2140 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 material 2 material 1 Fig. 2. The unperturbed outer domain o2. which the ligament of length / is neglected, i. e the crack impinges on the interface formally corresponding to the geometry obtained with E=0 denoted Q2(Fig. 2) The second term is a small correction to the first approximation, U(xix is a solution to a problem based on the same unperturbed structure n. The exact formu- lation of this second problem is a consequence of the matching rules as explained below. From a numerical point of view, these terms are quite easy to compute, since there is no small ligament. Finite elements do not necessitate a particularly refined mesh in this area to account for the geometry Indeed, such a solution Eq. (1) is valid outside a vicinity of the ligament(which is absent in the unperturbed structure); it is a so-called"outer"expansion. Thus, the precise knowledge of the solution requires another description near the ligament. It is obtained, first by stretching the initial domain by l] and then considering the limit domain as 8-0. It is an unbounded one Q, with a semi-infinite crack leaving a unit ligament between its tip and the interface(Fig. 3) On this domain spanned by the space variables y=x E, the second expansion, the (x1x2)=(eyy2)=1y2)+Fi(e)(iy2)+… where F(e0 as a-0. To define correctly the different terms of this expansion conditions at infinity must be prescribed. They derive from the matching rules Expansion Eq. (1)holds true outside a vicinity of the ligament and the other one q.(2)inside. There must exist an intermediate region in which both are valid: in material 2 nterface material 1 crack Fig. 3. The stretched inner domain Qn with a unit ligament2140 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 Fig. 2. The unperturbed outer domain V0 . which the ligament of length l is neglected, i.e. the crack impinges on the interface, formally corresponding to the geometry obtained with e=0 denoted V0 (Fig. 2). The second term is a small correction to the first approximation, U1 (x1,x2) is a solution to a problem based on the same unperturbed structure V0 . The exact formu￾lation of this second problem is a consequence of the matching rules as explained below. From a numerical point of view, these terms are quite easy to compute, since there is no small ligament. Finite elements do not necessitate a particularly refined mesh in this area to account for the geometry. Indeed, such a solution Eq. (1) is valid outside a vicinity of the ligament (which is absent in the unperturbed structure); it is a so-called “outer” expansion. Thus, the precise knowledge of the solution requires another description near the ligament. It is obtained, first by stretching the initial domain by l/e and then considering the limit domain as e→0. It is an unbounded one Vin, with a semi-infinite crack leaving a unit ligament between its tip and the interface (Fig. 3). On this domain spanned by the space variables yi =xi /e, the second expansion, the “inner” one, writes Ue (x1,x2)5Ue (ey1,ey2)5V0 (y1,y2)1F1(e)V1 (y1,y2)1 … , (2) where F1(e)→0 as e→0. To define correctly the different terms of this expansion, conditions at infinity must be prescribed. They derive from the matching rules. Expansion Eq. (1) holds true outside a vicinity of the ligament and the other one Eq. (2) inside. There must exist an intermediate region in which both are valid: in Fig. 3. The stretched inner domain Vin with a unit ligament