# Prediction of Transient Temperature Distributions for Laser Welding of Dissimilar Metals
## 2 Definition of the Model  
Two thin sheets of AISI 316L and Ti6 Al4V with a thickness of 2.5 mm were used for laser welding in a butt joint design, as shown in Figure 1. The size of the sheets considered for modeling was 20 mm \(\times\) 10 mm for both materials. A continuous mode (CW) fiber laser heat source of 300 W and laser beam radius of 200 \(\upmu\)m was used for the present study. No filler material was used in this study. It was also assumed that the laser was focused on the welding line and moving at a constant speed of 2 mm/s. The transient study COMSOL was used for the prediction of the temperature fields in the welded joints. Thermophysical properties of both materials were considered to be temperature dependent and the interpolation method was used to obtain the properties for intermediate temperatures. The heat source plane was assumed as the z = 1 mm plane and the top surface of the sheets as the z = 0 plane. The work planes and work samples were considered as the union for simulation in design and evaluation prospective using COMSOL software. For the present study, the following assumptions and features were used:  
1. Radiation and convection heat loss from the surface are considered for modeling. 2. The ambient temperature is 298 K and the system is within good thermal isolation from the environment. 3. The change in phase during the process is also taken into account. 4. The thermo physical properties of the materials change with the change of temperature. 5. The position of the laser beam is vertical to the surface. 6. Latent heat during a phase change is considered for simulation. Latent heat for melting of Ti6 Al4V and AISI 316L is 286 kJ/kg and 260 kJ/kg respectively. For modeling, the melting temperatures considered were 1703 K for AISI 316L and 1878 K for Ti6 Al4V. In transverse electromagnetic (TEM00) mode, a three dimensional conical heat source was studied using the Hermit-Gaussian spatially distributed model [44; 45]. As shown in Figure 2, for modeling the profile, a Hermit-Gaussian analytical function was used as it is one of the accurate profile model modes for laser welding. Previous research [46; 47; 48], that used this laser heat source model found that the results of numerical simulation agree well with experimental data, and provide a good match with the real weld pool shape with an error percentage as low as 7-8%, whereas other heat source models showed lower accuracy when compared against the experimental data [44; 45; 46; 47; 48; 49]. In the fusion welding process, a heat source can be modeled as either a point heat source or line heat source. The mathematical expression of the heat source model follows:  
\[an1=\exp\left[-\left\{\frac{(a-a0)^{2}}{2\times siga^{2}}\right\}-\left\{ \frac{(b-b0)^{2}}{2\times sigb^{2}}\right\}\right] \tag{1}\]  
Figure 1: Geometry of the model. The heat input equation of a moving heat source follows:  
\[ \begin{split} Qin=Q0\times(1-Rc)\times Ac\times\left(\frac{1}{\pi \times sigx\times sigy}\right)\times an1(x,x0,sigx,y,y0,sigy)\\ \times\exp\left\{-Ac\times abs(z)\right\}\end{split}  \tag{2}\]  
The origin of the new coordinate frame was not static with respect to the old frame. It moves at a constant velocity with respect to the old frame:  
\[d(t)=d0+tv_{no} \tag{3}\]  
where, \(v_{no}\) and \(d0\) are constant vectors. The vector \(v_{no}\) is the velocity of the origin of the new frame with respect to the old frame. \(d\) is the position vector of the origin of the new frame with respect to the old frame. The velocity of the origin of the new frame is the time derivative of \(d\):  
\[\frac{d}{dt}d=\frac{d}{dt}(d0+tv_{no})=v_{no} \tag{4}\]  
This vector gives the velocity of the origin of the new coordinate system with respect to the origin of the old coordinate system. In this model, only y axis velocity and t need to be considered in the new coordinate system for obtaining the boundary conditions.