A Variational Model for Non Linear Elastic Plates

A classical approach to the study of thin structures in elasticity consists in starting from three-dimensional models and deducing the behaviour of two-dimensional or onedimensional thin elastic bodies by passing to the limit when one or two dimensions go to zero. Though the two-dimensional linear model of an elastic plate has exhaustively been studied from different points of view (see [1], [3], [8], [9]), to our knowledge it doesn’t exist a rigorous theory which permits to deduce a non linear model of elastic plate as a limit (for instance, in the sense of [3]) of non linear three-dimensional thin elastic bodies. The most difficulty which arises in treating these problems is that, without some kinematical constraints on the deformations, no information about the compactness of minimizers of the energy functional of the three-dimensional elastic bodies can be expected and therefore it is very difficult to formulate any reasonable conjecture about the “limit” functional, that is the energy functional of the limit plate. The simplest method to avoid this difficulty is that of Kirchhoff, which consists in the linearization of the Green-St.Venant energy functional ∫