Reference

A111D1 777325

NBS

Publications

NBSIR 81-2445

NATL INST. OF STAND & TECH

AlHOt 2 Li 1 E Li

M

plication of Proof Testing to

Brittle Materials At High

Temperatures

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

December 1 981

Final Report

NBS-DOE Interagency Agreement EA-77-A-01 -601 0

Prepared for ~QC

100

,1156

31-2445

1931

apartment of Energy

ssil Fuel Utilization Division

ishington, DC

NATIONAL BUREAU

OF STANDARDS

LIBRARY

MAR 1 5 1982

NBSIR 81-2445

I ! i APPLICATION OF PROOF TESTING TO

BRITTLE MATERIALS AT HIGH

TEMPERATURES

N. J. Tighe and S. M. Wiederhorn

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

December 1 981

Final Report

NBS-DOE Interagency Agreement EA-77-A-01 -601 0

Prepared for Department of Energy

Fossil Fuel Utilization Division

Washington, DC

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
.? n- <":V

'• "-f; <.v.f:jAT( v-

• ‘

. ..
Table of Contents

Preface 1

Proof Testing of Ceramics: I Experiment 2

Proof Testing of Ceramics: II Theory 25

Application of Fracture Mechanics in Assuring

Against Fatigue Failure of Ceramic Components 62

In SiXu. Oxidation of -Doped Si

3

N

4

96

Effect of Strain Rate on the Fracture Toughness

of Silicon Nitride at 1400°C 106

Fracture Strength and Wei bull Distribution of

jB-Sialon 131

A Probabilistic Framework for Structural Design 138

Effects of Exposure on the Reliability of Silicon Nitride 171

Effects of Oxidation on the Surface and Near-Surface

Structure of Silicon Nitride 195

1

Preface

Papers published or submitted for publication during the

contract period of October 1978 to September 1981, are reproduced

in this report. This is the final report on NBS-DOE, EA-77-A01-6010. The principal investigators were Sheldon M. Wiederhorn

and Nancy J. Tighe.
NBSIR 79-1934

PROOF TESTING OF CERAMICS:

I. EXPERIMENT

J.E. Ritter, Jr. and P.B. Oates

Mechanical Engineering Department

University of Massachusetts

Amherst, MA 01003

E.R. Fuller, Jr. and S.M. Wiederhorn

National Bureau of Standards

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

National Bureau of Standards

Published in:

Journal of Materials Science

1!5, 2275 - 2281 (1980)
3

PROOF TESTING OF CERAMICS: I. EXPERIMENT

J.E. Ritter, Jr. and P.B. Oates

Mechanical Engineering Department

University of Massachusetts

Amherst, MA. 01003

E.R. Fuller, Jr. and S.M. Wiederhorn

National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

The effectiveness of proof testing as a method of improving component

reliability was studied by comparing the inert strength distributions of

soda-lime-silicate glass before and after proof testing. The effects of

unloading rate from the proof stress, hold time at the proof stress, and

proof-test environment were examined. The results indicate that the proof

test must be conducted with rapid unloading rates and good environmental

control to be effective. The theoretical implications of these results

are discussed.
4

I. INTRODUCTION

Proof testing is one means used to assure the mechanical reliability

of structural ceramics. In proof testing, ceramic components are subjected

to stresses that are greater than those expected in service in order to

break the weak components and thus truncate the low end of the strength

distribution. In this manner, weak components are eliminated before they

can be placed in service. Proof testing has been applied to spacecraft

12 3

windows, * electrical porcelain insulators,, and vitrified grinding

4

wheel s

.

5-7

Wiederhom, Evans, and Fuller have provided a mathematical foundation for the selection of the proof-test stress and the establishment of

proof-test conditions. Their analysis is based on the assumption that

failure of ceramics occurs from the growth of preexisting flaws. By

characterizing this crack growth and coupling crack growth parameters

with proof testing, they derived the strength after proof testing assuming

flaw growth during the proof-stress, load-unload cycle. The resulting

theory indicates that crack growth must be minimized to have effective

proof testing. This can be achieved by having rapid unloading from the

proof stress and good environmental control during the proof test.

Q

In support of the proof-test theory, Ritter has shown recently for

soda-lime silicate glass after proof testing that the inert strength distribution

and failure time under static stress agree with that predicted from theory.

Aside from this study, there has been no extensive experimental confirmation

of proof- test theory. The purpose of the present study, therefore, was

to conduct a detailed study of the proof- test technique and to assess the
5

validity of the theory in predicting the strength after proof testing.

In particular, the effectiveness of proof testing and the applicability

of proof-test theory was determined by comparing the inert strength

distributions of soda-lime silicate glass before and after proof testing

over a range of proof-test conditions. The major proof test variables

were unloading rate from the proof stress and the proof-test environment.

Soda-lime-silicate glass was chosen as the model material for this study

because it is readily available and because its subcritical crack behavior

is well characterized.

II. EXPERIMENTAL PROCEDURE

All specimens used in this study were soda-lime-silicate glass,

microscope slides* (7.62 x 2.54 x 0.10 cm). The slides were annealed at

500°C for 1 h, furnaced cooled, and then abraded in the center with a

standard blast of No. 240 SiC grit. After abrasion, the samples were

aged for 24 h in distilled water to normalize their strength and then

stored in a desiccator prior to testing. Over 2000 samples were prepared

in this manner and they were randomly selected for testing.

The inert strength measurements before and after proof testing were

made in liquid nitrogen using four point bending. The bending apparatus

had inner and outer supports of 2.54 and 5.08 cm., respectively. All

strength testing was done on a universal testing machinef using a constant

crosshead speed of 0.2 cm/min, corresponding to a stressing rate of 5.29 MPa/s.

The initial inert strength distribution was determined from 79 samples while

the after-proof, strength distributions were determined from about 30 samples.

+Corning No. 2947

•4

* Instron Corp., Canton, MA.
6

The bend apparatus was also used for proof testing. The samples were

loaded up to the proof stress at a stressing rate of 5.29 MPa/s and then

unloaded at various rates (132.33* 5.29* and 1.32 MPa/s). The time at the

proof stress varied from momentary (less than 0.5 s) to 60 s. The proof

test environments were liquid nitrogen* dry nitrogen gas (about 5% Relative

Humidity)* ambient air (55-65% Relative Humidity) and water. The last three

test environments were at room temperature, about 23°C. Groups of 50 samples

were proof tested for each proof-test condition and proof-test stresses were

chosen to break approximately 40% of a set of test specimens, leaving about

30 samples for the determination of the inert strength distribution after

proof testing.

III. RESULTS AND DISCUSSION

5-7

The strength after proof testing has been derived for 3 conditions:

no flaw growth during proof testing, flaw growth up to unloading, and

flaw growth during the entire proof-stress cycle. If no flaw growth occurs

during the proof test* then the inert strength for a given specimen before

and after proof testing are equal r however because the weak samples have

been eliminated from the initial distribution, the cumulative failure probability after proof testing (F a ) will have changed

a

where F is the cumulative failure probability before proof testing and F

p

is the cumulative failure probability of the proof test.

Assuming that a single power law relationship exists between subcritical

crack velocity and the stress intensity factor,

^jlf F is obtained by ordering a set of strength data. F is given by r/(N+l) where

N is the total number of datum points and r is the position of each point in

the ordered set. r = 1 for the lowest strength, r = 2 for the second lowest and

so forth.
7

V = A Kj

n (2)

where A and n are constants for a given material and test environment, the

inert strength after proof testing ( S can be derived accounting for crack

growth during the proof-stress cycle. Considering crack growth up to

5 6

unloading but not during unloading, the S^ distribution is given by: *

where n is the crack propagation parameter appropriate for the proof-test

environment, a

p

is the proof stress, and m and S

Q are the Wei bull shape and

scale parameters, respectively, of the initial inert strength distribution.

If flaw growth occurs during the entire proof stress cycle, the distribution

is now:**

For purposes of identification in the following discussion, the various

predicted distributions will be labelled:

Type I, based on Eq. (1) for no flaw growth.

Type II, based on Eq. (3) for no flaw growth on unloading.

Type III, based on Eq. (4) for flaw growth during entire proof

stress cycle.

For all three types of distributions it can be shown that is greater than

the initial strength at all levels of failure probability, provided m < n - 2.

For Type I and II distributions, the distribution is truncated at a

p

;

hence, a represents the minimum inert strength after proof testing. When

r flaw growth occurs on unloading (Type III distribution), the strength distribution is not truncated and no assurances of a minimum strength can be given.
8

The experimental inert strength distributions after-proof testing could

be divided generally into one of four distributions depending on the proof

test conditions:. First, under inert proof-test conditions (liquid nitrogen)

the strength distributions after proof testing agreed with that predicted

from either a Type I, II, or III distribution (Fig. 1). In this case the

Type II and III distributions were determined by assuming an n = 120.

As can be seen in Fig. 1 there was no significant difference between the three

predicted distributions within the range of the after-proof data. This

observation just reflects, the fact that for an inert environment such as

liquid nitrogen little or no crack growth can- occur because of the high n

value appropriate for this environment.

Second, with rapid unloading rates in the dry nitogen and air environments,

the strength distributions after proof testing agreed with that predicted

from either a Type II or III distribution where n - 18.4 * (Fig. 2). Since

the strength distributions after proof testing are shifted significantly to the

left of the Type I distribution,, it is evident that crack growth occurs during

proof testing in these "moist" environments; however, it could not be

determined if the after-proof strength distributions were truncated at a . r To conclusively show truncation, much larger sample size after proof testing

would have to be used (about 10000 samples).

* Finally, when good proof-test controls were not used, i.e. relatively

slow unloading rates and/or moist proof-test environments, the strength

distributions after proof testing either were not different significantly

from the initial distribution (Fig. 3) or were weaker than the initial

distribution (Fig. 4). In these cases none of the theoretical distributions

could explain the observed distributions after proof testing since the strengths

were much weaker than predicted from theory.
9

Table I summarizes all. the proof- test results. It is evident that

for effective proof testing crack growth must be minimized through use

of good proof test conditions, namely, rapid unloading rates and good

environmental control. In addition, under experimental conditions where

proof testing was not effective,

+ none of the theoretical, after-proof

distributions agreed with the data. There was only one exception to

these trends and that was for the proof test conducted in water at an

unloading rate of 5.29 MPa/s. It is not known why this one set of

results did not fit into the general pattern.

To provide additional support to the general conclusions made above

regarding the effectiveness of proof testing, a large group of asreceived microscope slides were proof tested in very dry nitrogen (less

than 0.1% R.H.) and in moist nitrogen (50% R.H.) using an unloading rate

of about 13.2 MPa/s. Because the dry nitrogen gas was a relatively

inert environment* it was used as the test environment for the strength

measurements before and after proof testing* Figure 5 and 6 shows that

these results are in agreement with the trends shown in Table I. When

good proof-test controls are used, proof testing is effective and the

strength distribution after proof testing can be characterized theoretically (Fig. 5)* On the other hand, when a relative slow unloading

rate is used in a moist environment, proof testing is not effective in

truncating the distribution and the after-proof strength distribution

does not agree well with the theoretical distributions (Fig. 6).

+ The reader is cautioned not to jump to the conclusion that when good proof

test controls are not used, proof testing will not be beneficial. It must be

remembered that our conclusion is based on a group of uniformly abraded

samples that could statistically be characterized by a single strength, i.e.

flaw, distribution. If in a set of samples to be proof tested, some samples

contain gross flaws, perhaps due to incorrect manufacturing, then proof

testing even without ideal conditions, would undoubtedly eliminate these weak

samples from the population, thereby, improving the reliability of the

remaining samples.
10

Based on the above proof test results, the question of why strengths

after-proof testing under non-ideal conditions are significantly weaker

than that predicted must be addressed. The use of Eq. (2) is a

key assumption in deriving the strength after-proof testing. Experiments

have characterized the dependence of crack velocity on stress intensity

factor by three principal regions of behavior (Fig. 7)J®

Regions I and II result from a stress corrosion reaction between the

glass and water in the environment. The rate of the reaction between

water and the glass controls crack motion in Region I, whereas diffusion

of water to the crack tip controls crack motion in Region II, where the

crack velocity is essentially constant. In Region III the stress intensity

factor is close to Kjq and crack velocity does not depend on water in the

environment. Since Eq. (2) represents Region I crack growth, the equations

derived for the strength after-proof testing are based on the assumption

that the strength is controlled exclusively by subcritical crack growth in

Region I. If the behavior shown in figure 7 is important, then Eq. (2) would

overestimate crack velocity when the stress intensity factor of the crack

was in the range that characterizes region II, and the theory would predict

failure when in fact, the samples may just pass the proof test. The samples

that just survive the proof test would then be weak, and the distribution

after the proof test would not be predicted from equation 3 or 4. Therefore,

an analysis that takes into account Region II crack growth is necessary to

account for the observed strength distributions after proof testing under

non-ideal conditions. An analysis of this sort is presented in Part II

of this paper.
ACKNOWLEDGEMENTS

Two of the authors (JER and PBO) are grateful for the support of

the National Science Foundation Grant No. DMR 77-05647 for this research;

the other (E..R. Fuller, Jr. and S. M. Wiederhorn) are greatful for the

support of the Department of Energy, Fossil Fuel Utilization Division.
12

REFERENCES

1. S.M. Wiederhorn, A.G. Evans and D.E. Roberts, "A Fracture Mechanics

Study of the Skylab Windows," pp. 829-41 in Fracture Mechanics

of Ceramics, Vol . 2 , Edited by R.C. Bradt, D.P.H. Hasselman and

F. F. Lange, Plenum Press, New York (1974).

2. S.M. Wiederhorn, A.G. Evans, E.R. Fuller and H. Johnson, "Application

of Fracture Mechanics to Space-Shuttle Windows", J. Am. Ceram. Soc.,

57, 319-23 (1974).

3. A.G. Evans, S.M. Wiederhorn, Melvin Linzer and E.R. Fuller, Jr.,

"Proof Testing of Procelain Insulators and Application of Acoustic

Emission", Am. Ceram. Soc. Bull., 54, 576-81 (1975).

4. J.E. Ritter, Jr. and S.A. Wulf, "An Evaluation of Proof Testing to

ASsure Against Delayed Failure", to be published Am. Ceram. Soc. Bull.

(1978).

5. A.G. Evans and S.M. Wiederhorn, "Proof Testing of Ceramic Materials - An Analytical Basis for Failure Predictions", Int. J. Frac. Mech. , 1_0, 379-92

(1974).

6. A.G. Evans and E.R. Fuller, "Proof Testing - The Effects of Slow

Crack Growth", Mat. Sci . and Engr. , T_9 , 69-77 (1975).

7. S.M. Wiederhorn,. "Reliability, Life Prediction, and Proof Testing of

Ceramics", pp 635-65 in Ceramics for High Performance Applications , Ed. by J. J. Burke, A.E. Gorum and R.N. Katz, Brook Hill Publishing Co.,

Chestnut Hill, MA. (1974).

8. J.E. Ritter, Jr., "Engineering Design and Fatigue Failure of Brittle

Materials", pp 667-686 in Fracture Mechanics of Ceramics, Vol. 4 , Edited by R.C. Bradt, D.P.H. Hasselman, and F.F. Lange, Plenum

Publishing Co., New York (1978).

9. P.B. Oates, "A Detailed Study of the Proof-Test Technique for Assuring

the Mechanical Reliability of Glass", M.S. Thesis in Mechanical

Engineering, University of Massachusetts (1976).

10. S.M. Wiederhorn, "Influence of Water Vapor on Crack Propagation in

Soda-Lime Glass", J. Am. Cera. Soc., 50, 407-14 (1967).
13

Table I. Suninary of the Proof-Test Data for Soda-Lime Silica Glass.

Proof-Test

Environment

Proof-Stress

Unload Rate (MPa/s

)

Hold Time at

Proof Stress (s)

After- Proof Inert

Strength Distribution

Liquid Nitrogen 132.22 < 0.5 I, II, III +

Liquid Nitrogen 5.29 < 0.5 I, II, III

Dry Nitrogen Gas 132.33 < 0.5 II, III

Dry Nitrogen Gas 5.29 < 0.5 Same as Initial

Dry Nitrogen Gas 1.32 < 0.5 Same as Initial

Dry Nitrogen Gas 5.29 60 Same as Initial

Air 132.33 < 0.5 II, III

Air 5.29 < 0.5 Weaker than Initial

Air 1.32 < 0.5 Weaker than Initial

Air 5.29 5 Same as Initial

Water 132.33 < 0.5 Same as Initial

Water 5.29 <0.5 II, nr

Water 5.29 5 Same as Initial

Type theoretical distributions that agreed with after-proof data.
14

FIGURES

Fig. 1. Inert strength distributions of soda-lime-silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in liquid nitrogen at unload

rate of 5.29 MPa/s and a

p s 127.5 MPa, F

p = 0.36, n - 120,

m - 8.19, S

Q = 137.4- MPa.

Fig.. 2. Tnert strength distributions of soda-lime-silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in air at unload rate of 132.3

MPa/s and a

p = 79.3 MPa, Fp = 0.33, n = 18.4, m - 8.19,

S

Q « 137.4 MPa.

Fig. 3. Inert strength distributions of soda-lime silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in water at unload rate of 132.3

MPa/s and a = 68.6 MPa, F = 0.37, n = 18.4, m = 8.19, S = 137.4 MPa

r r u

Fig. 4. Inert strength distributions of soda-lime silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in air at unload rate of 5.29 MPa/s

and a = 79.3 MPa, Fn = 0.47, n = 18.4, m = 8.19, = 137.4 MPa.

p p o

Fig. 5. Strength distributions for soda-lime-silicate glass in very dry

nitrogen gas before and after proof testing compared to the

theoretical, after-proof distributions. Proof testing was in very

dry nitrogen gas at unload rate of 13.2 MPa/s and a = 103.9 MPa,

r F

p = 0.30, and n = 120.
15

Fig. 6

Fig* 7

Strength distributions for soda-1 ime- silicate glass in very dry

nitrogen gas before and after proof testing compared to the

theoretical, after-proof distributions. Proof testing was in

50% R.H. nitrogen gas at unload rate of 13.2 MPa/s and a

p = 103.9 MPa,

Fp = 0.58, and n 3 18.4.

Crack propagation behavior of soda-lime-silicate glass in moist

nitrogen gas environments. Percent relative humidity is given

on the right hand side of diagram (after Wiederhorn, reference 10).
Strength- (MPa)

Fig. 1. Inert strength distributions of soda-lime-silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in liquid nitrogen at unload

rate: of 5.29 MPa/s and cr^ = 127.5 MPa, = 0.36, n = 120,

m = 8.19, So = 137.4 MPa.-
17

Stangtti (MPa)

100.0 1154) 130.0 1S04) 1754)

Fig. 2. Inert strength distributions of soda-1 ime-sili cate glass before

and after proof testing compared to the theoretical, after-proof

distributions. Proof testing was in air at unload rate of 132.3

MPa/s -and cr^ = 79.3 MPa* F

p = 0.33, n = 18.4, m = 8.19,

$

0 - 137.4 MPa.
18

Strength (MPa)

100.0 115.0 130.0 150.0 175.0

Fig. 3. Inert strength distributions of soda-lime silicate glass before

and after proof testing compared to the- theoretical , after-proof

distributions* Proof testing was in water at unload rate of 132.3

MPa/s and a

p = 68.6 MPa, F p = 0.37,. n = 18.4, m = 8.19, S

Q

= 137.4 MPa
19

Strength (MPa)

100.0 115.0 130.0 150.0 175.0

90.fr

70.0

50.0

30.0 F<%)

10.0

5.0

1.0

4.5S 4.78 4.98 5.18 Ln Strength (MPa)

Fig. 4. Inert strength distributions of soda-lime silicate glass before

and after proof testing compared to the theoretical, after-proof

distributions- Proof testing was in air at unload rate of 5.29 MPa/s

and ^

= 79.3 MPa, F

p * 0.47* n = 18.4, m = 8.19, S

Q = 137.4 MPa.
20

90.0

70.0

50.0

30.0 F (%)

10.0

5.0

1.0

Fig. 5. Strength distributions for soda-1 ime-silicate glass in very dry

nitrogen gas before and after proof testing compared to the

theoretical, after-proof distributions. Proof testing was in very

dry nitrogen gas at unload rate of 13.2 MPa/s and cr p

= 103.9 MPa,

Fp = 0.30, and n ~ 120.

Strength (MPa)

70 85 100 115 130
21

Strength (MPa)

Fig. 6. Strength distributions for soda-lime-silicate glass in very dry

nitrogen gas before and after proof testing compared to the

theoretical,, after-proof distributions. Proof testing was in

50% R.H. nitrogen gas at unload rate of 13.2 MPa/s and a

p = 103.9 MPa,

F

p = 0.58, and n = 18.4.
22

Fig. 7. Crack propagation behavior of soda-lime-silicate glass in moist

nitrogen gas environments. Percent relative humidity is given

on the right hand side of diagram (after Wiederhorn, reference 10).
>> I ls^ (Rl • 9*78*

U.S. DEPT, OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. 79-1934

i.ytv.:' : AL • ..i " ' * •

1 N<J.

;

4. TITLE AND SUBTITLE

Proof Testing of Ceramics: I. Experiment

5. Publication Date

I. Performing Organization Code

7. author(S) E> Ritterj j r>9 p> B> Oates, E> R> Fuller, Jr.

nd S. M. Wiederhorn

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

Unit Mo.

" * '

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, city, state. ZIP)

Department of Energy

Fossil Fuel Utilization Division

Washington, D. C.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

I | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas (actual summary of moat significant information. If document includes a significant bibliography or

literature survey, mention it here.) The effectiveness of proof testing as a method of improving component reliability

was studied by comparing the inert strength distributions of soda-lime-silicate glas< before .and after proof testing. The effects of unloading rate from the proof stress^ hold time at the proof stress, and proof-test environment were examined. The results indicate that the proof test must be conducted with rapid unloading rates and good environmental control to be effective. The theoretical implications of these results

are discussed.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Ceramics; crack growth; fracture; glass; proof testing; strength

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NBSIR 79-1944

PROOF TESTING OF CERAMICS:

II. THEORY

E.R. Fuller, Jr. and S.M. Wiederhorn

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

National Bureau of Standards

J.E. Ritter, Jr. and P.B. Oates

Mechanical Engineering Department

University of Massachusetts

Amherst, MA 01003

Published in:

Journal of Materials Science

15_, 2282 - 2295 (1980

Proof Testing of Ceramics: II. Theory

E. R. Fuller, Jr. and S. M. Wiederhorn

National Bureau of Standards

Washington, D. C. 20234

and

J. E. Ritter, Jr. and P. B. Oates

Mechanical Engineering Department

University of Massachusetts

Amherst, Mass. 01003
ABSTRACT

Theoretical estimates are made of strength distributions after

proof testing. Assuming that the crack velocity can be expressed as a

power function of the stress intensity factor, v = A K^

n

, an analysis of

the amount of strength loss during a load cycle is presented for singleregion crack propagation. For multi-region crack propagation, a numerical

analysis is used to describe strength loss. In both analyses, the

effects of environment and loading rate are studied. For single region

crack propagation, the strength after proof testing can be represented

by two Weibull curves: one with' a slope of m at high cumulative failure

probability levels; the other with a slope of n-2 at low failure probability levels. Truncation of the strength distribution always occurs

as the result of proof testing; the truncation strength depends on the

rate of unloading. Multi -region crack propagation results in a more

complicated strength distribution after proof testing. Bimodal strength

distributions occur as a consequence of region II type crack growth

(i.e. n=o). Theoretical results confirm experimental findings that

proof tests must be conducted at rapid unloading rates and with good

environmental control to be effective.
23

Proof Testing of Ceramics: II. Theory

1. Introduction

The first part of this two part study presented the results of an

experimental investigation of the effect of proof testing on the strength

of a set of soda-lime silicate glass microscopic slides. Effects of

stress cycle and test environment were investigated. Agreement between

experimental results and theoretical predictions of strength distributions

after proof testing were obtained when proof tests were conducted in inert

environments, or when rapid rates of loading and short hold times

were used in the proof tests. By contrast, when water was present in

the environment and when slow rates of loading were used, strength

distributions were obtained after proof testing that were not consistent

with simple theoretical predictions. At the conclusion of Part I of

this study, we noted that the lack of agreement between theory and

experiment may be a consequence of the complex crack growth behavior

associated with crack motion in moist gaseous environments. We also suggested

that additional theoretical work be conducted to explore the possibility of

complex crack growth effects on strength distributions. The second part

of this study explores this possibility.

In part II of this paper the effect of complex crack growth behavior on

the strength distribution after proof testing is explored by the use of a

modified fracture mechanics approach. Although fracture mechanics is used as

an basis for understanding proof testing, strength degradation is discussed

in terms of component strength and applied stress, a modification which permits

the process of strength degradation to be illustrated graphically in a relatively

simple manner. Because this approach is new, a review of the earlier literature

is presented to achieve a uniform and comprehensive picture of the subject.

Strength degradation resulting from simple crack growth behavior (i.e., one region
29

of crack growth) is treated analytically, while complex crack growth behavior

is treated numerically. This study illustrates the types of strength

distribution curves to be expected from a population of test specimens after

they have been proof tested. The complex stress distributions reported in

Part I of this paper for soda lime silicate glass are supported by the

results obtained in this part of the paper. The importance of environmental

control and rapid unloading during a proof test is also confirmed.

2» Strength and Crack Growth Behavior

The strength of ceramics is determined by the presence of small,

so-called Griffith cracks in the surface. When stresses reach a critical

value, crack propagation occurs and ceramics fail by brittle fracture.

The level of stress required for crack propagation depends on the test

environment, crack growth usually occurring more easily in the presence

of water. The effect of water on crack growth has been used to explain

the well-known phenomenon of static fatigue, or delayed failure, which

is a stress enhanced reaction in which water in the environment behaves

as the stress corrosion agent

J

The advent of fracture mechanics provided a method of developing

2

deeper insight into crack growth processes in ceramic materials.

Fracture mechanics also offers the possibility of relating fundamental

crack-growth data to strength data, so that techniques of assuring the

3 reliability of structural materials can be developed. Data obtained on

ceramic materials by fracture mechanics techniques provide a relation

between the crack growth velocity, v, and the driving force for fracture,

the stress intensity factor, Kj. In soda lime silicate glass, the

crack velocity is dependent on both the applied stress intensity factor,

Kp and on the amount of water in the environment. The three regions of

crack growth (shown schematically in figure 1) depend on the amount of

water in the environment and the, stress intensity level at the crack
30

tip. Regions I and II result from a stress corrosion reaction between

glass and water: Region I represents reaction rate limited stresscorrosion cracking; Region II represents transport rate limited stresscorrosion cracking. Region III represents environment independent

fracture that depends only on the structure of the glass, and is not

observed for all glass compositions. The scientific significance of this

type of data has been discussed at length elsewhere, and will not be

4

repeated here. For purposes of the present paper, this type of data

can be used to calculate the fracture strength of glass or other ceramic

materials after they have been subjected to a proof test.

Crack propagation data can be used to estimate the amount of crack

growth occurring in a component that is subjected to an applied stress.

Given an initial measure of the crack length, the change in crack length

can be estimated from the applied load, and a functional relation between

the crack velocity, v, and the stress intensity factor, Kj , (i.e. v = v(Kj)).

For a uniform applied stress, a, and a crack length a_, Kj is given by

Kj = a Y rr (1)

where Y is a geometric constant. If the relation between v and Kj has

the form v=AKj n

, then crack growth during a stress cycle can be determined

analytically and the strength of a component after a random stress cycle

can be obtained. If the relation between v and Kj does not have a simple form

(eq. fig. 1) then the strength of a component after a random stress cycle can be

obtained numerically. In either case, failure occurs during the stress cycle

when the stress intensity factor reaches a critical value, K^, for rapid fracture.

Quantitative estimates of the amount of crack growth during a

stress cycle can be used to evaluate the strength of a component at any

point in the cycle. The fracture strength , S, at any time during the cycle
31

can be defined in terms of the crack length, a_, and the critical stress

intensity factor K^, from Eq. (1):*

S - K

IC

/Y /T. (2)

If the stress, a, equals the strength, S, at any point in the cycle,

then K

X s K

IC*

and failure occurs. By evaluating the effect of a stress

cycle on strength, it is possible to calculate theoretical Wei bull probability

curves, which can then be compared with experimental ones obtained from

proof“test studies. Furthermore, by using a numerical procedure to

evaluate the strength after a load cycle, the effect of multi -region

crack propagation can be taken into account.

The strength degradation that occurs during a stress cycle can be

represented by a relatively simple diagram on which are plotted simultaneously

the strength of a component and the applied stress at any time during a

stress cycle. Figure 2a gives a schematic representation of a strength

degradation diagram which is based on the assumption that only one

region of crack growth contributes to reduction of component strength. The

diagram represents a typical proof test cycle. The applied stress on

the diagram is represented by the curve labelled a which consists of three

straight lines: one line representing a constant rate of stress increase

during the stress cycle; the second, horizontal line representing a hold

time during the proof test; the third line representing the stress

reduction as the proof test cycle is completed. Although the diagram

shown in figure 2a is for a simple proof test cycle, any stress cycle

can be represented in a similar way.

*As defined in equation 2, the strength is simply related to the length of the

critical flaw that causes failure. In operational terms, S is the inert strength

at any point of the stress cycle. It is the breaking stress that would be

measured if the component could be loaded to failure in such a way that subcritical

crack growth did not occur at all during the strength test.
32

The strength of components during a stress cycle is illustrated in

figure 2a by curves labelled, S-j , $

2

and S^. Each of these curves represents

components having different initial strengths (i.e. strengths before the

start of the proof test). As the stress level in figure 2a is increased

(curve a), cracks present in specimens increase in size, resulting in a

gradual decrease in strength. Strength degradations occurs most rapidly

as the stress, a, approaches the strength of the specimen. For specimens

with initial strengths, , much greater than the maximum applied stress,

little strength degradation occurs during the load cycle. However, when

the initial strength, $2

* is less than or approximately equal to the

maximum applied stress, the specimen breaks during the test, and the

breaking stress is less than that which would have been measured in the

absence of crack growth. Breakage during the proof test can also occur

when the initial strength, Sy is greater than the maximum applied

stress provided sufficient crack growth occurs during the stress cycle.

The effect of multi -region crack propagation can also be represented

on this type of diagram. The lines, labelled eq

, s-| and z 2

in figure 2b,

represent the strength of components at the boundaries that separate the

different regions of crack propagation (fig. 1). For a given applied

stress, these boundaries define the range of strengths that lie within

each region of crack growth. The position of the boundary lines in figure 2b

can be evaluated from the values of Kj in figure 1 that define the

limits of each region of crack growth (i.e. Kq, , l^). For example,

the curve Z-j , that corresponds to the boundary between regions I and II

in Fig. 1 , Kj = K-j » i s given in fig. 2b by E-j = a(Kj^/K^), which is obtained from

equations 1 and 2 by setting Kj - K-j and S=Z-| . Thus, the boundary lines in

figure 2b are proportional to cr, and since the proportionality constant.
33

Kjc/K-j » is greater than 1 , Z-j is greater than the applied stress, a. In a

similar way, the boundary lines for Kq is given by z Q = a (K^/Kg) and

that for l<w, is given by Zg “ cr (Kjg/l^).

During a proof test cycle, the value of the stress intensity factor at the

most severe flaw crosses from one region of crack propagation to the

next in figure 1 as the crack gets longer. At the same time the strength

goes from one region of crack growth to the next each time the curve in

figure 2b marked S crosses the boundary lines. At each boundary, both

the slope and value of the strength curve are continuous.* Again,

fracture occurs spontaneously when the strength curve touches the stress

cycle boundary (i.e. when a = S). In later sections of the paper,

specific examples of these diagrams will be given.

3. Single Region Crack Propagation

To understand the types of curves obtained on strength degradation

maps, it is worth exploring single-region crack propagation. An exploration of

this simple case is also useful for quantifying the basic types of

curves that are to be expected in failure distributions and strength

histograms. Some of the subject matter presented here has been discussed

5-1

1

earlier, but the treatment presented here is somewhat simplified and

the results are more complete. As will be shown, this simplification is the

result of considering subcritical crack growth from the point of view of

strength degradation rather than crack length. New insight into the strength

degradation process is attained by this approach.

*As is shown in equation 3, the slope of the strength curve depends on both the

crack length and the crack velocity. Since both of these are continuous throughout

the test region, the slope must be continuous.
J4

For single-region crack propagation, an analytical solution can be

obtained and used to plot strength degradation diagrams, Weibull diagrams

and strength histograms. In later sections of the paper, examples of

such diagrams will be presented for a variety of crack propagation

environments. These diagrams will be used both as a means of evaluating

the effect of stressing rate and stress cycle on strength and as a basis for

comparison with more complex multi -region crack propagation behavior.

The rate of strength degradation can be obtained by differentiating

equation 2 with respect to time,

dS/dt - -(K

IC

/2Y) a~ 3/2 (da/dt) - - (Y

2

/2K

IC

2

) S

3

v(K) (3)

Thus, the rate of strength degradation depends only on the strength, the crack

velocity and the materials parameter K

ic* Expressing the crack velocity

as a simple power function of the stress intensity factor

V = AK

x

n =AK

Ic

n (o/S)n (4)

the rate of strength decrease becomes:

dS/dt « -(A Y

2

K

Ic

n-2

/2) (o/S)

n

S

3

(5)

This equation gives the slope of the strength degradation curve at any

point on the strength degradation diagram (figure 2). Provided appropriate

values of A and n are used, equation 5 is applicable to each region of

behavior in multi -region crack growth exposure. As will be shown below

this finding has important implications in determining the minimum

strength of a set of components that have been proof- tested. For a

given value of a/S, the rate of strength degradation, -dS/dt, increases

as n increases. Note that when n=o (region 2 crack growth) dS/dt decreases

in absolute magnitude as the strength decreases, equaling zero as S

approaches zero. Consequently, dS/dt exhibits positive curvature when

n=0. For region I (n>lS) or for region III (n>_100) crack growth,

however, dS/dt exhibits strong negative curvature.

*Note, this conclusion holds for all n<3.
35

For any load cycle, equation (5) can be integrated analytically to

provide a relation between the initial strength, S. (i.e. before proof

testing), and the strength, S at any time, t, during the proof test cycle.

0 OT, 0

where (1/B) s (n-2)AY K

Jc / 2. The amount of strength degradation during a

stress cycle depends only on the integral given on the right hand side

of the equation, so that the strength degradation can be determined for

any stress cycle provided the applied stress* a, is known as a function

of time.

For a typical proof test, a component is loaded at a constant

loading rate, d-j , held at the proof load, a , for a time t

p

, and then

unloaded at a constant rate, d^. The times for loading, t-j , and unloading,

tu , the specimens are given by: t-, s a /d, ; t = a /d . Integrating

equation (6) for a typical load cycle then gives the following equation for

the final strength,

$ f

n

* 2 = S

i

n '2

-(1/B) O

p

n t

p + a

p

n+1

(1/a, H- l/a

u )/(n+l)] (7)

for any component that does not break during the proof test.

The development of adequate proof testing procedures for ceramic

materials requires the strength distribution after proof testing to be

well characterized, especially in the low strength regime. This characterization is especially necessary when fracture and/or crack propagation

occurs during the unloading cycle of the proof test. To characterize fully

the strength distribution after proof testing it is necessary not only to

know (eq. 7), but also to know the breaking stress of those specimens that fail

during the proof test. This step is needed for the determination of the

minimum survival strength after the proof test.
36

The breaking stress, a*» of those :hat fail during the unloading

portion of the proof test can be obtained from equation 6 by setting S=a*. The

upper limit of integration of equation 6 is then equal to t=a p/c-| +t

p+( a p”a*)/°

u

’

which is the total time to failure during the proof test. The equation

so obtained is given by

0*

n‘2

- (1/B) o,

n+1 /(n+l)a

u -

S. n'2

- (1/B) [a

p

n tp + a

p

n+1

(1/a-j + l/*

u )/(n+l ) ] (8)

We now have two equations, 7 and 8, that describe specimen behavior during

a proof test. Equation 7 gives the strength, S.p of components that pass the

proof test cycle, whereas equation 8 gives the breaking stress, a*, of components

that do not pass the proof test cycle. Both equations are expressed in terms of

the load cycle parameters (a , a, , a , t ), the crack propagation parameters

(B and n) and the initial component strength. Si, before the start of the proof

test cycle. If the initial strength. Si, is relatively high, the component will

pass the proof test; whereas if the initial strength is low, the component will

break during the proof test cycle (figure 3a). There is a critical initial

k k

strength. Si , that will separate components that break. Si <Si , from those

it that survive, Si>Si , the proof test cycle. The component that has an initial *

strength that is infinitesimally greater than Si will have a final strength,

Sfmin* that forms a lower bound for the strengths of components that pass the

proof test (figure 3a). S

frnin

1S important because it represents the lowest

possible strength of components that have been subjected to a proof test cycle.

S fm^ n is in effect the truncation strength for the strength distribution after

the completion of the proof test. By similar reasoning, the component that has

* an initial strength that is infinitesimally less than Si will break at a stress,

, that forms a lower bound to the breaking stresses of components that fail

during the unloading cycle (figure 3a).
37

A relationship between the minimum strength, , after the proof test

cycle, and the minimum breaking stress, a*^, can be obtained by setting Si =Si

in equations 7 and 8 and equating the left hand sides of these equations:

s£?n * -Sin C’-d/BM)) (<in/*u>3

Equation 9 can be simplified by expressing in terms of

the unloading rate. During unloading, the strength degradation curve

that characterizes . will approach within an infinitesimal distance

of the stress curve, but will fail to touch it. At the point of closest

approach of the two curves, the slope, dS/dt, of the strength curve will be

equal to the slope, -d , of the stress curve. By letting dS/dt = -d

and a = S = in equation 5, the following equation is obtained:

(9)

°*min ' CB<n-2)*u

]

1/3

By substituting this relation for d

u in equation (9) a simple

equation is obtained for the minimum strength after proof- testing:

( 10 )

Sfm1n

“ ff*min )]

1/<n' 2) <">

One concludes from this equation that the strength, S^. , of a component

that just survives the proof test is always less than the breaking stress, ,

of a one that just breaks during the proof test. Because n for most ceramic

materials is large, >10, S fmi n/a*min is normally greater than 0.85.

Equation 10 can be written in an alternative form that is useful when

relating the truncation strength, .j , to parameters that describe crack growth.

By expressing B in terms of the crack growth parameters A and n, and by

defining a limit crack velocity (fig lb), Vj£ = AKj^n

, the following alternative

representation of equation 10 is obtained:

%nin [(2Vic

2

) / < V

IC

y2 »

1/3 (12)

From equations 11 and 12 we note that the truncation strength, . ,

depends only on the unloading rate, d , the critical stress intensity
33

factor, and the limit crack velocity, Vj

C

, which is determined by the

crack propagation parameters, A and n. Th refore, the truncation strength

does not depend on the proof-test level, the time at load, or the loading

rate. If d

u >0, equations 11 and 12 imply that proof testing always

truncates the strength distribution . The higher the unloading rate, d , the greater

is the strength level, S.p

^

, at which strength truncation occurs. Fracture

during the unloading part of the cycle can be avoided provided the

unloading rate is sufficiently great that calculated from equations 10 or 12

is greater than the proof test load, cr^. This condition for truncation

7 was noted earlier by Evans and Fuller from other considerations.

However even if >_am

, strength degradation will still occur during

unloading so that will be less than a (eq. 11). Finally, since

the only crack growth parameters that influence are those that

occur when V=Vj^, equation 12 should also apply to multi-region crack

growth, for which the intersection of the region III curve with

(fig. lb) normally determines V jq.

4. Single Region Crack Propagation; Statistical Parameters

Since proof testing modifies the initial strength distribution, a discussion

of the shape of the distribution curve that results from proof testing is

important. The treatment of strength degradation given above provides all of

the necessary information to determine the effect of a stress cycle on the

strength distribution. The change in strength during the proof test can be

determined from equation 7, while the change in probability can be determined

from simple probability theory. Therefore, if the initial strength, Si, is

known as a function of the initial cumulative failure probability, Fi , the

strength after proof testing, S^, can be evaluated in terms of the cumulative

failure probability, F^, of the components that pass the proof test by the use

of simple transformations of both strength (equation 7) and failure probability.

This approach to estimating the effect of proof testing on the strength
39

distribution is illustrated below for a two -parameter Weibull distribution.

Although the approach is not limited to Weibull statistics, this form of extreme-val

statistics is easy to use and provides a relatively accurate description

of the strength distribution for brittle materials. Two-parameter

Weibull statistics can be easily incorporated into the present treatment

of proof- testing, so that strength distributions after proof-testing can

be expressed in terms of statistical parameters.^*^ 0 ’^

Weibull statistics relate the cumulative failure probability, F, to

* 12

the strength, S, by the following standard equation :

In (In (1-F)'

1

) * m In (S/S ) (13)

where m and S

Q are parameters that are determined from experimental data by a 13 least squares fit or by other methods of estimation. For simplicity

in succeeding calculations let ln(l-F)”^ = Q. Note that for small

values of the failure probability Q=F (F<0 . 01 ) . If Q.. is determined by

the initial strength distribution before proof testing, then equation (13)

can be simply expressed as:

Q

i » (S^Sg)1” • (14)

From probability theory the initial failure probability (as characterized by ) can

be related to the failure probability after proof testing (as characterized by Q f

)

and the failure probability of the component that just fails during the

proof test (characterized by Q

)."** F Qi=Qf + Qp

(15)

*F is obtained by ordering a set of strength data. F is given by r/(N+l)

where N is the total number of datum points and r is the position of each

point in the ordered set. r=l for the lowest strength value, r=2 for the

second lowest and so forth. + Equation 15 follows from the probability relation F^ = (F^ -F )/(l -F ) where F^ is

the failure probability after proof- testing, F- is the initial failure

1

1 o probability evaluated from eqn. 13 and F is the proof test failure probability.
40

By substituting equation (14) and (15) into equation (7), the strength

Sf, after a proof test can be expressed in terms of the cumulative failure

probability (represented by Qf )

:

n-2

(Sf/S

0

)"’ 2 . (Qf + Q

P

) m

- (1/8) S

0

' (n ' 2)

Co

p

n t

p + a

p

n+1

(l/ii+l/a

u )/(n+l)] (16)

Equation 16 can be simplified considerably by noting that for the specimen

that just survives the proof test cycle S

f " S fmin and Qf * 0. Substituting for

Sf and Qf in equation 16, the following expression is obtained for the last term

on the right of equation 16:

(l/B)So"^

n “ 2

^ [a

p

n t

p + O/a-j + l/a

u )/(n+l)]

n 2 -V^w*^2 (17)

Substituting this expression into equation 16 eliminates the term containing

the square bracket so that:

(ys

0

) n' 2 * (Qf + v

""’2

-V + <WV"'2

(13)

'P "P

Equation 18 illustrates the important conclusion that once Sfm<

. , S

Q

, m and n are

determined the strength distribution after a proof-test can be determined

simply by counting the number of specimens that break during the proof test.

For single region crack growth, equation 18 gives a complete description of

the type of probability curve expected for the strength distribution after proof

testing. It is straightforward to show that the strength distribution after proof

testing is given by a trimodal curve, each part of which corresponds to a

*Sf • approaches zero as the unloading rate becomes vanishingly small (type III

distribution'

7 discussed in part 1 of this paper), and has an upper limit when

Sfmin = a n

( i • e • infinite unloading rate, for which no crack growth occurs during

the proof test, type II distribution * discussed in part 1 of this paper.)
41

different range of initial strengths and corresponding failure probability

(figure 3b). When strengths before testing are high, little crack growth

occurs during the proof test, and as a consequence, strength degradation

resulting from the proof test is slight (fig. 2). In quantitative

terms, the strength after the proof test will be much greater than the

proof test stress (S f » a ), so that Qf » Q . If the final strength

is also much larger than the truncation strength (S^- >> Sfmi ) , then the

strength distribution after the proof test (calculated from eq. 18)

is given by ( S^./S

Q

) m

= Qf

, which is identical to the initial distribution

(eq. 14). This portion of the strength distribution curve is indicated by a

slope of m in figure 3b.

For an intermediate range of initial cummulative failure probabilities,

strength degradation resulting from crack growth significantly alters the

strength distribution after proof testing. If the final strength is

significantly greater than the truncation strength (S^ >> S^min ), and if

the failure probability after proof testing, Q^, is less than the failure

probability, Q , calculated from Q =(a /S ) m

(i.e. Q- « Q ), then the

r r r u u r strength distribution given by equation 18 can be simplified by expanding

(Q.p + Q

p

)^ n “ 2 ) /m

as a Taylor series. The final distribution for

the intermediate probability range is then given by:

Qf» [m/n-2)] Qp

( m+2 - n )/m

(Sf/S

Q

) (n_2)

, (19)

which implies that the strength distribution after proof testing can be

represented by a straight line with a slope m

1 = n-2 (figure 3b). Finally, in the

lowest probability range, the strength distribution must be truncated

so that S

f

= S

fniin , where S fmin is given by equation (11).
42

In summary, the strength distribut on after proof testing can be

described by two Wei bull curves (Figure 3b): one at higher probabilities

having a slope of m; the other at lower probabilities having a slope of

n-2. As indicated in figure 3b, the low probability curve is truncated

at a strength given by S^.. .

5. Single Region Crack Propagation: Examples of Strength Degradation

Maps and Wei bull Diagrams

The equations presented in the previous sections will now be illustrated

for the special case of single-region crack propagation in soda-limesilicate glass. Experimentally determined values of A and n were selected

to represent crack propagation in air (50% rh) and dry nitrogen gas

(0% rh). To illustrate the effect of unloading rate on strength, two values

(1 MPa/s and 1000 MPa/s) were used for each environment, the loading

rate was conveniently set equal to the unloading rate and the hold time was set

equal to zero. The strength/stress axes of the resulting strength degradation

maps shown in figures 4 and 5 are represented in terms of the reduced

variables S/S

Q

and a/S

Q

, where S is the strength, a is the stress and S

Q

is the Wei bull strength measured in an inert environment. The time axes

in these figures are plotted in terms of the reduced variable t/T where T is

the total time of the proof test cycle.

Figure 4a and b illustrate the predicted effect of moist air (50% r.h.)

on the strength of soda-lime-silicate glass specimens for the loading conditions

just specified. The strength degradation map for a loading rate of 1 MPa/s

(figure 4a) illustrates a number of features discussed in the previous sections:

at high initial strengths relatively little strength degradation occurs

during the stress cycle; at low initial strengths, the strength decreases

relatively rapidly once the stress reaches a sizable fraction of the

initial strength. Over a relatively narrow range of initial strengths that separate

those specimens that fail from those that survive the proof test, a dramatic
43

change in the shape of the strength degradation curve is observed. The

strength curve bends over as the unloading cycle begins, and the strength

appears to decrease at a relatively constant rate as the stress is

decreased further. Specimens with initial strengths that are just above

the critical value survive the proof cycle; initial strengths that are

just below critical fail the proof cycle. Note that significant strength

degradation occurs for specimens that just pass the proof test and that

a significant number of failures can occur during the unloading portion

of the proof test cycle. The truncation strength for conditions given for

figure 4a was S*. =0.035 S . fnnn o

The effect of higher loading rates on strength is shown in figure

4b. As can be seen from this figure increasing the loading rate to 1000 MPa/s

decreases the value of that just survives fracture. Also, the value

of the truncation stress = 0.35S

Q

is increased by one order of magnitude

as the loading rate is increased by three orders of magnitude. For tests

dry nitrogen (figures 4c and 4d) strength degradation curves are similar

to those just discussed, however because water is not present, strength

degradation is less for all levels of initial strength, and far fewer

components break during the proof test.

Figure 5 illustrates the effect of environment and loading rate on the

strength distribution after proof testing. For all test conditions, the

curves obtained for specimens that survive the proof test approaches the curve

that represents the initial strength distribution (the straight lines in figure 5).

At low probability levels the curves for the proof test survivors approach a

slope of n-2, as expected. For the test conditions selected, the truncation

strength occurs at a value too low to be represented on figure 5. The effect of

loading rate on the strength distribution after proofing can be discerned by

comparing figures 5a and 5b for the moist environment. Because far fewer specimens

break at the high loading in the moist environment, (^7 percent for 1000 MPa/s vs.
44

<\78 percent for 1 MPa/s), the low probability portion of the survival curve

shown in figure 5b has shifted to lower strength values relative to the curve

in figure 5a. This effect of loading rate on the survival curve is greatly

suppressed in dry environments » as can be discerned by the fact that virtually

the same survival curve is obtained at low and high loading rates (figure 5c and 5d).

For the same loading conditions, proof testing in dry environments greatly

reduces the number of failures during the proof test (-^2.9 percent at 1000 MPa/s

and ^4.5 percent at 1 MPa/s). Furthermore since the slope of the survival curve

increases at low failure probabilities, the probability of failure at a given

strength after proof testing is greatly reduced by using a dry environment for

the proof test, thus confirming the importance of environmental control during proof

testing.

6. Multi -region Behavior: Examples of Strength Degradation Maps and

Wei bull Diagrams

Multi-region crack propagation can, in principle, be handled by the

same mathematical techniques discussed in Sections 2 and 3. Equation

7 would be applicable within each region of crack growth, with appropriate

values of A and n being used to describe crack growth behavior. As the

strength degradation curves passes between two regions of crack growth, boundary

conditions require that both the strength and the slope of the strength degradation

curve (dS/dt) be continuous. Given the initial strength, equation 6 can be used to

estimate either the final strength (if the strength curve does not

intersect the boundary line between region I and II), or the strength at

intersection of the boundary. Determination of the strength at the

intersection with the boundary requires an iterative procedure to be

applied to equation 6, because both the limits of integration of equation

6 and the strength depend on the point of intersection of the strength

curve with the boundary. Once the strength at the boundary is determined.
45

this same procedure can be repeated to estimate the strength at other

boundaries of the strength degradation map. It is worth emphasizing

that* depending on the initial strength, a strength degradation curve can

either exit from a region of crack growth by going to the next region

(i.e., region II to region III), or by going back to the previous region

of crack growth, (i.e., region II to region I). Because of these complexities,

strength degradation maps are obtained more easily by direct numerical

integration of crack propagation data.

The computer routine used here to investigate multi region crack propagation is based on a direct integration of equation 3, assuming that the

rate of crack growth is controlled by equation 4. Given an initial

strength, the crack length can be determined from equation 2. The

stress for crack growth determined from the stress cycle is then used to

calculate the crack velocity from equation 4. For a time increment,

dt, the change in crack length, da, during dt is determined from the

crack velocity, da=vdt. Finally a new length, a+da, is used to calculate

a new- strength, which completes one iteration of the computer cycle. By

repeating the calculation many times, the rate of change of strength with

time can be determined. At the end of each cycle, a comparison is made

between the calculated strength and boundary-value conditions (see

figure 2b) to determine the appropriate crack growth parameters that are

applicable (i.e., the correct region of crack growth). If the condition S=a,

is attained fracture is assumed to occur. This numerical method was

used as a subroutine in a larger program to obtain strength degradation maps,

Wei bull diagrams, and strength histograms. These diagrams demonstrate the

effects of multi -region crack propagation on the strength of materials.

As will be shown in a comparison of multi -region maps with their counterparts

described in figure (4), region II crack growth behavior has a significant

effect on the appearance of these diagrams.
46

The main features of strength degradation in a moist environment

(air, 50 percent r.h.) are shown in figures 6-8. When specimens are

loaded and unloaded relatively slowly (1 MPa/s), both the strength

degradation map (figure 6a) and the Wei bull plot (figure 6b) look

similar to those obtained for single mode propagation in moist nitrogen

gas, (50 percent r.h., figures 4a and 5a). By contrast, at high loading rates

(1000 MPa/s), the strength degradation map (figure 7a) and the Weibull

diagram (figure 7b) are nearly identical to those obtained for single mode

propagation in dry nitrogen. Thus, the distribution after proof testing

seems to be determined mainly by region I crack growth when the rate of

loading is slow, and by region III crack growth when the rate of loading

is high.

At intermediate loading rates, region II crack growth behavior

dominates the shape of the strength distribution curves of specimens

that survive the proof-test cycle. As illustrated in figure 8a, the

effect is most pronounced when the slope of the strength degradation

curve is slightly greater than the slope of the line that represents the

boundary between region I and region II crack propagation. From equation

5, we see that once the strength curve enters region II, its curvature

changes from negative (large n) to positive (nso), so that the slope

decreases as S decreases, with the consequence that components with a

range of initial strengths that normally would have failed during the

a load cycle (for single region crack growth) now survive, but with a

greatly reduced strength. Because of this behavior, the Weibull curves

(figure 8b) are severely distorted. Instead of consisting of two

straight lines with a truncation stress at low probabilities (fig. 3b), the

Weibull curve exhibits a large plateau which indicates a constant cumulative

failure probability for a fairly wide range of final strengths, S^.

Therefore, the probability of finding specimens with strengths that lie
47

within the strength range of the plateau is ssmall. This plateau in the

Wei bull curve suggests a bimodal strength distribution with a small peak

in the distribution at low levels of strength. This type of distribution

is potentially dangerous for materials that are intended for structural

appl i cations.

The shape and position of the plateau shown in figure 8b, is

effected by both the stress cycle parameters and the crack propagation

parameters. Although the probability level of the plateau is found to

vary as a is changed, the truncation strength is unaffected by a , r r provided the stressing rate remains constant (figure 9). This result

is consistent with the discussion in section 2 and with equations 10 and

12, which are also applicable to multi-region crack growth. Increasing

the stressing rate almost completely eliminates the effect of region II

crack propagation (figure 7b). Conversely, decreasing the stressing

rate suppresses the plateau to low probability levels (so that it does

not appear in the diagram). The effect of region II crack propagation

on the strength distribution is enhanced as the width of region II is

increased. Here, the main effect of Region II crack growth on the Wei bull

curve is to increase the range of stresses over which the plateau occurs.

With regard to proof testing as a method of assuring component

reliability, a disturbing aspect of region II crack propagation is that,

for equivalent failure probabilities, portions of the strength distribution

after proof testing lie at lower strength levels than those given by the

initial distribution, resulting in significant probabilities of failure

at low stresses after proof testing. This effect occurs for both dry test

environments (fig. 10) and moist test environments (fig. 8b). The

effect can result in either a significant strength degradation or an apparently

unchanged strength distribution as a result of the proof test (high

probability region of figure 8b). Effects such as these have been
48

reported in Part I of this study and have a plausable explanation in

terms of region II crack propagation. For practical purposes, elimination

of these effects can be accomplished most easily by increasing the rate

of unloading during a proof test. Thus, even for a relatively moist

environment, rapid unloading (figure 7b) leads to a distribution after

proof testing that is sharply truncated and is better than the initial

distribution at all probability levels. Equation (10), with = a ,

provides a quantitative estimate of the unloading rate needed for effective

truncation of the strength distribution curve in a moist environment

such as air.

7

As a final comment, we refer to the discussion at the end of Part

I of this paper, and note that the theory presented in this part provides

the basis for understanding strength degradation of ceramic materials.

The theory has been used in Part I to explain the general shape of

strength distribution resulting from proof testing. The fact that

detailed agreement between theory and some experiments was not obtained

suggests that our understanding of subcritical crack growth in specimens

that contain small cracks (e.g. <10um) is not complete, and requires

further research. The discussions presented in this part of the paper

should help in interpreting any new data.

Acknowledgements

Two of the authors (E. R. Fuller, Jr. and S. M. Wiederhorn) are

grateful for the support of the Department of Energy, Fossil Fuel

Utilization Division; the other two (J. E. Ritter, Jr. and P. B. Oates)

are grateful for the support of the National Science Foundation, contract

DMR-77-05647.
References

1. S. M. Wiederhorn, "Mechanisms of Subcritical Crack Growth in Glass,"

pp 549-580 in Fracture Mechanics of Ceramics, Vol . 4, Edited by R.

C. Bradt, D. P. H. Hasselman, and F. F. Lange, Plenum Publishing

Co., New York (1978).

2. S. M. Wiederhorn, "Subcritical Crack Growth in Ceramics," pp. 613-646

in Fracture Mechanics of Ceramics, Vol. 2, Edited by R. C. Bradt,

D. P. H. Hasselman and F. F. Lange, Plenum Publishing Co., New York

(1974).

3. John E. Ritter, Jr., "Engineering Design and Fatigue Failure of

Brittle Materials," pp 667-686 in ref. 1.

4. S. M. Wiederhorn, "Influence of Water Vapor on Crack Propagation in

Soda-Lime Glass," J. Am. Ceram.. Soc. 5£, 407-414 (1967).

5. A. G. Evans, "Analysis of Strength Degradation after Sustained Loading,"

J. Am. Ceram. Soc., 57_, 410-411 (1974).

6. A. G. Evans and E. R. Fuller, "Crack Propagation in Ceramic Materials

under Cyclic Loading Conditions," Met. Trans. 5_, 27-33 (1974).

7. A. G. Evans and E. R. Fuller, "Proof Testing - The Effects of Slow

Crack Growth," Mat. Sci. and Engr., 1_9_, 69-77 (1975).

8. A. G. Evans and H. Johnson, "The Fracture Stress and Its Dependence on Slow Crack Growth," J. Mat. Sci. 1J1> 214-222 (1975).

9. A. G. Evans, "Slow Crack Growth in Brittle Materials under Dynamic

Loading Conditions," Inst. J. Fract., H[, 251-59 (1974).

10. A. G. Evans and S. M. Wiederhorn, "Proof Testing of Ceramic

Materials - An Analytical Basis for Failure Prediction," Int. J.

Fract. Mech., T£, 379-92 (1974).

11. S. M. Wiederhorn, "Reliability, Life Prediction and Proof Testing of

Ceramics," High-Performance Applications, J. J. Burke, A. E. Gorum, and

R. N. Katz editors. Brook Hill Publishing Co., Chestnut Hill, Mass. (1974).

12. W. Weibull, "A Statistical Distribution Function of Wide Applicability,"

J. Appl . Mech., 1_9, 293-97 (1951 ).

13. N. R. Mann, R. E. Shafer and N. D. Singpurwallee, Methods for Statistical

Analysis of Reliability and Life Data, John Wiley & Sons, New York (1974).
Dependence of crack velocity on applied stress intensity

factor. Schematic diagram of crack velocity dat£ indicating

three regions of crack growth. Boundaries of these three

regions are given by specific values of K^. (K^, , K0

, K^)

and by a critical value of the crack velocity (V )

.
2. Schematic diagram of strength degradation maps for

soda-lime-silicate glass. (a) Single region crack growth.

Curve labeled a gives the applied stress as a function

of time. Curves labeled S , S_ , S_ give the strength as

a function of time starting at different initial strength

levels. (b) Three regions of crack growth (as in figure 1)

Curves labeled £o> ^ 1 > ^2 an<^ 0 an this f-‘-Sure correspond

to K_, K , K , and K of figure 1. The curve labeled S gives the strength degradation as a function of time.
Single region crack propagation. (a) Definition of

critical initial strength, S*, minimum failure stress,

a*min> an<^ minimun strength, S£mj_n , after the proof test.

Specimens with initial strengths greater than S* will survive, while those with strengths less than S* will fail.

Equal loading and unloading rates during the proof test

are assumed in this diagram. (b) Theoretical strength

distribution after proof testing. The curve is a graphical

representation of equation 18. As discussed in the text,

the truncation strength depends only on the rate of

unloading, o u , the critical stress intensity factor, K-j-q

and the crack propagation parameters, n and A.
Time/Total Time Time/Total Time

4. Region 1 crack growth: schematic diagrams of strength

degradation maps for soda-lime-silicate glass. (a) loading

rate: 1 MPa/s. Test in air; 50% r.h. (n = 19.7, InA = -271.9)

(b) Loading rate: 1000 MPa/s. Test in air, 50% r.h. (c)

loading rate: 1 MPa/s. Test in dry air, 0.01% r.h. (n = 120.9,

InA = -1641.4) (d) loading rate: 1000 MPa/s^ dry air, 0.01%

r.h. . ^ m = 7.7; So = 126.9 MPa; K0 = 0 MPa-in 5

; KJr = 0.75

MPa-m 2

, a = Q.65So.
54

I I 1 L_ !——!——J _L—-I-

- 1.4 - 1.0 ' 0.6 -0.2 0 0.2

Ln (S/S0)

Region 1 crack growth: schematic diagram of strength

distribution after proof testing for soda-Ume-silicate glass.

loading rate 1 MPa/s. Test in air, 5(U r,h 0 (b) loading

rate 1000 MPa/s. Test in air 50% r.h. (c) loading rate 1 MPa.s.

Test in air 0.01% r.h. <d) loading rate 1000 MPa/s.

^ Tesi in air

0.01%r.h. Experimental constants used tor this ^igu^e cu_e given in figure 4.

5.
6. Multiregion crack growth: Air 50% r.h., loading rate:

1 MPa/s. (a) strength degradation map. (b) Strength distribution

after proof test. Crack growth parameters for figures 6 through

9: region 1, n = 19.7, InA s —271.9; region 2, v - 1 X 10

m/s; region 3, n = 120.9, In A - -1641.4. Weibull parameters

for figures 6 through 9: S Q = 137.9 MPa, m - 8.4. £ther

constants for figures 6 through 9: K

IC - 0.75 MPa-m 2

; K0

0 MPa-m 2

; a

p = 0.65S o

.
Ln <S/SQ)

Multiregion crack growth: Air 50% r.h. , loading

rate: 1000 MPa/s. (a) Strength degradation map,

(b) Strength distribution after proof testing.

7 .
Time/Total Time

Multiregion crack growth: Air 50% r.h., loading

rate: 100 MPa/s. (a) strength degradation map;

(b) strength distribution after proof testing.

8 .
Ln

Ln

Dependence of strength distribution after proof

test on proof test stress, o , indicated by the numbers

associated with each curve. Loading rate: 100 MPa/s.

Air, 50% r.h.

9 .
Ln (S/SQ)

10. Strength distribution after proof test: air 0.01%

r.h. ; loading rate, 1 MPa/s; Op = 0.82 S . Crack growth

parameters: region 1, n = 19.7, InA = -2?6.5; region 2, v = 1.5 X 10~^ m/s; region 3, n = 120.9, InA = -1641.4.

Weibull parameters: S

Q = 126.9 MPa/s; m * 7.7.
USCOMM-OC

62

NBSIR 80-2047

APPLICATION OF FRACTURE

MECHANICS IN ASSURING AGAINST

FATIGUE FAILURE OF CERAMIC

COMPONENTS

J.E. Ritter

Mechanical Engineering Department

University of Massachusetts

Amherst, MA 01003

S.M. Wiederhorn, N.J. Tighe, and E.R. Fuller, Jr.

National Bureau of Standards

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

U.S. Department of Commerce

Washington, DC 20234

Published in:

Ceramics for High Performance Applications - III Reliability

E.M. Lenor, R.N. Katz, and J.J. Barke, ed.

Plenum Publ. Corp., 1981

Army Materials Technology Conference Series

Vol . 6

APPLICATION OF FRACTURE MECHANICS IN ASSURING AGAINST FATIGUE

FAILURE OF CERAMIC COMPONENTS

J. E. Ritter, Jr.

University of Massachusetts

Amherst, MA. 01003

S» M. Wiederhorn, No Jo Tighe and E. R* Fuller, Jr.

Fracture and Deformation Division

National Bureau of Standards

Washington, D. C. 20234

INTRODUCTION

The use of ceramics in high performance applications offers

a challenge to scientists and engineers: the challenge of

designing structural components with brittle materials. Design

problems arise for two reasons when brittle materials are used

as structural components: 1. The strength of brittle materials

is not a well defined quantity, but can vary widely depending on the material; 2. The strength of brittle materials is time

dependent so that these materials often exhibit a time delay to

failure. This time dependence and scatter of strength so typical

of most ceramic materials occurs because of the presence of defects

such as cracks or crack-like flaws in these materials. When

subjected to an applied tensile stress, these defects act as stress concentrators and fracture occurs when the applied stress

intensity factor reaches a critical value. Scatter in the

strength of ceramic materials is a consequence of the scatter in

the size of the most critical defect in the ceramic. The time

dependence of strength results from subcritical crack growth

which gradually lengthens the crack until it reaches critical

dimensions, at which time failure occurs. The time delay to

failure is the time required for the crack to go from a subcritical

to a critical size.
64

Methods of dealing with design problems involving fatigue

of ceramic materials have been developed over the past 10 years

through the application of the techniques and concepts of, fracture

mechanics (1—3) . Since fracture mechanics techniques can be

used to characterize both the conditions for subcritical crack

growth and the conditions for crack instability, they can be

used for purposes of design to estimate the allowable applied

stress and the expected lifetime for a given component. This is

accomplished by estimating the initial crack size in a ceramic

component and the time required for the crack to grow from its

initial size to a final critical size. Several mutually independent techniques have been developed recently to provide these

types of estimates. Since these techniques promise to revolutionize the way in which structural components made of ceramic

materials are designed, a complete understanding of these techniques, their application, and their limitations will be necessary

to use them correctly.

This paper will review the application of fracture mechanics

theory to the prevention of delayed failure of ceramics. Three

successful applications of this theory of assuring the mechanical

reliability of ceramics are discussed in order to demonstrate

the viability of the theory for purposes of engineering design.

Finally, a description is presented of practical limitations of

the theory with regard to heat engine application. Methods of

overcoming these limitations through modification of test procedures, and application of statistical theory are then presented.

THEORY

Since fracture mechanics concepts can be used to characterize

both the conditions for subcritical crack growth and the conditions

for crack instability, they can be used to predict the failure

of ceramic materials under given service conditions. This is

accomplished by estimating the initial crack size in a ceramic

component and the time for the crack to grow from its initial

size to a final critical size. As the crack grows to its

critical size, the crack velocity (V) is assumed to be dependent

on the applied stress intensity factor (K^) by (1)

:

V s A Kj (1)

where A and N are crack growth constants that depend on the

environment and material composition. From Equation (1) and

relation a=K /Y/a, it can be shown that the time to failure

under constant applied tensile stress (a ) is (3)

:

t

f

N-2

B S

the

<t

£

)

( 2 )
2 N“2 where B = 2/(AY (n-2)Kj

C

), K = critical stress intensity

factor, and S = fracture strength in an inert environment where

no subcritical crack growth occurs prior to fracture. From its

definition, B is a crack propagation constant that depends on the

environment and material composition.

In Equation (2) , t^ represents the time required for a flaw

to grow from an initial, subcritical size, to dimensions critical

for catastrophic propagation; B and N are the constants that

characterize this subcritical crack growth. The initial flaw

size is characterized in Equation (2) by the fracture strength

in an inert environment*. From Equation (2) it is seen that the

time to failure decreases with increasing stress, i.e. fatigue

under a static stress.

Also from Eq. (1) , a relationship can be derived between

the fracture strength and stressing rate a. In this case

flaws grow from subcritical to critical size under constantly

increasing stress. The results of this analysis is (1-3):

N+l _ , n s JN-2 . a

f - B (N+l) S a (3)

where B and N are the same fatigue constants as in Eq. (2)

From Equation (3> it is seen that fracture strength decreases

with decreasing stressing rate since the flaws have more time to

grow. This behavior is known as dynamic fatigue , i.e. fatigue

under constant stressing rate conditions.

The probability of failure (F) for a given lifetime and

applied stress can be obtained from Equation (2) by expressing

the inert strength in terms of its failure probability distribution. Generally, the inert strength distribution of ceramics

can be approximated by the Weibull relationship (3):

In In ® In ~ (4) o where m and S are empirical constants evaluated by a fit of the

strength data. Likewise, the strength distribution at a fixed

stressing rate can be obtained by substituting Equation (4)

into Equation (3).

Because ceramics exhibit a wide spread in strength values

(m is typically 4-8) , the allowable stress in service is quite

*The initial crack size, a, can be calculated, if desire^, jjrgm

the well known fracture mechanics relationship: a = /Y S , where Y is a constant that is determined by the geometry of the

specimen and the crack. In subsequent discussion, a is assumed

to be small relative to component dimensions so that Y is a singl

valued constant.
66

low if low failure probabilities are required. Proof testing

offers one means of increasing the design stress for ceramics (1-3).

The value of proof testing is that it characterizes the largest

effective flaw possible in a tested component, since any larger

flaws would have caused failure during the proof test.

Assuming that flaw growth during the proof test cycle is

given by Equation (1) (i.e. one region of crack growth) and

that the initial inert strength distribution can be characterized

by Equation (4) , the inert strength after proof testing (S ) is

given by (4) : a N -2 (N -2)/m

(S JS ) P = (-ln(l-F )-ln(X-F ))

P a o a p

(N -2)/m N -2

-(-ln(l-Fp)) p + < WV P (5)

where' N is the crack propagation constant appropriate for the

proof tlst environment, F is the failure probability after

proof testing, F is the failure probability of the proof test

and S . is the Minimum inert strength of a sample that just

passes

1fhe proof test cycle, i.e. the truncation strength. It

is significant to note that the inert strength after proof

testing is truncated at S . and will be greater than the initial

inert strength at all levefs of failure probability if m < N -2 (4)

.

For a given material and proof test environment, S . is

determined only by the unloading rate (cr ) from the proofn

s tress

(4)*

min [cr B (N -2) ] 1/3 [3/(N +1)] u p p p

u

1/ (N -3)

( 6)

where B is the crack propagation constant appropriate for the

proof tlst environment. If cr^ > 0, Equation (6) shows that

proof testing always truncates the strength distribution. The

higher cr

, the greater is the strength level, S . , at which

truncation occurs. However, since S . cannot ?>engreater than

the maximum stress, a , during the proof test, a is an upper

bound for S „ . Substitution of a for S „ in Equation (6)

gives the minimum unloading rate fir whic^a is approximately

equal to the truncation strength. Equation ^6) also shows that

good proof test conditions (high N^) result in a high smin *

The failure probability of the proof test (F ) can be

determined directly from the number of specimens lhat break

during the proof test or can be predicted from (4)

:

-ln(l-F ) = (S V1

CD /B ) m/(Np'2)

P O P P

( 7 )
where D =

P

t N /v (t) dt represents the amount of strength

degradation during the proof stress cycle. For a typical proof

test, the component is loaded at a constant rate, 5^, held at

the maximum proof- stress , a , for a time t , and then unloaded

at a constant rate, 5 . Inlegrating the D

p equation for this

typical proof stress cycle gives: P

N N +1

p - [a P D = /(N +1)] (1/5. + 1/5, ) (8)

P

t +

P u

By coupling Equation (5) x?ith Equations (2) and (3) , the

and distribution after proof testing can be predicted for

any service environment (4) . Of particular interest is the

minimum lifetime (t . ) in service of a component after proof

testing. This is oSlained by substituting S . for S in Equation

(2) so that:

t . - BS . N“2 a mn min a

(9)

where now B and N are the appropriate crack parameters for the

service environment.

Equations (5) - (9) summarize failure predictions for

ceramic materials after proof testing. These failure predictions

are dependent on the crack propagation parameters B and N. These

parameters must be determined in a test environment that simulates

the appropriate proof-test and service conditions and can be

obtained from one of three type of experiments: crack velocity

experiments, stress rupture experiments, and stressing rate

experiments (3). From crack velocity experiments. Equation (1) is

used to determine A and N; K is determined in a separate

experiment. Stress rupture data (t^ vs. a ) and the inert

strength data are used to determine B and § from Equation (2)

,

whereas stressing rate data (S vs. 5) and median inert strength

data are used to calculate B and N from Equation (3). The stress

rupture and stressing rate techniques are commonly referred to as -static-fatigue and dynamic fatigue, respectively.

APPLICATION

The fracture mechanics principles described in the previous

section have been experimentally verified for soda-lime glass

(3) and alumina (5) in a moist environment. Specifically, these

studies showed that the three experimental techniques for

measuring the crack propagation parameters B and N give equivalent

results, that the time-to-failure and strength failure probability

distributions can be predicted from Equations (2) and (3) coupled

with Equation (4), and that strength distributions after proof
68

testing can be predicted on the basis of Equation (5) . The

purpose of this section is to describe three applications of

fracture mechanics principles in assuring against the fatigue

failure of ceramic components in service.

Failure Statistics

For alumina substrates fabricated by cofired metallurgy,

stress corrosion cracking can be a serious problem. The metals

most commonly co fired with alumina are either molybdenum or tungsten. Because these metals possess lower thermal expansion

coefficients than alumina, their cosintering can cause the

ceramic to retain residual tensile stresses of considerable

magnitude. Subsequent cleaning and plating baths, necessary to

render the metallurgy solderable, expose the substrate to

conditions that promote stress corrosion cracking in the alumina.

To statistically assess the susceptibility of polycrystalline

(MPa/SEC)

Figure 1. Dynamic fatigue data of polycrystalline alumina in a KOH cleaning solution at 100°C, in a gold-plating bath

at 78°C, and in moist air at 30°C and 80% RH (from

Ref. 6).

r
69

alumina to stress corrosion cracking under processing conditions,

the stressing rate technique was used to characterize the crack

propagation parameters for alumina in a KOH cleaning solution at

100°C and a gold plating solution at 78°C (6).

Figure 1 gives the experimental results of strength as a function of stressing rate of alumina in the KOH and gold

solutions (6). For comparison, the fatigue strength results

determined for the same alumina in a moist air environment are

included (5) . From this data and a knowledge of the inert

strength of the samples, the crack propagation parameters B and

N were determined from Equation (3) for the various environments.

To validate the applicability of fracture mechanics principles

in predicting failure probability in the processing solutions, the

actual fatigue strength distributions were compared to predictions

based on Equation (3) using the appropriate values for B and N and

the inert strength distribution of the samples. Figure 2 gives

these comparisons and it is evident that agreement between the

theoretical predictions and the actual fatigue strength distributions

3 (MPa) K>0 190 200 300

Ln 3 (MPa)

3 OtPa)

too SO 200 300

Ln 3 (MPa)

Figure 2. Comparison to theoretical predictions of the strength

distributions of polycrys talline alumina in a) KOH

cleaning solution at 100°C and b) gold plating bath at

78°C (from ref. 6).
is good. It should also be noted that these strength distributions

are bimodal since the low strength regime of the distribution

was caused by gross flaws induced in the samples during grinding.

The severity of the processing conditions on the alumina

substrates can be most easily assessed by means of a lifetime

prediction diagram (Figure 3) based on Equation (2) and the inert

strength distribution of the substrates. Although the strengths

of the substrates could not be measured directly, they were

estimated from the strength distribution of the samples and the

Weibull size relation?

S

X = S

2

<A

2

/A

i

)1/m

(10)

where and are the strengths of the substrates and samples,

respectively, and A- and A^ are the surface area of the substrates

and samples, respectively. Figure 4 shows the estimated inert

strength distribution of the substrates. It should be noted that

it was assumed that the inert strength distribution of the

Figure 3. Lifetime prediction diagram for polycrystalline alumina

in wet processing environments and moist air (from ref.

6 ).
71

20

10

5.0

4.0

3.0

2.0

1.0

0.8

0.6

0.4

0.2

4. 0 So 0 6. 0 In Sjj5MPa)

£

i

sa©

fiL Ul

OS3

Figure 4. Estimated Weibull inert strength distribution for

alumina substrates (from ref. 6)

substrates was not altered through any interaction between the

metal and alumina during sintering.

The residual tensile stress in a substrate having cofired

metallurgy is dependent on the thermal expansion mismatch

between the metallurgy and the ceramic, the temperature difference

between the set point of the system and room temperature, and the

geometry of the metallurgy in the system. In molybdenum-alumina

substrates residual stresses of 200 MPa can be generated. If

substrates having such stresses are subjected to the KOH cleaning

solution or the gold-plating solution for 30 minutes, respective

S /o ratios of 1.75 and 2.27 are predicted (Figure 3). Since

is known, S can be calculated and a failure probability determined

from Figure 4. With a 200 MPa residual stress, an 8% failure rate

would be expected for the substrates in the 10% KOH solution, and

40% of the substrates in the gold-plating bath would crack after

30 minutes'. Also, substrates having a residual stress of this

magnitude would have a failure rate of 8% in hot, humid air.

Clearly, failure rates of this magnitude are unacceptable.
The failure probability of the substrates can be lowered by:

(1) using a stronger alumina to make the substrates, (2) reducing

the processing times, or (3) reducing the residual stress in the

substrates. The obvious choice is to reduce the residual stress

in the ceramic. This can be accomplished by reducing the thermal

expansion mismatch between the metal and the ceramic, by altering

the geometry of the system, or by a combination of both. Since

the thermal expansion differential can be reduced only by altering

the metal configuration and/or the ceramic, this choice alone is

not particularly attractive. Similarly, if the stress is lowered

only through geometric redesign, the potential geometric density

of the metallurgy in the system becomes limited. Thus, a combination of both approaches is generally considered most effective in

reducing stress.

Obviously, the ideal solution would be to completely eliminate

any residual tensile stress in the substrates; this, however, is

not always possible without drastically altering the material

system. The problem then is to reduce the failure probability of

the substrates to a magnitude compatible with good product reliability.

Regardless of the technique used, if the stress in the ceramic

is lowered, for instance, to 50 MPa, the lifetime prediction diagram and the inert strength distribution can again be vised to

assess the probability of stress corrosion cracking of the substrates. If substrates having a residua] stress of 50 MPa are

placed in the KOH cleaning solution or in the gold plating bath

for 30 minutes, it can be determined from Figure 3 and 4 that the

failure probabilities are 0.2 and 0.34% respectively. While the

possibility of stress corrosion cracking of substrates during

processing is not completely eliminated by reducing the residual

stress to 50 MPa, the resultant failure rates are now low enough

so as to offer a minimal impact on product reliability.

The above examples of lifetime predictions serve to illustrate the importance of fracture mechanics theory in designing

electronic substrates fabricated by cofired metallurgy, namely,

that substrates of various designs can be compared for the probability of stress corrosion cracking due to the residual tensile

stress in the substrates. In this way, fracture mechanics theory

is useful in making more rational design decisions. These predictions can also be checked by fabricating substrates of several

designs with different residual stresses, and then subjecting the

substrates, for instance, to the gold-plating bath for a given

period of time. By comparing the incidence of stress corrosion

cracking with that predicted by fracture mechanics theory, the

theory can be experimentally validated. This was done in the present research program, and the experimental results generally

agreed with those theoretically predicted (6)

.
73

Proof Testing

Over speed proof testing is used in the grinding wheel

industry to assure individual wheel safety by providing protection

against excessive variability in wheel strength and fatigue

failure. Empirical standards have been established for overspeeding

on the basis of measurements of relative strengths and experience

from wheel usage. To place overspeed proof testing on a more

sound, fundamental basis, the fracture mechanics principles described in the previous section were applied to verifying the use of overspeed proof testing in assuring against the fatigue failure

of vitrified grinding wheels (7). Two' vitrified wheel specifications were used in this study: 17A90-L5-VX2 and 2A601-K4-V9676

For purposes of identification these two specifications will be

labelled VGW1 and VGW2, respectively.

To measure the crack growth parameters appropriate for VGW1

and VGW2 in a commercial coolant, stress rupture and stressing

rate experiments were used (7). Figure 5 compares minimum lifetime

predictions for these two vitrified grinding wheels based on the

two test techniques (in this case labelled static fatigue and

dynamic fatigue). This figure is based on Equation (9) where S „ is taken to be equal to a . Since the maximum applied stress in'

service for these two vit?ified grinding wheels was calculated to

be 1280 psi and the desired minimum lifetime in service was 6 weeks

(7) , it is evident from Figure 5 for a given vitrified grinding

wheel composition the a /a ratio based on the two test techniques

are not significantly dSfferent. From the a /a ratio the overspeed proof test could then be calculated. p

<$/%

Figure 5. Minimum lifetime predictions comparing static fatigue

(SF) and dynamic fatigue (DF) techniques for VGW1 and

VGW2 in aqueous coolant at 35°C (from reference 7)
Table 1

Prediction of the Spin-Proof-Ratio (SPR) Necessary to Insure

Against Delayed Failure in Service of Vitrified Grinding Wheel

for a 6 Week Lifetime at 1280 psi. (after Reference 7)

Material Test Technique a (psi) SPR

VGW1 Static Fatigue 2715 1.57

VGW1

‘ Dynamic Fatigue 2825 1.60

VGW2 Static Fatigue 2285 1.45

VGW2 Dynamic Fatigue 2280 1.45

Table I summarized these results where the spin-proof-ratio (SPR)

is defined to be the overspeed proof divided by operating speed.

These predicted SPR for the vitrified grinding wheels evaluated

in this study can be compared to those actually used: 1.80 for

VGW1 and 1.50 for VGW2. It is evident from these results that

the SPR actually used for these wheel specifications is greater

than that predicted on the basis of fracture mechanics theory.

Therefore, it is believed that the overspeed proof test currently

used for these vitrified grinding wheel compositions is adequate

for preventing fatigue failure in service. In support of this

conclusion it is significant to note that there has never been a known fatigue failure in service for these wheel specifications.

To further demonstrate the effectiveness of proof testing,

the distribution of initial wheel bursting strengths for the

wheel specification VGW2 was compared to that after overspeed

proof testing (7)

.

The tested wheels had dimensions 20 x 1 x 5

in. For the proof test the wheels were accelerated in air up to

the proof speed of 4550 RPM and then quickly decelerated. During

proof testing 3 of the 20 wheels burst. Figure 6 compares the

initial wheel bursting strengths to the after-proof strengths.

The wheel bursting strength was taken to be the tensile stress

present at the wheel arbor when reached its burst speed. It is

quite evident that proof testing eliminated the weak samples,

resulting in the after-proof strength distribution being stronger

than the initial distribution. It is also important to note the

good agreement between theory and experiment.

Characterization of High Temperature Fatigue Behavior

The increasing demand for materials that will perform under

temperature conditions too severe for metals has led to many

possible applications for ceramics. Thus, it is important to
75

S (Ml)

3000 WO 5000

Figure 6. Comparison of wheel bursting strengths before and after

proof testing for wheel specification VGW2 (from ref. 7).

characterize the strength and fatigue behavior of ceramics at

elevated temperatures to establish allowable stress levels for

component design. To study the high temperature strength and

fatigue behavior of high purity alumina (Wesgo AL 995) the

stressing rate test technique was used because of its simplicity (8)

The effect of temperature on the strength and fatigue behavior

of high purity alumina is given in Figure 7 (8) . The temperature

dependence of both strength and fatigue behavior is small up to

500°C. From 800 to 1100°C both the strength and fatigue resistance

decreases markedly. A convenient way to show the effects of

fatigue is to compare minimum lifetime predictions, based on Equation (9) taking S . to be equal to a , as a function of temperature. Figure 8 gives these predictioRs and clearly illustrates

that there is little difference in the fatigue behavior of high

purity alumina from 23 to 500°C; however at higher temperatures

fatigue effects become very important. For example, to assure a minimum lifetime of one year under a service stress of 100 MPa the

required proof stress ratio is 2.27 for a temperature of 23°C but

for 1000°C it is 6.90 and for 1100 c’C it is 14.91. These elevated

temperature proof stress ratios would result in such large proof

stresses that no sample would be able to pass the proof test. To

get more reasonable proof stresses, the expected minimum life or allowable stress in service must be decreased for these elevated

temperatures. For example, if the allowable stress is reduced to

30 MPa, then the proof stress to insure a minimum lifetime of 1

year at 1000°C is 179 MPa and 361 MPa at 1100°C.
TEMPERATURE (*F)

Figure 7. Median fracture strength of alumina as a function of

temperature and stressing rate (from ref. 8).

I 2 l 4 5 e 7 10

Figure 8. Minimum lifetime prediction diagram for alumina as a function of temperature (from ref. 8).
77

Scanning electron micrographs shoved that the fracture in

this high purity alumina changed from transgranular at 23°C to

intergranular at temperatures above 800°C (8) . In addition, a glassy phase was found at the fracture origins of the samples

tested at above 800°C, giving evidence that the glassy phase is

playing a major role in determining the strength and fatigue

behavior of high purity alumina at elevated temperatures. However,

proof test results (8) showed that the preexisting flaw population

is not altered at elevated temperatures. Figure 9 compares the

strength distribution of alumina at 23 and 1100°C before and after

proof testing at room temperature. It is evident that this proof

test eliminated the weak samples so that the after proof strength

distribution at low failure probabilities are stronger than the

initial distributions and that agreement between theory and experiment is good. On the other hand. Figure 10 shows that proof

testing alumina at 1000 °C is not effective in improving the

strength distribution at 1000°C; however, the good agreement

between experiment and theory gives evidence that fracture mechanics

theory applies . In this case weak samples are eliminated by proof

testing at 1000°C, but flaw growth during the proof test weakens

the survivors to such a degree that proof testing is not effective

in improving the strength distribution.

Figure 9. Effect of room temperature proof testing on the

strength distribution of alumina at 23°C and 1000°C

(from ref. 8)
78

SIRtSS (MPo)

ISO 175 200 225

CD

<4SBO

K

a.

Figure 10. Effect of proof testing at 1000°C on the strength

distribution of alumina at 1000°C (from ref. 8).

In summary, it is believed that at moderate temperatures

the fatigue failure of alumina is caused by moisture assisted

subcritical crack growth; whereas, at elevated temperatures it

is caused by subcritical crack growth enhanced by the presence

of a glassy phase. It is also thought that the same pre-existing

flaws control the strength both at room temperature and at

elevated temperatures.

LIMITATIONS OF THE LIFETIME PREDICTION METHOD

The lifetime prediction theory presented in this paper is

based on the assumption that subcritical crack growth from preexisting flaws is the only mechanism of strength degradation.

Using this assumption, component lifetime can be predicted once

the initial flaw population, the crack growth rate and the

critical crack size have been determined. The theory is deterministic because the element of chance is eliminated from the

lifetime prediction once these parameters have been evaluated.

The accuracy of the lifetime prediction will depend on both the

accuracy with which the pertinent parameters can be evaluated

(9-11) , and the adequacy of the crack growth equation (12)

.

This requirement does not detract from the basic deterministic

nature of the method. The prediction scheme can, however, be
79

invalidated when the basic assumptions of the theory are not

satisfied. Thus, fracture processes, other than subcritical

crack growth, can invalidate failure predictions based on this

theory. Unfortunately, other fracture processes do occur for

materials currently under consideration for use in heat engines,

so that alternate mechanisms of failure must be considered for

any scheme of predictive failure for heat engine components. As

will be shown, proper modeling of these factors permits extension

of the theory to handle fracture situations that are not currently

included in the method of lifetime prediction.

As with most other ceramic materials, silicon nitride and

silicon carbide fail at low temperatures from preexisting flaws.

As discussed by Rice et al. (13,14) and by Richerson and Yonushonis

(15) , these flaws are usually surface cracks introduced into

silicon nitride and silicon carbide components by machining when

the components are manufactured. For hot-pressed silicon nitride

at temperatures less than ^1000°C, fracture originates from

surface flaws in a completely brittle manner (13,14). At higher

temperatures, however, plasticity of components and effects

resulting from the reactive nature of the test environment

intervene to alter the mode of fracture (13) . The same types of

effects occur for hot-pressed silicon carbide at higher temperatures

(M.400°C). In oxidizing environments, the effects of temperature

on mechanical behavior can be attributed to: (1) enhanced

plasticity at the crack tip which results in subcritical crack

growth by creep fracture (16,17), and (2) surface oxidation which

results in the healing of surface machining cracks, and for some

materials, the generation of new flaws that act as new origins

for fracture (13-15, 18-20). Both processes have to be considered

to accurately predict the lifetime of high temperature structural

components. A variety of other processes (thermal shock, particle

impact, chemical corrosion, component contacts, etc.) can also

intervene to effect the fracture behavior of heat engine ceramics.

Although a complete theory of failure would require all of these

processes to be considered, such extensive considerations are clearly beyond the scope of this paper. The discussion in the

remainder of this paper will be more limited in scope, dealing

with pit formation, crack healing and crack growth. Materials

to be considered are magnesia-doped, hot-pressed, silicon nitride

and reaction bonded silicon nitride for which the processes that

occur at elevated temperatures are best understood.

High temperature oxidation has been shown to be particularly

effective in altering the initial flaw population in hot-pressed,

silicon nitride (NC132) . Investigations of the strength and

oxidation behavior of this material have shown that surface

induced machining flaws are reduced in severity by high temperature oxidation of the ceramic surface (13-15, 18-20). The
oxidation process is complex involving mass transport of impurities

from the matrix, and transport of oxygen and nitrogen through

the glass oxide layer that forms on the surface of the ceramic

(17,21). As the oxide scale grows, the initial silicon nitride

surface recedes, reducing the size of the surface cracks and

thus their severity (Figure 11) . If oxidation continues long

enough, the surface cracks can be completely removed from the

ceramic surface. In addition to crack removal by dissolution,

plastic deformation at elevated temperatures also permits blunting

of machining flaws and relief of surface stresses, thus enhancing

the strength of the component (22,23). The effectiveness of

oxidation in reducing flaw severity can be illustrated by the

introduction of fresh cracks into the surface of silicon nitride

by indentation (20,24). As illustrated in Table 2 on hotpressed silicon nitride (billet B), exposure at 1200°C in air

increased the strength from 408 MPa at room temperature to 488

MPa after 100 hours exposure, largely as a result of decreasing

the severity of the flaw associated with the indentation.

Because of the beneficial effect of oxidation, high temperature

exposure in air is being used as a means of improving the finish

of hot-pressed silicon nitride ceramic blades (15,23). Similar

strengthening results are observed for reaction bonded silicon

nitride (NC350) (23,24).

EFFECT OF OXIDATION ON

FLAWS IN Si 3 N4

J - OXIDE- —

*-$13^4-

(a) (b) (c)

Figure 11. Effect of exposure at 1200°C on flaws in silicon nitride

(a) surface crack and subsurface inclusion before ex- posure; (b) after exposure crack size is reduced by

surface dissolution and subsurface flaw reacts with

environment. At the same time, crack blunting or crack

healing occurs; (c) oxide growth has removed the subsurface crack and enlarged the subsurface flaw to produce a pit in the silicon nitride (from ref. 34).
Table 2.

Strength of Indented Specimens: 2kg Load (34)

Material

Test-Temp.

°C

Exposure

Conditions

Strength

MPa

Fracture

at indentation

NC132 25 as indented 397 + 12 Yes

Billet A 25 16 hr 1200°C 432 + 27 Yes

25 100 hr 1200°C 461 + 42 No

1200 1/2 hr 1200°C 438 + 15 Yes

1200 16 hr 1200°C 396 + 42 Yes

1200 100 hr 1200°C 402 + 12 No

NC132 25 as indented 408 + 17 Yes

Billet B 25 100 hr 1200°C 524 + 47 Yes

1200 1/2 hr 1200°C 441 + 10 Yes

1200 100 hr 1200 e G 488 + 11 Yes

NC350 25 as indented 115 + 24 Yes

25 33 hr 1200°C 181 + 22 Yes

25 100 hr 1200°C 225 + 38 Yes

1200 100 hr 1200®C 188 + 64 Yes

NCX-34 25 as indented 491 + 16 Yes

Y.0 o 25 100 hr 1200°C 616 + 51 No

2 3 1200 100 hr 1200e C 744 + 25 Yes

If flaw removal were the only process occurring during

oxidation then the lifetime prediction methods presented earlier

in this paper could be used as a conservative basis for component

design. In this case, the starting flaws would decrease in size

as oxidation occurred, and the component would become stronger,

not weaker. Research on hot-pressed silicon nitride (NC132) has

shown that flaw generation during long term (>100 hr.) exposure

results in a decrease in the strength of this material (15, 20,

22). The strength decrease results from pit formation at the

ceramic surface due to rapid localized oxidation. Although the

exact cause of localized oxidation is not fully understood, it

probably results from the presence of impurities in the ceramic

which increase the oxidation rate as the oxide interface approaches

the location of the impurity (Figure 11; 20,26). The formation

of glass-like oxide mounds over the pits with holes in the

center of the mounds suggests that rapid gas generation is

associated with the nucleation and growth of these pits (20)
Figure 12. Effect of oxidation at 1371°C (2500°F) on the strength

of silicon nitride (NC132) . Note that at time zero

the scatter in strength is considerably greater than

that occurring after substantial periods of exposure

(from ref. 25).

The effect of pit formation on the long term strength is

illustrated in Figure 12 which shows that after 1000 hours

exposure magnesia-doped, hot-pressed silicon nitride has lost

approximately 50 percent of its strength (25) . This strength

decrease seems to occur most rapidly during the initial stages

of exposure, suggesting rapid formation of pits initially, and

then stabilization of the pit structure after periods of about

1000 hours. Whether or not the pit structure has reached a state of static equilibrium, or is in a state of dynamic equilibrium has not yet been determined.

In freshly machined specimens the combined effect of crack

healing, and flaw generation should result in a strength enhancement followed by a strength degradation as pits form in the

material (20,24). This type of behavior has been confirmed by a number of investigators. Richerson and Yonushonis (15) have

shown, for example, that the strength of transverse ground

silicon nitride is increased by exposure to temperatures of

982°C or 1066°C for 50 hours (Table 3) . At higher temperatures

or for longer periods of time, a strength decrease is observed

as pits are formed in the surface. Wiederhorn and Tighe (20,24)

showed that this type of behavior depended on the particular

billet used for study, since some billets increased in strength
83

Table 3.

Room Temperature Strength of Hot-Pressed, Silicon Nitride (NC132)

Transverse Ground Specimens Tested in 4-Point Bending (15)

Exposure Strength and Samples Predominant Fracture

a C (°F) Standard Deviation Tested Region

time MPa (KS1)

Control 450 + 36

(63.0 + 5.0)

12 Grinding Groves

982 (1800)

50 hours

657 + 61

(92.0 + 8.6)

50 Surface (Flaws not

obvious)

1066,(1950)

50 hours

636 + 41

(89 + 5.7)

11 Surface (Flaws not

obvious)

1129 (2065)

140 hours

593 + 36

(83 + 5.0)

6 Surface Pits

1129 (2065)

240 hours

' 443 + 58

(62.0 _+ 8.1)

6 Surface Pits

1204 (2200)

24 hours

529 + 19

(74.0 _+ 2.6)

12 Surface Pits

1371 (2500)

24 hours

457 + 29

(64 + 4.1)

12 Surface Pits

with exposure at 1200°C (for times of 100 hr.) while others

decreased for the same exposure conditions. Richerson and

Yonushonis (15) have made similar observations on hot-pressed,

silicon nitride. More recently, the combined effect of flaw

healing and pit formation was demonstrated by Jones and Rowcliff

(22) , for silicon nitride exposed at temperatures of 1370°C for

periods up to 64 hours. For both indented (knoop; 1 Kg) and

transverse ground specimens, the strength first increased then

decreased as the exposure time increased (Figure 13)

Reaction bonded silicon nitride behaves in a manner quite

different from that of the hot-pressed material. Flaw healing

results in a significant strength increase in this material at

elevated temperatures. When the material is cooled to room

temperature, however, much of this increase is lost as a result

of surface cracking caused by a phase transformation of cristobalite in the oxide coat that forms on the surface at high temperatures (27)

The absence of strength degradation (at 1200°C)
1 2 3 4 1 8 7 8 9

(TIME AT TEMPERATURE)^ -hours'^

Figure 13. Effect of high temperature exposure on the strength

of silicon nitride (NC132) measured at room temperature (from ref. 22).

after sustained (1000 hr.) exposures (28), suggests that pit

formation and subcritical crack growth are not significant

factors limiting the lifetime of this material.*

LIFETIME ESTIMATE: EFFECT OF FLAW HEALING AND PIT FORMATION

From the above discussion, we conclude that flaw healing,

pit formation and subcritical crack growth must be considered

simultaneously in order to establish an accurate method of

lifetime prediction. The development of a failure prediction

method requires each of these processes to be evaluated independently. Once they have been characterized, mathematical techniques

can be used to determine the relative contribution of each of

these processes to the total failure time.

Subcritical crack growth can be characterized either by

strength or fracture mechanics techniques. However, when flaw

healing or pit formation is occurring, the use of strength

techniques for evaluating crack growth parameters may not be a good procedure, since both of these processes have an independent

effect on the material's strength. The feasibility of quantifying

the crack growth rate by fracture mechanics techniques has been

demonstrated by Evans and Wiederhorn (30) , who have shown that

*Recently, Richerson et al. (29) have observed pit formation in

heavily oxidized reaction bonded silicon nitride.
the crack velocity can be represented as a power function of the

applied stress intensity factor (Equation 1) . Details on the

application of fracture mechanics techniques to crack growth

data can be found in ref. 31.

A second method of collecting crack velocity data was

illustrated recently by Rowclif f (22) , who used 3 point bend

specimens to study the static fatigue of silicon nitride and

silicon carbide. The specimens were all indented to form wellcharacterized cracks in the tensile surface of the specimens.

The technique is based on the assumption that equation 1 is the

correct form of the crack velocity equation* By rearranging

Equation 2, the following equation is obtained for the failure

time, t, of a component that is subjected to a constant applied

load

t _ i!i (id

C (n-2)v

i

Thus, by measuring the initial crack size, a., and the time to

failure, t, the initial crack velocity, v„, can be determined.

The stress intensity factor thgt corresponds to v. is obtained

directly from a^ (K^, . - oYi/aT) .

Data on flaw healing or on pit formation can be obtained

from strength measurements. Data on magnesia-doped, hot-pressed

silicon nitride and reaction bonded silicon nitride suggest that

crack healing occurs relatively quickly and is completed before

pits begin to fore in the component surface. Therefore, strength

changes during the early stages of component exposure can be

used to characterize the crack healing process; strength changes

occurring after extended exposure can be used to characterize

the pit formation process. For normally machined surfaces,

quantification of both of these processes requires the collection

of extensive strength data so that the strength distribution can be determined accurately as a function of exposure time.

Data on hot-pressed silicon nitride suggests that both the

mean strength and its standard deviation change with exposure

time when flaw healing occurs (20,24,25). In terms of a Weibull

distribution, Equation 4, both the Weibull strength, S , and the

shape parameter, m, are a function of time. Data on reaction

bonded silicon nitride suggests that only S

q

is a function of

*The validity of the data obtained by this technique is limited by

the assumption that the crack size at failure is small relative to

the specimen dimensions. As has already been noted, this same limitation applies to earlier parts of this paper.
86

time (28). The effect of flaw healing on specimens that contain

large initial cracks can be determined from studies on specimens

that have been indented so that relatively large cracks are introduced into the specimen surface. Studies of this type have

been conducted recently by Tighe and Wiederhorn (20,24) and by

Cubicciotti et al. (22). The investigation by Singhal et al. (25) of

strength degradation after long term exposure suggests that for

both crack healing and pit formation S and m seem to approach

fixed values after a certain amount of time, so that both distributions eventually become stable with regard to further exposure.

Once crack growth, flaw healing, and pit formation have been

quantified, the combined failure probability, and the total time

to failure can be estimated. To obtain these estimates it is

necessary to develop a procedure to determine which mechanism

controls failure at a given probability level. Flaw healing and

pit formation as a combined failure mechanism will be discussed

first. For simplicity it is assumed that the Weibull slope, m^,

for crack healing is less than the Weibull slope, m^, for pit

formation, and that m^ and m

2 do not depend on time. The

Weibull strength for crack healing, S^, and that for pit formation,

S^j, are assumed to follow exponential decay curves that can be

fit to the following formulae:

(flaw healing) St=S + AS (1-exp (-at))

°1 °1

(12)

(pit formation) So=S - AS (1-exp (— 8

1

)

)

02 02

(13)

where S , S , AS , AS a and 8 are determined from experimental

strength1 data? The^nitiil Weibull strengths for flaw healing and

pit growth are S and S respectively, while AS and AS are the

changes in strength for ?ftese two processes af ter°txtended° 2 exposure

times, a and 8 determine the rate at which these changes in

strength occur. If Equations 12 and 13 are substituted into

Equation 4, time dependent strength distributions are obtained

that describe the effect of exposure time on strength.

The type of behavior expected from Equations 12 and 13 is

shown schematically in Figure 14. We see that during the initial

stages of exposure, failure is determined entirely by the initial

flaw population, while in the later stages of exposure, pit

formation dominates the strength distribution. At any point in

time, the combined failure probability, F, is given by the

following relation (32):

(1-F) = (1-F

1

) (1-F

2

) (14)

where F^ and F^ are the time dependent failure probabilities for

flaw healing and pit formation respectively. Thus, a unique
87

Figure 14. Schematic representation of the effect of high temperature exposure on the strength distribution: (a) initial

strength distribution. Curves are assumed to move in

the direction of the arrows as the exposure time is

increased; (b) distribution after the flaw structure

has reached a state of "equilibrium". Proof testing

or nondestructive evaluation can be used to improve

reliability at stresses lower than the point of intersection of the two lines, statistics can be used to

the right of the point of intersection.

equation is obtained that relates the failure strength, S, to

the failure probability, F.

With regard to lifetime prediction, the above discussion is

sufficient to draw several important conclusions with regard to

proof testing and nondestructive testing. At any time during

exposure, the strength of components to the left of the point of

intersection of the two curves shown in Figure 14 will be controlled by the initial flaw population. Those specimens that lie to

the left of the intersection will contain the largest flaws and

consequently will be the ones to he eliminated by proof testing

or nondestructive inspection. Since the initial flaw population

controls the strength in this regime, proof testing and nondestructive evaluation techniques can be used to truncate the population,

and the theory presented earlier in this paper can be used to

provide a conservative estimate of the component lifetime. The

intersection of the two curves shown in Figure 14 provides an upper limit to the proof test load that can be usefully employed

to truncate the initial flaw population. Above this load,

component strength depends not on the flaws initially present in
the component, but on those that are generated by exposure to

elevated temperatures (20,24).

After long periods of exposure, proof testing or nondestructive evaluation can be applied again over the entire range of

strengths provided the flaw distribution (pits and cracks) has

reached a static state. If, however, the distribution to the

right of the intersection in Figure 14 represents a dynamic

state of equilibrium, then new pits will be continuously generated

as the old ones are removed. As a consequence, nondestructive

evaluation and proof testing will not be a viable lifetime

prediction technique . For the dynamic equilibrium situation,

statistical techniques such as those described earlier in this

paper would have to be used fq^ purposes of lifetime prediction.

Pits generated by the oxidation process would serve as the

initial flaw distribution for the lifetime prediction.

The combined effects of subcritical crack growth and pit

formation on strength can be determined by considering the rate

at which each of these processes degrade the strength. As long

as strength degradation due to pit formation exceeds that

resulting from crack growth, subcritical crack growth will not

play a major role in the strength degradation process; when the

converse is true, crack growth will dominate the strength degradation process. The switch from one controlling mechanism to the

other will occur when the two strength degradation rates are equal.

The rate of strength degradation due to pit formation can

be determined by substituting S 2 of equation 13 for S of

Equation 4, and differentiating with respect to time;

dS/dt = -3 AS [exp(-Bt) ] [In(l-F)” 1

] 1/ra

(15)

°2

The rate of strength degradation due to crack growth has

been derived by Fuller et al. (4) and is given by the following

expression:

. dS/dt = - (AY

2

K

IGn“2

/2) (a/S) n

S

3

(16)

where A and n are defined by equation 1; Y is a geometric constant

a is the applied stress and S is the instantaneous strength.

Equating Equations 15 and 16, and substituting Equations 4 and 13

for S, an equation can be obtained which gives the time, t , at

which the mechanism of strength degradation changes from pi?

formation to crack growth. The total time to failure is then

equal to t , plus the time, t , for the crack to grow to a critical sSze. t can be determined from Equation 2, (using

Equations 4 and l5 to evaluate the strength at which the degradation mechanism changes from pit formation to crack growth)

.
87

Figure 14. Schematic representation of the effect of high temperature exposure on the strength distribution: (a) initial

strength distribution. Curves are assumed to move in

the direction of the arrows as the exposure time is

increased; (b) distribution after the flaw structure

has reached a state of "equilibrium" . Proof testing

or nondestructive evaluation can be used to improve

reliability at stresses lower than the point of intersection of the two lines, statistics can be used to

the right of the point of intersection.

equation is obtained that relates the failure strength, S, to

the failure probability, F.

With regard to lifetime prediction, the above discussion is

sufficient to draw several important conclusions with regard to

proof testing and nondestructive testing. At any time during

exposure, the strength of components to the left of the point of

intersection of the two curves shown in Figure 14 will be controlled by the initial flaw population. Those specimens that lie to

the left of the intersection will contain the largest flaws and

consequently will be the ones to be eliminated by proof testing

or nondestructive inspection. Since the initial flaw population

controls the strength in this regime, proof testing and nondestructive evaluation techniques can be used to truncate the population,

and the theory presented earlier in this paper can be used to

provide a conservative estimate of the component lifetime. The

intersection of the two curves shown in Figure 14 provides an upper limit to the proof test load that can be usefully employed

to truncate the initial flaw population. Above this load,

component strength depends not on the flaws initially present in
the component, but on those that are generated by exposure to

elevated temperatures (20,24)

„

After long periods of exposure, proof testing or nondestructive evaluation can be applied again over the entire range of

strengths provided the flaw distribution (pits and cracks) has

reached a static state. If, however, the distribution to the

right of the intersection in Figure 14 represents a dynamic

state of equilibrium, then new pits will be continuously generated

as the old ones are removed. As a consequence, nondestructive

evaluation and proof testing will not be a viable lifetime

prediction technique. For the dynamic equilibrium situation,

statistical techniques such as those described earlier in this

paper would have to be used fojc purposes of lifetime prediction.

Pits generated by the oxidation process would serve as the

initial flaw distribution for the lifetime prediction.

The combined effects of subcritical crack growth and pit

formation on strength can be determined by considering the rate

at which each of these processes degrade the strength. As long

as strength degradation due to pit formation exceeds that

resulting from crack growth, subcritical crack growth will not

play a major role in the strength degradation process; when the

converse is true, crack growth will dominate the strength degradation process. The switch from one controlling mechanism to the

other will occur when the two strength degradation rates are equal.

The rate of strength degradation due to pit formation can be determined by substituting S 2 of equation 13 for S of

Equation 4, and differentiating with respect to time:

dS/dt = -3 AS [exp(-St) ] [lnd-F)"1

] 1 ^

(15)

°2

The rate of strength degradation due to crack growth has

been derived by Fuller et al. (4) and is given by the following

expressions

. dS/dt « - (AY

2

K__

n“2

/2) (a/S) n

S

3

(16)

JLL where A and n are defined by equation 1; Y is a geometric constant

a is the applied stress and S is the instantaneous strength.

Equating Equations 15 and 16, and substituting Equations 4 and 13

for S, an equation can be obtained which gives the time, t , at

which the mechanism of strength degradation changes from pi?

formation to crack growth. The total time to failure is then

equal to t , plus the time, t , for the crack to grow to a critical sSze. t can be determined from Equation 2, (using

Equations 4 and l5 to evaluate the strength at which the degradation mechanism changes from pit formation to crack growth)

.
89

Although the mathematical scheme for obtaining the total time to

failure is straightforward, the equations are not included here

because of their complexity. Never-the-less , the procedure does

provide a rational method of obtaining the total time to failure

when pit formation plays a critical role in determining component

lifetime. It is worth noting that the final equation can be

expressed in terms of strength, probability and failure time.

This type of representation was first suggested by Davidge et

al. (33) as a method of design with ceramic materials.

SUMMARY

In this paper, a review is presented of current techniques

used to evaluate the lifetime capability of structural ceramics.

Three successful applications of these techniques are presented

and discussed in order to demonstrate the viability of the

techniques for purposes of engineering design. Finally, limitations of the techniques are discussed with regard to heat-engine

applications. An extension of theory is recommended to treat

crack healing and pit formation resulting from high temperature

exposure.

The theory presented in this paper is based on the theory

of linear-elastic, fracture mechanics. Failure of structural

ceramics is assumed to be the result of subcritical crack growth

from pre-existing flaws in the structural components. Lifetime

predications require evaluation of the initial flaw size and of

the rate at which cracks grow when subjected to applied loads.

The three techniques that can be used to evaluate the initial

flaw size are nondestructive evaluation, proof-testing and

statistics, while the three techniques that can be used to

evaluate the rate of crack growth are fracture mechanics techniques

,

static fatigue techniques (stress rupture) and dynamic fatigue

techniques (constant loading rate to failure) . The critical

equations required for purposes of failure prediction are presented

in the paper. Successful applications of the lifetime prediction

method are illustrated for electronic-substrate ceramics, for

vitreous grinding wheels and for high-purity aluminum oxide at

elevated temperatures.

Potential limitations in applying the technique to heat

engine ceramics, such as silicon nitride and silicon carbide,

are discussed. These limitations arise because of flaw generation

and flaw healing that is observed to occur in these materials at

elevated temperatures. Flaw generation can be caused by a number of processes: oxidation or other environmental attack

which results in surface pit formation, particle impact or component contact which results in localized stresses that

nucleate cracks, and thermal shock which results in transient

stresses that can result in crack formation. Flaw healing can
be attributed to surface dissolution caused by oxidation and

silicate glass formations These processes change the initial

flaw population so that failure no longer occurs from machining

flaws originally present in the ceramic. In this situation the

development of a model for failure prediction requires the

consideration of flaw generation processes.

Extension of the theory presented in earlier sections of

the paper is illustrated for the combined processes of flaw

healing, pit formation and subcritical crack growth. The method

independently evaluates these three processes and then determines

the relative contribution of each of these processes to the

total failure time. Conditions are established for proof testing

of as-received components. Because of the occurrence of pit

formation, a maximum value for the proof test load is established,

above which proof testing does not contribute to component

reliability. After long term exposure, proof testing or nondestructive evaluation can be applied again provided the flaw (pit)

distribution has reached a state of static equilibrium. If,

however, the flaw distribution has reached a state of dynamic

equilibrium, then statistical techniques must be applied for

lifetime prediction. Assuming that the strength distribution of

pits can be represented by a two parameter Weibull distribution,

the application of statistics is discussed for the combined

failure modes of pit formation and crack growth. For these

conditions, equations are developed for determining the total

failure time.

ACKNOWLEDGEMENTS:

One of the authors (JER) is pleased to acknowledge the

support of the Office of Naval Research, Metallurgy and Ceramics

Program, the other authors are pleased to acknowledge the

support of the Department of Energy, Division of Fossil Fuel

Utilization.

REFERENCES

1. A.G. Evans and S.M. Wiederhorn, ’’Proof Testing of Ceramic

Materials-An Analytical Basis for Failure Prediction," Int.

J. Fract., 10, 379-92 (1974).

2. S.M. Wiederhorn, "Reliability, Life Prediction, and Proof

Testing of Ceramics," pp. 635-65 in Ceramics for High

Performance Applications , ed. by J. J. Burke, A.E. Gorum,

and R.N, Katz, Brook Hill Pub. Co., Chestnut Hill, MA.

(1974).
91

3. J.E. Ritter, Jr., "Engineering Design and Fatigue Failure

of Brittle Materials," pp. 667-686 in Fracture Mechanics of

Ceramics

,

Vol. 4, ed. by R.C. Bradt, D.P.H. Hasselman, and

F. Lange, Plenum Press, NY (1978).

4. S.M. Wiederhorn, E.R. Fuller, Jr., J.E. Ritter, Jr., and

P.B. Oates, "Proof Testing of Ceramics: I. Experiment"

NBSIR 79-1934; "II Theory " NBSIR 79-1944 , (December 1979).

5. J.E. Ritter, Jr. and J.N. Humenik, "Static and Dynamic

Fatigue of Polycrystalline Alumina," J. Mat. Sci., JJ+, 626-

632 (1979).

6. J.N. Humenik and J. E. Ritter, Jr., "Susceptibility of

Alumina Substrates to stress Corrosion Cracking During Wet

Processing," to be published. Bull. Am. Ceram. Soc.

7. J.E. Ritter, Jr. and S.A. Wulf, "Evaluation of Proof Testing

to Assure Against Delayed Failure," Bull. Am. Ceram. Soc.

57, 186-190 (1978).

8. K. Jakus, T. Service, and J.E. Ritter, Jr., "High Temperature

Fatigue Behaviour of Polycrystalline Alumina," to be published,

J. Am. Ceram. Soc.

9. S.M. Wiederhorn, E.R. Fuller, Jr., J. Handel, and A.G.

Evans, "An Error Analysis of Failure Prediction Techniques

Derived from Fracture Mechanics," J. Am. Ceram. Soc. 59 , 403-11 (1976).

10. D. F. Jacobs and J.E. Ritter, Jr., "Uncertainty in Minimum

Lifetime Predictions," J. Am. Ceram. Soc. 5J3, 481-486

(1976)

11. P.N. Thorby, "Experimental Errors in Estimating Times to

Failure," J. Am. Ceram. Soc. 59., 514-16, (1976).

12. S.M. Wiederhorn, "Dependence of Lifetime Predictions on the

Form of the Crack Propagation Equation," pp, 893-901 in

Fracture 1977, Vol. 3, D.M.R. Taplin, Ed., University of

Waterloo Press, Waterloo, Ontario, Canada (1977).

13. R.W. Rice, S.W. Freiman, and J.J. Mecholsky, Jr., "Fracture

Sources in Si~N, and SiC," pp. 665 in Proceedings of the

1977 DARPA/NAVSEA Ceramic Gas Turbine Demonstration Engine,

J.W. Fairbanks and R.W. Rice, Eds., Metal and Ceramics

Information Center Report MCIC-78-36, Battelle Columbus

Laboratories, Columbus, Ohio (1978).

14. S.W. Freiman, C. Cm. Wu, K.R. McKinney, and W.J. McDonough,

pp 655-663 in ref. 13.

15. D.W. Richerson and T.M. Yonushonis, "Environmental Effects

on the Strength of Silicon-Nitride Materials," pp. 247-71 in

ref. 13.

16. F.F. Lange, "Evidence for Cavitation in Crack Growth," J.

Am. Ceram. Soc. 62 222-3 (1979).

17. N.J. Tighe, "Structure of Slow Crack Interfaces in Silicon

Nitride," J. Mater. Sci. 13, 1455-63 (1978).
92

18. S.W. Freiman, A. Williams, J. J. Mecholsky and R.W. Rice,

"Fracture of Si_N, and SiC," pp. 824-34 in Ceramic Microstructures

s 76, R.M. Fulrath and J.A. Pask, Eds., Westview Press,

Boulder, Colo. (1977).

19. S.W. Freiman, J.J. Mecholsky, W.J. McDonough, and R.W.

Rice, "Effects of Oxidation on the Room Temperature Strength

of Hot-Pressed Si^N^-MgO and Si^N. -ZrO^ ," pp. 1069-76 in

Ceramics for High Performance Applications," J.J. Burke,

E.M. Lenoe, and R.N. Katz, Brook Hill Publishing Co.,

Chestnut Hill, Mass. (1978).

20. S.M. Wiederhom and N.J. Tighe, "Proof-Testing of HotPressed Silicon Nitride," J. Mat. Sci. 13 , 1781-93 (1978).

21. D. Cubicciotti and K.H. Lau, "Kinetics of Oxidation of Hot

Pressed Silicon Nitride Containing Magnesia," J. Am. Ceram.

Soc. , 61 512-7 (1978)

.

22. D.D. Cubicciotti, R.L. Jones, K.H. Lau, and D.J. Rowcliffe,

"High Temperature Oxidation and Mechanical Properties of

Silicon Nitride," Interim Scientific Report 5522-2, Nov.

15, 1978, Prepared for Air Force Office of Scientific

Research/NE, SRI International.

23. T.M. Yonushonis and D.W. Richerson, "Strength of ReactionBonded Silicon Nitride," pp. 219-233 in ref. 13.

24. S.M. Wiederhorn and N.J. Tighe, "Effect of Flaw Generation

on Proof-Testing," pp 689-700 in ref. 13.

25. A.F. McLean, E.A. Risher, R.J. Bratton and D.G. Miller, pp

133-134 in Brittle Materials Design, High Temperature Gas

Turbine, technical report AMMRC CTR 75-28 to the Army

Materials and Mechanics Research Center, Watertown, Mass.,

October 1975.

26. J.A. Rubin, "Probable Causes of Pitting in Hot-Pressed

Si_N, Ceramics," pp. 739-743 in ref. 13.

27. R.W. Davidge, A.G. Evans, A.G. Gilling and P.R. Wilgman,

"Oxidation of Reaction Sintered Silicon Nitride and Effects

on Strength," in Special Ceramics, 5, 329-44 (1972).

28. N.J. Tighe, to be published.

29. D.W. Richerson, private communication.

30. A.G. Evans and S.M. Wiederhorn, "Crack Propagation and

Failure Prediction in Silicon Nitride at Elevated Temperatures,"

J. Mat. Sci., 9, 270-278 (1974).

31. A.G. Evans, "Fracture Mechanics Determinations," pp. 17-48

in Fracture Mechanics of Ceramics, Vol. 1, R.C. Bradt,

D.P.H. Hasselman and F.F. Lange, eds.. Plenum Press, New

York (1974).

32. N.R. Mann, R.E. Shafer and N.D. Singpurwalla, Methods for

Statistical Analysis of Reliability Data, John Wiley and

Sons, New York (1974).
93

33. R.W. Davidge, J.R. McLaren and G. Tappin, "Strength-ProbabilityTime (SPT) Relationships in Ceramics," J. Mat. Sci. .8,

1699-1705 (1973).

N.J. Tighe and S.M. Wiederhorn, "Fracture of Brittle Materials

at High Temperatures," Air Force Materials Laboratory

Technical Report: AFML-TR-78-83, July 1978.

34.
94

MBS-114A (REV. 9-78)

u.s. dept, of comm. | 1. PUBLICATION OR REPORT NO. BIBLIOGRAPHIC DATA § SHEET j NBSIR 80-2047

a.Gov'L Accession No. 1

S. Recipient's Accession No.

| 4. TITLE AND SUBTITLE

APPLICATION OF FRACTURE MECHANICS IN ASSURING AGAINST

FATIGUE FAILURE OF CERAMIC COMPONENTS

5. Publication Date June 1980

' ' , " -

7. author(S) E# r. Fuller, Jr.

J. E. Ritter, Jr., S. M. Wiederhom, N. J. Tighe, and

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

13. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP)

Department of Energy

Fossil Fuel Utilization Division

Washington, D.C.

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

|

j Document describes a computer program; SF-185, FIPS Software Summary, is attached.

IS. ABSTRACT (A 200-word or less factual summary of most signi ficant information. If document includes a significant bibliography or

literature survey, mention it here.) This paper will review the application of fracture mechanics theory to the

prevention of delayed failure of ceramics. Three successful applications of this

theory of assuring the mechanical reliability of ceramics are discussed in order

to demonstrate the viability of the theory for purposes of engineering design.

Finally, a description is presented of practical limitations of the theory with

regard to heat engine application. Methods of overcoming these limitations through

modification of test procedures, and application of statistical theory are then

presented.

17. KEY WORDS ( six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Dynamic fatigue; fracture mechanics; lifetime prediction; proof testing;

stress corrosion; subcritical crack growth.

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NBSIR 80-2075

IN SITU OXIDATION OF Y 2 0

3

-DOPED Si^

N.J. Tighe

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

National Bureau of Standards

U.S. Department of Commerce

Washington, DC 20234

K. Kuroda, T.E. Mitchell, and A.H. Heuer

Case Western Reserve University

Cleveland, Ohio

Published in:

Electron Microscopy 1980

Vol . 4 , Hi gh Vol t.age

97

In Situ Oxidation of Y 20 3 -doped Si 3 N4

N. J. Tighe, National Bureau of Standards

Washington, D. C.

K. Kuroda, T. E. Mitchell and A. H. Heuer

Case Western Reserve University

Cleveland, Ohio

Hot-pressed Si

3

N

4

+8-13%Y

2

0

3 compacts, which exhibited catastrophic

oxidation in air at 740°C but only passive oxidation at 1380°C, were

studied by HVEM using an environmental cell. The in situ experiments

were started as part of a larger program to establish a model for this

catastrophic oxidation process . The low temperatures required for

initiation of oxidation are within the capability of the hot stage and

the reaction products are sufficiently complex to require analysis by

electron diffraction. This yttria-doped silicon nitride ceramic is of

interest for use in ceramic turbine engine components because of its

good mechanical properties at temperatures above 1200°C [1] . However,

until the cause of the low temperature catastrohpic oxidation is identified

and corrected the material cannot be manufactured successfully for these

components. The low temperature oxidation is similar in appearance to

the so-called "pest condition" and occurs in some billets and not in

others which were manufactured under the same apparent commercial,

powder processing and hot pressing conditions. Lange et al . [2] found

that billets they prepared in the Si^N^-Si^N^O-Y^S^O^ compatibility

triangle of the phase equilibrium diagram, shown in Fig. 1, were oxidation resistant; and, that billets with compositions outside this triangle

were unstable in oxidizing conditions.

In this paper, the material designated -A- exhibited only passive

oxidation during heating in air from 600°C to 1400°C; while the material
98

designated -B- exhibited catastrophic oxidation during heating in air at

* 740 C. Both billets were from the same manufacturer [3] but were purchased

at different times. A bar of material -B- when heated in a gradient

furnace from 735°C to 1380°C for 20 hours oxidized catastrophically at

the low temperature end, and passively at the high temperature end. The

major crystalline phases in both materials were identified by powder

x-ray diffraction as SSi^N^, Y^Si^O^N^ (H phase) and WSi^ (there were

some unidentified lines) . The tungsten phase (3-4%) is a result of

contamination from WC balls used in ball-milling of the silicon nitride

powder, and thus is a common impurity phase in hot-pressed Si^N^. The

WSi^ phase was not present in the castrophically oxidized portion of the

material B, but was present in the portion oxidized at 1380°C. Although

the compositions of A and B were similar, there was considerable phase segregation and inhomogeneity in material B as seen in the light microscope.

Both materials appear to have compositions outside the compatibility

triangle discussed by Lange et al. [2].

The Swann enviromental cell in the CWRU 650kV electron microscope

was used with a platinum strip heater, in flowing oxygen. Specimens

of materials -A- and -B- were prepared as ion-thinned discs with one

flat side and one dished side. This configuration gave good contact

with the grid heater; and, resulted in better heat conduction to the

specimen.

The microstructure of the yttria -A- is shown in Fig. 2, and is

similar to that found in other billets [4,5]. The large tabular grains

are Si^N^, the triangular shaped phases surrounding the grain corners

are the + Si^N^ phases and the dark particles along the grain

boundaries are the WSi^ and/or WC phases. During hot-pressing at 1750-1800°C,

* The specific materials identified are not the only available material.
99

the yttrium oxynitride phases can r sit and form a series of solid solutions

with the impurity phases and with the silica in the starting powder [5]

Both crystalline and non- crystal line phases of variable composition can

fill the interstices around Si^N

4 grains.

Specimens were heated gradually to 750°C in 40 torr of 0^. Under

these conditions oxide platelets were seen to nucleate on the surfaces

of the BSi^N^ grains, around holes and at the yttria phases as in Fig.

3. The platelets, which were randomly oriented and incoherent with the

matrix, gave spotty ring patterns both at temperature and after cooling

to room temperature. The platelets in Fig. 4 were identified from these

ring patterns as a mixture of Si02 and S^^O. During the oxidation

considerable drift and flexure of the specimens occurred, as is evident

in the figures

After a few minutes of heating, cracks appeared along some grain

boundaries; and, as oxidation proceeded a film was observed to grow by

the merging of nucleating clusters. These clusters changed diffraction

contrast as they merged. It is assumed at this time, that this process

represented the gradual reaction and volatilization of the tungsten rich

phase. The film thickened and the crack widened to form a sizeable hole

as seen in Fig. 5. Iron rich and tungsten depleted phases around reacted

thin foil edges were detected with an EDX system on a 200 KV instrument [7]

The yttria-silicon oxynitride phases became porous during the in

situ oxidation. This type of reaction can be seen at Y in Fig. 5 and in

Fig. 6. The morphology of the phases suggests that the initial, yttriarich regions were multiphase or possibly an amorphous-crystalline mixture.

The phase in Fig. 6 was identified from its electron diffraction pattern

as YSiO^N or the K-phase of Fig. 1.
100 The in situ specimens were bent and slightly pink when removed

from the strip heates. Later TEH examination showed that the oxidation

had not occured uniformly over the entire thin area of the specimens.

Some of the differences reflect temperature gradient and oxygen partial

pressure differences between parts of the specimen that are exposed through

holes in the grid heater and other parts of the specimen that were under

the grid bars. In the region shown in Fig. 7 few oxide platelets

are seen, and the boundaries between Si^N^ grains have opened. The

yttria phase visible on one edge of the grain was identified as Y^S^O^^

or the J phase. The oxidation rate differences could reflect also the

inhomogenity of the phase distribution mentioned previously.

Additional in situ experiments are needed to verify the mechanism

of the loss of W and the observed reaction differences. Video or cine^

recording will be used to make a more complete record of the

observations.

The results of the in situ experiments demonstrate that: castrophic

low temperature oxidation in yttria-silicon material -B- started at

surfaces exposed to oxygen; volatilization of a W containing phase created

voids; continuous films were produced over some grains; yttria rich

phases became porous; and considerable strain was associated with the

oxidation.

References

1. G. E. Gazza, J. Amer. Ceram. Soc. 56, 662 (1973).

2. F. F. Lange, S. C. Singhal and R.C. Kuznicki, Westinghouse Report

76-904 (1976).

3. The Norton Co. Material Designation NCX-34.

4. D. R. Clarke and G. Thomas, J. Amer. Ceram. Soc. 60 491-5 (1978).
101

5. 0. L. Krivanck, T. M. Shaw and G Thomas, J. Amer. Ceram. Soc., 62_,

585-89 (1979).

6. K. H. Jack in Nitrogen Ceramics, F. L. Riley ed. Noordhoff, 1977,

p. 109-125.

7. G. Nord, USGS, Reston, Va.

This work is supported partially by DOE Office of Coal Utilization under

Task Order No. A0861PS and by AFOSR grant 49620-78C-0053. The material -Bwas supplied by J. Wimmer of Garrett AiResearch, Phoenix, Az.

102

Fig. 2. Microstructure of hot-pressed

Si 3 N 4 + 8%Y 2 0 3 showing 6 S

i

3 N 4 (3) , grains

¥ Si ON phase (Y) WSi particles (W).

Fig. 3. Material -B- at 750°C in 5200 Pa after

heating for %5 min., showing oxide platelets

(arrowed) and reacting yttria phase (Y).
103

Fig. 5. Specimen at 750°C in 5300 Pa O 2 . The hole (H) and the film around it (Ox) were produced in <30 min (Material -B-)

.

Fig. 4. 3Si 3 N 4 grain with Si02 and Si 2 N 2 platelets produced during in situ oxidation. Yttria phase at Y. (Material-B-)

Fig. 6. Material-B-. After oxidation

the yttria phase (Y) is porous and layered,

3Si 3 N4 gains (B) have oxide platelets.

Fig. 7. Portion of in situ specimen which

was under the grid, yttria phase (Y) is

Yi^Sl^OyNy.
N 3 --11 4a (REV. 9-78) 10 '

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHE~T

1. PUBLICATION OR REPORT NO. NBSIR 80-2075

4. TITLE AND SUBTl fLE

In Situ Oxidation of Y^-doped SigN^

5. Publication Date August 1980

WM*'

7. AUTHOR(S)

N.J. Tighe, K. Kurodo, T.E. Mitchell, A.H. Heuer

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

i

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

Department of Energy

Office of Coal Utilization

Washington, D.C.

13. Type of Report & Period Covered Interim Report

April 1980

15. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of moat significant information . If document includes a significant bibliography or

literature survey, mention it here,) Specimens of -doped SigN^ were oxidized in an environmental cell in a 600 kV

electron microscope. This ceramic oxidizes passively at tanperatures greater than

1000°C but oxidizes catastrophically at a/750 oC. The in situ experiments showed that

oxidation occurs by nucleation and growth of Si0

? and SiJO on the gSi hi surfaces and

by volatilization of W inclusions. The paper discusses the HVEM methodology and the

oxidation results

17. KEY WORDS (six to twelve entriee; alphabetical order; capitalize only the first letter of the first key word unleaa a proper name;

separated by semicolons)

Ceramic; high voltage electron microscopy; in situ ; oxidation; silicon nitride:

yttria-doped silicon nitride

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VA. 22161

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106

NBSIR 79-1960 (R) (DoE)

EFFECT OF STRAIN RATE ON THE

FRACTURE TOUGHNESS OF SILICON

NITRIDE AT 1400°C

R.J. Fields, N.J. Tighe, and E.R. Fuller, Jr.

Fracture and Deformation Division

National Measurement Laboratory

Center for Materials Science

National Bureau of Standards

U.S. Department of Commerce

Washington, DC 20234

October 1979

107

ABSTRACT

The critical stress intensity factor for fracture, K , was measured

i.U in Si^N^ at 1400°C as a function of strain rate. It was found that K^

c increased with decreasing rates of bending of notched bars. This effect

could be approximately predicted using a time dependent plastic zone

correction to K

IC

*
I
108

1. INTRODUCTION

At very high temperatures, ceramics can deform extensively by time dependent

deformation processes like creep. This permanent deformation is stress

and time dependent and is not predicted by elasticity theory. Whether

or not it is appropriate to apply linear elastic fracture mechanics to

creeping materials depends mainly on the extent of creep deformation

relative to certain specimen dimensions. With this in mind, the strain

rate dependence of the fracture toughness of silicon nitride was investigated

at 1400°C. 12 3 Work on silicon nitride, aluminum oxide, and silicon carbide

have shown that at high temperatures and at moderate to slow strain

rates, the fracture toughness is above the room temperature value by a

factor of two or more. Under the same conditions, these materials are

3 4 observed to creep. * It is believed that it is this creep deformation

which is mainly responsible for the increased toughness. Since the

stress at which creep deformation is observed is highly rate sensitive,

there should be a correspondingly strong dependence of the high temperature

on strain rate or loading rate. Figure 1 shows the toughness of

silicon nitride measured by various researchers as a function of temperature.

Two groups find a strong increase in with temperature, and two

groups find no increase or even a slight decrease. The fracture was in

all cases a fast fracture; that is, the specimen broke in two under its

own elastic strain energy and the load dropped immediately to zero,

faster than the loading or straining system could follow. The main

difference in these results is the rate at which the tests were carried

out. In impact loading studies with displacement rates greater than 100

mm/sec , no enhanced toughness was observed; whereas, increased toughness
Fig. 1. Effect of Temperature and STrain Kate oh the Fracture

Toughness of Silicon Nitride as Measured by Various

Researchers

.
no

was observed for slow bend tests or double torsion tests carried out at

a displacement rate of 1 tran/sec or less. Such a rate effect is observed

g in metals at much lower temperatures and is generally ascribed to the

rate dependence of the yield stress. In such cases, a plasticity

correction can be made to to predict with some success the effect of

9

rate on toughness. In the present paper, the effect of strain rate on

both the fracture toughness and the stress for the onset of creep is

measured for silicon nitride at 1400°C. The creep stress data is used

to predict the influence of strain rate on toughness.

2 . EXPERIMENTAL PROCEDURE

Bars having a rectangular cross-section were cut from a billet of

NCX34, yttria doped silicon nitride. A narrow notch was introduced using a 1/2 mm

thick diamond slitting wheel. Two sizes of bars were tested having the

approximate dimensions listed in Table (1). The tests were performed in

% four point bending (See Figure 2) using a rig made of silicon carbide.

The major span was 40 mm and the minor span was 10 mm. All tests were

performed in air in furnaces which were brought to temperature (1400°C)

in about 3 hours. The furnaces had feed-throughs in the top and bottom

for silicon carbide push rods. One push rod was attached to a load cell

and the other to the cross-head of a displacement-rate controlled testing

machine. The cross-head was driven at rates from 1.25 to 0.005 mm/minute.

In general, a subcritical crack initiated stably from the slit and

grew a few millimeters before fast fracture occurred. This avoided

ambiguities associated with the sharpness of the slit and eliminated the need

to precrack or fatigue crack the specimen. For fracture tests at rapid

displacement rates or at room temperature, where little or no subcritical

crack growth occurs, relatively long pre-cracks were sometimes grown in by

bending at 1400°C with a displacement rate of 0.0125 mm/min. After producing

these cracks, the specimens were kept at temperature unloaded for at least one hour

before cooling or testing.
Table 1

Nominal Specimen Dimensions (nun)

Large Specimen Small Specimen

Length, l 52 52

Width, W 10 5

Thickness, t 5 4.5

Slit depth, c 3 and 5 1.5
p

Fig. 2. Schematic of Four-Point Notched Bend Test.
After fracture, the crack length(a) was determined as the sum of slit

depth plus the extension resulting from the subcritical crack growth.

This, together with the load at the onset of fast fracture (P^) was used

to calculate according to the formula^

SPfCVV

2 B W

2 F(a/W)

where

F(a/W)

/2W~ tan Tra_ na 2W

4

0.923 + 0.199 (l-sin||)

cos (ira/2W)

( 1 )

and where 5,^ and are the major and minor spans of the four point

bending rig, B is the specimen thickness, and W is the specimen width.

This formula was used rather than the more common polynomial^ because

it is purported to be accurate for any crack length/ specimen width (a/W)

ratio. The polynomial expression is only good for (a/W) less than or

equal to 0.6.

Additionally, two other types of fracture tests were performed.

One test consisted of slowly growing a crack at a displacement rate of

0.0125 mm/min., waiting a time (At), and then fracturing at a faster

displacement rate of 1.25 mm/min. This test was performed for various

At's. The other type of test consisted of growing in a crack at 0.0125

mm/min and then cooling under load to room temperature and fracturing at

0.05 mm/min.

To describe the influence of displacement rate on the fracture

behavior, the creep behavior of this particular material was needed.

We made the assumption here that the creep behavior would correlate well with

the stresses at which the load-deflection curves became non-linear.

These proportional limit stresses were obtained for displacement rates

from 0.125 to 0.0125 mm/min. Small cubes (5 mm on a side) were cut
114

from some of the fracture bars. These were compressed between the silicon

carbide push rods. Cross-head displacements agreed with dimensional changes

measured on the specimens after the test. No attempt was made to reach

steady-state creep. It is known that creep behavior of ceramics is very

4 different in tension and compression. However, the primary stage of

creep is probably dominated by some viscous component like grain boundary sliding

rather than a cavitational component (which dominates tertiary creep) in the

12 material’s mechanical behavior. This viscous deformation should not

depend strongly on the sign of the loading.

To investigate the extent of creep deformation in the fracture

process, a fine array of Vicker's indentations was made on the side of a

flexural specimen (Fig. 3) which was then loaded at a displacement rate

of 0.0125 mm/min. The positions of the indentations were measured

before and after with a precision of 0.0075 mm.

3. RESULTS

The measured at room temperature for a specimen which was precracked at 1400°C was 8.3 MPa ^m. This high value is discussed in the next

section. The results of measuring at 1400 c C as a function of displacement

rate are shown in Fig. 4. This shows an increase in toughness from

about the room temperature value to more than 18 MPa i/m at the lowest

displacement rate of 0.0125 mm/min. Evidence that this toughening is

due to a time dependent process, is shown in Fig. 5. A specimen was

loaded slowly until a crack started growing slowly. The specimen was

unloaded rapidly and then reloaded at 1.25 mm/min. If the reloading

took place immediately, the toughness value is close to that obtained

by breaking the specimen at the slow rate. If the specimen was allowed

to remain unloaded for several minutes before reloading, the toughness

fell. The maximum effect of delayed reloading was obtained in about 60

minutes after which time the toughness no longer decreased. This effect
115

Fig. 3. Array of Vicker’s Indentations on Side of Flexural Specimen
116

Fig. Fracture Toughness of NCX34 as a Function of Cross-Head

Displacement Rate at 1400°C.
11

Fig. 5.

HiPhT High

36 in Hi§h TemPerature~ToughiTess with Time afteT Temperature Pre-Cracking.
was obtained in specimens which had been heated to 1400°C for several

hours as well as in the as-ground specimens.

In an attempt to "freeze-in" the cause of the toughening, a specimen

was loaded until a crack began to grow subcritically and then was cooled

under a slightly reduced load to room temperature. This resulted in a room

temperature K of 9.9 MPa /m which is a 20% increase in toughness. Tests

like this on alumina have obtained similar increments in room temperature

2 toughness

The results of the creep tests are shown in Fig. 6.* From this

data, the relation between the strain rate and the stress at the onset

of creep may be represented by

4 24

e = a

i /(2.04 GPa sec) (2)

where the pre-exponential term and the power were determined from a

least squares analysis of the data. The correlation coefficient was

0.96.

The extent of permanent creep deformation was determined approximately

from the flexural specimens on whose side faces an array of indentations

% had been made. Within the resolution of the technique (0.075 mm in 10 mm),

the permanent deformation was confined to a 1 to 2 mm region around

the crack.

4. DISCUSSION

The room temperature value for K^, 8.3 MPa /m, is higher than that

13 usually reported for NCX34. This could be due to test technique,

since four point bending usually gives higher toughnesses than double

torsion tests, and slow creep cracks are probably blunter than indentation

cracks. Alternatively, this particular batch of NCX34 might be especially

k

The least squares line has been drawn over the range of strain rates that occur in the bending tests.
sj§

mmm

rate

Proportional Limit Yield Stress as a Function of Strain Rate

in NCX34 at 1400°C.

Fig. 6.
120

strong as is suggested by other strength studies made on this billet.

In any case, regardless of the exact values reported here, the trends

with loading rate or delay time should represent general trends in hotpressed silicon nitrides.

Fig. 4 shows that there is a strong variation in toughness with

displacement rate. This raises a serious question as to the existence

of a unique fracture toughness which reflects a material's microstructural

resistance to fracture. If there is no unique value, then K^

c must be

measured for each strain rate in question. However, we feel that the

time dependent plastic response of this material at 1400°C is responsible for

lessening the intensification of the stress near the crack tip. Therefore,

the K measured at the highest rate will most nearly approach the purely

elastic case and ought to be the correct value for comparison of a material's

fracture properties. The average for four specimens measured at

1.25 mm/min (i.e. the fastest rate) and 1400°C is 7.9 + .6 MPa /m. This value is

slightly below the room temperature value in accord with the trend observed in

impact studies on NC132. While 1.25 mm/min is not an impact test, it is

very fast for a material with good creep resistance like NCX34 . It is

interesting to note that there was no appreciable difference in toughness

between pre-cracked specimens and as-slit specimens.

The measured at lower rates represents some mixture of a material's

fracture and deformation behavior. It is probably the more important

engineering quantity to specify since it is the effective toughness that

will be observed at a given, low strain rate.

Knowing both for the purely elastic case (rapid fracture with little

deformation) and the deformation behavior, allows one to predict the apparent

K for the case of fracture preceded by deformation. One of the simplest
adjustments is to recalculate assuming that the crack is longer than

the measured crack by an amount equal to the extent of the deformation

zone. Such an assumption works well for metals. It may be particularly

valid for silicon nitride in which cavitation is a major component in the

12 tensile creep deformation.

The radius of the deformation zone may be derived from the stress

distribution ahead of the crack tip, i.e.

a = K//2tTt (3)

by inserting the appropriate stress and solving for distance ahead of the

crack tip(r)

:

1

K

IC

w pl 2tt ’pi(A)

Note that this analysis relies on the assumption that the stress field

outside of the small plastic zone is described by K. The above distance

( rp£> fc ^en added to the crack length used in the highest strain rate

measurement of K^, (a=5.02 mm). is then recalculated from Eqn. 1

using the new crack length and the original load at fracture for the

high rate K^

c (471 N) . This is the predicted K^

c corresponding to the

strain rate at which a ^ was measured. pi

To carry this procedure out for the present data, the strain rate was

determined from the displacement rate in a bend test.^ The relation

. . 16 used is:

e max

6 (W-a) A

( W ( V2 V

(5)

where e is the strain rate in the outer fiber, A is the displacement

max *

rate, and are the major and minor spans, and (W-a) is the ligament

width. The assumption is made here that the strain rate at the crack tip
122

is equal to the outer fiber strain rate in an elastic bar having a width

equal to the remaining ligament width, dince most of the bar is

deformed elastically, Eqn. 5 seems more appropriate than an equation which

assumes a creeping or perfectly plastic material. At longer times than were

studied here, however, creep deformation occurs on a scale comparable to

the specimen’s dimensions and other relations would have to be used. In

this regime, however, the validity of measuring K would also be questionable.

In Table 2, the strain rates at which the tests were carried out are

listed in decreasing order in the first column. In the second column, the

corresponding yield stresses calculated from Eqn (5) are shown. These

yield stresses are then used (Eqn. (4)) to determine the plastic zone size.

Finally the corrected is calculated from the highest rate by altering

the effective crack length by an amount equal to the extent of the

deformation. These values may be compared with the experimentally determined

ones in the last column. This comparison is shown graphically in Fig. (7).

The agreement between the predicted and experimental toughness is as good

as the scatter between the experimental values. The predicted line tends

to be below the experimentally determined points. This could be due to

the measurement of the creep stress in compression rather than in tension.

Creep should occur more readily at lower stresses in tension. This would

tend to shift the predicted line upward and give better agreement.

The effect of the deformation or plastic zone on the toughness decreases

with time after unloading as shown in Fig. (5). It takes about 1 hour at 1400°C.

to recover fully. The effect can be frozen in by cooling to room temperature

under load. This is additional evidence that it is the time dependent

deformation which is responsible for increasing the apparent K^. by

reducing the crack tip stress field intensification. The delay time
123

CMO

JQ

as

</» 3

to

QJ c£4-obfO

CO • CM

CO "

«3 r— F=» O'* f— I""* CD E 00 f— ai

CO co o co 00 E CO co CO p—>

ra • « • • o LO r— CM oo o a. f"* 00 CO r—*

p*=° p—»

(P= r—

0=3 5" • t- CM

i- "** O r— r- 00 U E o o o CM o F=— cx> r-^ cr*

(P=> r— CM cn • • • • O (O « • • • « O o co

»-H Ci. CO CO 00 CO oo p— p= 0=

to3

•r» *o

as Od f o r—

•f— E cn CT> o CO o o r—

+J E co 00 cr* CO r-» CO LO LO r>* CO o o o ' co r-» cn • • CO f=“ r— r—

a. •

+-> •r—E

*r°

<oc (TJ o Q- CO r— o CM CM

•f— 2: co CO CO J''.. CO r— o o CO

+-> co CO CO CM r— T— 1— CO

s-o

Q.o

J- Ou

CD

+-> fO p—>

LO LO CO CO co

or. 1 i » 1 1 1 1 1 1 i a o o o o o o o o o c <u p»» j—>

e== r~ r--s— p— 1— f—

•r— CO ><

XXX

XX

X

X

X

<o r— cr> o CO cn LO LO CM

t- LO r^. CM o CO CM CO

+jto CO CO CO •— co r-^ CO CM
124

ir* j*ir§ ir1 3*i#

-1 10*4

3«io*4

STRAIN RATE (m 1

)

Fig. 7. Measured and Predicted K as a Function of Crack Tip Strain

Rate. •LU

126

several hours, no appreciable change in fracture properties was noted

when they were compared to the specimens that were tested at room temperature or

heated and tested immediately. Also, the microstructural instability cannot

simultaneously explain the delay time effect and the strain rate effect.

It is important to note that the range of strain rates over which

the plastic zone correction is applicable is limited. At very high strain

rates, the zone is so small that it makes a negligible correction to the

and the toughness there should be independent of rate. At very low

strain rates, the creep deformation extends throughout the whole -specimen

so that the correction would give effective crack lengths equal to or

greater than the specimen width. In such a case, K^. no longer describes

the crack tip stress field and it might be better to evaluate J in order

i. to predict the K that would be obtained in larger bodies.

JLLr

5. CONCLUSION

At temperatures where creep becomes an important component in the

deformation of silicon nitride, the apparent depends on the

rate at which the fracture test is carried out. This toughening is due

to the relaxation of crack tip stress intensification by high temperature,

time dependent plasticity rather than oxidation or microstructural

effects. It has been shown here that, when the extent of this deformation

is small, its effect on the toughness may be estimated by making a

plastic zone correction to the crack length.

Acknowledgments: The authors thank P. L. Land of Air Force Materials

Laboratory, Wr ight-Patterson AFB and R. deWit for helpful advice and

useful discussions.
127

References

1. A.G. Evans and S.H. Wiederhorn, J. Mater. Sci. 9_ (1974) 270.

2. T.D. Ketcham and R.M. Cannon, Amer . Ceram. Soc. Bui. 58_ (1979) 343.

3 . M. Srinivasan and R.H. Smoak, Amer. Ceram. Soc. Bui. 58_ (1979) 347.

4. D.G. Miller, C.A. Anderson, S.C. Singhal, F.F. Lange, E.S. Diaz, and

R. Kossowsky, AMMRC 76-32, ARPA Order No. 1849, Project Code No 1D10,

Agency Accession No. OD 4733, December 1976.

5. M.G. Mendiratta, J. Wimmer, and J. Bransky, J. Mater. Sci. L2 (1977) 212.

6. S.T. Gonczy and D.L. Johnson, "Fracture Mechanics of Ceramics, Vol. 13," edite

by R. C. Bradt, D.P.H. Hasselman, and F.F. Lange (Plenum Press, New York,

1978) p. 495.

7. J.J. Petrovic and L.A. Jacobson, J. Amer. Ceram. Soc. 59. (1976) 34.

8. S.T. Rolfe and J.M. Barsom, "Fracture and Fatigue Control in Structures;

Applications of Fracture Mechanics," (Prentice-Hall, Englewood Cliffs,

N.J., 1977).

9. G.R. Irwin, Appl. Mater. Res. 3_ (1964) 65.

10. H. Tada, P.C. Paris, and G.R. Irwin, "The Stress Analysis of Cracks Handbook",

(Del Research Corp., Hellertown, PA., 1973).

11. W.F. Brown, Jr. and J. Srawley, "Plane Strain Crack Toughness: Testing

of High Strength Metallic Materials", ASTM-STP 410 (Amer. Soc. Testing and

Mater., Phila; PA., 1966).

12. F.F. Lange, Met. Trans, (to be published).

13. S.W. Freiman, private communication.

14. N.J. Tighe and S.M. Wiederhorn, Air Force Mater. Lab. Tech. Rpt.,

AFML-TR-78-83 (1978)

.

15. P.L. Land, J. Mater. Sci. (to be published).

16. G.W. Hollenberg, G.R. Terwilliger, and R.S. Gordon, J. Amer. Ceram.

Soc. 54_ (1970) 196.

17. F.F. Lange, S.C. Singhal, and R. C. Kuznicki, J. Amer. Ceram. Soc. 60

(1977) 249.
128

IBS-114A (REV. 9-781

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 79-1960 (R) (DoE)

2. Gov't Accession No.

4. TITLE AND SUBTITLE

EFFECT OF STRAIN RATE ON THE FRACTURE TOUGHNESS OF

SILICON NITRIDE AT 1400°C

5. Publication Date

October 1980

6. Performing Organization Code

7. AUTHOR(S) R. J. Fields, N. J. Tighe, E. R. Fuller, Jr,

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP

)

Department of Energy

Washington, D.C.

13. Type of Report & Period Covered Interim

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less [actual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.)

The critical stress intensity factor for fracture, K^, was measured in

S

i ^ at 1400°C as a function of strain rate. It was found that increased

with decreasing rates of bending of notched bars. This effect could be

approximately predicted using a time dependent plastic zone correction to K^.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name

separated by semicolons)

Crack growth; creep fracture; high tenperature fracture; silicon nitride;

strain rate effect; toughness.

18. AVAILABILITY Unlimited 19. SECURITY CLASS

(THIS REPORT)

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PRINTED PAGES

For Official Distribution. Do Not Release to NTIS UNCLASSIFIED 23

Order From Sup. of Doc., U.S. Government Printing Office, Washington, DC

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U SCO MM- O C

FRACTURE STRENGTH AND THE WEIBULL DISTRIBUTION OF g-SIALON.

Kazuo Kobayashi, Seiki Umebayashi, and Kazushi Kishi

Goverment Industrial Research Institute, Kyushu

Shuku-machi , Tosu-shi 841

Nancy J. Tighe and Richard J. Fields

National Bureau of Standards

Washington, DC 20234

131

Sra SC • Paper

/MM to

/Jn ^ fP Jz • # IE ^ • # fH H • Nancy J. Tighe* • Richard J. Fields*

V* National Bureau of Standards/

7 p ^ (Si6 AlAN7 ) , 1200°C &LK 1400°C (D 4 17 -f 7/k

Lfc. Jpffil&ig p-fj 7 p v (4 Si02> A1 Rtf Si N2 5 X i ici <9 ftf&LtziiVX’hZ. Hl»(tM:aii^3.14T'feofc. tT

/3-f-d'TP yVIST'S) 6 iih comfit(4 450 MPa T*& D, 17 d' Y"/H&$G m (4 8.4 Xfcofc. 1200°C X(4,

b G &V'/J* »z fiti4 3.1 (rffiT Lfc. w 1i<0^lc/>f4|giE£Oj5T^ (c-iF^cthSELfc fcV 1' cDtzib b^X

hi1,-5. 1400°CX^(4^>L/t^, mf({4 4.0 iiitjR L fc . 3fi^cO{£Tf4^#l:@cO^'t4i; t“y

v^fcvm±\z£&.%fc±fo\z.\m\'tz\iv b vxy’MU, m*

1400°C XiiiP^-frfc'LOtffibtvS. 4 ilif/ -/f Ft-Ar7 h (-4 SfifciiEXtMf, Klc (4 |<j3.2MPa.v/m XfeD, 1400°C tfllt A/t'-£l?feo/i. (1981 *£1/16 Hgtt)

Fracture Strength and the Weibull Distribution of /9-sialon

Kazuo KOBAYASHI, Seiki UMEBAYASHI, Kazushi K1SH1

Nancy J. TIGHE* and Richard J. FIELDS*

(

Government Industrial Research Institute, Kyushu \ Shuku-machi, Tosu-shi 841

J * National Bureau of Standards, Washington D.C. 20234, U.S.A./

4-point bending strength and the Weibull distribution were measured for hot-pressed /)-
35 132

89 no] i98i 551

sialon (Si 5 Al,0,N7 ) al room temperature, 1200°C and 1400°C in air. The hot-pressed /3-sialon was fabricated from Si02 , Al and Si powders in N2 . The ap- parent specific gravity was about 3.14. The hot-pressed sample was consisted of single

/3-sialon as a crystalline phase by X-ray analysis, but glassy grain boundary phase was

detected by SEM observation. The room temperature strength was about 450 MPa as the maximum value and the Weibull

parameter, m, was 8.4. The m value decreased to 3.1 at 1200°C showing almost same maximum strength at room temperature. Decrease of m value was considered to be due to the formation of a few local pits which appeared on the surface. At 1400°C the strength

decreased, while the m value increased to 4.0. This strength degradation was due to viscous grain boundary phase and increase of pit formation. Appearance of pits on all over the surface and formation of oxidized layer seemed to form relatively homogeneous surface and

increased Weibull parameter, m, at 1400°C. The fracture toughness, Klc , measured by

4-point notched bend test was almost constant value of about 3. 2 MPa V*11 from room temperature to 1500°C. [Received January 6, 1981]

Key-words : /3-sialon, High temperature strength, Weibull distribution, Fracture toughness

1 • 3

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after polishing and etching.
134

89 [10] 1981 553

£ t bixtz. 0 2 31 -y f"> X Lfzfo t <D W&'&M Sr /Y'-f• xj/f> fitty&fttf'MW £*0,

Fig. 3. Weibull distribution curves between

failure probability and flexural strength for

(9-sialons with rough surface (O) and well-poli- shedsurface (®)

.

Fig. 4. Pore at fracture origin at room

temperature.

tXb H Xb>hC3L-y

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Fig. 5. Fracture surface at room temperature.

Fig. 6. Weibull distribution curves between

failure probability and flexural strength for /3- sialon at room temperature, 1200°C and 1400°C.
135

554 Yogyo-Kyokai-Shi 89 [10] 1981

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Fig. 7. A few local pits formation (arrows) appeared on the surface at 1200°C.

Fig. 8. Appearance of pits formed on the surface at 1400°C\

K. KOBAYASHI et al.

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Fig. 9. Fracture surface at 1400°C.

Fig. 10. The fractured surface at 1400°C

after etching.

° HS 130

0 NC 132 • SljAlONj A NC 330 Fig. 11. Fracture toughness, A'lc , of p-sialon

(Si 5 AhOtN,) and those of hot-pressed Sis N( (HS 130, NC 132) and reaction bonded Si 3 N4 (NC350) as a function of temperature.
136

/NBeW!

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ftft, » ISSr^fefclS*.

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89 [10] 1981 555

2) 1200°C T-ttlllltfftSOftftfitttt 450 MPa g;^

Ltzi\ yj-ffrWBf., «tl3.i t^Scofiti 91ST

Lfc. £*Ltt*ffifcSS£Lfctf, y h «to-cft*9, 3MEfclWS:Rl£Lfci ©

5.

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ftft, mlt 4.0 t 1200°C (Cit-<±i*Lfc. $g»<6

fc»4Lfc^ h

4) K,c 13^ 3.2 MPa y/m T-fe9,

Sfti»e» i5oo°c ^xmt—^xhotz.

x st

1) N. Claussen and C.P. Lahmann, Powder Metallurgy

International, 7, 133 (1975). 2) L.J. Gauckler, S. Prietzel, G. Bodemer and G. Petzow

“Nitrogen Ceramics” Edited by F.L. Riley 529

(1977). 3) h£ is. swum mMmm, »mm. m$ua.

mfa, 88, 298 (1980).

4) S. Dutta, Am. Ceram. Soc. Bull., 59, 623 (1980). 5) ftffi. 88. 469 (1980). 6) H. Tada, P.C. Paris and G.R. Irwin, “The Stress Analysis of Crack Hand Book” Del Reseach Corp., Hellertown, PA. (1973).
138

A PROBABLISTIC FRAMEWORK FOR STRUCTURAL DESIGN

Sheldon M. Wiederhorn

National Bureau of Standards

Washington, DC 20234

To be published in:

Fracture Mechanics of Ceramics, Vol . 5 & 6

R.C. Bradt, A.G. Evans, D.P.H. Hasselman, and F.F. Lange, eds.

Plenum Publishing Corporation, 1982.

A PROBABILISTIC FRAMEWORK FOR STRUCTURAL DESIGN

Sheldon M. Wiederhorn

National Bureau of Standards

Washington, D.C. 20234

ABSTRACT

Recent experiments on structural ceramics at elevated temperatures suggest that time dependent processes such as creep crack

growth, cavitation and pit formation have an important influence

on the long term reliability of these materials. Since these

processes are inherently stochastic, fracture mechanics based

theories of structural reliability are not as useful at elevated

temperatures as they are at low temperatures. In this paper, an alternative approach to structural design at elevated temperatures

is recommended. Although, the approach is probabilistic in nature

and suggests the use of probability density functions to describe

the time evolution of strength, concepts of fracture mechanics can be factored into the approach. The probabilistic concepts of

strength presented here are used for the construction of strength

degradation maps, which relates strength, failure time and failure

probability on a single diagram. These maps provide a general

picture of the failure behavior of a ceramic material and can be

used to establish engineering loads and temperatures that are consistent with material behavior.

INTRODUCTION

Modern structural ceramics are used in a wide range of applications in which thermal, mechanical and chemical loading results in

a gradual decay of the strength of these materials. At room

temperature the decay in strength is due primarily to subcritical

crack growth which initiates from preexisting cracks or flaws that are normally present in ceramic materials. At elevated temperatures

the decay in strength is determined by high temperature physical

and chemical processes that alter flaw populations and modes of
140

failure of these materials. Although materials scientists are

generally aware of the processes that determine long term susceptibility to strength degradation, the engineering techniques that

have been used to address strength degradation have been limited

either to empirically developed safety factors, or to fracturemechanics-based crack growth theories. Although both of these

approaches are of value for the prediction of structural reliability,

recent experience on experimental heat engines and heat exchangers

suggest that alternate procedures may be needed to assure structural

reliability in certain applications.

The use of safety factors in design requires extensive practical

familiarity with both the design concept and the material used for

the design. Safety factors are commonly used in the glass industry

for the design of structural glass products (1). For this application, safety factors run from 2.5 for plate glass windows to 8 for

underwater windows which support uniform water pressures. The use

of safety factors is an effective way of meeting design requirements when the material and the conditions of exposure are fully

understood. For new applications, however, the use of safety

factors entails a trial-and-error process to select a factor that

is sufficient to assure safety. In order to encompass all possible

exposure conditions, a considerable amount of empirical investigation is required for each design-material combination. Although

the use of safety factors is the most common method employed for

the design of structural ceramics, more effective methods of

design have been developed which are based on a detailed understanding

of the causes of failure in ceramic materials.

Newer techniques of design that have been developed over the

past 10 years are based on the science of fracture mechanics.

These techniques assume that fracture in ceramic materials originates

from preexisting flaws, which grow when subjected to excessive

stresses (2,3). Fracture mechanics theory provides a basis for

predicting the effect of these flaws on the strength and on the

lifetime of ceramics that are subject to mechanical stresses.

These theories of lifetime prediction are deterministic in nature,

in the sense that once the size of the flaws and the resistance of

the material to fracture have been determined, the- failure time can be predicted. These techniques have been applied successfully

to electronic substrate materials (4), windows for spacecraft

(5,6), grinding wheels (7), optical fibers (8), re-entrant heat

shields (9-11) and ceramics for electronic capacitors (12). The

theory has also been used to improve our understanding of complex

processes such as thermal shock (13) and erosion (14).

Fracture mechanics based theories of lifetime prediction work

very well as long as the mechanism of failure is^ in fact, due to

subcritical crack growth from pre-existing flaws. If structural

failure has its roots in other causes, then the fracture mechanics
141

based theory can give an erroneous prediction of lifetime. An

example of the problems that can be encountered when failure is

not the result of subcritical crack growth is illustrated by

attempts to predict the lifetime of magnesia-doped, hot-pressed

silicon nitride. Problems with this material arise as the result

of chemical reactions between the silicon nitride and oxygen at

elevated temperatures (15-28). At room temperature, the strength

is determined primarily by surface flaws introduced into the

material during machining and finishing operations. At high

temperatures (>1200°C), the machining flaws are altered as a result of chemical attack and increased atomic mobility that

results in plastic deformation and diffusion within the silicon

nitride. Because of these effects, the strength of hot-pressed

silicon nitride is observed to increase during the initial stage

of high temperature exposure, due to surface stress relaxation and

crack healing. After long periods of exposure to stress and

temperature, new flaws are formed and strength decreases as a result of creep cavitation (26,27) and localized corrosion at the

silicon nitride surface. Because of these factors, the fracture

mechanics based theory of lifetime prediction, which assumes

subcritical crack growth to be the sole cause of failure, is

clearly limited.

When estimates of lifetime are made, processes that alter the

type and density of flaws must be considered. Such considerations

are especially important when the initial flaw population is

eliminated as a cause of failure early in a particular structural

application. When this happens, flaws generated during use are

the prime causes of structural failure, and the time necessary for

the flaw generation may amount to a significant fraction of the

total expected lifetime. Processes that cause flaw generation at

elevated temperatures are cavitation due to creep, pitting due to

localized corrosion and phase transformations due to chemical

interactions with the environment. Processes that cause flaw

elimination are generalized corrosion, stress relaxation as a result of plastic flow, and crack healing as a result of high

temperature sintering. Adequate lifetime procedures require that

these processes be compared to determine which one controls the

strength at a particular instant in time. In addition, subcritical

crack growth, which is known to occur in some ceramics at elevated

temperatures, must be factored into the failure prediction scheme.

In this paper, a general approach to the prediction of component reliability is suggested. The approach is more probabilistic

in nature than either the fracture mechanics approach or the

safety factor approach, and has been used in a broad range of

applications to assure the reliability of both electronic and

structural components (29) . The approach is based on the assumption that both the applied stress, and the strength of a set of

components can be described by probability density functions. The
142

failure rate of the set of components is then determined by the

time evolution and interference of the two density functions. In

this paper, the concepts of fracture mechanics are factored into

this general approach to structural reliability. It is shown that

the fracture mechanics based crack growth theories can be used to

provide a mechanistic understanding of the time evolution of the

probability density function for strength. Thus, the fracture

mechanics approach and the probability approach are complimentary

in their use for improving the lifetime of structural ceramics.

GENERAL BACKGROUND (29,30)

In order to predict the probability of failure of a structural

component, both the applied stress, a, and the component strength,

S, must be known with some degree of certainty. When the strength

is less than the stress for a given component, failure occurs. The strength and stress are expressed in terms of probability

functions, which give the probability that component strength or applied stress lie within prescribed limits. Thus, if the probability density function for strength is given by f (S) , then the

probability, p(S ), of having the strength lie within a small

interval of strength, dS, is given by

p (S - dS/2 < S < S + d S/2) = F (S )dS (1) o — — o so

A similar definition of the probability density function is used

for the applied stress, f (a). It is worth noting that the

cumulative failure probabxlity (for a strength, S, less than some value S*) used by most materials scientists is obtained from

equation 1 by integration:

P(S<S*) f

s (S) dS ( 2 )

where f (S) is usually assumed to be a two parameter Weibull

probability density function:

f s(s) — exp [ - (S/S ) m

i] s m °

(3)

These concepts of probability can be integrated with concepts of

fracture mechanics by defining the strength in terms of the size,

a, of the most critical crack in the component, and the critical

stress intensity factor, K , of the component:

lw

S = K

IC/(Y ST) (4)

where Y is a factor that depends on the geometry of the crack.
143

By plotting the probability density functions for strength

and stress on a common set of axes, figure 1, the probability of

failure for a component can be estimated from the portion of the

figure where the two distribution function overlap: the crosshatched area in figure 1 (29). Within this area, there is a finite probability that some of the components will experience

stresses that are greater than their strength. Methods of estimating the total failure probability are given in references 29 and

30 for a variety of density functions. For purposes of the present

paper, the recognition that these functions and their time evolution

determine the failure probability is important.

The probabilistic approach to structural reliability concentrates on methods of evaluating and modifying probability density

functions for a given structural situation. The functions are

then used to establish component and system lifetime and rejection

criteria for structural applications (29,30). Since the main

interest of the present paper is in the material, the stress will

be assumed to be constant for the remainder of the discussion.

Although the importance of the stress distribution for lifetime

predictions is recognized, this assumption is made to simplify

further discussion and to concentrate on the properties of the

material. Using this assumption, the failure probability is given

by the area under the probability density function curve for which

the values of strength are lower than the applied stress, figure

2 .

RELIABILITY CLASSIFICATIONS

Reliability classifications for structural materials can be

made on the basis of the character of the strength distribution

function and its evolution with time (29,30). The classifications

of reliability can be divided into three types: deterministic;

random-fixed; random-independent. These classifications have

their basis in the precision with which the strength distributions

are known.

Deterministic Classification of Strength

The deterministic classification is based on the assumption

that the strength of the components and the evolution of their

strength with time is known precisely. Given these conditions,

then the strength of any component within the strength distribution

can be calculated as a function of time and the time-to-fallure

can be determined. With regard to design techniques developed for

ceramic materials, proof-testing and non-destructive evaluation

techniques fall into this category. Proof testing or non-destructive evaluation are used to truncate the probability density

function so that the weak components are removed from the strength
144

Area

Fig. 1. Illustration of the interference theory of reliability.

The interference area given by the overlap of the probability functions for stress and strength determines the

failure probability (from Kapur and Lamberson 29).

Broken

Fig. 2. Illustration of the interference theory of reliability,

assuming the applied stress to be a well defined constant.

The probability of failure, calculated from the shaded area under the strength curve, depends on time if the

probability density function is time dependent.
145

distribution, figure 3. The truncation serves to define a minimum

strength for the population of components, which serves as a basis

for estimating the minimum time-to-fallure for the set of components.

Strength degradation is usually assumed to result from subcritical crack growth, and the minimum time to failure is calculated on this basis (2,3). Fracture mechanics data suggest that the

crack velocity, v, can be expressed as a power function of the

applied stress intensity factor, K^.:

v = v (K /K ) n

(3) o I o where v and n are empirical constants determined by a least

squares fit of experimental data and K is an arbitrary constant

required for dimensional stability of ?he equation. Using equation

5 as a mechanistic basis for strength degradation, the following

equation has been derived for the minimum time to failure (3)

t B S

(n 2)

P -n

a ( 6 )

where B and n are parameters that characterize the crack growth

and S is the minimum strength of the truncated distribution.

Severll experimental techniques have been developed to measure B

and n, and are adequately discussed in the literature (3, 11).

The attraction of proof testing, or non-destructive evaluation

techniques for design lies in the precision with which these techniques can be used to make predictions of lifetime. If all of the

parameters in equation 6 are known, then the minimum lifetime can

be predicted accurately. Furthermore, the probability of failure

at a time less than the minimum lifetime will be zero. In practice,

errors of measurement always occur in the determination of the

minimum lifetime, and the precision' of the lifetime estimate is

reduced by an amount that depends on the measurement error.

The measurement errors that enter into the lifetime prediction

are of two types, those involved in the determination of B and n,

and those involved in the determination of S . Both types of error can be of sufficient magnitude to intrBduce a considerable

degree of uncertainty into the estimate of the minimum time-tofailure. The errors in B and n are common sources of uncertainty

to both the proof-test technique and the non-destructive testing

technique. Therefore, regardless of which technique is used to

truncate the probability density function, accurate crack growth

data are required to evaluate B and n (11,32,33).

The errors involved in evaluating S depend on the test technique used to evaluate S (i.e. whether B ro°f testing or nondestructive evaluation tichniques are used to truncate the probability density function). In the absence of subcritical crack
146

• Proof Testing

• NDE

Fig. 3. Truncation of the probability density function by proof

testing or non-destructive evaluation. The failure

probability is zero until the truncated portion of the

probability density function intersects the line of

constant stress.

Strength (MPa)

70 85 100 115 130

90.0

70.0

50.0

30.0 ^

Q- 10.0

5.0

1.0 '

4.2 4.4 4.6 4.8

In Strength (MPa)

Fig. 4. Strength distributions for soda-lime-silica glass in dry

nitrogen before and after proof testing; compared with

theoretical curves type I, II and III (from Ritter et

al. 49).
147

growth, the errors involved in determining S by proof-testing are determined by the error involved in establisRing the proof-test

load. This error can be very small as can be seen by the sharply

truncated distributions that have been obtained on number of

materials, figure 4. However, when subcritical crack growth

occurs, the errors in S are determined not by the proof-test

load, but by the errors^in B and n, and by errors in the unloading

rate, as can be seen from the following equation for S

p

(34)

S

p = [B(n-2) oJ

1/3 [3/(n+l)] 1/(n_2)

(7)

where a is the rate of unloading during the proof test and n and

B relate to the crack growth parameters under the conditions of

the proof test. These errors can be sizable and lead to considerable uncertainty in the predicted time-to-failure. Hence careful

environmental control and accurate evaluation of B and n are required for reliability in proof testing.

Non-destructive techniques for evaluating the strength of

materials are in a much earlier state of development than are

the proof testing techniques just discussed (35). Consequently,

the capability of the non-destructive techniques to truncate a strength distribution have not been thoroughly explored. With

regard to establishing the truncation strength, the errors of

measurement that are important are (1) those involved in measurement of the flaw size, and (2) those involved in characterization

of the flaw behavior. Both types of measurement error can be

sizable. The nature of these sources of error have been discussed

recently in a number of references (35-37) . However, there is at

present insufficient data to completely evaluate the effect of

these errors on the predicted time-to-failure.

Of the non-destructive analysis techniques that have been

developed, the surface analysis technique discussed by Khuri-Yakub

et al. (38,39) seems to be the most advanced. The technique is

capable of detecting surface cracks that range in size from ^50ym

to V500ym. Although systematic differences between measured and

actual crack size occur when this technique is used, the precision

of the technique seems to be of the same magnitude, as might be

expected from proof testing in the absence of crack growth.

Therefore, the surface ultrasonic technique offers promise of

being an effective method of truncating the probability density

function for strength. However, additional work is needed to

fully assess the value of this technique for reliability analysis.

Random-Fixed Classification of Strength

The random-fixed classification of strength is based on the

assumption that the time evolution of strength is well known, but

that at any instant of time, the strength is known only as a
function of probability. The position of the individual components

within the strength distribution is, however, fixed relative to

the other components within the distribution. Hence while the

distribution may change with time, the relative position of the

components in the strength distribution does not change with time.

Despite the fact that the strengths are fixed in the distribution,

the strength of a random component is determined only with regard

to the probability density function, and therefore, is known only

as a function of probability. Because of this property, the time

to failure of a given component can be determined only as a function

of probability.

In the absence of proof testing, or non-destructive evaluation,

design with most ceramic materials at room temperature falls into

the random-fixed classification. If subcritical crack growth

occurs, then the lifetime of a component subjected to a stress, a, is given by an equation that is similar to equation 6:

t B S^n 2 ') o n

( 8 )

where S is now the strength of components as given by the probability

density function. Quite often, the strength data for ceramic

materials can be described by a two parameter Weibull analysis:

S = S

Q

[ln (l-P)

_1

] 1/m

(9)

where P is the failure probability calculated from the probability

density function. By substituting equation 9 into equation 8 we

can see how failure time can be described in terms of failure

probability.

As with proof-testing and non-destructive evaluation, the

confidence in the lifetime prediction is sensitive to errors of

measurement of the crack growth parameters n and B. The lifetime

prediction is also sensitive to errors in the determination of

strength, S, as a function of probability. Of the two sources of

uncertainty, the error in the strength relationship, S = S(P), is

usually the greater source of uncertainty in the lifetime prediction. This source of uncertainty is illustrated ip figure 5 for

strength data taken on hot-pressed silicon nitride after 64 hours

of exposure at 1200° C in air. The central line in the figure

represents a graphical estimate of the relationship between S and P

based on a two parameter Weibull probability density function.

The two curves to each side of the central line represent 90 percent

confidence bands for the probability estimate, which was obtained

using order statistics and a median value assignment of the probability. As a result of the uncertainty in the probability assignment,

the uncertainty in strength ranges from approximately 100 to 200 MPa

depending on the probability level. These rather large uncertainties

in strength are further magnified in estimating the time-to-failure
149

Fig. 5. Two parameter Weibull plot of strength data for hot-pressed

silicon nitride (NC132) . Specimens were first annealed in air

at 1200°C for 64 hours under a stress of 250 MPa, then were

broken at 1200°C without cooling to room temperature. The

error bands represent 90 percent confidence limits on the

median line.

INERT SERVICE

Fig. 6. Proof Testing conducted on hot-pressed silicon nitride

(NC132) in air at room temperature: (a) broken at room

temperature immediately after the proof test; (b) broken

at 1200°C after being annealed for one-half hour at 1200°C.

(from Wiederhorn and Tighe 20)
150

because of the exponent, n, in equation 8. These sorts of uncertainties require the applied stress to be decreased substantially

to retain confidence in the lifetime prediction (11, 40). Recent

publications dealing with this source of uncertainty are given in

reference 33, 29 and 41.

Random-Independent Classification of Strength

The random- independent classification of strength is based on

the assumption that both the strength variable and the time evolution of the strength variable are random quantities. The time

evolution of strength is random in the sense that the strength of

a component at any instant in time (or during any load or thermal

cycle) is unrelated to the strength at any previous or subsequent

time. A consequence of this definition is that the strength of a component is not fixed relative to the strength of the other com- ponents in the strength distribution. Therefore, the position of

the components within the strength distribution can change with

time.

An implication of random- independent behavior is that proof

testing or non-destructive evaluation are no longer effective

means of design, since a distribution that has been truncated at one point in time may lose its truncated character because of the

reordering of components within the strength distribution. An

example of this type of behavior has been reported by Wiederhorn

and Tighe (20) for hot-pressed silicon nitride (NC 132) . A set of

specimens that had been truncated at room temperature by prooftesting was compared with a second set that was not truncated (20)

.

Both sets wete broken after an exposure of only one-half hour at

1200°C. The distribution of strengths of the two sets of specimens

were essentially identical, figure 6, indicating that the heattreatment had eliminated the truncated character of the set of

specimens that had been proof-tested . In this experiment, the

randomness was the result of uneven annealing of surface damage in

the two sets of components. Other examples of random- independent

behavior have been reported by Wiederhorn and Tighe (20) , and more

recently by Jakus et al. (42) for thermal cycling of hot-pressed

silicon nitride (NC 132)

.

Random independent strength behavior must be treated by

probabilistic techniques. A simple example of this type of behavior

can be given for thermal cycling. During thermal cycling the flaw

distribution of each component can be changed as a result of the

thermal stresses and the annealing that occurs during each cycle.

It will be assumed that at some point in the cycle a stress, o, is

applied to the components, and then removed. It is also assumed

that at that point in the cycle, the probability function for

strength is independent of the number of cycles. For this situation,

the position of the components within the distribution can change
151

even though the distribution function, f (S), is independent of

the number of cycles. The reliability, tl (probability of survival)

of the components after n cycles is given^by:

00 R

n

= [ f

f

s (S) dS] n

(10)

J o It is worth noting that as the number of cycles increases, the

chance of survival decreases. Such would not be the case for a random-fixed distribution. For the same situation, the reliability

of a random-fixed distribution would be given by:

*

11 ~fJn

f

s

( S ) dS (ID

Without a separate time dependence of f (S) , the number of components

that break from a random-fixed distribution would not depend on

the number of cycles; all the components would break during the

first cycle. A more general approach to reliability estimates

using the random-independent classification, requires a deeper

understanding of the way in which flaws are nucleated and annealed

in ceramic components as a function of time. This type of approach

to failure prediction has not been applied to ceramic components.

STRENGTH DEGRADATION MAPS

In earlier sections of the paper, the importance of probability

density functions for design purposes was emphasized. The time

evolution of these functions was discussed as a means of predicting

minimum operating lifetime, or lifetime as a function of probability.

In this section a graphical method of representing the time evolution

of the probability density function is presented. Although the

method is basically probabilistic in nature, crack propagation and

detailed flaw generation modes can be incorporated into the method

to give a more accurate picture of component lifetime under load.

The method centers about the construction of a strength degradation

map, which relates strength, failure time and failure probability

on a single diagram. These maps give a general picture of the

strength behavior of ceramics, and as such, are similar to the

deformation maps, and fracture maps first suggested by Ashby (43).

The discussion presented in this section will proceed by

example, starting with a simple map in which the probability

density function is time-invariant, and finishing with a map that

includes flaw generation, flaw healing and crack propagation.

This step-wise procedure is used to familiarize the reader with

each aspect of these maps so that the final map can be readily

understood. This procedure is especially important when concepts

of static fatigue and flaw generation are introduced.
152

Basic Definitions

The time evolution of strength distributions can be represented on a diagram in which the strength is plotted as the ordinate

and the time is plotted as the abscissa. A schematic view of such

a diagram is shown in figure 7 for the case in which neither

strength degradation nor strength enhancement occurs as a function

of time. To the left of the strength degradation diagram is a probability density function that represents the initial strength

distribution of a set of specimens or components. The application

of a stress will cause a fraction of the components to break.

This fraction is given by the area under the probability density

function indicated by the letter A; the fraction that survives is

indicated by the letter B.

On the strength degradation diagram a component fails when

its strength is less than the applied stress shown by the horizontal

line in the diagram. Hence, the area on the strength degradation

diagram that lies below the horizontal line labelled "applied

stress" represents the fraction of components that have broken.

The lines shown on the upper part of the diagram represent the

failure probability of components that survived the applied stress.

These failure probabilities are based on the initial distribution

of strength, so that the first probability line has a value given

by area A under the probability density function. Thus if the

fraction of components that failed on application of load is 0.1,

then the first probability line will be 0.1 and all other probabilities will be greater than this value.

From an experimental point of view, the failure probability

of components that survive the applied stress can be determined by

increasing the stress to break all of the components. Then, if

the initial strength distribution is represented by a two parameter

Weibull distribution, the components above the constant stress

line will be represented by a truncated Weibull distribution

having the same value of So and m as the original distribution,

figure 8. The probabilities given in figures 7 and 8 are determined by order statistics using the total number of components in

the initial distribution as a basis for calculating the failure

probability. For the case in which strength is invariant in time,

figure 7, the lines of constant probability are horizontal.

Crack Growth and Crack Healing

Crack growth from the initial flaw population will alter the

appearance of the diagram shown in figure 7. The growth of cracks

causes the strength of components to degrade so that the probability lines are no longer invariant in time. As the strength

decreases, each of the probability lines curves downward, gradually

approaching the line of constant applied stress. Intersection of
153

Fig. 7. Strength degradation diagram for time invariant distribution. Specimens represented by region A under the

probability density function are broken by the applied

stress. The strengths of the remainder of the population are invariant in time.

Fig. 8. Effect of proof stress on time invariant strength

distribution. Probability estimate is based on the

initial population size: (a) Distribution before

application of the proof load; (b) distribution after

application of the proof load.
154

the probability lines with the stress line represents an increasing

fraction of components broken as a result of the applied stress.

The diagram shown in figure 7 also will be altered in appearance

if crack healing (or any other stress relieving process) occurs.

As the time of exposure increases, the effective size of the

cracks in the components decreases causing an increase in component

strength. As a consequence, the probability lines move to higher

values of strength.

A simple illustration of the effect of crack growth and crack

healing can be given for the case in which crack healing and crack

growth occur simultaneously, figure 9. Since these processes have

an opposite effect on the strength, they will be competitive when

they occur simultaneously, and a boundary will exist on a strength

degradation map separating the two processes. This boundary is

similar to the static fatigue limit in glass which separates crack

growth from crack blunting when glass is subjected to a mechanical

stress in a moist environment. If crack healing occurs by generalized corrosion (i.e. hot-pressed, silicon nitride), then the

fatigue limit can be defined as the strength for a crack of length,

a , that is invariant with time (i.e. the rate of crack growth is

just compensated by the rate of crack healing) . If we assume that

crack healing is solely the result of crack shortening due to

corrosion of the surface, figure 10, then the fatigue limit

occurs when the rate of crack growth, v, is just equal to the rate

of surface recession, V, due to corrosion.* If the crack velocity

in ceramic materials can be represented as a power function of the

applied stress intensity factor (eq. 5), then the stress corrosion

limit is obtained when v = V is given by the following equation:

v v

o OWVn

( 12 )

where represents the stress intensity factor of the crack

which has a length that is invariant in time. If the rate of

generalized corrosion, V, and the crack growth behavior have been

characterized, then K can be determined from equation 12 for

a given experimental condition.

From fracture mechanics considerations it can be demonstrated

that

:

S

f - 0 ffic^W (13)

*The discussion presented here is for a material such as silicon

nitride which oxidizes at elevated temperatures.
155

Fig. 9. Strength degradation diagram for the case in which crack

healing and crack growth occur simultaneously. Specimens

in area A under the probability density function break

upon application of stress; specimens included in area B

break as a result of subcritical crack growth; specimens

in area C survive the load application.

V = v= v (K./K ) n

O I o

Fig. 10. A schematic model of crack growth accompanied by surface

dissolution: v is the crack velocity, V is the velocity

of surface dissolution.
where S^ is the strength that defines the boundary between crack

growth and crack healing on figure 9. Since K >K^g , is

larger than the applied stress, a. This result is consistent with

the fracture mechanics concept of a fatigue limit. As the crack

size gets smaller, the strength increases, and for a constant

applied stress, the stress intensity factor decreases. Hence,

crack growth does not occur for cracks smaller than the value a , which is determined for a given applied stress when K.j.=K . Therefore, crack growth on figure 9 does not occur for S>S^..

Crack healing and strength enhancement will occur for initial

strengths that are greater than the fatigue limit, whereas crack

growth and strength degradation will occur for initial strengths

that are below the fatigue limit, figure 9. Again, each of the curves on figure 9 represent equal probabilities of failure; the

diagram gives the time evolution of the strength distribution.

The intersection of the probability lines in the crack growth

region with the constant stress axis corresponds to component

failure. All of the components with initial strengths that are

less than the fatigue limit will eventually fail. These components

correspond to area B under the probability density function curve

in this figure.

The static fatigue limit will depend on the thickness of the

oxide scale that forms on the ceramic surface and on its ability

to protect the surface against further oxidation. If the surface

layer is not protective then the rate of surface attack, V, will

be a constant, and the static fatigue limit, will be given by:

S = o (K /K ) (V/v )~ 1/n

(14)

f IC o o If the surface layer is protective then V will not be a constant,

but will decrease with time as the thickness of the surface layer

increases*. As the surface layer becomes more protective, the

rate of crack healing will decrease, and the strength that is

equal to the fatigue limit will increase. For a parabolic oxidation law in which the rate of oxidation is limited by transport

through a protective layer:

V = V (t/t )" 1/2

(15) o o

Substituting equation 15 into equation 14, the following equation

for the fatigue limit is obtained:

*Changes in the oxidation rate of the surface may also occur as a result of chemical modification of the surface. Such modification

has been reported for hot-pressed silicon nitride (22) . In this case the effective barrier to corrosion will be the modified surface of

the ceramic and not the surface oxide layer.
157

S

f

S fo(t/t

o)1/2“

where S

fo=o(K /K

q

) (V

0 /vo>' 1/n

. for ceramic materxals at elevated

gradually increases with time.

(16)

Since n ranges from M.0 to 'vlOO

temperatures, the fatigue limit

The rate of healing and the rate of strength degradation

depend on the rate of crack growth relative to the free surface,

provided the oxide layer has zero strength (as it will for glass

or slag on a ceramic surface) . The strength is determined by the

Griffith equation (i.e. eqn. 4). If the crack growth rate v

exceeds the rate of recession of the ceramic surface, V, then the

crack length, a, will increase and the strength of the component

will decrease. Conversely if V>v, the crack length will become

shorter and the strength will increase. As flaws resulting from

machining and polishing are removed, and others typical of bulk

defects such as pores and inclusions take their place, the flaw

structure of the surface will be modified by corrosion.

Figure 9 presents a schematic representation of the type of

crack healing and strength degradation expected on ceramic materials

that are subject to corrosive attack. The fatigue limit is

represented as a line of constant strength (which assumes

V = constant) . In the crack healing regime the lines of constant

failure probability gradually increase in strength until all of

the initial flaws are removed and a new distribution of surface

flaws is obtained that is typical of the bulk material. Because

the high probability curves represent higher strength specimens

which contain smaller cracks, components that lie at the high

probability end of the initial strength distribution will reach

the final surface state first. Components from the initial strength

distribution that lie close to the fatigue limit will require a longer period of time to reach the final strength distribution

because they contain larger cracks. Finally since the distribution

of flaws that typify the final distribution may differ considerably

from the initial distribution, both Weibull moduli, S and m, may

change in time, so that the lines of constant probability in the

crack healing regime of figure 9 will not necessarily be parallel.

In the crack growth regime, the strength of components will

decrease until fracture occurs when S=a. Crack growth will be

slow for components that contain cracks that lie close to the

fatigue limit. Because of the strong dependence of crack growth

rate on K^., equation 5, crack growth will accelerate rapidly prior

to fracture, and the strength degradation curves will have a shape

similar to those shown in figure 9. Components that have strengths

that lie close to the applied stress line, will suffer rapid crack

growth from the initial application of load, and fracture will

occur very soon after load is applied to the specimen. Finally if
158

only crack healing and crack growth occur, the number of failures

that occur will gradually diminish with time until nearly all of

the specimens indicated by area B under the probability density

function have been broken. After that, no failure will occur.

The ideas presented above can be expressed quantitatively by

deriving a relation for the crack velocity relative to the corroding

interface. At any instant in time, the crack velocity, v, will be

given by equation 5, diminished by the rate of motion of the

specimen surface as a result of corrosion:

This equation is independent of the functional form of the law

assumed for the corrosion rate. For a real material the parameters

v , n and V may be time dependent. The parameters v and n can change as a result of chemical modification of the ceramic surface,

as it does for sialons, (45) while V will depend on the type of

barrier that is formed on the ceramic surface as a result of high

temperature exposure.

The rate of strength degradation can be calculated directly

from equations 4 and 17:

The most direct means of obtaining the strength trajectories shown

on figure 9 is by numerical integration of equation 18. For a real material, the regions of crack growth and crack healing on figure 9 may be missing. For example, if subcritical crack growth

does not occur, the fatigue limit line will be concurrent with the

applied stress line and the region for crack growth will be missing

from the strength degradation diagram.

Flaw Generation

The strength degradation diagram will become much more

complicated if new flaw populations are generated during the course of exposure. As illustrated in figure 11, -new flaws can be

generated in ceramic materials in a number of ways. The simplest

involves the dissolution of flaws due to machining, and the

simultaneous exposure of microstructural defects such as pores or

Michalske (44) has made this same observation for the static

fatigue of glass. Because of the existance of a fatigue limit in

soda lime silica glass, he predicts that fractures will not occur

under static loading conditions for times greater than 10^ sec. His prediction seems to be consistent with experimental findings.

(17)

dS/dt = ~[(Y/K

ic) 2 S

3

/2] [v

q

(Kj/K^

11 - V] (18)
159

~T * OXIDE-

—

-Si3

l\l 4

-.

(a) (b)

( C )

Fig. 11. Effect of oxidation on flaw generation in hot-pressed

silicon nitride: (a) initial surface; (b) dissolution

of machining flaw and oxidation of inclusion; (c) machining

flaw eliminated, pit remaining from localized oxidation

of inclusion.

(a) (b)

Fig. 12. Illustration of the evolution of strength distributions

during exposure to elevated temperatures and high stresses:

(a) Initial stresses distributions; (b) distributions

after the two initial distributions have intersected.
inclusions that are present in the bulk of the material. New

flaws may also be generated as a result of localized oxidation in

the surface of the ceramic caused by chemically reactive inclusions

within the solid, or by localized regions of activity in the oxide

surface layer that form during exposure. Such localized oxidation

has been reported to cause pits to form on the surfaces of magnesiadoped, hot-pressed silicon nitride (17-20,25). Another mechanism

of flaw formation occurs if the material undergoes creep at

elevated temperatures(26,27) . Unless creep can be accommodated by

deformation of the grains of the solid, grain boundary sliding

will result in the formation of small cavities in the solid, which

can then act as fracture nuclei.

Because of the stochastic nature of the flaws that are generated at elevated temperatures, the strength will have a statistical

distribution that differs from that due to machining flaws, which

are removed as a result of oxidation. The distribution of strengths

that represents the new set of flaws may have a value of S and m

that differ substantially from the initial set of surface ?laws.

Furthermore, the Weibull moduli, S and m, may be a function of

time. Therefore as the strength o? the components begins to be

influenced by the newly generated flaws, the Weibull distributions

of both kinds of flaws will overlap, and the combined distribution

must be used to determine the strength as a function of probability.

A schematic representation of the combined influence of two

Weibull distributions is shown in figure 12.* Initially, the

damage caused by machining determines the strength, because the

strength distribution determined by machining flaws has lower

strengths for all levels of probability. With time the two

distributions overlap, and the failure probability, P, at a given

value of strength is given by (46)

:

1-P = (1 - V

±

) (1 - P

2

) (19)

In this figure, we assume that the Weibull slope for both distributions is time invariant. The parameter S is assumed to depend

on time. For simplicity, the treatment given here is based on the

assumption that the two distributions are coherent. For a discussion

of the more complicated situation of partially coherent, or exclusive strength distributions the reader is referred to

reference 46.
161

where P and are the failure probabilities for the initial

flaws and the newly generated flaws respectively. The effect of

both strength distributions will be equal when P^ = (i* e * at the point of intersection of the two distributions) . At lower

values of P, the distribution with the smaller value of m will

control the strength, whereas at higher values of P, the distribution with the higher value of m will control the strength.

A boundary between two sets of flaws that control strength

can be established on a strength degradation diagram by setting P^

= P and by solving for the point of intersection of the two lines

that represent the Weibull distribution of the two sets of flaws.

In terms of the Weibull moduli the value of S at the point of

intersection is given by:

In S = (m^ In In s

2

) /(m^-n^) (20)

and the corresponding probability, P, is given by

(1-P) = (1-PX

) 2 (21)

k

since P^ = P^ . If the Weibull moduli for both distributions are known as a function of time then equations 20 and 21 define a boundary on a strength degradation diagram, figure 13, that separates regions in which the strength is controlled by a single

population of flaws. Although the above derivation was presented

in terms of an initial surface flaw population, and a newly

generated population, the results are applicable whenever one set

of flaws replaces another. Furthermore, the same analyses applies

when the strength distributions do not form a straight line on a Weibull type of plot. For this case, graphical methods must be

used to determine the line of intersection on figure 13.

To complete the strength degradation diagram, the condition

that defines the change from flaw generation to crack growth has

to be defined. The rate of strength degradation of flaws that are generated in the ceramic can be determined from the Weibull equation

that describes the strength distribution. If the Weibull moduli m

and S

q are a function of time then the strength, S-, of a component

will also be a function of time. The rate of strength degradation

resulting from crack growth can be determined from fracture mechanics

considerations (34), as was done in section 4.2. The dominant

process (flaw generation or crack growth) will be the one that

k

The subscripts in equation 20 correspond to the two probability

functions shown in figure 12.
162

Fig. 13. Strength degradation diagram indicating flaw generation

during exposure. Interpretation of the regions A, B and

C under the probability density function are given in

figure 9.

1 10 100 1000

Exposure Time (hours)

Fig. 14. Strength degradation diagram for magnesia-doped, hotpressed silicon nitride (NC-132) . The diagram illustrates

four regimes of strength degradation or strength enhancement .
163

gives the highest rate of strength degradation, and the boundary

between flaw generation and crack growth will be determined when

these two processes occur at an equal rate:

(dS/dt >fla„s

" (dS/dt) cracks (22)

This condition for the transformation from flaw generation to

crack growth is similar to that used to establish the fatigue

limit.

A hypothetic example of the type of boundary that occurs

between the regions of crack growth and flaw generation is given

in figure 13. The boundary is determined from the points of

inflection of the lines of constant probability; the boundary is an envelope of these points. The boundary that separates flaw

generation from crack growth will depend on the rate of strength

degradation within the region of flaw generation, and in general,

will not be contiguous with the fatigue limit established from

considerations of surface corrosion.

Empirical Methods of Establishing Boundaries

In principle, the boundaries on the strength degradation

diagram can be determined from a detailed theoretical and experimental analysis of the mechanisms of flaw generation within each

region of the diagram. The boundaries of each region are determined from the boundary conditions given by equations 13, 19 and

22. This is the same sort of procedure followed by Ashby in the

construction of fracture and deformation maps (43). Unfortunately,

the detailed mechanistic understanding needed for construction of

these boundaries is not available in the ceramics literature at

the present time. Although work has been carried out on the

theory of cavitation during creep (47) , our state of understanding

is not yet sufficient for the construction of strength degradation

diagrams from theoretical considerations. Furthermore, the theories

that have been developed have not been completely verified experimentally. Consequently, an alternative method is needed to

determine the structure of strength degradation diagrams.

Strength degradation diagrams can be evaluated directly from

strength measurements and microstructural analysis of ceramic

specimens that have been exposed to a temperature and load for

specified times. By breaking sets of specimens after fixed times

of exposure, the strength distribution can be obtained as a function of time. The failure probability is determined directly

from the strength distribution. By using the result of a series

of strength tests made as a function of time, the failure probability lines can be plotted on the strength degradation diagram.

If the strength data is obtained in sufficient detail, then the
spacing and slopes of the probability lines will tend to change at

the boundaries between the different regions of the strength

degradation diagram. The boundary of each region may be selected

by the use of equations 19 and 22. The mechanism of failure

within each region of flaw generation is determined by microstructural analysis of the specimens. A fracture surface analysis,

for example, can be used to determine the source of failure, eg.

pits, pores, inclusions, whereas a microstructural analysis can be

used to determine the extent of cavitation within the solid.

Although tedious in its application, this method of establishing

the strength degradation diagram is unequivocal with regard to

identifying the regions on the diagram. Furthermore, the data can be used to obtain a basic understanding of the mechanism of failure

of ceramic materials at elevated temperatures.

An example of the type of analysis just described is given in

figure 14 for magnesia-doped, hot-pressed silicon nitride (48)

.

Four-point, bend specimens of this material at a maximum fiber

stress of 250 MPa were exposed to air at a temperature of 1200°C.

The figure clearly illustrates that there are several regions to

the strength behavior. For periods of time less than 0.5 hours

exposure under load, stress relaxation as a consequence of high

temperature dislocation mobility results in a strength increase.

Between 0.5 hours and 2 hours of exposure only slight changes in

strength occur suggesting that residual stress relaxation in the

vicinity of the initial machining flaws has been completed.

Between 2 and 4 hours of exposure, pit generation in the ceramic

surface results in a radical change in the distribution of strength

for the test specimens. As indicated by the apparent discontinuity

in the probability curves, pit formation occurs quite rapidly and

in a random way between 2 and 4 hours of exposure, so that there

is no relation between the initial set of flaws caused by machining,

and the pits resulting from localized oxidation. Between 4 and 16

hours of exposure, the pits become less severe as sources of

failure and a plateau is reached in the probability curve suggesting that new pits are not generated after this time. At approximately 1 week exposure time, creep of the specimens under the

applied load increases, and creep-fracture takes over as the

dominant mode of failure. Thus, for magnesia-doped, hot-pressed

silicon nitride, there are at least 4 regions of strength control:

(1) subcritical crack growth; (2) machining flaws; (3) pits; and

(4) cavitation cracks. An advantage of a diagram such as that

shown in figure 14 over other methods of representing strength

data is that it gives perspective to the strength behavior of the

silicon nitride. As in the case of the deformation maps first

proposed by Ashby, (43) the diagram delineates those regimes of

stress that should be avoided by the designer, and the time frame

over which specific strength behavior is to be expected.
165

ACKNOWLEDGEMENT

The author is pleased to acknowledge the support of the

Department of Energy, Division of Fossil Fuel Utilization.

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13.

14.

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16.

17.

18.

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20 .

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.

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170

EFFECTS OF EXPOSURE ON THE RELIABILITY OF SILICON NITRIDE

Nancy J. Tighe and Sheldon M. Wiederhorn

National Bureau of Standards

Washington, DC 20234

ABSTRACT

Oxidation at 1200 °C affects the strength and reliability of

hot-pressed, magnesia-doped silicon nitride by modifying the flaw

population. Static load data, and microstructural examination

are used to identify the new flaw populations and to describe the

failure mechanisms. The analysis is displayed graphically by

means of a strength degradation map which delineates specific

regions for the failures due to pit formation, crack growth,

cavitation, and creep rupture.

INTRODUCTION

In order to use silicon nitride effectively as a structural

material in high temperature applications, it is essential to have

a means of assuring its reliability under stress. It is well

known that oxidation in air at 1200 °C [1,2] affects the strength

and reliability of the material by modifying the flaw population

through thermally activated processes and chemical reactions [3,

4]. At room temperature, the strength of hot-pressed silicon

nitride is determined primarily by surface flaws introduced during

machining and finishing operations. At high temperatures, these

machining flaws are modified by generalized corrosion of the sur- face due to oxidation and by strain relaxation as a result of

annealing. New flaws are generated by processes of localized cor- rosion which cause pitting and cavitation, near the surface, and

by creep which causes cavitation and crack growth. The strength

of hot-pressed silicon nitride is observed to increase during the

initial stage of high temperature exposure as a result of flaw

healing and strain relaxation. However, after long periods of

exposure the strength decreases as a result of the new flaws.

Techniques of lifetime prediction proposed during the past

ten years are based on the assumption that fracture in ceramic

materials originates from pre-existing flaws [4,5,6]. Strength is

defined in terms of flaw size using the Griffith equation:

S = K/y-v/a (1)

where S is the strength, K is the stress intensity factor and a is

the flaw size. The flaw size is defined usually in terms of an equivalent penny-shaped crack. The use of fracture mechanics

theory provides a basis for predicting the effect of such flaws on the strength and on the lifetime of ceramics that are subjected to

mechanical stress. The failure time can be predicted once the

size of the flaw and the resistance of the material to fracture

have been determined.

Proof testing and non-destructive evaluation techniques have

been used to predict the reliability of components in structural

applications. With these techniques, -components with flaws larger

than the minimum size required for a desired engineering stress

level are removed from the population. The remaining components

are expected to have the predicted lifetime.

Proof Test Evaluation

Extensive experiments were carried out to determine the

validity of proof- testing as a means of assuring reliability of

hot-pressed silicon nitride at 1200 °C [3,8]. A proof test should

truncate a strength distribution by breaking all specimens below a desired stress level. Surviving specimens should not break below

this stress level. For our proof test experiments, an initial

strength distribution was obtained for as-machined test bars.

From this initial distribution a stress was selected that would

break * 50 percent of the specimens on application of the load.

This stress was applied to a second set of machined test bars and

the survivors of this proof test were then broken. The survivors

of a third set of specimens subjected to a proof stress were

broken after different times of exposure at 1200 °C. It was found

that the proof test truncated the strength distribution of the

survivors effectively only when the survivors were broken at room

temperature without a 1200 °C exposure. The data for one billet

of hot-pressed silicon nitride are plotted in Fig. 1. The calculated distribution for survivors is drawn as a curved line in each

plot. In this figure the strength is represented as ordered

distributions plotted on Weibull diagrams [9] whereby the cumulative failure probability, P, is plotted as a function of the
FLEXURE STRENGTH. MPa FLEXURE STRENGTH. MPa ' Fig. 1. Effect of proof-testing and high temperature exposure on

the strength distribution of hot-pressed silicon nitride. Solid

curve gives theoretical strength distribution after proof testing,

arrow indicates proof stress. (a) distribution for as-machined

specimens and proof test survivors; (b) distributions for as- machined and proof test survivors after %-hour of exposure at

1200 °C. Ref. [3].

Fig. 2. Effect of heating in air at 1200 °C on the strengths of

hot-pressed, magnesia-doped silicon nitride, (billet B) as measured

by breaking specimens at temperature after given time periods [3].
strength, S. When a straight line is obtained for the distribution the slope, m, and the intercept, S

Q

, can be determined from

a linear regression analysis of the experimental data as fitted to

the equation:

In ln(l~P) 1 = m In (S/S

Q

). (2)

Results such as those in Fig. 1 demonstrate conclusively that

the flaw population which defined the initial as-machined strength

distribution had changed between the time of application of the

proof load and the application of the final stress after heating

at 1200 °C for even 30 minutes. Because the flaw population was

changed, lifetime predictions based on an "inert" or initial

strength would have been in error. Therefore, the proof test as described represents an unreliable assurance test for silicon

nitride for 1200 °C service exposures. It is clear also that a nondestructive evaluation based on an initial flaw population

would have similar consequences for lifetime assurance.

Flaw Population in Silicon Nitride

The flaws identified in silicon nitride test bars can be categorized into an initial population in the as-machined specimens;

and, a modified population in the specimens exposed at 1200 °C.

The initial population consists of cracks and strained grains re- sulting from the machining process, as well as inclusions and

porosity resulting from the billet fabrication process. The modified flaw population depends on the exposure conditions. The - inert flaw population can be modified during 1200 °C exposure by:

(1) annealing of the strained surface, (2) oxidation at the surface and near-surface, (3) diffusion of impurity elements and

oxygen in the binder phase, the grain boundary phase and the inclusions, and (4) thermal stresses produced by thermal expansion

anisotropy in the oxide scale and in the internal oxide and non- oxide inclusions.

Examples of the changes in strength observed for silicon

nitride after 1200 °C exposure during the course of our proof

testing program are shown in Fig. 2. It is apparent from these

diagrams that the strength distributions for billet B shifted

toward higher strengths, but that after 64 hours the distributions

stayed constant or dropped marginally. Examination of the surface

showed extensive pitting and the traces of machining damage were

gone. Thus a new flaw population had been generated.

In order to predict the reliability of silicon nitride, it is

necessary to obtain experimental data showing the strength changes

that occur during oxidation as a function of both time and stress.

The Weibull type representation of strength is not adapted easily
to the prediction of strength as a function of time because as shown in Fig. 2 a series of diagrams is needed to represent the

strength at given time intervals; and, there is no clear functional relationship between the separate diagrams except that both S

and m changed with time. 0

Clearly, the development of a diagram that represents the

time evolution of a Wei bull type distribution would be of value

for the design of ceramic structural components. The strength

degradation map discussed in another paper in this volume [10] is

such a diagram. This map relates the strength, the failure time,

and the failure probability to the mechanism of failure. This

type of representation is similar in principle to the deformation

maps suggested by Ashby [11]. The development of our maps was

based on the observations of the changing flaw population identified in silicon nitride. Strength data obtained from static load

experiments at 1200 °C for different times and applied stresses

were used to develop the map, and are presented in this paper.

The validity of the strength degradation map is demonstrated by

identifying specific mechanisms of failure in regions on the diagram.

EXPERIMENTAL PROCEDURE

Materials and Procedures

Static load data were obtained from one billet of hot-pressed

magnesia-doped silicon nitride designated NC 132 billet C. Specimens were machined from the billet by diamond sawing and were

ground to final size using a 180 grit diamond wheel. Specimens

were ground along their length and the edges were retained. Each

specimen was given a number according to its location in the

billet from which it was taken. The numbers were randomized

before testing to avoid systematic errors resulting from non- homogeneous billets or systematic variations in the cutting procedure. Three hundred and fifty specimens (3 x 4 x 50 mm) were cut

from the hot-pressed billet which was manufactured from powder lot

HNS having the following analysis: 93.3% Si^N^; 0.14% Al^O^;

0.04% Ca; 0.36% Fe; 1.34% 0; 3.4% W. Billet B (shown in Fig. 2) was from the same powder lot.

All bars were tested in four-point bending in air at a temperature of 1200 °C. Bend fixtures were made of silicon carbide

and had upper and lower spans of 10 mm and 40 mm, respectively.

Selection of test temperature was based on our previous studies

[3] which indicated that the onset plasticity and crack growth

occurred near 1200 °C in hot-pressed materials. Hence it was our

opinion that 1200 °C was still a reasonable maximum service temperature for these materials. Static fatigue tests were carried
out according to the following scheme: ~ 2 kg was applied to hold

the specimens in place; the furnace temperature was raised at the

rate of * 20° per minute until the final temperature, 1200 °C, was

attained; then, the specimens were equilibrated at 1200 °C for \-

hour. After equilibration, specimens were loaded to the desired

stress then soaked at temperature for times that ranged from h~

hour to 1000 hours. A record was kept of the time of fracture of

all specimens that broke during the soak period. After soaking at

temperature for the requisite period of time, specimens were fractured (without cooling to room temperature), using a relatively

rapid stressing rate. Strengths were measured also at room temperature to serve as a basis of comparison for the high temperature results. Specimens were marked according to their position

in the furnace so that the pieces could be reconstructed after

fracture.

Test Facility

Because of the large number of specimens tested and the long

periods of time required for testing, standard universal testing

machines were inadequate for the present study. Therefore, it was

necessary to design a test facility in which a large number of

specimens could be tested simultaneously, and which was relatively

inexpensive. The test facility developed for this study is shown

in Fig. 3. The system used pneumatic bellows to apply the load to

the specimens and an electronic load cell to measure the load.

The bellows employed a rolling diaphram which was almost friction

free; the force required to move the bellows at zero pressure was * 3 N. The bellows and the load cell were attached to a heavy

steel load frame and the load was applied to the foutpoint bend

fixture via silicon carbide rods. The furnace used MoSi« elements

and was capable of achieving 1600 °C. By monitoring theHoad as a function of time, we determined that the load on each specimen

could be held constant to * 1 percent of the maximum load. At the

end of the test period specimens were broken by increasing the air

pressure to the bellows.

Because of the compact nature of the load train, three specimens could be tested in each furnace. During the development of

the test facility, we were concerned that the shock of one specimen breaking during the soaking period would trigger other breakage within a furnace. However, among the dozens of specimens that

broke during the soak period we never observed this type of breakage. Apparently, the pneumatic bellows were able to absorb the

shock of fracture without transmitting it to the unbroken specimens. Thus, the problem of multiple breakage that could occur

when a constant direct weight is employed to stress specimens did

not occur when pneumatic loading was employed.
176

Fig. 3. Facility for carrying out high temperature static fatigue

tests showing positions of (1) pneumatic cylinders, (2) furnace,

(3) load cells, and (4) water cooled rod holders. Micrograph b

shows the interior of the furnace with the specimens in place.

Fig. 4. Strength distribution for the as-machined silicon nitride

specimens (a) Weibull plot for billet C, (b) histogram including

specimens from billet B and billet C (shaded).
We measured the deflection on some of the specimens initially

by attaching a dial-gauge to the silicon carbide push rod, and

measuring the displacement rate of the rod relative to the test

frame. Direct tests of creep on the Si C push rods indicated that

the rods did not creep at 1200 °C. At a later time, linear variable differential transformers (LVDT) were mounted in each load

train and the specimen deflection was measured continuously over

the 1000 hour test periods. The creep measurements obtained from

the dial gauges and from the LVDTs were consistent with the deflections measured on unbroken specimens that were removed from

the test facility after specified soak periods.

RESULTS

Static Load Data

The stress levels for the static load tests were chosen by

considering the strength distribution obtained after heating one- half hour at 1200 °C; and, then selecting a range of stresses

with a low probability of failure on loading. The initial

strength distribution is shown in the Weibull plot of Fig. 4a.

The twelve strength values for billet "C" were compared with the

strength distribution obtained under the same conditions for billet "B." This combined distribution for the two billets is shown

in Fig. 4b. Since the 12 strengths fell within the histogram of

the 40 strengths for billet "B" , it was considered unnecessary to

break additional specimens for the selection of applied stress.

The results of the static load tests are summarized in

Fig. 5. In order to establish a reasonable applied stress,

stresses of 400, 350, 250, and 200 MPa were applied to sets of six

specimens each for an initial test period of one week at 1200 °C.

In this figure, specimens that failed under load are indicated by

the downward pointing arrows; and, the number of specimens tested

at each stress is indicated in brackets on the figure. Most of

the specimens that survived the required test period were broken

at temperature in order to obtain the survivor distribution. In a few cases the furnace was cooled to room temperature with the

surviving specimens under load either intentionally or because of

a power failure.

The stress 250 MPa was chosen for more extensive testing

because at least one specimen broke within one week of soaking and

therefore some failures were expected over the six week period.

No failures occurred in one week with an applied stress of 200 MPa

and too many failures occurred with stresses of 350 and 400 MPa.

Further, the load 250 MPa would approximate the actual stress for

structural parts. The strength distributions obtained from breaking the specimens that survived the 250 MPa stress for specific
time intervals are plotted in Fig. 6. The Weibull parameters and

the mean strengths obtained for these sets of specimens are listed

in Table I. Both presentations of the data indicate that there

are changes in mean strength and in the scatter of the strength

distribution in the different time periods.

It is instructive to compare the strength distributions obtained after %-hour without load and after 1000 hours with a stress of 250 MPa. These two distributions as shown in Fig. 7 are clearly very similar although the flaw populations have different

origins. The flaw population in the 1000 hour distribution was generated during the exposure, while the flaw population for the

Vhour distribution represents the initial distribution resulting

primarily from machining damage. The straight line drawn in for

the 1000 hour distribution represents the average behavior of the

data. However, it is worth noting that there are two inflections

in the distributions. Such inflections generally indicate that

there is a mixture of flaw populations in the distribution [12].

The bulk of the strength distribution after 1000 hours lies at

lower values of strength than the distribution obtained after h~

hour.

Table 1

Data Obtained From Static Load Tests on Hot-pressed

Silicon Nitride

Static Stress

Time

Hr.

Strength

S, MPa

Weibull

S , MPa 0*

Parameters

m

250 MPa 1/2 678± 55 700 14.4

2 681± 38 699 19.6

4 682± 81 722 7.8

16 709± 73 743 9.4

64 7421103 760 6.5

168 6911108 738 6.4

330 6831 91 719 8.4

1000 5531150 578 3.6

20 MPa 1/2 5811126 646 4.1
179

Fig. 5. Plot of applied stress vs time for hot-pressed silicon

nitride billet C, failure times are indicated by arrows. Numbers

in brackets indicate total number of specimens tested at that

stress level.

Fig. 6. Weibull plots for strength distributions obtained for

hot-pressed specimens from billet C, after holding a static load

of 250 MPa for (a) 1/2 hour, (b) 2 hours, (c) 4 hours,

(d) 16 hours, (e) 64 hours, (f) 168 hours, (g) 330 hours, and

(h) 1000 hours. Specimens were broken at temperature without

removing the static load.
FAILURE

PROBABILITY.

*

FAILURE

PROBABILITY.

%

I

FAILURE

PROBABILITY.

%

250 350 450 550 650 750 850 950 250 350 450 550 650 750 850 950

FLEXURE STRENGTH. MP» FLEXURE STRENGTH. MP<
The specimens that were held under load of 250 MPa at 1200 °C

for more than a few days were noticably bent. The deflections

were measured from unbroken specimens and from the deflection

curves to obtain a measure of the total outer fiber strain. The

creep behavior during a 1000 hour period is indicated in Fig. 8.

This figure shows the curve obtained from the continuous recording

of the LVDT for one specimen. Data points were plotted for one hour intervals for the first 24 hours and for 24 hour intervals

for the next 41 days. It is apparent that three regimes of creep

are present in the figure. Measurements of the steady state deflections in periods II and III give an average creep rate of

6.9 x 10 /s and 2.3 x 10 /s respectively. The data in region I could not be represented by a straight line because of its curvature.

Microstructure and Flaw Modification

Fracture surfaces and oxide scale surfaces were examined

microscopically after the static fatigue tests in order to identify fracture origins. Unfortunately, many of the fracture

origins were lost when the broken specimen impacted the silicon

carbide fixture after fracture. Fragmentation of the fracture

interfaces occurred and cracks parallel to the fracture interface

extended across the tension surfaces for a distance of a few

tenths of a mm from the interface. The fragmentation was noted

especially in the high strength specimens. Specimens that could

be reconstructed did have identifiable fracture origin tracings . which indicated that fracture initiated at corners or along the

tension surface. Similar fracture origins traceable to machining

cracks or to oxidation pits were identified in the earlier proof

test programs in which specimens were oxidized for up to 100 hours

at 1200 °C [3].

Examination of the oxide scale on the specimen surfaces by

light microscopy and by X-ray powder diffraction showed results

similar to those we had observed during earlier proof test experiments [3]. A continuous film formed on specimens heated for h~

hour; and after 2 to 4 hours, distinct globules were distributed

over the oxide scale. After 16 hours the appearance of needles

indicated the presence of enstatite in the oxide scale and oxide

mounds were clearly visible in the scale. These features are shown in Fig. 9. The oxide scale on the 1000 hour specimens had

a grey color which resulted from oxide pillars that extended 10 to

20 pm above the surface of the more compact sintered scale. When

the oxide scale was removed by etching in a solution of HF + ^SO^,

pits 50 to 100 pm in diameter were found distributed over the

silicon nitride surface. The etched silicon nitride surfaces

as shown in Fig. 9c, d, exhibit an overall roughness from the oxidation reaction which mirrors the microstructure of the oxide
Fig. 7. Weibull plot comparing the strength distribution obtained

after %-hour exposure without static load and after 1000 hours with

a static load of 250 MPa.

Fig. 8. Plot of the outer fiber strain obtained for one specimen

and showing three regions of deflection rate change.
scale. The microstructure of an oxide scale from a specimen from

a static load test is shown in Fig. 10. The two transmission

electron micrographs show upper and lower surfaces of a tapered

section of the oxide scale that are separated by several microns.

Figure 10a shows a portion of the amorphous layer adjacent to the

matrix and Fig. 10b shows the sintered pyroxene/tri dymite phases.

From stereo micrographs the oxide scale in Fig. 10a appears to be

composed of globules with extensive porosity throughout the phase.

The amorphous diffraction pattern taken from this phase has not

been identified.

Specimens that were broken after soaking for times greater

than 100 hours, where it is apparent from the deflection measurements that creep was occurring, were examined by transmission

electron microscopy in order to identify microstructural changes

associated with deformation and failure. For this examination,

sections were cut from the tension and compression surfaces and

also from the transverse cross-section. The sections were mechanically and ion thinned to reveal different portions of these specimens.

Figure 11 illustrates the extent of bending and the microstructural changes associated with creep at 1200 °C at a stress of

250 MPa. The unbroken bar was cooled under load after 900 hours

of exposure. The letters b and c correspond to the compression

and tension surfaces respectively. The electron micrographs were

taken from the compression and tension sides of a similar specimen

that was broken after 1000 hours (S = 636 MPa). The compression

side b shows some cavitation at triple junctions (arrowed). The

microstructure is otherwise compact with dislocation arrays,

precipitates and some amorphous grain boundary phase. The microstructure of the tension side c shows extensive cavitation at

triple junctions and along grain boundaries, dislocation arrays

within the grains; unusually large grains indicative of grain

growth; and, oxide filling some of the cavitation space. The

region shown in the micrograph was contained within an area 100 pm

in diameter that was very porous compared with its surroundings.

This type of porous region would constitute a creep generated

flaw. Figure 11 thus illustrates the the extensive cavitation

associated with creep in region III of the curve in Fig. 7. It is

apparent that some grain growth and grain boundary sliding occurred

during the creep as expected.

Strength Degradation Map

The static load data is presented in the form of a strength

degradation map in Fig. 12. Basically, the deformation map provides a means of displaying the time evolution of the strength of

the silicon nitride. On this diagram the strength is plotted as
184

Fig. 9. Light micrographs of specimens heated (a,c) 16 hours and

(b,d) 1000 hours. Surface oxide in a,b and etched surface in c,d.

Fig. 10. Transmission electron micrographs of oxide on specimen

heated 1000 hours showing (a) amorphous phase near matrix and

(b) sintered silicate phase.
the ordinate and the time of exposure to a static load is plotted

as the abscissa. Lines of constant probability are plotted to indicate changes in strength as a function of time at a given probability level. The applied stress and the static fatigue limit are plotted as horizontal lines. On this diagram, a specimen fails

when its strength is less than the applied stress of 250 MPa. The

intersection of the constant probability curves with the applied

stress line indicates the fraction of the components that are ex- pected to break at the time given by the intersection. The static

fatigue limit indicates the strength limit for sub-critical crack

growth. The boundaries drawn on the upper part of the diagram indicate the dominant processes such as flaw healing, pit generation,

and creep that affect the strength.

The strength degradation map in Fig. 12 was prepared by con- structing lines of constant failure probability obtained directly

from Weibull curves such as those shown in Fig. 6. Since each of

these curves presents the failure probability as a function of

strength at a particular instant in time, a series of these curves

can be used to plot the relation between strength and time for a given failure probability level. _ Thus, the map represents the

strength behavior of the hot-pressed silicon nitride specimens

from billet C at a temperature of 1200 °C and an applied stress of

250 MPa. In cases where specimens broke during the test period

the specimen was included in the populations of the Weibull distribution for the map.

Changes in the slopes and spacing of the constant failure ' probability lines indicate the changes in the strength distribution

and determine the boundaries between the different regions. The

different regions of strength behavior indicated on the map in

Fig. 12 were identified and verified by microscopic examination of

the specimens. At low failure probability the dominant mechanism

is crack growth. The other boundaries separate regions we have

identified as representing crack healing, pit formation, and cavitation creep.

DISCUSSION

The various regions of behavior indicated on the strength

degradation map of Fig. 12 were determined and verified by microscopic examination of the test specimens from the present series

of tests and from the earlier series of proof tests on the same

material [3]. Microscopic studies of fracture origins [3,8,12-14]

have shown that fracture during the initial period of exposure re- sults from cracks introduced into specimens by machining and surface finishing. This damage consists of cracks and residual

stresses due to grinding; and, the increase in strength must

result, therefore, from the annealing of this surface damage.
136

Fig. 11. (a) Light micrograph of a specimen bent during a 900 hour

exposure at 1200 °C, applied stress 250 MPa; (b) cavitation and

grain deformation near compression side of specimen broken after

1000 hour exposure at 1200 °C, applied stress of 250 MPa; and

(c) extensive cavitation near tension side.

1 10 100 1000 ;

i Exposure Time (hours)

Fig. 12. Deformation map constructed from strength distribution

data obtained for billet C specimens that survived static load

test at 1200 °C, with a load of 250 MPa.
The sudden change in the spacing of the probability curves

between 2 and 4 hours of exposure under load suggests a change in

the type of flaw that causes fracture. Examination of oxidized

surfaces indicate that globules of glass form a pattern after

4 hours of exposure suggesting the possibility of localized corrosion and the beginnings of pit formation at the specimen surface.

This suggestion is supported by the surface analysis studies in

which the glass was removed by etching and pits were found beneath

the globules (Fig. 9). Earlier studies showed that these pits are sources of failure. Hence, the second region of strength control

was labelled pit generation. The fact that the strengths of the

bars increase after this initial formation of pits suggests that

new pits are not generated continually during exposure and that

some annealing of the pits occurs with continued exposure.

After an exposure time of ~ 100 hours, a general decrease in

strength is observed for all of the probability curves. This

decrease in strength is associated with our observations (Fig. 8)

that the creep rate of our specimens increases after approximately

100 hours of exposure. Microscopic examination of the specimens

shown in Fig. 11 indicate extensive formation of cavities at

triple junctions and along grain boundaries in the regime III

creep of Fig. 8. Thus, the region of decreasing strength on the

strength deformation map is labeled cavitation creep. In addition

to the three mechanisms of strength control just discussed, a region labeled sub-critical crack growth is indicated on Fig. 13

and a fatigue limit of 350 MPa is assigned as an upper bound. The

data shown in Fig. 7 indicate that some specimens in the sample of . 150 specimens tested at 250 MPa should have had strengths that

were as low as 350 MPa. Based on the crack growth data reported

by Wiederhorn and Tighe [2j, the calculated failure time for

350 MPa should have been 10 seconds. Therefore, some failures

should have occurred on loading or shortly after loading the

specimens if crack growth occurred. However, no failures were

observed during the first * 40 hours of the test. Thus, the data

suggest that at low levels of applied load subcritical crack

growth is not an important failure mechanism and that the static

fatigue limit is less than 350 MPa.

From the electron microscopy results it appears that creep in

the 1000 hour test period resulted from grain boundary sliding

associated with a viscous grain boundary phase and from cavitation

at triple junctions and along grain boundaries. In addition, the

grains deformed along their boundaries and grain growth and rear- rangement took place. The large (> 100 pm) porous regions such as that described for Fig. 11 are potential flaw sites and the cracks

that develop in these regions can grow large enough to cause

failure under the constant stress. As evidenced from the x-ray

analyses of particles on grain boundaries and of the oxide scale
considerable diffusional changes took place in the 1000 hour specimens. The results of a more complete analysis, of the deformation

microstructures will be published separately.

The static fatigue data presented in this paper demonstrate

the complex behavior that can be encountered when structural ceramics are subjected to the combined effect of environmental attack

and mechanical stress. For the hot-pressed silicon nitride billet

studied in this paper, at least three regions of strength behavior

were observed. The strength degradation map provides a convenient

way of condensing this behavior onto a single diagram which displays the time dependence of strength in mechanistic terms. However, it should be recognized that relatively large sets of data

are needed to define a map for each condition of engineering interest. Further a series of maps are needed to fully define mechanical behavior as a function of applied stress and temperature. In

principle, the work required to obtain such a set of maps can be

reduced considerably by the development of theoretical models that

describe the strength behavior in the various regions noted in

Fig. 12. Such models would permit the materials engineer to con- centrate on determining the constants of the model and the boundary

conditions predicted for the strength degradation map. Although a start has been made in this direction [15], modeling of mechanical

behavior at elevated temperatures still needs further development

for application to structural design.

SUMMARY

Under constant stress conditions, during heating at 1200 °C,

the strength of hot-pressed silicon nitride showed three timedependent regions of behavior: (1) crack healing, leading to

strength enhancement; (2) pit formation, a corrosion reaction

which changes the flaw population to a more uniform size distribution; and (3) creep, whereby extensive cavitation, grain deformation, and diffusional grain boundary sliding result in plastic

deformation and fracture from extensive flawed regions. These

regions were identified from microscopic observations and from the

plots of constant failure probability vs time on a strength degradation map. This is a new form of data presentation for silicon

nitride that displays the time dependence of strength according to

the failure mechanisms.

ACKNOWLEDGMENT

This study was supported by the Department of Energy, Office

of Coal Utilization, under contract EA-77A01-6010. The authors

would like to thank G. G. Garrett, M. Maurer, C. L. McDaniel, W.

Willard, and J. M. Thomas for their assistance in taking data and

modifying equipment.
189

REFERENCES

[1] R. W. Davidge, A. G. Evans, D. Gilling, and P. R. Wilyman,

Oxidation of reaction sintered silicon nitride and effects on strength. Special Ceramics 5: 329 (1972).

[2] R. Kossowsky, Creep and fatigue of Si

3

N

4

as related to microstructures in Ceramics for High Performance Applications II.

J. J. Burke, A. E. Gorum, and R. N. Katz, eds. , Brook Hill,

Mass. 1974.

[3] S. M. Wiederhorn and N. J. Tighe, Proof-testing of hot-pressed

silicon nitride, J. Mat. Sci. 13: 1781 (1978).

[4] N. J. Tighe, The structure of slow-crack interfaces in silicon

nitride, J. Mat. Sci. 13: 1455 (1978).

[5] A. G. Evans and S. M. Wiederhorn, Crack propagation and

failure prediction in silicon nitride at elevated temperatures, J. Mat. Sci. 9: 270 (1974).

[6] E. R. Fuller, S. M. Wiederhorn, J. E. Ritter, and P. B. Oates,

Proof testing of ceramics, Part 2. Theory, J. Mat. Sci. 15:

2282 (1980).

[7] J. E. Ritter, S. M. Wiederhorn, N. J. Tighe, and E. R. Fuller,

Applications of fracture mechanics in assuring against fatigue

failure of ceramic components, in Ceramics for High Performance

Applications II. Reliability, E. M. Lenoe, R. N. Katz, and

J. J. Burke, eds., Plenum Press, New York, 1981.

[8] N. J. Tighe and S. M. Wiederhorn, Fracture of brittle materials at high temperature, Technical Report AFML. TR-78-33,

1977.

[9] W. Weibull, A statistical distribution function of wide

applicability, J. Appl. Mech. , 18: 293 (1951).

[10] S. M. Wiederhorn, A probabilistic framework for structural

design, this volume.

[11] M. F. Ashby, A first report on deformation mechanism maps,

Acta Metal 1. , 20: 887 (1972).

[12] C. A. Johnson, Fracture statistics in design and application.

Report No. 79CRP212, G.E. Technical Information Series (1979).
190

[13] J. J. Mecholsky and S. W. Freiman, Determination of fracture

mechanics parameters through fractographic analysis of ceramics in ASTM, Special Technical Publication 678, 1980.

[14] R. W. Rice, S. W. Freiman, J. J. Mecholsky, R. Ruh, and

Y. Harada, Fractography of Si^N^ and SiC, in Ceramics for High

Performance Applications II. J. J. Burke, A. E. Gorum, and

R. N. Katz, eds. , Brook Hill, Mass. 1974.

[15] A. G. Evans and W. Blumenthal, High temperature failure in

ceramics, this volume.
EFFECTS OF OXIDATION ON THE SURFACE AND NEAR-SURFACE STRUCTURE

OF SILICON NITRIDE

Nancy J. Tighe

National Bureau of Standards

Washington, DC 20234

To be published in:

Journal of Materials Science

193

EFFECTS OF OXIDATION ON THE SURFACE AND NEAR-SURFACE STRUCTURE

OF SILICON NITRIDE

Nancy J. Tighe

National Bureau of Standards

Washington, D.C. 20234

INTRODUCTION

Silicon nitride is being evaluated for special components in diesel and

turbine engines because their use offers the possibility of higher engine

operating temperatures with greater efficiency and less pollution. The components will be exposed to temperatures above 1000 °C in engine fuel atmospheres

which are slightly oxidizing. Therefore, it is necessary to consider the

effects of an oxidizing atmosphere on component strength and durability. In

the course of an extensive program concerned with proof testing to access reliability of structural ceramics, we have shown that the strength changes

that occur during heating in an oxidizing atmosphere result because the flaw

population (by which strength is defined) changes during the exposure to the

testing conditions [1,2].

The flaw population in as-ground silicon nitride specimens includes

cracks produced during surface grinding; and, internal particles and pores

produced during powder processing and hot-pressing. This flaw population can change during exposure by: (a) flaw healing whereby the size and effectiveness

of initial flaws are reduced; and, (b) flaw generation whereby new flaws such

as surface pits, grain boundary cracks and internal oxide inclusions are produced. The new flaws have more unpredictable effects on the strength

distribution because they are produced in a time and stress dependent manner

and thus are changing continuously.

Additionally, billets produced with the same nominal composition showed

measurable differences in strength distribution after high temperature exposures.

For example, the Weibull plots of strength distribution in Fig. 1 show that

specimens from billets A and B have similar strength distributions after 1/2
194

hr. at 1200 °C. After heating for 100 hr. billet A specimens are weaker and

billet B specimens are stronger than they were after the 1/2 hr. tests. In

these test specimens, the initial ground surface was covered in 1/2 hr. at

1200 ®C by amorphous and crystalline silica. After 100 hrs. , the oxide scale was ~20 pm thick and was composed of Si^M^O, SiOg, MgSiO^ and amorphous phases.

In order to understand the strength and microstructural changes that are produced during oxidation, it is necessary to examine the oxide scale, the

oxide/silicon nitride interface and the silicon nitride below the oxide/matrix

interface. In the present study, these three interfacial layers were removed

and analyzed using transmission electron microscopy, light microscopy and

x-ray energy analysis and diffraction. Oxide scales were produced on hotpressed silicon nitride samples by heating in air at 1000 °C, 1200 °C, and

1400 Q C for 1/2 to 1000 hr. The phases in the oxide scale were found to occur

in layers that were ordered according to the phase diagrams for the oxide

mixtures [3,4,5]. Crystalline and amorphous phases were present in all speci- mens examined. The oxynitride and amorphous phases are present in as-pressed

billets at triple junctions and along grain boundaries [6-8]. The elements in

the amorphous phases have been identified using electron energy loss analysis

(EELS) and energy dispersive x-ray analysis (EDAX) [9] however (EELS) instrument

was not available for the present study. In this paper, the phases found in

the oxide scales are characterized and the relationships between the oxide

scale, the oxide/matrix interface and the mechanical properties are discussed.

EXPERIMENTAL PROCEDURE

Specimens examined during this study were made from billets of hot-pressed

magnesia-doped Si^N^ designated HS130 and NC132. Specimens thin enough for

examination in a 200 kV electron microscope were made from oxidized specimens

by sectioning them parallel to the oxide/silicon nitride interface, grinding

the section to include the desired portion of the oxide layer, polishing the

thin section to show the oxide phases, and then ion-thinning until the desired

area was electron transparent.

The oxidation treatments were carried out on thin foil specimens as well

as on bulk samples. Small specimens were held vertically in a silicon nitride
195

boat and larger bars or plates were supported at the ends on silicon nitride

or silicon carbide plattens. Thus, there was no direct contact with an oxide

ceramic or a metal foil which would cause an undesirable chemical reaction.

Specimens were weighed and measured before and after oxidation in order to

record the oxide scale growth. Specimens were oxidized by heating in air at

1000 °C, 1200 °C, and 1400 °C for times of 1/2 to 1000 hr., i.e. using times

appropriate to the mechanical test conditions. The oxide scale on some specimens

was dissolved in HF+H^SO^ in order to establish the depth of oxide penetration

into the matrix.

RESULTS AND DISCUSSION

STRUCTURE OF THE OXIDE SCALE

Examination of the surface and cross sections of oxidized specimens

showed that the oxide scale formed in layers that varied in composition and in

crystal structure depending on the temperature and time of exposure. The

different phases could be distinguished readily by light microscopy, but not

so readily by scanning electron microscopy. Plates 1 mm thick which were

oxidized at 1400 °C sagged as a result of the exposure indicating that creep

and plastic deformation had taken place. Edge cracks in bent specimens were

filled with oxide and could be seen easily in the polished sections. The

oxide scale on specimens that were heated cyclically at 1400 °C showed globules

of silicate which was considered evidence for melting (Fig. 2). Such melting

is attributed to the gradual change in composition of the oxide surface which

enhanced the oxidation rate compared with continuously heated samples.

Silicon Oxynitri de-Si

2

N

2q

[10,11]. When polished sections are examined in the

light microscope, the oxynitride can be distinguished from the silicon nitride

matrix by its darker grey color which results from the difference in reflectivity

(r), and index of refraction (n): Si^N^ r=12.2, n=2.1; Si

2

N

2

0 r=8.5, n=1.82

[12]. In the light micrograph of Fig. 2d, spherical oxide nodules are visible

below the matrix/oxide interface. The smallest nodules are oxynitride, and in

the center of the larger nodules the silicate phase is visible; thus, indicating
196

Fig. 1. Plots showing strength distribution obtained at 1200 °C for Si 3 N4 after

heating 1/2 hr. and 100 hr. at 1200 °C [2].

Fig. 2. Hot-pressed silicon nitride samples which were oxidized at 1400 (A)

specimen heated continuously for 100 hr., (B) specimen heated and cooled to room temperature several times during 96 hr., (C) polished section of cyclically heated

specimen note thickness variations in oxide scale -0- and in Si 3N4 -S-. (D) nodules

of Si 2 N 2 0 below the primary oxide-matrix interface in spec. B.
197

a more complete oxidation reaction. The position of the nodules below the

reaction interface is strong evidence for significant diffusion of oxygen into

the matrix at 1400 °C.

The electron micrographs in Fig. 3a, b show the structure of the oxynitride

phase in a thin foil specimen prepared from a section similar to m-m in Fig.

2c. The lower magnification picture of Fig. 3a shows two oxynitride grains

with a region of silicon nitride grains between them. In Fig. 3b the faults,

pores and inclusions within one of the oxynitride grains are imaged. It is

apparent that the initial silicon nitride grains were consumed by the growing

oxynitride.

Cristobal ite-pSiO^. Beta-cristobalite was found along the silicon nitride

interfaces in specimens oxidized at 1000 °C and throughout the oxide scale in

specimens oxidized at 1400 °C. The Cristobal ite was found in quenched samples

as well as in samples that were oxidized at 1000 °C and 1200 °C in a special

high temperature x-ray furnace. The phase coexists with an amorphous phase

which appears to be silica from its diffraction pattern.

Figure 4 shows p-cristobalite which was produced in a thin foil by heating

for one hour at 1000 °C. The oxide grains, which were contained within the

amorphous layer, were identified by their diffraction patterns. This figure

demonstrates the major difficulty associated with examining Si^ in an electron

microscope, namely the vitrification that occurs rather quickly [13]. As

shown in Fig. 4, the p-cristobalite transformed to a disordered orcristobalite

which then transformed to amorphous silica without changing shape. A similar

decomposition occurred in all of the samples regardless of the oxidation

conditions. The diffraction pattern of the transformed amorphous silica was used as a standard pattern for silica glass.

Magnesium Silicate-MgSiO^. The magnesium silicate phase, enstatite, was found

in sections of specimens that were oxidized at 1200 °C and 1400 °C. The

enstatite was found to coexist with 0-cristobal ite forming a well -sintered

layer. Figure 5 shows an example of the layer morphology. In this figure the

0-cristobal ite has vitrified as expected, while the enstatite remained unchanged

during long time irradiation. The enstatite grains were usually faulted and

had a microstructure similar to the natural mineral [14].
198

Fig. 3. Electron micrograph of sections from specimen heated 264 hr. at 1000 °C showing (A) 2 Si 2 N 2 0 grains within the Si 3 N 4 matrix, (B) enlargement of oxynitride grain

showing faults, pores and precipitates.

Fig. 4. Thinned Si 3 N 4 foil after heating 1 hr. at 1000 °C, thin area oxidized com- pletely to cristobal ite and glass; (A) Si02 grains, (B) vitrified Si02 , (C,D,E)

diffraction patterns of the transformation pSi02 -> aSi02 -> amorphous Si02

.
199

Both cristobal ite and enstatite can be produced by devitrification of

magnesium silicate glass at 1400 °C [5]. At 1400 °C the stable crystalline

phase is protoenstatite while cl inoenstatite forms on cooling below 1000 °C.

Enstatite belongs to the mineral class of pyroxenes which vary in structure

according to the impurities in them. The element analysis shows that the

magnesium silicate layer contains Fe, Ca, A1 and other impurities commonly

found in hot-pressed silicon nitride. The presence of these impurities in the

scale in quantities sufficient to form varients of the pyroxenes is evidence

for diffusion from the matrix to the oxide scale. In sampling specimens from

the mechanical test specimens, it was noted that the occurrence of enstatite

coincided with the rapid degradation of strength and the rapid dissolution of

the ground surface [1,2].

STRUCTURE OF THE DIFFUSION REACTION ZONE

The changes in strength of silicon nitride as a result of oxidation must

be considered with respect to changes produced in the matrix near the oxide

scale interface. Because of the overwhelming evidence for diffusion from the

matrix to the outer oxide interface in this paper and in others [3,4,6], this

region is termed the diffusion reaction zone. The microstructure of this zone was examined using thin sections of specimens oxidized at 1200 °C for >200 hr.

and at 1400 °C for 1 and 24 hr.

Oxidation in the diffusion reaction zone appeared to initiate at grain

boundaries as evidenced by either porosity or by thickening of a grain boundary

phase. Figure 6 shows a pore between grains in a sample oxidized at 1400 °C.

An example of thickening of the oxide phase between grain boundaries is shown

in Fig. 7 where the breadth of the phase is indicated by the moire fringe

pattern. At this temperature, the evidence for creep and plastic deformation

in the bulk specimens was shown by the extensive dislocation interactions in

the diffusion reaction zone. There is a significant increase in the occurrence

of grain boundary dislocations, sub-boundaries, and dislocation interactions

with inclusions as in Figs. 6 and 7b. Rapid growth of oxide at the grain

boundaries can produce considerable deformation; and, this extended phase can fracture on cooling to produce extensive grain boundary cracking and a new

flaw population.
200

Fig. 5. Oxide scale from bulk specimen heated at 1400 °C showing grains of Si02 and

MgSi03 , note the Si

0

2 grain has vitrified.

Fig. 6. Silicon nitride grains in specimen heated 24 hr. at 1400 °C, large pore is seen along grain boundary (arrowed).
201

Evidence for nucleation and growth of Si^^Q within the silicon nitride

grains is shown in Fig. 8. This foil was made from a specimen that was oxidized

for 24 hr. at 1000 °C and had an oxide scale composed of amorphous silica and

0-cristobal ite. The presence of such plates in the matrix grains could reduce

fracture toughness by providing easy intragranular fracture paths along the

fault planes.

Oxidation at the well-known inclusions from the WC grinding media can

lead to the formation of an oxide filled pit in the matrix and a mound or depression in the oxide scale. Such pits were shown to be active fracture

sites in specimens broken at 1200 °C; and the oxide scale at the pit was found

to have a high Fe content [1]. The grinding particle inclusions seen in foils

made from as-received specimens, as shown in Fig. 9, are usually polyhedral

and range in size from a few nanometers to a few micrometers. Energy dispersive

analysis shows Fe, Cr, Co, A1 , and W from the WC particles and the metal alloy

binder. In oxidized specimens these inclusions may be polyhedral or spherical

and examples are shown in Fig. 10. The inclusion in Fig. 10a is surrounded by

an amorphous film and is presumed to show an early oxidation reaction which

when complete would produce an oxide-filled pit. The spherical particles in

Fig. 10b show a high Fe and Cr but not W content. The different morphology

and composition indicate possible melting during the exposure at 1400 °C. The

oxide pits affect the strength in a time dependant mechanism because they

require an incubation time to be produced. The distribution of the grinding

particle inclusions varies somewhat from billet to billet hence, the growth of

the oxide pits will vary to change the flaw population.

SIGNIFICANCE OF THE INTERFACIAL ANALYSIS

The results presented here represent the starting point for further

analysis of the relationship between microstructure and high temperature

mechanical properties of silicon nitride. Because this ceramic oxdizes its

high temperature behavior as a structural material must be defined in terms of

its oxidation behavior.
202

Fig. 7. Section from bar heated cyclically for 96 hr. at 1400 °C showing: (A)

Silicon nitride grains in the diffusion reaction zone, moire fringes along the

boundary show position of the growing oxide film. (B) Dislocation interactions

with inclusions near area in (A).

Fig. 8 . Grain of Si 3 N4 containing plates of Si 2 N2 0 in the diffusion reaction zone of

a specimen oxidized 24 hr. at 1000 °C.
203

Fig. 9. Inclusions in hot-pressed Mg-doped Si 3 N4 (C) dark field micrograph showing

distribution and size, energy dispersive analysis of particles (A) and matrix (B).

Fig. 10. Inclusions in Si 3 N4 specimen heated at 1400 °C (A) in 1 hr. inclusion is

surrounded by porous and amorphous phase, indicating that oxidation has started. (B)

After 96 hr. of cyclic heating, inclusions are spherical and show no W.
204

The effect of flaws, such as machining cracks which intersect the surface,

on the strength can be changed by oxidation which blunts them or shortens

them. When the matrix is consumed by the growing oxide as shown in Fig. 2

then the crack size is reduced and the strength is increased. The time frame

for such crack healing is related to the transformation of oxide scale from

oxynitride and silica to the silica and silicate. A sharp crack can be blunted

when the interfacial matrix grains oxidize. A blunt crack requires a higher

stress intensity factor for propogation which leads to a higher strength.

Thus either of these two effects can cause a strengthening. We have shown

here that in the billets studied oxidation can change the strength by reducing

the effectiveness of old flaws and by generating new flaws. The specific

causes of billet to billet variation in strength and oxidation behavior have

not been determined. The oxide scales are similar in composition but have

grown at considerably different rates apparently as a result of minor impurity

differences.

The interaction of the different oxide phases with each other and with

the silicon nitride matrix is time and temperature dependent. As long as only

silicon oxynitride and crystalline and amorphous silica cover the exposed

surfaces, the oxidation rate is slow and the oxide scale is protective.

Diffusion of oxygen through the oxide scale is then rate controlling. When

sufficient diffusion of Mg and Ca impurities from the matrix has occurred to

permit growth of the pyroxene phase then the oxidation reaction can change

rapidly. The diffusion reaction zone illustrated in this paper is more complex

than the original matrix structure; and, more examples taken from oxidized and

deformed samples must be examined to fully characterize this zone. Recent

experiments with sialons (silicon-alumina-oxynitride) have shown that oxidation

at 1400 °C for 1000 hr. can be beneficial to the strength as a result of

diffusional changes in the microstructure [15]. Specimens of magnesia-doped

hot-pressed silicon nitride which have undergone such long annealing treatments

during static fatigue tests show some strengthening but these specimens have

not been characterized completely to identify the sources of the strengthening.
205

CONCLUSIONS

1) The oxide scale on magnesia-doped hot-pressed SigN^ forms in layers

composed of amorphous and crystalline Si0

2

, Si^N^O, and MgSiO^ (MgO-SiO^).

2) A diffusion reaction zone in the Si^N^ matrix is produced during oxidation;

and, structure changes in this zone affect strength.

3) Flaw healing and flaw generation occur as a consequence of the growth of

oxide phases at and below the Si^N^ ground surfaces during heating at

temperatures ^ 1000 ®C.

REFERENCES

1. S. M. Wiederhorn and N. J. Tighe, Proof-Testing of Hot-Pressed Silicon

Nitride, J. Mat. Sci. 13:1781 (1978).

2. S. M. Wiederhorn and N. J. Tighe, Effect of Flaw Generation on Proof

Testing, MCIC Report, March 1978 p. 689.

3. N. J. Tighe, Microstructure of Oxidized Silicon Nitride, Proc. EMSA p. 470-71 (1974) Claitor Publ. Baton Rouge.

4. S. C. Singhal, Thermodynamics and Kinetics of Oxidation of Hot-Pressed

Silicon Nitride, J. Mat. Sci. 11:500 (1976).

5. L. Atlas, The Polymorphism of MgSiO^ and Solid-State Equilibria in the

System MgSiO^-CaMgSigOg, J. Geol. 60:125 (1952).

6. D. R. Clarke and G. Thomas, Grain Boundary Phases in Hot-Pressed MgO

Fluxed Silicon Nitride, J. Amer. Ceram. Soc. 60:491 (1977).

7. A. H. Heuer, V. Lou, L. Ogbugji, and T. E. Mitchell, Lattice Resolution

Studies of Engineering Ceramics: Si C and Si^N^, J. Micros. Spect. Electron

2:475 (1977).

8. 0. L. Krivanek, T. M. Shaw, and G. Thomas, Imaging of Intergranular

Phases by HREM, J. Appl Phys. 50:4223 (1979).

9. 0. L. Krivanek, Electron Energy Loss Spectrometer, Proc. EMSA (1980)

Claitor Publ. Baton Rouge.

10. I. Idrestedt and C. Brosset, Structure of Si^O, Acta Chem. Scand.

18:1879 (1964).
11. C. A. Anderson, K. Keil, and B. Mason, Silicon Oxynitride: A Meteoritic

Mineral, Sci. 146:256 (1964).

12. 0. Taylor, Reflected Light Microscopy of Silicon Nitride and Oxynitride.

13. N. J. Tighe and J. M. Christie, Deformation Structures in Quartz Rocks,

Proc. EMSA p. 60-61 (1969) Claitor Publ. Baton Rouge.

14. P. E. Champness and G. W. Lorimer, Precipitation in an Orthopyroxene, J. Mat. Sci. 8:467 (1973).

15. M. H. Lewis and P. Barnard, Oxidation Mechanisms in Si-Al-O-N Ceramics,

J. Mat. Sci. 15:443 (1980).

Ac knowl edgments

This work was supported in part by the Office of Coal Utilization, Div.

of Heat Engines and Heat Recovery, Department of Energy Contract EA77-A01-6010

Task A086. Figure 9 was taken with the STEM instrument at M.I.T. by A. Garrett

Reed.
NBS-114A (rev. 2-80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2445

2. Performing Organ. Report No. 3. Publication Date

December 1981

4. TITLE AND SUBTITLE

Application of Proof Testing to Brittle Materials at High Temperatures

5. AUTHOR(S)

Nancy J. Tighe and Sheldon M. Wiederhorn

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Department of Energy

Washington, DC 20545

10. SUPPLEMENTARY NOTES

| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The report contains the publications prepared during the contract period. The

publications concern the theories of proof testing for assuring reliability

and present experimental results obtained on silicon nitride tested at 120O°C.

The important conclusion from the research is that the flow population which

causes failure changes during exposure and produces new population. A approach

which includes dynamic flow population was developed and is included in the

report. The experimental data is presented in the form of a map which displays

strength vs. time under load.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Deformation maps; High temperatures ; Proof testing; Reliability; Silicon

nitride; Structural ceramics

13. AVAILABILITY

[**U nlimited

| 1 For Official Distribution. Do Not Release to NTIS

n Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Kid Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

201

15. Price

$ 18.00

USCOMM-DC 6043-P80