Variational multi-scale formulation for modelling material failure in heterogeneous solids

P.J. SÁNCHEZ; P.J. BLANCO; S. TORO; A.E. HUESPE; S. GIUSTI; R. A. FEIJÓO

Barrcelona

Conferencia; Int. Conf. on Comput. Modeling of Fracture and Failure of Materials and Struct. (CFRAC-2011); 2011

ECCOMAS

In this contribution a variational multi-scale formulation for modelling material failure inheterogeneous solids, is presented. The model is based on the classical Unit Cell concept (oftencalled FE2 method in the context of the finite element approach) and considers two coupledmechanical problems at different physical length scales, denoted as macro and meso scale. Theproposed multi-scale methodology follows the kinematically consistent framework reported in aprevious contribution (Souza et al. 2006). At the meso-structural level, the mechanical response ofeach phase of the heterogeneous medium (a matrix phase with particulate aggregates) ischaracterized by means of dissipative theories which incorporate degradation phenomena (softening)as mechanisms for capture failure and the associated consequences: material instabilities and strainlocalization patterns in the Unit Cell. The generation, evolution and propagation of such localizedstrain modes can also induce material instabilities at the macro structural level. The formulation ofstandard multi-scale models (based on classical homogenization procedures) for micro-structuresundergoing strain softening reveals the lack of an objective RVE size (Gitman et al.2007). Somenovel 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 consistentredefinition of the kinematics at the meso scale and (iii) proper boundary conditions over thedeformation fluctuation field. The Hill-Mandel principle of Macro Homogeneity is used for obtainingthe homogenized response for the degraded meso-scale, it means a traction-jump constitutive law.