The equilibrium of a cube of incompressible, neo-Hookean material, under the action of three pairs of equal and oppositely directed forces f1, f2, f3, applied normally to, and uniformly distributed over, pairs of parallel faces of the cube, is studied. It is assumed that the only possible equilibrium states are states of pure, homogeneous deformation. It is found that (1) when the stress components in the deformed cube are specified, the corresponding equilibrium state is uniquely determined (this is shown in § 6 of Part I). (2) When the three pairs of equal and oppositely directed forces f1, f2 and f3 are specified, (a) the corresponding equilibrium state is uniquely determined, provided that one or more of the forces f1, f2 and f3 is negative, i.e. is a compressional force, or, if they are all positive, provided that f1f2f3 > (1/3E)3, where 1/3E is the constant of proportionality between the stress and strain components (analogous to the rigidity modulus of the classical theory of small elastic deformations of isotropic materials). (b) If f1, f2 and f3 are all positive and f1f2f3 > (1/3E)3, then the equilibrium state is not necessarily uniquely determined. The number of equilibrium states which exist depends on the values of f1, f2, f3 and 1/3E. The actual state of deformation which is obtained depends in general on the order in which the forces are applied.
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R. S. Rivlin (1948) studied this question.