Asymptotic Analysis of a Viscoelastic Flexural Shell Model

Abstract

We consider a family of linearly viscoelastic shells with thickness $2\varepsilon$, clamped along a portion of their lateral face, all having the same middle surface $S=\mathbfθ(\barω)\subset\mathbb{R}^3$, where $ω\subset\mathbb{R}^2$ is a bounded and connected open set with a Lipschitz-continuous boundary $γ$. We show that, if the applied body force density is $O(\varepsilon^2)$ with respect to $\varepsilon$ and surface tractions density is $O(\varepsilon^3)$, the solution of the scaled variational problem in curvilinear coordinates, $\mathbf{u}(\varepsilon)$, defined over the fixed domain $Ω=ω\times(-1,1)$, converges to a limit $\mathbf{u}$ in $H^1(0,T;[H^1(Ω)]^3)$ as $\varepsilon\rightarrow 0$. Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average $\bar{\mathbf{u}}= \frac1{2}\int_{-1}^{1}\mathbf{u} dx_3$, which belongs to the space $H^{1}(0,T; V_F(ω))$, where $$ V_F(ω):= \{ \mathbfη=(η_i)\in H^1(ω)\times H^1(ω)\times H^2(ω) ; η_i=\partial_νη_3=0 \ \textrm{on} \ γ_0, γ_{αβ}(\mathbfη)=0 \textrm{ in } ω\}, $$ 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.