Mathematical justification of a viscoelastic generalized membrane problem

Abstract

We consider a family of linearly viscoelastic shells with thickness $2\epsilon$, clamped along a portion of their lateral face, all having the same middle surface $S=\theta (\bar{\omega })\subset {\mathbb{R}}^3$, where $\omega \subset {\mathbb{R}}^2$ is a bounded and connected open set with a Lipschitz-continuous boundary $\gamma$. We show that, if the applied body force density is $\mathcal{O}(1)$ with respect to $\epsilon$ and surface tractions density is $\mathcal{O}(\epsilon )$, the solution of the scaled variational problem in curvilinear coordinates, defined over the fixed domain $\Omega =\omega \times (-1,1)$, converges in ad hoc functional spaces as $\epsilon \to 0$ to a limit $\mathbf{u}$. Furthermore, the average $\overline{\mathbf{u}(\epsilon )}=\frac{1}{2}{\int }_{-1}^1\mathbf{u}(\epsilon )d{x}_3$, converges in an \textit{ad hoc} space to the unique solution of what we have identified as (scaled) two-dimensional equations of a viscoelastic generalized membrane shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.