Multi-scale computational modeling of localization problems

Conventional continuum mechanics models of inelastic deformation processes are size scale independent since they do not possess intrinsic length scales in their constitutive description. In contrast, there is considerable experimental evidence that inelastic flow in crystalline materials is size-dependent. As soon as material failure dominates a deformation process, the material increasingly displays strain softening (localization) and the finite element computation is considerably affected by the mesh size and alignment and gives non-physical descriptions of the localized regions. Gradient-enhanced constitutive viscoplastic and damage equations that include explicit and implicit micro-structural length scale measures are presented in this work. However, the numerical implementation of the gradient-dependent theory is not a direct task because of the higher order of the governing equations. In this work a direct computational algorithm for the gradient approach is proposed. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches [1,2]. The method is validated by conducting various numerical tests. As a result, pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered.