Preprint
Inserted: 27 sep 2021
Last Updated: 27 sep 2021
Year: 2021
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 $
\nabla u
\leq g$, where $u$ denotes the displacement of the membrane and $g$ is a certain positive threshold. We then introduce the concept of weak solutions to the associated wave equation, and prove the existence of them for any initial data and homogeneous Dirichlet boundary conditions. The presence of the gradient constraint provides the existence of a Lagrange multiplier $\lambda$ related to the existence of a reaction term $\Upsilon$, 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 $\alpha u+\frac{\partial u}{\partial \nu}+\frac{\partial \dot u}{\partial \nu}+ \Upsilon \cdot \nu =0$, for any $\alpha\geq0$, and we show that these solutions tend, as $\alpha\rightarrow\infty$, to a solution of the homogeneous Dirichlet constrained problem.
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