Investigation of More Complete Constitutive Theories for Heat Conduction in Solids and for Deviatoric Stress Tensor in Incompressible Fluids

Abstract

This thesis presents numerical studies utilizing more complete constitutive theories for: (i) Heat vector in isotropic, homogeneous, incompressible, elastic solid continua and (ii) Deviatoric stress tensor for isotropic, homogeneous, incompressible, viscous fluids without memory. The derivation of the constitutive theories for heat vector in Lagrangian description for solid continua and for deviatoric stress tensor for incompressible fluent continua without memory in Eulerian description, using theory of generators and invariants, have been presented by Surana, Reddy, Eringen. These theories utilize integrity i.e. complete basis, hence are complete. A serious shortcoming of these theories is that they require too many material coefficients that must be determined experimentally. Due to the lack of availability of the material coefficients, these theories have not been used commonly in applications, instead their simplified forms requiring fewer material coefficients are currently being used. The purpose of this investigation is to study the influence of additional terms in the more complete constitutive theories derived using integrity that are routinely neglected to examine the influence of the additional physics that is introduced in the constitutive theories by their presence and their impact in applications. In specific, the first study focuses on constitutive theory for heat conduction in Lagrangian description for solid continua in which the argument tensors of heat vector are temperature gradient and temperature and the constitutive theory for heat vector is based on integrity and is derived using theory of generators and invariants. The second study considers incompressible, viscous fluids without memory in which the constitutive theory for the deviatoric Cauchy stress tensor is also based on theory of generators and invariants in which symmetric part of velocity gradient tensor and its square are combined generators of its argument tensors. 1D transient heat conduction in a rod, fully developed flow between parallel plates, square lid driven cavity and asymmetric expansion are used as model problems to illustrate the significance of the newer constitutive theories considered here.