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.