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 Gamma-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume similar to h(beta), where h is the thickness of the plate. This is in turn related to the strength of the applied force similar to h(alpha). Membrane theory, derived earlier by Le Dret and Raoult, corresponds to alpha = beta = 0, nonlinear bending theory to alpha = beta = 2, von Karman 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 [29] which states that for maps nu : (0,1)(3) -> R-3, the L-2 distance of del nu from a single rotation is bounded by a multiple of the L(2)supercript stop distance from the set SO(3) of all rotations.