Rate Constitutive Theories of Order Zero in Lagrangian Description for Thermoelastic Solids

Abstract

This article presents constitutive theories for the stress tensor and the heat vector for homogeneous, isotropic thermoelastic solids in Lagrangian description for finite deformation. The deforming solid is assumed to be in thermodynamic equilibrium during the evolution. Since conservation of mass, balance of momenta, and balance of energy are independent of the constitution of the matter, the second law of thermodynamics must form the basis for deriving the constitutive theories. We introduce the concept of rate constitutive theory and show that for thermoelastic solids the constitutive theories are, in fact, rate theories of order zero. These theories for stress tensor consider material derivative of order zero of the conjugate strain tensor as one of the argument tensors of the stress tensor established as a dependent variable in the constitutive theory. Generalization of this concept leading to higher order rate theories in Lagrangian description are considered in followup works [1, 2]. The conditions resulting from the entropy inequality in Helmholtz free energy density permit the derivation of constitutive theory for stress tensor in terms of conjugate strain tensor or material derivative of order zero of the conjugate strain tensor. In the work presented here, it is shown that, using the conditions resulting from the entropy inequality, the constitutive theory for the stress tensor can be derived using three approaches: (i) assuming Helmholtz free energy density to be a function of the invariants of the material derivative of order zero of the conjugate strain tensor and temperature θ and then using the conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants; and (iii) expanding Helmholtz free energy density in the material derivative of order zero of the conjugate strain tensor using Taylor series about a known configuration and then using the condition resulting from entropy inequality. The constitutive theories resulting from these three approaches are compared for equivalence as well as their merits and shortcomings. It is shown that the constitutive theory for heat vector can be derived: (i) using the conditions resulting from the entropy inequality; and (ii) using the theory of generators and invariants. In this approach, the argument tensors of the heat vector determine the specific form of the resulting constitutive theory. In this article, we use second Piola–Kirchhoff stress tensor and Green’s strain tensor (material derivative of the Green’s strain tensor of order zero) as a conjugate pair.