On the Cover:

Schematic of a silicon carbide ceramic in an active atmosphere.

Data evaluation must account for the complexity of polycrystalline

materials and the resulting dependence of material properties on

chemical composition, physical microstructure, and environmental

conditions.
i

NIST Recommended Practice Guide

Special Publication 960-11

Data Evaluation

Theory and Practice

for Materials Properties

Ronald G. Munro

Materials Science and

Engineering Laboratory

June 2003

U.S. Department of Commerce

Donald L. Evans, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Arden L. Bement, Jr., Director
ii

◆ Data Evaluation Theory and Practice

Certain commercial entities, equipment, or materials may be identified in

this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or

endorsement by the National Institute of Standards and Technology, nor is it

intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology

Special Publication 960-11

Natl. Inst. Stand. Technol.

Spec. Publ. 960-11

136 pages (June 2003)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2003

For sale by the Superintendent of Documents

U.S. Government Printing Office

Internet: bookstore.gpo.gov Phone: (202) 512–1800 Fax: (202) 512–2250

Mail: Stop SSOP, Washington, DC 20402-0001
iii

FOREWORD

Data evaluation is the process by which collections of data are assessed

with respect to reliability, completeness, and consistency. The present work

addresses data evaluation for materials properties as a scientific discipline that

evolves from the formal underpinnings of materials metrology. A theoretical

foundation for data evaluation is developed and then illustrated in a practical

application to the development of an operational protocol for materials property

data. An extensive collection of examples is used to examine, in succession,

the issues of accessibility, reproducibility, consistency, and predictability.

Distinctions are made among definitive relations, correlations, derived and

semiempirical relations, heuristic theories, and value estimates. Subtopics

include the use of properties as parameters in models, the interpretation of

ad hoc parameters, and the treatments of procedural properties, response

dependent properties, and system dependent data.

More information on the SP 960 series can be found on the Internet at

http://www.nist.gov/practiceguides. This website includes a complete

list of NIST Practice Guides and ordering information.

Foreword ◆
iv

◆ Data Evaluation Theory and Practice
v

Acknowledgments ◆

ACKNOWLEDGMENTS

The collective understanding of the importance of data evaluation, particularly

with respect to the use of materials property relations, has evolved significantly

during the last decade. The author is especially mindful of the benefits derived

during that time from numerous interactions with colleagues at NIST. A true

list of the names of all the individuals who have been kind to me in this effort

would, perhaps, comprise a veritable staff directory. At the risk of being

accused of a certain lack of perspicacity, I would like to acknowledge explicitly

a few of those people whose shared wisdom has been repeatedly useful to me:

S. Block, S. Dapkunas, S. Freiman, S. Hsu, G. Piermarini, G. Quinn,

J. Rumble, Jr., L. Smith, and J. Wachtman, Jr.
vi

◆ Data Evaluation Theory and Practice
vii

Table of Contents ◆

TABLE OF CONTENTS

Foreword .............................................................................................iii

List of Figures ......................................................................................x

Symbols and Notations....................................................................xiii

1. Introduction......................................................................................1

1.1 A National Priority ....................................................................2

1.2 Overview and Scope ...............................................................2

References Cited in Chapter 1 ..........................................................4

2. Formal Materials Metrology ...........................................................6

2.1 Fundamentals..........................................................................8

2.2 Taxonomy of Materials Metrology.........................................10

References Cited in Chapter 2 ........................................................16

3. Formal Considerations in Data Evaluation ................................17

4. Data Evaluation Levels for Materials Properties .......................20

References Cited in Chapter 4 ........................................................24

5. Operational Protocol ....................................................................25

5.1 General Considerations ........................................................25

5.2 A Protocol for Materials Property Data .................................26

5.2.1 Materials Specification ..........................................................26

5.2.2 Measurement Method ...........................................................28

5.2.3 Certified Reference Values ...................................................29

5.2.4 Correlations or Models ..........................................................29

5.2.5 Independent Values ...............................................................29

5.2.6 Related Properties (with adequate

measurement information) ....................................................30
viii

◆ Data Evaluation Theory and Practice

5.2.7 Peer Review.......................................................................30

5.2.8 Incompatible Properties.....................................................30

5.2.9 Manufacturer’s Data ..........................................................32

5.2.10 Survey Data.......................................................................33

5.2.11 Subsidiary Data .................................................................33

5.2.12 Related Properties (without adequate

measurement information) ................................................33

5.2.13 Interim Report....................................................................35

5.2.14 Reassessment ..................................................................35

References Cited in Chapter 5 ........................................................36

6. Issues and Practices in Data Evaluation.....................................37

6.1 Stage I: Data Collection (Issue: Accessibility) ..................37

6.2 Stage II: Basic Evaluation (Issue: Reproducibility) ...........39

6.2.1 Illustrative Example: Particle Size Measurement..............39

6.3 Stage III: Relational Analysis (Issue: Consistency) ..........43

6.3.1 Definitive and Derived Relations .......................................43

6.3.1.1 Mass Fraction, Mole Fraction, Volume Fraction ...............44

6.3.1.2 Density, Theoretical Density, Porosity ..............................45

6.3.1.3 Thermal Expansion ...........................................................47

6.3.1.4 Elasticity ............................................................................49

6.3.1.5 Heat Capacity and Specific Heat ......................................51

6.3.1.6 Thermal Transport .............................................................53

6.3.2 Correlations and Semiempirical Relations ........................57

6.3.2.1 Estimates of Unmeasured Quantities ...............................58

6.3.2.2 Properties as Parameters .................................................61

6.3.2.3 Interpreted Parameters .....................................................65

6.3.2.4 Procedural Properties........................................................72
ix

6.3.2.5 Response Dependent Properties ......................................79

6.3.2.6 System Dependent Data ...................................................82

6.4 Stage IV: Modeling (Issue: Predictability) .........................86

6.4.1 Derived Correlations..........................................................86

6.4.2 Harmonic Theories ............................................................88

6.4.3 Heuristic Theories .............................................................93

6.4.4 Dominant Mechanisms .....................................................97

6.4.5 Bounded Values ................................................................99

6.4.5.1 Voigt and Reuss Bounds .................................................100

6.4.5.2 Characteristic Ranges.....................................................102

References Cited in Chapter 6 ...................................................... 104

Additional Source References ...................................................... 110

7. Conclusion ................................................................................... 114

Table of Contents ◆
x

◆ Data Evaluation Theory and Practice

List of Figures

Figure 2.1 Schematic of the taxonomy of materials metrology ...................... 11

Figure 5.1. Schematic of the basic data evaluation procedure. The

node labeled [Other Assessment] is expanded in Figures 5.2

and 5.3; [Other Classification] is expanded in Figure 5.4 ............ 27

Figure 5.2. Expansion of the data evaluation branch introduced in

Figure 5.1 as [Other Assessment] ............................................... 31

Figure 5.3. Expansion of the data evaluation branch introduced in

Figure 5.2 as [Other Comparisons] .............................................. 32

Figure 5.4. Expansion of the data evaluation branch introduced in

Figure 5.1 as [Other Classification] ............................................. 34

Figure 6.1. Histogram of particle size distribution for the

yttria-stabilized zirconia powder studied in IEA Annex II ............ 42

Figure 6.2. Consistency and data optimization through property relations,

illustrating the use of multiple data sets to optimize the

correlation of thermal conductivity and diffusivity for

sintered α-SiC .............................................................................. 54

Figure 6.3. Thermal transport data for sintered α-alumina (99% purity) ....... 56

Figure 6.4. The dependence of the superconducting critical

temperature, Tc , on the oxygen content in YBa2Cu3Ox .............. 59

Figure 6.5. Lattice parameters, a, b, and c, for the orthorhombic

structure of YBa2Cu3Ox , varied linearly with oxygen

content .......................................................................................... 60

Figure 6.6. Schematic of a typical creep curve (strain vs. time) showing

three distinguishable regions termed, respectively, primary,

secondary, and tertiary creep ....................................................... 66

Figure 6.7. Flexural creep rate of TiB2 as a function of inverse

temperature for various values of applied stress .......................... 67
xi

Table of Contents ◆

Figure 6.8. Tensile creep of polycrystalline silicon nitride, showing

a stress dependence that is not well described by the

diffusion model............................................................................ 71

Figure 6.9. Tensile creep of polycrystalline silicon nitride over a large

range of temperature and applied stress, showing the

greater suitability of the cavitation model ................................... 72

Figure 6.10. The indentation size effect for a polycrystalline hot,

isostatically pressed silicon nitride .............................................. 75

Figure 6.11. The dependence of the Meyer exponent on specimen

porosity for a collection of seventeen specimens of single

crystal and polycrystalline silicon nitride ..................................... 76

Figure 6.12. The Vickers hardness of polycrystalline hot, isostatically

pressed silicon nitride varies with both indentation load

and specimen density .................................................................. 77

Figure 6.13. Reduction and analysis of the data from Figure 6.12 in

accordance with Equation (45) ................................................... 79

Figure 6.14. Schematics of median and Palmqvist crack formations.

Side views: (a) median, (b) Palmqvist. Top views, showing

indentations as indented (upper) and after polishing (lower):

(c) median, and (d) Palmqvist ..................................................... 80

Figure 6.15. A wear transition diagram illustrating the variation of the

dominant wear mechanism depending on the tribological

contact conditions ....................................................................... 84

Figure 6.16. Correlation of the coefficient of friction and the ratio of

sliding speed to contact stress in the case of TiB2 ................................ 85

Figure 6.17. The correlation of fracture toughness and fracture strength

may differ under the conditions of constant grain size and

constant density .......................................................................... 87

Figure 6.18. Schematics of two flexural tests: (a) three-point bend;

(b) four-point bend ...................................................................... 96
xii

◆ Data Evaluation Theory and Practice

Figure 6.19. An illustration of the porosity dependence of the bulk

modulus of polycrystalline alumina ............................................. 99

Figure 6.20. Schematics for estimating the effective elastic modulus

of a simple composite material consisting of two

elastic components: (a) Reuss approximation;

(b) Voigt approximation .............................................................100

Figure 6.21. An illustration of characteristic ranges, using values of the

dimensionless product of thermal expansion and melting

temperature, viewed as a function of material class .................102
xiii

SYMBOLS AND NOTATIONS

αL linear coefficient of thermal expansion

αc,L cumulative linear coefficient of thermal expansion

αV volumetric coefficient of thermal expansion

αc,V cumulative volumetric coefficient of thermal expansion

a lattice parameter

A fitting parameter

b lattice parameter

B bulk modulus

Bo bulk modulus at zero porosity

BT isothermal bulk modulus

c lattice parameter

cx crack extension, length

CP specific heat, at constant pressure [J kg–1 K–1]

CV specific heat, at constant volume [J kg–1 K–1]

C(M)P molar heat capacity, at constant pressure [J mol–1 K–1]

d linear dimension of indenter’s impression

D thermal diffusivity

Ds sliding distance

0 strain, fractional elongation

E elastic modulus, Young’s modulus

ER Reuss approximation for the elastic modulus

EV Voigt approximation for the elastic modulus

findent force vector for indentation

F normal force, load

T general purpose function

Symbols and Notations ◆
xiv

◆ Data Evaluation Theory and Practice

γ shear strain

g grain size

G shear modulus

S Planck’s constant divided by 2π, 1.05457266(63)×10–34 J s

H hardness

κ thermal conductivity

kB Boltzmann’s constant, 1.380658(12)×10–23 J K–1

KIc critical stress intensity factor, fracture toughness

KW wear coefficient

i Palmqvist crack coefficient, 0.018 ± 0.004

Λ1, Λ2 general purpose constants, fitting parameters

L length

μi mass fraction of component i, μi

 = mi/M

m Norton model, grain size exponent

m Weibull model, Weibull modulus

mi mass of component or particle number i

M mass or total mass, M = m1 + m2 + ...

MW molar mass, e.g., 100 g/mol

η Meyer law exponent

ν Poisson’s ratio

n fitting parameter, stress exponent, Norton model

n(r) flaw density

Na number of atoms

NA Avogadro’s number, 6.0221367(36)×1023 mol–1

Nf number of flaw sites
xv

π ratio of the circumference of a circle to the diameter of the circle,

3.141592653...

p exponent, attractive term of Φij

P pressure

- probability function

q exponent, repulsive term of Φij

Q apparent activation energy

ρ density

ρtheo maximum theoretical density

rij distance between ion number i and ion number j

R molar gas constant, 8.31451(7) J mol–1 K–1

σ stress, force per unit area

σ mean fracture strength

σo Weibull characteristic strength

Θ Debye temperature

τ shear stress

t time

T absolute temperature [K]

Tc critical temperature

φ volume fraction of porosity

Φ total potential energy function

Φij pairwise potential energy function

Φ(σ) unimodal probability stress function, Φ(σ) = ln(1 – -)

Symbols and Notations ◆
xvi

◆ Data Evaluation Theory and Practice

Éi volume fraction of component number i; Éi = Vi /V

vL velocity of sound, longitudinal mode

vm velocity of sound, mean, ([vL

–3 + 2vS–3]/3)–1/3

vS velocity of sound, shear mode

V volume

Vo volume at zero porosity

Vp volume of pores

ωD Debye cutoff frequency

W work, energy

χ renormalization scale factor

ψ cavitation model stress coefficient

ψ1, ψ2 coefficients, attractive and repulsive terms of Φij

Y linear fracture mechanics coefficient (dimensionless numeric factor)

ζ Meyer law coefficient

z number of formula units in a unit cell

H median crack coefficient, 0.016 ± 0.004
1

1. INTRODUCTION

The departure of a physical property from its expected behavior can be a

cause for scientific discovery if the departure occurs in a research exercise.

The same departure occurring in an engineering design can be ruinous.

In the former case, the departure is a novelty that may be the result of a

previous lack of comprehension or a previous insufficiency in the available

data. In the latter case, the departure is a failure that may be the result of

a lack of reliability or a lack of sufficient investigation of the data. In each

case, the pivotal issue underlying the conclusion is an assessment of the data.

Ascertaining the credibility of data is the function of data evaluation.

Whether the application is scientific discovery or engineering design, what

we know about the data has a direct influence on our perception of the results

and the interpretation of the data. In every case, the determining issue is one

of confidence. An extraordinary observation must be reconciled with our

previous experience, while routine observations may need to be brought into

conformity with common knowledge and known constraints, such as a trend

that should be followed by the data. In both application domains, whether

explicitly or implicitly, data evaluation is fundamental to our acceptance of

observations and technical designs.

The same conclusion follows if we consider this issue in the context of theory

and experiment. For theory, the challenge is to mold principles into conformity

with observations. If the observations are deficient or lacking in any manner,

the efforts to develop theoretical models may be highly frustrated. Conversely,

for experiment, the challenge is to make observations accurately and without

bias. If the measurement principles are unfounded, efforts to elucidate the

understanding of material behavior may be highly illusive. More subtly, if

measurements are undertaken to confirm a theory, the resulting observations

may be prejudicial a priori.

Data evaluation, as a discipline, uses the fundamental principles of theory

and the practical observations of experiment as complementary exercises to

ensure that data compilations reflect the best understanding currently available

for the given subject matter. The end result of this discipline is an assurance

of the harmonization of theory and experiment, and, by extension, of science

and engineering.

Introduction ◆
◆ Data Evaluation Theory and Practice

2

1.1 A National Priority

Early recognition of the national importance of materials property data

began to emerge from studies[1–3] on the availability of property data for

metallic systems and on the economic effects resulting from the corrosion

of metallic structures. Subsequently, national attention to the broader

issues of materials property data acquired a sharper focus when the

National Materials Advisory Board of the National Academy of Science

issued a report[4] tailored to that purpose. That report cited materials

information as a critical national need. Further studies on the economic

effects of fracture[5] and the status of materials property databases[6]

emphasized how pervasive the need for property data had become.

Recognition of the growing urgency of this need was echoed[7] by the

Director of the National Bureau of Standards who cited the need for

materials data as a national priority. It was not long thereafter that

several national and international studies on the status, issues, and

concerns regarding materials property databases followed.[8–13]

Among those studies were numerous discussions on the logical design

of databases[14] and their practical development.[15–21] So thorough

were those discussions that the mechanical aspects of constructing a

database need not be discussed further in the present work. Indeed,

ample discussion of that task has been well documented by the American

Society for Testing and Materials (ASTM). During the years from 1986

to 2000, ASTM Committee E-49 conducted extensive deliberations and

held five international conferences[15–19] to debate and resolve the issues

specifically associated with the computerization of materials databases.

Much of that work was summarized in two documents, one being a

collection of ASTM standards[20] and the other being a manual[21] about

constructing a database. Especially noteworthy in the present context

is a review[22] of the nomenclature and contextual significance often

associated with data evaluation.

1.2 Overview and Scope

The complexities of advanced materials and their designed applications

pose a difficult problem for practitioners of science and engineering

alike. The degree of specialization required to comprehend every

measurement nuance pertaining to a reported property value often

exceeds the breadth of knowledge of the general practitioner.

Consequently, the practitioner must rely on recommended values

or must apply other assessment guidelines.

This treatise is directed towards the vast segment of technology whose

scientists and engineers must use materials property data, at least on

occasion, without the benefit of direct measurement or other assessment
3

experience. The intent is to provide a thoughtful and systematic approach

towards establishing confidence in numeric materials properties. Throughout

the work, it will be found that this particular interest is addressed in a

manner that evolves into one basic theme. Stated with perhaps some

oversimplification, it is that our confidence is high when the data are sensible.

In spite of the pervasive significance of reliable data, there are relatively

few dedicated and sustained efforts to compile and disseminate fully

evaluated materials property data. The scope of such an undertaking

is usually too broad for commercial enterprises that necessarily are

focused on the requirements of their products and too diffuse for national

programs whose priorities must be focused on transient critical issues.

Most database efforts, therefore, tend to be necessarily delimited to a

small, but well-defined, scope. Superficially, these delimited efforts may

appear to be quite unrelated. Yet, as diverse as these efforts might be,

all of the efforts are confronted with numerous common issues that must

be addressed in the course of their respective developments. This treatise

takes the view that the solutions to the common issues derive from a small

number of underlying principles that may be developed generically and

applied systematically.

It will be seen that among these unifying principles are concepts that

largely have been ignored or unrecognized in the progressive evolution

of materials science. A greater understanding of the mutual relations

among properties, or of the controversies surrounding the lack of

expected relations, is gained by acknowledging these principles.

Consequently, two chapters are devoted to formal considerations: one

on materials metrology in general, and one on data evaluation per se.

Subsequent chapters illustrate the principles by means of practical

applications, each chosen to represent a different aspect of data

evaluation. In each chapter, the discussion is presented in a manner

that reflects a logical evolution of the concepts, proceeding from the

clearest, most well-defined conditions and progressing to conditions

that are less well defined.

All of the examples in this work are taken from actual exercises in

materials property data evaluation. These examples are meant to be

merely illustrative. In an undertaking as vast as data evaluation, no

pretense to an exhaustive dissertation would be supportable. It is

hoped, however, that the principles and practices set forth in this work

will guide and sustain a productive application of data evaluation across

many disciplines wherein considerations of numeric property data are

crucial to the interpretation and application of quantitative observations.

Introduction ◆
◆ Data Evaluation Theory and Practice

4

References Cited in Chapter 1

1. Gavert, R.J.B., R.L. Moore, and J.H. Westbrook. Critical Surveys of

Data Sources: Mechanical Properties of Metals. NBS SP 396-1.

Gaithersburg: National Bureau of Standards, 1974.

2. The Economic Effects of Metallic Corrosion in the United States.

NBS SP 511-1 and 511-2. Gaithersburg: National Bureau of Standards,

1978.

3. Mechanical Property Data for Metals and Alloys: Status of Data

Reporting, Collecting, Appraising, and Dissemination. Washington,

D.C.: National Academy Press, 1980.

4. Materials Properties Data Management — Approaches to a

Critical National Need. Committee on Materials Information Used in

Computerized Design and Manufacturing Processes, National Materials

Advisory Board, Report Number NMAB-405. Washington, D.C.:

National Academy Press, 1983.

5. The Economic Effects of Fracture in the United States. NBS SP 647-1

and 647-2. Gaithersburg: National Bureau of Standards, 1983.

6. Westbrook, J.H., and J.R. Rumble, Jr. Computerized Materials Data

Systems. Gaithersburg: Office of Standard Reference Data, National

Bureau of Standards, 1983.

7. Ambler, E. “Engineering Property Data — A National Priority.”

ASTM Standarization News, August, pp. 46–50, (1985).

8. Petten, J.R.C. Factual Material Data Banks. CEC Workshop,

14–16 November 1984, Commission of the European Communities,

Luxembourg, Netherlands, 1985.

9. Grattidge, W., J. Westbrook, J. McCarthy, C. Northrup, Jr., and J. Rumble, Jr.

Materials Information for Science & Technology (MIST): Project

Overview. NBS SP 726. Gaithersburg: National Bureau of

Standards, 1986.

10. Westbrook, J.H., H. Behrens, G. Dathe, and S. Iwata. Materials Data

Systems for Engineering. Proceedings of a CODATA Workshop

held at Schluchsee. Karlsruhe, Federal Republic of Germany:

Fachinformationszentrum, 1986.

11. Rodgers, J.R., and A.D. Mighell, eds. Role of Numeric Databases in

Materials and Biological Sciences. ACA Annual Meeting, American

Crystallographic Association, Philadelphia, Pennsylvania, 26 June 1988.
5

12. Computer-Aided Materials Selection During Structural Design.

Commission on Engineering and Technical Systems, National Materials

Advisory Board, Report Number NMAB-467. Washington, D.C.:

National Academy Press, 1995.

13. Materials and Processes Research and the Information Highway.

Committee on Materials and Processes Research and the Information

Highway, National Materials Advisory Board, Washington, D.C., 1996.

14. Fong, E.N, M.W. Henderson, D.K. Jefferson, and J.M. Sullivan.

Guide on Logical Database Design. NBS SP 500-122. Gaithersburg:

National Bureau of Standards, 1985.

15. Glazman, J.S., and J.R. Rumble, Jr., eds. Computerization and

Networking of Materials Data Bases. ASTM STP 1017. Philadelphia:

American Society for Testing and Materials, 1989.

16. Kaufman, J.G., and J.S. Glazman, eds. Computerization and Networking

of Materials Databases: Second Volume. ASTM STP 1106.

Philadelphia: American Society for Testing and Materials, 1991.

17. Barry, T.I., and K.W. Reynard, eds. Computerization and Networking

of Materials Databases: Third Volume. ASTM STP 1140.

Philadelphia: American Society for Testing and Materials, 1992.

18. Sturrock, C.P., and E.F. Begley, eds. Computerization and Networking

of Materials Databases: Fourth Volume. ASTM STP 1257.

Philadelphia: American Society for Testing and Materials, 1995.

19. Nishijima, S., and S. Iwata, eds. Computerization and Networking

of Materials Databases: Fifth Volume. ASTM STP 1311.

Philadelphia: American Society for Testing and Materials, 1997.

20. ASTM Standards on the Building of Materials Databases.

Philadelphia: American Society for Testing and Materials, 1993.

21. Newton, C.H., ed. Manual on the Building of Materials Databases.

ASTM Manual Series: MNL 19. Philadelphia: American Society for

Testing and Materials, 1993.

22. Barrett, A.J. “Data Evaluation, Validation, and Quality.” Manual on

the Building of Materials Databases, ASTM Manual Series: MNL 19,

pp. 53–67. Philadelphia: American Society for Testing and Materials,

1993.

Introduction ◆
◆ Data Evaluation Theory and Practice

6

2. Formal Materials Metrology1

It is, perhaps, profoundly obvious that measurement processes have been

an integral part of every civilization, whether primitive or advanced, and, thus,

that every civilization has, in some way, felt the impact of metrology. If we

understand “materials metrology” to mean the science of measurement applied

to the characterization, understanding, and development of materials, then this

derivative of metrology will be seen to be nearly as ancient as metrology itself.

Yet, upon careful examination and in spite of their common ancestry, it will be

found that there remains an intellectual chasm separating these two disciplines.

We explore that issue in the present section and present a resolution of the last

intellectual disparity between conventional metrology and materials metrology.

It is easy to imagine our earliest ancestors marveling at the shapes and colors

of rocks, assessing the advantages of their sizes and weights, or recognizing

that softer materials could be used for different purposes than harder materials.

Modern materials metrology differs in essence from those crude beginnings

by the deliberate attempt to understand the nature of the materials and to

use that understanding to improve their characteristics or to develop entirely

new materials.

We may also note that the rapid pace of technological innovation and

the sometimes alarming rate of consumption of natural resources create

unprecedented demands to use materials more wisely, more effectively, and

more strategically. Concurrently, while the world remains divided into distinct

geographical areas, each with its own sovereignty, there is a de facto unity

among nations resulting from instantaneous communications, short transit

times, and global economies. Because of that technological unity, there is a

common need and an obligation to pursue advances in materials metrology in

a coordinated international effort.

Fortunately, there has been a natural evolution of just such an international

effort propelled principally by standards organizations. While each country

must acknowledge responsibility for its own standards, economies and the

well beings of nations suffer when those standards are not compatible with

1 This chapter is based on an internal study, “Taxonomy of Materials Metrology,”

prepared by the author at the request of Dr. Leslie Smith, Director, Materials Science

and Engineering Laboratory, as part of a strategic planning exercise for MSEL.

As such, the author has had the benefit of comments from L. Smith, D. Hall,

K. Jewett, S. Freiman, E. Amis, C. Handwerker, F. Fickett, M. Rowe, S. Dapkunas,

J. Tesk, G. Quinn, T. Siewert, C. Ehrlich, and B. Taylor.
7

the prevailing markets. Consequently, numerous international organizations

have been established specifically to promote the harmonization of standards,

and there already exists a rich history of standardization activities.

The most significant agreement promoting international standards was the

Convention du Mètre[1] which was signed in 1875. This treaty established

the Conférence Générale des Poids et Mesures (CGPM), composed of

representatives from the member states, to provide general oversight for the

international efforts. The CGPM, meeting approximately every four years,

elects the members of the Comité International des Poids et Mesures

(CIPM) which, in its turn, provides supervision for the Bureau International

des Poids et Mesures (BIPM).[2] Together, these interrelated organizations

diligently coordinate the development and maintenance of the basic units of

weights and measures, which form what we may refer to as conventional

metrology. An additional organization, the International Organization

for Standardization (ISO),[3] aggressively promotes the harmonization of

measurement standards relating to all facets of physics, chemistry, and

materials. Other organizations, such as the International Union of Pure

and Applied Chemistry (IUPAC)[4] and the International Union of Pure

and Applied Physics (IUPAP),[5] also support and promote standardization.

In addition to those efforts, the Versailles Project on Advanced Materials

and Standards (VAMAS)[6] actively conducts prestandardization research,

in the form of international round robins, as a preliminary step towards the

subsequent development of new standards in support of advanced materials

and related high technology products.

Central to the success of all of these efforts is agreement on the meanings

of basic terms commonly used in all aspects of measurement science.

This important issue was addressed by seven independent international

organizations including BIPM, IEC (International Electrotechnical

Commission), IFCC (International Federation of Clinical Chemistry),

ISO, IUPAC, IUPAP, and OIML (Organisation Internationale de

Métrologie Légale). The product of this cooperative venture was the

Vocabulaire International des Termes Fondamentaux and Généraux

de Métrologie (VIM).[7]

Remarkably, it appears that no organization has undertaken the task of

formally applying and extending the VIM to materials metrology. We address

this task in the present work and, in the process, develop a taxonomy for

materials metrology that is consistent and compatible with the VIM.

Formal Materials Metrology ◆
◆ Data Evaluation Theory and Practice

8

2.1 Fundamentals

Metrology, like all scientific endeavors, is a paradigm that is built upon

a foundation of concise definitions of basic terms and rational relations

among derived quantities. The specialization to materials metrology

should be no less concise, and of course, much of what is required is

achieved instantly by adopting the VIM in its entirety. Consequently,

we need to focus only on those terms not treated by conventional

metrology, i.e., not currently addressed in the VIM.

It is most appropriate that we begin with a definition of materials

metrology since no speciation of metrology is considered in the VIM.

The rules of English language simply imply that materials metrology

means metrology with respect to materials. However, this interpretation

is not very enlightening. We suggest the following two terms to provide

a more appropriate definition of the scope of this discipline.

Material: An assemblage of matter delimited in space.

Materials Metrology: The science of measurement applied to

the characterization, understanding, and development of materials.

The latter definition acknowledges not only the role of measurement

science, but also that measurements on materials may be subservient

to specific purposes. The role of “purpose” is therefore very important

for materials and implies that the intended purpose may influence the

specification of the measurement standard.

The notion that measurements may be used to characterize a material is

critical to materials metrology and leads to the equally critical definition

of a material property.

Material Property: A quantifiable response of a material to an

externally applied stimulus.

This definition recognizes that the role of a property is to gauge how

a material interacts with its environment and implies that there is some

process by which a number can be assigned to the response that the

material makes to the external stimulus. It is the process of assigning

a number that begins to form the link between materials metrology and

conventional metrology. It should be noticed most particularly that the
9

definition of a material property is distinct from the definition of a

measurable quantity.2

(Measurable) Quantity: [VIM 1.1] An attribute of a phenomenon,

body, or substance that may be distinguished qualitatively and

determined quantitatively.

The critical aspect of this definition is that a measurable quantity is an

attribute of the body. The dilemma that this definition poses for materials

metrology is that there are important materials properties that are not

attributes of the body. For example, one may recognize qualitatively that

some materials are harder than others, but the technical material property

called hardness is defined only when a procedure is specified, i.e., the

process by which the number is assigned defines the property. Thus,

Knoop hardness, Vickers hardness, and Rockwell hardness are different

quantities. This situation may be contrasted with the measurement of a

quantity called length. Different procedures for measuring a length may

produce differing degrees of uncertainty in the value[8,9] assigned to the

quantity, but the identity of the quantity is unchanged by the particular

procedure. As a result of this conceptual difference, important material

properties such as hardness, fracture toughness, three-point bend strength,

four-point bend strength, and creep are inadmissible as measurable

quantities!

To resolve this logical and practical dilemma, we must introduce a

new term.

Procedural Quantity: A quantity whose identity is determined

concurrently as an attribute of a phenomenon, body, or substance

and an attribute of a measurement procedure, and that may be

distinguished qualitatively and determined quantitatively.

2 Revision of the VIM is under consideration. It has been proposed that the term

“measurable quantity” be replaced by a broader term, “quantity,” which would

accommodate any attribute to which more than one value (numeric or non-numeric)

could be assigned. The revised term would be supplemented by an additional term,

“kind of quantity,” which would distinguish four types of quantities: nominal (e.g.,

non-numeric values such as gender = male or female), ordinal (distinguishable by

order of magnitude), difference (e.g., gravitational potential), and ratio (e.g., relative

atomic mass, formerly known as atomic weight). While this generalization is more

inclusive than the current term, it does not resolve the logical dilemma encountered

in materials metrology.

Formal Materials Metrology ◆
◆ Data Evaluation Theory and Practice

10

This new term preserves the scope of conventional metrology while

encompassing all of the material properties that would otherwise be

excluded from the formal realm of measurement science. Upon this

term and those preceding it, we may build a formal taxonomy of

materials metrology.

2.2 Taxonomy of Materials Metrology

It is clear from the preceding subsection that materials metrology may be

divided into two parts consisting of measurable quantities and procedural

quantities. The measurable quantities of conventional metrology obviously

belong to the former part, while some material properties will be of the

former type, and some of the latter. It will be convenient to introduce two

additional terms to distinguish the material properties that belong to the

two parts.

Inherent Material Property: A material property that is a

measurable quantity.

Procedural Material Property: A material property that is a

procedural quantity.

It will be convenient also to distinguish a special class of measurable

quantities, called structures, that are of considerable importance to

materials metrology. For example, an ideal, monatomic crystal consists

of an assemblage of identical atoms whose equilibrium positions in space

are distributed on a periodic lattice in which the positions are related by

a well-defined group of symmetry operations. The volume of such a

lattice may be subdivided into smaller volumes of identical shapes,

called cells, any one of which may be used as a generator of the entire

volume through the use of symmetry operations. The cell selected as

the generator is called the unit cell, and the smallest such cell is called a

primitive cell. The lengths of the principal axes of the unit cell, commonly

denoted by the symbols a, b, and c, are measurable quantities. The set

of lengths and the set of angles between the axes collectively comprise

the lattice parameters. Further, the positions of the atoms within the

unit cell may be determined and expressed relative to a, b, and c.

The symmetry group, the lattice parameters, and the relative atomic

coordinates of the atoms are taken together as the specification of the

crystal structure. Thus crystal structure is a special construct used to

characterize materials, and it is desirable, for both compatibility and

convenience, to retain this special construct as a special class of

measurable quantities.
11

Table 2.1: Examples of Measurable Quantities of Conventional

Metrology

Category Quantities

Base Quantities length, mass, time, electrical current, temperature, amount

of substance, luminous intensity

Derived Quantities strain, area, volume, concentration, velocity, acceleration,

force, pressure, stress, voltage, charge, capacitance,

current density, resistance

Figure 2.1. Schematic of the taxonomy of materials metrology.

The basic taxonomic scheme for materials metrology that evolves from

these considerations is depicted graphically in Figure 2.1. This schema

may be readily expanded into a full classification of all the measurements

pertinent to materials metrology. Tables 2.1–2.4 illustrate how such

a classification scheme can be developed from the taxonomy. In the

present exercise, approximately 130 measurable and procedural quantities

pertinent to materials science and engineering have been classified

within the context of this new taxonomy for materials metrology.

Formal Materials Metrology ◆
◆ Data Evaluation Theory and Practice

12

Table 2.2: Examples of Measurable Quantities of Structures in

Materials Metrology

Category Subcategory Quantities

Unit Structures Crystallography lattice parameters, atomic coordinates

Noncrystallinity radial distribution function

Microstructure Grains and Pores size distribution, mean size, aspect

ratio, texture

Dendrites mean size, orientation, branch density

Phase Equilibria phase transition temperature, phase

transition pressure, glass transition

temperature, Curie temperature, Néel

temperature, triple point temperature

Interphase Region thickness

Magnetic Domains mean size, aspect ratio, orientation

Surfaces Topography asperity mean size, aspect ratio,

orientation; surface roughness;

crack size, orientation, depth

Films and Deposits thickness

Electromagnetic Charge Distribution valence, ionicity, covalency, spin,

space charge density
13

Table 2.3: Examples of Measurable Quantities of Inherent Material

Properties

Category Subcategory Quantities

Mechanical Elasticity Young’s modulus, shear modulus,

bulk modulus, compressibility,

Poisson’s ratio, elasticity tensor,

compliance tensor, sound velocity,

Debye temperature

Thermal Capacity specific heat, Gruneisen parameter

Transport thermal conductivity, thermal

diffusivity, emissivity, interfacial

resistance

Stability thermal expansion, residual stress

Electromagnetic Electricity resistivity, conductivity, thermoelectric

power, Hall coefficient, critical

current density

Magnetism susceptibility, coercivity, critical field

strength

Optics dielectric strength, index of refraction,

permitivity, transmisivity, reflectivity,

absorptivity

Formal Materials Metrology ◆
◆ Data Evaluation Theory and Practice

14

Table 2.4: Examples of Procedural Quantities of Procedural Material

Properties

Category Subcategory Quantities

Mechanical Plasticity ductility, brittle to ductile transition

point, hardness, creep rate, creep

activation energy, creep stress

exponent

Strength yield strength, proportional limit,

tensile strength, flexural strength,

shear strength, compressive strength,

ultimate strength, fracture toughness,

fracture energy, fatigue strength,

Weibull modulus, Weibull

characteristic strength

Thermal Stability flammability

Durability Thermomechanical thermal shock resistance

Adhesion adhesive strength

Tribology friction coefficient, wear rate, wear

coefficient, lubricity, machining rate

Chemical Aging Corrosion corrosion rate, activation energy

Hydration hydration rate

Interdiffusion diffusion rate

Physical Aging Delamination delamination rate
15

There is a certain useful insight regarding measurement technology

that evolves from this taxonomic view of materials metrology.

In all applications of measurement science, the reproducibility of a

measurement is unexceeded in importance by any other consideration

and is accorded, therefore, the highest priority for measurement

technology. This basic importance often leads to a vigorous effort to

standardize the measurement procedure. It is not unusual in such efforts

to consider alternate procedures or to compare different measures of

the same quantity. When it happens that two measures of a quantity

are significantly different, fairly intense discussions of the relative

merits of the two measurements may ensue to excess.

Recognizing the distinction between measurable quantities and procedural

quantities may help to alleviate some of the unproductive aspects of such

debate. When the measures in question effectively pertain to procedural

materials properties, the quantitative results may be significantly different,

even though the individual measurements are highly reproducible with

small measurement uncertainty. For procedural materials properties, it is

not necessarily possible, even in principle, to resolve the differences of

two such measurements, and expending resources trying to do so would

be an unproductive and futile exercise.

In summary, it is important to remember that materials metrology is an

extension, not a subset, of conventional metrology. The fundamentals

of conventional metrology provide the essential basis on which is built

the rational extension of measurement science that is tailored to the

intricacies and technological necessities of materials science and

engineering. Materials metrology extends that basis by including

measurement aspects not immediately compatible with conventional

metrology. The introduction of the notion of a procedural quantity as an

element of measurement science, however, resolves the disparity and

permits the discipline of materials metrology to be formally and logically

consistent and compatible with conventional metrology.

Formal Materials Metrology ◆
◆ Data Evaluation Theory and Practice

16

References Cited in Chapter 2

1. Convention du Mètre, 1875.

2. Le BIPM et la Convention du Mètre, Paris: Bureau International

des Poids et Mesures, 1995.

3. International Organization for Standardization. World Wide Web:

http://www.iso.ch/.

4. International Union of Pure and Applied Chemistry. World Wide Web:

http://www.iupac.org/.

5. International Union of Pure and Applied Physics. World Wide Web:

http://www.iupap.org/.

6. Versailles Project on Advanced Materials and Standards. World Wide

Web: http://www.vamas.org/.

7. International Vocabulary of Basic and General Terms in Metrology.

2nd edition. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML.

Geneva: International Organization for Standardization, 1993.

8. Guide to the Expression of Uncertainty in Measurement. Geneva:

International Organization for Standardization (1995).

9. Taylor, B.N., and C.E. Kuyatt. Guidelines for Evaluating and Expressing

the Uncertainty of NIST Measurement Results. NIST Technical Note

1297, 1994 Edition. Gaithersburg: National Institute of Standards and

Technology (1994).
17

3. Formal Considerations in Data Evaluation

Data evaluation implies an imposition of criteria. The criteria set forth the

conditions under which the data may be deemed “accepted.” Obviously,

the criteria depend on the purpose or function to be served by the data.

Consequently, it should be understood readily that some circumstances require

more stringent conditions than do others, and we should anticipate, therefore,

an evaluation scheme that allows multiple acceptance categories with differing

acceptance criteria.

Regardless of the number of acceptance categories, the end result of data

evaluation is that a set of data is either accepted or it is not accepted. Data

that are not accepted fail to satisfy the criteria of any allowed acceptance

category. All such data may be placed together in an additional category

labeled simply “unacceptable.”

The classification “unacceptable” is conclusive and is meant to result in the

immediate exclusion of the data from the database. In this case, the criteria

must reveal a significant deficiency in the data or its associated metadata to

such an extent that there is no perceived application of the data, in the context

of the given database, for which this data would be useful. In this respect,

note that “unacceptable” does not convey a connotation of “wrong.”

Example: Suppose a paper contains a report on the hardness of

two materials and that the report consists only of the statement,

“The hardness value of material A was 31, and for material B it

was 17, each with an extended measurement uncertainty of 1

using a coverage factor, k = 2.” We might well understand from

these values that material A was harder than material B, and if that

circumstance is relevant to our purpose, we might wish to include

this information as supplementary metadata. However, from this

report, we do not know what measure of hardness was used,

and we do not know what unit was associated with the numeric

value. Consequently, the reported values would be considered

“unacceptable” as independent entries in the database because

they are independently uninterpretable.

Data that are not “unacceptable” are syntactically “acceptable;” i.e., the

data satisfy the criteria of some allowed acceptance category. As a practical

matter, if we use a label to identify the separately allowed categories, then

we may interpret the label as a data quality indicator. In the present work,

we call that indicator the data evaluation level.

Formal Considerations in Data Evaluation ◆
◆ Data Evaluation Theory and Practice

18

For the purposes of this discussion, we shall assume that there are multiple

(two or more) acceptance categories and that no two categories are identical.

While the categories are required to be distinct, we may anticipate that there

will be both similarities and differences between any two sets of criteria

from any two acceptance categories; i.e., some of the criteria may be the

same, while at least one criterion must be different. To consider how these

similarities and differences are manifest in the decision process, let us focus

on the application of a single criterion.

When any individual criterion is applied to the data set, the result is that

the criterion is either satisfied or it is not satisfied. Consequently, we may

represent any criterion as a mathematical operator which returns one of

two allowed outcomes when it is applied to the data set. It is convenient to

represent the allowed outcomes by the numeric values 1 and 0. We may

choose 1 as the outcome when the criterion is satisfied, and 0 as the outcome

otherwise. If we now form an ordered collection of all distinct criteria from

all of the acceptance categories, then each acceptance category may be

represented as an ordered n-tuple of 1s and 0s. By construction, no two

of the category n-tuples are identical. Application of the ordered collection

of distinct criteria to a given set of data then yields a response, the data

evaluation n-tuple, which may be compared to the category n-tuples.

A matching category n-tuple uniquely determines the data evaluation

category for that set. If there is no matching category n-tuple, the data

set is unacceptable.

Automation of this evaluation scheme might not be possible for all types

of criteria because it is sometimes very useful to have criteria that are

subjective and require judgmental responses. In such cases, it may be

advantageous to consider an equivalent alternative approach in the form

of a deductive, hierarchal protocol. We have seen in the preceding formal

considerations that each criterion has only two responses and, thus, may be

viewed as a simple binary filter. In this view, the admission of a set of data

to a particular acceptance category can be viewed as the successful passage

of the data through a sequence of these filters. If at any filter the criterion is

not satisfied, the data set would be passed to a different filtering sequence.

Thus, the evaluation protocol would appear as a collection of linear chains

of filters interconnected via branches to alternate chains. This collection of

branching chains is a classical model for a decision tree.

To construct a decision tree representing the evaluation process, we need to

consider the nature of the individual filters. With respect to the qualitative

outcomes of the filters, there must be a set of allowed terminal outcomes, i.e.,

decisions which cause the decision process to terminate. If a filter operation
19

yields any other outcome, the decision process continues. Topologically,

therefore, we need to consider only three types of binary filters: decisive

(doubly terminating), semi-decisive (singly terminating), and branching

(non-terminating). When a filter has a decisive outcome, the decision process

is complete. Thus, a decisive filter terminates the chain no matter which

outcome of that filter occurs. A semi-decisive filter may either terminate

the chain or extend the chain to another filter. A branching filter represents

a bifurcation point in the tree and causes the initial chain to divide and extend

into two different branches.

Using only these three types of filters, it is possible to attain a complete

implementation of a decision tree for data evaluation. In the following two

sections, we will develop a rationale for the criteria that may be used in the

design of such filters, and then we will construct a practical illustration of a

decision tree for materials properties.

Formal Considerations in Data Evaluation ◆
◆ Data Evaluation Theory and Practice

20

4. Data Evaluation Levels for Materials Properties

There is no fundamental necessity to have more than one acceptance

category for data evaluation. However, convenience and the desire for

a logical, orderly classification scheme often dictate or, at least, favor the

decision to use multiple categories. Once that somewhat arbitrary choice is

accepted, an equally arbitrary task, choosing what categories should be used,

follows. Although this subsequent task is also arbitrary, we can cast it into

a fairly reasonable approach if we let the categories be determined by the

objectives served by the data system. We explore that approach in this

section by explicitly constructing a rational set of acceptance categories

for a materials property database used as a general reference system.[1]

We must begin by detailing a working model of the reference system.

For the present purposes, we assume that the reference system consists

of data extracted from published technical papers. In this case, the heritage

of the data, from laboratory to database, may be relevant to the context

in which we accept the data. In most cases, the relevant history begins

with the production or procurement of a material and the measurement

apparatus. Subsequently, specimens for testing are prepared, instrumental

and environmental parameters are set, and measurements are made.

After collecting and processing or analyzing the raw data, a report is

written describing the measurements and the results. The report may

be sent to a project sponsor, a conference, or a journal, and after being

reviewed, the paper is published in the appropriate medium.

Several natural considerations evolve from this basic scenario. Perhaps

the most insightful cursory observation regards the nature of the project.

Was the project conducted as a normal or regular research effort, or was the

project distinguished in some manner? For example, a project in an established

standards setting laboratory whose goal is to provide a reference material

for material property measurements might well be considered special. The

rigorous requirements established for such efforts are usually fully described

and documented, and the property values usually are certified in some explicit

manner by the measuring laboratory.[2] Given the important role of reference

materials in measurement calibration, certified values have a de facto

acceptance in the user community and warrant a special category.

A project may also be distinguished by measurements at the forefront of

the technology. In this case, the material or experiment may represent an

innovative achievement that is being reported for the first time. The results

might even be considered preliminary or tentative, but the desirability of the

results imparts sufficient urgency to their announcement that including those
21

values in the database would be welcomed. We might generally think of such

works as research in progress. A category by this name might serve to alert

the user to both the currency of the results and the possibly tentative nature

of the values.

A third category that follows rather naturally from the circumstances of

the measurements is that of commercial data. It is usually in the best

interest of materials producers to inform their clientele about their products.

Consequently, commercial manufacturers of materials usually take great care

to provide property data on the principal characteristics. The view often taken

is that these data serve as a guide to the useful application of the material.

In this respect, the manufacturer may indicate that the reported values are

typical of results obtained using specimens from several production batches

of the material. As such, the manufacturer may present the property values

in a tabular format providing nominal information about the measurement

conditions. It is not unusual for those values to be propagated into the

technical literature by users who report the values in subsequent publications,

usually without further discussion or assessment.

The largest sustained effort to report materials properties arises collectively

from the broad spectrum of independent materials research programs that

are dedicated to characterizing, understanding, and improving materials of

all types. The laboratory conducting the measurements usually has invested

a considerable amount of its resources to acquire, develop, or refine the

measurement apparatus, and the personnel in these efforts often have acquired

a notable amount of experience and expertise in the measurement technique.

These efforts, consequently, tend to focus on the one or two characteristics

or properties of the material that pertain to the laboratory’s area of capability

and expertise. Reports issued by such programs tend to follow what might be

called the classical research protocol. After a general introduction describing

the background and current objectives for the work, both the material and the

measurement technique are discussed in rather precise detail. The guideline

commonly used in this portion of the report is that enough detail should be

reported that any other reasonably experienced research team, comparably

equipped, could reproduce the experiment. Only when this degree of detail

and qualification has been achieved can the reader proceed to knowledgeably

address the results, analysis, and discussion of the measurements. Research

conducted and reported in this manner serves as the figurative bedrock of

evaluated databases. In recognition of that fundamental role, data confirmed

to be of this type might well be categorized as qualified data.

An important characteristic of the category of qualified data is that results from

many laboratories are determined and reported independently. However, it is

Data Evaluation Levels for Materials Properties ◆
◆ Data Evaluation Theory and Practice

22

sometimes desirable to have an overview of how a property varies over a

range of conditions or over a range of materials. A common method used

to construct the overview is to review the literature for similar studies and to

combine the results. In this manner, ranges of observed values, differentiated

by general material or measurement conditions, can be identified. It is often

possible to designate a value that is typical of the material or property in the

context of some noted condition. Typical values derived from literature

surveys are generally only representative of the trend of observed values and

are not necessarily appropriate for any particular specimen. The utility of

typical values is, firstly, that useful order of magnitude estimates typical of a

material class sometimes can be made and, secondly, that functional trends,

such as power laws or dependance on a type of chemical dopant, may be

identified.

Table 4.1: Data Evaluation Levels

Designation Comment

Certified Standard reference values, specific to known production

batches

Validated Confirmed via correlations and models

Qualified Basic acceptance criteria satisfied

Commercial Manufacturer’s data for specific commercial materials

Typical Derived from surveys of nominally similar materials

Research Preliminary values from work in progress

Unevaluated All other data
23

Collections of both qualified data and literature surveys also form the basis

of more in-depth analyses leading to the establishment of reliable models or

correlations among properties. Subsequently, these models and correlations

may be applied to determine optimized or refined estimates of property values

for specific materials, thereby enhancing the reliability of evaluated data.

Data validated in this manner are distinguished by their mutual consistency

with other properties for the same material.

Other data, that cannot be validated, qualified, or otherwise documented with

respect to the measurements that were conducted, can still serve a useful and

desirable purpose as supplementary information providing a relevant context

for other properties. For example, it is not unusual for density to be reported

with no indication of how the density was measured or even with no indication

of the origin of the reported value. In this case, the data are not amenable

to any useful assessment. Nevertheless, these data often are essential for

understanding the context of other property measurements contained in the

same report. Consequently, it is desirable to have a category for unevaluated

data to permit the recording of contextual information while clearly indicating

that the reported values cannot be assessed in any manner.

Clearly, other categories may be defined according to the special interests

served by the database. The present reasoning, by design, considers

relatively generic categories and allows the data contained in a materials

property database to be organized or classified into seven distinct categories,

summarized in Table 4.1. The derivation of this classification scheme, again

by design, uses only generic considerations to distinguish the data entries and

does not depend on any restriction to a particular material class. Hence,

the classification scheme of Table 4.1 is applicable to any type of material,

whether the general class be ceramic, metallic, polymeric, or biomaterial.

Data Evaluation Levels for Materials Properties ◆
◆ Data Evaluation Theory and Practice

24

References Cited in Chapter 4

1. Munro, R.G., and H. Chen. “Data evaluation methodology for high

temperature superconductors.” Computerization and Networking of

Materials Databases: Fifth Volume, ASTM STP 1311, pp. 198–210.

Ed. S. Nishijima and S. Iwata. Philadelphia: American Society for Testing

and Materials, 1997.

2. May, W., R. Parris, C. Beck, J. Fassett, R. Greenberg, F. Guenther,

G. Kramer, S. Wise, T. Gills, J. Colbert, R. Gettings, and B. MacDonald.

Definitions of Terms and Modes Used at NIST for Value-Assignment

of Reference Materials for Chemical Measurements. NIST SP 260-136.

Gaithersburg: National Institute of Standards and Technology, 2000.
25

5. Operational Protocol

Compilations of data always are plagued by one overriding concern: are

the values credible? This concern most commonly is focused on whether

or not the values may be used reliably. This particular focus emphasizes the

intrinsic worth of data evaluation and argues for an evaluation procedure that

is conducted with the utmost care and consistency. More emphatically, the

critical role of data in advanced technologies demands that the quality of

the data be of the highest caliber.

The foremost consequence of such considerations is that reliability must be

accorded the highest priority in the assessment criteria applied to the data.

However, it must be recognized that data comprise a remarkably expensive

commodity. The costs incurred by the need for suitable equipment, measurement

standards, materials, and expertise extend far beyond the immediate funds

expended to measure a particular set of data, and the latter expenditure can

be significant by itself. The evaluation process, therefore, should consider all

data as being potentially useful and should discard data

only with the greatest reluctance.

Consequently, at the base of the data evaluation paradigm, there are two

counterbalancing mandates: (1) ensure the reliability of the data; and

(2) do not reject data without cause.

These mandates can be made an integral part of the decision logic applied

to the evaluation of data. Using the data evaluation levels discussed in the

preceding chapter, we present here a suitable protocol for assessing materials

property data.

5.1 General Considerations

Perhaps the most widely observed and practiced method of

demonstrating the credibility of experimental data is to establish

reproducible results, i.e. the mutual agreement among independent

studies. We may think of this notion of reproducibility as a

one-dimensional assessment of the data because it focuses on

the replication of a single observation.

Higher-dimensional assessments, yielding correspondingly higher

levels of confidence, may be constructed using correlations with

other measurements and theoretical models. In this approach, multiple

independent observations of related properties are used together to

assess the reliability of any one of them. Since each property represents

a different measurement perspective, a higher level of confidence may

Operational Protocol ◆
◆ Data Evaluation Theory and Practice

26

be ascribed to these data collectively, owing to the consistency of the

observations across the multiple properties, than could be ascribed to

any one of the properties individually.

Such considerations illustrate the general themes of independence,

replication, and multiplicity that form the underpinnings of data evaluation.

These themes are also fundamental to the development of a data

evaluation protocol.

5.2 A Protocol for Materials Property Data

The evaluation protocol[1] developed for the NIST Structural Ceramics

Database (SCD)[2] and the High Temperature Superconductors (HTS)[3]

database is shown schematically in Figures 5.1 to 5.4. This protocol

adheres to the procedure discussed in Chapter 3 and uses the data

evaluation levels listed in Table 4.1 in Chapter 4. In the present section,

a brief commentary is given to illustrate how each node in the decision

process contributes to the assessment of the data.

5.2.1 Material Specification

Question: Is the material specification or identification sufficient

for the purpose?

The first consideration stems from the understanding that materials

properties are of no value without some identification of the material.

It follows rather simply from this observation that we should establish

what is known about the material at the very beginning of the hierarchy

of evaluation criteria.

Specific guidelines for identifying materials and for determining the

adequacy of the identification have been discussed extensively by

the ASTM.[4] The ASTM scheme recommends the use of seven

distinct categories of identification information plus a general category

for supplementary information. These categories distinguish among

primary identifiers (e.g., material class), widely recognized specification

codes (e.g., M-52 steel), characterization of the material (e.g., in

terms of its chemical composition), the source of the material and its

processing history, geometrical details for the specimens used in the

measurements, and any fabrication treatments (e.g., heat treatments)

and service history experienced by the specimen used in the measurements.

However, we need to recognize that the adequacy of the material

identification can be interpreted either broadly or narrowly, depending

on the purpose of the data. For most property data, a basic objective of
27

Figure 5.1. Schematic of the basic data evaluation procedure. The node labeled

[Other Assessment] is expanded in Figures 5.2 and 5.3; [Other Classification] is

expanded in Figure 5.4. The data evaluation levels given in the italic font are

described in Table 4.1.

Operational Protocol ◆
◆ Data Evaluation Theory and Practice

28

material identification is to present sufficient information that someone

working independently of the original research would be able to prepare

or obtain a similar specimen. However, not all sets of data are meant to

pertain to specific materials. Surveys, for example, commonly present

only typical values or ranges of typical values; merely identifying a

class or subclass of material might be adequate for that purpose.

Consequently, the first question in the protocol, while simple and direct,

should be addressed in the context of both the purpose of the data set

and the extent to which identifying information has been provided.

In all cases, a negative answer to this first question obviates any

need for further consideration.

5.2.2 Measurement Method

Question: Is the description of the measurement method sufficient?

A basic objective in describing a measurement method is to present

sufficient information that someone working independently would

be able to conduct a comparable measurement. For purposes of

comparability, it must be noted that all measurement techniques have

associated measurement uncertainties, and all techniques have limited

ranges within which they may be usefully applied. It is possible for

a measurement technique to be developed for a particular class or

subclass of material, sometimes exploiting a characteristic that is not

available in other material classes. Further, the measured values of

procedural properties (see Chapter 2) depend (often explicitly) on the

specific procedures that are followed in conducting the measurement.

A case in point is the difference between a Vickers hardness and a

Knoop hardness. The two methods provide measures of hardness that

are fairly similar in the mechanical steps. Each method uses an indenter

under normal loading to create an impression in the surface of the

material. The ratio of the applied force and the area of the remnant

impression yields a measure of the hardness of the material. These

two measures of hardness, however, are necessarily different because:

(1) the indenters used by the two methods have different shapes; and

(2) one technique uses the actual surface area of the impression,

while the other uses the projected surface area.

There is rarely, if ever, a good excuse for not identifying a measurement

technique, but such is often the case when supplementary data are

reported. It is not uncommon, for example, for the density of a specimen

to be reported only summarily, such as: “The density was 3.21 g/cm3.”
29

Such supplementary data may be critical to understanding or using other

property data in the same report. Consequently, lack of information

about the measurement procedure is not automatically sufficient to

cause a set of data to be unacceptable. This assessment protocol

considers other data evaluation levels when the measurement method

is not described sufficiently.

5.2.3 Certified Reference Values

Question: Does the set of data consist of certified reference

values?

Certified reference values generally are established by a recognized

national standards measurement laboratory.[5] A certified reference

value usually is a property value for a specific batch of a material. The

material usually is designated as a standard reference material on which

homogeneity and other quality assurance tests have been conducted.

Commonly, more than one measurement method is used to determine

the property value. The certified value is determined statistically and

reported with an analysis of the associated measurement uncertainty.

5.2.4 Correlations or Models

Question: Are the reported values consistent with or confirmed

by correlations with other properties or model calculations?

A material property may be known to be well described by a model or

may have well-established relations with other properties. For example,

the oxygen content in the high temperature superconductor Y:123 in

the orthorhombic crystal structure is known to be correlated to the

c-axis lattice parameter. In such cases, it may be possible to confirm

the measured values by means of the correlations, if the correlated

property data also are available, or by means of the established

model calculations.

5.2.5 Independent Values

Question: Do the current values agree with other values

reported independently for the same property and material?

Reproducibility is perhaps the most reassuring evidence that a proper

value has been determined for a property. Values are considered

reproducible when independent measurements are conducted and

the resulting measured values are found to agree within a reasonable

measurement uncertainty.

Operational Protocol ◆
◆ Data Evaluation Theory and Practice

30

5.2.6 Related Properties (with adequate measurement

information)

Question: Are the current values consistent with values estimated

independently using known related properties of the same

material?

It often happens that a material property has a theoretical relation to

one or more other properties that can be determined independently.

For example, the elastic (Young’s) modulus, the shear modulus, the

bulk modulus, and Poisson’s ratio are often used to characterize the

elasticity of isotropic materials. However, for isotropic materials, only

two of these quantities are independent. Given any two of the elastic

properties, the others may be calculated. Consequently, if the elastic

modulus is reported in one paper, the shear modulus in a second paper,

and Poisson’s ratio in a third paper, it is possible theoretically to verify

the mutual consistency of these values within the bounds of the

appropriate measurement uncertainty.

5.2.7 Peer Review

Question: Was the report published under independent peer review?

Journals and other publications usually take great care to ensure that

papers undergo independent review by at least one person who is

considered knowledgeable in the subject area. In such cases, it is

assumed that the reported values are reasonable within the experience

of the reviewer. The specific rationale used by the reviewer usually is

unknown. However, it may be assumed that no substantial objection

to the paper was established and, more importantly, that at least some

merit was found to support its publication.

5.2.8 Incompatible Properties

Question: Is there any apparent conflict between the present

data and other properties or trends that have been reported for

this material?

When no direct confirmation of a property value is available, it is

desirable to verify that the present result is not incompatible with

the general knowledge base for this material or this type of material.

This assessment includes the possibility that there are independent

measurements of this property that disagree with the present data in

some significant manner. Heuristics or “rules of thumb” also might

be applicable to this consideration. For example, it is a common
31

Figure 5.2. Expansion of the data evaluation branch introduced in Figure 5.1

as [Other Assessment]. The node labeled [Other Comparisons] is expanded

in Figure 5.3.

observation that if the hardness of a bulk material increases, the elastic

modulus of the bulk material increases also. Contradictory observations

would indicate a need to reassess the apparently conflicting reports.

Operational Protocol ◆
◆ Data Evaluation Theory and Practice

32

Figure 5.3. Expansion of the data evaluation branch introduced in Figure 5.2 as

[Other Comparisons].

5.2.9 Manufacturer’s Data

Question: Is the data set provided by the commercial manufacturer

or supplier of the material?

Commercial manufacturers and suppliers of materials usually take

great care to provide property data to their customers or as general

information about their products. Often the values are reported with

only a nominal or a brief indication of the measurement details. The

data also may be presented as being typical of results obtained using

specimens from several production batches of the material. These

values are sometimes reported subsequently in published works without

further discussion or assessment.
33

5.2.10 Survey Data

Question: Is the data set provided as a literature survey or an

overview?

It is sometimes desirable to have an overview of how a property varies

over a range of conditions or over a range of materials. A common

method used to construct the overview is to review the literature for

similar studies and to combine the results. Values derived from such

surveys generally are representative of the trend of observed values

but are not necessarily appropriate for any particular specimen.

Typical values are useful for making order of magnitude estimates

and for illustrating a functional trend such as a power law dependence

on temperature.

5.2.11 Subsidiary Data

Question: Is the data set provided as supplementary information?

It is not unusual for a paper to focus rather sharply on one property,

providing considerable details about its measurement, while other

properties are mentioned only briefly. These additional data often

are intended collectively as supplementary information that helps to

define the context of the principal measurement. Subsidiary data of

this nature often are reported without discussion of the measurement

details. Density is very often reported in this manner.

5.2.12 Related Properties (without adequate measurement

information)

Question: Are the current values consistent with independent

reports of this property or related properties for this material

or a similar material?

It sometimes happens that independent studies have been made on

the same property or on related properties for the same material or for

similar materials. Independent measurements of the same property

may be compared for mutual consistency. It also may be possible to

estimate a value for the current property from known values of related

properties. Such estimates provide a reasonable validity check on

the reported value but cannot be said to confirm the reported value

because of the lack of information on the measurement technique or

because of differences in the experimental specimens.

Operational Protocol ◆
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34

Figure 5.4. Expansion of the data evaluation branch introduced in Figure 5.1

as [Other Classification].
35

5.2.13 Interim Report

Question: Are the reported values considered preliminary data?

Preliminary results are sometimes reported in conference proceedings,

interim reports to sponsors, or other status reports. Results that are the

first of their kind might also be reported as a brief communication prior

to the completion of the study. In such cases, it is tacitly understood

that the reported values are subject to further study and may be

different in the final report of the work. The timeliness and interest in

the current status, however, warrant the reporting of the preliminary

values.

5.2.14 Reassessment

Question: If an apparent conflict was identified between the

present data and other results, was the conflict resolved favorably

for the present data?

Many apparent discrepancies among material properties have

rational explanations and are not true discrepancies. For example,

two specimens of polycrystalline sintered ceramics having identical

chemical compositions might be reported as the same material.

However, if the sintered specimens have different densities, porosities,

or grain sizes, the two measured property values could be significantly

different, with both values being correct. Differences in measurement

techniques can also result in apparent discrepancies. For example, the

flexural strength of a ceramic material measured in three-point bending

is generally different from the flexural strength measured in four-point

bending because the two methods sample different volumes of the

statistically distributed flaw sites.

Operational Protocol ◆
◆ Data Evaluation Theory and Practice

36

References Cited in Chapter 5

1. Munro, R.G., and H. Chen. “Data evaluation methodology for high

temperature superconductors.” Computerization and Networking of

Materials Databases: Fifth Volume. ASTM STP 1311, pp. 198–210.

Ed. S. Nishijima and S. Iwata. Philadelphia: American Society for Testing

and Materials, 1997.

2. Munro, R.G., and E.F. Begley. “Structural Ceramics.” Standard

Reference Database, Number 30, Standard Reference Data

Program. WebSCD: http://www.ceramics.nist.gov. Gaithersburg:

National Institute of Standards and Technology (1991, 1993, 1998, 2001).

3. Munro, R.G., and E.F. Begley. “High Temperature Superconductors.”

Standard Reference Database, Number 62, Standard Reference

Data Program. WebHTS: http://www.ceramics.nist.gov.

Gaithersburg: National Institute of Standards and Technology (1995, 1997).

4. Moniz, B. “Nomenclature and current standards for identificaton of

engineering materials.” Manual on the Building of Materials Databases.

Ed. C.H. Newton. ASTM Manual Series: MNL 19, pp. 34 – 44.

Philadelphia: American Society for Testing and Materials, 1993.

5. May, W., R. Parris, C. Beck, J. Fassett, R. Greenberg, F. Guenther,

G. Kramer, S. Wise, T. Gills, J. Colbert, R. Gettings, and B. MacDonald.

Definitions of Terms and Modes Used at NIST for Value-Assignment

of Reference Materials for Chemical Measurements. NIST SP 260-136.

Gaithersburg: National Institute of Standards and Technology, 2000.
37

6. Issues and Practices in Data Evaluation

It is virtually impossible to undertake the deliberate collection of property

data without some form of data evaluation be it primitive or highly

refined. The very act of designating the kind of data to be accumulated is

discriminatory and therefore constitutes a degree of evaluation. For design

purposes, for materials science, and for the development of measurement

standards, it is usually desirable to have a much higher degree of evaluation

than is provided by a primitive discrimination applied to the nature of the data.

Designers need to have values that pertain to the materials that will be used in

their products. Materials scientists need values that can be used to distinguish

the roles of underlying independent variables and to deduce relations and

correlations among properties. Measurement standards require data of

the highest attainable reliability. Between the basic needs of a primitive

classification and the advanced status of a fully refined uncertainty analysis

may be found a wealth of requirements that are suitable for diverse

intermediate purposes. Accordingly, it may be useful to consider the

deliberate evaluation of data in a series of progressively refined stages

of discrimination and analysis.

6.1 Stage I: Data Collection (Issue: Accessibility)

All design and analytical applications of property data are predicated

on the existence of a significant collection of values. Even cursory

applications using order of magnitude estimates require data collections

that are sufficient to establish ranges of typical values. The task of

amassing data, therefore, is fundamental to all efforts requiring data,

whether the requirement is for data of maximal reliability or for data

of minimal reliability.

Perhaps the most daunting obstacle impeding data collection, and the

most worrisome, is the immediate need to identify suitable sources of

data. With the advent of computerized information systems, this task

has been reduced in many cases to the development of an appropriate

search strategy.

Computerized information systems used for literature searches are

similar, with respect to content, to their printed ancestors. The

computerized versions, however, differ from the printed versions in two

important respects. First, computerization reduces both the time needed

to conduct the search and the tedium that must be endured during the

process. Second, most of today’s computerized routines allow searches

to be constructed from words commonly used in the subject area.

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38

In the past, the practical and economic incentives to keep the printed

search documents as small as possible often led to the use of cryptic

abbreviations or code sequences. The user had to expend additional

time deciphering those codes and abbreviations to determine which

search results were relevant. In contrast, today’s computerized search

results usually contain the full title of the paper, along with the names

of the authors and the complete journal citation. Frequently, keywords

and an abstract of the paper are provided also. This information usually

is sufficient to determine whether or not a paper is relevant to your

particular interests.

Regardless of the available tools, the success of a search depends on

the logistical aspects of the strategy that is used to conduct the search.

For the NIST databases, it has been found that comprehensive focused

searches can be achieved using a strategy that evolves in three phases.

The first phase is to construct a “random sampling” of the literature

by using basic keyword searches to probe the appropriate subset of

the literature. For example, if you were searching for data on the

elastic modulus of alumina, you might begin by specifying “alumina”

as a keyword. One such keyword search of the open literature

found 29866 papers involving that keyword. The papers in this

subset addressed a great many topics other than the elastic modulus.

Fortunately, most search routines allow you to specify compound

searches consisting of multiple terms. To find only those papers

pertaining to the elastic modulus of alumina, you might specify “alumina

AND elastic AND modulus” as the search condition. This compound

condition applied to the same bibliographic database reduced the number

of potentially relevant papers to 202. Further specialization to “alumina

AND elastic AND modulus AND porosity” reduced the count to 24,

a readily manageable number of papers.

The second phase of the search strategy expands the reference set using

a technique that might be called “progressive branching.” In this phase,

the papers found in the first phase serve as seed references (because

the reference set will grow from them). Progressive branching makes

use of the observation that an essential aspect of technical writing is the

proper accreditation of prior research. This characteristic of technical

writing allows you to anticipate that some of the papers that you have

identified in the first phase of your search have also been used as

references in research reported by other authors. This circumstance

leads quite naturally to a technique known as the “cited reference”

search. In this technique, you search for all the papers that have cited a

particular paper. Because these papers necessarily were presented after
39

the cited work, they represent a branching forward in time from the

cited paper.

The third phase of the search strategy consists of following the reference

trails of the papers already identified, a technique that might be called

“regressive branching.” Quite simply, each paper that you have collected

for your reference set has its own list of references, some of which may

differ from your current collection and are therefore candidates for

expanding your set. Because those references necessarily preceded

the current paper, they represent a branching backward in time.

Upon reflection, it should be clear that each new reference that is

collected may be used for both progressive branching and regressive

branching. Consequently, the iterative application of these two search

techniques has the potential of generating a collection of references that

encompasses all of the mutually related papers that are documented in

the bibliographic database.

6.2 Stage II: Basic Evaluation (Issue: Reproducibility)

The underlying criteria for Stage I: Data Collection are existence and

availability. The purpose of Stage II: Basic Evaluation is to augment

these criteria with conditions that address the central issue of credible

data, i.e., reproducibility. The underlying criteria added in Stage II are

identification, description, and comparability. These criteria were applied

in Chapter 5 in developing an operational protocol for data evaluation.

Their roles are discussed under the headings of Material Specification

(section 5.2.1), Measurement Method (section 5.2.2), and several of the

decision nodes relating to the issue of reproducibility and the comparison

of values from independent studies.

A special case of twofold interest to Stage II analysis is the round robin

technique applied to measurement science. Round robin studies commonly

address two concerns: the relative reliability of the measurement methods

and the quantitative reliability of the measurement results.

6.2.1 Illustrative Example: Particle Size Measurement

When multiple techniques can be used for alternative measurements

of the same property, the data evaluator needs to be aware of the

limitations of the different techniques. Those limitations sometimes

can be subtle. A round robin study provides an opportunity to examine

those limitations under controlled conditions. A case in point is the

measurement of the mean particle size of a ceramic powder.[1]

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40

The processing procedure that produces a polycrystalline ceramic

generally begins with a powder and proceeds through a series of

mixing, compaction, sintering, and other heat treatment processes.[2] The

mechanical, thermal, and electrical properties of the finished material

often depend on the resulting microstructure and, most particularly, the

grain size and porosity of the material. Characterization of the starting

powder is critical to both understanding the process and ensuring the

reproducibility of the process.

To address numerous issues in powder characterization, the

International Energy Association (IEA) organized a series of round

robin studies. These studies were exploratory in nature, and their

primary purpose was to provide a basis for discovering and resolving

potential problems that could affect international measurement

standards. Among the issues addressed was the determination of

the mean particle size of a starting powder.

The participants in the IEA studies voluntarily conducted a variety

of measurements for characterizing ceramic powders.[3] Twenty-five

laboratories from Germany, Sweden, and the United States participated

in this study, designated Annex II in the IEA program structure. Each

laboratory was supplied a set of five powders carefully derived from a

master supply that had been developed specifically for this purpose at

the National Institute of Standards and Technology (NIST). Previous

efforts on ceramic powder characterization had established the critical

importance of the sampling procedures used to prepare these sets

of powders.[4] As a result, in Annex II, the master supply for each

powder was subjected to extensive rifling, handling, and packaging

procedures to ensure that the samples were homogeneous in content.

The laboratories were requested to follow their usual procedures in

preparing their test specimens from the materials that were supplied

to them, and each laboratory was to conduct measurements according

to the methods and procedures normally used in its own work.

Included in the results[1] were two sedigraph techniques, two light

scattering methods, and two electron microscopy procedures. All of

these methods were expected to have size detection limits that would

be adequate for the powders used in the study. However, specimen

preparation procedures could and did vary significantly among the

methods and within a given method. The greatest concern in specimen

preparation was with respect to the dispersion and deagglomeration of

the particles. Deagglomeration can be affected strongly by such factors

as the choice of dispersant, the power level of the sonic waves used

to disrupt the agglomerates, and the duration of sonication.
41

Results[5] for the zirconia powder, Table 6.1, were quite dramatic.

The two sedigraph techniques differed by a factor of six, and the

two light scattering methods produced results differing by more than

a factor of ten. The microscopy results were of the same order of

magnitude, but the agreement was not especially good. Using Student’s

t-test at a confidence level of 99 %, the gravitational sedimentation (GS)

and photon correlation light scattering (PCLS) methods easily were

found to be distinctly different from the other four methods, while the

differences among the centrifugal sedimentation (CS), Fraunhofer–Mie

light scattering (FMLS), and the two scanning electron microscopy

(SEM) methods, SEM(WD) and SEM(MD), could not be judged to

be significant.

Table 6.1: Results for the mean particle size of the yttria-stabilized

zirconia powder (YSZ) in IEA Annex II, as determined by

the indicated methods.[5]

Standard

Test Number of Mean Value Deviation

Method* Laboratories [μm] [μm]

GS 6 0.21 0.03

CS 4 1.32 0.52

FMLS 5 2.33 1.29

PCLS 2 0.19 0.01

SEM(WD) 1 2.02 —

SEM(MD) 1 1.65 —

* The method abbreviations are:

GS = Gravitational Sedimentation

CS = Centrifugal Sedimentation

FMLS = Fraunhofer-Mie Light Scattering

PCLS = Photon Correlation Light Scattering

SEM(WD) = Scanning Electron Microscopy (Waddel Diameter)

SEM(MD) = Scanning Electron Microscopy (Maximum Diameter)

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42

It might be conjectured that if the only results showing an extraordinary

difference were the results from the two laboratories using the PCLS

method, then there would have been a strong inclination to declare those

results to be of a spurious nature. The concurrence of the GS method

and the consistency of the six laboratories using that method clearly

showed that the results should not be dismissed as spurious.

Figure 6.1. Histogram of particle size distribution for the yttria-stabilized

zirconia powder studied in IEA Annex II.[6] The horizontal bars identify the

actual ranges of values reported by the different methods. CS: centrifugal

sedimentation. GS: gravitational sedimentation. FMLS: Fraunhofer-Mie light

scattering. PCLS: photon correlation light scattering. SEM(MD): scanning

electron microscopy (maximum diameter). SEM(WD): scanning electron

microscopy (Waddel diameter).
43

Some insight into this puzzle may be gleaned from Figure 6.1,

a histogram constructed[6] from all of the individually reported

observations of the mean particle size for the zirconia material. This

figure reveals that the material had a bimodal particle size distribution.

The horizontal bars superimposed on the histogram show the ranges of

sizes actually reported by each measurement method. Apparently, the

various methods reported quite different results because the methods

sampled quite different portions of the bimodal distribution. Given that

all of the methods had theoretical detection limits that were much wider

than the observed intervals, it had to be concluded that the discrepancies

were consequences of the specimen preparation procedures. Thus,

the procedures used with the devices had to be considered, in addition

to the detection limits of the devices, in determining the limitations of

the measurements.

Observations of this nature illustrate how the depth of detail of a

measurement procedure can influence both the interpretation of the

results and the conclusions drawn from them regarding the general

applicability of a measurement method. Clearly, both of these issues

need to be examined critically in applied data evaluation efforts.

6.3 Stage III: Relational Analysis (Issue: Consistency)

In the preceding section, Stage II: Data Evaluation, the given data

set was examined in the one-dimensional context of reproducibility.

In Stage III, the data set is viewed through its relations to other data,

particularly through trends, correlations, and known material property

relations. The central focus of Stage III is on consistency.

6.3.1 Definitive and Derived Relations

Technical expressions often depend on semantics to ensure clarity of

meaning. An underlying requirement of proper semantics is the use

of basic definitions. In data evaluation, these basic definitions and the

rudimentary relations derived from them often can be used directly to

assess the consistency of independently reported measurement results.

In the technical literature, however, some terms are in such common

usage that there is sometimes a confusion regarding which terms are

definitive and which are derived. The basic importance of definitive

relations and the potential for confusion in the literature warrant a

brief review of some of these terms.

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44

6.3.1.1 Mass Fraction, Mole Fraction, Volume Fraction

Elemental, molecular, phase, and microstructural compositions

often need to be described in the reporting of technical results.

The mixture of initial powders; the addition of stabilizers or sintering

aids; the assessment of the composition of a material after processing,

heat treatment, or environmental reaction; and the quantitative

measure of a material’s microstructure represent some of the

more common situations requiring a quantitative expression of

a composition. The prescription for such expressions has been

provided as part of the International System of Units (SI),[7–9]

which currently represents the agreement of 51 member states

and 9 associates to the General Conference.[10] The prescription is

quite simple. For example, if a substance has n components whose

masses are denoted by mi for i = 1 to n, resulting in a total mass

M = m1+m2+...+mn, then the mass fraction, μi, of the ith component

in M is simply

(1)

where clearly, (μ1 + μ2 + ... + μn) = 1. Similar definitions apply

to mole fraction and volume fraction.

Mass fractions are frequently reported using the fraction’s unit, %.

For example, if the mass fraction of powder A in a mixture is

μA = 0.98, the value might be reported as 98 % A. There is a long

standing, very widespread practice in the literature of condensing the

identification of the quantity (mass fraction of A) and its value (98 %)

into an abbreviated expression (98 wt % A). Similar condensed

expressions occur for mole fraction (mol %) and volume fraction

(vol %). While this practice is discouraged by the SI, the condensed

expressions are still favored by many journals, even by some journals

that have expressly endorsed the SI.

One consequence of this common practice is a relatively rare

occurrence of confusion between the proper SI term, mass fraction,

and another commonly used term, fractional density. When a

polycrystalline material is processed, the density of the end product

is often compared to the maximum density that could be achieved by

an ideal single crystal of the same material. The ratio of the actual
45

density to the maximum density is the fractional density, also known

as the relative density. The popular jargon, percent theoretical

density, is also found in the literature. The two terms, mass fraction

and fractional density, are clearly distinct. Mass fraction relates

to composition while fractional density relates to the degree of

compaction or consolidation. The potential for confusion apparently

arises from use of the word “mass” in the term “mass fraction.”

Because density also is related to mass, “mass fraction” may

be interpreted incorrectly as a density fraction and, hence, as

“fractional density.” While the correct meaning is usually clear

from the context in which the two terms are used, it is useful to

be alert to the possibility of confusion. Terms related to density

are discussed more fully in the following section.

6.3.1.2 Density, Theoretical Density, Porosity

Among the basic physical characteristics of a solid material is its

mass density, defined as the mass of the specimen divided by the

volume of the specimen. Most polycrystalline materials contain

void spaces between material grains resulting in a density that is

less than the density that would be obtained for a perfect single

crystal of the same material. The density of a perfect single crystal

is the maximum attainable density for a given structural phase of

the material and is quite commonly known as the theoretical density.

Other terms often used for this same quantity include the crystal

density and the x-ray density of the material. If the material has a

molar mass MW, computed from its chemical formula, and the unit

cell of its crystal structure has volume V, computed from its measured

lattice parameters, then the theoretical density, ρtheo, of the material is

(2)

where z is the number of formula units in the unit cell, and NA is

Avogadro’s number (6.0221367(36)×1023 mol–1). The volume of the

unit cell depends on the symmetry of the crystal system. Appropriate

expressions[11] are given in Table 6.2 for the seven distinct crystal

systems.

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46

For polycrystalline materials, the measured density, ρ, is less than

ρtheo because the specimen’s volume, Vs, is occupied, in part, by

voids. If M is the total mass of the specimen, then the measured

density is ρ = M/Vs = M/(Vm+Vv) = ρtheoVm/(Vm+Vv), where Vm is

the ideal volume, free of voids, that would be occupied by an ideal

material component of mass M, and Vv is the total volume of the void

space in the real specimen. It follows that the volume fraction of the

material component in the specimen volume is the fractional density,

ρ/ρtheo. The complementary quantity, the volume fraction of void

space, commonly known as the porosity, φ = Vv/(Vv+Vm), may then

be computed as

(3)

This expression for porosity may be extended readily to a

multicomponent material. If the material consists of n material

components, then

(4)

where ρi

 is the theoretical density of the ith component and μi is

the mass fraction of the ith component in the material.

Table 6.2. Unit Cell Parameters and Unit Cell Volumes for the Seven

Crystal Systems[11]

 
47

It should be noted that the porosity computed in this manner is the

total porosity including contributions from intra-granular void space

as well as from inter-granular voids. When porosity is measured

experimentally by immersion or pyncnometry techniques,[12] only

a portion of the total void space, the “open porosity,” is accessible

to the diffusing species. Some of the pores are “closed,” i.e.,

inaccessible to the diffusing species, and, therefore, are not included

in the void space detected by these techniques. Consequently, the

measured open porosity is necessarily less than or equal to the total

porosity. This distinction can be important when comparing results

from independent studies or when the influence of porosity on

property values is being investigated.

6.3.1.3 Thermal Expansion

For most materials, an increase in the temperature of a substance

that is maintained at constant pressure is accompanied by a change

in the volume occupied by the material. The fractional amount

by which the dimensions of a material change per degree of

temperature change is called the coefficient of thermal expansion

(CTE). The volumetric CTE is defined as

(5)

where the subscript on the derivative denotes constant pressure.

In many practical applications, changes in linear dimensions are

more relevant. The linear CTE is defined as

(6)

where L is any linear dimension. For isotropic polycrystalline

materials such as ceramics, the magnitude of the CTE is in the range

from 10–6 K–1 to 10–5 K–1. For such small CTE values, it may be

found for isotropic materials that αV ≈ 3αL given that V is of order

L3 and αL is the mean linear CTE for the material.

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48

A more common measure of expansion is the cumulative linear

coefficient of thermal expansion,

(7)

in which expansion is expressed relative to a reference state of

dimension L0 at temperature T0. The continuous and cumulative

linear coefficients are related by

(8)

All of these measures of thermal expansion are related to changes in

volume, and any change in the volume of a material must produce a

change in the density when the mass of the material is conserved.

The definition of density requires

(9)

from which it is straightforward to find

(10)

with respect to the cumulative coefficient of volumetric expansion,

and

(11)

for isotropic materials when V is of order L3.
49

These relations are particularly useful for applications involving

elevated temperatures. Density is commonly determined only

at room temperature, while the expansion of a specimen with

temperature often is reported only as the relative change in a linear

dimension between room temperature and one or more higher

temperatures. In such cases, the value of the density at high

temperature must be calculated using these relations.

6.3.1.4 Elasticity

All solid materials will deform in response to external forces or

internal thermal stresses. The extent to which a material deforms

and the nature of that deformation depend on geometrical features

of the microstructure of the material, interactions occurring among

grains, inclusions, other secondary material phases, and the bond

strengths of elemental constituents. When the deformation is

recoverable, it is said to be elastic. More precisely, the deformation

produced by an external force acting on a solid is said to be elastic

when removing the external force returns the solid to its original

undeformed state.[13]

The phenomenon of elastic deformation is a familiar one, being

commonly seen in the stretching of a spring, the flexing of a beam,

the bending and twisting of a cable, and the swaying of a tree in

the wind. Quantitative observations made as early as the 1600s[14]

found that most solids exhibit a linear response when the applied

stress is not too large. The relation known as Hooke’s law, σ = E0,

defines a coefficient of elasticity relating the applied stress, σ (force

per unit area), to the strain, 0 (fractional elongation), produced by

the stress. The coefficient, E, is known as the elastic modulus or,

perhaps more commonly, as Young’s modulus. A similar coefficient,

G, the shear modulus, also known as Coulomb’s modulus, is defined

for angular or torsional deformation, γ, produced by a shear stress, τ,

such that τ = Gγ.

For crystals and anisotropic materials, these definitions need to be

generalized to account for the different responses of the material in

different directions, and the scalar coefficients are replaced by a

tensor. For isotropic materials, however, the two coefficients, E

and G, are sufficient to describe the macroscopic elastic behavior

completely. However, it is convenient to consider two alternate

parameters, Poisson’s ratio and the bulk modulus, that describe

specific attributes of the elastic behavior of isotropic materials.

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The bulk modulus, B, describes the particular circumstance when

stress is applied hydrostatically rather than axially. The hydrostatic

pressure, P, applied to an isotropic solid, causes the volume, V, to

decrease such that,

(12)

where the subscript T denotes constant temperature.

Poisson’s ratio, ν, describes the relation between axial and lateral

strains. For example, when a rod of uniform circular cross section is

elongated by a tensile stress along the rod axis, the cross section

undergoes a decrease in diameter. The negative of the ratio of the

lateral and axial strains is Poisson’s ratio,

(13)

It can be shown that

(14)

(15)

for isotropic materials. In principle, any two of the four quantities,

E, G, B, and ν, can be used to describe the elastic properties of an

isotropic material.
51

The elastic and shear moduli can be determined directly from the

slopes of the appropriate stress vs. strain curves.[15] In practice,

however, the measurement of elastic properties is most commonly

accomplished using sonic resonance or the propagation of ultrasonic

waves in the material.[16–23] There are two independent sound

wave velocities corresponding to the two independent coefficients

of elasticity for isotropic materials. These velocities commonly

are denoted vS for the shear mode and vL for the longitudinal mode.

The relations between the sonic velocities and the elastic moduli may

be written as

(16)

(17)

where ρ is the mass density of the material.

It should be noted that the explicit occurrence of density in these

expressions does not imply a linear dependence of the elastic moduli

on density. The sound velocities also vary with density, and it will

be shown in Section 6.4 that the density dependence of the elastic

moduli is generally nonlinear.

6.3.1.5 Heat Capacity and Specific Heat

When materials are used at elevated temperature, the presence of

temperature gradients, thermally induced strains, and the rate of

transport of thermal energy become special concerns. The properties

of immediate interest in this context are specific heat, thermal

conductivity, and thermal diffusivity. The specific heat is related to

the increase of temperature when a quantity of heat energy is added

to the specimen; the thermal conductivity characterizes the steady

state heat flow in the material; and the thermal diffusivity is related

to the transient response of the material to a heat pulse.

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The heat capacity of a material specimen is the amount of energy

that must be added to the specimen to raise its temperature by 1 K.

It is a fairly common observation that a larger specimen requires

more energy than a smaller specimen to produce the same change

in temperature. To compare results from different specimens, it is

necessary to normalize the measured values to a common amount

of substance. There are two such normalized measures of heat

capacity that are in common use. One is called the molar heat

capacity, and the other is commonly called the specific heat.

The molar heat capacity expresses the heat capacity per mole of

the substance. The specific heat expresses the heat capacity per

unit of mass of the substance. In practice, the heat capacity of

a specimen is determined experimentally[24–26] under isobaric

conditions, normalized to specific heat, CP, using the known mass

of the specimen, and then converted to molar heat capacity,

C(M)P, for more convenient use in thermodynamic relations.

A subtle point in the conversion of specific heat to molar heat

capacity is worthy of additional comment. The conversion

relation is simply

(18)

where MW is the molar mass of the substance. While the mass of

an atom is a fundamental characteristic of matter, the recommended

values of the atomic masses (a.k.a., atomic weights) of the elements

change from time to time as the necessary experimental techniques

are improved or the determinations of the values are refined. Older

technical papers used values for MW that differ from the values that

would be used today. Consequently, to calculate the specific heat

from a molar heat capacity reported in an older paper, the older

values of MW need be used for the calculation rather than the present

values. However, if molar heat capacity is also desired, then the

current value of MW must be used to correct for the changes in

atomic masses. Symbolically,

(19)
53

(20)

where it should also be noted that the distinction of “old” and “new”

is not needed for Cp because the value of Cp is derived directly from

the measurements of heat, mass, and temperature and is unaffected

by any change in MW.

Laboratory measurements to determine heat capacity are usually

performed under conditions of constant pressure, yielding the

specific heat Cp. In theoretical models and computations, it is often

more convenient to use the specific heat at constant volume, CV.

The two specifications yield slightly different values for the specific

heat, the difference being given by the thermodynamic relation

(21)

where BT is the isothermal bulk modulus, αV is the volumetric

coefficient of thermal expansion, T is temperature, and ρ is

density. This difference often is considered negligible even

at high temperature. For example, for high purity TiB2 ,

[27]

(Cp–CV)/Cp ≈ 3.7 % at 1000 °C, while for α-SiC[28] at the

same temperature, the relative difference is only 1.4 %.

6.3.1.6 Thermal Transport

The capacity of a material to store heat also has an influence on the

transport of heat energy through the material under nonequilibrium

conditions. For isotropic materials, thermal conductivity and thermal

diffusivity should be correlated through a relation involving the density

(ρ) and the specific heat at constant pressure (Cp); i.e., the transport

properties should be related by the expression

(22)

where κ is the thermal conductivity and D is the thermal diffusivity.

This simple relation is quite remarkable because it can serve as a

unifying factor imposing consistency among otherwise unrelated data.

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Figure 6.2. Consistency and data optimization through property relations,

illustrating the use of multiple data sets to optimize the correlation of thermal

conductivity and diffusivity for sintered α-SiC.[29]
55

The flow chart in Figure 6.2 illustrates how this consistency arises.[29]

The temperature dependence of the density is governed by the

thermal expansion of the specimen. Quantitatively, the temperature

dependence of the thermal expansion coefficient may be established

by means of lattice parameters and dilatometry results. These results

then may be combined [see section 6.3.1.3 Thermal Expansion] with

Archimedean measurements of the density at room temperature to

establish the variation of density with temperature. In separate work,

the temperature dependence of the specific heat may be established

independently, often combining the results of several studies to obtain

an optimum representation of Cp. Then (and only then), Equation (22)

can be used to obtain maximum consistency among the measured

transport values. In this latter step, it is very useful to have an

analytic expression representing the general trend of the transport

properties with temperature. In practice, both thermal conductivity

and thermal diffusivity have been modeled successfully at elevated

temperatures using expressions of the same form[27, 28, 30]

(23)

(24)

where the Di

 and the κi are adjustable parameters. In Figure 6.3,

results are shown[30] for sintered α-alumina (99 % purity).

The data points shown for thermal conductivity were reported

separately from the points shown for thermal diffusivity. The dashed

curves show the fits of the respective models when the data sets are

considered separately as standalone results. The solid curves show

the optimized fit when the data sets are considered simultaneously,

subject to the consistency constraint provided by Equation (22).

At the low temperature end of the observed range, the fully

evaluated results differ by more than 30 % from the standalone

results, a quantitative demonstration of the value-added benefit

of data evaluation.

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Figure 6.3. Thermal transport data for sintered α-alumina (99% purity). The

dashed curves show fits when the data sets are considered separately as standalone

results, while the solid curves show optimized fits constrained for consistency in the

manner of Figure 6.2.[30] [Data: See Additional Source References, Figure 6.3.]
57

6.3.2 Correlations and Semiempirical Relations

Materials science is fundamentally an observational discipline, and its

foremost objective is to gain from its observations an understanding of

material behavior. The process of subjecting a material to a variety of

stimuli and observing its responses leads naturally to the introduction

of properties or characteristics as quantitative measures of the behavior

of that material. The number of properties that can be introduced in

this manner is limited principally by the imagination of the observer.

In contrast, the effective number of degrees of freedom of a bulk

specimen is relatively limited. It should be expected, therefore, that

not all of the identified properties and characteristics are independent,

i.e., there must be subsets of properties that are interrelated.

Such relations are of considerable importance to data evaluation

because they provide some of the most compelling evidence for

consistency. When relations between different properties can be

applied successfully to data obtained from measurements that were

conducted independently, the result is a cross validation of the two

results. Further, the more these relations are found to be valid and

reliable, the more reasonable it becomes to use them to make estimates

of property values when direct measurement results are not available.

Correlations, in general, serve numerous purposes. Perhaps their

most obvious function is to provide reliable interpolation functions for

estimating property values. Under limited conditions, they might also

be used as a means to extrapolate the property estimates beyond the

range observed experimentally. Further, when materials behave in a

similar manner under similar stimuli, we may expect their properties

to exhibit functionally similar correlations. Hence, the analytical forms

of the correlations may serve as a means of classifying material

behavior. When semiempirical models are applied as part of the

correlation or analysis, the parameters of the model may be related

to specific physical characteristics, or the parameters may have

physically meaningful interpretations. In both cases, useful quantitative

estimates may be derived from fitting the model to the observed data.

Correlations and semiempirical models are also exceptionally useful in

assessing procedural properties, underlying variables, and the scope of

physical process mechanisms.

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6.3.2.1 Estimates of Unmeasured Quantities

{Example: Critical temperature, oxygen content, and

crystallography}

Quantitative applications of correlations are especially useful when

the mathematical concept of transitivity applies. Here, transitivity

means that if a quantity A is related to a variable C by a relation

A(C), and a quantity B is related to C by B(C), then a variation dA

of A implies a variation dB of B such that an estimate of dB can be

made from

(25)

given the value of dA. This situation is encountered, for example,

in studies of oxide superconductors.

The outstanding characteristic of a superconductor is that the

electrical resistance of the material vanishes when the temperature

of the material becomes less than a well-defined value known

as the superconducting critical temperature, Tc. In 1957, the

theoretical understanding of conventional superconductors was

established in what quickly became known as the BCS theory

of superconductivity.[31] When superconducting oxide ceramics

were discovered in 1987,[32] they were found to have characteristics

distinctly different from those described by the BCS theory.

An intensive research effort was launched immediately worldwide

to study the properties of this new class of materials. In less than

fifteen years from the date of their initial discovery, more than fifty

thousand papers were published on studies of this phenomenon.

While many materials were soon found to belong to this class of

unconventional superconductors, a major portion of the effort was

focused on YBa2Cu3Ox which quickly became known as Y:123 or

YBCO. It was established rather quickly that the superconducting

properties of Y:123 and other oxide superconductors had a strong

dependence on the oxygen content of the material. For Y:123,

Figure 6.4,[33] the dependence of Tc on the oxygen content was

nonlinear and reproducibly exhibited plateau-like regions. An

interpolation expression that mimics the plateau regions was

established empirically on the interval 6.4 ≤ x ≤ 6.95.
59

Figure 6.4. The dependence of the superconducting critical temperature, Tc ,

on the oxygen content in YBa2Cu3Ox .

[33] [Data: See Additional Source References,

Figure 6.4.]

(26)

giving the critical temperature (the solid curve in Figure 6.4) with a

combined standard uncertainty of 3 K for x ≥ 6.5 and 5 K for x < 6.5.

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Figure 6.5. Lattice parameters, a, b, and c, for the orthorhombic structure of

YBa2Cu3Ox , varied linearly with oxygen content.[33] [Data: See Additional Source

References, Figure 6.5.]
61

Concurrently, crystallographic studies also had found a highly

reproducible dependence on the oxygen content. On the

interval 6.35 ≤ x ≤ 7.0, the lattice parameters, a, b, and c, for

the orthorhombic structure of Y:123, varied linearly, Figure 6.5,[33]

with oxygen content. Simple linear least squares fitting of the data

from several sources found

(27)

(28)

(29)

with relative standard uncertainties of 0.13 %, 0.15 %, and 0.16 %

respectively.

Given that there is a significant dependence on the oxygen content,

these empirical correlations are especially useful when comparisons

of independent studies are made. For example, if one study includes

a value of Tc while another reports the lattice parameters, these

relations can be used to verify whether the two studies pertain

to similar compositions or to significantly different compositions.

The lattice parameter c, in particular, often has been used to

estimate the oxygen content of a specimen of Y:123.

6.3.2.2 Properties as Parameters

{Example: Isothermal equations of state}

Quantitative applications of semiempirical relations often are used

to deduce the absolute values of proper material characteristics that

are embedded in the relations. While this practice can be quite

reliable, there are pitfalls that may need to be avoided.

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Name P – Po

Murnaghan

Tait

Birch

Birch–Murnaghan

Bardeen

Born–Mie

Olinger–Halleck

Table 6.3: Semiempirical isothermal equations of state
63

Table 6.3, for example, lists seven different semiempirical

expressions for the isothermal equation of state (EOS) used

to described the compression of a solid while the temperature

is maintained constant.[34] Each of these expressions has a

well-defined material property, the bulk modulus, embedded in

the expression as a parameter whose value is to be determined

by fitting the expression to the observed compression data.

The expressions in Table 6.3 assume that the thermodynamic

relation between pressure, P, and volume, V, can be cast into

the form

(30)

where

(31)

(32)

such that B0 yields an estimate of the bulk modulus in the reference

state whose volume and pressure are respectively V0 and P0 at the

fixed temperature T. The additional parameter, B0', occurring in

some of the expressions is the isothermal pressure derivative of the

bulk modulus, B0' = (MB/MP)T,V=Vo. The intrinsic difficulty with this

approach is that different expressions for T(V) result in different

expressions for B0 with the end result that the various estimates

must intrinsically disagree. For example, the Murnaghan EOS is

equivalent to the assumption that the bulk modulus is exactly linear

in the pressure,

(33)

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In contrast, all other relations require the bulk modulus to be nonlinear

in the pressure. In the Tait model, a linear term is multiplied by a

pressure dependent volume,

(34)

while the Birch model is made nonlinear by means of an additive

term.

(35)

Clearly, the bulk moduli deduced from these differing models are

inequivalent.

The inequivalence of these models motivated the use of a simulation

study[35] to examine how reliably B0 could be estimated under

idealized conditions using the expressions in Table 6.3. That study

produced several observations of interest to data evaluation. Using

idealized data to which a normally distributed error component was

subsequently added either to the volume or the pressure, the study

concluded:

(a) The Birch EOS worked well only when the idealized data

were generated by the Birch EOS. In that case, the other model

expressions did well also. The converse, however, was not true.

The Birch EOS performed poorly in all other cases in the study.

(b) Models other than the Birch EOS determined B0 within 10 %

of the ideal value when the volume was known within 0.9 %

(i.e., the lattice parameter was known within 0.3 %), or the

pressure was known within 4 %.

(c) When the uncertainty in the lattice parameter was greater than

3 %, only the order of magnitude of the estimate was reliable.
65

Consequently, estimates of the bulk modulus obtained by means of

the Birch EOS should be treated with some caution, and preference

with respect to credibility should be given to the other models. In

this respect, it should also be noted that the Birch–Murnaghan EOS

reduces identically to the Birch EOS when the fitting parameter B0'

is constrained to be exactly 4. This constraint sometimes is applied

in the literature to simplify the least squares fitting routine. In this

particular case, identifying the expression as the Birch–Murnaghan

EOS should be treated as a misnomer, and it should be recognized

that the reliability of the result is merely that of the Birch EOS.

6.3.2.3 Interpreted Parameters

{Example: Creep deformation}

Many semiempirical models describe the net outcome of a complex

process in terms of a small number of the independent variables,

such as temperature or pressure. At the atomic or molecular scale,

a material may be undergoing numerous parallel and sequential

interactions distributed inhomogeneously throughout the material.

However, in many cases, the macroscopic behavior of the material

may be viewed as a statistical average of the accumulated

microscopic events. In such cases, a semiempirical model,

with perhaps a few adjustable parameters, can form a useful

representation of the average response of the material to changes

in the control parameters.

When semiempirical models are used in this manner, it is not

unusual that some of the adjustable parameters in the model may be

interpreted as characteristics of the physical mechanisms assumed

to be at work in the material. Activation energies often are used in

this manner. These interpreted parameters, in turn, provide a useful

means of classifying and comparing the behaviors of materials

subjected to similar conditions.

Creep deformation, for example, can be described by models of this

type.[36] In the absence of creep, a tensile stress, σ, applied to the end

of a rod of length, L0, will cause the rod to stretch to establish a new

mechanical equilibrium length, L(σ). Once equilibrium is established,

the length remains constant if the applied stress is unchanged, i.e.,

the strain at time, t, is 0(σ,t) = (L(σ,t)–L0)/L0 = 0(σ,0). When creep

deformation occurs, 0(σ,t) generally increases with time.

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For polycrystalline materials, a graphical plot of an isobaric

(constant σ) strain, 0(σ,t), as a function of time typically exhibits

a nonlinear curve, Figure 6.6, consisting of three distinguishable

regions termed, respectively, primary, secondary, and tertiary

creep.[37] The strain rate in secondary creep (variously termed the

secondary creep rate, the steady state creep rate, and the minimum

creep rate) is most often the focus of attention in creep studies

because the useful lifetime of a material undergoing creep

deformation is related to this particular rate.

When diffusion is responsible for the creep behavior, either in the

form of dislocation movement[38] or material transport along grain

boundaries,[39, 40] the Norton model[41] is commonly used to

described the secondary creep rate.

(36)

Figure 6.6. Schematic of a typical creep curve (strain vs. time) showing three

distinguishable regions termed, respectively, primary, secondary, and tertiary creep.[37]
67

where A is an amplitude factor; σ0 and g0 are fixed scale factors

defining, respectively, the unit of stress and the unit of grain size; n is

the stress exponent; g is the grain size; m is the grain size exponent;

Q is the apparent activation energy; R = 8.31451(7) J mol–1 K–1 is

the molar gas constant; and T is the absolute temperature (i.e., the

temperature on the Kelvin scale). An example of the application

of this expression is shown in Figure 6.7.[27] Note that when only

a single batch of a material is considered, the grain size factor is

effectively constant. In that case, the grain size factor is effectively

absorbed into the amplitude factor, and only the dependence on

temperature and applied stress are considered.

Figure 6.7. Flexural creep rate of TiB2 as a function of inverse temperature for

various values of applied stress. The dashed lines show the fit of Equation (36).[27]

[Data: See Additional Source References, Figure 6.7.]

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Activated Processes

Many physical phenomena have thresholds, usually with respect to

temperature, denoting the perceptible onset or alteration of the phenomena.

For thermally activated processes, such as creep deformation, the

temperature dependence of the process rate, y, often can be described

by a multiplicative exponential factor, y ~ exp[–β/T]. In this case, a plot

of log(y) vs. (1/T), with all other variables held constant, exhibits a straight

line whose slope determines the value of β. In the example of creep

deformation, the parameter, β, is written as Q/R, where R is the molar

gas constant. The quantity Q then has units of energy per mole and is

usually interpreted as the activation energy for the process.

The interpretation of Q as an activation energy assumes implicitly that

there is a specific mechanism or process that accounts for the observed

material behavior. If there is a change in the physical mechanism, then

there is likely to be a change in the associated activation energy. If the

change in mechanism is sufficiently abrupt, the change will be manifest

as a change in the slope of the apparent trend line in the plot of ln(y) vs.

(1/T). An example of this behavior is illustrated in the figure which shows

creep data for a sintered α-silicon carbide. The abrupt change in slope at

1600 °C indicates that a low temperature mechanism with an activation

energy of 442 kJ/mol is dominant in this material for T ≤ 1600 °C, while

a different process with a much higher activation energy, 944 kJ/mol,

dominates for T > 1600 °C.
69

The temperature dependence in this expression is captured

entirely in the exponential factor, e–Q/RT, which is a very common

representation for a thermally activated process. In isobaric studies

for a single batch of material, a plot of the logarithm of the creep

rate at various temperatures

(37)

will be linear in the reciprocal temperature. The activation energy,

Q, clearly is determined by the slope of such a plot. However, in the

practice of determining Q in this manner, there is potentially a point

of confusion, a pitfall into which many studies have fallen. It is not

unusual for studies of the temperature dependence to be conducted

at several values of the applied stress, in which case Q might be

determined for each value of the stress. When this practice is

followed, it should be recognized that the model of diffusive creep

given by Equation (36) is not being applied per se. Rather, the

modified expression

(38)

is used. The significance of this distinction is that in the modified

expression, the stress dependence is no longer a power law.

Similarly, applying the Norton model independently to data obtained

at different temperatures leads to an exponent n(T) rather than

the constant n, resulting again in a stress dependence that is not

a simple power law.

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As it happens, the power law dependence of the isothermal strain

rate on the applied stress is an important distinguishing feature of the

diffusive model of creep and serves as the characteristic signature of

diffusion creep. In a log–log plot of creep rate against applied stress,

(39)

the power law dependence results in a straight line with slope n.

It has been observed that for a material transport mechanism,[39, 40]

n ≈ 1, while for dislocation controlled creep,[38] n typically is in the

range from 3 to 6. Consequently, the stress exponent may form a

useful characteristic in classifying long-term material behavior.

Rather generally, the values of the parameters, n and Q, form a basis

for the comparison of material performance under creep conditions.

A larger value of the stress exponent is indicative of a material that

is more susceptible to creep under an applied stress, while a higher

activation energy is indicative of a material that is more resistant to

creep at a given temperature. Consequently, these parameters may

have useful applications in materials selection exercises.

In recent work, it has been recognized that the tensile creep rates of

some materials, such as polycrystalline silicon nitride, Figure 6.8, are

not well described by the diffusion model.[42] In particular, a power

law dependence on stress does not appear to be appropriate for these

materials, and there does not appear to be a significant dependence

on grain size. Instead, the tensile creep rate is strongly correlated

with the formation of discrete cavities in the material as the material

deforms. These cavities typically occur at multigrain junctions

where grains may more readily separate under tensile stress and

where intergranular phase materials may more readily diffuse

to accommodate the formation of a cavity.

When cavitation controls the creep behavior, the dependence of

the strain rate on the applied stress is quite distinct from that of the

diffusion model.[42]

(40)
71

The parameters, ψ and Q, for the cavitation model serve functions

similar to those of the parameters, n and Q, in the diffusion model

and may be useful in materials selection exercises in the same

manner. Note, in particular, that a plot of the logarithm of the

creep rate against the logarithm of the applied stress

(41)

will exhibit a nonlinear curve when this model is appropriate,

as in Figure 6.8, because the term linear in σ changes much more

rapidly than the logarithmic term, ln(σ/σ0). For that same reason,

a log–linear plot, logarithm of creep rate against linear applied stress,

will be approximately linear. Consequently, in the context of data

evaluation, these two plots together provide a convenient means of

distinguishing between the diffusion model and the cavitation model.

Figure 6.8. Tensile creep of polycrystalline silicon nitride, showing a stress

dependence that is not well described by the diffusion model.[42]

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The distinction between these models becomes even more convincing

when data over a large range of temperature and applied stress

can be examined, as in Figure 6.9.[42] In this example, the diffusion

model is clearly an inadequate representation of the simultaneous

dependence of the creep rate on temperature and applied stress,

while the three-parameter cavitation model adheres faithfully to

the trends of the data over the entire range.

Figure 6.9. Tensile creep of polycrystalline silicon nitride over a large range of

temperature and applied stress, showing the greater suitability of the cavitation

model.[42]

6.3.2.4 Procedural properties

{Example: Indentation size effect, Meyer’s law}

Correlations and semiempirical relations have an especially

important role to perform in understanding the measurement results

for procedural properties. It will be recalled from Chapter 2 that

the quantitative values obtained for procedural properties depend
73

intrinsically on the details of the specific measurement procedures

used in their determination.

The hardness of a material[43] is perhaps the most pervasive example

of a procedural property. Intuitively, materials that yield less when

you press against them are harder. This qualitative notion of

hardness can be quantified by specifying how to press against the

material and how to measure the material’s response. The quantified

results, in turn, can be used to verify the consistency of a material

production or a heat treatment process or to rank a collection of

materials according to their relative hardness values. The broad

utility of hardness measurements in quality control and material

characterization has led to applications ranging from cutting tool

bits, dies, and seals to dental restorations, bone implants, and to

a wide variety of other industrial, biomedical, and engineering

uses. Accommodating this variety of applications are numerous

measurement procedures. While the definitions of these

procedures are relatively straightforward, their applications

have several features of interest to data evaluation.

In the subclass of hardness measurements known as indentation

tests, an object, called an indenter, is pressed into a locally flat,

polished surface of a specimen. The load applied to the indenter is

chosen to be sufficiently large that the outline or impression of the

indenter remains in the surface of the specimen when the indenter

is removed. The ratio, F/A, of the applied force or load, F, and a

measure of the area of the residual impression, A, provides the

quantitative value of the hardness.

The simplicity of this definition is not without its pitfalls. The load

must be sufficiently large to produce an impression. When the

load is sufficiently small, the interaction of the indenter and the

specimen may be entirely elastic, and no impression will be formed.

An impression will be formed only when the applied load is sufficient

to cause plastic deformation in the specimen. The plastic response of

the specimen, however, will depend on the shape of the indenter and

the brittleness of the surface material. Indenters of different shapes

applied in otherwise identical manners result in different hardness

values because the force vectors, findent, resolve differently with

respect to the normal to the surface. Consequently, the work done

by the indenter, Windent, is different for indenters of different shapes.

(42)

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The brittleness of the material affects the measured value because

the stress intensities differ along the points of application of the

indenter. If the stress intensity is sufficient to generate cracks (at an

apex of the indentation, for example), then the irreversible work done

by the application of the indenter is divided between the formation

of the impression and the generation of cracks away from the

impression.

(43)

Further, different measures of the area of the impression (actual

versus projected surface areas, for example) clearly also affect the

numeric hardness values.

A more subtle concern, inherent to all procedural properties, can be

expressed as follows in the case of hardness. If a load F produces

an area A, does a load 2F produce an area 2A? If the answer is

“no,” then the quantitative definition of hardness will be useful only

with additional constraints or with additional understanding of the

relation between A and F. Such is the case, for example, in the

results shown in Figure 6.10 for a silicon nitride material.[44] This

situation, often called the indentation size effect, occurs for hardness

because the procedural definition involves both an independent

variable, F, and a response variable, A, which is itself a function of F.

In early studies of indentations,[45] the indentation size was often

found to be related to the applied load by a power law, often called

the Meyer Law, such that

(44)

where d is a linear dimension of the impression, d0 is a fixed scale

factor defining the unit of length, and ζ and η are fitting parameters.
75

For ceramic materials, the exponent is usually greater than one

(η > 1); hence, to double the linear dimension of the impression would

require more than double the load. With the latter observation, it may

be anticipated that the value of the Meyer exponent may depend on

the microstructure of the material, particularly the porosity. Less

force should be required to produce an impression of a given size

when the material has a greater porosity. This effect is seen in

Figure 6.10 where the hardness is smaller when the density is lower.

Consequently, the value of η should be smaller when the porosity

is larger. Exactly this result has been observed, for example, in a

study[46] combining the results for numerous specimens of silicon

nitride, Figure 6.11.

Figure 6.10. The indentation size effect for a polycrystalline hot isostatically pressed

silicon nitride.[44]

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Rather than allowing a fitting parameter to become a function of

porosity, it may be advantageous to preserve the Meyer relation and

to account explicitly for the dependence on the independent variables

density, grain size, and temperature. If the dependence on the

independent variables is amenable to the separation of variables,

then a useful expression consistent with the Meyer law might be[27]

(45)

where H0, To, τ, h1, h2, and h3 are adjustable parameters, and ρ0,

g0, and F0 are fixed scale factors.

The utility of this model can be illustrated by its application to a study

on the hardness of silicon nitride.[44] This particular study reported

results for silicon nitride produced using four different starting

powders, three different sintering aids, and several different

Figure 6.11. The dependence of the Meyer exponent on specimen porosity for a

collection of seventeen specimens of single crystal and polycrystalline silicon nitride.[46]
77

processing and post-processing conditions. The specimens were

viewed as two groups, one group with fractional density ρ/ρo . 0.85

and one group with ρ/ρo / 0.97. One exception to this grouping

occurred for a specimen with ρ/ρo = 0.950 which was prepared

from a starting powder having a much larger grain size than the

other starting powders. The study succeeded in showing, as in

Figure 6.12, that the specimens in the group with higher fractional

density generally had higher hardness values than the specimens

in the group with lower fractional density.

Closer examination of Figure 6.12, however, raises a further issue

worthy of discussion, in addition to the issue of grain size. Within

each group, the quantitative differences between results for a given

load are larger and more systematic than would be attributed to

measurement uncertainty. Consequently, it might be expected

that the ranking within each group would be similar to the ranking

between groups; i.e., specimens with higher fractional density should

have higher hardness. In this respect, the results do not, at first,

appear to be quite sensible.

Figure 6.12. The Vickers hardness of polycrystalline hot isostatically pressed

silicon nitride varies with both indentation load and specimen density.[44]

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With reference to Equation (45), we see that there are two

considerations yet to be applied. First, Equation (45) assumes that

the specimens are of the same composition. There is no reason,

a priori, to expect specimens of different composition to behave

quantitatively alike. Likewise, because hardness is a measure of

surface properties, post-processing oxidation of the surface can be

expected to alter the quantitative results for the same specimen.

Additionally, Equation (45) indicates that variation of hardness with

grain size is to be expected.

While no quantitative grain size information was available in this

study, the observation that the specimen with ρ/ρo = 0.950 had a

larger mean grain size allows this result to be compared qualitatively

with other studies. Given that the hardness for this particular case

of higher density was comparable to the results for the lower

density group, the implication is that the hardness may decrease

with increasing grain size. This qualitative behavior has been

noted for other high-hardness ceramics such as titanium diboride.[27]

Consequently, this case for silicon nitride is at least consistent with

results for other materials.

More convincingly, we can separate the remaining materials into

subsets processed from the same starting powders and having the

same mass fraction compositions. For each subset, the initial grain

size distribution is, by design, a constant across all the specimens

in the subset. The additional assumption required here is that the

subsequent grain-size distributions, resulting from the hot isostatic

pressing cycles that produced the different degrees of densification,

do not differ greatly or significantly among the final specimens.

With that assumption, there remain two sets of results that can

be examined in the context of Equation (45). One set consists of

undoped specimens, and the other has specimens that were doped

to have mass fraction compositions with 2 % Y2O3 and 1 % Al2O3.

Application of Equation (45) to each set separately, with the

assumption of constant grain size, produced the smooth curves in

Figure 6.13. In each case, ρo was taken to be the theoretical density

of the specified composition, accounting for the presence or absence

of sintering aids. As can be seen in Figure 6.13, the results of this

data evaluation exercise do, indeed, permit the rankings of the data

to be understood both qualitatively and quantitatively.
79

6.3.2.5 Response Dependent Properties

{Example: Fracture toughness, Palmqvist cracks vs. median

cracks}

Procedural properties may be subject to limitations that arise

implicitly when a designated measurement procedure anticipates

a particular response from the material, i.e., the procedure itself

becomes dependent on the material’s response to the procedure.

This unusual situation occurs, for example, in measurements that

require the generation of a crack using an indentation technique.

Most materials can sustain the presence of a small number of minor

cracks without loss of functionality. However, fracture failure may

occur, particularly in brittle materials, when the extension of a crack

in a solid body becomes unstable. Consequently, several procedures

have been developed to measure the resistance of a material to

Figure 6.13. Reduction and analysis of the data from Figure 6.12 in accordance

with Equation (45).

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Figure 6.14. Schematics of median and Palmqvist crack formations. Side views:

(a) median, (b) Palmqvist. Top views, showing indentations as indented (upper)

and after polishing (lower): (c) median, and (d) Palmqvist.

crack extension. The material property that assesses this resistance

is called fracture toughness. In principle, the work required to extend

a crack should be determined by the energy required to create the

necessary new surface area and, therefore, fracture toughness

should be a well-defined material property. In practice, however,

procedural definitions more amenable to direct application in

experiments have been adopted. As a result, attention must be

given to the method used in any estimate of toughness.[47, 48]

To determine fracture toughness using an indentation technique, the

indenter must be pressed into the material with a load sufficient to

generate cracks beyond, and away from, the remnant impression of

the indenter. Qualitatively, shorter cracks for a given load imply a
81

greater toughness. However, to quantify this measure of fracture

toughness, it is not sufficient to measure the size of the crack. It is

also necessary to determine the type of crack that has been produced

by the indentation.

There are principally two types of cracks, illustrated schematically

in Figure 6.14, that need to be considered. The more common type

of crack, known as a median crack, is shown in Figure 6.14(a).

Median cracks form beneath the indenter along the diagonals of

the impression and extend into the bulk of the material. A rather

different crack formation is shown in Figure 6.14(b). These cracks

extend only outward from the corners of the impression and are

known as Palmqvist cracks. In both cases, a top view of the

indented surface appears to indicate a crack extending from each

apex of the impression. However, the two types of cracks can be

distinguished experimentally by polishing the surface after indentation.

After polishing, median cracks, Figure 6.14(c), are reduced in size

but remain attached to the impression. In contrast, the Palmqvist

cracks after polishing appear to have separated from the impression,

Figure 6.14(d).

If the indentation study is done as a function of load, then the two

types of cracks can be distinguished also according to the load

dependence. For Palmqvist cracks, the crack extension length, cx,

is proportional to the load,

(46)

while for median cracks, the crack size, c0 = cx + δ/2, is proportional

to the two-thirds power of the load,

(47)

such that the slope of log(cx) vs. log(F) should be 1 for Palmqvist

cracks, while the slope of log(c0) vs. log(F) should be 2/3 for median

cracks.

Only when the type of crack has been determined can the fracture

toughness be computed from the appropriate expressions for the

critical stress intensity factor, KIc.

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82

(48)

for median cracks,[49] and

(49)

for cracks of the Palmqvist type.[50] The numeric factors, i and H,

have been determined empirically by averaging fitted values for a

wide range of ceramics. The means and standard deviations were

found to be 0.016 ± 0.004 for median cracks[49] and 0.018 ± 0.004

for Palmqvist cracks.[50]

When discrepancies arise among fracture toughness values

determined by indentation tests, use of an inappropriate expression

should be high on the list of suspected causes. A useful preliminary

guide to the appropriate expression is provided by the ratio of the

crack size and the indentation diagonal. It has been observed[50]

for a variety of materials that the expression for Palmqvist cracks is

often appropriate when 2c0/δ < 3. Using the wrong expression for

a given type of crack can result in an estimate of the fracture

toughness that is incorrect by as much as a factor of 2.[44]

6.3.2.6 System Dependent Data

{Example: Friction and wear}

In most considerations of materials properties, the measured

characteristics are regarded as pertaining exclusively to the material.

However, when different material bodies are allowed to interact, it is

possible to define characteristics or properties that are meaningful

only with respect to the assembly of interacting materials. The

coefficient of friction of one material sliding against another is a
83

classic example of such a property.[51, 52] The coefficient of

friction is meaningful only when there is a complete specification

of the interacting materials, the contact environment, the topological

characteristics of the interacting surfaces, and the contact conditions,

such as temperature, sliding speed, and contact stress. Properties

of this nature are said to be system dependent.

All tribological properties are system dependent. Consequently,

much effort has been made in the field of tribology to provide useful

representations of the wear and friction behaviors of materials, and

we may gain from that effort an indication of the considerations

important to the evaluation of system dependent properties.

A primary objective in the study of system dependent properties is

to identify what influence the intrinsic material properties have on

the system dependent interactions. In this context, the focus of the

analysis is often on the development of normalized measures of the

material responses. The dimensionless wear coefficient, KW, is a

case in point.[53, 54]

(50)

where VW is the wear volume, H is the hardness, Fn is the normal

force acting between the surfaces, and Ds is the total sliding distance.

This quantity is prone to having a very large measurement uncertainty

because the total wear volume may be influenced by the topological

changes at the wear contact as the test proceeds, including the shape

of the contact, the stress distribution, the temperature gradient, and

the presence of accumulated wear debris. Additional uncertainty

arises from the value of the hardness, a procedural variable, whose

value may depend on the stress level at the contact and the extent

and nature of microcracking around the contact (see section 6.3.2.4

Procedural Properties).

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In spite of this large numeric uncertainty, the order of magnitude

of the wear coefficient has proven to be useful in classifying the

wear behavior of materials into wear regimes distinguishing between

conditions of mild and severe wear. When this measure of wear is

coupled to the measurement of the coefficient of friction, it becomes

possible to construct useful wear mechanism maps[55, 56] or wear

transition diagrams,[57] Figure 6.15, as a function of the operating

conditions denoted by the contact temperature and stress. Such

Figure 6.15. A wear transition diagram illustrating the variation of the dominant

wear mechanism depending on the tribological contact conditions.[57]
85

diagrams then can serve as a guide to the specification of acceptable

operating conditions in tribological contacts and can help to avoid

conditions of potentially excessive wear that may shorten the useful

lifetime of the material components. In that respect, a potentially

useful correlation between the coefficient of friction and the ratio

of sliding speed to contact stress has been noted in the case of

TiB2,

[58] Figure 6.16. A power law trend is clearly perceptible along

each isotherm, while the data as a whole appear to separate into two

temperature dependent clusters. The latter observation suggests

the onset of a distinguishable tribological mechanism at elevated

temperature, perhaps through the formation of B2O3 in the wear

track.

Figure 6.16. Correlation of the coefficient of friction and the ratio of sliding speed

to contact stress in the case of TiB2.

[58]

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6.4 Stage IV: Modeling (Issue: Predictability)

A set of property data for a given material may be said to form a

representation of the behavior of that material. Numeric representation,

however, does not constitute conceptual understanding, and it is the

attainment of such understanding that forms the fundamental essence of

materials science. An expression of a conceptual understanding of an

observed material behavior is generally called a model. Models that may

be expressed mathematically are especially pertinent to data evaluation

because they form the basis for establishing or confirming the consistency

of independently measured property values, the adherence to common

trends, and the identification of exceptional behavior.

6.4.1 Derived Correlations

{Example: Toughness vs. strength}

Most theories of material behavior focus on one particular aspect

of an observed phenomenon and arrive at a relationship between

that observation and one or more other characteristics describing or

pertaining to the phenomenon. Such relations subsequently may form

the basis of correlations among more readily measurable quantities or

may assist in the interpretation of empirically observed correlations.

Linear fracture mechanics, for example, has a primary focus on the

development and propagation of cracks in a brittle material.[59–63]

An ideally brittle material is one that fractures as a result of the sudden,

catastrophic propagation of a crack. Clearly, the factors controlling the

propagation of the crack must include the stress at the tip of the crack

and the strengths of the bonds that must be broken to extend the crack.

The theory of linear fracture mechanics yields a general relationship

among three factors: the critical stress intensity factor (commonly

known as the fracture toughness), KIc, the fracture strength of the

material, σf

, and the size of the crack (or, more generally, the critical

flaw), co, such that

(51)

where Y is a dimensionless numeric factor that depends on the

location and shape of the flaw. This relation plays a central role in

understanding the fracture behavior of brittle materials. However,

there is a subtlety that should be considered when this relation is applied.
87

Empirically, it has been observed that a plot of fracture strength

against toughness often can be fit by a straight line, from which it is

concluded that the flaw size is constant. This conclusion generally is

not correct.[64] Properties such as fracture toughness and strength

are not among the fundamental, independent variables of the material

system. Rather, they are consequences of the interactions of the

constituent particles of the material system, and, as such, they are

themselves functions of other variables such as composition, bond

strength, and microstructure.

Figure 6.17. The correlation of fracture toughness and fracture strength may differ

under the conditions of constant grain size and constant density.[64] [Data: See

Additional Source References, Figure 6.17.]

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Formally, we may assume the existence of a complete set of

independent variables, {xi

}, which does not include KIc, σf, or co. If we

then consider the variation of KIc, σf, and co with respect to the set of

independent variables, it can be shown that for each xk in the set {xi

},

(52)

in which the subscript, [x ≠ xk], indicates that all independent variables

except xk are held constant. The significance of this derived relation

is twofold. First, the variation of toughness with strength depends

explicitly on the variation of the flaw size. Second, the variation

of toughness with strength depends on which independent variable

produces the variation in the properties. Thus, for example, the

variations under the conditions of constant grain size and constant

density are not necessarily the same, as seen in Figure 6.17. Further,

if we impose the empirical observation that the plot of KIc varies

linearly with σf, then we find explicitly that the flaw size is not constant,

but rather varies as

(53)

where Λ1 and Λ2 are constants. Consequently, when evaluating data

relating fracture toughness and strength, it is essential to distinguish

the conditions under which the data were obtained and to be wary of

conclusions dependent on constant flaw sizes.

6.4.2 Harmonic Theories

{Example: Debye theory of specific heat}

Harmonic theories are well developed in the literature and are treated

thoroughly in many text books that are widely available.[65, 66, 67]

As such, the reader is likely to be well versed in this subject or will

have ready access to thorough discussions of it. However, there are a

few points that have a special relevance to the evaluation of materials

property data, and an effort is made here to emphasize those points,

but only with as much discussion as is needed to preserve the logical

continuity of the topic.
89

In solid materials, the constituent atoms interact sufficiently strongly

that their positions are relatively well localized within the volume of

the material. Within crystals or the crystallite grains of polycrystalline

materials, the atoms are localized to well-defined positions in periodic

arrays. When an external static stress is applied to the bulk material,

the resulting strain is manifest by the displacements of the atoms by

a small amount from their unstressed equilibrium positions. Upon the

release of the external stress, the atoms return to their normal positions.

In a similar manner, the addition of thermal energy to the body is

manifest by the displacement of the atoms in vibrations of relatively

small amplitude about their equilibrium positions. When energy, whether

thermal or mechanical, is transported through the body, the propagation

of the energy is accomplished by the collective modes of these vibrations

called phonons. The simplified picture of the solid material that evolves

from these microscopic descriptions of the static and dynamic behaviors

of the material is that of a collection of mass points interconnected by

springs obeying an effective Hooke’s law, i.e., a collection of classical

harmonic oscillators. It should not be surprising, then, that numerous

theories of material properties have been based on various aspects

of this schematic picture, more generally called the harmonic

approximation.

For the evaluation of material property data, the most important

application of the harmonic approximation is probably the Debye theory

of specific heat. Einstein had established the utility of the harmonic

approximation[68] by considering a collection of independent oscillators,

all having the same frequency of oscillation. The Einstein model rather

successfully accounted for the temperature dependence of the specific

heat of relatively simple material compositions, such as copper and

diamond, except at relatively low temperature. The difficulty with the

Einstein model was that it considered only one mode of vibration and,

therefore, did not distinguish between the case at high temperature, in

which very high frequencies of vibration might occur, and the case at

low temperature, in which only very low frequencies of vibration would

be likely to occur.

Debye resolved this difficulty[69] by allowing all frequencies from

zero up to an accessible maximum cutoff value to occur. To treat this

frequency range, Debye assumed that the allowed vibrational modes

were acoustic with a constant phase velocity; i.e., the dispersion

relation was analogous to that of light waves, [speed of sound] =

[wavelength] × [frequency]. The cutoff frequency then was determined

by requiring that the total number of allowed vibration modes was 3N,

Issues and Practices in Data Evaluation ◆
the number of independent vibrational modes of N point particles

(i.e., particles lacking internal degrees of freedom) in a space of three

dimensions. As fully derived in many introductory texts on statistical

mechanics and solid materials, the resulting expression for the molar

heat capacity in the Debye model, at constant volume, was found to be

(54)

(55)

where Θ is a parameter known as the Debye temperature defined in

terms of the Debye cutoff frequency, ωD, such that kBΘ = SωD, and

S is Planck’s constant divided by 2π. The theoretical estimate of the

Debye temperature for a specimen of density ρ and molar mass MW

was found to be

(56)

where vm is the sound velocity. The dispersion relation of the

original Debye model implies that the material is elastically isotropic.

In practice, the velocities of the longitudinal and shear modes differ.

To allow for this difference, vm is taken as the mean sound velocity

calculated as

(57)

where the terms on the right side of the equation represent the one

longitudinal mode and the two shear (a.k.a., transverse) modes.

At high temperature, Θ/T tends towards zero, and the Debye function

easily is found to have the limit f

D(0) = 1. Consequently, the Debye

molar heat capacity at high temperature is the classical Dulong and

Petit limit, 3NAkB = 3R for a monatomic species. (For polyatomic

species, see the discussion in the sidebar.)

Of particular interest to the evaluation of material property data is

the limit at low temperature.

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Specific Heats of Composites and Compounds

A composite material consists of multiple, but separable, phases.

The total mass of the composite, Mc, is the sum of the masses of the

constituent phases,

To raise the equilibrium temperature of the composite by 1 K, the

temperature of each constituent phase must be increased by 1 K.

Consequently, conservation of energy requires

from which follows simply

where μi

 = mi /Mc is the mass fraction of phase number i in the

composite. It should be noted that a principal condition of this

derivation is that the phases are separable. Interactions between the

separable phases are assumed to contribute negligibly to the internal

energy of the composite and, hence, may be ignored in the energy

conservation relation. The same situation is approximately true for

compounds, although the approximation is somewhat less reliable for

compounds than for composites.

A compound is a single phase material consisting of multiple atomic

species. The interactions between species are not negligible and may

not be ignored. Consider, for example, the special case of a binary

compound in which the structure of the material is conceptually separable

into two interpenetrating sublattices of one pure species each. The

internal energy of the compound, Uc, would consist of three terms,

corresponding to the internal energies associated with the two sublattice

components, considered separately, and the energy from the interactions

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92

(58)

The proportionality to T 3 in the low temperature limit often is used as

a means of evaluating the Debye temperature and may be compared

for consistency with the theoretical estimate given by Equation (56).

For pure elements such as Al or Cu, the estimates based on the elastic

measurements, Equation (56), and the heat capacity measurements,

Equation (58), tend to be in good agreement.

For more complex materials, the more detailed analysis of lattice

dynamics[70] indicates that the Debye approximation does not

adequately represent the frequency dependent dispersion relation

and, hence, that the elastic and calorimetric estimates of the Debye

temperature may differ appreciably.

Specific Heats of Composites and Compounds (continued)

between the two sublattices. If the interaction is favorable to the

formation of a compound, then it would be expected that U1,2 would

reduce the total internal energy. Consequently, simple additivity of the

atomic specific heats would be expected to yield an overestimate of the

specific heat of the compound. For example, SiC at room temperature

has a molar heat capacity of approximately 26.9 J/mol • K, while the

additive contribution of the constituents would yield 28.5 J/mol • K.

In this case, the additive approximation is approximately 6% higher

than the measured value. In a survey of 75 compounds, the ratio of

the composed value to the measured value ranged from 0.9 to 1.6

with the average value being 1.15 with a standard deviation of 0.14.

In a similar manner, it is often assumed that the vibrations of the

atoms of a compound at sufficiently high temperature may be treated

as independent and uncorrelated oscillations, such that the extended

law of Dulong and Petit (sometimes known as the Kopp–Neumann rule

for compounds) would be expected to be obeyed approximately by

compounds. Thus, a compound of the form ApBq consisting of p moles

of A and q moles of B would be expected to have a high temperature

limit of approximately p • 3R + q • 3R = (p+q) • 3R. For Al2O3 at 1500 °C,

Cp(Al2O3) = 1330 J kg–1 K–1 yielding Cp

(M)(Al2O3)/(2+3) ≈ 3.3R, in fair

agreement with this expectation.
93

A further consequence of the approximate nature of the Debye model

is the observed temperature dependence of Θ. In the Debye model,

Θ(T) = constant. In practice, however, Θ usually has a minimum at low

temperature and rises approximately asymptotically to a constant value

at elevated temperature.

6.4.3 Heuristic Theories

{Example: Weibull distribution}

Materials are sufficiently complex bodies that the derivation of an

appropriate theory from fundamental principles often is untenable for

realistic material compositions. Conversely, simplifying assumptions

applied to make a model mathematically tractable often result in

material constraints that are unrealistic. A pragmatic alternative to

these two extremes may be possible when experimental observations

can be transcribed directly into a descriptive model of the material’s

behavior. Such descriptive modeling fulfills an intermediate role

between fundamental theories and empirical correlations. The Weibull

distribution of the fracture strength of a brittle material is, perhaps, one

of the most successful and enduring examples of this heuristic approach

to modeling.[71, 72]

An intrinsic characteristic of all polycrystalline materials is the presence

of microstructural inhomogeneities, commonly called flaws, that are

distributed throughout the material.[73–75] The fracture of a brittle

material inevitably originates at one of those flaws within the stressed

volume of the material, usually at the flaw that is most susceptible to

stress intensification. Weibull transformed this experimental observation

into a weakest link statistical model of fracture.

In this model, if any one flaw fails, the whole specimen fails. Suppose

the flaws in the specimen are numbered from 1 to Nf

, and let -k(σ) be

the probability that flaw number k fails at stress σ. Taking -(σ) to be

the probability that the whole specimen fails, then the probability that

the whole specimen survives, 1 – -(σ), is

(59)

which may be written more conveniently as

(60)

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With these relations, it is possible to define a mean population fracture

strength, σ, by noting[76] that the number of specimens that fail at

stress σ is the difference between the number of specimens surviving

at stress σ–δσ/2 and the number of specimens still surviving at stress

σ+δσ/2. Thus,

(61)

Expanding this expression to first order in δσ, the summation can be

converted to an integral yielding,

(62)

where -N(σ) = d-/dσ is the first derivative of -(σ) with respect to σ.

While this model can be developed further in considerable complexity,

we can obtain a useful result for the purposes of data evaluation by

considering a rather simple case. In particular, consider the case of

uniformly distributed, unimodal flaws in a specimen bearing a uniform

stress. Because the distribution is unimodal, we may assume that a

single function of the local stress governs the survival probability at

each site, which we may write as ln(1 – -k) = Φ(σk). Further,

because the stress is uniform, Φ(σk) = Φ(σ) for all k. Consequently,

(63)

If we now assert the two-parameter Weibull model, for which

Φ(σ) = –(σ/σo)m, where m is called the Weibull modulus and σo is

the Weibull characteristic strength, we can evaluate the integral for

the mean fracture stress.

(64)
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(65)

where Q(m) is a slowly varying function of the Weibull modulus;

Q(m=2) = π1/2/2 ≈ 0.89, and Q→1 as m→∞.

This result shows quite reasonably, and in complete accord with

experimental observation, that the mean strength decreases as the

number of flaws increases. Suppose further that we want to compare

the mean fracture strengths from two sets of measurements that are

identical except for the size of the stressed volume. Because the

number of flaws in each specimen is proportional to the volume of

the specimen, Nf (V) ~ V, it follows that

(66)

The practical significance of this result for data evaluation is

that fracture strength is a procedural property. When apparent

discrepancies occur among reported strength values, an assessment

of the differences requires a comparison of the measurement details.

Two determinations of the fracture strength of a given material on a

given apparatus may be quantitatively different, for example, if the

sizes of the specimens are different. Similarly, two determinations may

yield different results if two different techniques are used. For example,

even when identical specimens are used, the measured fracture strength

will be larger when determined by three-point bend tests, Figure 6.18(a),

than when determined by four-point bend tests, Figure 6.18(b), because

the four-point test stresses a larger volume.
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96

Figure 6.18. Schematics of two flexural tests: (a) three-point bend; (b) four-point

bend.
97

6.4.4 Dominant Mechanisms

{Example: Porosity dependence of bulk moduli}

Within the complexity of material behavior, it may happen that there

is a dominant mechanism that accounts for the major features of an

observed trend. When this situation occurs, it may be possible to focus

very crudely on one single aspect of the material composition to obtain

an analytical description of the principal trend. While myriad subtleties

may be lost in such descriptions, a more insightful understanding of the

physical behavior may be gained. Additionally, the resulting analytical

expressions may be directly useful for interpolation and extrapolation

purposes and may provide a basis for understanding other empirical

correlations. An example possessing these virtues is the effective

medium theory of the porosity dependence of bulk moduli.[77]

Porosity refers to the presence of holes or gaps in a material body.

In polycrystalline materials, porosity can result from the stacking of

misaligned grains or from vacancies and inclusions within grains.

The local disruption of material contiguity associated with these porous

features may diminish the capacity of the material to transmit forces or

to diffuse energy through its microstructural components. Consequently,

the elastic moduli tend to decrease with increasing porosity. Given the

nonuniformity of the distribution of pores and the irregularity of pore

shapes, it is readily understandable that detailed mathematical models

of the effects of pores in a material might be highly intractable.

However, we can gain some useful insight into the porosity

dependence by means of an effective medium theory.

There are three principal aspects to an effective medium theory.

First, there is a reference system, usually of an idealized nature, in

which all the relevant mathematics can be performed. Second, there is

a renormalization of a metric (a length scale in the present case) which

reflects the transformation of the ideal system into the nonideal system.

Third, there is a consistency condition that ensures that the effective

medium conforms to a known characteristic of the nonideal system.

The classical model of an ionic solid provides the essential reference

system for the present purpose. The ionic model consists of a simple

crystal lattice in which each lattice point is occupied by an ion. The ions

have pairwise interactions described by a potential energy function with

both attractive and repulsive terms,

(67)

Issues and Practices in Data Evaluation ◆
◆ Data Evaluation Theory and Practice

98

where rij is the distance between ion number i and ion number j.

For the ideal lattice, the sum over all interacting pairs can be performed,

and the result can be expressed in terms of a characteristic volume.

(68)

From the equilibrium condition (∂Φ/∂V = 0 at V=Veq) and the definition

of the bulk modulus (B = –V(∂2Φ/∂V2), an expression for the bulk

modulus can be obtained in closed form.

(69)

To introduce porosity into this ideal system, we note that porosity

causes the mean molar volume to increase and, hence, that mean

interparticle distances must be increased. To permit this effective

scaling, we renormalize the distance scale by a factor χ. The

renormalized bulk modulus then can be written as

(70)

where Bo and Vo are the values of the bulk modulus and the volume,

respectively, at zero porosity. For the porous system, if we now impose

the consistency condition that Veq = Vo + Vp where Vp is the additional

volume due to pores, we obtain

(71)

and

(72)

where φ = Vp/(Vo + Vp) is the volume fraction of porosity. The efficacy

of this result, illustrated in Figure 6.19, enables data evaluation to assess

the consistency of bulk moduli determined for specimens of differing

porosities or densities and to resolve apparent discrepancies among

reported values.
99

Figure 6.19. An illustration of the porosity dependence of the bulk modulus

of polycrystalline alumina. The dashed curve is the fit of Equation (72).[77]

[Data: See Additional Source References, Figure 6.19.]

Issues and Practices in Data Evaluation ◆

6.4.5 Bounded Values

{Example: Voigt and Reuss bounds for the elastic moduli of

composites}

Composite materials often present a special challenge to data

evaluation. Being composed of multiple phases, comparable studies

of similar materials often are difficult to find and possibly do not exist.

It is often the case, however, that the relevant property values are

known independently for the individual constituent phases. In this
◆ Data Evaluation Theory and Practice

100

case, it clearly would be desirable to have a reasonable estimate of the

composite property based on the corresponding values for the individual

constituents. The challenge is to determine what constitutes a

reasonable estimate.

6.4.5.1 Voigt and Reuss Bounds

We have seen one example of this situation already, the estimate of

the specific heat of a composite material (sidebar to section 6.4.2).

For the specific heat, a weighted average of the specific heats of the

individual components, weighted according to the mass fractions of

the components in the composite, was found to be an appropriate

estimate based on the principle of the conservation of energy. For

other properties, however, there may be some ambiguity in deciding

what criteria are appropriate. For example, there is more than one

reasonable estimate that can be made for the elastic modulus of a

composite material.

Figure 6.20. Schematics for estimating the effective elastic modulus of a simple

composite material consisting of two elastic components: (a) Reuss approximation;

(b) Voigt approximation.
101

We can see how two such estimates arise by considering two

fairly simple, but distinct, examples. In Figure 6.20(a), a composite

material, consisting of two components connected in series, is

subjected to a stress. The strain resulting from the uniform stress

of the composite is easily found to be

(73)

where 01 and 02 are the strains of the individual components.

Recalling that 0= σ/E, it follows readily that the effective modulus,

ER is given by

(74)

where Éi

 is the volume fraction of component number i.

This estimate using uniform stress is known as the Reuss

approximation.[78]

In Figure 6.20(b), the same composite is subjected to a

stress applied parallel to the interface of the two components

and resulting in a condition of uniform strain: 01 = 02 = 0.

Mechanical equilibrium requires σ1A1 + σ2A2 = σA, where Ai

is the surface area of component i and A is the total area. Using

σ = 0E and converting to volume fractions, the effective modulus is

(75)

This estimate using uniform strain is known as the Voigt

approximation.[79]

In a polycrystalline material with randomly oriented grains, it

would be expected that the effective modulus would have a value

intermediate to these two bounds, such that ER # E # EV. These

bounds on the elastic modulus have proven to be sufficiently useful

that the arithmetic average of the two values, known then as

the Voigt–Reuss–Hill average,[80] often is taken as a reasonable

estimate of the elastic modulus of a composite material.

Issues and Practices in Data Evaluation ◆
6.4.5.2 Characteristic Ranges

The full domain of solid materials is commonly subdivided into

classes such as ceramics, glasses, metals, polymers, and composites.

The rationale for this classification is that there exists recognizable

similarities among materials of the same class and clear distinctions

among the principal characteristics of different classes. It may be

expected that the similarities and differences should be manifested

quantitatively in the ranges of values observed for the respective

material properties. Given that the observed properties are

fundamentally consequences of interatomic interactions, many

of the properties for a given material should exhibit correlations.

It follows that it may be possible to form dimensionless products

and ratios of properties that characterize those correlations for

Figure 6.21. An illustration of characteristic ranges, using values of the

dimensionless product of thermal expansion and melting temperature, viewed as

a function of material class.[81]

102

◆ Data Evaluation Theory and Practice
each entire class of material. The distinctions between classes

would then occur as quantitatively different bounds on the values

of the normalized correlations. An example[81] of such bounds

and their distinctions between material classes is shown in

Figure 6.21 for the case in which a dimensionless quantity is formed

by the product of the coefficient of linear expansion and the melting

point temperature. In practice, numerous correlations may be

developed,[81] and the bounds on their values may be established

empirically through a statistical analysis[82] of the data compiled for

each material class. These bounds may then be used to provide a

useful validation of the order of magnitude of new results reported

subsequently.

Issues and Practices in Data Evaluation ◆

103
◆ Data Evaluation Theory and Practice

104

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Issues and Practices in Data Evaluation ◆
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Additional Source References for Data Used in the Figures

Figure 6.3, Thermal Conductivity and Diffusivity

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Figure 6.4, Critical Temperature

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Benischke, R., T. Weber, W.H. Fietz, J. Metzger, K. Grube, T. Wolf, and

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Kraut, O., C. Meingast, G. Bräuchle, H. Claus, A. Erb, G. Müller-Vogt, and

H. Wühl. “Uniaxial pressure dependence of Tc of untwinned YBa2Cu3Ox

single crystals for x = 6.5 – 7.” Physica C, 205, 139–146 (1993).

Figure 6.5, Lattice Parameters

François, M., A. Junod, K. Yvon, A.W. Hewat, J.J. Capponi, P. Strobel,

M. Marezio, and P. Fischer. Sol. St. Commun., 66, 1117–1125 (1988).

Parks, M.E., A. Navrotsky, K. Mocala, E. Takayama-Muromachi, A. Jacobson,

and P.K. Davies. “Direct calorimetric determination of energetics of oxygen

in YBa2Cu3Ox.” J. Sol. St. Chem., 79, 53–62 (1989).

Subba Rao, G.V., and U.V. Varadaraju. “Structure and superconductivity

in Y-123 and related compounds.” Chemistry of High Temperature

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Triscone, G., J.-Y. Genoud, T. Graf, A. Junod, and J. Muller. “Normal-state

susceptibility versus oxygen content in the YBa2Cu3Ox and Y2Ba4Cu7Oy

superconducting phases.” J. Alloys Comp., 195, 351–354 (1993).

Figure 6.7, Creep Data

Mandorf, V., J. Hartwig, and E.J. Seldin. “High Temperature Properties

of Titanium Diboride.” High Temperature Materials II, pp. 455–467.

Ed. G.M. Ault, W.F. Barclay, and H.P. Munger. New York: Gordon and

Breach (1963).

Figure 6.17, Toughness vs. Strength Data

Mah, T., and K.S. Mazdiyasni. “Mechanical Properties of Mullite.”

J. Am. Ceram. Soc., 66, 699–703 (1983).

Kanzaki, S., H. Tabata, T. Kumazawa, and S. Ohta. “Sintering and

Mechanical Properties of Stoichiometric Mullite.” J. Am. Ceram. Soc.,

68, C-6–C-7 (1985).

Ismail, M., Z. Nakai, and S. Somiya. “Microstructure and Mechanical

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7. Conclusion

Undoubtedly, there is a great deal of work yet to be performed towards

understanding property relations and their intrinsic role in data evaluation.

Some of the relations that have been employed in this work have already

stood a brief test of time. More will be joined to that list in the future,

while others will be refined, replaced, or otherwise disabused in short order.

The principles set forth in this work are amenable to such changes and,

indeed, clearly anticipate them. It is a certainty that every such advance in

our conceptual understanding of properties, procured by empirical observations

and captured in theoretical or heuristic models, will enhance our ability to use

data more wisely. Application of this evolving knowledge in the course of data

evaluation will contribute significantly to our ability to ensure the credibility of

materials property data. In this effort, striving to attain reliability, consistency,

and accountability shall remain the primary goal, the focal responsibility, and

the fundamental testament of data evaluation.
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