A multi-scale approach to model localized failure with softening

We present a computational strategy for strongly coupled multi-scale analysis of heterogeneous material undergoing localized failure with softening. The proposed method can nicely fit within the standard architecture of finite element codes, with the key idea to replace the standard computation of the element tangent stiffness matrices and the residual vectors by an assembly of micro-scale computations whose contributions are statically condensed at the coarser level. The micro-scale elements act as local kinematic enrichments which allow us to deal with localized failure mechanism in essentially the same manner as the classical embedded discontinuity element enhancement, with the benefit of accounting for true microstructure of heterogeneous material with softening behavior. The proposed multi-scale solution strategy also incorporates a cylindrical arc-length procedure at the micro level which allows the softening phenomena to be handled. Some numerical examples dealing with localized failure of heterogeneous materials are presented in order to illustrate very satisfying performance of the proposed methodology.