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Asymptotic analysis of linearly elastic shells. III: Justification of Koiter’s shell equations. (English) Zbl 0887.73040

In the third part again, a family of linearly elastic shells with thickness \(2\varepsilon\) is considered. It is supposed that all shells have the same middle surface \(S= \varphi(\overline\omega)\subset \mathbb{R}^3\), where \(\omega\subset\mathbb{R}^2\) is a bounded and connected open set with Lipschitz-continuous boundary, \(\varphi\in C^3(\overline\omega, \mathbb{R}^3)\), and the shells are clamped on a portion of their lateral face, whose middle line is \(\varphi(\gamma_0)\), where \(\gamma_0\) is a portion of \(\partial\omega\) with length \(\gamma_0>0\). Let \(a^{\alpha\beta\sigma\tau}\) be the components of the two-dimensional elasticity tensor of the surface \(S\), \(\gamma_{\alpha\beta}(\eta)\) and \(\rho_{\alpha\beta}(\eta)\) be the components of the linearized change of metric tensor and change of curvarture tensors of \(S\), and \(p^{i,\varepsilon}\) be the components of the resultant of the applied forces. And let, for all \(\varepsilon>0\), \(u^\varepsilon_i\) denote the covariant components of the displacement \(u^\varepsilon_i g^{i,z}\) of the points of the shell, obtained by solving the three-dimensional problem, and \(\xi^\varepsilon_i\) denote the covariant components of the displacement \(\xi^\varepsilon_i a^i\) of the points of the middle surface \(S\), obtained by solving the two-dimensional model of W. T. Koiter: \[ \xi^\varepsilon= (\xi^\varepsilon_i)\in V_K(\omega)=\Biggl\{\begin{matrix} \eta= (\eta_i)\in H^1(\omega)\times H^1(\omega)\times H^2(\omega)\\ \eta_i= \partial_v\eta_3=0\quad\text{on }\gamma_0\end{matrix}\Biggr\}, \] \(\varepsilon\int_\omega a^{\alpha\beta\sigma\tau} \gamma_{\sigma\tau}(\xi^\varepsilon) \gamma_{\alpha\beta}(\eta)\sqrt ady+ {\varepsilon^3\over 3} \int_\omega a^{\alpha\beta\sigma\tau} \rho_{\sigma\tau}(\xi^\varepsilon) \rho_{\alpha\beta}(\eta) \sqrt ady= \int_\omega p^{i,\varepsilon}\eta_i \sqrt ady\) for all \(\eta=(\eta_i)\in V_K(\omega)\).
It is proved that the fields \({1\over 2\varepsilon} \int^\varepsilon_{-\varepsilon} u^\varepsilon_i g^{i,\varepsilon} dx^\varepsilon_3\) and \(\xi^\varepsilon_i a^i\) on the surface \(S\), under the same assumptions as in part I, have the same principal part as \(\varepsilon\to 0\) in \(H^1(\omega)\) for the tangential components and in \(L^2(\omega)\) for the normal component. The same fields under the same assumptions as in part II have the same principal parts as \(\varepsilon\to 0\) in \(H^1(\omega)\) for all other components.

MSC:

74K15 Membranes
35Q72 Other PDE from mechanics (MSC2000)
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