Ordered rate constitutive theories in Lagrangian description for thermoelastic solids and thermoviscoelastic solids with and without memory using Helmholtz free energy density

The research work presented here considers development of constitutive theories in Lagrangian description for homogeneous, isotropic, compressible and incompressible thermoelastic solids, thermoviscoelastic solids without memory, and thermoviscoelastic solids with memory. Since conservation of mass, balance of momenta, and the first law of thermodynamics assume the existence of a stress field and heat vector without regard to how they are arrived at, the constitutive theories for the stress field and heat vector must be derived using the second law of thermodynamics to ensure thermodynamic equilibrium in the deforming matter during evolution. In the present work, we use the entropy inequality resulting from the second law of thermodynamics expressed in terms of the Helmholtz free energy density φ. The initial choice of dependent variables is directly from the entropy inequality: φ, second Piola-Kirchhoff stress tensor σ[0], entropy density η, and heat vector q. The argument tensors are established based on desired physics, i.e., choices are made depending on whether the solid matter is thermoelastic, thermoviscoelastic without memory, or thermoviscoelastic with memory. The use of the entropy inequality with the desired choices of argument tensors of the dependent variables allows us to determine the final choice of dependent variables in the constitutive theories as φ, σ[0], and q as well as their argument tensors (depending on the desired physics). In the case of thermoelastic solids, the entropy inequality provides conditions from which the constitutive theories for σ[0] and q can be derived. For thermoviscoelastic solids with and without memory, the conditions resulting from the entropy inequality do not permit derivation of a constitutive theory for σ[0]. Using a decomposition of σ[0] into eσ[0] and dσ[0] (equilibrium and deviatoric second Piola-Kirchhoff stress tensors), the constitutive theories for eσ[0] can be derived using the conditions resulting from the entropy inequality. However, the entropy inequality provides no mechanism to derive constitutive theories for dσ[0]. In the present work, we use the theory of generators and invariants to derive constitutive theories for dσ[0]. The constitutive theories for \heat consistent with σ[0] or dσ[0] in the sense of argument tensors are also derived using the theory of generators and invariants. It is shown that for thermoelastic solids, the constitutive theory for σ[0] and q are of rate zero in the Green's strain tensor ε (ε[0]), the constitutive theories for thermoviscoelastic solids without memory for dσ[0] and q are up to orders n in the Green's strain tensor, and the constitutive theories for thermoviscoelastic solids with memory for dσ[0] and q are of orders m and n in the second Piola-Kirchhoff stress tensor and Green's strain tensor. Many simplified forms of the rate theories are presented and compared with those used currently to demonstrate the merits of this research and severe limitations and shortcomings of the constitutive theories used currently for the type of solids considered here. Numerical studies are also presented using the theories presented here, and in some cases, the results are compared with the currently used theories. These rate theories provide a comprehensive framework of constitutive theories that permit description of complex material physics