## Ordered Rate Constitutive Theories in Eulerian Description

##### Issue Date

2012-12-31##### Author

Nunez, Daniel

##### Publisher

University of Kansas

##### Format

242 pages

##### Type

Dissertation

##### Degree Level

Ph.D.

##### Discipline

Mechanical Engineering

##### Rights

This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.

##### Metadata

Show full item record##### Abstract

This research work presents development of ordered rate constitutive theories in Eulerian description for homogeneous, isotropic, compressible and incompressible matter experiencing finite deformation using contravariant, covariant and Jaumann bases. The constitutive theories presented here are applicable to thermoelastic solids, thermofluids and thermoviscoelastic fluids. Due to the inability to monitor material point displacements, and hence strain measures in Eulerian descriptions, the constitutive theories for Cauchy stress tensor utilizing strain measures in Eulerian description are not useful, hence the need for ordered rate constitutive theories presented in this work. Covariant, contravariant and Jaumann bases identify deformed material lines in the current configuration, thus these bases are possible choices for the development of constitutive theories. Covariant Cauchy stress tensor, contravariant Cauchy stress tensor and Jaumann stress tensor are measures of stress in these bases while Green's strain tensor, Almansi strain tensor and Jaumann strain tensor are conjugate measures of finite strain. Even though strain measures are not defined in Eulerian description, their convected time derivatives in their respective bases are defined. Thus, convected time derivatives of various orders of the Green's strain tensor in covariant basis, convected time derivatives of various orders of the Almansi strain tensor in contravariant basis and likewise, Jaumann strain tensor in Jaumann basis are defined and measurable in Eulerian description. These convected time derivatives in their respective bases are symmetric tensors of rank two and are fundamental kinematic tensors, and hence they can be utilized in the derivations of the constitutive theories for the Cauchy stress tensors in the chosen bases. In addition, we also have convected time derivatives of various orders of the contravariant Cauchy stress tensor in contravariant basis, convected time derivatives of various orders of the covariant Cauchy stress tensor in covariant basis and convected time derivatives of various orders of the Jaumann stress tensor in Jaumann basis. These are also fundamental symmetric tensors of rank two. The ordered rate constitutive theories presented in this work utilize convected time derivatives of upto orders n and m of the strain and stress tensors (i.e. rates) in their respective bases. Thus, there are many possibilities for various rate theories depending upon the choices of the dependent variables in the constitutive theories and their argument tensors. Specific choices of these are made to address specific physics. 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.

##### Collections

- Engineering Dissertations and Theses [705]
- Dissertations [3317]

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