Issue 17

M. Paggi et alii, Frattura ed Integrità Strutturale, 17 (2011) 5-14; DOI: 10.3221/IGF-ESIS.17.01 10 The parameters of the average Mode I traction-separation curve, corresponding to an average value of l 2 =0.15 μm, were selected in order to obtain an average Mode I fracture energy of 0.18 N/mm and a peak stress of 500 N/mm 2 , as suggested in [6] for polycrystalline interfaces (see the corresponding curve in Fig. 8 with solid line). More specifically, 0.53 c c     μm, 500 e c     N/mm 2 , 3 2 110 10 E   N/mm 2 were set for the grains, and the parameter α was selected as 0.0035. Alternatively, the parameters of the nonlocal CZM can be tuned to fit MD simulations, as illustrated in [2]. Interestingly, the shape of the nonlocal CZM can be matched with a good approximation with the standard CZM proposed by Tvergaard [9]. It is superimposed to Fig. 8 with dashed line and has the same Mode I fracture energy as our model. Figure 8 : Shape of the nonlocal CZM (average curve) used for the fracture simulations and that of the Tvergaard [9] model with the same average fracture energy. To quantify the effect of using the nonlocal CZM with random properties instead of the Tvergaard CZM with the same properties for all the interfaces, we consider an elastic modulus of the grains equal to 3 1 110 10 E   N/mm 2 and a Poisson ratio equal to ν=0.3 for all the grains. The material microstructure shown in Fig. 5 is first simplified by augmenting the size of the grains, according to the procedure outlined in [2]. After this preliminary operation, each grain is meshed with constant strain triangular elements according to a Delauney triangulation. Then, zero-thickness interface elements governed by the thickness-dependent nonlocal CZM are placed along the grain boundaries. The size of the finite elements used for discretizing the continuum has been chosen in order to be one order of magnitude smaller than the process zone size, estimated according to the method suggested in [16]. The Dirichlet boundary conditions imposed on the model are selected to reproduce a tensile test, i.e., the nodes pertaining to the left vertical boundary are restrained to the horizontal displacements, whereas horizontal displacement are imposed on the nodes of the vertical boundary on the right. A Newton-Raphson solution scheme is adopted to solve the nonlinear boundary value problem at each step. The tolerance for the internal Newton-Raphson loop used to compute the residual and the tangent stiffness matrix of the interface elements is chosen as 1×10 −12 . In Fig. 9, the evolution of the crack pattern using the Tvergaard CZM with the same parameters for all the interfaces (pictures at the top) is compared with that obtained using the nonlocal CZM and random fracture properties (pictures at the bottom), for three different deformation levels,  =0.100, 0.115 and 0.124. The last deformation level corresponds to the final failure of the samples. Dashed lines correspond to fictitious cracks, i.e., microcracks where cohesive tractions are still acting. Solid lines correspond to the interfaces with D =1, i.e., real stress-free microcracks. The evolution of cohesive microcracks is widely distributed in both cases. At a certain point, when the microcracks coalescence into a single rough macrocrack, a phenomenon of strain localization takes place. The cohesive microcracks far from the main crack experience a stress relief, whereas the deformation accumulates on the main crack. This is well evidenced by the fact that microcracks (dashed segments) almost disappear at the deformation level of 0.124. The final crack pattern in case of uniform interface fracture properties appears to be characterized by a single main crack. On the contrary, using the nonlocal CZM with thickness dependent fracture properties, we obtain a separation of some grains and a more diffuse crack pattern, which is often found in experiments. The homogenized stress-strain responses of the composite cell are compared in Fig. 10. The homogenized stress is computed by summing the horizontal reactions of the constrained nodes on the vertical boundary on the right, and dividing it for its length. The use of the Tvergaard model (local CZM with uniform interface fracture properties) leads to a higher peak stress than using the proposed nonlocal CZM. This is mainly due to the prevalence of subvertical microcracks