Spherical inflation for a class of compressible thermoelastic materials

Abstract

In this paper we consider spherical inflation for two families of constrained thermoelastic materials. The two families correspond to different ideas of mechanical incompressibility. For the first family the material is assumed incompressible under mechanical loading in isothermal conditions, and thus the density is independent of the deformation gradient and is a function of temperature alone. The second family consists of materials that are incompressible under adiabatic deformations, in other words materials whose density is a function of the specific entropy alone, independent of the deformation gradient. The first family, with density a function of temperature, has been shown to exhibit instability. We introduce a modification of the constitutive equations for the first family which satisfies the stability criterion. We show that the solutions of thermostatic boundary value problems for the modified material are identical to the solutions obtained from the second family, where density is a function of entropy, with the appropriate identifications.