Issue 47

E. Grande et alii, Frattura ed Integrità Strutturale, 47 (2019) 321-333; DOI: 10.3221/IGF-ESIS.47.24 328 From the plots clearly emerges the importance of introducing in the model the influence of the damage of mortar on the contribution of the upper interface. Indeed, while the curves deduced from the approach 1 overestimate the experimental peak load, the curves derived by using the approach 2 provide a good approximation of the experimental outcomes. A PPROXIMATE SOLUTION IN CASE OF SOFTENING BEHAVIOR n the previous section a solution of the de-bonding problem has been presented by introducing simplified shear stress-slip laws for the interfaces assumed with a perfect brittle post-peak behavior. This has allowed to obtain the analytical solution of the problem and to easily analyze different damage stages. The majority of studies available in the current literature are instead based on the use of constitutive laws where the softening governs the post- peak phase of the interface (cohesive models). The use of this type of laws at the interface level generally does not lead to computational problems when a Finite Element approach, or other types of numerical techniques, are used. On the contrary, it can present some drawbacks for the derivation of an analytical solution, particularly in the case of FRCM systems where there are two interfaces interacting among them in the de-bonding process. For this reason, the analytical solution is here derived by approximating the linear descending post-peak branch of the shear stress-slip law throughout a step function, i.e. a piecewise constant function (Fig. 5), and following the approach presented in the first part of the paper. Moreover, taking into account the ‘approach 2’ presented in the first part of the paper, the softening behavior is only introduced for the lower interface, whilst a linear-brittle behavior is accounted for the upper interface by opportunely calibrating the corresponding bond strength. Figure 5 : Scheme used for approximating the softening branch of the shear stress-slip law. Also in this case, the system of equations governing the problem depends on the status of the two interfaces (see Fig. 6). Indeed, increasing the applied load P, the first phase is certainly characterized by the pre-peak stage of both the interfaces (Fig. 6.a). In this case, the system of the two equations governing the problem is:       2 1 2 2 2 2 2 2 0 0 i e e i i i e e e d s K s s dx d s d s K s dx dx                           (12) where: ( ) ( ) i i i i e e e e s G s s G s     (13) s 1 s 2 s 3 s 4  1  2  3  4  s I