Two-Dimensional Approximations of Three-Dimensional Models in Nonlinear Plate Theory

Abstract

The asymptotic expansion method, with the thickness as the parameter, is applied to the nonlinear, three-dimensional, equations for the equilibrium of elastic plates under suitable loads and appropriate boundary conditions. It is shown that the leading term of the expansion is a solution of a system of equations equivalent to a well-known two-dimensional nonlinear plate model, namely the von Karman equations. The existence of solutions of the two-dimensional problem is established in all cases (by contrast with the three-dimensional model, where no satisfactory existence theory is as yet available). It is also shown that the displacement and the stress corresponding to the leading term of the expansion have the specific form generally assumed a priori in the usual derivations of two-dimensional plate models. In particular, the displacement field is of Kirchhoff-Love type. This approach clarifies in particular the nature of the admissible three-dimensional boundary conditions for a given two-dimensional plate model. A discussion is also given regarding the class of admissible three-dimensional models.