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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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