On a class of controllable deformations of isotropic incompressible elastic solids with simple material inhomogeneity: II

In a preceding paper, a class of deformations involving one or two unknown (shear) functions was proposed for consideration as statically possible in all incompressible, isotropic elastic materials, which are heat conducting and may also have a specific form of latent material inhomogeneity due to an infinitesimally thin layered structure. Within three well-defined subclasses C1, C2 and C3, the existence of controllable deformations was established, if the free energy of the material is smooth as a function of the invariants of the deformation tensor and satisfies the constraints of the Baker–Ericksen (B–E) inequalities. A deformation of a body is said to be controllable if the boundary tractions necessary to maintain it in static equilibrium are known. Here the subclass C4 of deformations, which is the set complementary to those previously demonstrated to be controllable in all materials of the general type, is considered. The (controllable) existence of the elements of this subclass is established in the main if (i) the free energy of the material is twice continuously differentiable with respect to the first two invariants of the left Cauchy–Green deformation tensor when the two unknown shear functions take the values zero and (ii) the B–E inequalities apply. For the exceptional cases (controllable) existence is established if the E-inequalities replace the B–E inequalities. The results are achieved by use of the implicit function theorem. The smoothness assumptions are stronger than those of the preceding paper.