Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. 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 stress-free states, uniform expansions, and compressions 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. We also consider non-stress free equilibria and states with residual stress and derive an energy law for the corresponding time-dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement-traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two-dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate.