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

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.

我们考虑厚度为\（2 \ varepsilon \）的线性粘弹性壳族，它们都具有相同的中间表面\（S = {\ varvec {\ theta}}（{\ bar {\ omega}}）\ subset \ hbox {I} \！\ hbox {R} ^ 3 \），其中\（\ omega \ subset \ hbox {I} \！\ hbox {R} ^ 2 \）是Lipschitz-continuous的有界和连通开放集{\ mathcal {C}} ^ 3（{\ bar {\ omega}}; \ hbox {I} \！\ hbox {R中的边界\（\ gamma \）和\（{\ varvec {\ theta}} \ } ^ 3）\）。壳被夹紧在其侧面的一部分，其中间线是\（{\ varvec {\ THETA}}（\伽马_0）\） ，其中\（\伽马_0 \）是一个非空的部分\ （\ gamma \）。这项工作的目的是证明粘弹性Koiter模型是解决粘弹性壳体位移问题的最精确的二维方法。此外，Koiter模型\（{\ varvec {\ xi}} _ K ^ \ varepsilon =（\ xi _ {K，i} ^ \ varepsilon）\）的解在\（H ^ {1}（ 0，T; V_K（\ omega））\），带有\（\ xi ^ \ varepsilon _ {K，i}：[0，T] \次{\ bar {\ omega}} \ rightarrow {\ mathbb {R }} \）中间表面S和点S的点的位移字段\（\ xi _ {K，i} ^ \ varepsilon {{\ textit {\ textbf {a}}}} ^ i \）的协变分量哪里

其中\（\ partial _ \ nu \）表示沿\（\ gamma \）的外部法线导数。在与粘弹性椭圆膜问题相同的假设下，我们表明位移场\（\ xi _ {K，i} ^ \ varepsilon {{\ textit {\ textbf {a}}}} ^ i \），收敛于\（H ^ {1}中的\（\ xi _ {i} {{\ textit {\ textbf {a}}}} ^ i \）（粘弹性椭圆膜的二维问题的解）（0，T; H ^ 1（\ omega））\）为切向分量，而\（H ^ {1}（0，T; L ^ 2（\ omega））\）为正切分量，如\（\ varepsilon \ rightarrow 0 \）。在与粘弹性挠性壳问题相同的假设下，我们表明位移场\（\ xi _ {K，i} ^ \ varepsilon {{\ textit {\ textbf {a}}}} ^ i \），收敛到\（\ xi _ {i} {{\ textit {\ textbf {a }（}}}} ^ i \）（切向分量在\（H ^ {1}（0，T; H ^ 1（\ omega））\）中的粘弹性挠曲壳二维问题的解），并在\（H ^ {1}（0，T; H ^ 2（\ omega））\）中表示正常分量，如\（\ varepsilon \ rightarrow 0 \）。同样，我们获得了与粘弹性广义膜问题相同假设的类似结果。因此，我们证明了Koiter的二维粘弹性模型适用于所有类型的粘弹性壳。