Multi-scale micropolar model: Theory and bridging scale computations

A hierarchical multi-scale material model and a concurrent bridging scale method are developed for the analysis of a class of localization problems in which dynamic failure occurs only in a small number of localized regions. The proposed material model is based on the micromorphic continuum theory where micro-deformation fields are included in the governing equations in addition to the displacement fields. Material models of this kind successfully regularize the numerical solution of rate-independent strain-softening materials by incorporating a length scale in the constitutive law. By taking advantage of an averaging procedure over the so-called meso-cell in the derivation of both momentum balance equations and constitutive relations, the material constants for the micropolar material are derived and related to the material characteristic length scale. The concurrent bridging scale method starts with a discretization of the entire domain with both coarse- and fine-scale finite element meshes. The coarse-scale mesh is employed to capture the nonlinear response with long wave length outside the localized regions, whereas the fine-scale mesh captures the detailed physics of the localized deformation. For both coarse- and fine-scale meshes to coexist, a bridging scale term is constructed so that the information common to both scales is correctly subtracted. To achieve computational efficiency, the localized regions are first identified by a preliminary calculation and the fine-scale degrees of freedom (DOFs) outside the localized regions are mathematically represented by the construction of dynamic interface conditions applied to the edges of these regions. Hence, a large portion of the fine-scale DOFs are eliminated and the fine-scale equations are reduced to a much smaller set with the added dynamic interface conditions. The two-way coupled coarse-scale and reduced fine-scale equations are then solved by a mixed time integration procedure. The combination of the hierarchical multi-scale material model and the concurrent bridging scale method realize an accurate yet efficient analysis of localization problems.