A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼ h ^β , where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼ h ^α . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)³→ ³, the L ² distance of ∇. v from a single rotation is bounded by a multiple of the L ² distance from the set SO(3) of all rotations.