Isotherm penetration depth under a moving Gaussian surface heat source on a thick substrate

This paper presents novel explicit expressions for the penetration depth of an isotherm induced by a moving heat Gaussian source on a semi-infinite solid. This work is novel in its methodology of dimensional analysis, asymptotics, and blending of more than one parameter, and provides closed form expressions of high accuracy compared to the exact solution. The solutions obtained are based on the governing equations, not on empirical fittings, and are valid for any material and heat source that are captured by the original governing differential equations. Previous scaling laws for penetration depth of temperature due to moving heat sources were based on point heat sources, and predicted a finite penetration under all conditions, which is against observations that penetration becomes zero for heat sources that are too weak, too distributed, or too fast. This discrepancy is especially relevant for modern processes such as laser heat treatment, laser cladding, and additive manufacturing of metals using lasers; the analysis and expressions presented here account for all those effects. The dimensionless maximum isotherm penetration depth depends on two dimensionless groups: the heat source distribution parameter and the Rykalin number associated with the velocity of the heat source. Previous work on blending involved a single parameter, and a novel extension of the blending technique is presented here to tackle the increased complexity. Maximum isotherm penetration depth is determined for the first time with explicit closed-form expressions over a wide range of heat distributions and Rykalin numbers with an error below 9.7% against the exact solution. The proposed equations are validated against experiments reported in the literature. The expressions developed can be evaluated using a handheld calculator or a spreadsheet and are suitable for practitioners in industry to optimize process parameters and for researchers to validate numerical models.

Automated summary

This paper introduces a novel methodology based on governing equations for calculating the penetration depth of an isotherm induced by a moving heat Gaussian source on a semi-infinite solid, presenting accurate closed-form expressions and applicable to a wide range of conditions.