Analysis of temperature field calculation methods for a metal plate during surface laser hardening

The instability of laser hardening results necessitates a comprehensive analysis of methods for calculating the temperature field to optimize the process and improve the accuracy of predicting the depth of the hardened layer. Purpose. Conducting a comparative analysis of the accuracy of the main analytical methods for calculating the temperature field of a metal plate during surface laser hardening to predict the depth of the hardened layer. The study aims to identify the most adequate analytical approach by comparing it with the results of numerical modeling. Materials and Methods. Three common analytical methods were selected for comparison: solving differential heat conduction equations using the Laplace transform; a model based on Green's functions for a moving Gaussian heat source; and a method based on a modified Rosenthal equation for a point heat source. The accuracy of the methods was assessed by the criterion of predicting the hardening zone depth at a fixed temperature (Tz ≈ 820 °C for 40Kh steel). The results of numerical modeling performed using the author's computer program “Laser_hardening_of_metal” were used as a benchmark. Calculations were performed for a 40Kh steel plate at a laser power of 2000 W, a scanning speed of 10 mm/s, and a beam radius of 4.5 mm. Results. It was established that the analytical methods provide different estimates of the hardened layer depth: ≈ 1.3 mm (Laplace method), ≈ 0.6 mm (Green's function method), ≈ 0.8 mm (Rosenthal method). Numerical modeling yielded a value of 0.822 mm. The method based on solving the heat conduction equation using Green's functions for a moving Gaussian source showed the closest agreement with the numerical result. The method based on the modified Rosenthal equation showed a discrepancy with the numerical model of about 25 %, while the solution using the Laplace transform demonstrated the highest error (over 50 %), confirming its exclusively estimative nature. Conclusion. The comparative analysis allowed ranking the analytical methods by the accuracy of laser hardening depth prediction. For engineering calculations requiring high accuracy and computational resources, the method based on Green's functions is recommended. For rapid approximate estimates with acceptable error (about 20–25 %), the modified Rosenthal equation can be applied. The results of the work provide a rational basis for selecting a calculation method and optimizing the parameters of laser surface heat treatment.