Scale-Dependent Homogenization of Inelastic Random Polycrystals

Rigorous scale-dependent bounds on the constitutive response of random polycrystalline aggregates are obtained by setting up two stochastic boundary value problems (Dirichlet and Neumann type) consistent with the Hill condition. This methodology enables one to estimate the size of the representative volume element (RVE), the cornerstone of the separation of scales in continuum mechanics. The method is illustrated on the single-phase and multiphase aggregates, and, generally, it turns out that the RVE is attained with about eight crystals in a 3D system. From a thermodynamic perspective, one can also estimate the scale dependencies of the dissipation potential in the velocity space and its complementary potential in the force space. The viscoplastic material, being a purely dissipative material, is ideally suited for this purpose.