Ordered Rate Constitutive Theories in Eulerian Description

In this work we consider homogeneous, isotropic, compressible and incompressible matter with finite deformation, that is in thermodynamic equilibrium during evolution. Thus, conservation laws and thermodynamic principles provide the basis for deriving mathematical models and constitutive theories. Conservation of mass, balance of momenta and the first law of thermodynamics yielding continuity equation, momentum equations and energy equation hold regardless of the constitution of the matter, hence naturally they provide no mechanism for deriving constitutive theories for the stress tensor and the heat vector. Thus, the second law of thermodynamics (entropy inequality) must form the basis for deriving the constitutive theories for the stress tensor and heat vector. The choices of dependent variables in the constitutive theories are made using entropy inequality. The arguments (or eventually argument tensors) of the dependent variables in the constitutive theories are chosen based on the desired physics in conjunction with entropy inequality. When the convected time derivatives of the strain tensor (in a chosen basis) are argument tensors of the dependent variables in the constitutive theories, entropy inequality requires decomposition of the Cauchy stress tensor into equilibrium stress tensor and deviatoric Cauchy stress tensor. Constitutive theories for the equilibrium stress tensor using entropy inequality result in thermodynamic pressure for compressible matter and mechanical pressure for incompressible matter. The conditions resulting from the entropy inequality require that the work expanded due to the deviatoric Cauchy stress tensor be positive but provide no mechanism for deriving constitutive theories for the deviatoric Cauchy stress tensor. The conditions resulting from the entropy inequality also require the scalar product of the heat vector and temperature gradient to be negative which can be used for example to derive the Fourier heat conduction law.;The work presented here utilizes theory of generators and invariants to derive the ordered rate constitutive theories for the deviatoric Cauchy stress tensor and heat vector for homogeneous, isotropic, compressible and incompressible thermoelastic solids, thermofluids and thermoviscoelastic fluids in contravariant, covariant and Jaumann bases. General derivations of rate constitutive theories are specialized to show that (i) generalized hypo-elastic solids, hypo-elastic solids with variable material coefficients are a subset of the general ordered rate constitutive theories of order n for thermoelastic solids (ii) constitutive theories for Newtonian fluids, generalized Newtonian fluids with variable material coefficients such as power law, Carreau-Yasuda model for viscosity, power law, Sutherland law etc. for temperature dependent material coefficients are a subset of the general ordered rate constitutive theories of order n for thermofluids (iii) Maxwell model, Oldroyd-B model, Giesekus model etc with variable transport properties are a subset of the general ordered rate constitutive theories for thermoviscoelastic fluids of orders (m, n). The conditions resulting from entropy inequality, leading to restrictions on the material coefficients, are presented to ensure that the constitutive theories derived using the theory of generators and invariants ensure thermodynamic equilibrium during the evolution. All theories presented here consider finite deformation as well as thermal effects.;A significant aspect of the general theories presented here and the simplifications used to obtain commonly used constitutive theories is that we have clear understanding of the many assumptions employed in obtaining them, hence the possibilities and opportunities for developing better constitutive theories for more precise behaviors of the deforming matter experiencing finite deformation. (Abstract shortened by UMI.).