Variational Multi-scale Formulation for Modelling Material Failure in Heterogeneous Solids

PABLO SÁNCHEZ; PABLO BLANCO; SEBASTIAN TORO; ALFREDO HUESPE; SEBASTIAN GIUSTI; RAÚL FEIJÓO

Barcelona

Conferencia; International Conference on Computational Modeling of Fracture and Failure of Materials and Structures (CFRAC 2011); 2011

International Center for Numerical Methods in Engineering (CIMNE) y European Community in Computational Methods in Applied Sciences (ECOMAS)

ABSTRACT 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  FE method in the context of the finite element approach) 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 (Souza et al. 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 RVE size (Gitman et al.2007). Some novel ideas addressing this issue have been recently proposed (Verhoosel et al. 2010, Nguyen et al. 2010 a-b).  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.