A fully-relaxed variationally-consistent framework for inelastic micro-sphere models: Finite viscoelasticity - Govindjee, Sanjay; Zoller, Miklos J.; Hackl, Klaus (Journal of the mechanics and physics of solids, 2019)
The micro-sphere modeling framework provides a popular means by which one-dimensional mechanical models can easily and quickly be generalized into three-dimensional stress-strain models. The essential notion of the framework, similar to homogenization theory, is that one allows the microstructural kinematic fields to relax subject to a constraint connected to a macroscopic deformation measure. In its standard presentation, the micro-sphere modeling framework is strictly applicable to elastic materials. Presentations considering inelastic phenomena invariably, and inconsistently, assume an affine relation between inelastic macroscopic and microscopic phenomena. In this work we present a methodology by which one can lift this modeling restriction using two formally different approaches. In particular, we show how one can construct and apply a homogenization with Biot theory to generate fully-relaxed variationally-consistent macroscopic models for inelastic materials within the context of the micro-sphere model. The primary application example will be finite deformation viscoelasticity.