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Asymptotic analysis of linearly elastic shells. II: Justification of flexural shell equations. (English) Zbl 0887.73039
In the second part, a family of linearly elastic shells with thickness \(2\varepsilon\) is considered. 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, and \(\varphi\in C^3(\overline\omega, \mathbb{R}^3)\). 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 \(\gamma_{\alpha\beta}(\eta)\) be the components of the linearized change of metric tensor of the surface \(S\). It is assumed that the space of inextensional displacements \(V_F(\omega)= \{\eta=(\eta_i)\in H^1(\omega)\times H^1(\omega)\times H^2(\omega)\); \(\eta_i= \partial_v\eta_3= 0\) on \(\gamma_0,\gamma_{\alpha\beta}(\eta)\) in \(\omega\}\) contains non-zero functions.
Let \(f\) be the applied body force density which is \(O(\varepsilon^2)\) with respect to \(\varepsilon\), and \(u_i(\varepsilon)\) denote three covariant components of the displacement of the points of the shell given by the equations of three-dimensional elasticity. Then the field \(u(\varepsilon)= (u_i(\varepsilon))\), once scaled so as to be defined over the fixed domain \(\Omega= \omega\times[- 1,1]\), converges to a limit \(u\in H^1(\Omega)\) as \(\varepsilon\to 0\), which is independent of the transverse variable. The average \(\xi= {1\over 2}\int^1_{-1} udx_3\) belongs to the space \(V_F(\omega)\) and satisfies the two-dimensional equations of a “flexural shell” \({1\over 3}\int_\omega a^{\alpha\beta\sigma\tau} \rho_{\sigma\tau}(\xi) \rho_{\alpha\beta}(\eta)\sqrt ady= \int_\omega\left\{ \int^1_{-1} f^idx_3\right\}\eta_i \sqrt ady\) for all \(\eta= (\eta_i)\in V_M(\omega)\), where \(a^{\alpha\beta\sigma\tau}\) are the components of the two-dimensional elasticity tensor of the surface \(S\), \(\rho_{\alpha\beta}\) are the components of the linearized change of the curvature tensor of \(S\), and \(f^i\) are the scaled components of the applied body force.

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