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Asymptotic analysis of linearly elastic shells. I: Justification of membrane shell equations. (English) Zbl 0887.73038
In this three-part work, the authors analyze the asymptotic behaviour of the scaled three-dimensional displacement field of a linearly elastic shell. In the first part, a family of such shells with thickness $$2\varepsilon$$ clamped along their entire lateral face 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 $$\gamma$$ and $$\varphi\in C^3(\overline\omega,\mathbb{R}^3)$$. It is supposed that $$\gamma$$ and $$\varphi$$ are smooth enough, and that two principal radii of curvature are either both positive at all points of $$S$$, or are both negative at all points of $$S$$.
Let the applied body force density be $$O(1)$$ 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 as $$\varepsilon\to 0$$ to a limit $$u\in H^1(\Omega)\times H^1(\Omega)\times L^2(\Omega)$$ which is independent of the transverse variable. The average $$\xi= {1\over 2} \int^1_{-1} udx_3$$ belongs to the space $$V_m(\omega)= H^1_0(\omega)\times H^1_0(\omega)\times L^2(\omega)$$ and satisfies the two-dimensional equations of a “membrane shell” $$\int_\omega a^{\alpha\beta\sigma\tau} \gamma_{\sigma\tau}(\xi) \gamma_{\alpha\beta}(\eta)\sqrt{a} dy= \int_\omega \left\{\int^1_{-1} f^idx_3\right\} \eta_i\sqrt ady$$ for all $$\eta= (\eta_i)\in V_M(\omega)$$; here $$a^{\alpha\beta\sigma\tau}$$ are the components of the two-dimensional elasticity tensor of the surface $$S$$, $$\gamma_{\alpha\beta}$$ are the components of the linearized change of the metric 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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##### References:
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