in. <*

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

December 1979

Prepared for Department of Energy

Fossil Fuel Utilization Division

Washington, D.C.

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

December 1979

Prepared for Department of Energy

Fossil Fuel Utilization Division

Washington, D.C.

U S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Jr., Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

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 Kj

11 , 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.

2
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

3
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

4
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 *

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

Kj = a Y S~T (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

5
can be defined in terms of the crack length, a^ and the critical stress

intensity factor Kj

C

, from Eq. (1):*

S = K

IC

/Y /“a. (2)

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

then Kj = K^, 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 subcritic;

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

6
The strength of components during a stress cycle is illustrated in

figure 2a by curves labelled, S-j , and Sg. 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, S-j , much greater than the maximum applied stress,

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

the initial strength, Sg, 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, Sg, 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 Zq, z

^ and Zg 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. Kg, , Kg). For example,

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

in Fig. 1 , Kj = K-|»is given in fig. 2b by Z-| = cr(Kj

C

/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 a, and since the proportionality constant.

7
K IC

/K^ , is greater than 1, z-j is greater than the applied stress, a. In a

similar way, the boundary lines for Kg is given by Zq = a (Kjq/Kq) and

that for is given by z^ = a (k

ic/k2

).

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

5-11

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.

8
For single-region crack propagation, an analytical solution can be

obtained and used to plot strength degradation diagrams, Wei bull 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) = - (Y2/2K

I(

, 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^. Expressing the crack velocity

as a simple power function of the stress intensity factor

V = AK

I

n =AK

Ic

n (a/S)n (4)

the rate of strength decrease becomes:

dS/dt = -(A Y

2

K

Ic

n

" 2

/2) (a/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>15) or for region III (n>100) crack growth,

however, dS/dt exhibits strong negative curvature.

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

9
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.

o n — 9

where (1/B)=(n-2)AY Kj

c /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, cj, 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, <r , for a time t , and then

unloaded at a constant rate, d

u

. The times for loading, t-j , and unloading,

t

u

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

i r u r ^ equation (6) for a typical load cycle then gives the following equation for

the final strength,

S

f

n

' 2 = S,"

-2

-(1/8) [a

p

n t

p + ap

n+1

(1/6, + 1/6

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 S

.p (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.

10
The breaking stress, a*, ‘of those that 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 /d-,+t +{a -a*)/cr , H r r ^ which is the total time to failure during the proof test. The equation

so obtained is given by

0*

n '2

- (l/B) a*n+1 /(n+l)6

u

=

s^"2

- (l/B) [a

p

n

t

p + ap

n+1 (l/a

1 + l/du )/(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

f

, 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 , d-, , d , t ), the crack propagation parameters

r

' u r (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

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

* that survive, S i > S i , the proof test cycle. The component that has an initial

•k 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). Sfm-j n is important because it represents the lowest

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

S^

m^ 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

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

a*min 5

‘fr ° nTls a l°wer bound to the breaking stresses of components that fail

during the unloading cycle (figure 3a).

11
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

fniin = [MVB(n+l)) («^1n/4

u

)]

Equation 9 can be simplified by expressing in terms of

the unloading rate. During unloading, the strength degradation curve

that characterizes S^m

. 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

u

, of the stress curve. By letting dS/dt = -a

and o = S = cr*m .j n in equation 5, the following equation is obtained:

(9)

a* • = [B(n-2)d ]"^

*imn u v ' u

By substituting this relation for d

u in equation (9) a simple

equation is obtained for the minimum strength after proof- testing

00 )

Sfmin ' °*min [3/(n+1 )f

/(n ~ 2) (11)

One concludes from this equation that the strength, . , 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^- . /a^. is normally greater than 0.85.

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

relating the truncation strength, . , 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), Vjq = AK

Ic

n

, the following alternative

representation of equation 10 is obtained:

°*m in

* [(2*u K

IC

2

>/<V

IC

Y2 >]

1/3

< 12 >

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

depends only on the unloading rate, a the critical stress intensity

12
factor, and the limit crack velocity, V^, which is determined by the

crack propagation parameters, A and n. Therefore, 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, 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, a . This condition for truncation

r was noted earlier by Evans and Fuller'

7 from other considerations.

However even if cr*^ >_am

, strength degradation will still occur during

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

the only crack growth parameters that influence cr*^. 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

IC4. 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, Ff

, 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

13
distribution is illustrated below for a two-parameter Weibull distribution.

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

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. 3 » 7, 10, 11

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

1

3

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

1 /SQ

) m

(14)

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

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

)

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

proof test (characterized by Q ). +

r

Q-i = Qf + Q

p

(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 )/ (1 -F ) where F^ is

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

1 10 probability evaluated from eqn. 13 and F is the proof test failure probability.

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

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

probability (represented by Q

f

):

n-2

(S f/S

0

) n ‘ 2

= (Qf + Qp)

m

- (1/B) S

0

' (n ‘ 2) [a

p

n t

p + a

p

n+1

(1 /<J +1 /*

u

) / (n+1 )] (16)

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

that just survives the proof test cycle S.^. and -> 0. Substituting for

Sf and 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 + ap

n+1

(1/a-j + l/a

u )/(n+l)]

= ^-(Sfin/ So >"‘2

< 17 >

Substituting this expression into equation 16 eliminates the term containing

the square bracket so that:

(Sf/s

0

) n- 2 = (Qf + %)"* - Q

p

nm

2

+ (Sfmin/s

0

) n -2 08 ) 'P' 'P

Equation 18 illustrates the important conclusion that once S^.. , S

Q

, m and n are

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

•k 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

*S fmin approaches zero as the unloading rate becomes vanishingly small (type III

distribution7 discussed in part 1 of this paper), and has an upper limit when

S^m ^ n = a

p

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

the proof test, type II distribution 7, ^

discussed in part 1 of this paper.)

15
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 Q- » Q . If the final strength

r r is also much larger than the truncation strength (S^ » S^.

^

) , then the

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

is given by (S^/S

Q

) m

= Q^, 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^ >> . ) , and if

the failure probability after proof testing, Qf

, is less than the failure

probability, Q , calculated from Q D

= (c

D/SQ

) m

(i.e. << Q ), then the

strength distribution given by equation 18 can be simplified by expanding

(Q^r + Qp)^

n " 2 ^m

as a Taylor series. The final distribution for

the intermediate probability range is then given by:

Q

f

= [m/n-2)] Qp

( m+2 -")/m

(Sf/S

o

) (n " 2)

, (19)

which implies that the strength distribution after proof testing can be

represented by a straight line with a slope m' = n-2 (figure 3b). Finally, in the

lowest probability range, the strength distribution must be truncated

so that S^r = Sf . , where is given by equation (11).

16
In summary, the strength distribution 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 .

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

(IMPa/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 Weibull 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

17
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 . J fmi n 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, (<\J percent for 1000 MPa/s vs.

18
^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,

19
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.

20
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 Weibull 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 (n=o), 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

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

21
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 , P P

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 Weibull

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

22
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/

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. <10ym) 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.

23
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. 50_, 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., T_9_, 69-77 (1975).

8. A. G. Evans and H. Johnson, "The Fracture Stress and Its Dependence on Slow Crack Growth," J. Mat. Sci. 10_, 214-222 (1975).

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

Loading Conditions," Inst. J. Fract., 1_0, 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.,. 10., 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., 19_, 293-97 (1951 ).

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

Analysis of Reliability and Life Data, John Wiley & Sons, New York (1974).

24
Dependence of crack velocity on applied stress intensity

factor. Schematic diagram of crack velocity data indicating

three regions of crack growth. Boundaries of these three

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

?

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

.

-L Li
2.0

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 Eq» ^1» ^2 an<^ a an ^is f j-8ure correspond

to K-, K , K , and K of figure 1. The curve labeled S gives the strength degradation as a function of time.
Ln (S/S

Q

)

Single region crack propagation. (a) Definition of

critical initial strength, S«, minimum failure stress,

c*min» an<^ minimun strength, Sfmj_ 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, du , the critical stress intensity factor, Kj-q

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

i

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

; KT = 0.75

MPa-m 2

, Op = 0.65So.

f
Region 1 crack growth: schematic diagram of strength

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

(a) loading rate 1 MEa/s. Test in air, 50% r.h. (b) loa<* iag

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

_ a ®*

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

0.01%r.h. Experimental constants used for this figure are given in figure 4.
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 = -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 0 = 137.9 MPa, m - 8.4. £ther

constants for figures 6 through 9: K

IC = 0.75 MPa-m 2

; KQ -

0 MPa-m 2

; Up = 0.65S o

.
Ln { S/S0 )

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

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

(b) Strength distribution after proof testing.

7 .
0 0.2 0.4 0.6 0.8 1.0

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 .
Dependence of strength distribution after proof

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

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

Air, 50% r.h.

9 .
I

Ln (S/S0)

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

7 m/s; region 3, n = 120.9, InA = -1641.4.

Weibull parameters: S q = 126.9 MPa/s; m = 7.7.
NBS-114A (REV. 9-78)

U.S. DEPT. OF COMM.

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SHEET

1. PUBLICATION OR REPORT NO. 79-19442. Gov't. Accession No.

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4. TITLE AND SUBTITLE

Proof Testing of Ceramics: II. Theory

5. Publication Date

6. Performing Organization Code

7 - AUTH0R(S) P. B. Oates

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

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

ID. Project/Task/WorK Unit No.

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

Department of Energy

Washington, D. C.

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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 leas factual summary of moat significant information. If document includes a significant bibliography or

literature survey, mention it here.) 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 siigle-region crack propagation. Tor 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 distributio

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.

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 aemicolona)

Crack growth; fracture; fracture mechanics; proof testing; statistics; strength.

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