On constitutive macro-variables for heterogeneous solids at finite strain

TLDR

In heterogeneous solid mechanics, linking macro‑variables to volume averages of microfields is essential for bridging microscopic and macroscopic behavior. The study establishes and evaluates connexions between macro‑variables and microfield averages for unrestricted deformation and internal rotations, focusing on their advantages in averaging tensor products. The authors analyze general stress and finite strain measures, objective stress fluxes, and a bilinear differential form analogous to Poincaré’s integral invariant to study these connexions.

Abstract

In theories of the overall mechanical behaviour of polycrystals, composites, or other heterogeneous media, the transition from microscopic to macroscopic levels depends on finding connexions between suitably defined macro-variables and volume averages of microfields over a representative sample. Some connexions are established here for unrestricted deformation and internal rotations, w ithout regard to constitutive properties. General measures of stress and finite strain are considered, together w ith objective fluxes of stress. The comparative advantages of such variables are assessed, more especially in relation to the averaging of tensor products. A bilinear differential form, akin to Poincare’s integral invariant in Hamiltonian dynamics, plays an im portant role in the analysis.