Koiter shell model

Definition

Let $S$ be the middle surface of a shell of thickness $h$, with metric $a_{\alpha\beta}$ (the first fundamental form) and second fundamental form $b_{\alpha\beta}$. Let the material be isotropic and linearly elastic, with Young's modulus $E$ and Poisson's ratio $\nu$. For a displacement $u$ of $S$, let $a(u)$ and $b(u)$ be the two fundamental forms of the deformed surface. The change of metric is $\gamma_{\alpha\beta}(u) = \frac{1}{2}(a_{\alpha\beta}(u) - a_{\alpha\beta})$, and the change of curvature is $\rho_{\alpha\beta}(u) = b_{\alpha\beta}(u) - b_{\alpha\beta}$. For a symmetric tensor $e_{\alpha\beta}$ on $S$ put $$\mathcal{C}(e) = \frac{E}{1 - \nu^2} \left[ \nu \, (\operatorname{tr} e)^2 + (1 - \nu) \, e^{\alpha\beta} e_{\alpha\beta} \right],$$ with traces and raised indices taken with $a_{\alpha\beta}$. Let $f$ be the load per unit area. The Koiter energy is $$J(u) = \frac{1}{2} \int_S \left[ h \, \mathcal{C}(\gamma(u)) + \frac{h^3}{12} \, \mathcal{C}(\rho(u)) \right] dS - \int_S f \cdot u \, dS.$$ Admissible displacements are those that satisfy the support conditions. The Koiter shell model takes the displacement of the shell to be a stationary point of $J$ among them, in the stable case a minimiser. In the linear Koiter model, $\gamma$ and $\rho$ are replaced by their linear parts in $u$.