2146 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 while the second one vanishes B=0. The agreement between our calculations and He and Hutchinsons results is quite good. It is to be pointed out that HH criterion ha been first defined with the assumption B=0. It has been shown next by Martinez and Gupta(1994) that the influence of B on the criterion remains weak over the range B=-0.2, 0.2. The LS criterion, based on a numerical-analytical approach, can be carried out whatever B and extends without additional complexity to in-axis ortho- ropic substrates The right part a>0 of Fig. 6 corresponds to a weak singularity, when the main crack lies in the softest of the two components and the left one a<o corresponds to a strong singularity when the stiffest material is broken 5. Interface debonding ahead of the crack tip Herein, we do not propose an improved branching criterion but the analysis of the debonding mechanism ahead of the crack tip, disregarding at first any competition with the penetration event. This is highlighted by the forthcoming results involving only material 1 and interface toughnesses G@) and G, but not the second substrate The geometry includes a ligament ahead of the main crack and is first considered prior to any interface debonding. The aim of this section is to analyse the competition between the growth of this main crack(the shortening of the ligament)and a possible debonding of the interface ahead of the crack tip 5.1. The primary crack growth t We suppose that the main crack is growing within material I towards the interface Is consider an increment Sl<<I of the initial crack length L (remember that 7 is not the crack length but the ligament width). The corresponding dimensionless parameter is Se=Slle. The associated length in the stretched domain is SelE(Fig. 7) The energy release rate is minus the derivative of the potential energy W with material 2 interface stretched crack rack material 1 ig. 7. The main crack growth in the stretched domain Q2n.2146 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 while the second one vanishes b=0. The agreement between our calculations and He and Hutchinson’s results is quite good. It is to be pointed out that HH criterion has been first defined with the assumption b=0. It has been shown next by Martinez and Gupta (1994) that the influence of b on the criterion remains weak over the range b=20.2, 0.2. The LS criterion, based on a numerical-analytical approach, can be carried out whatever b and extends without additional complexity to in-axis ortho￾tropic substrates. The right part a.0 of Fig. 6 corresponds to a weak singularity, when the main crack lies in the softest of the two components and the left one a,0 corresponds to a strong singularity when the stiffest material is broken. 5. Interface debonding ahead of the crack tip Herein, we do not propose an improved branching criterion but the analysis of the debonding mechanism ahead of the crack tip, disregarding at first any competition with the penetration event. This is highlighted by the forthcoming results involving only material 1 and interface toughnesses G(1) c and G(i) c , but not the second substrate toughness G(2) c . The geometry includes a ligament ahead of the main crack and is first considered prior to any interface debonding. The aim of this section is to analyse the competition between the growth of this main crack (the shortening of the ligament) and a possible debonding of the interface ahead of the crack tip. 5.1. The primary crack growth We suppose that the main crack is growing within material 1 towards the interface. Let us consider an increment dl,,l of the initial crack length L (remember that l is not the crack length but the ligament width). The corresponding dimensionless parameter is de=dl/e. The associated length in the stretched domain is de/e (Fig. 7). The energy release rate is minus the derivative of the potential energy W with Fig. 7. The main crack growth in the stretched domain Vin