During both laser forming and laser-assisted pre-stress forming processes, laser is introduced as a moving heat source to achieve a local temperature rise and further a local plastic deformation for sheet metal forming. To assure the usability and the service performance of the formed sheet part, laser parameters are definitively restricted. As a consequence, understanding the relationship between laser parameters and the temperature field/variance induced by the laser is of an importance. In the recent study, based on the physical model of laser heating a finite slab, an analytical solution for the temperature field during a forming process is obtained to compute the temperature distribution under certain processing parameters such as the power density, the beam radius and the moving speed of the laser spot. This research is published online on Applied Mathematical Modelling (doi://10.1016/j.apm.2015.11.024).
As for the model for a temperature field introduced by a moving laser (Fig 1), efforts have been made to solve the temperature field via various approaches such as analytical methods, numerical simulations, and artificial intelligence techniques. The numerical simulations and the artificial intelligence techniques can be adopted for complicated situations, but they are generally time-consuming. Besides, a large data set must be built as training samples for artificial intelligence techniques. On the contrary, analytical solutions can be used to predict the temperature field in a very limited time. In this work, a three-dimensional heat transfer equation with convective boundary conditions is established in a finite region. And an analytical solution is obtained by directly solving such equation. First of all, Solving the corresponding eigenvalue problem enables a set of complete base functions (eigenfunctions). Second, expanding the non-homogeneous term in the heat conduction equation and the temperature function according to such base function set, with coefficients to be determined. Last, substituting the expanded series into the heat conduction equation gives the final form of the analytical solution (Fig 2). In practical process, due to the exponential terms, the analytical solution converges rapidly and cost for computation is limited. In addition, with numerical integration such as quasi-Monte Carlo method with low-discrepancy sequences, temperature distribution under arbitrary laser scanning path can be computed. Further, to verify the analytical solution, both experiments and FEM simulations were employed. During experiments, aluminum AA6061T6 sheets and a continuous wave YAG laser system were used. Temperature histories at point P1, P2 and P3 were logged. The results show that the proposed analytical model is consistent with both FEM model and experimental data.
This research is funded by National Science Foundation of China.
Fig. 1 Model for heating a finite slab with a moving laser source
Fig. 2 Analytical solution to the thermal conduction equation
Fig. 3 Comparison between analytical solution, experimental data and FEM predictions in the temperature history at point P1, P2 and P3