Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals

Abstract

The paper presents new continuous and discrete variational formulations for the homogenization analysis of inelastic solid materials undergoing finite strains. The point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro-structure as a generalized standard medium in terms of an energy storage and a dissipation function. Consistent with this type of finite inelasticity we develop a new incremental variational formulation of the local constitutive response, where a quasi-hyperelastic micro-stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. We specify the local variational formulation for a distinct setting of multi-surface inelasticity and develop a numerical solution technique based on a time discretization of the internal variables. The existence of the quasi-hyperelastic stress potential allows the extension of homogenization approaches of finite elasticity to the incremental setting of finite inelasticity. Focussing on macro-deformation-driven micro-structures, we develop a new incremental variational formulation of the global homogenization problem for generalized standard materials at finite strains, where a quasi-hyperelastic macro-stress potential is obtained from a global minimization problem with respect to the fine-scale displacement fluctuation field. It is shown that this global minimization problem determines the state of the micro-structure for finite increments of time. We consider three different settings of the global variational problem for prescribed displacements, non-trivial periodic displacements and prescribed stresses on the boundary of the micro-structure and develop numerical solution methods based on a spatial discretization of the fine-scale displacement fluctuation field. Representative applications of the proposed minimization principles are demonstrated for a constitutive model of crystal plasticity and the homogenization problem of texture analysis in polycrystalline aggregates.