RESUMEN

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.

RESUMEN

Films and molds of nematic polymer materials are notorious for heterogeneity in the orientational distribution of the rigid rod or platelet macromolecules. Predictive tools for structure length scales generated by shear-dominated processing are vitally important: both during processing because of flow feedback phenomena such as shear thinning or thickening, and postprocessing since gradients in the rod or platelet ensemble translate to nonuniform composite properties and to residual stresses in the material. These issues motivate our analysis of two prototypes for planar shear processing: drag-driven Couette and pressure-driven Poiseuille flows. Hydrodynamic theories for high aspect ratio rod and platelet macromolecules in viscous solvents are well developed, which we apply in this paper to model the coupling between short-range excluded volume interactions, anisotropic distortional elasticity (unequal elasticity constants), wall anchoring conditions, and hydrodynamics. The goal of this paper is to generalize scaling properties of steady flow molecular structures in slow Couette flows with equal elasticity constants [M. G. Forest et al., J. Rheol., 48 (2004), pp. 175-192] in several ways: to contrast isotropic and anisotropic elasticity; to compare Couette versus Poiseuille flow; and to consider dynamics and stability of these steady states within the asymptotic model equations.