2142 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 Two intensity factors k Eq. (3)and K Eq. (5)are involved in the previous pressions. Their computation is based on contour integrals y(Leguillon and San- chez-Palencia, 1987) (r) y(yPu) y(ru, r), pupU) For any fields U and y satisfying the equilibrium equation in a wedge and stress free boundary conditions on the edges, y is a contour independent integral which is defined by y(U,D=o(Unv-o(nnu)ds C is any contour in n2o to compute k or Din to compute K surrounding the origin and starting and finishing at, the primary crack stress free edges. The unit normal n to C points towards the origin Indeed UD and y Eq.(7)are a priori unknown, and must be replaced by the corresponding finite element approximations. Note that in Eq. (7)V can be used in place of y since y(,,p! 4. Application to fracture mechanics 4.I. Differential and incremental approaches on The previous expansions Eqs. ( 1)(6)express the effect of a small perturbation a solution to a structural problem. Above, the perturbation is a narrow ligament remaining between a main crack tip and an interface, but it can be also a short crack increment with small dimensionless length n. Roughly speaking, this increment is a forward growth while the ligament is a"backward"one. Replacing formally e with n in the inner and outer expansions and substituting these relations in the potential energy expression allow, at the leading order, the change in potential energy between the unperturbed(before crack growth) and perturbed (after crack growth) states to be defined(Leguillon, 1989) H(0)-W(n)=k2Km2+ The energy release rate is the driving force associated with n. It is the derivative of w with respect to this variable G=limW(O-W( limk2Kn2 n-07 0 Obviously, this limit is meaningful for the classical crack tip singularity characterized by 2=1/2. It vanishes if 2>1/2(weak singularity, a>0) and it tends to infinity if n<1/2(strong singularity, a<o). These situations are met in case of a crack2142 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 Two intensity factors k Eq. (3) and K Eq. (5) are involved in the previous expressions. Their computation is based on contour integrals C (Leguillon and San￾chez-Palencia, 1987) k5 C(U0 ,r−lu− ) C(rlu+ ,r−lu− ) , K5 C(Vˆ 1 ,rlu+ ) C(r−lu− ,rlu+ ) . (7) For any fields U and V satisfying the equilibrium equation in a wedge and stress free boundary conditions on the edges, C is a contour independent integral which is defined by C(U,V)5E C (s(U)nV2s(V)nU) ds. (8) C is any contour in V0 to compute k or Vin to compute K surrounding the origin and starting and finishing at, the primary crack stress free edges. The unit normal n to C points towards the origin. Indeed U0 and Vˆ 1 Eq. (7) are a priori unknown, and must be replaced by the corresponding finite element approximations. Note that in Eq. (7) V1 can be used in place of Vˆ 1 since C(rlu+ , rlu+ )=0. 4. Application to fracture mechanics 4.1. Differential and incremental approaches The previous expansions Eqs. (1)–(6) express the effect of a small perturbation on a solution to a structural problem. Above, the perturbation is a narrow ligament remaining between a main crack tip and an interface, but it can be also a short crack increment with small dimensionless length h. Roughly speaking, this increment is a forward growth while the ligament is a “backward” one. Replacing formally e with h in the inner and outer expansions and substituting these relations in the potential energy expression allow, at the leading order, the change in potential energy between the unperturbed (before crack growth) and perturbed (after crack growth) states to be defined (Leguillon, 1989) W(0)2W(h)5k 2 Kh2l 1 …. (9) The energy release rate is the driving force associated with h. It is the derivative of 2W with respect to this variable G5lim h→0 W(0)−W(h) h 5lim h→0 k 2 Kh2l−1 1…. (10) Obviously, this limit is meaningful for the classical crack tip singularity characterized by l=1/2. It vanishes if l.1/2 (weak singularity, a.0) and it tends to infinity if l,1/2 (strong singularity, a,0). These situations are met in case of a crack