Issue 17

M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01 7 of two grains sharing the same interface. A criterion of equivalence of cross-sectional areas is used to compute the diameter d of a polyhedral grain. Although the interface thickness l 2 is approximately one order of magnitude smaller than the grain diameter, as shown in Fig. 2(b), its simplification as a zero-thickness region, also called boundary layer , is not always possible. Numerical simulations considering an elasto-plastic behaviour of the interfaces suggest that the ductility of the material is strongly affected by these finite thickness regions [4]. Figure 2 : Dependency of the interface thickness on the grain diameter according to the relation proposed in [4]. In order to simplify the real material microstructure by considering a finite element discretization with zero-thickness interfaces, a suitable interface constitutive law has to be used (see [5-16] for a wide range of problems modelled using CZMs). Here, we consider the nonlocal CZM recently proposed in [2] for finite thickness interfaces. The traction- separation relations that describe the nonlinear response of the interface are the following: 1 T e e gD D      (1a) 1 N e e gD D      (1b) where  and  are the tangential and normal cohesive tractions. The parameters T g and N g denote, respectively, the tangential and normal anelastic displacements evaluated at the boundaries of the finite thickness interface. Finally, e  and e  are the threshold values of the cohesive tractions for the onset of damage that correspond to the global tangential and normal displacements e  and e  of the interface region. The damage variable D (0 1) D   is computed as follows: /2 2 2 c c w u D w u                        (2) where c w and c u are material parameters analogous to the critical opening and sliding displacements Nc l and Tc l used in standard CZMs [9] and  is a free parameter. The displacements u and w are given by: 2 2 T e l u g G      (3a) 2 2 N e l w g E      (3b) where 2 l is the thickness of the interface, 2 E and 2 G are the initially undamaged normal and tangential elastic moduli of the interface material. Changing the parameter  , different shapes of the CZM can be obtained, as shown in Fig. 3 for a d l 2 d = grain diameter l 2 = interface thickness