NBSIR 75-952 ~~ An Error Analysis of Failure

Prediction Techniques Derived

From Fracture Mechanics

S. M. Wiederhorn, E. R. Fuller, Jr., J. Mandel, and A. G. Evans*

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

•Science Center, Rockwell International

Thousand Oaks, California 91360

December 1975

Interim Report for Period July 1, 1975 through December 30, 1975

Prepared for Department of the Air Force

Materials Research Laboratory

Wright Patterson Air Force Base

Dayton, Ohio
II
NBSIR 75-952

AN ERROR ANALYSIS OF FAILURE

PREDICTION TECHNIQUES DERIVED

FROM FRACTURE MECHANICS

S. M. Wiederhorn, E. R. Fuller, Jr., J. Mandel, and A. G. Evans*

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

*Science Center, Rockwell International

Thousand Oaks, California 91360

December 1 975

Interim Report for Period July 1, 1975 through December 30, 1975

Prepared for Department of the Air Force

Materials Research Laboratory

Wright Patterson Air Force Base

Dayton, Ohio

U.S. DEPARTMENT OF COMMERCE, Rogers C.B. Morton, Secretary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Acting Director

ABSTRACT

Three principal methods of failure prediction for brittle materials

are analyzed statistically. Each method depends on fracture mechanics for

its predictive value and hence, the variance of the failure time is found

to depend on the scatter in the fracture mechanics data and the scatter

in the estimate of the initial size of the strength limiting crack. The

variance is used to calculate confidence limits for the prediction of

failure for two materials, glass and silicon carbide. Procedures for

the collection and analysis of data are discussed, and the implications

of the analysis for lifetime prediction are evaluated.

1. INTRODUCTION

Recent developments in the application of fracture mechanics to brittle

materials have provided improved design methods to assure structural reliability.

Since brittle materials generally fail from preexisting cracks, these design

methods are based on a characterization of crack size and subcritical crack

growth. Crack growth data obtained by fracture mechanics techniques are used

to construct design diagrams, from which predictions of lifetime can be made

for a given set of service conditions. Three types of diagrams have been

developed. Each depends on a different technique to estimate the size of the

crack that causes failure. One method requires direct measurement of the crack

size by nondestructive techniques. A second method uses proof testing to

eliminate specimens with cracks greater than a given maximum size. The third

method uses strength measurements to obtain a statistical description of the

crack size distribution in the body. Once the crack size has been determined

the time to failure and the strength can be estimated from the fracture mechanics

data.
2

The accuracy of lifetime predictions depends on the accuracy with which the crack

4

growth data and the initial crack size can be determined. Both of these are

experimental quantities, and as such are influenced by errors of measurement.

Both systematic and random errors are important. Errors of measurement result

in uncertainties in predicting the lifetime and strength of structural components.

Since it is not possible to completely eliminate errors of measurement, an error

analysis is needed to establish confidence limits for the application of fracture

mechanics to structural design. The purpose of this paper is to provide just

such an analysis. . ;

• ^

The uncertainty in the failure time is estimated from the variance (of both

the fracture mechanics data and the flaw size detection capability), using

standard statistical techniques. Then, for illustration purposes, confidence

limits are determined using fracture mechanics and strength data for two brittle

materials, glass and silicon carbide. Finally, the practical implications

of the analysis for data collection and for lifetime prediction are discussed.

2. THEORY r ,..: i,,v ^ : - no

2.1 Variance of Time to Failure / -

• '•'

Fracture mechanics techniques have been used to collect crack growth data

on a variety of brittle materials. In general, it has been shown that the quasistatic crack velocity, v, can be expressed as a power function of the stress

intensity factor, K^*, ,

.

, ,. ,,,,t

where v^ and n are empirical constants determined by a least squares fit of crack growth

data; is the value of Kj which is selected to express the stress intensity

*^nH'^'!fr.^T^'r"°" P^"^^^^'"^ to ^^tigue crack propagation, da/dN oc (ak)^

Quasistat r r.^'J^

analysis, which is equivalent to that presented here for quasistatic crack prnraafltmn. ran ho anni-iaw ^r.
factor in dimensionless form. has often been assigned the (arbitrary) value

5 6 As shown by the following equation, ' the time to failure, t, dep'rnds on the

applied stress, a , the crack velocity, v, the critical stress intensity factor,

Kj^, and the stress intensity factor, Kj^, at the most serious crack when it is

first subjected to a load:

t = (2/a^^^) j

(Kj/v)dKj (2)

where Y is a constant that depends on flaw and specimen geometry and relates the

applied stress and flaw size to the stress intensity factor (Ki = aaYv?)

.

By substituting equation (1) into equation (2), t can be expressed in closed

form. For n

>" 10, t is given, to a good approximation, by the following equation:

t = (2 KV(n-2) y^ya/) K. "^""^^

(3)

The error analysis is performed more easily if equation (3) is expressed in

logarithmic form; then znt has the following functional relationship to experimental and design variables:

Jin t = / (n, £.n v^, Jin K, . , in G ) (4) 0 i 1 a

equation

Using error propagation theory^the variance of £n t is given by the following

7

V(Ant) = (3f/9n)^ V(n) + 20f/3n) (9f/a£n v^) Cov(n,£nv^)

+ Of/3£n v^)^ V(£n v^) + (af/9£nKj.)^ V(£nKj.)

+ (9f/3£nag)^ V(£na^)+ 2(3f/9£nK, . ) (sf/9£naJCov(£nK. . , znc ) (5) •^1 u 11 a *Since f is a non-linear function, a linearized error propagation theory for the error in £nt has been used. This linearization is ;alid when the relative errors

in n and ;nK, . are of the order of 105' or smaller/ The validity of this approach

is supported By recent unpublished work by Ritter and Jacobs who show that

V(int) obtained by a Monte Carlo method is in substantial agreement with that
4

The first three terms to the right of the equality represent the contribution

to V(£nt) from uncertainties in the crack propagation data. A covariance term

is included because n and ^•n are not statistically independent quantities.

This term is needed to eliminate any dependence of V(£nt) on the set of axes

selected to represent the crack propagation data. The fourth term to the right

of the equality represents uncertainties in the value of in K^^. , while the fifth

term represents uncertainties in ino^. The sixth term accounts for possible

statistical interdependence between and zno^.

Substituting equations (3) and (4) into (5) we obtain;

^iant) = [)inKj.-JinK^+(n-2)'^ ]^V(n)+2[s,nKj .-£nK^+(n-2)'^]Cov(n,£nvQ)

+ \l{lny^) + (n-2)^ V(ilnKj.)+ 4 V(£na^)+4(n-2)Cov(JlnKj . , £na^) (6)

The variance terms of equation (6) can be evaluated from the crack propagation

data and from an analysis of the method used to determine in K, . . ino^ is a

Ii a design variable that is assumed to have a known value for purposes of this paper

(i.e. Viina ) = 0 and Cov(£nKT., ino) =0). In a real component, these variances

a n a are not zero, and their values depend on how close the design stresses approximate

those actually present in a component. .m;,

2.2 Variance From Crack Propagation Parameters

The parameters in the crack propagation terms of equation (6)

are determined by the classical method of least squares. The fit of the crack

propagation data is performed in Kj upon in v(see Appendix A);

in Kj = {in - ^ Jin v^) + ^ nn V, ' '

' (7)

where n and in v^ are given by,

N

'

• N
5

j,n = <in v> -n (< Jin Kj> -2,n K^) (9)

N

N where <ln v> = £n v^j/N; <£n Kj> = in K^^j/N and N is the total

j^l i=l

number of data points.

V(n) and V(2,nv^) are given by the following equations:

N V(nj = n^ V(6)/E(iin v . - <Jlnv>)^ (10)

V(£n Vq) = (<£n Kj>-£n K^)^ V(n) + n^V(6)/N (11)

where

^ V(6) = (^n - £n + ^ in - ^

£nv^. )^] / (N-2) (12)

j = l

Finally, Cov (n,Jlnv^) is obtained by methods given in Appendix b*

Cov(n,£nvQ) = -{<in Kj> -In Kq) V(n) (13)

Substitution equations 11 and 13 into equation 6, V (£nt) reduces to:

V(Jlnt) = [an Kj. - <Jin Kj>+ (n-2)"'']^ V(nj + n^V(5)/N

+ (n-2)^ V(£n Kj.) (14)

Note that V(Jlnt) does not depend on K*. Thus, the error in the time-to-failure

prediction does not depend on the set of axes, or the system of units, which are

used to represent the crack propagation data. As noted above, this result can

be attributed to the covariance term in equation (5).

The above result (equation 14) can be used only if the errors

of measurement are random, i.e. the data points must be statistically independent.

In crack growth studies statistical independence is usually obtained when each

data point is determined on a separate specimen. Then, errors resulting from

indeterminate characteristics of specimens are limited to only a single data

*Although V(£), as given by equation 12, appears to depend on Kq, substitution of

equation 9 into 12 shows that the variance of the fit V(6) is independent of K .
point. The errors do not propagate through an entire set of data as they would

if an entire K^-v curve was generated from a single specimen. Systematic shifts

of crack propagation data that have been reported in the literature are due to the

fact that complete sets of crack propagation data were collected from individual

specimens '. When statistically significant shifts occur between sets of K^-v

data, equation (14) is no longer applicable and other means of estimating V(£nt)

must be found.* One method of analyzing sets of K^-v data is to calculate the

slope and intercept for each data set, and treat these slope-intercept pairs as

estimates of the true slope and intercept, statistically independent from set

to set. If n" and ln\j^ are the slope and intercept obtained from equations

(8) and (9) for the a^*^ set of crack propagation data and there are J such sets

of data, then the mean values of the slopes and intercepts of the data are

+

J

given by the following equations

r J n

=

a=l

/J %r\ v. a=l

/J (15)

and the variances of n and ir\ Vq are

V(F) E (n^'- n )

= a=l

r J

/(J-1) J

V(iln v^)

0

E (^n v^ - ur^ r |/(j-i)j

a =1 ]

(16)

(17)

Finally, since n and in v^ are not statistically independent parameters.

the covariance of n and iln v^ are estimated from the following equation:

*However, if the shifts are not statistically significant (as determined by an analysis of variance^ >9) ^ there may be justification for pooling the data, in

which case equation (14) would be applicable.

+Note that the superscript a is an index and does not denote exponentiation.
7

J _

Cov (n, uT^ -- E (n^- n ) (in v^" - IPTIT )/(J-l)J (18)

^ a=l

An estimate of V(2.nt) can now be obtained by substituting equations (15) through

(18) into equation (6) (after using the transformation n-> n and inv^ 2.n ).

The equation obtained by this substitution is independent of K^, again demonstrating that the error in the predicted time to failure does not depend on the

set of axes chosen to represent the crack propagation data.

2.3 Variance From Initial Stress Intensity Factor

2.3.1. Direct Measurement

The initial stress intensity factor, K^^. , can be determined by

direct measurement of the initial crack length, a^ , which is then substituted

into • = a Y /aT . The accuracy of . then depends on the accuracy of the

i 1 a 1 i 1 crack length measurement;

V(]in Kj.) = (1/4) V(£n a.) (19)

For a small relative error in the crack length measurement equation (19)

becomes

;

V(iin Kj.) = V(a.)/4a^ (19a)

where V(a^. ) is obtained by performing several completely independent flaw size

determinations for each supercritical flaw (each scan contributes a single flaw

size measurement, per flaw).

Since the equation obtained by this substitution is more complex than equation

6, it is simpler to use equation 6 directly for an estimate of V (£nt).
8

2.3.2. Proof Testing ' ;

From fracture mechanics considerations it has been shown that the

initial stress intensity factor, Kj^. , is related to the proof test stress, a^,

. 3

by the following equation :

An estimate of the minimum time to failure, t^^^-^, can be obtained by substituting

relationship (20) into (3) and by using the equality sign of the relationship. V(£n K^^.

is now determined by the accuracy of the proof test ratio, a /a , and of the

P

3

critical stress intensity factor: - ^

'

V{Jln Kj.) = V (£n K^^) + V(Jln o^/o^) '

' (21)

If the errors of these quantities are small then the following approximation is

appropriate:

V(*n K,,.) = V(K,^)/k2^ + (Op/a^)-2 v(ap/aj (22)

2.3.3. Strength Measurements

Strength measurements can be used to provide a statistical value

of Kj^. . The strength in the absence of slow crack growth, a^^, is first determined on a series of ceramic components. The strength can then be related to

the cumulative failure probability, P, by the use of order statistics.'' If the

strength data fits a Wei bull type distribution,^^ then,

i_ £n Q E Iniln []/{]-?)]}= m In (a, /a ) .' (23)

I c o

where and m are empirical constants determined from the data , and Q Is defined

as 2,n[l/(l-P)]. A least squares fit of )lnQ upon In o^^^ can be used to determine

m and a . o ^
By substituting Kj^ = a^^Yv^ and Kj. = ^^gY/aT into equation (23), the

following equation is obtained for K^^.

:

^li = '"a/"o' ^Ic

(^''

V(JlnKj^. ) is now given by the following expression:

V(£n Kj.) = V(£n o^) + ^ (£nn)^V(m)

m

- ^ (jinQ) Cov(m,£na^) + \'{ln K^^) (25) m

The covariance term is introduced into equation (25) because and n are not

statistically independent, m and may be evaluated by the method of least

squares .*

M JI

^1 m =Xi [j^nQ^] [Jina^-<£na>] / {ina^-<Zna>) (26)

r=l

where

1 2,na = <S-na> <znQ> (27)

0 m

<ina> =(2 lno)/^^ (28)

r=l

M

<lnQ> = i^nQ^ /M (29)

r=l

and M is the total number of strength measurements. The cumulative failure

probability is given from order statistics as = r/(l+M) so that

= £n [1/(1-P^)].

*The subscript Ic has been dropped in these equations and a running subscript,

r, has been used for the suaimations.
V{m) and \l{ino^) are also obtained by the method of least squares

M

V(m) = V(e) / £ {lna^-<ino>) (30)

r=l

Vlima^) = [ <ilnQ>]2 V(m)/m^ + V(e)/m^M ^^^^

where V(e) is the standard deviation of the fit given by the following equation

V(e) = EUnQ^ - ni£n(a /a„)]^/(M-2) ^^^^

r=l r' c

The covariance of m and ina^ is derived in Appendix B:

Cov (m ,ilna^) = (<W> ) V(m)/ni^ : ^ (33)

By substituting equations (31) and (33) into equation (25) the following

expression is obtained for V(2.nKj^. );

V(JinKj.) = [iinQ - <£nQ>]^ V(m)/m^+V(e)/m^M+V(ilnKj^) ' (34)

2.4 Estimation of Confidence Limits :

'

' - ^

Once V(ilnt) has been obtained, confidence limits are calculated from

V(iint) and a Student's t^ table*. To find t^ from the table, the number of

a a

* t^ is not the time to failure, but is a number obtained from a Student's

aStatistical Table.
n

degrees of freedom, vs has to be estimated from the experimental data. A method

of estimating v

''S given by Welch]''who discusses "Student's" problems when several

different population variances are involved. For the direct flaw detection method

V is calculated from;

2

1/v = (1/v ) [V(v)/V(£nt)] + (1/. ) (n-2)'

4

2 2i~~J [V(a.)/V(iint)]^ (35)

where v and v are the number of degrees of freedom determined from the crack

V a velocity and flaw size data respectively, and V(v) is the component of V(£,nt)

due to the crack propagation data [the first three terms in equation (6)];

is given simply by N-2 (where N is the number of velocity measurenients*) ,

and V is given by the number of flaw-size measurements minus one.

a For the proof test method;

1/v = (l/v^)[V(v)/V(£ntj]^ + (l/v,<)[V(K)/V(£nt)]^ (36)

where V is the number of degrees of freedom determined from the critical stress

intensity factor data, (equal to the number of K^^ tests minus one), and V(K) is

the component of V(£nt) due to the critical stress intensity factor data

[V(K) = (n-2)2 nzn K^^)]

The number of degrees of freedom for the strength method is given by;

Vv = (l/vy)[V(v)/V(£nt)]^ + (l/vK)[V(K)/V(£nt)]^

+ (l/v^)[V(a)/V(iLnt)]^ (37)

where V(a) is the component of V(2.nt) resulting from the Weibull fit of

the strength data [(n-2) times the first two terms of Equation (34)]

and are the number of degrees of freedom (M-2) associated

If equations (15-18) are used to estimate V(v), is given by J-1 where

J is the number of sets of K-v data used to determine V(v).
12

with that fit.

Once V has been estimated, t can be obtained and cona_ fidence limits for iint can be calculated from the usual expression,

1 /2

± t [V()int)] . The limits can be plotted on design diagrams to give an

a estimate of the uncertainty of the predicted failure time.

^

'

-

3. APPLICATION

3.1 Data Analysis and Collection

In this section, the theory presented in Section 2 is applied

to two sets of experimental data; one set was collected on a low expansion silica

glass (7.5 w% Ti02, 92.5 w% SiO^);^'^ the other set was collected on hot-pressed

1 silicon carbide. ^ The K-v curves determined from these data are shown in Fig. 1.

The glass data were collected in water using the double cantilever

beam technique. Although there appears to be little scatter between the three

q curves, an analysis of variance shows them to be separate. Therefore, the

between specimen scatter is significantly greater than the within specimen

scatter, and V(n), V(Jlnv ) and Cov (n, in v ) have to be calculated from

0 0

equations (15) through (18). V(£,nt) is determined by using equation (6). Data

on the strength (Fig. 2a) and critical stress intensity factor are also available for

this glass, so that V(jinKj^. ) can be estimated by the proof testing method,

equation (21), and by the strength method, equation (34). An estimate of the

initial flaw size ('vlOnm)can be obtained from the strength data. Since the

small size of the strength impairing flaws in the glass precludes the use of

non-destructive detection with presently available techniques, the variance

calculations are not performed for this method.

The silicon carbide data were collected at high temperature (1400°C)

using the double torsion technique. Since seven specimens were used to collect

the data, equation (14) is appropriate for calculating V(£nt). Data for the
13

strength (Fig. 2b) and Kj^ are also available and variance calculations can be

performed for both the proof testing and strength methods. Again the small

size of the initial flaws ('^-l OOym) precl udes the use of non-destructive

detection techniques to estimate initial flav; sizes.

The pertinent statistical data for the estimation of V(Jlnt) are listed

in Tables 1 and 2; and the failure diagrams, including the 90% confidence limits,

are presented in Figs. 3-7. For discussion purposes, the failure prediction

methods are considered separately.

3.1.1 Direct Flaw Detection Method

The direct flaw detection approach is presently less definitive

than the other two methods because the available test techniques have not been

well quantified. Consequently, meaningful data which we can use to obtain V(a^)

are simply not available. For completeness, however, we have indicated qualitatively in Fig. 3 the type of information we would anticipate from a flaw size

determination based on the proviso that a1

1

flaws have been detected (this topic

is discussed in more detail in Section 3.2.2). There will be a lower size limit

of detection determined by the background scattering of the material , the test

technique, and the relative physical properties*, of the defect and the material.

For ceramic materials this lower limit is presently of the order of 100 ym, and

this limit is indicated on the failure diagram (Fig. 3 j. Also, as the flaw

size increases, the consistancy of detection generally (but not always) improves;

this will be reflected in a reduced V(a.) in equation 19.

3.1.2 Proof -Test Method

In the proof-test method the confidence limits depend on errors

in both the crack propagation and critical stress intensity factor data. Since

V()lnt) depends on 2-n Kj^-, increasing as In Kj^- decreases (see equation 14), the

width of the confidence bands are expected to increase as the proof test ratio is

* For example, the ratio of acoustic impedances determines the intensity of the
14

increased. This increase in width is observed in both the glass and the silicon

carbide diagrams (Fig. 4 and 5); it occurs because the lifetime calculations for

large proof test ratios require extrapolation outside of the range of the crack

propagation data. (This is discussed further in Section 3.2.2.) The broadening

of the confidence limits is more apparent for silicon carbide than for the

glass, reflecting a more accurate determination of Kj^ for the silicon carbide.

In fact the error in Kj^ for glass was sufficiently large that it was the

primary factor determing the width of the confidence interval.

The magnitude of the confidence band thus depends on both the

basic material variability and the number of test specimens. It is not possible,

therefore, to determine a priori the number of fracture mechanics tests and the

(stress intensity factor) data range needed to achieve an acceptable separation

of the confidence limits. These quantities must be re-established for each

material. However, the silicon carbide exhibits a variability typical of

ceramic polycrystals and we anticipate from the analysis of the silicon carbide

data that about ten specimen will generally be sufficient for K^^ determinations

and another ten for K, v determinations.*

3.1.3 Strength Method -

Examination of the failure diagrams (Figs 6 and 7) for the strength

method indicates that the width of the confidence band increases as the failure probably

+ decreases, and the confidence bands broaden as the failure times increase.

For fused silica, for example, thirteen tests to determine gave a value

of (n-2)'^V(Jin Kj^) equal to 0.054; this compares with a value of 0.687

for four tests on the titania glass, described above.

+ This effect is more apparent for the silicon carbide because the variance

in Kj^ is small compared to the other variance terms.
15

Again these effects occur because of the uncertainties that result from

extrapolation of the strength and crack propagation data beyond the data limits.

Inspection of the individual variance terms indicates that the

major contribution to the variance in the failure time emanates primarily from

the variance in K^^ at high failure probabilities, but from the variance

in the strength data at low failure probabilities. As discussed above, V(£riKj^)

can be substantially reduced by obtaining a reasonable number of fracture

mechanics data (about 10). We anticipate, therefore, that close control of

the confidence band will generally be possible at the high probability extreme,

with a modest test schedule. However, the variance in the strength data can

only be significantly reduced at low probabilities by performing large numbers

of tests. There are limits (of time and expense) to testing more than a few hundred

specimens, and it thus becomes unreasonable to expect that a close separation

of the confidence limits can be achieved at the low probability extreme. We

suggest, therefore, that about twenty to forty strength tests be performed and

that the resultant confidence limits should establ ish the lifetime expectancy

for the material

.

3.2 Life Prediction

For any well defined structural component, the service life, the

stress distribution, the tolerable failure probability and the level of confidence should be clearly delineated quantities. The role of the lifetime

prediction analysis is thus to identify test procedures which ensure that the

component satisfies these structural specifications. The failure diagrams

presented in Figs. 3-7 supply the pertinent information, at least in

principle, and these can be used directly for a preliminary selection of a

reliability test procedure. Certain limitations of the failure prediction

methods should then be examined before embarking on a final selection.
16

3.2.1 Preliminary Test Method Selection

The preliminary selection process comprises the following steps.

The failure diagrams are firstly developed for each test method from the fracture

mechanics and strength data. The confidence bands, for the confidence level

specified by the application, are then superimposed, as illustrated schematically in Fig. 8 • The intersection of the specified lifetime and stress

coordinates* on each diagram (Fig. 8 ) then designates the service requirements.

Commencing with the probability diagram (Fig. 8 a), the potential

for utilizing the material without 'a nondestructive evaluation procedure can be

assessed by comparing the location of the lower bound of the acceptable failure

probability with the service condition. For the example in Fig. 8 a, the material

would be acceptable for this application, without evaluation, if the tolerable

failure probability were ^^2' failure probabilities <P2,

each component must be subjected to an evaluation procedure.

For materials which require evaluation, we now proceed to the

proof test diagram (Fig. 8 b). The proof condition that ensures reliability is

selected from the diagram by comparing the service condition with the lower

confidence bound of the proof ratios. For the present example, (Fig. 8 b) the

proof ratio, R^, would suffice.* Next by referring again to the probability

diagram, the proportion of components that are likely to fail in the proof test

can be estimated by determining the intersection of the stress coordinate at

Hereafter termed the 'service condition'.

* The fact that the confidence band for each proof level extends over an order

of magnitude (or more) in time is not an impediment to the application of

the proof test technique, it simply requires that the proof ratio be set at

a slightly larger value (see Figs. 4 and 5) than anticipated by the analysis

that does not account for the data variability. The required increase in the proof

test ratio, /a , is given by: Aa /a = (cTn/^a^ ^^a ^^(^nt)' )/(n-2) ; which

can be derived from equations 3 and 20.
17

at the proof stress io^), and the time coordinate for the time at the peak proof

load (t ) (Fig. 8 a). If this proportion is unacceptable (for economic reasons)

P the material is inadequate and an alternative material should be explored,

this process should proceed until a material with a satisfactory

proof test failure proportion is identified.

Finally, if proof testing is unsuitable (due, for

example, to the complexity of the component)* the direct flaw detection technique

should be explored. The approach taken at this juncture is strongly material

dependent. An approximate value for the flaw size that must be detected to ensure

integrity is firstly obtained from the flaw size diagram, without accounting for

the extension of the confidence limits (a^ in Fig. 8c). Then the detectabi 1 ity

of such flaw types should be evaluated using available NDE techniques. At this

stage, confidence limits on flaw detection in the test material can be established

for each NDE technique (the dotted lines in Fig. 8c), and the flaw size detection

requirement can be more closely defined. The suitability of the material for this

application is determined by the capability of detecting all flaws (in the critical,

parts of the component) that exceed the specified size.

3.2.2 Limitations

bach of the failure prediction approaches have limitations

and these are briefly discussed here because they strongly influence the final

selection of a test procedure. The prerequisites for effective proof testing

have been discussed in detail in other publ ications^ and hence are only cursorily

presented here for completeness. The proof test must be devised such that the

stress in each element of the component exceeds the service stress in that same

element by an amount at least as large as the recommended proof ratio. The
18

proof test must be performed under controlled conditions of environment and

load cycle to validate the predicted lifetime. If these conditions can be

satisfied, then proof testing is undoubtedly the best failure prediction method,

because there are no unresolved aspects of the analysis and because the test

errors are minimal.*

The strength method is unequivocal provided that the strength

distribution function is determined on actual components (for each batch of

material) under conditions that simulate the service condition. This requirement often poses an intolerable economic constraint, because large numbers of

tests must be performed to achieve acceptable confidence. An alternative

application of the strength method uses strength data obtained on small

specimens machined randomly out of components selected from each batch. The

specimen strength distribution is then converted into the appropriate component

strength distribution using flaw size statistics. This approach is only valid

if the flaw size statistics are treated correctly and if the flaw distribution

function is invariant (i.e., the same for the components as for the specimen).

Recent statistical treatments have established a basis for handling flaw size

14 statistics correctly, but additional analytical developments (such as an allowance for the relative severity of internal and surface located flaws) are still

required. Additionally, the validity of the constant flaw distribution assumption

must be evaluated empirically for each material. Consequently, the strength

method has minimal current application; but the specimen testing method may

develop into a widely used approach if the uncertainties in the utility of the

flaw size statistics are satisfactorily resolved.

The direct flaw detection method is the least definitive ^for ceramic

* Each component is subjected to a single well-defined test, and the

confidence band due to test error should then be insignificant.
19

iraterials), at its present stage of development. The two principal limitations are,

the correctness of the predicted flaw size, and the requirement that al

1

flaws

larger than the specified size must be detected. A problem occurs with the

prediction of the flaw size, because the microfracture mechanics parameters,

that pertain to the extension of small preexisting flaws, can differ from the

macrofracture mechanics parameters, obtained using conventional fracture

mechanics specimens,* For certain materials, therefore, flaws smaller than

the size predicted by the failure diagram may need to be detected.

The requirement that all supercritical flaws be detected can be a major source

of test error. It is premature to attempt to ascertain the confidence limits

due to test errors, but unless a highly automated test routine can be

established, the confidence limits could be disturbingly expansive.

Finally, we discuss limitations in the analysis of lifetime prediction

that originate from the limited range over which crack propagation data can

be collected. If the initial stress intensity factor, K^^. , is less than the

minimum measured value of stress intensity factor, Kj^^.^, for the crack

propagation data, it is necessary to extrapolate the crack propagation data

beyond its range of measurement in order to obtain a lifetime prediction.

The analysis of variance in Section 2.2 gives an estimate of the uncertainty

involved in this extrapolation only if equation (1) is assumed to be valid

for the region of extrapolation for which Kj^. i Kj j< ^i^ir)'

manifestation

of this uncertainty is the broading of the confidence limits which was noted

The discrepancy between the micro- and macro- fracture mechanics parameters

(when it occurs) generally appears as a difference in the absolute value of

the stress intensity factor (i.e., in Kj^) with no effect on the slope, n, and this only has a significant effect on failure prediction using the flaw

size method.
20

in Section 3.1.2 for longer failure lifetimes.) However, since equation (1)

is an empirical fit of crack propagation data, there is no a priori reason

to expect such an extrapolation to provide an accurate prediction of failure

lifetimes. In fact, other empirical equations sometimes fit the crack

propagation data equally as well For example, an exponential dependence

of quasistatic crack velocity upon stress intensity factor, v = v-, exp (b Kj/K^),

has been used to describe crack propagation data in glass and other ceramic

materials. The power-function [equation (1)] and exponential -function

representation of crack propagation data are compared in Figure 9, where

2 t . a, IS plotted as a function of the proof-test ratio. The + in the min a

^

figure marks the lower limit of the crack propagation data, for which

°p^^a

~ '^Ic^'^Imin'

difference in lifetime prediction for larger prooftest ratios results from an extrapolation of the crack velocity data to

values of Kj less than Kj^^.^. This diagram illustrates the importance of

low-velocity data for effective failure predictions and the fact that

extrapolation below this limit should be viewed with some caution. This

limit can be easily incorporated in the failure diagrams. Substitution

of the minimum measured stress intensity factor, Kj^^.^, into equation (3)

for Kj^, gives the maximum lifetime which can be predicted for a given

applied stress without extrapolation of the crack propagation data. For

the proof-test diagram, this is equivalent to a proof-test ratio of

0 /a = Kt /Kt • . This same line applies to all three diagrams, as is

p a Ic Imin

illustrated schematically in Figure 10. Figure 10a also illustrates

a region of extrapolation for the strength measurements to lower

probabilities. Using order statistics, this limit is a probability of

P = 1/(M+1 ).
21

4. CONCLUSION

A method for the statistical treatment of data to be used for failure

prediction has been presented. The analysis of two sets of typical data show

that the 90% confidence bounds for the predicted lifetime can spread over two

to four orders of magnitude (depending on the method used to estimate the initial

flaw size). However, the stress allowables (or the conditions for the reliability

test) based on the lower confidence bound do not differ substantially from those

predicted from median values of the data, primarily because the slow crack growth

exponents for ceramic materials are relatively large.
22

APPENDIX A

Least Square Fit of the Crack Propagation Data

The constants in\j^ and n (equation 1) are determined from a least squares

fit of the crack propagation data. The method used to obtain this fit must give

the best possible representation of the crack propagation data in order to obtain

a reliable prediction of the time to failure. The classical method of applying a

linear least squares fit to a set of data is based on the assumption that one

of the variables of the fit (the independent variable) is free from error. The

entire error of the fit is assumed to lie in the other variable (the dependent

variable). Unfortunately, both variables of a least squares fit are often

derived quantities and are subject to error. In the present analysis, both v

and Kj are derived quantities and therefore are subject to error, v is often

determined by direct measurement and its accuracy depends on the accuracy of

measuring length and time. Kj is a calculated quantity that depends on measurements of specimen dimensions and crack length. In addition, other uncertainties such as crack tip shape and the orientation of the crack plane to

the specimen surface can result in serious errors in measured Kj. Thus, for

purposes of failure prediction it is necessary to demonstrate the applicability

of the classical least squares method to the crack propagation data and to

determine which variable (Kj or v) is to be assumed free from error.

An extensive discussion of the fitting of straight lines has been given

by Mandel^ He has described a method of least squares that can be used when

both variables are subject to error, provided independent estimates of the

relative error in the two variables can be obtained. The method is somewhat

more complicated than the classical method of least squares. Mandel has compared this more general method with the classical method and described the
23

conditions that must be met for the classical method to be applicable. For a

linear equation Y = a + BX, the parameter that must be considered to decide

on the applicability of the classical method is the sensitivity ratio.

where and are independent estimates of the error in y and x respectively.

If RS << 1 then the variables y and x should be fit y upon x (x is assumed to

be free of error). However, if RS » 1 then the fit should be x upon y.

By applying these arguments to crack propagation data, it can be shown

for most ceramic materials that K-v data should be fit £nKj upon £n v (£nv is * assumed to be free of error) . In glass for example, 15 < 6 = n< 100; o^^y^

0.03; a^^i^ ~ 0.03. Therefore, RS = n/i^^j^^y/aj^^i^) > 15, and the least square

fit should be JinK upon Jinv. The constants of the least squares fit can then

be calculated from the following equation.

RS = &/{a^/o^) (Al)

(A2)

which is just equation (1) rearranged.

* It should be emphasized that this conclusion may not be applicable for

metals for which n is not a large quantity.
24

: - APPENDIX B

Estimate of the Covariance

The relationship between Jlnv^ and n is given by equation (9):

In = <iinv> -n(<!lnKj>^nKQ) (Bl)

From this equation we see that the intercept, in v^, depends on the slope, n*

the origin of the axis for measuring Jin Kj, Hn K^, and the mean value of

jyiKj, <JlnKj>. Note that when <fi,nKj> = JlnK^, In v^ =<lnv>. Thus, if the

origin of InK^ axis is selected to equal <lnKj>, a regression analysis of

ilnKj upon 2,nv yields a slope and intercept that are statistically independent.

Using equation (Bl) the regression analysis of S,nKj upon !lnv, equation (7),

can be written as:

an K, = <lnK, >+- (Unv - <!lnv > ) (B2)

I In

where <lnK^ > is the intercept and 1/n is the slope of the fit. As already

noted oinKj > and 1/n are statistically independent. Therefore, Cov (n, <inK^ >)= 0.

Cov (n. In v^) can now be evaluated by substituting equation (Bl) for

Jin v„

Cov (n,JlnvQ) = Cov (n, <!lnv> - n[<JlnKj> -^nK^]). (B3)

But since n is not statistically dependent on <Jlnv> or <JlnK,> ,
25

Cov(n,£nvQ) = -(<ilnKj> -£nK^) Cov (n,n), (B4)

and since Cov (n,n) = V(n)

Cov(n,ilnvQ) = -(<!lnKj>- jlnK^) V{n) (B5)

which is the equation used in the text.

By similar arguments, it can be shown that for a Wei bull analysis of

strength data,

Cov (m, Jin cr^) = Cov (m,<5,na> - ^

<inQ> )

= - <S,nQ > Cov (m,l/m)

<S,nQ> V(m)/m^ (B6)
(r.VV '
Table 1. Summary of Statistical Data - High Silica Glass

Crack Propagation Data

(with = 1 Nm"-"^^)

Specimen Data Points n in V 0

1 19 37 .369 -498.68

2 19 34 .400 -460.66

3 22 35 .174 -470.09

n = 35.648; V (n) = 0.791

Jlnv^ = -476.48; V (£nv ) = 130.68

0 0

Gov (n.Jlnv^) = -10.165

Stress Intensity Factor Data (determined on 4 specimens)

<ir\K^^> = 13.454

V(<JlnK,p >) = 6.072 x 10"^

Strength Data

m = 6.222 (determined on 30 specimens)

V(m) = 4.217 X 10"^

V(e) = 3.9233 x 10'^

<£nQ> = -0.5362
Table 2. Summary of Statistical Data

Hot Pressed Silicon Carbide

Crack Propagation Data

(with Kq = 1 Nm"^/^)

n = 23.6326

V(n) = 0.7998

V(S) = 9.8238 X 10'^

< £nKj> = 14.528

' N = 21

iny^ = -353.67

V(£nv^) = 168.83

Cov (n.^nv^) = -11.620

Stress Intensity Factor Data (determined on 8 specimens)

<ilnKj^> = 15.146

V(<iinKj^>) = 9.2048 x 10"^

Strength Data

m = 9.401 (determined on 19 specimens)

V(m) = 9.8439 x 10"^

V(e) = 2.3150 X 10"^

<£nQ> = -0.5217
References

1 A G Evans and S.M. Wiederhorn, "Proof Testing of Ceramic Materials - An

Analytical Basis for Failure Prediction," Int. J. Fracture, [3], 379-392

(1974).

2 R W. Davidge, J.R. McLaren and G. Tappin, "Strength-Probability-Time (SPT)

Relationships in Ceramics," J. Mater. Sci., 8, 1699-1705 (1973).

3 S.M. Wiederhorn, pp. 633-663 in Ceramics for High Performance Applications,

eds*. J.J. Burke, A.E. Gorum, and R.N. Katz, Brook Hill Publ . Co., Chestnut

Hill, Mass. (1974).

4 B J S. Wilkins and L.A. Simpson, "Errors in Estimating the Minimum Time-toFailure in Glass," J. Am. Ceram. Soc, 57 [11], 505 (1974).

5. A.G. Evans, pp. 17-48 in Fracture Mechanics of Ceramics, Vol. 1, Eds.

R.C. Bradt, D.P.H. Hasselman and F.F. Lange, Plenum Press, New York (1974).

6. S.M. Wiederhorn, pp. 613-646 in Fracture Mechanics of Ceramics, Vol. 2, Eds.

R.C. Bradt, D.P.H. Hasselman, and F.F. Lange, Plenum Press, New York (1974).

7. J. Mandel , The Statistical Analysis of Experimental Data . Interscience,

New York (19647:

8. J.S. Nadeau and J.N. Hay, "An Index of Resistance to Delayed Failure,"

J. Can. Ceram. Soc, 43, 31-36 (1974).

9. O.L. Davies and P.L. Goldsmith, Statistical Methods in Research and Production,

Harper Publ. Co., New York (1972).

10. W. Weibull, "A Statistical Theory of the Strength of Materials," Ingenibrs

Vetenskaps Akademien Handlingar, No. 151, 1939.

11. B.L. Welch, "The Generalization of 'Student's' Problem When Several Different

Population Variances are Involved," Biometrika 34 , 28-35 (1947).

12. 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_ [7],

319-323 (1974).

13. A.G. Evans and F.F. Lange, "Crack Propagation and Fracture in Silicon

Carbide," J. Mat. Sci. in press.

14. J.R. Mathews, F.A. McClintock and W.J. Shack, "Statistical Determination

of Flaw Density in Brittle Materials," J. Am. Ceram. Soc, to be published.

USCOMM-NBS-OC
i h
Figure 1. Crack velocity as a function of applied stress intensity factor

(a) high silica glass tested in water

(b) hot pressed silicon carbide tested in air at 1400°C.
1 1 1 1 1 1 r

High Silica Glass

0.99

0.9

d 0.5

flO <

eaO

eci 0.2

DC< 0.1

0.05

0.02

140 160 180 200 220 240 260 280 STRENGTH, M Pa

500 550 600 650

STRENGTH, MPa

700

Figure 2 Strength as a function of cumulative failure probability, P,

(a) high silica glass tested in dry nitrogen

(b) hot pressed silicon carbide tested in air at Zb L.
lOOOMm 100/Jm lOMm l^ni

10 100 1000 SERVICE STRESS. MPa

Figure 3. Design diagram based on direct flaw size measurement,

silicon carbide. Size of flaws in this material,

'^'^00ym, precludes their detection and analysis by

statistical means, thus the lack of confidence bands

on this diagram.
SERVICE STRESS, M Pa

Figure 4. Design diagram based on proof testing, high silica

glass

.
1 10 100 SERVICE STRESS, M Pa

Figure 5. Design diagram based on proof testing, hot pressed

silicon carbide.
10-' 10" 10' lO-"

1 10 100 SERVICE STRESS. M Pa

Figure 6. Design diagram based on strength measurements, high silica

glass

.
Figure 7. Design diagram based on strength measurements, hot

pressed silicon carbide.
Figure 8. Composite design diagram.
1 r

1.5 2.0 2.5 3.0 3.5 4.0

Proof Test Ratio, c^/ojj

Figure 9. Proof test diagram comparing power function and exponential

function representations of crack propagation data, high

silica glass. The plus sign, +, in the figure marks the

lower limit of the crack propagation data, reference 12.
. Schematic representation of design diagrams indicating

regions of strength and crack propagation data extrapolation Uj strength method, contains two regions of extrapolationone due to strength data; the other due to crack propagation

data, (b) proof test method, (c) direct flaw measurement.
NBS.n4A (REV. 7-73)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

'• PUHLICATION OR REPORT NO. NBSIR 75-952

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE .\N'D SUBTITLE

An Error Analysis of Failure Prediction Techniques

Derived from Fracture Mechanics

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

S. M. Wiederhorn^ E. R. Fullpr, Jr., .1. Mandpl and A. f;.Fvan«;

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NA.ME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No.

3130459

12. Sponsoring Organization Name and Complete Address (Street. City, State, ZIP)

Air Force Materials Lab

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

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.) Three principal methods of failure prediction for brittle materials are analyzed statistically. Each method depends on fracture mechanics for its predictive

value and hence, the variance of the failure time is found to depend on the scatter

in the fracture mechanics data and the scatter in the estimate of the initial size

of the strength limiting crack. The variance is used to calculate confidence limits

for the prediction of failure for two materials, glass and silicon carbide. Procedures for the collection and analysis of data are discussed, and the implications

of the analysis for lifetime prediction are evaluated.

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 propagation; error analysis; failure prediction; failure prevention; fracture;

statistics; strength

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