Analysis and numerics of the mechanics of gels

Abstract

In this thesis a mathematical model of polymer gel dynamics is proposed and analyzed. This work is motivated by problems in biomedical device manufacturing. The goal of this thesis is to develop and analyze models of gels consisting of balance laws in the form of systems of partial differential equations and boundary conditions. The model based on mixture theory accounts for nonlinear elasticity, viscoelasticity, transport, and diffusion. The derived model includes as limiting cases incompressible elasticity, viscous incompressible fluid, and Doi's stress diffusion equations. Two classes of problems are considered. The first class addresses nonlinear problems in special domains and the second class addresses linear problems in arbitrary domains. Special emphasis is placed on linear problems with the goal of studying and implementing finite element methods. The first class of problems includes a one dimensional free boundary problem analyzed in terms of one dimensional hyperbolic theory. The second class includes coupled elasticity and fluid flow problems. One challenging issue is accounting for the fact that, although the gel may be incompressible, the polymer may experience large changes of volume. Numerical analysis of elastic solids and polymer gels is carried out. The Taylor-Hood algorithm for Stokes flow is applied to linearly visco-hyperelastic polymers. The simulations show the presence of stress concentrations at the boundary which relax over time. In the case of a gel, the conditionally stable mixed finite element method proposed by Feng and He for Doi's stress-diffusion coupling model is modified to handle the case of polymer viscosity. The modified numerical scheme is shown to be unconditionally stable and convergent.