From the classical to the generalized von Kármán and Marguerre–von Kármán equations

Abstract

In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre–von Kármán shallow shells. First, we briefly review the “classical” von Kármán and Marguerre–von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data. Following recent joint works, we then reduce these more general equations to a single “cubic” operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation.