Buckling and barrelling instabilities in finite elasticity

Abstract

In this paper we consider the uniaxial compression of a two dimensional elastic rectangle. The rectangle, henceforth referred to as a rod or bar, is assumed to consist of a general hyperelastic material satisfying appropriate ellipticity and growth hypotheses. We study the mixed displacement- traction boundary value problem for the deformation x of the bar, in which its deformed length is prescribed, the surface traction at its ends has no tangential component, and the lateral surfaces (i.e. those with normal perpendicular to the axis) are specified to be stress-free. We first give sufficient conditions to ensure that for each )~E(0, 1) there is a homogeneous deformation x~ satisfying the traction boundary conditions and having )~ as compression ratio; where by compression ratio we mean the ratio of the deformed to the undeformed lengths of the bar. The main result presented here is that x;~ will always become 'unstable' in some sense at a sufficiently low compression ratio 2~.. We show this by finding the set A of those ,~ for which x;~ fails to be a stationary point of the energy in a suitable Sobolev space, and rigorously prove that 2~ is the largest element of A. We also address the problem of finding the type of instability associated with 2~. as an element of A. The ways in which bodies are observed to start deforming inhomogeneously under compression are described in various ways in the literature, and to avoid confusion we shall stick to a single choice of terminology from now on we say that such an inhomogeneous deformation is of barrelling type if it is symmetric about the axis of compression, and call it a buckling deformation otherwise. We give necessary and sufficient conditions for barrelling to occur, and in particular we show that for a certain class of materials the bar will always become unstable by asymmetric buckling, no matter what its relative dimensions.