Deriving Constitutive Equations for a Simple Incompressible Mixture of an Elastic Solid and an Inviscid Fluid

摘要

The biphasic mixture theory has been shown to successfully represent the apparent viscoelastic behavior of various biological tissues, including articular cartilage and the annulus fibrosus. Finite deformation mixture theory was used to derive nonlinear constitutive equations by Mow et al. and Kwan et al. However, to reduce the complexity of these formulations the following assumptions were made: 1) the Helmholtz free energy functons (PSI~s, PSI~f) of both phases are equal and do not depend on the relative velocity a, and 2) the terms involving grad (rho~f=apparent fluid density) and gradC (C=right Cauchy-Green deformation tensor) could be ignored in the constitutive equation for the diffusive force pi. Our objective was to rederive these constitutive equations from the entropy inequality by first defining a "simple" mixture. Using this approach it followed that: 1) neglecting the relative velocity a form the free energy functions can be justified after assuming a linear dependence on a; 2) neglecting grad rho~f and gradC from the diffusive force pi can be justified; and 3) that the Helmholtz free energy functions of the two phases cannot be equal. Our results differ from Kwan et al. by an additional term in the fluid stress equation that is found in classical results, and which was contained in the original linear biphasic theory.