Micro- and macro-failure models of heterogeneous media with micro-structure

In this paper, the effectiveness of homogenization techniques for media with micro-structure subject to large deformations has been studied by comparing their micro- and macro-failure mechanisms. The material has been studied by considering its representative volume element (RVE) which entails all the geometric and constitutive information of the micro-structure. First, the formulation of the elastostatic problem governing the non-linear (large deformation and non-linear elastic) behavior of the structure of the RVE is presented. The RVE is subject to loading paths that produce uniform macroscopic strains. In this way it has been possible to use an homogenization procedure in order to simulate the overall behavior of the material, i.e. its constitutive tensor, at each point of the equilibrium path. Then, a macro failure surface has been defined as the locus of the points, in the macroscopic stretches space, corresponding to the loss of positivity of the macroscopic fourth order constitutive tensor in terms of the Biot stress [Encyclopedia of Physics, vol. 3(3), Springer-Verlag, Berlin]. Further, a micro-failure surface is defined as the locus of the points, in the overall stretches space, corresponding to the first critical point detected along the equilibrium path which can be characterized by an eigenmode compatible with the boundary conditions. Finally, a representative volume element, schematized by plane rods with strongly non-linear elastic constitutive behavior, is considered and the corresponding micro- and macro-failure surfaces are obtained in order to validate the proposed methodology.