Reliability evaluation of brittle solids under polyaxial stress states based on maximum energy release rate criterion

~)

Pergamon

Engineering Fracture Mechanics Vol. 51, No. 4, pp. 629-635, 1995 Copyright © 1995 ElsevierScienceLtd 0013-7944(94)00278-9 Printed in Great Britain. All rights reserved 0013-7944/95 $9.50+ 0.00

RELIABILITY E V A L U A T I O N OF BRITTLE SOLIDS U N D E R P O L Y A X I A L STRESS STATES BASED ON M A X I M U M E N E R G Y RELEASE RATE C R I T E R I O N M. ICHIKAWA Department of Mechanical and Control Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu City, Tokyo 182, Japan Abstract--In relation to reliability-based design of ceramic components, an investigation is made on evaluation of reliability of brittle solids under polyaxial stress states. Applying the maximum energy release rate criterion to the crack-like flaws in a brittle solid subjected to polyaxial stresses, the probability of failure Ps is calculated. As an analytical expression of the maximum energy release rate of a mixed mode crack, Gmax, a new expression derived by the present author is used. The result is compared with Pr calculated by using Lamon's expression of Gmax.

1. I N T R O D U C T I O N EVALUATION of the probability of failure is useful in design of ceramic components. From a practical point of view, the case when a component is subjected to polyaxial stresses is important. In order to treat such a case, consideration of fracture from a mixed mode crack is essential. Several criteria were proposed with respect to brittle fracture from a mixed mode crack. These involve the maximum tangential stress criterion [1], the energy density criterion [2], and the maximum energy release rate criterion. The last one was investigated in many papers including [3] which will be used in the present paper. In considering fracture under polyaxial stresses, influence of load biaxiality is also important. This problem has long been investigated by Eftis and Liebowitz since 1972 [4]. They showed that the normal stress applied in parallel to a crack affects fracture behaviour through many papers which were summarized in [5]. In the present paper, we focus our attention upon application of the maximum energy release rate criterion in relation to Lamon's work [6]. The maximum energy release rate criterion assumes that crack extension occurs when Gmax reaches a critical value, where Gmax is the energy release rate G for crack extension in such a direction that makes G maximum with respect to the angle ~k in Fig. 1. For simplicity, we refer hereafter to this criterion as Gmax criterion. Gmax has been calculated numerically by several investigators. To the author's knowledge, however, an analytical expression of Gmax has not appeared until recently. In 1988, Lamon [6] proposed an expression of Gmax based on the paper by Hellen and Blackburn [7] and made a calculation of the probability of failure, Py, of a ceramic component based o n Gmax criterion. However, the expression of Gmax used by Lamon is rather inaccurate as noted in [8]. The purpose of the present paper is to calculate P/using another expression of Gmax proposed by the present author [8], and compare the result with Ps calculated by using Lamon's expression of GmaX.

2. E X P R E S S I O N OF T H E MAXIMUM ENERGY RELEASE RATE Lamon [6] made a reliability analysis of ceramic components using the following analytical expression of the maximum energy release r a t e Gmax: Gmax = ~

1 ( K 4 --t-6 Ki2KH + KH4)1/2 2

(1)

where E* = E for plane stress, and E* = E/(1 - v 2) for plane strain. E and v are Young's modulus and Poisson's ratio, respectively. K~ and K, are the mode I and mode II stress intensity factors, EFM

51i4--H

629

630

M. ICHIKAWA

_+. Fig. I. Extension of a mixed mode crack.

respectively, at the tip of the crack before extension. In the previous paper [8], we showed eq. (1) to be rather inaccurate, and derived another analytical expression of Gmaxas follows (see Appendix): 1

Gm,x = 2E---;[K~ + 3K~, + Kj (K~ + 6 K~21)~/2].

(2)

Although eq. (2) is not rigorous, it gives almost accurate values as may be seen from Fig. 2, in which eq. (2) is compared with the accurate values given by the numerical analysis [9]. In the ordinate of Fig. 2, Gm,x is normalized by (K~ + K~I)/E*, which is the energy release rate for crack extension in the direction of ~ = 0 in Fig. 1. Tan-~(K./KO = 0 and 90 ° correspond to pure mode I and pure mode II, respectively. It is also seen from Fig. 2 that eq. (1) is rather inaccurate. Since eq. (1) underestimates G . . . . use of eq. (1) yields unconservative evaluation of reliability. When gll = 0, eq. (2) reduces to G m a x = K~/E*. According to the viewpoint of Eftis et al. [5], this seems to imply that eq. (2) is valid only under so-called fixed-grip and dead-load conditions. Examination of this point will be made in a separate paper. 3. CALCULATION OF T H E PROBABILITY OF FAILURE To calculate the probability of failure theoretically, we make the following assumptions. (1) A large number of crack-like flaws are contained in a solid, and the weakest link model holds. Hence, the asymptotic distribution of the smallest value can be applied. (2) In order to make the analysis two dimensional, flaws are modelled as two-dimensional cracks in an infinite solid. (3) Orientation of a crack is random, and follows a uniform distribution. (4) Uniaxial tensile fracture strength follows a two-parameter Weibull distribution. Let o~ and 02 be the principal stresses and define the angle 0 as shown in Fig. 3. The normal and shear stresses, 0. and % acting on the crack plane are given by o. = al sin 2 0 + a2 cos 2 0,

z. = (a2 - al)sin 0 cos 0.

2.0

Exact

1.8 1.6 .~

1./,

~9 ~_

1.2 1.o

T

2'o 3'o 4'o tan-1 ~Ktl

7;

8b 90

(degree)

Fig. 2. Comparisonof eqs (1) and (2) with the accurate value of the maximum energy release rate Gin,~.

(3)

631

Reliability evaluation of brittle solids

0"2

y

normal

Fig. 3. Orientation of a crack with respect to the principal stresses.

T h e stress intensity factors K~ and K . are given by K~ = a ~ / ~ ,

K. = z,v/~

(4)

where a is the half crack length. Substitution o f eq. (4) into eq. (2) yields

na [a~ + 3*2. + o..(o.~ + 6z2~)'/2].

(5)

Gmax- 2E*

F r o m eq. (5) and the m a x i m u m energy release rate criterion 2

2

2

2 1/2

o.. + 3z, + o.,(o., + 6z.)

Gmax >

Gc, we obtain

2E'G,

~>--

7za

(6)

where G,. is the critical value o f Gmax required for crack extension. Equation (6) is rewritten as o.oq >/o'er

(7)

where o.eq is the equivalent n o r m a l stress [10] defined by o.m = [~I {o..2 + 3z.2 + o..(o..2 + 6z~)'/2}] '/2

(8)

and o.c, is the critical value o f o'eq defined as

[ E ' G , "~1/2 o.cr = ~--"----~ac )

.

(9)

o.cr is a r a n d o m variable due to scatter of the crack size a. We assume that the left-hand tail of the distribution o f o.cr takes a form o f a power function. As will be shown below, this assumption is equivalent to a s s u m p t i o n (4) above. Let H(o.cr ) be the distribution function o f o.,. Then, as far as its left-hand tail is concerned, H(o.c~) is expressed as n(o.cr)

=

__..L

(lO)

where m is a non-dimensional exponent, and fl is a factor with a dimension of stress.

(a) The case of uniaxial tension (o.i = 0, o-2 = o.(>0)) In this case o.. = o. cos 0 2,

~. = o. sin 0 cos 0.

(11)

Then, we have aeq = cr cos 0[~{(1 + 2 sin 2 0) + cos 0(1 + 5 sin 2 0)1/2)] I/2.

(12)

632

M. ICHIKAWA

Consider a small volume element which contains only one crack, and let FI (a) be the distribution function of fracture strength of this volume element. Then, F~ (a) is obtained for its left-hand tail as follows. F l ( a ) = -n

dO

-./2

1 ('./2

/'O-"km 1

=-n 3|-:i2 tfl)[2c°s2 O{1 + 2 sin2 0 + cos O(1 + 5 sin20)}m/~]dO =

A

(13)

where A =

{ ½cos 2 0 (1 + 2 sin 2 0) + cos 0 (1 + 5 sin 2 0)}"/2d0

.

(14)

Since the number of flaws contained in a solid, n, is assumed to be very large (assumption 1), the distribution function of fracture strength of a solid, F(a), is governed by the left-hand tail of F~(a). Thus, F(a) = 1 -exp[-nFl(a)]

= 1 -exp

[ -n

A

.

(15)

This is a two-parameter Weibull distribution. Let a,. be defined as O'c = n -I/raft.

(16)

Then eq. (15) is rewritten as

,,o, 1 ex(

(17)

Equation (17) gives the probability of failure Ps in the case of uniaxial tension. (b ) The case o f equi-biaxial tension (trj = a2 = a( > 0)) In this case, t7 n =

tT,

Zn =

(18)

O.

Hence, O'eqis given by aeq = a.

(19)

The distribution function of the fracture strength of a small volume element containing only one crack, F~(a), is obtained for its left-hand tail as

Hence, the distribution function of the fracture strength of a solid containing a large number of flaws, F(a), is F(a)=l-exp

[ -n

=l-exp

-

(21)

where a,. is given by eq. (16). Equation (21) gives the probability of failure Pi in the case of equi-biaxial tension. (c) The case o f shear (al = --z, a2 = z( > 0)) a, and z, are given in this case by a, = r cos 20,

~, = z sin 20.

(22)

Reliability evaluation of brittle solids

633

From eqs (8) and (22), we obtain aeq

Z[½{1 + 2 sin E28 + COS28(1 + 5 sin 2 20)l/E}]1/2.

=

(23)

Assuming that only those cracks for which a, > 0 are relevant to fracture, the distribution function of the fracture strength of a small volume element containing only one crack, F~ (z), is obtained for its left-hand tail as

where 1 ./2 {½(1 + 2 sin 2 ~b + cos q~(1 + 5 sin z ~b))}m/2d~b B = ~-~ j0

.

(25)

Therefore, the distribution function of the fracture of a solid containing a large number of flaws, F(z), is F(z) = 1 -

exp[- (B ~']" 1 k \ o'UJ

(26)

where ac is given by eq. (16). Equation (26) gives the probability of failure Pj in the case of shear. Figure 4 shows eqs (17) and (26) on Weibull probability paper for the case of m = 15. This value of m is a typical value of Weibull modulus of fine ceramics. The factors A and B given by eqs (14) and (25), respectively, were evaluated by numerical integration. 4. DISCUSSION When eq. (1) is used instead of eq. (2) as the expression of case of uniaxial tension by F(~)=l-exp

Gmax,eq. (17) is replaced for the

[-( A' qq L

\

(27)

°dJ

where A' =

I2 j0I n/2{cos O(l + 4 sin 2 0 cos20)u4}mdO] l/".

, 9:9F °0t so.oF

0.5

/

0.6

,~

0.7 0.8 0.9 1

o/a~, "~/~

l.s

2

Fig. 4. Comparison of the probability of failure P! based on eq. (1) and that based on eq. (2).

(28)

634

M. ICHIKAWA 1.2

~(2) 0.6 0.4

0.8

, I

0.2 o

0

10

',~

\\\

,

. . . . . 20

30

40

"'~ 50

60

70

80

90

(9 (degree) Fig. 5. Comparison of eqs (I) and (2) in the case of uniaxial tension.

Also, eq. (26) is replaced for the case of shear by

[(

F(~)= 1 -exp

-

B'

(29)

where B" =

(1 + 4 sin 2 4~ cos 2 4~)~'/4d4~

(30)

j0 For the case of equi-biaxial tension, use of eq. (1) also gives eq. (21). This is due to the fact that K . = 0 for any crack in this case, so that eqs (1) and (2) coincide with each other. The probability of failure based on eq. (1) was compared with that based on eq. (2) in Fig. 4. It is seen that PI is underestimated except for the case of equi-biaxial tension when eq. (1) is used. That is, use of eq. (1) results in unconservative evaluation of reliability. It is also noted from Fig. 4 that the difference between Pz based on eq. (l) and that based on eq. (2) is more pronounced in the case of shear than in the case of uniaxial tension. In order to find a reason for this, eq. (1) was compared with eq. (2) for these two cases in Figs 5 and 6. It is seen from these figures that the difference between eqs (l) and (2) is much greater in the case of shear than in the case of uniaxial tension. This interprets the more pronounced difference of PI in the case of shear.

20f

1.8

E

q.(.~---~

1.6

z4 1.4

/, , . - - - . Eo#)

1 Oo

'

' ' 3'o' (9 (degree)

Fig. 6. Comparison of eqs (1) and (2) in the case of shear.

Reliability evaluation of brittle solids

635

5. CONCLUSIONS Applying the maximum energy release rate criterion and assuming that fracture strength under uniaxial tension follows a two-parameter Weibull distribution, the probability of failure PI of a brittle solid was calculated for several polyaxial stress states. As the equation of the maximum energy release rate G.... the expression by the present author (eq. 2 in this paper) was used. The result was compared with Pr calculated by using Lamon's expression of Gmax (eq. 1 in this paper). It was found that use of eq. (1) gives unconservative evaluation of reliability, especially in the stress state of shear. APPENDIX: REVIEW OF EQ. (2) Since eq. (2) was derived in a Japanese paper [8], its derivation will briefly be reviewed below for the convenience of readers. Gmax is a function of K~ and Kit, and is represented as a surface in K~-K.~3ma x space. The intersection of Gmaxsurface with the plane of Gmax = K w / E * gives the fracture locus on K t - K n plane by Gmax criterion. On the other hand, Palaniswamy and Knauss [3] showed that the fracture locus numerically obtained using Gm~~ criterion is well approximated by the following parabola: Kt

K,c

3 I'KltXX2

I-~,~)

= 1.

(A1)

We derived eq. (2) using eq. (Al) in the following way. Equation (A1) can be rewritten as: KI

3//

Ktt

"~2

=1

(A2)

where G m = K2c/E *. It is important to note that eq. (A2) holds for any value of Gw( > 0). Hence, replacing G~c in eq. (A2) by Gmax, we obtain ~

Kl + }

[

3[

~

Kn )

"X2

=1.

(A3)

Equation (A3) represents the intersection of Gmax surface with a plane Gm~x = const. From eq. (A3), an analytical expression of Gm~xcan be obtained as: 1

Gmax= 2 ~ [KI2+ 3K~, -{- KI(K ~ + 6K~0 '/21

(A4)

which is eq. (2) in this paper. Since eq. (A1) by Palaniswamy and Knauss is approximate, eq. (A4) is also approximate. However, eq. (A4) gives almost accurate values as shown in Fig. 2.

REFERENCES [1] E. Erdogan and G. C. Sih, On the crack extension in plates under plane loading and transverse shear. A S M E J. Basic Engng g5, 519-525 (1963). [2] G. C. Sih, Strain energy density factor applied to mixed crack problem. Int. J. Fracture 10, 305-321 (1974). [3] K. Palaniswamy and W. G. Knauss, On the problem of crack extension in brittle solids under general loading. Mechanics Today (Edited by S. Nemat Nasser) Vol. 4, pp. 87-148. Pergamon Press, New York (1978). [4] J. Eftis and H. Liebowitz, On the modified Westergaard equations for certain plane problems. Int. J. Fracture Mech. g, 383-392 (1972). [5] J. Eftis, D. L. Jones and H. Liebowitz, Load biaxiality and fracture: synthesis and summary. Engng Fracture Mech. 36, 537-574 (1990). [6] J. Lamon, Statistical approaches to failure for ceramic reliability assessment. J. Am. Ceram. Soc. 71, 106-112 (1988). [7] T. K. Hellen and W. B. Blackburn, The calculation of stress intensity factor for combined tensile and shear loading. Int. J. Fracture 11, 605-617 (1975). [8] M. Ichikawa, Proposal of an approximate analytical expression of maximum energy release rate of a mixed mode crack in relation to reliability evaluation of ceramic components. J. Soc. Mater. Sci. Jpn 40, 224-227 (1991), (in Japanese). [9] K. Kageyama and H. Okumura, Elastic analysis of infinitesimally kinked crack under tension and transverse shear and the maximum energy release rate criterion. Trans. Jap. Soc. Mech. Engnrs A48, 783-791 (1982), (in Japanese). [10] Y. Matuo, Probabilistic analysis of brittle fracture loci under biaxial stress state. Trans. Jap. Soc. Mech. Engnrs A46, 605-612 (1980), (in Japanese). (Received 20 December 1993)