## 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 section 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$_{3}$ < ($\frac{1}{3}$E)$^{3}$, where $\frac{1}{3}$E 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$_{3}$ > ($\frac{1}{3}$E)$^{3}$, 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 $\frac{1}{3}$E. The actual state of deformation which is obtained depends in general on the order in which the forces are applied. In the cases (1) and (2a), it is shown that the unique equilibrium state is one of stable equilibrium. In case (2b), it is shown that the possible equilibrium states are of eight different types. Two of these are mutually exclusive. Of these eight types of equilibrium state, four are inherently unstable. Of the other four types, one is inherently stable. The three remaining types of equilibrium state are not necessarily unique; i.e. more than one equilibrium state of each type may correspond to specified values of f$_{1}$,f$_{2}$ and f$_{3}$. However, if one or more equilibrium states of a particular type exists, then at least one of them is stable. If f$_{1}$ = f$_{2}$ = f$_{3}$ = f, then there is a unique state of stable equilibrium, provided that f < $\frac{1}{3}$E. If $\frac{1}{3}$E < f < ($\frac{1}{4}$)$^{\frac{1}{3}}$E, then there are three states of stable equilibrium, none of which is identical with the undeformed state of the cube. If ($\frac{1}{4}$)$^{\frac{1}{3}}$E < f < $\frac{2}{3}$E, there are four states of stable equilibrium, one of which is identical with the undeformed state of the cube. If f > $\frac{2}{3}$E, then there is only one state of stable equilibrium, and this is identical with the undeformed state of the cube.

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