Generalized von Kármán equations

In a previous work, the first author has identified three-dimensional boundary conditions "of von Kármán's type" that lead, through a formal asymptotic analysis of the three-dimensional solution, to the classical von Kármán equations, when they are applied to the entire lateral face of a nonlinearly elastic plate. In this paper, we consider the more general situation where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, while the remaining portion is subjected to boundary conditions of free edge. We then show that the asymptotic analysis of the three-dimensional solution still leads in this case to a two-dimensional boundary value problem that is analogous to, but is more general than, the von Kármán equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined solely from the data. © 2001 Éditions scientifiques et médicales Elsevier SAS.