Advancements on the FE·Meshless CH for the analysis of heterogeneous periodic materials

Over the last few years, the intrinsic role of different spatial scales in the mechanics of materials has been well recognized. Generally, two main different scales can be identified in the heterogeneous materials: the macroscopic level, which coincides with the global structural one, and the mesoscopic level, that is the scale at which the heterogeneities can be identified and where the most relevant nonlinear mechanical phenomena occur. In this framework, substantial progress has been made in the two-scale computational homogenization (CH). This method is essentially based on the on the fly assessment of the macroscopic constitutive behavior from the boundary value problem (BVP) of a statistically representative volume element, named as unit cell (UC). In the CH, the first-order method has now matured to a standard tool [1]. Several extensions that depart from the first-order method have been addressed in the literature: • a second-order CH [2], which takes into account higher order deformation gradients at the macroscale; • a continuous-discontinuous multiscale approach for damage, in which the coarse scale is en- riched by discrete localization bands (weak discontinuities), whereas the fine scale is modeled using a continuum [3]. In the present study, a first-order homogenization scheme based on a discontinuous-continuous ap- proach is presented. This means that at the mesoscopic level the formation and propagation of fracture is modeled employing a UC consisting of an elastic unit surrounded by elasto-plastic zero- thickness interfaces, characterized by a discontinuous displacement field. At the macroscopic level, instead, the model maintains the continuity of the displacement field. Hence, the fracture effects are taken into account in a smeared mode, introducing a strain localization band. Another key point is the numerical solution of the UC BVP, which is obtained by means of a more cost-effectiveness mesh-free model. A standard finite element discretization is conversely employed at the global level. Both linear and periodic boundary conditions have been tested and applied to the UC. The results are presented by means of experimentally validated examples. References [1] Giambanco, G., La Malfa Ribolla, E. and Spada, A., Meshless meso-modeling of masonry in the computational homogenization framework, Meccanica, in press (2017). [2] Kouznetsova, V., Geers, M.G. and Brekelmans, W.M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, In- ternational Journal for Numerical Methods in Engineering, 54, page 1235-1260 (2002). [3] Massart, T.J., Peerlings, R.H.J.. and Geers, M.G.D., An enhanced multi-scale approach for masonry wall computations with localization of damage, International Journal for Numerical Methods in Engineering, 69, page 1022-1059 (2007).