# Weld Metal Microstructure Prediction in Laser Beam Welding of Austenitic Stainless Steel
## 2 Materials and Methods
### Thermal Field Modelling and Analysis  
An experimentally-fitted multipoint-line heat sources approach previously presented [24] was used for thermal field modelling. This approach is based on the configuration and fitting of a generalized conductivity-based model, aimed to set up the system of types (1) and (3) sources, able to produce the same thermal effect of the actual interaction between the laser beam and the workpiece, without evaluating the complex energy-transfer processes that characterize keyhole mode laser welding. This effect is represented by the shape and boundaries of the weld bead on the plane orthogonal to the direction of source movement. The generalized model can be adapted to the specific case to be analyzed, on the basis of the geometrical features of the experimentally detected bead cross-section. By varying the configuration of the model, it is possible to simulate the thermal sources system capable of determining the laser welding beads of the most common shapes, from those typical of deep penetration laser welding, to those characterized by low penetration and parabolic cross-section beads. After the setup of the most suitable combination of heat sources, the multi-source model is defined in detail by fitting, on the shape of the bead cross-section experimentally detected, the parameters that characterize the geometric layout and strength of the sources: the length of the line source and the location depth of the point sources (z\({}_{\text{L}}\), z\({}_{\text{P1}}\),..., z\({}_{\text{P1}}\),... ), which define the layout of the model; the distribution coefficients (\(\gamma_{\text{L}}\), \(\gamma_{\text{P1}}\),..., \(\gamma_{\text{P1}}\),... ) of the laser's thermal power absorbed by the keyhole, which define the strength distribution among the sources. In the case under consideration, the thermal analysis was carried out by an analytic model based on the superimposition of two point sources and one line source, whose thermal fields, according to the conductivity-based modelling previously introduced, are expressed by Equations (1) and (3):  
\[\text{T}(\text{x},\text{y},\text{z})=\text{T}_{0}+\sum_{i=1}^{2}\frac{\text{ Q}_{\text{P1}}}{\text{c}_{i}\gamma_{\text{R}}\text{x}\text{r}_{\text{P1}}} \text{e}^{-\frac{\text{y}}{2\alpha}(\tau_{\text{P1}}+\text{x})}+\frac{\text{ Q}_{\text{L}}}{2\pi\text{k}}\text{e}^{-\frac{\text{y}}{2\alpha}\text{x}}\text{K}_{0} \Big{(}\frac{\text{v}_{\text{T}}}{2\alpha}\Big{)}, \tag{5}\]  
where T\({}_{0}\) is assumed equal to the room temperature (293 K). To simulate the full penetration welding and the keyhole effect, reproducing the experimental profile of the molten area in the best possible way, the thermal sources were located as follows:  
* First point source on the external surface, at the laser beam side (z\({}_{\text{P1}}\) = 0 mm);
* Second point source inside the bead where the beam is focused (z\({}_{\text{P2}}\) = \(\Delta\)z = 5.5 mm);
* Line source along the whole bead thickness (z\({}_{\text{L}}\) = 10 mm). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **C** & **Mn** & **Si** & **P** & **S** & **Ni** & **Cr** & **Mo** & **Fe** \\ \hline AISI 304L & 0.018 & 1.15 & 0.41 & 0.025 & 0.001 & 10.1 & 18.4 & - & Bal. \\ AWS 309L 1 & 0.010 & 1.65 & 0.33 & 0.020 & 0.010 & 12.4 & 24.5 & 0.47 & Bal. \\ AWS 308L 2 & 0.025 & 1.40 & 0.30 & 0.030 & 0.030 & 10.0 & 20.0 & - & Bal. \\ \hline \hline \end{tabular} \({}^{1}\) Filter material (FM) utilized for the experimental welding trials. 2 FM considered for simulation.  
\end{table} 
Table 2: Composition of the base metal and the filler (% by weight). This choice was made according to the morphology of the FZ profile in the welded section: the line source contributes to the formation of a regular shape of the FZ along the entire thickness, while the point sources give rise to the convexities that characterize the experimental profile, respectively, near the surface exposed to the laser beam and inside the joint, where the beam is focused.