Objective equations of heat conduction in deformable bodies

Abstract

The paper is devoted to the modeling of heat conduction in deformable media. The constitutive functions are in the form of rate equations which generalize the Maxwell–Cattaneo law. The purpose is to establish equations compatible with a finite speed, thermodynamically consistent, and objective. Both Lagrangian and Eulerian versions of the models are investigated. It follows that the time derivative of the Lagrangian description is associated with the Truesdell derivative of the Eulerian description. In a generic rate equation in the Eulerian description the stress turns out to be affected by the heat flux. The stress is unaffected if the derivative is the corotational one. Next, in accordance with recent literature, a time fractional derivative is also considered for the rate equation. As to wave propagation of jump discontinuities it follows that the time-fractional derivative is continuous even for discontinuities of the time derivative thus maintaining the paradox of infinite speed of propagation.