Kutz M. Handbook of materials selection

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3 FRACTURE MECHANICS 791

0 0.1 0.2 0.3

500

1000

(

MPa

)

(

)

mMPa10K

Fig. 2 Strength as a function of ﬂaw size for materials with different fracture toughness.

Fig. 3 Pores with annular crack.

of 781 MPa for a material with K

⫽ 10 MPa (e.g., a zirconia). For a ﬂaw兹m

size of 20

␮

m the corresponding values are 873 and 1747 MPa. In view of this

large effect of the ﬂaw size on the strength, a process technique that leads to a

reduction of the processing ﬂaws is desirable. Fracture may also be caused by

surface ﬂaws, introduced by grinding. Therefore, polishing the surface may also

increase the strength.

The type of ﬂaws also affects the strength. Applying Eq. 1 requires the ex-

istence of a crack. Three-dimensional ﬂaws such as pores are described as

spheres with a surrounding annular crack (Fig. 3). The stress intensity factor

then depends on the ratio of the size of the crack a to the radius of the sphere

R. The size of the crack may be in the order of the grain size.

792 FAILURE MODES

100

equivalent crack len

th a , mm

1000

500

–4

–3

–2 –1

Fig. 4 Strength of silicon carbide vs. crack size. (From Ref. 1.)

4 STRENGTH

Equation 1 can be applied to predict the strength, if the crack size is known. In

metals this relation is used, for instance, to predict the remaining strength of a

component with a fatigue crack, whose size has been determined by a nonde-

structive testing method and is in the order of millimeters. In ceramic materials

the ﬂaws are small in the order of 10–100

␮

m and difﬁcult to detect. Therefore

Eq. 1 has not yet been used to predict the strength or to reject components after

a nondestructive testing procedure. However, the relation shows the potential of

a material. A material with a high fracture toughness and a low strength may

have a potential for higher strength by a reduction in the ﬂaw size.

In Fig. 4 the strength of SiC is plotted versus the crack size in a log-log plot.

For cracks larger than about 100

␮

m a straight line with a slope of ⫺ is obtained

–

according to Eq. 1. For smaller cracks a deviation with an upper limit is ob-

served. The equivalent crack size is a length transformed to an internal crack in

a plate. The reasons for the deviation from Eq. 1 for small ﬂaw sizes shall not

be discussed here.

Because of the large dependence of the strength on the ﬂaw size for a given

material, the strength exhibits a large variation, depending on the processing

method of the material. Strength also can be varied for a given material by

varying the grain size and the microstructure in multiphase materials because

this may change the fracture toughness and the processing ﬂaw size. The values

given in Table 1 are typical values of dense materials. Generally, the tensile

strengths of ceramics are lower than those of high-strength metals. Silicon nitride

and some zirconia materials, however, have strength values of up to 1000 MPa.

5 DELAYED FAILURE

Under cyclic loading, but also under constant loading, ceramic materials may

fail after some time. This is called static fatigue for constant loading and cyclic

5 DELAYED FAILURE 793

fatigue for cyclic loading. This designation contrasts with that in metals, where

the designation of fatigue is restricted to the damage under cyclic loading. The

failure behavior is described by an S–N curve, where the stress is plotted versus

the lifetime under static loading and the stress range

⌬

␴

(twice the amplitude)

versus the number of cycles to failure under cyclic loading. In a log-log plot

straight lines are observed, which correspond to the relations:

t ⫽ (2a)

␴

N ⫽ (2b)

(⌬

␴

)

The phenomenon of delayed failure is caused by subcritical extension of the

ﬂaws. For static loading the crack growth rate depends on the stress intensity

factor K and for cyclic loading on the range of the stress intensity factor

⌬K.In

a large range of crack growth rate, the subcritical crack extension can be de-

scribed by the relations of

⫽ CK (3a)

⫽ C(⌬K) (3b)

where da/dN is the crack extension for one load cycle. Equation 3 can be

obtained by integration of Eq. 4 from an initial crack size a

to a critical crack

size a

. Because of the high values of n, the critical crack size can be set to

inﬁnity. The values of n in Eqs. 3 and 4 are the same for corresponding loading

situations (static or cyclic). The parameters A, C, and n are different for static

and cyclic loading, n

cyclic

being always somewhat smaller than n

static

. A and C for

cyclic loading depend on the mean stress, which conveniently is expressed by

the ratio of R

⫽

␴

⫽ K /K .

min max min max

The following relation was found:

da C

⫽ (⌬K) (4)

⫺

dN (1 ⫺ R)

Integration leads to

⫺

(1 ⫺ R)) 1

N ⫽ C* ⫽ C* (5)

npn

⫺

(⌬

␴

)(⌬

␴

)(

␴

⫹ ⌬

␴

/2)

Some values are given in Table 2. High values of n mean a large effect of the

stress or the stress range on the lifetime.

794 FAILURE MODES

Table 2 Some Values of n and p

Material nP

ZrO

(Y-TZP) 19 3

29 1.3

SiC

/Al

10.2 4.8

Summary

●

Delayed failure is described by a power law relation between crack growth

rate and the stress intensity factor for constant loading and by a power

law relation between crack extension for one load cycle and the range of

the stress intensity factor for cyclic loading.

6 SCATTER OF STRENGTH AND LIFETIME

6.1 Scatter of Strength

Because of the statistical distribution of the ﬂaw size, the strength is subjected

to considerable scatter. As a consequence, the design of ceramic components

cannot be based on an average strength and the application of a safety factor as

it is usually done for metallic components. The scatter is associated with an

effect of the size of a component and the stress distribution in the component

on the strength. Therefore, a statistical treatment of the scatter and a design

according to prescribed failure probability is necessary. In this section the

method of applying Weibull statistics to uniaxial loading is described. The design

applying multiaxial Weibull statistics will be presented in Section 7.

The scatter in the strength is described by the two-parameter Weibull distri-

bution. The failure probability, i.e., the probability of failure occurring below

the stress of

␴

⫽

␴

, is given by

␴

F ⫽ 1 ⫺ exp ⫺ (6)

冋冉冊册

␴

The Weibull parameter m is called the Weibull modulus and a measure of the

scatter. The Weibull parameter

␴

is related to the strength. Whereas m is a

material parameter,

␴

depends on the material, but also on the size of the

component and on the stress distribution in the component.

From Eq. 6 it follows that failure is possible also at very low stresses. There-

fore, a three-parameter Weibull distribution with a lower limit

␴

is considered

sometimes:

␴

⫺

␴

F ⫽ 1 ⫺ exp ⫺

␴

⬎

␴

(7)

冋冉冊册

␴

It is, however, not possible to determine

␴

with sufﬁcient accuracy when ap-

plying a limited number of test specimens. Hence, the two-parameter distribution

is used usually.

6 SCATTER OF STRENGTH AND LIFETIME 795

Failure can be caused by surface ﬂaws or volume ﬂaws. For a homogeneous

stress distribution in a component and failure by volume ﬂaws, Eq. 6 can be

replaced by

␴

F ⫽ 1 ⫺ exp ⫺ (8)

冋冉冊册

␴

0 V

where V is the volume and V

is the unit volume, which is introduced for nor-

malization. Now

␴

is a quantity independent of the size of the component. An

inhomogeneous stress distribution in a component is described by the relation

␴

(x, y, z) ⫽

␴

*g(x, y, z) (9)

where

␴

is a characteristic stress in the component and g(x, y, z) a geometry*

function. The failure probability then is given by

␴

eff

F ⫽ 1 ⫺ exp ⫺ 冕gdV ⫽ 1 ⫺ exp ⫺ (10)

冋冉冊册冋冉冊册

␴

0 V 0 V

where is the value of

␴

at fracture and

␴

V ⫽ 冕gdV (11)

eff

is the effective volume.

For surface ﬂaws the volume in Eqs. 8–11 has to be replaced by the surface

area.

If Eq. 6 is used to calculate the failure probability for a component,

␴

0,C

the component can be calculated from

␴

0,SP

of the specimen by

1/m

eff, SP

␴

⫽

␴

(12)

冉冊

0,C 0,SP

eff,C

Experimental results are presented in a Weibull diagram, where ln ln 1/(1 ⫺

F) is plotted versus ln

␴

. To correlate the measured strength values with the

failure probabilities, the strength values are ranked in increasing order and num-

bered from 1 to n. Then, the strength values

␴

are related to the failure prob-

ability F

according to

⫺ 0.5

F ⫽ (13)

The slope of the Weibull plot is m;

␴

is the strength for F ⫽ 0.632. Examples

are shown in Fig. 5.

796 FAILURE MODES

200

400 600 800

1000

–4

–3

–2

–1

ZrO

ZTA

SSN

RBSN

(MPa)

ln ln (1/(1 – F))

Fig. 5 Weibull plots of strength for several materials (RBSN: reaction bonded silicon nitride,

SSN: sintered silicon nitride, ZTA: zirconia toughend alumina).

Table 3 Ratios

/ ␴

and ␴

med

/␴

␴

m ⫽ 5 m ⫽ 10 m ⫽ 20 m ⫽ 30 m ⫽ 50

␴

0.918 0.951 0.974 0.982 0.989

␴

med

␴

0.929 0.964 0.982 0.988 0.993

The average strength and the median strength for

␴

med

(

␴

for F ⫽ 0.5)

␴

are related to m and

␴

1/m

␴

⫽

␴

⌫(1 ⫹ 1/m)

␴

⫽ (0.693) (14)

c 0med

where ⌫ is the gamma function. In Table 3 the ratios /

␴

and

␴

med

␴

are

␴

given for different values of m.

Equations 8 and 10 lead to a large effect of the size of the component on the

strength, which depends on the Weibull modulus m. They also show a large

effect of m on the failure probability. This effect is illustrated in Fig. 6 by the

strength at different failure probabilities, Weibull parameters m, and different

effective volumes. For a material with a mean strength of

⫽ 500 MPa, which

␴

was determined with four-point specimens having an inner span L ⫽ 20 mm,

width W

⫽ 4.5 mm, and height H ⫽ 3.5 mm. The effective volume of this

specimen is

LWH

V ⫽ (15)

eff

2(m ⫹ 1)

In Table 4 the parameter

␴

and the effective volume are given for different

failure probabilities and different values of m.

6 SCATTER OF STRENGTH AND LIFETIME 797

0 102030405060

100

200

300

400

500

(MPa)

100

1000

100

1000

F=0.1

F=0.0001

eff

mm,V

Fig. 6 Effect of Weibull modulus m and effective volume on the strength for two different fail-

ure probabilities for materials with the same average strength of 500 MPa.

Table 4 Strength at Different Failure Probabilities and Effective Volumes (Materials with

the Same Average Strength in Bending Tests)

eff

(mm

)

␴

F ⫽ 0.0001 F ⫽ 0.001 F ⫽ 0.01 F ⫽ 0.1 F ⫽ 0.5

510

100

1000

661

417

263

105

166

105

264

168

195

4222

266

168

615

388

245

10 10

100

1000

545

433

349

217

172

137

273

217

172

435

346

275

435

346

275

526

417

332

20 10

100

1000

506

451

402

319

284

253

358

319

284

402

358

319

452

403

359

496

442

394

50 10

100

1000

495

473

452

411

393

376

431

412

394

451

431

412

473

452

432

491

470

449

6.2 Scatter of Lifetime

The scatter of the lifetime under constant or cyclic loading is much larger than

that of the strength and can also described by a Weibull distribution. For static

fatigue the failure probability is

F ⫽ 1 ⫺ exp ⫺ (16a)

冋冉冊册

and for cyclic fatigue

F ⫽ 1 ⫺ exp ⫺ (16b)

冋冉冊册

If the strength and lifetime are caused by the same ﬂaws, then the Weibull

798 FAILURE MODES

parameters m*, t

, and N

can be related to the Weibull parameters m and

␴

the strength:

m* ⫽ (17)

⫺ 2

⫺

␴

t ⫽ (18a)

␴

⫺

␴

N ⫽ (18b)

(⌬

␴

)

where n is the crack growth exponent for static and cyclic crack extension,

respectively; B is given by

B ⫽ (19)

2 n

⫺

(n ⫺ 2)CY K

with C from Eq. (4).

In Eq. 18

␴

of the component has to be used. If

␴

as obtained from the

0,C 0,SP

specimens is known, the parameters t

and N

in Eqs. 16 are

⫺

2) / m

⫺

␴

0,SP eff, SP

t ⫽ (20a)

冉冊

0,C

␴

eff, C

⫺

2) / m

⫺

␴

0,SP eff, SP

N ⫽ (20b)

冉冊

0,C

(⌬

␴

) V

eff, C

Summary

●

The scatter of the strength is described by the two-parameter Weibull

distribution with the Weibull modulus m.

●

The average strength and the strength at given failure probability depend

on the size of the component and the stress distribution in the component.

●

The lifetime is described by a Weibull distribution; the parameters can be

obtained from the Weibull parameters of the strength, if strength and life-

time are caused by the same ﬂaws.

7 DESIGN APPLYING MULTIAXIAL WEIBULL STATISTICS

7.1 Strength Under Compression Loading

The strength of ceramic materials under compressive loading is much higher

than under tension loading. Usually compression tests are performed applying

cylindrical specimens. The cylinders are loaded with parallel stamps. Due to the

different lateral expansion of the specimen and the stamp under compressive

loading, radial stresses occur in the loaded specimen, which may lead to pre-

mature failure. The compressive strength measured therefore may depend on the

loading conditions. A detailed description of the compressive test is given in.

Some compressive strength results are presented in Table 1. The ratio between

compressive and tensile strength ranges between 2.5 and 18 depending on the

7 DESIGN APPLYING MULTIAXIAL WEIBULL STATISTICS 799

Mohr

modified

principal

stress

Fig. 7 Mohr’s hypothesis for biaxial loading.

material and the porosity. Dense materials have a higher ratio than porous ma-

terials.

7.2 Global Multiaxial Fracture Criterion

Whereas the strength of materials is measured under uniaxial tension or com-

pression loading, multiaxial stresses very often occur in components. This is the

case for uniaxial loading of notched components, for components under internal

pressure, or components under inhomogeneous temperature distribution. The de-

gree of multiaxiality is described by the ratios of

␴␴

␣

⫽

␤

⫽ (21)

␴␴

where

␴

, and

␴

are the principal stresses. Two different approaches exist

to assess multiaxial stresses: A global multiaxiality and a local multiaxiality

criterion.

A global multiaxiality criterion is a relation between the principal stresses

and one or two material properties. The most reliable criterion is Mohr’s hy-

pothesis, which is presented here for a plane stress situation. It is based on the

tensile strength

␴

and compressive strength

␴

. The criterion is shown in Fig.

7 and reads

␴

⫽

␴

for

␴

⬎ 0,

␴

⬎ 0

1ct 1 2

␴

⫽⫺

␴

for

␴

⬍ 0,

␴

⬍ 0 (22)

2cc 1 2

␴

⫽

␴

(1 ⫹

␴

) for

␴

⬎ 0,

␴

⬍ 0

1ct 2cc 1 2

7.3 Local Multiaxiality Criterion

The local multiaxiality criteria start with a ﬂaw type that may be a pore or a

crack. The local fracture criterion is a critical tensile stress for a pore or a critical

stress intensity factor for a crack. Details can be found in Ref. 2. In the pore

models only the tensile strength appears as a material property. For a spherical

pore the failure relation is

800 FAILURE MODES

=0.2

–1–2

–3

–4

–2

–1

–3

=0.3

Fig. 8 Failure diagram for a spherical pore.

–1

–2

–3

–5

–4

–6

–1

–2–3–4

–5

–6

b/a

0.1 , 0.3

0.5 , 0.3

0.5 , 0.2

Fig. 9 Failure diagram for an elliptical pore. (From Ref. 2.)

␯

⫺ 33⫹ 15

␯

␴

⫹

␴

⫺

␴

⫽

␴

(23)

123ct

27 ⫺ 15

␯

27 ⫺ 15

␯

From this relation the compressive strength follows as

⫺ 15

␯

␴

, for

␯

⫽ 0.2

␴

⫽

␴

⫽ (24)

再

cc ct

␴

, for

␯

⫽ 0.3

⫹ 15

␯

Under triaxial compression, no failure is expected because the stresses at the

pore are negative. In Fig. 8 the failure diagram for plane stress is shown. Results

for an ellipsoidal pore are represented in Fig. 9. From these results it can be