Some Restrictions on Constitutive Equations

Abstract

The condition for stability of a homogeneous state of deformation of an elastic body under dead-loading is derived on the basis of the Hadamard criterion. The strong-ellipticity condition then follows as a necessary condition. The pure homogeneous deformation is then discussed, of a cube of incompressible isotropic neo-Hookean elastic material, under dead-loading by three equal pairs of equal and opposite forces applied to the faces of cube. It is shown that the resulting state of pure homogeneous deformation is not uniquely determined. The implications of this result, with respect to the material stability conditions proposed by Coleman and Noll and by Truesdell and Toupin are discussed. Finally, some explicit restrictions on the strain-energy function are given, which result from the consideration that the velocities of propagation of plane waves in the pure homogeneously deformed material must be real.