Instability of an internally constrained hyperelastic material

The wave speed criterion is applied to discuss restrictions on the material response functions of an isotropic, Bell constrained material in order that small amplitude, plane waves may propagate in the material with real-wave speeds. The case of a body subjected to a biaxial stress is related to the elastic stability of a Bell constrained slab. We show that two cases among five reported by Beatty and Pan in study of the elastic stability of a thick plate cannot support such waves. These cases, therefore, are not possible and must be excluded as potential solutions. The question of the stability of the equibiaxial deformations of a Bell constrained material under an all-around Cauchy stress, suggested in primary work of Beatty and Hayes on homogeneous deformations of Bell materials, is investigated. It is shown that for a general Bell material model, the state of equibiaxial deformation under an all-around Cauchy stress is materially unstable, unless all stretches are equal to one, their value in the initial undistorted state.