On the non-linear buckling and post-buckling analysis of thin elastic shells

Abstract

In this paper the non-linear stability and post-buckling analysis of elastic structures is considered in the frame of a geometrically non-linear shell theory with moderate rotations. Using a total Lagrangian description variational statements as well as associated sets of shell equations including boundary conditions are derived to determine the fundamental equilibrium path, critical points of snap-through or bifurcation buckling and also the post-buckling deformations. To present these equations in a compact form a unified operator description is introduced. It also allows one to prove some important properties, which are needed to construct appropriate approximation procedures like finite element methods. A shell example is calculated numerically by using a simple Rayleigh-Ritz approximation. It is shown that for a two-parameter loading the collection of critical snap-through buckling points is a catastrophe of the ‘cusp’ type.