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A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses. [ABSTRACT FROM AUTHOR]

In the present article, a spectral model is developed for residually stressed electro-elastic bodies. The model uses a total energy function that depends on the right stretch tensor, residual stress tensor and one of the electric variables. Some boundary value results with cylindrical symmetry are given. Results for the inflation of a hollow sphere, where the residual stress is assumed to depend only on the radial position, are also given. The constitutive formulation contains spectral invariants that have an immediate physical interpretation which is useful in a rigorous construction of a specific form of the total energy function via an appropriate experiment [ABSTRACT FROM AUTHOR]

The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math., 48, 279-319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58, 309-317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A, 463, 357-367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy-Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green-St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A, 463, 357-367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures. [ABSTRACT FROM AUTHOR]

The implicit function theorem is used to obtain some necessary and sufficient conditions, for which an implicit constitutive relation can be solved in terms of the stresses and strains, in the sense of expressing, for example, the strains in terms of the stresses or vice versa. For an isotropic body some exact solutions for the strains in term of the stresses are presented, using the theory of tensor equations of the form AX + XA = H. For the exact solutions considered, conditions for the existence and the uniqueness are given. [ABSTRACT FROM PUBLISHER]