A Gaussian-regularized analytical framework for localized heat transfer and thermoelastic response in semi-infinite media

Localized heat transfer induced by non-uniform thermal loading is widely encountered in advanced thermal technologies such as laser processing, microwave heating, and geothermal engineering, where accurate characterization of spatial temperature distributions and thermoelastic responses is essential for understanding coupled heat-transfer and mechanical behavior. Classical analytical solutions for semi-infinite media commonly employ truncated thermal boundary conditions, which may produce an unbounded temperature gradient at the heating boundary and a corresponding finite discontinuity in surface thermoelastic stress. In this study, a Gaussian-regularized thermal boundary is introduced to represent the continuous spatial attenuation of localized surface temperature. Based on linear thermoelasticity and the zeroth-order Hankel transform, an analytical framework is developed for a semi-infinite medium subjected to a prescribed Gaussian surface-temperature distribution. The internal temperature and thermoelastic stress fields are expressed analytically in integral form, while explicit closed-form expressions are obtained for the surface radial and circumferential stresses. The Gaussian formulation provides a smooth representation of the thermal boundary and removes the associated surface-stress discontinuity of the classical formulation. The analytical results are compared with three-dimensional electromagnetic–thermal–mechanical simulations and microwave-heating experiments on granite. Results for different prescribed thermal states show that the characteristic Gaussian length scale governs the spatial localization of the thermal gradient and circumferential tensile stress, whereas the prescribed temperature-rise amplitude and thermo-mechanical properties primarily affect their magnitudes.