Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part I: Calculation of stress intensity factors

Journal of the Mechanics and Physics of Solids 49 (2001) 1399 – 1420

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Crack front rotation and segmentation in mixed mode I + III or I + II + III. Part I: Calculation of stress intensity factors V,eronique Lazarus, Jean-Baptiste Leblond ∗ , Salah-Eddine Mouchrif Laboratoire de Modelisation en Mecanique, Universite Pierre et Marie Curie, 8 rue du Capitaine Scott, 75015 Paris, France Received 30 June 2000; received in revised form 7 December 2000; accepted 2 January 2001

Abstract This paper and its companion are devoted to the theoretical analysis, compared with experiments, of crack front rotation and segmentation in brittle solids under mixed mode I + III or I +II + III loadings. A ;rst, indispensable task, which is the topic of the present paper, consists of evaluating the stress intensity factors along the front of a crack after rotation or segmentation of this front. The method of analysis combines three elements. The ;rst one is 3D matched asymptotic expansions. The second one is an analogy with the related problem of a planar crack, but with a slightly wavy front. The output of these two ingredients is the reduction of the search for the quantities of interest to the solution of some 2D, plane strain or antiplane elasticity problems. The third element consists of the use of Muskhelishvili’s complex potentials c 2001 Elsevier Science Ltd. All formalism and conformal mapping to solve these problems.
rights reserved. Keywords: A. Stress intensity factor; B. Elastic material; C. Analytic functions; C. Asymptotic analysis; Crack front segmentation

1. Introduction It has long been observed experimentally that if a brittle cracked body is subjected to some mixed-mode I + III loading, the crack front tends to rotate about the direction of propagation. This rotation may be sudden, notably if the mode III stress intensity factor (SIF) KIII is large enough compared to that, KI , of mode I; see, e.g. Palaniswamy and Knauss (1975). It may also occur in a progressive way as the crack front proceeds, if ∗

Corresponding author. Tel.: +33-1-44-27-39-24. Fax: +33-1-44-27-52-59. E-mail address: [email protected] (J.-B. Leblond). c 2001 Elsevier Science Ltd. All rights reserved. 0022-5096/01/$ - see front matter  PII: S 0 0 2 2 - 5 0 9 6 ( 0 1 ) 0 0 0 0 7 - 2

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Fig. 1. Crack front rotation (a) or segmentation (b) under mixed mode I + III.

KIII is suCciently small as compared with KI ; see, e.g. Sommer (1969) and Hourlier and Pineau (1979). We shall only consider the second case here, that is, progressive rotation of the crack front. If the thickness of the body, in the direction of the crack front, is small enough, as in the experiments of Yates and Mohammed (1994), the entire crack front rotates about the direction x1 of propagation, as schematically shown in Fig. 1a. On the other hand, if this thickness is large, as in the experiments of Sommer (1969) and Hourlier and Pineau (1979), the crack extension splits into many pieces each of which rotates individually, as depicted in Fig. 1b. One then talks about crack front “segmentation”, and the shape of the crack extension is depicted in a picturesque way as looking like that of a “factory roof ”. This paper and its companion are devoted to the theoretical analysis of these complex phenomena, also in the even more involved case of mixed mode I +II + III loadings, with some comparisons with experiments, either taken from the literature or new. The aim of the present Part I is to evaluate theoretically the SIF along the crack front, after some propagation over a small distance  giving rise to rotation or segmentation of this front, under general mixed mode loading conditions. This is an indispensable

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Fig. 2. Trace of the crack in a plane x3 = Cst.

prerequisite to the discussion of crack front rotation and segmentation to be presented in Part II. It is a rather technical matter, however, so that the presentation has been arranged in such a manner that the reader only interested in the discussion of Part II can safely omit most of the present part, admitting only the formulae recalled in this Introduction and the numerical results and small-order expansions for the operator P(’) given in Section 7. A common feature of the crack geometries obtained after rotation (Fig. 1a) or segmentation (Fig. 1b) of the front is that in each plane x3 = Cst., they look as indicated in Fig. 2. That is, the crack extension, of small length , makes an angle ’(x3 ) (the local kink angle) with the original direction x1 of the crack, which, as indicated by the notation, is a function of the position x3 along the front. Indeed it is this dependence of the kink angle upon x3 that generates the rotation of the crack front about the direction x1 (Fig. 1a) or the “factory roof” appearance of the extended crack surface (Fig. 1b). The SIF for this kind of crack con;guration were recently studied theoretically by Leblond (1999) and Leblond et al. (1999). These authors obtained the following three-term expansion of the SIF Kp (x3 ; ) (p = I; II; III) at the point x3 of the extended crack front in powers of the crack extension length : √ K (x3 ; ) ≡ K ∗ (x3 ) + K (1=2) (x3 )  + K (1) (x3 ) + O(3=2 ); (1) where K (x3 ; ) ≡ [KI (x3 ; ); KII (x3 ; ); KIII (x3 ; )] is the “vector of stress intensity factors” and the quantities K ∗ (x3 ); K (1=2) (x3 ); K (1) (x3 ) are similar vectors, the detailed expressions of which are provided below. First, K ∗ (x3 ) is given by K ∗ (x3 ) ≡ F(’(x3 )) · K :

(2)

In this expression K denotes the vector of initial (that is, prior to propagation of the crack over the distance ) SIF, which are assumed to be independent of x3 for simplicity. Also, F(’(x3 )) denotes some universal operator depending upon the local value ’(x3 ) of the kink angle. The word “universal” here means that this operator applies in every possible situation, whatever the geometry of the body and whatever the loading, and depends on the sole argument ’(x3 ). The values of the components of this operator are given by Leblond (1999), on the basis of some previous works

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devoted to the 2D, plane strain or antiplane case (Sih, 1965; Amestoy et al., 1979; Amestoy and Leblond, 1992). Second, the expression of K (1=2) (x3 ) is K (1=2) (x3 ) ≡ G (’(x3 )) · T:

(3)

The quantity T ≡ (TI ; TII ; TIII ) here represents the vector of initial non-singular stresses, which are again supposed to be independent of x3 ; TI represents a uniform stress ;eld 11 ≡ TI ; TII a uniform ;eld 13 ≡ TII and TIII a uniform ;eld 33 ≡ TIII . Also, G (’(x3 )) denotes some other universal operator, the values of the components of which are provided by Leblond (1999), again on the basis of the previous works of Sih (1965) and Amestoy and Leblond (1992). Finally, the expression of K (1) (x3 ) is more complex: d’ K (1) (x3 ) ≡ [K (1) (x3 )]’(x
)≡’(x3 ) + P(’(x3 )) · K (x3 ) 3 d x3
Z (; x3 ; x3 ) · [(F T · F)(’(x3 )) + F(’(x3 )) · PV T

F

− (F · F)(’(x3 ))] d x3 · K :

(4) (1)

In this formula, the ;rst term in the right-hand side represents the value of K (x3 ) for a uniform kink angle equal to ’(x3 ) (’(x3 ) ≡ ’(x3 ); ∀x3 ). This quantity is of non-universal character, that is, it depends on the entire geometry of the body and the crack considered and thus must be evaluated in each particular case. However its value is known at least for such interesting special cases as that of a tunnel-crack in an in;nite body loaded by uniform stresses at in;nity (Lazarus, 1997). In the second term of the right-hand side of Eq. (4), P(’) denotes some new universal operator, the components of which are unknown at present. Finally, in the last, integral term of the right-hand side, F denotes the same universal operator as in Eq. (2), F T its transpose, F the crack front, and Z (; x3 ; x3 ) some non-universal operator, which again means that it depends on the entire geometry of the body and the crack considered, as underlined by the argument “” which represents the domain studied. However the values of its components are known for instance for a semi-in;nite crack in an in;nite body, from the works of Rice (1985) and Gao and Rice (1986). These components diverge like (x3 − x3 )−2 for x3 → x3 , so that the integral exists as a Cauchy principal value (PV ). Thus, the sole unknown universal quantity in Eqs. (2) – (4) is the operator P(’), whereas the two non-universal quantities involved are known at least for signi;cant special cases. Therefore the problem of evaluating the SIF along the front of a small crack extension with the type of geometry represented in Figs. 1a and b is reduced to calculating the operator P(’). This is the precise object of this paper. The task will be more diCcult than for the operators F(’) and G (’), because these operators were basically of 2D nature, whereas P(’) is intrinsically 3D, since it appears multiplied by d’=d x3 in expression (4) of K (1) (x3 ). It should be noted that several other, previous works (Gao, 1992; Xu et al., 1994; Ball and Larralde, 1995; Movchan et al., 1998) have been devoted to the problem of the slightly non-planar crack. However the hypotheses made about the crack geometry were somewhat diPerent from ours so that the results obtained were at the same time

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more and less general than those derived here. Indeed, instead of perturbing the sole location and geometry of the crack front as in the present work, the small parameter considered being the length  of the crack extension, these authors perturbed the entire crack surface, the small parameter being then the distance from the actual crack surface to the reference Ox3 x1 plane. In this respect, their results were more general than ours. On the other hand, it was assumed that both slopes, in the x1 - and x3 -directions, of the crack surface with respect to the Ox3 x1 plane were small, so that the kink angle, if any, had to be small, whereas it can take arbitrary values here. In that other respect, our results are more general. However, a detailed comparison between these earlier works and the present one can be made for small values of the kink angle; such a comparison will be presented in due time. The method employed to evaluate the operator P(’) will combine three elements. The ;rst one will be matched asymptotic expansions in three dimensions. The treatment will be inspired from the paper of Leguillon (1993) devoted to the 2D case. The second one will be an analogy between the problem envisaged here of an extended crack with uniform extension length  and non-uniform kink angle ’(x3 ), and that of an extended crack with zero kink angle (planar crack) but non-uniform extension length (x3 ). Use will be made of the results of Rice (1985) and Gao and Rice (1986) concerning that other problem. Combining these two elements, we shall not only recover the results (Eqs. (1) – (4)) derived by Leblond (1999) and Leblond et al. (1999), but also derive a practical method for calculation of the operator P(’), which these works failed to provide. It will turn out that this calculation can be reduced to solving some 2D, plane strain and antiplane problems, the initial 3D nature of the operator looked for appearing only through the presence of body forces and modi;ed boundary conditions on the crack lips in these problems. The third element of the approach used will consist of solving these 2D problems with the aid of Muskhelishvili’s (1953) complex potentials formalism and conformal mapping. Since the method is classical, although quite involved in the present case, it will not be expounded in detail and we shall merely state the results obtained. 2. Generalities We basically follow the approach of Leguillon (1993), extending it to the 3D case. As in Fig. 1a or b, we thus consider, within a 3D body  loaded by prescribed displacements up and tractions T p on the portions @u and @T of its boundary, an initially plane crack C with a straight front lying along the Ox3 axis, endowed with a non-uniformly kinked extension E of uniform small length. The kink angle is denoted ’(x3 ), and the extension length, . The initial geometry, prior to extension of the crack over this length, and the loading are assumed to be invariant in the direction x3 , so that the problem is in fact initially 2D. The equations of the problem are div (x; ) = 0 ”(x; ) =

in ;

1 2 [∇u(x; )

+ (∇u)T (x; )]

(x; ) = C : ”(x; ) in ;

in ;

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u(x; ) = up (x) on @u ; (x; ) · n(x) = 0

(x; ) · n(x) = T p (x)

on @T ;



on C ∪ E ;

(5)

where u(x; ); ”(x; ) and (x; ) denote the displacement, strain and stresses, C the (isotropic) elasticity tensor and n(x) the unit normal vector to @T ; C or E . It was proved by Leblond (1999) that the mechanical ;elds are once (but not twice) di6erentiable with respect to  for  = 0+ . Thus they admit an “outer” expansion of the form u(x; ) ≡ u(0) (x) + u(1) (x) + O(3=2 ); ”(x; ) ≡ ”(0) (x) + ”(1) (x) + O(3=2 ); (x; ) ≡ (0) (x) + (1) (x) + O(3=2 );

(6)

where we admit, following Leguillon (1993), that only semi-integral powers of  are involved. In these expressions, u(0) (x); ”(0) (x); (0) (x) simply denote the mechanical ;elds in the absence of the crack extension E . Thus u(0) (x) admits the following classical expansion near the initial crack front, forgetting the rigid-body term hereafter u(0) (x) ≡ Kp Up(1=2) (x) + Tp Up(1) (x) + Bp Up(3=2) (x) + O(r 2 );

(7)

where the coeCcients Kp ; Tp ; Bp (p = I; II; III) are independent of x3 ; Up() (x) denotes the fundamental mode p displacement ;eld homogeneous of degree , Einstein implicit summation convention is used for the index p, and r ≡ (x12 + x22 )1=2 . With regard to the ;elds u(1) (x); ”(1) (x); (1) (x), they are readily shown, upon expansion of Eqs. (5) in powers of , to verify the following homogeneous equations: div (1) (x) = 0

in ;

”(1) (x) = 12 [∇u(1) (x) + (∇u(1) )T (x)]

in ;

(1) (x) = C : ”(1) (x) in ; u(1) (x) = 0

on @u ;

(1) (x) · e2 = 0

(1) (x) · n(x) = 0

on C:

on @T ; (8)

These equations are to be completed by some “matching conditions”, as will be seen. We also consider the “dilated” coordinates y1 ; y2 ; y3 and associated “inner” mechanical ;elds v(y; ); e(y; ); (y; ) de;ned by y1 ≡ x1 =;

v(y; ) ≡ u(x; );

y2 ≡ x2 =;

e(y; ) ≡ ”(x; );

y3 = x3 ;

(y; ) ≡ (x; ):

(9)

The dilated domain d spanned by the variable y, in the limit  → 0, is in;nite and contains a semi-in;nite plane crack Cd with a straight front endowed with a

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non-uniformly kinked (with kink angle ’(y3 )) extension Ed of unit length. Using Eqs. (5) and (9), one easily derives the equations veri;ed by the inner ;elds: ; (y; ) + 3;3 (x; ) = 0 in d ; 3; (y; ) + 33;3 (x; ) = 0 in d ; in d ;

e (y; ) = 12 [v; (y; ) + v; (y; )] e3 (y; ) = 12 [v3; (y; ) + v;3 (y; )]

in d ;

e33 (y; ) = v3;3 (y; ) in d ; (y; ) = C : e(y; ) in d ; (y; ) · n(y; ) = 0

on Cd ∪ Ed ;

(10)

where Greek indices take the sole values 1 and 2 and a semicolon denotes a derivative with respect to some yi coordinate. It is assumed that the mechanical ;elds admit an “inner” expansion of the form √ v(y; ) ≡ v(1=2) (y)  + v(1) (y) + v(3=2) (y)3=2 + O(2 ); √ e(y; ) ≡ e(1=2) (y)  + e(1) (y) + e(3=2) (y)3=2 + O(2 ); √ (y; ) ≡ (1=2) (y)  + (1) (y) + (3=2) (y)3=2 + O(2 ): (11) The absence of terms proportional to 0 = 1 here is a consequence of the matching conditions and will be justi;ed later. Expanding Eqs. (10) and the normal vector to Ed in powers of : n(x) ≡ n(y; ) ≡ n(0) (y) + n(1) (y) + O(2 );

(12)

one gets the equations satis;ed by the inner ;elds introduced in Eqs. (11): (in ) (y) = 0 ;

in d

(in ) 3; (y) = 0 in d (in ) (in ) (in ) e (y) = 12 [v; (y) + v; (y)]

in d

(in ) (in ) (y) = 12 v3; (y) in d e3 (in ) e33 (y) = 0 in d

(in ) (y) = C : e(in ) (y) in d (in ) (y) · e2 = 0

on Cd ;

(in ) (y) · n(0) (y) = 0

on Ed

for in = 1=2 and 1;

(1=2) d (3=2) ; (y) + 3;3 (y) = 0 in  ; (1=2) d (3=2) 3; (y) + 33;3 (y) = 0 in  ; (3=2) (3=2) (3=2) (y) = 12 [v; (y) + v; (y)] e

in d ;

(3=2) (3=2) (1=2) (y) = 12 [v3; (y) + v;3 (y)] e3

in d ;

(13)

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420 (3=2) (1=2) (y) in d ; e33 (y) = v3;3

(3=2) (y) = C : e(3=2) (y) in d ; (3=2) (y) · e2 = 0

on Cd ;

(3=2) (y) · n(0) (y) + (1=2) (y) · n(1) (y) = 0

on Ed : (14)

(out )

out

(x) of Let us now discuss matching conditions. We admit that each term u the outer expansion (6)1 itself admits an expansion in powers of r for r → 0; such an expansion is given by Eq. (7) for u(0) (x) for instance. The exponents of r involved are denoted out . Similarly, we suppose that each term v(in ) (y)in of the inner expansion (11)1 itself admits an expansion in powers of ≡ (y12 + y22 )1=2 = r= for → +∞. The exponents of are denoted in . We assume that the term proportional to out r out in the outer expansion must be identical to the term proportional to out ( )out = out +out out in the inner expansion. Thus the correspondence between exponents of identical terms reads in = out + out ;

in = out ⇔ out = in − in ;

out = in :

(15)

Furthermore, we shall make the following assumption, which is implicit in Leguillon’s (1993) work: the outer exponents of all non-zero terms verify the inequality out + out ≥ 1=2. 1 As a ;rst consequence, note that by Eq. (15)1 ; in ≥ 1=2; this is why there are no terms proportional to 0 = 1 in the inner expansions (11). Let us ;nally introduce some extra notations aimed at simplifying future equations. We shall denote (u(x)) the usual stresses deriving from the displacement u(x). Also, we shall denote ˆ(v(y)) ≡ C : e(y) ˆ the stresses associated to the “strain” e(y) ˆ de;ned by eˆ  (y) ≡ 12 [v; (y) + v; (y)]; eˆ 3 (y) ≡ 12 v3; (y); eˆ 33 (y) ≡ 0:

(16)

These equations are identical to those de;ning the usual strain except for the omission of terms containing a derivative with respect to y3 . 3. First two terms of the inner expansion With the compact notation just introduced, Eqs. (13) for v(1=2) (y) read ˆ; (v(1=2) (y)) = 0;

ˆ3; (v(1=2) (y)) = 0 in d

ˆ(v(1=2) (y)) · e2 = 0

on Cd ;

ˆ(v(1=2) (y)) · n(0) (y) = 0

on Ed :

(17)

This homogeneous system is incomplete insofar as it does not specify the behaviour of v(1=2) (y) for → +∞. Let us therefore look for the greatest possible exponent in 1

This assumption can be rigorously, and easily justi;ed for coplanar propagation. However, we are not aware of any general proof of it.

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of in the expansion of this ;eld. Since in = 1=2 for it, and since the exponents out of  in √ Eq. (6)1 are non-negative, in ≤ 1=2 by Eq. (15)3 . Thus the principal term of v(1=2) (y)  for → +∞ has exponent 1=2 and must, by Eqs. (15), be identical to the term (out ; out ) = (0; 1=2) of the outer expansion, that is by Eq. (7), Kp Up(1=2) (x) = √ Kp Up(1=2) (y)  (since Up(1=2) is homogeneous of degree 1=2). Therefore v(1=2) (y) ∼ Kp Up(1=2) (y)

for

→ +∞:

(18)

It follows from Eqs. (17), (18) and linearity that v(1=2) (y) may be written in the form v(1=2) (y) ≡ Kp vp(1=2) (y);

(19)

where the ;elds vp(1=2) (y) verify the following equations: ˆ; (vp(1=2) (y)) = 0;

ˆ3; (vp(1=2) (y)) = 0 in d ;

ˆ(vp(1=2) (y)) · e2 = 0

on Cd ;

vp(1=2) (y) ∼ Up(1=2) (y)

for

ˆ(vp(1=2) (y)) · n(0) (y) = 0

on Ed ;

→ +∞:

(20)

Note that these equations do not involve any derivative with respect to y3 . This means that they represent, instead of a single 3D problem, an in7nity of independent 2D problems posed each within a plane y3 = Cst. Since the sole datum depending on y3 in each problem is the local kink angle ’(y3 ), the solution also depends on this coordinate only through this kink angle. Similarly, Eqs. (13) for v(1) (y) read ˆ; (v(1) (y)) = 0;

ˆ3; (v(1) (y)) = 0 in d ;

ˆ(v(1) (y)) · e2 = 0

on Cd ;

ˆ(v(1) (y)) · n(0) (y) = 0

on Ed :

(21)

Reasoning as for v(1=2) (y), it is easy to show that v(1) (y) behaves like the term (out ; out ) = (0; 1) of the outer expansion for → +∞; that is, by Eq. (7), v(1) (y) ∼ Tp Up(1) (y)

for

→ +∞:

(22)

Again, it follows from Eqs. (21) and (22) that v(1) (y) may be written as v(1) (y) ≡ Tp vp(1) (y);

(23)

where the ;elds vp(1) (y) satisfy the following equations: ˆ; (vp(1) (y)) = 0;

ˆ3; (vp(1) (y)) = 0 in d

ˆ(vp(1) (y)) · e2 = 0

on Cd ;

vp(1) (y) ∼ Up(1) (y)

for

ˆ(vp(1) (y)) · n(0) (y) = 0

→ +∞:

on Ed (24)

Again, these equations represent an in;nity of independent 2D problems rather than a 3D one, so that the solution depends on y3 only through the local kink angle ’(y3 ).

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4. Second term of the outer expansion With the compact notation introduced at the end of Section 2, system (8) satis;ed by the second term u(1) (x) of the outer expansion reads ij; j (u(1) (x)) = 0 in ; u(1) (x) = 0

(u(1) (x)) · n(x) = 0

on @u ;

(u(1) (x)) · e2 = 0

on @T ;

on C:

(25)

In order to complete this homogeneous system, one must determine the asymptotic behaviour of u(1) (x) for r → 0, that is, the smallest exponent out of r in the expansion of u(1) (x). Since out = 1 for this function and out + out ≥ 1=2, the minimum value of out is −1=2. By Eqs. (15), the term (out ; out ) = (1; −1=2) of the outer expansion corresponds to the term (in ; in ) = (1=2; −1=2) of the √ inner expansion, √ that is the term proportional to −1=2 in the expansion of v(1=2) (y)  ≡ Kp vp(1=2) (y) . Now, by Eq. (20)5 , the function vp(1=2) (y) − Up(1=2) (y) is O(1) for → +∞. Neglecting the constant term representing some rigid-body motion, one concludes that this function asymptotically behaves like a linear combination of the fundamental ;elds Uq(−1=2) (y) homogeneous of degree −1=2. But these ;elds can be taken as identical to the deriva(1=2) tives Uq;1 (y) since the latter ;elds satisfy the same equations as the Uq(1=2) (y) while being homogeneous of degree −1=2. Therefore (1=2) vp(1=2) (y) ≡ Up(1=2) (y) + Aqp (’(y3 ))Uq;1 (y) + O(

−1

)

for

→ +∞;

(26)

where the Aqp (’(y3 )) are coeCcients depending only on the local kink angle ’(y3 ), just as the functions vp(1=2) (y) themselves. (The precise expression of these coeCcients will be determined later). It then follows from the matching conditions that u(1) (x) ∼ Kp Aqp (’(x3 ))Uq;(1=2) 1 (x)

for r → 0:

(27)

In contrast to the systems encountered so far, system (25), (27) de;nes a fully 3D problem, since it involves derivatives with respect to x3 . Now it will turn out compulsory in the sequel to know the second term of the expansion of u(1) (x) for r → 0, and especially its type of dependence upon the function ’(x3 ). Matched asymptotic expansion fail to say anything about this. We shall therefore resort to some other technique, which consists in exploiting an analogy with another problem the solution of which is known. As a preliminary, let us introduce the following notation: f(x3 ) denoting an arbitrary function, let Uq (x; {f}) denote the displacement ;eld de;ned by ij; j (Uq (x; {f})) = 0 in ; Uq (x; {f}) = 0

on @u ;

(Uq (x; {f})) · e2 = 0

(Uq (x; {f})) · n(x) = 0

on @T ;

on C;

Uq (x; {f}) ∼ f(x3 )Uq;(1=2) 1 (x)

for r → 0:

(28)

Comparison of Eqs. (25), (27) and (28) then reveals that u(1) (x) = Kp Uq (x; {Aqp ◦ ’}):

(29)

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We now consider the new problem of a coplanar crack extension with non-uniform length (x3 ) ≡ &f(x3 ), where & denotes some small parameter. This problem is studied in detail in Appendix A. The output is the following asymptotic expression of Uq (x; {f}): (1=2) Uq (x; {f}) = f(x3 )Uq;(1=2) (x) 1 (x) − Kpq (x3 ; {f})Up

−f (x3 )13 · Uq(1=2) (x) + O(r):

(30)

In this expression, 13 is the antisymmetric tensor de;ned by 13 ≡ e1 ⊗ e3 − e3 ⊗ e1 :

(31)

Also, K(x; {f}) denotes the operator de;ned by K(x3 ; {f}) ≡ KT ()f(x3 ) + N (0)f (x3 )
+ PV Z (; x3 ; x3 )[f(x3 ) − f(x3 )] d x3 ; F

(32)

in this equation, KT () represents some non-universal operator, depending upon the entire geometry of the body ; N (0) is a universal operator given by Eq. (A.5) of Appendix A; and Z (; x3 ; x3 ) is the same non-universal operator as in Eq. (4). Combining Eqs. (29) and (30), we obtain the following expansion of u(1) (x) for r → 0: (1=2) u(1) (x) = Kp Aqp (’(x3 ))Uq;(1=2) (x) 1 (x) − Kr Kpq (x3 ; {Aqr ◦ ’})Up

dAqp d’ (’(x3 )) (33) (x3 )13 · Uq(1=2) (x) + O(r): d’ d x3 √ This formula provides the second term, proportional to r, of the expansion of u(1) (x). This term is the sum of the second and third terms in the right-hand side here. − Kp

5. Third term of the inner expansion System (14) for v(3=2) (y) reads, with the notation ˆ(v(y)) introduced at the end of Section 2: (1=2) (3=2) ; (y) + 3;3 (y) = 0;

(1=2) (3=2) 3; (y) + 33;3 (y) = 0

in d ;

(1=2) ˆ (v(3=2) (y)) + ' v3;3 (y) in d ; (3=2)  (y) =  (1=2) (3=2) ˆ3 (v(3=2) (y)) + (v;3 (y) in d ; 3 (y) = 

(3=2) (y) · e2 = 0

on Cd ;

(3=2) (y) · n(0) (y) + (1=2) (y) · n(1) (y) = 0

on Ed ;

where ' and ( denote Lam,e’s coeCcients. Now it is easy to see that the unit normal vector to the crack extension is, to ;rst order in , n(x) ≡ n(y; ) ≡ −sin ’(x3 )e1 + cos ’(x3 )e2 − r(d’=d x3 )(x3 )e3 with r =  , so that, by Eq. (12), d’ n(0) (y) ≡ −sin ’(y3 )e1 + cos ’(y3 )e2 ; n(1) (y) ≡ − (y3 )e3 : (34) dy3

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

Expressing (1=2) (y) in terms of v(1=2) (y) with the aid of Eqs. (13)3 –(13)6 , using the fact that the latter ;eld depends on y3 only through the local kink angle ’(y3 ) and Eqs. (19) and (34)2 , we readily transform the preceding system into (1=2)

ˆ; (v(3=2) (y)) + (' + ()Kp

@vp3; d’ (y) = 0 (y3 ) dy3 @’

in d ;

ˆ3; (v(3=2) (y)) + (' + ()Kp

(1=2) @vp; d’ (y3 ) (y) = 0 dy3 @’

in d ;

  (1=2) (1=2) @v @v d’ p2 p3 (y)e2 + ( (y)e3 = 0 ˆ(v(3=2) (y)) · e2 + Kp (y3 ) ' dy3 @’ @’ ˆ(v

(3=2)

on Cd ;

  (1=2) (1=2) @vp3 @vp d’ (0) (0) (y)n (y)e3 (y3 ) ' (y)) · n (y) + Kp (y)n (y) + ( dy3 @’ @’

− Kp

(0)

d’ (y3 ) ˆ(vp(1=2) (y)) · e3 = 0 dy3

on Ed :

(35)

Again, this system must be completed by matching conditions. For each term of the expansion of v(3=2) (y)3=2 for → +∞, one has 3=2 = in = out + out ≥ out = in (see (15)). Therefore the three maximum possible values of in are 3=2, 1 and 1=2. Now for in = 3=2, the term considered in the inner expansion must be identical to the term (out ; out ) = (0; 3=2) of the outer expansion, that is, Bp Up(3=2) (x) (see Eq. (7)). For in = 1; (out ; out ) = (1=2; 1); but there is no such term in the outer expansion (6)1 . Finally, for in = 1=2; (out ; out ) = (1; 1=2) so that √ the corresponding term in the outer expansion is the second term, proportional to r, in expansion (33) of u(1) (x). Thus the behaviour of v(3=2) (y) at in;nity reads as follows: v(3=2) (y) = Bp Up(3=2) (y) − Kr Kpq (y3 ; {Aqr ◦ ’})Up(1=2) (y) − Kp

dAqp √ d’ (y3 )13 · Uq(1=2) (y) + o( ): (’(y3 )) d’ dy3

(36)

It follows from Eqs. (35), (36) and linearity that v(3=2) (y) may be written as d’ (y3 )vp(3=2) (y) + Kr Kpq (y3 ; {Aqr ◦ ’})vp(3=2) (y) + Bp vp(3=2) (y); v(3=2) (y) ≡ Kp dy3 (37) where the ;elds vp(3=2) (y); vp(3=2) (y); vp(3=2) (y) satisfy the following systems: ˆ; (vp(3=2) (y)) + (' + () ˆ3; (vp(3=2) (y)) + (' + ()

(1=2) @vp3;

@’

(y) = 0 in d ;

(1=2) @vp; (y) = 0 in d ; @’

V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

ˆ(vp(3=2) (y))

· e2 + '

ˆ(vp(3=2) (y))

(0)

(1=2) @vp3

@’

· n (y) + '

(y)e2 + (

(1=2) @vp3

@’

− ˆ(vp(1=2) (y)) · e3 = 0 vp(3=2) (y) ∼ −

(1=2) @vp2

@’

(y)e3 = 0

(y)n(0) (y) + (

on Cd ;

(1=2) @vp (y)n(0)  (y)e3 @’

on Ed ;

dAqp (’(y3 ))13 · Uq(1=2) (y) d’

ˆ; (vp(3=2) (y)) = 0;

ˆ3; (vp(3=2) (y)) = 0

ˆ(vp(3=2) (y)) · e2 = 0

on Cd ;

vp(3=2) (y) ∼ −Up(1=2) (y)

1411

for

→ +∞;

in d ;

ˆ(vp(3=2) (y)) · n(0) (y) = 0

for

(38)

on Ed ;

→ +∞;

ˆ; (vp(3=2) (y)) = 0;

ˆ3; (vp(3=2) (y)) = 0

ˆ(vp(3=2) (y)) · e2 = 0

on Cd ;

vp(3=2) (y) ∼ Up(3=2) (y)

for

(39) in d ;

ˆ(vp(3=2) (y)) · n(0) (y) = 0

on Ed ;

→ +∞:

(40)

Comparison of systems (20) and (39) immediately reveals that vp(3=2) (y) ≡ −vp(1=2) (y):

(41)

Thus the vp(3=2) (y) are not new functions. On the other hand, the problems de;ned by Eqs. (38) and (40) are new. Again, since they do not involve any derivative of the unknown ;eld with respect to y3 , each of them de;nes, rather than a 3D problem, an in;nity of 2D ones depending on y3 only through the value of the local kink angle ’(y3 ). 6. Stress intensity factors Let M 0 denote the projection of the point M (x) onto the crack front perpendicularly to e3 ; also, let (M 0 ; x1∗ ; x2∗ ; x3 ) denote the frame deduced from (O; x1 ; x2 ; x3 ) by a rotation of angle ’(x3 ) about e3 (Fig. 2). The associated basis vectors are e1∗ ≡ cos ’(x3 )e1 + sin ’(x3 )e2 ;

e2∗ ≡ −sin ’(x3 )e1 + cos ’(x3 )e2 ;

e3∗ ≡ e3 : (42)

An asymptotic study of the displacement ;eld near the point M 0 , presented in Appendix B, reveals that √ v(y; ) ≡ Kp∗ (y3 )Up∗(1=2) (y∗ )  + Kp(1=2) (y3 )Up∗(1=2) (y∗ )   d’ ∗ (1) ∗(1=2) ∗ ∗ ∗(1=2) ∗ + Kp (y3 )Up (y ) + Kp (y3 ) (y3 )23 (y3 ) · Up (y ) 3=2 dy3 + O(2 ) + O(

∗

)

(43)

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

for ∗ → 0. In this expression, Kp∗ (y3 ); Kp(1=2) (y3 ) and Kp(1) (y3 ) are the coeCcients of the expansion of Kp (x3 ≡ y3 ; ) in powers of  (see Eq. (1)). Also, the ;elds Up∗(1=2) are de;ned by (1=2) ∗ ∗ ∗ Up∗(1=2) (x∗ ) ≡ Upi (x1 ; x2 )ei

(44)

and the antisymmetric tensor ∗23 (x3 ≡ y3 ) by ∗23 (x3 ) ≡ e2∗ ⊗ e3 − e3 ⊗ e2∗ :

(45)

Finally, y∗ is tied to x∗ like y to x: y1∗ ≡ x1∗ =; y2∗ ≡ x2∗ =; y3∗ ≡ x3∗ ≡ x3

(46)

and ∗ ≡ (y1∗2 + y2∗2 )1=2 . Let us use these results to study the quantities K ∗ (y3 ) and K (1=2) (y3 ). Comparing Eqs. (11)1 and (43), one gets, for ∗ → 0: v(1=2) (y) = Kp∗ (y3 )Up∗(1=2) (y∗ ) + O( (1)

v (y) =

Kp(1=2) (y3 )Up∗(1=2) (y∗ )

+ O(

∗

); ∗

):

(47)

Now let Fpq (’(y3 )) and Gpq (’(y3 )) denote the SIF associated to the ;elds vq(1=2) (y) and vq(1) (y), that is the quantities de;ned by vq(1=2) (y) ≡ Fpq (’(y3 ))Up∗(1=2) (y∗ ) + O( vq(1) (y) ≡ Gpq (’(y3 ))Up∗(1=2) (y∗ ) + O(

∗ ∗

);

):

(48)

As indicated by the notation, these SIF depend on the sole local value ’(y3 ) of the kink angle, just like the ;elds vq(1=2) (y) and vq(1) (y) themselves. Combining Eqs. (19) and (23) with Eqs. (48) and comparing with Eqs. (47), one gets Kp∗ (’(y3 )) = Fpq (’(y3 ))Kq ; Kp(1=2) (’(y3 )) = Gpq (’(y3 ))Tq :

(49)

Eqs. (49) are equivalent to Eqs. (2) and (3) in component form, which were obtained by Leblond (1999) by another method. It is recalled that the components of the operators F(’) and G (’) here are known and given for instance by Leblond (1999). The study of the quantity K (1) (y3 ) is more technical than that of K ∗ (y3 ) and (1=2) K (y3 ) and therefore relegated to Appendix C. The ;nal result is Eq. (4), where the operator P(’) is given by ˜ P(’) ≡ P(’) + F(’) · N (0) ·

d(F T · F) (’); d’

(50)

V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

1413

˜ P(’) being itself de;ned through the asymptotic behaviour of the ;elds vq(3=2) (y) for ∗ → 0, namely, vq(3=2) (y) = P˜pq (’(y3 ))Up∗(1=2) (y∗ ) + Fpq (’(y3 ))∗23 (y3 ) · Up∗(1=2) (y∗ ) + O(

∗

): (51)

It is recalled that Eq. (4) was derived by Leblond et al. (1999) by another method. The major advantage of the present approach, however, is that it leads to an explicit evaluation of the unknown operator P(’). Indeed, to get the components Ppq (’), it suCces to (i) solve the 2D problems on the ;elds vp(3=2) (y) de;ned by Eqs. (38); (ii) evaluate the “SIF” P˜pq (’) from Eq. (51); (iii) use Eq. (50) to relate the Ppq (’) to the P˜pq (’), using the fact that the components of the operator N (0) are known, thanks to the works of Rice (1985) and Gao and Rice (1986) (see Eq. (A.5) of Appendix A).

7. Explicit evaluation of the operator P(’) The evaluation of the operator P(’) is thus essentially reduced to solving the 2D elasticity problems de;ned by Eqs. (38). These problems involve a “dilated”, in;nite body (d ) with a semi-in;nite straight crack (Cd ) endowed with a kinked extension of unit length (Ed ). They may be solved using Muskhelishvili’s (1953) complex potentials formalism combined with conformal mapping. This has already been done by Amestoy et al. (1979) and Amestoy and Leblond (1992) to calculate the operators F(’) and G (’). The sole, albeit considerable, additional complexities here arise from the presence of body forces and tractions exerted on the crack lips in system (38). In spite of these intricacies, the method employed is quite classical in principle, and for this reason we shall just concentrate on results here and refer the interested reader to Mouchrif’s (1994) and Lazarus’ (1997) theses for a full account of the treatment. Table 1 provides the numerical values of the non-zero components of the operator P(’) corresponding to the value / = 0:3 of Poisson’s ratio, for values of ’ ranging ◦ from 0 to 80 . This table is easily supplemented for negative values of the kink angle since because of symmetry reasons, PI; III (’) and PIII; I (’) are even, and PII; III (’) and PIII; II (’) odd, functions of ’. It is also possible to derive analytical expressions of the Ppq (’) for small kink angles. One thus gets PI; III (’) = −2 + O(’2 );   8 2/ − 3 ’ + O(’3 ); PII; III (’) = 1 + 2
2−/ 2(1 − /)2 + O(’2 ); 2−/   4 2(1 − /) 1 − 2/ + 2 (2 + /) ’ + O(’3 ): PIII; II (’) = − 2−/


PIII; I (’) =

(52)

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

Table 1 Values of the non-zero components of the operator P(’) ◦

’( )

PI; III (’)

PII; III (’)

PIII; I (’)

PIII; II (’)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

−2 −1:998 −1:992 −1:982 −1:968 −1:951 −1:930 −1:906 −1:878 −1:848 −1:816 −1:780 −1:743 −1:704 −1:663 −1:620 −1:576

0 −0:013 −0:025 −0:037 −0:048 −0:059 −0:068 −0:077 −0:084 −0:090 −0:095 −0:099 −0:101 −0:103 −0:104 −0:104 −0:103

0.576 0.574 0.565 0.552 0.533 0.510 0.483 0.452 0.418 0.381 0.343 0.303 0.263 0.224 0.185 0.147 0.110

0 −0:095 −0:189 −0:280 −0:366 −0:445 −0:517 −0:581 −0:634 −0:678 −0:711 −0:733 −0:743 −0:743 −0:731 −0:710 −0:679

A comparison with the results derived in the earlier works of Gao (1992), Xu et al. (1994), Ball and Larralde (1995) and Movchan et al. (1998) is in order here. As explained in the Introduction, these works made hypotheses on the crack geometry differing somewhat from ours, so that their results were inevitably presented in a diPerent format; in particular, the universal operator P(’) was not clearly identi;ed as such. However, it is possible to extract from these works the values of the components of this operator for a zero kink angle. The value of PIII; I (0) found by Ball and Larralde (who did not calculate PI; III (0)), and those of PI; III (0) and PIII; I (0) found by Movchan et al., agree with Eqs. (52)1 and (52)3 . On the other hand, the results of the works of Gao and Xu et al. are in conVict with ours, and furthermore in mutual disagreement. However, both of these works used previous results of Bueckner (1987) which unfortunately contained a misprint. It has been checked that when the necessary correction is brought into the results of Xu et al., they then agree with Eqs. (52)1 and (52)3 . Although Gao’s results have not been revised in a similar way, these elements seem suCcient to warrant that Eqs. (52)1 and (52)3 are correct. Appendix A. We study here the “auxiliary” problem of a coplanar crack extension with a nonuniform length (x3 ) ≡ &f(x3 ) where & is a small parameter (Fig. A.1). This problem was envisaged by Rice (1985) and Gao and Rice (1986) in the special case of a semi-in;nite crack in an in;nite body, and we shall use some of their results here. The various mechanical quantities in this new problem will be denoted with a prime if they diPer from their counterparts in the original one de;ned in Section 2. We wish to study the ;eld u(1) (x); which is de;ned as the derivative of the displacement ;eld

V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

1415

Fig. A.1. Coplanar crack extension with a non-uniform length.

u (x; &) with respect to &. In order to do so, we shall diPerentiate the expansion of u (x; &) near the extended front with respect to this parameter. The displacement u (x; &) at the point M (x) admits the usual expansion, analogous to (7), in the frame (M 0 ; x1 ; x2 ; x3 ) “adapted” to the extended crack front (see Fig. A.1): (1=2)  (1)  u (x; &) ≡ Kp (x30 ; &)Upi (x1 ; x2 )ei + Tp (x30 ; &)Upi (x1 ; x2 )ei (3=2)  (x1 ; x2 )ei + O(r 2 ); + Bp (x30 ; &)Upi

(A.1)

where x30 denotes the x3 -coordinate of the point M 0 and r  ≡ (x12 + x22 )1=2 . Now it is easy to see that x30 = x3 + O(&), so that, to ;rst order in &, x1 = x1 − &f(x3 );

e1 = e1 − &f (x3 )e3 ;

x2 ≡ x2 ;

e2 ≡ e2 ;

x3 = 0;

e3 = e3 + &f (x3 )e1 :

(A.2)

Kp(1) (x3 )

is de;ned as the derivative of Kp (x3 ; &) with respect to &. DiPerentiating Also, then expansion (A.1) with respect to this parameter, we get (1=2) (1) (1=2) u(1) (x) = −f(x3 )Kp Up; (x) 1 (x) + Kp (x3 )Up (3=2) + f (x3 )Kp 13 · Up(1=2) (x) − f(x3 )Bp Up; 1 (x) + O(r);

(A.3)

where 13 is the tensor de;ned by Eq. (31) of the text. The term involving this tensor arises from diPerentiation of the basis vectors ei . Now it has been shown by Leblond et al. (1999) that K (1) (x3 ) admits the following expression: K (1) (x3 ) ≡ [K (1) (x3 )]f(x
)≡f(x3 ) + N (0) · K f (x3 ) 3
+ PV Z (; x3 ; x3 )[f(x3 ) − f(x3 )]d x3 · K : F

(A.4)

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

In this expression N (0) ≡ N (’(x3 ) = 0) represents some universal operator, the non-zero components of which can be deduced from the works of Rice (1985) and Gao and Rice (1986): NII; III (0) = −

2 ; 2−/

NIII; II (0) =

2(1 − /) ; 2−/

(A.5)

where / denotes Poisson’s ratio. Also, the quantity Z (; x3 ; x3 ) represents the same non-universal operator (depending upon the entire geometry of the body ) as in Eq. (4). Finally [K (1) (x3 )]f(x
)≡f(x3 ) denotes the value of K (1) (x3 ) for a uniform extension 3 length equal to &f(x3 ) (f(x3 ) ≡ f(x3 ); ∀x3 ). This value in fact relates to some 2D problem. It has been calculated by Sumi et al. (1983) and Leblond (1989), who found the same, following result: [K (1) (x3 )]f(x
)≡f(x3 ) ≡ [I (0) · B + KT () · K ]f(x3 ); 3

(A.6)

where I (0) ≡ I (’(x3 ) = 0) denotes some universal operator, B ≡ (BI ; BII ; BIII ) and KT () represents some non-universal operator. (The precise values of I (0) and KT () do not matter here). Inserting now Eqs. (A.4), (A.6) into Eq. (A.3), and accounting for the fact that the terms proportional to the Bp must necessarily cancel out because Bueckner–Rice’s theory (see e.g. Gao and Rice, 1986) implies that u(1) (x) depends upon the loading only through the initial SIF Kp , we get (1=2) (x) u(1) (x) = −f(x3 )Kq Uq;(1=2) 1 (x) + Kpq (x3 ; {f})Kq Up

+ f (x3 )Kq 13 · Uq(1=2) (x) + O(r)

(A.7)

for r → 0, where K(x3 ; f) is the operator de;ned by Eq. (32) of the text. If we now consider only the ;rst term of expansion (A.7), proportional to r −1=2 , we get u(1) (x) ∼ −f(x3 )Kq Uq;(1=2) 1 (x)

for r → 0:

Comparison with Eq. (28)5 then reveals that u(1) (x) ≡ −Kq Uq (x; {f}): Eq. (A.7) then yields expression (30) for the ;eld Uq (x; {f}).

Appendix B. We study here the relations between the outer and inner ;elds and the SIF along the front of the extended crack. The “intermediary” frame (M 0 ; x1∗ ; x2∗ ; x3 ) (see Fig. 2) would be adapted to the ;nal con;guration of the crack front, were it not for the variation of the kink angle ’(x3 ) along that front. Because of this variation, the frame which is truly adapted is not (M 0 ; x1∗ ; x2∗ ; x3 ) but (M 0 ; x1∗ ; x2∗ ; x3∗ ) (Fig. B.1). Let x30 denote the x3 -coordinate of the

V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

1417

Fig. B.1. View of the crack front in the plane M 0 x2∗ x3 .

point M 0 . It is easy to see that x30 = x3 + O(), so that to ;rst order in , x1∗ ≡ x1∗ ; e1∗ ≡ e1∗ ;

x2∗ = x2∗ ;

x3∗ = 0; d’ e2∗ = e2∗ −  (x3 )e3 ; d x3

e3∗ = e3 + 

d’ (x3 )e2∗ : d x3

(B.1)

Near the extended crack front, u(x; ) admits the following asymptotic expression in the adapted frame, analogous to (7): (1=2) ∗ ∗ ∗ u(x; ) ≡ Kp (x30 ; )Upi (x1 ; x2 )ei + O(r ∗ );

where r ∗ ≡ (x1∗2 + x2∗2 )1=2 . Using Eqs. (B.1) and expanding the Kp (x30 ; ) in powers of  as in Eq. (1), we readily transform the preceding expression into √ u(x; ) ≡ Kp∗ (x3 )Up∗(1=2) (x∗ ) + Kp(1=2) (x3 )Up∗(1=2) (x∗ )    d’ ∗ (1) ∗(1=2) ∗ ∗ ∗(1=2) ∗ + Kp (x3 )Up (x ) + Kp (x3 ) (x3 )23 (x3 ) · Up (x )  d x3 + O(3=2 ) + O(r ∗ ):

(B.2)

In this equation, Up∗(1=2) (x) is given by Eq. (44) and represents the fundamental mode p displacement ;eld homogeneous of degree 1=2 adapted to the “intermediary” crack con;guration, ∗23 (x3 ) is the tensor de;ned by (45) and r ∗ ≡ (x1∗2 + x2∗2 )1=2 . The term involving ∗23 (x3 ) here arises from diPerentiation of the basis vectors ei∗ . Eqs. (9)4 and (B.2) then readily yield Eq. (43), where y∗ is related to x∗ through Eq. (46) and ∗ ≡ r ∗ =. Appendix C. The aim of this appendix is to study the quantity K (1) (y3 ).

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V. Lazarus et al. / J. Mech. Phys. Solids 49 (2001) 1399 – 1420

Comparing Eqs. (11)1 and (43), we get v(3=2) (y) = Kp(1) (y3 )Up∗(1=2) (y∗ ) + Kp∗ (y3 ) for

∗

d’ (y3 )∗23 (y3 ) · Up∗(1=2) (y∗ ) + O( dy3

∗

)

(C.1)

→ 0. Now, by Eqs. (41) and (48)1 ,

vq(3=2) (y) = −Fpq (’(y3 ))Up∗(1=2) (y∗ ) + O(

∗

):

(C.2)

Also, let Ipq (’(y3 )) denote the SIF associated to the ;elds vq(3=2) (y) verifying Eq. (40), de;ned by vq(3=2) (y) ≡ Ipq (’(y3 ))Up∗(1=2) (y) + O(

∗

)

(C.3)

∗

for → 0. Inserting Eqs. (C.2) and (C.3) into Eq. (37) and the result into Eq. (C.1), we get, using Eq. (49)1 : d’ Kq (y3 )vq(3=2) (y) dy3 = [Kp(1) (y3 ) + Kr Ksq (y3 ; {Aqr ◦ ’})Fps (’(y3 )) − Bq Ipq (’(y3 ))] ×Up∗(1=2) (y∗ ) + Fpq (’(y3 ))Kq

d’ (y3 )∗23 (y3 ) · Up∗(1=2) (y∗ ) + O( dy3

∗

): (C.4)

Eq. (C.4) shows that the expression d’ Kq (y3 )[vq(3=2) (y) − Fpq (’(y3 ))∗23 (y3 ) · Up∗(1=2) (y∗ )] dy3 is, up to some O( ∗ ) term, a linear combination of the fundamental ;elds Up∗(1=2) (y∗ ). Since the SIF Kq here are independent quantities, this means that for every value of q, the term [ : : : ] must be such a linear combination. Thus there are some quantities P˜ pq (’(y3 )), depending only on ’(y3 ) (just like the term [ : : : ]), such that Eq. (51) of the text be satis;ed. The quantities P˜ pq (’(y3 )) can be interpreted as the “SIF” associated to the ;eld vq(3=2) (y), with the remark that the asymptotic expression of this ;eld involves the additional term Fpq (’(y3 ))∗23 (y3 ) · Up∗(1=2) (y∗ ). This term, which is not a linear combination of the Up∗(1=2) (y∗ ), arises from the presence of body forces and tractions exerted on the crack lips in system (38). Inserting Eq. (51) into Eq. (C.4) and using de;nition (32) of K(x3 ; {f}), one gets Kp(1) (y3 ) = Ipq (’(y3 ))Bq − Fps (’(y3 ))KT sq ()Aqr (’(y3 ))Kr dAqr d’ d’ + P˜ pq (’(y3 )) (y3 )Kq − Fps (’(y3 ))Nsq (0) (y3 )Kr (’(y3 )) dy3 d’ dy3
− Fps (’(y3 )) PV Zsq (; y3 ; y3 )[Aqr (’(y3 )) − Aqr (’(y3 ))]Kr dy3 : F

(C.5)

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1419

Let us temporarily consider the case where the kink angle is uniform along the crack front: ’(y3 ) ≡ ’(y3 ); ∀y3 . Then expression (C.5) for Kp(1) (y3 ) ≡ [Kp(1) (y3 )]’(y
)≡’(y3 ) 3 reduces to its ;rst two terms. Furthermore the problem is 2D so that the result derived by Leblond (1989) in that case applies and reads [K (1) (y3 )]’(y
)≡’(y3 ) = I (’(y3 )) · B + F(’(y3 )) · KT () · (F T · F)(’(y3 )) · K : 3

(C.6)

Comparison between the two expressions immediately yields A(’) ≡ −(F T · F)(’):

(C.7)

Insertion of these results into (C.5) yields the ;nal expression (4) of K (1) (y3 ) ≡ K (1) (x3 ), in the general case where ’(x3 ) = Cst., where P(’) is given by Eq. (50). References Amestoy, M., Bui, H.D., Dang Van, K., 1979. D,eviation in;nit,esimale d’une ;ssure dans une direction arbitraire. C.R. Acad. Sci. Paris S,er. B 289, 99–102. Amestoy, M., Leblond, J.-B., 1992. Crack paths in plane situations — II: Detailed form of the expansion of the stress intensity factors. Int. J. Solids Struct. 29, 465–501. Ball, R.C., Larralde, H., 1995. Linear stability analysis of planar straight cracks propagating quasistatically under type I loading. Int. J. Fract. 71, 365–377. Bueckner, H.F., 1987. Weight functions and fundamental ;elds for the penny-shaped and the half-plane crack in three-space. Int. J. Solids Struct. 23, 57–93. Gao, H., 1992. Three-dimensional slightly nonplanar cracks. ASME J. Appl. Mech. 59, 335–343. Gao, H., Rice, J.R., 1986. Shear stress intensity factors for planar crack with slightly curved front. ASME J. Appl. Mech. 53, 774–778. Hourlier, F., Pineau, A., 1979. Fissuration par fatigue sous sollicitations polymodales (mode I ondul,e + mode III permanent) d’un acier pour rotors 26 NCDV 14. M,em. Sci. Rev. M,etall. 76, 175 –185. Lazarus, V., 1997. Quelques problYemes tridimensionnels de m,ecanique de la rupture fragile. Ph.D. Thesis, Universit,e Pierre et Marie Curie, Paris. Leblond, J.-B., 1989. Crack paths in plane situations — I: General form of the expansion of the stress intensity factors. Int. J. Solids Struct. 25, 1311–1325. Leblond, J.-B., 1999. Crack paths in three-dimensional solids. I: Two-term expansion of the stress intensity factors — Application to crack path stability in hydraulic fracturing. Int. J. Solids Struct. 36, 79–103. Leblond, J.-B., Lazarus, V., Mouchrif, S.-E., 1999. Crack paths in three-dimensional solids. II: Three-term expansion of the stress intensity factors — Applications and perspectives. Int. J. Solids Struct. 36, 105–142. Leguillon, D., 1993. Asymptotic and numerical analysis of crack branching in non-isotropic materials. Eur. J. Mech. A=Solids 12, 33–51. Mouchrif, S.-E., 1994. Trajets de propagation de ;ssures en m,ecanique lin,eaire tridimensionnelle de la rupture fragile. Ph.D. Thesis, Universit,e Pierre et Marie Curie, Paris. Movchan, A.B., Gao, H., Willis, J.R., 1998. On perturbations of plane cracks. Int. J. Solids Struct. 35, 3419–3453. Muskhelishvili, N.-I., 1953. Some Basic Problems of the Mathematical Theory of Elasticity. NoordhoP, Groningen. Palaniswamy, K., Knauss, W.-G., 1975. Crack extension in brittle solids. In: Mechanics Today, Vol. 4, Nemat-Nasser, S. (Ed.), Pergamon Press, New York, pp. 87–148. Rice, J.R., 1985. First-order variation in elastic ;elds due to variation in location of a planar crack front. ASME J. Appl. Mech. 52, 571–579. Sih, G.C., 1965. Stress distribution near internal crack tips for longitudinal shear problems. ASME J. Appl. Mech. 32, 51–58.

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Sommer, E., 1969. Formation of fracture “lances” in glass. Eng. Fract. Mech. 1, 539–546. Sumi, Y., Nemat-Nasser, S., Keer, L.M., 1983. On crack branching and curving in a ;nite body. Int. J. Fract. 21, 67–79. Xu, G., Bower, A.-F., Ortiz, M., 1994. An analysis of non-planar crack growth under mixed mode loading. Int. J. Solids Struct. 16, 2167–2193. Yates, J.R., Mohammed, R.A., 1994. Crack propagation under mixed mode (I+III) loading. Proceedings of the Fourth International Conference on Biaxial=Multiaxial Fatigue, Saint-Germain-en-Laye, France, Vol. II, pp. 99 –106.