A Nonlinear Constitutive Theory for Heat Conduction in Lagrangian Description Based on Integrity

Abstract

If the deforming matter is to be in thermodynamic equilibrium, then all constitutive theories, including those for heat vector, must satisfy conservation and balance laws. It is well known that only the second law of thermodynamics provides possible conditions or mechanisms for deriving constitutive theories, but the constitutive theories so derived also must not violate other conservation and balance laws. In the work presented here constitutive theories for heat vector in Lagrangian description are derived (i) strictly using the conditions resulting from the entropy inequality and (ii) using theory of generators and invariants in conjunction with the conditions resulting from the entropy inequality. Both theories are used in the energy equation to construct a mathematical model in R1 that is utilized to present numerical studies using p-version least squares finite element method based on residual functional in which the local approximations are considered in higher order scalar product spaces that permit higher order global differentiability approximations. The constitutive theory for heat vector resulting from the theory of generators and invariants contains up to cubic powers of temperature gradients and is based on integrity, hence complete. The constitutive theory in approach (i) is linear in temperature gradient, standard Fourier heat conduction law, and shown to be subset of the constitutive theory for heat vector resulting from the theory of generators and invariants.