A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues II: Solutions to the Biphasic Equations for a Multicell Spheroid

Abstract

This is the second paper in the series A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues. While the first paper in the series elaborated a general formulation for such a theory, the present paper develops a simple biphasic model of residual stress evolution in a growing multicell spheroid comprising a linear-elastic cellular phase and an inviscid interstitial fluid. Both isotropic and anisotropic growth are considered in this study, highlighting the necessity to incorporate stress relaxation in order to predict an evolution of stresses over a period of growth. The solutions to the biphasic equations are juxtaposed with the corresponding solutions to the single phase equations, illuminating the approximate nature of the single phase formulation for growing tissues. Moreover, the analysis demonstrates the significance of both interphase drag and the stress-relaxation characteristics of the solid phase in distinguishing between the single phase and multiphase paradigms.