Kutz M. Handbook of materials selection

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3 OVERVIEW 811

initial ﬂaw population and that the ﬂaw population in the test pieces mimics that

in the ﬁnal component. In addition, we assume that the ﬂaws change only

through the crack growth processes described in the model. It is essential, there-

fore, that the test specimens that are used to determine the initial ﬂaw distribution

be fabricated by the same processing conditions and prepared with the identical

surface preparation as applied to the ﬁnal components.

4,34

It is also essential that

care be taken to avoid damaging test specimens during fabrication, storage, and

handling. The care taken often includes, but is not limited to, chamfering test

specimens to reduce failures resulting from edge defects, individually wrapping

test specimens to prevent damage during storage and transportation, and avoiding

contact with the tensile surface of the specimens.

Strength

Because the ﬂaw population under stress deﬁnes the initial strength of a brittle

material, it is necessary to characterize this distribution or, equivalently, the

distribution of initial strengths. Almost always, the initial strength distribution

is easier to characterize. This can be done in three ways: (1) a statistical char-

acterization of the initial strength distribution, (2) an overload proof test to es-

tablish a minimum strength, or (3) direct nondestructive ﬂaw detection from

which strength can be calculated. Each of these characterization techniques,

described below, has speciﬁc advantages. However, each of the techniques also

contains stringent requirements and limitations. It is essential to bear in mind

that if these requirements are not met or the limitations are exceeded, the tech-

niques will lead to erroneous lifetime predictions.

Statistical Strength Distribution

The ﬁrst strength characterization technique to be discussed is the statistical

strength characterization approach. This approach is particularly useful to eval-

uate the suitability of different materials, processing procedures, and surface

treatments for a given application. However, because the approach is statistical,

it cannot guarantee that components will have strengths greater than some min-

imum value; rather, it provides a failure (or survival) probability for a given load

level.

Because it is the most critical ﬂaw in a component or test specimen that leads

to failure, it is not the distribution of ﬂaws itself that is important but rather the

distribution of the extremes of the ﬂaw distribution. Fortunately, distributions of

extremes, or extreme-value statistics, have been thoroughly studied.

11–13,35

Strength distributions are typically described by the Weibull distribution, one of

three extreme-value distributions. Procedures for estimating the parameters of

the Weibull distribution are well documented.

36–38

The generally accepted pro-

cedure is to use maximum likelihood estimators. Procedures for the two-

parameter Weibull distribution are described in the ASTM standard practice

guide, ASTM C 1239-95.

Occasionally, the strength data exhibits threshold-

strength behavior, and must be ﬁtted to a three-parameter Weibull distribution.

In such cases a number of difﬁculties are associated with the numerical tech-

niques used to estimate the Weibull parameters.

37,38

Typically a Newton-Raphson

iteration procedure is used (see Appendix F of Ref. 37), but a sequential search

technique with subsequent bisectional bracketing also works well.

812 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

The strength distribution obtained is unequivocal provided that it is deter-

mined on actual components (for each batch of material) under conditions that

simulate the service conditions. However, this requirement often poses an intol-

erable economic constraint, because so many tests must be performed to achieve

an acceptable conﬁdence level. Consequently, tests are often conducted on small

coupons whose processing history and surface treatment mimic those of the ﬁnal

component.

Some care must be exercised in collecting the strength data. Measurements

should be conducted at a high loading rate and in an inert environment, e.g., a

dry gaseous nitrogen environment, to avoid environmental effects discussed be-

low. In addition, the stress state,

␴

, used for the Weibull test must represent as

closely as possible the stress conditions the component will see in service. For

example, conducting the Weibull tests in uniaxial tension will not necessarily

provide meaningful data if the actual component will experience biaxial loading.

After the Weibull strength distribution has been obtained and analyzed, a

particular probability of failure, F, can then be associated with a corresponding

initial strength, S. The relationship between F and S, i.e., the spread of initial

strengths, is controlled by the distribution of initial ﬂaw sizes in the material.

Finally, if the tests have been conducted on test coupons rather than ﬁnal

components, it is essential to relate the area under load in the coupon to the

area under load in the component (see Appendix 1) because the probability of

encompassing a more severe ﬂaw is proportional to the area under load. There-

fore, the test specimen area under load must be normalized to the component

area under load. It should be borne in mind that, if failure results from subsurface

defects, the important parameter to normalize is the volume under load rather

than the surface area.

Minimum Strength Overload Proof Test

An overload proof test is a procedure for establishing the upper limit on the

most critical ﬂaw in a component or, equivalently, the lower limit on the initial

strength distribution.

4,7,39–43

The technique is expensive since real components

are tested. However, a properly conducted proof test completely eliminates the

problem of failure through statistical outliers that is unavoidable when the sta-

tistical strength distribution technique is used. Therefore, if the statistical

strength distribution analysis indicates that there is an unacceptably high prob-

ability of failure at the required load, the overload proof test procedure can be

used to eliminate the weak components and truncate the failure distribution. It

should be noted that, because the overload proof test is applied to ﬁnished com-

ponents, it is generally not a technique that is appropriate for material or design

selection. In addition, for a ﬁnal component with a complicated geometry, it can

be difﬁcult to apply controlled stresses that are equivalent in orientation to those

experienced in service.

In concept, the overload proof test is straightforward. One simply applies an

overload stress to each component, typically two to three times the service stress.

The ratio of the overload stress to the service stress is the proof-test ratio.

Components with critical ﬂaws larger than a predetermined size, or with

strengths less than a predetermined minimum (namely, the proof-test stress), will

break. Accordingly, such components are automatically eliminated from the dis-

tribution.

3 OVERVIEW 813

However, the requirements for a properly conducted proof test must be strin-

gently met. Speciﬁcally, the proof test must be devised such that the stress at

each location of a component exceeds the service stress in that same location

by an amount at least as large as the recommended proof-test ratio. The proof

test must be performed under controlled conditions of environment and proof-

test load cycle; the most critical aspect being to have conditions as inert as

possible and to unload from the proof stress as rapidly as possible. If these

conditions are satisﬁed, then proof testing is undoubtedly the best lifeprediction

procedure, because no aspects of the analysis are unresolved. If the conditions

are not met, then crack growth (discussed below) could occur during unloading,

obviating the desired truncation of the initial ﬂaw population and leaving the

components weaker than predicted by the proof test-analysis.

Nondestructive Flaw Detection

In principle, a good nondestructive evaluation (NDE) test for ﬂaw detection

offers several beneﬁts: 100% inspection, on-line inspection, in-ﬁeld health mon-

itoring, and quantitative critical ﬂaw detection. In practice, the approach is al-

most never used for brittle materials. The difﬁculties associated with using NDE

involve both ﬂaw detection and ﬂaw quantiﬁcation. Since the toughness of most

ceramics and glasses is on the order of 1–5 MPa

䡠 , critical ﬂaw sizes are less兹m

than approximately 50

␮

m for even low-strength materials. This is currently

beyond the detection limit of most available NDE procedures. However, mea-

surement techniques are rapidly improving and, within the foreseeable future,

instruments may be commonly available that will detect defects of micrometer

and sub-micrometer size. A more formidable task will then be relating the NDE

measurements to quantiﬁable reliability predictions.

As discussed above, brittle materials fail from the most severe, not the av-

erage, ﬂaws. The phrase ‘‘most severe ﬂaw’’ means the ﬂaw that generates the

largest stress intensity in the expected stress ﬁeld. Consequently, the ﬂaw size,

shape, and orientation relative to the stress ﬁeld must be quantiﬁed before results

from an NDE measurement can be used to predict strength.

Unlike metals, in which ﬂaws can grow to many times their original size

without component rupture while NDE techniques monitor the ﬂaw growth or

the concomitant material property degradation, brittle materials fail with very

limited ﬂaw extension and almost no property degradation. Consequently, unless

a speciﬁc NDE test has been clearly shown to predict failure strength accurately

for the same brittle material with the same surface ﬁnish and under the same

loading conditions of interest, it is better to use the statistical strength distribution

or the overload proof-test techniques described previously to determine strength

information.

Environmentally Enhanced Fracture

The second property inﬂuencing the reliability of brittle materials is the growth

of ﬂaws in the presence of stress. If the ﬂaws remained constant with time,*

there would be no need to proceed beyond the initial strength analysis. Based

*Throughout this discussion, we consider only the growth of preexisting ﬂaws over time and ignore

the possibility of generating a new ﬂaw population during use.

4,34

814 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

upon the Weibull distribution, the probability of failure would be known for any

stress condition and would be constant. Alternatively, based on an overload proof

test or NDE, a minimum strength is deﬁned such that failure would not occur

for stresses below these values. However, it is well known that ﬂaws grow under

stress in most brittle materials.

1,16–19,44,45

As the ﬂaws grow, the material weak-

ens, leading to a time-dependent mechanical failure. This process is known by

several names; three of the most common are ‘‘environmentally enhanced frac-

ture,’’ ‘‘delayed failure,’’ and ‘‘static fatigue.’’

Environmentally enhanced fracture is a process that is still not completely

understood. However, it is known that ﬂaws in materials under stress can react

with certain environments, resulting in bond rupture and ﬂaw growth at stresses

well below the nominal bond strength of the material. Water is particularly ef-

fective at enhancing fracture in many brittle materials. In addition, water is

almost ubiquitous in either liquid or vapor phase. Therefore, it is essential that

the effects of water-enhanced fracture be incorporated into any lifetime predic-

tion model.*

Because the process of environmentally enhanced fracture is only partially

understood, expressions, relating crack growth velocity, V, to stress intensity

factor, K

, are typically phenomenological rather than truly theoretical. Although

several crack-growth-rate expressions have been proposed, the one used in al-

most all lifetime predictions for brittle materials is given by:

2–4,6,7,39

V ⫽ V (1)

冉冊

where V

is a constant and N is called the environmentally enhanced crack

growth parameter, sometimes denoted the inverse crack growth susceptibility.

is the critical stress intensity factor at which fracture ensues catastrophically

but, for the present context, can be considered as a scaling parameter for the

stress intensity factor, K

. N denotes the extent to which the environment interacts

with the stressed crack to cause enhanced crack growth. For a given V

and K

the larger N is, the less effective the environment is at enhancing crack growth.

The fact that Eq. 1 is phenomenological rather than rigorously based on theory

should not be a concern as long as the equation describes the observed behavior,

particularly in the low stress region. Guaranteeing this condition is difﬁcult,

however, since the crack growth rates for a component in use are usually several

orders of magnitude smaller than the lowest crack growth rates that can be

determined experimentally. As long as any deviation from Eq. 1 is of the form

real

) ⬍ V

Eq. 1

), that is, the real velocity is less than the velocity predicted

from Eq. 1, lifetime predictions based upon Eq. 1 are conservative. However, if

real

) ⬎ V

Eq. 1

), lifetime predictions based on Eq. 1 are no longer conser-

vative.

At this point, it should be noted that there are a number of other possible

crack growth expressions that are statistically indistinguishable from Eq. 1 over

*Few environments are more effective than water in enhancing crack growth for most brittle mate-

rials. An exception is very basic solutions for silica-based glasses.

For such cases, the following

discussion still applies, but the tests must be conducted in the new environment.

3 OVERVIEW 815

the measurable crack velocity range.

Some of these expressions are more con-

servative, i.e., predict higher crack growth rates, at low K

than the power law.

A particularly important example is the expression V

⫽ B exp[bK

], which is

derived from chemical rate theory, and has been used by several authors to

parameterize environmentally enhanced fracture.

3,24,46–48

Although having a

stronger scientiﬁc basis than the power law (Eq. 1), the exponential expression

for crack growth rates has never developed the following obtained by the power

law expression. The dominance of the power law expression derives from the

fact that it allows closed form, analytical lifetime solutions to be obtained even

when the critical ﬂaws are surrounded by a residual stress ﬁeld, as will be

discussed in the section on constant loading rate experiments. With the advent

of powerful desktop computing, it is now possible to use numerical techniques,

rather than analytical derivations, to predict lifetimes. The approach outlined

above is still required, i.e., determination of the initial strength distribution

through Weibull statistics

36–38

or proof test

4,39

measurements, evaluation of crack

growth parameters through dynamic fatigue experiments,

4,49

and estimation of

conﬁdence limits through statistical analysis, e.g., bootstrap methods

21,22,50–52

discussed below. Consequently, it is now possible to make lifetime predictions

using crack growth relationships like the exponential expression. However, it is

much easier to demonstrate how the different measurements are combined to

obtain an expression for the lifetime if analytical expressions are used. Therefore,

for the purposes of this chapter, we will focus exclusively on the power law

expression for crack growth.

Constant Loading Rate Experiments

Conducting fracture mechanics crack growth experiments to evaluate the param-

eters governing environmentally enhanced crack growth is a process that requires

both a large amount of time and a thorough understanding of issues associated

with environmentally enhanced fracture. Consequently, a faster evaluation

method that is less prone to experimental subtleties is desirable. Equation 2

deﬁnes the constant loading rate or dynamic fatigue measurement relationship

(Appendix 2). In Eq. 2,

␭

is a ﬁtting parameter related to V

and K

and N⬘ ⫽

(3N ⫹ 2)/4, where V

, K

, and N are parameters in Eq. 1.

␴

⬘⫺

␴

⫽

␭

(2)

Although there are various possible geometries and several critical experimental

requirements for dynamic fatigue measurements (see Appendix 2), the measure-

ments are conceptually straightforward. A series of specimens is subjected to a

load that increases linearly with time. The log of the failure strength,

␴

, plotted

as a function of stressing rate,

␴

, gives the parameters N⬘ and

␭

. To obtain

conservative values of N

⬘ and

␭

, dynamic fatigue measurements need to be

carried out in the harshest environmental conditions under which the ﬁnal com-

ponent will be used. For example, if water vapor enhances crack growth, liquid

water could provide the limiting case of water-enhanced environmental fracture.

However, it is not necessary that the initial ﬂaw distribution be maintained. A

change in ﬂaw distribution will change

␴

, but not the slope,

␭

/(N ⫺ 1). There-

fore, dynamic fatigue experiments are usually conducted on indented specimens;

816 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

the indentations provide a uniform and repeatable critical ﬂaw.

53–57

Typically,

the resultant data have much less scatter than those obtained from the initial

ﬂaw distribution, thereby resulting in a more precise determination of the slope.

Inert Strength for Indented Specimens

The inert strength for indented specimens, S

, is determined using the same

procedure as that used to obtain the Weibull distribution: high loading rate and

an inert atmosphere. However, instead of using the natural ﬂaw population as

the strength limiting defects, S

uses the same level of indentations that were

used in the dynamic fatigue experiments. These data provide an upper limit on

the dynamic fatigue strength data.

Lifetime Prediction

Once the Weibull distribution, the dynamic fatigue data, and the inert strength

for the indented specimens have been obtained, the lifetime, t

, for any initial

strength, S, and any applied stress,

␴

, can be calculated using Equation 3

⬘⫺

␭

⫺

⬘

t ⫽

␴

(3)

冉冊

N⬘ ⫹ 1 S

Although Eq. 3 predicts a time to failure for a component under load, there is

still no way to estimate how statistically signiﬁcant the value is. To obtain the

most information from the lifetime prediction model, it is necessary to estimate

the conﬁdence limits surrounding t

Conﬁdence Limits

As with any experimentally determined quantities, there are uncertainties asso-

ciated with each of the parameters in Eq. 3. Some of the parameters may not

be amenable to standard error propagation analysis. For example, for large N

⬘,

e.g., N

⬘ⱖ60, the uncertainties in evaluating the slope of the dynamic fatigue

curve are much larger than the calculated slope itself. Under such conditions,

standard error propagation techniques are invalid. An alternative is the bootstrap

technique.

50–52

It is a statistical procedure that takes advantage of modern com-

puting power and makes it relatively straightforward to determine conﬁdence

limits (Appendix 3). It applies to any data set that is independent and identically

distributed (iid). The major assumption with the bootstrap technique is that the

data used to evaluate the parameters in Eq. 3 adequately represent the statistical

scatter in the experiments. If this condition is met, combining the bootstrap

technique with the data used to evaluate t

permits the user to estimate the time

to failure at whatever conﬁdence level is desired.

Life Prediction Process

The individual steps required for lifetime prediction have been laid out above.

If the steps are followed and careful attention is paid to the requirements and

restrictions at each step in the process, a lifetime prediction with a speciﬁc failure

probability and known conﬁdence limits for a speciﬁc applied load is obtained.

However, frequently, other questions are of interest. For example, the needed

lifetime may be an input parameter, and the limiting stress value may be the

3 OVERVIEW 817

Fig. 1 Design approach.

unknown parameter of interest. As another example, a required lifetime at a

known load may be the input parameters, and the unknown may be whether a

given material can be used and, if so, with what probability of failure. The

formalism outlined above is adequate to address any of these questions and, in

fact, can be used in an iterative manner to determine how variations in one group

of parameters affect others. In this section, failure diagrams will be introduced.

These diagrams, constructed from the data obtained using the techniques out-

lined above, graphically relate the parameters to show quantitatively how mod-

iﬁcations in one parameter, e.g., stress or lifetime, affect the others.

Figure 1a is derived directly from Eq. 3. Each of the solid lines maps the

applied stress,

␴

, onto the time to failure, t

, for a different value of the initial

strength or, equivalently, different probability of immediate failure upon the ap-

plication of the

␴

(see Appendix 1). The lines are all parallel with slope of ⫺N⬘.

The bands about each line represent the conﬁdence limit at which the position

of the line is known and can be determined by a statistical analysis tool such as

the bootstrap method. For conservative calculations, the left-hand conﬁdence

limit should be used instead of the centerline. Figure 1a immediately shows, in

a quantitative way, how all the parameters affecting lifetime predictions are

coupled if a statistical strength distribution is being used. Known, or required,

values of some of the parameters can be located on the plot and the range of

possible values for the remaining parameters can be taken directly from the

graph.

If the overload proof test, rather than the statistical strength distribution, ap-

proach is being used, Fig. 1b maps the applied stress onto the lifetime. This

ﬁgure is obtained from Eq. 3 by replacing the initial strength S divided by

␴

the proof-test ratio, R. This gives a linear relationship between log(

␴

) and log(t

)

with a slope of

⫺2. The different parallel lines in Fig. 1b are now determined

by changes in R. The larger R is, the larger t

is for a given

␴

, but at the cost

of more components being broken. To estimate the fraction of components that

will fail during the proof test, one refers again to the probability diagram, the

proportion of components likely to fail in the proof test can be estimated by

determining the intersection of the stress coordinates of the proof stress,

␴

⫽

␴

, and the time coordinate for the time at the peak proof load, t

. If this

proportion is unacceptable, e.g., most of the components fail in the proof test,

818 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

then the material, including ﬂaw distribution, is inadequate, and an alternate

material should be explored. This process should proceed until a material with

a satisfactory proof-test failure proportion is identiﬁed.

Finally, if proof testing is unsuitable, for example, as a result of the com-

plexity of the component, NDE techniques could be explored. The approach

taken at this juncture is strongly material dependent. Figure 1c relates the life-

time, t

, to the applied stress,

␴

, and the ﬂaw size, a. To use Fig. 1c, an ap-

proximate value for the ﬂaw size that must be detected to ensure integrity is

ﬁrst obtained from the graph, without accounting for the extension of the con-

ﬁdence limits. Then, the ability to detect such ﬂaw types should be evaluated

using available NDE techniques. At this stage, conﬁdence limits on ﬂaw detec-

tion in the test material can be established for each NDE technique, and the ﬂaw

size detection requirement can be more closely deﬁned. The suitability of the

material for this application is determined by the capability of detecting all ﬂaws

in the critical parts of the component that exceed the speciﬁed size. As discussed

previously, use of this approach requires that NDE techniques are available that

provide quantitative size and shape data regarding ﬂaws.

4 SUMMARY

In summary, the following procedure is required to predict lifetimes in isotropic,

homogeneous, brittle materials:

1. Fast fracture tests in an inert environment to determine the Weibull dis-

tribution of the initial ﬂaws/strengths and the probability of failure for

any arbitrary initial strength, or a proof test to weed out specimens below

a predetermined initial strength, or an NDE technique that guarantees a

lower limit on strength

2. Dynamic fatigue measurements on indented specimens to determine sus-

ceptibility to environmentally enhanced fracture

3. Fast fracture tests in an inert environment on indented specimens to de-

termine the inert strength (strength in the absence of environmental ef-

fects) of the indented material

4. Bootstrap Eq. 3, using the data in steps 1–3, to predict lifetimes and

determine conﬁdence levels for predetermined initial strength

It is essential for the user to remember that this lifetime prediction methodology

assumes that the ﬂaw/strength distribution determined from the Weibull analysis

is the same distribution to be found in the ﬁnal components (for the statistical

strength distribution approach), that the tests do not alter the ﬂaw distribution,

and that ﬂaws change over time only through environmentally enhanced fracture

as described by Eq. 1.

APPENDIX 1: WEIBULL TESTS

Since brittle failure is expected to occur from the most severe defect in the

component, the strengths are assumed to come from an extreme value distribu-

tion. A critical assumption in the following discussion is that the strength dis-

tribution in the strength test is the same as the distribution in the ﬁnal

components. Typically, either a two- or a three-parameter Weibull is used to

APPENDIX 1: WEIBULL TESTS 819

evaluate the distribution. Equation A1.1 gives the form of the three-parameter

Weibull expression

8,22,35

␴

(x) ⫺ S

(1 ⫺ F) ⫽ exp ⫺ (A1.1)

冉冊

S ⫺ S

冤冥

0 t

Two of the three parameters in Eq. A1.1 are the shape parameter, m, which

determines the shape of the probability distribution function, and the scale pa-

rameter, S

, which deﬁnes the scale of the strength axis. The shape parameter,

m, deﬁnes how quickly the probability of survival drops from a value of 1.0 as

the stress on the material increases. Low values of m correspond to a broad,

relatively slowly varying decrease in the probability of survival as a function of

increasing

␴

. In contrast, high m values generate a survival probability curve

approaching a step shape; the survival probability remains near 1.0 until

␴

nears

and then drops rapidly to zero. As implied by the previous sentence, S

related to the value of stress at which the probability of survival is near the

midpoint between 1.0 and 0. For S

⫽ 0, the probability of survival at

␴

⫽ S

is 1/e. In fact, in the limit of m approaching inﬁnity and S

⫽ 0, S

is identically

the value of stress for which the probability of survival drops from 1.0 to 0.

The third Weibull parameter, S

, represents a ‘‘threshold’’ value for strength

below which there is zero probability of failure. The form of the two-parameter

Weibull expression is identical with Eq. A1.1 with S

equal to zero; i.e., there is

no lower strength threshold. The decision as to whether S

is zero or ﬁnite is

not always easy to make. The question may become moot if it is borne in mind

that the two-parameter distribution is a special case of the three-parameter dis-

tribution. The decision whether to include a ﬁnite value for S

could be addressed

statistically by determining the signiﬁcance of a threshold value and uncertainties

related to its inclusion (i.e., with what conﬁdence can a ﬁnite value be attributed

to S

?) by the bootstrap simulation (Appendix 3). However, the implications of

assuming a two-parameter or a three-parameter Weibull distribution should be

kept in mind. A two-parameter distribution is always more conservative, and,

therefore, safer, since it implicitly assumes that there is a possibility for failure

at any load. However, if a threshold does exist, use of a two-parameter distri-

bution will have economic consequences because it will predict a ﬁnite failure

probability at loads for which no failures will occur.

Based on Eq. A1.1, a set of test specimens can be broken rapidly in bending

(Appendix 2) in an inert environment and the strength of each specimen plotted

against the survivability ranking, (1

⫺ F):

ln ln ⫽ m ln[

␴

(x) ⫺ S ] ⫺ m ln(S ⫺ S ) (A1.2)

冉冊

t 0 t

1 ⫺ F

冤冥

This results in a line of slope m and intercept of ⫺m ln(S

⫺ S

). Therefore, the

survival probability of a given test specimen could be predicted for a given load.

820 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

However, usually, the area under load on a component in use is not identical to

the area sampled in the bend test. Because the ﬂaws are assumed to be distributed

randomly in both the ﬁnal components and the test specimens, if the area under

load in the ﬁnal component is larger than the area sampled by the Weibull test,

the component is likely to sample larger ﬂaws than those sampled during the

tests. Therefore, the probability of failure at a given load will be greater for the

component than for the test specimens. The converse is true if the area sampled

in the ﬁnal component is smaller than that sampled by the test specimens. Thus,

a weighting factor is incorporated into the analysis to account for differences in

sampling area between the test specimens and the ﬁnal components.

This is handled in the following way. If an elemental area, dA, of a component

surface at position x experiences a tensile stress

␴

(x) and a survival probability,

⫺ F), then the survival probability of the component, (1 ⫺ F

), is the joint

survival probability of all the individual surface elements. In the limit of inﬁn-

itesimal elemental areas, this survival probability is

S ⫺ S

[1 ⫺ F (S)] ⫽ exp ⫺k(m, S ) (A1.3)

冉冊

S ⫺ S

冤冥

0 t

where F

is the failure probability of the component, S is the maximum tensile

stress in the component (i.e., the strength), and k(m, S

) is an area-scaling factor

given by

␴

(x) ⫺ S

k(m, S ) ⫽ 冕冕 dA (A1.4)

冉冊

AS⫺ S

0 t

If A

is chosen to make the k factor for the laboratory specimen unity, k(m, S

)

is the relative area-scaling ratio between the components and the laboratory

specimens.

Best Practices

●

Unless there is clear evidence that a threshold exists in the Weibull dis-

tribution, it is better (i.e., conservative) to assume a two-parameter Weibull

distribution.

●

A good rule of thumb is that 30 or more specimens should be used to

generate a Weibull plot.

●

It is frequently valuable to identify the failure source of specimens under

an optical microscope.

58,59

This procedure distinguishes failures from ma-

chining damage, inclusions, handling accidents, etc. A description of frac-

tography is outside the range of this document, but an excellent discussion

can be found at the web site: http://www.ceramics.nist.gov/webbook/

fracture/fracture.htm.

●

Measurements should be conducted in as inert an environment as possible.

Dry ﬂowing N

gas in a glove bag is usually a good choice.

●

Measurements should be made at the highest loading rate for which the

testing machine provides reliable data.