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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)
##### Keywords:
convergence; middle surface; Lipschitz-continuous boundary
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##### References:
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