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 ∼hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼hα. 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→ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.