Propagation of a plane wave to a materially uniform but inhomogeneous body

We produce the equations of small deformations superimposed upon large for materially uniform but inhomogeneous bodies and specialize to an isotropic material and to a homogeneous finite elastic deformation. By assuming the small deformation to be a plane wave, a set of equations for the amplitude of the wave is produced which is accompanied by an additional set of conditions. By requiring a non-trivial solution for the amplitude, we obtain the secular equation and from it a set of necessary and sufficient conditions for having a real wave speed. The second set of conditions that have to be satisfied is due to the materials inhomogeneity. Essentially, the present analysis enhances the approach of Hayes and Rivlin for materially uniform but inhomogeneous bodies. The outcome is that for such bodies the restrictions on the constitutive law for having real wave speeds for an isotropic material subjected to a pure homogeneous deformation involves the field of the inhomogeneity as well.