Variational Problems for Föppl--von Kármán Plates

Abstract

Some variational problems for a Föppl--von Kármán 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 $\Gamma$-convergence approach we show that critical points of the Föppl--von Kármán energy can be strongly approximated by uniform Palais--Smale sequences of suitable functionals: this property leads to identifying relevant features for critical points of approximating functionals, e.g., buckled configurations of the plate. The analysis for rescaled thickness is performed by assuming that the plate-like structure is initially prestressed, so that the energy functional depends only on the out-of-plane displacement and exhibits asymptotic oscillating minimizers as a mechanism to relax compressive states.