Kutz M. Handbook of materials selection

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7 DESIGN APPLYING MULTIAXIAL WEIBULL STATISTICS 801

concluded that the ratio of compressive to tensile strength may vary between

about 3 and 6, depending on the shape of the pore and the Poisson ratio. It also

shows that Mohr’s hypothesis is a good approximation to the more detailed pore

models.

For a given pore geometry (spherical or ellipsoidal with a given ration of

b/a), the strength is independent of the size of the pore. Consequently, it is not

possible to derive a statistical multiaxial failure relation. Such a relation can be

obtained for a crack model. In this model the ﬂaws are described as cracks with

a given shape, e.g., as circular cracks or semicircular surface cracks. Then, a

statistical multiaxial failure criterion can be developed in the following steps.

The cracks are assumed to be randomly orientated and have a size distribution

that leads to the Weibull distribution of the strength under uniaxial loading.

Depending on the orientation of the ﬂaw with respect to the principal stress axis

and the ratios of the principal stresses

␣

and

␤

, the stresses at the crack tip are

described by the mixed mode stress intensity factors K

, K

, and K

III

: K

is caused

by the normal stress on the crack area

␴

; K

and K

III

are caused by the shear

stress

␶

acting on the crack plane. The local failure criterion is a relation between

these three stress intensity factors and the fracture toughness K

. Different re-

lations are discussed in literature, some of them also include the fracture tough-

ness K

IIC

for mode II loading. An example is the criterion of constant energy

release rate:

22 2 2

K ⫹ K ⫹ K ⫽ K (25)

I II III IC

1 ⫺

␯

For any local stress state a critical orientation of the crack exists. If the normal

stress is compressive (

␴

⬍ 0), then K ⫽ 0 and K

and K

III

have to be calculated

with an effective shear stress:

兩

␶

兩 ⫹

␮␴

for 兩

␮␴

兩 ⬍

␶

⫽ (26)

再

eff

0 for 兩

␮␴

兩 ⬎

␶

with the friction coefﬁcient

␮

. As an example, the resulting failure criterion is

shown for circular cracks and a speciﬁc mixed mode failure criterion in Fig. 10.

If the scatter of the strength is included in the calculation procedure, the

resulting failure criterion also depends on the Weibull parameter m. For details

see Ref. 2.

For an inhomogeneous stress distribution the failure probability of a com-

ponent is given by the integral of

␴

F ⫽ 1 ⫺ exp ⫺ 冕冕h sin ⌽ d⌽ d⌿ 冕gdV (27)

冋冉冊

␲␴

The angles ⌽ and ⌿ characterize the orientation of the crack normal to the

principal stress axis. The quantity h depends on the stress ratios

␣

and

␤

and

on the mixed-mode fracture criterion;

␴

is a material-dependent critical stress.

802 FAILURE MODES

–1

–2

–3

–5

–4

–1

–2–3–4

–5

k=0.8

-1

-2

-3

-4

-1

-2-3

k=1.5

-4

Fig. 10 Failure diagram for circular cracks (k characterizes

the mixed mode criterion). (From Ref. 2.)

Load

(mechanical, thermal)

Geometry

crack normal

x,y,z

coordinates in

the component

crack model

e.g. circular crack

(

)

(

)

III

(

)

local fracture criterion

h(α,β,Φ,Ψ)

Failure probability

(σ

,α,β,ψ,Φ,x,y,z)

(σ

,x,y,z)

(x,y,z)

Weibull parameters

f(K

, K

III

)=const.

Ieq

(

)

Fig. 11 Flowchart for computation of failure probability for

multiaxially loaded components. (From Ref. 2.)

The ﬂowchart for the computation of the failure probability is presented in

Fig. 11. It shows the necessary input information:

●

The stress state in the component (usually from ﬁnite-element calculation)

●

A crack model

●

A local failure criterion

●

The material properties m and

␴

Postprocessor programs are available for the calculation of failure properties.

8 MATERIALS SELECTION FOR THERMAL SHOCK CONDITIONS 803

Summary

●

The failure probability of components under multiaxial loading can be

calculated applying multiaxial Weibull theory, where the ﬂaws are de-

scribed as randomly orientated cracks.

●

As a global multiaxial fracture criterion Mohr’s hypothesis can be applied.

8 MATERIALS SELECTION FOR THERMAL SHOCK CONDITIONS

Ceramic materials may fail during rapid cooling or rapid heating. Due to the

time-dependent transient temperature distribution in a component, the thermal

expansion depends on the location and has to be compensated by mechanical

deformation. Whereas in metals these strains lead to small local plastic defor-

mation, they may cause high local stresses in ceramics that may exceed the local

strength. The failure probability of a component under thermal shock conditions

can be calculated applying the multiaxial Weibull statistics described in Sec-

tion 7.

The resistance against thermal shock varies for different materials. It depends

on the following material properties: Thermal expansion coefﬁcient

␣

, elastic

properties Young’s modulus E, Poisson’s ratio

␯

, thermal conductivity

␭

, density

␳

, head capacity C

, and strength

␴

. Under different thermal conditions different

thermal shock parameters have been deﬁned, which characterize the thermal

shock sensitivity.

For perfect heat transfer from the medium to the component the thermal stress

at the surface after a thermal shock by

⌬T is

␣

⌬T

␴

⫽ (28)

max

1 ⫺

␯

The critical temperature for fracture under thermal shock therefore is

␴

(1 ⫺

␯

)

R ⫽ (29)

␣

For cooling with a constant heat transfer coefﬁcient h the maximum stress at

the surface increases with time and decreases after passing a maximum. The

maximum stress is given by

␣

⌬T

␴

⫽ ƒ(B) (30)

max

1 ⫺

␯

where ƒ is a function of the Biot modulus

B ⫽ (31)

␭

where

␭

is the thermal conductivity, and d a characteristic size parameter of the

804 FAILURE MODES

component. For small values of B the maximum stress is proportional to B.

Hence the critical temperature for failure is proportional to

␴␭

(1 ⫺

␯

)

R⬘ ⫽ (32)

␣

For a constant heating rate at the surface the maximum stress is

␣

␳

␴

⫽⫺C (33)

max

␭

(1 ⫺

␯

) dt

where

␳

is the density, C

the heat capacity, and C a constant depending on the

geometry. The critical heating rate therefore is proportional to

␴␭

(1 ⫺

␯

)

Rⴖ ⫽ (34)

␣␳

Whereas under mechanical loading a crack extends unstably after initiation,

a crack stops after some extension under thermal shock conditions. The remain-

ing strength, however, decreases considerably. If the strength of the component

is not of importance, a limited amount of crack extension may be tolerable.

Then, the condition for materials selection may be a small extension of an ex-

isting crack. It can be shown that the amount of crack extension during thermal

shock increases with increasing crack size. Consequently a material with a large

initial crack may be of advantage. The initial crack size follows from

␣

⫽ (35)

冉冊

␴

leading to another thermal shock parameter

Rⴖⴖ ⫽ (36)

␴

(a Rⴖⴖ is not considered here).

In Table 5 typical values of the thermal shock parameters are given.

Summary

●

The sensitivity to failure by thermal shock increases with increasing

Young’s modulus, increasing thermal expansion coefﬁcient, decreasing

strength, and decreasing thermal conductivity.

●

For a constant heating rate at the surface the sensitivity decreases further

with increasing density and increasing heat capacity.

9 FAILURE AT HIGH TEMPERATURES 805

Table 5 Thermal Shock Parameters for Different Materials

MgO ZrO

SiC

HPSN RBSN BeO Al

TiO

␣

(10 K

⫺

)

⫺

8 12 11 4 3.2 2.5 8 1.8

E (GPa) 400 270 200 350 300 180 360 30

␯

0.22 0.17 0.25 0.2 0.28 0.23 0.25 0.2

␭

(W m

⫺

) 30 30 2.5 100 35 10 300 2.5

␳

(g cm

⫺

) 3.9 3.5 6.0 1.0 0.7 0.7 1.3 0.7

C (J g

⫺

) 1.0 1.0 0.5 1.0 0.7 0.7 1.3 0.7

␴

(MPa) 300 180 950 360 660 200 180 65

(MPa m

1/2

) 4.5 3.0 10 4 7 2 4.8 —

R (K) 73 46 324 206 495 342 47 962

⬘ (kW m

⫺

) 2.19 1.38 0.81 20.6 17.3 3.42 14.1 2.41

ⴖ (W cm

⫺

K) 5.6 3.9 2.7 66 75 20 36 9.6

ⴖⴖ (mm) 0.23 0.28 0.11 0.12 0.11 0.10 0.71

9 FAILURE AT HIGH TEMPERATURES

Failure at high temperatures is caused by creep elongation and creep rupture.

Another failure mode is oxidation. Different mechanisms may contribute to creep

and creep damage. In many cases, creep is caused by grain boundary effects,

such as viscous ﬂow of an amorphous grain boundary phase, diffusion of va-

cancies along grain boundaries, formation and extension of cavities at grain

boundaries or dissolution, and reprecipitation of material through the glassy

phase. Creep within the grain is caused by the motion of dislocations or vacan-

cies.

Damage mechanisms leading to the rupture of a component can be the for-

mation of grain boundary cavities, coalescence to cracks, and growth of the

cracks until ﬁnal failure.

9.1 Creep Strain

After application of a load to a specimen, the same variation of strain with time

as in metals is observed. After an instantaneous strain

␧

, the creep curve can

be subdivided into three stages. In stage I—the primary creep—the strain rate

decreases and reaches in stage II—secondary creep—a constant value. After-

wards, in stage III—tertiary creep—the strain rate increases.

Primary creep can be described by one of the following relations:

␧ ⫽ At with m ⬍ 1 (36a)

␧ ⫽ A[1 ⫺ exp(⫺mt)] (36b)

The effect of the stress on the creep rate very is often described by Norton’s

law:

˙␧ ⫽ B

␴

(37)

Other proposed relations are

806 FAILURE MODES

60 80 100 150 200 300

–10

–9

–8

–7

–6

Slope=11

Slope=13

Slope=3.5

Slope=3.8

Tension

Compression

1300°C

(MPa)

(

in 1/s)

Fig. 12 Stress dependence of the secondary creep rate for reaction-bonded siliconized silicon

carbide tested under tension and compression (from Ref. 4).

˙␧ ⫽ B[exp(

␣␴

) ⫺ 1] (38)

␧ ⫽ B sinh(

␣␴

) (39)

These relations can be applied for the primary and secondary creep range with

possibly different parameters

␣

and n.

Creep under compression differs from creep in tension especially for second-

ary creep, where the creep rate in compression is lower than under tension. An

example is shown in Fig. 12. In this case the exponent n in Eq. 37 is similar in

tension and compression, the constant B, however, is different. Figure 12 also

shows that Eq. 37 cannot be applied for the whole range of the stress: The

exponent n for small stresses is lower than at higher stresses.

The effect of temperature usually can be described by

˙␧ ⫽ C exp ⫺ (40)

冉冊

where R is the gas constant and Q the activation energy for the leading creep

process. An example of Eq. 40 is shown in Fig. 13.

The problem of these relations is that the lowest creep rates measured are

usually higher than the allowable creep rate in a component. It is obvious from

Fig. 12 that the lowest creep rates measured in these investigations are in the

order of 10 /s. For this creep rate the creep strain is 3% in one year. Therefore

⫺

an extrapolation to lower creep rates often is necessary.

9.2 Creep Rupture

The mechanism of creep rupture differs from the mechanism of subcritical crack

extension at low temperatures. This can be seen in Fig. 14, where the stress is

plotted versus the lifetime for an alumina at 1100

⬚C in a log-log plot. Two ranges

can be distinguished. At high stresses, the failure is caused by the extension of

preexisting ﬂaws. The relation between stress and failure strain can be described

9 FAILURE AT HIGH TEMPERATURES 807

678

–7

–6

–5

–4

–3

–2

(

in 1/h)

678

=70.5 MPa

–4

1/ T (K

–1

)

Fig. 13 Secondary creep rate as a function of temperature for three hot-pressed

silicon nitrides. (From Ref 5.)

100

120

␴0

(MPa)

0.01

100

(h)

Fig. 14 Lifetime as a function of the (elastically calculated) outer ﬁber bending stress

for static bending tests on Al

with 4% glass content. Solid line predictions

from subcritical crack growth.

by Eq. 3a with n ⫽ 12 [the slightly curved line in Fig. 14 is obtained by applying

Eq. 3a for the bending tests taking into account small stress redistribution due

to creep]. At lower stresses, the observed lifetime is much lower than predicted

from Eq. 3.

Summary

●

Creep at high temperatures is described by relations between creep rate,

stress, and temperature.

●

The time to failure in the creep range is less stress dependent than at

lower temperatures.

808 FAILURE MODES

REFERENCES

1. S. Usami, I. Takahashi, and T. Machida, ‘‘Static Fatigue Limit of Ceramic Materials Containing

Small Flaws,’’ Eng. Fract. Mech. 25, 483–495 (1986).

2. D. Munz and T. Fett, Ceramics—Mechanical Properties, Failure Behaviour, Materials Selection,

Springer, Berlin, 1999.

3. G. Sines and M. Adams, Compression Testing of Ceramics, Fracture Mechanics of Ceramics, Vol.

3, Plenum, New York, 1978, pp. 403–434.

4. D. F. Caroll and S. M. Wiederthom, ‘‘High Temperature Creep Testing of Ceramics,’’ in Mechan-

ical Testing of Engineering Ceramics at High Temperatures, Elsevier Applied Science, London,

1989, pp. 135–149.

5. R. Kossowsky, D. G. Miller, and E. S. Diaz, ‘‘Tensile Creep Strength of Hot-pressed Si

,’’ J.

Mater. Sci. 10, 983–997 (1975).

809

CHAPTER 28

MECHANICAL RELIABILITY

AND LIFE PREDICTION

FOR BRITTLE MATERIALS

G. S. White

E. R. Fuller, Jr.

S. W. Freiman

Ceramics Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, Maryland

1 SCOPE 809

2 INTRODUCTION 809

3 OVERVIEW 810

4 SUMMARY 818

APPENDIX 1: WEIBULL TESTS 818

APPENDIX 2: STRENGTH AND

DYNAMIC FATIGUE TESTS 821

APPENDIX 3: CONFIDENCE

LIMITS 824

REFERENCES 825

1 SCOPE

This chapter is intended to provide the reader with a general understanding of

why brittle materials fail in a time-dependent manner in service and how to

estimate the lifetimes that can be expected for such materials. In addition, we

describe procedures to evaluate the conﬁdence with which these lifetime predic-

tions can be applied. Throughout this chapter, we assume that the material under

investigation can be treated as isotropic and homogeneous, and that microstruc-

tural inﬂuences on properties can be ignored. Therefore, more complicated is-

sues, such as crystallographic texture, grain-boundary phases, and R-curve

behavior, lie outside the scope of this discussion and will not be addressed

herein.

2 INTRODUCTION

Techniques to determine reliability of components fabricated from brittle mate-

rials (e.g., ceramics and glasses) have been extensively developed over the last

30 years.

1–7

Reliability is generally deﬁned as the probability that a component,

or system, will perform its intended function for a speciﬁed period of time.

Accordingly, the two overarching principles inﬂuencing reliability are the statis-

tical nature of component strength and its time-dependent, environmentally en-

HandbookofMaterialsSelection.EditedbyMyerKutz

810 RELIABILITY AND LIFE FOR BRITTLE MATERIALS

hanced degradation under stress. The statistical aspect of strength derives from

the distribution of the most severe defects in the components (i.e., the strength-

determining ﬂaws).

9–15

The time-dependent aspect of strength results from the

growth of defects under stress and environment, resulting in time-dependent

component failure.

16–19

These concepts have lead to a lifetime prediction for-

malism that incorporates strength and crack growth as a function of stress. Pre-

dicted reliability, or lifetime, is only meaningful, however, when coupled with a

conﬁdence estimate. Therefore, the ﬁnal step in the lifetime prediction process

must be a statistical analysis of the experimental results.

2,20–22

While phenomenological, the reliability methodology has been useful in pre-

dicting lifetimes for myriad applications, including all-glass aircraft windows,

spacecraft

23,24

windows, ﬂat panel displays,

optical glass ﬁbers,

porcelain in-

sulators,

vitreous grinding wheels,

and electronic substrates.

Moreover, al-

though developed for isotropic and homogeneous materials, the methodology

appears to be generally valid for most ﬁne-grain ceramic materials,

7,30–32

and

perhaps for single-crystal materials.

The lifetime prediction approach, described below, represents the currently

accepted procedures of reliability assessment for homogeneous, brittle materials.

There are, of course, a number of assumptions built into any technical procedure

and, frequently, these assumptions cannot be tested. The assumptions inherent

in steps associated with lifetime prediction are clearly stated in this chapter, and

implications that arise if the assumptions are violated are discussed at the end

of each section.

The structure of this chapter has been chosen to allow the user to access

easily regions of particular interest. It has been arranged to provide a general

overview of the processes involved in lifetime prediction of brittle materials

followed by a series of appendixes that describe the technical details and re-

quirements for each of the different stipulated experimental procedures. The

overview describes why the various measurement procedures are necessary and

how they ﬁt into an overall lifetime prediction model. For details associated with

the derivation of the models, the user is directed to the references at the end of

the chapter.

3 OVERVIEW

General Considerations

The most basic assumption made in this chapter is that the material whose

lifetime is of interest is truly brittle; that means there are no energy dissipation

mechanisms (e.g., plastic deformation, internal friction, phase transformations,

creep) other than catastrophic bond rupture occurring during mechanical failure.

It has been well documented that brittle materials fail from ﬂaws that locally

amplify the magnitude of stresses to which the material is subjected.

9,10,14,15,33

These ﬂaws, e.g., scratches, pores, pits, inclusions, or cracks, result from proc-

essing, handling, and use conditions. For a given applied or residual stress, the

initial ﬂaw distribution determines whether the material will survive application

of the stress or will immediately fail. Similarly, the evolution of the ﬂaw pop-

ulation with time determines how long the surviving material will remain intact.

Other key assumptions in reliability predictions for brittle materials are that

the experiments used to determine inert strength distributions do not alter the