Strain‐driven homogenization of inelastic microstructures and composites based on an incremental variational formulation

Abstract The paper investigates computational procedures for the treatment of a homogenized macro‐continuum with locally attached micro‐structures of inelastic constituents undergoing small 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 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 setting of smooth single‐surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables. The existence of the quasi‐hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity. Focusing on macro‐strain‐driven micro‐structures, we develop a new incremental variational formulation of the global homogenization problem 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 linear displacements, periodic fluctuations and constant stresses on the boundary of the micro‐structure and discuss their numerical solutions based on a spatial discretization of the fine‐scale displacement fluctuation field. The performance of the proposed methods is demonstrated for the model problem of von Mises‐type elasto‐visco‐plasticity of the constituents and applied to a comparative study of micro‐to‐macro transitions of inelastic composites. Copyright © 2002 John Wiley & Sons, Ltd.