Justification of Koiter’s Shell Model Using Gamma Convergence

Abstract

We consider a family of linear elastic shells of thickness 2e, all having the same middle surface \(S=\varphi(\omega)\subset \mathbb{R}^3,\) where \(\omega\subset \mathbb{R}^2\) is a bounded and connected open set with a Lipschitz-continuous boundary ∂ω. The shells are clamped on a portion of their lateral face, whose middle line is φ(γ0), where γ0 is a portion of ∂ω with length(γ0) > 0. Under the assumption that γ0 = ∂ω, the applied forces f i e , once appropriately scaled on a domain \(\Upomega=\omega\times (-1, 1),\) are independent of e and the middle surface S of the shell is uniformly elliptic we show using \(\Upgamma\)-convergence that \(\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}u^{\epsilon}_ig^{i,\epsilon}dx_3, \) where \(u^{\epsilon}_ig^{i,\epsilon}\) denotes the displacement of the shell and \(\zeta^{\epsilon}_ia^i,\) the displacement of the middle surface of the shell obtained by solving the two dimensional Koiter equation will have the same principal part as \(\epsilon\rightarrow 0\) in H 1(ω) for the tangential components, and in L 2(ω) for the normal components. We also show that if \(f^{\epsilon}_i=\epsilon^2f_i, \, f_i\in L^2(\Upomega)\) and if the space of inextensional displacement is non-zero then \(\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}u^{\epsilon}_ig^{i,\epsilon}dx_3 \) and \(\zeta^{\epsilon}_ia^i\) will have the same principal part as \(\epsilon\rightarrow 0\) in H 1(ω) for their tangential components and in H 2(ω) for their normal components.