Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 268 2 2 max max max max max 2exp(1) 1 exp exp N N T T T T T N N T g g p g g p g g g g                          (2) where g Nmax and g Tmax are the normal and tangential displacements corresponding to the normal and tangential cohesive strength p Nmax and p Tmax , respectively. The model is totally defined by four independent parameters, e.g. the fracture energies in the normal (mode-I) and tangential (mode-II) directions,  N and  T , and the cohesive strengths p Nmax and p Tmax , respectively. For pure modes I and II, the uncoupled stresses p N =p N (g N ,g T =0) and p T =p T (g N =0, g T ) are depicted in Fig. 1. The fracture energies for pure modes,  N and  T , are then given by max max 0 ( , 0) exp(1) N N N T N N N p g g dg p g       (3) max max 0 1 ( 0, ) exp(1) 2 T T N T T T T p g g dg p g       (4) (a) (b) Figure 1 : CZM1: (a) Pure mode I; (b) Pure mode II. A second exponential law is then considered, which is a modified potential-based XN model recently proposed by McGarry et al. [9]. This model, henceforth indicated as CZM2, avoids unphysical repulsive normal tractions and instantaneous negative incremental energy dissipation under displacement controlled monotonic mixed-mode separation when the work of tangential separation exceeds the work of normal separation. In addition, it provides a benefit of correct penalisation of mixed-mode over-closure in contrast to the XN model. The modified form of the XN potential function is derived from the potential 2 2 max max max max ( ) ( ) ( ) 1 ( , ) 1 exp 1 exp 1 1 T N T N T N T N T N N N N T f g g f g g f g g q g r q g g r q g g r g r g                                                                   (5) where 2 2 max ( ) 1 exp T T T g f g m m g           (6) Coupling between normal and tangential tractions are here defined by three coupling parameters: the traction-free normal separation following complete shear separation denoted as r , the parameter m , which controls the zone of influence of mode II behaviour for mixed-mode conditions, and the already defined parameter q . The interface interaction vector p =( p N , p T ) is obtained from p ( g ) = ∂  ( g )/∂ g , where g is the relative displacement vector g =( g N , g T ). Six characteristic parameters are necessary to define the model, e.g. the fracture energies  N and  T always defined by Eq. (4), (5), the cohesive strengths p Nmax and p Tmax and the coupling parameters r and m . The uncoupled relationships p N =p N (g N ,g T =0) and p T =p T (g N =0, g T ) are the same as in Fig. 1. We then analyze two common bilinear models proposed by Högberg [10] and Camanho et al. [11], which are largely used