Variational Inequalities For Heterogeneous Microstructures Based On Couple-Stress Theory

In this work, we view mesoscopic material volume elements consisting of heterogeneous microstructures as couple-stress continua to account for underlying length-scale dependence. We use a recently established self-consistent version of couple-stress theory that results in a skew-symmetric couple-stress tensor, along with the energy-conjugate mean-curvature tensor. Using this new theoretical framework, we establish a generalized Hill energetic equivalence relationship that leads to a homogeneous material representation at the macroscale point associated with the mesoscopic volume element. We identify the necessary and sufficiency conditions that enable the extension of the couple-stress continuum framework and its application to incorporate the mesoscale features into the macroscale continuum description. We establish the concept of a micromechanically consistent macroscopic elastic constitutive tensor within this paradigm and also propose special kinematically and statically uniform boundary conditions, analogous to previous work in classical elasticity. This then leads to determination of two suitable matrices that bound the matrix representation of the macroscopic elastic constitutive tensor in the positive definite sense. Similar bounds based on classical mechanics are found to be critical quantities in several aspects of multiscale material modeling. We envisage that the theoretical work presented here will be useful in analyzing coarse-grained heterogeneous microstructures with inherent characteristic length-scale features contained within the mesoscopic material volume element.