Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure, homogeneous deformation

Abstract

The equilibrium of a cube of incompressible, neo-Hookean material, under the action of three pairs of equal and oppositely directed forces f 1 , f 2 , f 3 , 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 f 1 , f 2 and f 3 are specified, (a) the corresponding equilibrium state is uniquely determined, provided that one or more of the forces f 1 , f 2 and f 3 is negative, i.e. is a compressional force, or, if they are all positive, provided that f 1 f 2 f 3 > (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 f 1 , f 2 and f 3 are all positive and f 1 f 2 f 3 > (1/3E) 3 , then the equilibrium state is not necessarily uniquely determined. The number of equilibrium states which exist depends on the values of f 1 , f 2 , f 3 and 1/3E. The actual state of deformation which is obtained depends in general on the order in which the forces are applied.