Weakened conditions of admissibility of surface forces applied to linearly elastic membrane shells
Résumé
We consider a family of linearly elastic shells of the first kind (as defined in [Ciarlet, 2000]), also known as non inhibited pure bending shells [Sanchez-Hubert, Sanchez-Palencia, 1997]. This family is indexed by the half-thickness $\varepsilon$. When $\varepsilon$ approaches zero, the averages across the thickness of the shell of the covariant components of the displacement of the points of the shell converge strongly towards the solution of a ”2D generalized membrane shell problem” provided the applied forces satisfy ”admissibility” conditions [Ciarlet-Lods, 1996]. The identification of the admissible applied forces usually requires delicate analysis. In the first part of this paper we simplify the general admissibility conditions when applied forces \textbf{h} are surface forces only, and obtain conditions that no longer depend on $\varepsilon$ [Luce-Poutous-Thomas, 2007] : find $h^{\alpha \beta} = h^{\beta\alpha}$ in $L^2(\omega)$ such that for all $\eta = (eta_i)$ in $V(\omega)$, $\int_{\omega} h^i\eta_id\omega = \int_{\omega} h^{\alpha\beta}\gamma_{\alpha \beta}(\eta)d\omega$ where $\omega$ is a domain of $^2$, $\theta$ is in $C^3(\bar{\omega},\mathbb{R^3})$ and S = \theta(\bar{\omega}) is the middle surface of the shells, where $(\gamma_{\alpha\beta}(\eta))$ is the linearized strain tensor of $S$ and $\mathbf{V}(\omega) = {\eta \in \mathbf{H}^1(\omega), \eta=0 \mbox{ on } \gamma_0$}$, the shells being clamped along $\Gamma_0 = \theta(\gamma_ 0)$. In the second part, since the simplified admissibility formulation does not allow to conclude directly to the existence of $h^{\alpha \beta}$, we seek sufficient conditions on \textbf{h }for $h^{\alpha \beta}$ to exist in $L^2(\omega)$. In order to get them, we impose more regularity to $h^{\alpha \beta}$ and boundary conditions. Under these assumptions, we can obtain from the weak formulation a system of PDE with $h^{\alpha \beta}$ as unknowns. The existence of solutions depends both on the geometry of the shell and on the choice of \textbf{h}. We carry through the study of four representative geometries of shells and identify in each case a special admissibility functional space for \textbf{h}.
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