Stability of elastic bodies under omnilateral compression

Abstract

We have analyzed here the stability of the equilibrium of a simply connected isotropic compressible body with the elastic potential of arbitrary form and under uniform omnilateral deformation. A survey has been given here of earlier results obtained by other authors. The basic celations have been stated in a general form covering the theory of finite subcritical strains and two variants of the theory of small subcritical strains. For the latter theory new relations have been rigorously derived from which perturbations of “tracking” surface loads can be calculated, on the basis of corresponding expressions in the theory of finite subcritical strains. It has been proven that the sufficient conditions for the applicability of the static method of analysis are satisfied when the same boundary conditions are given over the entire body surface, as well as in several cases of different boundary conditions given at different segments of the boundary surface. It has been shown in a general form, for the theory of finite subcritical strains and for two variants of the theory of small subcritical strains, that the equilibrium of an elastic body under omnilateral deformation is stable, if a “tracking” load, is given over the entire boundary surface. As an example of problems with different boundary conditions at different segments of the boundary surface, we have considered the conventional problem concerning the stability of a bar on hinge supports and under uniform omnilateral deformation. It has been rigorously proven that in this case the equilibrium is stable when “tracking” loads are given at the lateral surfaces and is unstable when “dead” loads are given at the lateral surfaces. These conclusions apply to the theory of finite subcritical strains as well as to the theory of small subcritical strains, and they represent the complete version pertaining to compressible bodies.