A multiscale analysis of growth and diffusion dynamics in biological materials

Abstract

We study a growing biological tissue as an open biphasic mixture whose phases undergo exchange interactions. We assume that both the solid- and fluid-phase are composed of several constituents allowed to be transferred from one phase to the other. Because of growth and exchange, or transfer, source terms must be accounted for in balance laws. We relate the source terms which are relevant for our purposes with thermodynamic quantities defined at the pore scale of the tissue. This procedure, carried out through the Theory of Homogenization, aims to give growth a pore scale justification. Particular attention is given to the exploitation of the Clausius–Duhem inequality and the kinematics of growth. Since the mixture under investigation has to satisfy restrictions, we provide a modified Clausius–Duhem inequality that, following Liu’s theorem, accounts for constraints through the Lagrange multiplier technique. Constraints, and related Lagrange multipliers, are also introduced in the definition of Helmholtz free energy densities in order to include constitutive laws for solid- and fluid-phase mass densities less strict than incompressibility. We perform an analysis of our constrained Clausius–Duhem inequality in the neighborhood of thermodynamic equilibrium. This enables us to obtain Onsager relations that generalize some results found in the literature about a thermodynamically consistent procedure for determining an evolution law for growth and mass transfer. We show that the driving mechanism for mass transfer and growth is related to a generalized Eshelby-like tensor, which accounts for chemical potential. Effective stiffness tensor is derived by means of the self-consistent effective field method using the analogy with Levin’s method of effective thermal expansion.