ON A CLASS OF DEFORMATIONS OF COMPRESSIBLE, ISOTROPIC, NONLINEARLY ELASTIC SOLIDS

We consider deformations of unconstrained, isotropic hyperelastic soli ds 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 possibl e in any of the considered materials, (ii) that in the case when the b ody fills the whole space, it is composed of a compressible neo-Hookea n material and it is subjected to relatively moderate loads, these def ormations are necessarily homogeneous and (iii) that for boundary cond itions of place and relative to a certain sub-class of the class of co nsidered materials, these deformations are globally stable, in the sen se that they are minimizers for the total energy with respect to smoot h variations that are compatible with the boundary conditions.