Alternate forms of thermodynamic laws for thermoelastic solids and the constitutive theories

ABSTRACT

For thermoelastic solids, rate of mechanical work equilibrates with the rate of kinetic energy and rate of strain energy. In this article, this aspect of the physics is utilized to: (i) derive an alternate form of the energy equation based on the first law of thermodynamics and (ii) derive an alternate form of entropy inequality based on the second law of thermodynamics, both free of rate of strain energy. This alternate form of entropy inequality strictly contains physics related to the rates of entropy. This form of entropy inequality is essential to show that the constitutive theories for stress tensor for thermoelastic solids are free of any thermodynamic restrictions and thus can be derived independently of the entropy inequality. In this article, we explore both forms of entropy inequality, one containing rate of strain energy and the alternate form that is free of rate of strain energy in deriving the constitutive theories for the stress tensor. It is shown that the alternate for entropy inequality free of rate of strain energy provides greater flexibility in deriving the constitutive theories for the stress tensor as it places no thermodynamic restrictions on the constitutive theories for the stress tensor. Both forms of the entropy inequality are examined for establishing dependent variables and their arguments for deriving constitutive theories for such solids in Lagrangian description. The solid matter is assumed to be homogeneous, isotropic, compressible, as well as incompressible, but the deformation and the strains can be finite.

KEYWORDS:

• first and second laws; constitutive theories; thermoelastic solids; generators and invariants

Acknowledgment

The authors are grateful to Dr. J. Myers, Program Manager, Scientific Computing, ARO.

Funding

This research was supported by a grant from ARO, Mathematical Sciences division under the grant number W-911NF-11-1-0471(FED0061541) to the University of Kansas, Lawrence, Kansas and Texas A & M University, College Station, Texas.