One-dimensional thermo-elastic wave analysis for dynamic thermoelasticity coupled with dual-phase-lag heat conduction model

Abstract

Mathematically, the well-known heat conduction equation is a parabolic partial differential equation, which defines the temperature at a point as proportional to the difference in the average of the surrounding temperatures. Marin indicated that the associated equation assumes that heat propagates at an infinite speed, which does not satisfy Einstein's special theory of relativity. Later, Landau discovered the existence of heat waves in liquid Helium II by observing the anomalous thermal conductivity at certain temperatures. Recently, it was reported that heat waves also occur in carbon nanotubes. To eliminate such physical inconsistencies, researchers have attempted to improve the associated equation in various ways, such as by introducing a relaxation time. In this study, we focused on the heat conduction equation, in which two relaxation times were introduced by Tzou, and attempted to couple it with the dynamic thermoelastic equation so that heat and stress waves could be generated and propagated simultaneously, but at different speeds. As a simple example, a one-dimensional bar problem was investigated and solved numerically using the Laplace transform technique. The results showed that in the conventional heat conduction equation, the temperature was diffusely distributed from the heating point in the depth direction, whereas the spike-shaped compressive stress propagated at a constant speed. However, in the heat conduction equation with relaxation times, the temperature distribution has discontinuities propagating at a constant speed, confirming that the heat wave can be simulated. However, the compressive stress results showed that the thermoelastic coupling effect increased the period of stress occurrence and reduced the peak stress.