# Weld Metal Microstructure Prediction in Laser Beam Welding of Austenitic Stainless Steel
###### Abstract
In any case, these numerical simulations require increasing computing capacity and time, according to the degree of accuracy of the mesh into which the joint is subdivided [18], and needs to be validated by experimental measurements which, by their nature, are specific of the welding conditions considered [19]. A less complex approach, from the point of view of the thermal field calculation, is given by the phenomenological laws of the heat conduction, which are based on the integration of the Fick's second law and consist in analytical solutions developed starting from the well-known equation proposed by [PERSON] [20]. Considering a reference system (x,y,z) whose origin is fixed to the thermal source that moves along the welding axis x with speed v (m/s), in the case of a point heat source Q\({}_{\text{P}}\) (W) the temperature T(x,y,z) in a generic point is given by the following equation:  
\[\text{T(x,y,z)}=\text{T}_{0}+\frac{\text{Q}_{\text{P}}}{\text{c \,\pi\,k \,r_{\text{P}}}}\text{e}^{-\frac{\text{v}}{2\alpha}\left(\text{r}_{\text{P}}+ \text{x}\right)}, \tag{1}\]  
where T\({}_{0}\) is the initial temperature, c a numeric coefficient, k (W/mK) the thermal conductivity, \(\alpha\) (m\({}^{2}\)/s) the diffusivity, r\({}_{\text{P}}\) (m) the radial distance from the point source:  
\[\text{r}_{\text{P}}=\sqrt{\text{x}^{2}+\text{y}^{2}+\left(\text{z}-\text{z}_{ \text{P}}\right)^{2}}, \tag{2}\]  
where z\({}_{\text{P}}\) is the depth of the mobile source respect to the axes' origin, which is located on the body surface. The numeric coefficient in the denominator of Equation (1) is c = 2 when z\({}_{\text{P}}\) = 0 (point source on the body surface) and c = 4 when z\({}_{\text{P}}\) > 0 (point source inside the body). In the case of a mobile source uniformly distributed on a line along the body thickness (z axis), [PERSON] gives the following equation for the temperature fields in the plane xy [20]:  
\[\text{T(x,y)}=\text{T}_{0}+\frac{\text{Q}_{\text{L}}}{2\,\pi\,\text{k}}\text{ e}^{-\frac{\text{v}}{2\alpha}\text{x}}\text{K}_{0}\left(\frac{\text{v}\,\text{r}_{ \text{L}}}{2\alpha}\right), \tag{3}\]where Q\({}_{\rm L}\) (W/m) is the power per unit of the source length, K\({}_{0}\) the modified Bessel function of the second kind of order zero, \(r_{\rm L}\) (m) the radial distance from the source in the plane xy:  
\[r_{\rm L}=\sqrt{x^{2}+y^{2}}. \tag{4}\]  
As was exposed in a recent review [21], subsequent works, starting from those of [PERSON] and [PERSON] [22] and [PERSON] and [PERSON] [23], have offered further simulations of the temperature fields generated by the advancement of mobile heat sources of various geometries with surface energy distributions. However, in the case of high-power laser beam welding, a modelling based only on heat conduction does not take into account the complex fluid dynamics phenomena inside the keyhole, which give rise to thermal distributions that are very difficult to simulate analytically. So, in order to compensate for the simplification inherent in the modelling carried out according to the laws of heat conduction, in this work the authors consider an experimentally-fitted multipoint-line system of the thermal sources, which was proposed in a previous paper [24]. The actual heat input generated by the \"keyhole\" and its effects on the melt pool and on the weld cross sections are simulated through the adoption of parameters which define the sources layout and the power distribution between them. Due to such peculiarities the fitted model is suitable to analyze in detail the overall thermal field due to a moving heat source that simulates the laser beam welding effects, also focusing on the temperature profiles, as time varies, in fixed detection points of the workpiece. In this work the analytical model was further implemented by developing the derivatives of the temperature-time curves with the aim of evaluating the cooling rate, which is a crucial parameter for the weld microstructure.