Issue 49

F.J.P. Moreira et alii, Frattura ed Integrità Strutturale, 49 (2019) 435-449; DOI: 10.3221/IGF-ESIS.49.42 440 Mesh bias effects were employed in the models to grade the elements’ size, enabling to reduce the total number of elements but without compromising the accuracy. This was done by considering a more refined mesh near the adhesive layer and towards its edges, and a coarser mesh in the zones with less stress variations [28]. The elements’ side dimensions in the adhesive layer for the XFEM simulations were 0.2×0.2 mm along the bondline, i.e., only one element through-thickness was considered. On the other hand, the models for the stress analysis were comprised of 0.02×0.02 mm elements at the overlap edges. The boundary conditions consisted of fixing the base edges to simulate gripping in the testing machine, applying symmetry at the middle of the specimen and pulling the curved adherends’ edges in peel. XFEM formulation As an extension to the conventional FEM, the XFEM is based on the integration of enrichment functions in the FEM formulation [29]. These functions allow modelling the displacement jump between crack faces that occur during the propagation of a crack. The Abaqus ® XFEM formulation enables the user to create a pre-crack, or it can initiate cracks in un-cracked regions by using initiation criteria. In this last scenario, considered in this work, damage initiates and subsequently propagates during the simulation at regions experiencing stresses and/or strains higher than the corresponding limiting values. Six crack initiation criteria are available in Abaqus ® . The MAXPS (maximum principal stress) and MAXPE (maximum principal strain) criteria are based on the introduction of the following functions (by the respective order) 0 or max max 0 max max f f                   (1)  max and  0 max represent the current and allowable maximum principal stress. The Macaulay brackets indicate that a purely compressive stress state does not induce damage.  max and  0 max represent the current and allowable maximum principal strain. Crack growth for the MAXPS and MAXPE criteria is software defined as orthogonal to the maximum principal stress/strain direction. As a result of this, and due to the inherent mixed-mode loading of these joints, the crack grows fast towards the adherends. For these two criteria, the maximum load ( P m ) estimation was thus considered to take place at the time of first cracking in the adhesive layer. The MAXS (maximum nominal stress) and MAXE (maximum nominal strain) criteria are represented by the following functions, respectively 0 or n n s s 0 0 0 n s n s max , max , t t f f t t                   (2) t n and t s are the current normal and shear traction components to the cracked surface. t n 0 and t s 0 represent the respective limiting values. The strain parameters have identical significance. The quadratic nominal stress (QUADS) and quadratic nominal strain (QUADE) criteria are based on the introduction of the following functions, respectively or 2 2 2 2 n n s s 0 0 0 0 n s n s t t f f t t                                 (3) For the MAXS, MAXE, QUADS and QUADE criteria the user can select between horizontal or vertical crack growth (in this work horizontal growth, i.e., along the adhesive layers’ length, was selected). All the six aforementioned criteria are fulfilled, and damage initiates, when f reaches unity. For damage growth, the fundamental expression of the displacement vector u , including the displacements enrichment, is written as [30]     1 N i i N x H x         u u a i i (4) N i ( x ) and u i relate to the conventional FEM formulation, corresponding to the nodal shape functions and nodal displacement vector linked to the continuous part of the formulation, respectively. The second term between brackets, H ( x ) a i , is only active in the nodes for which any relating shape function is cut by the crack and can be expressed by the product of the nodal enriched degree of freedom vector including the mentioned nodes, a i , with the associated