Stability of pure homogeneous deformations of an elastic cube under dead loading

Abstract

1. Introduction. In a previous paper [1], the problem was considered of the pure homogeneous deformation of a unit cube of incompressible neo-Hookean elastic material by three pairs of equal and opposite forces acting normally on the faces of the cube and distributed uniformly over them. It was found that, for certain specified values of the forces, more than one equilibrium state of pure homogeneous deformation can exist. The stability of each of these states was investigated, with respect to superposed infini-tesimal pure homogeneous deformations, with the same principal directions as the equilibrium state. It was found that for certain ranges of values of the applied forces, more than one equilibrium state of pure homogeneous deformation which is stable in this sense can exist. Which of these stable states is actually attained in practice will depend on the order in which the forces are applied. As a special case, the situation was considered in which all three pairs of forces are the same and it was found that, even in this case, for certain values of the applied forces, more than one stable equilibrium state is possible. We denote by A1 , A2 , A3 the extension ratios for the pure homogeneous deformation and by W the strain-energy per unit volume. Then, for an incompressible neo-Hookean material, W is given by