Effects of permeability and viscosity in linear polymeric gels

TL;DR

This paper develops a mathematical model for polymeric gels considering permeability and viscosity, analyzing stability, existence of solutions, and implications for medical device design.

Contribution

It introduces a rigorous analysis of gel mechanics incorporating permeability and viscosity effects, including stability conditions and solution existence under various boundary conditions.

Findings

Derived stability conditions for gel equilibrium solutions.

Established existence of weak solutions for different boundary conditions.

Numerical simulations reveal pressure concentration effects.

Abstract

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body-implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi-static approximation to the dynamics. We focus on the linearized system about relevant equilibrium solutions, and derive sufficient conditions for the solvability of the time dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. The fact that some equilibrium solutions of interest are not stress free brings additional challenges to the analysis, and, in particular, to the derivation of the energy law of the systems. It also singles out the special role of the rotations in the analysis. From the point of view of applications, we point out that the conditions that guarantee stability of solutions also provide criteria to select material parameters for devices. The boundary conditions that we consider are of two types, first displacement-traction conditions for the governing equation of the polymer component, and secondly permeability conditions for the fluid equation. We present a rigorous study of these conditions in terms of balance laws of the fluid across the interface between the gel and its environment. We also consider the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component. We establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present numerical simulations to study pressure concentration on edges (debonding).