Multiscale, hybrid mixture theory for swelling systems .2. Constitutive theory

In this second part of a three-part paper we derive constitutive theory for a multiphase, multicomponent, three-scale, swelling system which includes interfaces. In Part I, the governing field equations and the definitions of all mesoscopic and macroscopic variables therein were defined in terms of microscopic variables. In this paper, we choose the independent variables and derive constitutive restrictions for two cases of a dual-porosity multiple-component swelling media: one which assumes no interfacial effects, and one which includes interfacial effects. For each case, the entropy inequality is fully derived using a Lagrange multiplier technique. Novel definitions for macroscopic pressures and chemical potentials are given, and generalized Darcy's and Fick's laws are presented. Although these models are developed with a clay soil in mind, the results hold for any medium which assumes the same set (or subset of) independent variables as constitutive unknowns, e.g. swelling biopolymers.