2138 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 1997). The mechanical response of such materials can differ substantially from that of bulk materials due to the presence of interfaces. To enable a non brittle mode of failure, the interface strength between adjoining phases must be adjusted to decrease the stress concentration effect of any crack that occurs in a phase, reducing the likelihood that it will propagate into the next phase The capability of an interface to deflect a crack is usually analysed in terms of competition between deflection and penetration for a stationary crack terminating at the interface at a normal or an oblique angle. This approach provides a crack defec tion condition involving a strength ratio or a toughness ratio to be compared with a critical ratio depending on the elastic mismatch between the two phases(He and Hutchinson, 1989; Gupta et al., 1992). This criterion has been applied to describe the crack deflection observed in various combinations of brittle bimaterials(Kumar and Singh, 1998; Liu et al., 1998; Martin et al., 1998). In spite of the uncertainties related to the determination of the properties of micron sized materials, the criterion sometimes fails to predict interfacial deflection(Warrier et al., 1997; Kovar et al 1997)or interfacial penetration(Ahn et al., 1998). Among the various reason invoked to explain such discrepancies, a different mechanism of crack deflection can be postulated( Cook and Gordon, 1964): interfacial failure occurs ahead of the main crack and deflection results from linking between the interfacial crack and the pri- mary crack. This mechanism has been experimentally evidenced in some bimaterial systems(Theocaris and Milios, 1983; Warrier et al., 1997; Zhang and Lewandowski, 1997; Majumdar et al., 1998)or suggested as a result of a micromechanical analysis (Pagano, 1998). Further, Lee et al. (1996)and Clegg et al.(1997) considered crack deflection as an interaction between the primary crack and the growth of an interface defect. It must be pointed out that the modelling of this deflection mechanism has al ways been restricted to an interface between similar materials The purpose of this paper is to establish a crack deflection model taking into account the interface debonding ahead of a primary crack. The analysis is performed within the framework of two dimensional linear elasticity. Asymptotic expressions are used to derive the change in potential energy induced by crack nucleation growth. The competition between the growth of the main crack and the interface debonding(crack nucleation) is first investigated. The derived criterion is compared with the experimental results obtained by Lee et al. (1996)on cracked brittle lami- nates submitted to a four-point bending test Further, the competition between the interface debonding and the interface pen- etration ahead of the primary crack is examined. The approach leads to an energy multi-criterion involving the ligament width, the various crack extensions along and across the interface and the elastic mismatch of the bimaterial components 2. The notched bimaterial specimen The analysis is restricted to plane strain elasticity. We consider(Fig. 1)a bima- terial specimen( thickness e) made of two isotropic layers bonded together along an interface with perfect transmission conditions: displacements and tractions are2138 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 1997). The mechanical response of such materials can differ substantially from that of bulk materials due to the presence of interfaces. To enable a non brittle mode of failure, the interface strength between adjoining phases must be adjusted to decrease the stress concentration effect of any crack that occurs in a phase, reducing the likelihood that it will propagate into the next phase. The capability of an interface to deflect a crack is usually analysed in terms of competition between deflection and penetration for a stationary crack terminating at the interface at a normal or an oblique angle. This approach provides a crack deflec￾tion condition involving a strength ratio or a toughness ratio to be compared with a critical ratio depending on the elastic mismatch between the two phases (He and Hutchinson, 1989; Gupta et al., 1992). This criterion has been applied to describe the crack deflection observed in various combinations of brittle bimaterials (Kumar and Singh, 1998; Liu et al., 1998; Martin et al., 1998). In spite of the uncertainties related to the determination of the properties of micron sized materials, the criterion sometimes fails to predict interfacial deflection (Warrier et al., 1997; Kovar et al., 1997) or interfacial penetration (Ahn et al., 1998). Among the various reasons invoked to explain such discrepancies, a different mechanism of crack deflection can be postulated (Cook and Gordon, 1964): interfacial failure occurs ahead of the main crack and deflection results from linking between the interfacial crack and the pri￾mary crack. This mechanism has been experimentally evidenced in some bimaterial systems (Theocaris and Milios, 1983; Warrier et al., 1997; Zhang and Lewandowski, 1997; Majumdar et al., 1998) or suggested as a result of a micromechanical analysis (Pagano, 1998). Further, Lee et al. (1996) and Clegg et al. (1997) considered crack deflection as an interaction between the primary crack and the growth of an interface defect. It must be pointed out that the modelling of this deflection mechanism has always been restricted to an interface between similar materials. The purpose of this paper is to establish a crack deflection model taking into account the interface debonding ahead of a primary crack. The analysis is performed within the framework of two dimensional linear elasticity. Asymptotic expressions are used to derive the change in potential energy induced by crack nucleation or growth. The competition between the growth of the main crack and the interface debonding (crack nucleation) is first investigated. The derived criterion is compared with the experimental results obtained by Lee et al. (1996) on cracked brittle lami￾nates submitted to a four-point bending test. Further, the competition between the interface debonding and the interface pen￾etration ahead of the primary crack is examined. The approach leads to an energy multi-criterion involving the ligament width, the various crack extensions along and across the interface and the elastic mismatch of the bimaterial components. 2. The notched bimaterial specimen The analysis is restricted to plane strain elasticity. We consider (Fig. 1) a bima￾terial specimen (thickness e) made of two isotropic layers bonded together along an interface with perfect transmission conditions: displacements and tractions are