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MiS Preprint
7/2005
A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence
Gero Friesecke, Richard D. James and Stefan Müller
Abstract
We derive a hierarchy of plate models from three dimensional nonlinear elasticity by $\Gamma$-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 $\thicksim h^\alpha$. 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 $v:(0,1)^3 \rightarrow \mathbb{R}^3$, the $L^2$ distance of $\nabla v$ from a single rotation is bounded by a multiple of the $L^2$ distance from the set $SO(3)$ of all rotations.
