D. Leguillon er al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2141 other words the outer terms behaviour approaching the ligament area must match with the inner terms"behaviour going at infinity More precisely, U(x1-X2) has a singular behaviour approaching the crack tip LP(x1x2)=0,0)+kr丝()+ where r and 0 are the polar coordinates with origin at the crack tip. The exponent is positive, smaller than I and, depending on the relative stiffness of the substrates can be larger, equal to or smaller than 1/2(the classical crack tip singularity)(Zak and Williams, 1963). Two modes are usually involved in such a situation, but it is assumed here that the applied loads only generate a symmetrical one in Eq (3)(three or four-point bending tests on notched specimens for instance). Coefficient k is the intensity factor. We avoid the expression"stress"intensity factor which we reserve for the usual fracture mechanics crack tip singularity. The explanation for the index + will be found below This leads to Lv1y2)=((0,0)F1(e)=kex,y(y1y2)=p2t(0)+(0v1y2) where p=r/E. Here, PIGi,2)is the solution to a well-posed problem with decaying conditions at infinity and it is once more a singular behaviour which is involved p(13y2)Kp-y(6)asp→∞, where K denotes the corresponding intensity factor. It is independent of the applied loads, of the global geometry of the structure and of the actual length ee of the perturbation. It depends only on local material properties and geometry; more pre- cisely it depends on the shape of the perturbation not on its size. Function a singular behaviour at infinity whereas the other, with a positive exponent expresses a singular behaviour at the origin. This entails the complementary matching f(e)=kKe,(x1x2)=ry(6)+C(x1x2) where U(x1x2) is the solution to a well-posed problem in the unperturbed structure no. The outer expansion Eqs. (1),(3)and(6) provides a formulation for the remote fields whereas the inner expansion Eqs. (2),(4)and(5) plays the same role for the near field The different terms in the asymptotics as well as the singular modes and their dual counterparts can be evaluated as indicated in Leguillon and Sanchez-Palencia (1987). The far(outer) fields are obtained by classical finite element computations in the domain n2o. The computation difficulty of the inner expansion terms comes from the infinite domain ( n. It is necessary to bound it artificially at a large distance from the perturbation(ligament with a length unity) to perform computations. From a theoretical point of view, y Gi, y2) has a finite energy while Vvi J2)has not, but his difference disappears on a bounded domain and vv1y2) can be computed directly with prescribed displacements Put(@)along the artificially created bound- ary. Such a domain must be large but the problems under consideration are inde pendent of the applied loads and can be solved once and for allD. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2141 other words the outer terms’ behaviour approaching the ligament area must match with the inner terms’ behaviour going at infinity. More precisely, U0 (x1,x2) has a singular behaviour approaching the crack tip U0 (x1,x2)5U0 (0,0)1k rlu+ (q)1 … , (3) where r and q are the polar coordinates with origin at the crack tip. The exponent l is positive, smaller than 1 and, depending on the relative stiffness of the substrates, can be larger, equal to or smaller than 1/2 (the classical crack tip singularity) (Zak and Williams, 1963). Two modes are usually involved in such a situation, but it is assumed here that the applied loads only generate a symmetrical one in Eq. (3) (three or four-point bending tests on notched specimens for instance). Coefficient k is the intensity factor. We avoid the expression “stress” intensity factor which we reserve for the usual fracture mechanics crack tip singularity. The explanation for the index + will be found below. This leads to V0 (y1,y2)5U0 (0,0),F1(e)5ke l ,V1 (y1,y2)5rlu+ (q)1Vˆ 1 (y1,y2), (4) where r=r/e. Here, Vˆ 1 (y1,y2) is the solution to a well-posed problem with decaying conditions at infinity and it is once more a singular behaviour which is involved Vˆ 1 (y1,y2)|K r−lu− (q) as r→`, (5) where K denotes the corresponding intensity factor. It is independent of the applied loads, of the global geometry of the structure and of the actual length ee of the perturbation. It depends only on local material properties and geometry; more pre￾cisely it depends on the shape of the perturbation not on its size. Function r−lu− (q) is the dual mode to rlu+ (q); the former, with a negative exponent expresses a singular behaviour at infinity whereas the other, with a positive exponent expresses a singular behaviour at the origin. This entails the complementary matching f1(e)5k K e2l ,U1 (x1,x2)5r−lu− (q)1Uˆ 1 (x1,x2), (6) where Uˆ 1 (x1,x2) is the solution to a well-posed problem in the unperturbed structure V0 . The outer expansion Eqs. (1), (3) and (6) provides a formulation for the remote fields whereas the inner expansion Eqs. (2), (4) and (5) plays the same role for the near fields. The different terms in the asymptotics as well as the singular modes and their dual counterparts can be evaluated as indicated in Leguillon and Sanchez-Palencia (1987). The far (outer) fields are obtained by classical finite element computations in the domain V0 . The computation difficulty of the inner expansion terms comes from the infinite domain Vin. It is necessary to bound it artificially at a large distance from the perturbation (ligament with a length unity) to perform computations. From a theoretical point of view, Vˆ 1 (y1, y2) has a finite energy while V1 (y1,y2) has not, but this difference disappears on a bounded domain and V1 (y1,y2) can be computed directly with prescribed displacements rlu+ (q) along the artificially created bound￾ary. Such a domain must be large but the problems under consideration are inde￾pendent of the applied loads and can be solved once and for all