Variational Problems for Foppl-von Karman plates

Abstract

Some variational problems for a Foppl-von Karman plate subject to general equilibrated loads are studied. The existence of global minimizers is proved under the assumption that the out-of-plane displacement fulfils homogeneous Dirichlet condition on the whole boundary while the in-plane displacement fulfils nonhomogeneous Neumann condition. If the Dirichlet condition is prescribed only on a subset of the boundary, then the energy may be unbounded from below over the set of admissible configurations, as shown by several explicit conterexamples: in these cases the analysis of critical points is addressed through an asymptotic development of the energy functional in a neighborhood of the flat configuration. By a $Γ$-convergence approach we show that critical points of the Foppl-von Karman energy can be strongly approximated by uniform Palais-Smale sequences of suitable functionals: this property leads to identify relevant features for critical points of approximating functionals, e.g. buckled configurations of the plate. Eventually we perform further analysis as the plate thickness tends to 0, by assuming that the plate is prestressed and the energy functional depends only on the transverse displacement around the given prestressed state: by this approach, first we identify suitable exponents of plate thickness for load scaling, then we show explicit asymptotic oscillating minimizers as a mechanism to relax compressive states in an annular plate.