Cauchy’s Mathematical Formulations on Thermoelasticity and Analytical Solutions to Pennes’ Bioheat Transfer Equations in Living Human Tissues under Extreme Thermal Stress
Author : Dr. Sangita B Pimpare
Abstract :
This paper presents a rigorous study of thermal stresses induced within biological human tissue during exposure to severe external thermal environments. Drawing directly from Cauchy’s fundamental stress principles and the generalized equations of thermoelasticity, we map how thermal gradients translate into structural mechanical loading within living tissue systems. To describe the underlying spatiotemporal temperature distribution, we apply the classical Pennes' Bioheat Transfer Equation (PBTE), which models tissue heat conduction alongside blood perfusion and metabolic heat generation. The mathematical governing equation is solved analytically using the method of separation of variables under standard physiological and boundary conditions. A complete mechanical analysis is then coupled to the thermal solution via Duhamel-Neumann constitutive relations to compute the induced Cauchy thermal stresses. Quantitative profiles indicate that rapid external heating creates highly localized, compressive thermal stress gradients near the skin surface, presenting a high risk for irreversible thermal injury. The results emphasize the critical role of blood perfusion as a thermal sink and pressure stabilizer.
Keywords :
Bioheat Transfer, Pennes' Bioheat Equation, Cauchy Thermal Stress, Thermoelasticity, Thermal Stress.
