Issue 29

R. Dimitri et alii, Frattura ed Integrità Strutturale, 29 (2014) 266-283; DOI: 10.3221/IGF-ESIS.29.23 271 or alternatively as   2 2 2 2 . ,max 2(1 ) 0 ( ) 0 T N T N N m u N T m N T Tu N K for g g K K g K K g for g                                     (22) Seven independent parameters describe the model, e.g. the fracture energies  N and  T , the cohesive strengths p Nmax and p Tmax , the relative separations g Nmax and g Tmax , and the material parameters  or  depending on the debonding criterion chosen for propagation. A NALYTICAL ASSESSMENT OF THE COUPLING EFFECTS his section analyzes the mixed-mode debonding behaviour as predicted by the aforementioned coupled CZMs. A detailed assessment of the influence of the coupling on stresses and energy dissipation is performed to more deeply understand the performance of each model. As mentioned earlier, an adequate coupling between the normal and tangential directions is necessary in a CZM to describe the physically occurring interface behaviour realistically. The coupling parameters, if not well defined, may yield to physical inconsistencies such as, for example, the existence of a residual non-zero traction or energy dissipation, when the load bearing capacity of the interface is lost [7]. Coupling effect on stresses Fig. 3-7 show the dimensionless traction-separation curves of each cohesive model in the normal and tangential directions for varying degrees of mode mixity expressed by means of the quantities g T p Tmax /  T and g N p Nmax /  N, respectively. Pure mode I is obtained by imposing g T p Tmax /  T =0 while pure mode II corresponds to g N p Nmax /  N =0. Convex domains are always obtained for varying mode-mixities by using the CZMs 1 and 2, as visible in Fig. 3, 4. The exponential models provide smooth traction-separation curves, thus enabling the continuity of derivatives which is a desirable feature in the numerical implementation. However, they do not allow control of the initial stiffness in the ascending branch without affecting the cohesive strength. This aspect may be considered as a drawback of CZMs1 and 2. The maximum tangential and normal traction reponses are largely affected by the mode mixity, i.e. a gradual reduction in cohesive strengths p Nmax and p Tmax , is obtained for increasing separation values in the other debonding mode. Here CZM2 shows the first inconsistency depending on the value of the coupling parameter m . E.g. for m=2 unphysical values (i.e. values higher than the single-mode cohesive strength) are reached by the mode-I cohesive strength for increasing mode-mixity (Fig. 4a). A flexible control of the initial stiffness is enabled by the bilinear CZMs 3 and 4, which are however characterized by discontinuous derivatives at the onset of the softening stage, depending on the mode-mixity (Fig. 5-7). A second inconsistency is found in that the traction-separation curves of CZM3 are not always convex but show some concavities and/or hardening effects (Fig. 5a,b) under some mixed mode conditions. Convex stress domains are obtained, instead, with CZM4 for a PL debonding propagation criterion, due to the decrease in cohesive strenghts for increasing mode mixities. Some changes of concavities can be noticed, however, in the softening branch, depending on the cohesive and material parameters (i.e. g Nu /g Nmax , g Tu /g Tmax ,  ). This dependency can be noticed by comparing Fig. 6a,b and 6c,d. Further inconsistencies can be also noticed for CZM4 when a BK criterion defines the debonding propagation stage. Most curves in the normal and tangential directions feature stress values higher than those in pure mode for wide ranges of mode mixity values (see Fig. 7a,c,d). This leads to residual stresses when g N =g Nu or g T =g Tu in mode I or II respectively, (i.e. when g N p Nmax /  N or g T p Tmax /  T reach the respective maximum values in pure modes), whose entity varies with the material parameter  as well as with the cohesive parameter ratios g Nu /g Nmax and g Tu /g Tmax ,. The same CZMs considered thus far for debonding problems can be also employed to simulate contact in the range of negative normal relative displacements ( g N <0 ) and stresses ( p N <0 ). It is worth noting as contact modelling is incorporated into the debonding model for CZMs 1 and 2, while it can be controlled independently from debonding-related parameters for CZMs 3 and 4. In the first two models the continuity of the derivative at g N =0 is guaranteed, however a load-varying penetration error is introduced which is affected by the debonding-related parameters and by the coupling. This disadvantage can be avoided by using a coupling-independent contact formulation with CZMs 3 and 4, which are characterized by discontinuous derivatives. T