Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 274 Coupling effect on energy dissipation The behaviour of the coupled CZMs is now studied by analyzing the normal, the tangential and the total work of separation, W N , W T , and W , respectively, for different loading paths. A similar evaluation was performed by van den Bosch [7] when evaluating the consistency of the exponential model by Xu and Needleman [8]. Two non-proportional loading paths are taken into account. In path 1, the interface is first loaded in the normal direction up to a maximum normal displacement g * N , and subsequently is loaded in the tangential direction up to total failure (Fig. 8a). Conversely, in path 2, the interface is first loaded in the tangential direction up to g * T , and then completely broken in the normal direction (Fig. 8b). In each case, the total work-of-separation is computed as ( , ) ( , ) N T N N T N T N T T W W W p g g dg p g g dg         (23) where  is the selected separation path. (a) (b) Figure 8 : Loading paths for debonding process: (a) Path 1; (b) Path 2. Analytical predictions are also compared with finite element results, as described in detail in the following Section. In the analytical and numerical studies, all the models are assumed to be characterized by the same fracture energy, cohesive strength, and peak relative displacement in both directions. Three examples are here analyzed for different combinations of fracture energies (i.e.  N <  T ,  N =  T ,  N >  T ), whereas the fracture energies are set to 100 N/m or 200 N/m . The cohesive strength is arbitrarily set to 6N/mm 2 in both directions, but it does not affect significantly the behaviour of CZMs, as often emphasized in the literature. The values of g Nmax and g Tmax in CZMs 3,4 are such that the secant stiffness at the peak defined by CZM 1 is maintained constant. This means that g Nmax =0.006mm, g Tmax =0.020mm when  N =100N/m ,  T =200N/m; g Nmax =0.006mm, g Tmax =0.010mm when  N =  T =100N/m ; g Nmax =0.012mm, g Tmax =0.010mm when  N =200N/m ,  T =100N/m. Within the CZM4, a coupling parameter  =0.5 or  =1.75 is assumed for debonding propagations dictated by a PL- or BK-criterion, respectively. These values agree with results of experimental investigations on composite materials [11]. Example 1 (  N <  T ) The normal, the tangential and the total work of separation as computed analytically by means of each model under non- proportional loading paths are shown in Fig. 9,10, along with the respective numerical results. When the CZM1 is applied, the limiting cases (failure in pure modes I and II) are consistently captured for each loading path and the transition in between is smooth and monotonic, as expected due to the corrections already applied by van den Bosch et al. [7] (Fig. 9a, 10a). For non-proportional path 1, this means that W=W T =  T when g * N =0 (i.e. the loading is completely driven by shear), and W=W N =  N when g * N tends to an infinite value (i.e. when the loading is completely driven by normal separation). A monotonic decrease of W from  T to  N is correctly observed for intermediate values of g * N (Fig. 9a). Conversely, for path 2, W=W N =  N when g * T =0 and W=W T =  T when g * T tends to an infinite value, while W smoothly increases from  T to  N for intermediate values (Fig. 10a). By repeating the same analytical investigation for CZMs 2, 3, and 4, some unphysical results can be observed under some mixed-mode loading conditions. The pure-mode cases are consistently captured by the three models but the transition is not always smooth and monotonic (see Fig. 9c, 10c for CZM3, and Fig. 9d, 10d for CZM4-PL criterion), whereas monotonic variations of W are obtained for CZM2 under path 1 (Fig. 9b), as well as for CZM4-BK criterion under paths 1,2 (Fig. 9e and 10e). For CZM2 with path 2, W N changes from  N to zero with increasing values of g * T from zero up to a certain value  * T , while W T monotonically increases from zero to  T for the same values of g * T (Fig. 10b). As a result, W is constant for g * T values up to  * T and afterwards follows the pure tangential contribution.