A non-equilibrium model for rapid finite deformation of hydrated soft biological tissue in uniaxial confined compression

Abstract

The viscoelastic behavior of hydrated soft biological tissue is strongly influenced by the mechanical interaction of its interstitial fluid with the solid phase. The classical uniaxial steady biphasic theory for soft tissue with high moisture content under confined compression is extended by replacing the Darcy steady flow assumption with an unsteady internal fluid flow model that involves an unsteady flow coefficient based on non-equilibrium thermodynamics. The finite deformation, non-equilibrium, unsteady biphasic viscoelastic analysis derived from conservation of linear momentum, which includes the mass terms neglected in some steady models, produces a nonlinear hyperbolic partial differential equation for the solid phase displacements, with a finite propagation velocity of the disturbance from the plunger in contrast to the instantaneous propagation of the classical parabolic steady description for cartilage. For confined compression of a small cylinder of hydrated soft biological tissue, the new hyperbolic equation is solved for the solid phase local displacements during both loading and relaxation as a function of time and position by a finite difference scheme of Shampine. A parametric study shows the influence of the unsteady coefficient on the predicted behavior, for both a linear elastic and a linear viscoelastic solid phase and for a constant permeability, by a comparison of the respective solid phase displacement distributions at the end of a constant rate plunger displacement at 0.001/s, 1/s and 1,000/s. Models assuming the Darcy relation predict physically realistic responses only for slow rates. The concavity of the stress–time curve for the solid phase at the plunger during a constant displacement of the plunger even at slow rates may be adjusted from softening to hardening by increasing the unsteady coefficient to increase the drag force exerted by the fluid on the solid phase. Because of unsteady fluid flow, overshoot of the solid phase displacement may occur initially during relaxation and also again near the end of the approach to equilibrium. A Zener standard linear viscoelastic solid phase slows the relaxation to equilibrium in comparison with a linear elastic solid phase.