There are six known families of deformations which are (isothermally) universal and controllable for incompressible, isotropic hyperelastic solids. It is shown here that, independently of material, the first five of these families and a subclass of the sixth, no others, are compatible with non-uniform temperatures in the sense that a pressure exists to ensure static equilibrium, provided the temperature gradient is parallel to the divergence of the left Cauchy-Green strain tensor.
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