Implicit formulation of homogenization method for periodic elastic-viscoplastic solids

In this study, to determine incremental, perturbed displacement fields in periodic elastic-viscoplastic solids, an incremental homogenization problem is fully implicitly formulated using a linearized constitutive relation, a micro/macro-kinematic relation, and a stress balance equation. It is shown that the homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increments in unit cells, and that the present formulation allows versatility in the initial setting of strain increments in contrast to Terada-Kikuchi (2001) and Miehe (2002). This homogenization algorithm is then examined by analyzing a holed plate, with an elastic-viscoplastic micro-structure, subjected to tensile loading. It is thus demonstrated that the convergence in iteratively solving the homogenization problem strongly depends on the initial setting of strain increments in unit cells, and that quick convergences can be attained if the initial setting of strain increments is appropriate.