On the Justification of Koiter’s Equations for Viscoelastic Shells

Abstract

We consider a family of linearly viscoelastic shells with thickness $2\varepsilon$, all having the same middle surface$S={\varvec{\theta }}({\bar{\omega }})\subset \hbox {I}\!\hbox {R}^3$, where $\omega \subset \hbox {I}\!\hbox {R}^2$ is a bounded and connected open set with a Lipschitz-continuous boundary $\gamma$ and ${\varvec{\theta }}\in {\mathcal {C}}^3({\bar{\omega }};\hbox {I}\!\hbox {R}^3)$. The shells are clamped on a portion of their lateral face, whose middle line is ${\varvec{\theta }}(\gamma _0)$, where $\gamma _0$ is a non-empty portion of $\gamma$. The aim of this work is to show that the viscoelastic Koiter’s model is the most accurate two-dimensional approach in order to solve the displacements problem of a viscoelastic shell. Furthermore, the solution of the Koiter’s model, ${\varvec{\xi }}_K^\varepsilon =(\xi _{K,i}^\varepsilon )$, is in $H^{1}(0,T;V_K(\omega ))$, with $\xi ^\varepsilon _{K,i}: [0,T]\times {\bar{\omega }}\rightarrow {\mathbb {R}}$ the covariant components of the displacements field $\xi _{K,i}^\varepsilon {{\textit{\textbf{a}}}}^i$ of the points of the middle surface S and where

$$\begin{aligned} V_K(\omega ):=\{ {\varvec{\eta }}=(\eta _i)\in H^1(\omega )\times H^1(\omega )\times H^2(\omega ); \eta _i=\partial _{\nu }\eta _3=0 \ \text {in} \ \gamma _0 \}, \end{aligned}$$

with $\partial _\nu$ denoting the outer normal derivative along $\gamma$. Under the same assumptions as for the viscoelastic elliptic membranes problem, we show that the displacement field, $\xi _{K,i}^\varepsilon {{\textit{\textbf{a}}}}^i$, converges to $\xi _{i}{{\textit{\textbf{a}}}}^i$ (the solution of the two-dimensional problem for a viscoelastic elliptic membrane) in $H^{1}(0,T;H^1(\omega ))$ for the tangential components, and in $H^{1}(0,T;L^2(\omega ))$ for the normal component, as $\varepsilon \rightarrow 0$. Under the same assumptions as in the viscoelastic flexural shell problem, we show that the displacement field, $\xi _{K,i}^\varepsilon {{\textit{\textbf{a}}}}^i$, converges to $\xi _{i}{{\textit{\textbf{a}}}}^i$ (the solution of the two-dimensional problem for a viscoelastic flexural shell) in $H^{1}(0,T;H^1(\omega ))$ for the tangential components, and in $H^{1}(0,T;H^2(\omega ))$ for the normal component, as $\varepsilon \rightarrow 0$. Also, we obtain analogous results assuming the same assumptions as in the viscoelastic generalized membranes problem. Therefore, we justify the two-dimensional viscoelastic model of Koiter for all kind of viscoelastic shells.