## 3 Results and Discussion  
### Effects of the Beam Diameter and Hatch Spacing  
Several factors simultaneously contributed to the formation of a dome plate, including the beam parameters, temperature at the top surface, plastic strain, and microstructure, which were analyzed in this study. For each beam diameter, the plastic strains were analyzed with the change in hatch spacing. Figure 5a-e show the plastic strains calculated by using Equations (4) and (5). Under almost all conditions, similar trends were observed, i.e., \(e_{spacing}\) increased with the hatch spacing, while \(e_{hunting}\) decreased. Figure 6a shows the total strain calculated by using Equation (3). The largest sum of the plastic strains was obtained at the smallest beam diameter (Table 1). The obtained plastic deformations and deformed sample are presented in Figure 6b,c, respectively. Case \(d\) produced the largest plastic  
Figure 4: **(a)** Simulation of the deformed sample; **(b)** definition of \(e_{hunting}\) and \(e_{spacing}\) in the strain field; and (c) definition of the in-plane and out-of-plane strains.  
deformation, followed by that in case \(e\). Despite the largest plastic deformation, the strains on the spacing between scan tracks, \(e_{spacing}\) was smaller than that of the sample with the larger hatch spacing presented in Figure 5d. Figure 6a,b show that the large plastic strain did not necessarily produce a large plastic deformation. The larger total plastic strain in Figure 6a might be attributed to the higher temperature on each path and counter bending on the unaffected areas between the scanning paths, although it decreased the total plastic deformation, as shown in Figure 6b. As is already understood, in laser forming based on geometrical and laser beam parameters, three key mechanisms called temperature gradient mechanism (TGM), buckling mechanism (BM) and upsetting mechanism (UM) have been identified [3]. TGM can be activated when the diameter of the laser beam has a size smaller than or equal to that of the sheet thickness, i.e., in the samples with \(D_{b}\) = 0.25 and 0.5 mm, in both \(h_{s}\) of 1.0 and 0.5 mm. In this mechanism the sheet was bending away from the laser beam during heating, because of the high thermal expansion on the top surface. During cooling, because of the thermal contraction at the top surface, the sheet was bending towards the laser beam. Since in this mechanism the temperature generated by the laser beam is very high in the surface, a quick decay of the temperature occurred. It was assumed that in a higher hatch spacing of \(h_{s}\) = 1.0 mm, the previously counter bent area between the scanned paths could not be bent towards the laser beam, due to rapid cooling that was followed by insufficient surface shrinkage, resulting in lower deformations in these samples. While with the decrease in hatch spacing to \(h_{s}\) = 0.5 mm, the previously counter bent area between the scanned paths could probably be bent towards the laser beam fairly easily, due to the larger temperature distribution at the top surface and the subsequent shrinkage.  
Figure 5: \(Cont.\)  
Accounting for the total strains generated in the forming process was not sufficient to predict the deformation behavior [19]. This implied that the temperature variations and microstructures of the bent samples should be also considered.