Asymptotic Analysis of a Viscoelastic Flexural Shell Model

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 $S=\mathbf{\theta}(\bar{\omega})\subset\mathbb{R}^3$, 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 $O(\varepsilon^2)$ 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 $\Omega=\omega\times(-1,1)$, converges to a limit $\mathbf{u}$ in $H^1(0,T;[H^1(\Omega)]^3)$ 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 $H^{1}(0,T; V_F(\omega))$, where $$V_F(\omega):= \{ \mathbf{\eta}=(\eta_i)\in H^1(\omega)\times H^1(\omega)\times H^2(\omega) ; \eta_i=\partial_\nu \eta_3=0 \ \textrm{on} \ \gamma_0, \gamma_{\alpha \beta}(\mathbf{\eta})=0 \textrm{ in } \omega \},$$ 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.