MiS Preprint Repository

We derive a hierarchy of plate models from three dimensional nonlinear elasticity by $\mathrm{\Gamma }$-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume $\sim h^{\beta }$, where $h$ is the thickness of the plate.

This is in turn related to the strength of the applied force $\sim h^{\alpha }$. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to $\alpha =\beta =0$, nonlinear bending theory to $\alpha =\beta =2$, Föppl von Kármán theory to $\alpha =3$, $\beta =4$ and linearized vK theory to $\alpha >3$. Intermediate values of $\alpha$ lead to certain theories with constraints.

A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [31] that for maps $v:(0,1{)}^3\to {\mathbb{R}}^3$, the $L^2$ distance of $\mathrm{\nabla }v$ from a single rotation is bounded by a multiple of the $L^2$ distance from the set $SO(3)$ of all rotations.