Recent Progress in the Two-Dimensional Approximation of Three - Dimensional Plate Models in Nonlinear Elasticity

Abstract

The asymptotic expansion method, with the thickness as the parameter, is applied to the equilibrium and constitutive equations of nonlinear three-dimensional elasticity. Then the leading term of the expansion can be identified with the solution of well-known two dimensional nonlinear plate models, such as the von Kármán equations. Recent progresses in the application of this method, such as the extension to more general constitutive equations and boundary conditions, the effect of the assumption of polyconvexity, the application to one-dimensional rod models, etc…, are presented. Various open problems, regarding in particular the existence of corresponding three-dimensional solutions and the nature of admissible three-dimensional boundary conditions, are also discussed.