A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation

Abstract

In this paper a theoretical framework for the study of residual stresses in growing tissues is presented using the theory of mixtures. Such a formulation must necessarily be a solid-multiphase model, comprising at least one phase with solid characteristics, owing to the fundamental role played by the incompatibility of strains in generating residual stresses. Since biological growth involves mass exchange between cellular and extracellular phases, field equations are presented for individual phases and for the mixture as a whole which incorporate this phenomenon. Appropriate constitutive equations are then deduced from first principles, appealing to the second law of thermodynamics. The analysis shows that the distinguishing feature of multiphase models involving mass exchange is the necessity to propose an additional constitutive postulate between the variables in the mass-balance equation in order to close the model. In particular, the defining characteristic of a solid-multiphase model which describes biological growth is a constitutive postulate which relates the process of interphase mass exchange (cell proliferation/cell death) with the expansion or contraction of the solid phase. Thus, the framework presented here represents a new class of mathematical models which extends the concepts of poroelasticity to accommodate continuous volumetric growth. A set of modelling equations is then proposed for the simplest case of a solid-multiphase model, being a biphasic mixture of a linear-elastic solid and an inviscid fluid.