Thermoelasticity and Irreversible Thermodynamics

TLDR

Generalized free energy functions as a thermoelastic potential, coupled with a new dissipation function defined via the time derivative of entropy displacement. The study unifies thermoelasticity by extending irreversible thermodynamics methods. General thermoelastic laws are formulated variationally under a minimum entropy production principle, yielding Lagrangian equations, a thermal force defined by virtual work, and solutions for heat conduction via matrix algebra and Papkovitch‑Boussinesq potentials. Entropy density follows a diffusion‑type law, and the generalized coordinate method allows calculation of thermoelastic internal damping in elastic bodies.

Abstract

A unified treatment is presented of thermoelasticity by application and further developments of the methods of irreversible thermodynamics. The concept of generalized free energy introduced in a previous publication plays the role of a ``thermoelastic potential'' and is used along with a new definition of the dissipation function in terms of the time derivative of an entropy displacement. The general laws of thermoelasticity are formulated in a variational form along with a minimum entropy production principle. This leads to equations of the Lagrangian type, and the concept of thermal force is introduced by means of a virtual work definition. Heat conduction problems can then be formulated by the methods of matrix algebra and mechanics. This also leads to the very general property that the entropy density obeys a diffusion-type law. General solutions of the equations of thermoelasticity are also given using the Papkovitch-Boussinesq potentials. Examples are presented and it is shown how the generalized coordinate method may be used to calculate the thermoelastic internal damping of elastic bodies.