CONICET | Buscador de Institutos y Recursos Humanos

In this contribution a variational multi-scale formulation for modelling material failure in heterogeneous solids, is presented. The model is based on the classical Unit Cell concept (often called FE2 method) and considers two coupled mechanical problems at different physical length scales, denoted as macro and meso scale. The proposed multi-scale methodology follows the kinematically consistent framework reported in a previous contribution (de Souza Neto and Feijoo, 2006). At the meso-structural level, the mechanical response of each phase of the heterogeneous medium (a matrix phase with particulate aggregates) is characterized by means of dissipative theories which incorporate degradation phenomena (softening) as mechanisms for capture failure and the associated consequences: material instabilities and strain localization patterns in the Unit Cell. The generation, evolution and propagation of such localized strain modes can also induce material instabilities at the macro structural level. The formulation of standard multi-scale models (based on classical homogenization procedures) for micro-structures undergoing strain softening reveals the lack of an objective representative volume element size (Gitman et al., 2007). Some novel ideas addressing this issue have been recently proposed (Verhoosel et al., 2010; Nguyen et al., 2010b,a). In our contribution, the objectivity respect to the size of the Unit Cell is recovered by introducing: (i) a new kinematical field which represents a weak discontinuity at the macro level, (ii) a consistent redefinition of the kinematics at the meso scale and (iii) proper boundary conditions over the deformation fluctuation field. The Hill-Mandel principle of Macro Homogeneity is used for obtaining the homogenized response for the degraded meso-scale, it means a traction-jump constitutive law.

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