On a Class of Deformations of Compressible, Isotropic, Nonlinearly Elastic Solids

We consider deformations of unconstrained, isotropic hyperelastic solids which satisfy the condition that the determinant of the deformation gradient is constant. In the absence of body forces, it is shown (i) that a certain deformation in this class (which describes the bending of rectangular blocks into annular cylindrical sectors) is not possible in any of the considered materials, (ii) that in the case when the body fills the whole space, it is composed of a compressible neo-Hookean material and it is subjected to relatively moderate loads, these deformations are necessarily homogeneous and (iii) that for boundary conditions of place and relative to a certain sub-class of the class of considered materials, these deformations are globally stable, in the sense that they are minimizers for the total energy with respect to smooth variations that are compatible with the boundary conditions. This revised version was published online in August 2006 with corrections to the Cover Date.