Pseudomonotone Operators, Bifurcation, and the von Kármán Plate Equations

Abstract

In this chapter we consider a plate which is clamped at the boundary. Our method of proof, however, can also be applied to other boundary conditions. We use the following tools:

• (I) Implicit function theorem (Theorem 4.B).

• (P) Main theorem about pseudomonotone operators (Theorem 27.A).

• (B) Main theorem of bifurcation theory for potential operators (Theorem 45.A).

The historical development of nonlinear plate and shell theories has not been as felicitous as that of rod theories up to the creation of Kirchhoff’s theory. Kirchhoff (1824–1887) established a satisfactory linear theory of plates begun by Navier (1785–1836). A popular nonlinear plate theory was finally developed by von Kármán in 1910.

These equations are “derived” by introducing, in the standard procedure of mechanics, a number of geometric approximations that are roughly analogous to the replacement of sin φ in the rod equation

$\varphi" \, + Sin\,\varphi \, = \,0$

by its cubic approximation φ − φ ³/6. Moreover, the von Kármán’s equations are based upon a certain linear stress-strain law.

Stuart Antman (1977)

The number of papers about plates is infinitely large, and gets larger and larger.

Mathematical folclore