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

A class of deformations involving one or two unknown functions is proposed for consideration as statically possible in isotropic, incompressible elastic materials which are heat conducting and may also have a specific form of material inhomogeneity due to the material having a layered structure, the layers being infinitesimally thin. Two forms of Fourier's law of heat conduction which allow the determination of the temperature to be independent of the unknown functions are considered. The unknown functions must be determined using semi-inverse methods to solve a system of two non-linear ordinary differential equations arising from the equations for static equilibrium for specific materials. Three subclasses of the general class of deformations are defined. Within these subclasses, existence of controllable solutions to the equations of static equilibrium involving the unknown functions is established for neo-Hookean materials using conventional inequalities, and more generally for materials satisfying the constraints of the well known Baker–Ericksen inequalities.