A note on the Burger’s method

Abstract

The existence and uniqueness of the solutions to an infinite system of nonlinear equations arising in the  dynamic analysis of large deflection of a plate are discussed. Under the assumption that the rim of the plate is prevented from inplane motions, explicit equation for the coupling parameter is given.

Keywords: Existence, uniqueness, Berger’s method.1991 Mathematics Subject Classification. 46(Functional analysis),73(Mechanics of solids)

Introduction

In the analysis of classical mechanics problems, there are cases where linear mathematical model can not fully describe the phenomena. If the deflection of the plate is of order of magnitude of its thickness, the differential equations for the deflection and displacements can be written in terms of nonlinear equations. These nonlinear equations are usually difficult to obtain the solution. Thus several attempts have been tried to obviate the difficulties.
Among these attempts, it was Berger’s method which drew much attention.
Berger[1] derived a simplified nonlinear equations for a plate with large deflections by assuming that the strain energy due to the second invariants of the middle surface strains can be neglected when deriving the differential equations by energy method. Berger restricted his analysis to static and isotropic cases. Later, his procedure was generalized to dynamics of isotropic plates by Nash and Modeer [2] and to dynamic phenomena in anisotropic plates and shallow shells by Nowinski [3]. Berger’methods is dealt in recent books [5, 6].
There many are papers giving explicit solutions to various cases, however the search for the existence and uniqueness of the solution is rare, thus it is the purpose of this paper to discuss this matter.
The governing equations are

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