## 3 Results and Discussion
### Microhardness  
In general, the microstructure affected the hardness of the scanned track. The mean value of Vickers hardness (average of three measurements) at the top surface along the half-length of the lateral slice through the middle pass is shown in Figure 11. An increase in measured hardness was observed with the decrease in hatch spacing and increase in beam diameter. All samples exhibited hardnesses comparable to the initial hardness of AISI 316 (147.6 HV0.1). During the plastic deformation, the dislocation movement turns into dislocation accumulation when the grain boundaries act as obstacles against the dislocation flow [24]. Consequently, the stress concentration and strength of the material at the grain boundaries are increased. When the beam diameter was increased to 1.0 and 1.3, the microstructures of the heat-affected areas of the scanned tracks included equiaxed fine grains (Figure 8e,f). The microstructural results showed the larger numbers of grain boundaries in the above samples. This implied that small spherical grains were replaced with inhomogeneous austenite grains. The grain boundary strengthening effect contributed to an increase in the microhardness at the scanned track, owing to the reduction in grain size and increase in the number of grain boundaries. The relation  
Figure 10: Optical micrographs of the overlap areas of the samples obtained at \(D_{b}\) = 1.3 and hatch spacings of (**a**) 1.0 mm and (**b**) 0.50 mm, which show the back-transformed grains (within the dashed ovals). between the hardness and the number of grain boundaries could be expressed by using the Hall-Petch equation [28]. \[h=h_{0}+\frac{k}{\sqrt{d}}, \tag{6}\]  
where \(h\) is the hardness of the material, \(h_{0}\) and \(k\) are the material strengthening constants, and \(d\) is the grain size, which is inversely proportional to the number of grain boundaries. The larger hardnesses of the samples obtained at \(D_{b}\) = 0.7, 1.0, and 1.3 mm (\(h_{s}\) = 0.5 mm) (approximately 209, 218, and 215 HV, respectively) could be attributed to the larger strain hardening induced by the bending strain (Figure 6b). The analysis at the hatch spacing of 1.0 mm showed that the smaller deformation was not strictly associated with the decrease in hardness. Notably, increases were observed at the smaller beam diameters of 0.25, 0.5, and 0.7 mm. Therefore, the cooling rates of the simulated models were analyzed through the following equations. The thermal gradient (\(G\)) and solidification rate (\(R\)) were the fundamental parameters determining the cooling rate (C), which could be calculated by [29]  
\[G=\sqrt{\left(\frac{\partial T}{\partial x}\right)^{2}+\left(\frac{\partial T }{\partial y}\right)^{2}+\left(\frac{\partial y}{\partial z}\right)^{2}}, \tag{7}\]  
\[R=v\cos(\alpha), \tag{8}\]  
\[C=GR, \tag{9}\]  
where \(\frac{\partial T}{\partial x}\), \(\frac{\partial T}{\partial y}\), and \(\frac{\partial T}{\partial z}\) are the temperature gradients along the \(x\), \(y\), and \(z\) directions, respectively, \(v\) is the laser scan speed, and \(\alpha\) is the angle between the scan and growth directions of the grains. For the top surface and cross section, \(cos(\alpha)\) in a specific direction (\(x\), \(y\), or \(z\)) could be expressed by [29]  
\[\cos(\alpha)_{i}=\frac{-\frac{\partial T}{\partial i}}{G}, \tag{10}\]  
where \(i\) is the specific direction (\(x\), \(y\), or \(z\)). The obtained cooling rate is shown in Figure 12. For the solidification rate, the mean value (\(R_{\text{avg}}\)) of the three directions was calculated. Figure 11: Average microhardnesses of the samples, indented along the scan direction outlined in the lateral slice image. When the beam diameter was in the range of 0.25-1.0 mm, the cooling rate was in the range of approximately 6,660 up to the maximum of 7,147 for the sample with a hatch spacing of 1.0 mm.