Inequality Problems in the Theory of Thin Elastic Plates

Abstract

This chapter is devoted to the study of certain classes of static and dynamic inequality problems pertaining to the theory of thin elastic plates. Here we consider the model proposed by von Kármán for plates undergoing large deflections relative to their thickness. Kirchhoff’s theory for plates with small deflections constitutes a special case of von Kármán’s theory. The latter permits a rational treatment of the problem of plate buckling and therefore is preferred in this chapter. The first section, in which some static unilateral problems are formulated and studied, is based on papers by G. Duvaut and J. L. Lions [87], J. Franču [102], P. Hess [140], O. John [154], J. Naumann [225], M. Potier-Ferry [268], [269] among others. In the second section, to study the buckling problem, the corresponding eigenvalue problem is discussed. With reference to this problem, we give some general propositions concerning the eigenvalue problem for variational inequalities. This is a new area, only recently developed, and for this reason many questions are still open; we rely herein on the papers of A. F. Abeasis, J. P. Dias and A. Lopes-Pinto [1], M. Berger [22], M. Berger and P. Fife [23], A. Cimetière [43], [44], C. Do [71], [72], [73], [74], M. Kučera, J. Nečas and J. Souček [165], E. Miersemann [203], J. Naumann and H. Wenk [226], and M. Potier-Ferry [270], [271]. The third section concerns dynamic unilateral problems of von Kármán plates and is based on [88] and [248].