ON THE PECULIARITIES OF THE LOSS OF STABILITY OF A NONLINEAR ELASTIC RECTANGULAR BAR

The stability of equilibrium of a compressed rectangular bar made of a n isotropic incompressible elastic material that satisfies the Hadamar d condition is investigated. It is found that the qualitative behaviou r of the bar depends on whether the material belongs to one of three c lasses, conventionally referred to as materials of low, moderate, and high stiffness. For materials of low stiffness the equilibrium of an a rbitrarily thick bar undergoes a bifurcation at a finite value of the critical deformation. For materials of moderate stiffness the critical deformation increases without limit as the relative thickness of the bar increases. For materials of high stiffness a ''limiting'' thicknes s exists, above which no bifurcation of the equilibrium is possible. S imple criteria are obtained which enable one to determine the class to which the specific material belongs. It is established that the propo sed classification of incompressible elastic materials is complete and consistent. Necessary and sufficient conditions for the existence of symmetric and antisymmetric modes of stability loss are found for mate rials of moderate and high stiffness. It is pointed out that, in some cases, symmetric bifurcation occurs prior to the antisymmetric one. Fo r materials of low stiffness it is found that (for certain values of t he relative thickness) double critical values of the deformation param eter may exist corresponding to two distinct buckling modes, namely, s ymmetric and antisymmetric ones. A sufficient condition for a symmetri c bifurcation to be preceded by an antisymmetric one is stated. Specif ic models of incompressible elastic materials are considered. It is po inted out that the method developed in this paper is also effective in analysing the axisymmetric instability of a circular plate compressed by a uniform side pressure.