A General Analytical Solution for Two-Dimensional Columnar Crystal Growth during Laser Beam Welding of Thin Steel Sheets

A technique for calculating the main solidification parameters for a two-dimensional columnar crystal growth during complete penetration laser beam welding of thin steel sheets was developed. Given that the weld pool interface is described by Lamé curves (superellipses) within the horizontal plane of growth, general analytical solutions were derived for the geometry of the crystal axis and the corresponding growth rate and cross-sectional area of the crystal. A dimensionless analysis was performed to provide insights on the dependence of the solidification parameters on the shape and dimensions of the rear part of the weld pool boundary. The derived solutions were applied for the case of complete penetration laser beam keyhole welding of 2 mm thick 316L austenitic chromium-nickel steel sheets. It was shown that the reconstruction of the weld pool boundary with Lamé curves provides higher accuracy and flexibility compared to results obtained with elliptical functions. The validity of the proposed technique and the derived analytical solutions was backed up by a comparison of the obtained solutions to known analytical solutions and experimentally determined shapes and sizes of the crystals on the top surface of the sheet. The dimensions of the calculated crystal axis correlated well with the experimentally obtained results.