Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 278 N UMERICAL IMPLEMENTATION he results illustrated thus far were found analytically. Only the integrals in eq. (23) were computed numerically. This section is devoted to the numerical verification, which is performed through the finite element simulation of a simple mixed-mode debonding problem, i.e. a mixed-mode patch test (see Fig. 15). A 10x10 mm 2 plate is pulled apart by applying uniform vertical and horizontal displacements ( v and u ) to the top and right sides, respectively, until complete failure. The normal, tangential and total works of separation are evaluated once again for non proportional loading paths 1 and 2, by varying the mixed-mode conditions between pure modes, or equivalently the loading time histories, in terms of u and v . In the numerical model, all the interface laws analytically described in the previous sections have been implemented into a contact element based on the node-to-segment strategy as employed in [12] and generalized to handle cohesive forces in both the normal and tangential directions. Each element contribution for the cohesive/contact forces is added to the global virtual work equation as follows      C N N T T W F g F g (24) where  W C is the interface contribution to the virtual work, and F N and F T denote, respectively, the normal and tangential cohesive/contact force. Plane-stress four-nodes isoparametric elements have been used to model the two adherends which are characterized by an elastic modulus E=150GPa and a Poisson’s ratio  =0 . The cohesive strengths and fracture energies are the same as in the analytical investigation. The non-linear problem is solved with an iterative Newton- Raphson procedure, where the global tangent stiffness matrix is properly obtained by the consistent linearization of Eq. (24). The discretization is refined appropriately to yield mesh-independent results. The model is implemented in the finite element code FEAP (courtesy of Prof. R.L. Taylor). A good agreement between the analytical and numerical results is always obtained in terms of tangential, normal and total energies, as visible from Fig. 9-14, confirming all the physical inconsistencies previously discussed, and the path dependency of the energy dissipation. (a) (b) (c) Figure 15 : Geometry of mode I (a) , mode II (b) and mixed mode (c) patch test. T HERMODYNAMICALLY CONSISTENT REFORMULATION OF THE CZM1 MODEL Fundamentals s shown in the previous sections, the CZM1 model is the only one not leading to inconsistencies in the behavior of interfaces subjected to mixed-mode loading. This law allows for different values of the fracture energy in the normal and tangential directions, as measured experimentally. However, the law as formulated originally from van den Bosch et al. [7] is not based on a potential. As shown in [15], non-potential-based models do not satisfy the symmetry requirements for the tangent matrix which has to be verified at least for elastic unloading between interfaces, and introduce independent models for loading and unloading (in fact, unloading was not explicitly dealt with in [7]). The last feature is in contrast with thermodynamical requirements, for which loading and unloading are uniquely defined by means of the same Helmholtz energy [16]. Therefore, belonging to the category of non-potential based laws, CZM1 should be not thermodynamically consistent. In this second part of the paper, a suitable reformulation of the CZM1 is elaborated. Through this reformulation, it is demonstrated that this model is indeed thermodynamically consistent, which also agrees T A