NBSIR 74-486

Reliability, Life Prediction and Proof

Testing of Ceramics

S. M. Wiederhorn

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

May 1974

Interim Report for Period July 1, 1973 through June 30, 1974

Prepared for

Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

NBSIR 74-486

RELIABILITY, LIFE PREDICTION AND PROOF

TESTING OF CERAMICS

S. M. Wiederhorn

Inorganic Materials Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

May 1974

Interim Report for Period July 1, 1973 through June 30, 1974

To be published in Proceedings of Conference, "Ceramics

for High Performance Applications," Hyannis, Mass., Nov. 13-16, 1973.

Prepared for Department of the Navy

Office of Naval Research

Arlington, Virginia 22217

-

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

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

RELIABILITY, LIFE PREDICTION AND PROOF TESTING OF CERAMICS

By

S. M. Wiederhorn

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

Abstract

A critical review is presented of the use of proof testing as a

design method for assuring the reliability of structural components.

The advantage of proof testing over the statistical approach used for

design lies in the insensitivity of the proof testing method to the

detailed history of handling or processing of structural components.

Methods are presented for developing and using proof test diagrams to

assure component lifetime after proof testing. Procedures of proof

testing and precautions that must be follov/ed during proof testing are

discussed. Provided these precautions are followed, proof testing offers a general method for assuring the reliability of structural

/ components under stress.

/

1

1. Introduction

Because of its value as a method of assuring

structural reliability, proof testing is often used in the design of

structural ceramics. Proof testing is especially valuable in applications in which ceramics are subjected to tensile stresses. In proof testing, ceramic components are siibjected to stresses that are greater

than those expected in service in order to break the v/eak components and

thus to truncate the low end of the strength distribution. In this

manner, weak components are eliminated before they can be placed in

service. Proof testing has been applied to windows for spacecraft

and experimental aircraft, ^ to electrical porcelain insulators that

4 are expected to support tensile loads for long periods of time, and to

5 ceramic pressure vessels. Proof testing also finds application m more

complex situations in which complete engineering assemblies are tested.

In this way, proof testing is used to assure the performance of aircraft

engines and may find similar application in gas turbines for energy

generation.

Despite its great value for assuring reliability, proof testing has

had no firm theoretical basis until very recently. Using trial-anderror estimates, proof test loads for ceramic materials were generally

set at two to six times the expected load. Whereas such estimates have

usually resulted in reliable structural ceramics, a more systematic method of

1
selecting a proof test load offers the advantage of either lighter

structural components or fewer rejects due to breakage during the

proof test. A more systematic method of selecting a proof test load

also eliminates the possibility of selecting a test load too small to

assure structural reliability. We gain these advantages because a

more exact value of the proof test load can be selected for a given

application. - - •

Recognizing the need for greater insight into proof testing,

6-9 Wiederhorn, Evans and Fuller provided a mathematical foundation for

the selection of the proof test load and for the establishment of proof

test conditions. Their analysis is based on the fact that failure of

ceramic materials occurs mainly from the growth of preexisting cracks.

By characterizing crack growth, and coupling crack growth parameters

with proof testing, they have demonstrated how to construct design

diagrams that relate the expected failure time to the maximum design

stress. Their analysis provides a rational guide for the selection of

both proof test load and test conditions. Since these are the keys to

a successful proof test, the analysis by Wiederhorn, Evans and Fuller

is the basis of a general method that can be incorporated into the

design of structural ceramics.

This paper presents a review of the mathematical foundation for

proof testing. For clarity of presentation, most of the important

equations are derived from first principles. Examples are given of how

2
these equations may be used to assure structural reliability. The

proof testing technique is compared with the technique of structural

design that is based on Weibull statistics, and the superiority of the

proof testing technique is demonstrated.

In developing the mathematical foundation of proof testing, an

estimate is first presented of the time to failure due to crack growth.

The time to failure is shown to depend critically on the size of the

largest flaw contained within the structural component. By using a

proof test method, or a statistical analysis of strength to estimate

the critical flaw size, design diagrams are obtained that relate the

failure time to the service stress. The value of proof testing ceramic

components prior to use is discussed and precautions are presented for

conducting proof tests on brittle materials. It is hoped that this

paper will provide the background needed to develop better design

techniques for structural ceramic materials.

2. FAILURE TIME UNDER STRESS

The time required for the failure of a ceramic component under tensile

stress can be calculated by using fracture mechanics concepts. For most

materials, the stress intensity factor, , at a flaw is related to the

applied stress, a , and flaw size, a, by the following equation:

a

K = a Y/i (1) I a

3
Y in the above equation is a geometric constant that depends on both the

shape and location of the flaw, and the type of stress applied to the material.

For surface flaws (cracks) which are usually the most critical flaws in ceramic

materials, Y~-/t\. For a constant applied stress the derivative of both

sides of Eq. (1) with respect to time yields the following equation,

where v is the crack velocity.

dK /dt = {o^Y^/2K^)^ (2) T a -L

By separating the variables of Eq. (2) and rearranging the equation,

the following expression is obtained for the total time-to-failure under

constant load, where K is the critical stress intensity factor and . IC Ii

is the initial stress intensity factor at the most serious flaw in the

component

.

K

IC

t = (2/a^Y^) r (K /v)dK^ (3) a I II

K . Ii

Evaluation of Eq. (3) requires information on the critical stress

intensity factor, the initial stress intensity factor at the most serious

flaw, and the functional relationship between the stress intensity factor

and the crack velocity. The critical stress intensity factor is obtained

by fracture mechanics techniques. Fracture mechanics specimens are

broken in an inert environment and K^^ is calculated from the dimensions

of the specimen, the crack length, and the failure load. A critical review

4
of techniques commonly used to determine K^^ on ceramic components may

be found in Ref. 11.

The relationship between the stress intensity factor and the crack

velocity in a given environment can also be found by fracture mechanics

techniques. Usually crack velocity measurements are made on specimens

similar in shape and size to those used for critical stress intensity

factor measurements. Again, the stress intensity factor is calculated

from the crack length, load and specimen dimensions. Although fracture

mechanics techniques are commonly used to collect crack growth data,

these data can also be obtained by measuring the strength as a function

of loading rate. A critical review of these methods has been presented

in Ref. 12. For the purposes of the present paper, it should be

recognized that both the critical stress intensity factor and the relationship between the crack velocity and the applied stress intensity factor

are easy to measure.

For many ceramic materials, the crack velocity is a power function

of the stress intensity factor, v = Ak!^, in which case Eq. (3) is integrated to give (to a good approximation) the following equation for the

time to failure:

t = 2K^T^/Aa^Y^(n-2) (4) Il a

However, other analytical expressions relating to v may also be used,

or if no simple expression is available, Eq. (3) can be integrated

niimerically

.

5
Because of its dependence on flaw size, K^_^ in Eqs. (3) and (4) is not

easily measured on real components. Flaws in most ceramic materials are less

than 1 mm in size and their dimensions are not easily measured by nondestructive

means. Therefore, indirect methods must be used to estimate the size of

flaws in ceramic materials.

There are two indirect methods for determining the initial size of the most

serious flaw in ceramic materials. One method uses statistics to characterize

the strength of the ceramic component as a function of the cumulative

probability for failure. The flaw size is then estimated on a

statistical basis by siibstituting the results of the statistical analysis

into Eq. (1) . The second method employs a proof test to determine the

maximum size flaw in the component at the time of the test. Once an

estimate of is available (by either method) the initial stress

intensity factor is deteinnined from the equation: K^_.= CT^Y/aT- The

expected service life of the component can then be calculated by

substituting K^^ into Eq. (3) or (4)

.

3. STATISTICAL APPROACH

The statistical approach usually used to describe the strength of

materials was originally developed by Weibull,"^^ This technique uses a

form of extreme value statistics that is particularly sensitive to the

* By contrast, flaws in metals are larger and can be measured by

a variety of nondestructive techniques: X-ray radiography, ultrasonics,

dye penetrants, etc.

6
low strength ©nd ©f tht itrsngth diitributien. Weibull's rtlationship

* bttwitn th© cumulativ© failure probability, P, and th© str©ngth 0^^ is

given by the following equation:

P = 1 - exp -[(Cfj^ - ^e^/^o^"

where o^, a^, and m are empirical constants. is the strength measured under

the condition that no subcritical crack growth occurs prior to failure, and should

be determined in an inert environment using rapid loading rates. By arranging the

strength measurements in order of increasing strength, P may be determined

from the equation: P = j/(j + i) , where j is the position of the jth strength

measurement in the ordered set of strengths and J is the total number

of measurements, a^, the lowest possible strength, is usually set equal

to zero, is a scaling parameter and m is a shape parameter, m and

are determined by fitting a straight line to the strength data which

have been plotted as log log (1/1 - P) versus log a^^ (Fig. 1) ; the

slope of the line is m whereas the intercept is m log a^.

In practice the Weibull theory is easy to apply because once m and

have been evaluated the strength is uniquely characterized on a

statistical basis. However, some precautions must be exercised when the

Weibull theory is used for strength analysis, m and may vary from

one component to the next due to the susceptibility of strength to

variations in the manufacturing process. Because of these variations,

m and determined on laboratory specimens may not be correct for

* The cumulative failure probability gives the fraction of specimens that will

break at a given stress level.
structural components, m and 0^ are also subject to inaccuracies which

result from the statistical uncertainty encountered whenever a straight

line is fitted to a set of experimental data. If only a small number

of strength measurements are used to determine m and O , the statistical

o uncertainty of these parameters can cause substantial errors in estimates

of the strength at low failure probability. As shown in Fig. 1 this

uncertainty in strength increases as the failure probability decreases.

Finally, even if O and m are accurately determined, errors in estimating

o the strength may occur because strength depends on component size. Because of the greater probability of larger flaws, large components

are usually weaker- than small ones. Although scaling equations are

used to account for size effects, these equations are sometimes difficult

to obtain and may be complex, especially for complex stress distributions.

Furthermore the location of the flaw that causes failure must be identified,

because scaling equations differ depending on whether the crucial flaws

lie at the surface or in the volume of the component. Because of uncertain

information regarding the flaw location, scaling equations must be tested fo

accuracy by using laboratory test specimens of varying size and shape.

All of these considerations decrease the value of Weibull statistics

as a method of determining strength. Consequently, design values of

strength must be more conservative than those given by equation 5.

Although the method used by Weibull did not consider strength as a

time dependent variable, a time dependence due to crack growth can be

8
easily introduced into the Weibull method by expressing K^^ in terms of

7 14

the failure probability. ' At failure, the critical stress intensity factor,

K^^, the fracture strength, CT-j-^* ^'^^ the crack length, a^, are related

by: K = O y/aT. At the service load the initial stress intensity

IC IC 1

^

factor, K . , is related to the service stress, Q , and the crack length

Ii a by: K . = a y/aT- From these two equations, an equation for K, . is obIi a 1 Ii

tained

:

By siibstituting Eq. (5) (with 0^ = o) into this equation, an equation

may be obtained relating K^^ to the cumulative failure probability:

K = K (a /a^) [log (1/1 - P)] ^/"^ (7)

This equation may now be substituted into Eq. (3) or (4) to obtain an

equation that relates the failure time to the cumulative probability for

failure. The functional form of this equation is:

t = a""«f (P) (8) a

For each value of P the failure time is a power function of the

applied stress. Therefore a series of straight lines of slope -n can be

9
obtained from a logarithmic plot of t versus a (Fig. 2) , each line

representing a different value of P. This type of diagram completely

describes the failure characteristics of a material on a statistical

basis. Similar diagrams are often used to represent static fatigue

data. However their value is more limited for design purposes because only

the average or median strength is used as a plotting parameter. By

contrast, diagrams such as that given in Fig. 2 represent the general

fatigue behavior of a material and are more useful for design purposes.

To illustrate how the diagram can be used for design purposes, refer

to the data of Fig. 2 which was obtained on a glass that is being

3 considered for the space shuttle. During its lifetime the shuttle

will undergo many stress cycles. The windows must not fracture during

the critical period of each cycle when the spacecraft is in orbit. To

assure reliable performance, the windows should be able to survive for

at least one year under load while subjected to the most severe environmental conditions expected in each cycle. Since the spacecraft will be

periodically exposed to moist air, and since water causes stress corrosion

in glass, the most adverse environment for the proposed space shuttle

glass would be a moist environment. For this reason, the diagram shown

in Fig. 2 has been based on crack propagation data collected on glass

specimens that were immersed in water. From Fig. 2 we see that for a -5 failure probability of 10 the windows can be subjected to a stress of

no more than 2,000 psi if one year's survival is desired. This stress

value is approximately one-half the present design load, 4,000 psi. If

10
a design load of 4,000 psi is used the probability of failure for these

-3

windows increases to approximately 2 x 10 , a probability much too high

for safety. Hence, we see that the statistical approach leads to ley design

values of the load if low failure probabilities are required.

The design load suggested from Fig. 2 has not been scaled for the

fact that laboratory specimens were much smaller than the spacecraft

windows. The laboratory specimens used to evaluate m and were disks

approximately one inch in diameter, while spacecraft windows will be

2 trapezoids approximately 1000 in . If the high stress regions of these

two sets of specimens are assumed to be proportional to their areas,

then to a first order approximation the Weibull approach predicts a reduction of strength proportional to the ratio of the two areas raised to the

1/m power, which reduces the strength of glass by a factor of 3 for all

probability levels. This reduction limits the use at the 10

^

probability level to a strength of approximately 700 psi instead of the

2000 psi shown in Fig. 2. A further reduction in stress may be required

because of uncertainties in the assumption of equal distribution of flaws

in large and small components and because of the very simplified treatment used to get the correction ratio.

4. PROOF TESTING APPROACH

Proof testing removes many of the difficulties present in the

statistical approach to failure. Proof testing truncates the strength

distribution, eliminating those specimens with the largest probability

for failure. As a result, components can be used at higher stress levels

with greater assurance of reliability than can be obtained from the

statistical approach.

* For effective proof testing, all components intended for service must be

tested.

11
" To develop the mathematical basis for proof testing, one must

recognize that a proof test limits the maximum flaw size that can be

*15 . . present in the specimen after the proof test. All specimens containing

flaws larger than this critical value will fail during the proof test,

because the stress intensity factor at the tips of the flaws will exceed

the critical stress intensity factor, K^^. For flaws that are smaller

than the critical size, the stress intensity factor will be less than

the critical stress intensity factor and the components will survive

the proof test. Therefore, if failure does not occur during a proof

test, K > K = a Yv/a7, where K is the stress intensity factor at the

IC P P i P

^

largest flaw during the proof test, and 0^ is the stress applied during

the proof test. When the component is first used in service, the relationship between the initial stress intensity factor, ^jj^' the applied stress,

a , and the flaw size is given by . = a Y/a7. From these two equations,

a

'^ Ii a 1

the following equation is obtained:

This estimate for the initial stress intensity factor can be substituted

into Eq. (3) or (4) to obtain an estimate for the minimum time to failure.

Because the limits of integration of Eq. (3) depend only on the ratio of

the proof test load to the applied load, the minimum time to failure,

^min' """^ given by the following functional relationship:

In this section it is assumed that crack growth does not occur during the

proof test. Crack growth during the proof test is discussed in section 5.4.

12
mm a Pa (10)

Thus, for any given proof test ratio, 0/0 , the minimum time to failure

is inversely proportional to the square of the applied stress. A logarithmic plot of the minimum time to failure versus the applied stress

gives a straight line with a slope of -2, the position of the straight

line depending on the proof test ratio.

A design diagram incorporating these ideas is presented in Fig. 3,

which is a logarithmic plot of the time to failure versus the applied

stress. Lines derived from the statistical approach to failure are

also included in this diagram for comparison. For a survival time of

one year at a stress of 4000 psi, a proof test ratio of 2.6 will be necessary for this glass. Specimens passing the proof test will survive the

minimum time to failure with zero probability of earlier failure. By

comparison, the failure probability is 2 x 10 at the same stress levels

in the absence of a proof test. To pass the proof test, the glass

windows will have to sustain a load of approximately 11,000 psi.

Referring now to the Weibull diagram for these glasses

(Fig. 1) , it is observed that at a stress level of 11,000 psi only one

specimen in a thousand v/ill be broken during the proof test. If the

reduction in strength is taken into account because the glass windows

are much larger than the disks tested in this experiment, a failure

rate of one in ten is reached. Thus, by taking the chance that one

window out of ten will break during the proof test, adequate performance

of the remaining windows is assured.

13
Ag ghewn in an inalygii fey ivani and fuller > pm^S t@§tinf @in alg©

3

be applied to cyclic loading. If failurt involved subcritieal crack

growth, the failure time due to cyclic loading, t^, is proportional to

the failure time, t . , due to static loading: t = g "'"t . , where g min c min

is a proportionality constant that can be evaluated by a numerical

integration for any periodic load cycle. For square wave, sinusoidal or

saw-tooth type of loading, values of g have been evaluated analytically

and are available in diagrams that express g

''' as a function of the

exponent n and the ratio O /O , where a. is the stress amplitude and Q

1 a 1 a

is the average applied stress. (Fig. 4) . ''^^^j^ determined from a design

diagram (Fig. 3) by setting the service stress equal to the average

applied stress for the cycle. The analysis by Evans and Fuller permits

a direct comparison of experimentally determined crack growth data and

failure times for cyclic and static loads. Agreement between these

two types of data indicates that failure is most likely due to subcritical crack growth. However if these two types of data do not agree,

failure is most likely the result of some mechanism besides subcritical crack

growth due to static loading. Crack propagation and strength results on

Q porcelain, alumina and glass at room temperature and on silicon nitride

and alumina at high temperature,"*"^ give no indication of an enhanced effect

of cyclic loading on slow crack growth. Consequently, by using g t design

diagrams developed for failure predictions for static loading are also

applicable to cyclic loading of these materials.

14
The main advantage of the proof test procedure is its relative

insensitivity to the particular flaw distribution associated with a given

component. Since the flaw size distribution is determined by the component'

history, it follows that the proof test method of assuring reliability is

independent of the component's history prior to the time of the test.

By contrast, diagrams obtained from statistical considerations are

extremely sensitive to the flaw size distribution and therefore to the

component's history. A further advantage of the proof test method is its

insensitivity to the particular arrangement of flaws in a component. Thus

even though the analysis of failure assumes that the most serious flaw

lies perpendicular to the maximum stress, a deviation from this condition leads to a more conservative estimate of failure time.^

5. PRECAUTIONS IN PROOF TESTING

In this section a discussion is presented of some of the precautions

to be exercised when proof testing is used for design. These precautions

are concerned with the service stresses, the accuracy and applicability

of the crack growth data, the design of proper proof test procedures and

the possibility of strength degradation as a consequence of the

proof test. This discussion is necessary since an incorrect assessment

of any of these precautions could invalidate the proof test and result in

premature component failure.

5. 1 Stress Equivalence During Proof Testing

Proof testing is based on the assumption that all parts of a

structural component are subjected to the required proof stresses during th

proof test. At any point in the component these stresses must exceed the

15
service stresses by a specified amount, otherwise portions of the component

will remain untested and predictions of component lifetime will not be

meaningful. Therefore proof tests must duplicate the expected service

stress distribution as closely as possible. For simple components such

as windows for pressure vessels or electrical porcelain suspension

insulators, service stresses are easily duplicated by applying loads that

exceed the service load by the required amount. However, for other types

of components duplication of the service stress distribution may be

more difficult because of complicated component geometry or stress

distribution. In some cases a double proof test may be necessary to

test different types of loading expected in service. For example, proof

tests for turbine rotors must duplicate the stresses resulting from both

the centrifugal force and the gas pressure on the vanes of the rotor.

In general, each structural component must be considered individually

to decide on the appropriate proof test.

5 . 2 Parasitic Stresses

The results of a proof test are sometimes invalidated by parasitic

stresses that cause strength deterioration or crack growth over and above

that predicted by the proof test. Parasitic stresses arise from unintended

features of the proof test, for example, if the proof test apparatus inadvertently applies loads that differ from the service loads. An example of

component deterioration due to parasitic stresses was recently observed in

4 large porcelain insulators meant to support radio towers. The proof test

16
consisted of slowly increasing the load on the insulators to 2.5 x 10

pounds, and then slowly decreasing it to zero load. During the proof

test , acoustic emission transducers that were attached to the insulator

indicated high rates of emission at approximately two million pounds as

the load on the insulator was decreasing . The emission was due

to the growth of a large internal crack during the unloading

* procedure. The presence of this crack was confirmed by an ultrasonic

examination of the insulator during the test. The stresses that caused

the crack to grow resulted from unexpected slippage of the porcelain in

** its metallic end caps. As a consequence of this observation, it may

be concluded that proof test procedures should be designed to avoid

parasitic stresses.

5. 3 Equivalence of Crack Propagation Data and Failure Mechanism

The proof test method of assuring reliability depends critically

on an accurate evaluation of the crack propagation parameters that control

fracture. If proof test methods are to be of value, these parameters

must represent the failure mechanism. Therefore, information on the

factors that determine the crack propagation parameters is essential.

These factors include: the crack tip environment; the size of the strength

limiting flaw relative to the micros tructure of the component; and the

functional dependence of crack velocity on stress intensity factor.

* The crack growth detected by acoustic emission transducers does not

necessarily result in mechanical failure for compressive loading. These

porcelain insulators are not acceptable for service if they contain large

cracks because the cracks cause the insulators to become electrically unstable.

** We suspect that this type of failure may be characteristic of some kinds

of compressive loading.
For many materials (ceramics, metals and polymers) enviromaent

is the dominant factor that controls crack growth. Many structural

ceramics are sensitive to water in the environment, which causes stress

12 corrosion cracking. As water penetrates to the crack tip, crack motion

results from a stress enhanced chemical reaction between the water and the

material at the crack tip. This reaction can occur even in relatively

17 18 dry gases or in 'organic liquids . In aqueous solutions the reaction

19

IS sensitive to the pH of the solution. In glass for example crack

propagation curves are steeper in high pH solutions than in low pH

3 19 solutions ' (Fig. 5) . Finally, the crack tip environment may differ

from the bulk environment as a result of the chemical reaction at the

19 crack tip. Therefore, to use crack propagation data for failure

prediction purposes, one has to be certain that the crack tip environment is the same for the crack propagation measurements as for the component in service. -

;

Another factor that influences the crack growth is the size of the

strength limiting flaws relative to features of the microstructure such

as grain size, pore size, or pore spacing. The relative flaw size is

important because of a possible dependence of crack propagation rate on

microstructure. In crack growth studies, the flaw is artificially produced and is always larger than characteristic dimensions of the microstructure. However in some materials, the flaws that limit strength

are comparable in size to the microstructure. Therefore, crack growth

behavior of these flaws will not necessarily be described by crack

growth data obtained on fracture mechanics specimens. As a consequence.

18
failure predictions may not be valid if they Axe based only on design diagrams

derived from this growth data. Fortunately, for many materials the flaw size

large relative to the microstructure and the assumptions used to construct

proof test diagrams are valid. In hot-pressed silicon nitride for example the

20

flaw size, lOOym, is approximately 100 times the grain size.

Similarly, cracks in glass are usually much larger than the inhomogeneities

that may be present in this material. This assiimption of large crack

size is not valid for other materials. For example, the crack size in

21

aliiminum oxide is approximately equal to the grain size. For these

materials, proof test diagrams should be used only if it can be shown on

a laboratory scale that material strength can be predicted from these

diagrams. Agreement between strength measurements and crack propagation

data in laboratory studies would permit these diagrams to be used for

design purposes.

The functional dependence of crack velocity on stress intensity

factor is another variable that must be considered in the design of a

proof test. Knowledge of this functional dependence is important when

large values of the proof test ratio are used for design. In this case

the crack propagation data have to be extrapolated to low values of the

velocity, and the accuracy of the extrapolation determines the validity

of the time- to- failure prediction. This point is illustrated in Fig. 6

which gives a logarithmic plot of f (a Pa/a ) from Eq. (10) versus a /O ~ Pa

for two representations of the crack velocity data, (a logarithmic and a

power function representation). A^f (0^0 ) and a /a exceed the limits

19
of the experimental data, given by the cross in the figure, the two

curves diverge, resulting in some uncertainty in the selection of

an appropriate proof test ratio. For the proposed space shuttle

glass this uncertainty was not large since the power function representation of Fig. 6 suggested a proof test ratio of 2.6 to 1 for a year's

lifetime at 4,000 psi, while the exponential representation suggested

a proof test ratio of just over 2.8 to 1 for the same stress. However,

for other materials this uncertainty might be crucial. To avoid this

source of error, crack velocity data should be collected at as low a

* velocity as is feasible.

One way to assure the applicability of crack growth data to failure

predictions is to demonstrate an agreement between the strength and

crack growth data. This agreement has been demonstrated for the glass

3 proposed for the space shuttle (Fig. 7) and for hot pressed silicon

22 nitride (Fig. 8) . The points in Fig. 7 represent the mean value

of the strength of the space shuttle glass obtained for each value of the

loading rate; the brackets give 95 percent confidence limits for the

mean values. The solid straight line is a least squares fit of all of

the strength data, while the dashed line is the predicted strength

from crack propagation data. Since the dashed line falls within the 95

percent confidence limits of the strength data, one may conclude that

for this glass the crack propagation data and the proof test diagram

Since approximately one month is required to measure a crack velocity of 10

meters per second, this velocity appears to represent a lower practical limit

for crack propagation data.

20
derived from it can be used effectively for design purposes. Similar

agreement is found for the silicon nitride data given in Fig. 8 where

23

a comparison is presented between strength data obtained at 1400° C

and crack propagation data (the solid lines) obtained at the same temper22

ature. The lines fall close to the measured strength values suggesting

that crack propagation data on hot pressed silicon nitride gives adequate

predictions of strength, and that proof test diagrams can be used for

design purposes for this material.

For general application of the proof test method, additional

comparisons between crack propagation data and strength data will be

necessary. An equivalence between crack propagation data and strength

data should be demonstrated on a laboratory scale for each material

so that proof test diagrams can be used confidently for design purposes.

5. 4 Loading Procedure During Proof Test

In section 4 the minimum time to failure, t . , was estimated on mm

the assuraption that crack growth did not occur during the proof test. In

addition, failure was assumed to occur instantaneously if the critical

stress intensity factor was exceeded at some flaw in the component. As

a consequence of these assumptions, the failure probability was found to

be zero for all periods of time that were less than the minimum failure

time, t . . While these assumptions are valid for some commercially mm

important materials (silicon nitride and silicon carbide tested in air at

room temperature;^'^ silica, chemical Pyrex and the space shuttle glass tested

21
in vacuum ) , they are not valid for others. Soda lime silicate type

25

glasses, for example, exhibit subcritical crack growth even in a vacuum.

If crack growth occurs during a proof test, then the failure probability

is no longer zero, but has a finite value that is determined by the proof

test conditions and the test environment. Fortunately, the failure

probability can be reduced to a vanishingly small quantity by judicious

selection of the proof test conditions.

A theoretical basis for selection of the proof test conditions

9

has been developed recently by Evans and Fuller who have examined the

* effect of crack growth on the failure probability after proof testing.

Evans and Fuller assumedthat constant rates of loading, O^, and unloading,

•a^, were used in the proof test, and that the proof test load, o^, was

held on the component for a period of time, t^. They also assumed that

crack growth was controlled by a single fracture mechanism so that the

crack growth rate could be expressed as a power function of the stress

intensity factor, v = AK^. Using these assumptions, they found that the

crack length after a proof test could be described by the following

equation:

* —n-2

^

n-2

a = a [1 - (a /a ) ^

] (11) o o o

* Their theory is fully developed in the appendix for the interested reader,

22
where a is the crack length after completion of the proof test and a^ is

* the initial crack length, a , the critical crack length for failure

o during the proof test, is the initial length of a crack that will just

grow to failure during the proof test. As shown by the following equation

* a^ is completely determined by the proof test conditions

a ={[{AOY (n-2)/l][(t + (t„ + t )/(n+l)]}

O P p 36 U

^2

n-2 (12)

where t^ = ^ is the time required to load the component to the proof

test load, CJ , and t =0/0 is the time required to unload the component.

P u P u The dependence of crack length a on a^ is depicted in Fig. 9 for

n = 20 and n = 100. As indicated by the abrupt increase in a/a^

as a^ approaches a^, crack growth is significant only when a^ is close in

value to a^. Because of this behavior, strength degradation is serious

* only for components that contain cracks for which a - a . o o The failure probability after a proof test is determined by the

range of initial crack lengths that result in significant crack growth (R

of Fig. 9) : The narrower the range of the crack lengths, the lower the

failure probability. Since this range is small for large values of n,

test environments should be selected so that n is large. Thus, proof

tests on soda lime silicate glass should be conducted in dry nitrogen

for which n - 100 rather than in air for which n - 20. This conclusion

is supported by a detailed calculation of the failure probability after a

proof test.

23
An estimate of the probability for failure after a proof test

can be obtained by using the crack length, a, of equation n to evaluate K^^

of Eq. (4) . After rearranging the eauation and using a Weibull analysis

to describe the initial crack size distribution (see appendix) , the following equation is obtained for the failure probability, P^f after a proof

test:

" n-2

\ '—

P = SL- ,e /a (VK,^)"-"-'(t/t (13) a n-2 P o P IC mm

* where : m and O^ are Weibull parameters ; n and n are crack propagation

* equation exponents, n referring to the test environment and n referring

to the service environment; and t . is the minimum failure time calculated

mm

from Fig. 3 which is based on the assumption that crack growth does not

occur during the proof test. t is the time in service after the proof

* test. Kp is the stress intensity factor calculated for a crack of length

* * /*" * a ; K = a Y/a - K would be the stress intensity factor for this crack

o P P o P

^

if crack growth did not occur during the proof test.

P is a useful engineering quantity because it gives the fraction

a of components that break in service in a period of time t. P is a function

a of the time at load, increasing with service time, t. By a proper selection of the

proof test procedure, P , can in principle be made arbitrarily small. Since n is ! a * * usually much larger than n , the failure probability is very sensitive to

the ratio t/t . . Therefore by selecting the service time to be less than mm

t . , the value of P can be reduced without limit. In practice this mm a

24
requires the proof test load to be increased over the value estimated if

crack growth did not occur during the proof test. This increase of proof

test load is usually not excessive for t/t . - 0.1.

^ mm

* Kp is completely determined by the proof test conditions and can

be evaluated from the following equation which relates the crack propagation data to the proof test conditions

(K*)Vvp = [apY^(n-2)/2] [t^ + ( t^+t^) / (n+1) ] (14)

The right hand side of this equation (14) is easily evaluated from the

proof test load, a^, the crack propagation parameter, n, the loading and

unloading times, t- and t , and the time, t , that the component is exposed

36 u. P

* to the proof test load. To evaluate from the above equation, a plot of

2 * K^/v versus is useful, since can be easily evaluated from such a

* 2

* diagram once (K^) /v^ has been determined from Eq. 14.

The use of Eq. (13) will now be demonstrated for abraded soda

lime silicate glass, for which a Weibull plot is given in Fig. 10 and the

2 crack propagation data is given in Fig. 11. A plot of K^/v versus

calculated from the crack propagation data is given in Fig. 12. For

illustrative purposes suppose that the glass is required to support a

2 5 load of 4 MN/m (~570 psi) for a period of 10 seconds in a wet environment. By selecting t . to be 10^ seconds a proof test ratio of 3 is mm

* * n

v^ E A(K ) .

25
2 estimated from Fig, 13; thus Op = 12 MN/m . This proof test ratio is

larger than the ratio of 2.6 that would be required if crack growth did

not occur during the proof test. If the proof test is conducted in a

moisture-free environment, n - 100 while n - 20. K for soda lime

26 3/2 silicate glass is 0.75 MN/m and from Fig. 10 m = 2.63 and = 45

2

MN/m . If the glass components are loaded and unloaded in a relatively

short time, ~10 sec, and if the proof load is applied for ~1 sec, then

* 2 16

from Eq. (14) (K^) /v^ is calculated to be 2.66 x 10 . From the vacuum

* 3/2 curve in Fig. 12, is estimated to be 0.71 MN/M . By substituting

these values into Eq. 13 a value of 1.6 x 10 is obtained for P , which

a for practical purposes is equal to zero. The fraction of specimens that

* m

would have been broken in the proof test, given by =

^'^p^jQ^^o^P^

(see appendix) , is 0.03. If the glass had been used without prior proof

testing the failure probability from Fig. 13 would have been approximately

-3 2 5

5 X 10 at the service conditions (O =4 MN/m ; t = 10 sec) . Thus by

breaking only 3 percent of the components during the proof test, a much

greater assurance of reliability is obtained.

The assurance of reliability is not nearly as great if the proof

* test is conducted in a wet environment, for which n = n =20. For the

* 2 15 , same loading conditions and proof load, (K^) /v^ is 7.95 x 10 , and

* 2 K calculated from Fig. 11 (the curve for water) is 0.58 MN/m . P is now

-5

8.7 X 10 and the fraction breaking during the proof test is approximately

0.061. Although the failure probability in a wet environment can be improved

26
by increasing K^/v^ (by increasing for example) the values are never

as small as t±iose obtained in a dry environment using the same test

conditions. Thus, whenever possible, inert environments should be used

for proof tests.

From the above results one may conclude that proof testing is

a valuable method of assuring the reliability of stmactural ceramics even

if crack growth occurs during the proof test. To apply the method, a

statistical description of the component strength and crack propagation

data in both an active and inert environment are needed. However, these

types of data are easy to obtain, and once these data have been obtained,

they may be used to establish proof test conditions that assure a low

failure probability after proof testing.

5. 5 Protection of Components after Proof Testing

Although proof testing assures a minimum lifetime at the time

of rhe test, it loses value if subsequent damage occurs. This can be a

serious problem for materials which fail from surface flaws, since these

flaws are easily introduced into the material by rough handling. This

problem can also arise in materials that fail primarily from internal

flaws if the structure changes in service. Therefore some considerations

have to be given to protection of components after the proof test and

to periodic retesting of components during use.

For components that fail from surface flaws, caution in handling

or the adoption of. some protective measure after proof testing is necessary

to assure reliability during service. For glass, which is particularly

27
susceptible to surface damage, this means that the surfaces have to be

protected, perhaps by plastic coatings which would prevent abrasive damage

of the surface after proof testing. In the space shuttle windows, damage

after proof testing is avoided by the design of the windows. These are

triple paned, the tensile stress surfaces being on the interior of the

panes. Thus further surface damage by handling cannot occur.

The possibility of strength degradation in service due to the

initiation of new flaws must also be considered. This degradation could

occur by chemical reactions within the material, by phase transformations,

or by pore growth at high temperatures. Strength degradation of this kind

is not considered in the equations used to predict the time to failure.

Perhaps the best way to eliminate such degradation is by designing the

material so that new flaws will not generate during use. However, if this

is not possible, periodic proof testing may be needed to assure minimiom

periods of service life.

6. SUMMARY AND GENEiE^L CONCLUSIONS

In this paper, proof testing is considered as a general method of

assuring reliability of ceramic components for structural uses. A review

of current work in this field is presented. The proof test technique is

compared with the older statistical approach for design, and it is

demonstrated that many of the uncertainties inherent in the statistical

approach are eliminated by the use of proof testing. It is shown for

example that lifetime predictions after proof testing are not affected by the

particular flaw size distribution contained within the specimen, whereas

28
predictions obtained by the statistical approach are affected by the

flaw distribution. In proof testing, prtdictions of lifetim© diptnd more

on material properties, such as the value of the critical stress intensity

factor and the relationship between the stress intensity factor and the

rate of crack growth in a given environment. Since these both can be

determined to high precision, proof test diagrams derived from these

parameters should be reliable for the prediction of design lifetimes.

However, certain precautions must be exercised in the application of proof

testing. The proof test must duplicate the actual stresses expected in

the component. Otherwise, the most serious flaw in the specimen might

not be subjected to a proof test load. Crack propagation data used to

develop design diagrams for proof testing should represent the failure

mechanisms that occur in service. To ensure an equivalence between the

proof test diagram and the actual failure mechanism, crack propagation

data should be compared with data obtained from strength measurements.

An agreement between these two types of measurements will lend assurance

that the proof test diagram does in fact describe the failure mechanism.

The loading procedure during proof test is also important since crack

growth during the proof test may result in weak components. To eliminate

this problem, the proof test should be conducted in an environment that

is relatively inert. An analysis of the loading procedure demonstrates

that low probabilities of failure after proof testing can be achieved even

if siabcritical crack growth does occur during the proof test. The importance

of parasitic stresses as a limitation to the proof test method is also

discussed. Here it is noted that unexpected stresses may arise during the

29
proof test. These stresses may in fact damage the specimen to an extent

not predicted by the proof test technique. In some cases, the damage

may be unavoidable and proof testing may not be a viable procedure.

Finally, the necessity of protecting the specimen after proof testing is

emphasized since subsequent structural damage may degrade the specimen to

a greater extent than is predicted by the proof test. Periodic proof

testing is recommended for materials that change structurally or develop

damage in service. Provided these precautions are followed, proof testing

can be a practical method for assuring the structural reliability of

ceramic materials.

Acknowledgment

Portions of this work were supported by the Office of Naval Research,

Contract NAonr 5-73, NR032-535. .

30
APPENDIX

PROBABILITY OF FRACTURE AFTER PROOF TESTING

The failure probability after a proof test was calculated by Evans

9 and Fuller by considering the growth of cracks during the proof test.

The length to which a crack grows during a proof test can be calculated

from the definition of crack velocity, v=da/dt, and the functional

dependence of crack velocity on stress intensity factor, v = AK^.

Combining these equations and using the expression = a(t)Y/a, the

following differential equation is obtained:

da/dt = Aa'^(t)Y'^a"^^ (lA)

where the stress, Q(t), is a function of time. If the stress, o it) , is

known, this equation can be integrated after rearranging the variables.

Evans and Fuller assumed that constant stress rates were used to load and

unload the component and that the proof stress was held for a time, t^.

The total time to load the components to the proof test load, O^, is given

by t^ = C?^/a^ where is the stress rate. Similarly the total tirrie to

unload the component is given by t^ = CTp/O^. By using these values of time

as integration limits, equation (lA) can be integrated to give the following

equation for the crack length, a, after the proof test:

* ~2~ ""^

• a = a [1 - (a /a ) ] (2A) o o o

31
where is the crack length before the proof test and a^ is the critical

* crack length that will juit result in failure during the proof test, a^ is

given by the proof test conditions:

a* ={[(AapY"(n-2)/2) j [tp + (t^ + t^)/(n+ 1)]}"'^ (3A)

Since a is the crack length after the proof test, it is also the initial

crack length when the component is placed in service. Therefore, when the

component is placed in service, the initial stress intensity factor K^^ is

given by: . = a Y/a. The time to failure after the proof test is given

Ii a

* t = 2K.^~^ /A*a^Y^(n*-2) (4A) Ii a

* * Eq. (4A) is identical to Eq. (4) of the text. A and n represent the crack

propagation parameters for an active environment while A and n used in Eqs.

(2A) and (3 A) represent these parameters in the proof test enviro.nment. This

distinction is made because the proof test environment may differ from the

test environment. By substituting Eq. (2A) and the equation for K^_j^ into

Eq. (4A) , the following equation is obtained for the failure time:

*

* n-2 n -2

t = \2{a LaoY/a*)^"^ /A*0^ Y(n* - 2)1 [ (a /a*) ^ - ll (5A) a JLoo J

* * /T

By using a to define a proof test stress intensity factor: K =0 Y/a ; o P P o

* * and by using the identity: = ^j-q^\/'^jq^ ' the following equation is

obtained for the failure time:

32
* n;^ n -2

The first term in brackets of Eq. (6A) is the minimum time to failure,

t . , used to compute proof test diagrams. Therefore Eq. (6A) can be min

simplified:

* n-2 n -2 *

* n -2 , , , *. 2 , ,n-2 t/t . = (K /K ) ^ [(a /a ) " -1]" " (7A) mm IC P o o

* By expressing the crack lengths, a^ and a^, of Eq. (7A) in terms of

the fracture strength, ^j^r and by using the Weibull equation to relate the

strength to the failure probability, the failure time of Eq. (7A) can be

expressed in terms of the failure probability. The crack lengths and the

strengths are related by the following equation:

* 1/2

(a /a ) = cr /a^^ (8A) o o IC IC

since =

^-^q.^^^o

^"^^ "^IC ~

'^IC'^^''^^" ^'^^ ^ given strength distribution,

the cumulative probability for failure, P, is related to the strength, cr^^,

by the Weibull equation:

In £n (l/l-P) = m £n(a /a ) (9A)

IC o

where m and are empirically determined constants. For sufficiently

small values of P (P < 0.1): £n(l/l-P) = P. Therefore P = (a^„/a and

IC o

33
* m

p = (Cf /CT ) , from which the following equation is obtained:

whtra is ©qual to th@ fraction of specimens broken during the proof

test. Both P and Pp refer to the initial flaw distjribution, the distribution

before the proof test. Crack growth alters the initial flaw distribution

so that after the proof test the probabilities for failure, P^, is given

by the following equation:

P = (P - P )/(l - P )

If only a small fraction of the components tested are broken, P^ is a

small niimber and to a good approximation:

P = P - P^ (12A) a P

By substituting Eqs. (12A) , (lOA) , and (8A) into Eq. (7A) , an equation

relating the failure time, t, to the failure probability, P , is obtained;

a

* n-2 n -2 *

^/'rain

= 'hcK'" '"^d H- P^/P^, " -1, "'^ (13A)

n-2

For a proof test to be of any value P^ << P^, in which case (1 + P^/^p)

34
can be expanded by the binomial theorem to give: (1 + P /P ) Si ir

n-2

m

1 + (n-2) P /n»P . Sxibstituting this equation into Eq. (13A) we obtain

the following equation relating the failure probability P to the failure

a time, t:

n-2

By eliminating Pp, Eq. (14A) can be expressed in terms of the Weibull

parameters and crack propagation parameters. For small values of P^:

m * /IF

Pp = K ^ = OYVa ; ^^ic'^'^o^ ' fJ^oni the definitions of a^^ and Kp (K^^ = O^^

Kp = apY/a^) : '^jQ~ ^p^^ic'^^p^ * Substituting these two equations into

Eq. (14A) we obtain the relationship between the failure probability, P ,

and the Weibull and crack propagation parameters:

n-2

. ^ ia_/a (k!/k..)"-"'-2 (t/t . ) n-2

m and of this equation are determined by a Weibull fit of strength

data (measured in an inert environment) ; n and n are crack propagation

exponents measured respectively in an inert and active environment; and

t . is determined from a proof test diagram, mm ^ ^

* Kp is determined by the proof test conditions. By substituting Eq. (3A)

* * fir

into the definition of K (K = a Y/a ) we obtain the following equation

P P p o for K*: .

35
-1

K* = apY[(AapV(n - 2)/2) (tp + (t^ + tj/n + 1)] ^"^ (16A)

Eq. {16A) can be expressed in a simpler form by defining a crack

velocity v^: '^p - ^^p"' Physically, Vp is the velocity of a crack of

* length a^ if the proof test load is applied instantaneously so that crack

growth does not occur during the loading period. After rearranging Eq. (16A)

and using the definition of v^ the following equation is obtained:

(KJ^/^T, = [<^^V(n - 2)/2] [t^ + (tp + t )/(n + 1)] (17A)

P P P P ic u

The left-hand side of this equation is determined by the crack propagation

data, while the right-hand side is determined by the loading conditions.

* To determine K^, the right-hand side of Eq. (17A) is evaluated from the

* proof test conditions , and Kp is determined from an experimental plot of

2 K^/Vp versus K^. This procedure is illustrated for abraded soda lime

silicate glass in the main body of the text.

36
References

1. A. F. Shoemaker, Corning Glass Works, private communication.

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

Study of the Skylab Windows," in Fracture Mechanics of Ceramics, Vol. II,

pp. 829-841, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, eds.,

Plenum Press, New York (1974).

3. S. M. Wiederhorn, A. G. Evans, E. R. Fuller and H. Johnson, "Application of Fracture Mechanics to Space Shuttle Windows," to be published

in the J. Am. Ceram. Soc. # August 1974.

4. A. G. Evans, S. M. Wiederhorn, M. Linzer and E. R. Fuller, "Proof

Testing of Porcelain Insulators—And the Application of Acoustic

Emission," to be published in the J. Am. Ceram. Soc.

5. W. C. Clark, Jr. , Westinghouse Electric Corp., private communication.

6. S. M. Wiederhorn, "Prevention of Failure in Glass by Proof-Testing,"

J. Am. Ceram. Soc. 56, 227 (1973)

.

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

An Analytical Basis for Failure Prediction," NBS Report No. NBSIR 73-147,

to be piiblished in Intl. J. Fracture/ 1974.

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

Under Cyclic Loading Conditions," Met. Trans. 5^, 27 (1974).

9. A. G. Evans and E. R. Fuller, "Proof Testing—The Effects of Slow Crack

Growth,"' to be published in the J. Am. Ceram. Soc.

10. W. F. Brown and J. E. Srawley, ASTM Special Tech. Publ. No. 410 (1966)

.

11. A. G. Evans, "Fracture Mechanics Determinations," in Fracture Mechanics of

Ceramics, Vol. I, pp. 17--48, R. C. Bradt, D. P. H. Hasselman, and F. F. Lange,

eds.. Plenum Press, New York (1974).

12. S. M. Wiederhorn, "Subcritical Crack Growth in Ceramics," ibid.

Vol. II, pp. 613-646.

13. W. Weibull, "A Statistical Theory of the Strength of Materials, " IngeniBrs Vetenskaps Akademien, Handlingar No. 151 (1939)

.

14. R. w. Davidge, J. R. McLaren and G. Tappin, "Strength-probability-time

(SPT) Relationships in Ceramics," J. Mat. Sci. 8, 1699 (1973).

37
15. C. F. Tiffany and J. N. Masters, pp. 249-77 in Fracture Toughness

Testing and Its Applications, ASTM Spec. Tech. Publ. No. 381 (1965).

16. A. G. Evans, "Ceramics for High Performance Applications," this volume.

17. s. M. wiederhorn, "Influence of Water Vapor on Crack Propagation in

Soda-Lime Glass," J. Am. Ceram. Soc. 50, 407 (1967).

18. S. W. Freiman, "The Effect of Organic Alcohols on Crack Propagation

in Glass," to be published in the J. Am. Ceram. Soc, August 1974.

19. S. M. Wiederhorn and H. Johnson, "Effect of Electrolyte pH on Crack

Propagation in Glass," J. Am. Ceram. Soc. 56, 192 (1973).

20. R. Kossowsky, "The Microstructure of Hot-Pressed Silicon Nitride,"

J. Mat. Sci. 8, 1603-15 (1973).

21. E. M. Passmore, R. M. Spriggs, and T. Vasilos, "Strength-Grain Size

Porosity Relations in Alumina," J. Am. Ceram. Soc. 4£, 1-7 (1965).

22. A. G. Evans and S, M. Wiederhorn, "Crack Propagation and Failure

Prediction in Silicon Nitride at Elevated Temperatures," J. Mat. Sci.,

1974. -

.

23. F. Lange, to be published.

24. A. G. Evans, private communication.

25. S. M. Wiederhorn, H. Johnson, A. M. Diness and A. H. Heuer, "Fracture

of Glass in Vacuum," to be published, J. Am. Ceram. Soc. , July 1974.

26. S. M. Wiederhorn, "Fracture Surface Energy of Glass," J. Am. Ceram.

Soc. 52, 99-105 (1969).

38
Figure Captions

1. Weibull plot of the strength data for a low expansion glass being

considered for the Space Shuttle, The hyperbola gives 95% confidence

limits for the best straight-line fit of the data, m = 5.29; a = 250 MN/m^. After Wiederhorn, Evans, Fuller and Johnson, reference 3.

2. Statistically based design diagram. For a given value of the failure

probability, each line on this diagram relates the failure time to

the applied stress. The cross-hatched area gives 95 percent confidence

limits for the position of each line.

3. Design diagram based on proof testing. For a given value of the

proof test ratio, O-p/O^, each line on this diagram relates the

minimum failure time to the applied stress. The niombers over each

line give Op/a^. The probability lines from figure 2 are included

here for comparison.

4. Curves to estimate g"-^ for sinusoidal load. After Evans and Fuller,

reference 8.

5. The effect of pH on crack propagation in the low expansion glass being

considered for the Space Shuttle. After Wiederhorn, Evans, Fuller and

Johnson, reference 3.

6. Proof test diagram comparing power function and exponential function

representations of the crack propagation data. The cross, +, marks

the lov/er limit of the crack propagation data. After Wiederhorn,

Evans, Fuller and Johnson, reference 3.

7. Strength as a function of loading rate. The brackets represent 95

percent confidence limits for the mean strengths. The solid line

is a least squares fit of the strength data. The dashed line was calculated from the crack velocity data for the low expansion glass

being considered for the Space Shuttle. After Wiederhorn, Evans,

Fuller and Johnson, reference 3.

8. A comparison of the stress rate dependence of the strength predicted

from crack velocity data with experimental data obtained by Lange

for similar data. After A. G. Evans and S. M. Wiederhorn, reference 22.

9. Dependence of the crack length, a, on the initial crack length, a , and crack propagation exponent n. A significant amount of crack

growth occurs when - a*. For a 5 percent increase in crack length,

a = 0.998 a* if n = 100; a = 0.90 a* if n = 20. o o o °

39
10. Weibull plot of strength data for abraded soda-lime silicate glass.

After Evans and Wiederhorn, reference 7.

12

11. Crack propagation data for soda lime silicate glass in water and in vacuum. 25

2

12. Crack propagation data from figure 11 replotted as K^/v versus K^.

13. Design diagram based on proof testing. Data for soda lime silicate

glass. After Evans and Wiederhorn, reference 7.

40
Fig. 1. Weibull plot of the strength data for a low expansion glass being

considered for the Space Shuttle. The hyperbola gives 95% confidence

limits for the best straight-line fit of the data, m = 6.29; a =

9 O 250 MN/m'*. After Wiederhorn, Evans, Fuller and Johnson, reference 3.

41
Ultra-low Expansion Glass

l.O 10.0

Log (a ) , MN/m

Fig. 2. ^v^^^K^j''^;^^^ ^^^^^ "^^^^^^ diagram. For a given value of the failure probability, each line on this diagram relates the failure time to

s

42
a>CO

9

3

Ultra-low Expansion Glass

3.5

Log (o-q), MN/m

Fig. 3. Design diagram based on proof testing. For a given value of the

proof test ratio, Op/aa, each line on this diagram relates the

minimum failure time to the applied stress. The numbers over each

line give ap/d^. The probability lines from figure 2 are included

here for comparison.

43
Fig. 4. Curves to estimate g for sinusoidal load. After Evans and Fuller,

reference 8,

44
0.3 0.4 (

Stress Intensity Factor, Kj, MN/m^^^

Fig- 5. The effect of pH on crack propagation in the low expansion glass being

considered for the Space Shuttle. After Wiederhorn, Evans, Fuller and

Johnson, reference 3.

45
^9- 6. Proof test diagram comparing power function and exponential function

representations of the crack propagation data. The cross, +, marks

the lov/er lim.it of the crack propagation data. After Wiederhorn,

Evans, Fuller and Johnson, reference 3.

46
Fig. 7. Strength as a function of loading rate. The brackets represent 95

percent confidence limits for the mean strengths. The solid line

is a least squares fit of the strength data. The dashed line was calculated from the crack velocity data for the low expansion glass

being considered for the Space Shuttle. After Wiederhorn, Evans,

Fuller and Johnson, reference 3.

47
600

500

PREDICTED STRENGTHS

STRESS RATE (Nsec"')

Fig. 8. A comparison of the stress rate dependence of the strength predicted

from crack velocity data with experimental data obtained by Lange

for similar data. After A. G. Evans and S- M. Wiederhorn, reference 22.

48
-J I I I I J L

0.94 0.95 0.96 0.97 0.90 0.99 1.0

* a /a

Dependence

and

of the crack length, a, on the initial crack length, a , crack propagation exponent n. A significant amount of crack ° r-Tpor^^'-f - aj. For a 5 percent increase in crack length, a^- 0.998 a^ If n = 100; a = 0.90 aj if n = 20.

49
Fig. 10. Weibull plot of strength data for abraded soda-lime silicate glass.

After Evans and Wiederhorn, reference 7.

50
51
0.3 0.4 0.5 0.6 0.7

STRESS INTENSITY FACTOR. K,. MN/m

Fig. 12. Crack propagation data from figure 11 replotted as K /v versus K .

52
0.1 I 10 I02

STRESS (MNm"^)

Fig. 13. Design diagram based on proof testing. Data for soda lime silicate

glass. After Evans and Wiederhorn, reference 7.

53
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Office of Naval Research

Branch Office

219 South Dearborn Street

Chicago, Illinois 60604

Director

Office of Naval Research

Branch Office

1030 East Green Street

Pasadena, California 91101

Commanding Officer

Office of Naval Research

San Francisco Area Office

50 Fell Street

San Francisco, California 94102

Conmanding Officer

Naval Weapons Laboratory

Attn: Research Division

Dahlgren, Virginia 22448

Organiza tion

Director

Naval Research Laboratory

Attn: Technical Information Officer

Code 2000

Washington, D, Co 20390

Director

Naval Research Laboratory

Attn: Technical Information Officer

Code 2020

Washington, Do C. 20390

Director

Naval Research Laboratory

Attn: Technical Information Officer

Code 6000

Washington, Do C. 20390

Director

Naval Research Laboratory

Attn; Technical Information Officer

Code 6100

Washington, Do C. 20390

Director

Naval Research Laboratory

Attn J Technical Information Officer

Code 6300

Washington, Do Co 20390

Director

Naval Research Laboratory

Attn: Technical Information Officer

Code 6400

Washington, Do Co 20390

Direc tor

Naval Research Laboratory

Attn: Library

Code 2029 (ONRL)

Washington, Do C. 20390

54
Commander

Naval Air Systems Command

Department of the Navy

Attn: Code AIR 320A

Washington, D. C. 20360

Commander

Naval Air Systems Command

Department of the Navy

Attn: Code AIR 5203

Washington, D„ C= 20360

Commander

Naval Ordnance Systems Command

Department of the Navy

Attn: Code ORD 033

Washington, Do C. 20360

Commanding Officer

Naval Air Development Center

Aeronautical Materials Divo

Johnsville

Attn: Code MAM

Warminster, Pao 18974

Commanding Officer

Naval Ordnance Laboratory

Attn: Code 210

White Oak

Silver Spring, Maryland 20910

Ccsnmander

Naval Ship Systems Command

Department of the Navy

Attn: Code 0342

Washington, D= Co 20360

Commanding Officer

Naval Civil Engineering Laboratory

Attn: Code L70

Port Hueneme, California 93041

Ccmmander

Naval Ship Engineering Center

Department of the Navy

Attn: Code 6101

Washington, D. C» 20360

Naval Ships R^D Laboratory

Annapolis Division

Attn: Code A800

Annapolis, Maryland 21402

Ccsnmanding Officer

Naval Ships R^D Center

Attn : Code 747

Washington, D. C. 20007

U. S. Naval Postgraduate School

Attn: Department of Chemistry

and Material Science

Monterey, California 93940

Commander

Naval Weapons Center

Attn: Code 5560

China Lake, California 93555

Commander

Naval Underseas Warfare Center

Pasadena, California 92152

Scientific Advisor

Commandant of the Marine Corps

Attn: Code AX

Washington, Do Cc 20380

Commanding Officer

Army Research Office, Durham

Box CM, Duke Station

Attn: Metallurgy i Ceramics Div

Durham, North Carolina 27706

Office of Scientific Research

Department of the Air Force

Attn: Solid State Divo (SRPS)

Washington, D. Co 20333

Defense Docvnnenta tion Center

Cameron Station

Alexandria, Virginia 22314

55
National Bureau of Standards

Attn: Metallurgy Division

Washington, D„ C. 20234

National Bureau of Standards

Attn: Inorganic Materials Divo

Washington, Do C„ 20234

Atomic Energy Commission

Attn: Metals i Materials Branch

Washington, D. Co 20545

Argonne National Laboratory

Metallurgy Division

Po 0. Box 299

Lemont, Illinois 60439

Brookhaven National LaboratoryTechnical Information Division

Attn: Research Library

Upton, Long Island, New York 11973

Library

Bldg. 50, Room 134

Lawrence Radiation Laboratory

Berkeley, California 94720

Los Alamos Scientific Laboratory

Po Go Box 1663

Attn: Report Librarian

Los Alamos, New Mexico 87544

Commanding Officer

Army Materials and Mechanics

Research Center

Attn: Reso Programs Office (AMXMR-P

Watertown, Massachusetts 02172

Director

Metals & Ceramics Division

Oak Ridge National Laboratory

P . 0 . Box X

Oak Ridge, Tennessee 37830

Commanding Officer

Naval Underwater Systems Center

Newport, Rhode Island 02844

Aerospace Research Laboratories

Wright -Patters on AFB

Building 450

Dayton, Ohio 45433

Defense Metals Information Center

Battelle Memorial Institute

505 King Avenue

Columbus, Ohio 43201

Army Electronics Command

Evans Signal Laboratory

Solid State Devices Branch

c/o Senior Navy Liaison Officer

Fort Monmouth, New Jersey 07703

Commanding General

Department of the Army

Frankford Arsenal

Attn: ORDBA-1320, 64-4

Philadelphia, Pennsylvania 19137

Executive Director

Materials Advisory Board

National Academy of Sciences

2101 Constitution Avenue, N, W,

Washington, D. C. 20418

NASA Headquarters

Attn: Code RRM

Washington, D„ Co 20546

Air Force Materials Lab

Wright-Patterson AFB

Attn: MAMC

) Dayton, Ohio 45433

Air Force Materials Lab

Wright -Patters on AFB

Attn: MAAM

Dayton, Ohio 45433

Deep Submergence Systems Project

Attn: DSSP-00111

Washington, D. Co 20360

56
Advanced Research Projects Agency

Attn: Director, Materials Sciences

Washington, Do Co 20301

Army Research Office

Attn: Dro T., Eo Sullivan

3045 Columbia Pike

Arlington, Virginia 22204

Department of the Interior

Bureau of Mines

Attn: Science i Engineering Advisor

Washington, D. C. 20240

Defense Ceramics Information Center

Battelle Memorial Institute

505 King Avenue

Columbus, Ohio 43201

National Aeronautics i Space Adm.

Lewis Research Center

Attn: Librarian

21000 Brookpark Rd.

Cleveland, Ohio 44135

Naval Missile Center

Materials Consultant

Code 3312-1

Point Mugu, California 93041

Ccsnmanding Officer

Naval Weapons Center Corona Labs.

Corona, California 91720

Commander

Naval Air Test Center

Weapons Systems Test Div. (Code OlA)

Patuxent River, Maryland 20670

Director

Ordnance Research Laboratory

Po Oo Box 30

State College, Pennsylvania 16801

Director

Applied Physics Laboratory

Johns Hopkins University

8621 Georgia Avenue

Silver Spring, Maryland 20901

Director

Applied Physics Laboratory

1013 Northeast Fortieth St.

Seattle, Washington 98105

Materials Sciences Group

Code S130ol

271 Catalina Boulevard

Navy Electronics Laboratory

San Diego, California 92152

Dr. Waldo K. Lyon

Director, Arctic Submarine Laboratory

Code 90, Building 371

Naval Undersea R&D Center

San Diego, California 92132

Dr. R. Nathan Katz

Ceramics Division

U.S. Army Materials & Mechanics

Research Center

Watertown, Mass. 02172

57
SUPPLEMENTARY DISTRIBUTION LIST

Professor Ro Roy

Materials Research Laboratory

Pennsylvania State University

University Park, Pennsylvania 16802

Professor Do Ho Whitmore

Department of Metallurgy

Northwestern University

Evans ton, Illinois 60201

Professor Jo A, Pask

Department of Mineral Technology

University of California

Berkeley, California 94720

Professor D, Turnbull

Divo of Engineering and Applied Science

Harvard University

Pierce Hall

Cambridge, Massachusetts 02100

Dr. To Vasilos

AVCO Corporation

Research and Advanced Development Div.

201 Lowell Street

Wilmington, Massachusetts 01887

Dro Ho Ao Perry

Naval Ordnance Laboratory

Code 230

Silver Spring, Maryland 20910

Dr. Paul Smith

Crystals Branch, Code 6430

Naval Research Laboratory

Washington, Do C„ 20390

Dr. Ao Ro Co Westwood

RIAS Division

Martin -Marietta Corporation

1450 South Rolling Road

Baltimore, Maryland 21227

Prof. M. H. Manghnani

University of Hawaii

Hawaii Institute of Geophysics

2525 Correa Road

Honolulu, Hawaii 96822

Dr. Ro H. Doremus

General Electric Corporation

Metallurgy and Ceramics Labo

Schenectady, New York 12301

Professor Go R. Miller

Department of Ceramic Engineering

University of Utah

Salt Lake City, Utah 84112

Dr. Philip Lo Famsworth

Materials Department

Bat telle Northwest

P. 0. Box 999

Richland, Washington 99352

Mr. G. H. Haertling

Ceramic Division

Sandia Corporation

Albuquerque, New Mexico 87101

Mro lo Berman

Army Materials and Mechanics

Research Center

Watertown, Massachusetts 02171

Dr. Fo Fo Lange

Westinghouse Electric Corporation

Research Laboratories

Pittsburgh, Pennsylvania 15235

Professor H, Ao McKinstry

Pennsylvania State University

Materials Research Laboratory

University Park, Pa o 16802

Professor T, A. Litovitz

Physics Department

Catholic University of America

Washington, Do Co 20017

Dro Ro Jo Stokes

Honeywell Corporate Research Center

10701 Lyndale Avenue South

Bloomington, Minnesota 55420

Dr. W, Haller

Chief, Inorganic Glass Section

National Bureau of Standards

Washington, D.C. 20234

58
Dto Harold Liebowitz

Dean of Engineering

George Washington University

Washington, Do Co 20006

Dro H, Kirchner

Ceramic Finishing Company

P. 0. Box 498

State College, Pennsylvania 16801

Professor A. He Heuer

Case Western Reserve University

University Circle

Cleveland, Ohio 44106

Dro Do Eo Niesz

Battelle Memorial Institute

505 King Avenue

Columbus, Ohio 43201

Dro Fo A. Kroger

University of Southern California

University Park

Los Angeles, California 90007

Dr. Sheldon M. Wiederhorn

National Bureau of Standards

Inorganic Materials Division

Washington, D.C. 20234

Dro Co Oo Hulse

United Aircraft Research Labs

United Aircraft Corporation

East Hartford, Connecticut 06108

Stanford University

Depto of Materials Sciences

Stanford, California 94305

Dr. R. K. MacCrone

Department of Materials Engineering

Rensselaer Polytechnic Institute

Troy, New York 12181

Dro D. Co Mattis

Belfer Graduate School of Science

Yeshiva University

New York, New York 10033

Professor R. Bo Williamson

College of Engineering

University of California

Berkeley, California 94720

Professor Ro W. Gould

Department of Metallurgical

and Materials Engineering

College of Engineering

University of Florida

Gainesville, Florida 32601

Professor Vo S. Stubican

Department of Materials Science

Ceramic Science Section

Pennsylvania State University

University Park, Pennsylvania 16802

Dro Ro Co Anderson

General Electric R and D Center

P. Oo Box 8

Schenectady, New York 12301

Dr. Bert Zauderer

MHD Program, Advanced Studies

Room L-9513 - VFSC

General Electric Company

Dr. Stephen Malkin P. 0. Box 8555

Department of Mechanical Engineering Philadelphia, Penna . 19101

University of Texas

Austin, Texas 78712 Prof. C. F. Fisher, Jr.

Department of Mechanical and AeroProf. H. E. Wilhelm space Engineering

Department of Mechanical Engineering University of Tennessee

Colorado State University Knoxville, Tennessee 37916

Fort Collins, Colorado 80521

59
NBS-114A (KLv

U.S. DE PT . OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. l'i:UI.I( A 1 ION OK Ri:i'OR r NO. NBSIR 74-4862. ( Tov 'l Aci r s.s ion No.

3, Kri ipiciu's Alccssloll No.

4. n 11.1- AN\^ .SUM 1 1 11. 1-:

. Reliability, Life Prediction and Proof Testing of Ceramics

3. r iibl u at ion 1 'at c

6. Pcrformitig Organization ( oiic

7. AU rHOR(S)

S. M. Wiederhorn

8. I'orforniing Org.in. Report No. NBSIR 74-486

9. I'l- KFORMINC, C)Rc;ANI/,A TION NAMI{ ANH AIIIIR I'.SS NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. I^ro|c-ct/ I a.sk/Work llnrt No. 3130453

11. Contract /Cirant No. NR-032-517

12. Sponsoring Organ i/.,u ion Name aiul < Umpkic AJiirtss (Street, City, State, ZIP)

Department of the NavyOffice of Naval Research

Code 471, 800 N. Quincy Street

Arlington, Virginia 22217

13. Ivpt' of I^c Dor t & J'^t' r 1 otl

( f.tc.rcd Interim

7-1-73 thru 6-30-74

14. Spcinsoring Agency ( ode

15. SUPPLEMENTARY NOTi;S

16. ABSTRAC 1 (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) A critical review is presented of the use of proof testing as a design method for

assuring the reliability of structural components. The advantage of proof testing

over the statistical approach used for design lies in the insensitivity of the proof

testing method to the detailed history of handling or processing of structural

components. Methods are presented for developing and using proof test diagrams to assure component lifetime after proof testing. Procedures of proof testing and precautions that must be followed during proof testing are discussed. Provided these

precautions are followed, proof testing offers a general method for assuring the

reliability of structural components under stress.

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 semic olons ) Ceramics; crack propagation; delayed failure; fracture; proof testing; Weibull

analysis.

18. AVAILABILITY [X, Unlimited 19. SECURITY CLASS 21. NO. OF PAGES

(THIS REPORT)

60

r_ Eor Official Distribution. Do Not Release to NTIS UNCLASSIFIED

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