Analytical Solutions for Thermoelastic Stress Fields Under Exponential and Step Pulse Heating
Hussain Al-Qahtani
The present study presents comprehensive analytical solutions for the thermoelastic response of semi-infinite media subjected to exponentially decaying volumetric heat sources with two distinct temporal profiles, namely exponential decay and rectangular step pulses. The governing coupled thermoelastic equations are nondimensionalized using characteristic thermal diffusion scales, revealing the fundamental parameter γ = c12 / (α2 δ2) that governs the relative importance of elastic wave propagation to thermal diffusion. Closed-form solutions are derived for both temperature and stress fields using Laplace transform techniques, yielding expressions involving complementary error functions with complex arguments. The stress solutions are decomposed into five distinct physical components representing source effects, diffusive relaxation, wave propagation, and causal elastic response. A novel compact operator formulation is introduced for the step pulse solution, demonstrating the principle of superposition and revealing the self-similar nature of the thermoelastic problem. The solutions show that maximum thermal stresses do not necessarily coincide with maximum temperatures in space or time, particularly for rapid heating cases where elastic waves propagate ahead of the thermal front. The analytical results provide fundamental insights into the selection of optimal pulse parameters for applications ranging from laser material processing to thermal barrier coating evaluation, where control of thermal stress is critically important for preventing material damage.
Keywords: Laser pulse heating, thermal stress, temperature distribution, steel