Issue 51

F. Clementi et alii, Frattura ed Integrità Strutturale, 51 (2020) 313-335; DOI: 10.3221/IGF-ESIS.51.24 319 NSCD for masonry structures modelling The discrete modelling permits to describe the masonry as the interaction between block, assuming that the properties of the bodies and their contact points govern the model. Moreover, the contact is supposed to be punctual and not an area of the interfaces to simplify the process. Other relevant hypotheses assumed in this method are that the bodies are rigid, with the strain applied to the contact points, and the contact forces are provided by the strain at the punctual contact, with independent interactions between bodies. The approach of the Non-Smooth Contact Dynamics involves first the contact detection, then the contact problem, i.e. the derivation of the contact forces for local scale, and at last the individuation of bodies displacement, for global scale. In fact, in the framework of the NSCD, to compute the multi-contact problem it is necessary to resolve the local unknows, due to the interactions, and the global unknows, due to the bodies. These two sets of unknows are bounding by a mapping (see Fig. 7). Figure 7 : Global and local mapping in the NSCD algorithm At the contact , a linear mapping ఈ allow to obtain the global resultant forces related to the local forces r ஑ , with the equation:   R q r     H . (1)   q  H is a mapping with the local information of the contactors and q is the vector of generalised coordinates of the rigid displacement. Hence, to achieve the global resultant contact forces exerted on bodies: R R     . (2) Moreover, to calculate the velocity u  relative to contact in relation with the velocity of the blocks, it can be used the transposed T  H as in this equation:   T u q v    H , (3) with v is the time derivative of q . Figure 8 : Contact at the interface between blocks (a), Signorini’s impenetrability condition (b), and friction Coulomb’s law (c)