Rigorous formulation of crack path in two-dimensional elastic body

Of 1"1f111111 ELSEVIER

Mechanics of Materials 26 (1997) 1-14

Rigorous formulation of crack path in two-dimensional elastic body Muneo Hod *, Navarathnam Vaikuntan University of Tokyo, Earthquake Research Inst., 1-1, Yayoi 1-chome, Bunkyo-ku, Tokyo 113, Japan Received 25 July 1996; revised 24 January 1997; accepted 5 February 1997

Abstract

A rigorous formulation is proposed for the prediction of a path along which a crack propagates smoothly in a two-dimensional elastic body. In this formulation, the curvature and length of a small crack extension are determined when an increment of external load is given. The major features of the proposed formulation are to clarify conditions for the change in the displacement due to the crack growth and to explicitly pose a boundary-value problem for the change in the displacement. It is proved that the effects of the crack growth on the field variables are equivalent to a pair of concentrated forces acting at the crack tip. The change in the displacement yields an explicit expression for the change in the stress intensity factors in terms of the curvature and length of the extension, and these geometrical parameters can be determined from the assumed fracture criteria. This differentiates the analysis based on the proposed formulation from a conventional analysis which takes a trial-and-error approach to determine the crack extension path. The validity of the present formulation is examined by studying crack problems which have analytic solutions. The present formulation is applied to a problem of arbitrary plural crocks which grow smoothly while interacting with each other. © 1997 Elsevier Science Ltd.

1. Introduction

Propagation of a crack is a phenomenon that leads to brittle failure of materials. Therefore, the analysis of crack growth has been a major subject of fracture mechanics since Griffith's celebrated work (Griffith, 1921); see, for instance, Rice (1968), Sih (1972), and Erdogan et al. (1973). In particular, it is often necessary to predict the transit from stable to unstable growth or the occurrence of kinking and branching. A great deal of research has been done for various situations and materials; Lo (1978) studied branched

* Corresponding author. Tel.: +81-3-38122111; fax: +81-338161159.

cracks; Cotterell and Rice (1980) and Karihaloo et al. (1981) analyzed a curving and kinking crack in isotropic materials, and Gao and Chiu (1992) for anisotropic elastic solids; Sumi (1986) proposed a general formulation of a branched or curved extension (see also Palaniswamry and Knauss (1978), Nemat-Nasser et al. (1978), Sumi et al. (1980)); and Xu and Keer (1992) found an asymptotic solution of fields near moving crack tip (see also Freund, 1990; Willis and Movchan, 1995; Movchan and Willis, 1995). The key issue that is common in these analyses is the accurate prediction of a path along which a crack propagates, since advanced numerical computation techniques enable us to evaluate field variables and resulting fracture parameters to any desired accuracy, once the configuration of the crack is speci-

0167-6636/97/$17.00 Copyright © 1997 Elsevier Science Ltd. All rights reserved. PI! S 0 1 6 7 - 6 6 3 6 ( 9 7 ) 0 0 0 0 8 - 2

2

M. Hori, N. Vaikuntan/Mechanics o f Materials 26 (1997) 1-14 crack length a

crack length a+da

u given

u given

displacement field u

displacement field u+du

Fig. 1. Crack O before and after extension.

fled. The target of this paper is such a crack-path problem, the prediction of the path of a propagating crack. A crack-path problem is usually solved in an incremental manner, i.e., a small extension of a crack is determined for a given increment of external loads, and this process is repeated until the crack becomes sufficiently long or grows in an unstable manner; see, for instance, Sumi (1986). The major difficulty in this procedure is that fracture parameters depend on the geometry of the small extension only in an implicit manner. These parameters must be evaluated from numerical computation, since it is impossible to solve a crack problem analytically such that the field variables are explicitly expressed in terms of the extension geometry. Hence, a trialand-error approach is usually taken, and a most suitable small extension is found by solving boundary-value problems for various extensions and examining fracture criteria for the fracture parameters. Besides the accuracy of the prediction, such a trialand-error approach requires a great deal of numerical computation. In particular, the amount increases exponentially ~ as the number of cracks being analyzed increases. In this paper, we propose a rigorous formulation of general crack-path problems. The curvature and

i If M sets of geometrical parameters are examined for each of N cracks, N M cases must be computed to find the most suitable extension.

length of a small extension can be determined without taking a trial-and-error approach. In the proposed formulation, a boundary-value problem is posed for the change in a displacement field that is induced by the crack extension. To demonstrate the existence of such a boundary-value problem, we consider a twodimensional elastic body V, which contains a crack 2 /2; see Fig. 1. The tip of O, denoted by C, moves smoothly as it propagates. The elasticity tensor 3 and the crack length is a. Assuming that V is subjected to surface displacement ~ on the boundary OV, and the surfaces of O are traction-free, we have a boundary-value problem for displacement u, given as

CijklUk,li ( X )

= 0

X in VXx~2,

u,(x) = ~,(x)

x on a v ,

ti( x ) =O

x on O ,

(1)

where t = n . tr is the traction with n being the outer unit normal. As the crack propagates, the displacement changes accordingly. Denoting this change by d u, we have another boundary-value problem for

2 12 stands for a region of the crack, or its upper and lower surfaces, depending on the context. 3 A tensorial quantitity is denoted by a bold-face letter, and its component by the corresponding letter with subfix; for instance, u and o" are displacement and sl~ess, and u i and trij are components. For symbolic notation, inner product is denoted by.. For index notation, the summation convention is employed and an subfix following a comma stands for partial differentiation.

M. Hori, N. Vaikuntan/ Mechanics of Materials 26 (1997) 1-14

the displacement after the extension, u + d u, given as

Ci~l( u,,.( x) + duk,.( x)) = 0

x in V \ ( O +

Ui( X) + dui( X) = "ui(X) t a x ) + dry(x) = 0

x on OV, x o n /2+ d ~ ,

2. Displacement rate field

where d O stands for the extension of length da, and dt = n • d o ' , is the change in the traction associated with d u. The following equations for d u are derived from the difference between Eqs. (2) and (1): x in V \ ( O +

dui( x ) = 0

x on OV,

dti( x ) = 0

X on /2,

while interacting with each other and to a circular hole which evolves in an axi-symmetric manner.

dO),

(2)

Ci#ldUl~,li(X ) -~0

3

To explore the non-trivial condition on the crack extension, we focus our attention on traction there. To this end, we first define a displacement rate, the rate of the displacement change 5 with respect to the increasing crack length. In the example shown in Section 1, the displacement rate, d , is defined as

dl2),

1 (3)

If d t vanishes on the extension d O , the displacement change becomes trivial. Therefore, we can expect that there exist some non-trivial conditions on d O . As d a decreases, such conditions are reduced to a condition at the outer crack tip. The objectives of this paper are to explore the non-trivial condition for the change in the displacement due to the crack extension, and to rigorously formulate a crack-path problem using this displacement change. For simplicity, we consider 4 a two-dimensional elastic body in a quasi-static state and assume ordinary fracture criteria, though the configuration of the body as well as the geometry of the smoothly propagating crack are arbitrary. The content of this paper is as follows. The non-trivial condition for the displacement change is identified in Section 2, and it is shown in Section 3 that displacement change satisfies a boundary-value problem with this non-trivial condition. A set of linear equations for the curvature and length of a crack extension is derived from the assumed fracture criteria. In Section 4, the validity of the proposed formulation is demonstrated by studying crack-path problems which have analytic solutions. Finally, in Section 5, the present formulation is applied to plural cracks which grow

4 The formulation can be applied to a more general setting; it is directly applied to a non-linear elastic or elasto-plastic body, or to a case when inertia and body forces are included. A three-dimensional crack-path problem requires some modification in treating the geometry of crack surface extension.

d(x) =

lim

da--* 0 ~aa

du(x)

(4)

The stress associated with u' is denoted by o.'. Next, we specify the configuration of the propagating crack and determine the unit normal on the extension. As shown in Fig. 2, the crack path and the extension can he expressed in terms of a common vector-valued function ~b, as

12=(q~(s)l-a
dl2={dp(s)D
Without loss of generality, we can set ~bI = s for 0 < s < d a and choose the x t, x2-coordinate system such that the origin coincides with the crack tip C, and the x : a x i s becomes parallel to the orientation of the crack at C, i.e., [~:l

:

[ (jl~l1,l :

[0],

at s - - 0 . Note that 0 ' stands for a derivative with respect to s. In terms of 4~2 the unit tangent and normal, s and n, are expressed as 1

1

[-:] •

(63

a The displacement rate is the same as the derivative of di., placement with respect to the crack extension, which was consic ered by Sumi et al. (1980). While they studied a straight crac extension, the present study considers an extension of an arbitra ily smooth configuration.

4

M. Hori, N. Vaikuntan/Mechanicsof Materials26 (1997) 1-14 xl

Now, we consider traction on the extension. Since the stress after the extension is or + or'da + . . . , the traction just to the left of the tip of O + d/2 (s = d a), i.e., at x = qb(da) + rs(da), is written as

[K'XAr

xl infinitesimally small extension

ni(da)nj(da)( orij( x ) + o'~j( x) da + . . . ) . We expand this expression with respect to da, as n,(O) nj(O) ,~,j(x) - (,,,(0) ,,j(O) o',j.A x)(e,'AO )

_

L r stress singularity oft -~ linear to [K,.K,,]r

+ rS'p(O) ) + 2nti(0) nj(O) O'ij(X)

-{- ni(O ) rlj(O) % ( x ) ) da + . . .

l

In view of Eqs. (5) and (6), the term within the bracket is ~( - 2 ~ 1 ( x ) + r,~22,,(x)) + ~ 2 , 1 ( x ) + ~ h ( x ) , (7) where K = d24,2/ds2(0 +) = 4,~(0 +) is the curvature of the extension, and superscript + on C or 0 emphasizes that the quantity is measured at the outer tip of the crack 12. Similarly, the tangential component of traction at the outer crack tip, ni(da)sflda)(o'ij(x) + o'i~(x)da + . . . ), is expanded, and the coefficient for da is K(or22(X

) -- O ' I I ( X )

+rcr21,1(x)) + cr21,1(X )

+

(8)

It should be noted that Eqs. (7) and (8) include terms such as o-2j,l, which are added to or'2j due to the change in the n and s. If the mode I and II stress intensity factors at the tip crack are K l and K~I, the leading terms of o-2j,/ near the crack tip are

=

(

[K, l

~

tF/'(s (o )o~(tP(s))

C

d[2

x~

Fig. 2. C r a c k path and traction on extension.

crack driving' " forces.'~-[-'-~ V

r

-

[P2.P,i linear to

[KnKu]"

Fig. 3. N o n - t r i v i a l condition at outer c r a c k tip a n d c r a c k d r i v i n g

forces at inner crack tip. This [O'21,1, O'22,1] has a singularity 6 of Xl 3/2, which is not physically admissible, in the sense that it procedures and unbounded strain energy at the tip of the extended crack. The high singularity ought to be canceled out by the third term in Eqs. (7) and (8). Therefore, the following non-trivial condition for the displacement rate is derived from Eqs. (7) and (8): l i m , "3/2[°'~l] r--*0 + "

1 =

[Kxx] g I

(9)

"

To examine Eq. (9) in detail, we consider a semi-infinite straight crack in an unbounded body; see Fig. 3. The crack lies on the negative xl-axis, with the tip being located at the origin, and the crack surfaces are traction-free. If the stress intensity factors are K 1 and K n, this problem has an analytic solution, and Westergaard's potentials (Westergaard (1939)) are [Z l, Zn] T = 1 / ~ / 2 ~ z [ K l, Kn] T, where z = Xl + ¢x2 with L= ~/- 1. When the crack propagates on the xl-axis by da, the corresponding potentials become [ Z l ( z - d a), Zu( z - d a)] T. Hence, tak-

fundamental

6 T h i s h i g h singularity is the s a m e as that o f the W i l l i a m s (1957); see also S u m i et al. (1980).

solution o f

M. Hori, N. Vaikuntan / Mechanics of Materials 26 (1997) 1-14

ing the derivative with respect to da, we can obtain the following potentials:

8(g- z-

[KH"

While this [Z[, Z~I]T is derived from the differentiation, it is associated with particular boundary conditions. Indeed, the boundary conditions are a pair of concentrated forces of the same magnitude but the opposite direction, which act at the upper and lower surfaces of the inner tip of the semi-infinite crack; see Fig. 3. To show this, we use Westergaard's potentials for a crack with concentrated forces + P acting at - b

z,

[e,l.

PI ]T with -- ~ [ K 1, Kn] T, we can express [Z~, Z~i] given by Eq. (10) in terms of these potentials, as Replacing [P2,

--

tZ~l]

-

,im

1 -,/2( +b)-,[ K'] z

b--* 0 8---~-'~ z

Kn

•

Therefore, [Z~, Z~I] is for a semi-infinite crack with a pair of concentrated forces acting at the inner tip. It should be noted that the magnitude of the forces P diverges. These diverging forces are called crack driving forces in this paper. Since the fields near the tip of /2 coincide with those of the semi-infinite crack with the same stress intensity factors, it is seen that the non-trivial condition, Eq. (9), is equivalent 7 to the crack driving forces, i.e. the limit of concentrated forces P at ~b(-b) as b ~ 0. The magnitude of P diverges while satisfying

5

Using the crack driving forces instead of Eq. (9), we can pose the following boundary-value problem for the displacement rate: CijklUlk,li( X) -~ 0

X in V \ / 2 ,

Uti( X) = 0

X o n 0V,

4"(x) = 0

x on/2\ C-,

ti( X ) ' = Pi

X at C-.

(12)

This boundary-value problem has a unique solution; see Appendix A.

3. Formulationof crack path problem Now, we consider a formulation of general crack-path problems using the displacement rate. Due to the smooth propagation, the direction s of the crack before and after extension must be continuous, i.e., ~b[(0)= 0; see Appendix B for an analysis of kinking which admits the discontinuity of ~b[. To determine the crack configuration 4,2, therefore, we consider the curvature ~b~(0+) = r , as well as the length d a. The crack-path problem is then stated as follows: Let /2 be a crack in a body V. When /2 propagates smoothly due to some increment of the external loads, determine the curvature and length of the crack extension. Fig. 4 illustrates this problem. For simplicity, the following boundary conditions are set on OV on /2,

ui(x) =fii(x; T)

x on dV,

ti(x) = ~ ( x; T)

x on /2,

(13)

Kx ]"

where fi and .f are external displacement and traction, and T stands for a loading parameter. A simple

7 AS seen, we derive the crack driving forces in a purely mathematical manner, and they are different from the configurationalforces which were considered by Gurthin and Podio-Guidugli (1996).

s Conventional analyses use the orientation of the small line extension. While the present formulation uses the curvature, the orientation change after the extension is naturally expressed in terms of this curvature.

P2

b-',0 It 8b

6

M. Hori, N. Vaikuntan/Mechanics of Materials 26 (1997) 1-14 fred curvature and length of extension d.Q

stress intensity factors of the extended crack are given as + Kill d a 1

"ni(da)nj(da)(°'iJ(X) q" (rij(x)da) ] ni(da)sj(da)( ~rlj( x) + 6"ij( x)da ) J '

= lira ~

r-*O

where x = 4)(0) + rs(O). In the left side, [K'I, K'n]v is the stress intensity factor rate due to the crack extension. Taking the limit of da ~ O, we can compute [K'x, K'II] v a s u0,) given

K,n = K

Fig. 4. Crack path problem.

1 I ~K

(16)

+ [K,,*, I '

where [ K I * ' , KI~' ] is defined 1o as = lira ~ fracture criterion 9 is set for the Mode I stress intensity factor at the crack tip, i.e., K I = Kc,

(14)

where K c is the fracture toughness. The target of the formulation is to clarify the relation between the change in the fracture parameters and the geometrical parameters. The effects of the crack extension on the displacement change are represented by the displacement rate. Since fi and j? are given, the displacement rate satisfies the same boundary-value problem as Eq. (12), i.e.,

in V \ g2

CijklUPk,li( X ) = 0

X

di( x) = 0

x on OV, x on O \ C - , x at C-.

4(x) =0

fi( x) = Pi

(15)

While r does not appear in Eq. (15), it changes the singularity of the stress near the crack tip after the extension. Indeed, in view of Eqs. (7) and (8), the

[ KI~' j

r--*0

°'22a + o.22

In Eq. (19), the first term, ~ [ ( - 3/2)KII, ( 1 / 2 ) K I ] T, represents the change in the stress intensity factors due to curving, while the second term, [Kl*', KI~']T, gives the contribution of the crack driving forces. It should be noted that Kli and K I appear K~ and K~, respectively. If the crack does not propagate, the effects of the increase of the external loads on the displacement change are easily evaluated by considering a case when ~ e boundary conditions change by ~ d , / o n OV and f d y on ~ . The resulting increment of the displacement can be expressed as t i d y , where ti, called displacement increment, satisfies the following boundary-value problem:

Cijklf~k.t( X) = 0

X in V,

f~i(x) = ~ i ( x ; T)

x on OV,

ii(x) =3~.( x; T)

x on g-2.

(18)

Here, a dot emphasizes that the quantities are associated with the increment of the boundary conditions. The increment of the stress intensity factors is defined from 6- associated with ~, as = lira ~ /~11

9 The present formulation is applicable to more complicated boundary conditions on 3V or ~ . It is also applicable to another ordinary fracture criterion, such as a critical value for an energy release rate (G = G¢) or a J-integral ( J = Je), if it uses fracture parameters determined from the stress intensity factors.

(17)

O"21,1 "[- O.21 "

r'~ 0 +

(19) 6-21(X)

"

t0 The limit in Eq. (17) is bounded since the singularity of - 3 / 2 is canceled.

M. Hori, N. Vaikuntan / Mechanics of Materials 26 (1997) 1-14

Since the effects of the external load increment and the crack extension are separately represented by the displacement increment ti and the displacement rate d , we can obtain the total change in displacement, d u, in terms of ti and u'. Denoting the speed of the crack extension with respect to y by ti = l i m d y _ , 0 d a / d y , we can express d u as

du(x)=it(x)d2/+d(x)(?t('y)d'y).

(20)

The change in the stress intensity factors, [dK~, dKn] r, is then determined from Jig I, /(H] r and [K'I, K'u ]r in the same manner as in Eq. (20), i.e., dK I =

+

K'

LK".i or

=

dKll

/i~tl

dy-

l ~KI

(Ktidy)

[KI*'] + [ KI;,] (~idT).

(21)

Eq. (21) provides a relation between the stress intensity factor change and the geometrical parameters of the extension. It should be noted that Kfid T ----K d a gives the change in the crack orientation before and after propagation. The following condition for K and t~ is immediately derived from Eq. (14): /~1-- ~KII 3 ( K a ) + Ki*'ti ------0.

(22)

Another condition is required to determine two unknowns, K and d. In conventional analyses, the condition of (I) m a x i m u m K I or (2) zero K u is used to determine the orientation of the line segment; see, for instance, W u 0979) and Hayashi and Nemat-Nasser (1981). Since d K I is linear to x, the first condition ~I cannot be met. The second condition leads to I

/~II -[" ~ K I ( K t J ) + KI~'t~ = 0.

(23)

t t Appendix C presents an analysis which uses higher-order perturbation terms. It is shown that the stxess intensity factor change can be maximized/minimized with respect to K if we consider terms linear to (dy) 2.

7

Eqs. (22) and (23) are a set of linear equations for K and ti, and the coefficients, [/~l, /~n] T and [KI*', KI~'] r, can be obtained by solving the two boundary-value problems, Eqs. (18) and (15), without taking any trial values on x and ti. If K n = 0 is used, therefore, we can uniquely determine these geometrical parameters from Eqs. (22) and (23); see Appendix D for an analysis of determining Kti and ti which considers the stability of the crack growth.

4. Crack path problem with analytic solution To demonstrate the validity of the proposed formulation, we consider crack-path problems which have analytical solutions; V and /2 are an infinite body and a slit crack of length 2 a, simple boundary conditions are prescribed on OV and /2, and /2 propagates without curving. The effects of curving or the curvature are not examined in this paper, since it requires numerical computation to solve a boundary-value problem for a curved crack. An accompanying paper (Hori and Vaikuntan, 1996) reports on the numerical analysis of a crack forming a smooth arc, and shows that the displacement rate and the stress intensity factor rate determined from the proposed formulation coincide with those which are numerically computed by using a finite-difference technique; see also Vaikuntan (1996). 4.1. Crack with traction f r e e surfaces

As the simplest example, we consider a case when the surfaces of /2 are traction-free and V is subjected to the far-field stress o'~; see Fig. 5. Westergaard's potentials are ZI =

z

0022 + 2 00

ZII

00n ~

0012

__

°'22 oo

.

Due to the symmetry, both tips of O propagate the same amount, and the corresponding potential rate is obtained by taking the derivative with respect to a, i.e.,

]=az(z P'I,J

2

-

a2

)-3/2L 00,2j

.

(24)

First, we show that [ P~, P~I] given by Eq. (24) are the potentials for a crack which has crack driving

8

M. Hori, N. Vaikuntan/Mechanics of Materials 26 (1997) 1-14

infinitesimallysmallextemion

forces acting at both tips. When ~ is subjected to a pair of concentrated forces P at b, Westergaard's potentials are

ZII

7r(Z-- b ) ~

oncentratedforces

P1 "

Since the stress intensity factors of ~ are [K I, KII]T = ,1["~'[O.~2' Oq2] " T, the magnitude of the crack driving forces is determined from Eq. (11), and the potentials for the crack driving forces are given as lira

- - ~

8(a-b)

b-.a ¢ r ( z - b ) ~

+ lim

~-b2

{

b-~-a ¢ r ( z + b ) ~

Kn "x [ KI] 8 ( a + b ) Kn

crackdriving forc~ Fig.6.Crackwithpairofconcentrated forces.

K,,, ] . = a z ( z 2 - a 2 ) - 3 / 2 iqr1a [ K These potentials coincide with Eq. (24). Using the displacement rate determined from [ P~, P[I IT, we study the stress intensity factor rate. On the xl-axis near the crack tip, o'2j,i + 6-2i is computed as

O"21,1+~21] ~ l

_1/21[KIll 2¢T;" 2a[K,] + "

By definition, Eqs. (16) and (17), the stress intensity factor rates are evaluated as Kn] = ' ~ a

K. "

These rates coincide with the derivatives of [K I, K , ] T = ~q"~a-[tr~2, o~2]r with respect to a, i.e.,

infinitesimallysmallextension

-7 dr/ G d.O / tracti°nfreesurfaces

4.2. Crack with concentrated forces on surface

Next, we examine a crack with a pair of equilibrating concentrated forces acting on its surface; see Fig. 6. The solution of this problem 12 serves as Green's function for a body with one crack. If the magnitude and location of the forces are P and b, Westergaard's potentials are given by Eq. (25). When propagates from, say, the right tip, the potential

crackdrivingf o r c e ~

Fig. 5. Crackwith tractionfree surfaces.

12A crackwithdistributedforcescan be solvedby superposition of these concentratedforces. Furthermore, a case when V has plural cracks can be solvedby using a similarsuperposition;see Horii and Nemat-Nasser(1983).

M. Hori, N. Vaikuntan/ Mechanics of Materials 26 (1997) 1-14 rate is obtained by taking the corresponding derivative 13, as

P;,]

(z-a)~z2--a

2 2~rV a - b

P2 " (27)

Since the stress intensity factors at the right tip of Oare

¢

[;:]'

making use o f Eqs. (I1) and (25), we can obtain potentials for the crack driving forces as lim

8(a----c)

( z - a)z vrSY-J-a

Kt,

K.

As is seen, these potentials coincide with those given by Eq. (27). Furthermore, the stress intensity factor rate is computed from Eq. (17) as

--a-

produces /¢i > , the crack considered in Section 4.1 grows in an unstable manner, while the crack considered in Section 4.2 grows in a stable manner. It should be emphasized that the stress intensity factor rate can be computed from the displacement rate without taking any trial for the extension geometry. The accuracy o f the computation of the stress intensity factor rate could not be worse than that of the stress intensity factor, when the same numerical techniques are applied to solve the boundary value problems of the displacement rate and the displacement and to compute the stress intensity factor rate and the stress intensity factor, respectively.

5. Application of proposed formulation

. . . .

c~a "Ir( z - - c ) ~

rn ]

9

One essential feature in the proposed formulation is to identify the non-trivial condition for the displacement rate, Eq. (9), which is derived by taking perturbation ~4 of the traction at the crack extension. The perturbation technique can be applicable when a body contains plural cracks, or even to the evolution of other defects. This section presents two examples o f such applications of the proposed formulation.

-b=)-s/2(a+b) 5.1. Plural cracks

X(a2-2ab-b2)

[ PP2 '] "

(28)

which coincides with the derivative of [ K I, K n ]r. The two crack problems considered in Sections 4.1 and 4.2 are typical examples of unstable and stable growth; the stress intensity factor rates are positive in Eq. (26) and negative in Eq. (28). Hence, if a small far-field tensile stress is applied and

is Denoting the fifi_~tand left tips by a r = + a and a I = - a, we can rewrite 1/a2 - b 2 and l~z2 - a 2 in Eq. (25) as ¢-(ar-b)(al-b) and ¢ ( z - - a t ) ( z - - a l ) . Hence, the derivative with respect to a r is given as

[zq z~] = -(1/~/-g)( ¢ ( a ' -

d)/2 )

We consider a case when V contains N plural cracks, {O ~} ( a = 1, 2 . . . . . N), which interact with each other (Fig. 7). For simplicity, we set the same boundary conditions on £2a's as considered in Section 3. Each O ~ may propagate smoothly as the external loads increase, and the curvature and length of its extension need to be determined. As a whole, there are 2 N unknown geometrical parameters, {x ", ti ~} (a = 1, 2 . . . . . N). Therefore, a conventional analysis which takes a trial-and-error approach is not suitable, since there are numerous combinations for the geometrical parameters; see Footnote 1 in Section I. According to the present formulation, we can pose a boundary-value problem for the displacement rate field that is due to the extension of say, the a th

× ( Z - al)- l /2 ( z - ar)- s/2[ KIKll] . If a r and a j are taken as 'a and - a, respectively, this [Z~, Z[I]"r coincides with Eq. (27). Note that [Z~, Z[I]r can Ix: used to determine the displacement rate when O is finite.

~4See also Appendix C for more general perturbation of field variables.

10

M. Hori, N. Vaikuntan / Mechanics of Materials 26 (1997) 1-14

threerate fieldsforextensionofthreeeracks

the total change t5 in the stress intensity factors of g2 ~ is obtained as dKl

Kl =

-7

I1

d'y-

,~.~ (k

~gl *'a']

a

dT)

•

+ ~ [K,;,~Pl(at3 d T ) .

If the fracture criteria for each ~ `~ are, say, [K~, K~] T = [Kc, 0] T. a set of 2 N linear equations for {K '~, fi~} is immediately derived from Eq. (30). Therefore we can determine these geometrical parameters, making a stability analysis for the growth of each g2 a, which is essentially the same as that shown in Appendix C.

u(~,)given Fig. 7. Propagation of interacting plural cracks. crack, ~ % This displacement rate is denoted by u' % Using Eq. (11), we can determine the crack driving forces which act at the tip of ff]~, denoted by C '~, in terms of the stress intensity factors there, [ KT, KI~ ]T. Then, the boundary-value problem for u '~ is CijklUtkali ( X) = 0

X) = 0

x in V \

u~2 a,

x on OV, x on U , O ~ \ C

=0

t;a( X) = Pi ~

~-,

(29)

x at C a - ,

where t '~ = n • or '~ is the traction associated with d a, and P~' is the crack driving force. A boundary condition for the displacement increment ~ is essentially the same as Eq. (18). The singularity of 0 "'~ at the tip of OP, denoted by C t3, is r - x / 2 for / 3 q : a or r - 3 / 2 for / 3 = a . Therefore, or '~ contributes the change in the stress intensity factors at C a , as follows:

[[[KI*'O"] = [Kti'Oa J

limr~0 2V/2V/2V/2V/2V/2V/2t°'2~J ~ ~ l i m , , o 2~'-~r

[[ O' 22'1+ O'6~] o.2,., + o.~7 J

(30)

for/3~a, for/3= a,

5.2. E v o l v i n g c a v i t y

The perturbation technique which is used in Section 2 can be applicable to the traction at a boundary which moves smoothly, and a non-trivial condition for the associated rate field can be identified. As the simplest example, we examine the evolution of a circular hole. For the displacement rate due to the evolution, we identify a non-trivial condition at the hole boundary, and pose a boundary-value problem. To use an analytic solution, we consider that a hole H is located in an infinite body B, assuming that B is subjected to far-field hydrostatic tension tr o, and H has traction-free surface; see Fig. 8. Due to the axi-symmetry, the radial displacement is 16 given as =

--

r

,

(31)

where a is the radius of H, and r is the distance from the center of H. When H evolves with increasing a, the rate of the radial displacement is obtained by taking the derivative of u r with respect to a, i.e., Or _U_ _ . . _ ( )

=

--

(1 - v ) o -° 2 a

Oa - r -

where the components are measured in the local coordinate of $2 t3 (the local x r a x i s is set parallel to the orientation of /'2 P). The stress intensity factor increment of ~'~, denoted by [ / ~ , /~i~]T, can be computed from u' in the same manner as Eq. (19). Therefore, if the extension of f2 '~ is d a '~= fi'~dy,

r-

E

E

r

~5The total change in displacement is du = ~idy +Ea d~(a~d~).

16Plane stress state is assumed.

(32)

M. Hori, N. Vaikuntan / Mechanics o f Materials 26 (1997) 1-14

infinitesimally small ovolution

11

The displacement rate then satisfies

d2

d

dr 2 Pur(r ) + r - l -~rU'r(r) - r-2UPr(r)~O

a< r

trr'r( r ) -- 0

r -o oo

2 ~r'r(r) m - or o _ a

r ~ a +.

(34)

non-trivialc o n d i t i o n ~ at outerb o u n d a r y ~ Fig. 8. Evolving circular hole.

Now, we consider a boundary-value problem for the displacement rate ti, which is due to the evolution of H. When the radius becomes a + da, the change in the radial traction at r = a + d a is expressed as nr( a + d a ) ( trrr( a + d a ) + trr'r( a + d a ) d a ) ,

where t r ' is the stress rate associated with u'. Expanding this traction at r - - a, and taking the limit as d a -* O, we can compute the traction change as O'rr.r( a + r ) + trr'r( a + r ) .

This traction change must vanish at r = 0, if the surface of the evolving cavity remains traction free. Therefore, using O'rr= tr°(1 -- ( a / r ) 2 ) , we can derive a non-trivial condition 17 for the displacement rate, as follows: 2 O'/r = -- tr o _ a

r = a +.

(33)

t7 This condition at the hole boundary corresponds to the nontrivial condition at the outer crack tip, Eq. (9).

The solution of this boundary-value problem (Eq. (34)), l~r(r) = --((1 -- u ) ~ r ° / E ) 2 a / r , coincides with Eq. (32). It is demonstrated that the perturbation technique which identifies the non-trivial condition is applicable to the above simple example of an evolving hole. Principally, we can apply this technique to other problems which involve a moving boundary; for instance, the evolution of defects such as phase transformations (see Abeyaratne and Knowles, 1993), or the extension of regions which reach an inelastic state in an elasto-plastic body. These problems belong to the Stephan problem or the moving boundary problems. Therefore, the present formulation, which determines a boundary-value problem for the rate fields with non-trivial conditions at the moving boundary, can provide a new tool to solve this class of problems.

6. Concluding remarks Compared with conventional analyses of a crack path problem, the proposed formulation is more efficient since it only requires solution of the boundaryvalue problems for the displacement rate and the increment fields; an explicit relation between the change in the fracture parameters and the curvature and length of a crack extension is derived, and a set of linear equations for these geometrical parameters is obtained by using the assumed fracture criteria. Stability analysis of the crack growth becomes transparent and straightforward, since no trial-and-error is needed for the crack extensions. Therefore, the proposed formulation is particularly suitable to solve crack-path problems for plural cracks which grow while interacting with each other. One essential feature of the proposed formulation is to identify non-trivial conditions for the rate field

12

M. Hori, N. Vaikuntan / Mechanics of Materials 26 (1997) 1-14

by taking the perturbation for the traction which acts at the surfaces of a propagating crack. This perturbation technique can be applicable to other problems which involve a moving boundary, such as the growth of cavities of phase transformations. The analysis based on the proposed formulation could provide a new tool for solving this class of problems.

Appendix A. Boundary condition of crack driving forces

When a boundary-value problem (Eq. (12)) is solved numerically, some attention needs to be paid to the high singularity that is caused by the diverging crack driving forces at C-. Westergaard potentials defined by Eq. (10), [Z~, Z~I]r, can be used to account for the high singularity. Choosing the branch cut for z -3/2 to coincide with /2, we can compute the displacement due to these potentials, u ¢. Superposing fields associated with this u ~ on Eq. (12), we have ordinary boundary conditions for u ' - u *, i.e.,

( d i - u~)( x) -- --u~( x) f i - t~( x) = -t~( x)

x o n aV

trivial condition for ti replaces Eq. (9), and the crack driving forces equivalent to this condition are found, although the form becomes slightly different from Eq. (11). In the same manner as Eq. (19), we can compute the stress intensity factor rate using the above traction rate. The resulting rate, which becomes non-linear to tan 0 and x, does not depend on the length of the kinked crack extension. This is consistent with Azhdari and Nemat-Nasser (1996), who showed from numerical computation that the stress intensity factors at the tip of a kinked crack take on constant values for an extension of sufficiently small length.

Appendix C. Higher-order perturbation

To determine the value of K that maximizes/ minimizes the stress intensity factor rate, we take higher-order terms in the perturbation of the traction. The second derivative of the displacement change with respect to the crack extension is denoted by d', and the second derivative of the unit normal is

x on f2,

where t c is traction associated with u c. Since t ~ is finite on O, this boundary value problem for d - u c has a unique solution. Therefore, we can determine i/ uniquely from Eq. (12). For a finite crack, Westergaard's potentials for the crack driving forces are obtained in Section 4.

n'~ =

--K2--p '

at s = 0, where p = d3tk2/ds3(0+). Since the stress at x = tk(da) + rs(da) is expanded as

trij + (o'ij,p ( dp; + rs'p) + o'i~)da "Jt-l((orijopq(t~p

+re, l ( , ; + rS'q)

+o'iy,e ( qb'~+ re;)) + o'i~

Appendix B. Analysis of kinking

+ re; ) ) ( d a) 2 + The discontinuity in the crack orientation corresponds to kinking. If ~b[ is discontinuous at s--O, i.e., ~b[(O-) = 0 and ~ ( 0 +) #= O, then, this tk[(O +) gives the angle of kinking, as O= tan-lqb[(O+). In this case, n' becomes n'l 1 = ( 1 + ( th[)2)-3'2[

-tk~

....

we can compute terms linear to d a 2 in this traction. A non-trivial condition for u" is derived from this expression such that the high singularity of the stress derivatives is canceled out by o-~.

] Table 1 Crack growth state for K a = 0

and the traction rate at C + will become more complicated. In particular, the high singularity of o'2z 1 in Eqs. (7) and (8) is replaced by that of O'gj,1 + ~ ( O ' l j , i -- 0"2j,2) dr (~)20"1j,2 , and that this singularity must be canceled out by tr6j - th[ 6"tj. This non-

KI*' < 0 Kl° = 0 K~*'> 0

/71 < 0 no growth

/(l = 0 RI > 0 no growth stable growth not determined unstablegrowth

no growth stablegrowth unstablegrowth unstablegrowth

M. Hori, N. Vaikuntan / Mechanics of Materials 26 (1997) 1-14

13

Table 2 Crack growth state for K n ¢=0

K I Kr*' - K u KI]' < 0

K l KI*' - K n Kl~' = 0 K ! K;*' - K n K;~' > 0

KI/~l - Kn/~II < 0

K1/(t - Ku Kn = 0

KI g l -- KII gll > 0

no growth no growth stable growth

no growth not determined unstable growth

stable growth unstable growth unstable growth

As [ K [ ' , KI~'] T is computed from Eq. (17), we can compute the second derivative of the stress intensity factor using the above perturbation of the traction. This second derivative, denoted by [K~', K ,liJ1r' is a polynomial of the second order for x; see the last two terms in the expanded traction. Therefore, we can maximize/minimize [K[', K~] x with respect to K. As mentioned in Section 3, r d is regarded as the change of the crack orientation during the extension, and this x gives the direction that maximizes/minimizes the Mode I stress intensity factor.

(2) case of K [ ' = 0: for /~l < 0, the crack does not propagate since the external loads do not induce the crack growth; and f o r / ( i > 0, there is no d and r , and hence this may correspond to kinking of the crack which induces the discontinuity of the crack orientation. Tables 1 and 2 summarize the results of this discussion for the cases of K n = 0 and KII 4: 0, respectively.

References Appendix D. Stability of crack growth Eqs. (22) and (23) form a set of linear equations for Kd and d , from which the following equations are derived: (glgl *t-

gllglr t )(l + (gl/~I -gllt~ll) = 0,

( K I K,*' - KII KI; t) ( K a ) =/r~ll g l *t -- ~[I K I ; "

Since fi cannot be negative, it requires some investigation for the first equation. For simplicity, we assume K n = 0, and rewrite the first equation as K { ' d + R I = O. Since KI*' > 0 and tr~i > 0 correspond to the increase of K t due to the crack growth and the external loads, respectively, we can determine t¢ and d as follows: (1) case of K [ ' . ~ 0: for K [ ' < 0 and /~ > 0, or for KI *~ > 0 and K~ < 0, the crack growth is stable and ti and x are given by

/~1

/r~lKII t --/r~ll g l *t

K[' '

KcR I

'

for KI*'> 0 and /ft > 0, the crack growth is unstable since both the extemal loads and the crack extension accelerate the growth; and for K [ ' < 0 and /~i < 0, the crack does not propagate.

Abeyaratne, R., Knowles, J.K., 1993. A continuum model of thermoelastie solid capable of undergoing phase transitions. J. Mech. Phys. Solids 41,541. Azhdari, A., Nemat-Nasser, S., 1996. Energy-release rate and crack kinking in anisotropic brittle solids. J. Mech. Phys. Solids 44, 929. Cottereil, B., Rice, J.R., 1980. Slightly curved or kinked cracks. Int. J. Fract. 16, 155. Erdogan, F., Gupta, G.D., Cook, T.S., 1973. Numerical solution of singular integral equations, in: Sih, G.C. (Ed.), Methods of Analysis and Solutions of Crack Problems, Noordhoff, Holland, p. 368. Freund, L.B., 1990. Dynamic Fracture Mechanics, Cambridge University Press, Cambridge. Gao, H., Chiu, C., 1992. Slightly curved or kinked cracks in anisotropic elastic solids. Int. J. Solids Struct. 29, 947. Griffith, A.A., 1921. The phenomenon of rupture and flow in solids. Philos. Trans. R. Soc. A 221, 163. Gurthin, M.E., Podio-Guidugli, P., 1996. Configurational forces and the basic laws for crack propagation. J. Mech. Phys. Solids 44, 905. Hayashi, K., Nemat-Nasser, S., 1981. Energy-release rate and crack kinking. Int. J. Solids Struct. 17, 107. Hori, M., Vaikuntan, N., 1996. Numerical analysis of smoothly growing displacement discontinuity, Deformation Charact. Model. Mater., to appear. Horii, H., Nemat-Nasser, S., 1983. Estimate of stress intensity factors for interacting cracks, in: Yuceoglu, U., Sierakowski, R.L., Glasgow, D.A. (Eds.), Advances in Aerospace, Structure, Materials and Dynamics, Vol. AD-06. ASME, New York, p. 117.

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Karihaloo, B.L., Keer, L.M., Nemat-Nasser, S., Oranratnachai, A., 1981. Approximate description of crack kicking and curving. J. Appl. Mech. 48, 515. Lo, K.K., 1978. Analysis of branched cracks. J. Appl. Mech. 45, 797. Movchan, A.B., Willis, J.R., 1995. Dynamic weight function for a moving crack: II shear loading. J. Mech. Phys. Solids 43, 1369. Nemat-Nasser, S., Keel L.M., Parihar, K.S., 1978. Unstable growth of thermally induced interacting cracks in brittle solids. InL J. Solids Struct. 14, 409. Palaniswamry, K., Knanss, W.G., 1978. On the problem of crack extension in brittle solids finder general loading, in: NematNasser, S. (Ed.), Mechanics Today, Pergamon Press, Oxford, p. 97. Sumi, Y., Nemat-Nasser, S., Keel L.M., 1980. A new combined analytical and finite element solution method for stability analysis of the growth of interacting tension cracks in brittle solids. Int. J. Eng. Sei. 18; 211. Sumi, Y., 1986. A note on the first order perturbation solution of a

straight en crack with slightly branched and curved extension under a general geometric and loading condition. Eng. Fract. Mech. 24, 479. Rice, J.R., 1968. Mathematical analysis in the mechanics of fracture, in: Liebowitz, H. (Ed.), Fracture, Vol. 2, Academic, New York, p. 191. Sih, G.C., 1972. A special theory of crack propagation, Mechanics of Fracture, Vol. 1, Noordhoff, Leiden. Vaikuntan, N., 1996. Analysis method of initiation and propagation process of earthquake faults, Ph.D. dissertation, Department of Civil Engineering, the University of Tokyo. Westergaard, H.M., 1939. Trans. ASME, J. Appl. Mech. A6, 49. Williams, M.L., 1957. J. Appl. Mech 24, 109. Willis, J.R., Movchan, A.B., 1995. Dynamic weight function for a moving crack: I mode I loading. J. Mech. Phys. Solids 43, 319. Wu, C.H., 1979. Explicitly asymptotic solution for the maximum energy-release-rate problem. Int. J. Solids Struct. 15, 561. Xu, Y., Keel L.M., 1992. Higher order asymptotic field at the moving crack tip. Int. J. Fract. 58, 325.