Asymptotic Analysis of a Viscoelastic Flexural Shell Model

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 $O({\epsilon }^2)$ with respect to $\epsilon$ and surface tractions density is $O({\epsilon }^3)$, the solution of the scaled variational problem in curvilinear coordinates, $\mathbf{u}(\epsilon )$, defined over the fixed domain $\Omega =\omega \times (-1,1)$, converges to a limit $\mathbf{u}$ in $H^1(0,T;[H^1(\Omega ){]}^3)$ as $\epsilon \to 0$. Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average $\bar{\mathbf{u}}=\frac{1}{2}{\int }_{-1}^1\mathbf{u}d{x}_3$, which belongs to the space $H^1(0,T;V_F(\omega ))$, where $$V_F(\omega ):=\{\eta =({\eta }_i)\in H^1(\omega )\times H^1(\omega )\times H^2(\omega );{\eta }_i={\partial }_{\nu }{\eta }_3=0\text{on}{\gamma }_0,{\gamma }_{\alpha \beta }(\eta )=0\text{ in }\omega \},$$ satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic flexural shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

32 pages. arXiv admin note: text overlap with arXiv:1604.02280