D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2143 impinging on an interface between two elastic materials, depending if the crack lies on the soft side (weak singularity) or on the stiff side(strong singularity). These two particular cases play an important role in fracture mechanics of heterogeneous materials(Leguillon and Sanchez-Palencia, 1992) If G=0 the critical value G(material toughness) is never reached and the griffith criterion cannot be fulfilled whatever the applied loads. This is a drawback of the classical differential theory which may be overcome with the help of the following incremental condition derived from expansion Eq (9) w(0)-W(n)=G n= (or-(n-k"Kn2-IeG (11) n Eq.( 1)is a necessary condition for fracture and provides an incremental criterion which coincides with the differential theory when it holds true(=1/2)and which can still bc used when this theory fails. Compared to the usual Griffith criterion, it contains the additional unknown parameter n. This incremental approach will be used in the next sections On the opposite, if G-o, the criterion is violated for any non-zero applied load as small as it can be. Eq. (10)implicitly assumes that the derivative -aw/an exists but this existence is clearly questionable. A modified criterion is examined by the authors(Leguillon et al., 1999)in this special case. It is based on the principle of maximum decrease of the total energy as suggested by Lawn(1993) 4.2. He and Hutchinson criterion for interface deflection He and Hutchinson(1989)have considered the problem of a crack impinging(n ligament) on an interface between two isotropic elastic materials: the crack lies in one material and can either penetrate the other one or branch along the interface For simplicity, we limit here the comparison to a crack normal to the interface and a penetration(dimensionless length np)or a double symmetric deflection along the interface(total dimensionless length 2nd)(Fig. 4). From Eq.(I1), deflection is pro- moted if Kp( np (12) where G@ and G2) are the respective toughness of the interface and of material 2 and where Kd and K, are the intensity factors Eq. (5)extracted from the term yo1V2)of an inner expansion considering a unit penetration or a unit deflection (F1g.5) In addition to the respective toughness of material 2 and of the interface, such a criterion Eq(12)requires the knowledge of the elementary increments in the two directions(or at least their ratio) except if 2 =1/2, but in that case the problem turns to be a classical crack branching one in a homogeneous material(Leguillon, 1993) with anisotropic toughness. The cracks' extension lengths should be related to the material and the interface microstructure. Moreover such increments' sizes should differ and often remain unknownD. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2143 impinging on an interface between two elastic materials, depending if the crack lies on the soft side (weak singularity) or on the stiff side (strong singularity). These two particular cases play an important role in fracture mechanics of heterogeneous materials (Leguillon and Sanchez-Palencia, 1992). If G=0 the critical value Gc (material toughness) is never reached and the Griffith criterion cannot be fulfilled whatever the applied loads. This is a drawback of the classical differential theory which may be overcome with the help of the following incremental condition derived from expansion Eq. (9) W(0)2W(h)$Gch⇒ W(0)−W(h) h 5k 2 Kh2l−1 $Gc. (11) Eq. (11) is a necessary condition for fracture and provides an incremental criterion which coincides with the differential theory when it holds true (l=1/2) and which can still bc used when this theory fails. Compared to the usual Griffith criterion, it contains the additional unknown parameter h. This incremental approach will be used in the next sections. On the opposite, if G→`, the criterion is violated for any non-zero applied load as small as it can be. Eq. (10) implicitly assumes that the derivative 2∂W/∂h exists but this existence is clearly questionable. A modified criterion is examined by the authors (Leguillon et al., 1999) in this special case. It is based on the principle of maximum decrease of the total energy as suggested by Lawn (1993). 4.2. He and Hutchinson criterion for interface deflection He and Hutchinson (1989) have considered the problem of a crack impinging (no ligament) on an interface between two isotropic elastic materials: the crack lies in one material and can either penetrate the other one or branch along the interface. For simplicity, we limit here the comparison to a crack normal to the interface and a penetration (dimensionless length hp) or a double symmetric deflection along the interface (total dimensionless length 2hd) (Fig. 4). From Eq. (11), deflection is pro￾moted if Kd Kp S 2hd hp D 2l−1 $ G(i) c G(2) c , (12) where G(i) c and G(2) c are the respective toughness of the interface and of material 2 and where Kd and Kp are the intensity factors Eq. (5) extracted from the term V1 (y1,y2) of an inner expansion considering a unit penetration or a unit deflection (Fig. 5). In addition to the respective toughness of material 2 and of the interface, such a criterion Eq. (12) requires the knowledge of the elementary increments in the two directions (or at least their ratio) except if l=1/2, but in that case the problem turns to be a classical crack branching one in a homogeneous material (Leguillon, 1993) with anisotropic toughness. The cracks’ extension lengths should be related to the material and the interface microstructure. Moreover such increments’ sizes should differ and often remain unknown