the grain

boundary region and hence have a much greater

effect than if these same imperfections were

dispersed throughout the grains. If enough were

known about the degree of crystallinity of the

boundary region and the types and amounts of

impurities and imperfections concentrated there,

it should be possible to estimate the thermal

conductivity of the grain boundary region. To

my knowledge, no one has attempted this.

For a reasonably pure ceramic material, the

grain boundary thickness is probably of the order

of 1 nm. At room temperature the thermal conductivity of glasses is about 1 Wm_1K_1 while

that of single crystal ceramics would typically be

less than 100 Wm_1K

_1

. Inspection of figure 42

shows that for X 6/Xg =0.01 and g/b=1000 (corresponding to an average grain size of 1 nm) the

effective conductivity would be about 91 percent

of that of a single crystal. Evaluation of eq (93)

for an average grain size of 10 /um yields a corre- sponding value of 99 percent. Thus, at room

temperature and above, one would not expect

grain size to have much effect on the thermal

conductivity of a dense, pure ceramic unless the

grain size were less than 10 /mi. However, if porosities or impurities are present, grain size

effects might occur even at larger grain sizes.

One case where microstructure can very significantly affect thermal conductivity is in materials that have different coefficients of thermal

expansion in different directions or, in the case of

poly-phase materials, between different phases.

Microstresses in such materials can cause microcracks in the grain boundaries, thus severely

reducing the effective conductivity of the grain

boundary region (see e.g., Kingery [241]).

There has been very little definitive work on

the effect of microstructure on thermal conductivity. A few pertinent references include [241,

281, 285, 288, 439]. Three references to the general

subject of microstructure are [440-442].

As an example of the effect of microstructure on the thermal conductivity of a ceramic, we show,

in figure 43, the data of Moore, et al. [317] on

thoria samples prepared by two different processes.

The samples prepared from thoria obtained by

the sol-gel process consisted of large particles of

fairly high density materials suspended in a low

density matrix of fine particles; the samples prepared from thoria obtained by an oxalate-precipitate process contained only small pores located

predominantly within the grains of a homogenous

material. The values of j8 shown on this figure

correspond to eq (89).

The thermal history and the method of fabrication of a ceramic material can greatly affect the

thermal conductivity since the composition and

structure of the ceramic depend on such factors as type of pressing or casting used, maximum firing

temperature, time of firing, and, in some cases, rate of cooling. There has been very little work

directly relating thermal conductivity with method

of fabrication or thermal history. The thermal

conductivity of a ceramic is a function of its present

composition and structure so that we need not

concern ourselves unduly with how that composition and structure were attained. It is useful, however, to list some of the factors affecting thermal

i.o

.9

X

1 1 G ORNL

r OXALATE THORIA

0RNL SOL-GEL THORIA (air)

ORNL SOL-GEL THORIA (H2

)

p=o

—~

p=l.2

i

1

I

1 0.05 0.10 0.15 0.20

Figure 43. Effect of porosity on the thermal

conductivity (at 75 °C) of thoria samples

having different microstructures (see text)

.

The oxalate thoria samples and one of the sol-gel thoria samples were sintered in air at 1650 °C while the other

sol-gel thoria sample was sintered in hydrogen at 1750 °C

(after Moore, et al. [317]).

94
conductivity which depend on the fabrication and

thermal history: — quantity and distribution of impurities and

whether impurities are in solution or dispersed as additional phases, —stoichiometry, —crystallinity, —quantity and distribution of other phases,

including pores, —grain size and boundary thickness, —quantity and distribution of imperfections

(e.g., dislocations, microcracks)

.

The most obvious effect of heat treatment is to

control the final density, and hence porosity, of a ceramic. Two general references to the sintering

process are [443-444].

In figure 43, we saw the effect of a different

starting material on the thermal conductivity of

thoria. To illustrate the effect of heat treatment,

I have taken several examples from Austin [262].

Figure 44 shows the effect of firing time at 1350 °C

on the thermal conductivity of an initially unfired

magnesia brick in which the porosity decreased

as a result of sintering; the thermal conductivity

increased, asymptotically approaching a value

corresponding to an infinitely long firing time.

Figure 45 shows the effect of firing time at 1350 °C

on the thermal conductivity of an initially unfired

4 0

1.5 1 1 1 1 1 1 1 1 1 0 40 80 120 160 FIRING TIME AT 1350 °C, Hr

Figure 44. Increase with firing period

of the thermal conductivity at 800 °C

of a brick in which the chief reaction

during firing was sintering (after

Austin [262]).

0 40 80 120 FIRING TIME at 1350 °C, Hr

Figure 45. Decrease with firing

period of the thermal conductivity

at 800 °C of a brick in which the chief reaction during firing was

the removal of combustible or

volatile matter (after Austin

[262]).

1200 1400 1600 MAXIMUM FIRING TEMPERATURE, °C

Figure 46. Influence of maximum temperature

attained during firing on

the thermal conductivity at 800 °C of fire clay, and of magnesite blocks with dif- ferent size starting material

(after Austin [262]).

brick in which the porosity increased as a result of

burning off some of the material of the brick; the

thermal conductivity decreased, again asymptotically approaching a value corresponding to an

infinitely long firing time. As a final example of the

effect of heat treatment, figure 46 illustrates the

effect of firing the same material at different

maximum temperatures.

6. Reliability of Existing Data

In general the accuracy of thermal conductivity

values found in the literature is not good. Thermal

conductivity values reported for ceramics are, if anything, less accurate than values reported for

metals or thermal insulating materials. The

reasons for this unhappy state of affairs are several

1. Thermal conductivity is a difficult property

to measure accurately. This is particularly

so for materials having intermediate values

of thermal conductivity (i.e., neither good

conductors nor good insulators)

.

2. Thermal conductivity reference standards are not readily available nor is the proper use

of such reference standards well known or understood.

3. Adequate characterization of thermal conductivity samples seems to be the exception

rather than the rule. This is probably particularly so for ceramic materials where the

porosity is not always reported, the purity

is rarely reported, and the microstructure

almost never reported.

Even a cursory examination of the available

data compilations [e.g. 238-240] covering thermal

conductivity values reveals that the spread in

the existing data on a given material is very large,

often of the order of 20 to 40 percent. Since many

of the samples tested were uncharacterized, it is

95
usually difficult, if not impossible, to separate the

effects of sample differences from the effects of

measurement errors. It is quite apparent, however,

that both effects are important. Some of the

examples cited in previous sections of this paper

clearly illustrate the effects of sample differences,

such as purity and porosity, on the thermal conductivity of a given material. As for measurement

errors, I have seen cases where reputable, and

reasonably experienced, investigators have re- ported thermal conductivity values (obtained,

for example, by an ASTM Standard Method of

Test) which were in error byfactors of two, three, or even five.

In the above discussion, I do not mean to

appear to be an alarmist. There are many good

data in the literature. However, to distinguish

between good and bad data is not always easy. If

you wish to find a suitable value for the thermal

conductivity of a particular material, you should

approach the available data on similar materials

cautiously and critically. Was the specimen adequately characterized? Has the investigator utilized the same equipment to measure the thermal

conductivity of reasonably well-defined materials

(having a thermal conductivity similar in magnitude to that of the material in question) and, if so, how do the results compare with those of other

investigators?

In summary, as regards literature values for

thermal conductivity, especially when material

characterization data are lacking, caveat emptor!

7. Measurement Methods

The technique selected for measuring the

thermal conductivity of a particular material

depends upon:

1. The thermal conductivity of the material.

2. The temperature range.

3. The physical nature of the material.

4. The accuracy which is required.

5. The availability of existing equipment.

6. Cost.

7. The experience, skills, knowledge, and prejudices of the investigator.

In this section we shall discuss a number of

different methods for measuring thermal conductivity. The methods being discussed were selected

to illustrate the principles and problems involved

as well as to cover, if only in a somewhat cursory

manner, many of the types of methods in common

use. Where possible, apparatuses used at NBS

were selected as examples since I am more familiar

with the advantages and disadvantages of our equipment.

In most methods of measuring thermal conductivity, the experimental arrangement is designed to force the flow of heat to be in the direction of one of the coordinate axes. Most methods

utilize a cylindrical specimen geometry and force

the heat flow to be in either the longitudinal or the radial direction. For anisotropic materials, it

is usually preferable to have the heat flow be along

one of the principal crystallographic axes.

7.1. Longitudinal Heat Flow

a. Mathematical Development

The total longitudinal heat flow through a specimen of constant cross sectional area is

Q=~XA

Vz'

(94)

where v is the temperature relative to an arbitrary

datum plane, and where it is to be understood

that X now refers to the thermal conductivity in

the 2-direction. If the temperatures are independent of time, if the specimen is properly

insulated or guarded so that there is no trans- verse gain or loss of heat, and hence Q is a constant, if there is no heat generation within the

specimen itself, and if, further, the specimen is homogeneous in the 2-direction so that X does

not depend on z, then (94) is easily integrated

to yield

«=fj>*- (95)

where Vi and v2 are the values of v at positions

z=l and 2=0, respectively.

Equation (95) is valid under the following

restrictions

1. The specimen is opaque to thermal radiation.

2. The specimen is homogeneous in the direction

of heat flow.

3. There is no internal heat production (or

absorption).

4. All temperatures in the system are independent of time.

5. The cross-sectional area is constant at all 2-positions between z=0 and z=l.

6. The planes 2=0 and z=l are isothermal

surfaces.

7. There are no transverse heat gains or losses.

Immediately following eq (94), v was defined as the temperature above an arbitrary datum plane.

For reasons which will shortly become obvious,

it is convenient to designate this datum plane as T0 so that v=T—T0 , where T is the temperature

on a fixed scale (e.g., T is expressed on the Celsius

or Fahrenheit scale). The temperatures at z=l and

2=0 are T1 = T0 +v1 and T2 =T0 +v2 , respectively.

In order to use eq (95), it is convenient to

assume a particular form for the temperature

dependence of the thermal conductivity so that

the integration can be carried out. Three cases

will suffice:

A. The thermal conductivity is independent of

temperature so that

X=X0 . 06)

96
In this case eq (95) becomes, after integration and rearrangement,

Ql

A{T2 -T,) (97)

B. The thermal conductivity is a linear function

of temperature, so that

X=Xo(l+j9p). (98)

Substitution of eq (98) into eq (95) yields

(2=XoA^=^[l+|(2,

1 +r2 -2r0)]- (99)

If we define T0 as

To=^±^2

> (100)

then the term in eq (99) involving 0, the

temperature coefficient of thermal conductivity, vanishes so that eq (99) reduces

exactly to eq (97). For a material having a thermal conductivity which is a linear function of temperature, the thermal conductivity, X0 , given by eq (97) corresponds to

that at the temperature, T0 , given by (100),

C. For many crystalline dielectric solids, the

thermal resistivity, or reciprocal of the

thermal conductivity, is a linear function

of temperature over a considerable temperature range so that

l+7» (101)

Substitution of eq (101) into eq (95) yields

Q=\0A

T2 -Tx

I l_7(»2-

In the present case, the coefficient, y, is not

so amenable to elimination by choice of a

particular reference temperature as was the

coefficient, /J, in eq (99). If we retain the

definition for T0 given by eq (100), then eq

(102) becomes

Q=XoA

[^..g,) arctanh

(103)

+!(M)V..} (,04,

For sufficiently small values of y{T2 — Tt ), eq (104)

reduces to eq (97). Consider a case where 7=.003

°C-\ a rather high value; if (Ta-T^lOO °C,

the second term in the square brackets in eq (104)

represents a correction of 0.75 percent; if {T2 —Ti) =300 °C, this term represents a correction of

almost 7 percent.

In computing thermal conductivity values for a material whose thermal conductivity is expected

to have a temperature dependence such as that

in eq (101), one may use eq (100) and eq (97) to

compute a mean temperature and the approximate

thermal conductivity at that temperature. The

first approximations to the thermal conductivity

thus obtained for several tests at different mean

temperatures can be used to compute values of

the coefficient, y which can then be used in eq

(103) or (104) to obtain more refined values for

thermal conductivity. The resultant values of X0 can be used to obtain better values of y etc. The convergence is quite rapid in most practical

cases, so that only one or two iterations will

normally be required.

If eq (97) is used to compute thermal conductivity, the maximum fractional error in the

resultant thermal conductivity value is given by

AXr

X0 + +

+ \\dTj,

I

A T0

A(r2 -ro

T2 -Tx +

a<2

Q

+

errors due to failure to meet

requirements of eq (97)

(105)

where | AA/A\ is the absolute value of the fractional

error in the area, etc.; the term in AT0 is the fractional error in the thermal conductivity due to

the error in the mean temperature; the last term

represents errors due to test conditions departing

from the assumptions made in deriving eq (97),

e.g. temperatures varying with time.

b. Experimental Techniques

The first concern in most techniques for measuring thermal conductivity is to force the heat

flow to be unidirectional, which, for purposes of

the present discussion, means longitudinal. Since

heat flow is proportional to a geometric factor, a thermal conductivity, and a temperature difference, the direction of heat flow must be controlled

by controlling one or more of these variables. 16 The experimenter's freedom in adjusting these

parameters is constrained, sometimes severely,

by the often-conflicting requirements of being

able to accurately measure total heat flow, area, and temperature differences. Very practical considerations, such as available specimen size,

frequently constitute severe constraints on apparatus design.

16 This statement is strictly true only for conductive heat transfer. In some

apparatus, transverse heat flows would be by radiation or, rarely, convection.

If these transverse heat exchanges are small, as is desired, they usually can be

represented as being proportional to an "effective" thermal conductivity.

97
It is not possible to directly measure the heat

flow in a specimen; one must, rather, measure

the heat flow into a specimen or out of a specimen.

This necessitates that not only must transverse

heat losses or gains of the specimen be prevented

or accounted for but, further, there must be no

unaccounted-for losses or gains between the specimen and the location at which the heat flow is measured. The most common method of measuring

heat flow into a thermal conductivity specimen

is to measure the electrical power dissipated in a heater at the hotter end of the specimen. The

heat flow out of a specimen is sometimes measured

by a "flow calorimeter," with which one observes

the temperature rise and flow rate in a circulating

liquid of known heat capacity, or by a "boil-off

calorimeter," with which one observes the boil-off

rate of a fluid of known heat of vaporization.

Another common method of measuring the heat

flow into or out of the specimen is by use of a "heat-flow meter," with which one observes the

temperature drop across a known thermal con- ductance. As most commonly employed, the heatflow meter is a solid material of known thermal

conductivity placed in series with the specimen

so as to, hopefully, carry the same heat current.

Similarly, it is not possible to directly measure

the temperature gradient in a specimen; one must,

rather, measure the temperature difference between

two or more locations and then compute the

gradient. Consider a specimen held between a heat

source and a heat sink. If the total thermal

resistance of the specimen is large compared with

the thermal contact resistances between the source and the specimen and between the specimen

and the sink, then the temperature drop across the specimen can be taken as equal to the temperature of the source minus the temperature of the

sink and it is not necessary to install temperature

sensors in the specimen. However, if the specimen

has a low thermal resistance such that thermal

contact resistances are not negligible, it is neces- sary either to correct for these contact resistances

or, what is usually done, to install temperature

sensors in the specimen.

The apparatus illustrated in figure 47 has re- cently been used at NBS to measure the thermal

conductivity of a specimen of microcrystalline

glass over the temperature range 100 to 330 K.

The thermal conductivity of the specimen was

measured by a steady-state longitudinal heat

flow method in which the thermal conductivity

was computed from the measured power input to a heater at one end of the specimen, the measured

longitudinal temperature distribution along the

sample, the measured thermocouple separation,

and the cross-sectional area. The specimen (A) was fastened to a heavy

copper "cup" (C) which was thermally connected

by three brass legs (E) to a liquid-nitrogen-cooled

copper heat sink (F). A heater (I) wound around

the copper cup (C) was used, in conjunction with

a thermopile and controller, to maintain the

Figure 47. An apparatus utiliz- ing longitudinal heat flow

to measure thermal conductivity over the temperature

range 100 to 830 K. The

components are identified in

the text.

copper cup at any temperature from about 90 to

330 K.

A copper block (B) with an internal heater

winding provided the measured heat flow down

the specimen. The specimen was held between

thin copper caps (shown in black) fastened to the

adjustable-temperature cup (C) and the heater

block (B). In order to achieve good thermal contact, a eutectic alloy of gallium and indium (freezing point 15.7 °C) was used in the interfaces

between the specimen and the thin copper caps.

This assembly was pressed together by means of a loading screw, spring, and stainless steel spider, as shown at the top of figure 47. The spring was

necessary to accommodate differential thermal

expansion; it also provided considerable resistance

to heat loss from the heater block.

A copper guard cap (H) and radiation shields

(G) surrounded the specimen and heater block.

Using the heater (J) on the guard cap in conjunction with a thermopile and controller, the

temperature of the guard cap (H) was automatically controlled to closely match the temperature

of the heater block (B) . The space between the

specimen and the guard cap and radiation shield

was packed with glass wool to minimize radiative

heat losses. The entire apparatus was contained

in a bell jar which was evacuated to less than

10~5 torr to further reduce losses from the

specimen and the heater block.

Current leads from the specimen heater were

thermally grounded to both the heater block

and the guard cap so as to minimize conduction

of heat along the leads. Because the temperature

of the guard cap matched quite closely that of the

heater block, the current leads to the specimen

heater could be made large enough for heat

generation in these leads to be negligible.

98
The temperature distribution along the specimen was measured by three butt-welded thermocouples pressed into transverse slits in the convex

surface of the specimen. Additional thermocouples, designated by X in figure 47, were used

to monitor other temperatures in the apparatus.

All thermocouples were fabricated from calibrated

0.013 cm diam copper and constantan wire.

Care was taken to thermally temper all thermocouple leads by wrapping them around the specimen in an isothermal region for several centimeters

from the junctions. All thermocouple leads were

wrapped around and cemented to a copper

tempering block (D) to prevent heat from being

conducted to the measuring junctions. The

thermocouple leads passed through epoxy vacuum

seals to individual ice junctions from which copper

wires lead to the emf-measuring circuitry.

Power to the specimen heater was provided

by a regulated dc power supply operating in a constant voltage mode. The power input was

determined by measuring the current through the

heater using a standard resistor and the voltage

drop across potential taps to the heater using

a high-resistance voltbox. Voltages from the

standard resistor and the voltbox were read to

five significant figures using a precision potentiometer. The emfs of the specimen thermocouples were read to 0.1 uV. The emfs of other thermocouples were read to 1 mV.

Each data point was obtained from two tests

at essentially the same mean temperature but

with different power inputs to the specimen

heater in order to minimize systematic errors, especially in temperature measurement.

The thermal equilibriums obtained were such

that the specimen temperatures did not drift

more than 0.01 K over a half-hour period.

Several tests were run with a rather large

temperature difference between the specimen

heater block and the guard cap in order to

evaluate the thermal conductance between these

regions. Most of this conductance was due to

the screw-spring-spider arrangement used to

press the heater against the specimen. Using

the thermal conductance values thus obtained,

all tests were corrected for heat exchanges due

to small temperature differences between the

specimen heater block and the guard can. The thermal conductivity values obtained using

this apparatus exhibited an estimated standard

deviation from a smooth curve of less than onehalf of 1 percent. The uncertainty in the smoothed

thermal conductivity values was believed to be

not more than 2 percent

The apparatus just described utilizes several

features which are not necessary at lower temperatures or in making measurements on materials

having higher thermal conductivity values than

did the microcrystalline glass (X = 5Wm_1 °C_1

). Further comments are also in order for several

other features of the apparatus.

If the heater were wound directly onto the

specimen or if the heater block could be clamped

or soldered to the specimen, the heater holddown assembly shown in figure 47 would not be

necessary. A procedure used by some investigators

is to place the heater between two similar specimens as shown in figure 48. We have recently

fabricated an apparatus of this general design

for use in measurements on glass samples.

The gallium-indium alloy between the heater

and the specimen and between the specimen and

the adjustable-temperature cup in the apparatus

shown in figure 47 was for the purpose of providing uniform thermal contact and hence promoting

plane isotherms in the specimen. If the specimen

is longer, relative to its diameter, so that all thermocouples can be placed one to two specimen

diameters from both the heat source and the heat

sink, the method of introducing and extracting

heat from the specimen is not critical since these

longer relative lengths allow plane isotherms to

become established near the thermocouple locations even if the isotherms are distorted near the source or sink.

At lower temperatures, or in the case of highly

conductive specimens, it is not necessary to sur- round the specimen and heater with thermal insulation—a good vacuum will suffice to preclude

unwanted heat exchanges. Similarly, in some

applications, particularly below liquid nitrogen

temperatures, guard caps or shields are not necessary, it being sufficient to surround the specimen

and heater with an essentially isothermal environment not too different in temperature from the

temperatures in the specimen. In our application

of the apparatus shown in figure 47, it was neces- sary to use a guard cap and shield, to install

insulation around the specimen, and to evacuate

the insulation (to reduce its thermal conductivity)

in order to render extraneous heat exchanges

essentially negligible. At higher temperatures, as we shall see, it is not always possible to reduce

extraneous heat exchanges so as to be negligible

and it becomes necessary to evaluate these ex- changes and make corrections.

-HEAT SINK

SPECIMEN THERMOCOUPLE

HEATER |—GUARD HEATER

GUARD SHIELD

Figtjbe 48. An apparatus to measure

thermal conductivity by longitudinal

heat flow through two glass specimens

with a heater sandwiched between

them.

99
The method of controlling the temperature of

the heat sink varies considerably among different

investigators. Most apparatuses designed for operation below room temperature are immersed in a

fluid bath (e.g., liquid nitrogen, liquid helium)

rather than having a cryogenic fluid circulated

through the apparatus. I feel that the ease-ofaccessibility to the apparatus shown in figure 47

more than offsets the slightly higher costs due to

increased liquid nitrogen consumption. Many investigators control the temperature of a cryogenic

bath by using a pump to reduce the vapor pressure

above the bath. The temperature of the bath is determined from the heat balance condition that

at equilibrium the rate of heat flux into the bath

equals the latent heat of the fluid times the mass

rate of exhaust. The minimum temperature available for a given fluid depends on the pumping

speed and the magnitude of the heat input. The

temperature of the cryogenic fluid can be raised

by decreasing the speed of the pump, which usually

is not practical, by increasing the pressure drop

between bath and pump, or by increasing the heat

flow into the bath. The general problems of providing a controlled-temperature heat sink at cryogenic temperatures are discussed in, for example,

the very useful book by White [445].

The apparatus shown in figure 47 utilized copper

versus constantan thermocouples fabricated from

.013 cm diam wire. The cross-sectional area of

one wire was only 1/40,000 of the area of the

specimen. However, the thermal conductivity of

copper is about 100 times that of the microcrystalline glass specimen. Thus the thermal con- ductance per unit length of one thermocouple wire

was about 0.25 percent of that of the specimen.

For the accuracy desired, this high a thermal con- ductance for thermocouple wires is barely tolerable. The options are to use finer wire, which is very delicate to handle; to use an alloy in each

leg of the thermocouple so as to have a lower

thermal conductivity, with perhaps some sacrifice

in the homogeneity of the wire; or to lead the

thermocouple wires radially and attach each of

them thermally to the guard shield at a point

which is at the same temperature as the thermocouple junction on the specimen, a not-too-easy

procedure in many apparatuses.

The apparatus shown in figure 49 was used by

Ditmars and Ginnings [446] to measure the thermal conductivity of beryllium oxide over the

temperature range 40 to 750 °C. This apparatus

is similar in principle to, but different in detail

from, the apparatus (figure 47) previously discussed. Measured electrical heat, introduced in

the sample heater at the top of sample, flowed

down the sample and its "adapter" to a heat sink.

The sample heater consisted of six small helices of

fine wire located in holes in the top of the sample.

Anhydrous boric oxide was used to give good

thermal contact between the adapter and the sink.

The sink was cooled with either water or air, depending on the temperature range, and was

Sil-O-Cel

Nichrome

heater

Tap

fcL

v

i n.

.O.OIO" Inconel

tube

'Container

-Shield heater

Powdered

alumina

Differential

couple

-Temp ring

"Sample heater

-Guard heater

-Control couple

-Guard tube

-Sample couple

-Guard couple

-Sample(BeO)

-Differential

couple (Pt-Rh)

"B 2 0 3 -Container

"Adapter

"B 2 03

-Sink heater

-Alundum tube

-Sink cooling

Scale: o 2 4 6 cm 1—1—i—i———-

Figure 49. An apparatus utilizing longitudinal heat

flow to measure theimal conductivity of beryllium

oxide over the temperature range Jfi to 750 °C

(after Ditmars and Ginnings [446)]-

equipped with a heater and a thermocouple so that it could be automatically kept at a constant

temperature.

The temperatures along the sample were

measured with three thermocouples (0.013 cm

diam platinum-rhodium versus platinum) having

reference junctions at 0 °C and principal junctions

on the sample at the three levels shown in figure

49. In addition, a differential thermocouple was

used to measure the temperature difference between the upper and lower levels on the sample.

All of the thermocouples on the samples were made

with junctions peened into small holes (about 0.06

cm in diameter and depth) in the cylindrical

100
surface of the sample. In order that the temperature difference measured on the sample would

correspond to the electric heat put into the top of

the sample, precautions were taken to minimize

radial heat loss from the sample. For this purpose

the sample was surrounded by a metal guard tube

provided with thermocouples. The top of the

guard tube was automatically controlled at the

desired temperature using the guard heater. The

top portion (tempering ring) of the guard tube was

made of thick nickel to which the electrical leads

to the sample were thermally connected, so that

it served as a tempering region for bringing the

leads to the temperature of the top of the sample,

and thus to reduce heat conduction along them.17 A thermal shield and heater, placed above the

sample and tempering ring, also served for this

purpose and to prevent heat transfer upward

through supports and insulation.

The whole assembly, supported by rods ex- tending down to the bottom of the guard, was

filled with fine alumina powder for insulation and

enclosed in a thin metal tube which served to hold

the powder. The apparatus, which was filled with

argon during operation, was maintained at the

chosen temperature by the surrounding furnace.

Two basic types of experiments were performed

for each measurement of thermal conductivity. In

the first type, which Ditmars and Ginnings called a conductivity experiment, the furnace temperature was controlled to the desired value, a known

constant electric power was put into the sample

heater, the temperatures of the guard and shield

were adjusted to match those on the sample as

closely as possible, and the sink temperature was

adjusted to a constant value. In the second type

of experiment, which Ditmars and Ginnings

called a calibration experiment, no power was put

into the sample heater but in other ways the

experimental procedure was similar to that in the

first type of experiment. The purpose of this

experiment was to correct for errors that did not

depend on the power transmitted through the

sample. Errors of this type are those due to

differences in thermocouples, and those resulting

from unknown heat leaks, which presumably were

the same in both types of experiments.

Ditmars and Ginnings estimated that their

results on beryllia were in error by less than about

3 percent. Laubitz [447] feels this possible inaccuracy could be reduced with only minor

modifications.

Laubitz [447] has carefully analyzed extraneous

heat flows in thermal conductivity measurement

apparatus utilizing longitudinal heat flow methods

such as those discussed to this point in this

section. His chapter should be required reading

for anyone contemplating building an apparatus

of this nature. Laubitz and his co-workers at

the National Research Council in Canada (see

references given in [447]) and our group at the

National Bureau of Standards (e.g., see Flynn

and O'Hagan [243]) are among the few workers

who have undertaken detailed mathematical

analyses of this type of apparatus in order to

evaluate extraneous heat flows. As Laubitz' recent

work [447] clearly illustrates, most apparatus

would benefit enormously from such an analysis.

Other useful survey papers pertinent to the

type of apparatus we have been discussing

include White [445, 448], Drabble and Goldsmid

[14], and Mikryukov [449]. The discussion of error sources by Bauerle [450] is also very helpful.

The apparatuses described so far can be used

when there is an available insulation, suitable

for use between the specimen and guard, which

has a thermal conductivity value much lower

than that of the specimen. However, if the

specimen has a low thermal conductivity, such

as is found for example in very porous materials

(fibrous or granular materials, insulating firebrick), there may be no insulation which has a

significantly lower thermal conductivity. Then

the specimen is often used as its own insulation.

The first example of an apparatus of this type

which we shall discuss is the guarded hot plate.

Since this is an ASTM Standard Method of Test,

which is adequately described elsewhere [451], I shall only briefly describe the apparatus in

order to illustrate the concepts involved.

As shown in figure 50, a guarded hot plate

apparatus consists of a heated metering plate,

which may be square or circular, separated by

a narrow insulating gap from a surrounding

coplanar guard plate. Similar specimens are placed on either side of the hot plate; the outside

surfaces of the specimens are held between

constant temperature cold plates. In operation,

the electrical power input to the guard plate is adjusted, usually automatically, so that a mulGUARD PLATE

GUARD GAP

METER PLATE

SPECIMENS

COLD PLATES

" Laubitz, who recently analyzed this apparatus, [447] feels that the tempering ring caused unwanted heat exchanges between the sample and the guard

and hence should not be used.

Figure 50. A guarded hot

plate apparatus for measuring the thermal conductivity of relatively good

insulating materials using

ASTM Standard Method

of Test C177.

101
tiple-j unction differential thermocouple spanning

the guard gap has zero output, indicating that

there is no temperature difference across the

guard gap. Thus the electrically generated heat

input to the metering plate flows perpendicularly

from both sides of the plate through the specimens

to the cold plates. The average thermal conductivity of the two specimens is determined

from the measured power to the metering plate,

the temperature drops across the specimens, and

the geometry.

In using a guarded hot plate, measurement

errors can arise from imperfect guarding, inaccurate determination of temperature differences, or failure to achieve thermal equilibrium, as

well as from possible errors in reading instruments

or measuring specimen thickness or in thermocouple calibrations.

While it is relatively straightforward to ac- curately measure the electrical power dissipated

in the metering plate heater, it is not at all easy

to ascertain that all of this heat is going where it

is supposed to. The hot plate must be well designed

in order that there be no significant heat ex- change between the metering and guard plates.

The thermal resistance across the guard gap should

be large, the differential thermocouple junctions

must be installed so as to appropriately sample

the temperature on either side of the gap, and

the power to the guard heater must be controlled

so as to maintain a sufficiently small temperature

difference across the guard gap. If the conductivity

of the specimen is high, a corresponding gap may

be cut part way or all the way through the specimens. If the guard is too small, or its surfaces

are not isothermal, the guarding will not be adequate. In order to ensure adequate guarding, it

is common to use edge insulation around the

periphery of the guard plate and specimens and/

or to effectively increase the area of the guard

plate by using a second (outer) guard plate or by

controlling the ambient temperature around the

apparatus to a value near the mean temperature

of the specimens.

If the thermal resistance of the specimen is sufficiently high, determination of the temperature

gradient is easy. The temperature drops between

the metering plate and the cold plates are measured

using thermocouples (or other temperature-measuring devices) embedded in the metering and

cold plates. Contact resistances are neglected

and the gradients are computed from these

temperature drops and the specimen thicknesses

(which preferably should be the same and should

be measured under the conditions of testing).

If thermal contact resistances may not be neglected, the determination of temperature gradients

becomes more difficult. Thermocouples may be

installed in the specimen in such a manner as not

to disturb the uniform flow of heat. It is frequently advisable, as in the case of rigid specimens,

to install a thin resilient pad (rubber, asbestos

paper, etc.) on either side of each specimen

(i.e., cold plate, resilient pad, specimen, resilient pad, hot plate, resilient pad, etc.). The

philosophy here is that if you cannot get good

contacts it is better to have uniformly poor

ones so that heat flow through the specimens will

be uniform. One technique which we have found

very useful is, using the thermocouples in the hot

plate and cold plates, to measure the effective

thermal resistance of the pad-specimen-pad as- sembly and then, separately, the resistance of

the resilient pads without the specimens. By

subtraction, the resistance of the specimen is found, providing the contact resistance between

the specimen and resilient pads is negligible. In

this manner we have obtained good data on, for

example, 2.5-cm thick glass specimens using

0.3-cm gum rubber pads.

If contact resistance problems can be handled,

a well-designed guarded hot plate will yield results

having uncertainties less than 2 percent near room

temperature with the uncertainties increasing at

both lower and higher temperature. ASTM Committee C-16 is currently studying the problems

involved in using a guarded hot plate apparatus

to make thermal conductivity measurements at

low and high temperatures.

Because of the difficulties of guarding, and be- cause of contact resistance problems, a guarded

hot plate is usually not appropriate for measurements on materials having thermal conductivities

much higher than that of, say, fused silica.

Useful references to the use of the guarded hot

plate, as well as to its associated possible errors, include [452-455].

In all of the thermal conductivity measurement

methods we have discussed thus far, the heat flow

through the sample has been determined by measuring the electrical power dissipated in a heater.

We shall now discuss methods in which the heat

flow is determined by other means.

In ASTM Standard Method of Test C201-47,

the heat flow through a ceramic specimen, such

as a firebrick, is determined by measuring the

temperature rise in a stream of water passing at a known rate through the meter section of a guarded water-flow calorimeter. Typically the calorimeter is double-guarded and the water-flowrate through each guard must be carefully adjusted

if the guarding is to be effective. This test method

is adequately described elsewhere [456] so we will

not discuss it further here.

We have been using a somewhat similar test

method at NBS intermittently for about 20 years.

The NBS guarded steam calorimeter apparatus,

which we believe to be simpler and more expeditious in use that a water flow calorimeter apparatus, has not previously been described in the

open literature.

Figure 51 shows the arrangement of the as- sembled apparatus, which consists of a high temperature furnace, the disk-shaped 15.2-cm diam

specimen, and the steam calorimeter. The specimen is supported slightly above a circular silicon

102
SILICON CARBIDE

SLAB

FIRE BRICK (I200°C)

Figure 51. The NBS steam calorimeter apparatus for

measuring the thermal conductivity of refractories.

carbide plate heated by the furnace, and is heated

by transfer of heat across the space between them.

At high temperatures, the heat transfer is chiefly

by radiation, with a relatively small temperature

drop. Heat passes through the specimen, and is transferred from its upper surface, by radiation

and conduction, across an air space, to the underside of the steam calorimeter, which is supported

above it. The use of thin air spaces on both sides

of the specimen is for the purpose of avoiding,

or reducing, problems of thermal contact with

solid surfaces that arise when a specimen is warped

or distorted by expansions resulting from the temperature gradient established in it. The heat flow through the specimen is determined by means of the steam calorimeter, the

metering area of which is the 7.6-cm diam area covered by the metering chamber bottom. The

guard chamber of the calorimeter, which operates

at the same temperature, acts to protect the

metering area of the specimen from edge effects.

The temperature drop through the specimen is measured by means of thermocouples imbedded in

its surfaces, as indicated. The relatively high

thermal conductivity of the silicon carbide plate

promotes substantial uniformity of temperature at

its top surface. To promote uniformity of radiant

heat emission and absorptance, the surface of the

calorimeter bottom, and either one or both surfaces of the specimen, may be painted with a high

temperature paint of high emittance.

The high temperature furnace has a fixed cylindrical base, with a removable annulus resting on

it which surrounds the specimen and the calorim^

eter bottom. The base and annulus are fabricated

from 1200 °C insulating firebrick, but each has an

inner liner capable of withstanding 1600 °C. The

heat source consists of six silicon carbide electrical

resistance elements located in the circular chamber

in the furnace base. The chamber opening is covered by a 1-cm thick by 18.4-cm diam plate of

self-bonded silicon carbide, capable of withstanding

1600 °C in an oxidizing atmosphere. A heater

ring, not shown in figure 50, surrounding the specimen is used to minimize radial heat flow.

A detailed illustration of the steam calorimeter

is given in figure 52. Under test conditions, the

heat entering the calorimeter base boils the water

in both the metering chamber (A) and guard

chamber (B). Steam from the metering chamber

rises through the stack (H) from which it passes

to the condenser (I) through a connecting tube

with a 10 deg downward slope. The condensate is collected in the flask (F). By determining the

time required to fill the known volume of the flask

between index marks with condensate, it is possible

to calculate the heat flow rate represented by the

steam condensed. The guard steam passes up on

both sides of the radiation shield (G), through

the orifice (K), and into the condenser (L). The

condensate drains from the base of the condenser

back to the guard chamber through the external

return tube (E).

u

F

Figure 52. Detailed schematic of the NBS steam calorimeter. The components are identified in the text.

103
The wetted surfaces of the 0.6-cm thick copper

bottoms are grossly roughened by means of 0.3-cm

cubes of brass distributed over and soldered to the

surfaces, to aid boiling. The diameter of the

metering chamber bottom is 7.6 cm; the gap to the

inner diameter of the guard chamber bottom

averages 0.02 cm. Both steam condensers are operated at atmospheric pressure, and boiling in the chambers

occurs at substantially this pressure. However,

to preclude the possibility of condensation of the

metering chamber steam before it reaches the

downward sloping tube to the condenser, the

temperature of the guard chamber steam is automatically kept about 0.1 °C higher than that of

the metering chamber steam by means of the

small pressure drop occurring across the orifice

(K), which can be changed in size. Metallic connection of the sloping thin stainless steel tube to

the guard chamber wall, and to the brass radiation shield (G), also tends to prevent cooling of

the stack (H) by the condenser (I).

An auxiliary electrical heater (C) is located on

the outer wall of the guard chamber, for use when

necessary in tests at very low flux rates to assure an adequate flow of steam through the orifice

(K). Two thermocouples are installed in the bottoms of the metering and guard chambers for

monitoring purposes.

In preparation for tests, the two chambers are

filled with an excess of water, which is then

withdrawn by suction to the level of the ends of

the filler tubes (J, D), which are self-trapping.

The condensate collected from the flask (F) is returned manually to the metering chamber

through the filler tube (J). Flasks of a few different volumetric capacities are used, depending on

the rate of condensate collection. The volumes

between index marks are determined to within

0.01 cm3

. Since the water in both the guard and meter

sections boils at nearly the same temperature,

the guard section temperature automatically

matches that of the meter section without the

need for critical adjustments such as are required

in using ASTM C201.

We intend to make several changes in the steam

calorimeter apparatus in order to improve the

accuracy and render it more convenient in operation. These changes include: (a) putting a second

guard, of about 30-cm diam, around the existing

guard to extend the 100 °C isothermal surface

and thus improve the guarding action, (b) building a new furnace, of lighter construction so as to

attain equilibrium more rapidly, with a 30-cm

diam hot plate, and (c) investigating the possible

advantages of changing to an all-glass condenser

for the meter section. Even without these modifications, the apparatus is very convenient in

operation and yields thermal conductivity values

estimated to be uncertain by not more than, say,

3 or 4 percent at specimen hot-side temperatures

of 1200 °C or higher.

The last longitudinal heat flow method we

shall discuss is the type of comparative method

which is variously known as a comparator, cutbar method, or divided-rod method. Ideally, in

this method the same heat flow is caused to pass

through both the specimen, of unknown thermal

conductivity, and a reference specimen, of known

thermal conductivity, which acts as a heat flow

meter and hereafter will be referred to as the

"meter bar". For constant properties, the heat

flow is given by

QJhAAT1=\mAj,Tm j (1Q6)

' s I'm

where A, AT, and / refer to cross-sectional area,

temperature difference, and length, respectively,

the subscript s designates the specimen, and m

designates the meter bar. The thermal conductivity of the specimen, under these ideal conditions, is given by

Xs=x-^B^" (io7)

As normally used with ceramic materials, the

cut-bar method is subject to essentially the same

uncertainties in determining temperature gradient

and area as would be present in other types of

longitudinal heat flow methods. However, we are involved with two areas, two lengths, and two

temperature differences, so in principle, it would

be possible for the total error from these sources

to be roughly twice as large. (In practice, certain

systematic errors would have the same sign in

both the specimen and the meter bar and hence

would partially cancel.) In addition, one must

add the uncertainty in the thermal conductivity

of the "known" meter bar. The additional uncertainties introduced via \m, Am, lm, and ATm

are almost invariably larger than the error would

have been if the heat flow were determined by

measuring the electrical power dissipated in a

heater.

Probably the largest uncertainty in comparative

methods arises from failure to meet the basic

underlying requirements of eqs (106) and (107),

namely that the heat flow must be the same

in the specimen and the meter bar. In the somewhat vain hope of ascertaining compliance with

this requirement, many investigators have sandwiched their specimen between two meter bars

in a configuration such as that shown in figure 53.

The lower meter bar supposedly measures the

heat flow into the specimen while the upper meter

bar hopefully measures the heat flow out of the

specimen. Agreement between the readings obtained with the upper and lower meter bars is a

necessary, but not sufficient, condition for the

heat flow through the specimen to be the same as that through the meter bars. The heat-flow

indications provided by the two meter bars will

104
z = L/2—

MIDPLANE-

-L/2-

Figure 53. A comparative, or "cut-bar"

apparatus in which the heat flow

through the unknown specimen is determined from the temperature gradients and thermal conductivities of the "meter-bars" on either side of the specimen.

<r

UJ>

tr o

-J LU

UJ D_

LU

-L/2

METER SPECIMEN METER

1

\ \1 \

\\\\ \

\\

1

1

1

1

1

1

1

1

1

1

1

1

V !

1 \\ \ \\ \

1

L/2

POSITION

agree provided the following rather general

conditions are met:

(a) The apparatus is geometrically symmetrical

about its midplane.

(b) The two meter bars have the same thermal

conductivity.

(c) The temperature distribution along the

inner surface of the guard cylinder and

along the surfaces which bound the outer

faces of the meter bars (e.g., the planes

2=±Z/2 and the cylinder r=b shown in

figure 53) possesses odd symmetry with

respect to the temperatures at the midplane

of the apparatus. By this, I mean that

[T (r, 6, z) + T (r, d, -z)]/2=T (r, 6, 0),

where T is temperature and r, 6, and z are

cylindrical coordinates (z being measured

relative to the midplane of the apparatus),

for all points on these bounding surfaces.

To illustrate the above, let us suppose that the

apparatus shown in figure 53 meets conditions

(a) and (b). We further suppose that the planes

z=±L/2 are each isothermal such that condition

(c) is satisfied for these two planes. All of these

suppositions are very reasonable and, in fact, are goals which investigators often strive for in designing this type of apparatus. In figure 54 we

show three possible temperature distributions

that might exist along the inner surface of the

guard cylinder, r=b. The solid curve represents

"matched guarding", where the temperature

distribution along the guard is the same as that

which would be along the meter bar/specimen/

meter bar assembly if the heat flow through this

assembly were constant (which is required if eq (106) is to be valid). The dashed curve represents "linear guarding", where the guard

temperature varies linearly with longitudinal

position from end to end. The hatched curve

represents "isothermal guarding", where the

guard is isothermal at the specimen mean temperature except very near the ends where it changes

rapidly to match the temperatures of the isoFigtjre 54. Three possible types of

guard temperature distribution which

will result in agreement between the heat-flow indications of the two

meter bars even though the heat

flow through the specimen may be

different.

thermal surfaces (z=±L/2) which bound the

outer faces of the meter bars. All three of these

guarding conditions (plus an infinite number of

others) satisfy condition (c) given above so that

the two meter bars will indicate the same heat

flow. However, the heat flowing in the specimen

can differ considerably from the heat flow indicated by the meter bars.

In some unpublished work that H. E. Robinson

and I carried out in this laboratory in 1961 and

1962, it was shown that the fractional power

change, from that power which would flow through

the meter bars and specimen if they were perfectly

insulated, can be represented quite closely, for

matched or linear guarding, by

e(2 )=\i (i~).JV(g) (108)

where Xs , Xm and X4 are the thermal conductivities

of the specimen, meter bar, and insulation, respectively, and F(z) is a geometrical factor which depends on the type of guarding. Figure 55 shows

F(z) for the case of matched guarding in the apparatus shown in figure 53. It is informative to

evaluate eq (108) for a not-too-unreasonable set of

values for the thermal conductivities. Assume we

wish to use dense aluminum oxide meter bars

(X=36 Wmr1 °C_1

) to measure the thermal conductivity of a borosilicate glass sample (X= 1.1

Wm"1 °C_1

) near room temperature. As insulation,

we use fine aluminum oxide powder in air (\ =0.1

Wm-1 °C~1 corresponds to very low density

powder). Substitution of these values into eq

(108) yields #(2) =0.09 F(z); at the midplane of

the apparatus, F=2.0 so that #(0)=0.18, or in

323-655 O - 69 - 105
-L/2 L/2

POSITION

Figure 55. The geometrical factor, F(z), to be used in

eg. (108) for the case of matched guarding in the cut-bar apparatus shown in figure 53.

other words, the heat flow through the midplane

of the specimen is 18 percent less than it would

be if the specimen and meter bars were perfectly

insulated. The error if eq (107) were used to

compute Xs would be less than this but still of the

order of ten percent. Since the temperature distribution along the guard is assumed to be the same

as that along the specimen and meter bars, there

are no radial heat exchanges with the guard. For

matched guarding the error arises simply due to

shunting of heat around the specimen through

the insulation. For other than matched guarding,

there will be radial heat exchanges between the

specimen-meter bars assembly and the guard

cylinder in addition to the longitudinal shunting

heat flow in the insulation. The attendant error in

Xs , if it is computed using eq (107), can be very

large.

For matched guarding, an upper limit for the

geometrical factor for shunting heat flow is given

where a is the radius of the specimen and meter

bars and b is the inner radius of the guard cylinder.

For any other type of guarding, there will be a

contribution due to radial heat exchanges.

As revealed in eq (108), the error due to shunting

heat flow in the insulation is proportional to the

difference between the thermal resistivities of the

specimen and the meter bars. This error can,

therefore, best be minimized by using meter bars

having a thermal conductivity as similar as

possible to that of the specimen. This also makes

guarding easier since no sharp breaks are required

in the longitudinal temperature distribution along

the guard. A further advantage to having similar

thermal conductivities is that the temperature

drops in the specimen and meter bars can be made

large enough to facilitate accurate measurement

while the total temperature drop is kept small.

References which may be consulted for discussions of various aspects of guarding of cut-bar

apparatuses including Laubitz [447, 457-458],

Bauerle [450], Petrov [459], Didion [460], and

Minges [461]. Apparatuses of the type we have

been discussion are described by, for example,

Francl and Kingery [462], Francis and Tinkelpaugh [463], Mirkovitch [464], and Mahmoodi

[465].

A rather different configuration for a comparative cut-bar apparatus is typified by ASTM C408-

58, Standard Method of Test for Thermal Conductivity of Whiteware Ceramics [466]. In this apparatus the specimen is soldered between two

rather long copper meter bars or "thermodes".

There is no guard—the apparatus is contained

within an evacuated chamber so that heat loss can

occur only by radiation. No thermocouples are placed in the specimen—the temperature drop

across the specimen and the contacts with the

thermodes is determined by extrapolation of

temperatures in each thermode. No correction is made for thermal contact resistance between the

specimen and the thermodes. I feel that thermal

conductivity values obtained using ASTM C408,

in its present form, are liable to large errors. For a

material, such as beryllia, which has a high thermal

conductivity, errors may arise due to thermal contact resistance. For a material, such as mullite,

which has a low thermal conductivity, errors may

arise due to heat losses. These factors are considered for a somewhat similar apparatus in the report

by Moore, et al. [317].

A number of error sources can be eliminated if a comparative thermal conductivity apparatus is calibrated using specimens of known thermal conductivity. If this is done, it is not necessary to

know the thermal conductivity of the meter bars.

These considerations led us to suggest an apparatus, which Laubitz [447] has since called an interpolator, such as that shown in figure 56. The meter

bars and specimen are each of length equal to

THRUST

INSULATION

1 1 1 •

•

SPECIMEN S ~A T-) \

METER 'C- <- t=- \

•

•

•

•

i i

6a

r = b

Figure 56. A cut-bar apparatus in which the guard diameter

is sufficiently large that the detailed temperature distribution

along the guard is unimportant.

Such an apparatus can most accurately be calibrated by the use of substitute specimens.

106
twice their radius, a. Thick disks of conductive

material of radius 6= 7a, approximately, are used

to provide substantially isothermal surfaces at

the outer faces of the two meter bars. The "guard",

constructed of conductive material, is positioned

on poorly conducting supports carried by the lower

disk.

With a guard of such large diameter, shunting

heat flows are independent of the details of the

temperature distribution along the guard. We

have calculated the heat exchanges between the

meter bars and the insulation and between the

specimen and the insulation and found that these

exchanges are essentially identical for the three

guarding conditions shown in figure 54. Thus in

this apparatus all such extraneous heat exchanges

are constant for given meter bars and specimen

provided only that the rather general condition

(c) described previously is met (i.e., the guard

temperature distribution possesses odd symmetry

about the midplane of the apparatus). This is easily assured, for example, by making the guard

conductive enough for it to be essentially isothermal.

In using this apparatus, the conductivity of the

meter bars is assumed to be unknown. A number

of specimens of known and differing thermal conductivities should be measured in the apparatus

over the intended temperature range. The apparent thermal conductivity, Xm, of the meter bars

should then be computed from eq (107) for each

calibration specimen tested. The several Xm values

obtained will differ somewhat due to shunting

heat flows in the insulation surrounding the specimen and meter bars. A plot should be made of

these apparent XTO values as a function of the

known Xs values. This constitutes the calibration curve for the apparatus. To compute a thermal

conductivity value from data obtained on an

unknown specimen, use eq (107) with an approximate value of Xm to compute a first approximation

of Xs . Using this value of \s , enter the calibration curve and obtain a better value of Xm which then is used in eq (107) to compute an improved Xs

. One

or two such iterations should suffice.

There are, of course, alternate methods of

treating the data. The main point to be made is that the measurements on unknown specimens

are made under the same guarding conditions as existed when the apparatus was calibrated using

known specimens. In this way, certain systematic

errors cancel out and the attainable accuracy can

be significantly improved.

Some obvious precautions or arrangements

desirable in the use of the envisaged apparatus

include the following:

(a) The insulation of conductivity X 4 should be

installed, and if necessary, renewed from a uniform stock, so that X* is substantially

the same in all tests at the same temperatures.

(b) Contacting surfaces of the meter bars and

specimens should be clean and quite flat to

avoid perturbations of the calibration and

tests due to altered thermal resistance at

contacts.

(c) The reference specimens and the test

specimens should always have the same

dimensions and thermocouples should be

installed at the same locations and in the

same manner.

7.2. Radial Heat Flow

Although a few investigators have utilized

radial heat flow in spherical or spheroidal specimens, the great majority of thermal conductivity

measurements utilizing radial heat flow have been

carried out on circular cylindrical specimens and

we will restrict our discussion to these.

a. Mathematical Development

The total radial heat flow through a cylindrical

element of axial length, L, is

c dr (HO)

where v is the temperature relative to an arbitrary

datum plane, and where it is to be understood that

X now refers to the thermal conductivity in the

r-direction. If temperatures are independent of

time, if the specimen is properly insulated or guarded so that there is no axial gain or loss of

heat, and hence. Q is constant, if there is no heat

generation within the specimen itself, and if, further, the specimen is homogeneous in the

^-direction so that X does not depend on r, then

(110) is easily integrated to yield

where va and vb are the values of v at r=a and r=b,

respectively. Equation (111) is valid under restrictions analogous to those listed following eq(95).

By analogy with the development of eqs (98)

to (100), if X is a linear function of temperature,

we obtain

n_2tL\0 ( Ta —

T

b ) nio\

where Ta and Tb are the temperatures at r=a

and r=b, respectively, and X0 corresponds to the

temperature T0 =(Ta -{-Tb)/2. For materials

having a thermal resistivity which is proportional

to temperature, expressions analogous to eqs

(103) and (104) are easily obtained.

If eq (112) is used to compute thermal conductivity, the maximum fractional error in the

resultant thermal conductivity value is given by

AXo

X0 <

Ab

+

b In (b/a)

A(Ta —Tb)

+

Aa

Ta -Th +

a In (b/a)

AQ

+1x1

Q

+ (113)

107
+ errors due to failure to meet requirements

of equation (112)

where the terms are analogous to those following

eq (105).

b. Experimental Techniques

Many of the problems and techniques used in

radial heat flow methods for measuring thermal

conductivity are similar to those we have discussed

in conjunction with longitudinal heat flow

methods. Accordingly, I will only briefly discuss

the configuration usually used in radial heat flow

methods and mention a few of the problems

encountered with these methods. A very comprehensive discussion of radial heat flow methods is given by McElroy and Moore [467].

A generalized thermal conductivity apparatus

utilizing radial heat flow is shown in figure 57.

The specimen, in the form of a right circular

cylinder, is supported coaxially within a furnace

which is used to control the mean temperature

of the specimen. A coaxial central hole in the

specimen contains either a heater which produces

metered power flowing radially outward through

the specimen or a heat sink which measures

(e.g., by means of a water-flow calorimeter) the

power flowing radially inward through the specimen. Temperatures within the specimen, usually

at or near the midplane position, are measured

by thermocouples or by an optical pyrometer.

Guard heaters may or may not be used to control

longitudinal heat flows.

The worst errors usually arise from the difficulties of measuring small temperature differences

and from nonradial heat flows. Each thermocouple

THERMOCOUPLES

OR PYROMETER

SIGHTING HOLES

GUARD HEATER

HEAT SOURCE

OR SINK

SPECIMEN

FURNACE

GUARD HEATER

Figure 57. A typical configuration

for a thermal conductivity apparatus which utilizes radial heat

flow in a right circular cylinder.

or pyrometer sighting hole should run for some

distance along an isotherm. It is usually preferable

to have at least three temperature measuring

stations at each radius so that the effects of

possible eccentric isotherms can be averaged out.

Consideration should be given to errors arising

from distortion of the heat flow pattern by the

finite holes used for temperature sensing. It is good

practice to conduct measurements at two different

power levels to reduce the effects of certain

systematic errors in thermocouples or due to nonradial heat flow. If guard heaters are not used, the

specimen should be well-insulated at the ends and

should be sufficiently long that the heat flow is radial near the midplane of the specimen. All of

these points are discussed by McElroy and Moore

[467].

8. Reference Standards

Standards and standard reference materials are the basis of a consistent and accurate measuring

system. The need for standard reference materials

in thermal conductivity measurements is twofold.

In the first place, such materials are required for

comparative measurements in which the thermal

conductivity of the material under test is determined in terms of that of the standard reference

material. Secondly, such materials are required in

evaluating the accuracy of apparatus designed for

direct, as opposed to comparative, thermal conductivity measurements. The degree to which the

measured value of the thermal conductivity of the

standard reference material agrees with the ac- cepted value is a check on the accuracy of the

measurement process.

The basic requirements for any standard

reference material are that it be stable, reproducible, and appropriate for the measurements at

hand, and that the property in question be uniform

throughout the material. The ideal thermal conductivity reference standard should meet the

following criteria (also, see [468])

1. The thermal conductivity of the reference

material should be similar to that of the

materials to be measured. This is true whether

the reference standard is being used to

calibrate a comparative type of apparatus or

to check an absolute type of apparatus.

2. The reference material should be homogeneous and isotropic. Preferably it should be

reproducible. Four categories of uniformity,

in decreasing order of desirability, are as follows

(a) Reproducible materials

The most obvious of these would be

materials of quite high purity—usually

either chemical elements or compounds.

A disadvantage here is that the thermal

conductivities of such materials are usually markedly affected by the first

traces of impurity, at least at low and

moderate temperatures. Another, perhaps less obvious, type of reproducible

108
material is one in which the manufacturing process is sufficiently refined that one

may procure a sample of X from manufacturer Y with confidence that the

thermal conductivity of this sample is substantially identical to that of any

other sample of X procured, at least

from Y, at another time.

(b) Stocked materials of proven uniformity:

batch samples.

These are materials known, or found, to

be quite uniform in thermal conductivity

within a batch or lot, so that measurements made on a relatively few samples

of it are sufficient to define the conductivity of the entire batch.

(c) Materials whose thermal conductivity correlates well with some other more easily

measured physical property.

An example of this type of material would

be certain metal alloys, for which thermal

conductivity values may be computed

from electrical resistivity data using an

empirical correlating equation. A disadvantage of this procedure is the large

amount of work that may be necessary in

order to adequately define the correlating

equation.

(d) Materials not sufficiently uniform to forego

need for measuring each reference speci- men.

This procedure, although often necessary,

is expensive and places quite a work load

on the calibrating laboratory, which must

maintain an adequate facility indefinitely. In addition, lead-time to obtain a specimen may be onerous.

3. The thermal conductivity of the material

should not be affected by age or by environmental or temperature experience; i.e., it must be stable under the conditions of use. Phase transformations, even if completely

reversible, are not desirable.

4. The material should be usable over as broad

a temperature range as possible, preferably

in air, inert gas, or vacuum.

5. The material should be substantially opaque

to thermal radiation over the entire temperature range of interest.

6. The material should not react chemically

with materials commonly used in constructing thermal conductivity apparatus, especially

not with thermocouples.

7. The material should have suitable mechanical properties to permit reasonably economic

specimen fabrication while still possessing

sufficient strength and hardness to permit

normal handling during installation and re- moval from an apparatus.

8. The cost should not be excessive. "Excessive,"

however, appears to be defined rather differently by various individuals. Usually the cost

of the material is much less than the cost of

the measurement.

9. All other things being equal, an electrical

insulator offers a slight advantage over an

electrical conductor in that thermocouple

leads do not need to be insulated from the

specimen.

8.1. Current Status

With regard to the four categories of uniformity

listed above under item 2, a few comments are in

order. There are a few hopefully reproducible materials, for which thermal conductivity is reason- ably well known over a limited temperature range,

that can be used as reference materials. We will

discuss these shortly. As regards stocked materials,

I know of no source where a stock of material of

known thermal conductivity is maintained and

samples are sold. Efforts to make such materials

available are currently underway at the National

Bureau of Standards.

Powell [469] has summarized the correlations

which exist between thermal and electrical

conductivity for a number of alloy systems.

For a number of copper alloys having thermal

conductivities in the range 50 to 350 Wm_1 K_1

, it appears that thermal conductivity can be

predicted to within a few percent from measured

electrical conductivity values, at least in the

temperature range from room temperature to

about 500 K. Other systems that exhibit relatively good correlation include aluminum alloys,

magnesium alloys, certain steels, and certain

nickel-chromium alloys. In as-yet-unpublished

work, we have developed a somewhat more

complicated correlation expression for ferritic

iron-nickel alloys that seems to be good within

three percent from 200 to 800 K.

One approach to obtaining thermal conductivity

reference materials is to have a particular sample

calibrated by an experienced, reliable laboratory.

We have carried out a number of such ad hoc

measurements in the past and can continue to

do so whenever the workload permits.

Returning to the category of reproducible

materials, you will find listed below a number of

materials that may be used as reference standards

on an interim basis. The materials are listed

in two categories, metals and nonmetals, in

decreasing order of room temperature thermal

conductivity within each category. Unless otherwise stated, materials of at least 99.99 percent

purity are implied.

a. Metals

Copper (X=400 Wm-'K"1 at 300 K)

Powell, Ho, and Liley [238] give recommended

values for the thermal conductivity of pure

copper. They estimate a probable uncertainty

in these values of "±3 percent near room tem109
perature increasing to ±5 percent at low and

high temperatures."

Their recommended values are in excellent

agreement with the very recent experimental

data of Laubitz [71] and Moore, McElroy, and

Graves [72].

Aluminum (A = 240 Wm-'K"1 at 300 K)

Powell, Ho, and Liley [238] give recommended

values for the thermal conductivity of pure

aluminum which "are thought to be accurate

to within ±4 percent below room temperature

and ±2 percent to ±3 percent above." I feel

that this uncertainty estimate may be too

optimistic.

Tungsten (A = 180 Wm-'K"1 at 300 K)

Powell, Ho, and Liley [238] state that while

their recommended values for pure tungsten

"are thought to be within ±3 to ±5 percent of

the true values, further confirmatory work on

tungsten still seems to be required." As will be

briefly discussed at the end of this section,

further work on tungsten is currently in progress.

Powell and Ho [470] show some additional data

that were not given in [238].

Iron (A = 70 to 80 Wm-'K"1 at 300 K)

Commercially pure iron, usually produced by

the American Rolling Mill Company and known

as Armco iron, has been used for many years as a thermal conductivity reference material. The rec- ommended values for Armco iron by Powell, Ho,

and Liley [238] "are thought to have an accuracy

of ±3 percent below room temperature, ±2 percent to about 1000 K, the uncertainty probably

increasing to about ±8 percent at 1600 K". C. F.

Lucks, of the Battelle Memorial Institute Columbus Laboratories, has been investigating the effects

of heat treatment on the conductivity of Armco

iron and should be publishing his findings in the near future. In principle, pure iron would be more

reproducible and probably less subject to heat

treatment effects and hence would be a preferable

standard. Powell, Ho, and Liley have also given

recommended values for pure iron which they feel are similar in reliability to the values for Armco

iron (see also Powell and Tye [476]).

Platinum (X=70 Wm-'K'1 at 300 K)

The "very tentative" values given by Powell,

Ho, and Liley [238] for pure platinum should not

be used. Recent measurements by Laubitz and

van der Meer [471], Martin Sidles and Danielson

[472] _ and Flynn and O'Hagan [243] have led to a

revision of the recommended values (see Powell

and Ho [470, figure 16]). Until additional measurements on platinum, currently in progress at the

National Bureau of Standards and elsewhere, are completed, I suggest using the averages of the

values reported by Laubitz and van der Meer

[471] and by Flynn and O'Hagan [243].

Lead (A = 35 Wm-'K"1 at 300 K)

The use of lead as a thermal conductivity ref- erence standard at the National Bureau of Standards [473, 474] has led to its use in a number of

other investigations. Ho, Powell, and Liley [239]

give recommended values which "are thought to

be accurate to within ±3 percent of the true

values at moderate temperatures ±5 percent at

high temperatures and ± 10 percent at low temperatures . . .". The recent work of Lucks [475]

indicates that the thermal conductivity of lead

may be very sensitive to small amounts of impurity.

Stainless Steel (A = 15 Wm-'K"1 at 300 K)

Powell and Tye [476] have recently reported

thermal conductivity values for nine samples of

stainless steel of somewhat similar composition

(nominally 18 %_ Cr, 8% Ni). The differences

found among their samples indicate that the thermal conductivity of an arbitrary sample of 18/8

stainless steel would probably be within ±10

percent of their mean value at a given temperature.

Inconel 702 (\ = 12 Wm-'K"1 at 300 K)

Measurements of the thermal conductivity of

specimens of age-hardened Inconel 702 18 from a

single lot of material reported by Laubitz and

Cotnam [477], Watson and Flynn [478], and

McElroy and Moore [467], as well as other unpublished measurements at the National Bureau

of Standards, all agree within ±2 percent of a mean curve through these data. As far as I know,

there have been no measurements on samples of

this alloy from another lot of material. As discussed

by Laubitz and Cotnam [477] and Powell and Tye

[476], Inconel 702 is subject to changes in conductivity resulting from heat treatment so care is indicated in using this material.

b. Nonmetals

Beryllium Oxide (A =270 Wm-'K-1 at 300 K)

Powell, Ho, and Liley [238] give recommended

thermal conductivity values for 99.5 percent pure,

98 percent dense polycrystalline beryllium oxide.

The uncertainty of the recommended values "is

though to be within ±8 percent at temperatures

from 500 to 1000 K and increases to ±15 percent

below 300 K and above 1800 K".

Silicon (A — 150 Wm-'K"1 at 300 K)

Silicon is potentially a good reference material

due to its availability in large sizes of high purity.

Ho and Powell [266] show a plot of a recommended

curve for pure silicon. They do not tabulate their

recommended values but do state that the proposed curve was drawn intermediately between

the data of Glassbrenner and Slack [479] and of

Fulkerson, et al. [480].

18 A proprietory alloy of the International Nickel Company; the material used in these measurements was 79.3 percent Ni, 17 percent Cr, 2.5 percent Al, plus other elements in smaller quantities.

110
Magnesium Oxide (X = 50 Wm-'K"1 at 300 K)

Powell, Ho, and Liley [238] give recommended

values for 99.5 percent pure, 98 percent dense

polycrystalline magnesium oxide. The uncertainty

in these values "is though to be within ±8 percent

at temperatures from 500 to 1000 K and increases

to ±15 percent below 300 K and from 1650 to

2000 K. The uncertainty increases to about ±20

percent above 2100 K".

Aluminum Oxide (X=40 Wm-'K"1 at 300 K)

The uncertainty in the recommended values of

Powell, Ho, and Liley [238] for 99.5 percent pure,

98 percent dense polycrystalline aluminum oxide

"should be within ±8 percent at temperatures

from 500 to 1000 K and increases to about ±15

percent below 250 K and above 1800 K."

Crystalline Quartz (X = 10 Wm-'K-1 parallel to c- axis and X = 6 Wm-'K"' perpendicular to c-axis at 300 K)

Powell, Ho, and Liley [238] give recommended

values for the thermal conductivity of quartz

single crystals for heat flow both parallel and

perpendicular to the c-axis. The uncertainty for

both sets of values "is thought to be within ±5

percent at temperatures from 300 to 500 K and

increases at lower and higher temperatures up to

± 10 percent at 40 and 800 K". Because of infrared transmission of energy, I would not recommend

using clear quartz as a reference material above

about 600 K.

Pyroceram Code 9606 (X=4 Wm-'K"' at 300 K)

Powell, Ho, and Liley [238] state that their

recommended values for Pyroceram Code 9606 19 "should be accurate to within ± 5 percent at

temperatures from 200 to 1000 K. The uncertainty increases to ± 10 percent at 100 K and

1400 K". Of the six sets of data used in deriving

these recommended values, two sets were produced by direct measurement of thermal conductivity in two different apparatuses in our

laboratory; the other four sets were derived from

thermal diffusivity data obtained in three other

laboratories. Two of these sets of diffusivity data

were obtained in another laboratory at the National Bureau of Standards on material from the

same lot of material as was used in our conductivity measurements. Thus at least four of the

six sets of data were obtained on the same lot

of material. In more recent measurements, using

a third, and different, apparatus,20 we have

obtained thermal conductivity values on a specimen of Pyroceram Code 9606 from a different

lot of material; these values, obtained over the

temperature range 100 to 300 K, are about 5

percent higher than our previous data which

were used as a basis for the recommended values

given in [238]. We are not certain whether the

w A product of Corning Glass Works; the material is a "microcrystalline

glass."

20 The apparatus shown in figure 47.

differences observed are due to experimental difficulties or to sample differences but believe the

differences do reflect a variation between samples.

Further work is clearly indicated.

Fused Quartz (X-1.4 Wm-'K-' at 300 K)

Powell, Ho, and Liley [238] give recommended

thermal conductivity values for high-purity clear

fused quartz; the uncertainty "should be within

±4 percent at temperatures from 200 to 500 K,

and increases to ±8 percent at 50 K and 900 K

and up to ±15 percent below 10 K and at 1400

K". Carwile and Hoge [481] present independently

derived recommended values for the thermal conductivity of "vitreous silica" over the temperature

range 80 to 660 K. They state that "the tabulated

values of X between 150 and 450 K are believed

to be accurate to ±7 percent; near room temperature they may be somewhat better. Below

150 K the uncertainty increases and may reach

±12 percent. Likewise above 450 K the uncertainty increases and may be as large as ±15

percent at the upper limit of the table". These

two sets of recommended -values agree to within

less than 1 percent from 200 to 600 K. I would

not recommend using clear fused quartz as a thermal conductivity reference standard at temperatures above 600 K.

Pyrex Glass Code 7740 (X = l.l Wm-'K-' at

300 K)

The recommended thermal conductivity values

given by Powell, Ho, and Liley [238] for Pyrex

Glass Code 7740 21 are "thought to be accurate

to within ±5 percent at moderate temperatures

and within ± 10 percent below 200 K and above

700 K". Again, I would not recommend this as a standard above about 600 K.

8.2. Ongoing Activities

In order to make better thermal conductivity

reference materials available, the National Bureau

of Standards has efforts underway both in our Washington and in our Boulder laboratories. In

order that these efforts may be properly directed,

you are urged to communicate your needs in this area to R. L. Powell at Boulder (cryogenic temperature range—liquid helium to ambient temperatures) or to me at Washington (liquid nitrogen

to high temperatures).

The Air Force Materials Laboratory is sponsoring a program at Arthur D. Little, Inc. to establish

high-temperature ( above about 1000° C) thermal

conductivity and thermal diffusivity standards.

Part of this work has already been described

[468, 482] ; most, if not all, of the work will be

included in the Proceedings of the Eighth Conference on Thermal Conductivity held at Purdue

University in October, 1968. The materials being

investigated are polycrystalline sintered aluminum

2 ' A borosilicate glass produced by Corning Glass Works.

Ill
oxide and thorium oxide, arc cast tungsten, and

two commercial grades of graphite.

In a program under the auspices of AGARD

(Advisory Group for Aerospace Research and

Development of the North Atlantic Treaty Organization), samples of gold, platinum, tungsten,

and aluminum oxide are being distributed to a number of laboratories for the purpose of measuring thermal conductivity and ultimately establishing these materials as reference standards.

Several laboratories, notably the groups under

M. J. Laubitz at the National Research Council

of Canada and under D. L. McElroy at Oak Ridge

National Laboratory, have active programs to

study the thermal conductivities of well-defined

materials over broad temperature ranges. Measurements such as these will contribute greatly to

establishing accurate thermal conductivity values

on reproducible materials that are suitable as reference standards.

9. Conclusions

Many inaccurate thermal conductivity data

have been reported on poorly characterized materials—such data are almost useless and, if used in

critical applications, may even be harmful. Inaccurate data on well-characterized materials are not much, if any, more useful. Accurate data on

poorly characterized materials may be very useful

in applications where the identical material is used but are not of much use otherwise. Accurate

thermal conductivity data on well-characterized

samples, the ultimate desiderata, are characterized

by their paucity.

Admittedly, thermal conductivity is a relatively

difficult physical property to determine. Accurate

thermal conductivity values are expensive to

obtain—but not much, if any, more expensive than

inaccurate values. And when one considers the

possible costs in fuel, efficiency, or safety due to

the use of inaccurate thermal conductivity values

in designing a system, one is led to the conclusion

that the overall cost of an inaccurate measurement is usually larger than the overall cost would

have been if a little more effort had been expended

to obtain an accurate value. If thermal conductivity is worth measuring, it is worth measuring

well.

Similar arguments hold true regarding sample

characterization. Adequate identification of a given sample may be expensive, but if everyone

did properly characterize his samples, it would

eventually be possible, in many cases, to correlate

thermal conductivity with density, pore size,

impurity content, etc. Such correlations could

eliminate the need for many thermal conductivity

measurements at a later date. A seemingly unnecessary investment in sample characterization

might thus yield future dividends.

In selecting a suitable apparatus for use in

measuring thermal conductivity, there are a

variety of factors, such as those listed at the

beginning of section 7, which need be considered.

I strongly urge that anyone contemplating the

construction of a thermal conductivity apparatus

study the book "Thermal Conductivity" [483]

prior to committing himself to a given type of

apparatus. The thirteen chapters in this book,

written by some of the top workers in the field,

contain a veritable wealth of useful information.

A comprehensive error analysis (particularly helpful references include [243, 447, 450, 467]) prior

to construction of an apparatus can save many

difficulties later.

Once an apparatus has been constructed, it is absolutely essential to undertake measurements on

a material, such as one of those listed in the

previous section, whose thermal conductivity is well known. Ideally the material selected should

be as similar as possible, in thermal conductivity

and other physical properties, to the material

or type of material for which the apparatus was

designed.

Prior to undertaking a measurement of the

thermal conductivity of a particular material it is important to consider (and, if any doubts exist,

check) whether any chemical reactions or material

changes might occur during the measurement.

Many specimens could contaminate thermocouples

or react with apparatus components, especially at

elevated temperatures. Before fabricating specimens for conductivity measurements, the sample

should be checked for homogeneity to be sure representative specimens are tested.

The material should be characterized as completely as possible. This would include finding out

the method and details of material production and

any heat treatments or other special treatments

given to the material. Minimal characterization

should include a statement of the principal

chemical composition, the general type of microstructure (single-crystal, crystalline orientation,

polycrystalline, amorphous, etc.), the density, and,

whenever possible, a detailed chemical analysis.

Materials having noncubic structure should be

checked for anisotropy. A detailed microstructural

analysis should be made to determine grain size,

porosity, presence of other phases, and other

factors which conceivably could affect the thermal

conductivity (see Picklesimer [484]).

If there is any possibility of an electronic contribution to the thermal conductivity, the electrical

resistivity should be measured over the temperature range of interest. If there is any possibility of

significant radiative heat transfer, it would be

helpful to measure optical absorption and scattering coefficients, even if this can only be done at

room temperature. Other thermal and mechanical

properties (e.g., thermal expansion, thermoelectric

power, specific heat, sound velocity) can be very

helpful in understanding the results of thermal

conductivity measurements, particularly if a phase

change or other unusual behavior occurs. In carrying out high temperature determinations, it is good practice to make measure112
merits at 50 or 100 deg intervals in increasing order

of temperature with an adequate number of repeat

measurements at lower temperatures to check for

changes in the specimen or in some component of

the apparatus. If there is any question of change

in the specimen, post-test characterization is indicated.

During testing, especially with new apparatus,

it is important to make systematic variations in

the test conditions (temperature gradients, guard

temperatures, type of gas in insulation surrounding

specimen, etc.) to enable a more meaningful error

analysis. All test conditions which conceivably

could affect the results should be recorded.

A detailed error analysis [243, 467] is essential

in order to arrive at meaningful estimates of the

uncertainties in thermal conductivity values; the

estimated uncertainties should be stated clearly

and unambiguously [485-486].

10. References

[1] Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, 2nd ed. (Clarendon Press, Oxford,

1959).

[2] Schneider, P. J., Conduction Heat Transfer (Addison-Wesley, Reading, Mass., 1955).

[3] Arpaci, V. S., Conduction Heat Transfer (AddisonWesley, Reading, Mass., 1966).

[4] Seitz, F., Modern Theory of Solids (McGraw-Hill

Book Co., Inc., New York, 1940).

[5] Peierls, R., Quantum Theory of Solids (Clarendon

Press, Oxford, 1955).

[6] Wannier, G. H., Elements of Solid State Theory

(Cambridge University Press, Cambridge, 1959).

[7] Ziman, J. M, Electrons and Phonons (Clarendon

Press, Oxford, 1960).

[8] Rosenberg, H. M., Low Temperature Solid State

Physics (Clarendon Press, Oxford, 1963).

[9] Klemens, P. G., Thermal conductivity of solids at low temperatures, Handbuch der Physik 14, 198

(1956).

[10] Klemens, P. G., Thermal conductivity and lattice vibrational modes, Solid State Phys. 7, 1 (1958).

[11] Klemens, P. G., Theory of the thermal conductivity

of solids, ch. 1 in Thermal Conductivity, edited by

R. P. Tye (Academic Press, London, in press).

[12] Appel, J., Thermal conductivity of semiconductors,

Progress in Semiconductors 5, 143 (1960).

[13] Carruthers, P., Theory of thermal conductivity of

solids at low temperatures, Rev. Mod. Phys. 33,

92 (1961).

[14] Drabble, J. R., and Goldsmid, H. J., Thermal

Conduction in Semiconductors (Pergamon Press,

London, 1961).

[15] Krumhansl, J. A., Thermal conductivity of solids,

in Theory and Fundamental Research in Heat

Transfer, edited by J. A. Clark (The Macmillan

Co., New York, 1963).

[16] Brillouin, L., Wave Propagration in Periodic Structures (McGraw-Hill Book Co., New York, 1946;

reprinted with corrections and additions by Dover

Publications, New York, 1953).

[17] Louck, J. D., Exact normal modes of oscillation of a

linear chain of identical particles, Amer. J. Phys.

30, 585 (1962).

[18] Born, M., Lattice dynamics and x-ray scattering, Proc. Phys. Soc. (London) 54, 362 (1942).

[19] Born, M., Theoretical investigation on the relation

between crystal dynamics and x-ray scattering,

Rept. Progr. Phys. 9, 294 (1942).

[20] Pines, D., Elementary Excitations in Solids (W. A. Benjamin, Inc., New York, 1963).

[21] Loudon, R., Excitations in crystals, Sci. Prog.

(Oxf.) 54, 325 (1966).

[22] Clem, J. R., and Godwin, R. P., Dynamical properties

of a one-dimensional "crystal" with free ends,

Amer. J. Phys. 34, 460 (1966).

[23] Blackman, M., The specific heat of solids, Handbuch

der Physik 1/1, 325 (1955).

[24] de Launay, J., The theory of specific heats and

lattice vibrations, Solid State Phys. 2, 219 (1956).

[25] Keesom, P. H., and Pearlman, N., Low temperature

heat capacity of solids, Handbuch der Physik 14,

282 (1956).

[26] Leibfried, G., and Ludwig, W., Theory of anharmonic effects in crystals, Solid State Phys. 12,

276 (1961).

[27[ Mitra, S. S., Vibration spectra of solids, Solid State

Phys., 13, 2 (1962).

[28] Cochran, W., and Cowley, R. A., Phonons in perfect

crystals, Handbuch der Physik XXV/2a, 59 (1967).

[29] Leibfried, G., and Schlomann, E., Heat conduction

in electrically insulating crystals, Nachr. Akad.

Wiss. Gottingen Math-physik Kl. 2a, 71 (1954).

[30] Pieirls, R., Zur kinetischen Theorie der Warmeleitung in Kristallen, Ann. Phys. 3, 1055 (1929).

[31] de Haas, W. J., and Biermasz, Th., Die Warmeleitfahigkeit von Kristallen bei tiefen Temperaturen,

Physica 5, 320 (1938).

[32] Casimir, H. B. G., Note on the conduction of heat in

crystals, Physica 5, 495 (1938).

[33] Knudsen, M., Die Gesetze der Molekularstromung

und der Reibungsstromung der Gase durch Rohren, Ann. Phys. 28, 75 (1909).

[34] Smoluchowski, M., Zur kinetischen Theorie der

Transpiration und Diffusion verdiinnter Gase,

Ann. Phys. 33, 1559 (1910).

[35] Berman, R., Simon, F. E., and Ziman, J. M., The

thermal conductivity of diamond at low temperatures, Proc. Roy. Soc. A220, 171 (1953).

[36] Berman, R., Foster, E. L., and Ziman, J. M., Thermal conduction in artificial sapphire crystals at low temperatures I. Nearly perfect crystals, Proc. Roy. Soc. A231, 130 (1955).

[37] Herring, C, Theory of the thermoelectric power of semiconductors, Phys. Rev. 96, 1163 (1954).

[38] Geballe, T. H., and Hull, G. W., The size dependence

of thermal conductivity and Seebeck effect in p- tvpe germanium, Conf. de Phys. des basses Temps.,

p. 460, (Paris, 1955).

[39] Thacher, P. D., Effect of boundaries and isotopes on

thermal conductivity of LiF, Phys. Rev. 156, 975

(1967).

[40] Gereth, R., and Hubner, K., Investigations of surface

reflection of relevant phonons in silicon using the

transmitted phonon drag effect, Solid State Electronics 10, 935 (1967).

[41] Kazakov, A. E., and Nagaev, E. L., Consideration of partial phonon reflection from boundaries in the

calculation of the low temperature thermal conductivity of dielectrics, Soviet Phys.-Solid State

9, 1255 (1967).

[42] Guyer, R. A., and Krumhansl, J. A., Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals, Phys.

Rev. 148, 778 (1966).

[43] Hurst, W. S., and Frankl, D .R., Thermal conductivity of silicon at low temperatures, Bull. Am.

Phys. Soc. (II) 13, 170 (1968).

[44] Callaway, J., Model for lattice thermal conductivity

at low temperatures, Phys. Rev. 113, 1046 (1959).

[451 Callaway, J., Low-temperature lattice thermal conductivity, Phys. Rev. 122, 787 (1961).

[46] Carruthers, P., Thermal conductivity of solids. III. Modification of three-phonon processes by isotopic

scattering, Phys. Rev. 126, 1448 (1962).

[47] Nettleton, R. E., Foundations of the Callaway

theory of thermal conductivity, Phys. Rev. 132.

2032 (1963).

113
[48] Berman, R., The thermal conductivity of dielectric

solids at low temperatures, Adv. Phys. 2, 103

(1953).

[49] Klemens, P. G., The thermal conductivity of dielectric solids at low temperatures (Theoretical), Proc. Roy. Soc. (London) A208, 108 (1951).

[50] Ziman, J. M., The general variational principle of

transport theory, Can. J. Phys. 34, 1256 (1956).

[51] Krumhansl, J. A., Thermal conductivity of insulating crystals in the presence of normal processes, Proc. Phys. Soc. (London) 85, 921 (1965).

[52] Guyer, R. A., and Krumhansl, J. A., Solution of the

linearized phonon Boltzmann equation, Phys.

Rev. 148, 766 (1966).

[53] Holland, M. G., Analysis of lattice thermal conductivity, Phys. Rev. 132, 2461 (1963).

[54] Klemens, P. G., Thermal resistance due to point

defects at high temperatures, Phys. Rev. 119, 507

(1960).

[55] Greenberg, I. N., Thermal conductivity in solids,

USAELRDL Technical Report 2361 (May, 1963),

AD 422664.

[56] Klein, M. V., Phonon scattering by lattice defects

II, Phys. Rev. 141, 716 (1966).

[57] Berman, R., Heat conduction in non-metallic crystals, Sci. Prog. (Oxford) 55, 357 (1967).

[58] Ranniger, J., Thermal conductivity in nonconducting crystals, Ann. Phys. (N. Y.) 45, 452 (1967).

[59] Tavernier, J., General problems of lattice thermal

conductivity in solids (in French), J. Phys. 28

CI, Supp. 2, 40 (1967).

[60] Sham, L. J., and Ziman, J. M., The electron-phonon

interaction, Solid State Phys. 15, 223 (1963).

[61] Spector, H. N., Interaction of acoustic waves and

conduction electrons, Solid State Phys. 19, 291

(1966).

[62] Makinson, R. E. B., The thermal conductivity of

metals, Proc. Cambridge Phil. Soc. 34, 474 (1938).

[63] Sondheimer, E.H.,The theory of the transport phenomena in metals, Proc. Roy. Soc. (London) A203,

75 (1950).

[64] Ziman, J. M., The electrical and thermal conductivities of monovalent metals, Proc. Roy. Soc. (London) A226, 436 (1954).

[65] Cezairliyan. A., Prediction of thermal conductivity

of metallic elements and their dilute alloys at cryogenic temperatures, Purdue University,

TPRC Rept. 14 (1962).

[66] Cezairliyan, A., and Touloukian, Y. S., Generalization and calculation of the thermal conductivity

of metals by means of the law of corresponding

states, High Temperature 3, 63 (1965).

[67] Cezairliyan, A., and Touloukian, Y. S., Correlation

and prediction of thermal conductivity of metals

through the application of the principle of cor- responding states, in Advances in Thermophysical Properties at Extreme Temperatures and

Pressures, p. 301 (ASME, 1965).

[68] Berman, R., and Macdonald, D. K. C, The thermal

and electrical conductivity of copper at low temperatures, Proc. Roy. Soc. (London) A211, 122

(1952).

[69] White, G. K., The thermal and electrical conductivity of copper at low temperatures, Australian J. Phys. 6, 397 (1953).

[70] Powell, R. L., Roder, H. M., and Hall, W. J., Low

temperature transport properties of copper and

its -dilute alloys: Pure copper, annealed and colddrawn, Phys. Rev. 115, 314 (1959).

[71] Laubitz, M. J., Transport properties of pure metals

at high temperatures I. Copper, Can. J. Phys. 45,

3677 (1967).

[72] Moore, J. P., McElroy, D. L., and Graves, R. S., Thermal conductivity and electrical resistivity

of high purity copper from 78 to 400 °K, Can. J. Phys. 45, 3849 (1967).

[73] Mendelssohn, K., and Rosenberg, H. M., The

thermal conductivity of metals at low temperatures, Solid State Phys. 12, 223 (1961).

[74] Mott, N. F., and Jones, H., The Theory of the

Properties of Metals and Alloys (Clarendon

Press, 1936; reprinted by Dover Publications,

New York, 1958).

[75] Wilson, A. H., The Theory of Metals, 2nd. ed.

(Cambridge Univ. Press, Cambridge, England,

1965).

[76] Olsen, J. L., Electron Transport in Metals (Interscience (Wiley), 1962).

[77] Schleef, D. L., Review of modern theory of the

thermal conductivity of metals, in Thermodynamic and Transport Properties of Gases, Liquids

and Solids (ASME, 1959).

[78] Austin, J. B., The Flow of Heat in Metals (Amer.

Soc. Metals, 1942).

[79] Greenberg, I. N., Heat conduction in solids, Technical Report ECOM-2799 (1967), AD651229.

[80] Steigmeier, E. F., Thermal conductivity of semiconducting materials, ch. 10 in Thermal Conductivity, edited by R. P. Tye (Academic Press,

London, in press).

[81] Madelung, O., Halbleiter, Handbuch der Physik 20,

1 (1957).

[82] Planck, M., The Theory of Heat Radiation (Dover,

1959).

[83] Harrison, T. R., Radiation Pyrometry and Its Underlying Principles of Radiant Heat Transfer

(John Wiley & Sons, Inc., New York, 1960)

.

[84] Sparrow, E. M., and Cess, R. D., Radiation Heat

Transfer (Brooks/Cole, 1966).

[85] Bellman, R., Kagiwada, H., Kalaba, R., and

Ueno, S., Inverse problems in radiative transfer:

layered media, Icarus 4, 119 (1965).

[86] Cess, R. D., The interaction of thermal radiation

with conduction and convection heat transfer, in

Advances in Heat Transfer, Vol. 1, p. 1 (Academic

Press, Inc., New York, 1964).

[87] Chen, J. C, Simultaneous radiative & convective

heat transfer in an absorbing emitting, & scattering

medium in slug flow between parallel plates, A. I. Ch. E. Journal 10, 253 (1964).

[88] Chu, C. M., and Churchill, S. W., Numerical solution of problems in multiple scattering of electromagnetic radiation, J. Phys. Chem. 59, 855 (1955).

[89] Costello, F. A., and Schrenk, G. L., Numerical solution to the heat-transfer equations with combined

conduction and radiation, J. Comput. Phys. 1,

541 (1967).

[90] Edwards, R. H., and Bobco, R. P., Radiant heat

transfer from isothermal dispersions with isotropic

scattering, J. Heat Transfer 89C, 300 (1967).

[91] Einstein, T. H., Radiant heat transfer with flow and

conduction, NASA TR R-154 (1963).

[92] Engelmann, F., and Schmidt, H. E., Heat transport

by radiation in solids, Nuclear Sci. Engng. 24,

317 (1966).

[93] Goryainov, L. A., Methods of separation of composite heat transfer, Trudy. Mosk. In-ta Inzhen.

Zh.—D. Transporta (189), 59 (1965).

[94] Hottel, H. C, Sarofim, A. F., Evans, L. B., and Vasalos

I. A., Radiative transfer in anisotropically scat- tering media: allowance for Fresnel reflection at the boundaries, J. Heat Transfer 90C, 56 (1968).

[95] Hsia, H. M., and Love, T. J., Radiative heat transfer between parallel plates separated by a nonisothermal medium with anisotropic scattering, J. Heat Transfer 89C, 197 (1967).

[96] Landram, C. S., and Greif, R., Semi-isotropic model

for radiation heat transfer, AIAA J. 5, 1971 (1967).

[97] Leung, A. T., Simultaneous radiation, conduction,

and convection in a spectrally selective, absorbing,

emitting, and scattering porous bed, M. S. Thesis

in Engineering, University of California, Los Angeles, California, June 1965.

[98] Leung, A. T., and Edwards, D. K., Simultaneous

radiation, conduction, and convection in a spectrally selective, emitting and scattering porous

bed, J. Heat Transfer 88, 231 (1966).

114
[99] Lick, W., Energy transfer by radiation and conduction, Proc. 1963 Heat Transfer and Fluid Mechanics Institute, p. 14 (Stanford Univ. Press,

Palo Alto, California, 1963).

[100] Lick, W., Transient energy transfer by conduction

and radiation, Intern. J. Heat Mass Transfer 8, 119 (1965).

[101] Love, T. J., An investigation of radiation heat transfer in absorbing, emitting, and scattering media, Ph.D. thesis (Purdue University, 1963) ; also ARL63-3, Aeronautical Research Laboratories Report,

Office of Aerospace Research, U.S.A.F.

[102] Love, T. J., and Grosh, R. J., Radiative heat transfer in absorbing, emitting, and scattering media,

J. Heat Transfer C87, 161 (1965).

[103] Mori, Y., and Kurosaki, Y., Heat transport by

radiation and other transport mechanisms (1st

Rep., Couette flow with simultaneous radiation

and conduction), Bull. Jap. SME 9, 560 (1966).

[104] Nixon, J. A., and Monroe, S. G., Heat transfer

through diathermanous materials, J. Spacecraft

Rockets 2, 907 (1965).

[105] Oppenheimer, A. K., and Bevans, J. T., Geometric

factors for radiative heat transfer through an

absorbing medium in Cartesian coordinates, J. Heat Transfer C82, 360 (1960).

[106] Sidorov, Yu. P., Heat transfer by heat conduction

and radiation in an absorbing medium, Trudy

Mosk. In-ta Inzhen. Zh.D. Transporta (189),

109 (1965).

'

[107] Surinov, Yu. A., The integral equations of the theory

of radiation transfer in an anisotropically scatter- ing medium (for the generalized formulation of

the three-dimensional problem), High Temperature 3, 385 (1965).

[108] Surinov, Yu. A., Integral equation of radiation trans- port theory for an absorbing and anisotropically

scattering medium, High Temperature 5, 293

(1967).

[109] Tien, C. L., and Abu-Romia, M. M., Perturbation

solutions in the differential analysis of radiation

interactions with conduction and convection

AIAA J. 4, 732(1966).

[110] Viskanta, R., Heat transfer in thermal radiation

absorbing and scattering media, Ph.D. thesis

(Purdue University, 1960); also ANL-6170.

[Ill] Viskanta, R., Interaction of heat transfer by conduction, convection, and radiation in a radiating

fluid, J. Heat Transfer C85, 318 (1963).

[112] Viskanta, R., Heat transfer by conduction and

radiation in absorbing and scattering materials,

J. Heat Transfer C87, 143 (1965).

[113] Viskanta, R., and Grosh, R. J., Heat transfer in a thermal radiation absorbing and scattering medium, International Heat Transfer Conference,

Boulder, Colorado, 1961, p. 820.

[114] Viskanta, R., and Grosh, R. J., Effect of surface

emissivity on heat transfer by simultaneous

conduction and radiation, Intern. J. Heat Mass

Transfer 5, 729 (1962).

[115] Wang, L. S., Differential methods for combined

radiation and conduction, Ph. D. thesis (Uniri1 , versity of California, Berkeley, 1965).

[116] Werner, M. M., and Edwards, D. K., Simultaneous

conduction, convection, and radiation in a porous

bed, Proceedings of the 1963 Heat Transfer and

Fluid Mechanics Institute, p. 236 (Stanford

M1 „ „7Univ- Press, Palo Alto, California, 1963.

U17J Wang, Zhu-xi, Theory of heat conduction in the

presence of radiation, Scientia Sinica 11, 185

(1962).

[118] American Institute of Physics Handbook, 2nd ed.,

edited by D. E. Gray (McGraw-Hill Book Co.,

Inc., New York, 1963).

U19] Gardon, R., A review of radiant heat transfer in nom Slass> J- Am- Ceram. Soc. 44, 305 (1961).

U^UJ Glaze, F. W., Transmittance of infrared energy by glasses, Bull. Amer. Ceram. Soc. 34, 291

(1955).

[121] Cleek, G. W., Villa, J. J., and Hahner, C. H.,

Refractive indices and transmittances of several

optical glasses in the infrared, J. Opt. Soc. Am.

49, 1090 (1959).

[122] Folweiler, R. C, Thermal radiation characteristics

of transparent, semi-transparent, and translucent

materials under non-isothermal conditions, ASDTDR-62-719 (April 1964).

[123] Folweiler, R. C, and Mallio, W. J., Thermal

radiation characteristics of transparent, semitransparent and translucent materials under

non-isothermal conditions, ASD-TDR-62-719,

Part II (June 1964).

[124] Hobbs, H. A., and Folweiler, R. C, Thermal

radiation characteristics of transparent, semitransparent and translucent materials under

non-isothermal conditions, ASD-TDR-62-719,

Part III (March 1966).

[125] Chapin, E. J., and Howe, D. G., An exploratory

study of radiant energy transport to BeO, Report

No. NRL-6195 (March 1965).

[126] All papers in J. Am. Ceram. Soc. 44, No. 7 (July

1961).

[127] Anonymous, The application of high temperature

radiative thermal conductivity of minerals and

rocks to a model of lunar volcanism, NASA

CR-916 (November 1967).

[128] Ambartsumyan, V. A., Theoretical Astrophysics

(Pergamon, 1958).

[129] Barnes, R. B., and Bonner, L. G., The Christiansen

filter effect in the infrared, Phys. Rev. 49, 732

(1936).

[130] Bosworth, R. C. L., Heat Transfer Phenomena

(John Wiley and Sons, Inc., New York, 1952).

[131] Busbridge, I. W., The Mathematics of Radiative

Transfer (Cambridge Univ. Press, London,

England, I960).

[132] Busch, G., Frohlich, C, Hullinger, F, and Steigmeier,

E., Struktur, elektrische und thermoelektrische

Eigenschaften von SnSe2, Helvetica Physica

Acta 34, 359 (1961).

[133] Chandrasekhar, S., On the radiation equilibrium of a stellar atmosphere, XXIV, Astrophys. J. 108,

92 (1948).

[134] Chandrasekhar, S., Radiative Transfer (Oxford

University Press, 1950; reprinted by Dover

Publications, New York, 1960).

[135] Chandrasekhar, S., Elbert, D., and Franklin, A., The X and Y-functions for isotropic scattering,

Parts I and II, Astrophys. J. 115, 244 (1952).

[136] Chen, J. C, and Churchill, S. W., Radiant heat

transfer in packed beds, Amer. Inst. Chem.

Eng. J. 9, 35 (1963).

[137] Chromey, F. C, Evaluation of Mie equations for colored spheres, J. Opt. Soc. Am. 50, 730 (I960).

[138] Clark, S. P., Jr., Absorption spectra of some silicates in the visible and near infrared, Amer. Mineralogist 42, 732 (1957).

[139] Clark, S. P., Jr., Radiative transfer in the earth's

mantle, Trans. Amer. Geophys. Union 38, 931

(1957).

140] Cline, D., and Kropschot, R. H., The thermal properties of powder insulators in the temperature

range 300 ° -4 °K, in Radiative Transfer from

Solid Materials, edited by H. Blau and H. Fischer,

p. 61 (The Macmillan Co., New York, 1962).

[141] Collins, R. J., and Fan, H. Y., Infrared lattice

absorption bands in germanium, silicon and diamond, Phys. Rev. 93, 674 (1954).

[142] Czerny, M., and Genzel, L., Energy flux and temperature distribution in the glass baths of melting

tanks resulting from conduction and radiation,

Glastech. Ber. 25, 387 (1952).

[143] Czerny, M., and Genzel, L., On the depth to which

diffuse radiation penetrates glass, Glastech. Ber.

25, 134 (1952).

[144] Czerny, M., Genzel, L., and Heilmann, G., On the

flow of radiation in the interior of glass baths,

Glastech. Ber. 28, 185 (1955).

115
Damkohler, G., Der Chemi-Ingenieur, Eucken-Jacob

Vol. Ill, Part 1, 445 (Akademische Verlaeregresellschaft MBH, Leipzig, Germany, 1937).

Davison, B., Neutron Transport Theory (Oxford

Univ. Press, Oxford, England, 1957).

Devyatkova, E. D., Ya, M. B., and Smirnov, I. A., Thermal conductivity of tellurium containing

different concentrations of additional elements in the temperature range 80-480 °K, Solid State

Phys., USSR 1, 555 (1959).

Eckhardt, G., Warmeleitung und Warmestrahlung

in Glasern zwischen 100 ° and 1200 °C, Glastech.

Ber. 32, 373 (1959).

Everest, A., Glaser, P. E., and Wechsler, A. E.,

Thermal conductivity of non-metallic materials,

Summary Report, Contract NAS8-1567, Arthur

D. Little, Inc., Cambridge, Mass. (April 1962).

Ewing, C. T., Spann, J. R., and Miller, R. R.,

Radiant transfer of heat in molten inorganic

compounds at high temperatures, J. Chem. Eng.

Data 7, 246 (1962).

Fritz, W., and Poltz, H., Absolutbestimmung der

Warmeleitfahigkeit von Flussigkeiten I - Kritische

Versuche an einer neuen Plattenapparatur, Int.

J. Heat Mass Transfer 5, 307 (1962).

Fritz-Schmidt, M., and Gehlhoff, G., Warmeleitfahigkeitsmessungen an feuerfesten Baustoffen,

Glastech. Berichte 8, 206 (1930).

Gardon, R., Emissivity of transparent materials, J. Am. Ceram. Soc. 39, 278 (1956).

Gardon, R., Calculation of temperature distribution

in glass plates undergoing heat-treatment, J. Am.

Ceram. Soc. 41, 200 (1958).

Geffcken, W., The transmission of heat in glass at high temperatures, Part I, Glastech. Ber. 25, 392

(1952) . Geffcken, W., Zur Forleitung der Warme in glas bei

hohen temperaturen, Glastech. Ber. 29, 42 (1956).

Geffcken, W., Radiale Strahlungsleitung in Langen

Zylindern, Glastech. Ber. 32K, VII 28 (1959).

Genzel, L., Measurement of the infrared absorption

of glass between 20 °C and 1360 °C, Glastech Br.

24, 55 (1951).

Genzel, L., Calculation of the radiation conductivity

of glasses, Glastech. Ber. 26, 69 (1953).

Genzel, L., The role of heat radiation in the heat

conduction process, Z. Physik 135, 177 (1953).

Genzel, L., and Neuroth, N., Alternating illumination

method for the measurement of infrared absorption at high temperatures, Z. Phys. 134, 127

(1953) . Godbee, H. W., Ph. D. thesis, Georgia Institute of

Technology (July, 1963).

Godbee, H. W., and Ziegler, W. T., Thermal conductivities of MgO, A12 03 , and Zr02 powders to 850 °C. I. Experimental, J. Appl. Phys. 37, 40

(1966).

Godbee, H. W., and Ziegler, W. T., Thermal conductivities of MgO, A12 03 , and Zr02 to 850 °C. II. Theoretical, J. Appl. Phys. 37, 56 (1966).

Grove, F. J., and Charnock, H., Thermal conductivity of molten glass, Glastech. Ber. 32K, VII 24

(1959).

Grove, F. J., and Jellyman, P. E., The infrared trans- mission in glass in the range from room temperature to 1400°, J. Soc. Glass Technol. 39, 3 (1955).

Gryvnak, D. A., and Burch, D. E., Optical and

infrared properties of A12 03 at elevated temperatures, J. Opt. Soc. Am. 55, 625 (1965).

Gumprecht, R. O., and Sliepcevich, C. M., Scattering

of light by large spherical particles, J. Phys.

Chem. 57, 90 (1953).

Hager, N. E., Jr, and Steere, R. C, Radiant heat

transfer in fibrous thermal insulations, J. Appl.

Phys. 38, 4663 (1967).

Hamaker, H. C, Radiation and heat conduction

in light-scattering material I. Reflection and

transmission, Philips Res. Rep. 2, 55 (1947).

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

Hamaker, H. C, Radiation and heat conduction

in light-scattering material II. General equations

including heat conduction, Philips Res. Rep. 2,

103 (1947).

Hamaker, H. C, Radiation and heat conduction

in light-scattering material III. Application of

the theory, Philips Res. Rep. 2, 112 (1947).

Hamaker, H. C, Radiation and heat conduction

in light-scattering material IV. Various extensions

and a generalized theory, Philips Res. Rep. 2,

420 (1947).

Henry, R. L., The transmission of powder films in the infra-red, J, Opt. Soc. Am. 38, 775 (1948).

Hill, F. B., and Wilhelm, R. H., Radiative and

conductive heat transfer in a quiescent gas-solid

bed of particles: Theory and experiment, A.I. Ch. E. Journal 5, 486 (1959).

Huhmann, I., The transport of energy in glass

melts, Part II, Huttentechnische Vereingung der

Deutschen Glassindustrie (1958).

Joff6, A., Properties of various semiconductors, J. Phys. Chem. Solids 8, 6 (1959).

Jones, T. T., The effect of thickness and temperature on heat transfer through foamed polymers, Proc. Seventh Conference on Thermal Conductivity,

edited by D. R. Flynn and B. A. Peavy, p. 737

(NBS Special Publication 302, 1968).

Kellet, B. S., Steady flow of heat through hot glass,

J. Opt. Soc. Am. 42, 339 (1952).

Kellet, B. S., Transmission of radiation through

glass in tank furnaces, J. Soc. Glass Technol.

36, 115 (1952); for correction see J. Soc. Glass

Technol. 37, 268 (1953).

Klein, J. D., Heat transfer, by radiation, in powders,

Ph.D. Thesis, Mass. Inst. Tech. (June 1960).

Kohler, M., Einfluss der Strahlung auf den Warmetransport durch eine Fliissigkeitsschicht, Z. Angew. Phys. 18, 356 (1965).

Kourganoff, V., Basic Methods in Transfer Problems

(Oxford University Press, Oxford, England,

1952).

Krager, C, and Eligehausen, H., Uber das Warmeleituermogen des einschmelzenden Glasgemengls,

Glastech. Ber. 32, 362 (1959).

Krumhansl, J. A., Thermal conductivity in semiconductors, J. Phys. Chem. Solids 8, 343 (1959).

Kubelka, P., and Munk, F., Ein Beitrag zer Optik

der Farbanstriche, Z. tech. Phys. 12, 593 (1931).

Larkin> B. K., and Churchill, S. W., Heat transfer

by radiation through porous insulations, Amer.

Inst. Chem. Eng. J. 5, 467 (1959).

Laubitz, M. J., Thermal conductivity of powders,

Can. J. Phys. 37, 798 (1959).

Lee, D. W., and Kingery, W. D., Radiation energy

transfer and thermal conductivity of ceramic

oxides, J. Am. Ceram. Soc. 43, 594 (1960).

Lucks, C. F., Matolich, J., and Van Velzor, J. A., Experimental measurement of thermal conductivities, specific heats and densities of metallic,

transparent, and protective materials, Part III,

U.S. Air Force Tech. Rept., No. 6145, Wright

Air Development Center (1954).

McConnell, R. K, Jr., McClaine, L. A., Lee, D. W.,

Aronson, J. R., and Allen, R. V., A model for

planetary igneous differentiation, Rev. Geophys.

5, 109 (1967).

McMahon, H. O., Thermal radiation from partially

transparent reflecting bodies, J. Opt. Soc. Am.

40, 376 (1950).

Milne, E. A., Radiative equilibrium in the outer

layers of a star, Monthly Notices Roy. Astron.

Soc. 81, 361 (1921).

Monroe, J. E., Jr., Infrared transmission and radiation conductivity of alumina from room temperature to 1800 C, in Study of Heat Transfer of

Ceramic Materials, Part 2, pp. 38-66 (Alfred

Univ., Jan. 1955 to April 1957).

116
[195] Neuroth, N., The effect of temperature on the

spectral absorption of glasses in the infrared,

Glastech. Ber. 25, 242 (1952).

[196] Poltz, H., Einfluss der Warmestrahlung auf die

Isolierwirkung poroser Dammschichten, Allg.

Warmetechnik 11, 64 (1962).

[197] Poltz, H., Die Warmeleitfahigkeit von Fliissigkeiten

II. Der Strahlungsanteil der effektiven Warmeleitfahigkeit, Int. J. Heat Mass Transfer 8, 515

(1965).

[198] Poltz, H., Die Warmeleitfahigkeit von Fliissigkeiten

III. Abhangigkeit der Warmeleitfahigkeit von der

Schichtdicke bei organischen Fliissigkeiten, Int.

J. Heat Mass Transfer 8, 609 (1965).

[199] Poltz, H., Some aspects concerning thermal conductivity data of liquids and proposals for new standard reference materials, Proc. Seventh Conf. on Thermal Conductivity, edited by D. R. Flynn and

B. A. Peavy, p. 47 (NBS Special Publication 302,

1968).

[200] Poltz, H., and Jugel, R., The thermal conductivity of liquids IV—Temperature dependence of thermal

conductivity, Int. J. Heat Mass Transfer 10, 1075

(1967).

[201] Reynolds, M. M., Brown, J. D., Fulk, M. M., Park,

O. E., and Curtis, G. W, Vacuum-powder insulation, in Advances in Cryogenic Engineering, Vol. 1, p. 216 (Plenum, 1954).

[202] Richards, P. I., Multiple isotropic scattering, Phys.

Rev. 100, 517 (1955).

[203] Rolinski, E. J., and Purcell, G. V., Investigation of radiation and conduction heat transfer in fibrous

high temperature insulation, AFML-TR-67-251

(December, 1967).

[204] Rosseland, S., Astrophysik auf Atom-Theoretischer

Grundlage (Julius Springer Verlag, Berlin, Germany, 1931).

[205] Rosseland, S., Theoretical Astrophysics (Oxford

Univ. Press, Oxford, England, 1936).

[206] Russell, H. W, Principles of heat flow in porous

insulators, J. Am. Ceram. Soc. 18, 1 (1935).

[207] Schotte, W., Thermal conductivity of packed beds, A. I. Ch. E. Journal 6, 63 (1960).

[208] Schuster, A., The influence of radiation on the trans- mission of heat, Phil. Mag. 5, 243 (1903).

[209] Schuster, A., Radiation through a foggy atmosphere, Astrophys. J. 21, 1 (1905).

[210] Slack, G. A., and Glassbrenner, C., Thermal conductivity of germanium from 3 °K to 1020 °K,

Phys. Rev. 120, 782 (1960).

[211] Sobelev, V. V., A Treatise on Radiative Transfer

(D. Van Nostrand Co., New York, 1963).

[212] Spitzer, W. G., and Fan, H.Y., Infrared absorption

in indium antimonide, Phys. Rev. 99, 1893 ( 1955)

.

[213] Strong, H. M., Bundy, F. P., and Bovenkirk, H. P., Flat panel vacuum thermal insulation, J. Appl.

Phys. 31, 39 (1960).

[214] Van De Hulst, H. C, Light Scattering by Small

Particles (John Wiley & Sons, Inc., New York,

1957).

[215] Van De Hulst, H. C, Scattering in a planetary

atmosphere, Astrophys. J. 107, 220 (1948).

[216] Van der Held, E. F. M., The contribution of radiation to the conduction of heat, Appl. Sci. Res. A3,

237 (1952).

[217] Van der Held, E. F. M., The contribution of radiation to the conduction of heat II. Boundary conditions, Appl. Sci. Res. A4, 77 (1953).

[218] Van Zee, A. F., and Babcock, C. L., A method for the measurement of thermal diffusivity of molten

glass, J. Am. Ceram. Soc. 34, 244 (1951).

[219] Verschoor, J. D., and Greebler, P., Heat transfer by

gas conduction and radiation in fibrous insulations,

Trans. ASME 74, 961 (1952).

[220] Walther, A., Door, J., and Eller, E., Mathematical

calculation of the temperature distribution in a

glass melt due to conduction and radiation,

Glastech. Ber. 26, 133 (1953).

[221] Wechsler, A. E., and Glaser, P. E., Investigation of

the thermal properties of high-temperature insulation materials, ASD-TDR-63-574 (July 1963),

AD 420193.

[222] Wray, K. L., and Connolly, T. J., Thermal conductivity

of clear fused silica at high temperatures, J. Appl.

Phys. 30, 1702 (1959).

[223] Yagi, S., and Kunii, D., Studies on effective thermal

conductivities in packed beds, A. I. Ch. E. Journal

3, 373 (1957).

[224] Hirschfelder, J. O., Curtiss, D. F., and Bird, R. B.,

Molecular Theory of Gases and Liquids, Second

edition (John Wiley & Sons, Inc., 1964).

[225] Lawson, A. W., On the high temperature heat

conductivity of cylinders, J. Phys. Chem. Solids

3, 155 (1957).

[226] Jamieson, J. C, and Lawson, A. W., High-temperature heat conductivity of some metal oxides, J. Appl. Phys. 29, 1313 (1958).

[227] Whitmore, D. H., Excitation processes in ceramics

and the anomalous increase in thermal conductivity

at elevated temperatures, J. Appl. Phys. 31, 1109

(1960).

[228] Whitmore, D. H., Excitation processes in ce amies

and the anomalous increase in thermal conductivity

at elevated temperatures, WADD TR 60-866

(July, 1962), AD 285283.

[229] Jerome, D., Rice, T. M., and Kohn, W., Excitonic

insulator, Phys. Rev. 158, 462 (1967).

[230] Zittartz, J., Transport properties of the "excitonic

insulator" I. Electrical conductivity, Univ. of

Calif. (San Diego) Technical Report No. 47

(Contract NONR-2216-(ll)), AD 657439.

[231] Zittartz, J., Transport properties of the "excitonic

insulator" II. Thermal conductivity, Univ. of

Calif. (San Diego) Technical Report No. 48

(Contract NONR-2216-(ll)), AD 657440.

[232] Berman, R., Heat transfer in solids, Bull. I.I.F./

I.I.R., 23 (Annexe 1965-2).

[233] Kingery, W. D., Klein, J. D., and McQuarrie, M. C.,

Development of ceramic insulating materials for high temperature use, Trans. Amer. Soc. Mech.

Engrs. 80, 705 (1958).

[234] Ruh, E., and McDowell, J. S., Thermal conductivity

of refractory brick, J. Am. Ceram. Soc. 45, 189

(1962).

[235] Williams, W. S., High-temperature thermal con- ductivity of transition metal carbides and nitrides,

J. Am. Ceram. Soc. 49, 156 (1966).

[236] Taylor, R. E., Thermal conductivity and expansion

of beryllia at high temperatures, J. Am. Ceram.

Soc. 45, 74 (1962).

[237] Kittel, C, Interpretation of the thermal conductivity

of glasses, Phys. Rev. 75, 972 (1949).

[238] Powell, R. W., Ho, C. Y., and Liley, P. E., Thermal

conductivity of selected materials, NSRDS-JNBo

8 (November 1966).

[239] Ho, C. Y., Powell, R. W., and Liley, P. E., Thermal

conductivity of selected materials, Part 2, NSRD&

NBS 16 (February 1968).

[240] For a listing of these, contact the Thermophysical

Properties Research Center, Purdue University,

2595 Yeager Road, West Lafayette, Indiana

47906.

[241] Kingery, W. D., The thermal conductivity of ceramic

dielectrics, in Progress in Ceramic Science, Vol. 2, edited by J. E. Burke, p. 181, (Pergamon Press,

Inc., New York, 1962).

[242] Young, R. C, Hartwig, F. J., and Norton, C. L.,

Effect of various atmospheres on thernaal con- ductance of refractories, J. Am. Ceram. Soc. 47,

205 (1964).

[243] Flynn, D. R., and O'Hagan, M. E., Measurements of

the thermal conductivity and electrical resistivity

of platinum from 100 to 900 °C, J. Research NBb

71C (Engr. and Instr.) No. 4, 255 (1967).

117
244] Bridgman, P. W., The thermal conductivity and

compressibility of several rocks under high presssures, Amer. J. Science, VII, 81 (1924).

[245] Bridgman, P. W., The Physics of High Pressure,

(The Macmillan Co., New York, 1931).

[246] Hughes, D. S., and Sawin, F., Thermal conductivity

of dielectric solids at high pressure, Phys. Rev.

161, 861 (1967).

[247] Drickamer, H. G., Lynch, R. W., Clendenen, R. L., and Perez-Albuerne, E. A., X-ray diffraction

studies of the lattice parameters of solids under

very high pressure, Solid State Physics 19, 135

(1966).

[248] Berman, R., Simon, F. E., and Wilks, J., Thermal

conductivity of dielectric crystals: the "Umklapp"

process, Nature 168, 277 (1951).

[249] Webb, F J., Wilkinson, K. R., and Wilks, J., The

thermal conductivity of solid helium, Proc. Roy.

Soc. (London) A214, 546 (1952).

[250] Webb, F J., and Wilks, J., The thermal conductivity

of solid helium at high densities, Phil. Mag. 44,

664 (1953).

[251] Drickamer, H. G., The effect of high pressure on the

electronic structure of solids, Solid State Physics

17, 1 (1965).

[252] Bridgman, P. W., The effect of pressure on the thermal

conductivity of metals, Proc. Amer. Acad. Arts

Sci. 57, 77 (1921).

[253] Starr, C, The pressure coefficient of thermal conductivity of metals, Phys. Rev. 54, 210 (1938).

[254] Lawson, A. W., The effect of hydrostatic pressure on the electrical resistivity of metals, in Metal

Physics, Vol. 6, edited by B. Chalmers and R. King, p. 1 (Pergamon Press, Inc., New York,

1956).

[255] Swenson, C. A., Physics at high pressure, Solid

State Physics 11, 41 (1960).

[256] Bundy, F. P., and Strong, H. M., Behavior of metals

at high temperatures and pressures, Solid State

Physics 13, 81 (1962).

[257] Paul, W., The electrical properties of metals and

semiconductors, in High Pressure Physics and

Chemistry, Vol. 1, edited by R. S. Bradley, p. 299

(Academic Press, Inc., New York, 1963).

[258] Landwehr, G., Electrical properties of metals and

semiconductors under high pressure, in Physics

of High Pressures and the Condensed Phase,

edited by A. Van Itterbeek, p. 556 (NorthHolland Publishing Co., Amsterdam, Netherlands

1965).

[259] Bridgman, P. W., The effect of tension on the thermal

and electrical conductivity of metals, Proc. Amer. Acad. Arts Sci. 59, 119 (1923).

[260] Lees, C. H., and Calthrop, J. E., The effect of. torsion on the thermal and electrical conductivities of metals, Proc. Phys. Soc. (London) 35, 225

(1923).

[261] Calthrop, J. E., The effects of torsion on the thermal

and electrical conductivities of metals, Proc. Phys. Soc. (London) 36, 168 (1924).

[262] Austin, J. B., Factors influencing the thermal con- ductivity of non-metallic materials, Symposium

on Thermal Insulating Materials, p. 3 (Amer. Soc.

for Testing Materials, 1939)

.

[263] Patrassi, E., Die Warmeleitfahigkeit von Urandioxid

bei sehr hohen Temperaturgradienten, J. Nucl.

Mater. 22, 311 (1967).

[264] Fulkerson, W., McElroy, D. L., Moore, J. P., Comments on a paper by E. Patrassi, Die Warmeleitfahigkeit von Urandioxid bei sehr hohen

Temperaturgradienten (The thermal conductivity

of U02 at very high temperature gradients), J. Nucl. Mater. 26, 223 (1968).

[265] Kaspar, J., and Zehms, E. H., Thermal diffusivity

of carbons and graphites in the temperature

range from 1800 to 3300 °K, SSD-TR-67-116

(June, 1967) AD 655786.

[266] Ho, C. Y., and Powell, R. W., The state of knowledge regarding the thermal conductivity of the

non-metallic elements—those solid at normal

temperature, Proc. Seventh Conference on Thermal

Conductivity, edited by D. R. Flynn and B. A. Peavy, p. 33 (NBS Special Publication 302, 1968).

[267] Nordheim, L., Die Theorie der thermoelektrischen

Effekte, Sci. et Ind., No. 131 (Paris, 1934).

[268] Linde, J. O., Elektrische Eigenschaften verdunnter

Mischkristallegierungen, Ann. Phys. 10, 52 (1931);

15, 219 (1932).

[269] Berman, R., and Brock, J. C. F., The effect of iso- topes on lattice heat conduction, I. Lithium

fluoride, Proc. Roy. Soc. 289A, 46 (1965).

[270] Aldrich, R. E., Burke, W. J., and McCarthy, K. A., Resonant phonon scattering in colored sodium

chloride crystals, Solid State Commun. 5, 899

(1967).

[271] Baumann, F. C, Harrison, J. P., Pohl, R. O., and

Seward W. D., Thermal conductivity in mixed

alkali halides: KCLLi and KBr:Li, Phys. Rev.

159, 691 (1967).

[272] Baumann, F. C, and Pohl, R. O., Scattering of

phonons by monatomic impurities in potassium

halides, Phys. Rev. 163, 843 (1967).

[273] Kutasov, V. A., and Smirnov, I. A., Effect of surplus

antistructural Bi on the thermal conductivity of

crystal lattices of Bi2 Te3 , Bi2 Te3-Sb2 Te3 and

Bi2 Te3-Bi2 S3 with iodine admixtures, Phys. Status

Solidi 18, 479 (1966) (in German).

[274] Mirlin, D. N., and Smirnov, I. A., Thermal conductivity of additively colored KC1 crystals

Soviet Phys. Solid State 9, 1452 (1967).

[275] Schwartz, J. W., and Walker, C. J., Thermal conductivity of some alkali halides containing divalent

impurities, I. Phonon resonances, Phys. Rev.

155, 959 (1967).

[276] Schwartz, J. W., and Walker, C. J., Thermal conductivity of some alkali halides containing divalent

impurities, II. Precipitate scattering, Phys. Rev.

155, 969 (1967).

[277] Smirnov, I. A., Thermal conductivity of NaF

single crystals containing K, Li, and CI impurities

(80-400 °K), Soviet Phys. Solid State 9, 1454

(1967).

[278] Walton, D., Scattering of phonons by a square-well

potential and the effect of colloids on the thermal

conductivity, I. Experimental, Phys. Rev. 157,

720 (1967).

[279] Walton, D., and Lee, E. J., Scattering of phonons

by a square-well potential and the effect of col- loids on the thermal conductivity, II. Theoretical,

Phys. Rev. 157, 724 (1967).

[280] Worlock, J. M., Thermal conductivity in sodium

chloride crystals containing silver colloids, Phys.

Rev. 147, 636 (1966).

[281] Ross, A. M., Dependence of the thermal conductivity

of uranium dioxide on density, microstructure,

stoichiometry, and thermal-neutron irradiation,

Atomic Energy of Canada, Ltd. Report No.

CRFD-817 (September 1960).

[282] May, J. E., and Stoute, R. L., Observations on the

thermal conductivities of certain partially-reduced

ceramic oxides, Atomic Energy of Canada, Ltd.,

Report No. AECL-2169 (January 1965).

[283] Godfrey, T. G., Fulkerson, W., Kollie, T. G.,

Moore, J. P., and McElroy, D. L., Thermal

conductivity of uranium dioxide from —57 ° to 1100 °C by a radial heat flow technique, J. Am.

Ceram. Soc. 48, 297 (1965).

[284] Thurber, W. R., and Mante, A. J. H., Thermal

conductivity and thermoelectric power of rutile

(Ti02), Phys. Rev. 139, A1655 (1965).

[285] Hetzler, F. J., and Zebroski, E. L., Thermal conductivity of stoichiometric and hypostoichiometric uranium oxide at high temperatures,

Trans. Amer. Nucl. Soc. 7, 392 (1964).

[286] Pohl, R. O., Thermal conductivity and phonon

resonance scattering, Phys. Rev. Letters 8, 481

(1962).

118
[287] Eucken, A., and Kuhn, E., Ergebnisse neuer Messungen der Warmeteilfahigkeit fester krystallisierter Stoffe bei 0 und — 190C, Z. physik.

Chemie 134, 193 (1928).

[288] Kingery, W. D., Thermal conductivity: XIV, con- ductivity of multicomponent systems, J. Am.

Ceram. Soc. 42, 617 (1959).

[289] Abeles, B., Lattice thermal conductivity of dis- ordered semiconductor alloys at high temperatures,

Phys. Rev. 131, 1906 (1963).

[290] Abdurakhmanova, A. A., and Aliev, M. I., On the

mechanism of thermal conductivity in (GaSb) 3l-

(GajTej),-, and (GaDSb),-(GaTe) 1 -2r solid so- lutions, Phys. Status Solidi 20, 777 (1967).

[291] Aliev, M. I., and Dzhangirov, A. Yu., Investigation

of the thermal and electrical properties of InSbIn2 Te3 alloys, Soviet Phys.-Solid State 6, 1916

(1965) . [292] Bhandari, C. M., and Verma, G. S., Thermal conductivity of Ge-Si alloys at high temperatures,

Phys. Rev. 138, A288 (1965).

[293] Drabble, J. R., The lattice thermal conductivity of

alloys, Adv. Energy Conv. 3, 441 (1963).

[294] Hockings, E. F., Kudman, I., Seidel, T. E., Schmelz,

C. M., and Steigmeier, E. F., Thermal and electrical transport in InAs-GaAs alloys, J. Appl.

Phys. 37, 2879 (1966).

[295] Kelemen, F., Neda, A., Niculescu, D., and Cruceanu,

E., On the thermal conductivity of CdTe and some

CdTei-zSn* and CdTei-xSex solid solutions, Phys.

Status Solidi 21, 557 (1967).

[296] Kudman, I., Lattice thermal conductivity of semiconductor alloys, Proc. Seventh Conference on Thermal Conductivity, edited by D. R. Flynn and

B. A. Peavy, p. 97 (NBS Special Publication 302,

1968).

[297] Kudman, L, Ekstrom, L., and Seidel, T., Hightemperature thermal and electrical properties of

GaSb-InSb alloys, J. Appl. Phys. 38, 4641 (1967).

[298] Parrott, J. E., The high temperature thermal conductivity of semiconductor alloys, Proc. Phys.

Soc. (London) 81, 726 (1963).

[299] Remoissenet, M., and Godefroy, L., Thermal diffu- sivity of barium titanate and of strontium titanate

solid solutions, Comt. Rend. 262B, 56 (1966).

[300] Spitzer, D. P., Castellion, G. A., and Haacke, G.,

Anomalous thermal conductivity of Cd3 As2 and

the Cd3 As2-Zn3 As2 alloys, J. Appl. Phys. 37, 3795

(1966) . [301] Williams, W. S., Phonon scattering in KCl-KBr solid

solutions at low temperatures, Phys. Rev. 119,

1021 (1960).

[302] Zeinalov, S. A., and Aliev, M. I., Investigation of the

thermal conductivity of InSb-GaSb solid solutions,

Phys. Status Solidi 22, 153 (1967).

[303] Zerfang, R., The lattice thermal conductivity through

dilution in alloys at low temperatures, Phys.

Status Solidi 24, 221 (1967).

[304] Bar'yakhtar, V. G., Savchenko, M. A., and Tarasenko, V. V., Relaxation and thermal conductivity

in magnetic substances with dislocations, Soviet

Physics—JETP 24, 623 (1966).

[305] Bross, H., The influence of defect ordering on the

lattice heat conductivity of insulators, Z. Phys.

189, 33 (1966).

[306] Brown, R. A., Resonance scattering and the electrical and thermal resistivities associated with

extended defects in crystals, Phys. Rev. 156, 692

(1967) . [307] Caldwell, R. F., and Klein, N. V., Experimental and

theoretical studies of phonon scattering from

simple point defects in NaCl, Phys. Rev. 158,

851 (1967).

[308] Callaway, J., Localized defects in semiconductors,

Phys. Rev. 156, 860 (1967).

[309] Moss, M., Effect of plastic deformation on the

thermal conductivity of CaF and LiF, J. Appl.

Phys. 36, 3308 (1965).

[310] Moss, M., Scattering of phonons by dislocations,

J. Appl. Phys. 37, 4168 (1966).

[311] Payton, D. N., Ill, Rich, M., and Visscher, Wm. M.,

Lattice thermal conductivity in disordered harmonic and anharmonic crystal models, Phys. Rev.

160, 706 (1967).

[312] Petrov, A. V., Scattering of phonons by large lattice

distortions, Soviet Phys. Solid State 8, 2988

(1966).

[313] Smirnov, I. A., The effect of plastic deformation on

the thermal conductivity of NaCl and KC1 single

crystals, Soviet Phys. Solid State 8, 21 (1966).

[314] Taylor, A., Alhers, H. R., and Pohl, R. O., Effect

of plastic deformation on the thermal conductivity

of various ionic crystals, J. Appl. Phys. 36, 2270

(1965).

[315] Zavt, G. S., A heat conduction theory of dielectrics at low temperatures. A case of arbitrary interaction between phonons and defects, Trudy In-ta

Fiz. i Astronom, Akad. Nauk Estonsk. SSR (29),

95 (1965).

[316] Berman, R., Foster, E. L., and Rosenberg, H. M.,

The thermal conductivity of irradiated dielectric

crystals at low temperatures, in Rept. Conf. Defects in Crystalline Solids, p. 321 (Physical Society, London, 1955).

[317] Moore, J. P., Kollie, T. G., Graves, R. S., and

McElroy, D. L., Thermal conductivity measure- ments on solids between 20 and 150 °C using a

comparative-longitudinal apparatus: results on MgO, BeO, Th02 , ThxLVxOj+y and A1-U02 cermets, ORNL-4121 (June 1967).

[318] Cooper, M. K., Palmer, A. R., and Stolarski,

G. Z. A., The effect of neutron irradiation on

the thermal conductivity of beryllium oxide, J. Nucl. Mater. 9, 320 (1963).

[319] Hickman, B. S., The effect of neutron irradiation on

beryllium oxide, AAEC/E99, (October 1962)

.

[320] Berman, R., Simon, F. E., Klemens, P. G., and

Fry, T. M., Effect of neutron irradiation on the

thermal conductivity of a quartz crystal at low

temperature, Nature 166, 864 (1950).

[321] Berman, R., The thermal conductivities of some

dielectric solids at low temperatures (Experimental), Proc. Roy. Soc. A208, 90 (1951).

[322] Crawford, J. M., Jr., and Cohen, A. F., Effect of

fast neutron bombardment on the thermal conductivity of silica glass at low temperature, p. 43

in ORNL-2614 (November 1958).

[323] Cohen, A. F., Low-temperature thermal conductivity

in neutron irradiated vitreous silica, J. Appl. Phys.

29, 591 (1958).

[324] Albany, H. J., and Vandevyver, M., Effects of

electron irradiation on the thermal conductivity

of n- and p-type germanium, Phys. Rev. 160,

633 (1967).

[325] Albany, H. J., and Vandevyver, M., Low-temperature thermal conductivity of fast-neutron irradi- ated silicon and germanium, J. Appl. Phys. 38,

425 (1967).

[326] Berman, R., Foster, E. L., Schneidmesser, B., and

Tirmizi, S. M. A., Effect of irradiation on the

thermal conductivity of synthetic sapphire, J. Appl. Phys. 31, 2156 (1960).

[327] Clough, D. J., and Sayers, J. B., The measurement

of the thermal conductivity of U02 under irradiation in the temperature range 150-1600 °C,

AERE-R-4690 (United Kingdom Atomic Energy

Authority, Harwell, December, 1964).

[328] De Goer, A. M., and Dreyfus, B., 7-irradiation effect on the thermal conductivity of A12 03 with Cr or Mn impurities at low temperatures, Phys. Status

Solidi 22, 77 (1967).

[329] Freeman, R. J., Thermal diffusivity measurements

on pre- and post-irradiated BeO, GEM P-452

(General Electric Company, NMPO, Cincinnati,

Ohio, November 1966).

119
[330] Henson, R. W., and Reynolds, W. N., Lattice

parameter changes in irradiated graphite, Carbon

3, 277 (1965).

[331] Hunter, L. P., Effect of fission recoil fragments on

the thermal conductivity of graphite, J. Appl.

Phys. 30, 1969 (1959).

[332] Jaeger, R. E., The thermal conductivity of neutron

irradiated A12 03 , BeO, and MgO at low temperatures, Ph. D. Dissertation, Rutgers University,

March 1967.

[333] Jaeger, R. E., Thermal conductivity of neutron

irradiated A12 03 , BeO, and MgO at low temperatures, Paper No. 13-N-67F, Nuclear Division of

Amer. Ceram. Soc. (September 19-22, 1967).

[334] McDonald, D. L., Low-temperature irradiation of

beryllium oxide, Appl. Phys. Letters 2, 175 (1963).

[335] Pryor, A. W., Tainsh, R. J., and White, G. K.,

Thermal conductivity at low temperature of

neutron-irradiated BeO, J. Nucl. Mater. 14, 208

(1964) . [336] Seyfert, P., Influence of neutron irradiation at the

temperature of 80 °K on the thermal conductivity

of silicon, Compt. Rend. 265B, 609 (1967).

[337] van Dong, Nguyen, Effect of fast neutrons on thermal conductivity and the thermoelectric power of

germanium, J. Phys. (Paris) 27, 128 (1966).

[338] Vook, F. L., Thermal conductivity of electronirradiated germanium, Phys. Rev. 138, A1234

(1965) . [339] Vook, F. L., Thermal conductivity of electronirradiated silicon, Phys. Rev. 140,*A2014 (1965).

[340] Vook, F. L., Thermal conductivity of electronirradiated InSb, Phys. Rev. 149, 631 (1966).

[341] Vorozheikina, L. F., Igitkhanishvili, D. D., Politov,

N. G., and Kvachadze, V. G., Electric and thermal

conductivities and optical absorption of neutron

irradiated alkali halides, Bull. Acad. Sci. USSR,

Phys. Ser. 29, 75 (1965).

[342] Billington, D. S., and Crawford, J. H., Jr., Radiation

Damage in Solids (Princeton University Press,

Princeton, New Jersey, 1961).

[343] Kirchner, J. F., and Bowman, R. E., Effects of

Radiation on Materials and Components

(Reinhold Publishing Corp., New Yprk, 1964).

[344] Strumane, R., Nichoul, J., Gevers, R., and

Amelinckx, S., The Interaction of Radiation with

Solids (North Holland Publishing Co., Amsterdam,

Netherlands, 1964).

[345] Nuclear Science and Technology for Ceramists,

NBS Misc. Publ. 285 (May 1967).

[346] Powers, A. E., Conductivity in aggregates, KAPL2145, Knolls Atomic Power Laboratory (March

1961).

[347] Meredith, R. E., and Tobias, C. W., Conduction in

heterogeneous systems, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 2, edited by C. W. Tobias, p. 15 (John Wiley & Sons,

Inc., New York, 1962).

[348] Argo, W. B., and Smith, J. M., Heat transfer in

packed beds, Chem. Eng. Prog. 49, 443 (1953).

[349] Awbery, J. W., Note on heat-flow through a granulated material, Phil. Mag. 12, 1152 (1931).

[350] Babanov, A. A., Methods for calculation of thermal

conduction coefficients of capillary-porous materials, Soviet Phys—Tech. Phys. 2, 476 (1957).

[351] Baldwin, D. E., Jr., Beckman, R. B., Rothfus, R. R.,

and Kermode, R. I., Heat transfer in beds of

oriented spheres, I/EC Process. Des. Dev. 5, 281

(1966) . [352] Barrett, L. B., Heat transfer in refractory insulating

materials, Part I. Texture and insulating power,

Trans. Brit. Ceram. Soc. 48, 235 (1948-49).

[353] Baxley, A. L., and Couper, J. R., Thermal conductivity of two-phase systems: Thermal conductivity

of suspensions, Univ. Arkansas Eng. Exp. Station

Research Report No. 8 (November 1966).

[354] Baxley, A. L., Nahas, N. C, and Couper, J. R.,

Thermal conductivity of two-phase systems, Proc. Seventh Conf. on Thermal Conductivity, edited

by D. R. Flynn and B. A. Peavy, p. 685 (NBS

Special Publication 302, 1968).

[355] Bischoff, K. B., Axial thermal conductivities in packed beds, Can. J. Chem. Eng. 40, 161 (1962).

[356] Boegli, J. S., and Deissler, R. G., Measured effective

thermal conductivity of uranium dioxide powder

in various gases and gas mixtures, NACA RM

E54L10 (March 1955).

[357] Brailsford, A. D., and Major, K. G., The thermal

conductivity of aggregates of several phases,

including porous materials, Brit. J. Appl. Phys. 15,

313 (1964).

[358] Bretsznajder, S., and Ziolkowski, D., The effective

thermal conductivity of granular beds, Intern. Chem. Eng. 6, 85 (1966).

[359] Brodie, D. E., and Mate, C. F., Heat transfer in

gas-filled powders at low temperatures, Can. J. Phys. 43, 2344 (1965).

[360] Chaudhary, D. R., Bhandari, R. C, Heat transfer

through dispersed system having irregular pore

sizes, Indian J. Pure Appl. Phys. 5, 413 (1967).

[361] Chapoton, A., Wacrenier, J. M., and Lebrun, A., Contribution to the determination of conductivities and permittivities at the level of the

cavities of porous solids, Compt. Rend. 265B, 989

(1967).

[362] Luikov, A. V., Shashkov, A. G., Vasiliev, L. L., and

Fraiman, Yu. E., Thermal conductivity of porous

systems, Int. J. Heat Mass Transfer 11, 117 (1968).

[363] Deissler, R. G., and Boegli, J. S., An investigation

of effective thermal conductivities of powders in

various gases, Trans. ASME 80, 1417 (1958).

[364] de Vries, D. A., Het warmtegeleidingsvermogen van

grand, Mededel. Landbouwhogeschool Wageningen/Nederland 52, 1 (1952).

[365] Dul'nev, G. N., Heat transfer through solid disperse

systems, Inzh.-Fiz. Zh. 9, 399 (1965).

[366] Dul'nev, G. N., Heat conduction of statistical

mixtures, Inzh.-Fiz. Zh. 9, 538 (1965).

[367] Dul'nev, G. N., and Komkova, L. A., Analysis of

the experimental studies of thermal conductivity

of solid porous systems, Inzh.-Fiz. Zh. 9, 517

(1965).

[368] Dul'nev, G. N., and Zarichnyak, V. P., Thermal

conductivity of multicomponent mixtures, Inzh.-

Fiz. Zh. 12, 419 (1967).

[369] Dul'nev, G. N., and Sigalova, Z. V., The thermal

conductivity of granular systems, Intern. Chem.

Eng. 5, 218 (1965).

[370] Dunsky, V. D., and Tamarin, A. I., On heat conductivity of disperse material in vacuum, in Heat

and Mass Transfer in Dispersed Systems, p. 160,

(Minsk, 1965).

[371] Eucken, A., Allgemeine Gesetzmassigkeiten fur das

Warmeleitvermogen verschiedener Stoffarten und

Aggregatzustande, Forsch. Gebiete Ingenieurw.

11, 6 (1940).

[372] Euler, F., Simple geometrical model for the effect of porosity on material constants, J. Appl. Phys.

28, 1342 (1957).

[373] Francis, R. K., and Tinklepaugh, J. R., Thermal

conductivity in ceramic-metal laminates, J. Am.

Ceram. Soc. 43, 560 (1960).

[374] Francl, J., and Kingery, W. D., Thermal conductivity: IX, Experimental investigation of effect of porosity on thermal conductivity, J. Am.

Ceram. Soc. 37, 99 (1954).

[375] Gopalarathnan, C. D., Effective thermal conductivity in packed beds, A. I. Chem. E. J. 7, 249

(1961).

[376] Gorring, R. L., and Churchill, S. W., Thermal conductivity of heterogeneous materials, Chem. Eng.

Prog. 57, 53 (1961).

[377] Grekila, R. B., and Tien, T. Y., Conductivity dis- continuity in a two-phase system, J. Am. Ceram.

Soc. 48, 22 (1965).

[378] Hamilton, R. L., and Crosser, O. K, Thermal

conductivity of heterogeneous two component

120
systems, Ind. Eng. Chem. Fundamentals 1, 187

(1962).

[379] Jakob, M., Heat Transfer, Vol. I, p. 88 (John Wiley

& Sons, Inc., New York, 1950).

[380] Jefferson, T. B., Thermal conductivity determinations for suspensions of graphite in water and oil, Ph. D. thesis (Purdue University, 1955).

[381] Jefferson, T. B., Witzell, 0. W., and Sibbitt, W. L., Thermal conductivity of graphite-silicone oil and

graphite-water suspensions, Ind. Eng. Chem. 50,

1589 (1958).

[382] Joy, F. A., Thermal conductivity of insulation containing moisture, Symposium on Thermal Conductivity Measurements and Applications of Thermal

Insulations, ASTM STP 217 (1957).

[383] Kaganer, A. M. G., Contact heat transfer in granular

material in vacuum, Inzh.-Fiz. Zh. 10, 30 (1966).

[384] Kaganer, A. M. G., and Semenova, R. I., Thermal

conductivity and thermal diffusivity of insulating

powders at atmospheric pressure and in vacuum

studied by various methods, Inzh.-Fiz. Zh. 13, 24

(1967).

[385] Kannuluik, W. G., and Martin, L. H., Conduction

of heat in powders, Proc. Roy. Soc. (London)

A141, 58 (1933).

[386] Kharlamov, A. G., Thermal conductivity of packing

of ceramic pellets, Inzh.-Fiz. Zh. 9, 48 (1965).

[387] Keller, J. B., Conductivity of a medium containing a dense array of perfectly conducting spheres or

cylinders or nonconducting cylinders, J. Appl.

Phys. 34, 991 (1963).

[388] Keller, H. B., and Sachs, D., Calculations of the

conductivity of a medium containing cylindrical

inclusions, J. Appl. Phys. 35, 537 (1964).

[389] Kimura, M., Effective thermal conductivity of

wetted packed beds, Kagaku Kogaku (Chem.

Eng., Japan) 23, 502 (1959).

[390] Krischer, O., and Esdorn, H., Die Warmeiibertragung in feuchten, porigen Stoffen verschiedener

Struktur, Forsch. Gebiete Ingenieurw. A22, 1 (1956).

[391] Krupiczka, R., Analysis of thermal conductivity in granular materials, Intern. Chem. Eng. 7, 122

(1967).

[392] Kunii, D., and Smith, J. M., Heat transfer characteristics of porous rocks, A. I. Ch. E. Journal 6,

71 (1960).

[393] Lees, C. H., On the thermal conductivity of single

and mixed solids and liquids and their variation

with temperature, Proc. Roy. Soc. (London)

A191, 339 (1898).

[394] Levine, H., The effective conductivity of a regular

composite medium, Stanford University TR-6

(July 1965), AD 622870.

[395] Loeb, A. L., Thermal conductivity: VIII, A theory

of thermal conductivity of porous materials,

J. Am. Ceram. Soc. 37, 96 (1954).

[396] Little, R. C, Carpenter, F. G., and Deitz, V. R.,

Heat transfer in intensively outgassed powders,

J. Chem. Phys. 37, 1896 (1962).

[397] Masamune, S., and Smith, J. M., Thermal conductivity of beds of spherical particles, Ind. and Eng.

Chem. Fundamentals 2, 136 (1963).

[398] Miller, J. V., Predicting thermal conductivity of

tungsten-uranium dioxide dispersions, NASA TM

X-1145 (1965).

[399] Miller, J. V., Estimating thermal conductivity of

cermet fuel materials for nuclear fuel application, NASA TN D-3898 (April 1967).

[400] Nahas, N. C, Griffin, C. L., and Couper, J. R.,

Thermal conductivity of two-phase systems: A

survey and critical evaluation of the literature,

Univ. Arkansas Eng. Exp. Station Research Report No. 5 (April 1964).

[401] Nahas, N. C, and Couper, J. R., Thermal conductivity of two-phase systems: Thermal conductivity

of emulsions, Univ. Arkansas Eng. Exp. Station

Research Report No. 7 (March 1966).

[402] Neimark, L. A., Kittel, J. H., and Hoenig, C. L., Irradiation of metal-fiber-reinforced thoria-urania,

Report No. ANL-6397 (December 1961).

[403] Nusselt, E. K. W., Warmeubertragg. b.d. BoneSchnabelfeuerg, Z. Bayer, Revis.-Ver., (1913).

[404] Pliat, Sh. N., Extension of Odelevskii's theory of

generalized conductivity to thermal conductivity,

Soviet Physics—Tech. Phys. 2, 2588 (.1957).

[405] Preston, F. W., Mechanism of heat transfer in unconsolidated porous media at low flow rates,

Ph. D. thesis, The Pennsylvania State University

(1957), (see also Dissertation Abstracts 17, 2593

(1957)).

[406] Rabinowitz, P., Calculations of the conductivity

of a medium containing cylindrical inclusions by

the method of orthogonalized particular solutions,

Technical Report, Brown University, (June 1965),

AD 618 052.

[407] Ribaud, M., Conductibilite thermique des materiaux

poreaux et pulverulents, etude theorique, Chaleur

& Ind. 18, 36 (1937).

[408] Rieman, G. M. H. W., Die Partiellen DifferentialGleichunger der Mathematischen Physik, Band 1, p. 474 (F. Vieweg und Sohn, Braunschweig, 1919).

[409] Schroeder, J. B., Thermal conductivity in two-phase

alloys, J. Appl. Phys. 32, 1798 (1961), (see discussion by R. W. Ure, Jr., J. Appl. Phys. 33, 2290

(1962)).

[410] Shimokawa, J., Deionization processes with ion ex- change and with electromigration, Japan Atomic

Energy Research Institute, JAERI 1038 (June

1962).

[411] Shudnovskii, A. F., Thermophysical Characteristics

of Disperse Materials (Fizmatgiz (USSR), 1962).

[412] Somerton, W. H., Some thermal characteristics of

porous rocks, J. Petrol. Technol. 10, 61 (May

1958).

[413] Strong, M., II, Bundy, F. P., and Bovenkirk, M. P., Flat panel vacuum thermal insulation, J. Appl.

Phys. 31, 39 (1960).

[414] Sugawara, A., and Yoshizawa, Y., An investigation on the thermal conductivity of porous materials and its application to porous rocks, Australian

J. Phys. 14, 469 (1961).

[415] Sugawara, A., and Yoshizawa, Y., An experimental

investigation on the thermal conductivity of consolidated porous materials, J. Appl. Phys. 33,

3135 (1962).

[416] Swift, D. L., The thermal conductivity of spherical

metal powders including the effect of an oxide

coating, Intern. J. Heat Mass Transfer 9, 1061

(1966).

[417] Tadokro, Y., On the effect of porosity upon thermal

conductivity, diffusivity, and heat capacity at high temperatures, Science Report, Tohuku Imp.

Univ. I, 15, 567 (1926).

[418] Tsao, G. T.-N., Thermal conductivity of two-phase

materials, Ind. Eng. Chem. 53, 395 (1961).

[419] van Bueren, H. G., Imperfections in Crystals,

(North Holland Publishing Company, Amsterdam,

Netherlands, 1961).

[420] Vasil'ev, L. L., and Fraiman, Iu. E., Investigation

of the thermophysical properties of the chamotte

ceramics in the 80-1200 °K range. Inzh-Phys.

Zh. 9, 762 (1965) (Translation TT-66-61303

available from Clearinghouse for Fed. Sci. and

Tech. Inf.).

[421] Waddams, A. L., Flow of heat through granular

materials, J. Soc. Chem. Ind. 63, 337 (1944).

[422] Wakao, N., Omura, S., and Fukuda, M., Effective

thermal conductivity of packed beds in low

pressure range, Chem. Engng. Japan 30, 1119

(1966).

[423] Wang, D. I-J., Multiple layer insulation, in Aerodynamically Heated Structures, edited by P. E. Glaser (Prentice-Hall, Inc., New York, 1962).

[424] Warren, J. E., and Messmer, J. H., Thermal conductivity of two-phase materials, Ind. and Eng.

Chem. Fundamentals, 1, 222 (1962).

323-655 O - 69 - 9 121
[425] Watson, K., The thermal conductivity measure- ments of selected silicate powders in vacuum

from 150 °-350 °K, Ph, D. thesis (California

Inst, of Tech., 1964).

Webb, J., Thermal conductivity of soil, Nature 177,

989 (1956).

Wechsler, A. E., Glass beads—a standard for the

low thermal conductivity range, Proc. Seventh

Conf. on Thermal Conductivity, edited by D. R.

Flynn and B. A. Peavy, p. 89 (NBS Special

Publication 302, 1968).

Weiner, J. H., Transient heat conduction in multiphase media, Brit. J. Appl. Phys. 6, 361 (1955).

Whitmore, D. H., Apparent thermal conductivity

of a heterogeneous ceramic body under isothermal

conditions, J. Am. Ceram. Soc.'43, 281 (1960).

Wilhelm, R. H., Johnson, W. C, Wynkoop, R., and

Collier, D. W., Reaction rate, heat transfer, and

temperature distribution in fixed-bed catalytic

converters-solution by electrical network, Chem.

Eng. Prog. 44, 105 (1948).

Willhite, G. P., Kunii, D., and Smith, J. M., Heat

transfer in beds of fine particles (heat transfer

perpendicular to flow), A. I. Ch. E. Journal 8,

340 (1962).

Woodside, W, Calculation of the thermal conductivity of porous media, Can. J. Phys. 36, 815

(1958).

Woodside, W., and Messmer, J. H., Thermal conductivity of porous media. I. Unconsolidated

sands, J. Appl. Phys. 32, 1688 (1961).

Wyllie, M. R. J., and Southwick, P. F., An experimental investigation of S. P. and resistivity

phenomena in dirty sands, J. Petroleum Tech.

6, T. P. 3750 (1954).

Jackson, J. L., and Coriell, S. R., Transport coeffi- cients of composite materials, J. Appl. Phys. 39,

2349 (1968). (See also p. 4733).

Koenig, J. H., Ceramic Bodies with High Thermal

Conductivity, Progress Report No. VI (Rutgers

University, 1952) ATI 163519.

Koenig, J. H., Ceramic Bodies with High Thermal

Conductivity, Progress Report No. II (Rutgers

University, 1953) AD 13154.

Kingery, W. D., Thermal conductivity: XII, Temperature dependence of conductivity for singlephase ceramics, J. Am. Ceram. Soc. 38, 251 (1955).

Charvat, F. R., and Kingery, W. D., Thermal

conductivity: XIII, Effect of microstructure on

conductivity of single-phase ceramics, J. Amer.

Ceram. Soc. 40, 306 (1957).

Amelinckx, S., and Dekeyser, W., The structure and properties of grain boundaries, Solid State

Physics 8, 327 (1959).

Microstructure of Ceramic Materials, NBS Misc.

Publ. 257 (April 1964).

Kingery, W. D., Effect of microstructure on the

properties of ceramics, in Physics and Chemistry

of Ceramics, edited by C. Klingsberg, p. 286

(Gordon and Breach Science Publ., Inc., New

York, 1963).

Coble, R. L., and Burke, J. E., Sintering in ceramics,

in Progress in Ceramic Science, Vol. 3, edited by

J. E. Burke (The Macmillan Co., New York,

1963).

Burke, J. E., The science and technology of sintering,

in Physics and Chemistry of Ceramics, edited by

C. Klingsberg, p. 165 (Gordon and Breach

Science Publ., Inc., New York, 1963).

White, G. K., Experimental Techniques in LowTemperature Physics (Oxford Univ. Press, Oxford,

1959).

Ditmars, D. A., and Ginnings, D. C, Thermal con- ductivity of beryllium oxide from 40 ° to 750 °C,

J. Research NBS 59, No. 2, 93 (1957) RP2775.

Laubitz, M. J., Measurement of the thermal conductivity of solids at high temperatures using

steady-state linear and quasi-linear heat flow, ch. 3

in Thermal Conductivity, edited by R. P. Tye

(Academic Press, Inc., London, in press).

[448] White, G. K., Measurement of solid conductors at low temperatures, ch. 2 in Thermal Conductivity,

edited by R. P. Tye (Academic Press, London,

in press).

[449] Mikryukov, V. E., Teploprovodnost i Elektroprovodnost Metalov i Splavov (Metallurgi?dat, Moscow,

1959).

[450] Bauerle, J. E., Thermal conductivity, in chapter on Measurements of properties of thermoelectric

materials, p. 285 in Thermoelectricity, edited by

R. R. Heikes and R. W. Ure (Interscience

Publishers, Inc., New York, 1961).

[451] Standard method of test for thermal conductivity of

materials by means of the guarded hot plate,

ASTM Designation: C177-63, in 1967 Book of

ASTM Standards, Part 14, p. 17 (American

Society for Testing and Materials, Philadelphia,

1967) . [452] International comparative measurements of thermal

conductivity, Bull. I.I.F./I.I.R. XLVIII, 29

(1968).

[453] Rouselle, J.-C, A guarded hot plate apparatus for measuring thermal conductivity from —80 to + 100 °C, Proc. Seventh Conf. on Thermal Conductivity, edited by D. R. Flynn and B. A.

Peavy, p. 513 (NBS Special Publication 302, 1968).

[454] Orr, H. W., A study of the effects of edge insulation

and ambient temperatures on errors in guarded

hot-plate measurements, Proc. Seventh Conf. on Thermal Conductivity, edited by D. R. Flynn and

B. A. Peavy, p. 521 (NBS Special Publication 302,

1968) . [455] Pratt, A. W., Heat transmission in thermal insulating

materials, ch. 6 in Thermal Conductivity, edited

by R. P. Tye (Academic Press, London, in press).

[456] Standard method of test for thermal conductivity

of refractories, ASTM Designation: C201-47, in 1967 Book of ASTM Standards, Part 13, p. 170

(American Society for Testing and Materials,

Philadelphia, 1967).

[457] Laubitz, M. J., On measurement of thermal conductivity at high temperatures, Can. J. Phys. 39,

1029 (1961).

[458] Laubitz, M. J., On the series comparator methods of measuring thermal conductivity, Proc. Black

Hills Summer Conference on Transport Phenomena, p. 8 (South Dakota School of Mines and

Technology, 1962).

[459] Petrov, A. V., Methods of measuring the thermal

conductivit}^ of semiconductors at high temperatures, in Thermoelectric Properties of Semiconductors (Proc. First and Second Conferences on

Thermoelectricity), edited by V. A. Kutasov,

p. 17 (Consultants Bureau, 1964).

[460] Didion, D. A., An analysis and design of a linear

guarded cut-bar apparatus for thermal conductivity measurements (Catholic University of

America, 1968) AD 665789.

[461] Minges, M. L., Heat losses in a cut-bar apparatus:

experimental-analytical comparisons, Proc. Seventh Conf. on Thermal Conductivity, edited by

D. R. Flynn and B. A. Peavy, p. 197 (NBS

Special Publication 302, 1968).

[462] Francl, J., and Kingery, W. D., Thermal conductivity: IV, Apparatus for determining thermal

conductivity by a comparative method; Data for Pb, A12 03 , and MgO, J. Am. Ceram. Soc. 37, 80

(1954).

[463] Francis, R. K, and Tinkelpaugh, J. R., Thermal

conductivity in ceramic-metal laminates, J. Am.

Ceram. Soc. 43, 560 (1960).

[464] Mirkovitch, V. V., Comparative method and choice

of standards for thermal conductivity determinations, J. Am. Ceram. Soc. 48, 387 (1965).

[465] Mahmoodi, P., Apparatus for determining thermal

conductivity, U.S. Patent No. 3,263,485 (August

2, 1966).

122
466] Standard method of test for thermal conductivity of

whiteware ceramics, ASTM Designation: C408-58,

in 1967 Book of ASTM Standards, Part 13, p. 348

(American Society for Testing and Materials,

Philadelphia, 1967).

[467] McElroy, D. L., and Moore, J. P., Radial heat

flow methods for the measurement of the thermal

conductitity of solids, ch. 4 in Thermal Conductivity, edited by R. P. Tye (Academic Press,

Inc., London, in press).

[468] Development of high temperature thermal conductivity reference standards (Arthur D. Little, Inc., Cambridge, Mass.,), Tech. Report AFMLTR-66-415 (January 1967).

[469] Powell, R. W., Correlation of metallic thermal and

electrical conductivities for both solid and liquid

phases, Int. J. Heat Mass Transfer 8, 1033 (1965).

[470] Powell, R. W., and Ho, C. Y., The state of knowledge

regarding the thermal conductivity of the metallic

elements, Proc. Seventh Conf. on Thermal Conductivity, edited by D. R. Flynn and B. A. Peavy,

p. 1 (NBS Special Publication 302, 1968).

[471] Laubitz, M. J., and van der Meer, M. P., The

thermal conductivity of platinum between 300

and 1000 °K, Can. J. Phys. 44, 3173 (1966).

[472] Martin, J. J., Sidles, P. H., and Danielson, G. C,

Thermal diffusivity of platinum from 300 to 1200

°K, J. Appl. Phys. 38, 3075 (1967).

[473] Shelton, S. M., and Swanger, W. H., Thermal conductivity of irons and steels and some other

metals in the temperature range 0-600 °, Trans.

Amer. Soc. Steel Treating 21, 1061 (1933).

[474] Van Dusen, M. S., and Shelton, S. M., Apparatus for measuring thermal conductivity of metals up to 600 C, NBS J. Research 12, No. 4, 429 (1934)

RP668.

[475] Lucks, C. F., Pyrometric standard lead as a thermal

conductivity reference material, J. Appl. Phys. 38,

1973 (1967).

[476] Powell, R. W., and Tye, R. P., New measurements

on thermal conductivity reference materials, Int.

J. Heat Mass Transfer 10, 581 (1967).

[477] Laubitz, M. J., and Cotnam, K. D., Thermal and

electrical properties of Inconel 702 at high temperatures, Can. J. Phys. 42, 131 (1964).

[478] Watson, T. W., and Flynn, D. R., Thermal conductivity of four heat-resistant alloys, Trans.

Metall. Soc. AIME 242, 844 (1968)

[479] Glassbrenner, C. J., and Slack, G. A., Thermal

conductivity of silicon and germanium from 3 K

to the melting point, Phys. Rev. 134, A1058

(1964).

[480] Fulkerson, W., Moore, J. P., Williams, R. K.,

Graves, R. S., and McElroy, D. L., Thermal

conductivity, electrical resistivity, and Seebeck

coefficient of silicon from 100 to 1300 °K, Phys.

Rev. 167, 765 (1968).

[481] Carwile, L. C. K., and Hoge, H. J., Thermal conductivity of vitreous silica: selected values, Proc. Seventh Conf. on Thermal Conductivity, edited

by D. R. Flynn and B. A. Peavy, p. 59 (NBS

Special Publication 302, 1968).

[482] Wechsler, A. E., and Minges, M. L., Development of

high temperature thermal conductivity standards,

Proc. Seventh Conf. on Thermal Conductivity,

edited by D. R. Flynn and B. A. Peavy, p. 77

(NBS Special Publication 302, 1968).

[483] Tye, R. P., (editor), Thermal Conductivity (Academic Press, Inc., London, in press).

[484] Picklesimer, M. L., Microstructural Examination,

ORNL-TM-1338 (January, 1966).

[485] Eisenhart, C, Realistic evaluation of the precision

and accuracy of instrument calibration systems,

J. Research NBS 67C, (Engr. and Instr.) No. 2, 161 (1963).

[486] Eisenhart, C, Expression of the uncertainties of

final results, Science 160, 1201 (1968).

123

Thermal Radiation Properties of Ceramic Materials

Joseph C. Richmond

Institute for Basic Standards,

National Bureau of Standards, Washington, D.C. 20234

The thermal radiation properties of materials are defined, and the basic physical laws

governing the amount of thermal radiation emitted by a blackbody radiator, and its geometric and spectral distribution, are reviewed. Equations are given relating the thermal

radiation properties of emittance, reflectance, transmittance, and absorptance to the optical

properties of index of refraction, absorption coefficient, and scattering coefficient. The

effect of impurities and processing variables such as grain size and porosity on the optical

properties is discussed. The influence of thermal gradients normal to the surface of ceramic

specimens on the observed thermal radiation properties is reviewed. Several methods of measuring thermal radiation properties of ceramic materials are discussed briefly, and

representative normal spectral emittance data for alumina, thoria, magnesia, and zirconia are given.

Key Words: Alumina; ceramics; emittance; magnesia; reflectance; temperature coeffi- cient of emittance; thermal radiation properties; thoria, zirconia.

1. Introduction

Thermal radiation is electromagnetic radiation

that is generated by the thermal motion of the

particles of which a material is composed, such

as molecules, atoms, ions, and electrons. All

matter is constantly emitting thermal radiation,

and in turn is constantly being irradiated by

thermal radiation from its surroundings. Both

processes occur simultaneously, and the net rate

of heat transfer to or from a body is determined

by the relative rates of the two processes.

The thermal radiation properties of a body are those properties that determine the rates of

absorption and emission of thermal radiation, and

hence the rates of heat transfer to or from the

body by radiation. When thermal radiation is incident on a body, it is either reflected, absorbed

or transmitted. Nothing else can happen to it. Hence, we can write

(1)

where * is flux, the time rate of flow of radiant

energy, and the subscripts r, a, t and i designate

the reflected, absorbed, transmitted, and incident

flux, respectively. If we divide both sides of

equation (1) by $i and define the reflectance, p, as 3>

T I$U the absorptance, a, as $a/$< and the

transmittance, r, as we get

P+«+t=1. (2)

For an opaque specimen, t=0, hence

P+«=l. (3)

Kirchhoff's law states that the absorptance, a,

is equal to the emittance, e,

(4)

hence, for an opaque body the reflectance and the

emittance sum to one

P+€=l. (5)

We thus see that for an opaque body the thermal

radiation properties are fully defined by either

the emittance or the reflectance.

Fluorescent and phosphorescent materials absorb radiant energy at one wavelength and re- emit it at another, usually longer, wavelength.

Because of this behavior, they may appear to

deviate from the above relationships when irradiated with heterochromatic flux, since at the

wavelength of fluorescent or phosphorescent

emission the reflected or transmitted flux may

exceed the incident flux. For such materials it is necessary to measure the spectral reflectance or transmittance by a technique in which the specimen is irradiated by monochromatic flux, and the

reflected or transmitted flux is measured by means

of a detector that senses only flux of the same

wavelength as the incident flux. The flux emitted

by phosphorescent or fluorescent emission is not

considered to be reflected flux.

2. Definition of Terms

Up to this point we have described thermal

radiation, thermal radiation properties, and flux.

It will be appropriate at this time to more rigor125
ously define a number of additional terms that

we shall be using. 1 Reflectance, p, is the fraction of the incident

flux that is reflected without change in frequency.

Absorptance, a, is the fraction of the incident

flux that is absorbed and converted to another

form of energy.

Transmittance, r, is the fraction of the incident

flux that is transmitted without change in

frequency.

All of these properties are influenced by the

wavelength, direction of incidence, and degree and

direction of polarization relative to the plane of

incidence of the incident flux, and, for reflectance

and transmittance, by the size and direction of

the solid angle over which the reflected or transmitted flux is collected for measurement. They

may also vary somewhat with the temperature of

the specimen. Hence, these terms are modified to

indicate these conditions.

Spectral means at a given wavelength, or in a narrow wavelength band centered about a given

wavelength.

Directional means in a given direction, usually

measured from the normal to the surface.

Normal is a special case of directional, and

means in a direction normal to the surface.

Specular, referring to reflectance, means in the

direction of mirror reflection.

Diffuse, or hemispherical, referring to incident

flux, means incident uniformly from all possible

directions.

Hemispherical, referring to reflected, transmitted, or emitted flux, refers to flux summed

over a hemisphere.

The terms reflectance^ absorptance, and transmittance are defined as fractions of incident flux,

and are dimensionless numbers. The numerical

value, being a ratio, is independent of the magnitude of the incident flux, but for equations (2)

and (3) to be valid, the conditions of irradiation

must be the same for each term in the equation.

This means that the spectral or wavelength

composition and state of polarization of the

incident flux and its direction of incidence must

be the same for each term.

A blackbody radiator absorbs all radiant

energy incident upon it, and emits radiant energy

at the maximum possible rate for any body at

its temperature. The amount of flux emitted by

a blackbody radiator is given by the StefanBoltzmann equation, its spectral distribution by

the Planck equation, and its geometric distribution by Lambert's cosine law. These are the

fundamental laws of thermal radiation.

The Stefan-Boltsmann equation is M=aTi (6)

1 There has been much confusion in nomenclature in this area. Judd [1] has published a recommended system of nomenclature for reflectance terms.

( Figures in brackets indicate the literature references at the end of this paper)

.

where M is the radiant exitance, or total flux

emitted per unit surface area by a blackbody

emitter in units of Wm-2

, a is the StefanBoltzmann constant, 5.6697 x lO-8 Wm^K"4

, and T is the absolute temperature in kelvins.

Lambert's cosine law may be expressed as follows

It =I0 cos 6 (7)

in which I9 is the intensity per unit area in units

of Wsr-1cm-2

of a blackbody radiator in a

direction 6, 6 being the angle between the given

direction and the normal to the emitting surface,

and In is the intensity per unit area of the blackbody

radiator normal to its surface.

Planck's radiation equation is Mx =Cl\-5 [exp (c2/\T)—l]

-1

(8)

where Mx is the spectral radiant exitance of a blackbody radiator per unit wavelength interval

at wavelength X, in units of watts per square

meter and meter wavelength interval, cx is the

first radiation constant, 3.7415 x 10-16 Wm2

, X

is the wavelength in meters, c2 is the second

radiation constant, 1.43879 x 10~ 2 mK, and T is the absolute temperature, in kelvins.

No real material is a true blackbody radiator;

hence, for all real materials, the flux emitted is less than that emitted by a balckbody radiator.

However, the radiation laws can still be used for

real materials by using the emittance as a proportionality factor.

Emittance, e, is defined as the ratio of the radiant

exitance of a specimen (radiant flux per unit area

emitted) to that of a blackbody radiator at the

same temperature and under the same conditions.

Emissivity is defined as the emittance of a specimen having an optically smooth surface and

a thickness sufficient to be completely opaque,

and is a property of the material of which the

specimen is composed.

Emittance varies with the direction of emission

relative to the surface of the specimen, the wavelength of the emitted radiation, and the temperature of the specimen. The emitted radiation may

be elliptically polarized, particularly if emitted

at large angles from the normal to the surface of

the specimen. The terms directional, normal,

hemispherical and spectral apply to emittance as well as to all of the other terms previously defined.

The term total, referring to emittance, means at

all wavelengths. This term has no specific meaning for reflectance, absorptance, and transmittance, unless the spectral distribution of the

incident flux, or source, is specified. For emittance,

however, it is only necessary to specify the temperature, since the spectral distribution of flux

emitted by a blackbody radiator is given by the

radiation laws as a function of temperature.

For Kirchhoff's law, equation (4), to be rigorously correct, 2 it is necessary that the spectral

2 See reference [2] for a proof of Kirchhoff's law.

126
composition, degree of polarization, and direction

of incidence of the incident flux for the absorptance evaluation should be the same as the spectral

composition, degree of polarization, and direction

of emission for the emitted flux in the emittance

evaluation. This would appear to impose a severe

restriction upon the general applicability of Kirchhoff's law. However, there are no polarization

effects for normally incident flux, and the effects

become of appreciable significance only for specularly reflecting materials at angles near the

Brewster's angle. This point will be discussed in

more detail later. For spectral incident flux the

spectral distribution within the narrow wavelength interval approximates that of blackbody

radiation sufficiently closely that the effect of

the deviation is negligible. Hence equation (3)

applies to normal spectral values of absorptance

and emittance, and can be used to convert one

to the other.

Normal spectral emittance over a wide wavelength range, at a given temperature, can be con- verted to total emittance at the same temperature

by multiplying the spectral values, wavelength

by wavelength, by the value of the Planck distribution function at the same wavelength to

produce a curve of emitted flux as a function of

wavelength. This curve is integrated to obtain

the total emitted flux, which is then divided by

the total integrated flux for a black body at the

same temperature to obtain the total normal

emittance. For most nonconducting materials,

the emittance, either total or spectral, does not

change appreciably with angle of emission at

angles of emission less than about 50° from the

normal. For conductors such as metals, the effect

begins to appear at angles greater than 10 to 15°

from the normal.

Spectral emittance and absorptance are not

strong functions of temperature, hence values

obtained at one temperature can be used to

predict values at other temperatures with only

small errors, if the difference in temperature is not great. Total emittance, on the other hand,

may or may not vary appreciably with temperature

depending upon the shape of the spectral emittance

curve. Figure 1 shows the variation with temperature

of the total hemispherical emittance of Inconel

coated with two different ceramic coatings, and

figure 2 shows the variation with temperature of

total hemispherical emittance of electropolished

and of oxidized type 321 stainless steel. Figure 3

shows variation with temperature of normal

spectral emittance of alumina, thoria, magnesia

and zirconia.

Reflectance is usually easier to measure than

absorptance, and spectral absorptance is frequently computed from spectral reflectance by

use of equation (3). In this case, the reflectance

must be measured under conditions of hemispherical viewing, and the resulting absorptance is for

1.00 <

i-^ .9 0

<o

E

IdX

Q_CD

UJI<O

.80

10

.90

.8 0

.70

1 1 1 ——

T

INCONEL A- 418 COATING

o specimen oi 1-0.036 mm

- -o- SPECIMEN 012-0.041 mm

"v.

-Ov

-«o.

(i^_^^

INCONEL N-143 COATING

- o SPECIMEN 09- 0.056 mm - -o SPECIMEN OIO-0.063mm

-

-O- o ' m

200 400 600 800

TEMPERATURE, °C

1000

Figure 1. Total hemispherical emittance of composite specimens consisting of Inconel coated

with NBS coating A-418, top curve, and NBS

coating N-143, bottom curve. Open points represent values obtained during heating;

closed points, during cooling.

1.0

0.9 -

< 0.8

,0.7

I 06

0.5

0.4

0.3

0.2

O.I

32I S.S. OXIDIZED IN UR.

AT lOOO°C

H SPECIMEN C-l, OXIDIZED I/2 HOUR

-If- SPECIMEN C-3.OXI0IZED 3 HOURS

v SPECIMEN C-4, OXIDIZED 5 HOURS

ELECTROPOLISHED TYPE 32I STAINLESS STEEL

Hut

A SPECIMEN I

H SPECIMEN D

4- SPECIMEN 7

v SPECIMEN 9

200 400 600 800 TEMPERATURE °C

I000 I200

Figure 2. Total hemispherical emittance of type

321 stainless steel specimens in the electro- polished condition, bottom curve, and after oxidation in air at 1000 °C for various times,

top curves. Open points represent values obtained during heating; closed points, during cooling.

the same conditions of irradiation as the reflectance.

Thermal radiation properties are frequently

referred to as surface properties. This is obviously

127
LU 0

1 1 1 1 1 1 1

ALUMINA

'

4.5/A 0

I 1 1 > i i i

THORIA

- •

Z

P- o

MAGNESIA

o—

0—~"°

ZIRCONIA

5.5 fL

4.5/i ^o——°

9~

~~0

^'

i i i i i i I

1 1 1 1 I 1 I

TEMPERATURE -°K

Figure 3. Temperature dependence of normal spectral

emittance of four polycrystalline ceramic oxides.

a gross approximation, and is true only to the

extent that the thermal radiation properties can be markedly affected by the surface condition of

the specimen being measured, particularly its

roughness. The processes of absorption and

emission involve interaction of the electromagnetic waves with material particles, and hence

cannot occur at a surface, which is only a twodimensional interface. While in one sense reflection occurs at an interface, even this process involves

penetration of the incident electromagnetic waves

beyond the interface to a depth that may be

several times the wavelength.

The function of the surface, or interface, in

determining thermal radiation properties is that

it affects the amount and geometric distribution

of the reflected flux, and the amount and distribution of the flux that penetrates the interface

to where it can be absorbed and transmitted. It

also affects the fraction of the internally generated

flux that penetrates the interface and is emitted,

and to a lesser extent, its geometric distribution.

This will be discussed in more detail later.

Surface contamination, even in amounts that

are difficult to detect by normal chemical means,

can also significantly affect thermal radiation

properties.

For metals, all of which have a very high

absorption coefficient over most of the wavelength range of thermal radiation, the processes

of emission, absorption and reflection occur in a very thin surface layer, perhaps only a few hundred

atomic diameters in thickness. Hence for metals,

it is a fair approximation to say that the thermal

radiation properties are surface properties. Ceramic

materials, on the other hand, usually have much

lower absorption coefficients than do metals, at

least at some wavelengths, and for such materials

the thermal radiation properties must be considered to be volume properties.

The thermal radiation properties of a specimen

can be affected by the contour of its surface. The

specimen will have the highest reflectance or the

lowest emittance if its surface is completely

smooth. Increasing surface roughness will decrease

reflectance and increase emittance. In the case

of a metal, the effect can be large. For instance

the total hemispherical emittance of type 321

stainless steel at 400 °C increased from about 0.2

in the electropolished condition to about 0.55

when sandblasted [3]. The normal spectral emittance of polycrystalline ceramics, on the other

hand, has been shown to be nearly independent

of surface roughness [4].

Bennett [5] has derived the following equation

relating reflectance of a metal at normal incidence

to surface roughness:

p/p0 =exp- (47r<r/\) 2 +327r>/\)4 (Aa/m) 2

, (9)

in which p is the reflectance of a rough metal

surface for normally incident flux reflected into

an optical system having an acceptance solid

angle with a half angle Aa (Aa=7r/2 for hemispherical collection), p0 is the reflectance of a

perfectly smooth surface of the same metal under

the same conditions, a is the root-mean-square

(rms) roughness of the surface, X is the wavelength of the incident radiation, and m is the rms

slope of the surface.

The first term, exp— (4tt(t/X) 2

, accounts for

interference effects, and predominates when a/\

<<1.0. The second term accounts for geometric

effects, and predominates when cr/X>>1.0.

The treatment becomes almost hopelessly complex at oblique incidence, because of polarization

effects. If such effects are neglected, equation (9) can still be used by substituting for a the term a

cos 0, where 0 is the angle of incidence, measured

from the normal to the gross plane of the surface.

3. Wavelength Range of Interest

The spectral distribution of the radiance (the

flux per unit wavelength interval, unit solid angle

and unit area of source projected normal to the

direction of emission) of a blackbody radiator is given by the Planck equation:

Zx=1.19096X-5

x[exP (l^syg/x^-i^xio-16, (io)

where Lx is the radiance in watts per square

meter, steradian, and meter wavelength interval

(Wm"2sr

_1 )m_1

, X is wavelength in meters and

T is temperature in kelvins. A plot of L\ as a function of X for a blackbody at 1000 K is shown

in figure 4. The nature of equation (10) is such that there is some energy emitted at all wavelengths between

zero and infinity, but the amount at very short

and very long wavelengths is so small that it can

be neglected. At any temperature, 99 percent of

128
Figure 4. Relative spectral radiance of a blackbody radiator, plotted as a function

of XT in cmK, curve A, and percentage increase in spectral radiance produced

by a one percent increase in temperature, plotted as a function of XT in cmK,

curve B.

the energy is emitted between wavelengths represented by values of XT of 1322 and 29372 umK,

and about 25 percent of the energy is at wavelengths shorter than the peak. At room temperature (300 K) 99 percent of the energy is at

wavelengths between 4.4 and 97.9 jum. As a

practical matter, most determinations of thermal

radiation properties cover the wavelength range

of 1 to 15 pim, with others covering the extended

range from about 0.25 to 40 p.m. There is considerable interest in spectral emittance at 0.65

fj.m, the wavelength often used in optical pyrometry. Most of our discussion will be confined to the

1- to 15-jum range. Figure 5 shows how the wavelength limits for 99 percent of the radiant energy

change with temperature.

4. Theoretical Considerations

When two transparent media of different indices

of refraction are separated by an optically smooth

interface, a beam of radiation striking the interface will be refracted according to Snell's law, and

reflected according to Fresnel's law. Snell's law is expressed as:

rix sin 6i=n2 sin 0r , (11)

where %x is the index of refraction of the medium

from which the beam is incident, d t is the angle of

incidence, measured from the normal to the interface, n2 is the index of refraction of the medium

into which the beam is refracted, and 0r is the angle

of refraction, again measured from the normal to

the surface. The equation is valid over the entire

range of 6x when n2 ^>ni. When nx^>n2 , the equation

is still valid at values of 0, less than the critical

angle. At angles greater than the critical angle,

total internal reflection will occur. The critical

angle, 6C , is defined by the equation

0c =sin- 1 (n2 /nx). (12)

Fresnel's law, in its general form, is expressed as:

_ , cos dj—

n

2 cos 6r ~V

p

\_nx cos Qi+rh cos 8 rj

Vn2 cos dj—nx cos 0rT

\_n2 cos d t -\-nx cos 0 rJ

where p is the reflectance of the interface for

unpolarized incident flux. Equation (13) can be

reduced, by substituting for nx or n2 from equation

(11), to

fsin2 Ran2 fo-m (

, P ~ 1/2

Lsin2 (e i+6r)r1/2Ltan2 (0 i+0,)J

(14)

In equations (13) and (14) the first term gives

the reflectance for radiation polarized perpendicular to the plane of incidence, and the second

term that for radiation polarized parallel to the

plane of incidence. The place of incidence is the

plane containing the direction of incidence and

the normal to the interface (surface). There is

some disagreement in the literature as to what

constitutes the plane of polarization. In this paper

it is defined as the plane containing the E (electric)

vector.

It can be seem from equation (14) that when

(d t +dT ) = Tr/2, tan (0i+0r ) goes to infinity, and the

reflectance for radiation polarized parallel to the

plane of incidence goes to zero. The angle at

129
SOLAR range WAVELENGTH, MICRONS

WAVELENGTH, MICRONS

Figure 5. Wavelength limits encompassing 99 percent of the flux emitted by a blackbody radiator at the indicated

temperature, for high temperatures, top curve; and low

temperatures, bottom curve.

which this occurs is known as the Brewster's

angle, 6B, and is defined as:

0B=tan-1

n2AH (15)

Equations (11) through (15) were derived for a perfect dielectric that is optically homogeneous.

Such a material has no absorption. Although no

real material conforms completely to this model,

the equations can be applied with no significant error to transparent dielectric materials in general.

For materials of which the absorption coefficient

is not negligible, the complex index of refraction,

n*, must be substituted for the simple index of

refraction, n, in the equations.

n*=n-\-ik (16)

where i=V—1, and k is the absorption index.

k=a\/4:w (17)

where a is the absorption coefficient, and X is wavelength.

The absorption coefficient, a, is defined by the

equation

$x =$o exp(-aa:) (18)

where <i>z is the flux remaining after traversing a thickness x of nonscattering material, and <£0 is the flux at x=0.

Let us assume that we have an optically homogeneous specimen of a dielectric material, having

two parallel faces, A and B, that are optically

smooth and essentially infinite in extent, so that

edge effects can be neglected. We will further

assume that interference effects can be neglected.

This is a reasonable model for a sheet of polished

plate glass, or a single crystal of quartz or sodium

chloride, for instance. When a beam of radiation

is incident on surface A of such a specimen, it is partly reflected and partly refracted, according to

the Fresnel equation (13) or (14). The radiation

that is refracted is transmitted undeviated until

it hits the second surface, B, where it again undergoes reflection and refraction according to the

Fresnel equation. In its course through the material, it is partially absorbed according to equation (18). By following the beam until it is completely absorbed or has left the specimen, we can

compute the reflectance, transmittance, and absorptance of the specimen. All of the flux initially

reflected by surface A plus that leaving the specimen through surface A makes up the reflected

portion, that leaving through surface B makes up

the transmitted portion, and that absorbed makes

up the absorbed portion. These portions divided

by the incident flux are the reflectance, transmittance, and absorptance.

Gardon [6] gives a very complete account of

the emittance of isothermal transparent materials.

He derives a spectral volume emissive power, j\,

defined by the equation

j,=axn

2MB^

(19)

where MB,\ is the spectral exitance of a blackbody

radiator at the temperature of the specimen.

MB,x can be obtained from the Planck equation.

His final equation for the spectral radiance on

the inside of the surface, L.x{pXj at wavelength X and angle <p for a specimen of thickness X is

L^=n2 [1— exp (—aX/cos <p) cos <p], 7T

(20)

and for the internal spectral irradiance on the

inside of the surface, jSJiXX) at wavelength X for a specimen of thickness X,

EiXx =n2Mb ,x[2- exp(- a*X)- (ax X)2 -£i(- axX)]

130
where the exponential integral £*(— p) is denned

as — = e~"q~ 1dq. The flux that is incident

internally on the surface of the specimen undergoes reflection and refraction. All of the flux that

is incident at angles greater than the critical angle

is totally reflected. For a specimen thick enough

to be opaque, this is a fraction 1 — 1 jn2 of the total

flux. For a transparent specimen, the fraction is

larger. The critical angle, 9 C , is defined by equation (12). The fraction of the internally incident

flux that is incident at angles less than 6 C is re- fracted to fill the entire hemisphere on the

external side of the surface, and the fraction transmitted at any angle can be computed from equation (14) by setting d r as the internal angle of

incidence and 6 { as the external angle of refraction.

For transparent birefringent single crystals

each ray appears as two plane polarized components with their planes of polarization normal to

each other, and the fraction of each component

is determined by the orientation of the ray relative

to the crystal axes of the specimen. For such a specimen the index of refraction will be different

for the two portions of the ray, and because the

two portions are plane polarized, the two parts of

equation (14) must be used separately. As a further complication, unless the plane of incidence

happens to include a crystallographic axis of the

crystal, neither of the planes of polarization will

be in the plane of incidence. The situation where

there is a thermal gradient normal to the surface

of the specimen has been treated in a second paper

by Gardon [7]. For an optically inhomogeneous specimen, the

situation becomes so complex that Gardon's approach is no longer valid. A single ray undergoes

so many internal reflections and refractions that

it soon loses its identity. Practically all polycrystalline materials fall into the category of

intensely scattering materials, which have been

treated by Kubelka [8] and by Hamaker [9]. The

scattering occurs by reflection and refraction

whenever the index of refraction of the material

changes along the path of the ray, primarily at

pores, grain boundaries, or inclusions. For materials of this type, an entirely different mathematical treatment is required.

Consider a specimen of very large area, so that

edge effects may be neglected, that is irradiated

diffusely over its entire area. Under steady-state

conditions, the incident flux not reflected at the

surface will penetrate into the specimen, diminishing with the depth of penetration; the amount

of the diminution at any point along a plane

parallel to the surface will be a function only of

the distance of the plane from the surface. Under

these conditions a one-dimensional analysis can be

used. All flux lost by lateral scattering will be

compensated by an equal gain through similar

scattering from the surrounding area.

The flux is separated into two portions, /,

traveling outward from the interior of the specimen, and J, traveling inward from the surface,

both traveling normal to the plane of the surface.

Both are attenuated by scattering and absorption.

A diffuse absorption coefficient, K, is defined by

equating KI dx to the reduction in I by absorption within an infinitesimal layer dju. It should be

noted that the diffuse absorption coefficient, K,

is exactly twice the absorption coefficient, a, referred to previously. A diffuse backscattering

coefficient S is similarly defined by equating SI dx

to the flux scattered backwards from / and added

to J in the layer dx.

Within the distance dx the flux / will be diminished by absorption and backscattering, and

reinforced by the flux backscattered by J. Hence

we can write:

dI/dx=-(K+S)I+SJ, and (22)

dJ/dx=(K+S)J-SI. (23)

These are the basic differential equations used by

Kubelka [8], Hamaker [9], and others.

Klein [10] derives the following equations for

those materials where there is essentially no

specular reflection at the surface. These equations

should apply to most unglazed ceramic specimens,

particularly if the surface is not polished:

(1—ft,

2

) sinh<r0Z)

p ~(l+,82

) sinh (ToD+2/30 cosh cxJD

2ffo (Bp sinh o-0J>+cosh a0D—1)

a-

(l+(82o) sinh <toD+2I30 cosh aJD

T = (24)

(l+ft>2) sinh <r0D+2l3o cosh aJ)

V '

in which p0 =[K/(K+2S)] 1/2

,

a0 =[K(K+2S)] 1/2

, and

D=thickness of specimen.

Equation (24) has been further developed [11]

to include the effects of external specular reflectance of the specimen and internal specular

reflection at both surfaces of the specimen. The

final equation is

p(27r;2x)=pe +(l-Pe)(l-p! )

( 1 -ft,) Afe"Q°- (1 +&,) Oe-'o0

, X ?)

MNe°oD-OPe-°oD K '

where p(27r;27r) is the hemispherical reflectance

of the specimen for completely diffuse irradiation,

pe is the specular reflectance of the specimen front

surface for externally incident diffuse flux, p, is the specular reflectance of the specimen front

surface for internally incident diffuse flux, ps is

131
the specular reflectance of the specimen back

surface for internally incident diffuse flux, and

Af=(l+/30)-A.(l-jS0), AT=(l+/3o)-p*(l-/3o),

O=(l-j8o)-p,(l+.j8„),

P=(l-/30 )- Pi (l+/3o).

For a specimen thick enough to be completely

opaque, equation (27) reduces to:

nt9-.<hr\ —n | (1 —Pe) (1 —Pi) (1 —ft)) p(2rr,2,)- Pe + (28)

and for such a specimen:

e(27r) = l-p(27r;2ir), (29)

which states that the hemispherical omittance is equal to one minus the hemispherical reflectance

for diffuse irradiation.

The absorption coefficient, either a or K, is primarily a function of composition, and is usually

not greatly affected by the structure. However,

for some impurities, the effect on the absorption

coefficient will depend on whether they are present

in solution in the crystals of the primary constituents or as separate crystallites at grain

boundaries, and if present as crystallites, by their

size. The scattering coefficient, on the other

hand, is markedly affected by the structure of the

specimen, particularly the grain size and porosity.

A few years ago we observed an interesting

example of the effect of the form in which impurities are present on the thermal radiation properties

of a ceramic material. We are making some measurements on sintered magnesium oxide. The

material as received was a dirty brown color,

which we assumed was due to contamination by

iron oxide. A specimen was treated with hydrochloric acid in an effort to remove the contaminant,

then washed and heated with a torch to drive off

any remaining chlorides. After this treatment

the specimen was white, and we assumed that

the contaminant had been removed. However,

when the specimen was heated in a furnace for

measurement of its emittance, it regained its

dirty brown color on cooling to room temperature.

We then discovered that the color change was

reversible. When the specimen was cooled rapidly

from a high temperature, it was white; when

cooled slowly, it was dirty brown. The explanation

offered was that the impurity, presumably iron

oxide, went into solution in the MgO crystals at

high temperature, and remained in solution on

rapid cooling, where it had little effect on reflectance. On slow cooling, the impurity precipitated

out at grain boundaries, where it had an appreciable

effect on reflectance.

Typical values of j8 0 , 0o> & and K are as follows.

For a highly opaque white porcelain enamel with

a reflectivity of about 0.9, /3 0 =0.05 and o- 0 =50

mm"1

, K=2.5 mm"1 and £=500 mm-1

. For a

polycrystalline ceramic with a reflectivity of

about 0.82 that is essentially opaque at a thick- ness of 3 mm, the values would be /3 0 =0.1,

0-0=1.67 mm"1

, #=0.167 mm"1 and S=8.25

mm-1

. For a material with a reflectivity of about

0.6 that is essentially opaque at a thickness of

25 mm, /3 0 =0.25, o- 0 =0.2 mm-1

, K=0.05 mm"1 and #=0.375 mm-1

. A material such as an

opalescent glass might have these properties.

In general, K can vary from near zero for materials

composed of highly transparent crystals, to a value as high as 50,000 mm-1 for some metals at

some wavelengths. S will vary from near zero

for an optically homogeneous material such as an

optical glass or a pure metal to a high of perhaps

1000 mm-1 for some white paints. It should be

emphasized that for light-scattering materials

the reflectivity is related to /3 0 , which is a function

of the ratio of K to S, and not to the absolute

value of either K or S, while the extinction

coefficient, which determines the opacity of a

material, is related to the product SK.

The usual laws of reflection and refraction at

an interface between two materials of different

indices of refraction, equations (14) and (11)

above, apply only if there is a finite thickness of

material on each side of the interface. This finite

thickness is of the same order of magnitude as the

wavelength of the radiation involved. The phenonemon can be explained quantitatively in terms

of frustrated internal reflectance [12]. If two 45°

totally reflecting prisms are placed so that their

totally reflecting surfaces are parallel, and are then brought slowly together, the totally reflecting

surfaces will begin to transmit as they approach

each other. The transmission for polarized radiation with its electric vector in the plane of incidence is given by

where p is the reflectance of the interface, r is its

transmittance ( p +t=1), iVis the index of refraction ratio, d is the distance separating the prisms

and X is the wavelength of the radiation involved.

For these conditions, p/t=% that given in equation (30) when the electric vector is normal to the

plane of incidence. The index of refraction ratio is

N=n-il,n2, where nx is the index of the prisms and

n2 is that of the layer. When the layer is vacuum,

N=nu When it is air, N=n1 with an error of

about 3 parts in 10,000.

Because of this phenomenon, there is a marked

effect of the size of particles such as inclusions,

crystallites, or pores on the scattering efficiency

of each particle. The scattering efficiency per

particle will decrease markedly with diameter

132
below a critical diameter. As a rough approximation, the total scattering power will be proportional to the total interface area, and from this

standpoint the scattering particles should be as small as possible. The net result of the two effects

is that there is an optimum size for scattering

efficiency. A study of rutile pigments in paint

[13] has shown that in this case the optimum

diameter of the pigment particles is about 0.4

times the wavelength of the light being scattered.

The scattering power of a particle diminishes as the distance separating it from its nearest neighbors becomes less than the critical distance

mentioned above. On this basis the maximum

scattering power per particle would occur when

particles of the optimum size are separated by a distance at least equal to their diameter. If

spherical particles were arranged in a cubic array,

this would occur at volume concentrations of

particles of less than about 6.6 percent. Actually

the total scattering for all particles increases as the number of particles increases, because of the

increase in total interface area, until the particles

get so closely packed that the decrease in effective

interface area is greater than the increase in total

interface area. Reference [13] indicates that for

the rutile pigment studied, the maximum total

scattering occurred at a pigment volume concentration of about 25 percent.

Radiant heat transfer within and from optically

inhomogeneous materials with no specular reflection at the surface is covered by Hamaker [9] and

Klein [10], for the case where there is a thermal

gradient in the material.

5. Thermal Gradient Problems

The general physical properties of ceramic

materials combine to make thermal gradients a

serious problem in the evaluation and use of

thermal radiation properties of ceramics. Ceramic

materials in general tend to be somewhat translucent, and hence emit and absorb thermal radiation within a surface layer of appreciable thickness.

Ceramic materials in general also have low thermal

conductivity and high total emittance as compared

to metals. These properties combine to produce

th ermal gradients within a heated specimen unless

careful precautions are taken to minimize such

gradients. The gradients tend to be normal to a

surface that is emitting or absorbing thermal

radiation. As a further complication, the gradients

tend to be nonlinear near such a surface.

When a specimen is emitting from a surface

layer of appreciable thickness with a large thermal

gradient normal to the surface, it has no unique

temperature, and it is difficult to define an effective

temperature for the emitting layer. Emittance is denned as the ratio of the flux emitted by a specimen to that emitted by a blackbody radiator at

the same temperature and under the same conditions. It is thus necessary to define an effective

temperature for the nonisothermal specimen

before its emittance can be evaluated. If the

effective temperature is defined as that of the

surface, a specimen with a positive thermal

gradient (surface cooler than interior) will emit

at a greater rate than an isothermal specimen at

the same temperature, and in some cases may even

have an "emittance" greater than 1.0. If the

gradient is negative (surface hotter than interior)

it will emit at a lesser rate. If the "effective

temperature" is defined as that of an isothermal

specimen that emits at the same rate as the nonisothermal specimen, we find that the "effective

temperature" is difficult to evaluate even if the

extinction coefficient and thermal gradient are accurately known. Also, if spectral emittance is desired, we find that the extinction coefficient,

and hence the thickness of the emitting layer,

varies with wavelength, and we have the anomolous

situation of a specimen whose effective temperature varies with wavelength. There is no completely satisfactory solution to this problem.

The Planck equation does not apply to a nonisothermal specimen, and any attempt to apply it to such specimens will result in errors. Because of

these considerations, all measurements of emittance of ceramic specimens should be made under

conditions where thermal gradients within the

specimen are as small as possible.

6. Methods of Measurement

The properties of reflectance, absorptance, and

transmittance for most materials, and particularly

ceramic materials, are not strong functions of temperature, and hence these properties can be measured on nonisothermal specimens with but little error. However, such measurements are not easy

to make on hot specimens. First, all of the reflected or transmitted flux must be measured, which involves collection over a hemisphere. Such measurements are difficult at wavelengths beyond about

2.5Mm, where conventional integrating sphere coatings decrease markedly in reflectance. Second, the

small amount of reflected or transmitted flux must

be measured in the presence of a much larger

amount of flux emitted by the hot specimen. Such

measurements require the use of chopped incident

flux and synchronous amplification of the signal

from the detector.

Because of the difficulties mentioned above,

most measurements of thermal radiation properties of ceramic materials have been direct emittance measurements, where the flux emitted by a hot specimen is measured and compared to that

emitted by a blackbody reference at the same

temperature.

Any temperature difference between the specimen and reference blackbody will cause an error

in the measured emittance that is much larger

than the percentage difference in absolute temperature. For total emittance measurements, the

percentage error in emittance is approximately

four times the percentage difference in tempera133
ture. This is illustrated in figure 4 for a one percent

difference in temperature. As seen in figure 4, curve B, the error in spectral emittance can be

much larger at wavelengths shorter than the peak

in the Planck curve, A.

In most cases where direct emittance measurements of ceramic specimens have been made, an

effort has been made to heat the specimen

under conditions where the thermal gradients are kept as small as possible. This may involve alternate heating and viewing of a moving specimen.

The specimen is enclosed in a furnace where it is heated then moved in front of a cooled port where

it is viewed briefly. It may also involve alternate

heating by radiation and viewing of a stationary

specimen, where the periods of heating and viewing are controlled by rotating shutters. Movable

specimens have been tested in the form of a rotating disc [14, 15, 16, 17], arotating cylinder [18,19], or as discs mounted on an oscillating beam [20].

Stationary specimens have been heated in an arc- image furnace [21, 22, 23, 24] and in a solar

furnace [25].

In the case of a moving specimen that is alternately heated and cooled, the specimen will reach

a steady-rate temperature, and if the motion is rapid enough that the viewing time is short, the

temperature fluctuation of the viewed surface will

be so small that it can be neglected.

Photon conduction does not generate a thermal

gradient in a material. The photons pass through

the material without affecting it in any way until

they are absorbed. As a result, a specimen that is heated by radiation tends to have a small thermal

gradient normal to the heated surface. The radiation is absorbed within the surface layer from

which the emitted radiation originates, and after

the specimen has reached a steady state condition,

there is little phonon conduction within the sur- face layer. If the time periods during which the

specimen is alternately heated and viewed are

brief, the temperature fluctuation of the specimen

will be small. Usually these periods are only a

fraction of a second. With both moving specimens

and radiantly heated stationary specimens the

thermal gradients normal to the surface can be

reduced to a point where they do not introduce a

significant error into the measurement.

The rotating cylinder method used at the National Bureau of Standards [18] will be described

briefly. The specimen was a hollow cylinder

25 mm (one inch) in diameter and 25 mm (one

inch) high, with an outer surface ground to be

round to about .05 mm (0.002 in). A cross sec- tional view of the furnace is shown in figure 6. The specimen was mounted on top of an alumina

pedestal, which revolved inside a platinum-wound

resistance furnace. The winding of the furnace

was designed so that there was no axial thermal

gradient along the specimen. The specimen re- volved in front of a water-cooled viewing port. A

theoretical analysis [26], confirmed by temperature

measurements at a point near the surface of a rotating specimen, indicated (1) that the temperature drop at a point on the surface of the rotating

specimen as it passed in front of the viewing port

was inversely proportional to the speed of rotation

of the specimen, and was only about 2 K at a speed

of 50 rmp, and (2) that the temperature measured

by a stationary, radiation-shielded, thermocouple

located at the center of the hollow rotating specimen was the same within about 1 K as that

measured by the thermocouple imbedded near the

surface of the rotating specimen.

The flux from the hot specimen was focused on

the entrance slit of one beam of a double beam

infrared spectrometer, and the flux from a laboratory blackbody furnace was focused on the

entrance slit for the second beam of the spectrometer. The blackbody furnace was controlled to the

same temperature as the specimen. The spectrometer automatically scanned over the spectral

range of 1 to 15 jum, and plotted the ratio of the

signals from the two beams, which after correction

for spectrometer errors, was the normal spectral

emittance of the specimen.

Typical data at 1400 K for alumina, thoria,

magnesia and zirconia are shown in figures 7, 8, 9,

and 10. The curves are similar in several respects.

The emittance is low at short wavelengths, where

the absorption coefficient for these materials is low, and high at long wavelengths, where the

absorption coefficient is high. The location of the

absorption edge, the steep part of the curve

connecting the high and low portion, and its

slope, is different for different materials. The height

of the low-emittance and high-emittance portions

of the curve is also different for the different

materials. The effect of temperature on the

spectral emittance at several wavelengths for

these materials is shown in figure 3. In general,

the spectral emittance increases with an increase

in temperature, and the percentage increase is

greater at short wavelengths, where the emittance

is low, than at long wavelengths, where the

emittance is high. The absorption edge also tends

to move toward a shorter wavelength as the temperature is increased, but the change is not large.

7. Summary

The terminology in the field of thermal radiation

properties and radiant heat transfer has been

reviewed, and the equations relating the thermal

radiation properties of ceramics to their optical

properties of index of refraction, absorption coefficient, and scattering coefficient have been given.

The problems involved in measuring the thermal

radiation properties of ceramic materials have

been discussed, one specific method has been

briefly described, and data on alumina, thoria,

magnesia, and zirconia presented.

134
WATER

BOOSTER

COIL

ALUMINA

, SUPPORT

WATER

T.C. LEADS

PLATINUM

RADIATION SHIELD

COPPER

VIEWING PORT

PROVISION FOR

NaCI WINDOW

STAINLESS : STEEL SPINDLE

SCALE : h- I" MOTOR DRIVE

(VARIABLE SPEED)

Figure 6. Cross-sectional view of the rotating cylinder specimen furnace used in measuring

normal spectral emittance of ceramic materials.

135
t I

i i

i

i

i

i

i

i—i—

r

j L

Figure 7. Normal spectral emittance of commercially pure

alumina at 1400K.

z

-1—1—1—1—1—1—1—1—1—1—1—1—1—

r

X- MICRONS

Figure 10. Normal spectral emittance of commercially pure

zirconia at 1400K.

.<

r ~i -1

[

1 1 r

-1 1 1 1 1 I 1 I 1

I 1I 2 4 6 8 10 12 14 X- MICRONS

Figure 8. Normal spectral emittance of commercially pure

thoria at HOOK.

I I I _l I I I

I I 10 12 X- MICRONS

Figure 9. Normal spectral emittance of commercially pure

magnesia at 1400K.

8. References

[1] Judd, D. B., Terms, definitions, and symbols in re- flectometry, J. Opt. Soc. Am. 57, No. 4, 545-552

(April 1967).

[2] Kneissl, G. J., and Richmond, J. C, A laser-source

integrating sphere reflectometer, NBS Tech. Note

439, 1968.

[3] Richmond, J. C, and Harrison, W. N., Total hemispherical emittance of coated and uncoated Inconel

and types 321 and 430 stainless steel, J. Res. NBS

66C (Engineering and Instrumentation), No. 3, 261-269 (1962).

[4] Richmond, J. C, Effect of surface roughness on emittance of nonmetals, J. Opt. Soc. Am. 56, No. 2, 253-254 (Feb. 1966).

[5] Bennett, H. E., Influence of surface roughness, sur- face damage, and oxide films on emittance, Proceedings of Symposium on Thermal Radiation of

Solids, San Francisco, 1964, ed. by S. Katzoff

(NASA Special Publication SP-55, Washington,

Superintendent of Documents, GPO 1965) pp. 145-

152.

[6] Gardon, R., The emissivity of transparent materials,

J. Am. Ceram. Soc. 39, No. 8, 278-287 (August

1956).

[7] Gardon, R., A review of radiant heat transfer in glass,

J. Am. Ceram. Soc. 44, No. 7, 305-312 (July 1961).

[8] Kubelka, P., New contributions to the optics of

intensely light-scattering materials, J. Opt. Soc. Am. 38, No. 5, 448-457 (May 1948).

[9] Hamaker, H. C., Radiation and heat conduction in light scattering materials, Philips Research Reports

2, 55-67, 103-111, 112-125, 420-425 (1947).

[10] Klein, J. D., Heat transfer, by radiation, in powders,

Doctoral Dissertation, Massachusetts Institute of

Technology (1960) Available from University

Microfilms.

[11] Richmond, J. C., Relation of emittance to other

optical properties, J. Res. NBS 67C (Engineering

and Instrumentation), No. 3, 217-226 (1963).

[12] Strong, John, Concepts of Classical Optics, p. 124

(W. H. Freeman and Co., San Francisco, 1958).

[13] Bruehlman, R. J., Thomas, L. W., and Gomek, E., Effect of particle size and pigment volume concentration on hiding power of titanium dioxide, Federation of Societies for Paint Technology Official

Digest, 33, No. 433, 252-267 (Feb. 1961).

[14] Slemp, W. S., and Wade, W. R., A method for measuring the spectral normal emittance in air of a variety

of materials having stable emittance characteristics,

Proceedings of Symposium on Measurement of

Thermal Radiation Properties of Solids, Dayton,

136
O., 1962, ed. by J. C. Richmond (NASA Special

Publication SP-31, Washington, Superintendent of

Documents, GPO, 1963) pp. 433-439.

[15] Clayton, W. A., A 500° to 4500° F thermal radiation

test facility for transparent materials, Proceedings

of Symposium on Measurement of Thermal Radiation Properties of Solids, Dayton, 0., 1962, ed. bv

J. C. Richmond (NASA Special Publication SP-31,

Washington, Superintendent of Documents, GPO,

1963) pp. 445-460.

[16] Kjelby, A. S., Emittance measurement capability for temperatures up to 3000 °F, Proceedings of Symposium on Measurement of Thermal Radiation

Properties of Solids, Dayton, O., 1962, ed. by J. C. Richmond (NASA Special Publication SP-31,

Washington, Superintendent of Documents, GPO,

1963) . [17] McMahon, H. O., Thermal radiation characteristics

of some glasses, J. Am. Ceram. Soc. 34 No. 3, 91-96 (1951).

[18] Clark, H. E., and Moore, D. G., A rotating cylinder

method for measuring normal spectral emittance

of ceramic oxide specimens from 1200° to 1600° K,

J. Res. NBS 70A (Physics and Chemistry), No. 5, 393-415 (1966).

[19] Folweiler, R. C, Thermal radiation characteristics of

transparent, semi-transparent and translucent material under non-isothermal conditions, Air Force

Systems Command, Wright-Patterson Air Force

Base, Aeronautical Systems Division Technical

Documentary Report ASD-TDR-62-719 (April,

1964) . [20] Betz, H. T. Olson, O. H., Schurin, B. D., and Morris,

J. C, Determination of emissivlty and reflectivity

data on aircraft structural materials, Part II, Techniques for measurement of total normal emissivity, solar absorptivity and presentation of results,

Wright Air Development Center Technical Report

TR 56-222 Part II (Oct. 1958).

[21] Comstock, D. F., Jr., A radiation technique for determining the emittance of refractory oxides

Proceedings of Symposium on Measurement of

323-655 O - 69 - 10

Thermal Radiation Properties of Solids, Dayton,

O., 1962, ed. by J. C. Richmond (NASA Special

Publication SP-31, Washington, Superintendent of

Documents, GPO, 1963) pp. 461-468.

[22] Evans, R. J., Clayton, W. A. and Fries, M. A very

rapid 3000 °F technique for measuring emittance of

opaque solid materials, Proceedings of Symposium

on Measurement of Thermal Radiation Properties

of Solids, Dayton, O., 1962, ed. by J. C. Richmond

(NASA Special Publication SP-31, Washington,

Superintendent of Documents, GPO, 1963) pp. 483-488.

[23] Null, M. R., and Lozier, W. W., Measurement of

reflectance and emittance at high temperature with

a carbon arc image furnace, Proceedings of

Symposium on Measurement of Thermal Radiation

Properties of Solids, Dayton, O., 1962, ed. by J. C. Richmond (NASA Special Publication SP-31,

Washington, Superintendent of Documents, GPO,

1963) pp. 535-539.

[24] Wilson, R. G., Hemispherical spectral emittance of

ablation chars, carbon and zirconia (to 3700 °K),

Proceedings of Symposium on Thermal Radiation

of Solids, San Francisco, 1964, ed. by S. Katzoff

(NASA Special Publication SP-55, Washington,

Superintendent of Documents, GPO, 1965) pp. 259-275.

[25] Laszlo, T. S., Gannon, R. E. and Sheehan, P. J., Emittance measurements of solids above 2000 °C,

Proceedings of Symposium on Thermal Radiation

of Solids, San Francisco, 1964, ed. by S. Katzoff,

(NASA Special Publication SP-55, Washington,

Superintendent of Documents, GPO, 1965) pp. 277-286.

[26] Peavy, B. A., and Eubanks, A. G., Periodic heat flow- in a hollow cylinder rotating in a furnace with a viewing port, Proceedings of Symposium on Measurement of Thermal Radiation Properties of

Solids, Dayton, O., 1962, ed. by J. C. Richmond,

(NASA Special Publication SP-31, Washington,

Superintendent of Documents, GPO, 1965) pp. 553-563.

Elastic Deformation of Ceramics and Other Refractory Materials

J. B. Wachtman, Jr.

Institute for Materials Research,

National Bureau of Standards, Washington, D.C. 20234

Elastic deformation of ceramics is reviewed first from a continuum, thermodynamic

point of view; nonlinear effects and pressure dependence of elastic moduli are included.

The results of several important technological problems, including wave propagation,

resonance, and the effects of gross microstructure on elastic moduli, are discussed in terms

of continuum elasticity. The atomic theory of elastic moduli is then briefly sketched from

the lattice dynamics point of view and correlations of other physical properties with elastic moduli are discussed. Finally the measurement of elastic moduli and the characterization

of ceramics with respect to elastic moduli are considered.

Key Words: Ceramics; elastic constants; lattice dynamics; porosity; resonance; wave propagation.

CONTENTS

Page

1. Introduction 139

2. Continuum elasticity 142

2.1. Thermodynamics and simple pressure

volume relations 142

2.2. Stress 143

2.3. Strain . 144

2.4. Linear elasticity , 145

2.5. Nonlinear elasticity 148

2.6. Generalizations of linear elasticity 149

3. Specific problems involving elastic deformation. 150

3.1. General discussion 150

3.2. Elastic waves in isotropic materials 150

3.3. Elastic waves in anisotropic materials 151

3.4. Attenuation of elastic waves by scattering

and nonlinear effects 152

3.5. Resonance of isotropic materials 152

3.6. Resonance of anisotropic materials 152

3.7. Effect of porosity on elastic moduli of iso- tropic materials 153

3.8. Relation of elastic moduli of polycrystalline

materials to the moduli of the single-crystal

grains 153

5.

6.

Page

3.9. Elastic moduli of composites 154

Atomic theory of elasticity 154

4.1. Lattice statics 154

4.2. Linear lattice dynamics: Relation of elastic constants to sound velocity, specific heat,

optical absorption, and dielectric constant. 156

4.3. Nonlinear lattice dynamics: Thermal ex- pansion and temperature dependence of

elastic constants 157

4.4. Effect of solid solution and phase changes on elastic constants 157

4.5. Internal friction and the AE effect 159

Special topics 160

5.1. Correlations of other properties with elastic constants 160

5.2. Elastic moduli of graphite 160

5.3. Measurement of elastic moduli 160

5.4. Characterization of ceramics with respect

to elastic moduli 161

References 162

1. Introduction

Anyone reading in modern elasticity theory

might well obtain the impression of a vast and

complicated field with little contact with laboratory experiment or technological practice. Yet

much of that theory is based on sound analysis.

Its applicability and relevance of some of its

results cannot be seriously doubted. The more

experimentally inclined and the more technologically interested should be given a clear statement of the principal conclusions derived. This

then is such an attempt at a review of elastic

theory for the ceramist.

Elastic properties of ceramics are of interest to a variety of scientists and engineers in relation

to a wide range of problems. They provide one of

the most direct measurements of chemical bonding

and accordingly are needed as basic data in calculations of cohesive energy, surface energy, theoretical

strength of crystals, a wide range of properties

such as specific heat which are interrelated by

lattice dynamics, point defect properties, and dislocation properties. They are needed for direct

engineering use in calculating the stress distribution and deflection of structures under static load

and their resonances under dynamic loads, the

values of thermal stresses, and the stress distribution in composites. They are needed in designs of

electromechanical transducers and of various

devices employing ultrasonic wave propagation.

139
Elastic properties are needed to interpret wave

propagation in the earth as used in fundamental

studies of its composition, in prospecting, and in

study of soils. Elastic moduli, or quantities

directly dependent upon elastic moduli, are sometimes measured as a quality control technique. We seek here to provide sufficient background knowledge on the elastic properties of

ceramics and the factors upon which they depend

to facilitate their use in such applications. We

hope also to convey some impressions of the

interesting and active research going on in many

areas of elasticity.

Solids respond to the application of stress by

undergoing deformation which may be either

elastic (i.e. recoverable) or plastic (i.e. not re- coverable). This definition of elastic deformation

includes not only instantaneous deformation which

is linearly proportional to the stress, but also

includes deformation which is time dependent

and/or nonlinearly related to stress, provided the

deformation is fully recoverable in time, however

long. The subject of elasticity is frequently developed by introducing empirical relations between

stress and strain, but may also be developed from

the point of view of thermodynamics in which a generalized form of the Helmholtz free energy is written as a function of the strain and temperature.

The usual stress-strain relation at constant temperature is then obtained by differentiating the free

energy; the second derivatives are the isothermal

elastic constants. This thermodynamic approach

is useful in treating the relation of the elastic

properties of solids to other properties; solids also

respond to stress by changes in their thermal,

electrical, and magnetic properties. The latter are not properly a part of the subject of elasticity,

but a number of equations can be derived from

thermodynamics which express relationships between the value of an elastic property measured

under one condition, such as constant electric

field, to the value of the same property measured

under a different condition, such as constant

electric displacement.

In the usual thermodynamic development of

elasticity one treats a solid as a continuous

material and the resulting treatment is strictly

applicable only to volumes containing many

atoms. Within this limitation many important

scientific and technical problems can be treated.

Section 2 of the present paper outlines the development of continuum elasticity including a brief

discussion of single crystal effects (anisotropy) and

nonlinear effects.

In section 3 the use of the continuum theory to

solve various problems of technological importance

is considered. The application of this theory to

resonance, wave propagation, and the relation of

the texture and microstructure of ceramics to

their elastic properties is briefly discussed.

Section 4 discusses the relation between the

atomic structure of solids (including defects) and

their elastic properties. On one level this involves

lattice dynamics in which each atom is treated

as a point mass (or as a point mass and concentric

shell) connected by springs to other atoms; the

resulting theory leads to useful relations between

elastic properties and a variety of other properties such as specific heat, index of refraction,

dielectric constant at low frequency, cohesive

energy, and surface energy. If nonlinear springs

are used, relations to anharmonic properties such

as thermal expansion, thermal conductivity, and

high frequency scattering result. On a deeper

level elastic properties should be derivable by

detailed calculation starting from fundamental

constants; recent work in this area is beyond the

scope of this paper. Some physical properties,

such as phase changes and solid solution effects, are discussed in section 4. The effect of structural

defects in producing anelastic (time dependent

but recoverable) deformation is then discussed.

In section 5 special topics including elastic

moduli of graphite and the correlation of other

physical properties with elastic moduli are discussed. Experimental techniques for determining

elastic properties have been greatly improved by

modern electronic instrumentation. These techniques are described and estimates of their

accuracy are given.

The paper is concluded with a discussion of

the features of composition and microstructure

upon which elastic properties of ceramics depend.

Throughout the paper, qualitative features are described and just enough mathematics is introduced to indicate the manner in which the quantitative treatment is developed; the mathematic

details are left to the references as much as

possible. The subject, however, demands the introduction of an appreciable number of symbols

and the use of a subscript notation; the symbols

approved by IRE [1]

1 are used (except that we

retain £ and T without subscript as entropy and

absolute temperature respectively), and are summarized in table 1. Table 2 lists the International

System of Units [2] and their equivalents in some

of the customary engineering units.

Tables listing numerical values of various elastic properties are given as the properties are introduced. In selecting data for each table, the

contemporary definition of ceramics as inorganic,

non-metallic materials was used, and accordingly

data on some important semiconductor materials

is included, but data on alkali halides are not

included because these are extensively covered

elsewhere [3]. Data on some refractory materials (here considered as materials melting above 1000 °C)

other than ceramics are also included because

one of the major applications of ceramics is at

high temperatures so that a comparison with

other high temperature materials is of interest.

The literature on single crystal elastic constants

through 1964 has been surveyed in a pair of

1 Figures in brackets indicate the literature references at the end of this paper.

140
Symbol

A=U-TS

A,A*

A(ijk)

a=(ai, a2 , a3) B

Cijk imn

Cijkl .

Cij

C

d

E

3

G=H-TS

G= ji H= U+pV Kn

k

i_

m

n

P

P

R

rS

Si

T

T

Table 1. Symbols

Quantity

Helmholtz free energy per unit mass. Anisotropy factors.

Constants in nonlinear stress- strain relation.

Initial position vector.

Bulk modulus.

Third-order elastic constants in

tensor notation.

Third-order elastic constants

in matrix notation.

Second-order elastic constants

in tensor notation.

Second-order elastic constants

in matrix notation.

Velocity of light.

Specific heat at constant pressure

or volume.

Diameter of cylinder.

Thermal energy.

Force per unit area. Frequency.

Gibbs free energy per unit mass. Shear modulus.

Enthalpy per unit mass. Thickness correction factor for

longitudinal vibration of a slender cylinder.

Boltzmann's constant, also wave

number.

Unit normal to plane.

Length of rod or bar.

Reduced mass. Index of refraction, also mode

number for resonance.

Porosity.

Pressure, also number of atoms

per molecule.

Gas constant.

Interatomic spacing.

Entropy per unit mass.

Infinitesimal strain in tensor

notation, also spheroidal mode

of sphere.

Infinitesimal strain in matrix

notation.

Second-order elastic compliances

in tensor notation.

Second-order elastic compliances

in matrix notation.

Absolute temperature, also cor- rection factor in flexural

vibration of slender bar or rod.

Stress components in tensor

notation, also torsional modes

of sphere.

Stess components in matrix

notation.

Thickness of plate.

Table 1. Symbols—Continued

Symbol

tij

uu

u=(uuu2 ,u3 ) V

v

Y

zea

7

8

€0

i=G

V

"0

pX

CO

Quantity

Thermodynamic tension in

tensor notation.

Thermodynamic tension in

matrix notation.

Internal energy per unit mass. Motion vector.

Volume.

Velocity of sound; v { for longitudinal, vs for transverse.

Final position vector.

Young's modulus of elasticity.

Ionic charge; e= electronic charge.

Linear thermal expansion coefficient.

Volume thermal expansion coefficient.

Gruneisen's constant.

Griineisen's second constant.

Finite strain components in tensor

notation and spatial description

(sometimes called an Eulerian

description).

Finite strain components in matrix

notation and spatial description.

Static dielectric constant.

Finite strain components in tensor

notation and material description (sometimes called a Lagrangian description).

Finite strain components in matrix

notation and material description.

Lame second-order elastic constant for elasticially isotropic

material, also wavelength.

Lame second-order elastic constant for elastically isotropic

material.

Poisson's ratio.

Long wavelength limit of the

frequency of the transverse optical mode of vibration.

Density.

Compressibility.

Angular frequency.

Table 2. Units

Quantity

Inter- national

unit Value in other commonly used units

Volume m 3 10« cm3 =35.314 ft 3 =6.1024X10< in3

Pressure N/m2 10 dyn/cm2 =1.450X10-< psi=1.020X10-«

kg/cm2

Energy J 0.239 Cal (gm)

141
excellent papers [3, 4] which draw on a number

of previous surveys [5-9]. We have accordingly

concentrated on listing values published since

1964, but have included some of the older data

for comparison. The engineering data on elastic

modidi of polycrystalline ceramics has been

surveyed [10] and several other useful compilations

of elastic properties and related properties of

ceramics have recently been published [11-13].

2. Continuum Elasticity

2.1. Thermodynamics and Simple PressureVolume Relations

The simplest type of elastic behavior is the

change of volume (and sometimes of shape) in

response to pressure (i.e., to isostatic stress).

If the solid is isotropic (i.e., if its properties do

not vary with direction), its shape will not change

under pressure and its equation of state is simply

its volume as a function of pressure and temperature. The exact functional form of the equation

of state is not known for any solid; a power

series approximation is commonly used. Overton

[14] has discussed the equivalence of Bridgman's

form of the equation of state,

V(T,p)=V(T,p=0) [\-al {T)p+a2 {T)p>+..\, (D

to that of Slater,

V(T,p)=V(T=0,p=0) [l+A0 (T)-a\(T)p

+a2

8 (7V+..], (2)

where V—volume, p= pressure, T= absolute temperature, superscript s refers to Slater's parameters, a0 is a function of temperature which gives

the volume expansion at zero pressure, and the

remaining a's are empirical functions of temperature which express the elastic properties. In

general, contribution of higher terms is small.

To give a complete thermodynamic description,

these elastic properties are needed together with

the thermal expansion and specific heat. These

data permit the internal energy, U, and the

entropy, S, to be calculated from which the

enthalpy, H= U-\-pV, the Helmholtz free energy,

A= U— TS, and the Gibbs free energy,

G= U-\-pV— TS can be calculated.

Randomly oriented polycrystalline ceramics and

glass are isotropic so that either equation (1) or (2)

provides a complete description of their elastic

response to isostatic pressure, but the behavior of

anisotropic ceramics such as single crystals re- quires the introduction of strain and stress with

a corresponding increase in complexity of notation.

Even for isotropic ceramics, however, strain and

stress must be introduced to describe their elastic

response in other situations such as in bending.

Before going on to do so it is helpful to use the

conceptually simple pressure-volume relationship

to illustrate some points concerning the distinctions between adiabatic and isothermal values

and between linear and nonlinear behavior.

The isothermal bulk modulus, BT , is defined as the reciprocal of the isothermal compressibility,

Xt, defined by

From equation (1) the compressibility is given by

xT ( T,p) = [a, ( T) -2a2 (T)p)V( T,p =0)/V( T,p)

.

(4)

For small pressures, the second term can be

neglected and the volume ratio taken to be unity

so that xAT,p=0)=ai(T). This is the quantity

commonly measured in static tests at relatively

low pressures. Even at modest pressures the second

term can be determined, however, by using highaccuracy ultrasonic velocity measurements and

the results used to predict behavior at high pressure. The calculations involve some complexity in

converting from transit time data to elastic

properties which we will not consider here, and

also necessitate careful handling of the values of

bulk modulus measured at different pressures

because a pressure dependent volume enters into

equation (4). Another complication arises because

ultrasonic measurements give adiabatic rather

than isothermal values. The conversion equation

from the adiabatic to the isothermal compressibility is XT=Xs+W/pCp (5)

where /3 is the volume thermal expansion coefficient, p is the density, and Cp is the specific heat

at constant pressure [15]. These latter values are

all to be taken at the pressure of measurement.

Cook [16] has derived equations which permit one

to combine specific heat and density measured as a function of temperature at a single, fixed pressure

with ultrasonic data measured as a function of

pressure and temperature to calculate Xr{T,p). A

modified form of these equations, due to McSkimin

[17] was used by Anderson [18] to obtain the

isothermal compressibility of vitreous silica over

the range 0 to 300 K and 1 atm to 8000 atm.

Thurston [15, 19] has reviewed the calculations

necessary if the solid being studied is anisotropic

as well as nonlinear. Anderson [20, 21] has followed Murnaghan in assuming that the bulk

modulus is a linear function of pressure given by

B(p,T)=B0 (T)+B'0 (T)p (6)

and has applied the resulting Murnaghan logarithmic equation [21]

ln(Vo/V) = (5'c)-

1 ln[B' 0 (p/B0) + 1] (7)

and a similar equation due to Birch to a variety

of metals and ceramics. He evaluated the param142
eters B0 and B'0 from ultrasonic data taken in the

pressure range 0 to 4 kbar and found that the

resulting equation was in reasonable agreement up

to quite high pressures, for example up to 1500

kbar for A12 03 as shown in figure 1

.

Simple thermodynamic manipulation leads to

the relation

This is a useful expression for the isothermal bulk

modulus which involves only the internal energy

as a function of volume and does not involve the

pressure explicitly. Similar expressions give various

other isothermal elastic constants as second

derivatives of the Helmholtz free energy with

respect to various strain components, as we shall

see. These expressions are useful in discussing the

coupling of elastic properties to electrical and

magnetic properties. For the elastic properties to

be accurately defined, the electrical and magnetic

state must be specified during the deformation.

This may be accomplished by holding the electric

and magnetic fields constant, or, as one of several

other possibilities, by holding the electric polarization and the magnetic field constant. The resulting

elastic properties will generally have slightly

different values; equations relating the two sets of

0-60 1 1 I I I

| 0 250 500 750 1000 1250 1500 PRESSURE (KBARS)

Figure 1. Compressibility of AI2O3.

Solid line calculated from data taken at pressures below

2 kbar using the Murnaghan equation (7). See Anderson [21] for details of calculations and references

to experimental work.

values can be derived by using the appropriate

expressions similar to equation (8) as described in

detail by Nye [22].

In the following sections we sketch the mathematics necessary to extend the foregoing to

stresses other than simple pressure and to deformations which involve change of shape as well as change of volume. It is simplest to first introduce

the stress and strain in a form suitable for description of small deformations; this is the subject of infinitesimal elasticity and is the level on

which engineering problems in elasticity are usually solved. A complication arises because

the strains and stresses are second rank tensors

and so would normally be written with two subscripts, but are instead usually written with one

subscript because they possess a symmetry which

allows this to be done. A second complication

arises from the fact that some of the engineering

strains differ by a factor of two from the strains

in tensor notation commonly used in modern

treatments of elasticity because of the convenience of the latter in expressing the orientation

dependence of elastic moduli. For elastically

isotropic bodies, such as glass and randomly

oriented polycrystalline ceramics, the equations

relating stress and strain take a simple form. For

all single crystals, including cubic crystals, the

equations take on a similar form but have more

terms which increase in number for less symmetric

crystals. Another complication arises for single

crystals; the value of the elastic constants depends

on the orientation of the coordinate axes used to

describe the deformation. In using single crystal

elastic constants one must bear this in mind.

When the strain is large and accurate treatment

is required, finite elasticity must be used. A new

definition of strain must be introduced and a distinction must be made between the use of coor- dinate axes fixed in space (giving a spatial or Eulerian description) and coordinate axes which

move with the deforming material (giving a material or Lagrangian description). The stress

previously defined in infinitesimal elasticity is appropriate to a spatial description; for a material

description a related concept, the thermodynamic

tension, is appropriate. Throughout the treatment

of either infinitesimal or finite elasticity the

distinctions previously mentioned concerning the

difference between adiabatic and isothermal behavior and the specification of electrical and

magnetic state remain necessary.

2.2. Stress

Stress is simply the force per unit area with

which the material on one side of a chosen surface

acts on the material on the other side. We see from figure 2 that two vectors are required to

specify stress for a particular surface. One vector

is required to describe the orientation of the surface

and is drawn perpendicular to the surface; the

other describes the force per unit area on the

143
z = x,

Figure 2. TTie force, F, acting on a surface with unit normal

vector 1.

surface and can have any orientation. The surface

can be chosen through a given point in an infinite

number of orientations: for each there will be a

different value of the force vector even though

the same stress is acting at the point. It can be

shown [22] that the stress acting at a point is completely specified by giving six numbers; the

six stress components Tn , T22 , T33 , Tl2 = T2l , T13 =T31 , and T23 =T32 . The force (with components Fx , F2 , F3 ) acting on a surface with unit normal (with

components lu l2 , l3 ) is given by [22]

Fi= Tnli+ Tl2l2 -\- T13l3 , F$= T21I1 + T22l2 -\- T23/3,

F3 = T3i li-\- T32l2

~\~ ^33^3-

The same equation can be written

and

F^J^TJj, i=l,2,3.

i=i

(9)

(10)

The stress components have the following physical

meaning ; Ttj is the force in the xt direction acting

on a unit surface area whose normal is in the x, direction.

These stress components written with two subscripts are said to be in tensor notation. It is convenient to introduce a single subscript notation, called matrix notation, by denning Ti=Tn, T2 =T22 , T3 ~T33 , Ti=T23 , TB =T13 , and T6 =T12 .

2.3. Strain

The components of strain at a chosen point in a

solid body are introduced to specify the deformation of a small region around the point. The strain

components are deliberately chosen to specify

only the deformation of this region and do not

include translation or rigid body rotation. Figure

3 is an illustration of the deformation of a body

which carries a portion initially at A0B0 CoD0 into

ABCD. We specify the position of A0 by the

vector a; after being moved along the vector u, A0 becomes A at position x. If the whole portion

AqBqCqDo moves by the same amount u, the

motion is simply one of rigid body translation.

We are therefore interested not in u itself, but in

how u varies as we examine different points, such

as D0 , near A0 and so we wish to know the values

of the set of 9 quantities at A0 (i,j=l,2,Z).

The same information is also provided by the set

of 9 quantities ^ which have the same value as ^

4 if iy^j and are greater by one if i=j; these

quantities are called the deformation gradients

and the whole description of strain can be written

in terms of them as an alternative to using

^—1 [19, 23]. Either set of quantities is satisfactory

in that rigid body translation has no effect on

their values, but both sets are still affected by

rigid body rotation. We can, however, take combinations of either set which will exclude rigid

body rotation and so give a proper description of

strain.

In most engineering applications of ceramics,

the elastic strain is small enough to justify the

use of infinitesimal elasticity although some ex- ceptions exist for very high strength materials,

for isostatic pressure applications, for devices

which exploit nonlinear behavior, etc. A strain is considered infinitesimal if the largest of the

+ du C

Figure 3. Deformation of a portion A0 B0 CoDo of a body into ABCD by a motion vector u which

is a function of position.

144
quantities is small (typically less than 0.01)

so that powers and products of the quantities

but ~ can be neglected. Then the infinitesimal strain

oa,j ° component Si; is defined by

The same symmetry holds here as for the stress so that S12 =S2i, S13 =S3U and S23=S32 - As in the

case of the stress, these strain components written

with two subscripts are said to be in tensor notation. As with the stress, it is customary to introduce

a single subscript matrix notation, but factors of

2 are introduced into the definition of the shear

strains in matrix notation so that the whole set

of strain components in matrix notation is just

the usual engineering strains: Si =Sn, S2 =S22> S3=S33 , *S

,4=2iS,2 3, S5 =2Sl3 , and S6 =2S12 . The physical meaning of the infinitesimal

strains is illustrated in figure 4. The strain Si

refers to an increase from initial length I to l-\-Al in

the Xi direction with other dimensions remaining

constant. Its value is Si = Al/l. The strains $2 and

S3 refer to the same type of deformation along the

x2 and X3 axes respectively. The strain S6 refers to

a shear in the XiX2 plane and also has the value

Al/l, but now Al is measured at a right angle to

I. It thus corresponds to changing a square into a parellelogram as shown. Si and $5 refer to the

same type of deformation in the x2 x3 and xx x3 planes respectively.

These definitions of infinitesimal strain suffice

for infinitesimal elasticity and hence for most

engineering problems. When large deformations

or nonlinear effects must be treated, the definition

of strain in equation (11) is replaced by defining

the finite strain components given by

(12)

In this equation, the last term contains a repeated

subscript k, which is understood to mean a sum;

that is buk du]s= Sy duk buk (13)

1 *2

1 * 2

I AX

t

i I

1 *

1

1

1

1

1

1

1 Ml

S| = S|| S 6 =2S I2

=

x 2S 2I

Figure 4. Illustration of the strain components Si and Si.

When the partial derivatives are small, this term

can be neglected and the finite strain of equation

(12) reduces to the infinitesimal strain of equation

(11). A new feature arises here, however. The

finite strain of equation (12) is called the Lagrangian strain in a material description because the

independent variables, a { refer to points fixed in

the material. An alternate and equally valid

definition of finite strain is that of the Eulerian

strain or strain in a spatial description which is defined by =1 fduj buj buk buk\

(

As the partial derivatives become small

bUi

dx,

approaches so that the strain of equation

(14) also approaches the infinitesimal strain of

equation (11). When the strain is finite, however,

77 y is not the same as e u. Either can be used, but

one must carefully distinguish which is being

used as we shall see. The physical meaning of

these strain components in terms of the local

rotation and stretch of line elements in a deforming

body is discussed elsewhere [19, 23].

2.4. Linear Elasticity

For stresses small enough to allow use of infinitesimal strains, it is found that strain is proportional to stress to a good approximation (Hooke's

law) . Thus a stress Tx will produce six strain

components for a body with no symmetry and all can have different constants of proportionality

which we call the elastic compliances, s i} . These

strains are given by

S'i =Sn2i,

S2 =s2 iTi,

$3 =531 >

iS

,4=s4iT1 , Ss=sb iTi,

and S6 =sSiTu (15)

If all six independent stress components are acting

simultaneously, these equations become the generalized Hooke's law, which is

Si =Si 1 Ti +sn T2 +si3 T3 +su T± +S15 T5 +s i6 T6 , S2 =s2 iTi-\-s22 T2 -\- s2 3 T3 +s2i Ti +

s

25 T5 +

s

26 T6 , S3 =

s

3 i Tx +

s

32 T2 +S33 T3 +

s

34 Ti +

s

35 T5 +

s

36 T6 , Si =

s

41 Ti +

s

42 T2 +

s

43 T3 +S44 Ti +S45 T5 +

s

46 T6 , S5 =s51 Ti +

s

52 T2 +

s

53 T3 +

s

54 Ti +

s

55 T5 +

s

56 T6 , and (16)

S6 =s61 Ti +

s

62 T2 +

s

63 T3 +

s

6 4 Ti +

s

65 Tb +

s

68 T6

.

These equations can all be written in one line as

St=S ««2V, i=l, 2 . . . 6. (17)

These are six equations giving the six strains in

terms of the six stresses. Alternately we can solve

145
these equations for the stresses in terms of the

strains and get

6 rf =X) CtjSj, i=l, 2, 6, (18)

where c {j are called the elastic constants and can be calculated from the s tj and conversely [22].

For any material, it can be shown that sw=Sj4 and Cij=cn. This means that for a crystal of very

low symmetry (triclinic) there are 21 independent

elastic compliances or constants. Fortunately this

number is much reduced for higher symmetry

crystals being only 3 for cubic crystals and 2 for

isotropic materials. For each of the 32 crystal

classes, the scheme of elastic constants and compliances has been worked out from the symmetry

elements possessed by the class and the results are

listed in many texts [3, 5, 7, 22]. In table 3 the

numerical values of the constants and compliances

are listed for a number of ceramics and some other

refractory materials. These data are the fundamental description of the linear elastic properties

of crystals. They are independent except that

conditions of crystal stability require some to be

larger than others [24, 25].

For cubic crystals the generalized Hooke's law

becomes

S1 =su T1 +s12 T2 +s12 T3 +0-\-0+0,

S2=Sl2T1 +.Sll T2 +s12 T3 +0+0-\-0,

S3 =s12 T1 +s12 T2 +snT3 +0+0+0,

St = 0+O+O+S44T4+O+O,'

S5 = 0 +0 +0 +0 +S44 7'

8 +0,

and S6 = 0+0+0+0 +0 +s44 7V

(19)

Thus, for example, the application of a tensile

stress, Tu along the xx axis causes a tensile strain,

Si, along the xx axis and perpendicular tensile

strains, S2 and S3, which make up the contraction

expressed by Poisson's ratio. This ratio is defined

by v=—S2/S1 when only Tx is acting and so is given by —s12 /sn . It is important to note that the

form of equation (19) depends on the reference axes xx x2 x3 being chosen along the usual axes of a cubic crystal. If the axes are rotated equation

(19) goes over into equation (16) (usually written

with primes on the s {j to indicate a rotated system

of coordinates) even for a cubic crystal; in this case each of the s'

tj of (16) is a complicated trigonometrical function of the orientation of the

axes and the three independent compliances

^44; and S12 of equation (19). For example:

s' u=Sn—[2(su—S12) —S44] [sin20cos20+ .25sin40sin22</>] (20)

where 6 and <f> express the position of the x'3 axis

in the usual spherical polar angles. The compliance

Table 3. Single crystal elastic constants of some nonpiezoelectric ceramics and some metallic refractory

crystals

In units of 10 10 N/m2 for c„ and 10-i2m2 /N for

Substance Subscripts of c, or «,, Refer- ence*

Cubic

11 44 12

C (diamond) c8

102.0

1. 12

49.2 2.07 25.0 -0. 22

[3]

Fe--_- C 23.0 11.4 13.5 13) s 7.65 8.78 -2. 82

Garnet: YA1G 33 3 11.

5

11.

1

T31 s 3. 60 8.69 -0.90 YGaG. c 29.0 9. 55 11.7 [3] s 4.49 10.5 -1.29 YFeG c 26.9 7. 64 10.8 [3]

[26]

[26]

[27]

EuFeG—

8

C

C

4.83 26.8

25.1

13.

1

7. 66 7.62

—1.38

11.1 10.7 Natural Almandite- nvrnnp

garnet C 30.6 9.27 11.2

Ge c 13.

0

6. 70 4.90 [3] s 9. 72 14.9 -2.66 MgO ... c 29.6 15.6 9.51 [28] Mo

sc

3.99

47.0

6.41

10.7

-0. 97

16.8 [29] g

c

s

2. 62

44^1 2.92

9.36

12.2 8.22

—0. 69

17.2 -0. 82

[30]

Nb c 24.6 2. 87 13.4 [3] 8

3

6.60

23.

5

6. 54

34.8

2.82

35! 45

-2. 33

1. 21 -2. 22

[31]

Si 16.

6

7.96 6 4 [31 8 7^68 12! 6 -2. 14

Spinel (MgO

2.61 AI2O3) Th ....

C

C

29.86

5. 152

7! 53

15. 76 6.347 4^78

15. 37 —1.751

4.89

[32]

[3] 3 27.2 20.9 -10.7 Th02 C 36.7 7.97 10.6 [3] 3 3. 13 12.5 -0.70

TJC — C 32.0 6.47 8.4 [3] 3 3. 51 15.4 -0.73

UO2 C 39.5 6. 41 12.1 [3] 3 2.96 15.6 -0.70

ViOt C 31.9 4.1 12.1 [33]

v...

8

C

3.96

22.8

24.4

4.26 -1.09

11.9 [3] 3 6.83 23.5 -2. 34

w_. C 51.5 15.6 20.4 [3] 8 2. 50 6.40 -0. 71

ZrC C 42.3 14.6 4.08 [34]

11

Hexagonal

33 44 12 13

Beryl c 28.2 24.8 6.86 9.94 6.59 [3] 3 4.20 4.51 14.6 -1.28 -0.82

Re C 61.2 68.3 16.2 27.0 20.6 [3] 8 2. 12 1.70 6. 16 -0.80 -0.40

11

Trigonal

33 44 12 13 14

Corundum

(AI2O3) c 49.7 49.8 14.7 16.4 11.1 -2. 35 [35] 3 2. 35 2. 17 6.94 -.716 -.364 .489

c 49.0 49.0 14.5 16.5 11.3 -2. 32 [36] Hematite

(Fe [3] 2 03) c 24.2 22.8 8.53 5.49 1. 57 -1.27

8 4.41 4.43 11.9 -1.02 -0.23 -0.79

See footnote at end of table.

146
Table 3. Single crystal elastic constants of some nonpiezoelectric ceramics and some metallic refractory

crystals—Continued

Substance Subscripts of c,-j or s,-,- Refer- ence*

Tetragonal, higher symmetry

11 33 44 66 12 13

Rutile (Ti02 )_..- c 27.3 48.4 12.5 19.4 17.6 14.9 [37] S 6. 55 2.59 8. 00 5. 16 -3. 76 -0. 86

c 26.6 47.0 12.4 18.9 17.3 13.6 [38]

s 6.79 2.59 8.07 5.30 -4. 02 -0. 80

Zircon c 7.35 4. 60 1.38 1.60 0. 90 -0. 54 [3] s 13.9 22.1 72.0 62.0 -1.6 1.4

Tetragonal, lower symmetry

11 33 44 66 12 13 16

CaMo04 c 14. 47 12. 65 3.69 4.51 6.64! 4.46 1.34 [39] s 9. 89 9.29 27. 10 24. 68 -4. 32 -1.96 -4.22

c 14. 50 12. 82 3. 67 4. 56 6. 18 4. 96 -1.46 [40] 8 9. 74 9.58 27. 22 24. 71 - -3. 80 -2. 30 4.33

•References used by Hearmon [3] are not repeated in our list of references except in a few cases where comparison with other data on the same material

is involved.

Sn is the reciprocal of Young's modulus (defined

as the ratio of an applied tensile stress to the

resulting parallel tensile strain) and s'n continues

to have this meaning when the coordinates are rotated. Thus, Young's modulus of cubic crystals

varies with orientation as do all the other elastic

moduli. It is therefore of importance to avoid

misunderstanding in dealing with the elastic

moduli of crystals, including cubic crystals, to

refer all measurements to a standard set of axes

[41,42].

If 2(sn —S12) is equal to s44 , Young's modulus

and all other elastic moduli become independent

of orientation. Zener [43] has defined A=2(sn —

Si2)/s44, which is also equal to 2c44/(cn —Ci 2 ), as the

anisotropy factor for cubic crystals. It has the

physical significance of expressing the ratio of the

resistance to shear deformation in the [010]

direction on the (100) plane to that in the [110]

direction on the (110) plane. A modified anisotropy factor for cubic crystals, A*, has been

introduced by Chung and Buessem [44]. The

factor A is found to vary with composition from

0.25 for Rbl to about 8 for Li [44].

For elastically isotropic materials, the generalized Hooke's law takes the form

S1 =sn T1 +s12 T2 +s12 T3 +0+0+0,

S2 =Sl2 Tx +Sil T2 +sn T3 +0 +0 +0,

s3 =s12 T1 +snT2 +s11 T3 +o+o+o,

S4 =0+0 +0 +2(s11 -s12 )7

74 +0+0,

S5 =0+0+0+0+2(su -s12) r5 +0,

and S6 =0+0+0+0+0+2(s11 -s12 )T6 . (21)

That is, for isotropic materials s44 =2(sn —s12 ). It is, however, more usual to describe an elastically

isotropic body in terms of any two of the five

quantities Y= Young's modulus, G =shear modulus (sometimes called one of Lame's constants

and designated by p.), i/=Poisson's ratio, B=bulk

modulus, and X=the other Lame constant. Table

4 gives the relations between these isotropic

moduli; for example, if B and G are chosen as the independent pair, the remaining moduli are given by Y=9BG/(3B +G), v=(3B-2G)/{QB+

2G), and \=B—(2G/3). A connection to the

compliances for elastically isotropic material is provided by Y=l/su and G = l/s44 = l/2(su —

s

12 ), v——s12/sn , i?=l/3(s1 i +2si2 ), and a connection to

the elastic constants by 6r=c44 , X=ci2 .

Table 4. Expressions for various isotropic elastic moduli in

terms of various pairs of moduli

Independent Pair of Moduli

Y, G=y. B, G=M B, v X, G=ii Y Y 9BG/(3B+G) ZB(\-2v) G(3X+2G)/

(X+G)

G=„ G G 3B(l-2^)/

2(l+x)

G

B VG/3(3G-K) B B X+(2G/3)

V (V/2G0-1 (3B-2G)/

(6B+2G)

V X/2(X+G)

X (Y-2G)GI

(3G-Y)

B-(2G/3) 3vB(l+v) X

Both glass (treated elsewhere in this volume

[45]) and randomly oriented, fine grained polycrystalline ceramics are iso tropically elastic. Values

of isotropic moduli for polycrystalline ceramics

are presented in table 5. This table gives values

for pore-free ceramics obtained by measuring at

various porosities and extrapolating to zero

porosity. The elastic moduli of many porous

ceramics have been measured [10]; we defer a

discussion of the effect of porosity on elastic

moduli to section 3.7.

Table 5. Elastic moduli of isotropic polycrystalline ceramics

at or near zero porosity

Substance Porosity

Y G B

V Reference

100 bar = 10'0 N/m*

AI2O3-- 0.0

.0

.0

.004

.0.0

.0

.0.0

.0

.0

.0.0

.0

41.02

40.68

40. 20

39.86

40. 39

17.05

18.63 30.50

31. 77 31.00

26. 10 28.42

13.83

12. 35

[46]

[47, 48]

[12] [12 [49

[50

[50

[51

[52

[12

[53

[49 [54

[49

AI2O3 -

AI2O3

AI2O3* AI2O3 Dy2 03

16. 33 16.40

16. 14

16. 35

(26. 65)

(24. 42) 25.02

(25. 42)

(0. 246)

(. 226)

(. 235)

(. 235)

Er2 03 MgO

MgO

MgO

Th02 -._:.. T1O2

Y2O3

12.90

13.96

13. 15

10. 06 11.15

(15.99)

(14. 62)

(16. 08)

(21. 45)

(21. 00)

( 182)

(. 138)

(. 179)

(. 297)

(.274) ZnO 4.56 (14. 11) (.354)

Values in parentheses calculated from other values using expressions in

table 4; B value is quite sensitive to error in Y or G. •Values for all other materials extrapolated to zero porosity. These values taken at p =3.972 g/cm3 on alumina containing trace of MgO; density of pure alumina from x-ray lattice constants is 3.986 g/cm3

.

147
2.5. Nonlinear Elasticity

There has been much work done in recent years

on nonlinear mechanics and on nonlinear elasticity

in particular. The subject is large and complex;

we can only give a brief introduction supported

by references for further reading. Much of the

recent work is directed toward attempting to describe the properties of fluids; Truesdell and

Toupin [23] have given a general survey and

Fung's book provides a readable introduction

[55]. Within the very general field on nonlinear

continuum mechanics we shall restrict our attention to the finite elasticity of solids.

In discussing finite strain we introduced the

Lagrangian strain, y\ u and the Eulerian strain, as being appropriate for use with coordinate

systems fixed in the material and in space respectively. Finite elasticity can be developed in terms

of either but most of the experimental work is reported in terms of Lagrangian strain. The

stress, Tk, which we discussed is referred to axes

fixed in space. The corresponding quantity for axes fixed in the material is called "the thermodynamic tension," f <; in tensor notation or t { in

matrix notation with the same rule relating t t to as was given for relating Tt to Ti}. The thermodynamic tension is related to the stress by [19]

(22)

where J= determinant W,7 For infinitesimal

strains ttj becomes equal to Tu. In introducing infinitesimal elasticity we considered the stress to be applied and developed

equation (17) to express the resulting strain.

Much of the basic data on nonlinear elasticity is determined in plane wave propagation experiments in which a strain is imposed by the wave

motion and the resulting stress (for infinitesimal

elasticity) would be given by equation (18).

The literature on nonlinear elasticity of solids is accordingly written in terms of the c w of equation

(18) and their generalization rather than the s i} of

equation (17).

To generalize equation (18) one first replaces the

infinitesimal stress and strain by their Lagrangian

counterparts for finite strain

««=Z)CW (23)

One then adds a nonlinear term of the form

JZMij^C^^ (24)

j, /c = l

to express the nonlinear contribution to the

thermodynamic tension. The A(ijk) are numerical

constants which various authors have introduced

dfferently leading to definitions of the third order

elastic constants, Cijk, which differ from one

another by numerical factors. The definition which

is usually used today is due to Brugger [56]. It

is easiest to express Brugger's definition by re- calling the relation that the stress is the derivative

of the internal energy per unit volume with respect

to the strain and using the corresponding relation

for finite strain

:po (i!)s =po G£t' (25)

The internal energy is taken per unit mass for

finite strain and the factor p0 (which is the density

of the unstrained solid) is introduced to give the

proper limit of energy per unit volume for infinitesimal strain. The internal energy is then

given by [56]

1

6 16

* = l O k = l

and the elastic constants are given by

/ b

2 U \

CiJ -po Wdr,Js

and °TM=*(*M*d:

(26)

(27)

(28)

By using (25) and (26) one obtains the generalized

form of (23) including the term (24) and the

numbers A(ijk). In using the original literature

in which the third order elastic constants were

introduced and enumerated for various crystal

symmetries [6, 57, 58, 59, 60] care must be used

because of the different definitions used for the

c m. The relations between the different definitions are summarized by Brugger [56].

For cubic crystals there are three independent

second-order elastic constants as mentioned earlier: cu =c22 =c33 , c12 =c13 =c23 , c44=C55=c66 and all others zero. The five cubic crystal classes split

into one set of three with six independent thirdorder constants and a set of two with eight independent third-order elastic constants. Cubic

crystals with point group mZm, 432, or 432 have

the following six independent third-order elastic

constants: Cm=6222=^333, C144 =£7255=^366, Cm

£223= Ci33= Cu3 = Ci22= £233, Ci55= ^244= £344 = C166 = C2m =C355 , Cm, Cib6 , and all others zero. These third-order constants are determined ex- perimentally by measuring the variation of the

velocity of ultrasonic waves with application of

homogeneous static stress as we shall discuss in a later section. This procedure causes some complexity in the analysis of the data; the experimental results give the isothermal variation of

combinations of adiabatic effective second-order

elastic constants with pressure. The resulting

"mixed derivatives" must be corrected either to purely adiabatic or purely isothermal conditions

148
using appropriate thermodynamic relations [61,

62, 63, 64]. The term "effective second-order

constants" is used to make a distinction from the

thermodynamic second-order elastic constants which

are defined as second derivatives of the energy.

The effects of pressure enter the thermodynamic

formulation through the third-order and higherorder elastic constants all of which are defined by

(27) and (28) evaluated at zero strain and so are pressure independent. One may, however, choose

to express the effect of pressure by allowing the

second-order thermodynamic coefficient to be

evaluated as a function of strain and so giving its

pressure derivative. On the other hand, the effect

of pressure may be described by using only the

second-order constants which enter wave velocity

expressions and allowing these to vary with

pressure; in this case they are termed effective

elastic constants. One may calculate either set

of data given the other complete set, but the

calculations are quite compex. This type of

calculation is considered for various crystal

symmetries and types of experiments in many

papers; we refer to some of the more recent from

which earlier references can be obtained [65, 66,

67]. Both types of data are useful. One may be

interested in the response of solids to isostatic

pressure in which case a table of dc^/dp is useful.

Data for a number of solids is presented in table 6. For isotropic materials, as mentioned previously, the number of independent second-order

elastic constants is reduced from three for cubic

crystals to two. The three constants cn , c44, and

C12 are no longer independent; c11 = 2c44 -|-c1 2. The independent pair are sometimes given the

symbols x and tx and are termed the second-order

Lame constants with X = cI2 and m=c44 . Similarly

the six independent third-order elastic constants

for cubic crystals reduce to three which are sometimes [65, 74] written vu v2 , ?3 and called

the third-order Lame constants with vi=Cm,

V2=CUi, f3=C456, C

rii2 = fl +2y2 , Ci55=f2 +2l/3 , aild

Cm= ^i +6j>2 +8^3. Some numerical values of

third-order elastic constants for ceramic materials

are given in table 7 and isotropic materials in

table 8. Recent developments in nonlinear elasticity

theory include the calculation of lattice parameter

changes with pressure in terms of second and thirdorder moduli [76]. Ghate [77, 78] has introduced

fourth-order constants and calculated the pressure

dependence of the effective second-order constants

in terms of these; Chung has also considered

fourth-ordered effects for cubic crystals [79].

Truesdell and Toupin [80] have considered conditions on elastic constants in finite elasticity similar

to those previously mentioned for infinitesimal

elasticity [24, 25].

Table 7. Third-order elastic constants of some ceramic

crystals

All djk in units of 10"> N/m2.

Substance Cm Cm Cm Cm Cm C«e Refer- ence

GaAs -62.2 0.2 -38.7 -26.9 -5.7 -3.9 [70] Ge -71.0 -2.3 -38.9 -29.2 -1.8 -5.3 [71] Ge -71.6 -5.3 -40.3 -31. 5 -1.8 -4.7 [72] MgO

Si

-489. 5 11.3 -9.5 -65.9 -6.9 14.7 [72] -82.5 1.2 -45.

1

-31.0 -6.4 -6.4 [71] YFeG -233 -14.8 -71.7 -30.6 -3.3 -9.7 [73]

Table 8. Third-order elastic constants of isotropic materials

All Cuk in units of 10>» N/m2.

Substance "\ = Cm n=Cui "3= Cm Reference

Polycrystalline Mo 1 19.4 -39.8 -22.7 [75 Polycrystalline W ' -42.9 -25.8 -26.7 [75 Vitreous Si02 . 54 93 -11 [72

1 Specimens were sintered and had some porosity.

2.6. Generalizations of Linear Elasticity

Mindlin and Tierstein [81, 82] have proposed a

generalization of linear elasticity in which couplestresses acting on a unit area are taken into

account as well as the usual stress. A proportionality is assumed between couple-stress (couple

per unit area) and curvature (gradient of rotation)

.

The ratio of couple-stress to curvature is then a new type of modulus of elasticity with the dimensions of force. The theory has been generalized to

take into account nonlinear effects [83, 84]. The

linear theory has been applied to the calculation

of stress distribution around a hole [82] and to the

bending and torsion of bars [85]. Schijve [86] has

Table 6. Dependence of effective second-order elastic constants upon pressure

Cn 8 for Some Ceramics at 26 °C.

Superscript S indicates constant entropy.

at zero pressure and their values,

Substance c S

/dcii s \

V dp / t

/dcus \

\ dp / t V dp / t V dp / t Reference

GaAs.._ _ lOiW/ro2 10'W/m2 lO'W/m2 lO'W/ra2 11.877 4.63 5.944 1. 10 5. 372 4. 42 7.540 4.49 [68] Si 16. 577 4. 33 7.962 0.80 6. 393 4. 19 9. 788 4.24 [69] Spinel 1 29. 857 4.9 15. 758 0. 85 15. 372 3.9 20. 200 4.2 [32] Garnet 2 30. 619 7.48 9.266 1. 31 11.246 4. 41 17. 704 5. 43 [27] MgO 29. 647 9. 477 15. 589 1.160 9. 507 1.992 16. 220 4.497 [28]

' MgO-2.61 AI2O3.

2 Almandite-pyrope type.

149
carried out bending experiments which suggest

that the new couple-stress elastic moduli for

aluminum are small (he could not detect them)

and that stress distributions around holes are not

likely to be in significant error if couple-stress

effects are neglected. It should be noted that the

couple-stress theory appears to be on a firm

theoretical foundation; only the size of the effect

is in question.

A different attempt to generalize linear elasticity

was previously made by Laval [87] and by Raman

and his co-workers [88-90]. This point of view has

been developed by LeCorre [91] and by Joel and

Wooster [92]. Some data from the literature were

examined [90, 93] and it was claimed that the

classical theory of infinitesimal elasticity was

inadequate. Recently, however, a careful experimental test of Raman's version [89] of this theory

was made [94]; the results support the classical

theory. In any event, it appears that the proper

way to introduce rotations into elasticity is that

discussed in the preceeding paragraph and that

the effects are small, at least for small strains.

Further discussion of the Laval-Raman approach

can be found in Huntington's review paper [7].

3. Specific Problems Involving Elastic

Deformation

3.1. General Discussion

So far we have been concerned with elasticity as a way of describing a property of material; we

have assumed the stress and strain to be uniform

throughout the material. In many practical situations the stress and strain vary from point to

point in a specimen being tested for its properties

or in a material in service. Many materials are inhomogeneous so that the stress varies within

the body even though the external loading may

be uniform. We must therefore be concerned with

some problems involving stress distribution within

materials even though it is the properties of materials and not the solution of elasticity problems

that is our subject.

The propagation of elastic waves is the basis

for the most accurate methods of determination

of elastic properties as well as being of great

importance to the study of many other physical

properties (physical acoustics), to nondestructive

testing, to military applications, and to geological

studies. Accordingly, we sketch some of the

elementary results of the theory of wave propagation in solids.

Resonance of solid shapes is another technique

used to measure elastic properties. Resonance

is also a behavior of considerable engineering

importance. We therefore give the relations

between resonance frequencies and elastic properties for a few simple shapes.

Polycrystalline ceramics are aggregates of

elastically anisotropic crystals and frequently

include pores. The effect of porosity and the

relation between bulk elastic properties and the

properties of the grains are therefore discussed.

Ceramics are increasingly being fabricated as composites so that the elastic properties of

composites are also of interest.

3.2. Elastic Waves in Isotropic Materials

The simplest and most widely used type of

elastic wave is a plane wave in an infinite medium

(one whose dimensions are many times the wavelength). The general result (as derived, for

example, by Kolsky [95]) is that two types of

plane wave can be propagated in an isotropic

medium. The first is called longitudinal (or

irrotational or dilational) and has its direction

of vibration parallel to the direction of propagation. The second is called transverse (or equivoluminal or distorsional) and has its direction of

vibration perpendicular to the direction of propagation. For each wave the velocity squared times

the density is equal to an elastic modulus or combination of elastic moduli. Table 9 lists

expressions for pv2 in terms of various choices of

elastic moduli; see Anderson et al. [12, 96] for

expressions in terms of the pairs B, Y, and Y, v as well as most of the present pairs. If the velocities and density are known, one can, for example,

solve for G and for \+2G (see the last column of

table 9) and then obtain other elastic moduli by

using the expressions in terms of X and G given

in table 4.

When the body is not large compared with the

wavelength, these expressions no longer hold; the

velocities become dependent on frequency and

approximation methods must be used in most

cases. One case which can be solved exactly is that of a very thin cylindrical rod (circular cross

section). The equations for pv

2 are unchanged but

the longitudinal waves have a different velocity,

termed the rod velocity, vr , which is given by

pv

2 =Y. If the diameter of an initially thin rod

is continuously increased, the value of pv

2 =Y

should change to Pv l2 =Y(l-v)/(l +v)(l-2v) which is equivalent to the forms of table 9. The

transition is not continuous, however, surface

waves are involved; the two values of pv

2 are

correct for the limits of small and large diameters,

respectively [97-102]. The same transition occurs

if the dimensions of a slender rod are kept con- stant but the frequency is changed from a low

value to a high one. At sufficiently low frequency

the wavelength (given by \=v/j where / is the

frequency) will be large compared with the diamTable 9. Expressions for pv2 for Longitudinal (vO and

Transverse Waves (vt ) in an Infinite, Isotropic Medium

Independent Pair of Moduli

Y, G=M B, G=n B, v X, G=M

pfl 2 G(4G-Y)I(3G-Y)

G

B+(4G/3)

G

3B(l-x)/(l+")

3B(1-2f)/(2+2f)

\+2G

G

150
eter of the rod and pv

2 should approach the value

Y. It is therefore convenient in the resonance method of determining Young's modulus to use

slender rods; correction factors for finite thickness

are discussed in a subsequent section. At sufficiently high frequency, the wavelength becomes

small compared with the rod diameter and pv

2 should approach pv

2

. In the high-frequency

methods of determining elastic moduli it is therefore convenient to use cylinders of large diameter.

Experimental work [102] supported the theoretical

prediction that as the diameter-to-wavelength

ratio increased the bulk velocity was approached;

for ratio values above five no difference in velocity

was detected. Modern, high-precision methods are capable of precision of velocity measurement ex- ceeding one part in 105

, however, and diameter-towavelength ratios of 50 or more appear necessary

when working to this precision. For aluminum

oxide Vi is about 104 m/sec so that a diameter of

5 cm would be required for a frequency of 10 Mhz

or 0.5 cm for 100 MHz for high precision work.

The propagation of elastic compressional waves

in rectangular bars [103], in impulsively loaded

cylindrical rods [104], and in explosively loaded

plates [105] has also been studied. Truesdell [106]

has discussed the propagation of waves in an

isotropic elastic material subject to a finite static

strain.

3.3. Elastic Waves in Anisotropic Materials

Single crystals are elastically anisotropic as are most composite materials. The symmetry of

many composite materials is the same as that

of certain crystals so that the results derived for

single crystals can sometimes be applied to cer- tain composites. For this reason a qualitative

understanding of the elastic behavior is useful

in discussing behavior of engineering materials

even though single crystals may not be directly

involved.

A simple example of wave propagation in an

elastically anisotropic material is that of plane

wave propagation along the [110] direction in a cubic crystal. It is found [8] that a longitudinal

wave (vibration in the [110] direction) travels

with velocity given by

Pv l 2 =(2cii +cn +c12 )/2 (29)

In addition, there are two transverse waves:

v n with vibration in the [110] direction and v a with vibration in the [001] direction. Their

velocities are given by

pv a*=(cn-c12)/2, (30)

and pv^^cu. (31)

This example illustrates one feature of a general

result for small-amplitude elastic waves in an

unstrained body. For any given direction of propagation there are three waves whose directions

of vibration are at right angles to each other.

In the above example one direction of vibration

coincides with the direction of propagation so that one wave is purely longitudinal and the

other two are purely transverse. In general,

however, one wave is quasi-longitudinal (vibration at a small angle to the direction of propagation) and the other two are quasi-transverse. If

the wave is propagating along an axis of rotational

symmetry one wave is required by symmetry to

be purely longitudinal; the other two are accordingly purely transverse. In addition to these puremode directions, each type of crystal symmetry

permits some additional special directions whose

number and orientation depend on the numerical

values of the elastic constants. These special directions have been tabulated for all crystal symmetries except monoclinic and triclinic by Borgnis

[107] and Brugger [108]. These directions are useful

because a plate cut perpendicular to such a direction will vibrate in a pure thickness mode [107]

and because the use of these directions for the

propagation of waves simplifies the determination

of the third-order elastic constants [65, 66]. A

second class of special directions is one for which

a pure transverse mode may be propagated but

for which the other two modes are quasi-transverse

and quasi-longitudinal; equations for these directions have been derived for several classes of

crystal symmetry [109].

The expressions for the velocity of propagation

in directions for which crystal symmetry requires

pure longitudinal and transverse wave propagation are easily derived and many have been tabulated [8]. Unfortunately these do not provide

enough information to completely determine the

elastic constants for low-symmetry crystals. A

complete determination for low-symmetry tetragonal crystals has, however, been carried out [39]

and equations for monoclinic crystals have been

derived [110].

An interesting feature of wave propagation in

anisotropic crystals is the fact that the ray direction need not be parallel to the direction of propagation of a plane wave [111-115]. This effect occurs for light waves as well as sound waves. Thus, if a mechanical transducer (such as a quartz

crystal) is cemented on an arbitrary flat face of a crystal specimen, three plane waves (one quasilongitudinal and two quasi-transverse) will be

excited in the crystal. Each will have its direction

of propagation normal to the face of the specimen

but the energy will travel along inclined directions

(ray direction) and each of the three rays will

reach the opposite side of the specimen at a region not directly opposite the transducer. It is desirable when possible to avoid this complication

by choosing directions for which the ray direction

coincides with the wave normal, but even when

this is not done the velocity expressions mentioned previously are still correct. One must

arrange ' the experiment to measure the wave

151
velocity even though the ray is traveling at an

angle [116]. A method of calculating the elastic

constants by successive approximations using

velocities measured in an arbitrary direction has

been developed [117, 118].

3.4. Attenuation of Elastic Waves

by Scattering and Nonlinear Effects

Elastic waves lose amplitude by gradual conversion of their energy into heat. This occurs by

many mechanisms; this subject is generally termed

"Physical Acoustics" and is a large subject in

itself [119, 120]. This attenuation in solids frequently is caused by nonelastic behavior associated

with impurities and defects; these effects are

briefly discussed in a subsequent section. Attenuation also takes place in purely elastic materials

by scattering in inhomogeneous materials and as a result of nonlinearity. Mason and McSkimin

[121] have discussed scattering in polycrystals;

the process of Rayleigh scattering which takes

place when the wavelength is greater than three

times the grain size gives a loss proportional to

grain volume. At higher frequencies the type of

loss process changes and becomes a diffusion

process with the loss inversely proportional to

grain diameter when the wavelength is one third

the grain diameter or less. In pure and highly perfect single crystals the

attenuation is typically very low at frequencies

of a few megahertz but increases with increasing

frequency due to interaction between the elastic

waves and thermal phonons through the thirdorder elastic constants [122-124].

The velocities of elastic waves in rocks and

minerals have been compiled [12, 125] and the

attenuation observed in seismology has been

discussed [126].

3.5. Resonance of Isotropic Materials

A simple type of resonance is the thickness

vibration of a thin plate. The motion can be

described by a pattern of standing waves resulting

from the reflection of traveling waves at the large

flat surface of a plate. Constructive interference

will occur whenever the thickness of the plate

equals an integral number of half wave-lengths;

t=n\/2=nv/2f. If the lateral dimensions of the

plate are large, the velocity of waves in an infinite

medium can be substituted. Thus, for thicknesslongitudinal resonance of a thin flat plate of large

extent

\+2G=4Pfl 2 t

2/n2 (32)

and for thickness-shear resonance

G=4:f>ft H

2 /n2

. (33)

For a cylinder of circular cross section the last

equation holds unchanged and results from an

analysis of torsional resonance so that a cylinder

of any diameter-to-length ratio is suitable for the

determination of the shear modulus from torsional resonance. The longitudinal resonance frequency

of a very slender cylinder can also be deduced from

the condition for standing waves and the resulting

equation is Y=4pjl H

2/n2Kn (34)

where Kn approaches one exactly as the diameterto-length ratio approaches zero. Rayleigh derived

an approximate expression for Kn valid for small

diameter-to-length ratios which is Kn=l-nWd2/8P. (35)

More accurate Kn values can be obtained from

Bancroft's [97] velocity results; a table is given by

Spinner and Tefft [127] who also summarize theoretical and experimental resonance equations for

torsional, longitudinal, and flexural vibrations of

cylinders and bars of rectangular cross section.

For the fundamental mode of flexure of a cylinder

Y=1.2Q1886Pl

ifT/d2 (36)

where T is a correction factor [128-130] which

approaches one exactly as d/l approaches zero and

which is tabulated up to d/l=0.Q0 [128]. Equations

for the vibration of thick-walled hollow cylinders

have also been derived [131]. Lucey [132] has proposed approximate equations for thick discs and

thick cylinders derived by requiring the first order

corrections for thin discs and slender rods to be

compatible.

Resonance equations for bars of rectangular

cross section are available [128] but are accurate

only for slender bars.

A sphere displays two types of vibrational

modes [133, 134]; one, designated Tnm is torsional

and involves only shear deformation so that the

shear modulus can be calculated from a frequency

measurement. The other type, designated Snm, is spheroidal and involves both shear and dilation

so that a frequency measurement combined with

a knowledge of the shear modulus completely

determines the properties of an isotropic material.

All of the discussion of resonance refers to free

vibration. In practice care must be taken not to

constrain the vibration if the equations for free

vibration are to be used.

3.6. Resonance of Anisotropic Materials

The resonance of thin plates of large extent can be treated as before by using t=nv/2j but for

anisotropic materials the velocity is a function of

the orientation and the single crystal elastic

constants as previously discussed. Many single

crystals are piezoelectric so that the electrical

boundary conditions must be taken into account.

Tiersten treats this problem [135]; vibration of

plates has been used to determine the elastic constants of several piezoelectric crystals [136].

152
The vibration of slender single crystal cylinders

has been treated by Voigt [137] and Goens [138];

the use of this theory has been clarified by Kimura

[139] and Brown [140]. The general result is that

the flexural and torsional modes are coupled except

when the axis of the cylinder lies along certain

special orientations in the crystal. This coupling

changes the flexural and torsional resonance

frequencies in general. If, however, the cylinder

is very slender, the flexural resonance frequency

is unchanged and the torsional frequency is shifted

by an amount which can be calculated. Measurements of flexural and torsional resonance frequencies can therefore be used to determine the

single crystal elastic constants as has been done,

for example, for CaMo04 [40].

3.7. Effect of Porosity on Elastic Moduli of

Isotropic Materials

Most polycrystalline ceramics are porous; we

shall define porosity by P=fractional pore volume.

Elastic moduli (bulk, shear, or Young's modulus)

all decrease with increasing porosity; the initial

decrease is rapid but becomes less rapid at higher

porosity. Many authors have proposed equations

to account for this effect; these equations are

listed in table 10. Equations 2, 3, 6, 7, 13, and 14

of this table are theoretical equations and should

agree at small porosity values. In fact they do;

all reduce to the form G/G0 =l-bp and similarly

for B and Y at small porosity with the same values

of b from any of the equations for the same modulus

[53]. For a Poisson's ratio value ^=0.20 all give

6=2.00 [149]. The predicted slope agrees with

experiment for alumina with large, artificially

made spherical pores [144] and for glass with

spherical pores [149]. At larger values of porosity

Table 10. Equations proposed by various authors for

porosity dependence of elastic moduli of isotropic materials

Subscript zero indicates value at zero porosity.

Number Equation Reference

1 G=G0 [l-5P/3] [1411 2 l/B=l/Bo(l-P)+3P/4G0 (l-P)+terms of [142 order P3

3 G=Go[l-5P(3Bo+4Go)/(9Po+8Go)+terms [1421 of order P2

] 4 dY/dP= -2.36 [143 5°___ G=Gotl-5P(3Bo+4Go)/(9Bo+8G0)+AP2] [144 6. B=Bo[l-P(l+3B/4G)] [145 7 G=Go[l-5P(3B+G)/(9B+8G)] [145 8' y=Foexp(-6P) [47 9& G=G0 exp (-6P) [48 10* y=y0 exp (-60Po-6 c Pc ) [146 11" B=Boll-AP/(l-(A+l)P)] [147 12" G=G0 [1-APK1-(A+1)P)] [147 13 ____ B=Bo(l-P)/[l+(l+*o)/2P(l-2*o)] [148 14 G=Go(l-P)/[l+2(4-5r0 )/P(7-5^o)] [148 15 y=yo(l-P)/[l+(l+»o) (13-15xo)/ [148

16« _ 2P(7-5-o)]

G=G0 (l-6P+cP2) [53] 17« Y=Yo(l-bP+cP2) [53] 18 G=G0[l-15P(l-xo)/(7-5^o)] [153] 19 K=Yi>[l-3P(9+5..<,)(l-*o)/2(7-5»o)] [153]

« Parameter A to be determined by setting G=0 at P=l.

' Parameter 6 to be determined experimentally.

c Parameters 60 and 6 e to be determined experimentally. Subscripts o and

c refer to open and closed pores.

d Parameter A to be determined empirically.

• Parameters b and c to be determined experimentally.

the slope becomes smaller in absolute value. The

equations of Hashin as modified by Hasselman

[147], equations 11 and 12 of table 10, and by

Weil [148], equations 13 and 14 of table 10, appear

preferable in combining a theoretical foundation

with a proper form, but the empirical equations

8, 9, 16, and 17 appear to give a satisfactory fit to the data out to 40 or 50 percent porosity.

Although the theoretical equations correctly

predict the slope at low porosity for specimens

with spherical pores, the slope in normal polycrystalline ceramics is larger numerically [46].

Table 11 gives slope values for a number of polycrystalline ceramics; the values vary widely for

some materials and generally depart from the

theoretically expected value of about 2.0. Spriggs

[146] proposed that open and closed pores affect

elastic moduli to a different extent (see equation 10

of table 10). It has long been suspected that b

values greater than 2.0 (see table 11 for examples)

are the result of pores not being spherical. Hasselman and Fulrath [151] used the solution by

Hashin and Rosen [152] for cylindrical pores

perpendicular to the direction of measurement

and found that the data on alumina compiled by

Knudsen [46] fall within the upper and lower

bounds predicted by Hashin and Rosen's theory.

Table 11. Slope of elastic modulus versus porosity curve at

zero porosity

Material Modulus

Type of equation

fitted to determine

slope

Slope =6 References

AI2O3- Y A 1. 61 to 4. 35 [47] AI2O3 Y A 3. 95 [46 AI2O3-- G A 1. 72 to 3. 26 [146 BeO _. Y B 1. 99 to 2. 81 [150 BeO G B 2. 00 to 2. 80 [150 Dy2 03 ---- Y C 2.44 [50 Er2 Os Y C 2. 36 [50 Glass* Y C 2.06 [149' Glass* G C 1.94 [149 MgO Y A 4.74 [52' MgO G A 3.90 [52 Th02 Y D 2. 34 to 2. 68 [53 Th02 G D 2. 52 to 2. 88 [53 Y2O3 Y C 2.49 [54

A: V=K0 exp (-6P).

B: y=Yo[l-6P/(l-(6+DP)].

C: y=y0 (i-&P). D: y=Yb(l-6P+cP2 ). •16% Na2 0, 14% B2 03) 70% Si0 2 .

3.8. Relation of Elastic Moduli of Polycrystalline Materials to the Moduli of the

Single-Crystal Grains

The elastic moduli of isotropic aggregates of

anisotropic crystals in terms of the moduli of

the crystals has been calculated by a number of

writers. There are two general approaches. In

the first approach upper and lower bounds are rigorously derived; in the second approach approximations are introduced to calculate an

average value whose reliability is then difficult to

assess. The first approach was introduced by Hill

[153] who showed that averages previously in323-655 O - 69 - 11 153
troduced by Voigt and Reuss were in fact upper

and lower bounds respectively; i.e., Bv >B>Br and Gc >G>Gr . The values for these bounds are given in table 12. Hashin and Shtrikman developed

a variational method giving closer bounds and

used it to obtain specific expressions for cubic

crystals [154-156]. Peselnick and Meister [157,

158] have used this method to determine bounds

for crystals of hexagonal, higher trigonal, and

higher tetragonal symmetry. The Hashin-Shtrikman bounds are always better and sometimes

much better than the Voigt-Reuss-Hill bounds as shown in a number of compilations [11, 13, 157,

158].

Table 12. Expressions for Bounds and Estimates on Elastic

Moduli of an Isotropic Aggregate of Anisotropic Crystals

Crystals of Any Symmetry

B,>B>B, and (Voigt-Reuss-Hill) [7, 152] G,>G>Br where: -B.= (Cll+C22+C33+2Cl2+2C23+2Cl3 )/9 G,= (Cll+C22+C33-Cl2-C23-Cl3+3C44+3C55+3C6e)/15

-Br =[«ll+S22-|-«33 +2Sl2+2S23+2Sl3]-1 0«=[(4sii-|-4S22+4S33— 4«i2— 4«23— 4si3+3s«+3s55+3see)/15]-i

Cubic Crystals

-B=(cii+2ci2)/3 (Exact expression)

G lies between Gi* and G2* (Hashin and Shtrikman as expressed by

Simmons [13]) Where:

Gi, =Gi+3[-4^i+5/(<?!-(?i)]-i

<?2*= G2+2[-6|S2+5/(Gi-G2)]-i

ft = -3(B+2Gi)/5Gi(3B+4Gi)

02= -3(B+2G2 )/5G2(3B+4G2)

Gi = (Cn-ci2)/2 G2 =cu G is conjectured by Kroner [160] to be exactly equal to a root of the equation:

we)H«WO)! -6Wfi)-(=o

where:

a=Zn(k+iv)/6ki>

b=it(,9k+ir)l6kv c=WI3kv

fc=(Cll+2Cl2)/3 M =C44 »=(Cll-Cl2)/2

The second approach was followed by Hershey

and by Kroner [159, 160] who used a self-consistent

method. Kroner et al. [161] have shown that his

solution for the shear modulus of an aggregate of

cubic crystals lies within the Hashin-Shtrickman

bounds and Kroner suggests that his result is exact provided the orientations are completely

random. Kroner [160] subsequently derived an- other expression for the shear modulus using the

correlation function method; this answer is also

said to be exact but is too complex to give here.

Walpole [162, 163] has given another proof of the

Hashin-Shtrickman bounds and confirmed that

Kroner's first solution does lie within the HashinShtrikman bounds for cubic crystals.

Polycrystalline ceramics sometimes display sufficient preferred orientation to significally affect

elastic moduli. Extruded BeO shows varying

degrees of preferred orientation depending on the

starting materials and the degree of sintering [150].

The variation of Young's modulus with degree of

preferred orientation has been used to draw conclusions about the single crystal elastic constants

[164]. Rocks usually have a preferred orientation

of the grains; the linear compressibility has been

studied as a function of preferred orientation [165].

3.9. Elastic Moduli of Composites

The engineering importance of a wide variety of

composite materials has led to extensive work on

means of describing and measuring their bulk

properties and on the relationship between these

bulk properties and the properties of their constituents. One important type of composite is typified by a plywood sheet. This is usually treated

as a problem in two-dimensional elasticity and

orthorhombic symmetry is assumed; the material

is said to have orthotropic symmetry. Formulas

for bending, twisting, etc. are available [166].

Another important class of composites consists

of slender fibers dispersed in a matrix. The case in

which all the fibers are aligned parallel to each

other and are so long that end effects can be

neglected has been treated in detail. Such a composite has five independent elastic moduli and is equivalent to a single crystal with hexagonal

point group. A convenient description of elastic

properties is provided by using the plane strain

bulk modulus, the tranverse shear modulus, the

axial shear modulus, the axial Young's modulus,

and the corresponding Poisson's ratio. Upper and

lower bounds in terms of the elastic moduli and

volume fraction of the fibers and matrix have

been derived [167, 168]. Hill [169] has derived

estimates for these five moduli, using the selfconsistent method, which lie within these bounds.

Similar estimates have been given for aligned

ellipsoids or spheres in a matrix [170-171]. Comparisons with some of these theoretical results

have been made for the systems graphite in zirconium carbide [172], alumina in glass [149], and

tungsten in glass [173].

A number of special problems involving the

elastic response of composites have been treated

with varying degrees of approximation. We mention concrete [174], plaster and procelain [175],

and rocks containing cracks [176-178].

4. Atomic Theory of Elasticity

4.1. Lattice Statics

A treatment of the internal energy of a crystal

in terms of elastic constants and a simple atomic

model was given by Born in 1918 [179; 180].

The theory has subsequently been elaborately

developed but the original simple treatment still gives insight into the dependence of elastic constants on interatomic distance and on a unique

characteristic of many ceramic oxides. In this

simple model one assumes the energy is given by

U=-A/rm+B'/rn (37)

154
where r is the interatomic spacing. The first term

is the contribution due to attractive force; for

ionic crystals having ions with charge z the ex- ponent m is one and the addition of all the Coulomb forces gives A= a(ze) 2 where a is a numerical

constant (a— 1.7476 for MgO) obtained from

this sum. The second term is the contribution of

the repulsive forces; the constants B' and n must

be determined by fitting the equation to experimental data in the following manner. We have

from thermodynamics.

dU

dv

and B = -v

dp

dv

(38)

(39)

Since v— constant times r

3

, we have Svdr—rdv

and so obtain

and

p=(-A/r+nB'/rn)/3v

B=p +(-A/r+n2B'/rn)/9v.

We now use r=r0 at p=0 and obtain

nB'/r0 n =A/r0 which gives

B=p(l +n/3) + (n- l)A/9rv.

(40)

(41)

(42)

(43)

The parameter n is determined by setting p—0

and requiring this expression to give the measured

value of the bulk modulus. The resulting value of

n ranges from 6 to 12 for various alkali halides and

from 6 to 9.5 for a small number of oxides considered [181]. The internal energy calculated

from n usually agrees with the measured cohesive

energy within 10 percent. This simple theory

has a further success in predicting how the bulk

modulus varies with lattice parameter in a family

of ionic compounds with the same crystal structure.

Taking p=0 and r=rQ we have

B=a(ze2

) (n-l)/9r0v0

. (44)

Since r0vQ is proportional to a

4 (where a is the

lattice parameter) we expect plots of log B

as a function of a to have a slope of —4 and plots

of both the bulk modulus and the shear modulus

do show this slope for alkali halides [182, 183].

The above treatment also provides a basis for

equation (6) which is the basic assumption needed

for the successful Murnaghan equation, number

(7).

. Oxides do not generally obey the predicted

1/a4 variation of bulk modulus, however. The

variation is more nearly described by 1/a9 [183,

184], see figure 5. Anderson [183] has given a

qualitative explanation in terms of a modification

of the above model in which another term of

5000

4 000

3000

2000

CDo

o

aO

10

1000

800

600

500

400

- 300

200

100

0*-}STISHOVIT£

-\- DIAMOND

vV oTlC

\ \\CARBIC

w

ES

rc

^SEMICON DUCTO RS

^(•F \oxiDES \ \(M/P~2l) >

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^Gasb

LOjfttLAss)

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ybl

\

10 20 30 40 50 60 70 VOLUME PER ION PAIR (Cm'/MOLE)

Figure 5. The variation of the bulk modulus with

volume for several classes of materials.

(Oxides have a slope of —3 on this plot.) After Anderson [183].

the form C/Rm is introduced to account for the

repulsion of the large anions which will be in

direct contact with each other for a cation-anion

radius ratio less than 0.414. Many oxides have

smaller ratio values and are essentially structures

made up of close packed oxygen with small

cations in the interstices. The attractive forces

draw the oxygen into a state of compression and

it is the form of the oxygen-oxygen repulsive

term which causes the 1/a9 variation of the bulk

modulus. Anderson [185] has proposed that the

change of bulk modulus of oxides with temperature

or pressure is approximately accounted for by

the change of volume through the 1/a9 law. That

is, the bulk modulus of a pore-free oxide made

up of close packed oxygen ions can be calculated

approximately simply from its specific volume,

even though changes of volume may result from

change of cations, of pressure, or of temperature.

General approximation formulas are given [183]

for the large class of oxides (including most

oxides of technological interest) having a mean

atomic weight of about 21 ; these permit estimation

of averaged B, 6, and related elastic properties

from the density.

155
4.2. Linear Lattice Dynamics: Relation of

Elastic Constants to Sound Velocity,

Specific Heat, Optical Absorption, and

Dielectric Constant.

The foregoing static model permits the calculation of the internal energy, U, and the enthalpy,

U+pV, as a function of pressure. By considering

the vibrations of the atoms one can introduce

temperature as a variable and calculate many

properties. The subject of lattice dynamics is large

and mathematically complex. We shall quote a few

results to show how some thermal and electrical

properties depend on the elastic constants and

other quantities. The simplest lattice dynamics

treatments are based on the MgO structure or other simple structures; the resulting equations

cannot be expected to apply exactly to other

structures. The simplest treatment assumes linear

springs between atoms (the harmonic approximation) and successfully treats sound velocity, specific heat, optical absorption, and ionic contributions to the dielectric constant. We defer to the

next section a discussion of such properties as

specific heat which depend on the introduction

of nonlinear springs (the third-order elastic

constants)

.

We have already given the relation of sound

velocity to elastic constants in discussing continuum mechanics. The equations for v t and vs are compatible with the atomic model for crystals and

hold unchanged for wavelengths large compared

to atomic spacings. As the wavelength decreases

the velocity eventually begins to decrease also.

This effect is usually shown on plots of frequency

as a function of wave number (k=2ir/\). This plot

has two curves emerging from the origin with

slopes equal to the velocity of lingitudinal and

transverse waves respectively as shown in figure 6.

WAVE NUMBER

Figure 6. Frequency-wave number plot for a three-dimensional isotropic body.

After Anderson [186].

There are also two optical branches of this plot

corresponding to vibrations in which ions of opposite sign move in opposition. It is possible to

choose light of such a frequency (usually in the

infrared) that it equals v0 of figure 6 with the

result that a strong absorption occurs. Szigeti

[187, 188] has derived an equation giving this

frequency in terms of the isothermal bulk modulus,

BT, the lattice parameter, a, for the MgO structure,

the static dielectric constant, e 0 , the index of re- fraction, n, and the reduced mass, m:

2TM0 = [6 (n2 +2)BTa/m(e 0 +2)]

1 /2

. (45)

Anderson and Glynn [189] have used the measured

absorption frequency to calculate the bulk modulus successfully for MgO.

The dielectric constant at low frequency (below

10 10 Hz) is the sum of a contribution due to

electronic motion and ionic motion; its value at

optical frequencies (which equals the square of

the index of refraction, n) is due to electronic

motion alone. Szigetti has shown that the difference, Ae, due to the ionic contribution is given by

Ae=(n2 -f2)V/6Wa3 m (46)

for the MgO structure. We see that this depends

on the bulk modulus through v 0

. The specific heat of a crystal is obtained from

counting up all the modes of vibration using the

frequency-wave number plot mentioned above.

Debye approximated this complex problem by

taking the sound velocity to be independent of

wave number and obtained a useful approximate

equation giving the specific heat as a function

(tables of which are available [190]) of T/6 where

6 is the Debye temperature and is given by

e= (h/k) (3pNP/4:TrM) V3vm (47)

where A=Planck's constant, &=Boltzmann's constant, p=number of atoms per molecule, N=

Avogadro's number, p= density, and M=molecular weight. The velocity vm is a special type of

average over all directions in the crystal given by

«m =[ave. of [(l/^3

) + (l/^ (1 3) + (l/^3 )]/3]- 1/3 (48)

The problem of calculating vm and hence 9 and the

specific heat at any temperature is simple in

principle; for any direction the three velocities can be calculated from the elastic constants and

a numerical average over a large number of

directions can be taken. Alers [191] has reviewed

a number of procedures for doing this. Anderson

has shown that approximating a single crystal by

an isotropic material with elastic constants given

by averaging the Voigt and Reuss bounds leads

to a simple expression for vm which agrees with

the accurately calculated value within a few

percent for many crystals [192]. The calculation of

Debye temperatures from elastic constants continues to be an active field [193-196].

156
Detailed treatments of the vibrational spectrum

and related properties have been given for a number of structures besides MgO including

rutile (Ti02 ) [197, 198] and the fluorite structure

(U02 , Th02 , etc.) [199, 200].

4.3. Nonlinear Lattice Dynamics: Thermal

Expansion and Temperature Dependence of

Elastic Constants

The forces between atoms in a crystal are not

simply linear functions of displacement but include nonlinear terms as well. This leads to great

complexity in the treatment of lattice vibrations

and properties dependent upon these vibrations

[9, 179]. Some properties, such as thermal ex- pansion and temperature dependence of the elastic

constants, depend for their very existence upon

the nonlinear terms in the interatomic force laws.

These properties are usually treated in the so- called quasi-harmonic approximation in which

the forces are taken to be linear but with a vibrational frequency, co, for the atoms which

depends upon the average interatomic distance.

This dependence is expressed by a number y, called Griineisen's constant, defined by y=dln

a/dln V which is assumed to be the same for all vibrations and in the simplest treatment is independent of temperature. It can be shown

[179, 201, 202] that the volume thermal expansion

coefficient, 0, is given by

p=yc,,/BTV (49)

where c„ is the specific heat at constant volume.

The use of an integrated form of this equation

has been discussed [203] and it has been shown

[204] that y is not truly constant but may vary

by about 10 percent as the temperature is increased from absolute zero to the melting point.

We shall leave further discussion of thermal

expansion to the companion paper in this volume

on thermal expansion and simply point out that

in principle one can calculate the thermal ex- pansion from knowledge of elastic properties

alone. We have already discussed the calculation

of cv from elastic constant data; y can be calculated

using both the second-order and third-order

constants [205, 206].

The elastic constants change with temperature

because of the presence of nonlinear terms in the

force expressions. Leibfried and Ludwig [9] give

B=B0 (1-DE +dp) (50)

where B0 is the value in the absence of nonlinear

terms and D and d are complicated functions of

the nonlinear terms. The temperature dependence

comes in through the thermal energy, E, which

increases with the temperature. Anderson [207]

has derived the following equation for the temperature dependence of the bulk modulus

B=B0 -3RySEIVo (51)

where R is the gas constant, subscript zero indicates value at absolute zero and 5 is a second

Griineisen constant (approximately independent

of the temperature) defined by

S=-(lfa)(blnB/bT) v . (52)

This proportionality of the temperature dependent

part of the bulk modulus to the thermal energy

leads to a variation proportional to T4 at low

temperature (where the specific heat goes as T3 in

the Debye theory) and proportional to T at high

temperature (where the specific heat is approximately constant). It is sometimes said that

elastic constants vary with temperature because

of thermal expansion and this is true in the sense that both depend on nonlinear terms in the

potential energy. However, Stern [208] points

out that thermal expansion depends only on the

odd terms (corresponding to third-order elastic

constants) but the temperature dependence of

elastic constants depends on both odd and even

terms (corresponding to third-order and fourthorder elastic constants). The transition from zero slope at absolute zero to a linear decrease with

increasing temperature predicted by theory is found in plots of Young's modulus or the shear

modulus for many crystalline oxides and is shown

in figure 7 for A12 03

. This behavior is well described

by the empirical equation

Y=YQ -bTexV (-T0/T) (53)

in which Yn is Young's modulus at absolute zero and b and T0 are empirical constants [209]. Anderson [207] has shown that at high temperatures T0 corresponds approximately to one-half the Debye

theta. Tables 13 and 14 give temperature coefficients for elastic moduli of various single-crystal

and polycrystalline ceramics respectively. These

coefficients are actually functions of temperature,

usually changing smoothly from a value of zero

at low temperature to a constant at high temperature, but the value calculated from any temperature interval above 0/2 will generally approximate

the high temperature limit. An order-of-magnitude

rule is that at high temperatures, elastic moduli of

crystalline ceramics decrease one percent with each

increase of 100° C. Extrapolations of elastic moduli

to very high temperatures has been given for

several oxides [215].

Barker [216] has shown that the quantity a

2 Y

is approximately constant for 68 of 79 materials

examined. This implies a correlation between the

anharmonic properties and the elastic moduli.

4.4. Effect of Solid Solution and Phase

Changes on Elastic Constants

The effect of solid solution on elastic moduli

has been studied in a number of metal alloys

but little appears to have been done in ceramic

systems. Slagle and McKinstry [217] have studied

the system KCl-KBr and found that plots of

157
4700

100 500 600 TEMPERATURE, *K

1000

Figure 7. Young's modulus of single-crystal aluminum, oxide as a function of

temperature.

After Wachtman et al. [209].

Table 13. Temperature coefficients for some single crystal

elastic constants

Material

Al2 03 YA1G

(garnet) MgO UC

Quantity

(ABslAT)/Bs (ACu/AD/Cu . (AC44/AD/C44

(AC12/AD/C12 (i>Bs/dT)/Bs

(ACu/ATJ/c,,

(AC44/AD/C44

(Ac,2 /AT)/c12

Value

-0.68 -0.9

-0.7

-0.5

-0. 97 -1.0 +2.0

-0. 75

Temperature range K

300 to 800

200 to 300

200 to 300

200 to 300

300

150 to 300

150 to 300

150 to 300

Refer- ence

[2101

[211]

[211]

[2111

[28] [212] [212

[212]

Debye

theta K

1045

946

Refer- ence

[209]

[190]

Table 14. Temperature coefficients for elastic moduli of some polycrystalline ceramics

Material

AUO,

Dy2 0; Er2 03 MgO.

Mg2 Si04.-_

(forsterite). Mg2 Si03 .._

Th02 . Y2 03 .

Quantity

(dY/dT)/Y

(dY/dT)/Y

(dG/dT)/G

(dB/dT)/B

(dY/dT)/Y

(dY/dT)/Y

(dYldT)IY

(d^ldT)IY

(dG/dT)/G

(dB/d T) /B

(dY/dT)/Y

(dGldT)IG

(dB/d T)/B

(dY/dT)/Y

(dG/dT)/G

(dB/d T)/B

(dY/dT)/Y

(dY/dT)/Y

Value

io-*/k -1.4

-1.3

-1.4

-0. 84 -1.1

-1.1 -1.9

-1.8 -2.0

-1.1 -1.6

-1.7

-1.3

-1.2 -1.2

-1.2

-1.4

-1.1

Temperature range K

M300-1400

300-1400

300-1400

300-1100

300-1100 W300-1100

300-1100

300-1100

300-900 300-900

300-900

300-900

300-900

300-900 M300-1300

Refer- ence

[209 [213

[213

[213'

[50; [50 [209

[213;

[213; [213 [214

[214; [214 [214

[214

[214 [209

[54]

Debye

theta K

1045

946

6363

Refer- ence

[209]

[209]

[209]

"Fit of B in Y=Y0 -BT exp (-T0 /T) over 78 to 873 K.

6 Incorrectly listed as 782 rather than 363 in reference

[209].

cu and c44 fall below the straight line predicted

by a simple method-of-mixtures rule; i.e. the curves are concave upward. Examples of departures from straight line behavior in both

directions (either concave upward or concave

downward) are known in various metal systems

[7] so that no simple rule can be given.

The behavior of elastic constants near the

melting point of materials with the MgO structure has been extensively studied [217]. It is proposed that melting occurs when vibrations

corresponding to the shear associated with cuC12 become so large that the structure becomes

unstable. One might therefore expect cn-Cn to

approach zero or at least to become small as the melting temperature is approached. The most

recent measurements show a smooth decrease of

C11-C12 with temperature to a value just before

melting of about one-quarter of the value at

25 °C for NaCl, KC1, KBr. In the system KC1-

KBr the values of cn-c12 measured at 700 °C vary

with composition in the same way as the melting

point.

In solid-to-solid phase changes the elastic constants change as would generally be expected

for physical properties in a first-order phase

change. In second-order phase changes the usual

thermodynamic state functions are continuous

but their derivatives, including elastic constants,

may be discontinuous. Second-order phase changes

are important in connection with order-disorder

processes, change from nonferroelectric to ferroelectric state, etc. Some phase changes of this

type were first discovered by measuring elastic

158
constants as a function of temperature; this was

the case for SrTi03 [218]. Typically some of the

elastic constants may show a smooth change with

temperature while others show a large variation

and may approach zero. Presumably the latter

correspond to a type of vibration whose amplitude is becoming large as the transition temperature is approached and which corresponds to the

type of atomic rearrangement involved in the

phase change. Thus the phase change of first or second order may be signaled by the behavior

of the elastic constants at temperatures well

removed (e.g., 20 °C) from the transition temperature. Examples are the change at 184 K in

KMnF3 [219], the paramagnetic-antiferromagnetic transition at 30.8 K in U02 [2201, and the

order-disorder transition at 243 K in NH4 CI [221].

4.5. Internal Friction and the AE Effect

We have previously emphasized that elastic

deformation (in our definition) is characterized by

complete recovery when the stress is removed

even though the recovery may require a long time.

Time dependent behavior of this type causes a portion of the energy used in producing the

deformation to be converted into heat. This loss

is said to be due to internal friction and is usually

a function of temperature and of frequency in

cyclic deformation. Loss also occurs in cyclic

plastic deformation and as a result of other

irreversible processes (such as domain wall motion)

induced by stress. Only the type of internal friction

associated with recoverable deformation is an

appropriate subject for discussion in a treatment

of elasticity but the two types are sometimes

difficult to separate experimentally and even

conceptually for certain processes which have

long recovery times. Extensive reviews of internal

friction are available [222-225] and we limit our treatment to a brief discussion of two types which

are particularly useful in throwing light on the

properties of point defects and dislocations

respectively.

A point defect (such as a vacancy, an interstitial, or an impurity atom) generally causes a distorsion of its neighborhood. If this distorsion

has a symmetry lower than the point group of the

crystal, it can occur in more than one way. The

different ways of occurring will have equal energy

and equal probability in the absence of stress or other influence such as electric field which lowers

the symmetry of the crystal. Under stress the

different ways can become inequivalent leading

to the development of a preferred distribution if time is allowed for the necessary thermally activated jumps. This jumping is accompanied by an

additional strain (causing the appropriate elastic

modulus to decrease) and an energy loss. This

subject has been worked out in detail [226-238].

Measurement of this internal friction can be used

to obtain several different types of information

concerning defects in crystals. First, the variation

of the amplitude of the internal friction peak

with type of stress applied gives information on

the symmetry of the defect and can sometimes be

used to reduce the number of possible models for

an unknown defect. This type of investigation is most useful when it can be combined with measurements of other properties such as dielectric loss,

optical spectra, and electron spin resonance which

also give information on the symmetry of the

defect. Second, study of the temperature dependence of the internal friction with varying

stress components at constant frequency can

give information on the number of relaxation

times involved in point defect motion which can

then be interpreted in terms of a specific atomic

model of the defect. Third, the anisotropy of

distortion around a defect can be deduced,

although one other piece of information such as the lattice parameter dependence on defect concentration is needed to completely specify the

strain caused by a defect. Fourth, studies of the

size of the internal friction peak in quenched

specimens as a function of annealing temperature

can sometimes be used to measure the association

energy for pairing of point defects. Fifth, the

jump frequency for the motion involved in diffusion can sometimes be measured so that diffusion

measurements can then be made at temperatures

much lower than is possible with direct diffusion

measurements [239]. Ceramics in which point

defect properties have been studied by internal

friction measurements include CaF2 [240, 241],

Ce02 [2421, MgO [243, 2441, NiO [245], Si02 (quartz) [246], Th02 [231], Ti02 [247-250], TJ02 [251], and Zr02 [252].

Dislocations in crystals are usually pinned at

various points along their length by impurities

or point defects; they are pinned more firmly at

intersections with other dislocations. Under small

stress the free segments bow out, but the ends

remain attached to the pinning points if the stress

is not too great; the dislocations return to their

original positions when the stress is removed. This

inelastic, recoverable deformation gives rise to a frequency-dependent but amplitude-independent

internal friction and a corresponding decrease in

elastic moduli. If greater stress is used, the dislocations break away from their less firm pinning points

producing a loss which increases with increasing

amplitude. These effects have been extensively

studied in metals in which plastic deformation may

. produce such a high density of dislocations that

reduction of the elastic moduli by a few percent

occurs. This is termed the AE effect. The elastic

moduli can be returned to their original value

by annealing or by pinning the dislocations by

radiation damage. The theory of dislocation

damping has been extensively developed [253]

and can be used to determine the resistance to

dislocation motion if the other parameters of the

theory are determined. The theory has been

applied to explain internal friction in Al2 03 [254,

255], MgO [254, 256, 257], and Ti02 [258].

159
Polyc^stalline ceramics may exhibit damping

caused by point defects and dislocations, but other

effects such as porosity, viscous phases, and phase

changes are frequently predominant in causing

damping [259].

5. Special Topics

5.1. Correlations of Other Properties With

Elastic Constants

Several physical properties in addition to those

already discussed are related to elastic moduli.

The presence of interstitials and vacancies should

cause a change in the bulk modulus [260]; accurate

measurements of bulk modulus might be useful

as a measure of high densities of Frenkel defects.

Solid solution causes a change of lattice parameter; the lattice parameter of ruby has been calculated as a function of chromium content in

terms of elastic moduli [261]. The change of lattice

parameter is usually proportional to the concentration of the solute in a solid solution (Vegard's

law) , but deviations occur which have been treated

in terms of elastic moduli [262]. The elastic

constants of many valley semiconductors depend

somewhat on the population of electrons in the

conduction band and hence on the doping [263].

We have previously mentioned the relation between

elastic moduli and volume for isostructural

compounds; a similar correlation involving both

the volume and cohesive energy has been proposed

[264]. Finally, a correlation between elastic

moduli and strength is often noted as, for example,

for a series of whitewares [265]. There is an

apparent justification for this relation; approximate

treatments of theoretical strength typically give

values of the order of one-tenth of Young's

modulus. Empirical correlations, however, usually

involve much lower strengths under circumstances

in which fracture starts at some flaw. The correlation of strength with elastic moduli is often due

to having both bulk elastic moduli and strength

affected in the same way by a variation of microstructure such as porosity. This type of correlation

must not be expected to hold in general, however.

A series of alumina bodies showed a variation in

strength of a factor of 500 [266]; their elastic

moduli vary by only about a factor of 2.

5.2. Elastic Moduli of Graphite

Graphite is not a unique material but a family

of materials differing in crystal structure, microstructure, and preferred orientation. Single crystals are usually hexagonal with a layer structure and

have a very much greater stiffness parallel to the

base plane than perpendicular to it. Spence [267]

gives the single crystal elastic constants as di = 116, c33 =4.7, c44 =0.23, c12 =29 and c13 =10.9

X1010 N/m2

. A typical extruded polycrystalline

graphite has Young's modulus values of 1.06 and

0.47 X1010 N/m2 parallel and perpendicular respectively to the direction of extrusion [268]. Measurements of Young's modulus on various pyrolytic

graphites gave values ranging from 56 to 0.8 X1010 N/m2 where dislocations were pinned by light

neutron irradiation [269] to eliminate the AE

effect which is large in graphite [270-273]. The

temperature dependence of Young's modulus

from 25 to 1000 °C has been reported as a function

of various heat treatments at higher temperatures

[274].

5.3. Measurement of Elastic Moduli

Elastic moduli were originally determined by

static tests in which relatively large loads were

applied and the resulting deflection was measured.

These static tests are less accurate than dynamic

tests but are still of great engineering importance.

They are covered in the paper on mechanical

testing in this volume. The dynamic tests are of

two basic types: measurement of the velocity of

propagation of waves, and measurement of the resonance frequency. We have previously discussed the relation of elastic constants to wave

velocity or resonance frequency; it remains now

to briefly sketch the experimental procedures.

The principles of wave velocity measurement

and the calculation of elastic moduli from the

velocities have long been understood. The availability of good megahertz oscilloscopes and other

electronic apparatus following World War II has

led to much work in this field. We shall mention

a few papers which are important to the development of this technique to its present highly

refined state and some which consider its application specifically to ceramics.

The basic technique is to bond a piezoelectric

generator onto one flat face of a specimen cyrstal,

excite sound waves, and measure the transit time

to a flat and parallel opposite face. Price and

Huntington [275] measured the echo time for

ultrasonic pulses in single crystals and studied

some of the phenomena of acoustical _ birefringence, mode coupling, and beam spreading. Eros

and Keitz [276] discussed the problem of transit

time error arising from change of shape of the

pulse upon reflection from the interface between

two media. McSkimin [277, 278] has summarized

(and greatly developed) methods in which re- flected but unrectified pulses are compared with

an identical later pulse whose delay is controlled

by a continuous wave oscillator. This is essentially

an interference technique capable of a precision

approaching one part in 106

. Application of

accurate methods to high temperatures [279] and

low temperatures [280] has been discussed. The

application of these techniques to rocks and

minerals has been reviewed [12, 281]. Neighbours

and Schacher [282] have presented a general

perturbation method for calculating second-order

elastic moduli from measurements made in directions other than those for which pure modes are propagated.

160
The accuracy of elastic constant measurements

generally much less than the precision with which

measurements can be repeated in the same apparatus or with which changes of elastic constants

with temperature or pressure can be measured.

Einspruck and Truell [283] conducted a survey in

which nine laboratories made measurements on

the same vitreous silica specimen; seven of the

nine agreed within 0.3 percent in values of transit

time. An error in absolute value of 0.3 percent is much greater than precision of 1 in 106 achievable

in relative measurements but is still better than

the accuracy of static tests which is rarely better

than a few percent. In addition to experimental

problems, the comparison of dynamic and static

test results may be affected by the different stress

level and the different effects of defects such as cracks [284].

The resonance method of determining elastic

moduli is capable of achieving absolute accuracy

of 0.1 percent or better when care is taken to

choose a shape for which the vibrational equations

are accurately known and when the dimensions can be held to a corresponding tolerance. It can be used to measure changes with temperature of

1 in 104 or better when care is taken to minimize

coupling to the apparatus and the atmosphere.

Methods of exciting and measuring resonances have been extensively reviewed [127, 285, 286]

and specialized apparatus has been described for use at low temperature [287], high temperature

[288], constant strain amplitude [289], very low

strains [290], and for materials with very high

damping [291]. Techniques for direct mechanical

excitation and detection of resonance up to 300

kHz have been developed [292]. Resonance of

spheres at frequencies up to 2 MHz has been

used [134].

A method of measuring elastic constants using

diffuse reflection of x-rays has been developed

[293, 294] but accuracies of only 5 to 45 percent

were reported in a recent evaluation of the method

[295]. A technique for detecting mechanical

resonance and measuring sound velocity by

nuclear magnetic resonance techniques has been

described [296].

Measurement of the elastic moduli of libers

presents special problems ; experimental techniques

have been reviewed [297].

5.4. Characterization of Ceramics With

Respect to Elastic Moduli

It is sometimes desired to reproduce a given

physical property closely in a whole series of

ceramic parts. The manufacturer should then seek

to control the aspects of composition and microstructure upon which the property of interest

depends. If property reproducibility is not sufficiently well achieved, it is then of interest to

characterize the specimens, i.e. to measure

appropriate aspects of composition and microstructure, and to determine which are varying.

For elastic moduli the characterization required

is usually relatively straightforward for single

crystals, but more complicated for polycrystals

and composites. Our whole discussion of elastic

moduli has been aimed at explaining the factors

upon which elastic moduli depend; in conclusion

we briefly summarize the more important ones. For single crystals the presence of impurities

is usually of little consequence to elastic moduli

provided the solid solution limit is not exceeded.

Within the range of solid solution a simple linear

method-of-mixtures calculation, though not exactly

correct, will give a resonable estimate. If the solid

solution limit is exceeded and precipitate particles are included within the crystal the system may

be treated as a composite using one of the theories

previously discussed provided the shape of the

precipitate corresponds to the model assumed.

Care is necessary, however, because if the precipitate shrinks more than the crystal during cooling

and breaks away from the matrix the specimen

may actually be a single crystal with pores so far

as elastic behavior is concerned. If the precipitate

shrinks less than the crystal, internal cracking or

dislocation generation may occur so that the elastic

moduli are reduced more than would be expected

from the volume of second phase. Single crystals are usually sufficiently pure, however , that these

problems do not occur. A more common problem

is lack of accurate measurement of orientation and

sometimes lack of understanding that elastic

moduli of single crystals depend on orientation even for cubic crystals and that for trigonal and

tetragonal crystals they depend on the angle around

the optic axis as well as the angle to the optic axis.

The elastic moduli of relatively ductile ceramics

such as single crystal MgO can be reduced by the

presence of a high density of unpinned dislocations

through the AE effect.

In single-phase polycrystalline ceramics the

factors most likely to affect elastic moduli at room

temperature are porosity (including cracking from

thermal stresses) and preferred orientation. As

mentioned previously equations are known which

correctly predict the change in elastic moduli as a function of a small volume fraction of spherical

pores, but porosity usually is not even approximately spherical and causes a greater reduction in

elastic moduli. Polycrystalline ceramics prepared

by sintering of a powder made up of equiaxed

particles are likely not to have a significant degree

of preferred orientation. Ceramics prepared by

extrusion or hot pressing or ceramics prepared

from unequally axed powder particles are likely

to show appreciable preferred orientation and

this can have a pronounced effect on elastic moduli.

Polyphase polycrystalline ceramics are subject

to the same sources of variability, but may in

extreme cases of thermal mismatch exhibit severe

internal cracking causing a large reduction in

elastic moduli. Such a body may undergo a degree

of crack healing when heated and develop a new

161
pattern of cracks upon recooling. This type of

body has no well defined values of elastic moduli;

the measured values will change erratically as a

result of thermal cycling.

Ceramic composites may possess any of the

above sources of variability but the degree of

orientation of the reinforcing fibers or plates and

the degree of bonding between components are

likely to be the predominant factors.

6. References

[1] Mason, W. P., Crystal Physics of Interaction

Processes, Academic Press, New York (1966).

[2] Chisholm, L. J., Units of Weight and Measure,

NBS Misc. Publ. 286, U.S. Government Printing

Office (1967).

[3] Hearmon, R. F. S., The Elastic Constants of Nonpiezoelectric Crystals, Landolt-Bornstein, New

Series, Group III, Volume 1 (1966).

[4] Beckmann, R., The Elastic, Piezoelectric, and

Dielectric Constants of Piezoelectric Crystals,

Landolt-Bornstein, New Series, Group III, Volume

1 (1966).

[5] Hearmon, R. F. S., The Elastic Constants of

Anisotropic Materials, Rev. Mod. Phys. 18,

409 (1946).

[6] Hearmon, R. F. S., The Elastic Constants of

Anisotropic Materials, II, Advan. Phys. 5, 323

(1956).

[7] Huntington, H. B., The Elastic Constants of

Crystals, Solid State Phys. 7, 214 (1958), edited

by F. Seitz and D. Turnbull, Academic Press.

[8] Aleksandrov, K. S., and Ryzhova, T. V., The

Elastic Properties of Crystals, Soviet Phys.

Cryst. 6, 228 (1961). Translated from Kristallografia 6, 289 (1961).

[9] Leibfried, G., and Ludwig, W., Theory of Anharmonic Effects in Crystals, Solid State Phys. 12,

276 (1961), edited by F. Seitz and D. Turnbull,

Academic Press.

[10] Lynch, J. F., Ruderer, C. G., and Duckworth,

W. H., Engineering Properties of Ceramics,

published by the American Ceramic Society,

Columbus, Ohio (1966).

[11] Simmons, G., Single Crystal Elastic Constants and

Calculated Elastic Properties, Journal of the

Graduate Research Center, Southern Methodist

University Press, 34, 1 (March 1965).

[12] Anderson, O. L., and Lieberman, R. C, Sound

Velocities in Rocks and Minerals, Vesiac State-ofthe-Art Report 7885-4-X, The University of

Michigan.

[13] Simmons, G., Hashin Bounds for Aggregates of

Cubic Crystals, Journal of the Graduate Research

Center, Southern Methodist University Press,

36, 1 (January 1967).

[14] Overton, W. C, Jr., Relation Between Ultrasonically Measured Properties and the Coefficients

in the Solid Equation of State, J. Chem. Phys.

37, 116 (1962).

[15] Thurston, R. N., Ultrasonic Data and the Thermodynamics of Solids, Proc. IEEE 53, 1320 (1965).

[16] Cook, R. K., Variations of Elastic Constants and

Static Strains with Hydrostatic Pressure: A

Method for Calculation from Ultrasonic Measurements, J. Acoust. Soc. Am. 29, 445 (1957).

[17] McSkimin, H. J., Elastic Moduli of Single Crystal

Germanium as a Function of Hydrostatic Pressure,

J. Acoust. Soc. Am. 30, 314 (1957).

[18] Anderson, 0. L., An Accurate Determination of the

Equation of State by Ultrasonic Measurements,

Proceedings of the Conference on High Pressure

Research, p. 225, E. Bundy, Editor, John Wiley

and Sons, Inc., New York (1963).

[19

[20

[21

[22

[23

[24

[25

[26

[27

[28

[29

[30

[31

[32

[33

[34

[35

[36

[37

[38

[39

[40

[41

[42

Thurston, R. N., Wave propagation in fluids and

normal solids, p. 1 in Physical Acoustics, Volume

I A, W. P. Mason, editor, Academic Press, New

York (1965).

Anderson, O. L., Two methods for estimating compression and sound velocity at very high pressures Proc. Natl. Acad. Sci. 54, 667 (1965).

Anderson, O. L., The use of ultrasonic measurements

under modest pressure to estimate compression

at high pressure, J. Phys. Chem. Solids 27, 547

(1966) . Nye, J. F., Physical Properties of Crystals, Oxford

University Press, Oxford, England (1957).

Truesdell, C, and Toupin, R. A. The classical field theories, in Handbuch der Physik Volume III/l,

S. Flugge, editor, Springer, Berlin (1960).

Bechterew, P., Analytical investigation of the general Hooke's Law. Use of the rule of potential

energy and the principal of minimum work, Z. Krist. 64, 373 (1926).

Alers, G. A., and Neighbours, J. R., Crystal stability

and elastic constants, J. Appl. Phys. 28, 1514

(1957).

Bateman, T. B., Elastic moduli of single-crystal

europium iron garnet and yttrium iron garnet, J. Appl. Phys. 37, 1294 (1966).

Soga, N., Elastic constants of garnet under pressure

and temperature, J. Geophys. Research 72, 4227

(1967) . Anderson, O. L., and Andreatch, P., Jr., Pressure

derivatives of elastic constants of single-crystal

MgO at 23 °C and -195.8 °C, J. Am. Ceram. Soc. 49, 404 (1966).

Bolef, D. I., and de Klerk, J., Elastic constants of single-crystal Mo and W between 77 °K and 500

°K, J. Appl. Phys. 33, 2311 (1962).

Featherstone, F. H., and Neighbours, J. R., Elastic

constants of tantalum, tungsten, and molybdenum, Phys. Rev. 130, 1324 (1963).

Wasilewski, R. J., Elastic constants of Nb by reso- nance at sonic frequencies, J. Phys. Chem. Solids

26, 1643 (1965).

Schreiber, E., Elastic moduli of single-crystal spinel at 25 °C and to 2 kbar, J. Appl. Phys. 38, 2508

(1967) . de Batist, R., Room temperature elastic constants

of single crystal U4O9, Materials Research Bulletin

1, 75 (1966).

Lowrie, R., quoted by H. L. Brown and C. P. Kempter, "Elastic properties of zirconium carbide"

Phys. Stat. Sol. 18, K21 (1966).

Wachtman, J. B., Jr., Tefft, W. E., Lam, D. G., Jr., Stirichfield, R. P., Elastic constants of synthetic

single crystal corundum at room temperature",

J. Res. NBS 64A (Phys. and Chem.) No. 3, 213

(1960) . Bernstein, B. T., Elastic constants of synthetic

sapphire at 27 °C, J. Appl. Phys. 34, 169 (1963).

Birch, F., Elastic constants of rutile—a correction to a paper by R. K. Verma, Elasticity of some highdensitv crystals., J. Geophys. Res. 65, 3855 (1960).

Wachtman, J. B., Jr., Tefft, W. E., and Lam, D. G.,

Jr., Elastic constants of rutile (Ti02), J. Res.

NBS 66A, (Phys. and Chem.) No. 6, 465 (1962).

Alton, W. J., and Barlow, A. J., Acoustic-wave

propagation in tetragonal crystals and measure- ments of the elastic constants of calcium molybdate, J. Appl. Phys. 38, 3817 (1967).

Wachtman, J. B., Jr., Brower, W. S., and Farabaugh,

E. N., Elastic constants of single crystal calcium

molvbdate (CaMo04 ), J. Am. Ceram. Soc. 51, 341

(1968) . J. G. Brainerd (Chairman of Standards Committee),

IRE standards on piezoelectric crystals, 1945,

Proc IRE 37, 1378 (1949).

H. Jaffe (Chairman of Subcommittee), IRE standards on piezoelectric crystals: Measurement of piezoelectric ceramics, 1961, Proc IRE 49, 1161

(1961) .

162
[43] Zener, C, Elasticity and Anelasticity of Metals, [67

University of Chicago Press, Chicago, Illinois (1948).

[44] Chung, D. H., and Buessem, W. R., The elastic anisotropy of crystals, J. Appl. Phys. 38, 2010 [68

(1967).

[45] De Macedo, P. B., Viscoelasticity of glasses, p. 169,

this volume.

[46] Knudsen, F. P., Effect of porosity on Young's [69

modulus of alumina, J. Am. Ceram. Soc. 45, 94

(1962).

[47] Spriggs, R. M., Expression for effect of porosity on [70

elastic modulus of polycrystalline refractory

materials, particularly aluminum oxide, J. Am.

Ceram. Soc. 44, 628 (1961). [71

[48] Spriggs, R. M., and Brissette, L. A., Expressions for shear modulus and Poisson's ratio of porous re- factory oxides, J. Am. Ceram. Soc. 45, 198 (1962). [72

[49] Chung, D. H., and Buessem, W. R., The VoigtReuss-Hill (VRH) approximation and the elastic moduli of polycrystalline ZnO, Ti02 (rutile), and [73 a - A12 03 , J. Appl. Phys. (in press).

[50] Manning, W. R., Marlowe, M. O., and Wilder, D. R.,

Temperature and porosity dependence of Young's [74

modulus of polycrystalline dysprosium oxide and

erbium oxide, J. Am. Ceram. Soc. 49, 227 (1966).

[51] Chung, D. H., Elastic moduli of single crystal and [75

polycrystalline MgO, Phil. Mag. 8, 833 (1963).

[52] Spriggs, R. M., Brissette, L. A., and Vasilos, T., Effect of porosity on elastic and shear moduli of polycrystalline magnesium oxide, J. Am. Ceram. [76 Soc. 45, 400 (1962).

[53] Spinner, S., Knudsen, F. P., and Stone, L., Elastic

constant-porosity relations for polycrystalline

thoria, J. Research NBS 67 C (Engr. and Instr.) [77

No. 1, 39 (1963).

[54] Marlowe, M. O., and Wilder, D. R., Elasticity and [78

internal friction of polycrystalline yttrium oxide,

J. Am. Ceram. Soc. 48, 227 (1965).

[55] Fung, Y. C, Foundations of Solid Mechanics,

Prentice-Hall, Inc., New York, (1965).

[56] Brugger, K., Thermodynamic of higher order elastic rsn

coefficients, Phys. Rev. 133, A1611 (1964). l8U

[57] Murnaghan, F. D., Finite deformations of an elastic

solid, Am. J. Math. 59, 235 (1937).

[ 81

[58] Birch, F., Finite elastic strain of cubic crystals,

Phys. Rev. 72, 809 (1947).

[59] Murnaghan, F. D., Finite Deformation of an Elastic [82

Solid, John Wiley and Sons, Inc., New York

(1951).

[60] Hearmon, R. F. S., Third-order elastic coefficients, [83 Acta. Cryst. 6, 331 (1953).

[61] Skove, M. J., and Powell, B. E., Symmetry of mixed [84

third-order elastic constants, J. Appl. Phys. 38,

404 (1967).

[62] Powell, B. E., and Skove, M. J., Relation between [85

isothermal and mixed third-order elastic constants, J. Appl. Phys. 38, 404 (1967).

[63] Barsch, G. R., Adiabatic, isothermal, and inter- [86

mediate pressure derivatives of the elastic constants for cubic symmetry, I. Basic formulae, [87

Phys. Stat. Sol. 19, 129 (1967).

[64] Barsch, G. R., and Chang, Z. P., Adiabatic, iso- [88

thermal, and intermediate pressure derivatives of

the elastic constants for cubic symmetry, II. Numerical results for 25 materials, Phys." Stat. [89

Sol. 19, 139 (1967).

[65] Thurston, R. N., and Brugger, K., Third-order

elastic constants and the velocity of small ampli- [90

tude elastic waves in homogeneously stressed

media, Phys. Rev. 133, A1604 (1964). See also R. N. Thurston, Effective elastic coefficients for [91 wave propagation in crystals under stress, J. Acoust. Soc. Am. 37, 348 (1968).

[66] Brugger, K., Determination of third-order elastic [92

coefficients in crystals, J. Appl. Phys. 36, 768

(1965).

Chung, D. H., The first pressure derivatives of polycrystalline elastic moduli: Their relation to singlecrystal acoustic data and thermodynamic rela- tions, J. Appl. Phys. 38, 5104 (1967).

McSkimin, H. J., Jayaraman, A., and Andreatch,

P., Jr., Elastic moduli of GaAs at moderate pres- sures and the evaluation of compression to 250

kbar, J. Appl. Phys. 38, 2362 (1967).

McSkimin, H. J., and Andreatch, P., Jr., Elastic

moduli of silicon vs. hydrostatic pressure at 25.0

°C and -195.8 °C, J. Appl. Phys. 35, 2161 (1964).

McSkimin, H. J., and Andreatch, P., Jr., Thirdorder elastic moduli of gallium arsenide, J. Appl.

Phys. 38, 2610 (1967).

McSkimin, H. J., and Andreatch, P., Jr., Measurement of third-order moduli of silicon and ger- manium, J. Appl. Phys. 35, 3312 (1964).

Bogardus, E. H., Third-order elastic constants of Ge, MgO, and fused Si02 , J. Appl. Phys. 36,

2504 (1965).

Eastman, D. E., Measurement of third-order elastic moduli of yttrium iron garnet, J. Appl. Phys. 37,

2312 (1966).

Toupin, R. A., and Bernstein, B., Sound waves in

deformed perfectly elastic materials. Acoustoelastic effect, J. Acoust. Soc. Am. 33, 216 (1961).

Smith, R. T., Stern, R., and Stephens, R. B. W.,

Third-order elastic moduli of polycrystalline

metals from ultrasonic velocity measurements,

J. Acoust. Soc. Am. 40, 1002 (1966).

Thurston, R. N., Calculation of lattice-parameter

changes with hydrostatic pressure from third-order

elastic constants, J. Acoust. Soc. Am. 41, 1093

(1967).

Ghate, P. B., Fourth-order elastic coefficients, J. Appl. Phvs. 35. 337 (1964).

Ghate, P. B., Effect of hydrostatic pressure on the

elastic behavior of cubic crystals, Phys. Stat. Sol. 14, 325 (1966).

Chung, D. H., The elastic constants of a cubic

crystal subjected to moderately high hydrostatic

pressure, J. Phvs. Chem. Solids, 29, 417 (1968).

Truesdell, C, and Toupin, R., Static grounds for inequalities in finite strain of elastic materials,

Arch. Rati. Mech. Anal. 12, 1 (1963).

Mindlin, R. D., and Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Rati.

Mech. Anal. 11, 415 (1962).

Mindlin, R. D., Influence of couple-stresses on stress concentrations, Experimental Mechanics, 3, 1 (1963).

Toupin, R. A., Elastic materials with couple-stresses,

Arch. Rati. Mech. Anal. 11, 385 (1962).

Green, A. E., and Rivlin, R. S., Multipolar con- tinuum mechanics: Functional theory I, Proc. Roy. Soc. (London) A284, 303 (1965).

Koiter, W. T., Couple-stresses in the theory of

elasticity, Proc. Kon. Ned. Akad. V. Wetensch

B67, 17 and 30 (1964).

Schijve, B., Note on couple-stresses, J. Mech. Phys.

Solids 14, 113 (1966).

Laval, J., "Elasticity des cristaux, Compt. Rend.

232, 1947 (1951).

Raman, C. V., and Viswanathan, K. S., The elastic behavior of isotropic solids, Proc. Indian Acad.

Sci. 42, 1 (1955).

Raman, C. V., and Viswanathan, K. S., On the

theory of the elasticity of crystals, Proc. Indian

Acad. Sci. 42, 51 (1955).

Raman, C. V., and Krishnamurti, D., Evaluation of

the four elastic constants of some cubic crystals, Proc. Indian Acad. Sci. 42, 111 (1955).

LeCorre, Y., Les densites de couple et les pseudorotations dans la theorie de P elasticity de Laval,

J. phys radium 19, 36 (1958).

Joel, N., and Wooster, W. A., Number of elastic constants required in crystal elasticity, 182, 1078

(1958).

163
[93] Joel, N., and Wooster, W. A., On the validity of the

classical theory of crystal elasticity as applied to ammonium di-hydrogen phosphate, Acta. Cryst.

11, 575 (1958).

[94] Swartz, K. D., and Granato, A. V., Experimental

test of the Laval-Raman-Viswanathan theory of

elasticity, J. Acoust. Soc. Am. 38, 824 (1965).

[95] Kolsky, H., Stress Waves in Solids, Oxford University Press, Oxford, England (1953).

[96] Soga, N., and Anderson, 0. L., Simplified method

for calculating elastic moduli of ceramic powder

from compressibility and Debye temperature data,

J. Am. Ceram. Soc. 49, 318 (1966).

[97] Bancroft, D., The velocity of longitudinal waves in

cylindrical bars, Phys. Rev. 59, 588 (1941).

[98] Hudson, G. E., Dispersion of elastic waves in solid

circular cylinders, Phys. Rev. 63, 46 (1943).

[99] Cooper, J. L. B., The propagation of elastic waves

in a rod, Phil. Mag. 38, 1 (1947).

[100] Davies, R. M., A critical study of the Hopkinson

pressure bar, Trans. Roy. Soc. (London) A240,

375 (1948).

[101] Stanford, E. G., A contribution on the velocity of

longitudinal elastic vibrations in cylindrical rods,

and on the relationship between Young's modulus

and temperature for aluminum, Nuovo Cimento

supplement to Volume 7, 332 (1950).

[102] Tu, L. Y., Brennan, J. N., and Sauer, J. A., Dispersion of ultrasonic pulse velocity in cylindrical rods,

J. Acoust. Soc. Am. 27, 550 (1955).

[103] Morse, R. W., The velocity of compressional waves

in rods of rectangular cross section, J. Acoust. Soc. Am. 22, 219 (1950).

[104] Miklowitz, J., The propagation of compressional waves in a dispersive elastic rod, J. Appl. Mech.

24, 231 (1957).

[105] Lee, E. H., Finite-strain elastic-plastic theory with

application to plane-wave analysis, J. Appl.

Phys. 38, 19 (1967).

[106] Truesdell, C., General and exact theory of waves in

finite elastic strain, Arch. Rati. Mech. Anal. 8,

263 (1961).

[107] Borgnis, R. E., Specific directions of longitudinal wave propagation in anisotropic media, Phys. Rev.

98, 1000 (1955).

[108] Brugger, K., Pure modes for elastic waves in crystals,

J. Appl. Phys. 36, 759 (1965).

[109] Khatkevich, A. G., Special directions for elastic waves in crystals, Soviet Phys.—Cryst. 9, 579

(1965). Translated from Kristallografiya 9, 690

(1964).

[110] Aleksandrov, K. S., An ultrasonic pulse method of

measuring the elastic moduli of monoclinic crystals,

Soviet Phys.—Cryst. 3, 630 (1958).

[Ill] Musgrave, M. J. P., On the propagation of elastic waves in aeolotropic media. I. General principles, Proc. Roy. Soc. (London) A226, 339 (1954).

[112] Musgrave, M. J. P., On the propagation of elastic waves in aeolotropic media. II. Media of hexagonal

symmetry, Proc. Roy. Soc. (London) A226, 356

(1954).

[113] Miller, G. F., and Musgrave, M. J. P., On the propagation of elastic waves in aeolotropic media. III. Media of cubic symmetry, Proc. Roy. Soc. (London) A236, 352 (1956).

[114] Musgrave, M. J. P., On whether elastic wave surfaces possess cuspidal edges, Proc. Cambridge

Phil. Soc. 53, 897 (1957).

[115] Musgrave, M. J. P., On an elliptic cone of internal

refraction for quasi-transverse waves in tetragonal

crystals, Acta Cryst. 10, 316 (1957).

[116] Markham, M. F., Measurement of elastic constants

by the ultrasonic pulse method, Brit. J. Appl.

Phys., Supplement No. 6, 56 (1957).

[117] Neighbours, J. R., and Smith, C. S., An approximation method for the determination of the elastic constants of cubic single crystals, J. Appl. Phys.

21, 1338 (1950).

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

Neighbours, J. R., An approximation method for

the determination of the elastic constants of

single crystals, J. Acoust. Soc. Am. 26, 865 (1954).

Herzfeld, K. F., Fifty years of physical ultrasonics,

J. Acoust. Soc. Am. 39, 813 (1966).

Mason, W. P., Physical Acoustics, a series of

volumes published by Academic Press, New

York.

Mason, W. P., and McSkimin, H. J., Energy losses

of sound waves in metals due to scattering and

diffusion, J. Appl. Phys. 19, 940 (1948).

Fitzgerald, T. M., Chick, B. B., and Truell, R.,

Ultrasonic attenuation in AI2O3 at ultrahigh

frequencies and low temperatures, J. Appl. Phys.

35, 2647 (1964).

Maris, H. J., The absorption of sound waves in perfect dielectric crystals, Phil. Mag. 12, 89

(1965).

Oliver, D. W., and Slack, G. A., Ultrasonic atten- uation in insulators at room temperature, J. Appl. Phys. 37, 1542 (1966).

Simmons, G., Velocity of shear waves in rocks to 10 kilobars, I, J. Geophys. Res. 69, 1123 (1964).

Knopoff, L., and MacDonald, G. J. F., Attenuation

of small amplitude stress waves in solids, Rev.

Modern Phys. 30, 1178 (1958).

Spinner, S., and Tefft, W. E., A method for determining mechanical resonance frequencies and for

calculating elastic moduli from these frequencies, Proc. ASTM 61, 1221 (1961).

Pickett, G., Equations for computing elastic

constants from flexural and torsional resonant

frequencies of vibration of prisms and cylinders, Proc. Am. Soc. Testing Mater. 45, 846 (1945).

Pickett, G., Flexural vibration of unrestrained

cylinders and discs, J. Appl. Phys. 16, 820 (1945).

Tefft, W. E., Numerical solution of the frequency

equations for the flexural vibration of cylindrical

rods, J. Research NBS 64B (Math. Sci.) No. 4,

237 (1960).

Gazis, D. C., Exact analysis of the plane-strain

vibrations of thick-walled hollow cylinders, J. Acoust. Soc. Am. 30, 786 (1958).

Lucey, G. K., Jr., Resonant and antiresonant

frequencies of thick discs and rods, J. Acoust.

Soc. Am., to be published.

Love, A. E. H., A treatise on the mathematical

theory of elasticity, Reprinted by Dover Publications, New York & London (1944).

Frazer, D. B., and LeCraw, R. C, Novel method of measuring elastic and anelastic properties of solids,

Rev. Sci. Inst. 35, 1113 (1964).

Tiersten, H. F., Thickness vibrations of piezoelectric plates, J. Acoust. Soc. Am. 35, 53 (1963).

Warner, A. W., Onoe, M., and Coquin, G. A., Determination of elastic and piezoelectric constants for crystals in class (3m), J. Acoust. Soc. Am. 39, 42 (1967).

Voigt, W., Lehrbuch der kristallphysik, Teubner,

Berlin (1928).

Goens, E., Uber die Biegungs-und Drillungsschwinigungen eines diinnen kreiszylindrischen Kristallstabes von beliebiger kristallographischer Orientierung, Ann. Physik 15, 455 (1932).

Kimura, R., On the elastic moduli of ferromagnetic

materials , Part I. Dynamical measurements of the

elastic moduli of iron crystals, Proc. Phys.-Math.

Soc. Japan 21, 686 (1939).

Brown, W. F., Jr., Interpretation of torsional fre- quencies of crystal specimens, Phys. Rev. 58, 998

(1940).

Dewey, J. M., The elastic constants of materials

loaded with non-rigid fillers, J. Appl. Phys. 18,

578 (1947).

Mackenzie, J. K., The elastic constants of a solid

containing spherical holes, Proc. Phys. Soc. B63,

2 (1950).

Gatto, F., Influence of small cavities on velocity of

sound in metals, Allumino 19, 19 (1950).

164
[144] Coble, R. L., and Kingery, W. D., Effect of porosity on physical properties of sintered alumina, J. Am.

Ceram. Soc. 39, 377 (1956).

[145] Kerner, E. H., The elastic and thermo-elastic properties of composite media, Proc. Phys. Soc. (London)

69B, 808 (1956).

[146] Spriggs, R. M., Effect of open and closed pores on

elastic moduli of polycrystalline alumina, J. Am.

Ceram. Soc. 45, 454 (1962).

[147] Hasselman, D. P. H., On the porosity dependence of the elastic moduli of polycrystalline refractory

materials, J. Am. Ceram. Soc. 45, 452 (1962).

[148] Weil, N. A., Parametric effects governing the mechanics of ceramic materials, Chapter in "Proceedings of an International Symposium on High

Temperature Technology", Butterworths, London

(1964) . [149] Hasselman, D. P. H., and Fulrath, R. M., Effect of

small fraction of spherical porosity on elastic moduli of glass, J. Am. Ceram. Soc. 47, 52 (1964).

[150] Fryxell, R. E., and Chandler, B. A., Creep, strength,

expansion, and elastic moduli of sintered BeO as a function of grain size, porosity, and grain orientation, J. Am. Ceram. Soc. 47, 283 (1964).

[151] Hasselman, D. P. H., and Fulrath, R. M., Effect of cylindrical porosity on Young's modulus of polycrystalline materials, J. Am. Ceram. Soc. 48, 545

(1965) . [152] Hashin, Z., and Rosen, B. W., Elastic moduli of fiber-reinforced materials, J. Appl. Mech. 31, 223

(1964).

[153] Hill, R., The elastic behavior of a crystalline aggregate, Proc. Phys. Soc. (London) 65A, 349 (1952).

[154] Hashin, Z., The elastic moduli of heterogeneous

materials, J. Appl. Mech. 29, 143 (1962).

[155] Hashin, Z., and Shtrikman, S., On some variational

principles in anisotropic and nonhomogeneous

elasticity, J. Mech. Phys. Solids 10, 335 (1962).

[156] Hashin, Z., and Shtrikman, S., A variational approach to the theory of the elastic behavior of polycrystals, J. Mech. Phys. Solids 10, 343 (1962).

[157] Peselnick, L., and Meister, R., Variational method

of determining effective moduli of polycrystals:

(A) hexagonal symmetry, (B) trigonal symmetry,

J. Appl. Phys. 36, 2879 (1965).

[158] Meister, R., and Peselnick, L., Variational method

of determining effective moduli of polycrystals

with tetragonal symmetry, J. Appl. Phys. 37, 4121

(1966) . [159] Hershey, A. V., The elasticity of an iostropic aggregate of anisotropic cubic crystals, J. Appl. Mech.

21, 236 (1954).

[160] Kroner, E., Elastic moduli of perfectly disordered

composite materials, J. Mech. Phys. Solids 15, 319

(1967) . [161] Kroner, E., Datta, B. K., and Kessel, D., On the

bounds of the shear modulus of macroscopically

isotropic aggregates of cubic crystals, J. Mech.

Phys. Solids 14, 21 (1966).

[162] Walpole, L. J., On bounds for the overall elastic moduli of inhomogeneous systems—I. J. Mech.

Phys. Solids 14, 151 (1966).

[163] Walpole, L. J., On bounds for the overall elastic moduli of inhomogeneous systems—II. J. Mech.

Phys. Solids 14, 189 (1966).

[164] Sjodahl, L. H., and Chandler, B. A., Single-crystal

elastic constants of BeO from polycrystalline

measurements, J. Am. Ceram. Soc. 46, 351 (1963).

[165] Brace, W. F., Relation of elastic properties of rocks

to fabric, J. Geophys. Res. 70, 5657 (1965).

[166] Hearmon, R. F. S., Applied Anisotropic Elasticity,

Oxford University Press, Oxford, England (1961).

[167] Hill, R., Theory of mechanical properties of fibre- strengthened materials; I. Elastic behavior, J. Mech.

Phys. Solids 12, 199 (1964).

[168] Hashin, Z., On elastic behavior of fibre reinforced

materials of arbitrary transverse phase geometry,

J. Mech. Phys. Solids 13, 119 (1965).

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

Hill, R., Theory of mechanical properties of fibre- strengthened materials—III. Self consistent model,

J. Mech. Phys. Solids 13, 189 (1965).

Hill, R., A self-consistent mechanics of composite

materials, J. Mech. Phys. Solids 13, 213 (1965).

Budiansky, B., On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids 13, 223

(1965).

Hasselman, D. P. H., Experimental and calculated

Young's modulus of zirconium carbide containing a dispersed phase of graphite, J. Am. Ceram.

Soc. 46, 103 (1963).

Hasselman, D. P. H., and Fulrath, R. M., Effect of spherical tungsten dispersions on Young's modulus

of a glass, J. Am. Ceram. Soc. 48, 548 (1965).

Hirsch, T. J., Modulus of elasticity of concrete

affected by elastic moduli of cement paste matrix

and aggregate, J. Am. Concrete Inst. 59, 427

(1962).

Rath, J., Modulus of elasticity of a two-component

material, Silikaty 10, 4 (1966).

Walsh, J. B., The effect of cracks on the compressibility of rock, J. Geophys. Res. 70, 381 (1965).

Walsh, J. B., The effect of cracks on the uniaxial

elastic compression of rocks, J. Geophys. Res. 70,

399 (1965).

Walsh, J. B., The effect of cracks in rocks on Poisson's

ratio, J. Geophys. Res. 70, 5249 (1965).

Born, M., and Huang, K., Dynamical Theory of Crystal Lattices, Oxford University Press, Oxford,

England (1954).

Born, M., Reminiscences of my work on the dynamics

of crystal lattices, in Lattice Dynamics, edited by

R. F. Wallis, Pergamon Press, New York (1965).

Seitz, F., The Modern Theory of Solids, McGrawHill, New York (1940).

Gilman, J. J., The strength of ceramic crystals, in The Physics and Chemistry of Ceramics, ed. by

C. Klingsberg (Gordon and Breach, New York,

1963).

Anderson, O. L., Determination and some uses of isotropic elastic constants of polycrystalline aggregates using single-crystal data, in Physical

Acoustics, edited by W. P. Mason, Academic

Press, New York (1965).

Anderson, O. L., and Nafe, J. E., The bulk modulusvolume relationship for oxide compounds and

related geophysical problems, J. Geophys. Res. 70,

3951 (1965).

Anderson, O. L., A proposed law of corresponding

states for oxide compounds, J. Geophys. Res. 71,

4963 (1966).

Anderson, O. L., Lattice dynamics in geophysics,

Trans. N.Y. Acad, of Sci. 27, 298 (1965).

Szigeti, B., Polarizability and dielectric constant of

ionic crystals, Trans. Faraday Soc. 45, 155 (1949).

Szigeti, B., Compressibility and absorption frequency

of ionic crystals, Proc. Roy. Soc. (London) A204,

51 (1950).

Anderson, O. L., and Glynn, P., Measurement of

compressibility in polycrystalline MgO using the

reflectivity method, J. Phys. Chem. Solids 26,

1961 (1965)

Gray, D. E., Coordinating Editor, American Insti- tute of Physics Handbook, McGraw-Hill, New

York (1963).

Alers, G. A., Use of sound velocity measurements in

determining the Debye temperature of solids, in

Physical Acoustics IIIB, edited by W. P. Mason,

Academic Press, New York (1965).

Anderson, O. L., A simplified method for calculating

the Debye temperature from elastic constants, J. Phys. Chem. Solids 24, 909 (1963).

Fedorov, F. I., The Debye temperature of a crystal,

Soviet Phys.—Cryst. 10, 125 (1965). Translated

from Kristallografiya 10, 167 (1965).

Fedorov, F. I., and Bystrova, J. G., Debye temperatures of cubic crystals, Soviet Phys.—Cryst. 11,

165
333 (1966). Translated from Kristallografiya 11,

368 (1966).

[195] Robie, R. R., and Edwards, J. L., Some Debye

temperatures from single-crystal elastic constant

data, J. Appl. Phys. 37, 2659 (1966).

[196] Ali, M., and Nagels, P., Evaluation of the Debye

temperature of thorium dioxide, Phys. Stat. Solidi

21, 113 (1967).

[197] Dayal, B., and Appalonarasimham, N., The evaluation of the elastic constants of rutile from spectroscopic data, J. Sci. Research of the Benares Hindu

University 1, 26 (1950).

[198] Pandey, H. N., The theoretical elastic constants and

specific heats of rutile, Phys. Stat. Solidi 11, 743

(1965).

[199] Dolling, G., Cowley, R. A., and Woods, A. D. B., The crystal dynamics of uranium dioxide, Can. J. Phys. 43, 1397 (1965).

[200] Axe, J. D., and Pettit, G. D., Infrared dielectric

dispersion and lattice dynamics of uranium dioxide

and thorium dioxide, Phys. Rev. 151, 676 (1966).

[201] Griineisen, E., The state of a solid body, NASA

republication RE 2-18-59W of a translation of

"Zustand des festen Korpers", Handbuch der

Phys. Bd. 10, pages 1-52, Springer, Berlin (1926).

[202] Slater, J. C, Introduction to Chemical Physics,

McGraw-Hill (1939).

[203] Hume-Rothery, W., On Griineisen's equation for thermal expansion, Proc. Phys. Soc. (London) 57,

209 (1945).

[204] Barron, T. H. K., On the thermal expansion of solids at low temperatures, Phil. Mag. 46, 720 (1955).

[205] Sheard, F. W., Calculation of the thermal expansion

of solids from the third-order elastic constants,

Phil. Mag. 3, 1381 (1958).

[206] Swartz, K. D., Anharmonicity in sodium chloride,

J. Acoust. Soc. Am. 41, 1084 (1967).

[207] Anderson, O. L., Derivation of Wachtman's equation

for the temperature dependence of elastic moduli

of oxide compounds, Phys. Rev. 144, 553 (1966).

[208] Stern, E. A., Theory of anharmonic properties of

solids, Phys. Rev. Ill, 786 (1958).

[209] Wachtman, J. B., Jr., Tefft, W. E., Lam, D. G., Jr., and Apstein, C. S., Exponential temperature

dependence of Young's modulus for several

oxides, Phys. Rev. 122, 1754 (1961).

[210] Tefft, W. E., Elastic constants of synthetic single

crystal corundum, J. Research NBS 70A (Phys.

and Chem.) No. 4, 277 (1966).

[211] Alton, W. J., and Barlow, A. J., Temperature

dependence of the elastic constants of yttrium

aluminum garnet, J. Appl. Phys. 38, 3023 (1967).

[212] Graham, L. J., Nadler, H., and Chang, R., Elastic

constants of UC, J. Appl. Phys. 34, 1572 (1963).

[213] Soga, N., and Anderson, O. L., High-temperature

elastic properties of polycrystalline MgO and

A12 03 , J. Am. Ceram. Soc. 49, 355 (1966).

[214] Soga, N., and Anderson, O. L., High-temperature

elasticity and expansivity of fosterite and steatite,

J. Am. Ceram. Soc. 50, 239 (1967).

[215] Soga, N., Schreiber, E., and Anderson, O. L., Estimation of bulk modulus and sound velocities

of oxides at very high temperatures, J. Geophys.

Res. 71, 5315 (1966).

[216] Barker, R. E., Jr., An approximate relation between

elastic moduli and thermal expansivities, J. Appl.

Phys. 34, 107 (1963).

[217] Slagle, O. D., and McKinstry, H. A., Temperature

dependence of the elastic constants of the alkali

halides. I. NaCl, KC1, and KBr. II. The solid

solution KCl-KBr. III. CsCl, CsBr, and Csl,

J. Appl. Phys. 38, 437 (1967).

[218] Bell, R. O., and Rupprecht, G., Elastic constants of strontium titanate, Phys. Rev. 129, 90 (1963).

[219] Aleksandrov, K. S., Reshchikova, L. M., and Beznosikov, B. V., Behavior of the elastic constants of

KMnFj single crystals near the transition of

puckering, Phys. Stat. Solidi 18, K17 (1966).

[220] Brandt, O. G., and Walker, C. T., Temperature

dependence of elastic constants and thermal

expansion for U02 , Phys. Rev. Letters 18, 11

(1967).

[221] Garland, C. W., and Jones, J. S., Elastic constants of

ammonium chloride near the lambda point, J. Chem. Phys. 39, 2874 (1963).

[222] Leurgans, P. J., Editor, Conference on internal

frictidn due to crystal lattice imperfections, Acta

Met. 10 [4] (1962).

[223] Lazan, B. J., and Wyman, L. L., Symposium Chairman, Internal friction, damping and cyclic

plasticity, ASTM special technical publication

No. 378, 1965.

[224] Mason, W. P., Editor, Physical acoustics, IIIA,

Effect of imperfections, Academic Press, New

York (1965).

[225] Mason, W. P., Editor, Physical acoustics, IVA,

Applications to quantum and solid state physics,

Academic Press New York (1966).

[226] Haven, Y., and van Santen, J. H., Electrically and

elastically active relaxation modes, J. Chem.

Phys. 22, 1146 (1954).

[227] Haven, Y., and van Santen, J. H., Dielectric re- laxation of lattice defects in crystals, Nuovo

Cimento 7 (suppl.), 605, (1958).

[228] Wachtman, J. B., Jr., and Peiser, H. S., Symmetry

conditions for internal friction caused by jumping

of point defects in crystals, Appl. Phys. Letters 1,

20 (1962).

[229] Nowick, A. S., and Heller, W. R., Anelasticity and

stress-induced ordering of point defects in crystals, Advan. Phys. 12, 251 (1963).

[230] Franklin, A. D., Relaxation modes for trapped

crystal point defects, J. Research NBS, 67A

(Phys. and Chem.) No. 4, 291 (1963).

[231] Wachtman, J. B., Jr., Mechanical and electrical

relaxation in Th02 containing CaO, Phys. Rev.

131, 517 (1963).

[232] Wachtman, J. B., Jr., Peiser, H. S., and Levine,

E. P., Symmetry splitting of equivalent sites in

oxide crystals and related mechanical effects, J. Research NBS 67A (Phys. and Chem.) No. 4, 281

(1963) . [233] Bhagvantam, S., and Pantulu, P. V., Point defects

and relaxation phenomena in crystals, Proc. Indian Acad. Sci. 58, 183 (1963).

[234] Chang, R., Mechanical relaxation associated with

paired point defects in cubic lattices of O h point

group symmetry, J. Phys. Chem. Solids 25, 1081

(1964) . [235] Franklin, A. D., Shorb, A., and Wachtman, J. B., Jr., Relaxation modes of trapped crystal

point defects: The three-neighbor shells model in

NaCl, J. Research NBS 68A (Phys. and Chem.)

No. 5, 425 (1964).

[236] Nowick, A. S., and Heller, W. R., Dielectric and

anelastic relaxation of crystals containing point

defects, Advan. Phys. 14, 101 (1965).

[237] Wachtman, J. B., Jr., and Peiser, H. S., Symmetry

conditions on jump rates occurring in relaxation

times associated with point defect motions between equivalent general sites in crystals, J. Phys.

Chem Solids 27, 975 (1966).

[238] Nowick, A. S., Dielectric and anelastic relaxation

of crystals containing point defects. II, Advan.

Phys. 16, 1, (1967).

[239] Wert, C, Determination of the diffusion coefficient

of impurities by anelastic methods, in Physical

Acoustics, Vol IIIA, edited by W. P. Mason,

Academic Press, New York (1966).

[240] Johnson, H. B., Tolar, N. J., Miller, G. R., and Cutler,

Ivan B., Dipole relaxation in CaF2 doped with

NaF, J. Am. Ceram. Soc. 49, 458 (1966).

[241] Southgate, P. D., Anelastic and dielectric loss in

yttrium-doped calcium fluoride, J. Phys. Chem.

Solids 27, 1623 (1966).

166
[242] Lay, X. W., and Whitmore, D. H., Dielectric and

mechanical losses due to calcium-oxygen vacancy

pairs in cerium dioxide, Proc. Fifth Rare Earth

Conference, Ames, Iowa (1965).

[243] Southgate, P. D., Mechanical relaxation of a point

defect in magnesium oxide, J. Appl. Phys. 36,

2696 (1965).

[244] Southgate, P. D., Internal friction due to chromium

and iron in magnesium oxide, J. Phys. Chem.

Solids 27, 1263 (1966).

[245] van Houten, S., Mechanical losses in Li-doped NiO

semiconductors, J. Phys. Chem. Solids 23, 1045

(1962).

[246] Dodd, D. M., and Frazer, D. B., The 3000-9000

cm-1 absorption bands and anelasticity in crystalline a-quartz, J. Phys. Chem. Solids 26, 673

(1965).

[247] Carnahan, R. D., and Brittain, J. O., Point-defect

relaxation in rutile single crystals, J. Appl. Phys.

34, 3095 (1963).

[248] Wachtman, J. B., Jr., and Doyle, L. R., Internal

friction in rutile containing point defects, Phys.

Rev. 135, A276 (1964).

[249] Carnahan, R. D., and Brittain, J. O., Low-temperature internal friction peak in rutile, J. Appl.

Phys. 37, 1808 (1966).

[250] Wachtman, J. B., Jr., Spinner, S., Brower, W. S., Fridinger, T., and Dickson, R. W., Internal

friction in rutile containing Ni or Cr, Phys. Rev.

148, 811 (1966).

[251] Sochino, G., de Batist, R., and Gevers, R., Relaxation processes in single-crystal UO2, Proc. British

Ceram. Soc. 9, 73 (1967).

[252] Wachtman, J. B., Jr., and Corwin, W. C, Internal

friction iii Zr02 containing CaO, J. Research

NBS 69A (Phys. and ChemJ No. 5, 457 (1965).

[253] Granato, A. V., and Lucke, K., The vibrating string

model of dislocation damping, in Physical Acoustics, Vol. IVA, edited by W. P. Mason, Academic

Press, New York (1966).

[254] Chang, R., Dislocation relaxation phenomena in oxide crystals, J. Appl. Phys. 32, 1127 (1961).

[255] Huber, R. J., Baker, G. S., and Gibbs, P., Hightemperature kilocycle internal friction in AI2O3

single crystals, J. Appl. Phys. 32, 2573 (1961).

[256] Dahlberg, P., Carnahan, R. D., and Brittain, J. O.,

dislocation damping in magnesium oxide crystals at low frequencies, J. Appl. Phys. 33, 3493 (1961).

[257] Southgate, P. D., Mendelson, K. S., and DePerro

P. L., Kilocycle-range dislocation damping in

magnesium oxide, J. Appl. Phys. 37, 206 (1966).

[258] Carnahan, R. D., and Brittain, J. O., Dislocation

damping in rutile single crystals, J. Am. Ceram.

Soc. 48, 15 (1966).

[259] Astbury, N. F., and Davis, W. R., Internal friction

in ceramics, Trans. Brit. Ceram. Soc. 63, 1 (1964).

[260] Melngailis, J., A model for the change in elastic modulus of a solid caused by point defects,

Phys. Stat. Solidi 16, 247 (1966).

[261] Jan, J. P., Steinemann, S., and Dinichert, P., The density and lattice parameters of ruby,

J. Phys. Chem. Solids 12, 349 (I960).

[262] Munoz, R., On the elastic model for the deviations

from Vegard's law, Phil. Mag. 14, 1105 (1966).

[263] Csavinsky, P., Effect of the thermal population of

the (000) and (100) conduction bands on the

elastic constants of heavily doped n-type germanium, J. Appl. Phys. 37, 1967 (1966).

[264] Plendl, J. N., Mitra, S. S., and Gielisse, P. J., Compressibility, cohesive energy, and hardness

of non-metallic solids, Phys. Stat. Solidi 12,

367 (1965).

[265] Kalnin, I. L., Strength and elasticity of whitewares:

Part I, Relation between flexural strength and

elasticity, Bull. Am. Ceram. Soc. 46, 1174 (1967).

[266] Wachtman, J. B., Jr., Mechanical properties of

ceramics—an introductory survey, Bull. Am.

Ceram. Soc. 46, 756 (1967).

[267] Spence, G. B., Extended dislocations in the anisotropic elastic continuum approximation, Proc. of

the Fifth Conference on Carbon, Volume 2,

Pergamon Press, New York (1961).

[268] Reynolds, W. N., The mechanical properties of reactor graphite, Phil. Mag. 11, 357 (1965).

[269] Price, R. J., Young's modulus of pyrolytic carbon

in relation to preferred orientation, Phil. Mag.

12, 561 (1965).

[270] Baker, C, and Kelly, A., The effect of neutron

irradiation on the elastic moduli of graphite

single crystals, Phil. Mag. 9, 927 (1964).

[271] Kelly, B. T., The elastic constants of polycrystalline

carbons and graphites, Phil. Mag. 9, 721 (1964).

[272] Woolley, R. L., The elasticity of polycrystalline

graphite, Phil. Mag. 11, 475 (1965).

[273] Summers, L., Walker, D. C. B., and Kelly, B. T.,

The effect of fast neutron irradiation on the

c33 elastic modulus of highly oriented pyrolytic

graphite, Phil. Mag. 14, 317 (1966).

[274] Mason, I. B., and Knibbs, R. H., Variation with

temperature of Young's modulus of polycrystalline

graphite, Nature 188, 33 (1960).

[275] Price, W. J., and Huntington, H. B., Acoustical

properties of anisotropic materials, J. Acoust. Soc. Am. 22, 32 (1950).

[276] Eros, S., and Reitz, J. R., Elastic constants by the

ultrasonic pulse echo method, J. Appl. Phys. 29,

683 (1958).

[277] McSkimin, H. J., Notes and references for the

measurement of elastic moduli by means of ultrasonic waves, J. Acoust. Soc. Am. 33, 606 (1961).

[278] McSkimin, H. J., Variations of the ultrasonic

pulse-superposition method for increasing the

sensitivity of delay-time measurements, J. Acoust.

Soc. Am. 37, 864 (1965).

[279] McSkimin, H. J., Measurement of ultrasonic wave

velocities and elastic moduli for small solid

specimens at high temperatures, J. Acoust. Soc. Am. 31, 287 (1959).

[280] Alers, G. A., The measurement of very small sound

velocity changes and their use in the study of

solids, in Physical Acoustics IVA, edited by W. P. Mason, Academic Press, New York (1966).

[281] Simmons, G., Ultrasonics in Geology, Proc. IEEE

53, 1337 (1965).

[282] Neighbours, J. R., and Schacher, G. E., Determination of elastic constants from sound-velocity meas- urements in crystals of general symmetry, J. Appl.

Phys. 38, 5366 (1967).

[283] Einspruch, N. G., and Truell, R., Results of an

ultrasonic measurement survey, J. Appl. Phys.

33, 3087 (1962).

[284] Simmons, G., Comparison of static and dynamic

measurements of compressibility of rocks, J. Geophys. Res. 70, 5649 (1965).

[285] Fine, M. E., Dynamic methods for determining the

elastic constants and their temperature variation

in metals, ASTM Bulletin No. 181, page 20 (1952).

[286] Kesler, C. E., and Chang, T. S., Review of sonic

methods for the determination of mechanical

properties of solid materials, University of Illinois T. and A. M. report 54, September 1954.

[287] Fine, M. E., Apparatus for measuring the elastic moduli and internal friction of solids from 1.7 to above 77 °K and some values for a-quartz, Rev.

Sci. Inst. 25, 1188 (1954).

[288] Fine, M. E., Apparatus for precise determination of

dynamic Young's modulus and internal friction at elevated temperatures, Rev. Sci. Inst. 29, 643

(1957).

[289] Thompson, D. O., and Glass, F. M., Elastic constant-internal friction spectrometer, Rev. Sci. Inst. 29, 1034 (1958).

167
[290] Vernon, E. V., An apparatus for the determination

of dynamic elastic moduli at low strains, J. Sci. Inst. 35, 28 (1958).

[291] Sinnott, K. M., Apparatus for the measurement of shear modulus and internal friction between 4.2

and 100 °K, J. Appl. Phys. 29, 1433 (1958).

[292] Watts, C. R., and Carnahan, R. D., Determination

of elastic constants of solids, Aerospace Corporation report TR-1001 (2250-10)-2, Air Force re- port number SSD-TR-67-4, Sept. 1966.

[293] Ramachandran, G. N., and Wooster, W. A., Determination of elastic constants of crystals from

diffuse reflexions of x-rays. I. Theory of method,

Acta. Cryst. 4, 335 (1951).

[294] Ramachandran, G. N., and Wooster, W. A., Determination of elastic constants of crystals from

diffuse reflexions of x-rays. II. Application to some

cubic crystals, Acta. Cryst. 4, 431 (1951).

[295] Sumer, A., and Smith, J. F., A comparison of the

elastic constants of chromium as determined from

diffuse x-ray and ultrasonic techniques, J. Appl.

Phys. 34, 2691 (1963).

[296] Mahler, R. J., and Tanttila, W. H., Ultrasonic

determination of crystalline resonances and sound

velocities, using NMR techniques, J. Acoust. Soc. Am. 38, 429 (1965).

[297] Ross, J. H., Approving Officer, Evaluation techniques for fibers and yarns used by the Fiberous

Materials Branch, Nonmetallic Materials Division, Air Force Materials Laboratory, Air Force

Materials Laboratory, Technical Report AFML—

TR-67-159, Sept. 1967.

168
Viscoelasticity of Glasses*

Pedro Buarque de Macedo

The Catholic University of America, Washington, B.C. 20017

and

Institute for Materials Research,

National Bureau of Standards, Washington, B.C. 20234

Viscosity measurement techniques such as rotating cylinder, beam bending, fiber elongation, etc., are reviewed and compared to each other as to their viscosity range and experimental uncertainty. Thermal expansion coefficient and density measurements are discussed.

Since the high-temperature melt density values in the literature do not agree with each other

to within their respective standard deviations, a detailed discussion of such measurements is given. The high-temperature ultrasonic interferometer used to make sound absorption and

velocity measurements is described along with its elaborate electronic circuitry.

Viscoelasticity provides a linear mathematical framework to analyze both the time and

the frequency dependence of the mechanical parameters in the glass transition. The deformation, annealing, and strain points are analyzed as to the particular physical process (viscous

flow, volume relaxation, shear relaxation) involved.

The limits of validity of the empirical (technological) experiments used to characterize

these processes are considered. Volume relaxation is distinguished from shear relaxation.

Non-Newtonian flow is considered to be due to immiscible glasses.

Since the viscosity governs most manufacturing processes, such as the ability to stir a melt or to shape an object, its temperature dependence is of utmost importance. Not only is the least squares method to fit data to the Fulcher equation discussed, but also the various

theories which lead to the equation. The validity of these theories is tested with recent data on B2O3, and they are found to be deficient.

Key Words: Annealing density; glasses; glass transformation; modulus; viscoelasticity;

viscosity.

1. Introduction

In the first part of this paper a review of the

best techniques for making viscoelastic measurements in molten glasses will be made. There will be

a detailed description of viscosity measurements,

since viscosity plays such a critical role in the

manufacture and fabrication of glass. A shorter

discussion of density, sound absorption, and phase

velocity measurements will also be given.

Secondly, a review of annealing and relaxation

theory as well as a discussion of the interpretations

of experimental data which are still controversial,

such as the role of microstructure in viscous flow,

will be made. These reviews are planned to help

both the quality control man and the researcher.

Thirdly, a review of viscosity theories will be

made. The material in this section is taken from

a recent paper by Napolitano and Macedo [l]. 1 Since this area is highly controversial, the opinions

expressed here are those of the author and may

or may not be widely accepted throughout the

scientific community. Unlike the first two sections,

this last section will be of primary interest to the

researcher, but the engineer should be aware of

existing viscosity theories.

•Work ft Otholic University of America supported by Air Force Office

of Scientific Research Grant No. AFOSR-68-1376.

1 Figures in brackets indicate the literature references at the end of this paper.

2. Measurement of Physical Properties

of Molten Glasses

2.1. Viscosity

Between the boiling point and the glass transition a liquid may exhibit viscosities ranging from

10~2 to 10+15 poise. [One poise is the cgs unit of

viscosity. A newton-s/m2, the mks unit, is equal

to 10 poise.] In most molten silicate glasses, the

difficulty in readily obtaining temperatures in excess of 1500 °C rules out the measurement of

viscosities much below 102 P. Thus, for molten

glasses one is interested in measuring viscosities

between 102 and 1015 poise. At present this cannot

be done with a single technique. Table 1 gives the

most common techniques used for molten oxides, as well as the range of viscosities and the approximate error. Most of the techniques require calibration of the equipment with liquids of known

viscosity at room temperature. In fact all the

high-temperature viscosity measurements below

107 poise rely on the calibration made with NBS

standard viscosity oils. The most reliable high-temperature viscometer

is the rotating cylinder one [2, 3, 4, 5, 6, 7]. The

latest version of this type of apparatus is described

in detail by Napolitano et al. [2], and can measure

323-655 O - 69 - 12 169
Table 1. Viscosity measurement techniques

Technique Viscosity Error a References range

Poises Percent Rotating cylinder 10° • -1010 2 2, 3, 4, 5, 6, 7 Counterbalance-sphere 101 - -10' 5 8,9 Penetrometer 105 - -10° 5 10, 11 Fiber elongation 10> - -10" 10 3, 4, 12, 13, 14 Beam bending 10' -101* 10 15, 16 Softening point.. ~10?-« 2°C 3, 19 Annealing point ~10'3 2°C 3, 16, 19 Strain point ~10H.S 4°C 3, 16, 19

"The estimated proportional standard deviation of viscosity will vary

between various authors using the same technique.

viscosities from the centipoise range up to 1010

poise. The instrument is capable of measuring

viscosities to within 1 percent, but the reported

errors are at least double that because of the inability to eliminate temperature gradients and

fluctuations during measurements. The rotating

cylinder viscometer is based on the general

equation of motion:

where

r d

2e dd

dt

+K2e=K3i (1)

I =moment of inertia,

0= angular displacement,

t=time,

r]= viscosity,

Ku K2 , and K3 are constants of the apparatus;

and i is current through the coil. Thus the first

term of the equation is the torque from inertia.

The second term is the torque from viscous drag.

The third term is an elastic torque due to an

angular deflection of the suspension, and the last

term is the magnetic torque applied to the system.

Method 1. Over the lowest viscosity range log

t? =0 to log 77 = 5 (log stands for log i 0) the outer

cylinder is rotated at a fixed angular velocity. The

viscous drag of the liquid produces a torque on

the inner cylinder. A measure of this torque is made by applying an appropriate current to the

coil such that the resulting magnetic torque will

bring the inner cylinder back to its rest position

at 0=0.

Therefore the first term of eq (1) is zero, because

there is no angidar acceleration in the system

when measurements are being made. The second

term is present and significant. It is retained in

the equation of motion for Method 1, but since

the inner cylinder is held at rest, the angular

velocity of the outer cylinder is used in this term.

The third term is zero when the system is balanced,

since the angle across the suspension is zero. Even

so, it controls the sensitivity. By changing sus- pensions the value of K2 can be changed and with

it, the sensitivity. Since the fourth term is significant, it is kept. Thus the equation of motion

for Method 1 is -nK^=K}i (2)

or

where Ci =

|^ an<i W=W ( angular velocity). Cx in eq (3) was obtained by calibrating the viscometer

with NBS standard oils. Thermal expansion cor- rections on Ci, known for room temperatures,

have to be made for accurate viscosity measurements at temperature, T, up to 1400° C according

to the following equation:

G(D=clM. c a+/yr-25])-3 (4)

where /3 P is the linear expansion coefficient of

platinum.

Method 2. For viscosities from about log

77=4.5 to log 77=7.5 the aperiodic method [4] is used. In this decay mode of operation, the outer

cylinder remains at rest while the inner cylinder

is displaced by an angle 0 and allowed to return

to its zero position under the force of the torsion

wire. Since the system is an over-damped harmonic

oscillator, one observes an exponentially decaying

angular velocity as the inner cylinder approaches

its equilibrium position. The term involving the

moment of inertia in eq (1) is extremely small

and can be neglected, giving rise to the following

equation

where

Integrating

we have

C2 =K2IKX

.

Je l 0 J,

j

Cidt

_ C2 (t2 —£1) V ~ ln0,/02

' (6)

The calibration constant (C2) for Method 2

can be calculated once the viscometer has been

calibrated by Method 1. First an independent

measurement of the current required for a given

angidar deflection of the inner cylinder is made.

This gives us:

K2 i = CA- (7)

From eqs (3), (5), and (7) one may obtain C2 from Ci and C4 according to the following

c2 =(a/ct ). (8)

n=C\%lu> (3)

Method 3. For viscosities from about log

77=5.5 to log 77= 10 the following method was used.

The outer cylinder again remains at rest whfle a constant torque is applied to the inner cylinder by

means of the magnetic field interaction, and the

angular velocity is measured about 0=0. Thus the

term K2 0 is negligible and can be omitted. Since

170
measurements are made in a steady state the

acceleration is zero, and we have the following:

de K3 . n . (9)

integrating

we have

rsi ft

r,de=\ C3 i dt

n=03i d2 —6\ (10)

C3 and Ci are equal.

An easier way of obtaining the torque involved

in the measurement is with a Brookfield viscometer after adapting it for high temperatures.

This was done by Eppler [5], Gruber and Litovitz

[6], and Tiede [7]. Eppler [5] not only used the

Method 1 operation, which is the one recommended by the manufacturer, but also Method 2,

with a similar increase in viscosity range.

For viscosities higher than 1010 poise two techniques are commonly used, the fiber elongation

and the beam bending techniques. Both of the

techniques represent absolute methods of measuring viscosity and, unlike the rotation methods,

require no calibration.

The fiber elongation technique has been used

by many authors [3, 4, 12, 13, 14]. The recent

work of Napolitano and Hawkins [3] gives one

of the best descriptions of the apparatus described below:

Weights ranging from 5 to 1000 g, depending on

the test temperature and viscosity of the glass,

were used to apply tension to fibers of uniform

diameter (~0.75 mm) and 10 cm long. An optical

lever with a multiplication factor of 50 was used

to observe the resulting elongation with a telescope equipped with a vertical scale. The viscosity,

v, can be calculated from [14].

v=Fl/(3Adl/dt) (11)

where

F=extending force (dyne)

1=fiber length (cm)

A=fiber cross section area (cm2

) dl

dt =elongation rate (cm/s)

Some glasses cannot be made into fibers because

of devitrification, phase separation, surface volatilization, etc. For many of these, samples can be

cut and ground to form beams of simple geometry.

The beam-bending method, devised by Jones [15]

and perfected by Hagy [16], gives viscosity results

derived from the rate of viscous deflection of a beam according to the equation

gU

2ALv \m- pALl

1.6 J

(12)

where

g— acceleration due to gravity (cm/s2

) Ic =cross section moment of inertia (cm4

) v

midpoint deflection rate (cm/min)

M=centrally applied load (g)

p= density of glass (g/cm3

) A=cross section area of beam (cm2

) Z=suport span (cm).

The second term within the brackets accounts for

the contribution of the weight of the beam to

bending and is usually neglected.

Both techniques can measure between 108 and

1015 P. Figure 1 of Napolitano and Macedo [17]

shows an Arrhenius plot (log n versus l/T) of the

viscosity of Ge02 glass in the annealing region. As

can be seen from the scatter of points both fiber

elongation [17] and the beam bending [18] data

not only give the same values but also have about

the same uncertainty.

Table 1 also shows other viscosity measurement

techniques, and the interested reader can look up

the appropriate reference. When building a viscometer it is highly recommended that the apparatus

be tested at or about the temperature one plans to use it. For this, one can now buy NBS Standard

Reference Materials with certified viscosities.

In industry complete viscosity runs are made

only to evaluate a new glass. For quality control

TEMPERATURE °C x 10"

o

CL

COO

co

CDO

6.0 5.9 5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8

15.00

1 >

/

/ / /

14.50

/ X s /

/ / A/

/ /

14.00

13.50

13.00

ozy y

? /

1250 yy /

12.00 yf'

'° y

° /

/

1150 /

11.00

11.10 .20 .40 12.00 .20 13.00 .20 .40 .60

Kf HO"'

T

Figure 1. Arrhenius plot of viscosity of Ge02 . F & P ref. [18] • Flame drawn fibers

| from N & M ref. [17] O Fiber drawn from crucible | least squares fit to F & P data

least squares fit of combined data

. extrapolation of high temperature Arrhenius least square

fit of F & P data

171
one uses fixed point measurements such as softening point, annealing point, and strain point. These

three points represent empirical measurements

which not only involve viscosity but also other

physical quantities. They are described in the

ASTM part 13 [19]. NBS also provides five refer- ence glasses for the calibration of furnaces used in

measuring these fixed points.

2.2. Density

Density measurements of glasses are usually

made by Archimedes' principle either by direct

weighing or by flotation. These measurements are now routine and require no further discussion.

The expansion coefficient methods are also generally accepted and can be made either by interferometric technique [20] or by dial technique [19].

Measurements of the density of molten glasses,

on the other hand, have not been standardized

and the disagreement between various authors

[21, 22, 23, 24, 25, 26] is outside their respective

standard deviations. The technique universally

used is to weigh a restrained platinum sphere

suspended by a platinum wire in a melt. The

possible sources of error are:

(a) homogeneity of the melt; if there are seeds

(bubbles) in the melts, the measured density will

be lower.

(b) difficulty in finding the null point, because

of viscous drag.

(c) the role of surface tension on the suspension

wire; it has either been ignored or miscalculated.

Shartsis [8] made an error in the calculation of

the surface tension effect which was extensively

copied by other authors.

(d) convection currents caused by an updraft

from the furnace may give erroneous weighings.

I will describe the technique Napolitano et al. used in a previous publication [21] to measure

the density of B203. The density was measured by Archimedes'

principle using a restrained sphere. Equilibrium

was obtained by letting a one-arm deflection

balance settle. Thus it was necessary only to adjust

the weights to 0.1 g, whereas the next decimals

were obtained by deflection and required only

periodic observations. Densities were measured

on glasses with viscosities up to 106 poise. At the

aforementioned value, the balance took 92 hr to

come to equilibrium. To estimate the uncertainty

of these measurements after the first equilibrium

was obtained, the cup was lowered 1 mm so that

an upper uncertainty limit could be obtained in

the weight; then it was raised 1 mm for the lowered

limit. It took 92 hr for the balance to come to

equilibrium at each of the subsequent weighings.

These limits were within ±0.001 g/cm3

. To calculate the density from the data, the

effects of surface tension on the suspension wire

had to be considered. Surface tension force was

not reproducible at room temperature with a platinum wire 0.081 cm in diameter because of

variation in contact angle; therefore, a goldplated tungsten wire 0.010 cm in diameter was

used. The volumes of three spheres (2.0, 1.5, and

1.0 cm in diameter), as well as the platinum

suspension wire (0.081 cm in diameter), were

calculated from the difference in weight in air

and in 4 C distilled water. The combined effects

of surface tension and buoyancy of the tungsten

wire were computed to be less than 1 mg in weight

and therefore were neglected.

At high temperatures the surface tension effect

was reproducible in molten B203

. This was demonstrated by weighing each sphere at three depths.

From the difference in volume and weight between

the spheres an absolute calculation of the density can be made. In order not to make measurements

at each temperature with three balls, the effect of

surface tension was calculated assuming 100 percent wettability and disregarding the Wilhelmy

[27] correction (for the effect of surface tension on

thin wires). The authors found that Wilhelmy

gave experimental evidence that this correction,

reported in the early part of his publication, was

not correct. Thus the density was calculated

according to the following:

d=(»r

al rWR |2W#.(l+^)! r

1 (13)

where W&\ T —weight of sphere in air (g)

W"m =weight of sphere in melt (g)

V0 =volume of sphere in distilled water

at 4° C (cm3

) r=radius of suspension wire (cm)

0-= surface tension of B203 (dynes/cm2 ) g=gravitational constant (cm/s2

)

/3p =linear expansion coefficient of platinum/0^ T=temperature (°C)

Density values from this calculation, using

spheres 2.0 and 1.5 cm in diameter, fall within

±0.002 g/cm3 of the average values obtained by

the difference in volume calculations.

In comparing the data [21] in figure 2 with other

authors, our data approximately parallel those of

Mackenzie [23] and of Shartsis et al. [25] but are about 1 percent higher. Our data are more nearly

in agreement with the results of Li et al. [26] and

Riebling [22]. I believe that most of the disagreement lies in the surface tension correction. The

best way to minimize such errors is to have as large a sphere as possible, since in this case the

surface tension correction is a minimum.

A more difficult problem is to make density

measurements on melts having viscosities higher

than 106 poise. In this region it becomes prohibitively long to obtain balance by the Archimedes

method. Thus a new technique was developed by

Macedo, Capps, and Litovitz [18] to operate at

high temperatures. A picnometer was fashioned

which had a ground-glass ball joint, which made

it possible to add or remove large specimens without affecting the internal volume of the picnometer. Cylindrical specimens of boron trioxide glass

172
T

0 200 400 600 800 1000 1200 1400

TEMPERATURE (°C)

Figure 2. Specific volume versus temperature for vitreous B2O3.

Data above 400 C taken from NMH, ref. [21], below 400° C from MCL, ref. [28].

were cast with a diameter of about 2 cm and a length of about 7 cm. A Teflon stopcock in the

filling tube made it unnecessary to use any grease.

Mercury was used as the picnometer fluid which

on heating was allowed to overflow into a weighing

bottle outside of the furnace. The overflow spout

was immersed below the mercury level in the

weighing bottle, permitting intake of mercury

when the system contracted on cooling. The bottle

was removed from the assembly for weighing.

Temperature changes were made in discrete steps.

After thermal equilibrium was reached at a new

temperature, the overflow bottle was weighed and

then replaced; all of the overflow accumulated

from step to step, and the total amount was always

weighed to prevent an accumulation of weighing

errors. This weight gain or loss of the picnometer

system between temperatures represented the

amount of mercury displaced by the expanding

or contracting system. Comparing the expansion

rate of the mercury system with the expansion

rate of the mercury-plus-sample system enables

one to determine the density p of the sample alone.

Figure 3 shows the weight of mercury displaced

in both runs. The solid points represent the picnometer filled with mercury. They are numbered

in chronological order to show reproducibility.

The open circles represent the picnometer with

B2 03 sample and Hg. The first three points are high due to the fact that the original sample was

highly quenched. Upon heating we have annealed our sample, and only points after No. 4 were used

in our calculation. The agreement of Nos. 6 and 11

shows that once the sample is annealed the agreement is comparable to that of the mercury curve.

28

24

o 20 z

u

Q_o 16

12

-4

1

1 1

1 1 1

1 1 1

9/

/ AA 4,

I

5

y

7 ~

3

2 / o 10

6

2

0

—

\ /

10

i I 1

1 1

1

1

1 1

1 40 80 120 160 200 240

TEMPERATURE (°C)

280 320

Figure 3. Weight of mercury displaced from picnometer as a function of temperature.

•, picnometer filled with mercury; O, picnometer Med with sample

plus mercury. , roB, Symbol 1 is common to both curves. Figure taken from MCL, rel. L^JThe greatest source of error comes from the

degassing of mercury at high temperatures and

can be estimated from the scatter of the data

points about the smooth curve. In both cases this

173
scatter is less than 0.5 g which corresponds to a

fractional uncertainty in the volume, of 0.15

percent. Thus the maximum error of the reported

density or specific volume is ±0.2 percent. This

data was used together with that of Napolitano,

Macedo, and Hawkins [21] to produce a complete

volume versus temperature curve in figure 2. Interpolation was necessary between 325 and 411

°C, but this was easily carried out because of the

smooth nature of the specific volume versus temperature curve. An upper limit of 325 °C for picnometer experiments was a practical limit. Degassing of the

mercury was a troublesome and imperfect process

as some of the bubbles released would cling

tenaciously to the vessel walls. This problem

increased with temperature. Under ideal conditions the upper limit would be the boiling point

of the mercury (356-357 °C) but because it has a high vapor pressure well below this, safety considerations also forced a compromise. This technique can be used at higher temperatures if one

substitutes for mercury other molten metals such

as Wood's metal or tin.

2.3. Ultrasonic Propagation

Mason's volumes on physical acoustics [29]

describe ultrasonic measurements very well. Therefore, I will only describe a new technique which

has not been included yet in this series. It was

developed by Litovitz [30], Tauke [31], Simmons

[32], and myself. Prior to this work high-temperature ultrasonic measurements were made in

such a way that the sound was generated at

room temperature and transmitted to the hightemperature section of the oven through fused

silica or polycrystalline alumina rods. The difficulties of this technique for measurements in

corrosive liquids such as the molten oxides have

been clearly outlined by Bockris and Kojonen

[33]. Two main problems appear: firstly, the

temperature range desired is above the softening

point of fused silica (1100 °C) and secondly, both

fused silica and alumina are soluble in the molten

oxides. To overcome this difficulty Bockris and

Kojonen glued molybdenum crucibles on the ends

of alumina rods. This technique has two disadvantages: firstly, there is a large acoustic loss at

the junction of the crucible and rod, and secondly,

polycrystalline alumina exhibits large grain boundary loss at high temperatures. The ultrasonic

attenuation in the bond was so high in this system

that the top alumina rod had to be directly immersed in the liquid. The resultant rapid corrosion

caused considerable experimental difficulty.

To avoid these difficulties we [30] used molybdenum acoustic transmission lines. This metal is

superior to alumina for several reasons. It is not

corroded by molten oxides and melts at a much

higher temperature (2700 °C). It withstands high

thermal gradients and is mechanically strong.

JM]

'I

7^

Figure 4. High-temperature interferometer.

1, melt sample; 2, water cooled shear transducer holders; 3, single crystal molybdenum rods oriented with their axes along (110) and 1.27 cm in diameter by 20 cm in length; 4, furnace wound with molybdenum wires;

5, micrometer screw; 6, water cooled alignment columns; 7, metal belljar used as both a shield against stray pickup of electrical signals and a chamber for a controlled atmosphere of 95% N2 and 5% H2 gas; 8, base

plate and support; 9, tuned cathode follower (from ref. [32]).

An obvious disadvantage of molybdenum is its

high thermal conductivity. This was surprisingly

easy to overcome. The heating element was taken

from the walls of the oven and put right on the

rods. Polycrystalline molybdenum has a higher

value of ultrasonic absorption than quartz owing

to scattering from crystalline boundaries. To

diminish this loss, arc cast molybdenum was used

because of its smaller crystal size with upper

frequency limit of 15 MHz. Single crystals of

molybdenum are readily available and permit

measurements up to 85 MHz.

The alumina rods used by Bockris and Kojonen

have a disadvantage compared with either polycrystalline or single crystals of molybdenum in

that they have a significantly higher absorption at

temperatures about 1100 °C due to grain boundary losses. This temperature-dependent increased

loss is not observed in molybdenum.

This interferometer shown in figure 4 can be

used for both shear and longitudinal ultrasonic

measurements in the megahertz range (1 to 85

MHz). All that is required to change from one

measurement to another is to replace the quartz

transducers, which convert electromagnetic en- ergy to accoustical energy and back to electromagnetic.

174
a. Electronic Apparatus

Tauke, Litovitz, and Macedo [31], using the

standard pulse echo technique (described below)

with a 50-dB dynamic range (difference between

maximum signal and noise), were able to make

shear measurements through a melt of B203 , only in cases of absorption per cycle less than

2 Np/cycle. This restricted their measurements

to the region wr>l (r is relaxation time), so that they could obtain only half of the relaxation

curve. In order to make shear transmission

measurements over a larger range of cor, other

methods were investigated by Simmons and

Macedo [32].

The use of a lock-in amplifier was considered

to reduce noise. Normally such an amplifier is used with "continuous wave" operation and

reduces the frequency bandpass of the receiver to

less than 1 Hz with much lower noise levels.

Unfortunately the dynamic range gained at the

receiver by the "continuous wave" operation,

in the case of the high-temperature interferometer, is lost at the transmitter. Due to the con- tinuous excitation of the crystal, the peak power

must be lowered to keep the average power

level constant and avoid damaging the crystal

and bond. In addition crosstalk, which is the

receiving of transmitted signals by electrical

pickup rather than ultrasonic transmission, cannot

be discriminated against in the "continuous

wave" application of the lock-in amplifier.

In pulse operation however, crosstalk is separated by time discrimination since such signals

travel with the speed of light and arrive at the

receiver promptly, while the ultrasonic signal is delayed. Simmons and Macedo [32] showed how

one could advantageously combine the pulse

technique and the lock-in amplifier technique,

while still using a commercial amplifier. This is done by using the triggering rate of the transmitter

in pulse operation as the reference frequency for

the lock-in amplifier.

(i) Standard Pulse Operation. The standard

equipment for pulse-echo measurements in wide

use today is shown in figure 5. The transmitter

generates a radiofrequency pulse of the order of

1 kW peak power which is applied to the lower

transducer through a matching network. The

signal after passing through both rods and the

liquid is received by the upper transducer, amplified using a tuned cathode follower at the crystal

holder, and then fed into a narrow band preamplifier.

It is added to the comparison signal and then fed

into the wide-band amplifier. The amplifier has a wide enough band to allow the use of the comparison signal generator at its lowest frequency (10

MHz) to measure signals of much lower frequencies without distortion. This was checked

using a step attenuator. The video output from

the amplifier is displayed on an oscilloscope.

(u) Pulsed Phase-Lock Detection. Figure 6 shows

the circuit diagram of Simmons and Macedo [32]

CATHODE

FOLLOWER

PREAMPLIFIER

VARIABLE ATTENUATOR

9

MATCHING SOX

COMPARATOR

SIGNAL

GENERATOR

DELAYED

TRIGGER

Figure 5. Standard equipment for pulse-echo measure- ments showing both the water and high temperature inter- ferometers (from ref. [82]).

for the operation of the pulsed ultrasonic system

with the lock-in amplifier. The details of the

ultrasonic system are shown in figure 5. It should

be noted in examining figure 6 that since the lockin amplifier detection system is used on the video

signal from the first amplifier, it can be added to

any ultrasonic apparatus in which the rf signal is rectified.

The triggering system provides a series of pulses

which alternately trigger the ultrasonic system

and the comparator pulse. These rf pulses are combined, amplified, and rectified in the receiver.

Five hundred times a second the following train

of video pulses leaves the receiver: the first pulse

is due to the electronic crosstalk (it carries no

ultrasonic information) ; the second pulse represents the straight through transition of sound

energy, which has the desired ultrasonic information; subsequently a series of echoes arrive, which

for this experiment are not interesting. After the

NBS STANDARD

FREQUENCY SOURCE TRIGGERING SYSTEM

(1000 Hz

DELAYED]

ULTRASONIC

SYSTEM

FUORIER ANALYZER

COMPARATOR

SIGNAL

GENERATOR

REFERENCE SIGNAL

FOURIER

ANALYZER

10CHN

AMPLIFIER

Figure 6. Modifications of the pulsed ultrasonic equipment

necessary for use of the phase-lock detection system.

For simplicity, the section of figure 5 between the transmitter and

amplifier, which remains unchanged, has been included into one "ultra- sonic system." (from ref. [32]).

175
echoes have died out (have been attenuated in

time till their amplitude is much less than the

noise of the system) approximately 1 ms later the

comparison pulse arrives. The gate width and

trigger are chosen to allow transmission of only

the maximum amplitude sections of the echo of

interest and the comparator pulse. In this manner

the information fed into the Fourier analyzer and

the lock-in amplifier is made up of two nearly

square pulses. The Fourier analyzer passes only

the 500 Hz component of these pulses. This not

only cancels out the gate pedestal which has a 1000 Hz repetition rate, but also subtracts the

comparison signal from the echo since they are out

of phase. The output C at 500 Hz is used as comparison signal for the lock-in amplifier, which

therefore analyzes the echo pulse as an in-phase

signal and the comparison pulse as an out-of-phase

signal, thus changing phase by 180° according to

which signal is largest. Intensity measurements

are made by matching both signals and obtaining

a null.

The lock-in amplifier uses its narrow band pass

with an integration network and produces an en- hancement in signal to noise ratio of around 25

dB over the video display of the pulse. The null

detector feature of this system permits comparison

with a high degree of sensitivity (less than 0.1 dB)

provided the original signal is 10 to 15 dB above

noise.

b. Measurement Technique

Methods used for measuring absorption and

velocity of ultrasonic waves in liquids are discussed below using the apparatus described in the

preceding sections. I am summarizing previously

used methods [30, 31] as described by Simmons

et al. [32].

i. Absorption. The absorption measurements are made by stepwise decreasing the path length in

the melt with a micrometer screw and recording

the amplitude of the comparison pulse necessary

to produce a null at the lock-in amplifier for each

step.

ii. Velocity. Depending on the magnitude of the

absorption coefficient and the viscosity of the melt,

various techniques can be used to measure the

ultrasonic velocity in the sample. For low absorption and viscosity below 2X105 poise, the signal

coming through the high-temperature ultrasonic

system is beaten against a pulse coming through

a water interferometer at room temperature. The

position of the rods in the water remains fixed. As

the high-temperature path length is changed, a

series of maxima and minima can be observed on

the oscilloscope. The slope of a plot of minima

versus rod separation in the melt is the ultrasonic

wavelength, X, in the sample. Knowing the frequency /, the velocity, V=\f, can be calculated.

As the absorption increases, this measurement

becomes progressively more difficult as fewer and

fewer minima are measured, and the accuracy

consequently goes down.

The technique used in the high attenuation

regions is similar to the preceding one, in that

two ultrasonic systems are used in parallel, one

with water, the other with the sample at high

temperatures. The smallest detectable signal is found by closing the gap between the two rods

in the high-temperature path. Then an attenuator

in the water path must be adjusted so that the

intensities of both signals match each other.

By adjusting the path length in the water, the

signals can then be superimposed, a minimum

attained by phase cancellation; and the respective

positions of the interferometers recorded. The

melt gap is then decreased by steps of 0.0C5 or 0.010 cm, depending on the available range of

travel, and the gap in the water is reduced until a minimum is again observed. After the positions

of the interferometers are again recorded the

above procedure is repeated (see fig. 7). To obtain

good phase cancellation one must adjust the

attenuator in the water line until both signals

have approximately the same strength. Whenever

this is not possible (i.e., large melt signal) the

comparison pulse must be adjusted to cancel out

the excess signal and emphasize the minimum

at the lock-in. Since the velocity of sound in

water is known, the velocity in the sample can be calculated by the relation

V, (14)

where Vs is the velocity in the sample,

the velocity in water, and Xn2 o the separation in

water corresponding to Xs in the melt.

For viscosities above 2X 105 poise (temperatures

below 840 °C in the alkali borosilicate glass tested)

it becomes difficult to move the two rods with

200 300 400

DISTANCE IN MELT./J.

Figure 7. Velocity measurements at 15 MHz in alkali

borosilicate glass over the temperature range 1003 °C to 1264 °C and having absorption per wavelength between

1.88 Nplcycle and 4.87 Npjcycle from ref. [32].

176
respect to each other without noticeable slippage

in the micrometer screw driving linkage. In this case, Simmons et al. [32] devised a new technique

for measuring the sound velocity. The temperature

induced change in transit time of the shear wave

through the two transmission lines separated by

a fixed distance in liquid is first measured. The

change in transit time in the liquid alone is then

obtained by subtracting the temperature-induced

change in transit time of the wave through the

transmission lines with the two rods touching.

Since the measurements are made for a fixed

distance of travel in the liquid, the temperatureinduced change in transit time can be related to a temperature dependence of the velocity.

This technique was used on the molten sodaborosilicate glass below 830 °C. The first echo of

the transmitted wave in the sample was beaten

against the water echo, but in this case the travel

distance in the water (whose temperature was

kept constant) was adjusted to remain out of

phase with the sample echo while the sample

temperature was varied and the sample distance

remained fixed. Thus, one could keep track of

the temperature-induced changes in transit time

by recording the changes in the travel distance

in water necessary to keep the two echoes out of

phase. Plots of the temperature dependence of

the travel through the water are shown in figure

8 [32]. Curve (a) represents the distance traveled

in water versus temperature when the two molybdenum rods are touching. Curve (b) is the same

plot except that the rods are separated by 0.254

cm of melt, and the added slope represents the

temperature dependence of the sound velocity in

the sample. The water temperature was kept

600

500

400

P, 300 -

P 200

< 100 -

0 -

750 760 770 780 790 800 810 temperature in degrees c

820 830 840

Figure 8. Temperature dependence of distance traveled in water necessary to keep the water echo out of phase with the sample echo in the measurements of velocity at viscosities above 2 X 10s poise. Curve (a) is the temperature dependence corresponding to the trans- mission lines only, and curve (b) is the dependence corresponding to the transmission lines and the melt from ref. [32].

constant; thus the change in transit time could

be directly related to a change in travel distance

in the water through the known sound velocity in

water. The shear velocity outside the relaxation

region (temperatures lower than 830 °C in this

liquid sample) was then calculated by adding the

changes in velocity to a directly measured value

at a higher temperature. This was done using the

following equation:

V(T) = V0 [l AV0 xmv, (To-T)] (15) m V H2O

where V0 , is the velocity of the melt at T0 , Xm is the fixed distance of travel in the melt, Vh2o the

velocity in water, and A the difference between

the slopes of curves (b) and (a) in figure 7.

3. Viscoelastic Considerations

The fundamental question in viscoelastic behavior is : What and where is the glass transition?

In figure 9 from ref. [17], the increase in length

is plotted versus temperature for a Ge02 glass

sample. In this experiment the temperature was

raised at 2 °C/min. The region where the expansion coefficient increases drastically is known as the transformation range. Above this range one

has a liquid in metastable thermal equilibrium.

By metastable it is meant that the state of lowest

free energy is the crystalline, but this transformation requires nucleation and as long as there

are no nuclei present the liquid is in thermal

equilibrium. The larger expansion coefficient above

Tg is characteristic of the liquid state and is due

to the change in order as one changes temperature.

This order has been characterized by various authors as the "number of holes," or "free volume,"

or "excess entropy," or "number of broken bonds,"

etc.In a glass the degree of order does not change

as one raises or lowers the temperature; thus all the second-order thermodynamic parameters

(thermal expansion, (3=(l/V)(c)V/dT) p ; specific

heat Cp=(dH/bT) v ; compressibility K,= —

(dV/5P) s ) have a value lower than the corresponding parameters in the liquid. This lack in change

of the degree of order makes the glass in a thermodynamic non-equilibrium or unstable situation.

If one observes a glass for long periods of time

one would expect to see the degree of order change

as it tries to approach equilibrium. This process

is known as relaxation and measurements of its

time dependence are commonly called annealing.

In order to mathematically describe the annealing process, one has to study the time dependence

of the approach to equilibrium as a function of

both temperature and one or more order parameters. This work can be separated into two parts

firstly, the study of the time dependence of the

structural rearrangements, which will be characterized in terms of a spectrum of relaxation

177
70 I

1 1 i 1

i I

1 r

-10

-180 -100 0 100 200 300 400 _ 500 600

TEMPERATURE , °C

Figure 9. Thermal expansion of vitreous Ge(>2 showing the critical point and the deformation

point.

Figure taken from N & M, ref. [17].

times; secondly, the temperature and structure

dependence of this spectrum.

3.1. Measurement of the Spectrum

of Relaxation Times

Let us define as linear an experiment where the

time dependence of the normalized stress or strain

relaxation is independent of the initial stress or

strain. Of course, a nonlinear experiment is one

which is not linear. One of the major problems is to design an annealing experiment which is linear.

Ultrasonics provides probably the best solution

to this problem. Sound waves can be generated

and detected (see sec. 2) with stresses and strains

which may be treated as infinitesimal. From the

sound absorption, a, and velocity, V, the real and

imaginary parts of the complex modulus can be

calculated according to

pF2 (l-(aF/co) 2) m*=m' +im" =

(l + («T7co)T

+i 2pV2 (aV/u>) (i+m2

) 2 (16)

where m* can either be the complex shear modulus

G*, or the complex longitudinal modulus M*,

depending whether the shear or longitudinal sound

absorption and velocity are used. The longitudinal

modulus can be calculated from G* and the complex compression modulus, K*, according to

M*=K*+4/3 G* (17)

Figure 10(a) shows a plot of the real and imaginary

components of the longitudinal modulus, M*,

versus frequency. At low frequencies where the

structure of the liquid has plenty of time to re- arrange itself between successive crests of the

sound wave, M* is real and frequency independent

and represents the modulus of compression of the

liquid (the reciprocal of the liquid compressibility).

As the sound period becomes shorter and shorter

the structure cannot keep up with the cycling

pressure, and the liquid becomes progressively less

and less compressible. Finally, at high frequencies

the liquid exhibits glasslike moduli, (M a> =Ka> +4/3 where Km is the unrelaxed modulus of

178
n'-

4 / -At

Figuee 10. (a) Plot of real and imaginary parts of modulus versus log frequency.

(b) Reduced plot.

(c) Plot of shear modulus versus log frequency.

This plot is exaggerated to show all three relaxation mechanisms:

(1) entropy or polymer like relaxation not observed in molten oxides; (2) pseudo glass transition; (3) delay elastic effects; in silicate and borate melts

the relative strengths are 0%; 90%; ~10% respectively.

(d) Spectrum of relaxation times.

compression and is the shear modulus). By

definition liquids do not exhibit equilibrium shear

rigidity; thus the high-frequency behavior is definitely a solidlike property. Actually, it is the

glass modulus that one is observing, and by

changing the frequency one can go through the

glass transition. Associated with the dispersion of

the real part of the modulus there is an absorption

of the sound wave which is related to the imaginary

part of the modulus. Just as the low-frequency

real part of the modulus was the liquid modulus

of compression (reciprocal of the compressibility),

the low-frequency imaginary part of the modulus

is given by urjL , where co is the angular frequency

and vl is the longitudinal viscosity. In a liquid

there are two elementary and independent stress- strain processes, one associated with shear and the

other with compression. The shear has the wellknown shear viscosity, vs , the measurement of

which is described in section 2, while the compression exhibits an analogous volume viscosity, riv

. The longitudinal viscosity represents the following

sum of these viscosities:

ifc=^+4/3 ^. (18)

By measuring the propagation of both shear and

longitudinal sound waves, one can subtract G*

from M* and obtain the relaxation curves for pure

compression. By calculating the reduced modulus

for compression

m

K'-K,-• 777"

orforshear TU^G'/G^; Tfl"=G"\Gm (19')

at each frequency one can compare the experimental curve with theoretical results. For a

single relaxation time, r, one has

K'

(19)

m+im" (cor)

2 COT

l +(cor) 2 ' l +(cor)

2 (20)

In general, the data do not fit, and one has to

assume a distribution of relaxation times #(r).

Thus, one has integrals of the form

m'+im"=f"\j M2 +(«r) 2

cor

l +(«r) fj#(r)d7

(21)

179
where for molten oxides g(r) is taken to be

g(r)dT ={bl^K) exp[-62 In2 (r/r')] (flnr (22)

where r' is the most probable t, and b is the width.

Schematically g(r) is given in figure 10(d). From

eqs (21) and (22) one obtains a very useful expression

7?s =lim (—)=Gm f=Ga Crg{T)dT (23)

wt->0 \ W / JO

where r is the average shear relaxation time.

This is the reason why t? s is used to define the

annealing range (annealing point and strain point)

.

Working with glass one needs to specify the

time-temperature relationship for three basic

processes (deformation, annealing, and strain

points). The upper limit of the annealing range is determined by the deformation point, where

within minutes the glass will sag under gravity.

This process is directly related to the viscosity

itself and can be well characterized by viscosity

measurements. Next, one is interested in the glass

transition itself, where the volume relaxation

time becomes comparable with the effective

experimental times. A priori the volume relaxation

time has nothing to do with the shear viscosity,

and a measurement of shear viscosity would be

useless to characterize this process. Fortunately,

there are some relationships which we found to

hold for silicate glasses between the shear and

volume relaxation times. The average volume

relaxation time at constant pressure and temperature is about six times larger than the average

shear relaxation time. Also the shear modulus of

rigidity Gm is about 2X1011 dynes/cm2

. Thus,

using eq (23) for a viscosity of 10 13 poise one

calculates the average shear relaxation time to

be ^50 sec and the average volume relaxation

time ~5 min. If one measures the expansion

coefficient of a glass by rate heating at 2 °C/min,

the first kink (critical point in fig. 9) corresponds

to a temperature having an average volume relaxation time of a few minutes. For silicate glasses the

sample has ~1013 poise at the critical point, but

this is not a viscosity measurement; and for the

same volume relaxation time the shear viscosity

varies from 1014 poise for Ge02 to 1012 poise for

B2 03

. A second and sometimes worse error is the

use of the average relaxation times rather than

the spectrum, since the very small relaxation times

often play the predominant role in annealing.

This introduces at least a factor of 10 error in

estimates of annealing times.

To better understand the effects of a spectrum

of volume relaxation times let us examine figure

11 [17]. In the relaxation curves shown, the

structure changes enough to affect the relaxation

time spectrum during an approach curve (nonlinear effect). The existence of both nonlinear

effects and a distribution of relaxation times make

the calculation of the relaxation time spectrum

very difficult, if not impossible. In order to circumvent this problem the crossover technique

[34, 35] was developed. The thermal history

involved in the crossover can best be understood

if one first understands the mathematical model

used to represent the spectrum. The analysis is simplified by assuming that the spectrum is given

by two relaxation mechanisms of equal strength,

each having a single relaxation time. Thus the

average index N is given as a function of time by

the following relation

N=y2 [Ml (t)+M2 (t)] (24)

where Mt (t)=N^ J

r Nie-tlTi

, in which iV» is the

equilibrium value of index; N{ , the initial deviation

of index from iV» associated with mechanism i having relaxation times r t

. This formulation can

be thought of as describing the relaxation (change

of volume) to a final equilibrium value following

a change in temperature (analogous to an isothermal recovery function) in terms of two

separate mechanisms, the first of which relaxes

appreciably faster than the second. Ni and 7V2 are measures of the total relaxation undergone by

the two separate mechanisms in reaching equilibrium. If the starting point is one of equilibrium (Mi (0)=M2 (0)), the two mechanisms

will have to relax the same total amount Ni = N2 =(Na,(Tl )—Na>(T2 )). However, when starting

from a nonequilibrium approach curve, Ni and

N2 will depend on the particular approach curve

being followed. Figure 12 due to Macedo and

Napolitano [34] shows schematically the time

dependence of N, Mu and M2 at T2 for a glass

initially at equilibrium at Tx , where T^>T2

. As

the structure rearranges itself to the new equilibrium, Mi will rise faster since it is associated

with the shorter relaxation time while M2 will

change more slowly. Thus, around the bend of

the approach curve, there will be the largest

difference between Mx and M2 in the glass. As

time progresses, Mi will reach its equilibrium value

for T2 first (r2 >ri) and stay there. From then on

the spread between Mi and M2 narrows because

the slow relaxation process continuously approaches the same equilibrium value at a slower

rate. Finally, when the structure is completely,

at the new equilibrium the spread becomes zero and one has again an equilibrium glass (My=M2 ). The crossover experiment involves taking a

glass having index, a (see fig. 12) from the approach curve and introducing it into a furnace

whose temperature is Tx . Tx is preselected such

that the equilibrium index at Tx , iV^T*), is equal

to a. Thus at a crossover, a, the fast relaxation

time (n) corresponds to an index Mx higher than

the (average) measured value, N, and the slow

relaxation time (r 2 ) corresponds to an index M2 lower than N.

If the index at the point of crossover, a, is equal to the equilibrium index at the crossover temperature Tx, then N2 =—Ni. A prerequisite

for a minimum in the crossover experiment is

180
i r i i I I l l I I I I I I I I I l I I l I

= 1.60700

P .60500

I I I I I I I I I I I I I I

23 00

13 16

TIME.hr

24 2!

Figure 11. Index of refraction versus time isotherms for vitreous GeOi.

Figure taken from N&M, ref. [17].

~i 1 1 r

CROSSOVER EQUILIBRIUM INDEX-X

20 30

TIME, HOURS

40 50

Figure 12. Illustration of the two relaxation time model

{ref. 34) represented by the upper and lower dashed

curves. The average or measured index is given by solid line. The crossover equilibrium index (x) and the transfer point (a) are also shown.

that Ni and N2 have opposite signs. This is fulfilled when N has a value between A and B

in figure 12. Figure 13, from Macedo and Napolitano [34], shows Mx (the upper curve), M2 (the

lower curve) and N the solid line (calculated index).

The agreement of the calculated curve with the

actual experimental points for a borosilicate

crown glass is seen to be good.

From this discussion there are two points one

shoidd consider. First, because of the spectrum

of relaxation times, two glasses with the same

composition and same index of refraction or Active temperature can have vastly different

structures (compare glasses in the beginning and

end of figure 13). Second, the two relaxation time

model shoidd only be considered as a mathematical convenience; the spectrum of relaxation times

physically can be represented better as a gaussian.

This explains the existence of a very small number

of relaxation times, orders of magnitude smaller

than the average relaxation times. These very fast

relaxation times are responsible for the annealing

observed at 351 °C in figure 11.

181
1.51600

75-

50

25

2 1.51500 G 1.51493 <

or 1.51475

rr

50

25

1.51400

1.51375

• l.5l493 + .OOI03exp.(-t/7.77)

Crossover A

Equil. Index Line

Average Calculated Curve

I.5I493-.00III exp.(-t/64.4)

30 60 90 120 150 180 TIME- MIN.

Figure 13. Crossover A for BSC glass (ref. 34,

35) , showing the average of the fast and slow

relaxation processes.

3.2. Strain Point and Viscosity in

Nonequilibrium Structures

Glasses that are cooled through the glass transformation region with a temperature gradient

across the sample have frozen (thermally arrested)

strains. These strains produce optical birefring- ence. Such glasses are brought to the "bottom"

of the annealing range and there strain relaxation

will eliminate birefringence. This relaxation is due to shear flow and shear viscosity controls

relaxation rates. The problem is that glasses are not at thermal equilibrium, because complete

volume relaxation takes too long at this temperature region. The viscosity is not a unique function

of temperature, but also depends on the structure.

Thus, in order to characterize strain relaxation

one needs to know how to extrapolate the viscosity

to lower temperature for a glass at fixed structure.

This area is in a major controversy between

theoretical arguments, experiments, and technical

applications. The theoretical arguments will be

discussed in section 4. The problem with making

a viscosity measurement as a function of volume

relaxation is that viscosity can not be measured

instantaneously. Upon applying a load the elongation rate is much larger than one would expect

from viscous flow. This occurs because glasses are complicated materials and there are several

different relaxation mechanisms acting at the

same time. If the load is removed there will be a time-dependent contraction, but when the process

is completed not all of the deformation will have

been recovered. Those relaxation processes whose

deformation can be recovered are called delayed

elastic effects, while the irreversible deformations

are associated with viscous flow. In fact one does

not have to remove the load and measure the

permanent part of the deformation, but one need

only wait till all the delayed elastic effect has

worked itself out and the elongation rate is no

longer a function of time. Zijlstra [36] made

measurements of elongation rate versus time after

temperature jumps and erroneously associated

all the elongation rate with a viscous process.

From these measurements he concluded that the

viscosity of the nonequilibrium frozen structure

glass was almost temperature independent (fig.

14). On the other hand, Kurkjian [37] showed that

by the time that the delayed elastic effects are worked out the glass is at or almost at thermal

equilibrium (volume relaxation is almost completed). Kurkjian not only questioned Zijlstra's

work, but also indicated that he considered that

type of measurement would be impossible to make.

Lillie [13], studying the technical problem of

strain relaxation in glass, had in effect solved the

theoretical problem 30 years before it was proposed. He loaded the sample at high temperature

where both volume relaxation and delayed elastic

eflects take a reasonable time. Then the sample

was cooled at a constant rate. For awhile volume

relaxation kept up with the cooling rate but as volume relaxation time became longer and longer

its structure lagged the equilibrium structure.

That could be seen from the viscosity-temperature curve as it changed slope. However the change

in slope was only 25 percent, far less than Zijlstra's

value (95 percent). This rate cooling measurement

is the now Annealing-Strain Point ASTM procedure. As it will be seen in the next section, this

relatively small change in slope contradicts most

modern theories of viscosity. A sample run of the

viscosity versus temperature at constant cooling

rate is shown in figure 15 for NBS Standard glass

710 due to Hagy [15].

3.3. Nonlinear Viscoelasticity

All the viscoelastic theory thus far presented

rested on the fact that one can linearize the equa16

14 .sl2

o

I, 2

/A---. 4/

/ /

' / » 3/

/ * /

jy

>-

(curves 3 & 4) 0-6 0-8 1-0 1-2 1-4 1-6

0-6 0-8 10 1-2 14 (curves 1 & 2) Reciprocal of absolute temperature x 10'

Figure 14. Viscosity-temperature curves taken from Zijlstra [36].

equilibrium viscosity

initial viscosity of quenched fibers (erroneous data)

.

182
13

o 12

\V« \—

\ \

\ —\V540 560 580

Temp(°C)

Figure 15. Data for NBS Standard Glass No.

710 from Hagy [15], sample cooled at J, °C

per minute for determination of annealing and

strain points.

0 observed data O data corrected by 2 °C for thermocouple log. Equilibri- um viscosities for glass shown by solid line.

tions. For viscosity that means that the shear

strain rate is proportional to the shear stress. Recently authors [38, 39, 40] have shown experimental proof of nonlinear behavior in the annealing

region of several glasses. Since the experimental

uncertainty of these measurements were comparable with the effect, the results were generally

attributed to bad data and ignored. However,

the effect is probably real and must be accounted

for before it destroys our whole theoretical framework. Bartenev [38] states that the nonlinear

effects are most pronounced when the glasses

were observed to phase separate. Lack of linear

behavior of two liquid-phase composites is ex- pected, and does not contradict our theoretical

structure.

This existence of microphase in melts and

glasses does cause unexpected and otherwise unexplained behaviors. Whenever one is working

with a glass, one should keep track of its microstructure by electron microscopy. Glasses must

now be characterized not only by composition

but also by detailed thermal history, since it will

affect both its volume relaxation (Active temperature) and its microstructure.

4. Temperature Dependence of the Viscosity and Relaxation Time

4.1. Fulcher Equation

In order to interpolate and extrapolate viscosity

measurements, one would like to have an analytical expression for the temperature dependence

of the viscosity. The most widely used expression,

is an empirical equation proposed by Fulcher [41],

log r,=A+B/(T-T0) (25)

in which A, B and T0 are arbitrary constants. In

order to fit this equation to data by the least

squares method, it has to be linearized to the form

T log V =B-AT0 +AT+ To log n. (26)

One needs a program subroutine for least squares

fitting a function of three variables (vectors)

Yi=CiXu-\-C2 X2 i+C3X3U (27)

where Yt are the unknowns, XH are the variables,

and Ci are the desired coefficients. Note "i"

differentiates each set of data. Matching equations

(26) and (27) one has

Yi=Ti log ri i Xu=l

X2 i= Tt -ST3< =log Vi

and A=C2 T0 =C3 B=C1 + C2 C3

. We have minimized the error on T log v which

favors the high-temperature data. This is the usual

procedure since the low viscosity measurements at

high temperatures are more accurate than the high

viscosity data at low temperatures. Because of the

linearization of the Fulcher equation the standard

deviations calculated for Cx , C2 and C3 are not

applicable to A, B, and T0

. If one wants the

standard deviation of the data about the Fulcher

curve, a, one can calculate it

0-2= (jV-3)-%(log w-log lea.,)

2 (28)

where N is the number of data points. If, on the

other hand, the uncertainties on A, B, and T0 are

desired, a second calculation has to be performed.

This time we assume a set of parameters (A', B'

,

and Tq from the above calculation) and proceed

to calculate an improved set {A, B, and T0) by the

following iterative procedure. A difference in log v can be caused by:

Alog^S^AX^

-f(T-ro)-1 Aff+Tr^yp ATo- (29)

183
Matching eq (29) and (27) we have

Fi=log 7/iexp— log i?ical=log 77jexp

-A'-B'iTt-To')- 1 (30)

^^(T.-To)"1 X3i =B (Ti—T0)~2

.

After the least squares fit one has: AA=Cy \ AB=C2 ; AT0 =C3

. The best values then become

A=A' +AA=A' + d

B=B' + AB=B' + C2

T0 =T0

' +AT0 =T0

' +a

and the standard deviations of A, B, and T0 are those of Cu C2 , and 03 respectively. If the initial

values of the parameters are taken from a least

squares fit to eq (26), the Ct are comparable to

their standard deviations and only one iteration

is necessary. Notice, that eq (29) minimizes the

error in log 77 rather than T log 77. If the reader is interested in the latter he can multiply all the

equations by T. The rest of this section is taken

from Macedo and Napolitano [1]. The extensive measurements carried out at the

National Bureau of Standards and several participating laboratories on standard glasses No. 710

[3] and 711 [42] have shown that the Fulcher

equation does indeed fit the viscosity of these

glasses within experimental accuracy between

102 and 1012 poise. A plot of the deviations between

the experimental points and a least squares fit to the Fulcher equation in this range for both of

the standard glasses is shown in figure 16 from

reference [2]. For a comparison a plot of the deviations from

a least squares fit to the Fulcher equation for a

BSC glass (Ritland's) [43] where viscosity measurements have been made to 10 14 72 poise is shown in

figure 17 from reference [1]. The deviations are

well outside experimental error but still reasonable. In general, if the data are limited to viscosity

values below 10 12 a good fit can be obtained to

the Fulcher equation [41]. In extrapolating this

fit to (low temperature) higher viscosity values,

the Fulcher equation predicts higher values than

the actual measured values, whereas, if (as shown

in fig. 17) a least squares fit is made to the Fulcher

equation with all of the data (up to 10 14 72 poise)

an "s" shape type of curve is obtained for the

deviations.

In view of the great empirical success of the

Fulcher equation, many different theoretical

bases have been proposed for it. In all of these

models approximating assumptions had to be made

to reduce the theoretical equation to the form of

the Fulcher equation. In the hope that one of

these theoretical models will be an improvement

over the Fulcher equation we will review theories

in terms of all the available viscosity data in

vitreous B203. For the sake of presentation, the

smooth data taken from reference [1] will be used

and plots will be given of the differences between

the predicted and experimental values.

These approximating assumptions are especially

bad for B203 because of the peculiar temperature

dependence of the expansion coefficient [44] and

the specific heat [45]. Thus, it can be expected that

the Fulcher equation will be a poor fit to the B2 03 data. In fact, as shown in figure 18, curve A, the

equation is a very poor fit, having a proportional

standard deviation of 40 percent and deviating by

as much as a factor of 2.2. The calculated conTEMPERATURE "CxlO 2

5.3 6.0 7.0 8.0 10.0 13.0

TEMPERATURE - °C * I0~2

6.5 70 7.5 8 9 10 II 1 2 13 14 15

d .05

FT 0

<S> Q

I

05

2 -.10 o

3 05


 o

-.05

-.10

n 1

1 1 1 1—I I I I

ST'D GLASS NO. 710 *»*.*o o »» --_ n 0 °° 0°0 0 °

o

12 II 10 98765432

L0G ^OBS.

. a ST'D GLASS NO. 711

I

I 5.0 5.5 6.0 65 7.0 7.5 8 9 10 II 12 13 14 TEMPERATURE - °C * I0" 2

Figure 16. Deviations between the observed log vis- cosity values and those calculated from the least squares fit to the Fulcher equation for NBS standard

glasses Nos. 710 and 711 [2, 3, 42].

o

<o

p-Qoo

p- - I oo

o

-.2

O

o O Oo

o

o o

o.o

0

10 8 6 LOG io f] -POISES

Figure 17. Deviations between the observed log viscosity

values and those calculated from the least squares fit to the Fulcher equation for BSC glass [1].

184
stants for this equation and those that follow are given in table 2.

Table 2. Calculated constants for each equation for B2 03 glass from ref. [1].

Constants 1" 2" 3" 4" 5"

AOogio) B

-0. 176

1825

138

-5. 239

1. 160 -0. 668 562.1 -5.334 -8. 634

115, 500 ToVo(cm3/g.) 0. 492 0.480 £(kcal/mol) 11.9

5. 68 6.70

0. 147

A£(kcal/mol)

AS(cal/k/mol) AV(cmVg.)

"1—Fulcher equation (ref. 41). 2—WLF equation (ref. 46). 3—RRE equation (ref. 51). 4—Hybrid equation (ref. 52)

.

5—A & G equation (ref. 56)

.

4.2. The Free Volume Approach

a. Doolittle Equation

Williams, Landel, and Ferry (WLF) [46] have

shown that the Fulcher equation can be considered

identical (assuming the expansion coefficient is

TEMPERATURE-°CxlOZ

2.6 3.0 3.5 4 5 7 10 14

16 14 12 10 8 6 4 2 L0G|07y- POISES

Figure 18. Deviations between the observed log

viscosity values and those calculated from the least squares fit to several models for B2O3 glass [1]. • A—Fulcher [41]. O B—WLF [46] C—Ree, Ree and Eyring [51] D—Macedo and Litovitz [52] A E—Adam and Gibbs [56]

temperature independent) to the free volume

equation (in the Doolittle [47] equation) where

log V =A+B/(V-V0) (31)

in which V-V0 is the free volume per unit mass,

and A, B, and V0 are arbitrary parameters. This

equation was later derived by Cohen and Turnbull

[48] and Bueche [49]. Even though a constant ex- pansion coefficient would be a good assumption

for most liquids, it would be especially bad for

B203 since 0 changes by a factor ~20 between

270 and 1300 °C. Curve B, figure 18, represents a

fit to the free volume equation, and, as can be seen even though the deviations are well above experimental uncertainty, the fit is much better than the

Fulcher equation fit. The free volume equation not only predicts

the temperature dependence of the viscosity

through the volume but also the pressure. In fact,

it predicts that if one changes the pressure

appropriately to maintain constant volume, the

viscosity should be temperature independent.

Sperry and Mackenzie [50] have observed strong

temperature dependence of the viscosity at

constant volume, and found the activation energy

at constant volume to be not only large as in most

other liquids but almost equal to the activation

energy at constant pressure. Thus, the free

volume approach can not explain the viscositypressure dependence, suggesting that the success

in fitting the viscosity-temperatue dependence

is probably fortuitous.

b. Significant Structure Theory

In an effort to introduce a temperature dependence of the viscosity at constant free volume

Ree, Ree, and Eyring [51] proposed an equation

of the form

log n =A+B/[T(Y-V0)] (32)

in which the activation energy [B/(V—V0)] is a function of the free volume. Curve C, figure 18,

shows the deviations of the least squares fit of this

equation to the data.

c. Hybrid Equation

In the hybrid equation [52] the probability of a molecule obtaining simultaneously enough energy

to break its bonds and free volume to perform a jump is calculated. This equation has the form:

In V =A+V0/(V-V0) +E/(RT) +ln T. (33)

It became evident that this equation, similar to

the free volume equation (31), overestimates the

pressure dependence of the viscosity. Among

investigators in this field there has grown a general

sentiment that VQ is pressure dependent. For

example, Matheson [53] solved a similar problem

with the viscosity of glycerine by making the

323-655 O - 69 - 13 185
reasonable assumption that V0 is temperature and

pressure dependent. Such a calculation has been

made for B2 03 [54] where a two-state model for

the free volume of B203 was assumed. In this case the hybrid equation takes the form

In r,=A+(V-V,))V,+E,/(RT) (34)

in which Vf is the free volume given by a two-state

calculation,

Vf =AV. (1 +exp [(AH—TAS)/RT\) (35)

where AV, AS, and AH (AH=AE+PAV) are

respectively the differences in volume, entropy,

and enthalpy between states 1 and 2. It was also

shown that these equations not only predicted

the temperature dependence of the volume and

viscosity, but also the pressure dependence of the

viscosity as measured by Dane and Birch [55].

Figure 19 shows a plot of d log n/dP versus temperature where the two extreme data points are from Dane and Birch, while data between the

extremes is the more recent work of Sperry and

Mackenzie [50]. As can be seen in figure 19 the

two sets of data agree well with each other, and

since the hybrid equation was shown to be consistent with the Dane and Birch data it also agrees

with the Sperry and Mackenzie data.

In figure 18, curve D, the hybrid equation is tested with Macedo and Napolitano's [1] new lowter .^erature data and it also fails by predicting a too fast increase in viscosity in the glass transition

region. The failure can be attributed to either eqs

(33) and (34) (viscosity theory) or eq (35) (improper calculation of the free volume). This point

is discussed at length in ref [1], the conclusions of

which are given in section 4.4.

70

10 -

o I i I i I i I i i l_

360 400 440 480 520 TEMPERATURE -°C

Figure 19. Plot of dlogVdp versus temperature from ref. HI

#—Dane and Birch [55] O—Sperry and Mackenzie [50]

4.3. Configurational Entropy Model

Adam and Gibbs [56] proposed that in order to

have viscous flow a flow unit has to overcome a

potential barrier. In addition, depending upon

the configurational entropy, more than one unit

may have to perform the jump at a time. This

gives an equation of the form

log r,=A+B/[T (S-S0)] (36)

in which {S—S0) is the configurational entropy,

Se . Araujo [57] equating the temperature dependence of Sc to the difference between the liquid

and the crystal temperature dependence of the

entropy, was able to fit the viscosity data for

B203 between 106 and 10 1

4

poise very well.

Unfortunately, there must have been an arithmetical mistake since over this temperature range

the extrapolation of crystal specific heats of

Kelly [58] and Thomas and Parks [45] (which

Araujo claims to use) crosses the liquid values, see figure 20. This would predict increasing viscosity with increasing temperatures at a temperature above the crossover point but still within

the measurement range. This phenomenon was

not observed. Thus, one must assume, the specific

heat extrapolation is incorrect. The viscosity

data was cut off at 500 °C and refitted where T0 , A, and B were varied for best results. Curve E,

figure 18, shows the deviation from the fit. The

fit is still poor, but better than average. This

appearance of improvement is due to a much

smaller temperature range.

4.4. Conclusions of Equation of Viscosity

Having considered the best known viscosity

theories, in view of their new additional viscosity

data, and finding no reasonable fits, Macedo and

Napolitano [1] examined the validity of the basic

assumptions underlying these theories. They found

that the temperature dependence of the viscosity

is not controlled by structural effects such as free

6O1 1 1 1 1 1 1 11 r

50-

fc

1 20-

Q. o

10 -

ol 1 I 1 1

I 1 1 1 -I 300 500 700 900 1 100 1300 TEMPERATURE-°C

Figure 20. Plot of specific heats versus temperature from

Kelly and Thomas and Parks for B2 03 from ref. [1].

186
volume, configurational entropy, etc., but by

activation energy effects which are probably

caused by the necessity of bonds to be broken in

the viscous flow process. Further, the onset of

the non-Arrhenius region is a direct consequence

of the appearance of a distribution of relaxation

times and/or a distribution of activation energies

(as shown by Tauke, Litovitz, and Macedo [31]).

The appearance of a symmetric distribution of

relaxation times (activation energies) cannot be

explained by existing viscosity theories.

In the annealing range all the theories are in error in that they predict that the viscosity

should go to infinity much faster than experimental data shows ($E/b(l/T) should be an

ever-increasing function and it is not (see fig. 21)).

In fact, B203 viscosity was again Arrhenius at low

temperatures for over five decades. Therefore, one

must conclude that the present theories of viscosity are inadequate, in particular, to fit the viscosity data and, in general, to give physical

significance to the calculation of shear relaxation

and its spectrum times.

A naive, yet better, model for the viscosity of

B2 03 is one similar to that of Ottar and Ruigh

[59] in which the liquid has a "disassociated"

state at high temperature (T>800) in the Arrhenius region. Then, there is an intermediate

region where association takes place, finally,

almost completely associated, the liquid again

becomes Arrhenius. This model has to be made

much more specific in order to explain the distribution of relaxation time behavior, and to be

generalized for other liquids. However, this must

await the more extensive data on the distribution

of relaxation times.

TEMPERATURE -°C X 100

14 II 9 8 7 6 5 4.5 4 3.5 3 26

1001 I I I

1 1 1—I

1 1 1 r~

Ol I I I I I I 6 8 10 12 14 16 18 20

!0_

4 -°K-'

Figure 21. Plot of apparent activation energies versus temperature for B2 03 glass from ref. [1].

5. References

[1] Macedo, P. B., and Napolitano, A., J. Chem. Phys. 49,

1887-95 (1968).

[2] Napolitano, A., Macedo, P. B., and Hawkins, E. G.,

J. Res. NBS 69A (Phys. and Chem.), No. 5, 449-55

(1965).

[3] Napolitano, A., and Hawkins, E. G., J. Res. NBS

68A (Phys. and Chem.), No. 5, 439-448 (1964).

[4] Robinson, H. A., and Peterson, C. A., J. Am. Ceram.

Soc. 27, 129 (1944).

[5] Eppler, R. A., J. Am. Ceram. Soc. 49, No. 12, 579-80

(1966).

[6] Gruber, G., and Litovitz, T. A., J. Chem. Phys. 40,

13-26 (1964).

[7] Tiede, R. L., J. Am. Ceram. Soc. 43, 537 (1959).

[8] Shartsis, L., and Spinner, S., J. Res. NBS 46, No. 3, 176-194 (1951).

[9] Riebling, E., Rev. Sci. Instr. 34, 568-72 (1963).

[10] Kelley, J. E., Roberts, T. D., and Harris, H. M., Bur.

Mines Report No. 6358 (1963).

[11] Boow, J., Phys. Chem. Glasses 8, 45-55 (1967).

[12] Poole, J. P., J. Am. Ceram. Soc. 32, 215 (1949).

[13] Lillie, H. R., J. Am. Ceram. Soc. 14, 502 (1931).

[14] Lewis, W. K., L. Squire, and G. Broughton, Industrial

chemistry of colloidal and amoprhous materials,

p. 24 (MacMillan Co., New York, 1942).

[15] Hagy, H. E., J. Am. Ceram. Soc. 46, 93-97 (1963).

[16] Jones, G. O., J. Soc. Glass Tech. 28, [130] 432 (1944).

[17] Napolitano, A., and Macedo, P. B., J. Res. NBS

72A (Phys. and Chem.), No. 4, 425-33 (1968).

[18] Fontana, E. H., and Plummer, W. A., Phys. Chem.

Glasses 7, 139-146 (1966).

[19] 1966 Book of ASTM Standards Part 13—Softening

point page 305, Annealing and strain points page

295, Average Linear Expansion of Glass page 301.

[20] Saunders, J. B., J. Res. NBS 35, 157-186 (1945) RP

1668.

[21] Napolitano, A., Macedo, P. B., and Hawkins, E. G.,

J. Am. Ceram. Soc. 48, 613-616 (1965).

[22] Riebling, E. F., J. Am. Ceram. Soc. 49, 19-23 (1966).

[23] Mackenzie, J. D., J. Phys. Chem. 63, 1875-78 (1959).

[24] Mackenzie, J. D., Rev. Sci. Instr. 27, 297-99 (1956).

[25] Shartsis, L., Capps, W., and Spinner, S., J. Am.

Ceram. Soc. 36, 35-43 (1953).

[26] Li, P.-C, Ghose, A. C, and Su, G.-J., J. Am. Ceram.

Soc. 45, 89-91 (1962).

[27] Wilhelmy, L., Pogg. Ann. 119, 177-217 (1863).

[28] Macedo, P. B., Capps, W., and Litovitz, T. A., J. Chem Phys. 44, 3357-64 (1966).

[29] Mason, W. P., Physical Acoustics, Vol. I Methods and

Devices; Vol. IIA Properties of Gases Liquids and

Solutions; Vol. IIB Properties of Polymers Glasses

and Nonlinear Acoustics (Academic Press, New

York and London, 1965).

[30] Macedo, P. B., and Litovitz, T. A., J. Phys. Chem.

Glasses 6, 69-80 (1965).

[31] Tauke, J., Litovitz, T. A., and Macedo, P. B., J. Am.

Ceram. Soc. 51, No. 3, 158-163 (1968).

[32] Simmons, J. H., and Macedo, P. B., High temperature shear ultrasonic interferometer using sensitive

phase-lock detection system, J. Acoust. Soc. Am.

43, 1295 (1968).

[33] Bockris, J. O'M., and Kojonen, E., J. Am. Chem. Soc. 82, 4493 (1960).

[34] Macedo, P. B., and Napolitano, A., J. Res. NBS 71A

(Phys. and Chem.), No. 3, 231-38 (1967).

[35] Spinner, S., and Napolitano, A., J. Res. NBS 70A

(Phys. and Chem.), No. 2, 147-152 (1966).

[36] Zijlstra, A. L., Phys. Chem. Glasses 4, 143-151 ( 1963).

[37] Kurkjian, C. R., Phys. Chem. Glasses 4, 128-136

(1963).

[38] Bartenev, G. M., Physics of Non-Crystalline Solids

(Delft Conference, edited by J. A. Prins) (NorthHolland Publ. Corp., Amsterdam, 1964).

[39] Stevels, J. M., Verres et Refractaires 7, 91 (1953);

Glass Ind. 35, 657 (1954).

187
[40] Tarasov, V. V., New Problems in the Physics of Glass

(Moscow, USSR, 1963).

[41] Fulcher, G. S., J. Am. Ceram. Soc. 8, 339-55 (1925).

[42] Napolitano, A., and Hawkins, E. G., NBS Misc.

Publ. 260-11, 24 pp (1966).

[43] Ritland, H. N., J. Am. Ceram. Soc. 39, 403-6 (1956).

[44] Napolitano, A., Macedo, P. B., and Hawkins, E. G.,

J. Am. Ceram. Soc. 48, 613-16 (1965).

[45] Thomas, S. B., and Parks, G. S., J. Phys. Chem. 35,

2091 ( 1931).

[46] Williams, M. L., Landel, R. F., and Ferry, J. D.,

J. Am. Chem. Soc. 77, 3701-07 (1955).

[47] Doolittle, A. K, J. Appl. Phys. 22, 1031-35 (1951);

J. Appl. Phys. 22, 1471-75 (1951); J. Appl. Phys.

23, 236-39 (1952).

[48] Cohen, M. H., and Turnbull, D., J. Chem. Phys. 31,

1164-69 (1959).

[49] Bueche, F., J. Chem. Phys. 30, 748-52 (1959).

[50] Sperry, L. L., and Mackenzie, J. D., Phys. Chem.

of glasses 9, 91 (1968).

[51] Ree, T. S., Ree, T., and Eyring, A., Proc. Natl.

Acad. Sci. U.S. 48, 501-16 (1962).

[52] Macedo, P. B., and Litovitz, T. A., J. Chem. Phys. 42,

245-56 (1965).

[53] Matheson, A. J., J. Chem. Phys. 44, 695-699 (1966).

[54] Macedo, P. B., Capps, W., and Litovitz, T. A., J. Chem. Phys. 44, 3357-64 (1966).

[55] Dane, E. B., and Birch, F., J. Appl. Phys. 9, 669-74

(1938).

[56] Adam, G., and Gibbs, J. H., J. Chem. Phys. 43,

139-146 (1965).

[57] Araujo, R. J., J. Chem. Phys. 44, 1299-1300 (1966).

[58] Kelley, K. K., U.S. Bur. of Mines Bull. 584 (1960),

Contributions to the data on theoretical metallurgy,

XIII. High temperature heat-content, heat capacity, and entropy data for the elements and inorganic

compounds.

[59] Ottar, B., and Ruigh, W. L., Phys. Chem. Glasses 3,

95-8 (1962).

188
Inelastic Deformation of Oxide Ceramics

Richard M. Spriggs*

Physical Ceramics Laboratory, Materials Research Center, Lehigh University,

Bethlehem, Pennsylvania 18015

The current understanding of the inelastic deformation of oxide ceramics is reviewed.

Special emphasis is given to the permanent deformation caused by stress-enhanced diffusional

creep, dislocation motion, and grain-boundary sliding. Brief reference is also made to recoverable inelastic deformation caused by point defect motion, dislocations, and grain boundary

sliding. It is recognized that the observed permanent deformation is usually the result of more than one process, with complicating interactions frequently occurring, and with the

relative contribution of each mechanism depending upon a host of structural, compositional,

and experimental factors. Detailed information is available for only a few oxide ceramics

and the need for additional research is evident.

Key Words: Anelasticity; creep; deformation; dislocations; grain boundaries;

inelastic (plastic); oxide ceramics; point defects.

1. Introduction

When ceramic materials are subjected to a

stress at ambient or elevated temperatures, they

respond by deforming. This deformation frequently is the result of more than one deformation

process. Furthermore, interactions among simultaneous processes often occur, making it difficult

to assess quantitatively the effects of each process.

However, the individual processes can frequently

be separated or indentified and qualitatively or semiquantatively evaluated.

The deformation experienced by a ceramic material may be elastic and hence recoverable (i.e.,

strain which is fully and instantaneously recovered

when the stress is removed). Alternatively, the

deformation may be inelastic and may or may not

be recoverable. Recoverable inelastic deformation can be considered to be time-dependent strain

which is fully recovered when the external load

is removed, even though recovery may take infinite time, i.e., observations over experimentally

practical times indicate complete recovery upon

extrapolation [1] Nonrecoverable inelastic deformation, on the other hand, is permanent strain or deformation which is not recovered in any length

of time when the load is removed ; it is frequently

referred to as plastic deformation.

The subject of the present discussion is re- stricted to inelastic deformation, primarily the

permanent deformation caused by phenomena

such as the introduction of microcracks, disloca-

*The author is professor of metallurgy and materials science, director of the Physical Ceramics Laboratory, and associate director, Materials Research Center, Lehigh University, Bethlehem, Pennsylvania 18015.

1 Figures in brackets indicate the literature references at the end of this paper.

tion motion, grain boundary sliding, and stress

enhanced diffusional creep. Furthermore, it deals

largely with oxide ceramics. Brief reference is also

made to recoverable inelastic deformation. The

reader is referred to other papers in this Symposium

for a complete description of elastic deformation

of crystalline ceramics and other refractory materials [2], and inelastic deformation of non-oxide

ceramics [3]. An unusually complete introductory

survey by Wachtman [1] on the mechanical

properties of ceramics serves as a starting point

for much of the following discussion.

2. Recoverable Inelastic Deformation

Wachtman [1] cites several important examples

of recoverable inelastic deformation which_ are summarized below. Included are anelasticity

caused by point defect motion, anelasticity in

amorphous glasses, and recoverable inelastic deformation caused by either dislocations or grain

boundary sliding. In each of these instances, it is

important to keep in mind that recoverable inelasticity implies that when a load is applied to a specimen, the free energy is increased in such a way that when the load is removed the deformation is eventually fully recovered as the free energy

returns to a minimum under the action of thermal

vibrations [1].

2.1. Anelasticity Caused by Point Defect

Motion

Small inelastic deformation which is linear in

the stress, i.e., anelasticity, is produced by the

189
stress-driven motion of point defects such as

interstitial ions or vacancies. Anelastic deformation is usually and more easily studied in dynamic

experiments in which the stress varies periodically

with time and the internal friction (fractional

energy dissipated per cycle) is measured.

While most of the work on anelasticity caused

by point defects has been conducted with metals,

some work has been done on ceramics and such

studies promise to be quite useful in identifying

types of point defects in ceramic crystals and

measuring their jump rates [1]. Wachtman [1]

cites work which has been reported on MgO [4, 5],

Zr02 [6, 7], Th02 [8], Ti02 [9-11], CaF2 [12], and

Si02 [13]. Under certain circumstances, electronic

motion can also contribute to anelasticity in

ceramics, as in lithium-doped NiO semiconductors

[14]. In addition, typical internal friction values

have been tabulated for commercial ceramics [15];

such values tend to be variable, however, and

depend upon factors such as trace impurities,

phase changes, etc. [1].

2.2. Anelasticity in Amorphous Glasses

According to Wachtman [1], internal friction

measurements in glass may show either the motion

of alkali ions or the viscoelastic behavior of the

glass network, or even a combination of the two

effects [16, 17]. The motion of alkali ions gives

rise to a pronounced "elastic after effect" in

chemically strengthened glass; an anelastic strain

equal to 11 percent of the elastic strain has been

observed after 15 min at 25 C [18]. The viscous

relaxation of the glass network itself at high

temperatures gives rise to large differences in

elastic moduli measured at frequencies low enough

to permit relaxation [1]. For B203 at 800 C, for

example, the longitudinal modulus at high frequency is three times the low-frequency value [19].

2.3. Recoverable Inelastic Deformation Caused

by Dislocations

The movement of dislocations is the most important mechanism of inelastic deformation in

metals at room temperature and in many glassfree ceramics of large grain size at high temperatures [1]. Dislocation mobility is thus very important to an understanding of the deformation

of ceramics and depends strongly on the presence

of other imperfections such as point defects, precipitates, and other dislocations [20].

Under lower stresses, segments of pinned dislocation bow outward while the ends remain

attached to the pinning points. At greater stresses,

the dislocation breaks away from some of the

pinning points. With the former situation, where

bowing only occurs, the dislocations return to

their original positions when the stress is removed

this inelastic, recoverable deformation gives rise to a frequency-dependent but amplitude-independent

internal friction. By contrast, the latter occurrence,

where dislocations break away, produces amplitude-dependent internal friction [1]. Dislocation

mobility can be related to measurable internal

friction by theory [21-23], but other factors such

as dislocation loop length and dislocation density

are also involved.

Chang [24] analyzed data for MgO and A12 03 , but Carnahan and Brittain [25] found that for

Ti02 (rutile) the experimental results showed

qualitative disagreement with the theory, as was

also found for MgO [26] and Cu-Ge alloys [27].

Wachtman [1] concluded that while the study of

internal friction caused by dislocations in ceramic

oxides is a promising area, the analysis is not yet

completely reliable.

The ordering of point defects in the stress field

of a moving dislocation should represent still another source of inelastic deformation in ceramics,

but apparently has not yet been investigated [1].

2.4. Recoverable Inelastic Deformation Caused

by Grain Boundary Sliding

Inelastic deformation due to grain boundary

sliding appears to be a general phenomenon in

polycrystalline ceramics [1], and the role played

by impurities is probably of critical importance.

However, grain boundary properties and their

effect on the deformation behavior of ceramics are not well understood at present. The frequent occurrence of concurrent recoverable and nonrecoverable phenomena tends to make it difficult

to separate effects. Some inelastic deformation

effects attributable to grain boundary sliding,

however, have been cited by Wachtman [1]. A rapid decrease in Young's modulus of polycrystalline alumina above 950 C, accompanied by

an increase in internal friction over the single

crystal values, has been attributed to grain

boundary sliding [28]. Other studies of alumina

show qualitatively the same effects [29-31], with

increasing purity increasing the temperatures at

which these effects occur by as much as 400 C.

Thus the introduction of grain boundaries appears

to raise the damping capacity (i.e., internal

friction) in a certain characteristic temperature

range; this range is broadened and usually ex- tended to lower temperatures by the addition of

impurities.

Creep measurements with Cr203-doped alumina

at 1000 to 1200 C have revealed decreasing creep

rates which were slowly recovered when the load

was removed [32, 33]. This recovery was attributed

to viscous grain boundary sliding accompanied by

elastic deformation of the grains. The elastically

deformed grains reversed the grain boundary

sliding when the external load was removed.

Higher stresses, higher temperatures, or longer

times permit plastic deformation within grains, so that only part of the inelastic strain can then be

recovered [1].

190
3. Nonrecoverable Inelastic

Deformation

The deformation of a ceramic material may, in

certain circumstances, be completely recoverable

(e.g., a small amount of grain boundary sliding

accompanied by elastic deformation), partly re- coverable, or completely plastic, depending upon

the mechanisms operating [1]. The latter behavior

is of particular interest in this section, especially

creep deformation under constant load.

At elevated temperatures, two factors become

important to change the deformation behavior of

ceramics. First, dislocations become more mobile

on existing slip systems and new slip systems

become possible, and second, the diffusion of

point defects is enhanced to the point where they

begin to contribute to the deformation [34].

Unfortunately, the temperature range where a

multiplicity of slip systems becomes available in

most ceramic oxides is so high that additional

complications come into play and the occurrence

of nonrecoverable inelastic deformation by the

simple movement of dislocations is never observed

[34]. Elevated temperature deformation is always

enhanced by the diffusion of point defects and by

the onset of viscous sliding at grain boundaries.

The diffusion of vacancies (or interstitial ions)

contributes to each of the three basic mechanisms

considered responsible for creep: vacancy migration under stress, dislocation motion, and

grain boundary sliding. (The first and third processes, it should be noted, apply specifically to

polycrystalline materials.) In the sections that

follow, each of these mechanisms is discussed.

Additional factors are considered, including the

viscous deformation of glass and combined

deformation processes.

The mechanisms responsible for deformation

depend upon a host of structural and compositional

factors as well as stress, temperature, and other

test parameters. Structural variables include

crystallography, grain size, subgrain structure,

and porosity. Insofar as possible, the dependence

of deformation on each of these factors is reviewed.

3.1 Vacancy Migration Under Stress

(Diffusional Creep)

In vacancy migration under stress, deformation

results from diffusional flow within each grain

away from those boundaries where there is a high local compressive stress toward those boundaries

having a high local tensile stress. This is generally

referred to as Nabarro-Herring creep [35, 36] and

has been found to be the primary deformation

process for many ceramics at high temperatures,

relatively low stresses, and fine grain sizes, with

a temperature dependence similar to that for

volume diffusion.

In Nabarro-Herring creep, self-diffusion within

the grains of a polycrystal^ne solid permits the

solid to deform under a shear stress, leading to

viscous flow with the diffusional viscosity, -q, given by:

where k is Boltzmann's constant, T is the absolute

temperature, r is the average grain radius, Dl is the bulk self-diffusion constant, and 12 is the

volume of the diffusing species. The diffusion

coefficient, Dlt is exponentially related to temperature as follows:

D, =D0 exp ( - AH/R T). (2)

Under a tensile stres, a, the tensile creep rate

is given in terms of the average grain diameter,

GS, by [37]:

€= 13. ZaDfr/kTiGSy. (3)

In a similar mechanism proposed by Coble

[38], the changes in grain shape are attributed to

grain boundary diffusion rather than lattice diffusion, and the creep rate is described by:

t=±7*DbwQ/kT(GS)3

, (4)

where Db is the diffusion coefficient for the slowest

moving species in the grain boundary and w is

the width of the grain boundary.

In a very recent brief review of diffusional creep

mechanisms, Gifkins [39] compared both NabarroHerring creep and Coble creep with the suggestion that creep can occur by grain boundary

sliding, also controlled by diffusion. Gifkins creep

leads to movement along the boundary of grainboundary protrusions and can be accommodated

by diffusion around triple points. The original

diffusion-controlled mechanism for grain-boundary

sliding was proposed by Gifkins and Snowden

[40] and is based on the idea of the movement of

double ledges or protrusions along the boundary.

The rate of sliding, for a grain boundary at

45 deg to the applied stress is then:

Sb~aaDbQ/LkT, (5)

where a is a geometrical constant, approximately

equal to 2; L is the length of the grain boundary

protrusion (~10-6 cm in tungsten [31]); and a, Db , fi, k, and T have their former meanings.

The predicted creep rates, utilizing eq (5), have

been 100 times too fast to match certain experimental results [42]; this has been attributed to

factors which prevented continuous sliding on all boundaries. An equation for the accommodation

of sliding by diffusion around triple points, one

possible reason for sliding pauses, has also been

derived by Gifkins [43]; the rate of accommodation, Sa , for a boundary at 45° to the applied

stress, is:Sa~40QDb <Tw/(GS) 2kT, (6)

191
where w is the grain boundary width and GS is the

grain diameter.

By combining the mechanisms represented by

eq (5) and (6) with some grain boundary migration,

Gifkins envisions creep extension which occurs

entirely by diffusional mechanisms but which

does not result in permanent change of grain shape

[39], as is required by Nabarro-Herring and Coble

creep. The predominating mechanism would

depend upon grain size; coarse grain sizes would be

controlled by accommodation (eq (6)), while fine

grain sizes would be controlled by sliding (eq (5)).

To compare eqs (5) and (6) with NabarroHerring creep and Coble creep, Gifkins [39] makes

certain further approximations and assumptions

and derives the proportionalities between creep

rate and stress, temperature, grain size, and numerical constants shown in table 1. The suggestion by Gifkins [39] of diffusioncontrolled grain-boundary-sliding-type creep is too recent to have been critically examined in

relation to the creep of polycrystalline ceramic

oxides. The most promising evidence offered to

date has been the very recent work of Hensler

and Cullen [42] on grain shape change during the

creep of 98 percent dense MgO at 1200 to 1500 C

and 1000 to 6000 psi. Hensler and Cullen found

no change in grain shape after compressive creep

deformation up to 30 to 44 percent compression;

these results where qualitatively interpreted in

terms of Gifkins grain boundary sliding creep [39].

As previously mentioned, Nabarro-Herring

intragranular diffusional creep has been found to

be the primary deformation process for many

ceramics at high temperatures, relatively low

stresses, and fine grain sizes; eq (3) has been

confirmed for A1203 [45-49], MgO [50], BeO [51-

53], U02-BeO mixtures [54], and NaCl [55].

For A1203 , eq (3) is obeyed using Dt for aluminum

in aluminum oxide despite the fact that D, for

oxygen is smaller and should be rate controlling;

this is thought to be due to enhanced diffusion of

oxygen along the grain boundaries [56].

A brief summary of the major findings concerning AI2O3, BeO, MgO, and SiC is presented below.

a. Creep of Alumina

As cited by Vasilos and Passmore [57], the available creep data at high temperatures for polycrystalline alumina generally support the Nabarro

Herring diffusional creep model over a range of

stress (100 to 4,000 psi) and temperature (1200 to

1800 C). Studies of high-density, fine-grained

AI2O3 (LUCALOX) by Warshaw and Norton

[46] and Folweiler [45] illustrate this point. Both

found the creep rate to vary linearly with stress

between 1500 and 1800 C, and the activation

energy was measured to be 130 kcal/mole. Similarly, Chang [58] also found a viscous behavior,

although with a higher activation energy (200

kcal/mole) in this temperature range.

Recently, Passmore et al. [49, 59] determined

the variation of creep rate in alumina with stress

and temperature by using the incremental stress

and temperature method introduced by Dorn [60]

and illustrated in figure 1. Specimens were first

crept to a strain of 0.32 percent at the relatively

high stress of 9000 psi ±50 psi; the stress was

then decreased to a lower level. After further creep

at constant temperature and the lower stress, the

3 00

rr1

rrLd

CL

i?

•<j/

Ld

crz

<

cr

CLLd

Ld

CCO

1 1 1 r- 1 r r

0.30

0.20

0.15

O.O

o.oe

0.06

0.05

cr 8.950

II

AH = 184.000 CALS/MOL

1H= V.

159.500 CALS/MOL

175.400 CALS/MOL

A H 1 170.000 CALS'MOL

AH = 174.200 £ALS^MOL

1 cr.- 1 .990 psi -1 I50t '1200'C-J—I250X—+C006^— laSO'C-rGOOti

Al2°3 SPECIMEN NO I23T-I

0.0 0.2 0.4 0.6 OB 1.0 1.2 14 16 1.8 2.0 2.2 CREEP STRAIN.%

Figure 1. Typical creep rate-creep strain curve used in the evaluation of the stress and temperature dependence of

creep for pure, dense, fine-grained aluminum oxide.

(After Vasilos and Passmore. Courtesy of Syracuse University Press)

Table 1. Dependence of creep rate (e) on variables in equations based on diffusion-controlled creep mechanisms [39]

Creep type Mechanism characteristic Stress Temp.

Grain size Numerical

constants

(A)i (B)2

Nabarro-Herring [36] Grain elongation. <r

a

a

a

D,Db Db D„

(GS)-2

(GS)-3

(GS)-i (GS)-2

(GS)-'

13.3

3 i7w

«2/x

40w

Coble [38] Grain elongation _ _

Gifkins-Snowden [40]

Gifkins [43]

Grain-boundary sliding; some grain elongation

Grain-boundary sliding, accommodated; no grain elongation

"Assuming L in eq (5), grain boundary protrusion length, is independent of (OS) for Gifkins-Snowden creep. 'Assuming L depends on (OS).

3 w (grain boundary width)~:10-7 cm.

4 x is a factor introduced to account for pauses other than the necessity for triple point accommodation, and may have a value of

-100.

192
temperature was rapidly changed to a new constant level. This process was repeated several

times with periods of creep at constant temperature being alternated with rapid temperature

changes. This procedure yields several estimates

of the activation energy for creep. Rapid stress

changes at constant temperature can likewise be

employed to determine the creep rate dependency

on stress. In contrast to the observation of Warshaw and

Norton [46], Passmore et al. [49, 59] observed no

region of steady-state creep. As indicated in

figure 1, the creep rate continued to decrease with

increasing creep strain up to strains of 1.5 to 2.0

percent. Warshaw and Norton, on the other hand,

reported constant, steady-state creep in bending

at strains of 0.2 percent and below. Vasilos and

Passmore [57] concluded that the different creep

rate-strain dependences may have resulted from

the different temperatures employed. At the high

temperatures (1600 to 1800 C) used by Warshaw

and Norton, recovery processes may have been

rapid enough to offset any strain hardening,

whereas the latter process predominated at the

lower temperatures used by Passmore et al. Creep rates determined graphically at a strain

of 0.32 percent are shown as a function of stress

(OOO 2000 3000 4000 6000 9000

STRESS. PSI

Figure 2. Effect of stress on the temperature-compensated

creep rate of dense, fine-grained alumina at constant strain

of 0.32 percent.

(After Vasilos and Passmore. Courtesy of Syracuse University Press)

in the range of 1,000 to 9,000 psi in figure 2. After

Passmore et al., creep rates for the two temperatures used, 1150 and 1250 C, are expressed in

terms of the temperature-compensated creep

rate, commonly called the Zener-Hollomon parameter, Z=e exp(AH/RT), using a value of 141

kcal/mole for AH. At low stresses (up to about

2000 psi), a linear relation between creep rate

and stress was observed. At higher stresses, howeve,, the creep rate dependence of stress increased

to a square dependence, which suggests a transition from predominantly viscous creep mechanisms of the Nabarro-Herring type to a more

complex one, probably involving plastic flow. A

nonlinear stress dependence (eoco-4

) was also

observed by Warshaw and Norton [46] in the

650 to 1,300 psi stress range for coarse-grained

(50 to 100 yum) alumina. The results of Passmore,

et al. [49, 59] appear to be consistent with the

interpretation of Warshaw and Norton and are considered more definitive, because possible effects

of variations in substructure and intergranular

separation were minimized or eliminated by

carrying out the creep-rate experiments at low

stresses immediately after creep at a constant

level of higher stress (9000 psi) to a constant

creep strain (0.32 percent). Vasilos and Passmore

[57] thus believed that the enhanced creep rates

observed above 2,000 psi by Passmore et al. were,

in fact, indicative of a transition in the creep

mechanism.

The Nabarro-Herring creep mechanism (eq

(3)) predicts an inverse proportionality between

creep rate and grain size, i.e., iccl/(6S) 2

. This has

been confirmed by several investigators, the first

for alumina being Folweiler [45]. Thus, largegrained material has a higher creep resistance

than fine-grained material.

Table 2 summarizes the high-temperature

deformation behavior of pure and doped polycrystalline alumina with regard to temperature

range, rate-controlling mechanism, and activation

energy [57].

Several investigators have compared diffusion

coefficients calculated from creep rates using eq

(3) 'with directly measured diffusion coefficients,

but there has been no general agreement. The

creep-diffusion coefficients have generally been

greater than self-diffusion values, although Paladino and Coble [56] have reported very good

agreement with aluminum ion diffusion, which is the faster moving species. Enhanced diffusion of

oxygen along grain boundaries has been invoked

to explain this finding [56].

b. Creep of Beryllia

Chang [51] reported good agreement between

creep deformation rate and measured diffusion

coefficients for BeO, thus supporting the diffusional

creep model. In further work on the compressive

creep behavior of beryllia, Vandervoort and

Barmore [52] also found that the creep rate was

193
Table 2. High temperature deformation and creep of polycrystalline alumina (after Vasilos and Passmore)

Result No.

Material Temperature

range

Rate-controlling deformation mechanism Energy Investigators

1 High-density Lucalox, S-lO/im grain size.

2... High-density Lucalox, 5-35^m GS

3 97% dense, 50-100>m GS

4 High-density, 100/im GS

5 Same as 4 plus Crj03

6 Polycrystalline

7 Same as 6 plus 1% CraCM

8 Same as 6 plus 1% L.j03'_

9 99% Dense, 2 /im GS

10 Same as 9 plus 1 m/o CrzCh2

Centigrade kcallmole 1600-1800 Vacancy migration 130

1600-1800 Vacancy migration 130

1600-1800 Grain-boundary sliding 185

1750-1900 Grain-boundary sliding 280

1750-1900 Grain-boundary sliding 280

1400-1700 Vacancy migration plus grain-boundary sliding 200

1400-1700 Vacancy migration plus grain-boundary sliding 200

1400-1700 Vacancy migration plus grain-boundary sliding 200

1200-1375 Vacancy migration 154

1300-1376 Vacancy migration and formation 151

Warshaw and Norton [46].

Folweiler [45]. Warshaw and Norton [46].

Coble and Guerard [47].

Coble and Guerard [47]. Chang [51]. Chang [51]. Chang [51].

Passmore et al. [49 , 59].

Passmore et al. [49, 59].

1 o/o=weight percent.

2 m/o =mole percent.

linearly dependent on the applied stress (up to

6,000 psi) over the temperature range of 1370 to

1510 C, with an apparent creep activation energy

of 96 kcal/mole. In addition, a comparison of

calculated diffusion coefficients with directlymeasured beryllium ion coefficients revealed fair

agreement. In work of Fryxell and Chandler [53]

on the compressive creep of BeO, a linear relation

was also found between stress and creep rate up

to a stress of 6,000 psi at 1200 C. Above 6,000 psi,

however, the creep rate depended upon stress

raised to a power greater than unity. In addition,

for small-grained specimens, they observed a

creep rate which decreased with both time and

strain considerably more than for larger (10 fim

and larger) grain sized material. These latter

observations are consistent with the previously

mentioned findings of Passmore and Vasilos for

fine-grained alumina.

c. Creep of Magnesia

As is the case with alumina and beryllia, the

available creep data for polycrystalline magnesia

at low stresses also generally support a diffusional

creep model [50, 61]. Perhaps because of the more

plastic behavior of magnesia, however, the re- sults for magnesia are more complicated, with a creep activation energy which is not invariant

with grain size, a grain size dependence which is intermediate between the Nabarro-Herring and

Coble creep models, and a change in the stress- creep rate dependency from unity (viscous creep)

to 1.5 below 5 jtm grain size.

Passmore et al. [61], in a comprehensive investigation, recently determined the creep of

MgO over a range of temperature, stress, and

grain size. A typical creep rate-strain curve for

dense, fine-grained (2 to 3 jum) magnesia is shown in

figure 3 [61]. The creep behavior of MgO is similar

to other polycrystalline oxides in that the strain

rate decreases with increasing strain. Typical

changes in creep rate corresponding to changes in

temperature, are also shown in figure 3. Vasilos

and Passmore [57] attributed the changes in

creep rate in figure 3 solely to temperature

changes, since stress was maintained constant

and substructural effects were eliminated by

extrapolating creep rates to the same value of

strain for each temperature.

The activation energy values from Passmore'

s

work [61] are plotted versus grain size in figure 4

for grain sizes from 2 to 20 fim. As seen, AH at

first decreases sharply with increasing grain size

from about 96 kcal/mole for a grain size of 2^im

to 54.1 kcal/mole at 5.5Atm and then remains

constant with further grain size increases. Passmore et al. [61] observed no variations in AH

with stress, strain, or temperature. The observed

energy of 54.1 kcal/mole for the larger grain sizes

was held to be consistent with a diffusional creep

mechanism in which the creep rate is controlled

by extrinsic oxygen ion diffusion (Ai?=62.4

kcal/mole according to Oishi and Kingery [62]).

For grain sizes below 5.5jum, the increasing AH

may indicate a transition to control by intrinsic

magnesium ion diffusion (AH=70 kcal/mole ac- cording to Lindner and Parfitt [63]).

Passmore et al. [61] observed a linear relationship between creep rate and stress for grain sizes

of 5.5 to 20jum. There was, however, a transition

to a higher-order stress function, i.e., from 1.0 to

1.5, with decreasing grain size, which Passmore

took as further evidence of a transition in creep

mechanism for the finer grain sizes.

The grain size dependence of the creep rate

below 5.5Mm could not be determined [61] be- cause of the variations in AH with grain size, which

made it impossible to compensate the creep rates

satisfactorily for variations in temperature over

a significant range of grain sizes. The observed

grain size dependence for the range from 5.5 to

20 jum was described by an exponent of —2.5

(i.e., iocl/(GS) 2 - 5), which is intermediate between

those predicted by the Nabarro-Herring and

Coble models for diffusional creep.

194
0.000 0.002 0.004 0.006 0.008 0.010 0,0i2 0.014 0.016 CREEP STRAIN.

€

Figure 3. Typical creep rate-creep strain curve for dense, fine-grained

{2 to 3 fim) magnesium oxide.

(After Vasilos and Passmore. Courtesy of Syracuse University Press)

100

90

UJ

_iO

<U 80

i

<1

I

70

UJz

2 60 <

>

50 -

40

5 10 15 GRAIN SIZE, MICRONS20

Figure 4. Effect of grain size on the activation energy

for creep of dense magnesia.

(After Vasilos and Passmore. Courtesy of Syracuse University Press)

Impurities such as 1 m/o Zr02 or CaO added

to the MgO had no significant effects on creep

rate but did increase the activation energy by

20 percent [57].

Passmore et al. [61] attributed creep in MgO

in part to a stress-directed diffusional mechanism

controlled by extrinsic oxygen ion diffusion in the

5.5 to 20 nm grain size range, although the calculated diffusion coefficients were higher than those

predicted by the Nabarro-Herring model. It was

suggested that this discrepancy may be due to a vacancy formation mechanism, which was consistent with the observed formation of dislocation

substructure and preferentially distributed porosity during creep, as well as with the observed

decrease in creep rate with increasing creep

strain [61].

d. Creep of Silicon Carbide

Farnsworth and Coble [37] concluded that

diffusional creep was also the most probable ratecontrolling mechanism for the deformation of

polycrystalline silicon carbide. Viscous flow was

observed in dense SiC at temperatures from

1900 to 2200 C, at stresses from 3,000 to 30,000

psi, and at strain rates up to 10_3/min.

3.2. Creep by Dislocation Motion

The deformation of polycrystalline ceramic

oxides by plastic flow processes is less common

than deformation by stress-directed diffusion.

This fact results largely from the lack of sufficient

slip modes in most polycrystalline oxides. Taylor

[64] has shown that for plastic deformation to occur by dislocation movement in polycrystalline

bodies, slip must occur on five independent slip

systems to allow a general change in shape of

individual grains without void formation. While

metals, in general, meet the foregoing criterion,

many ceramics do not, especially at reasonable

temperatures. For example, Groves and Kelly

[65] have examined the observed crystallographic

slip systems for a number of common ceramic

materials; their results are shown in table 3.

195
It is concluded that none of the common ceramics

is capable of a great amount of plastic deformation

at low temperatures, although at elevated temperatures materials with the sodium chloride and

calcium fluoride structures should possess the

potential for such deformation. At brief summary

of results for oxides such as magnesia and alumina

is given below.

Table 3. Number of independent slip systems for some common ceramic crystal structures [after Groves & i\ elly]

Structure Crystallographic Number of

slip systems independent

systems

NaCl (low temps.) (e.g., MgO) (110) <ilo> 2 NaCl (high temps.) (e.g., MgO). (110) <110> and 5

(0011 <110>

Hexagonal (e.g., AI2O3, graphite). (0001|<1120> 2 CsCl 1100) <010> 3 CaF2 (low temps.) (e.g., UO2) (001) <110> 3 CaF2 (high temps.) (001) <110> and 5

(110) <110>

Ti0 2 _ (101) <101> and 4

(110) <001>

a. Magnesia

As cited by Vasilos and Passmore [57], the

requirements for plastic flow in magnesia are relatively severe as compared to simple halides. The

stress required to promote slip in MgO on the

{001} <110> system at room temperature is

approximately 100 times higher than that to cause

slip on the {110} <110> system, and extremely

high stresses are required even for local deformation. Therefore, polycrystalline magnesia exhibits

virtually no plastic deformation at room temperature, even though dislocations are quite mobile

and equivalent single crystals are ductile. The

difference in stress level for slip on the two sets of

systems decreases with increasing temperature,

and at 1000 C is down to a factor of 10 (equivalent to LiF at room temperature) . A slight amount

of plastic deformation in MgO has been observed

by Copley and Pask [66] in compression at temperatures up to 1000 C. At room temperature,

the strain at fracture was about 0.6 percent for

polycrystalline MgO and about 2 percent above

600 C [66].

Slip in. magnesia also tends to be nonuniform.

Slip bands are usually few and randomly spaced

before yielding at room temperature. Further

straining causes the widening of these already

existing slip bands, rather than the appearance

of new ones. Such slip localization into wide bands

causes stress concentration and the formation and

growth of crack nuclei where slip bands intersect

grain boundaries.

An important additional consequence of non- uniform slip is that a grain cannot change shape

as required. For such a change to be possible,

slip must occur on closely spaced planes, the

amount of shear varying continuously from one

phase to the next. With nonuniform slip, therefore, large elastic strains necessarily occur in the

grains of a deformed polycrystalline specimen.

Such strains, again, favor the growth of crack

nuclei and help propogate cleavage fracture [57].

As the temperature is increased, conditions

become less favorable for cleavage fracture to occur. Self-diffusion becomes more rapid and, as a

result, stress-induced climb can occur, increasing

the difficulty of nucleating cracks. Dislocations

also become increasingly mobile at higher temperatures and stress relaxation by plastic deformation becomes more important [57].

Day and Stokes [67] have shown that plastic

deformation of polycrystalline magnesia under

tension is possible only when the temperature

exceeds 1700 C. Below 1700 C, an insufficient

number of the available independent slip systems

are capable of interpenetrating and operating concurrently within a given volume of material to

satisfy the Taylor- Groves-Kelly criteria for deformation of polycrystalline bodies [67]. Therefore,

grain boundary constraints arise which, when re- laxed by intergranular sliding, lead to intergranular flaws and eventually intergranular rupture.

Day and Stokes [67] observed a slight amount

of slip below 1700 C in the vicinity of the fracture.

Passmore, et al. [61] also observed a slight amount

of slip at temperatures near 1400 C in bending

creep studies. Above 1700 C, Day and Stokes [67]

concluded that sufficient independent slip systems

were operating concurrently for continuity to be

maintained between adjacent grains. The additional ability of the material to polygonize and

recrystallize permitted a continuous relaxation of

internal stresses and contributed to both the

overall plasticity and change of shape under

tension [67]. Deformation was found to proceed in

fully dense specimens without intergranular separation and fracture occurred in a ductile manner

[67]. In the presence of porosity, however, intergranular rupture was observed at temperatures as high as 2100 C, although plastic deformation

occurred at temperatures below 1900 C [67].

b. Alumina

It has been considered for some time that

polycrystalline alumina creeps by a diffusioncontrolled mechanism rather than by dislocation

motion, as a consequence of the limited number

of independent slip systems available. It now

appears that at temperature above 1400 C, the

creep deformation of alumina is caused by a composite mechanism consisting partly of stressdirected diffusion, grain-boundary sliding and

dislocation motion [1, 571. As long as basal slip

predominates (cf. table 3), there are only two

independent slip systems in alumina, and even

taking into account the additional flexibility

afforded by dislocation climb, polycrystalline

alumina should not be able to deform by slip

without the generation of intergranular constraints [57]. Several experimental results, however, show that extensive inelastic nonrecoverable

deformation is possible with dense, fine-gain

material.

196
Plastic deformation, with associated strain

hardening, has possibly been observed by Passmore, et al. [57], in the range of 1300 C to 1400 C.

Furthermore, Vasilos and coAvorkers have been

able to extensively deform polycrystalline alumina

at temperatures above 1800 C by press forging

with a heavy resulting crystallographic texture.

These observations, coupled with the recent discovery of [1011] dislocations, suggest that limited

to extensive ductility is possible in alumina at very

high temperatures. Contributions due to diffusional creep, grain-boundary sliding, and dislocation motion all appear likely, with the relative

contribution of each changing with changing

experimental conditions.

c. Steady- State Creep by Dislocation Motion

In addition to the foregoing qualitative discussions concerning the role of dislocation motion

in the creep of polycrystalline ceramics, a number

of theories have been advanced to describe the

steady-state creep of crystals by dislocation

motion. As opposed to diffusional creep and

grain boundary sliding, it should be noted that

these dislocation motion theories do not apply

specifically to polycrystalline materials and do

not contain explicit grain size terms. Limited use

of these theories has been made, however, as described below, to explain certain creep behaviors

of polycrystalline material.

The most familiar of the so-called "microdynamical" dislocation motion theories of plasticity is that due to Weertman [68, 69], and relates

to dislocation climb. Dislocation climb occurs when vacant lattice sites condense on or escape

from edge dislocations, causing them to be

displaced vertically in a direction perpendicular

to the slip plane [34]. The climb process permits

unlike dislocations to annihilate and also permits

dislocations to detour around obstacles in the slip

plane. Thus, dislocations trapped in a pileup

against obstacles such as subgrain or grain

boundaries can escape by climb and rearrange

themselves; the relaxation of the back stress

enables the dislocation source to resume operation

and deformation continues [34]. When the rate

of dispersion of dislocations is approximately

equal to the rate at which they are pumped into

the pileup, a steady-state creep rate is established.

The rate-controlling step in this process is vacancy

diffusion, a fact which, according to Stokes [34],

has been well established for all metals for which

accurate data are available [60].

In a detailed analysis of the dislocation climb

mechanism, Weertman [68, 69] has shown that

the creep rate (e) varies with the applied stress

(a) and temperature (T) according to the following relationship [34]:

where U is generally the activation energy for

self diffusion. In ceramics, the presence of two

ion species having different self-diffusion activation energies, and the possibility that diffusion

may occur in pairs, makes the interpretation of

U more difficult [34].

It is seen in eq (7) that the creep rate depends

on a high power of the stress (4.5) in contrast

to the first powder dependence of diffusional

creep (eqs (3), (4), (5), and (6)).

A theory similar to Weertman's, but due to

McLean, also involving dislocation climb, has

been applied to Ti02 (rutile) [70] and gave a creep rate dependence on stress to the 1.7 power

Chang [71] reviewed creep data for single

crystals and large grain size polycrystals of UC,

TiC, AI2O3, and BeO, all of which give creep

rates proportional to the stress raised to a powder

from 4 to 6; Chang proposed a theory in which

the creep rate is controlled by the dissolution by

diffusion of debris left behind by moving dislocation [1]. Chang's theory gives a dependence of creep

rate according to the fourth or fifth power of stress,

depending upon which mechanism of dissolution

is assumed. Weertman [72] has also derived creep

rate equations for the case where the motion of

individual dislocations is rate controlling. Other

recent work on creep equations derived from

dislocation dynamics models is also cited by

Wachtman [1]. Stokes [34] outlined the work of Rogers et al.,

Kronberg, and Chang on the deformation of

single-crystal alumina at elevated temperatures

which has been related to dislocation motion.

Briefly, in the temperature range of 900 to

1700 C, basal slip by simple slip on the basal

plane, with an activation energy of 85 kcal/

mole, appears to be the rate-controlling mechanism, whereas dislocation climb, with an activation energy of 180 kcal/mole and a stress dependence of 4.5, appears to be rate controlling at

1800 C for single crystals of alumina.

3.3. Creep by Grain Boundary Sliding

Grain boundary sliding has been inferred from

internal friction measurements and dynamic

elastic modulus measurements at high temperatures [1]. Very little direct evidence exists,

however, of the quantitative nature of grain

boundary sliding in ceramics. It has usually

been thought of as a separate process, due possibly

to either the presence of an impure glassy phase

or controlled by vacancy diffusion or even dislocation motion through the disordered boundary

region [34]. Stokes [34] cites that the creep rate

due to this process should be roughly proportional

to the applied stress and inversely proportional

to the grain size, i.e.,

eoca/GS. (8)

197
On the other hand, Dorn [60] has shown that

grain-boundary creep in metals is not a distinct

mechanism but should follow the same stress

and temperature relationships as intragranular

creep. The recent efforts of Gifkins [39] to describe a stress-directed diffusional creep mechanism

for grain boundary sliding have also been described in an earlier section (3.1).

Prior to Gifkins' work [39], it had generally been

considered that grain boundary sliding could not

contribute to deformation of a continuous nature

without auxiliary processes, since intergranular

deformation should occur coincidentally m order

to maintain geometrical continuity between the

grains [57]. Under such conditions, it has been

argued that pores and eventually cracks would

form at triple points, leading to overall fracture.

Clearly, considerable additional study of grain

boundary sliding mechanisms in ceramics is required.

3.4. Creep of Glasses

As cited by Wachtman [1], oxide glasses at high

temperatures deform viscously. In this case, there

is no detailed atomic model comparable to matter

transport by vacancy motion in crystals undergoing Nabarro-Herring creep. However, viscosity

in glass is related to self-diffusion and the flow is enhanced by an increase in free volume in a manner analogous to an increase in the NabarroHerring creep rate by an increase of vacancy concentration [1]. A theory taking into account both

free volume and thermal activation effects has

been advanced by Macedo and Litovitz [73]. The

resulting equation correctly describes viscosity as a function of temperature over many orders of

magnitude in viscosity. An alternate approach for

associated liquids has been given by Adams and

Gibbs [74].

3.5. Effect of Porosity on Creep

Insofar as possible, the major influences of

structural factors, such as grain size, have been

covered in the preceding sections. The influence

of porosity, however, has not been mentioned and

is briefly considered here.

An increase in the creep rate with porosity has

long been recognized. Wygant [75], for example,

found that the creep rate of a 12 percent porosity

slip cast magnesia specimen was 10 times faster

than a 2 percent porosity magnesia specimen; both

specimens were tested at 1300 C and 1800 psi.

Coble and Kingery [76] studied the effect of porosity on the creep rate of sintered alumina with up

to nearly 50 percent porosity; their results re- vealed that the torsional creep rate increased from

1.5X10 -6 in/in/hr to 7.0X10 -5 in/in/hr for an in- crease from 8 to 50 percent porosity under a stress

of 600 psi at 1275 C.

Several functional relationships for the relation

between creep rate and porosity have recently

been suggested, largely as an outgrowth of attempts to describe hot pressing as a diffusional

creep process. For example, Spriggs and Vasilos

[77] suggested a linear relation between creep rate

and porosity (1/(1-P2/3)) when applied to the data

of Coble and Kingery [76]. However, when applied by Vasilos and Passmore [57] to magnesia

with porosity from 1 to 17 percent, a nonlinear

relationship was observed. It is concluded that

further study is required before specific comments

can be made about observed differences and suggested relationships between creep rate and

porosity.

3.6. Combined Deformation Processes

As suggested earlier, the observed deformation

is frequently the result of more than one process

and interactions also frequently occur. Deformation of a polycrystalline ceramic at high temperature is thus a complex process with_ several of the

deformation processes previously discussed probably operating simultaneously.

At small grain sizes, low stresses, and high

temperatures, Nabarro-Herring creep and grain

boundary sliding probably take place together in

most ceramics of commercial purity [l]. At larger

grain sizes and higher stresses, a combination of

grain boundary sliding and dislocation deformation may occur [1]. The transition with increasing

grain size in alumina from creep controlled by the

Nabarro-Herring mechanism to that controlled

by dislocation glide has been suggested _ [46].

Creep of polycrystalline thoria has been interpreted as grain boundary sliding plus dislocation

motion [78]. Other examples have also been given.

It has generally been considered that [1],

when creep occurs by more than one mechanism

operating simultaneously, the rate will be controlled by the slower process if the faster process

is unable to provide all the degrees of freedom

needed to maintain contact between adjacent

grains; too great a disparity between the rates

will generally lead to the development of high

tensile stresses of some grain boundaries and

will favor the development of pores with attendant increase of creep rate and loss of strength [1].

An alternate approach to compound deformation has recently been suggested by Backofen

et al. [79], in an effort to devise a comprehensive

model to explain the phenomenon of superplasticity. Basically, a three-element mechanical

model has been proposed with two elements in

parallel (one of which is Newtonian in nature) and

a third in series with that pair. A strongly developed Bingham-type behavior at low strain rates

has been interpreted as a result of the parallel

operation of diffusional creep and grain-boundary

shear. The third element operates at high stresses or high strain rates and may be associated with

plastic slip creep, e.g., dislocation climb. The

three mechanisms in competition would _ thus

produce the full strain-rate hardening behavior of a superplastic material and be capable of describing

198
the three-branch log stress-log strain rate curve

frequently observed for such materials.

4. Summary

The state-of-the-art of understanding the inelastic deformation of oxide ceramics has been

briefly reviewed. Special emphasis was given to

the permanent deformation caused by phenomena

such as stress-enhanced diffusional creep, dislocation motion, and grain-boundary sliding. Brief

reference was also made to recoverable inelastic

deformation, especially anelasticity caused

by point defect motion, and recoverable inelastic

deformation caused by both dislocations and

grain boundary sliding. The information given

dealt largely with polycrystalline oxide ceramics.

The primary mechanism by which most polycrystalline oxide ceramics appear to deform at

high temperatures, relatively low stresses, and

fine grain sizes was cited as Nabarro-Herring diffusional creep. Under most experimental conditions, however, it was recognized that the observed deformation is usually the result of more

than one process, with complicating interaction

frequently occurring, e.g., grain boundary sliding

and Nabarro-Herring creep. At larger grain sizes

and higher stresses, combinations such as grain

boundary sliding and dislocation deformation may

occur, but considerably less data are available to

substantiate this.

A host of structural and compositional factors, as well as stress and temperature, influence the

responsible mechanisms. Information concerning

such influences exists for only a few ceramics, e.g., A1203 , MgO, BeO, SiC, and in no case is the information complete and unambiguous. The need for

additional studies in essentially every area considered is clearly indicated.

5. References

[1] Wachtman, J. B., Jr., Mechanical properties of ceramics: An introductory survey, Am. Ceram.

Soc. Bull. 46 (8) 756-774 (1967).

[2] Wachtman, J. B., Jr., Elastic deformation of crystalline ceramics and other refractory materials, this

Syposium.

[3] Hollox, G. E., Inelastic deformation of non-oxide

ceramics, this Symposium.

[4] Southgate, P. D., Mechanical relaxation of a point

defect in magnesium oxide, J. Appl. Phys. 36, 2966

(1965).

[5] Southgate, P. D., Mendelson, K. S., and DePerro,

P. L., Kilocycle-range dislocation damping in

magnesium oxide, J. Appl. Phys. 37, 206 (1966).

[6] Chang, R., The elastic and anelastic properties of refractory materials for high-temperature applications, Chapter 14 in Mechanical Properties

of Engineering Ceramics, Interscience, 1961.

[7] Wachtman, J. B., Jr., and Corwin, W. C, Internal

friction in Zr02 containing CaO, J. Res. NBS 69A,

457 (1965).

[8] Wachtman, J. B., Jr., Mechanical and electrical

relaxation in Th02 containing CaO, Phys. Rev.

131, 517 (1963).

[9] Carnahan, R. D., and Brittain, J. 0., Point-defect

relaxation in rutile single crystals, J. Appl. Phys.

34, 3095 (1963).

[10] Wachtman, J. B., Jr., and Doyle, L. R., Internal

friction in rutile containing point defects, Phys.

Rev. 135, A276 (1964).

[11] Wachtman, J. B., Jr., Spinner, S., Brower, W. S., Fridinger, T., and Dickson, R. W., Internal friction

in rutile containing Ni or Cr, Ibid. 148, 811 (1966).

[12] Johnson, H. B., Tolar, N. J., Miller, G. R., and

Cutler, I. B., Dipole relaxation in CaF2 doped with

NaF, J. Am. Ceram. Soc. 49 (8) 458 (1966).

[13] Cook, R. K., and Breckenridge, R. G., Anelasticity

of quartz, Phys. Rev. 92, 1419 (1953).

[14] Van Houten, S., Mechanical losses in Li-doped NiO

semiconductors, Phys. Chem. Solids 23, 1045

(1962).

[15] Astbury, N. F., and Davis, W. R., Internal friction

in ceramics, Trans. Brit. Ceram. Soc. 63, 1 (1964).

[16] Kirby, P. L., XlX-Internal friction in glass. Part II, flexural and torsional vibrations, J. Soc. Glass

Technol. 38, 383 (1954).

[17] Blum, S. L., Some physical factors affecting the

internal dumping of glass, J. Am. Ceram. Soc. 38

(6) 205-10 (1955).

[18] Kerper, M. J., and Scuderi, T. G., Mechanical

properties of chemically strengthened glasses at elevated temperatures, Ibid. 49 (11) 613-18 (1966).

[19] Macedo, P. B., and Litovitz, T. A., Ultrasonic

viscous relaxation in molten boron trioxide, Phys.

Chem. Glasses 6, 69 (1965).

[20] Pratt, P. L., The role of point defects in mechanical

properties, Proc. Brit. Ceram. Soc. 1, 177 (1964).

[21] Niblett, D. H., and Wilks, J., Dislocation damping in

metals, Advan. Phys. 9, 1 (1960).

[22] Granato, A. V., Internal friction studies of dislocations

paper in Internal Friction, Damping and Cyclic

Plastic, ASTM S.T.P. No. 378, 1965.

[23] Seeger, A., and Schiller, P., Kinks in dislocation lines and their effect on the internal friction in crystals;

p. 361 in Physical Acoustics, Vol. IllA, Academic

Press, 1966.

[24] Chang, R., Dislocation relaxation phenomena in

oxide crystals, J. Appl. Phys. 32, 1127 (1961).

[25] Carnahan, R. D., and Brittain, J. O., Dislocation

damping in rutile single crystals, J. Am. Ceram.

Soc. 48 (1) 15-19 (1966).

[26] Southgate, P. D., Mendelson, K. S., and DePerro,

P. L., Kilocycle-Range Dislocation Damping in Magnesium Oxide, J. Appl. Phys. 37, 206 (1966).

[27] Fiore, N. F., A test of viscous dislocation damping

theory, J. Appl. Phys. 37, 878 (1966).

[28] Wachtman, J. B., Jr., and Lam, D. G., Jr., Young's

modulus of various refractory materials as a

function of temperature, J. Am. Ceram. Soc. 42

(5) 254-60 (1959).

[29] Crandall, W. B., Chung, D. H., and Grey, T. J., Mechanical properties of ultra-fine hot-pressed

alumina, Chapter 22 in Mechanical Properties of

Engineering Ceramics, Interscience, 1961.

[30] Turnbaugh, J. E., Effect of additions on the hightemperature grain-boundary relaxation of alumina;

Sc.D. Thesis, MIT, 1962.

[31] Stuart, J., Internal friction and creep in polycrystalline

alumina; Task 5 in Studies of the Brittle Behavior

of Ceramic Materials, Report ASD-TR-61-628,

Part III, June 1964.

[32] Wachtman, J. B., Jr., and Maxwell, L. H., Factors

controlling resistance to deformation and mechanical failure in polycrystalline (glass-free) ceramics,

WADC Tech. Rept. No. 57-526, ASTIA Document

No. AD 142245, 1957.

[33] Beauchamp, E. K, Baker, G. S., and Gibbs, P., Impurity dependence of creep aluminum oxide,

Tech. Documentary Rept. No. ASD TR 61-481,

University of Utah, April 1962.

[34] Stokes, R. J., Correlation of mechanical properties

with microstructure, Chapter 4 in Microstructure

199
of Ceramic Materials, NB5 Miscl. Publ. 257,

April 1964.

[35] Nabarro, F. R. N., p. 75 in Report of a Conference on

the Strength of Solids. Physical Society, London,

1948.

[36] Herring, C, Diffusional viscosity of a polycrystalline

solid, J. Appl. Phys. 21, 437 (1950).

[37] Farnsworth, P. L., and Coble, R. L., Deformation behavior of dense polycrystalline SiC, J. Am. Ceram.

Soc. 49 (5) 264-68 (1966).

[38] Coble, R. L., Model for boundary diffusion controlled

creep in polycrystalline materials, J. Appl. Phys.

34, 1679 (1963).

[39] Gifkins, R. C, Diffusional creep mechanisms, J. Am.

Ceram. Soc. 51 (2) 69-72 (1968).

[40] Gifkins, R. C, and Snowden, K. U., Mechanism for "viscous" grain-boundary sliding, Nature 212

(5065) 916-17 (1966).

[41] Ryan, H. J., and Suiter, J. W., Grain boundary topography of tungsten, Phil. Mag. 10, 727-29 (Oct. 1964).

[42] Gifkins, R. C, and Snowden, K. XL, Stress sensitivity

of creep of lead at low stresses, Trans. AIME

239 (6) 910-15 (1967).

[43] Gifkins, R. C, Superplasticity during creep, J. Inst. Metals 95 (12) 373-77 (1967).

[44] Hensler, J. H., and Cullen, G. V., Grain shape change

during creep in magnesium oxide, J. Am. Ceram.

Soc. 50, No. 11, 584-85 (1967).

[45] Folweiler, R. C, Creep behavior of pore-free polycrystalline aluminum oxide, J. Appl. Phys. 32,

773 (1961).

[46] Warshaw, S. J., and Norton, F. H., Deformation

behavior of polycrystalline aluminum oxide, J. Am.

Ceram. Soc. 45, No. 10, 479-86 (1962).

[47] Coble, R. L., and Guerard, Y. H., Creep of polycrystalline aluminum oxide, Ibid. 46, No. 7, 353-54

(1963) . [48] Coble, R. L., and Ellis, J. S., Hot pressing aluminamechanisms of material transport, Ibid. 46, No. 9. 438-41.

[49] Passmore, E. M., and Vasilos, T., Creep of dense,

pure, fine-grained aluminum oxide, Ibid. 49, No. 3, 166-68 (1966).

[50] Vasilos, T., Mitchell, J. B., and Spriggs, R. M., Creep

of polycrystalline magnesia, Ibid. 47, No.4, 203-204

(1964) . [51] Chang, R., High temperature creep and anelastic

phenomena in polycrystalline refractory oxides, J. Nucl. Mater. 1, 174 (1959).

[52] Vandervoort, R. R., and Barmore, W. L., Compressive

creep of polycrystalline beryllium oxide, J. Am.

Ceram. Soc. 46, No. 4, 180-84 (1963).

[53] Fryxell, R. E., and Chandler, B. A., Creep, strength,

expansion, and elastic moduli of sintered BeO as a function of grain size, porosity, and grain orientation, Ibid. 47, No. 6, 283-91 (1964).

[54] Vandervoort, R. R., and Barmore, W. L., Compressive

creep of beryllium oxide-uranium dioxide mixtures,

Ibid. 48, No. 11, 587-90 (1965).

[55] Kingery, W. D., and Montrone, E. D., diffusional

creep in polycrystalline NaCl, J. Appl. Phys. 88,

2412 (1965).

[56] Paladino, A. E., and Coble, R. L., Effect of grain

boundaries on diffusion controlled processes in aluminum oxide, J. Am. Ceram. Soc. 46, No. 3, 133-36 (1963).

[57] Vasilos, T., and Passmore, E. M., Effect of microstructure on creep mechanisms in ceramics, Chap-

[58

[59

[60

[61

[62

[63

[64

[65

[66

[67

[68

[69

[70

[71

[72

[73

[74

[75

[76

[77

[78

[79

ter 6 in Strengthening Mechanisms, Metals and

Ceramics, Syracuse Univ. Press, 1966.

Chang, R., High temperature creep and anelastic

phenomena in polycrystalline materials, J. Nucl.

Matls. 1, 174 (1959).

Spriggs, R. M., Passmore, E. M., Brissette, L. A.,

and Vasilos, T., Microstructure studies in polycrystalline refractory oxides, Progress Report, Bureau of Naval Weapons, Contract NOw 64-02 17-d

(1964) . Dorn, J. E., Creep and Recovery, Am. Soc. Metals,

Cleveland (1957).

Passmore, E. M., Duff, R. H., and Vasilos, T., Creep

of dense, polycrystalline magnesium oxide, J. Am.

Ceram. Soc. 49 (11) 594-600 (1966).

Oishi, Y., and Kingery, W. D., Oxygen diffusion in periclase crystals, J. Chem. Phys. 33, 905-906

(1960).

Lindner, R., and Parfitt, G. D., Diffusion of radioactive magnesium in magnesium oxide crystals,

Ibid. 26, 182 (1957).

Taylor, G. I., Plastic Strain in Metals, J. Inst. Metals 62, 307 (1938).

Groves, G. W., and Kelly, A., Independent slip

systems in crystals, Phil. Mag. 8, 877 (1963).

Copley, S. M., and Pask, J. A., Plastic deformation

of MgO single crystals up to 1600° C, J. Am.

Ceram. Soc. 48 (3) 139-145 (1965).

Day, R. B., and Stokes, R. J., The mechanical

behavior of magnesium oxide at high temperatures,

Ibid. 47, 493 (1964).

Weertman, J., Theory of steady-state creep based on

dislocation climb, J. Appl. Phys. 26, 1213 (1955).

Weertman, J., Steady-state creep through dislocation

climb, Ibid. 28, 362 (1957).

Hirthe, W. M., and Brittain, J. O. High-temperature

steady-state creep in rutile, J. Am. Ceram. Soc. 46

(9) 411-17 (1963).

Chang, R., Dislocation theories of the high-temperature creep of crystalline solids; paper in The

Physics and Chemistry of Ceramics, Gordon and

Breach (1963).

Weertman, J., Steady-State creep of crystals, J. Appl. Phys. 28, 1185 (1957).

Macedo, P. B., and Litovitz, T. A., On the relative

roles of free volume and activation energy in the

viscosity of liquids, J. Chem. Phys. 42, 245 (1965).

Adam, G., and Gibbs, J. H., On the temperature

dependence of cooperative relaxation properties in glass-forming liquids, Ibid. 43, No. 1, 139-46

(1965) . Wygant, J. F., Elastic and flow properties of dense,

Pure oxide refractories, J. Am. Ceram. Soc. 34, 374

(1951).

Coble, R. L., and Kingery, W. D., Effect of porosity on physical properties of sintered alumina, Ibid.

39, 377 (1956).

Spriggs, R. M. and Vasilos, T., Functional relaxation

between creep rate and porosity for polycrystalline

ceramics, Ibid. 47, No. 1, 47-48 (1964).

Poteat, L. E., and Yust, C. S., Creep of polycrystalline thorium dioxide, Ibid. 49 (8) 410-14 (1966).

Backofen, W. A., Azzarto, F. J., Murty, G. S., and

Zehr, S. W., Deformation processing of anisotropic

metals-superplasticity, Final Report on Contract

N000 19-67-C-0089, Sept. 1967.

200
Plastic Deformation of Carbides

Graham E. Hollox*

Research Institute for Advanced Studies (RIAS),

Martin Marietta Corporation, 1450 South Rolling Road, Baltimore, Maryland 21227

A significant research effort is being directed towards obtaining technologically useful

materials for high-temperature structural applications. The refractory metal carbides are of particular interest since (a) they include the materials having the highest melting point, (b)

they are extremely strong, and (c) they deform plastically in a manner similar to the face

centered cubic metals. This paper reviews the current understanding of the mechanical

behavior of the carbides. It concentrates on evaluating the more meaningful information

obtained from studies of single crystals rather than that from sintered polycrystalline materials, where inadequate characterization may have led to confusion in the literature. The

present knowledge of the behavior of these materials suggests that they may be of technological importance in the future.

Key Words: Carbides; high-temperature; mechanical behavior; microstructure; plastic

deformation.

1. Introduction

The need for improved materials for hightemperature structural applications has stimulated

research into the mechanical behavior of high

melting-point ceramics—carbides, borides, nitrides, aluminides, beryllides, silicides, etc. The

properties of these materials have been discussed

by, for example, Schwartzkopf and Kieffer [l],

1 and Kendall [2]. Additionally, several tabulations

of properties are available [3, 4, 5]. Material characteristics such as microstructural details have

recently been clearly recognized as variables

important in determining the properties of ceramics

[6]. Many of the materials used in past work on

the carbides have lacked sufficient characterization and the results therefore need careful evaluation if an understanding of their properties is to

be reached. It is evident that more scientifically

meaningful information has been obtained from

studies of single crystals of the carbides than has

been obtained by some 50 years research on

sintered, polycrystalline material. Although the

value of this latter work in producing technologically useful material is not to be underrated

or denied, the prospects for significant future

developments of structural carbides will be considerably enhanced by an understanding of the

relationship between microstructure and mechanical behavior.

The study of plastic deformation in ceramics is comparatively new, but some cubic materials,

•Present address: Brown Boveri Research Center, Baden, Switzerland.

1 Figures in brackets indicate the literature references at the end of this paper.

for example, MgO, NaCl, and LiF have been

studied in recent years [7, 8, 9]. Much of this work

is applicable to the even newer field of plastic

deformation in the carbides. With the exception

of the hexagonal carbide, WC [10, 11, 12], plastic

deformation studies appear to have been restricted

to single crystals of cubic monocarbides. No

equivalent effort on similarly characterized material has been directed towards understanding the

other non-oxide ceramics, although single crystals

of several diborides have been obtained and their

structure examined [13, 14]. Consequently, in this

review the relationship between structure and

properties of the carbides will be emphasized.

The occurrence of these materials has been well

documented [1, 15], and it is summarized in

figure 1. The titanium-carbon equilibrium diagram, figure 2, which is representative of the

equilibrium between Group IV metals and carbon,

illustrates the typical wide range in composition

observed in these materials [15].

2. Mechanical Behavior of Carbide

Single Crystals

2.1. Titanium Carbide

Single crystals of TiC deformed in compression

or bending at temperatures below about 800 °C

are completely brittle [16]. No macroscopic plastic

flow has been detected by etch-pit or slip line

observations. As expected, the observed fracture

strength is very dependent upon surface condition,

but values as high as 800,000 psi (0.4 of the shear

201

323-655 0 - 69 - 14
ScC fee TiC fee VC fee

V2C ortho -hex

Cr 23 Cg cubic

Ct-tC.-i hp*

Cr 3 C2 ortho

Y3C cubic YC

"2^3 YC 2 tefrag

ZrC fee NbC fee

Nb-,C ortho £ -hex

Mo2 C ortho -hex

MoC hex

-cubic

HfC fee TaC fee

Ta2C ortho

- hex

W2C hex

a -wo nex

/3-WC fee

LANTHANIDES Me3C -cubic ACTINIDES MeC -f.ee. (ThC.UC, PuC)

Me2C3

- b.c.c. MeC -be. tet

OTHERS... Fe3C-cubic B4C -hex

a-SiC-hex

/3-SiC-f.c.c.

Figure 1. The occurrence of refractory carbides.

This review will be primarily concerned with the mechanical behavior

of the cubic refractory carbides and these are indicated in heavier print

.

5300

CARBON: TITANIUM RATIO

Figure 2. The titanium-carbon equilibrium diagram.

The TiC phase exists over a range In composition by incorporating vacancies in the carbon sublattice (after Storms [15]).

modulus) have been measured on carefully

electropolished samples. Failure occurs by cleavage

on {100} planes initiated at surface or internal

defects, although the possibility that microcracks

are produced by dislocation interactions cannot be

completely eliminated. Evidence for some dislocation motion at room temperature has been

obtained from the surface markings close to Knoop

microhardness indentations [17] or close to friction

tracks [18], and these are consistent with slip

on { 111 } planes. Also, Williams [17] has shown that

the microhardness number is dependent on the

orientation of the indenter with respect to the

active slip planes in TiC, suggesting that dislocations participate in room temperature

deformation.

At testing temperatures above 800 to 900° C,

plastic deformation takes place on the {111}

<110> system [16, 19]. The critical resolved

shear stress for slip is very dependent upon carbon

content, figure 3. For example, at 900° C it in- creases linearly from about 12 kg/mm2 for TiC 079,

to about 22 kg/mm2 for TiC 0.97 [20].

Vacancies have been commonly recognized as a cause of hardening in crystal lattices. For example,

the yield strength of quenched aluminum is greater

than annealed aluminum as a result of dislocationvacancy interactions [21]. Also, nonstoichiometric

Ti02 is stronger than the stoichiometric composition because of the interactions between dislocations and vacancies [22, 23] or clustered

vacancies [24]. In TiC, however, the decrease in

strength as the concentration of vacancies in- creases is attributed to a decrease in the contribution made by carbon atoms to cohesion in

TiC. The nature of the electronic interactions

between constituent atoms in the lattice has been

deduced from studies of the band structure of

this material. Lye [25] has shown that the predominant contribution to the bonding is from

covalent metal-metal bonds, the strength of which

increase with carbon content because (i) carbon

atoms donate electrons to crystal states derived

from metal atom wave functions and increase the

number of 3c?-electrons available for metal-metal

bonding, and (ii) the presence of carbon atoms in

overlap regions of neighboring metal atom 3d0.8 0.9 1.0 CARBON: TITANIUM RATIO

Figure 3. The critical resolved shear stress for

slip in TiC as a function of carbon content.

(After Williams [20].)

202
HYPOTHETICAL fee.

f.C.C. TITANIUM TITANIUM CARBIDE 0 Titanium • Carbon

Figure 4. Schematic representation of the bonding between ir-oriented 3-d functions in

TiC compared with that of hypothetical

f.c.c. titanium.

This illustrates the position of carbon atoms in the over- lap region between orbitals on neighboring atoms (after Lye [26]).

orbitals introduces a potential that increases the

strength of the metal-metal interactions, figure 4. The observations of a {111} <110> slip system is consistent with the view that the structure should

be considered as that of a f.c.c. metal rather than

a rocksalt ionic compound.

The onset of bulk plastic flow in TiC is a gradual

transition which involves an increasing strain at

failure as the testing temperature is increased

rather than the sharp brittle-to-ductile transition

normally associated with bec metals at low

temperatures. It has been suggested that this

gradual transition indicates that diffusion may

play a role in determining the mobility of dislocations in TiC [27]. Some evidence to support

this concept may be obtained by considering both

the structure of the slip plane and the temperature

dependence of the yield stress of TiC.

Figure 5 shows a plan of the Ti and C atoms on

the {111} planes of the TiC [28]. Following consideration of the synchro-shear process suggested

by Kronberg [29], Rowcliffe [27] points out that

for a unit of slip from Bx to B3 , the titanium atom

at Bi must pass very close to that at Ax

. This re- quires a large lattice expansion normal to the slip

plane. Alternatively, if the carbon atom at C\ can move in a <112> direction to B2 or B3 at the

same time as the titanium atom moves to Ci, then

the unit displacement Bx to B3 can be accomplished

by movements of partial dislocations by slip from

Bx to Ci and Ci to B3

. The motion of the carbon

atom cannot be described by a single shear vector

and will require the diffusion of carbon atoms to

allow deformation to take place.

A3

O Ti below plane

# Ti above plane • C atoms in plane

Figure 5. The slip plane

of TiC.

(After Kelly and Rowcliffe [28].)

The temperature dependence of the yield

strength of TiC [20] is shown in figure 6. In TiCn.97,

the critical resolved shear stress ar for slip on

{111} <110> systems decreases from about 22

kg/mm2 at 900° C to 2 kg/mm2 at 1600° C. Over

limited temperature ranges, the relationship:

B

o> oc exp j,

is applicable (fig. 8) . This suggests that B may be

represented by -U/k where U is the activation

energy for plastic flow, and k is the Boltzmann

constant. Significantly, a characteristic change in

the slope of this function occurs at a critical

temperature [20] which is dependent on the carbon

content of the TiC [19] i.e., it is approximately

TEMPERATURE , °C

Figure 6. Temperature dependence of the critical resolved shear stress of TiC as a function of carbon content.

(After Williams [20].)

203
1150° C for TiCo.gs., 1305° C for TiC0

. 95 , etc. Williams [17] has indicated that:

aTm oc exp ^

(where m is the exponent in the relationship

between dislocation velocity, v, and stress, a; v=<rm

[8]) describes the variation in yield stress of

TiC with temperature. Since the slope of the line

in figure 7 must be multiplied by m, he suggests

that the activation energy for plastic flow above

about 1180° C in TiC is about 3.0 eV [17]. Below

the critical temperature the slope is almost

halved, this being consistent with an activation

energy for flow of about 1.5 eV.

Values of about 2.7 eV and 5.1 eV have been

measured for the activation energy of carbon and

titanium diffusion respectively in TiC [30, 31],

carbon having the lower activation energy consistent with its smaller size, interstitial position,

and the presence of a large number of vacancies

in its sublattice. Above about 0.5 Tm, creep

deformation appears to be controlled by titanium

diffusion [33] and so it may be expected that the

high-temperature deformation would exhibit an

activation energy for flow of about 5.0 to 6.0 eV.

This, however, is not observed, and as suggested

by the mechanism proposed for dislocation motion

[27], diffusion of the atomic species in the dislocation cores may have to be considered. In this

context, Williams [17] has correlated the activation

energy for flow ^3.0 eV with that for pipediffusion [32] of titanium atoms in TiC. Any

correlations with the bulk activation energy for

10

4/ToK

Figure 7. The temperature dependence of the yield

stress of TiCo.83-

(After Williams [20].) The change in slope—close to 0.475 Tm—

corresponds to a change in the mechanism controlling the deformation.

diffusion of C and Ti in TiC must be regarded as tentative until more accurate measurements of

these energies are available, particularly as a function of carbon content and in single crystal

material.

Annealing of plastically deformed TiC is accompanied by the coalescence of vacancy dislocation loops, figure 8 [32]. The initial stages

appear to be associated with the formation of

trails of small loops, figure 8(b), produced from

dislocation dipoles. The dislocation loop density

as a function of isochronal annealing temperature

for TiC0.97 and TiCo.88 is shown in figure 9. The

"self-diffusion temperature," TD , defined as the

temperature at which loops disappear completely

in a short time (15 min) anneal, is 1400° C for

TiCo.97, and 1270° C for TiC0

. 88 . The relationship

TD — 0.475 Tm [32] is obeyed in these alloys as it is in f.c.c. metals, and the activation energy for the

annealing process involving the diffusion of vacancies is 5.25 eV for TiC0 . 97 and 4.88 eV for TiC0 . 88 . The only reported value of the activation energy

for the diffusion of titanium atoms in TiC, 5.1 eV,

is in close agreement with these values [31].

It is significant to note that the "self-diffusion

temperatures," 0.475 Tm , are close to the values

of the critical temperatures observed for the

change in strength controlling mechanism. Above

this temperature, therefore, it appears that some

vacancy assisted process, involving dislocation

climb, is controlling the deformation behavior of

the material [19].

Measurements of the creep rate of large grained,

polycrystalline TiC0 . 97 over the temperature

range 1600 to 1700 °C have indicated an activation

energy of between 5 and 7 eV [33]. Steady-state

creep deformation is thought to be controlled by

the climb of dislocations away from obstacles, a process governed by the activation energy for

the diffusion of vacancies. This value is in reasonable agreement with the energy determined from

loop annealing, and is consistent with titanium

self-diffusion in TiC.

Williams [17] has applied Haasen's analysis

[34] to the creep rate of TiC0 . 93, and deduces an

activation energy of 3.3 eV in the temperature

range 1400 to 1700 °C. It would seem more rea- sonable to expect that the creep rate be governed

by an activation energy of about 5 eV in this

temperature range, as titanium diffusion appears

to control the strength in this temperature range,

and consequently these results are not fully

understood.

It has been mentioned already that the deformation characteristics of TiC are similar to those

of a f.c.c. metal. In addition, this material has a high stacking-fault energy, since fringe contrast

or dissociation of dislocations into partials has

not been observed [19] (although this may occur

within

o the width of the dislocation image,

~100A) . Moreover, the elongated dislocation loops

204
/

I

Figure 8. Dislocation structures in TiC.

(a) Early stage of deformation showing elongated dislocation loops, (b) annealing of an elongated

loop, A, by pipe-diffusion along dislocation cores, (c) formation of vacancy dislocation loops on

annealing at 1300 °C, (d) final annealed structure—a hexagonal dislocation network as observed in

f.c.c. metals (after Hollox and Smallman [19, 23]).

observed in the early stages of deformation, figure

8(a), and also cell structures in more heavily

deformed samples, are thought to result from a

relative ease of cross slip in the material. The

strong directional bonding in TiC probably

accounts for the high stacking-fault energy of

TiC, since the hexagonal symmetry in stacking

faults in the f.c.c. structure requires a change in

the bond directions. An alternative explanation

relies on the observation that metals with a filled

(Z-band have a lower stacking-fault energy than

those with partially filled bands [35, 36]. The

latter case is applicable to TiC, but no quantitative

estimate of the stacking-fault energy has yet been

made.

2.2. Vanadium Carbide

Although it is far from complete, the most

detailed behavior of a carbide other than TiC

has recently been established for VC [37, 38, 39].

When VC, having a carbon-to-metal ratio of

0.84, is deformed in compression, the slip plane is {111}, exactly as is observed for TiC. Slip lines

show a wavy appearance characteristic of a high

stacking-fault energy material, and this also suggests that dislocation structures and slip mechanisms of VC0

. 84 are similar to those observed

in TiC. On the basis of these observations, it was

thought that the behavior of TiC might be representative of the behavior of all the cubic transition

metal carbides. However, a generalization of this

nature is invalid! Three significant differences

in the behavior of this material and TiC of equivalent stoichiometry have been found (fig. 10)

:

(i) the brittle-to-ductile transition temperature

occurs between 1200 and 1250 °C, this being about

300 to 400 °C higher than that reported in TiC

;

(ii) in the temperature range 1200 to 1500 °C, the

critical resolved shear stress for slip in VC0 .g4 is greater than that of TiC; and (iii) the decrease

205
1000 1100 1200 1300 140C ISOCHRONAL ANNEALING TEMPERATURE, °C

Figure 9. The variation in dislocation loop density as a

function of isochronal annealing temperature in TiC.

(After Hollox and Smallman [32].)

i l i l i i 1200 1400 1600

TEMPERATURE, °C

Figure 10. The critical resolved shear stress for slip in

VCo.84 and TiCo.84 as a function of temperature.

(After Hollox and Venables [38].)

in critical resolved shear stress with increasing

temperature is much more rapid in VC0

. 84 than

that observed for other carbides, falling from about

25 kg/mm2 (26,000 psi) at 1250 °C to about 3 kg/

mm2 (4,200 psi) at 1500 °C.

The higher strength and increased brittle-toductile transition temperature of VC0

. 8 4 are

surprising in view of the relative melting points

of these carbides, since that of VC0

. 84 (2650 °C)

is some 600 C below that of TiC (3250 °C). Lye

[37] has suggested that the strength of the metalmetal d-d interactions in VC0

. 84 is greater than in

TiC and this may account for the higher strength

of this material. However, Venables et al. [39]

have shown an important difference between TiC

and VCn. 84 is that carbon atoms in VC0. 84 are ordered in a hexagonal superlattice. When the

crystals are viewed in polarized light, a colored

domain structure can be observed. This corresponds

to the several possible orientations of the anisotropic superlattice within the metal lattice.

Metallographic observations, figure 11, suggest

that disordering of the carbon superlattice occurs

at some temperature between 1250 and 1300 °C

[38]. Although the precise role of this superlattice

is not fully understood, it significantly affects

mechanical behavior, since the order-disorder

transformation temperature corresponds closely

with the brittle-to-ductile transition temperature.

Using nuclear magnetic resonance techniques,

Froidevaux and Rossier [40] have obtained

evidence for the ordering of carbon vacancies

throughout the complete composition range of

VC. de Novion et al. [41] have shown that a consequence of carbon vacancy ordering in

VC0 .s9 is the formation of a superlattice of cubic

symmetry. Venables et al. [39] have suggested

that the symmetry of the superlattice cell depends

on carbon concentration, and, as such, smgleFigure 11. The change in domain structure of

VCo.84 after an anneal at 1800 °C.

Tbe upper micrograph shows the structure before annealing, and the lower one the final structure. No change is observed in a similaranneal at 1250 °C. These observations

are consistent with disordering of the carbon superlattice between 1250 and 1300 °C (after Hollox and Venables [38])

.

206
phase VC might be more correctly described as a

series of ordered compounds V8 C7 (VCo.ss),

V6C5 (VCo.m), etc. The occurrence of ordered

structures at different compositions would be

expected to have different effects on the strength

of VC. Consequently, in comparing the properties

of VC with TiC, composition is a very important

variable. In this context, two other compositions

of VC, close to VC0.75 and VC0 .89, both appear to

be weaker than VC0 .g4 [42]. This suggests that the

strength of VC passes through a maximum as carbon content is increased.

2.3. Other Cubic Carbides

Very little data is available for other carbides.

Williams [20] has compared the properties of

ZrCo.gg and NbC0 .76 with TiC of varying carbon

content. He observed that ZrC0 . 8 8 compressed in

<001> directions exhibited greater strength than

TiCo.95, while both these were weaker than

NbCo.76- The greater strengths of ZrC and NbC

were attributed to an increased number of electrons in the bonding scheme. The variation in

strength with carbon content of these other

carbides has not been reported.

Lee and Haggerty [43] have measured the

strength of ZrC0 9 as a function of crystal orientation. They observed slip on { 111 } <110>, {110}

<110>, and {001} <110> systems when the

crystal orientation was chosen such that the

Schmidt factor favored slip on these systems.

These results confirm that ZrC0 . 9 is stronger than

TiC0 9. In_addition, the resolved shear stress for

{110} <110> slip is slightly lower than for { 111

}

<C110> slip. This cannot be explained at present.

These workers have also investigated the creep

behavior of single crystal ZrC and they obtain an

activation energy of about 4.8 eV for the steadystate creep. This value is slightly lower than that

expected for the diffusion of zirconium in ZrC

(5.7 eV) [31] but is in fair agreement with the

behavior of TiC.

2.4. Binary Carbide Alloys

Pseudo-binary solid solutions between carbides,

nitrides, and oxides have been reported for many

years. Schwartzkopf and Kieffer [1] state that

complete solid solution occurs between these

compounds except in cases where the lattice

parameters differ by more than about 15 percent

(e.g. VC-ZrC), reminiscent of the empirical

relationships deduced by Hume-Rotherey for

metallic solid solutions [44]. In addition, extensive

solid solubility, for example, up to 70 w/o (weight

percent) WC in TiC, has been reported for noncubic carbides in the cubic carbides [l].

Single crystals of binary carbides have been

prepared only recently [45] and their mechanical

behavior studied [42]. The preliminary results,

figure 12, however, are particularly interesting,

for three reasons: (i) the alloys show considerably

higher strength than the parent carbides; (ii) the

x 60

t—

£ 20

-60

,VC-25%TiC

VCo.84

\ r

VC-50% TiC V

Co.84

\ V

-TiC-25%VC

\

\ \

' - _] 1 L

40

20

800 1000 1200 1400 TEMPERATURE

,

1600 1800

Figure 12. The yield strength (<00^> compression) as a

function of temperature for some TiC-VC alloys, compared

with that of TiCo.84 and VCo.84.

temperature dependence of the yield strength of

the VC-25 a/o (atomic percent) TiC and VC-50

a/o TiC compounds appears to have a form different from that of TiC-25 a/o VC and the pure

carbides; (iii) there is evidence for a small amount

of ductility (~ 1 percent) at about 1000° C

in VC-25 a/o TiC and VC-50 a/o TiC.

It is likely that these effects are dependent on

stoichiometry and they have yet to be interpreted,

but Venables [46] has shown that the VC-25

a/o TiC alloys are two-phase, one of which is ordered. The details of this phase separation is

not fully understood, but the fine scale precipitation of a hard ordered phase in a "ductile"

matrix may explain the mechanical behavior of

this alloy [46].

2.5. Effect of Boron on the Structure and

Properties of TiC and VC

Boron has been shown to significantly increase

the strength of TiC and VC. For example, Williams

[47] observed a tenfold increase in strength at

1600° C in TiC crystals, figure 13, after they have

been heated in contact with boron powder at

2000° C. Similarly, the critical resolved shear

stress of boron-doped VC0 . 84 is about 19 kg/mm2

(27,000 psi) at 1500° C, compared with 3 kg/mm2

(4,200 psi) in the undoped carbide of the same

composition, figure 13 [38]. These increased

strengths have been correlated with the formation

of boride precipitates in the carbides.

In TiC, precipitates form on {111} planes,

figure 14(a), and Williams [47] has suggested

that they are TiB2 . Using transmission electron

microscopy, Venables [48] has confirmed this

and shown that the precipitates are nucleated

heterogeneously at dislocation nodes, figure 15.

Of particular importance is the fact that the

precipitates form at boron concentrations of

the order 100 ppm. An autoradiographic technique

was used to show that the boron is almost totally

207
40

E

TEMPERATURE , °C

Figure 13. The critical resolved shear stress for slip as a

function of temperature in boron-doped TiC (after Williams

[49] and VCo.84 {after Hollox and Venables [38]).

associated with the precipitates, and little or none

remains in solid solution in the lattice [49].

This technique relies on the observation by

Fleischer et al. [50] and Armijo et al. [51] that the

damage produced in thin films of cellulose nitrate

by a-particles resulting from the B10 (n,a) reaction can be detected by etching with 6N NaOH.

A metallographic map, figure 14(a), and a boron

map ("alphagraph") of the same area are shown

for comparison in figure 14(b).

When VC0 .s4 is doped with boron by diffusion

at 1800° C for 1 hr, precipitates form on {111}

planes in the surface layers where a high concentration of boron would be expected, but the plane

of precipitation changes to {100} with increasing

distance from the surface, figure 16 [38]. The composition of the precipitates has not been conclusively established, although the presence of

boron has been demonstrated in both types by

means of the alpharadiographic technique. It

might be expected that the precipitates which

form on {111} planes would be similar to those

which have been observed in boron-doped TiC.

The identical structures and similar atomic mismatch in the vanadium compounds therefore

suggests that the precipitates on {111} planes of

VC are probably of VB2 . It is possible that the precipitates on {100}

planes would be one of the lower borides of vanadium, since they appear in the more boron deficient

regions of the sample. In order of increasing boron

content, these borides are: (i) V3B2 (tetragonal,

U3Si2-type structure), (ii) VB (orthorhombic,

CrB-type structure,) and (hi) V3B4 (orthorhombic

Mn3B4-type structure) [52]. Consequently a different plane of precipitation may be the result of a change in composition and structure of the

precipitate.

3. Mechanical Behavior of Polycrystalline Carbides

3.1. Characterization

Before examining the wealth of information in

the published literature concerning the behavior

of polycrystalline carbides, it is worthwhile to

consider which variables may be critical in affecting the properties of such materials. From the

behavior of single crystals, it is clear that to compare their properties, at least three variables are important:

a. Stoichiometry

The observed dependence of the strength of

TiC and VC on carbon content has particular

significance when properties of carbides are compared. Before precise conclusions about their

Figure 14. Precipitates on [111 \ planes of TiC after boron doping at 2000 °C.

A direct comparison between a metallographic map and an alphagraph (see text) indicates that most of the boron in the lattice is associated with these precipitates (after Venables [49]).

208
*

Figure 15. TiB2 precipitates nucleated at dislocation nodes.

Their diffraction contrast resembles that of extrinsic stacking faults

(after Venables [48]).

Figure 16. A [100\ section through a sample of boron-doped

vcw

In the surface layers where a high concentration of boron is expected,

precipitates form on {111) planes (b), but the plane of precipitation

changes to (1001 with increasing distance from the surface (c). (After Hollox and Venables [38].)

relative behavior may be drawn, a full survey of

mechanical properties as a function of carbon

content is necessary. A comparison at equivalent

carbon-to-metal ratios may be inadequate since

ordering may occur in one carbide and not in the

other, as in comparisons between the properties

of TiC and VC.

b. Purity

The dramatic increase in strength produced by

<~100 ppm of boron in TiC single crystals illustrates an important influence of chemical composition on the properties of the carbides. In this case, the impurity forms a second phase having a different crystal structure. No mechanism for significantly changing the properties of the carbides can be visualized at present when impurities are in

small quantities as long as they form solid solutions. This may be applicable to solutions of, for

example, TiN and TiO in TiC, although considerable differences in properties must be expected for

appreciable quantities of these isomorphs in solid

solution. In hardness tests, Cadoff et al. [53] have

shown that polycrystalline TiC containing 45 a/o

carbon and 5 a/o oxygen is more brittle than pure

TiC. The impurity effects may be additionally

important in polycrystalline samples and in sintered materials where equilibrium conditions may

not be reached or where grain boundary segregation may be important.

c. Temperature

The observations on the high-temperature

strength of VC and TiC are contrary to the room

temperature microhardness of these compounds.

At all loads, the room-temperature hardness of

VCn.84 appears to be below that of TiC of equivalent stoichiometry [42]. Gilman [54] has shown a dependence of hardness on the elastic constant

C44 in a group of materials of similar structure.

The simple concept that the stiffer the material

at room temperature the stronger it will be at

high temperatures cannot be applied. Chang and

Graham [55] have shown that C44 is lower for

ZrCn.94 than for TiCn.91, yet the former material

is stronger at high temperatures. In this case, the

rate of decrease of the elastic constants of TiC

with temperature is more rapid than that of

ZrC [55]. Over the temperature range of 0 to

300 °C, UC, which is isomorphous with these car- bides [56], has a positive temperature coefficient

for its elastic constants [55] and its hardness

should therefore increase with temperature if Gilman's [54] correlations are valid, although

these data are not available. The temperature

dependence of the elastic constants in these materials would be a significant contribution to

understanding their behavior.

Three additional variables which have to be

considered in polycrystalline materials are:

d. Crystal Structure

The presence of grain boundaries in a crystal

imposes a barrier to the propagation of slip, and

209
unless each grain conforms to the deformation of

its neighbors, grain boundary separation will occur. Taylor [57] and von Mises [58] have shown

that for a polycrystalline aggregate to deform, five

independent slip systems are necessary. As Groves

and Kelly [59] have pointed out, most ceramic

materials do not meet these requirements. In the

cubic oxide ceramics, for example MgO, slip takes

place on {110} <O"l0> systems. The number of

independent slip systems is two, and ductility is not shown in polycrystalline material until slip is activated on additional systems [60]. Since the

cubic carbides have the five independent slip systems, they are expected to be ductile in polycrystalline form at temperatures not far above the

brittle-to-ductile transition temperature found in

single crystals, although the grain boundaries

could be weak due to the atomic mismatch in

these directionally bonded materials. The hexagonal carbides and other similar ceramics are likely

to possess more limited ductility.

e. Porosity

The yield strength of ceramics has been shown

to decrease exponentially with increasing porosity

[61, 62]. Most of the published results compare the

strengths of given carbides at different porosity

levels, and this is very difficult to interpret on a fundamental basis. At low temperatures, porosity

gives rise to increased brittleness since the pores

act as crack nuclei, while at high temperatures

the pores may increase the brittleness by restricting grain boundary sliding [63].

f. Grain Size

It is well known that the yield strength and

fracture strength of metals and ceramics increases

as the grain size decreases. The behavior of TiC

is likely to follow this pattern. The room-temperature modulus of rupture of TiC has been shown to

increase from about 31 kg/mm2 (44,000 psi) at a

grain size of 12/xm to 51 kg/mm2 (73,000 psi)

when the grain size is 4^ [64]. These data cannot

be accurately described because of the variation

in porosity of intermediate grain sizes. For metals,

an Orowan-Petch [65] type of relationship, i.e.,

Oy a d~ 1/2

, is observed, while in some ceramics, for

example BeO [66], the variation is more accurately

described by ay a d~ l

.

3.2. Discussion of Properties

Very little use can be made of data which compare the properties of the carbides unless some

attention is paid to the variables discussed in

section 3.1. Unfortunately, nearly all the data in

the literature has been obtained on inadequately

characterized material, and although technologically useful, it does not readily lend itself to

contribute to an understanding of the properties

of the carbides.

The most recently available results on the hightemperature mechanical properties of polycrystalline carbides are those of Kelly and Rowcliffe

[67], who have shown that hot-pressed TiC0 .75 , VC0 .8„ NbCo.gg, NbCo.95, TaC, and WC, all behave

in a ductile manner in bending tests performed at

above 0.5-0.6 Tm . Their results are represented in

figure 17. At similar porosity levels, NbC0 . 95 exhibited higher strength than NbC0 . 88 , indicating

an increase in strength with carbon content in

this range of composition and at these temperaatures. The strength of TiC0 . 75 was also shown to

be greater than that of NbC0 . 95 , over the above

temperature range, which is in direct contrast to

the results obtained by Williams [20], who found

NbCo.75 single crystals to be stronger than those

of TiCo.97. However, in this work, the porosity level

was 97.5 percent for TiC0 . 75 and 92.5 percent for

NbCo.95. The weakest carbide of the group was

shown to be VC0 . 6 i- However, incipient melting of

the sample was observed above 2000 °C , suggesting

that the carbide was probably in the two-phase

(V2 C+VC) region, the former carbide melting at

about 2165 °C [15]. Hence the properties of this

lower melting, two-phase mixture probably cannot

be compared directly with TiC and the other

carbides.

Several attempts to determine the hot hardness

(a measure of the yield strength) of carbides have

been made. Westbrook [68] presents the microhardness of a number of carbides as a function of

temperature (fig. 18) but no details of composition are given and it is difficult to evaluate the

results. These results do suggest that the temperature dependence of the elastic constants is an

important property, as suggested earlier in this

section.

0

\ ^NbCo.95 - y

V's,v^/' NbC088

\v WC

1 1 7

s- VCo.61

1 1 1 1 1 1100 1500 1900 2300

TEMPERATURE,°C

Figure 17. Yield strength of refractory

carbides as a function of temperature.

(After Kelly and Rowclifle [67].)

210
4000

3000

04| 2000

v.o>

1000

111 z

Q

Or<I

O

try 300

500 400

200

"TiC HfC-\

ZrCwcA^;

_v

. ^V^^1 \ HfC

>^ v^ vc \^NbC

TiC

- ZrC

i i i i i i 0 400 800 TEMPERATURE, °C

1200

Figure 18. Microhardness of carbides as a

function of temperature.

(After Westbrook [68].)

Atkins and Tabor [69, 70] have overcome the

problems of softening and graphitization of the

diamond indenter at high temperature by using

a mutual indentation hardness technique. Their

technique provides a measure of the plastic

yield stress of the material, and measurements

for single crystal TiC are in close agreement with

those obtained in conventional compression test- ing. The hardness variation of a number of

carbides—TiC, VC, ZrC, NbC, WC—as a function

of temperature was also reported, but again a constructive comparison is denied by variable

porosity and composition. This mutual indentation

technique has also been used to determine the

activation energy for creep, which is again consistent with self-diffusion of the metal species in

the carbide. This type of behavior seems to be

established for the cubic carbides [71] including

uranium carbide [72].

A hot-microhardness device for use up to

2000° C has been used by Koester and Moak [73]

with various oxides and carbides. Using a B4C

indenter, a linear dependence between log hardness

and temperature was shown for TiC and NbC.

NbC is the stronger carbide at all temperatures

above room temperature, but no composition

specifications were given. This method has the

inherent disadvantage of possible specimen con- tamination by boron at elevated temperatures,

this impurity significantly affecting the strength

of the carbides.

A particular example where specimen characterization may have played an important part in

contributing to confusion in the literature is suggested by considering the properties reported

for TaC, figure 19.

Santoro [74] has shown a maximum in the hardness and a minimum in the room-temperature

strength at the same composition. He correlated

these changes with several other physical changes

2800 KHNM 2400

( 100 gm

)

„ , , 2000 Kg/mm'

KHN (o) 2800

(100 gm) 2400

Kg /mm2

" 2000

1600

_i i i_

\

1900°C

2000°C\

LOG

30 CREEP

3 RATE ( »

)

«/hr

0.3

150 TRANSVERSE .„ RUPTURE STRENGTH (•:

1,0 Kpsi 90

BEND

STRENGTH

Kpsi

0.7 0.8 CARBON METAL

0.9 RATIO

.a STEINITZ (1966)

>o SANTORO (1963) ° JOHANSEN 8 CLEARY (1966)

Figure 19. Mechanical behavior of tantalum

carbides as a function of carbon content.

in the material, notably a change in color, and

a maximum in the melting point. There is no

simple explanation for such conflicting mechanical

properties, although it may be suggested that

measurements of tensile strengths in brittle

materials at room temperature give results which

are open to question because of stress concentrating effects which lead to premature failures.

Qualitatively, the harder the material, the more

susceptible it is to cracking, and this may be of

some significance. In addition, a microhardness

measurement implies measurement of the internal

material properties, whereas the bulk tensile

properties are more likely to reflect the properties

of the grain boundaries in the carbide.

Measurements of the bending strength of TaC

at high temperatures by Johansen and Cleary

[75], however, support the trend shown by the

room temperature hardness measurements [74].

The material has a minimum strength at about

TaCo.g. In addition, the brittle-to-ductile transition temperature increases linearly with increasing

carbon content from compositions within the

TaC-r-Ta2 C two-phase field across the TaC region

to the two-phase TaC-f C field. However, if compositions with the TaC single-phase region are

considered, which would appear more justifiable,

then their data indicate a minimum in the transition temperature corresponding exactly with that

of the bending strength.

Steinitz [71], in the most recent data, indicates

that the room temperature hardness decreases and

similarly the creep rate at high temperatures

increases with carbon content over the composition range TaC0 . 8 to TaCi 0

. He suggests that this

decrease in strength with increasing carbon content is due to dislocation-vacancy interactions.

This may not necessarily be true! The increase

in strength of TiC with increasing carbon content

211
has been interpreted as being due in part to an

increase in the occupancy of the electron states

which are responsible for cohesion in the lattice.

It is possible that in a different carbide, the carbon

atoms may contribute electrons to anti-bonding

states in a given compound as a function of composition, and this change in electronic structure

would give rise to a decreased strength of the

material [26].

All these investigations have utilized TaC

produced either by hot pressing with 4 percent

Co as a binder [71, 75] or by carburization of Ta

wires [74]. Although fully dense, and apparently

of constant radial composition, the latter showed

a large variation in grain size. It would seem that

with these specifications, the material lacked

metallurgical integrity and that these most confusing data must be evaluated by careful investigation of the properties of TaC single crystals. The

possibility of ordering to give irregular variations

in mechanical strength, as in VC, must also not be

ignored. Ordering has not been detected in other

carbides except in TiC [76] at low carbon-to-metal

ratio 2 (0.5-0.6). It is likely that further application

of neutron diffraction, electron diffraction, and

nuclear magnetic resonance will compensate for

the lack of sensitivity of the conventional x-ray

techniques in detecting ordering in these

compounds.

It will therefore be appreciated that the present

degree of understanding of the mechanical behavior of the carbides is very incomplete. TiC

appears to deform in a manner similar to f.c.c. metals rather than rock-salt structure ceramics,

and the strength arises from the complex electronic interactions between constituent atoms in

the lattice. Diffusion of the atom species is important in determining the plastic behavior of this

material. Very little is known about other Group

IV carbides, although ZrC is stronger than TiC,

and there is some indication that the strength of

HfC increases with carbon content [77] similar to

that observed in TiC. The behavior of Group V

carbides is even less clear. Ordering in the carbon

sublattice has been shown to significantly affect

the behavior of VC, and initial results suggest

that the strength of this carbide passes through a maximum as carbon content is increased. Some of

the results presented for polycrystalline TaC

suggest a similar trend [71], although this is uncertain.

The author feels that it is very difficult to base a fundamental understanding of the behavior of

the carbides on the presently available data, much

which has been accumulated on poor, inadequately

characterized material. The situation would be

clarified by experiments performed as single

crystals produced from the melt [45, 78, 79, 80,

81]. In this way, undesirable specimen characteristics such as porosity could be eliminated.

2 In the Ti+TiC region according to some phase diagrams [15].

In particular, it would appear that measurements of such aspects of the mechanical behavior

as activation energies and volumes involved in the

flow process, in conjunction with diffusion ex- periments as a function of carbon content, could

lead to a more complete understanding of the

Peierls stress in these materials. With this knowledge, it would then be appropriate to discuss the

effect of controlled grain size, strain-rate, temperature, and composition, etc., on the mechanical

behavior. However, technological developments

are not likely to wait for these data. Our present

knowledge of these materials, although elementary,

does enable an evaluation of their properties for

high temperature structural applications to be

made, and this will be discussed in section 4. The carbides appear to be a particularly appropriate field of study in which to relate electronic structure to mechanical behavior. Lye [25]

has suggested that alloying may be expected to

change the relative occupancy of the electron

states which are responsible for cohesion in the

carbides. By appropriate alloying, therefore, it may be possible to adjust the relative occupancy

of bonding and anti-bonding electronic states such

that differences in strength, ductility and brittleto-ductile transition temperature may be produced. Such a phenomenon may also account for

the observed variations in properties of TiC and

TaC as a function of carbon content, where

carbon vacancies may be considered as the alloying

constituents. The electronic structure of the

carbides may be controlled by alloying with solid

solutions of other carbides, nitrides, and oxides.

Although ordering may introduce complications

to these studies, a concurrent investigation of both

electronic and mechanical properties should provide valuable information on the fundamental

properties of these materials.

The present understanding of carbide base alloys

is even less than that for the component materials.

The mutual solubility between carbides was summarized by Schwartzkopf and Kieffer [1], but

recent work suggests that many of these observations are incorrect [82]. One characteristic which

has been reported in many of these binary systems is that of a melting-point maximum [1]. Examples of the maximum stability compositions

include TaC-20 percent HfC, TaC-20 percent

ZrC and TiC-50 percent TiN. The TaC-20 percent HfC has the highest reported melting temperature for any material, 4000 °C [83] and this

has been associated with a hardness maximum

[84]. These maxima suggest that ordering may be

present in the solid state but this has yet to be

detected. The application of more sensitive experimental techniques to determine phase equilibria

and to generally characterize these materials is necessary. The scope for study of the properties

of these materials also clearly is vast.

212
4. Carbides as High-Temperature

Structural Materials

The carbides are of particular interest for hightemperature structural applications for three rea- sons [85] : (i) they include the materials exhibiting

the highest melting points ; (ii) they are extremely

strong; and (iii) at high temperatures, they deform in a manner similar to the ductile f.c.c. metals.

Because of their high brittle-to-ductile transition temperature, fabrication of the carbides

currently presents problems. If a relatively ductile carbide could be developed by alloying to

control electronic structure it would be of great

technological importance. A thermomechanical

treatment involving prestrain of a ductile carbide

solid solution (perhaps in some forming process)

followed by controlled high-temperature doping

should produce an extremely strong structure.

Venables [48] has suggested that precipitate

density and size, and hence the mechanical

strength, may be controlled by varying the dislocation node density prior to or during doping.

The notable effect of boron in increasing the hightemperature strength of TiC and VC must be

considered as "casual" observations, since no control of precipitate size to optimize these properties has yet been attempted. However, the

strength-to-density ratio of doped single crystals

of VCo.84 is 280,000 in at 1500 °C (fig. 20). TiCVC alloys show values in excess of this. In particular, that of the VC-50 a/o percent TiC alloy

appears to exceed 600,000 in at 1000 °C. It is possible that these values are greater than those

of any conventional materials, and are of particub

1000 1200 1400 1600 1800 2000

TEMPERATURE, °C

Figure 20. The strength-to-density ratio of VC-50

a/o TiC and boron-doped VC0 .s4 as a function

of temperature, compared with some other hightemperature materials.

lar interest because they exceed the target set by

the U.S. Bureau of Naval Weapons in 1964 [86].

Additionally, considerably higher values may be

obtained when these alloy carbides are hardened

by boron-induced precipitation. It is significant

to note that if the strength-to-density ratio is a

criterion for utilization, then it may not be the

carbides of highest melting point (NbC, TaC,

HfC) which will prove of greatest usefulness, since

these carbides are of considerably greater density

than TiC or VC, and would therefore be required

to be very much stronger.

If a relatively ductile alloy-carbide cannot be

developed by controlling the electronic structure,

then cladding with a ductile metal may be re- quired to facilitate forming operations. Alternatively, cast shapes can be produced, as has already

been demonstrated [87]. In recent years, one of

the major advances in turbine blade technology

has been the development of directionally solidified "monocrystaloys" of nickel-base alloys [88].

The elimination of grain boundaries in these

materials has produced marked improvements in

creep rupture properties and thermal shock

resistance in comparison with equiaxed or columnar grained structures. It would appear that

similar techniques could be applied with advantage to alloyed carbides.

Ductility has been observed in polycrystalline

TiC at temperatures above the brittle-to-ductile

transition in single crystals. For example, at

1500 °C, in coarse-grained TiC of about 2 mm

grain size, about 30 percent ductility has been

observed, the yield strength being about 10 times

that of single crystals of the same composition at

the same temperature [42]. Other polycrystalline

cubic carbides may behave similarly, and additional strength may result from grain refinement.

Conversely, porosity reduces strength and in- creases brittleness. It appears, therefore, that

fine-grain, fully-dense, alloyed, polycrystalline

carbides could provide useful mechanical strength

and ductility at high temperatures, and may

therefore be of significant technological importance

in the future.

The author is indebted to R. G. Lye, C. M.

Preece, J. D. Venables, and A. R. C. Westwood

for their helpful discussions and constructive

criticism of this manuscript. Special thanks

are due also to W. Precht for developing a crystal

growth technique and to D. L. Novak and

R. D. Huntington for their assistance in the

experimental programs on which some of this

work was based. The financial support of the

Research Division of the National Aeronautics

and Space Administration under Contract NASw1290 and the Army Research Office (Durham)

under Contract DA-31-124-ARO-D-467 is gratefully acknowledged.

213
5. References

[1] Schwartzkopf, P., and Kieffer, R., Refractory Hard

Metals, Macmillan, New York (1955).

[2] Kendall, E. G., in Ceramics for Advanced Technologies, John Wiley & Sons, New York, p. 143

(1965) .

[3] Shaffer, P. T. B., Materials Index, Plenum Press (1964).

[4] Samsonov, G. V., Properties Index, Plenum Press

(1964).

[5] Lynch, J. F., Duckworth, W. H., and Ruderer, C. G.,

Engineering Properties of Selected Ceramic Materials, Am. Ceram. Soc. (1966).

[6] Report of the Materials Advisory Board Committee

on Ceramic Processing, to be published (1968).

[7] Stokes, R. J., in Strengthening Mechanisms of Ceramics, Syracuse Univ. Press (1966).

[8] Gilman, J. J., Prog. Ceram. Sci. 1, 146 (1961).

[9] Gilman, J. J., and Johnston, W. G., Solid State Phys.

13, 147 (1962).

[101 Takahasi, T., and Freise, E. J., Phil. Mag. 12, 1 (1965).

[11] Corteville, J., Monier, J. C, and Pons, L., C.R. Acad.

Sci Paris, 6, 260, 2773 (1965).

[12] French, D. N., and Thomas, D. A., Trans. AIME

238, 950 (1965).

[13] Vahldiek, F. W., Mersol, S. A., and Lynch, C. T.,

AFML Tech. Rept. No. AFML-TR-67-321,

July 1967.

[14] Haggerty, J. S., and Lee, D. W., Arthur D. Little

Corp., Cambridge, Mass., private communication.

[15] Storms, E. K., The Refractory Carbides, Academic

Pi ess (1967).

[16] Williams, W. S., and Schaal, R. D., J. Appl. Phys. 33,

955 (1962).

[17] Williams, W. S., presented at Intern. Conf. SemiMetallic Compounds, Orsay, France, September

1965, to be published.

[18] Brookes, C. A., Special Ceramics, Academic Press,

London (1962).

[19] Hollox, G. E., and Smallman, R. E., J. Appl. Phys.

37, 818 (1966).

[20] Williams, W. S., J. Appl. Phys. 35, 1329 (1964).

[21] Maddin, R., and Cottrell, A. H., Phil. Mag. 46, 735

(1966) . [22] Ashbee, K. H. G., and Smallman, R. E., Proc. Roy.

Soc. A274, 195 (1963).

[23] Hollox, G. E., and Smallman, R. E., Proc. Brit.

Ceram. Soc. 6, 317 (1966). . [24] Eikum, A., and Smallman, R. E., University of

Birmingham, private communication.

[25] Lye, R. G., Atomic and Electronic Structure of Metals, A.S.M., Cleveland (1967).

[26] Lye, R. G., RIAS, private communication.

[27] Rowcliffe, D. J., Ph.D. dissertation, The Mechanical

Properties of Transition-Metal Carbides, University of Cambridge (1965).

[28] Kelly, A., and Rowcliffe, D. J., Phys. Stat. Sol. 14,

K29 (1966).

[29] Kronberg, M. L., Acta Met. 5, 507 (1951).

[30] Tobin, J. M., Adeslberg, L. M. Cadoff, L. H., and

Brizes, W. F., in Nuclear Applications of Nonfissionable Ceramics, Am. Nuclear Soc, Illinois (1966).

[31] Harrod, D. L., and Fleischer, L. R., Proc. Intern. Symp. Anisotropy on Single-Crystal Refractory

Compounds, Dayton, June 1967, to be published.

[32] Hollox, G. E., and Smallman, R. E., Proc. Brit.

Ceram. Soc. 1, 211 (1964).

[33] Keihn, F., and Kebler, R., J. Less Common Metals

6, 484 (1964).

[34] Haasen, P., in Proc. Conf. Relation between Structure and Strength, H.M.S.O., London, 587 (1963).

[35] Seegar, A., in Dislocations and Mechanical Properties

of Crystals, John Wiley & Sons, New York (1957).

[36] Henderson, B., J. Inst. Metals 92, 55 (1963).

[37] Lye, R. G., Hollox, G. E., and Venables, J. D., In Anisotropy in Single Crystal Refractory Com-

[38

[39

[40

[41

[42

[43

[44

[45

[46

[47

[48

[49

[50

[51

[52

[53

[54

[55

[56

[57

[58

[59

[60

[61

[62

[63

[64

[65

[66

[67

[68

[69

[70

[71

[72

[73

[74

[75

[76

[77

[78;

[79

pounds, Ed. F. W. Vahldiek and S. A. Mersol,

Plenum Press, New York, p. 445, (1965).

Hollox, G. E., and Venables, J. D., Proc. Intern. Conf. Strength of Metals and Alloys, Tokyo,

Supp. Japan Inst. Metals 9, 295 (1968).

Venables, J. D., Kahn, D., and Lye, R. G., Phil.

Mag. 18, 177 (1968).

Froidevaux, C, and Rossier, D., J. Phys. Chem.

Solids 28, 1197 (1967).

de Novion, C. H., Lorenzelli, R., and Costa, P., C.R. Acad. Sci. Paris 263, 775 (1966).

Hollox, G. E., Mater. Sci. Eng. 3, 121 (1968).

Lee, D. W., and Haggerty, J. S., Arthur D. Little

Corp., Cambridge, Mass., private communication.

Hume-Rotherey, W., Atomic Theory for Students of

Metallurgy, Inst. Metals, London (1960).

Precht, W., and Hollox, G. E., Proc. Inter. Conf.

Crystal Growth, Birmingham, to be published in

J. Crystal Growth, 1968.

Venables, J. D., RIAS, private communication.

Williams, W. S., Trans. A.I.M.E. 236, 211 (1966).

Venables, J. D., Phil. Mag. 16, 143 (1967).

Venables, J. D., and Westwood, A. R. C, unpublished

work. Also, Venables, J. D., RIAS Tech. Rept.

No. 67-2, March 1967.

Fleischer, R. L., Price, P. B., and Walker, R. M.,

Phys. Rev. A133, A1443 (1964).

Armijo, J. S., and Rosenbaum, H. S., J. Appl. Phys.

38, 2064 (1967;.

Rudy, E., and Winisch, S., Aerojet General Corp.,

Tech. Rept. No. AFML-TR-65-2, Part 1, Vol.

X, May 1966.

Cadoff, I., Nielsen, J. P., and Miller, E., Plansee

Procs. 2, 50 (1955.)

Gilman, J. J., NBS Mono. 59, p. 79 (1963).

Chang, R., and Graham, L. J., J. Appl. Phys. 37,

3778 (1966).

Frost, B. R. T., J. Nucl. Mats. 10, 265 (1963).

Taylor, G. I., Proc. Roy. Soc. A145, 362 (1934). von Mises, R., Z. anger. Math, und Mech. 8, 611

(1928).

Groves, G. W., and Kelly, A., Phil. Mag. 8, 877

(1963).

Stokes, R. W., in Materials Science Research, Vol. 1,

Plenum Press, New York (1963).

Glaser, F. W., and Ivanick, W., J. Metals 4, 387

(1952).

Coble, R. C, in Ceramic Fabrication Processes,

M.I.T. Press (1963).

Day, R. B., and Stokes, R. J., J. Am. Ceram. Soc. 49,

345 (1966).

Harrison, W. B., Honeywell Research Centre, Tech.

Rept. AD 612253, Feb. 1965.

Orowan, F., Repts. Prog. Phys. 12, 186 (1949).

Hyde, C, Battelle Mem. Inst., in ref. [62]. Kelly, A., and Rowcliffe, D. J., J. Am. Ceram. Soc. 50, 253 (1967).

Westbrook. J. H., and Stover, E. R., GE Rept. No.

60-RL-2656M (1960).

Atkins, A. G., and Tabor, D., Brit. J. Appl. Phys. 16,

1015 (1965).

Atkins, A. G., and Tabor, D., Proc. Roy. Soc. A292,

441 (1966).

Steinitz, R., in Nuclear Applications of Non-fissionable Ceramics, Am. Nuclear Soc, Illinois (1966).

Chang, R., J. Appl. Phys., 33, 858 (1962).

Koesler, R. D., and Moak, D. P., J. Am. Ceram. Soc.

50, 290 (1967).

Santoro, G., Trans. A.I.M.E. 227, 1361 (1963).

Johansen, H. A., and Cleary, J. G., J. Electrochem.

Soc. 113, 378 (1966).

Goretzki, H., Phys. Stat. Sol. 20, K141 (1967).

Adams, R. P., and Beall, R. A., U.S. Bureau of Mines

Report of Investigations No. 6304 (1963).

Kieffer, A. D., Linde Co., private communication in

ref. [16]. Haggerty, J. S., Wencus, J. F., and Lee, D. W., Proc. 3d Intern. Symp. High Temperature Technology,

Asilomar, Calif., Sept. 1967, to be published.

214
[80] Bartlett, R. W., and Halden, F. A., Stanford Research

Institute, Final Report on Contract NASr-49(19),

June 1967.

[81] Tobin, J. M., and Fleischer, L. R., Proc. 3d Intern. Sympo. High Temperature Technology, Asilomar,

Calif., Sept. 1967, to be published.

[82] Brukl, C. E., and Harrison, D. P., Aerojet General

Corp., Tech. Rept. No. AFML-TR-65-2, Part II, Vol. IV, Feb. 1966.

[83] Agte, C, and Alterthum, H., Z. Tech. Physik 11, 182

(1930).

[84] Deadmore, D. L., and Zaplatynsky, I., NASA TN

D-2768, April 1965.

[85] Hollox, G. E., Proc. 3d Intern. Symp. High Temperature Technology, Asilomar, Calif., Sept. 1967, to be

published.

[86] U.S. Bureau of Naval Weapons Target, 1964.

[87] Leeds, D., Kendall, E., and Ward, J. F., Aerospace

Corp., El Segundo, Calif., Tech. Rept. No. 55DTDR-63-216 (1963).

[88] Piearcey, B. J., and VerSnyder, P. L., Pratt and

Whitney Aircraft Co., Report No. 66-007, February

1966.

215

Fracture of Ceramics

S. M. Wiederhorn

Institute for Materials Research,

National Bureau of Standards, Washington, D.C. 20234

In this paper the fracture of ceramics is considered primarily from a fracture mechanics

point of view. The maximum theoretical strength of ceramics is discussed. The ideas of a

stress intensity factor and a fracture surface energy are developed. Finally, examples are presented of the use of these parameters to describe strength, thermal shock resistance

and stress corrosion of ceramic materials.

Key Words: Ceramics; delayed failure; fracture; fracture mechanics; fracture surface

energy; static fatigue; strength; stress corrosion; stress intensity factor;

thermal shock.

CONTENTS

Page

1. Introduction 217

2. Theoretical cohesive strength 217

3. Flaw theory of fracture 219

4. Continuum approach to fracture 220

4.1. Stress intensity viewpoint 220

4.2. Surface energy viewpoint 222

4.3. Measurement methods 224

5. Experimental fracture surface energy values 225

5.1. Glass 225

5.2. Single crystals 226

5.3. Bicrystal fracture energy measurements 226

5.4. Polycrystalline ceramics 226

6. Applications of fracture mechanics to ceramics 227

6.1. Strength of glass 228

6.2. Strength of single crystals 228

6.3. Strength of polycrystalline ceramics 228

6.4. Thermal fracture 233

6.5. Environmental effects 234

7. Conclusions 238

1. Introduction

Mechanical failure of ceramic materials is almost completely controlled by brittle fracture,

which means that the failure occurs with negligible

plastic deformation and without prior warning.

This brittle behavior, combined with the presence

of surface flaws leads to low strengths in ceramics

and, consequently, severely limits the mechanical

use of these materials. Investigations over the

past decade on the deformation and fracture

properties of ceramic materials have indicated

that brittleness is an inherent property of ceramics;

consequently, any improvement of their mechanical properties will necessitate designing around

this brittle behavior by understanding the causes and effects of brittle behavior and by developing

the necessary criteria for failure.

The primary purpose of this paper is to survey

some aspects of brittle fracture that may be of use in the design of ceramic structures. Particular

emphasis will be placed on the flaw theory of

fracture and its generalization to the continuum

approach of fracture mechanics. The ideas of a

surface energy and a stress intensity factor in

fracture will be developed and specific examples

of their uses will be given. The effect of plastic

deformation and environment on fracture will

also be discussed.

2. Theoretical Cohesive Strength

The theoretical cohesive strength of a material

depends on the physical and chemical forces

holding it together and, as such, gives a meaningful

measure of the stresses that must be overcome

during fracture. All other things being equal, the

material with the higher cohesive strength will

be stronger. Thus the cohesive strength provides

a means of comparing the potential strengths of

different materials and gives the maximum strength

to be expected from a given material.

This section will deal with a calculation of the

theoretical strength of two ceramic materials,

sapphire and glass, and will compare the calculated with the measured strengths of these

materials. The theoretical cohesive strength will

be shown to be approximately 100 times the

normal engineering strength of these materials.

The method of calculation is that described by

Gilman [l]. 1 Other discussions of the theoretical

cohesive strengths of materials may be found in

references [2-4].

Fracture is the process of separating a solid into

two parts, each of which becomes bounded by a new surface called the fracture surface. During the

separation process attractive forces act across the

incipient fracture surfaces, first increasing, then

1 Figures in brackets indicate the literature references at the end of this paper.

323-655 O - 69 - 15 217
decreasing with distance of separation. The maximum force corresponds to the theoretical cohesive

strength of the solid. This behavior may be

approximated by a potential energy function that

is characteristic of the material under study and

which relates potential energy per unit area of

fracture surface, to distance of separation. The

shape of a typical potential energy function and

its first derivative is given in figure 1. Two typical

potential energy functions are the Morse potential

and the Born potential, eqs (1) and (2).

?7=£70 [e-2a <*-*°>-2e-a <*-*o>] (1)

where U represents the potential energy per unit area of the fracture surfaces separated by a distance x. The stress on these two surfaces is equal to bU/dx and the cohesive strength is calculated from the fact that the stress is a maximum

when d

2 £7/dx2 =0.

The constants in eqs (1) and (2) determine the

shape and position of the potential function. They

are evaluated from measurable material properties

such as the surface energy, y, which determines

the depth of the potential minimum; the zero

stress separation distance, x0 , which determines

the position of the potential minimum; and

Young's modulus, E, which determines the curvature at the potential minimum. The general shape

of the potential function at points away from the

Figure 1. Binding potential function and first derivative.

potential minimum is largely determined by the

type of function selected to represent the potential.

Consequently, to calculate the maximum stress,

which occurs at the point of inflection of the potential energy curve, a potential function characteristic of the material under study should be selected.

In this regard, a Born-type potential would be

selected for ionic crystals, because it contains the

long-range coulomb energy term characteristic of

these materials, while a Morse-type potential

would be selected for covalent solids which exhibit

only short-range forces.

The use of the Morse and Born potential

functions to calculate cohesive strength will now

be illustrated for silica glass and sapphire respectively. The constants of eqs (1) and (2) must

first be evaluated. At equilibrium displacement,

x=x0 , the stress between the planes is zero. Consequently, (dU/dx) x = Xo =0. By definition, the

surface energy of a solid is equal to work necessary

to form a unit area of surface. Therefore,

27 = f a(x)dx=f (dU/dx)dx =u]'° =-U(x0 ) Jxq Jxq _]x0

where U(x0) is the value of the potential minimum.

At equilibrium displacement, x0, the change in

stress is directly proportional to the change in

strain, Hooke's law, do-=Edx/x0, from which

E=x0 (d2 U/dx2

) x=Xo. Finally, the maximum stress occurs when d

2 U/dx2 =0. Using these four conditions, it is easily shown that the maximum stress

for the Morse potential is

emax =^yE/±x0 . (3)

An equationn of this type was first derived by

Orowan [3] using a linear approximation for the

potential. The maximum stress calculated from

the Born potential isEV 2 f+Vn-l

aTM*=2 U+TJ ' (4)

where n=x0E/2y.

The theoretical cohesive strength of silica glass,

calculated from eq (3) using experimental values

ofE, y, and x0 , table 1, is 2.2X.1010 N/m2 (3.2 X106

psi) while that of sapphire, calculated from eq (4),

is 3.1X1010iV/m2 (4.5X106 psi). While these

values compare well with the highest strengths

measured on these materials, it may be concluded

that they are about 100 times larger than engineering strength values normally accepted for

these materials, table 2. This conclusion leads

naturally to questions of why ceramics are weak,

how they can be made stronger, and how their

strength can be characterized for design purposes.

These questions will be discussed in succeeding

sections of this paper.

218
Table 1. Theoretical strength of silica glass and sapphire,

(1010) plane

Parameter Fused silica Refer- ence Sapphire Refer- ence

7 4.56 J/m2 Table 3 7.3 J/m" Table 4 Xo 1.62X10-'° m [7] 2.74X10-'° m [6] E 7.2X101° N/m2 42.5X10'° N/m2 m

Cm&x 2.2X10'° N/m2 eq(3) 3.1 X10'° N/m2 eq (4)

In fused silica, the oxygen-silicon distance was taken

as i0 - In sapphire, the oxygen-oxygen distance was taken

as xa . If half the oxygen-oxygen distance is taken as xa in sapphire, then "max — 4.6 X1010 N/m2

.

Table 2. Measured strengths of silica glass and sapphire

Material Strength Reference

Flame-polished silica, -196* C.

1.35X10'° N/m2 (1.96X10° psi) [8, 9]

Silica glass fibers, -269 C ... 1.47X10'° N/m2 (2.1X10° psi) [101 Bulk silica glass 1X108 N/m? (15X103 psi) [14 Flame-polished sapphire 7X10' N/m2 (1X10° psi) [13 Sapphire whiskers 1.1X10'° N/m2 (1.6X10° psi) I"

Sapphire whiskers 1X10'° N/m2 (1.5X10° psi) [12 Sapphire, single crystals 3.5-10X10« N/m2 (50-150X103 [14 psi).

3. Flaw Theory of Fracture

The low strength and brittle behavior of

ceramics are almost entirely due to limitations on the

amount and type of plastic deformation of these

materials. Ceramics either do not deform plastically or possess an insufficient number of independent active slip systems to undergo generalized

plastic deformation. An important consequence

of this behavior is the fact that stress concentrations at crack tips or flaws cannot be relieved by

generalized plastic flow. As a result any flaw

contained within the body acts as a stress concentrator and is a potential nucleation site for

fracture.

This section will deal with the flaw theory of

fracture as developed by Griffith [15-17], who

assumed that small cracks in the body of brittle

materials were causing their low strength. Griffith

was able to prove this by investigating the elastic

stress distribution about an elliptically shaped

crack in a uniform tensile stress field, figure 2. Using the stress analysis of Ingles [18], he was

able to show that the tensile stress at the tip of

such a crack was

o* vv =2SjLTp, (5)

where L is half the length of the crack, p is the

radius of curvature at the crack tip, S is the

applied stress, and i/p»l. If one assumes the flaw

length to be of the order of 10~6 m and the radius

of curvature of the crack tip to be 10~10 m, then

the stress concentration is of the order of 100.

Therefore, materials that ordinarily rupture at a

stress of 104 psi would have crack-tip stresses of

the order of 106 psi, which is approximately equal

to the theoretical cohesive strength found pre2L

ii S

Figure 2. Elliptically shaped crack in an infinite two- dimensional plane.

viously. Thus, the flaw theory seems to adequately

account for the weakness of ceramic materials.

A second approach to the strength of materials

was also conceived by Griffith [15, 16] by using a mechanical energy balance to determine the condition for material fracture. At incipient failure,

the external work, dW, equals the change in energy

stored in the material, dV, plus the energy neces- sary to form two new surfaces, d£ =4*ydL, where

7 is the surface energy.

dW=dV+d£ (6)

From the elastic solution of the crack problem,

d(W-V) = [()(W-V)/bL]dL = (2irLS2/E)dL=iydL (7)

or S=^2Ey/irL, (8)

where S is now the critical applied stress for

failure. Equation 8 is the famous Griffith criterion

for failure. Note the similarity between eqs (8)

and (3) . If one were to assume the crack length in

the material to be of the order of the interatomic

spacing, the rupture stress calculated by eq (8)

would be of the order of the theoretical strength.

In addition, the strength calculated for a crack of

219
length 10~6 m is approximately one-hundredth of

the theoretical strength, as found previously.

Therefore, the energy balance approach for calculating weakening effects of cracks in completely

brittle solids is consistent with the stress concentration approach in that they both predict the

same degree of material weakening.

At first glance, the Griffith criterion for fracture

might seem limited because it deals only with

two-dimensional cracks and ignores the cohesive

forces that must be acting between the fracture

surfaces at the crack tip. In fact, the approach is applicable to a great variety of fracture problems.

Three-dimensional crack problems that have been

solved yield equations not greatly different from

eq (8) [19, 20]. Griffith recognized the fact that

forces act across the crack tip, but avoided

discussing them by assuming a finite radius of

curvature at the crack tip. Crack problems with

cohesive forces at the crack tip have been treated

by Barenblatt [21], Elliot [22], and others [23,

24], with the conclusion that stresses are everywhere finite and the crack tip is cusp-shaped.

These treatments are compatible with the Griffith

assumption of a finite radius of curvature provided that radius is assumed to be of atomic

dimensions. Therefore, Griffith's equation is valid provided an appropriate radius of curvature

is assumed.

Several authors [25, 26] have discussed the

relative merits of the stress concentration and

energy balance methods as criteria for fracture,

eqs (5) and (8) respectively. The stress concentration method assumes that a maximum stress must

be overcome for fracture, eq (5) , while the energy

balance method assumes fracture results after

the creation of maximum amount of stored energy.

The two approaches are equivalent for crack

radii of the order of the interatomic spacing;

however, for a large radius of curvature, the stress

concentration approach, eq (5), is more fundamental since it always presents the criterion for

fracture in terms of the maximum cohesive

strength, which means that fracture always re- sults when crvy ^> o-max . Unfortunately, this more

fundamental approach is not too useful since the

crack root radius cannot be measured in most

cases. In contrast, the energy approach can be used

to establish fracture criteria provided it is recognized that the surface energy term reflects the

chemical and mechanical processes occurring at

the crack tip during fracture. The development of

the Griffith ideas into useful fracture criteria will

be discussed in the next section.

4. Continuum Approach to Fracture

4.1. Stress Intensity Viewpoint

The basic reason for the weakness of brittle

materials is that cracks tend to redistribute internal stress fields, causing high stress concentrations at crack tips. Fracture occurs when the stress

fields near the crack tip reach a critical value.

This local criterion for fracture depends entirely

on the stress fields at the crack tip. To determine

the maximum load at fracture it is necessary to

relate the crack tip stress field to the crack shape

and applied load. Therefore, some understanding

of the stress distribution around a crack tip is essential.

The most general form for the stress distribution near a crack tip was given by Irwin [27] and

Williams [28], who noted that the stress field at

any point near a crack tip, figure 3, is related to the

distance, r, from the crack tip, the angle of inclination, 0, and a constant, K, called the stress

intensity factor. The radial dependence is always

inversely proportional to the square root of the

distance from the crack tip, r~ 1/2

, regardless of the

type of loading, crack geometry, or magnitude of

the load. The angular dependence is independent

of the magnitude of the load and flaw geometry,

but does vary with stress component and type of

loading. The stress intensity factor, K, is always

proportional to the magnitude of the applied

load and is related to the crack geometry and type

of load, but is independent of inclination angle 0. Before giving examples of stress field distributions,

three general types of loading will be discussed.

Stress fields near a crack tip can be related to

three basic modes of deformation, figure 4. Mode

I, the opening mode, is associated with displacements perpendicular to the fracture surface.

Figure 3. Coordinate relationship near the crack tip

After P. C. Paris and G. C. Sin [29].

Figure 4. Three modes of loading

After P. C. Paris and G. C. Sin [29].

220
Modes II and III are associated with displacements parallel to the fracture surface. Mode II

is called the edge sliding mode and is characterized

by shear displacements perpendicular to the crack

tip. The tearing mode, mode III, is characterized

by shearing displacements parallel to the crack

edge. The superposition of these three modes is sufficient to describe the most general type of

crack tip deformation in a general stress field.

It is necessary to distinguish among these three

modes for a particular application or test because

the stress intensity factor and angular distribution

depend on the mode of load application. Because

ceramic materials fail in tension, mode I is the

most important type of loading. Consequently,

most fracture tests in ceramics are designed for

mode I failure. Torsion tests on ductile metals

are an example of mode III loading.

The stress and displacement fields for mode I loading are listed below for the case of plane strain

deformation [29]. The notation used is given in

figure 3.

<rxx= [KI/(2Trr) 1/2

] cos 6/2 [1 -sin 6/2 sin 36/2] (9)

<x1/v =[KI/(2Tr) 1/2

] cos 6/2 [1 +sin 6/2 sin 36/2] (10)

<rxv =[KI/(2irr) 1/2

] sin 0/2 cos 6/2 cos 36/2 (11)

^zz =v(<Xxx +<Tyy), <TXi =<JVi =0 (12)

u=(KI/G)(r/27r) 1/2 cos 6/2 [l-2j/+sin2 6/2] (13)

v= (Kj/G) (t/2t) 1/2 sin 6/2 [2 -2f-cos2 6/2] (14)

w=0 (15)

Kx is the stress intensity factor for mode I loading,

u, v, and w are displacements in the x, y, and z directions, and G and v are the shear modulus

and Poisson's ratio, respectively. Equations for

mode II and mode III deformation may be

found in reference [29]. Stress intensity factors

for these types of loading are denoted by Ku

and KmThe most significant feature of eqs (9) through

(15) is the separation of the stresses and displacements into radial, angular, and stress intensity dependent portions. The angular and

radial distribution of stress about the crack tip are identical for a given mode ot loading. The

level of stress, or stress intensity, depends entirely on the stress intensity factor which, in

turn, is a function of the crack geometry and

method of loading. In fact, the stress intensity

factor is given by K^azS, where S is the applied

load or applied stress and aT is a geometric factor, in this case for opening mode failure. The

geometric dependence of the stress intensity

factor can be obtained by solving the appropriate

elasticity equations, see reference [29]. Figures 5

through 7 give important examples of test specimen geometries for which the stress intensity

factor has been solved [30-32]. In all cases, Kz

tan 7TLw

Figure 5. Center-notched plate tension specimen con- taining a crack of length 2L.

After J. E. Srawley and W. F. Brown [30].

p / /

w

l c /

t /

2L = 8W

2 2 2 r t K

I =(-7-) —~ 31.7 c/W - 64.8(c/W) 2 +211 (c/W) 3

V I ' yy2 <- J

Figure 6. Edge-cracked bend specimen containing a crack of

length c. For four-point loading see ref. [30]. After J. E. Srawley and W. F. Brown [30].

is dependent on the applied load S, the crack

length and the specimen dimensions. A wide

variety of other geometries is given in reference

[29].

The radial dependence of the stresses in eqs

(9) through (11) leads to the conclusion that the

stresses are infinite at the crack tip, r==0. This

conclusion is, of course, erroneous and arises from

the assumption of a true continuum in deriving

221
2t <A\\\\\\\X? 7\

y — Lo" — (X

Figure 7. Double-cantilever specimen, Ki= (PL/Wt1 - 5

) (3.^7 +^.3/ j t/L).

eqs (9) through (11). Since matter is not continuous, but discrete, deviations from eqs (9)

through (14) will occur in brittle materials at

distances from the crack tip of the order of several

atomic dimensions, say 10-9 m. In ductile materials crack blunting occurs and deviations from

these equations result over even greater distances,

depending on the amount of plastic flow. If the

discrete nature of matter or size of the plastic zone is regarded as a disturbance in the continuum, then it can be concluded from St. Venant's

principle that the effect of the disturbance is

negligible at large distances with respect to the

size disturbance and that the real stress fields

approach those given by eqs (9) through (11) for

sufficiently large distances. It is usually assumed

in the continuum approach to fracture that the

disturbance is small and is contained within the

limits of applicability of eqs (9) through (11).

This assumption is quite adequate for ceramics

which exhibit little plastic relaxation at crack

tips.

Because the stress fields near the crack tip of a real material approach those given by eqs. (9)

through (11), a one-to-one relationship will exist

between the maximum tensile stress at a crack tip

and the value of the stress intensity factor at

fracture. The strength of a material can therefore

be characterized by a critical value of the stress

intensity factor, usually denoted KiC for openingmode failure. KIC is a material constant dependent

on physical processes occurring at the crack tip

during fracture, but independent of crack geometry, specimen shape, or test procedure. It characterizes the inherent difficulty of propagating a crack in a material and provides information that

is independent of the flaw distribution in the

material. For a given load condition and crack

geometry, the maximum load at fracture is obtained by equating KY to K1C - A valuable use of K1C is to predict the size of a

plastic zone at a crack tip. The estimation of the

plastic zone size presented below is based on the

Dugdale [33, 34] model of plastic flow at a crack

tip. The plastic zone is assumed to extend a distance R in front of the crack, figure 8. The length

of the zone is R=(ir/8)(Klc/<rv) 2

, (16)

where K1C is the critical stress intensity factor

and <j v is the yield stress of the material. When

the plastic zone extends from the crack tip a distance R, the sides of the crack become displaced a distance 2V(C) at the crack tip.

V(C)=4ayR/vE. (17)

Equations (16) and (17) are valid provided the

breaking stress is much less than the general yield

stress of the material, a condition that holds for

most ceramics. Equations similar to (16) and (17)

have been derived for other plastic flow models

[35]. Equations (16) and (17) are valuable because

they relate the extent of plastic flow at the crack

tip to macroscopic measurements of yield stress

and critical stress intensity factor at fracture.

4.2. Surface Energy Viewpoint

Another method of treating fracture problems

is the surface energy method, which is based on

the fact that work is associated with the creation

of new surfaces. As envisioned by Griffith, this

energy was considered to be the thermodynamic

free energy for the surface. In most cases, however,

the energy associated with a fracture surface is not the surface free energy, but is much greater

than the free energy of the surface. Irwin [17]

and Orowan [3, 26] first noted that the surface

energy in eq (8) should be written y=ys -\-yp

where ys is the surface free energy and yP represents the amount of plastic work going into the

formation of the surface. A number of other surface

energy terms may enter into eq (8) . For example,

if crack propagation were accompanied by a chemical reaction a term yc, representing the

energy of reaction per unit area of surface formed,

would have to be subtracted from 75. Similar

surface energy terms would be added for other

processes and the total fracture surface energy y

could be larger or smaller than the surface free

energy 75. In metals, for example, yP is typically

104 J/m2 while ys is 1 J/m2

. Consequently, 7~7p.

CRACK 2V(C) PLASTIC

ZONE

Figure 8. Dugdale model of plastic flow near a

crack tip.

After G. R. Hahn and A. R. Rosenfield [34].

222
A general relationship between the fracture

surface energy and experimental parameters can be derived using a mechanical energy balance

[17, 37]. Consider a crack in a solid, subjected to a constant load P, figure 9. As the crack extends

the work, dW, performed on the solid by the ex- ternal load must equal the change of internal

strain energy, dV, j lus the change of surface

energy, d£, dW=dV+d£- The work by the

external load is Pdu, where du is the differential

displacement of the ] ad P as the crack extends

a small amount. Th< total elastic strain energy

of the body is V-X Pdu. For a linear elastic

body, the displacement is proportional to the

applied load, u=\P, where A is called the compliance of the system. It therefore follows that

the total elastic energy is V=u2/2\=T>u/2, and

at constant load dV=%Pdu—}idW. The change

in surface energy is 2ydA where 2dA is the amount

of surface formed as the crack propagates. As

mentioned above, the surface energy per unit

area of surface formed, y, includes contributions

from chemical and mechanical processes associated

with crack motion. Substituting expressions for

dW, dV, and d£ into the energy balance, the

following equation is obtained

7= (P/4) (bu/dA) P = (P2/4) ®\/bA) P. (18)

'////////////////////A

Figure 9. A body containing a crack that is extended a

distance, dL, by a force P.

Displacement of the point of load application is du. After P. C. Paris and G. C. Sin [29].

For specimens of constant thickness, t, dA=tdL

where dL is the increase in crack length. Equation

(18) then becomes

y=(P2/4t)(d\/dL) P. (18a)

As discussed below, eq (18a) provides a general

experimental means of measuring y. Irwin originally expressed many of his results in

terms of a strain energy release rate, Q, which is

just equal to twice the fracture surface energy,

y. In parallel to the stress intensity factor definitions, Qic, Que, and Qnic represent the critical

strain energy release rates for crack motion, modes

I, II, and III respectively. They are material

constants dependent on the physical processes

occurring at the crack tip during fracture, but

independent of crack geometry, specimen shape

or test procedure. Finally, it should be noted that

eq (18) is valid for fixed grip as well as fixed load

conditions.

The relationship between the surface energy

and stress intensity approaches to fracture can be easily proven [17, 27, 37]. The calculation will

be given for opening mode failure and then the

general relationship between stress intensity factors and strain energy release rate values will be

presented. As a result of these relationships, the

two fracture viewpoints may be used interchangeably.

Consider the crack shown in figure 10 and

assume that the crack faces are gradually loaded

in some reversible manner to cause the crack to close a distance a. Prior to closing the crack, the

surfaces are stress free and the contour of the

crack surfaces is described by eqs (13) through

(15). After closing the surfaces, the stresses are described by eqs (9) through (12) and the crack

surfaces are coplanar with the surface y=0.

The reversible work to close the fracture sur- faces is

Qlc ={2la)^(^<rvvv^dx. (19)

Other possible energy terms containing (axxu)/2

or (axyv)/2 all equal zero. The stress o-yv is given

by eq (10) with r=x and 0=0. The displacement

v is given by eq (14) with r=a-x and Q=ir. Substituting eqs (10) and (14) into eq (19) and integrating, one finds,

Qxc—

{l-v)Klc* (l-v2)KIC

2 26 E

Values for other modes of displacement are

SO (1 —V

2) K 2 y IIC— 55 XV E

5*- IIIC Jji

IIC >

K 2 -^MIIC •

(20)

(21)

(22)

223
a

Figure 10. Closure of a crack a

distance a by applying the

stresses of equations (9)— (11) to the free surface of the crack.

After P. C. Paris and G. T. Sih [29].

For combined modefailure Q=2y=Qi-\-QII -\-QIn, which constitutes a generalized Griffith condition

for crack propagation. Equations (20) and (21) are

for plane strain. Equations for plane stress are obtained by dropping v

2 from eqs (20) and (21).

Equations (20) through (22) are given for isotropic media. Other types of media are discussed

in the literature [29] and relationships between the

stress intensity factors and strain energy release

rates values are available.

4.3. Measurement Methods

There are three experimental methods for

determining the critical fracture parameters, KiC

and Qic- In each of these methods K1C and QIC are

related through eq (20). One method requires a solution of the crack problem for the specimen

geometry under consideration so that KT can be

expressed as a function of applied load, crack size,

and sample dimensions. K1C is obtained from a strength measurement on the material being

studied. At failure KIC =Ki. Therefore, KIC is calculated from the specimen dimensions, crack

size and critical load at failure. For a given

set of physical circumstances, KIC should be independent of test method.

Several practical test geometries are shown in

figures 5 through 7. The center notched plate

tension specimen (fig. 5) and the edge-cracked

bend specimen (fig. 6) have been used extensively

in the testing of metals and the dimensions suggested are for metallic materials. The double

cantilever specimen (fig. 7) also known as the

infinite crack-line loaded edge-crack or Majoine

specimen, has been used in various forms for both

metals and ceramic materials.

The second method is called the compliance

method and requires the use of eq (18a) to determine Q. 1C - A series of specimens is prepared containing a range of crack sizes. As the specimen is loaded, the relative displacements of the loading

points, u, must be measured as a function of P

for each crack length. Since \=u/P, X can be

plotted as a function of L and dX/dL can be determined from the slope of the plot at any crack

length. Because the derivative of X is important in

the determination of Qic, displacements and loads

must be measured to a high degree of accuracy.

The main limitation of this method is the displacement measurements. The method has the

great advantage of being applicable to any specimen shape and can be used to check elastic solutions

of crack problems.

In the third method of measurement, the work

to propagate a crack over a complete cross section

is determined [38, 39]. The test is conducted in

three point bending and edge cracked bend specimens are used (fig. 6). This type of test requires a testing machine with a large spring constant.

Specimens are slowly deformed at a continuous,

constant rate throughout the test, and the test is continued until specimens have been completely

fractured. The method depends on the fact that

as the crack propagates, the compliance, X, decreases, and crack propagation occurs continuously and stably with gradually decreasing load.

The force-displacement curve during the test is as depicted in figure 11, and the area under the curve

is equal to the total work performed during the

experiment. There is no elastic energy stored in

the testing machine at the completion of the test,

since the machine is load free once the crack has

passed through the specimen. Consequently, all of the work performed during the experiment has

gone into the fracture process and the strain

energy release rate, Qic, can be determined by

dividing the total work by half the total fracture

area. The three methods of measurement just described are based on three different criteria for

fracture. Method 1, the stress intensity method,

is based on the assumption that crack motion

occurs when the stress intensity in the vicinity of

the crack tip has reached a certain level. The

critical stress intensity will depend on the physical

224
5.1. Glass

DISPLACEMENT, U Figure 11. Force displacement curve for the work of fracture

method of determining the fracture surface energy.

processes occurring at the crack tip, but not on

crack size or shape.

In the second method of measurement, the

compliance method, fracture occurs when the net

mechanical energy input into the body equals the

surface energy required for the production of new

fracture surfaces. The method is differential in

that energy values are expressed in the limit as the amount of new fracture surface area approaches zero. Thus, the second method, as the

first, is only concerned with initial crack motion.

The equivalence of the two methods is proven by

eqs (20) through (22).

The third method, the work of fracture method,

is also based on an energy criterion for fracture.

It differs from the second method in that the

fracture surface energy is determined by an integral technique. In other words, the surface energy

is determined by an averaging procedure over the

entire fracture cross section. Fracture surface

energy values obtained by the third method are related more to crack propagation than crack

initiation. This difference has been emphasized

recently by Davidge and Tappin [40], who classify fracture surface energy measurements with

reference to crack initiation, yT , or crack propagation, yF. Their experiments on graphite, alumina,

polymethylmethacrolate and glass show that ji

and yF are not necessarily equal, but depend on

physical processes occurring during fracture.

5. Experimental Fracture Surface

Energy Values

The available fracture surface energy data will

be reviewed in this section, first for glass, then

for single crystals and bicrystals, and finally for

polycrystalline ceramics.

Glass provides an excellent material for testing

the various ideas presented in previous sections.

It is elastically isotropic so that linear elastic

theory is applicable. It can be made homogeneous

so there is no concern with the interaction of the

crack tip with internal flaws or grain boundaries.

Finally, glass is one of the most brittle materials

known, so there is little concern with plastic

flow at crack tips. The one limitation of glass

is its extreme susceptibility to stress corrosion.

The data presented will be that for which there

was little or no stress corrosion.

Table 3 presents a summary of data for various

glasses tested in various environments. Most of

the data were obtained by the double cantilever

method; however, two sets were obtained by

other means. Agreement among the various test

methods is good. Variations in measured fracture

energy due to differences in composition and

temperature were as high as 50 percent, suggesting a maximum difference of about 25 percent

in the inherent cohesive strength of silica based

glasses. The surface energy values of table 3

can be substituted into eq (3), yielding maximum

tensile strengths of approximately 2X1010 N/m2 (3X106 psi), which agree within a factor of 2

with data available in the literature, table 2.

Table 3. Surface energies of glass, J/m2

Double cantilever specimens, method I

Glass Toluene-C02

Fused silica _ 25 C4. 32

4.00

3.82

4.75

4.65

3. 50

-196 C

4. 56

4. 17 4.53 4.80

5. 21 4.11

-60 C

4. 86 4.60

4. 38

96 percent silica... Soda lime . . Borosilicate Aluminosilicate - High lead...

Plate glass 3.4-5.2, 25 C, ref. [38], modified edge crack bend specimen,

method III. Soda-lime glass 4.06, Vacuum, ref. [41], center notch plate tension specimen,

method I.

It is of some interest to use eqs (16) and (17) to

estimate the size of the plastic zone at the crack

tip. The yield stresses for soda-lime glass and

fused silica at liquid nitrogen temperature are 1X1010 N/m2 and 1.95X10 10 N/m2 respectively

[42]. Data for the critical values of the stress

intensity factor are given in table 3. The lengths,

of R, the plastic zones for soda-lime glass and

silica glass are 2.6X10~9 m and 6.4 X10~10 m

respectively. The displacements, V(C), for the

same glasses are 4.5 X10-10 m and 2.2 X10-10 m

respectively. The very small value of the calculated plastic zones satisfies all of the assumptions

of fracture mechanics and, in addition, dramatizes

the reason for the very brittle nature of glass. In

the absence of large amounts of plastic flow

there is no mechanism for absorbing energy during

crack motion. As a result, a crack once started

tends to propagate catastrophically.

225
5.2. Single Crystals

The fracture of single crystals depends on the

crystallographic orientation of the fracture plane

and occurs more easily along those planes having

the lowest fracture surface energy [4]. In extreme,

but common, cases, this behavior leads to cleavage

fracture, where fracture occurs only on a single

crystallographic plane characterized by a fracture

surface energy considerably lower than any other

plane in the crystal. In less extreme cases, an

orientation dependence of the fracture strength

is observed, as in the case of sapphire [43]. Thus,

a complete characterization of the fracture behavior of single crystals requires the determination

of y, Qi, or Kic as a function of the orientation

of the fracture plane.

Most fracture surface energy determinations on

single crystals have been made on crystals that

cleave. Thus, the energy values published are

characteristic only of the cleavage planes in the

crystal. Since many crystals that cleave exhibit

fracture on no other crystal planes, these fracture

surface energy values completely characterize the

fracture in these materials.

A summary of some of the fracture surface

energy data on crystals that cleave is given in

table 4. The data obtained by the double cantilever

technique have been recalculated by the author

according to the equation in figure 7, which was

not available at the time of the original publications. The data presented in table 4 were obtained on specimens tested in dry environments or in

vacuum. The test conditions were such as to

eliminate environmental effects and to reduce

plastic flow at crack tips. Since the results presented in table 4 were close to those expected

theoretically, plasticity and environmental effects

may have been eliminated.

Table 4. Surface energies of single crystals

Crystal

Mica, vacuum, 25 C

Mica, vacuum, 25 C

LiF, N2U), -196 C

MgO, N2 (l), -196 C

CaF2 , N2 (l), -196 C

BaF2 , N2 (l), -196 C

CaC03 , N2 (l), -196 C

Si, N2O), -196 C

MgO, vacuum, —175 C

NaCl, N2U), -196 C

NaCl, vacuum, —175 C

Sapphire, (1123) plane, -196 C

Sapphire, (1011) plane, -196 C

Sapphire, (2243) plane, -196 C

Sapphire, (1123) plane, 20 C...

Sapphire, (1011) plane, 25 C...

Sapphire, (1010) plane, 25 C.__

Reference

[44]

[45]

[46]

[46] [46]

[46]

[46] [46]

[47] [47]

[47]

[48]

[48]

[48]

[48]

[122] [122]

Data from references [46] and [47] were recalculated using the equation in

figure 7. The values given represent averages of the three smallest surface energy values in each group. Data from reference [48] were obtained on

center-notched plate tension specimens. Data from the other references were obtained on double cantilever specimens. Morphological indices are used for sapphire, c/a=1.365.

Fracture surface energy measurements on noncleavable single crystals have been obtained only

on sapphire, a—A12 03

. The rather meager available data are presented in table 4. Energy values

obtained by Petch et al. [48] are about 10 times

that expected theoretically. The authors explain

the high values as due either to energy absorption

by plastic deformation or to the possible fact that

the drilled hole did not offer a suitably sharp

crack from which fracture could initiate. Considering the low fracture energy values obtained

by Wiederhorn [122], it is improbable that plastic

deformation plays a role in increasing the surface

energy for fracture. Mechanical twinning induced

by fracture has been observed by several investigators and may be playing a role in the (1011)

fracture energy measurements [49-52].

5.3. Bicrystal Fracture Energy Measurements

Bicrystals occupy an intermediate place between

single crystals and polycrystalline solids. They

contain a single grain boundary, which is characterized by the relative orientation of the crystals

on each side of the boundary. Because polycrystalline ceramics fracture primarily along grain

boundaries, a measurement of grain boundary

fracture energies is important. In theory, the

grain boundary fracture energy can be obtained

on specimen geometries such as those given in

figures 5 through 7, provided the boundary is coplanar with the crack. Class and Machlin [53]

have performed this type of experiment on twist

boundaries of KC1 using the double cantilever

method for their study. Their results, figure 12,

indicate that grain boundaries are weaker than

single crystals. For large angle grain boundaries,

the fracture surface energy is less than 50 percent of that for normal cleavage. These first ex- periments suggest the necessity for additional

work on bicrystals.

5.4. Polycrystalline Ceramics

Fracture surface energy values obtained on

polycrystalline ceramics, table 5, are an order of

magnitude greater than those obtained from single

crystals. In addition, for small grain sizes the

fracture surface energies of polycrystalline MgO

and A1203 increase with increasing grain size,

table 5. Since the strength of a material is ex- pected to be proportional to the square root of

the fracture surface energy, eq (8), polycrystalline

ceramics should be stronger than single-crystal

ceramics and strength should increase with increasing grain size. From the best available

strength data on polycrystalline ceramics [56-58],

these conclusions are erroneous. Single crystals are stronger than polycrystals and strengths increase

with decreasing grain size. Consequently, it appears there is no correlation between strength and

fracture energy measurements for polycrystalline

ceramics. As will be discussed in a later section

226
160

I I I I I I

I I I I 0 10 20 30 40 50

Twist Angle (degrees)

Figure 12. Fracture surface energy of KC1 bicrystal twist boundaries.

The upper curve gives the fracture surface energy of the (001) surface and the twist angle refers to the direction of crack propagation on the (001) surface. The lower curve gives the fracture

surface energy of the twist boundary. The angle of twist is given by the ordinate. After W. H.

Class and E. S. Mechlin [53].

of the paper, this lack of correlation suggests

that the strength of polycrystalline materials

depends more on single crystal and grain boundary

fracture surface energies than on polycrystalline

values of fracture surface energy. The polycrystalline values of surface energy are valuable for some

types of mechanical and thermal shock.

The high fracture energy of polycrystalline

ceramics is due to the tortuous nature of the

crack path in these materials [54, 59], figure 13.

Crack perimeters must extend through and around

many grains and can be pinned at many points

along the boundary by high-energy obstacles

consisting of poorly alined grains and grain

boundaries. As a result, the crack must overcome

all of these obstacles to propagate, requiring an

amount of work greater than is necessary to

propagate in single crystals. Some energy absorbing processes for fracture have been discussed

recently by Clarke et al. [54, 59] and include

secondary crack formation, plastic flow, area in- crease due to roughness of the fracture surface, and

step formation which leads to high-energy tearTable 5. Surface energies of polycrystalline ceramics

Material Density Grain size Energy Reference

Firebrick, Douglas X

metenXlO-11 J/m2

30 [54] MgO _ 3. 48 10 16 [54] MgO _ 3. 51 50 19.5 [54 MgO __ 3. 56 100 35 [54 MgO, theoretical density. 3.58 7 4. 2 [54 MgO , theoretical density. 3.58 13 8.9 [54 MgO, theoretical density. 3. 58 23 16 [54 MgO, theoretical density. 3.58 38 17 [54 MgO, theoretical density _ 3. 58 130 14 [54 MgO, theoretical density. 3.58 150 7.9 [54; BeO 2.79 5 15 [54] Graphite 100 [54; SiN 2.2

3. 97

3. 97

3. 97

20

1827

46

[54; [55

[55 [55

AI2O3, Lucalox

AI2O3, Lucalox

AI2O3, Lucalox

1030

45

f

Figure 13. Crack propagating through polycrystalline A12 03

.

Note the random nature of the crack path. After E. L. Coble, ref. [14].

p. 623.

ing, mode III fracture. The exact role played by

each of these processes has not yet been elucidated.

6. Applications of Fracture Mechanics

to Ceramics

Discussions of previous sections were phenomenological in outlook. No attempt was made to

relate the fracture process to fundamental phenomena such as microstructure, chemical composition, and ability to deform plastically. Instead,

fracture was assumed to be controlled by three

experimental parameters, fracture surface energy,

Young's modulus, and flaw size. From an engineering point of view, this approach is valuable

because it provides a convenient means of classifying materials. The fracture mechanics approach

will be continued in this section where it will be

227
applied to problems of strength, thermal shock,

and stress corrosion. Where applicable, the phenomenological approach will be correlated with

the microscopic approach. The effect of plastic

deformation, grain size, chemical composition,

and crystal structure on the fracture surface

energy will be discussed. For an excellent coverage

of the microscopic aspects of fracture, the reader

is referred to the literature, references [60-62].

6.1. Strength of Glass

Materials that are homogeneous, isotropic, and

completely brittle with absolutely no plastic

flow, are the simplest to classify mechanically.

Crack propagation in these materials is not influenced by the presence of grain boundaries,

cleavage planes, plastic deformation, or stress

inhomogeneities due to anisotropic thermal expansion. Fracture surface energies are isotropic, but

may depend on environment. Many inorganic

glasses fit this category and in the absence of

environmental effects have surface energies given

in table 3. Since these energies are nearly independent of composition, as is Young's modulus

for these glasses, fracture is almost completely

controlled by flaws contained in these glasses

[63, 64]. Experiments have shown that flaws are almost entirely limited to the external surfaces.

It follows that glass can be strengthened either

by eliminating surface flaws or by preventing

them from propagating. The latter approach has

been used in the recent development of surface

toughened glasses [65]. The surfaces of these

glasses are put into a state of compressive stress

either by chemical or thermal treatment. Surface

flaws and cracks then cannot result in fracture

until the surface stress has been overcome by the

applied load. In this manner, the strength of

ordinary glass has been increased from 10,000 psi

to 70,000 psi, an increase of sevenfold. The ideas

of surface strenghtening have been extended to

recrystallized glasses with even more impressive

results [65].

6.2. Strength of Single Crystals

As noted by Stokes [62], the mechanical behavior

of ceramics may be divided into three categories.

Depending on the amount and type of plastic

deformation that can occur, ceramic crystals are completely brittle, ductile, or semi-brittle.

Completely brittle crystalline ceramics cannot

be deformed plastically as dislocations are immobile and all deformation is elastic. As a result,

failure occurs by brittle fracture. Generally,

covalent crystals at less than 0.5 Tm are in this

category where Tm is the melting temperature.

Since inorganic glasses fit this category, the conclusions and statements concerning the strength

of glass are equally valid for completely brittle

single crystals. Fracture is governed by accidental

surface flaws, the elastic constants and the surface

energy, the latter two depending on crystallographic orientation of the fracture plane. In the

absence of surface flaws, single crystal strengths

exceed 106 psi.

Ductile crystals can be deformed quite easily.

Dislocations move readily with a great deal of

flexibility in choice of slip plane. Ductile crystals can be deformed into arbitrary shapes by applied

loads, and stress concentrations can always be

relieved by plastic flow. As a result, stresses never

exceed the cohesive strength of the solid and

failure occurs by plastic instability, in which the

solid necks to a point and then separates. Plastic

behavior is demonstrated by the silver halides [66]

at low temperature and by many ionically bonded

materials at high temperatures [67, 68], greater

than 0.5 Tm. Semi-brittle crystals can be deformed plastically

but, in contrast to ductile solids, cannot be deformed into arbitrary shapes by external loads.

As a result, stress concentrations cannot be

relieved by random shape changes and fracture occurs when local stresses exceed the cohesive

strength of the material. This behavior results

from the fact that the number of active slip

systems in these materials is insufficient for ductile

behavior. In addition, screw dislocations in semibrittle crystals are limited to specific slip planes

and cannot maneuver from one plane to another

by cross slip [62]. Semi-brittle crystals fail by

fracture and are sensitive to the presence of surface

cracks and notches [66, 69]. Fracture is also caused

by stress concentrations due to slip band interactions with grain boundaries [70-74] and other

slip bands [75]. Semi-brittle crystals consist of

ionic materials below 0.5 I

1 , and covalent crystals

above 0.5 Tm. The type of failure of a solid may be qualitatively related to its plastic properties via the

Von Mise criterion [76-78], which states that at

least five independent slip systems must be active

for ductile behavior. Any number less than this

results in semi-brittle behavior and the complete

absence of slip results in completely brittle behavior. The plastic behavior of several ceramic

materials is given in table 6. Most ceramics are

semi-brittle or completely brittle at room temperature. As the temperature is increased above 0.5

Tm , ceramics become semi-brittle or ductile.

For further discussion of the topics presented

in this section, references [61, 62] are suggested.

6.3. Strength of Polycrystalline Ceramics

In this section, a discussion will be presented of

some of the factors that effect the strength of

polycrystalline ceramics. Crack nucleation due to

grain boundary blocking of plactic deformation

will be discussed first. Then a theory of crack

nucleation due to anisotropic thermal expansion

will be presented. Finally, the effect of grain

boundaries on crack propagation will be discussed.

228
Table 6. Plastic behavior of ceramics

Crystal

Crystallographic

slip systems No. of independent

systems

Category of mechanical behavior

Low temperature

NaCl (e.g., NaCl, LiF, MgO)

CsCl, (e.g., CsCl, CsBr)

(110) (110) (1001 (010) (001) (110) 1110) (110) (001) (110) (111) (HO)

23

3

5

Semibrittle.

Semibrittle. Semibrittle. Ductile. CaF2 (e.g., CaF2 , BaF2 ) AgCl (e.g., AgCl, AgBr)...

High temperature

Hexagonal (e.g., graphite, AI2O3, BeO) - TiOj -

(0001) (1120) (101) (101) (110) (001) 1110) (110) (001) (HO) (HI) (110) (001) (110) 1110) (110)

2

4

5

5

High-temperature semibrittle. High-temperature semibrittle.

High-temperature ductile.

High-temperature ductile.

NaCl (e.g., NaCl, LiF, MgO)

CaF2 , (e.g., CaF2 , BaF2 , U02 )

Taken from ref. [79]. Original sources, refs. (62, 78, 121].

Crack nucleation due to grain boundary blocking

of plastic deformation was first suggested by Zener

[80], developed by Stroh [81] and most recently

discussed by Smith and Barnby [82]. Suppose a

dislocation slip band has impinged upon a grain

boundary as depicted in figure 14. Since the slip

planes on either side of the boundary are not

alined, the dislocations will be blocked at the

boundary, setting up a stress field at the point of

impingement. The stress field is described by an

equation very similar to that obtained for a freely

slipping crack under a shear stress [81], mode II

deformation,

o-={o--ol)(Llryi*j{6) (23)

where as is the resolved shear stress along a slip

band of length L and o- t is the shear resistance of

the material referred to as the friction stress. To

a first approximation the friction stress is the

minimum stress for dislocation motion.

The stress field at the crack tip will nucleate

plastic deformation in the adjacent grain and the

stress field will be relieved by plastic flow if the

material is ductile. If the material is not ductile,

but semi-brittle, stress will build up at the grain

boundary and fracture will occur when the stresses

at the boundary exceed the cohesive strength of

the solid. For a Morse type potential the following

equation is obtained as a condition for failure,

Vt^/4x0 = (a- Cj-) (L/ry*f(B) . (24)

Assuming L is proportional to the grain size, D,

and cr

s is equal to half the applied tensile stress,

<r s =y2 S, eq (24) can be rearranged to give a Petch-type relationship [83],

S=<t0 +JcD-1 /2 (25)

Figure 14. Slip band of length, L, impinging on a grain boundary to nucleate a

crack of length 2C.

Applied tensile stress is S and resolved shear stress along the slip plane is as

which relates the stress to nucleate fracture, S,

to the grain size, D, and the minimum applied

tensile stress, a0 , for dislocation motion.

The constant, k, can be evaluated by obtaining

a relationship between L and the grain size D

and by evaluating j{d) . Eshelby et al. [84] give

L=D/4: while the maximum value of jiff) is approximately 1. Substituting these values into eq

(24) and setting r=x0 , k=(4:Ey) l/2 is obtained.

The exact value of k depends on assumptions made

in the derivation of eq (25). Using an energy

balance and a more exact procedure Stroh [81]

obtained k=[3iryE/(l — v

2)}

1 '2

, which is about 1.7

times that given above.

A second mechanism for crack nucleation was

given by Clarke [85] and is based on the fact that

internal stresses occur in polycrystalline ceramics

229
due to anisotropic thermal contraction or radiation

induced anisotropic expansion. Fracture is assumed

to initiate at grain boundary pores and then to

propagate along the grain boundary, figure 15,

absorbing energy from both the external and

internal strain fields. These fields arise, respectively, from the applied stresses and the expansion

anisotropy. A Griffith type energy balance is used

to derive the critical stress for fracture

<j={E[2yb -Et\l-c0 )l\2{\-v2)]/irc o yi2

, (26)

where e is the grain boundary strain; the grain

size is 21; yb is the grain boundary surface energy;

and c 0 is the grain boundary pore diameter.

Equation (26) is essentially the Griffith equation

with an additional term to account for the thermal

expansion strain energy release per unit area of

crack formed. The equation predicts spontaneous

fracture when the term inside the square brackets

equals zero,

2y b =Ee2 (l- CQ)/12(l-v2

) (27)

or for c 0 <cZ,

e=(.24y b/Eiy<2

. (28)

The validity of this last equation has been demonstrated recently by Davidge and Tappin [86] on

polycrystalline BeO.

Once a crack has been nucleated, it must be

able to propagate for complete fracture to occur

[87]. Obstacles, such as grain boundaries, may

cr

Figure 15. Model for crack initiation at a grain boundary

-pore due to thermal stresses.

After F. J. P. Clarke [85].

hinder crack growth and thus prevent complete

fracture. The increase in stress required for a crack to abruptly change directions at a grain

boundary has been recently calculated by Gell and

Smith [88], and by Barnet and Tetelman [60].

The method of treatment is identical to that used

for dislocation pileups at grain boundaries.

Consider a crack that has been momentarily

arrested at a grain boundary, figure 16. Crack

motion is assumed to reinitiate either at some tilt angle, 6, to the original crack or at a twist angle, <f>, The stress distribution around the arrested crack,

given by eqs (9) through (12) may be resolved

along the projected plane of propagation by the

appropriate tensor transformation. For the case

of a two dimensional Griffith crack reinitiating

along a tilt plane, 0, these stresses are,

on =Ksl{2irryi* (29)

o-22 =KT/(2irry2 (30)

where Ks and KT are given by,

Ks =-S(way<2 sin 6/2 cos2 6/2 (31)

KT =S(wd) l/2 cos

3 6/2 (32)

0-12 and o-2 2 are respectively the shear and tensile

stress on the projected plane of propagation. The

stresses on these planes are identical to those

obtained on the projection of the original crack

plane provided combined mode I and mode II

deformation is assumed with stress intensity

actors KX =KT and Kn =Ks . Therefore, the condition for crack reinitiation for the case of plane

strain is given by the sum of eqs (20) and (21),

Ks

2 +KT

2 =2Ey/{\-v2). (33)

The criterion for reinitiation on the new plane

therefore becomes

a = [2Ey/Td(l-v2)Y/

2 sec2 6/2. (34)

Note that the condition for crack reinitiation along a new direction is the Griffith equation modified

by the angular dependence sec20/2. This conclusion

is also true for other crack shapes.

Figure 16. Change of crack plane orientation as crack

traverses a grain boundary.

a) tilt propagation, b) twist propagation. After M. Gell and E. Smith

[88].

230
The treatment by Gell and Smith [88] considers

the nucleation of a new crack at the edge of a penny

shaped crack of radius d. The equations for propagation on a plane ot tilt 6, or twist <p, figure 16, can be obtained by methods identical to those

given above. The reinitiation stress required for

the tilt and twist case respectively are

cr=[7r7#/2d(l-?2 )]

1/2 sec

2 6/2 (35)

<T=[^El2d{l-v2)]

1 ' 2 sec2

<p. (36)

Again, the Griffith equation for crack motion is modified by trigonometric functions of 6 and <p. Equations (34) through (36) give the condition for

reinitiation of crack propagation and are rigorous

only in this limit, since changes in the stress field

due to the reinitiated crack are not considered.

The increase in stress for crack reinitiation

depends on the angle of tilt or twist as would be

expected. For fracture along grain boundaries, tilt propagation is most important, since all grain

boundaries meet at tilt angles. If a polycrystalline

aggregate is assumed to be a nested assembly of

truncated octahedra (ref [14], page 410) the angle

of tilt will always be 60°. In a real material,

however, the angle of tilt scatters around this

value and ranges from 0 to 90°. Since the larger

angles will control the ease of fracture, it might

be expected that tilt angles ranging from 60° to

90° will account for the more important obstacles

to crack propagation. Tilt angles in this range are equivalent to surface energy increases of from 1.8

to 4 times the single crystal or grain boundary

surface energy value. Similar increases in measured

surface energies can occur for twist propagation,

which plays an important role in limiting transgranular crack propagation [88].

In the above discussions of crack nucleation

and propagation, a critical length and a surface

energy parameter were always involved in establishing the criteria for fracture. In general, both of

these parameters must be considered to decide

which of the possible mechanisms controls fracture.

In the dislocation pileup mechanism, eq (25) the

length parameter is the grain diameter, while the

surface energy parameter is the grain boundary or

single crystal surface energy, table 4. For the

thermal expansion mechanism, eq (26), the parameters are the pore diameter and the grain boundary

fracture energy. For the crack propagation

mechanism, the critical surface energy depends on

the size of the fracture nucleus. If the starting

crack is several grain diameters in size, the critical

surface energy will be the same as that measured

on polycrystalline specimens, table 5. If the starting crack is of the order of a grain diameter in

size, then the critical surface energy will be the

grain boundary or single crystal fracture surface

energy, possibly modified by a trigonometric

function as in eqs (34) through (36). These

mechanisms will now be discussed with regard to

experimental results.

Conclusive experimental evidence of the pileup

mechanism in MgO at room temperature has been

given by several people. Ku and Johnston [71]

proved the validity of the Petch equation, eq (15),

for bicrystals of MgO. D was shown to be the

distance between the origin of the glide band and

the grain boundary, a0 was shown to be the stress

for dislocation multiplication at room temperature

rather than the minimum stress to move dislocations, and k gave a qualitative measure of the

cohesive strength. The value of the friction stress,

(To, was 5X107 N/m2 (7.25 XlO3 psi).

Stokes and Li [70] investigated the effect of

mobile dislocations on the strength of MgO and

found that bicrystals possessed a tensile strength

of over 7 X108 N/m2 (100,000 psi) in the absence

of dislocation mobility, but only 7 XlO7 N/m2 (10,000 psi) in the presence of dislocation mobility.

The reduction in strength resulted from an interaction of the dislocations with the bicrystal grain

boundary. Stokes and Li also showed that dislocation mobility reduces the strength of 100 percent dense polycrystalline MgO from 20X 107 N/m2 (30,000 psi) to between 10 and 14X107 N/m2 (15-20X103 psi).

Recently Rice presented evidence for the dislocation pileup mechanism in hot pressed and hot

extruded MgO [89]. Fracture data on both types

of material were found to give a straight line on a Petch type plot. The calculated friction stress was

4X107 N/m2 (6,000 psi), which is close to the

value obtained by Ku and Johnston. Fracture was

transgranular so that the initial source of fracture

could always be identified. Microscopic examination of the sources of fracture gave evidence of

slip band pileups at the source of fracture. Thus

there can be no doubt as to the importance of

dislocation mobility to the strength of MgO at

room temperature or of the importance of the

pileup mechanism to crack nucleation in polycrystalline MgO.

The expansion anisotropy mechanism of crack

nucleation has been demonstrated on polycrystalline

BeO and also may be important in A12 03

. The

experimental evidence on BeO was obtained by

Davidge and Tappin [86], who used neutron

irradiation to develop expansion anisotropy strains

in BeO. Spontaneous cracking along grain boundaries was observed when eq (28) was satisfied,

with 76 assumed to be about 1 J/m2 (103 ergs/cm2). Further evidence for fracture induced by thermal

expansion anisotropy was presented by Coble

[90] who demonstrated spontaneous cracking in

completely dense A1203 , Lucalox, for grain sizes

greater than 100 /xm.

Crack propagation as a limiting mechanism

failure has been suggested by Davidge and

Tappin [86], who have shown that polycrystalline

BeO does not exhibit significant weakening until

cracks ot the order of 1 to 3 grain diameters have

appeared. Consequently, strength in BeO is believed to be limited by crack propagation from

231
flaws 1 to 3 grain diameters in size with a surface

energy of about 10 J/m2. An interesting correlation for fracture of MgO,

BeO, and A1203 was presented by Carniglia

[91, 92] who plotted published strength data of

several authors [56, 93-95] as a function of D~1/2 and obtained bimodal curves such as the one

shown in figure 17 for MgO. Fracture data from

the upper portion of this curve was attributed to

plastic flow and could be represented by the

Petch relationship, eq (25) . The friction stress can be obtained by extrapolating the upper portion

of the curve to infinite grain size and should be

of the same order of magnitude as the yield

strength of the material. Fracture behavior

depicted by the steeply sloped portion of the curve was attributed to elastic crack propagation

from internal cracks or flaws of the order of a

grain size in dimensions and could be represented

by a Griffith type relationship, eq (8) where L is proportional to the grain size. At high temperatures the upper portion of the curves for BeO

and AI2O3 had negative slopes that were attributed

to grain boundary sliding. Thus, failure of polycrystalline BeO, AI2O3, and MgO is believed to

be limited by crack growth for large grain size,

by crack nucleation due to plastic flow or grain boundary sliding for small grain size.

Although Carniglia's explanations of fracture

behavior of ceramics might seem reasonable,

additional research is necessary to prove his views,

since the shapes of the fracture curves are not

sufficient evidence in themselves to prove his

viewpoint.

One drawback to Carniglia's correlation is its

failure to predict friction stresses for MgO, A1203 ,

CO 200 100 50 20 10 5 2

10 20 30 40 50 60 70

(D)-' /2 (cm" 1/2)

Figure 17. Petch -plot of MgO bending data.

After S. C. Carniglia [91, 92]

and BeO that agree with other data in the literature. For example, the friction stress for MgO

obtained from figure 17 is 2.35X108 N/m2 (34,000

psi). This value is about five times the friction

stress obtained by Ku and Johnston, 5X107 N/m2 (7.25X103 psi) on bicrystals of MgO; four times

the stress required to initiate yielding in MgO

single crystals [96], 5.5 X 107 N/m2 (8X103 psi);

and six times the friction stress obtained by Rice

on polycrystalline MgO, 4X107 N/m2 (6X103 psi).

Similarly, friction stresses of 2.9 X108 N/m2 (42 X

103 psi) and 2.5X108 N/m2 (37X103 psi) are obtained for AI2O3 and BeO at room temperature

despite the fact that single crystals of these

materials are not known to deform plastically at

room temperature [43, 123].

Some comment should be made on the relationship between the fracture energy measurements presented in tables 4 and 5 and the observed

strength of MgO and A1203 . It is possible to use

the Griffith equation for a penny shaped crack

[19], a=^irEy/D(l — v

2), to calculate the strength

of fracture of polycrystalline MgO and A1203 from the data in table 5. Calculated strength

results for 100 percent dense MgO and A1203 are

in agreement with measured values provided

crack nuclei are assumed to be 10 grain diameters

for MgO and 10 to 20 grain diameters for A1203

. Nucleation cracks of this magnitude have never been reported in these materials, therefore, it is reasonable to conclude that the fracture surface

energy values listed in table 5 are not to be

associated with the strength of these materials.

In contrast, a similar calculation on the strength

of MgO and A1203 using surface energies characteristic of the single crystal, table 4, gives a

predicted flaw size of approximately one to three

grain diameters, which is in better agreement

with that expected. It therefore appears that the

strength of these materials is controlled either

by crack nucleation or propagation from nuclei

the order of a grain size in diameter, In either

case, the surface energy is of the order of the

single crystal value.

Discussions of the preceding sections have been

limited to three materials, alumina, beryllia, and

magnesia, and to mechanisms that are important

at low temperatures. It is felt that the mechanisms

presented have general applicability to many other

types of ceramics at low temperatures. At high

temperatures, where diffusion can occur, fracture

is affected by other phenomena such as grain

boundary sliding and Nabarro-Herring creep [97].

Figure 18 is illustrative of the type of failure that can occur at high temperatures during creep. This

type of behavior has not yet been treated by

methods of fracture mechanics, however, there

may be important applications of fracture mechanics in this direction.

232
Figure 18. Creep fracture of initially dense AI2O3 which has

been deformed at high temperatures in a bend test.

After R. L. Coble, reference 14, p. 619.

6.4. Thermal Fracture

Fracture is often observed to occur in ceramic

materials after a rapid change in temperature

known as thermal shock. The fracture results

from stresses that arise from nonlinear temperature gradients or restraints to thermal expansion.

The thermal stresses causing fracture depend on

the rapidness of the temperature change and on

the shape of the body being shocked. To determine

the onset of fracture, it is necessary to solve the

heat transfer equations and obtain the temperature distribution in the structure from which the

stress distribution can be calculated. A review of

the results of such analysis has been given by

Kingery [98]. Failure is deemed to have occurred

once the thermal tensile stresses exceed the tensile

strength of the material, as in the case of static

loading. Thus a maximum stress criterion is used

to describe thermal failure of ceramic materials.

(Table 6.)

Hasselman [99] and Clarke et al. [54] have suggested an exception to the maximum tensile

strength criterion for thermal shock failure. Their

criterion for failure is based on the availability of

strain energy for crack propagation. During crack

propagation, strain energy imparted to the structure by thermal shock is converted into fracture

surface energy. The amount of strain energy

available for fracture is limited by the severity

of the shock. If the shock is not too severe, the

conversion of strain energy into fracture surface

energy may be complete before structural integrity

is lost. Structural materials may crack in a limited

way due to thermal shock, but still may be able to serve their original design function. Thermal

failure therefore contrasts with constant load

failure for which the available elastic strain

energy is always sufficient for complete failure

once a critical load is exceeded. Mechanical shock

and failure under fixed grip conditions is similar

to thermal shock in that the initial elastic strain

energy is fixed so that crack propagation may

cease before structural usefulness has been destroyed. In the following paragraphs the approach

taken by Clarke et al. [54] will be used to illustrate

thermal shock.

In engineering practice it is always useful to

have a number that can be used to classify materials for a given end use. For the shock conditions described above, a useful parameter is the

ratio of the fracture surface energy to the elastic

strain energy available for fracture, V. Since this

ratio may be large or small, the logarithm of the

ratio will be more manageable for engineering

purposes. Thus, the thermal shock resistance can be defined by a parameter r, where

T =logl0(y/V). (37)

This definition is essentially the same as the one

suggested by Clarke et al., differing by an additive

and multiplicative constant. Larger values of r indicate greater resistance to thermal shock.

To evaluate r , the surface energy, 7, and the excess

elastic strain energy must be determined. The

fracture surface energies can be determined by

methods described earlier in the paper. The excess

strain energy can be estimated as follows. The

strain imposed by thermal shock at a free surface

of a solid is always of the form e=AaAT [98],

where a is the thermal coefficient of expansion, AT

the temperature difference between the initial

body temperature and that of the environment and

A is a constant that depends on the rate of heat

transfer, thermal conductivity, and the dimensions

of the structure. The value of A is usually less

than 1. The stress at the free surface of the material

is proportional to the thermal strain up to the

fracture stress. At higher strains, cracks form so that the surface stress cannot exceed the fracture

stress of the material. For purposes of approximation, it is assumed that crack formation releases

strain energy in such a manner that the surface

stresses remain constant as the material fractures.

The stress-strain relationship of a region near the

surface of the structure is assumed to be as depicted

in figure 19. The excess strain energy for fracture

is given by the shaded portion of figure 19. In

quantitative terms,

V= I <xde =(o-/E)(EAaAT-a

J aIE

) (38)

where a is the strength of the material and E is Young's modulus. From eq (38) it may be con- cluded that fracture does not initiate unless

a=EAaAT, which is Kingery's maximum stress

criterion for thermal failure [98]. Therefore, the

assumption used in the derivation of eq (38) are consistent with the treatment given by Kingery.

233
<

li.

A Q AT

SUFACE STRAIN , €

Figure 19. Assumed stress-strain curve in the surface layer

of a thermally shocked ceramic.

Substituting eq (38) into eq (37), the following

equation is obtained for thermal shock resistance,

r=log10 [yE/ff(EAaAT—<r)]. (39)

For identical amounts of overstressing, EAaAT —cr= constant>0, the resistance to thermal

fihock increases with increasing Young's modulus

and surface energy, but decreases with increasing

strength. The increased thermal shock resistance

with decreasing strength has been noted previously [99].

Clark et al. have shown that a thermal shock

ennntion similar in form to eq (39) compares

favorably with other methods of determining shock

resistance. Data presented by Clarke et al. are shown in tables 7 and 8. A comparison of r with

the British Standard Thermal Shock Test, table

7, for commercial alumina ceramics shows that

the thermal shock resistance of the various ceramics occupy similar positions on both scales. The

fracture surface energy can also be used to measure

the thermal shock resistance for these ceramics,

table 7. For a broader range of materials, the

fracture surface energy cannot be used as the only

criterion for fracture, as can be seen from table 8. The high thermal shock resistance of firebrick,

table 8, is due to its low strength.

6.5 Environmental Effects

Up to this point fracture has been assumed to occur in the absence of environment, controlled

only by flaw geometry and inherent physical

properties of materials. This assumption is very

poor for ceramic materials, which despite their

chemical inertness, are very sensitive to stresschemical effects in external environments. In

this section, the effect of environment on strength

will be discussed and the role of Kj and y will be

described.

Perhaps the first evidence for strength reduction

by external environment was made by Grenet [100]

who observed a time dependence in the strength

of glass. Glass lathes loaded rapidly were stronger

than those loaded slowly. In addition, a time

delay to failure was observed in which glass lathes

would support a given load for a period of time

before failure. All of these observations were

caused by water vapor in the atmosphere, although Grenet did not realize this at the time.

The phenomenon first observed by Grenet has

been termed static fatigue or delayed failure and

is common to many ceramic solids.

Table 7. Thermal shock of commercial alumina

10 6 1,18 14 „ 19 13 20 9

I M I/" Ml/

b 10

1

I T parameter t

|

i

t (10~3

)

YF (erg /cmf

104)

AT (°C).

1

6 18 1,11 2 14 9 13 19,20

-3

10 6 18ii4 2tl

f 2 10

t -. it

150

1319 r

10 11 6 18

I \l

2 14 1

] —

3

1 6,13 4—

L

170 + —I190

-2

20 131? 9 1^1 , 4 5 6

14,20 9,18 11 19

Comparison of a series of commercial aluminas, a) and b) are two different methods of calculating

t. c) represents the strain that can be accommodated before a fresh amount of cracking can occur,

d) is the fracture surface energy determined by method 3. e) is the British Standard thermal shock test. The numbers refer to the specimens in ref. [120]. After F. J. P. Clarke, H. G., Tattersall, and G. Tappin [54].

234
Table 8. Thermal shock of ceramics

Material ( 10" ergs cm' 2

)

of

cm 2

)

a H0 » °c->)

Young's Modulus

(I0' 2 dyn. cm 2

) r

Firebrick' 3 01 60 0 13 + i

Polycrystalline alumina" No. 9 No. 10

5-3 21

2- 2

3- 45

7-55 7,

3-5 3-9 -2-9

-4-5

Single-crystal alumina 111 1-2 3-5 7 0 30 -4-8

Polycrystalline magnesia 1 *

3-48 g.cm--1

, lO^m

3-51 g.cm 50fim

3-56 g.cm lOO^m

Theoretical density. 7finV

1 6

1-95

3-5 042

U *+ 0-7 10

2*5

1

2

1212

1 Z

30

30

3 0

3-0

-317

-3-5 -3-2 — 6-9

Single-crystal magnesia11 <d-2 10 12 3 0 -6-8

Polycrystalline beryllia*" 1 1-5 20 7-0 3-4 -4-2

Range for commercial

Zero porosity

7(i Magnesia

Sapphire

Single crystal Magnesia

aluminas

Beryllia Magnesia

100H 50m | r«H

Morganite

firebrick

-4 -3 -2 -1 Toughness Parameter, T.

Comparison of some ceramics in respect to their resistance to thermal

shock. Specimens are the same as those given in table 5. After F. J. P.

Clarke, H. G., TattersaU, and G. Tappin [54].

More recent experimental results on glass

clearly showed, the effects observed by Grenet to

be a form of stress corrosion caused by water

vapor in the environment. Reviews of the subject

may be found in references [63] and [100-102].

Most studies were made using rods or lathes in

three of four point bending from which stress- time-to-failure curves were obtained as a function

of temperature and envhohment, figure 20. From

such studies it was concluded that delayed failure

did not occur in the absence of water vapor. A

lower stress limit for failure was inferred from the

long-time strength limit of the stress-time-tofailure curve. This limit, called the static fatigue

limit, was approximately 20 to 30 percent of the

short-time strength [64, 105]. Charles was able to

show that delayed failure was an activated process

with an activation energy of 18.8 kcal/mole* [106],

about the same as that obtained for the corrosion

of glass by water [107]. Finally Mould and Southwick [108] were able to reduce data from specimens

of various surface treatments to a single curve,

called the universal fatigue curve. Theories of

delayed failure must satisfy the universal fatigue curve to be acceptable.

Recently Pearson [109] and Charles and Shaw

[58, 110] were able to demonstrate a moisture induced, time-dependent weakening of single and

16

£ cr|2

c —

jc O Q

!§CD —

- \

-

6600

hr. |

1 IN 1 III 1 III 1 III 111 1 III 1 III

I0< io- 10°

Duration of Load ( sec

Figure 20. Stress time to failure curve for annealed

soda-lime glass rods, }{ in diam, tested in bending.

After E. B. Shand [64].

polycrystalline alumina. The experimental observations were very similar to those on glass and

it is reasonable to conclude that the static fatigue

of sapphire is also caused by a stress corrosion

process. The activation energy for the processes

is 14 kcal/mole and the static fatigue limit is about

50 percent of the short-term strength for single

crystals. Static fatigue on other ceramics has also

been observed and is apparently quite common

[111].

The theory that has been most successful in

explaining static fatigue in ceramic materials is the one proposed by Charles and Hillig [112, 113].

They assume that static fatigue is caused by a

stress enhanced chemical reaction that occurs most

rapidly at crack tips where stresses are high.

Cracks extend at subcritical stresses under the

influence of the chemical reaction until the Griffith

condition is satisfied, at which point rapid fracture

ensues. Because of the general nature of this

theory, it is worthwhile examining some of its

underlying premises and conclusions.

Chemical reactions are well described as activated processes, in which reactants combine to

form intermediate unstable chemical complexes

that decompose to form the final reaction products

[114]. The rates of reaction are controlled by the

rate of formation of the activated complex. If

AF* is equal to the free energy of formation of

the activated complex from the reactants and

AFP * is the free energy of formation of the activated

complex from the reaction products then according

to the absolute rate theory, the rate of reaction

will be

K= (k T/h) [exp ( - AF*/RT) -exp ( - AF*/RT) ] ,

(40)

where k is Boltzmann's constant, h is Planck's

constant, R is the ideal gas constant, and T is the

temperature. If the reaction is assumed to be a heterogeneous chemical reaction between a ce- ramic surface and a corrosive environment, then

the rate or velocity of recession of the surface, v, will be proportional to the reaction rate K. Finally,

if AFT *^>AFP

*, the velocity of recession is

•The thermochemical calorie denned as 4.184 J Is used here. v=v0 exp(-AFr */RT), (41)

235
which is the starting point for the Charles-Hillig

theory.

The activation energy, AFr *, is the difference

between the free energy of formation of the activated complex and that of the reactants. Since

free energies of formation of materials are stress

dependent [115], it follows that activation energies

of chemical reactions are normally stress dependent. For simplicity it will be assumed that

the stress dependent activation energy may be

expanded into a Taylor's series as a function of

stress,

AF*(<r) =AF*(a=0) + I a +

Oct (t=o

(42)

geometry on stress corrosion process. Their

criterion for the static fatigue limit is that the

ratio of the crack tip stress to the applied load

<t/S, should be time independent, which occurs when d(L/p)/dt=0. Charles and Hillig obtain the

following equation for the static fatigue limit:

S*=(8yVM)/(4V*JTp), (45)

which differs from that given above by a numerical

factor of 3/2.

Equation (45) may be generalized for any

crack geometry, mode I failure, by expressing it in terms of the stress intensity factor KIt which

for an elliptically shaped crack is S^f^^L. Therefore,

eq (45) becomes

where only the first two terns of the expansion

have been retained. To simplify notation let

K/=(37T^/4F*)V^. (46)

AF*(a=0) be E0* and

bAF*(<r)

5<r be -V*.

<r = 0 E* is the activation energy of the chemical reac- tion in the absence of stress and V* has units of

volume per mole and has been called the activation volume by Charles and Hillig.

In a stress corrosion process, the chemical reac- tion is most rapid at the roots of cracks, which

act as stress concentrators. Since crack roots are curved, the effect of curvature on the activation

energy must be considered. From thermodynamics,

the free energy difference between a flat and

curved surface is yVM/p where VM is the molar

volume of the solid and p is the radius of curvature

of the surface [116]. This free energy difference

will tend to oppose the motion of the crack and is therefore subtracted from eq (42) . With curvature

taken into account, the velocity of recession of

the crack tip under the combined effects of stress

and chemical reaction is

v=v0exp-[Eo*-V*<r+yVM/p]/RT. (43)

Equation (43) may be used directly to obtain

an approximation for the static fatigue limit.

A reasonable criterion for the static fatigue limit

is that the velocity of the crack tip should be

equal to the general rate of surface recession, a

condition that is satisfied when

V*a=yVM/p. (44)

If the crack tip is assumed to be elliptical in shape

then the'crack tip stress, a, is related to the applied

stress, S, by the Inglis relation [18], a=^L/p/S, which

when substituted into eq (44) gives S*=(yVM)/

(2V*^Lp) for the static fatigue limit.

A more exact procedure followed by Charles

and Hillig accounts for the dependence of crack

The critical stress intensity factor for the static

fatigue limit is seen to depend inversely on the

square root of the crack tip radius of curvature.

Charles and Hillig [112] have shown that during

stress corrosion, dp/dL~— 2V*a/RT= — 10, which

means that during stress corrosion a small change

in crack length is accompanied by a decrease in

flaw tip radius. The flaw tip radius cannot de- crease indefinitely and will be limited by the

atomic nature of the solid. Therefore, p will be of

the order of 10~ 9 to lO -10 m.

A good stress corrosion theory should not only

describe the stress dependence of material failure,

but also the environmental dependence. The

Charles-Hillig theory is particularly well suited

for this purpose since the activation energy, E*,

can be related to the difference between the

thermodynamic chemical potential of the various

reactants [117]. For example, consider a gaseous

reaction at the tip of a moving crack, where

species A is reacting with species B in the glass to

form an activated complex B*. B may be considered to be proportional to the number of bonds

in the solid so that each time the reaction occurs,

the crack tip advances a small distance. Therefore, the rate of crack advancement is proportional

to the rate of reaction. Based on 1 mole of reacting

B, the equation for the chemical reaction is

nA+B^B*,

and the change of free energy is

EZ=n%—ixB —np.A ,

(47)

(48)

where is the chemical potential of the activated complex and p.B and p.A are the chemical

potentials of species B and A respectively. At

constant total pressure and temperature, nB

and ;uf should be constant, since they depend

only on the structure and state of the solid to

236
which they are attached. fxA , by contrast, depends

on environment. Assuming A to be a perfect gas,

»A =RT[*{T)+]npA ] (49)

where $(T) is some function of temperature,

T, and pA is the partial pressure of species A

immediately adjacent to the crack tip. Substituting eqs (49) and (48) into (43) , the following

equation is obtained for the crack velocity dependence on environment and crack tip stress; v=AP2 exp [V*a-VMylp\IRT, (50)

where all of the constant terms have been collected

into the constant A.

The important aspects of the Charles-Hillig

theory have been verified by several experiments.

The theory successfully predicts the universal

fatigue curve developed by Mould and Southwick

[108]. Using the data of Mould and Southwick,

Hillig and Charles [113] calculated the static

fatigue limit to be 0.17 of the critical stress intensity, Kf=0.\lKIc. They also find that

7=0.5 J/m2 (500 ergs/cm2

) and V* = lXl0_6m7

mole, which can be compared with an activation

volume of 4.9 X 10_6m3/mole for alkali diffusion

in a typical soda lime glass.

A more direct confirmation of the CharlesHillig theory comes from the stress-corrosion

experiments of Wiederhorn [117, 118]. Crack

velocities were measured as a function of temperature, environment and KIf using the double cantilever configuration. The pertinent experimental

evidence is shown in figures 21 and 22. In figure

21, the logarithm of the crack velocity is plotted

as a function of K7 for environments of several

different water vapor concentrations. Each water

vapor concentration is represented by a tri-modal

curve. In region I of these curves, crack velocity

depends exponentially on Ki and exhibits a water

vapor pressure dependency. In region II, the

plateau region, crack velocity is nearly independent of applied load, but depends on environment.

In region III, crack velocity is again dependent

on applied load, but is independent of environment. The data of figure 21, regions I and II, can be adequately explained using the theory of

Charles and Hillig [117]. In region I the crack

velocity is limited by the kinetics of the chemical

reaction, while in region II the crack velocity is

limited by rate of transport of water vapor from

the surrounding environment to the crack tip.

Figure 22 presents crack velocity data obtained

in water at four different temperatures [118].

The data were found to satisfy the following

equation

v= 1.2 X 102 exp- (19,500-1.92X lO-^/^T7

,

(51)

which is similar in form to that proposed by

Charles and Hillig. The activation energy, E0 *=

^ i i r

STRESS INTENSITY FACTOR, K, . N/mJ/z X IO»

Figure 21. Dependence of crack velocity on stress intensity factor, Ki, in soda-lime-silica glass.

The percent relative humidity for each set of runs is given on

the righthand side of the diagram. The Roman numerals identify the different regions of crack propagation.

STRESS INTENSITY FACTOR - N/M X 10

Figure 22. Dependence of crack velocity on stress intensity factor, Ki, and temperature in sodalime-silica glass.

237
19.5 kcal/mole, compares favorably with values

of 20 kcal/mole and 18.8 kcal/mole obtained by

Charles respectively for fracture and corrosion

of glass in saturated steam [106, 107].

An estimate for V* may be obtained by equating

V*a to the second term in the bracket of eq (51).

Since V*a=V*2S^LfP =KI2V*l4^9, the following equation is obtained for V*

F*=4.02X10-2 V^P, (52)

where a conversion factor of 0.239 has been used

to convert from calories to joules. Assuming a reasonable radius of curvature of 5X10-10 m,

y* = 1.6XlO-6 m3 (1.6 cm3 ), which compares well

with the value obtained by Hillig and Charles.

Equation (46) may now be simplified with the

aid of eq (52) and from the fact that K2/2E—y,

obtaining =32^X10-73^. (53)

For the soda lime glass used in the above ex- periments, FM=24.2X10-6 m3 (24.2 cm3) and E=

7.35X1010 N/m2

. With a crack tip radius of

5X10-10 m, a value of 2.2 X105 N/m3 /2 is obtained

for Ki*, which is 28 percent of KIC in liquid nitrogen and agrees quite well with experiment.

One of the most significant aspects of the equations presented in this section is their utility in

predicting the relative resistance of glass to static

fatigue. The crack velocity data necessary for the

estimate require less time to obtain than comparable stress-time to failure data. Consequently,

fracture mechanics techniques have a clear advantage over the usual strength techniques for this

purpose.

Data on the stress corrosion of sapphire are not as plentiful as those on glass. Nevertheless,

from the data available, the processes in sapphire

seem to be very similar to those occurring in glass.

Static fatigue curves [58, 110] and crack velocity

data, figure 23, bear a strong resemblance to

those obtained on glass. Consequently, it is reasonable to conclude that delayed failure in

sapphire is also caused by a water-induced stress

corrosion process.

Data on other ceramic materials are sparse;

nevertheless, there is good evidence that delayed

failure is a general property of ceramic materials

and that the strength of these materials is limited

by stress corrosion [111]. The effect of water vapor

on the strength of several ceramic materials is given in table 9, where it is evident that even a complex material such as granite can exhibit

static fatigue. To establish the maximum usable

strength of each of these materials, it will be

necessary to determine their static fatigue limit

either through strength time to failure studies or by fracture mechanics techniques. Considering

the superiority of the latter, fracture mechanics

techniques should be used where possible.

J 1 i_

1.0 1.5 2.0 STRESS INTENSITY FACTOR , K t ,N/m3 ' 2 X I0

6

Figtjbe 23. Dependence of crack velocity on

stress intensity factor, Ki, in sapphire.

The percent relative humidity for each set of runs is given on the right hand side of the diagram. Crack

propagation occurred on the (1011) rhombohedral plane.

Table 9. Effect of water vapor on the strength of ceramic

materials

Test conditions, N/m^XlO' (psiXlO3)

Material Liquid Dry Saturated water

nitrogen nitrogen vapor -195 C 240 C

At 240 C At 25 C

Soda-lime-silica glass (tr) 1.51 0.757

(22. 0) (11. 0) Fused silica glass (c) 4.52 4.44 2.52 3.83

(65. 7) (64. 5) (36. 6) (55. 6) Granite (c) 2.57 1.36 0. 414 1.62

(37.4) (19.7) (6.01) (23. 5) Spodumene (c) 6. 55 3.94 3. 14 2. 65

(95. 2) (57.3) (45. 7) (38. 5) Brazilian quartz (c).. 5.61 4. 38 2. 46 3.59

(81. 6) (63. 8) (35. 8) (52. 2) MgO crystal (c) 2. 10 1.83 0.550 0. 977

(30. 5) (26. 6) (8.0) (14. 2) AI2O3 crystal (tr) 10.5 7.99 4. 70 7.57

(152. 0) (116. 5) (68. 3) (110. 0)

Compressive (c) or transverse (tr) strengths under conditions indicated with machine-loading rate of 0.005 in 1 min. Table taken from ref. [14], page

617. Original data from ref. [111].

7. Conclusions

The main purpose of this paper has been to

discuss some aspects of brittle fracture that may be

of use in the design and manufacture of ceramic

materials. Strong emphasis has been placed on the

238
flaw theory of failure and its generalization into a continuum approach to fracture. The stress intensity factor and fracture surface energy have been

described and the relationship between the two

has been given. One of the main reasons for using

these parameters to describe fracture is their independence of flaw size. They are material properties dependent on microstructure and environment, but independent of structure shape or flaw

distribution. Examples have been given of the use

of these parameters in describing strength, thermal shock resistance, and environmental effects on

strength. It may be concluded that the continuum

approach to fracture should provide a valuable

adjunct to the usual strength measurements.

Readers interested in a more intensive survey of

this subject are referred to the recent book by

Tetelman and McEvily [60], which should appeal

to both the materials scientist and the structural

engineer. More theoretical aspects of the subject

will be found in references [17, 29], while engineering aspects of the subject are discussed in

references [30, 119].

Support for this work by the U. S. Army

Research Office—Durham, N.C. and the Advanced Research Projects Agency is gratefully

acknowledged.

8. References

[1] Gilman, J. J., The strength of ceramic crystals, in

The Physics and Chemistry of Ceramics, C.

Klingsberg (ed.), Gordon and Breach, New York

(1963), p. 240.

[2] Theoretical Strength of Materials, Materials Advisory Board, National Academy of Sciences, Publication MAB-221-M, Aug. 1966.

[3] Orowan, E., Fracture and strength of solids, Repts.

Prog. Phys. 12, 185-232 (1948).

[4] Gilman, J. J., Cleavage, ductility and tenacity in

crystals, in Fracture, Averbach, B. L., Felbeck,

D. K., Hahn, G. T., and Thomas D. A., (eds.),

John Wiley & Sons, New York (1959), pp. 193-222.

[5] Wachtman, J. B., Jr., Tefft, W. E., Lam, D. G., Jr., and Stinchfield, R. P., Elastic constants of synthetic single crystal corundum at room temperature, J. Research NBS 64A, 213-228 (1960).

[6] Kronberg, M. L., Plastic deformation of single crystals of sapphire: Basal slip and twinning, Acta

Met. 5, 507-524 (1957).

[7] Warren, B. E., and Biscoe, J., Fourier analysis of

x-ray patterns of soda-silica glass, J. Am. Ceram.

Soc. 21, 259-265 (1938).

[8] Hillig, W. B., The factors affecting the ultimate

strength of bulk fused silica, in Symposium on

Mechanical Strength of Glass and Ways of Improving It, Florence, Italy, Sept. 25-29, 1961,

Union Scientific Continentale du Verre, Charleroi,

Belgium, pp. 206-236 (1962).

[9] Hillig, W. B., Strength of bulk fused quartz, J. Appl.

Phys. 32, 741 (1961).

[10] Proctor, B. A., Whitney, I., and Johnston, J. W.,

The strength of fused silica, Proc. Roy. Soc. A297,

534-557 (1967).

[11] Mehan, R. L., and Feingold, E., Strength and struc- ture of very fine sapphire whiskers, pp. 355-406 in Strength Mechanisms, Burke, J. J., Reed, N. L., and Weiss, V. (eds.), Syracuse Univ. Press, Syracuse, N.Y. (1966).

[12

[13

[14

[15

[16

[17

[18

[19

[20

[21

[22

[23

[24

[25

[26

[27

[28

[29

[30

[31

[32

[33

[34

[35

[36

Brenner, S. S., Mechanical behavior of sapphire

whiskers at elevated temperatures, J. Appl. Phys.

33, 33-39 (1962).

Mallinder, F. P., and Proctor, B. A., The strengths

of flame-polished sapphire crystals, Phil. Mag.

197-208 (1966).

Kingery, W. D., Introduction to Ceramics, p. 610,

John Wiley & Sons, New York (1960).

Griffith, A. A., Phenomena of rupture and flow in

solids, Phil. Trans. Roy. Soc. (London) 221A,

163-198 (1920).

Griffith, A. A., Theory of rupture, Proc. First Intern. Congr. Appl. Mechanics, Delft, pp. 55-63 (1924).

Irwin, G. R., Fracture, in Encyclopedia of Physics,

Vol. VI, Springer, Heidelberg (1958).

Inglis, C. E., Stresses in a plate due to the presence

of cracks and sharp corners, Trans. Inst. Naval

Architects (London) 55, 219 (1913).

Sneddon, I. N., The distribution of stress in the

neighborhood of a crack in an elastic solid, Proc. Roy. Soc. (London) A187, 229 (1946).

Sadowski, M. A., and Sternberg, E. G., Stress concentration around a triaxial ellipsoidal cavity,

J. Appl. Mech. 16, 149 (1949).

Barenblatt, G. I., The mathematical theory of

equilibrium cracks in brittle fracture, pp. 55-126

in Advances in Applied Mechanics, Vol. VII, H.

L. Dryden and T. von Karman (eds.), Academic

Press, New York (1962).

Elliot, H. A., An analysis of the conditions for rupture due to Griffith cracks, Proc. Phys. Soc. (London) 59, 208-223 (1947).

Willis, J. R., A comparison of the fracture criteria of Griffith and Barenblatt, J. Mech. Phys. Solids

15, 151-162 (1967).

Cribb, J. L., and Tomkins, B., On the nature of the

stress at the tip of a perfectly brittle crack, J. Mech. Phys. Solids 15, 135-140 (1967).

Hillig, W. B., Sources of weakness and the ultimate

strength of brittle amorphous solids, pp. 152-194

in Modern Aspects of the Vitreous State, Vol. 2,

J. D. Mackenzie (ed.), Butterworths, London

(1962).

Orowan, E., Energy criteria of fracture, Weld. J., London 34, 157s-160s (1955).

Irwin, G. R., Analysis of stresses and strains near

the end of a crack traversing a plate, J. Appl.

Mech. 24, 361 (1957).

Williams, M. L., On the stress distribution at the

base of a stationary crack, J. Appl. Mech. 24,

109-114 (1957).

Paris, P. C, and Sih, G. C, Stress analysis of cracks, pp. 30-81 in Fracture Toughness Testing and its Applications, ASTM Special Technical Publication No. 381.

Srawley, J. E., and Brown, W. F., Fracture toughness testing methods, pp. 133-196 in Fracture

Toughness Testing and its Applications, ASTM

Special Technical Publication No. 381.

Srawley, J. E., and Gross, B., Stress intensity

factors for crackline-loaded edge-crack specimens,

Mat. Res. Stds. 7, 155-162 (1967).

Wiederhorn, S. M., Shorb, A. M., and Moses, R. L., A critical analysis of the theory of the double

cantilever method of measuring fracture surface

energies, J. Appl. Phys. 39, 1569-1572 (1968).

Dugdale, D. S., Yielding of steel sheets containing

slits, J. Mech. Phys. Solids 8, 100 (1960).

Hahn, G. T., and Rosenfield, A. R., Local yielding

and extension of a crack under plane stress, Acta

Met. 13, 293-306 (1965).

McClintock, F. A., and Irwin, G. R., Plasticity

aspects of fracture mechanics, pp. 84-113 in

Fracture Toughness Testing and its Applications,

ASTM Special Tech. Pub. No. 381.

Irwin, G. R., and Kies, J. A., Critical energy rate

analysis of fracture strength, Welding Research

Suppl. 33, 193s-198s (1954).

239
[37] Irwin, G. R., Fracture mechanics, pp. 557-592 in Structural Mechanics, Goodier J. N., and Hoff,

N. J. (eds.), Pergamon Press, New York (1960).

[38] Nakayama, J., Direct measurement of fracture

energies of brittle heterogeneous materials, J. Am.

Ceram. Soc. 48, 583-587 (1965).

[39] Tattersall, H. G., and Tappin, G., The work of

fracture and its measurement in metals, ceramics

and other materials, J. Mat. Sci. 1, 296-301 (1966).

[40] Davidge, R. W., and Tappin, G., The effective

surface energy of brittle materials, J. Mat. Sci.

3, 162-173 (1968).

[41] Berdennikov, W. P., Messung der Oberflachenspannung von Festen Korpern, Soviet Phys. Z.S. 4,

397-419 (1933); Zhur. Fiz. Khim. 5(2-3), 358

(1934). (A summary of these articles is given in Surface Energy of Solids by V. D. Kuznetsov,

Her Majesty's Stationery Office, London (1957),

pp. 224-234. Error in the original data corrected

by S. M. Wiederhorn in Materials Science Research, Vol. 3, W. W. Kriegel and H. Palmour

III (eds.), Plenum Press (1966), p. 514.)

[42] Marsh, D. M., Plastic flow in glass, Proc. Roy. Soc. A279, 420-435 (1964).

[43] Wachtman, J. B., Jr., and Maxwell, L. H., Plastic

deformation of ceramic-oxide single crystals,

J. Am. Ceram. Soc. 37, 291-299 (1954).

[44] Orowan, E., The tensile strength of mica and the

problem of the technical strength, Z. Physik. 82,

235-266 (1933).

[45] Bryant, P. J., Taylor, L. H., and Gutshall, P. L., Cleavage studies of lamellar solids in various gas

environments, pp. 21-26 in Transactions of the

Tenth National Vacuum Symposium, American

Vacuum Society (1963).

[46] Gilman, J. J., Direct measurements of the surface

energies of crystals, J. Appl. Phys. 31, 2208-2218

(1960).

[47] Gutshall, P. L., and Gross, G. E., Cleavage surface

energies of NaCl and MgO in vacuum, J. Appl.

Phys. 36, 2459-2460 (1965).

[48] Petch, N. J., Congleton, J., Hardie, D., and Perkins,

R. N., Effect of surface energy, Study of the

Brittle Behavior of Ceramic Materials, Technical

Documentary Report No. ASD-TR-61-628, Parts

I, II, III.

[49] Judd, J. W., On the structure planes of corundum,

J. Mineralogical Soc. 11, 49-55 (1895).

[50] Conrad, H., Mechanical behavior of sapphire,

J. Am. Ceram. Soc. 48, 195-201 (1965).

[51] Stofel, E., and Conrad, H., Fracture and twinning

in sapphire (alpha-Al2 03 crystals), Trans. AIME

227, 1053-1060 (1963).

[52] Heuer, A. H., Deformation twinning in corundum,

Phil. Mag. 13, 379-393 (1966).

[53] Class, W. H., and Machlin, E. S., Crack propagation

method for measuring grain boundary energies in

brittle materials, J. Am. Ceram. Soc. 49, 306

(1966).

[54] Clarke, F. J. P., TattersaU, H. G., and Tappin, G.,

Toughness of ceramics and their work of fracture, Proc. Brit. Ceram. Soc. 6, 163-172 (1966).

[55] Gutshall, P. L., and Gross, G. E., Observations and

mechanisms of fracture in polycrystalline alumina,

to be published in the Proc. Nat. Symp. Fracture

Mechanics, Lehigh University, June 19-21, 1967.

[56] Vasilos, T., Mitchell, J. B., and Spriggs, R. M.,

Mechanical properties of pure, dense magnesium

oxide as a function of temperature and grain

size, J. Am. Ceram. Soc. 47, 606-610 (1964).

[57] Passmore, E. M., Spriggs, R. M., and Vasilos, T.,

Strength-grain size-porosity relations in alumina,

J. Am. Ceram. Soc. 48, 1-7 (1965).

[58] Charles, R. J., and Shaw, R. R., Delayed failure of

polycrystalline and single-crystal alumina, General

Electric Research Laboratory Report No. 62-

RL-3081 M, July 1962.

[59] Clarke, F. J. P., and Wilks, R. S., Effects of irradiation on mechanical properties of nonfissionable

ceramics, pp. 57-74 in Proc. Conf. Nuclear Applications of Nonfissionable Cermaics, Boltan, A.,

and Handwerk, J. H., (eds.), sponsored by American

Ceramic Society and American Nuclear Society,

May 9-11, 1966.

[60] Tetelman, A. S., and McEvily, A. J., Jr., Fracture

of Structural Materials, John Wiley & Sons,

New York (1967).

[61] Johnston, T. L., Fracture of poly crystals, pp.

1-22 in Fundamental Phenomena in the Material

Sciences, Vol. 4: Fracture of Metals, Polymers,

and Glasses, Bonis, L. J., Duga, J. J., and Gilman,

J. J. (eds.), Plenum Press, New York (1967).

[62] Stokes, R. J., Fracture of ceramics, pp. 151-175,

in Fundamental Phenomena in the Material

Sciences, Vol. 4: Fracture of Metals, Polymers,

and Glasses, Bonis, L. J., Duga, J. J., and Gilman,

J. J. (eds.), Plenum Press, New York (1967).

[63] Mould, R. E., The strength of inorganic glasses, pp. 119-149, in Fundamental Phenomena in the

Material Sciences, Vol. 4: Fracture of Metals,

Polymers, and Glasses, Bonis, L. J., Duga, J. J., and Gilman, J. J. (eds.), Plenum Press, New

York (1967).

[64] Shand, E. B., Experimental study of fracture of

glass: I, The fracture process, J. Am. Ceram.

Soc. 37, 52-60 (1954).

[65] Stookey, S. D., Strengthening glass and glassceramics by built-in surface compression, pp. 669-680 in High-Strength Materials, Zackay, V. F.

(ed.), John Wiley Sons, New York (1965).

[66] Johnston, T. L., Li, C. H., and Stokes, R. J., The

strength of ionic solids, pp. 341-374 in Strengthening Mechanisms in Solids, American Society of Metals, Cleveland (1962).

[67] Stokes, R. J., Mechanical properties of polycrystalline sodium chloride, Proc. Brit. Ceram. Soc.

6, 189-207 (1966).

[68] Day, R. B., and Stokes, R. J., Grain boundaries

and the mechanical behavior of magnesium

oxide, pp. 355-386 in Material Science Research, Vol. 3, Kriegel, W. W., and Palmour, H.,

Ill (eds.), Plenum Press, New York (1966).

[69] Stokes, R. J., Johnston, T. L., and Li, C. H., Environmental effects on the mechanical properties

of ionic solids with particular reference to the

Joffe effect, Trans. AIME 218, 655-662 (1960).

[70] Stokes, R. J., and Li, C. H., Dislocations and the

tensile strength of magnesium oxide, J. Am.

Ceram. Soc. 46, 423-434 (1963).

[71] Ku, R. C, and Johnston, T. L., Fracture strength

of MgO bicrystals, Phil. Mag. 9, 231-247 (1964).

[72] Westwood, A. R. C, On the fracture behaviour of

magnesium oxide bi-crystals, Phil. Mag. 6,

195-200 (1961).

[73] Clarke,. F. J. P., SambeU, R. A. J., and TattersaU,

H. G., Cracking at grain boundaries due to dislocation pile-up, Trans. Brit. Ceram. Soc. 61,

61-66 (1962).

[74] Johnston, T. L., Stokes, R. J., and Li, C. H., Crack

nucleation in magnesium oxide bicrystals under

compression, Phil. Mag. 7, 23-34 (1962).

[75] Stokes, R. J., Johnston, T. L., and Li, C. H., The

relationship between plastic flow and the fracture

mechanism in magnesium oxide single crystals,

Phil. Mag. 4, 20-32 (1959).

[76] von Mises, R., Mechanik der plastichen formanderung von kristallen, Zeit. Angew. Math, und

Mech. 8, 161 (1938).

[77] Taylor, G. I., Plastic strain in metals, J. Inst. Metals 62, 307-324 (1938).

[78] Groves, G. W., and Kelly, A., Independent slip

systems in crystals, Phil. Mag. 8, 877-887 (1963).

[79] Wachtman, J. B., Jr., Mechanical properties of

ceramics: An introductory survey, Bull. Am.

Ceram. Soc. 46, 756-772 (1967).

240
Zener, C, Micro-mechanism of fracture, pp. 3-31

in Fracturing of Metals, Am. Soc. Metals, Novelty,

Ohio (1948).

Stroh, A. N., A theory of the fracture of metals,

Advances in Physics 6, 418-465 (1957).

Smith, E., and Barnby, J. T., Crack nucleation in crystalline solids, Metal Sci. J. 1, 56-64 (1967).

Petch, N. J., The cleavage strength of polycrystals,

J. Iron Steel Inst. 174, 25-28 (1953).

Eshelby, J. D., Frank, F. C, and Nabarro, F. R. N.,

The equilibrium of linear arrays of dislocations,

Phil. Mag. 42, 351 (1951).

Clarke, F. J. P., Residual strain and the fracture

stress-grain size relationship in brittle solids,

Acta Met. 12, 139-143 (1964).

Davidge, R. W., and Tappin, G., Internal strain

and the strength of brittle materials, J. Mat.

Sci. 3, 297-301, (1968).

Cottrell, A. H. Theory of brittle fracture in steel and

similar metals, Trans AIME 212, 192-203 (1958).

Gell, M., and Smith, E., The propagation of cracks

through grain boundaries in polycrystalline 3%

silicon-iron, Acta Met. 15, 253-258 (1967).

Rice, R. W., Strength and fracture of dense MgO,

presented at Third Intern. Materials Symp.,

Ceramic Microstructures—Their Analysis, Significance and Production, Univ. of California,

Berkeley, Calif., June 13-16, 1966.

Coble, R. L., Personal communication.

Carniglia, S. C, Grain boundary and surface in- fluence on mechanical behavior of refractory

oxides—experimental and deductive evidence,

pp. 425-471 in Materials Science Research, Vol.

3, W. W. Kriegel and H. Palmour III (eds.),

Plenum Press, New York (1966).

Carniglia, S. C, Petch relation in single-phase oxide

ceramics, J. Am. Ceram. Soc. 48, 580-583 (1965).

Spriggs, R. M., Mitchell, J. B., and Vasilos, T., Mechanical properties of pure, dense aluminum

oxide as a function of temperature and grain size,

J. Am. Ceram. Soc. 47, 323-327 (1964).

Fryxell, R. E., and Chandler, B. A., Creep, strength,

expansion and elastic moduli of sintered BeO as a function of grain size, porosity and grain orientation, J. Am. Ceram. Soc. 47, 283-291 (1964).

Bentle, G. G. and Kniefel, R. M., Brittle and plastic

behavior of hot-pressed BeO, J. Am. Ceram. Soc. 48, 570-577 (1965).

Stokes, R. J., Johnston, T. L., and Li, C. H., Effect

of surface condition on the initiation of plastic

flow in magnesium oxide, Trans. AIME 215,

437-444 (1959).

Kingery, W. D., Introduction to Ceramics, p. 581,

John Wiley & Sons, New York (1960).

Kingery, W. D., Factors affecting thermal stress resistance of ceramic materials, J. Am. Ceram.

Soc. 38, 3-15 (1955).

Hasselman, D. P. H., Elastic energy at fracture and

surface energy as design criteria for thermal

shock, J. Am. Ceram. Soc. 46, 535-540 (1963).

Grenet, L., Mechanical strength of glass, Bull. Soc. Enc. Industr. Nat. Paris (ser. 5), 4, 838-848

(1899). Partial translations of this article may be

found in Glass Ind. 15, 277 (1934) and J. Am.

Ceram. Soc. 18, 219 (1935).

Charles, R. J., A review of glass strength, pp. 1-38

in Progress in Ceramic Science, Vol. 1, J. E. Burke

(ed.), Pergamon Press, New York (1961).

Hillig, W. B., Sources of weakness and the ultimate

strength of brittle amorphous solids, pp. 152-194

in Modern Aspects of the Vitreous State, Vol. 2,

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

J. D. Mackenzie (ed.), Butterworth and Co.

(London) (1962).

Phillips, C. J., Strength and weakness of brittle materials, Am. Scientist 53, 20-51 (1965).

Phillips, C. J., Mechanical strength of glass, report

Research Laboratory, Corning Glass Works, 1937

(see also ref. [64] above)

.

Holland, A. J. and Turner, W. E. S., The effect of

sustained loading on the breaking strength of

sheet glass, J. Soc. Glass Technol. 24, 46T (1940).

Charles, R. J., Static fatigue of glass II, J. Appl.

Phys. 29, 1554-1560 (1958).

Charles, R. J., Static fatigue of glass I, J. Appl.

Phys. 29, 1549-1553 (1958).

Mould, R. E., and Southwick, R. D., Strength and

static fatigue of abraded glass under controlled

ambient conditions: II Effect of various abrasions

and the universal fatigue curve, J. Am. Ceram.

Soc. 42, 582-592 (1959).

Pearson, S., Delayed fracture of sintered alumina,

Proc. Phys. Soc. (London) 69B, 1293-1296 (1956).

Charles, R. J., Static fatigue: delayed fracture, pp.

469 in Studies of the Brittle Behavior of Ceramic

Materials, N. A. Weil (ed.), Technical Report

ASD-TR-61-628, Part II (April 1963); see also

refs. [113] and [58]. Charles, R. J., The strength of silicate glasses and

some crystalline oxides, pp. 225-249 in Fracture,

Averbach, B. L., Felbeck, D. K., Hahn, G. T.,

and Thomas, D. A. (eds.), John Wiley & Sons,

New York (1959).

Charles, R. J., and Hillig, W. B., The kinetics of glass failure by stress corrosion, pp. 511-527 in

Symp. Mechanical Strength and Ways of Improving It. Florence, Italy, Sept. 25-29, 1961, Union

Scientifique Continentale du Verre, Charleroi,

Belgium (1962).

Hillig, W. B., and Charles, R. J., Surfaces, stress- dependent surface reactions and strength, pp.

682-705 in High-Strength Materials, V. F. Zackey

(ed.), John Wiley & Sons, Inc., New York (1965).

Glasstone, S., Laidler, K. J., and Eyring, H., Theory

of Rate Processes, McGraw-Hill Book Co., New

York (1941).

Callen, H. B., Thermodynamics, pp. 213-237, John

Wiley & Sons, New York (1960).

Adamson, A. W., Physical Chemistry of Surfaces,

p. 56, Interscience Publishers (1960).

Wiederhorn, S. M., Influence of water vapor on

crack propagation in soda-lime glass, J. Am.

Ceram. Soc. 50, 407-414 (1967).

Wiederhorn, S. M., Effects of environment on the

fracture of glass, pp. 293-317 in EnvironmentSensitive Mechanical Behavior of Materials, A. R. C. Westwood and N. S. Stoloff (eds.),

Gordon and Breach, New York (1966).

Brown, W. F., Jr. and Srawley, J. E., Plane strain

crack toughness testing of high strength metallic

materials, ASTM Spec. Tech. Publ. 410, Am. Soc. Testing Mats. (1967).

Binns, D. B., and Popper, P., Mechanical properties

of some commercial alumina ceramics, Proc. Brit.

Ceram. Soc. 6, 71-82 (1966)

Stokes, R. J., "Mechanical behavior of polycrystalline ceramics", Third Intern. Materials Symp.,

Ceramic Microstructures—Their Analysis, Significance and Production, Univ. of California,

Berkeley, Calif., June 13-16, 1966.

Wiederhorn, S. M., To be published.

Bentle, G. G., and Miller, K. T., Dislocations, slip,

and fracture in BeO, J. Appl. Phys. 38, 4248-57

(1967).

241

Strength Testing Of Ceramics—A Survey

Leonard Mordfin and Matthew J. Kerper

Office of Aerospace Research, 1400 Wilson Boulevard, Arlington, Virginia 22209

Mechanical test methods for brittle ceramics are reviewed, and the state-of-the-art is presented for tensile and compressive strength determinations. Included are various indirect

tension, flexure, and torsion test methods and the relatively new stress wave technique.

The emphasis is on technological factors, particularly on improvements achieved since

1963, but the importance of the human element is also stressed. References are provided

for more detailed study and to guide the selection of an appropriate method for a given

set of circumstances.

Key Words: Brittleness; ceramics; mechanical testing; strength.

1. Introduction

The unique combinations of properties offered

by ceramics make them attractive for many

specialized structural applications where metals

are inadequate. On the other hand, the inherent

brittleness of most ceramics makes the design

and development of reliable ceramic structural

members exceedingly difficult. This problem has

been discussed perceptively by Harmsworth [l]. 1 One of the difficulties is the determination of

reliable strength data. Such determinations have

frequently been characterized by excessive variability, or scatter, due to testing errors and

material variability. The reduction in testing

errors is the subject of this paper. We assume

some understanding of this problem on the part

of the reader, particularly as it applies to brittle

materials, but where this is not the case we

recommend Duckworth's clear exposition [2] as preliminary reading.

In 1963 Shook [3] prepared an extremely

comprehensive survey of mechanical property

test methods for brittle materials. It is our

intention here to update his work insofar as strength-test methods are concerned. In both

Shook's work and our own, the emphasis is on

technological improvements in testing techniques.

Nevertheless, it may be appropriate to say a few

words on the human element involved. It cannot

be denied that much of the unreliable strength

data on ceramics have been generated as a result

of improper testing practices at the hands of

inexperienced or incompetent personnel. All of

the improvements in testing apparatus are for

nought if not utilized by experienced, capable

personnel. Obtaining strength measurements on a new brittle material, or an old one under a new

1 Figures in brackets indicate the literature references at the end of this paper.

set of conditions, is anything but a routine

operation. It should be treated as a research

experiment.

Our point is that mechanical testing is not a degrading occupation. Good people and good

equipment are both needed to obtain reliable,

useful data on brittle materials as well as to

develop good test methods.

Although many of the strength test methods for

ceramics were originally developed for concrete,

most of the recent advances in these methods have

been accomplished by the Air Force; principally

the Air Force Materials Laboratory and its

contractors, the IIT Research Institute, the

Stanford Research Institute, the Southern Research Institute, and the Battelle Memorial

Institute, to name a few. This is not surprising.

Most of the major innovations in structures

technology, for several decades now, were born

from aerospace requirements. Unfortunately, much

of this work has not found its way into the open

literature, for a variety of reasons. We have,

therefore, found it necessary to reference many

Air Force reports in preparing this paper. In

many cases we have included the AD number, by

which these reports are cataloged by the Defense

Documentation Center, Cameron Station, Alexandria, Virginia 22314. Copies of these are readily

available to qualified requestors. Many of the

reports are similarly available to the general

public from the Clearinghouse for Federal Scientific and Technical Information, 5285 Port Royal

Road, Springfield, Virginia 22151.

2. Direct Tension Test

In order to evaluate the fracture strength of a

material, it is necessary to have a means of relating

the load on a specimen, at the instant of fracture,

to the stress at the fracture location. For certain

243
loading and specimen configurations, the loadstress relationship is directly obtainable from

equilibrium considerations. The most obvious

example is the ideal tension test, wherein a prismatic rod is subjected to uniformly distributed,

axial end loads. The stress is then everywhere

equal to the force intensity, or the load divided by

the cross sectional area. The simplicity of this concept has made the tension test the principal tool

for evaluating the strengths of materials in general.

In practice, it is virtually impossible to apply

tensile load uniformly over the end of a rod. Instead, the various gripping methods available,

such as threads, button heads, tapered shoulders,

collets, etc., apply loads chiefly through a shear

mechanism.2 However, recourse to Saint-Venant's

Principle still enables our objective to be fulfilled.

That is, if the resultant of the applied load is pure

tension, and if the specimen is long enough, then

the central portion of the specimen will still be in

the required state of tension.

Unfortunately, even this approach is not realistic in practice. Friction, imperfect machining,

inhomogeneity, and other factors make the attainment of a purely tensile resultant force on the end

of a rod specimen impossible. Instead, the resultant consists of a tensile force plus a moment,

however small. The effect of the moment is to introduce bending stresses into the central portion

of the specimen in addition to the desired tensile

stresses.

If, through careful testing technique, the bending moment is kept reasonably small, its effect on

strength determinations can be ignored when

testing most metals, plastics, and other ductile

materials. The deformation capabilities of ductile

materials enable specimens to respond to small,

superposed bending moments in such a way that

the bending stresses in the central portion are

relieved, and a nearly pure state of tension is achieved prior to fracture.

In most ceramics and other brittle materials, on

the other hand, this deformation capability does

not exist, and fracture is initiated at a point in the

specimen where the sum of the tensile and bending

stresses is greater than the tensile stress alone.

The above discussion is not intended to create

the impression that satisfactory tension testing of

brittle materials is impossible, but rather that it

can be achieved only with special care and techniques that reduce the maximum bending stress to

less than, say, 2 percent of the tensile stress. Above all, we are trying to show why tensile test- ing of brittle materials is not a routine procedure.

To illustrate the point, Barnett et al. [5] subjected an instrumented tensile specimen to 67

consecutive cycles of loading and unloading, and

recorded the ratio of maximum bending stress

to average tensile stress each time. Between cycles

the specimen was removed from the grips, cleaned,

reinserted in the grips and carefully alined. The

1 An exception applicable to large, low-strength specimens uses thick steel plates bonded with epoxies to the ends of concrete specimens [4].

data obtained are shown in figure 1. It may be

seen that the bending stresses amounted to as much as 80 percent of the tensile stresses. Furthermore, the nearly linear nature of the cumulative

distribution indicates that high bending stresses are just as likely to occur as lower stress values.

It is not surprising, therefore, that relatively large

amounts of scatter are obtained in results from

tests of this kind.

In another study, which involved even more

careful alinement of specimens in grips and the use of crushable shims, Bortz [6] still obtained

bending strains as high as 10 percent of the total

strain.

The only solution is to instrument all test

specimens and to reject or re-start all tests which

produce excessive bending. The intent of the

recommendations and descriptions which follow

is merely to reduce the difficulties encountered in

initiating a successful test. We may also reiterate,

at this point, that there is no substitute for ex- perience and diligence on the part of the testing

personnel.

The objective is to make the applied force co- incident with the centerline of the specimen. This

involves all elements of the load train, which includes the specimen, the grips, and any extension

rods. All of these elements must be accurately

machined for concentricity and precisely alined.

Specimens should preferably be round, for easier

machining and elimination of corner effects, and

have a reduced, uniform diameter over the gage

length. Bortz and Wade [7] recommend a gage

length of at least four times the gage diameter

to reduce the effects of fillets. The collet type

of grip is widely considered the best for repeatedly

and reliably holding the necessary alinement.

°- 20

'1

?

m

/H~

/ Vf- N = 67 SECTION =

X 1/4 in.

1

-

(PI/ 1

CROSS

1/4 in

1 0 10 20 30 40 50 60 70 80 90 100

BENDING STRESS TO TENSILE STRESS RATIO, CT"B /CTT

Figure 1. Occurrence of errors due to eccentric loading in

tension tests.

244
Carefully machined, threaded grips have also

proved satisfactory where the specimen material

permits it. Digesu and Pears [8, 9] developed a tapered

collet grip much like those used in precision

lathes (see fig. 2). The grooves at each end of the

specimen that receive the grips are machined

concentrically about the axis of the gage section.

In fact, the specimen configuration was designed

to allow machining of all surfaces to a common

centerline without intermittent removal of the

specimen from the lathe, thus permitting concentricity to within 0.0005 in. All of the other

elements of the load train were similarly designed

and fabricated to the same tolerance. To eliminate

other sources of bending, the assembled load train

was statically balanced and instrumentation

leads were routed through the center of the ex- tension rods.

In using this collet grip, the three-piece split

ring is pressed into the groove by advancing the

compression nut. This provides the required

gripping force, and also forces the ground end

faces of the specimen and the pull rod together,

to provide axial alinement.

Having achieved satisfactory alinement in the

load train, the next problem requiring consideration is the attachment of the load train to the

testing machine crossheads. If the movable

GRAPHITE OR STAINLESS STEEL

PULL ROD

MATING SURFACE FLAT AND

PERPENDICULAR TO 0.0005"

3- PIECE SPLIT RING

COMPRESSION NUT

TRUE , CONCENTRIC , AND

PARALLEL TO 0.0005"

SPECIMEN

crosshead is constrained to move precisely parallel

to the load train centerline, then the attachments

may be rigid. Unfortunately, conventional testing

machines are often inadequate in this respect, so the attainment of uniaxial loading requires a

frictionless universal bearing. Mechanical universal

joints are hardly frictionless under load. Chains

are better. The best attachment available at this

time is the gas-bearing universal (fig. 3) developed

by Digesu and Pears [8, 9]. This apparatus provides a 0.0005-in-thick gas film between the

extension rods and the machine crossheads at

each end of the load train. With 9-in diam bearings

and a gas pressure of 300 psi in the contact area, the universals transmit less than 1 in-lb of torque

and essentially no bending moment to the specimen, at an applied load of 15,000 lb. Since the elimination of bending stresses permits

greater tensile stresses to be applied to a specimen

before fracture, improvements in tension testing

-techniques manifest themselves in higher strength

determinations as well as in reduced scatter.

Through the use of the above innovations, Pears

obtained tensile strength data which were at

least 30 percent higher than those previously

reported for the same materials. His values agree

more closely with reported flexural strengths.

An entirely different approach to the alinement

problem was proposed by Todd [10]. He compensated for the effects of eccentric loading by

applying an external moment over the middle

third of the tensile specimen. While good results

were apparently obtained, this technique is rather slow and tedious and requires very strict

control.

Electric resistance strain gages appear to be a convenient means of monitoring alinement during

TUBULAR FURNACE

HEATER

UPPER 9 SPHERICAL

BEARING

PRECISION LOAD CELL

FLAGS FOR STRAIN ANALYZER

(ROTATED 90°)

SPECIMEN

PRESSURE

GAS LOWER 9 FLAT

BEARING

LOAD YOKE

MECHANICAL SCREW

LOAD APPLICATION

Figure 2. Precision collet grip for tensile specimens.

Figure 3. Schematic of gas bearings for attachment of load

train to testing machine crossheads.

245
tests, and also provide the strain data needed for

the determination of stress-strain characteristics.

At elevated temperatures or in special environments, for which the direct tension test is well

suited, other strain-measuring instrumentation

may be more desirable. Examples are the optical

strain analyzer used by Digesu and Pears [8, 9],

and the photographic strain-measuring technique

[11]. Mechanical extensometers should, perhaps,

be avoided. Bortz and Wade [7] found that they

caused a 13 percent reduction in strength. This

has been attributed to surface damage produced

by the extensometer, which magnifies the load

eccentricity, however small it may be initially.

In the determination of elastic moduli from

tensile stress-strain measurements, abnormally

large scatter is obtained with certain materials

even when proper testing techniques have been

observed. This may be explained by the possibility

that the deformation is not truly elastic but

includes the creation and propagation of numerous

microcracks.

Recognition of the problems associated with

proper tension testing has caused this method to

achieve less acceptance for brittle materials than

it has enjoyed for ductile materials. Instead,

numerous other tests have been used or proposed

for evaluating tensile strengths. These are called

indirect tension tests, and will be discussed in

following sections.

(We have omitted any discussion of tension

tests for ceramic fibers. This area of endeavor is

still suffering growing pains and, while the

literature on it is rapidly increasing, a comparative

evaluation of available methods may be premature

at this time.)

3. Pressurized Ring Test

This test was developed by Sedlacek and Halden

in 1962 [12]. The specimens are carefully machined

in the form of short, thin-walled tubes, and the

loading is applied by means of an internal pressurized bag. Like the direct tension test, this

test permits the stresses to be evaluated solely

from equilibrium considerations, providing the

specimen wall thickness does not exceed one- tenth of the radius.

Numerous improvements in the testing technique have been made since this method was

introduced. Cornish and Broutman [13] used an

apparatus involving two hardened tool steel

platens with dowel pins which facilitate the

alinement of one platen above the other. A

neoprene bulb, which is formed to the internal

dimensions of the test specimen, serves as the

pressure diaphragm. The inlet to the bulb passes

through a tapered plug centrally located in the

upper platen. The bulb is approximately 0.010-in

thick.

The cylindrical test specimen is positioned in

the apparatus and the platens are mounted between the bed and crosshead of a testing machine,

which prevent separation of the platens due to the axial force exerted by the pressure bulb.

Spacers maintain a gap of approximately 0.00 1-in

between the specimen and the top platen to permit

radial expansion of the specimen.

Sedlacek [14] has provided details of suitable

equipment for pressure generation, sensing, monitoring, calibrating, and recording.

This type of test has many advantages for the

determination of tensile strengths. It is relatively

simple to perform, the loading is self-alining, and

the extraneous stresses caused by gripping in

direct tensile tests are eliminated. The uncomplicated stress distribution is highly reproducible

and failure cannot occur outside of the gage

section.

Babel [15] examined the bending stresses that

are introduced by possible variations in the wall

thickness of the specimen, and by imperfect

contact between the specimen and the pressure

bulb at all points. His analysis showed that for

ceramics these stresses are negligible. Differences

between measured and calculated hoop strains

were found to be less than 2 percent. Babel also

concluded that frictional effects, if any are present,

do not significantly affect the fracture strength

of a specimen.

These advantages, relative to direct tension

testing, have resulted in the measurement of

tensile strengths which exceed previously reported

values and approach flexural values. Most important is that a low scatter of results is observed.

On materials such as alumina, individual groups

of specimens have yielded coefficients of variation

as low as 1 or 2 percent. This compares favorably

with the value of 5 or more that is frequently

associated with flexural strength data.

It is entirely feasible that, by proper positioning

of strain gages, determinations of stress-strain

characteristics for a wide range of stress rates can be made during strength tests. As mentioned

earlier, however, wide scatter in results will obtain

when investigating materials that are prone to

microcrack development. On Boride Z, for ex- ample, which is a multi-phase material that is believed to develop microcracks on cooling after

the sintering process, strength tests reportedly

gave a coefficient of variation of 35 percent [16].

The principal disadvantage of the pressurized

ring test is the temperature limitation of bulb

materials, which is presently about 600 F. In an

attempt to overcome this limitation, a modified

test technique, called the floated ring test (fig. 4),

has been proposed. In this approach, the specimen

ring is floated between two conical end pieces

that permit a controlled gas leakage through the

gaps between the specimen and the end pieces.

The specimen ends are unrestrained, and the

temperature limitation introduced by a pressure

bulb is removed. A disadvantage is that large

power inputs are required to preheat the gas in

order to achieve high-temperature operation.

246
Tapered End Plug

-Clearance Gap

Torque Sensitive

Gap-Positioner

II

0. 040"

to 0. 080"

1

High Pressure

Gas

Note: Requires 40 scfm at 160 psig for 4000 psi

ultimate on ATJ

Graphite with 40 mil

wall

Torque Sensitive

Gap-Positioner

High Pressure

Gas

SHORT SLEEVES

Figure 4. Schematic of gas-floated ring test setup.

Several other interesting variations of the

pressurized ring method have been proposed. In

one of these [17], whioh is shown in figure 5, a universal testing machine is used to apply a compressive force to a floating piston. This force

is transferred to the silicone rubber plug which,

reportedly, acts as an incompressible fluid and

applies a hydrostatic force to the ring specimen.

A tough rubber bladder protects the soft rubber

plug. The pressure developed within the plug as a

function of applied load was measured by means

of a small liquid reservoir within it, which was

tapped with a pressure gage. No significant error was observed, but it still remains to verify that

the plug acts isotropically. If the bladder, which

gets chewed up, is replaced after every tenth test,

coefficients of variation of about 4 percent are apparently attainable with alumina specimens.

This apparatus is less expensive than gas-pressured

arrangements, and has been recommended for

field or routine production control uses.

Still another approach to pressurized ring testing

was introduced by Malhotra et al. [18, 19]. The

modification here is the use of thick-walled cylindrical specimens instead of thin-walled ones. The

Figure 5. Modified Sedlacek-Halden pressurized

ring test setup.

247
application is to concrete, where the size of the

aggregate may not permit the fabrication of thinwalled members. Tests yielded strength values

which exceed those obtained by conventional

tensile tests but which are lower than those from

flexure tests. Coefficients of variation were reported

to be about 5 percent.

The principal disadvantage of this method is that the load-stress relation is no longer statically

determinate. The maximum tensile stress is developed on the inner periphery of the ring and it was

calculated by assuming linear stress-strain behavior of the specimen material to fracture. Since

concrete normally exhibits some inelastic behavior,

the fracture strengths determined from the test

results are probably higher than the actual stresses

at fracture and may not exceed the values obtained

by conventional tension tests. In the tests, 6-in ring specimens were placed in a jig consisting of two 11-in-diam mild steel plates

held together by five tie bolts. Pressure was

applied through a specially molded bladder made

of }£-in-thick nitrile rubber.

4. Theta Specimen Test

The direct tension test and the thin-walled

pressurized ring test, described above, have the

advantage of providing nominally uniform tensile

stresses that may be calculated from equilibrium

considerations, irrespective of the nature of the

stress-strain characteristic of the specimen material. In the remaining indirect tensile tests which

will be discussed the tensile stresses are not

statically determinate. As is the case with the

thick-walled pressurized ring test, the determination of the load-stress relationship in these cases requires an assumption regarding the nature of

the stress-strain characteristic. Since we are con- cerned with brittle materials it is most convenient

to assume a linear stress-strain characteristic all the way to fracture. This assumption includes the

condition that the elastic moduli in tension and

compression are equal. If the specimen material

does not, in fact, behave in this manner the

fracture strengths obtained from these tests are accordingly in error. One test that falls in this category was mentioned by Daniels and Weil [20]. It consists of a diametrally loaded disk containing two holes

separated by a narrow strip of material (fig. 6).

Photoelastic studies reportedly show [3] that a uniform tensile stress area exists in the strip

under the diametral compressive load. This test

has not received wide acceptance but it apparently

led to the theta specimen, which has been investigated in several studies.

The theta specimen was introduced by Durelli,

Morse, and Parks [21] and Daniels and Weil [20] as a means for determining tensile strengths at

elevated temperatures in an oven, where gripping

and alinement problems are more difficult to

contend with. The specimen (fig. 6) is intended

to be loaded between two parallel platens with

the central bar parallel to the platens. Outward

motion of the ring on the horizontal axis subjects

the central bar to tension. The recommended

design (fig. 7) is the final choice of over 60 shapes

based on the same principle. Most of the other

designs were rejected because they failed in the

ring rather than in the bar. The recommended

design generally fails only in the bar.

Photoelastic studies snowed that (1) the maximum tensile stress occurs in the bar; (2) the

stress in the bar is uniform; and (3) the fillets at

the ends of the bar do not produce unacceptable

stress concentrations.

The principal advantage of this specimen is that it does not require excessive precautions to

achieve alinement. Alinement by eye is said to be

adequate to produce the desired symmetrical

stress distribution. A major disadvantage is obviously the considerable precision machining

that the specimen requires.

Other factors to be considered in evaluating this

test method are apparently not so well understood.

Figure 6. Loading configurations for theta specimen (top) and two-holed disk speci- men (bottom).

248
Minimum Thickness

to Avoid Buckling

will depend on Modulus of Elasticity

(About . 250)

1. 775

Figure 7. Theta specimen geometry.

The photoelastic studies indicated the following

load-stress relationship:

kP

where

o-= stress in bar

P= applied load

D=outer diameter

t= thickness of specimen

and k is a constant equal to 13.8 (according to

Durelli et al.) or 16.4 (according to Daniels and

Weil). If the deformations of the specimen are

small, and if the material is elastic, isotropic, and

homogeneous, this relationship should hold for any

model with' plane dimensions proportional to those

in the figure. (The thickness need not be proportional.) Dimensional accuracy is apparently not

critical [21] except, perhaps, in the width of bar

where significant deviations produce correspondingly significant changes in the stress [20]. Further

study of this test method is apparently required

before it can warrant wide acceptance.

5. Truss Beam Tensile Test

This test, proposed by Bortz in 1963 [22], uses a bending mechanism to achieve a uniform,

uniaxial, tensile stress. The specimen, figure 8,

consists of a heavy upper beam and a comparatively thin lower beam. Flexural loading of the

specimen results in bending of the upper beam

and a nominally uniform tensile stress throughout

most of the lower portion, which acts like a tension

tie rod.

The advantage of this test method is that it provides a uniform tensile stress over a relatively

large gage section, which makes it attractive for

the evaluation of stress-strain characteristics.

Measured strains in the lower beam of the specimen

agreed with theoretical predictions within 1.4

percent [23]. However, Bortz [7] recently con- cluded that the specimen is, in fact, subject to

stress conditions other than those predicted, and

recommended further study of the method.

249
Figure 8. Loading configuration for truss beam

tensile specimen.

6. Stress Wave Technique

An entirely new approach to the tensile strength

testing of brittle materials has been developed by

Cornish and Abbott [24, 25]. The method produces a dynamic state of pure tension, free of most

of the difficulties characteristic of direct tension

testing. It is applicable to many brittle materials

such as alumina which does not experience a change in tensile strength with changes in strain

rate at room temperature.

It is impractical for us to describe the technique

in detail here, and the reader is referred to one

or the other of the referenced descriptions, which

are written in a clear, interesting fashion. Briefly,

however, consider a long, slender, freely supported

bar. Applying an impulse to one end of the bar

causes an unchanging longitudinal compressive

wave to travel down the bar. At the other end

this wave is reflected as a mirror-image tensile

wave which propagates back up the bar toward

the first end. This is illustrated schematically in

figure 9. Since the tensile strength of a ceramic

is small relative to the compression strength, a ceramic specimen can be made to fail in tension

under the action of the reflected wave.

In a test, a strain gage is mounted longitudinally near the loaded end of the bar in order to determine its "weakest link" tensile strength. On the

basis of previous information, or from a trial

and error process, a compressive impulse is applied whose reflected value exceeds the tensile

strength of the specimen by a small amount.

The reflected wave travels back up the bar until

it reaches a point where the tensile strength of

the bar is less than the peak value of the wave.

At this point fracture occurs. That portion of

the tensile wave which is behind the fracture

section is lost, but the ^amount of strain which

caused fracture continues to travel up the bar

and is measured by the strain gage.

In their experiments Cornish and Abbott used

}4-in square alumina bars supported in a rubbercoated chemical clamp and loaded with an electrically ignited charge of lead azide. Their results are noteworthy. The average tensile strength

they obtained differed by less than 2 percent from

average values obtained by the pressurized ring

test and by the gas-bearing tensile test. The coefficient of variation was essentially the same for

both the stress wave technique and the pressurized ring test, about 8 percent. (The coefficient

for the direct tension test was not available.)

The stress wave technique appears to be applicable to elevated temperature use although the

strain rate sensitivity of the specimen materials

will first have to be evaluated.

7. Flexure Test

Because of the problems associated with direct

tension testing, the ceramics industry promoted

the well-known flexure test as a means for evaluating tensile strengths. This test, which is also

referred to as the bend test or the transverse test,

avoids the need for placing the entire cross section

of a specimen in uniform tension, and places only a portion of the cross section in tension. The maximum tensile stresses occur only in the extreme

outer fibers on one side of the specimen, with the

tensile stresses decreasing to zero at the neutral

plane and being balanced by compressive stresses

in the specimen on the other side of the neutral

plane. Thus, while the test is intended to provide

a tensile strength value, it is classified as a bending test because it involves a non-uniform tensile

IMPULSIVELY LOADED SURFACE

COMPRESSION PULSE

i Z Z Z Z Z I FREE BOUNDARY

FREELY SUPPORTED BAR

I // z z

REFLECTED TENSION PULSE

Figure 9. Schematic of stress wave reflection at a free boundary.

250
stress field instead of the uniform field considered

in our discussion of tension test methods.

The flexure test at first appears to be simplicity

itself. As shown in figure 10, a specimen is supported close to each end and on the side opposite

the supports a uniformly increasing load is applied

at a single point centrally located between the

supports (also called mid-point, center point, or three-point loading) or at two points (four-point

loading) equidistant from the supports. The load

is increased until the specimen fractures. At

present, ASTM recommends three-point loading

for the strength testing of brittle materials [26].

The maximum stress at fracture is defined as the flexural strength, and is frequently referred

to as the modulus of rupture, or MOR. The

maximum stress is obtained by using the following

equation:

MC

where a=extreme fiber stress

M=bending moment

C=distance from the neutral plane of the

specimen to the extreme tension fiber I =moment of inertia of the cross section

of the specimen about the neutral axis.

Some assumptions have to be made for the

equation to be valid. These are:

(1) The material obeys Hooke's Law.

(2) The stress is distributed linearly across the

bent beam and is directly proportional to the

distance from the neutral plane.

(3) Bending occurs in a plane of symmetry of

the beam.

(4) The original transverse plane remains a plane after bending occurs and is perpendicular

to all longitudinal fibers after bending.

(5) There are no shear stresses present in the

area where the maximum fiber stress is to be

measured.

In order to justify the assumptions, the flexure

test loses its simplicity and becomes more complicated. Some of the more important factors

1

1

1 V \

i

1

1

f V

> c <

< T >

) \ 1 V

Figure 10. Diagram of a bar flexure specimen.

The specimen is supported near the ends and loaded centrally (solid arrow at top) or in four-point bending (dashed arrows at top). The

dashed horizontal line is the neutral axis. During a test the specimen

is in tension below this line and in compression above it.

involved in designing a flexure test to give valid,

reliable results will be discussed.

The specimen should be of a size that is large

enough to be representative of the material for

which it is serving as a sample. It should also be

of a size convenient to handle but not too large

or expensive to fabricate so as to reduce the number of available specimens for test to a point

where meaningful results cannot be developed. It

is felt that the specimen shape should be kept as simple as possible and, where necessary, the test

assembly made complicated in order to meet the

test requirements.

The most commonly used shapes for specimens

are bars, cylinders, and enlarged-end specimens,

and a specialized enlarged-end specimen called

the dog-bone specimen. These are shown in

figure 11. The bar and cylindrical specimens have

the advantage of economy, ease of preparation,

and generally being representative of the material,

particularly the surfaces and edges, as found in use. Care must be taken not to damage the surface

and edges of the bar specimen that will be in

tension when tested. The cylindrical specimen

has the disadvantage of having only a small part

of the specimen under maximum stress. In fourpoint loading enlarged-end and dog-bone specimens have the advantage of eliminating con- centrated loads and shear stresses in the reduced

gage length, while the dog-bone specimen, which

is loaded and supported along the neutral plane,

has the further advantage of minimizing errors introduced by frictional forces.

To minimize torsional stresses arising from

warped specimens or nonplanar test beds, rockers

are generally used on the bottoms of the supports

(which are commonly knife edges) and loading is often done through spheres.

Frictional forces should be reduced to a minimum. Since these forces act by grabbing and

slipping they produce an extremely erratic effect

on the results. These forces are generally reduced

Figure 11. Common flexure specimens.

The bar and cylinder specimens are probably the most common types while the enlarged-end and dog-bone specimens can provide an area of uniform stress in their reduced sections.

251
by using rounded or rolling knife edges and

lubricants with the bar, cylindrical, and enlargedend specimens, and by loading along the neutral

plane in the dog-boned specimen. A picture of a

flexure test assembly is shown in figure 12. The

long loading arms allowed the specimens to be

tested in various atmospheres.

Some materials that are classed as brittle

nevertheless show some ductility. Using the

regular beam equation will give results higher

than they should be. A correction by Nadai [27]

will compensate for this departure from Hookean

behavior before fracture.

Three-point (or center-point) loading is not the

preferred flexure test method. There may be a wedging action due to the concentrated load which

could lead to erroneous results. In addition, with

four-point loading a much larger area of the specimen is under maximum stress, giving a more

representative value of the usable strength of the

material. Figure 13 [28] shows that center-point

loading tends to "force" the glass to fail near

the loading point which is the point of maximum

stress. In four-point loading it can be seen that

as the loading knife edges are moved farther apart

the failure origins fall between them in a fairly

uniform manner. The strengths indicated along

the sides of the specimen representations in the

figure show that as the area under uniform

maximum stress increases the strength decreases.

This is representative of the size effect. Figure

14 shows photoelastically the large area placed

under uniform stress when four-point loading is used.

8. Large Area Bending Tests

One of the problems in the flexure test is that

even a modestly large specimen may not be

Figure 12. Four-point bending test of a glass

specimen.

One support knife-edge fixes the plane of the speci- men while the other has a rocker bottom to compensate for twist in the specimen. The loading knife- edges have rocker tops and are mounted in a fixture which is loaded through a fifth knife-edge. These

precautions minimize the possibilities of nonvertical load application. The long arms on this apparatus

permit testing of specimens in furnaces and cold boxes.

n m7i i v'.

15. 700 psi

t ' 1

/

r

JZL n Ft

19, 800 psi

f

n n JZL mmTIX

20, 600 psi

Figure 13. Locations of fracture origins and strengths for 10 X 1% X }i-in

annealed soda-lime-silica glass specimens.

Slashed squares and hollow squares indicate fractures originating on the surface and at the edge, re- spectively. Specimens were supported on an 8-in span and tested in three-point bending (bottom

diagram), four-point bending with loads 2 in apart (middle diagram), and four-point bending with

loads 4 in apart (top diagram). The fracture origins cluster around the point of maximum stress for three-point bending, and are uniformly dispersed throughout the maximum stress regions in four-point bending. As the area under maximum stress increases, the strength decreases.

252
Figuee 14. This composite photoelastic print shows the central portion of a flexure specimen in four-point

bending.

The dark lines are lines of uniform stress which indicate that the tension

face of the specimen experiences uniform stress over a length slightly longer than the distance between loading points.

representative of large applications such as glass

windows. Pressure tests have been used with some

success to minimize this problem. In this test the

specimen, generally a flat plate, is held near the

edges and a uniform load is applied over the

entire surface of the specimen. There are no

recognized standards and this test is generally

designed to meet specific conditions. This test method has the advantage of being able to place

large areas of a material under load and under

conditions similar to those found in practice

(e.g., glass windows under wind load or water

pressure). It has the disadvantage of placing only

a small part of the specimen under maximum

stress. A test method presently being considered by

several laboratories to place large areas of a material under uniform stress is the concentric

ring test. In this test a flat specimen is supported

on a ring and loaded through a concentric ring of

appropriately smaller diameter than the support

ring. There is much that needs to be learned

before this method can be used with confidence,

such as the effects of specimen overhang, specimen

deflection, and the ratio of support ring diameter

to loading ring diameter. Advantages of this test method are its ability to place large areas under

uniform stress and to test only the surface of the

specimen and thus eliminate the influence of

edges on the results.

9. Diametral Compression Test

In this test method a cylindrical specimen is compressed diametrically between two flat platens.

Under proper conditions fracture occurs along the

loaded diameter (AB in fig. 15) as a result of

tensile stresses which develop across this diameter.

Like the flexure test, this test is used for the

determination of tensile strength although it does not develop a uniform uniaxial state of

tensile stress. The method is also known as the splitting

tension test, or the Brazilian test, and was originally developed in Brazil and Japan around 1943

(a) LOADING (b) MAXIMUM TENSILE STRESSES

DEVELOP ACROSS LOADED

DIAMETER , A B

Figure 15. Diametral compression of a solid cylindrical

specimen.

for concrete [19]. Figure 16 shows a diametral

compression test on an asphalt specimen [29].

This test has gained considerable favor because

of its relative simplicity. The specimen alines itself between the platens and can be easily tested

without elaborate fixtures.

If the specimen is not long in comparison with

its diameter, the diametral loading sets up a biaxial stress distribution in the cylinder. Frocht

[30] evaluated the stress distribution using elasticity theory and T. S. Love [31] verified this

solution photoelastically. The maximum tensile

Figure 16. Diametral compression test of an asphalt specimen at the University of Texas.

253
stresses which act uniformly across the loaded

diameter are given by:

2P

where P=applied load

D=specimen diameter

£=specimen thickness.

This analysis assumes that the specimen is isotropic and homogeneous as well as elastic, and

that the compression loads are restricted to lines

of contact between the cylinder and the platens.

In a real situation, local deformation of the cylinder and platens distributes the compression load over an area, but Rudnick, Hunter, and Holden

[32] have shown that this does not significantly

alter the stress distribution near the center of the

specimen. In fact, it is this feature of the problem

that makes the test feasible. It is considered good

practice to place narrow bearing pads of suitable

material between the specimen and the platens to

compensate for surface irregularities and to reduce

the infinite compressive stresses immediately

under the applied load. Figure 17 shows the

elastic stress distribution along the loaded diameter in a specimen having a length equal to one- tenth of the diameter, and assuming uniform load

distribution over a contact width of one-tenth of

the diameter. ox is the maximum tensile stress

(which acts normal to the loaded diameter), <r2 is the maximum compressive stress (which acts

along this diameter), and r is the maximum shear

stress (which is inclined to the diameter). It may

be seen that very high compressive stresses act

at the loaded surfaces and that high shear stresses

exist immediately below these surfaces. Thus, in

order to achieve the desired tensile failure the

compressive and shear strengths of the specimen

must be several times higher than its tensile

strength, and the applied load must be distributed over an adequate area. The selection of the proper bearing pad material

is the key to a successful test. As Rudnick, Hunter,

and Holden have pointed out, if the pad is too

hard the load application will approach line contact and fracture may initiate in compression or shear rather than in tension. If the pad is too soft

and the load is thereby distributed over too great

an area the tensile stresses in the central portion

of the specimen are reduced and the apparent

tensile strength is increased. The selection of the

proper pad material is done experimentally. One

criterion is that it produce tensile, rather than compressive or shear, failures. (Compressive failures

appear as local crushing, shear failures start at an

angle to the loaded diameter, and tensile failures

break the specimen along the loaded diameter.)

Another factor involved is friction between the

Figure 17. Stress distribution along loaded diameter for a diametrally compressed cylinder, assuming uniform

contact pressure over the contact area.

254
specimen and the pads. This tends to increase the

apparent tensile strength [33]. Blotting paper

and punch cards are two pad materials currently

used for tests on ceramics, while plywood is standard for concrete [34].

The most critical alinement factor in the

diametral compression test is the parallelism of

the platens in the direction of the axis of the

cylinder. In order to maintain this as loading

progresses, it is important that the specimen be

centered with respect to the loading heads of the

testing machine. Ivey and Buth [35] found that a longitudinal displacement of the specimen by

8 percent of its length caused an apparent strength

reduction of 12 percent. On the other hand,

parallelism of the platens in the direction normal

to the cylinder axis was not found to be critical.

An angularity of 6 degrees reduced the apparent

strength by only 4 percent. If the bearing pads

are narrow and are not centered with respect to

the loaded diameter, strength reductions of the

order of 8 percent may be expected. Any kind of

misalinement increases the coefficient of variation

of the test results beyond the 5 percent value

obtainable with homogeneous materials and good

testing technique.

Another possible source of error in the diametral

compression test was described by Bortz and

Wade [7]. If the tensile strength of the specimen

material is high, and its elastic modulus is low,

considerable flattening can develop at the loading

areas before failure is attained. This distortion

invalidates the theory used to develop the loadstress relation and results in lowered values of

apparent tensile strength.

Finally, it should be mentioned that the diametral compression test is not amenable to the

direct determination of stress-strain characteristics.

On the other hand, the test has numerous advantages besides simplicity. It is suitable for

elevated temperature testing and may be used for

very small specimens. It is less affected by surface

conditions of the specimen than any of the other

test methods. Only at the ends of the cylinder

do maximum tensile stresses appear at a surface.

Shook [3] has described a means of eliminating

even this surface interaction. If the ends of the

specimen are reduced in diameter by a gradual

tapering, the applied load and resulting tensile

stresses will be smaller in this region. However,

the effects of this non-uniform load distribution

on the stress distribution near the mid-length of

the specimen are not immediately obvious.

10. Brittle Ring Test

The brittle ring test is a variation of the diametral compression test and employs a thickwalled, hollow, cylindrical specimen instead of a

solid cylinder. Loading is accomplished by compressing the specimen diametrally between flat

platens. A steep stress gradient (fig. 18) is produced

in the ring similar to that obtained in the flexure

T

p

Figure 18. Stresses developed in a ring subjected to diametral compression.

test. The maximum tensile stress appears on the

inner surface of the ring, at the loaded diameter,

in the tangential direction. Fracture always

initiates here [23].

This test method has many of the advantages

of the diametral compression test method. The

specimen alines itself between the parallel platens

without any special gripping apparatus. Furthermore, the test is easily adaptable for elevated

temperature use. The principal deficiency of the method is the

determination of the load-stress relationship. This

problem has been studied theoretically by Timoshenko [36], photoelastically by Frocht [37], with

brittle coatings by Durelli et al. [38], and with

strain gages by Bortz and Lund [39]. From a comparison of available data, the latter investigators concluded that the relationship between the

maximum tensile stress and the applied load

depends only on the inner and outer radii of the

specimen, independent of the elastic constants. In

his review of this work Shook [3] expressed a less

optimistic viewpoint. It appears that more work

is required to evaluate the suitability of this test method insofar as numerical strength determinations are concerned. The statistical variation of

test results with this method, however, is reportedly better than that obtained with standard

tensile specimens [39].

Because of the steep stress gradient, only a small volume of the specimen experiences the

maximum tensile stress. Rudnick and Duckworth

[40] have pointed out that with large-grained

materials the highly stressed region may be

confined to a single grain, in which case transgranular fracture would be encouraged. With

small-grained materials fracture at the grain

boundary would be encouraged. Properly interpreted, the test can be valuable in both cases.

255
Bortz and Lund [39] have observed that with this

test method there is no strength variation with

volume variation such as occurs with other test

methods. This, too, is probably due to the maximum tensile stress being confined to a small

volume of material.

11. Torsion Test

In most of the test methods discussed thus far,

the state of stress at the fracture location is uniaxial. The exceptions are the pressurized ring test,

the large area bending tests, and the diametral

compression test. Classical theory holds that

fracture is dependent only on the maximum tensile

stress, i.e., that a transverse stress component does

not affect longitudinal strength. While there are

insufficient data available to categorically verify

this principle for brittle ceramics, it is equally true

that evidence to contradict it is likewise lacking.

In a torsion test of a circular cylindrical specimen the stresses are zero at the cylinder axis and

increase linearly with the radius. The maximum

tensile stresses occur on the surface at a 45-degree

angle to the axis. Numerically equal compressive

stresses act perpendicular to these tensile stresses. The maximum shear stresses, which are also

numerically equal to the maximum tensile and

compressive stresses, act parallel and perpendicular to the axis of the specimen. In a brittle material, which has lower strength in tension than in

shear, fracture occurs normal to the maximum

tensile stresses, i.e., at 45-degrees to the specimen

axis. An interesting example of such a fracture is shown in figure 19, which shows a torsion failure

of a human leg bone, similar to those experienced

in skiing accidents [41].

The torsion test has not been used much for

determining tensile strengths in brittle materials.

An often quoted objection to it [2, 42] relates to

the consequences of improper gripping. If the

specimen is not straight, or if the torque axis is not coincident with the specimen axis, undesirable

bending stresses are introduced. However, this

alinement problem is not as severe as the one in

the tension test, and if specimens having large

slenderness ratios (i.e., length/diameter) are used,

visual alinement should be satisfactory. In any

case, the presence of bending stresses can be

checked with strain gages.

Another objection to the torsion test is similar

to that raised in connection with the flexure test. If the material is not, in fact, linearly elastic to

failure then the magnitude of the fracture stress

is somewhat obscured. The possibility of some

plastic deformation prior to failure is greater

in the torsion test than it is in the tension or

flexure tests because the shear stress is higher

here. But this deficiency can be overcome by

using a thin-walled tubular specimen instead of a solid cylinder. In this case the load-stress

relation is statically determinate, regardless of

the shape of the stress-strain curve. Furthermore,

Figure 19. Spiral fracture of a tibia resulting

from a torsion test.

with the thin-walled tube the stresses through the

thickness may be considered uniform; more so, in

fact, than in the pressurized ring specimen [43].

The torsion test is readily adaptable to elevated

temperature use.

12. Compression Test

All of the test methods discussed thus far are

used, to some extent at least, to determine the

tensile strengths of brittle materials. The compression test, on the other hand, is used to

evaluate "compression strengths." However, it is realized that the values obtained are not truly

representative of the compressive strength of the

materials since, even in this case, failure occurs as a result of induced tensile stresses [44].

"In the strength testing of ceramics . . . analyses indicate that with proper and known

precautions specific to each test method, the use of

bending, torsion, direct tension, and diametral

compression can allow the stress that causes

failure to be determined without serious doubt.

However, with direct compression, the actual

failure stress is a subject of doubt . . ." [2]. The major problem, in compression testing, is related to the tendency of the specimen to expand

laterally under load. Frictional forces are created

between the specimen ends and the load plates

due to differences in geometry and Poisson's

ratio. The forces can be reduced but not eliminated

by lubrication, and the nominally uniaxial compressive stress distribution is thereby distorted

in the end portions of the specimen.

256
Another problem in compression testing involves

alinement. Specimen ends should be ground perfectly flat and parallel, and normal to the axis of

the specimen. These ends should be concentrically

mated with the flat and parallel loading plates in

the testing machine. Deviations from ideality

contribute to eccentric loading which, like the

friction forces, superpose undesirable bending

stresses on the uniaxial compressive stress distribution.

In order to reduce the effects of end friction,

Bortz and Wade [7] recommend the use of enlarged

end specimens in which the length-to-diameter

ratio of the reduced section is at least 2. For uniform specimens this ratio should be increased to

2.5. This exceeds standard requirements [45].

Of course, too large a ratio must also be avoided

since this may cause buckling. Specimens with

round cross sections are favored because their

transverse expansion under load tends to be radial

and uniform. However, friction forces cannot be

relied upon to behave uniformly. The use of a

lubricating pad (e.g., Teflon) is usually recommended, but the high compressive strengths of

some ceramics require that end bearing plates be

made of very hard material such as tungsten

carbide [13]. In this case specimen ends may be

lubricated with graphite and oil. Lynch, Ruderer, and Duckworth [42] recommend using three specimens in series to eliminate

the effects of end restraints on the middle one. This appears to be a remedy for the case where

specimen lengths are limited for some reason;

however, it will have to be demonstrated that

failure initiates in the middle specimen.

The use of a universal, or a spherically seated,

loading block in the testing machine is sometimes

recommended as a means of minimizing load

eccentricity [46]. Such devices are never entirely

friction-free and are, therefore, only a poor substitute for precision machining and accurate alinement.

An interesting approach to good compression

testing technique was recently described by Babel

[15, 47]. This approach employs hollow cylinders

for the specimen and end cylinders (see figs. 20

and 21). The test cylinder has a curved exterior

surface which provides a reduced thickness in the

gage area and thus precludes edge-initiated fail- ures. The specimen and the end cylinders have

conical ends which are lapped together to minimize

stress discontinuities at their interfaces. (A better,

although more expensive, design would include

the end cylinders as integral parts of the specimen.)

This design limits bending stresses in the specimen

to less than 5 percent and is not susceptible to

buckling. Nevertheless, it yields greater scatter in

compression test results than does the pressurized

ring test using the same specimen. It is recommended that strain gages be used in every test to

monitor the uniformity of stress. It is interesting

to compare a typical compression failure obtained

with this specimen design (fig. 22) with that

obtained with a solid cylinder (fig. 23) [48].

The compression test is adaptable to elevated

temperature use although, like the direct tension

test, limiting accessibility to the specimen usually

is conducive to alinement problems. Evans [49]

has described a high-temperature compression

testing apparatus capable of use to 2,200 C (see

fig. 24) . Testing is carried out in a helium atmosphere using a tantalum heating element. To avoid

damaging the ends of the rams, flat disks were

placed between the specimens and the rams. These

disks were removed after each test, ground to

LAP THIS SURFACE WITH END CYLINDER

THE THICKNESS MAY VARY WITHIN THE LIMITS STATED FROM SPECIMEN TO SPECIMEN,

BUT ON ANY ONE SPECIMEN THE THICKNESS MUST NOT VARY MORE THAN ± 0.0002.

Figure 20. Hollow compression specimen.

257

323-655 O - 69 - 17
Figure 21. End cylinder for use with the hollow compression specimen.

Figure 22. Typical failure of hollow zirconia specimen in compression.

yield a flat surface, and re-used. In tests of magnesia and alumina, tungsten rams and disks were

found to be adequate to 1600 C, but for higher

temperatures alumina was required.

The Griffith criterion for fracture of brittle

materials specifies a compressive strength eight

times the tensile strength. This is based on the

assumption of infinitely sharp cracks in the specimen, distributed with random orientations. Babel

and Sines [50] extended the Griffith analysis to

cracks of finite sharpness and arrived at compressive strengths from three to eight times the tensile

strength, depending on the sharpness of the cracks.

Measured compressive strength data for ceramics

are limited. They are often higher than reported

tensile strengths by a factor of 6 to 10, although

factors of 4 to 5 are reported for some amorphous

materials [44].

258
Figure 23. Explosive failure of standard

concrete cylinder specimen in compression

test by Soiltest, Inc.

SELF ALIGNING SEAT

LOAD CELL O-RING SEALS FURNACE BODY TUNGSTEN PLATENS SPECIMEN

SIGHT GLASS ASSEMBLY TUNGSTEN COMPRESSION RAMS

HEATER

SPACER COLUMN

1 1. JACK PRESSURE PAD

12. JACK

13 RADIATION SHIELDS

Figure 24. High-temperature compression testing

apparatus.

13. General Considerations

There are a great many factors that can influence the results of a strength test. A good testing

program is designed so that only one property is

measured at a time. In other words, a strength test

program may be initiated to evaluate strengths for a particular set of conditions, or it may be intended

to evaluate the influence of some other single

factor on strength. In either case, the possible

effects of other factors must be considered. Some

brittle materials are particularly sensitive to

ambient conditions, surface and edge conditions,

strain rate, or size effects. With a new material,

precautions should be taken until it is understood.

Only when the sensitivity of the material to these

factors has been studied can the required degrees

of control over them be established.

An example of the importance of not treating a new material in a routine manner was shown when

the chemically strengthened glasses were first

tested in flexure. Peculiarities were noted in some

of the strain gage readings. Further investigation

showed these glasses had considerable delayed

elastic effects. This is shown best in figure 25

[51] in which dead weights were used. As can be

seen, the chemically strengthened glasses showed

considerable deflection and recovery long after the

load was applied and removed—a rather unusual

behavior for a brittle material like glass. Clearly,

in tests of this material it is important that loads

be applied in a controlled manner and rate and

that these conditions should be well documented.

This precaution is as applicable to quasi-static

and dynamic strength tests (e.g., fatigue, stress

rupture) as it is to static strength testing. (We

might mention, at this point, that our omission

of these types of tests from this survey is intentional. The requirements for these tests that are peculiar to brittle materials are the same as they

are for the static tests, e.g., alinement. On the

other hand, the equipment for generating cyclic

loads or for maintaining constant loads is essentially the same regardless of whether the specimen

material is brittle or ductile.)

LOAD OFF

2 0.2

a 0.1 -

TIME IN MINUTES

Figure 25. Deflection and recovery of two chemically

strengthened glasses, as a function of time.

259
When evaluating the effects of some environmental condition on strength, a different specimen

geometry is often used under the special condition

than is used under normal conditons. A comparison

should be made of different types or sizes of specimens under identical conditions before comparisons

of results under different conditions are attempted.

An example of the effect of size on the strength of

glass rods is shown in figure 26 [52], and it can be

seen that there is a considerable variation in

strength with size.

In this connection it is instructive to consider

Rudnick's [32] view. "The relationship between

tensile strengths of brittle materials determined

by any two test techniques is not a universal

constant. It depends on the stress distributions

developed by each technique, the specimen sizes

used, and, particularly, the nature of the material

itself. According to the statistical concepts of the

strength of brittle materials, it is theoretically

possible to achieve any desired value of strength,

within limits, from any test procedure simply

by varying the specimen size. Thus, while the

bend strength is higher than the uniaxial tensile

strength if specimens of equal size are used,

this order can be reversed by employing very

small tensile specimens and very large bend

specimens."

An example of the effect of surfaces and edges on the strength is shown in table 1 [28]. This table

presents strength test results for annealed, ground,

and polished soda-lime-silica glass from two

different manufacturers. The table gives the

number of fractures that occurred at the edge or on the surface of the specimen and the average

modulus of rupture. The data indicate that glass

B is stronger than glass A and that glass B has

less severe flaws in the surface than glass A.

Also shown, for both glasses, is that the average

22 "

20 -

S 12

.6 .8 1.0 1.2

ROD DIAMETER, INCHES

1.4 1.6

strength of those specimens that fracture from

the edge is consistently lower than that of those

fracturing from the surface. The table also shows

another example of the size effect, for as the area under uniform stress increases the strength

decreases.

Table 1. Effects of surfaces and edges on flexural strength

Area Glass A Glass B

under max.

stress Fracture Number of Mod. of Fracture Number of Mod. of origin 1 fractures rupture origin 1 fractures rupture

in« psi psi

0 S 37 19 308 S 30 21 572 E 13 16 550 E 20 19 091

3 S 34 17 546 S 33 20 712 E 16 15 016 E 17 17 994

G S 33 15 260 S 32 19 215 E 17 13 341 E 18 14 919

12 S 19 13 365 S 21 16 968 E 31 11 942 E 28 14 788

Figure 26. Average fracture strength versus rod

diameter for annealed Pyrex glass, showing how

strength decreases as rod diameter increases.

1 S=surface. E=edge.

14. Selecting a Test Method

The criteria involved in selecting a test method

are more complex than immediately meets the eye,

and are beyond the scope of this paper. The

reader faced with this problem is referred to

Rudnick and Duckworth [40], who have described

the principles guiding the selection of a test

method exceptionally well. The principles are based upon the intended use of the test results.

If it is intended to determine, as closely as possible,

the "true" tensile strength of a brittle material,

the work of Bortz and Wade [7] merits careful

study. On the other hand, the designer concerned

with a particular piece of hardware would be wise

to follow Boland and Walton's [44] suggestion,

i.e., he should see that the materials evaluated

for his program are tested in such a manner that

stress fields are similar to those encountered in

service. To use data obtained under near-ideal

conditions for an application in which the material

will be used in a very non-ideal situation would

almost guarantee failure.

To this we add that no test method is better

than the experience and competence with which

it is used.

15. References

[1] Harmsworth, C. L., Design criteria and test techniques, Air Force Materials Symposium/1965

Technical Papers, AF Materials Lab Technical

Report No. AFML-TR-65-29, 831 (1965).

[2] Duckworth, W., Strength testing of ceramics, Pre- sented to Electronics Div, Am. Ceram. Soc,

Chicago (Sept. 1966).

[3] Shook, W. B., Critical survey of mechanical property

test-methods for brittle materials, AF Materials

Lab Technical Documentary Report No. ASDTDR-63-491 (July 1963).

260
[4] Hughes, B. P., and Chapman, G. P., Direct tensile

test for concrete using modern adhesives, RILEM

Bull. 26, 77 (Mar. 1965).

[5] Barnett, R. L., Costello, J. F., Hermann, P. C, and

Hofer, K. E., Jr., The behavior and design of

brittle structures, AF Flight Dynamics Lab

Technical Report No. AFFDL-TR-65-165 (Sept.

1965) AD 475040.

[6] Bortz, S. A., Effect of structural size—the "zero

strength," Studies of the Brittle Behavior of

Ceramic Materials, AF Directorate of Materials

and Processes Technical Documentary Report No.

ASD-TR-61-628, 36 (April 1962) AD 277659.

[7] Bortz, S. A., and Wade, T. B., Analysis and review

of mechanical testing procedures for brittle materials, Presented at Army seminar on Analysis of Mechanical Testing Procedures for Brittle Materials, Chicago (Mar. 1967).

[8] Pears, C. D., Evaluation of tensile data for brittle materials obtained with gas bearing concentricity,

AF Directorate of Materials and Processes Technical Documentary Report No. ASD-TDR-63-245

(May 1963) AD 410770.

[9] Digesu, F. J., and Pears, C. D., Some parameters of the mechanical behavior of graphites, Presented

to Am. Ceram. Soc, Los Angeles (Oct. 1963).

[10] Todd, J. D., The determination of tensile stress/strain curves for concrete, Proc. Inst. Civil Engrs.

(London) 4, Part 1, 201 (1955).

[11] Mordfin, L., and Robusto, T., Jr., Photographic

strain measuring technique for use above 3000F,

Proc. Instrument Soc. Am. 17, Part 1, 3.4.62-1

(1962).

[12] Sedlacek, R., and Halden, F. C, Method for tensile

testing of brittle materials, Rev. Sci. Instr. 33, 298

(1962).

[13] Cornish, R. H., and Broutman, L. J., Effects of

polyaxial stresses, Studies of the Brittle Behavior

of Ceramic Materials, AF Materials Lab Technical Documentary Report No. ASD-TR-6 1-628,

Part 3, 289 (June 1964) AD 444829.

[14] Sedlacek, R., Tensile strength of brittle materials,

AF Materials Lab Technical Report No. AFMLTR-65-129 (Aug. 1965).

[15] Babel, H. W., Biaxial-fracture strength of brittle materials, AF Materials Lab Technical Report

No. AFML-TR-66-51 (Mar. 1966).

[16] Dally, J. W., Design data for materials employed in thermal protective systems on advanced aerospace

vehicles, AF Materials Lab Technical Documentary

Report No. ML-TDR-64-204, Vol. 3 (Aug. 1965)

AD 476964.

[17] Woldokoff, V. E., Briggs, D. D., and Moss, J. R.,

Modified Sedlacek-Halden ring test, Presented to Am. Ceram. Soc, New York (May 1967).

[18] Malhotra, V. M., Zoldners, N. G., and Woodrooffe,

H. M., Ring test for tensile strength of concrete,

Materials Research & Standards 6, 2 (1966).

[19] Malhotra, V. M., and Zoldners, N. G., Comparison

of ring-tensile strength of concrete with compressive, flexural, and splitting-tensile strengths, J. Materials 2, 160 (1967).

[20] Daniels, I. M., and Weil, N. A., Effect of non-uniform

stress fields, Studies of the Brittle Behavior of Ceramic Materials, AF Directorate of Materials

and Processes Technical Documentary Report No.

ASD-TR-61-628, 119 (April 1962) AD 277659.

[21] Durelli, A. J., Morse, S., and Parks, V., The theta

specimen for determining tensile strength of brittle materials, Materials Research & Standards 2, 114

(1962).

[22] Bortz, S. A., A means for obtaining axial tension

from a bending test, Presented to Am. Ceram.

Soc, Pittsburgh (April 1963).

[23] Barnett, R. L., Utilization of refractory nonmetallic

materials in future aerospace vehicles, AF Flight

Dynamics Lab Technical Documentary Report

No. FDL-TDR-64-123, Part 1, Review of structural design techniques for brittle components

under static loads (Sept. 1964) AD 607977.

[24] Cornish, R. H., and Abbott, B. W., Strain rate effects,

Studies of the Brittle Behavior of Ceramic Materials, AF Materials Lab Technical Documentary

Report No. ASD-TR-61-628, Part 3, 77 (June

1964) AD 444829.

[25] Abbott, B. W., and Cornish, R. H., A stress-wave

technique for determining the tensile strength of

brittle materials, Experimental Mechanics 5, 148

(1965) . [26] Standard method of flexure testing of glass, ASTM

Designation C158-43 (1965), 1966 Book of ASTM

Standards, Part 13, 113 (Am. Soc. Testing Materials, Philadelphia, 1966).

[27] Nadai, A., Plasticity, 128 (McGraw-Hill Book Co.,

New York, 1931).

[28] Kerper, M. J., and Scuderi, T. G., Comparison of

single-point and two-point loading for determining

the strength of flat glass, Proc. Am. Soc. Testing

Materials 64, 1037 (1964).

[29] Anon., Versatility is the watchword, Closed Loop 1,

No. 5, 11 (1966).

[30] Frocht, M. M., Photoelasticity, Vol. 2 (John Wiley &

Sons, New York, 1948).

[31] Love, T. S.,' Structural analyses using photoelastic

techniques, General Dynamics/Convair Applied

Research Report No. 11, 131 (1967).

[32] Rudnick, A., Hunter, A. R., and Holden, F. C, An

analysis of the diametral-compression test, Materials Research & Standards 3, 283 (1963).

[33] Pincus, G., and Gesund, H., Evaluating the tensile

strength of concrete, Materials Research & Standards 5, 454 (1965).

[34] Standard method of test for splitting tensile strength

of molded concrete cylinders, ASTM Designation

C496-66, 1967 Book of ASTM Standards, Part 10,

356 (Am. Soc. Testing Materials, Philadelphia,

1967).

[35] Ivey, D. L., and Buth, E., Splitting tension test of

structural lightweight concrete, J. Materials 1, 859

(1966) . [36] Timoshenko, S., and Goodier, J. N., Theory of elasticity, 2d ed., 121-123 (McGraw-Hill Book Co.,

New York, 1951).

[37] Frocht, M. M., Strength of materials, 175-177

(Ronald Press Co., New York, 1951).

[38] Durelli, A. J., Phillips, E. A., and Tsao, C. H., Intro- duction to the theoretical and experimental analysis

of stress and strain, 441-455 (McGraw-Hill Book

Co., New York, 1958).

[39] Bortz, S. A., and Lund, H. H., The brittle ring test, Mechanical Properties of Engineering Ceramics

(Kriegel and Palmour, eds.), 383 (Interscience Publishers, New York, 1961).

[40] Rudnick, A., and Duckworth, W. H., Considerations

in testing brittle materials, Presented at Army

seminar on Analysis of Mechanical Testing Procedures for Brittle Materials, Chicago (Mar. 1967).

[41] Outwater, J. O., and Woodard, M. S., Skiing forces

and fractures, Mech. Eng. 89, No. 12, 26 (Dec.

1967).

[42] Lynch, J. F., Ruderer, C. G., and Duckworth, W. H.,

Engineering properties of ceramics, AF Materials

Lab Technical Report No. AFML-TR-66-52 (June

1966) AD 803765.

[43] Mordfin, L., Torsion creep of circular and noncircular

tubes, J. Research NBS 71C, 209 (1967).

[44] Boland, P., and Walton, J. D., Jr., Aerospace ce- ramics—characteristics and design principles, AF

Materials Lab Technical Report No. AFML-TR65-171 (June 1965) AD 469154.

[45] Standard method of test for compressive (crushing)

strength of fired whiteware materials, ASTM Designation C407-58, 1966 Book of ASTM Standards,

Part 13, 349 (Am. Soc. Testing Materials, Philadelphia, 1966).

261
[46] Standard method of test for compressive strength of molded concrete cylinders, ASTM Designation C39-

66, 1967 Book of ASTM Standards, Part 10, 27

(Am. Soc. Testing Materials, Philadelphia, 1967).

[47] Babel, H. W., and Sines, G., An improved method for uniaxial and biaxial testing of brittle materials,

Douglas Aircraft Co. Paper No. 4244 (Mar. 1967)

submitted to Am. Soc. Testing Materials.

[48] Test Engineering 18, No. 10, 38 (Oct. 1967).

[49] Evans, P. R. V., Effect of microstructure, Studies of the Brittle Behavior of Ceramic Materials, AF

Directorate of Materials and Processes Technical

Documentary Report No. ASD-TR-61-628, Part

2, 164 (April 1963) AD 408925.

[50] Babel, H. W., and Sines, G., A biaxial fracture cri- terion for porous brittle materials, Douglas Aircraft

Co. Paper No. 4221 (Feb. 1967), submitted to J. Appl. Mechanics.

[51] Kerper, M. J., and Scuderi, T. G., Mechanical properties of chemically strengthened glasses at high

temperatures, J. Am. Ceram. Soc. 49, 613 (1966).

[52] Kerper, M. J., and Scuderi, T. G., Relation of the

fracture stress to the fracture pattern for glass rods

of various diameters, Bull. Am. Ceram. Soc. 45,

1065 (1966).

262
SUBJECT INDEX

A

Absorptance, 74, 126

Absorption coefficient, ultrasonic, 74, 174,

178

Alloying, effect on thermal conductivity, 85

Alumina, 2, 3, 15, 69, 111

crack nucleation, 232

creep, 192, 196

elastic constants, 146, 147, 153

emittance, 136

fracture, 232

fracture surface energy, 226

strength, 2, 3, 232, 246, 247, 250, 258

American Ceramic Society, 1, 5, 260-262

Anelasticity, 159, 189, 190, 259

Anisotropy, 146, 83, 90

effect on elastic moduli, 146, 154

effect on elastic waves, 151

effect on resonance, 152

Anisotropic grains, 15, 41, 42, 45

Annealing, 177, 180, 204

effect on emittance, 132

effect on thermal conductivity, 95

Annealing point, 180

Applications of ceramics, 1, 3, 13, 16, 2

213, 260

ASTM, 1, 5, 9, 101, 102, 251, 261

B

Bend testing (see Flexure testing), 15, 210,

211

BeryIlia, 110

crack nucleation, 232

creep, 193

fracture, 232

strength, 210, 232

thermal conductivity, 110

Binary systems, 3, 24, 87-92, 207, 212

Blackbodies, 38, 126

Bonding, 202, 211, 212

Borides, 201, 208, 246

Brillouin zone, 66

Brittleness, 13, 14, 243

effect on design, 13, 14, 243, 260

Bulk modulus (see Compressibility), 142,

147, 149, 155

C

Carbides, 201, 257

Ceramic industry, 1, 2, 15, 250

Ceramics, 1, 2, 3

Ceramic society, 1 Ceramics, IV

applications, 13

ASTM activities, 9

relation to materials technology, 1 Character, IV, 14-16, 41

Characterization, 3, 14-16, 112, 208

with respect to elastic moduli, 161

Chemical composition, 3, 14-16, 41, 83

Clapeyron equation, 22

Coble creep, 191

Cohesive strength, 217

Compliance method, 223

Component, 19

Composites, 154

elastic moduli, 154

thermal conductivity, 87-92

thermal expansion, 55

Composition, 2, 3, 16

effect on thermal conductivity, 3, 14,83

effect on thermal expansion, 55

effect on elastic constants, 161

Compressibility (see Bulk modulus), 142,

147, 149, 155

Compression testing, 247, 248, 250, 253,

255, 256

Compromises in standards, 5

Conductivity

electrical, 70-73

thermal, 3, 63-123

Congruent melting, 212

Consensus, 10

Continuing education, 1, 2

Crack nucleation, 229, 231, 246

Crack propagation, 230, 235, 246

Crack tip, 220

nature of, 220

plastic deformation at, 222, 225

Creep, 189, 191-192, 195, 197-198, 204,

211

Critical point, 177

Crystal structure, 42, 83-84, 209

effect on reflectance, 132

Crystallinity, effect on thermal conductivity, 85
D

Deformation, 189, 198, 201, 243

Deformation point, 177

Degree of freedom, 19, 77

Delayed failure, 234

Density, 16, 41, 95

measurement, 172

Design techniques, 14-15

Dielectric constant, 156

Dielectric strength, 3

Diffusion, 64, 78, 191, 204

Dislocations, 2, 70, 202, 203, 204

motion, 195, 197, 204

E

Edge and surface effects, 15, 159, 246,

251-253, 255, 259, 260

on thermal radiation properties, 127

Education (see Continuing education), 1, 2

Elastic constants, 14, 139, 209, 246, 248,

255, 256

glasses and melts, 177

isotropic body, second order, 147

isotropic body, third order, 149

measurement, 160

porosity dependence, 153

pressure dependence, 149

single crystal, second order, 145

single crystal, third order, 148

temperature dependence, 157, 209

Electrical conductivity, 70-73

Electrical properties, 2, 3

Electrical resistance, 3

Electromagnetic fields, effect on thermal

conductivity, 81

Electromigration, 78

Electron microscopy, 204, 205, 207, 212

Electronic conduction, 70-72

Electronic structure, 70, 83, 202, 212

Electrons, 70-72

free, effect on expansivity, 45

Emittance, 37, 73-76, 126

Engineering, 2, 3

Exciton, 78

Extinction coefficient, 74, 132

F

Fermi function, 70

Firing temperature, effect on thermal

conductivity, 95

Firing time, effect on thermal conductivity,

95

Flaw theory, 14, 219

Flexure testing, 248-253, 255, 256, 259

Fourier's law, 64

Fracture, 14, 217, 243

continuum approach, 220

grain boundaries, 226, 231

surface energy approach, 222

Fracture mechanics, 14, 217

Fracture surface energy, 217

bicrystals, 226

glass, 225

measurement methods, 224

polycrystalline ceramics, 226

single crystals, 226

Friction effects in testing, 244-246, 251,

254, 256, 257

Fulcher equation, 183

Fused silica, 80, 85, 111

Fusion curve, 22, 23, 25

G

Gas conduction, 77, 81

Gifkins creep, 191

Glass, 1, 2, 14, 53, 85-86, 111, 177, 190,

198, 217, 225, 228, 234, 252, 253, 259,

260

Glass-crystal system, 54, 55, 111

Glass transformation, 177

Glass transition, 177, 180, 182

Grain boundaries, 2, 3, 14, 189, 210, 226,

255

effect on thermal conductivity, 93-95

Grain boundary sliding, 190, 197

Grain size (or growth), 2, 14, 15, 16, 210

effect on expansivity, 41, 55, 210

Graphite, elastic constants, 160

Griffith theory, 219

Griffith flaws, 14, 219

H

Hard metals, 201

Heat capacity, 64

Heat conduction, 63-123

Heat transfer, 125, 63-123

Heat treatment, effect on thermal

conductivity, 95

Heterogeneous materials, 87

emittance of. 131

thermal conductivity of, 87-95
Holes, 72

I

Imperfections, effect on thermal conductivity, 86-87

Impurities, 16

effect on expansivity, 41

effect on thermal conductivity, 84-85

Index of refraction, 73, 75-76, 156

Inelastic deformation (see Plastic deformation), 189, 190, 191, 201, 228, 231,

248, 252

Insulators, spark plug, 1,3

Interferometer, 48

L

Lattice conduction, 65-70

Lattice dynamics, 44, 65-67, 156

Lorenz function, 71-72

M

Magnesia, HI

crack nucleation, 232

creep, 194, 196

elastic constants, 146, 147, 149, 150,

153

emittance, 136

fracture, 232

plastic deformation, 201, 231

strength, 231, 232, 258

Materials, 5, 14

Mechanical behavior, 1, 2, 3, 14, 189,

201, 243

elasticity, 139

fracture, 217, 243

plastic deformation, 189, 201

viscoelasticity, 169

testing, 15, 243

Mechanical properties ( see Mechanical

behavior), 1, 2, 3, 14, 189, 201

Mechanical testing (see Methods of testing),

15, 47, 96-108, 133, 243

Melting point, 19

methods, 33

Metastability, 20, 23

Metastable equilibria, 23

Methods of testing, 15, 47, 96-108, 133,

243

brittle ring, 255

compression, 256

diametral compression, 253

flexure, 250

large area bending, 252

pressurized ring, 246

stress wave, 250

tension, 243

theta specimen, 248

torsion, 256

truss beam, 249

Microcracks, 189, 202, 246, 258

effect on expansivity, 55

effect on thermal conductivity, 94

Microcrystalline glass, 111

Microhardness, 3, 209, 210, 211, 212

Microstructure (see Character), IV, 1, 2,

3, 15, 210, 255

effect on thermal conductivity, 93-

94

effect on viscoelasticity, 182

Modulus of rupture, 210, 251, 260

Moduli of glasses and liquids, 178

Multi-component system, 28, 55, 87-93

N

Nabarro-Herring creep, 191

National Bureau of Standards, II, m, IV,

1, 111

National Research Council, IV, 2, 112

Neutron irradiation, effect on thermal

conductivity, 86-87

O

One component system, 20

Optical absorption, 156

Optical comparator, 49

Optical thickness, 75

Ordered compounds, 206, 212

P

Petch relationship, 210, 229

Phase diagrams, 2, 201

Phase rule, 19

265
Phases, 2, 19, 87-93

Phonons, 65-69

Photoelasticity, 248, 249, 252, 253, 255

Photons, 73-77

Planck's law, 36, 73, 128

Plastic deformation (see Inelastic deformation), 189, 190, 191, 201, 228, 231, 256

at crack tip, 222

crack nucleation due to, 228, 229

in crystals, 189, 228

in glass, 225

Point defects, 70, 159, 189

Polycrystalline ceramics, 2, 3, 208

crack nucleation, 228, 229

creep, 189

elastic moduli, 147, 153

fracture surface energies, 226

plastic deformation, 228

strength, 3, 228

Porosity, 2, 3, 14-16, 210

effect on creep, 198

effect on elastic moduli, 153

effect on expansivity, 41, 53

effect on thermal conductivity, 90-93

Predictability, 14, 16

Pressure

effect on elastic moduli, 142, 149, 155

effect on thermal conductivity, 81-82

Pressure testing, 246-250, 253, 257

Processing, IV, 14-16

Properties, 2, 3

Pyrometry, 36

Q

Quartz, 44, 54, 80, 85, 111

R

Radiation "conduction", 73-77

Raoult-Van't Hoff equation, 26

Reference standards

glass, 172

thermal conductivity, 108-112

thermal expansion, 53

Reflectance, 126

Relaxation time, 178

average, 180

Reliability, 13, 15, 95-96

Resonance, 152

S

Sapphire, 69

fracture surface energy, 226

strength, 217

Scattering coefficient, 74, 131

Semiconductor, 72-73

Silica, 44, 54, 80, 85, 111

Silicon, thermal conductivity, 110

Silicon carbide, creep, 195

Simon equation, 22

Single crystals, 201

elastic constants, 145

fracture surface energy, 226

plastic deformation, 197, 201

strength, 228

Sintering, 2, 85

Size effects, 252, 256, 259, 260

Slip systems, 209

Solid solution, effect on elastic constants,

157

Soret Effect, 78

Sound velocity, 150, 156

Spark plug insulators, 1, 3

Specific heat, 64

relation to elastic constants, 156

Spectrum of relaxation times, 178, 179

Stable equilibria, 20

Standards, 5,9, 37, 38, 53, 108-112, 253,

255, 257, 261

glasses, 172

thermal conductivity, 108-112

thermal expansion, 53

Static fatigue, limit, 236

Stoichiometry, 202, 208

effect on expansivity, 41

effect on thermal conductivity, 85

Strain, 144, 202, 244-250, 255, 256

Strain energy release rate, measurement

methods, 223

Strain gages, 245, 246, 250, 255-259

Strain point, 182

Strain rate effects, 204, 250, 259

Strength, 2, 3, 13-15, 201, 243

alumina, 2, 3, 232, 246, 247, 250, 258

cohesive, 217

effect on environment on, 14-15, 234

glass, 225, 228, 252, 253, 259, 260

magnesia, 231, 232, 258

polycrystalline ceramics, 3, 228

266
Strength-Continued

single crystals, 228

theoretical, 13, 217

Stress, 13-16, 81-82, 143, 243

effect on expansivity, 56

effect on thermal conductivity, 81-82

Stress corrosion, 14-15, 217, 225, 234,

235

activation energy for, 236, 237, 238

activation volume for, 236, 238

in alumina, 235, 238

in glass, 14, 235, 237, 238

Stress intensity factor, 217, 220

measurement methods, 224

Structral applications, 1, 13, 16, 201

213, 243

Structural ceramics, i, 213, 243

Structural relaxation, 178

Superplasticity, 198

Surface, 14

Surface effects (see Edge and surface

effects), 14, 15

Surface energy (see Fracture surface

energy), 217

Surface flaws, 201

Surface topography, 16

T

Technology, 1, 5

Temperature, 36

effect on elastic constants, 157

effect on expansivity, 44

effect on thermal conductivity, 78-81

mechanical testing at elevated, 202,

246, 248, 250, 255-259

Temperature coefficient of emittance, 127,

128

Temperature scales, 36

Tension testing, 244, 246, 248-253, 255-

257

Tentative standards, 9

Ternary systems, 28, 32

Test data (see also Methods of testing), 14-

15, 243

Test procedures (see also Methods of

testing), 14-15, 133, 243

Thermal activation, 203

Thermal conductivity, 3, 63-123

measurement methods, 96-108

Thermal diffusivity, 64

Thermal expansion, 41, 94

Thermal expansion anisotropy, stresses

due to, 94, 230

Thermal history, effect on thermal conductivity, 94-95

Thermal properties, IV, 1, 2, 3, 13-14,

41, 63

melting points, 19

thermal conductivity, 63

thermal expansion, 41

thermal radiation properties, 76-

77, 125

thermal shock, 217, 230, 233

Thermal radiation, 73-77, 125

Thermal radiation properties, 76-77, 125

Thermal shock, 3, 217, 230, 233

crack nucleation, 217, 230, 233

Thoria, 112, 136, 190

Torsion testing, 256

U

Ultrasonic propagation

electronic apparatus, 175

measurement technique, 176

Umklapp process, 67-68

V

Variability, 13-15, 243-248, 250, 255-257

Viscoelasticity, 169, 177

Viscosity, 169, 191, 198

definition of shear and volume viscosity, 179

measurement of longitudinal and shear

viscosity by ultrasonic propagation,

174

measurement of shear viscosity

by rotating cylinder, 169

by fiber elongation, 171

by beam bending, 171

non-equilibrium viscosity, 182

non-linear viscosity, 182

temperature dependence of, 183

theories of, 185

Viscous relaxation, 178

Vitreous-silica dilatometer, 47

Voluntary standardization process, 5, 9

Voluntary standards, 5, 9

267
w

Wave propagation, 65-67, 150, 174, 250

Wave vector, 66-67

Weight efficiency, 13

X

X-ray diffractometer, 50

Y

Yield strength, 202

Z

Zirconia, 93, 136, 190, 258

emittance, 136

268
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standards adopted and promulgated under the Public Law 89-306, and Bureau of the Budget Circular

A-86 entitled, Standardization of Data Elements

and Codes in Data Systems.

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Clearinghouse

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U.S. GOVERNMENT PRINTING OFFICE : 1969 O—323-655