On the generalized von Karman equations and their approximation

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Original languageEnglish
Pages (from-to)617-633
Journal / PublicationMathematical Models and Methods in Applied Sciences
Issue number4
Publication statusPublished - Apr 2007


We consider here the "generalized von Kármán equations", which constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions "of von Kármán type" only on a portion of its lateral face, the remaining portion being free. As already shown elsewhere, solving these equations amounts to solving a "cubic" operator equation, which generalizes an equation introduced by Berger and Fife. Two noticeable features of this equation, which are not encountered in the "classical" von Kármán equations are the lack of strict positivity of its cubic part and the lack of an associated functional. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J. L. Lions and on Brouwer's fixed point theorem. This convergence proof provides in addition an existence proof for the original problem. © World Scientific Publishing Company.

Research Area(s)

  • Brouwer's theorem, Finite element method, Nonlinear plate theory