Obstacle problem for von Kármán equations
This work is devoted to the study of free boundary variational problems in the theory of thin elastic plates with obstacles. In the formulation of this free boundary value problem a plate is considered in possible contact with a rigid obstacle such that the plate is subjected to a distributed load and the shape of the obstacle is given by a prescribed strictly concave function that belongs to some Sobolev space. The boundary of the set of points for which the vertical displacements coincide with the shape of the obstacle (the free boundary) is a priori unknown. A special nonlinear relation is also considered between the strain tensor and deformations, and usual symmetry and ellipticity properties for the elasticity tensor are assumed. The authors prove that any solution of the former problem fulfills some variational inequality, but in the geometric nonlinear case they prove that the variational inequality is not equivalent to the corresponding variational extreme problem. A complementary method is developed here in order to show that the primal nonlinear variational problem is equivalent to a semiquadratic dual optimization problem defined on the statically admissible space. Here the Rockafellar's duality theory is generalized. The authors prove that the uniqueness and existence of the solution depend directly on the gap energy function.
scientific article; zbMATH DE number 91493
scientific article