Issue 12

M. Angelillo et alii, Frattura ed Integrità Strutturale, 12 (2010) 63-78; DOI: 10.3221/IGF-ESIS.12.07 75 (a) (b) Figure 17 : Kinking of a straight crack in mixed mode I & II before kinking (a) , after kinking (b) . The elastic domain is discretized with a non-structured triangulation, as shown in Fig. 17a w here also the stress level is represented in a gray scale. The given displacement is discretized into 100 steps and the deformation at each step is determined by minimizing the energy with a descent algorithm. Denoting M the total number of triangles in the mesh and N the total number of nodes, the energy to be minimized is a function of (2) 3 M nodal displacements u 1 (n), u 2 (n) (n=1,2,..,3M) and of 2 N nodal positions in the reference configuration x 1 (m), x 2 (m) (m=1,2,..,N). The energy is a complex, non convex function of its arguments with a lot of local minima. Fig. 17a shows the deformation and stress at the onset of propagation. The propagation of the crack just before dynamical instabilization is shown i n Fig. 17b. The kink angle predicted by our model is 30.46°. From the deformed state of Fig. 17 the SIF’s K 1 , K 2 can be derived approximately by computing the J-integral. The values obtained for K 1 , K 2 are K 1 = 1.3176, K 2 = 0.4288 For these values the kink angle predicted by MTS and GMax criteria can be computed from (7) and (8). In Tab. 2 the theoretical and numerical values obtained in this case are compared. MTS GMax Numerical 30.84° 30.48° 30.46° Table 2 : Kinking angle with MTS, GMax criteria and from FEM analysis. We see that our analysis reproduces closely the GMax-criterion. The last results we present are concerned with the convergence analysis relative to the kinking angle assessment. The problem is essentially the same of Fig. 17, except for the signs of the given displacements: S H =(-1)  and V H =(1)  . The “loading” history is again discretized into loading steps but the analysis is stopped as soon as the crack propagates. The domain  is discretized with structured meshes of decreasing mesh size h: h= L/12 2 , L/16 2 , L/20 2 , L/48 2 . The results of these three analysis are reported in Figs. 18, 19, 20, 21. From these pictures we see that the kinking angle goes from 38.30° degrees for the coarser mesh to the value of 31.18° of the finest one. The value obtained converges toward the value predicted by GMax. Notice that a star shaped region of different size, centered at the crack tip, is always created during the four analyses. Such regions develop before crack propagation, that is in the elastic phase, and are formed by a fixed number of elongated triangles (6). A similar behaviour is detected also in the case of Fig. 17, also if it is less evident. Elongated triangles are usually hill-behaved in FEM analysis, but the point here is that such shapes are chosen through minimization and are the best possible in order to approximate the large stress that develops around the crack tip.