Decomposition of the deformations of a thin shell. Nonlinear elastic models

36 pagesWe investigate the behavior of the deformations of a thin shell, whose thickness $\delta$ tends to zero, through a decomposition technique of these deformations. The terms of the decomposition of a deformation $v$ are estimated in terms of the $L^2$-norm of the distance from $\nabla v$ to $SO(3)$. This permits in particular to derive accurate nonlinear Korn's inequalities for shells (or plates). Then we use this decomposition technique and estimates to justify a nonlinear bending model for elastic shells for an elastic energy of order $\delta^3$