Rate constitutive theory for ordered thermoelastic solids
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When the mathematical models for the deforming solids are constructed using the Eulerian description, the material particle displacements and hence the strain measures are not known. In such cases the constitutive theory must utilize convected time derivatives of the strain measures. The entropy inequality provides a mechanism for determining constitutive equations for the equilibrium stress with the additional requirement that the work expanded due to the deviatoric part of the Cauchy stress tensor be positive, but provides no mechanism for establishing the constitutive theory for it. In the development of the constitutive theory in the Eulerian description for thermoelastic solids, one must consider a coordinate system in the current configuration in which the deformed material lines can be identified. Thus the covariant, contravariant and Jaumann convected coordinate systems are natural choices for the development of the constitutive theory. The compatible conjugate pairs of convected time derivatives of the stress and strain measures in these bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of the constitutive theory for thermoelastic solids. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theory must satisfy the conditions resulting from the entropy inequality to ensure thermodynamic equilibrium of the deforming matter. This paper presents development of rate constitutive theories for compressible as well as incompressible, homogeneous, isotropic solids. The density, temperature, and temperature gradient in the current configuration and the convected time derivatives of the strain tensor up to any desired order in the chosen basis are considered as the argument tensors of the first convected time derivative of the deviatoric Cauchy stress tensor and heat vector. The thermoelastic solids described by these constitutive theories are termed ordered thermoelastic solids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to any desired order, the highest order defining the order of the solid. 2012 Springer-Verlag.