Dynamics of a viscoelastic membrane with gradient constraint

Abstract

Taking into account inertial and viscosity effects, we consider the dynamics of a two dimensional membrane subjected to an unilateral constraint on its deformation gradient. Specifically, due to the constitutive law, we assume that higher deformations lock the material, leading to the inequality |∇u|≤g, where u denotes the displacement of the membrane and g is a certain positive threshold. We then introduce the concepts of weak and generalised solutions to the associated wave equation, and prove the existence of them for rather general data and homogeneous Dirichlet boundary conditions. The presence of the gradient constraint provides the existence of a Lagrange multiplier λ related to the existence of a reaction term ϒ, which corresponds to a strongly nonlinear term in the wave equation. We then extend the existence result to a weak form of the Neumann type boundary condition αu+∂u∂ν+∂u˙∂ν+ϒ⋅ν=0, for any α≥0, and we show that these solutions tend, as α→∞, in a certain sense to a solution of the homogeneous Dirichlet constrained problem.