NBSIR 73-147

Proof Testing of Ceramic Materials -An

Analytical Basis for Failure Prediction

A. G. Evans and S. M. Wiederhorn

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

March 1973

Interim Report

Prepared for Office of Naval Research

Department of the Navy

Arlington. Virginia 22217

NBSIR 73-147

PROOF TESTING OF CERAMIC MATERIALS --AN

ANALYTICAL BASIS FOR FAILURE PREDICTION

A. G. Evans and S. M. Wiederhorn

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

March 1973

Interim Report

This report is to be superseded by a future publication

which will receive general distribution and should be

cited as a reference. Please consult the NBS Office of Technical Information and Publications to obtain the proper citation.

Prepared for Office of Naval Research

Department of the Navy

Arlington, Virginia 22217

U. S. DEPARTMEMT OF COMMERCE. Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

ABSTRACT

An analysis is presented which permits the accurate prediction of

component lifetimes after proof testing. The analysis applies to crack

propagation controlled fracture but can be used as a conservative

prediction when crack initiation is predominant. The analytical predictions are confirmed in a series of time-to-failure measurements.

1
1. INTRODUCTION

1-4

In many ceramic systems of structural importance, slow crack

growth precedes fast fracture and this leads to a time dependence of

strength. The successful structural exploitation of these materials

requires, therefore, a detailed understanding of the time dependent

behavior so that accurate failure predictions can be made. The accuracy

of failure prediction is very substantially enhanced by incorporating a

component proof test prior to service. It is generally considered,

therefore, that effective proof testing is an essential prerequisite for

the successful structural application of ceramic materials. The primary

objective of this paper is to present an analysis based on fundamental

principles which enables proof test conditions to be accurately selected,

thereby ensuring the "in-service" component lifetimes demanded by a

particular application.

The proof test analysis considers a rapid proof test (which does

not lead to any significant slow crack growth in the unbroken components)

and then a more practically realistic "slow" proof test (which may permit

slow growth, and hence, lead to strength degradation in the unbroken components) . The predictions of the analysis are then verified in a series

of critical experiments. Finally, some general considerations of time

dependent fracture in brittle materials are developed, which enable techniques for the rapid evaluation of the important crack propagation

parameters to be established.

2
2. FUNDAMENTAL ASPECTS OF TIME DEPENDENT FAILURE

Fracture involves two independent series processes, flaw initiation

and flaw propagation. One of these processes is usually predominant,

although it is important to recognize that sometimes both processes contribute in a significant way to failure . In most ceramic

materials of structural importance, there are preexisting sharp cracks^

so that flaw propagation is usually the predominant failure process. A

crack propagation analysis will thus predict the time to failure for most

ceramic materials (this will be a conservative underestimate when flaw

initiation is also necessary).

2.1 Crack Growth Characteristics in Ceramics

12 6 7 It has been established by a number of investigators ' ' ' that

for a given system (environment, temperature, material, etc.) there is a

unique relation between the crack velocity v and the crack tip stress

intensity factor, K^. For example, in many ceramics the amount of water

1 2 in the environment has a strong influence on crack propagation. ' Trimodal Kj-v curves (see figure 1) are then frequently obtained; Regions I

and II of the crack propagation curve result from the stress corrosion

caused by water in the environment, while Region III is independent of

environment.

For most ceramic systems. Region II occurs at a sufficiently

high crack velocity that the- crack propagation time is controlled

almost exclusively by crack growth in Region I. The crack velocity in

Region I can generally be expressed as a power function of the stress

g intensity factor

3
V = AK^.

(1)

where n and A are constants, n for ceramic materials is always a large

number; 9 for Si^N^ C1400° C)^;15-50 for glass"*- ' 30-40 for porcelain.'

In contrast to metals, therefore, large changes in velocity result from

relatively small changes in K^. Also, it should be noted that the slow

growth limit, K , occurs at such low velocities in ceramic materials,

o -10 6,10

< 10 m/s, that its existence has not been generally proven.

2.2 An Estimate of Time'-to-^Fai.lure from Crack Growth Kinetics

The crack growth kinetics (fig 1) can be used for predictions

of time-to-failure under constant load. The time, t, required for a

crack to propagate from subcritical to critical size is easily derived

from the definition of crack velocity, da/dt = v (where a is the crack

length), and the usual relationship between stress intensity, applied load,

a , and crack length; K = a Y/a (where Y is a geometric factor) This

3, J_ 3. 6

gives;

t = (l/aV)'J (K /v)dK

a 11

where . and K^^ are the initial and final values of the stress intensity

Ii If

factor. Using the relationship for crack velocity from eqn (1);

t = 2(k2:^ - K^-'')kn-2)AaV

Ii It a

(3)

4
Furthermore, since failure is essentially instantaneous when K^^ = K.^^

,

then the time to failure, T, is given by,

T = 2(4"- - 4-^)/[(n - 2)Aoy'^ (^)

2 Ti 2 n Also, since 9 < n < 50 for ceramic materials, K << K for the usual range

and

of load application (K < 0.9 K ),/the following equation holds as a

good practical approximation:

ii

2 2 (n - 2) Aa Y

a

(5)

Thus, the time-to failure is determined provided K.^^ , the initial stress

intensity factor at the largest flaw, and tKe - v curve are known.

3. FAILURE PREVENTION BY PROOF TESTING

3.1 Analytical Predictions

3.1.1 Time-to-Failure After Proof Testing

12 13 As demonstrated by Tiffany and Masters, Wiederhorn, and

14

Tetelman, an upper limit to K^^ and, consequently, a lower limit to the

time-to-failure can be obtained by proof testing. Survival of the proof

test guarantees that the stress intensity factor at the tip of the most

serious flaw does not exceed K , otherwise failure would have occurred.

Thus, K^./a = (X ) ./a < K^^/a . Substituting K . < a K^^/a into

Ii a I proof p IC p Ii a IC p

eqn (5) as the maximum value for th,e initial strass Intensity factor, the

follov/ing equation is obtained for the. minimum time to failure

T . min

= 2(0 /a )^-2/[n - 2)Ao^y\''-^] (6) pa j_ a IC

5
It is apparent that x . depends on the proof stress/ to applied mm

stress ratio and independently upon the magnitude of the applied stress (for

a given system). It is thus possible to represent the minimum failure time

for any system in graphical form as a series of parallel lines on logarithm

time, stress coordinates, with the position of the line depending only on

the ratio a /a (for a given system) . An example for soda-lime glass in

p a water (using - v data from ref 10) is shown in fig 2. It is immediately

apparent from the diagram what level of proof stressing is needed to

guarantee no failures within a specified time at the operating load.

More importantly, diagrams of this type can be used to select

materials for a specific application. For example, if a lifetime of 10^

seconds at an operating stress of 100 I'^Im Cl'i,000 psi) are the requirements,

this can only be assured using soda-lime glass (fig 2) if the components are

-2

proof tested at 400-HNm /'^ = A); the soda-lime glass component should

p a -2

therefore have a fast fracture stress of 400 Ml^m at an acceptable failure

probability (e.g. < 1%) so that excessive breakages during the proof test

are avoided. If this is not possible, then that application should be regarded as inadmissible for soda-lime glass, and another material should be

considered.

The minimum time to failure predicted in this way may, in

fact, be substantially lower than the observed failure time for most components, due to the wide distribution of strengths, and hence failure times.

When a certain level of failure during service is acceptable, therefore, it

is often possible to allow the components to remain in service for periods

much longer than x . . A formal treatment of this can be developed as mm ^

follows, by combining fast fracture strengths with. th,e time-to-failure

parameters (c.f, section 4.1),

6
directly

The time to

from eqn (5) > and

failure in excess

is given by,

of the minimum may be obtained

mm. \ P /

where refers to the component ;^ra,cture stress after proof testing. Values of T

usefully larger than T . are thus obtained for ^ 1.5 a . The strengths ° min f p

after proof testing may be found quite simply from the original strength

t distribution. Consider the typical distribution shown in fig 3 . Proof testing

to a stress, o , eliminates all components with lower strengths. This attenP

uates the distribution of the remaining strengths, as shown by the dotted line

in fig 3. The failure probability P for the attenuated distribution is

15 derived from,

where is the initial failure probability and Pp is the failure probability

at the proof stress. It is apparent from fig 3 that the magnitude of t/t . mm

depends essentially on the ratio of the number of components tkat can be permitted

to break in service, P, to the number that break during the proof test. Pp.

If only a small number of component failures can be permitted, such that

(see Ri)

P < P_,, then G. ~ O (fig 4) , R ~ l/ , and T . is the only acceptable

P f p mm

failure time. If a larger number of component failures are allowed,

V P ~ Pp, R >> 1 (see R^) and T can be larger than T^^.^ by several orders

' Weibull axes"*"^ are used for convenience of presentation; this does not mean,

of course, that the analysis is confined to strengths that fit this type of

distribution.

7
of magnitude. Finally, for P > Pp, (see R^) the significance of T^^^

is lost because T is many orders of magnitude larger and proof testing

is no longer a useful prerequisite to component application. These

conclusions are expressed in analytical form for a Weibull strength,

distribution in Appendix II,

3.1.2 Strength After Proof Testing

Thus far, it has been considered that proof testing is

not accompanied by slow crack growth. This can only be achieved if the

proof test is conducted vary rapidly, or in the absence of the slow

growth medium. Normally, however, some slow crack growth is expected to

occur during the proof test. An analysis of the extent of this slow crack

growth and its effect on strength and time to failure is clearly needed

to develop a complete appreciation of the consequences of proof testing.

The degree of weakening due to crack growth is easily obtained

from equation (3). Defining K as

K^-^ E Cn - DAaVt/l (9)

A ^

eqn (3) becomes

K^-^ = K^:^ - kJ:^ (10)

A Ii If

For K^^ = , the condition for failure becomes

(4i)'"" = K^-- + (11)

8
where K is the critical (i.e. the smallest) value of initial stress

li

intensity factor that results in failure during the proof test. Substituting eqn (11) into (10), the relative change in crack tip stress

intensity factor during loading is given by

(Kj./K^p"-2 = 1 - (K,,/k;.)"-^1 - (K*./K^,)"-'l (12)

The relative change in specimen strength due to proof

testing can be determined from eqn (12)

.

' Since the stress during proof

testing, a„, is constant, K^./K . = /a. /a. , where a. and a. are the flaw

P Ii If if 1 f

sizes before and after proof testing. Also, since fast fracture occurs

when K = K , the fast fracture strengths after proof testing are given

* —

by, Cfj/cJj = /a^/a^ , where is the strength if no slow crack growth had

occurred during proof and is the actual strength after proof. Similarly,

* * A it may be shown that K /K = o^/Or:, where a is the equivalent fast fracture proof stress, and K /K = o^lo^. These various stress parameters

are shown in fig 4 using strength data for E-glass fibers'!"^ Substituting

stresses for stress intensity factors in eqn (12) , the following equation

is obtained for the relative strength degradation

n-2 /_ ..x n-2 r . , v n-2

(?) ' -(¥) 1 -I (13)

'The strengths referred to here are fast fracture strengths, whereas the

strengths measured in practice will often be lower than the fast fracture

strength due to slow crack growth during the test. This difference is

dependent on the strain rate used for strength measurements, and the effect

can be predicted with good accuracy from the - V diagram for the system,

Csee section 4.2).
The strength distribution of the components that remain after

proof testing can thus be obtained quite simply from the original distribution of fast fracture strengths. Values of calculated for the E-glass

fibers (obtained for n = 12) are shown in fig 4. The predicted curve is

in very good agreement with actual strength measurements made after proof

t testing.

In addition, it should be noted that the strength after

proof testing is larger at all levels of failure probability than the

original strength prior to the proof test. This result can be expressed

in analytical form for a Weibull distribution (Appendix I) by; >

(the original strength) when m < (n-2) , where m is a constant related to

16

the strength distribution (small m corresponding to a wide distribution).

This condition will be satisfied for most ceramic systems of structural

importance.

The slow crack growth during the proof test also has an

effect on the time to failure. The minimum failure time is not affected

because the condition that K^^ < K^^ after proof testing still applies.

The strength CJ^ must be used however to evaluate T > T . , bv substituting

f min

for in eqn (7).

tThe data shown in fig 4 have been used to show that no slow growth occurs

during the proof test}^ This conclusion can only be correct if the flaws

that control fast fracture do not contribute in any way to the time

dependent fracture—a most unlikely situation.

10
3.2 Experimental Verification

Experiments are designed to verify the lifetime predictions

provided by the preceding analysis. In these, ground specimens of sodalime glass are proof tested in water and the times to failure after proof

testing are measured. Both fast fracture and constant load proof testing

are used.

3.2.1 Fast Fracture Proof Test

For the fast fracture proof test, the return and gauge

length controls of an Instron testing machine are preset so that the load

will increase rapidly to a predetermined value and then reduce immediately

to a fixed proportion of the maximum. The subsequent time to failure

is then measured on the chart. A proof stress /applied stress ratio of 1'35

is selected for convenience. The results are plotted on a proof test

diagram (fig 5) where they are compared with the predicted minimum

failure time. All of the measured times lie above the predicted x , line, mm

which verifies the position of the line as, at least, a conservative

estimate of observed x , . A more precise test of the analytical predlc- mm

tions is provided by estimating failure times on a probability basis and

comparing these with measured times. Values for x/x . are obtained mm

from the distribution of fast fracture strengths (fig 6) in conjunction

with the analysis in Appendix II. A total of 50 specimens were tested at

-2

each of two stress levels, 45 and 9 MNm , so that the weakest specimen in

each series has a failure probability of -0.02. The magnitude of x is thus

evaluated for P = 0.02, and plotted in fig 5. The measured time to failure

11
of the weakest specimen at each stress is in close agreement with the

predicted time to failure, showing that the analytical predictions are

of good accuracy and not excessively conservative.

3.2.2 Constant Load Proof Test

In the constant load proof test, the proof stress was applied for 5 minutes and then the load reduced to the applied stress level,

again using apa/a = 1*35. The measured failure times are shown in fig° 5

-2

for a ~ 15 MNm . The data lie above the predicted T , . Using apa ^ mm ^ propriate values of and n,a^ (and hence t) may be calculated. The

value of T obtained (fig 5) for P = 0.02 is again entirely consistent with

the measure of failure times at equivalent probability.

4. CRACK PROPAGATION PARAMETERS

The value of proof testing as a means of assuring reliability depends

on the accuracy of both the crack propagation parameters and the statistical

parameters needed to describe strength. Statistical parameters are used

to predict probability of failure under static loading (see section 3.1.1),

and for lifetime predictions of parts that have been subject to momentary

t overloads during use. Statistical parameters can be obtained experimentally

^ Here, the overload is treated as the proof test load and the treatment of

section 3.1.1 is followed.

12
from strength data that has been treated, for example, by the normal or

Weibull type distributions. Xn practice, it is important

that these experimentally determined parameters reflect the materials being

used if proof test predictions are to be reliable.

Crack propagation parameters can be determined in a number of ways.

The most direct method is to measure the crack velocity as a function

of stress intensity factor. When using crack velocity methods to determine

crack propagation parameters, one must be sure that environmental conditions at the tips of large cracks are identical to those at tips of the

small cracks normally present in ceramic materials. The crack

propagation parameters can also be determined using strength measurement

techniques. These have the advantage that the parameters are determined

for the same flaws that control strength; their disadvantage is that crack

propagation parameters have to be inferred from the data without actually

viewing the crack motion; in addition, detailed information on K-v curves

cannot be easily obtained. Nevertheless, when agreement is obtained between the crack propagation and strength techniques of measuring crack

growth parameters, one is assured of an accurate prediction of lifetimes

under load.

Both time-to-failure measurements at constant load^^'"^^ and the strain

8,18

rate dependence of strength can be used to obtain the crack propagation

parameters. Conventionally, these have entailed a laborious series of

measurements to obtain average values of time to failure or strength,

making some assumption concerning the strength distribution. Modified

methods for handling the data enable much more efficient procedures to

13
be developed, which take proper account of the statistical variations

in strength. These methods require a consideration of probability

relationships between fast and slow fracture and the application of

these to crack growth data, as described below.

4.1 Probability Aspects of Time Dependent Failure

controlling strength are usually identical for both fast fracture and

time dependent fracture. It is quite acceptable, therefore, to combine

an analysis of failure probabilities, P, measured for fast fracture

with independent measurements of slow crack growth in order to describe

the statistical nature of failure times. This may be expressed in

mathematical form as:

where A(a ) is a function of the fast fracture strength, cr . The time

to failure is related to the fast fracture strength and the applied stress,

When crack propagation controls fracture, the flaws

P = A(a^^) (14)

o^, by

a

a

= T(t)

a

IC (15)

14
where T(t) is a dimensionless function of time. Combining eqns (14)

and (15) at constant Q gives the probability of failure in time t;

(16)

4,2 Constant Strain Rate TecPmique

The princioles developed in section 4,1 will now be applied

to the constant strain rate technique of determining crack propagation

parameters. The fracture stress a at constant strain rate e is given

g for relatively small cracks by

K

a =

o

1 +

2YE£(n+l)C

V K (n-2)

o o

3/2 l/(n+l)

(17)

This reduces to

„*l/(n+l) (n-2)/(n+l)

a = BE a

IC (18)

where B is a constant. The ratio of th.e strengths of specimens tested

at a strain rate to the strengths at a strain rate £^ is thus.

^ l/(n+l)/a_

1 \ / 1

(n-2)/(n+l)

(19) a

IC,

15
At equivalent failure probabilities, a

t so that eqn (19) reduces

ICi ^"^102

to:

log = log - log

(20)

This relationship suggests a very simple experiment in which N strength

measurements are made at each of two separate strain rates, and the strengths

ranked in increasing order (the lowest strength first, the next lowest second,

etc.) The strengths for constant rank are then plotted on logarithmic axes.

This should yield a linear plot with a slope of unity and an intercept that

gives n. Results obtained for ground soda-lime glass tested in water are

plotted in fig 7. An attractive feature of this technique (apart from its

simplicity and rapidity) is that it permits an immediate comparison of the

strength distributions in the two series of specimens. If the data

generate a plot with a slope that differs significantly from unity, this

shows that the two distributions are not comparable (due to differences

in flaw size distributions), thereby invalidating the experiment, and suggesting a closer control of specimen preparation. This, therefore, eliminates

misleading results due to variables in specimen preparation and, equally

important, enables good data to be obtained with the minimum of testing.

The experiments on glass, for example, indicated that 20 specimens at each

strain rate are adequate.

The equivalence is not exact because a certain confidence level must be

applied to each, probability, and this depends on tKe total number of testsperformed .

16
A value for B is found from an additional N fast fracture strength

measurements to obtain O . These measurements must be performed either

at high strain rate, or in the absence of the corrosive environment,

to eliminate all slow crack growth prior to fast fracture. Then a plot

of log a vs log a at equivalent probability enables B to be evaluated

from:

Again, _ measurements of the slope, (n-2)/(n+l), can be used to check the

validity of the experiment.

Values for n obtained for ground soda-lime glass in water,

using this technique (fig 7) , can be compared with crack velocity

J i-if- .1.1-1 .. ^ 10 > 6 data obtained tor an identical glass composition and environment.

There is good correspondence (n = 15 in both cases) indicating that

time dependent fracture is predominately propagation controlled in

this system.

4.3 Constant Stress Techniques

When constant strain-rate equipment is not available, the

relevant information can be obtained using constant stress experiments,

although this does involve much longer test times for data acquisition.

log e + log B (21)

17
A complete set of time-to-failure parameters can be obtained by evaluating

times to failure for two series of N specimens at two separate stress

levels, a , and a „ , in conjunction with fast fracture strength measureai az ments on a further series of N specimens. Equating failure probabilities

at the two stress levels gives (from eqn (5)):

A plot of the logarithm of the stresses should thus generate a curve

with a slope of unity and an intercept which gives a value for n. Then

a plot of failure times against fast fracture strengths, at constant

probability, gives a value for the constant A in eqn Cl).

5. SUMMARY

An analysis is presented which enables accurate predictions to be

made of component lifetimes after proof testing, when fracture is crack

propagation controlled. It is envisaged that the analysis will be used

to design proof tests that will guarantee a minimum lifetime for a component

in service.

In most ceramic materials of structural importance, it appears that

fracture is predominantly propagation controlled, so that the analysis can

be used directly. If examples of fracture controlled by sharp crack

initiation occur, then the analysis can be used to give a conservative

estimate of lifetime.

Application of the analysis requires values of the crack propagation

parameters, n and A /for the prospective material. These can be obtained

log a

^2

= log a + (22)

18
directly from crack velocity measurements. It is suggested, however,

that the parameters be confirmed from strength measurements (at two

separate strain rates) to ensure that the crack tip environment is equivale

for the small preexisting cracks and the large through cracks used for

velocity measurements.

Experimental confirmation of the accuracy of the analytical lifetime

predictions after proof testing is obtained from a series of time-tofailure experiments.

19
APPENDIX I

THE STRENGTH AFTER PROOF TESTING

It is intuitively expected that the relationship between the strength

after proof testing and the initial strength will depend on the amount of

slow crack growth during the proof test and the width of the strength

distribution. A formal analysis of this effect is described here.

Consider that the strength data can be fitted to a Weibull distribution"''^ so that the parameter m can be used as a measure of the strength

19 variation. The initial distribution is given approximately by:

where O, are the initial strengths and J is a constant. The strengths

after proof test, O^, will be larger than the strength prior to proof, a_.

when the following condition is satisfied (see fig 4)

(Al)

1

£n[ - £n (1 - P.)] - £n [ - £n (1 - P )

]

1 a

> m (A2)

£na^ - £na

Substituting for P from eqn (8)

£n[l - £n (1 - Pp)/£n (1 - P^)]

> m (A3)

£n(a^ /op

20
Substituting for O from eqn (13),

£n[l - iln (1 - ?J/in(l - P.)]

J

^ > m (A4)

^ in [1- (a /Qj''"^]

(n-2) " P

' f

From eqn (Al)

,

- £n(l - P) = aV (A5)

Therefore

,

* V m

iln (1 - P

a) (A6) in (1 - P.

Substituting for P in eqn (A4) gives,

* m

£n [1 - (a /a ) ] _

1

^ > (A7)

Inil - (a /a j"" ^] ^

^ P f

Since CJ CJ , this condition is always satisfied provided

P f

m < (n-2) (A8)

The validity of this condition can be easily verified, using the data in

fig 4 (where m = 10) by calculating for various n. This shows that

0^ > a, for n > 11 and < O, for n < 11, which is entirely consistent

f 1 f 1 with the predictions of eqn (A8)

.

21
APPENDIX II

TIME-TO-FAILURE PROBABILITIES

1. TIMES TO FAILURE WITHOUT PROOF TESTING

If the time to failure (eqn 5) is combined with the fast fracture

probability (eqn Al) , it is possible to obtain the time to failure, T^,

on a probability basis. From (eqn 5),

T =

o

2 a.

1

n-2

n,,2 , „ , ,,n—

2

AO Y (n-2)K_

a iC

(A9)

Substituting for from eqn (Al) , and taking logarithms,

(n-2) , , / 1 \ = log il^ -n log a 1- log

a m A(n-2)Y\^^^

(AID)

For P. < 0.1, this reduces to.

log T = log P. - n log a - — + log

o m 1 am A(n-2)Y\J^2

(All)

The values of time to failure without proof testing can thus be plotted on the

proof stress diagrams (fig 2) for various P^ as a series of parallel lines

(with slope, -n) , as shown in fig 8 for ground soda-lime glass. It is immediately

apparent from this type of diagram the range of stress levels in which proof

testing is a useful prerequisite to component application. It is Important

to note, however, that the strength parameters must be those applicable to

22
the component, i.e. a volume correction should be applied if strengths are

measured on smaller specimens.

2. TIMES TO FAILURE AFTER PROOF TESTING

An analytical expression for the time to failure after proof testing

in excess of the minimum, t/t . , can be obtained by combining eqn (7) with

mm

the fast fracture probability eqn (Al) . Since the probability of fracture

after proof testing, P , tends to zero as the strength, a , tends to the

S. -L proof stress, 0^, substitution for P^ from eqn (8) into eqn (Al) gives,

log

log (1-P )- log (1-P^)

a r •log (1-Pp)

(A12)

Substituting for O /o from eqn (7), this reduces for small P , P (< 0.1)

L p 3. -t to

log

min

(n-2) m

log 1 + 1(A13)

The time to failure is only significantly in excess of the minimum, therefore, for P Pp> and provided (n-2)/m is a relatively large number. For

ground soda-lime glass (n-2)/m ~ 5, so that the effect is substantial, as

shown in fig 8 for the specific example P = 10 O /o =2. When P >> P a pa a eqn (A13) reduces to eqn (All) so that T ^ T ; wKen P„ >> P„, eqn (A13)

Q F P shows that T ^ T . , as deoicted in fig 8. mm ^

23
REFERENCES

1. S. M. Wiederhorn, J. Am. Ceram. Soc. 50, 407 (1967).

2. S. M. Wiederhorn, Intl. Jnl. Frac. Mech. _4, 171 (1968).

3. F. F. Lange, to be published.

4. P. Marshall, Special Ceramics 4-, 191 (1968).

5. See for example, A. G. Evans and G. Tappin, Proc. Brit. Ceram. Soc.

20, 275 (1972).

6. A. G. Evans, Intl. Jnl. Frac. Mech. - to be published.

7. K. Schonert, H. Umhauer and W. Klemra, "Fracture - 1969" (Ed. P. L.

Pratt), Chapman & Hall, London (1969).

8. A. G. Evans, submitted to Intl. Jnl. Frac. Mech.

9. A. G. Evans and M, hijize-X , to be published,

10. S. M. Wiederhorn and L. H. Bolz, J. Am. Ceram. Soc. 53., 543 (1970).

11. W. F. Brown and J. E. Srawley, ASTM, STP410 (1967).

12. C. F. Tiffany and J. N. Masters, ASTM, STP-381 (1964) p. 249.

13. S. M. Wiederhorn, J. Am. Ceram. Soc, to be published.

14. A. S. Tetelman, U.C.L.A. - Eng. 7249, July 1972 (Presented at U.S.-

Japan Joint Symposium on Acoustic Emission, Tokyo, Japan).

15. W. Weibull, Ing. Veternskaps Akad. APYEA, Handlinger 15, 153 (1939).

16. W. Weibull, J. Appl. Mech. 18, 293 (1951).

17. G. K. Schmitz and A. G. Metcalfe, I&EC Product Research and Development 5, 1 (1966).

18. J. E. Ritter and C. L. Sherburne, J. Am. Ceram. Soc. 54., 601, (1971),

19. See for example N. A. Weil and I. M. Daniel, Jnl. Am. Ceram. Soc.

47, 269 (1964).

24
FIGURE CAPTIONS

Fig, I. Schematic representation of the effect of crack tip stress

intensity, Kj, on crack velocity, V, during slow crack growth.

Fig. 2. A proof test diagram for soda-lime glass in water. The numbers

below each line refer to the proof stress/applied stress ratio

(cr /a ) corresponding to that line, pa

Fig. 3. A typical strength distribution for a ceramic material plotted

on Weibull axes, is the proof stress, is the applied

stress and Pj, P^, and are different levels of component

failure probability. The quantities Ri , R2 , and R3 are related

to the time to failure ratio, t/t^-,- ^

.

Fig. 4. The strength of E glass fibers before and after a constant stress

proof test at CJp. Also shown are the strengths after proof

testing when no slow crack growth occurs. Of, and the predicted

strengths due to slow crack growth, CJ.. Data obtained from

ref (17).

^

Fig. 5. A comparison of failure times after proof testing for ground sodalime glass in water, with the predicted minimum failure time for

the soda-lime glass/water system (full line) , and the times in excess of the minimum for a probability of 0.02.

Fig. 6. The flexural strength distribution of ground soda-lime

glass.

Fig. 7. A logarithmic plot of the strengths, 0, and O2 at^equivalent

probability at two separate strain rates, £1 and £2 (where

El 7^2 = 600). The intercept gives a value for n of 15 ±2.

Fig. 8. A complete time to failure diagram for ground soda-lime glass

immersed in water, under a flexural load. The minimum time to

failure after proof, T^j^^^' t° failure without proof,

Tq, and the time to failure after proof, T, are shown for

various P and a /a . a pa

25
26
2. A proof test diagram for soda-lime glass in water. The numbers

below each line refer to the proof stress/applied stress ratio

(a /a ) corresDonding to that line.

27
STRESS (MNm"2)

Fig. 3. A typical strength distribution for a ceramic material plotted

on Weibull axes. Cfp' is the proof stress, is the applied

stress and Pi, P2, and P3 are different levels of component

failure probability. The quantities Ri , R2 , and R3 are related

to the time-to-failure ratio, t/t^^^.

28
300 400 500 600

FRACTURE STRESS (psi x |0^)

Fig. 4. The strength of E glass fibers before and after a constant stress

proof test at Qp. Also shown are the strengths after proof

testing when no slow crack growth occurs, Of, and the predicted

strengths • due to slow crack growth, Of. Data obtained from

reference (17)

.

29
Fig. 5. A comparison of failure times after proof testing for ground

soda-lime glass in water, with the predicted minimum failure

time for the soda-lime glass/water system (full line) , and

the times in excess of the minimum for a orobability of 0.02.

30
ffi <

ffi o

a.

bJK

0.0001

0.01 -

STRESS (MN m~2)

Fig. 6, Th.e flexural strength, distrtb.uti^n of ground sodarO-ijna glass.

31
1.62 -

1.60

1.58

1.56 Log O" 2

1.54

1.52

1.50

1.48

146

/ • /

/ 7

/ • /

/' /« / /

/ - /

. / /

;' /

I L

1.30 1.34 1.38 142 146 1.50 1.54 Log cr I

Fig. 7, A logarithmic plot of the strengths, cr, and <5i at equivalent

probability at two separate strain rates, ei and £2 (where

ei/e2 = 600). The intercept gives a value for n of 15 ±2.

32
Fig. 8. A complete time-to-failure diagram for ground soda-lime glass

immersed in water, under a flexural load. The minimum time to

failure after proof, Tminj the time to failure without proof,

To, and the time to failure after proof, x, are sho•^^m for

various and Qp/a^.

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I
FORM NBS.114A (1-71)

2. Gov c Accessioa No.

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

PUBLICATION OR REPORT NO.

NBSIR 73-147

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Proof Testing of Ceramic Materials—An Analytical Basis

for Failure Prediction

5. Publication Date March 1973

6. Perfonning Organization Code

7. AUTHOR(S)

A. G. Evans and S. M. Wiederhorn

8. Performing Organization

NBSIR 73-147

I U. Pro ject/Task/Work Unit No.

I n3n4S3

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No. NR 032-517

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Office of Naval Research

Department of the Navy

Arlington, Virginia 22217

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Covered

In terim

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) An analysis is presented which permits the accurate prediction of component lifetimes

after proof testing. The analysis applies to crack propagation controlled fracture

but can be used as a conservative prediction when crack initiation is predominant.

The analytical predictions are confirmed in a series of time-to-failure measurements.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Ceramics; failure probability; minimum time-to-failure; proof stress diagrams;

proof testing.

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