## 3 Dynamic Model for Low Aspect Ratio Keyholes or High Welding Speeds
### Results of the Generalized Piston Model for High Welding Speeds
For similar reasons, it is also important to notice that the variable \(\mathrm{A_{F}(R)P/(dR)}\), which can be estimated for any aspect ratio R and is also reported in Figure 5, follows the same scaling law with (V\({}_{\mathrm{w}}\)d). Accordingly, one can write:  
\[\frac{\mathrm{A_{F}(R)P}}{(\mathrm{dR})}=\mathrm{a_{1}}+\mathrm{b_{1}}(V_{w} d). \tag{12}\]  
Using Equation (11), which defines the threshold \(\mathrm{P_{t}}\), Equation (12) can be simply rewritten as:  
\[\mathrm{R}=\frac{\mathrm{A_{F}(R)P}}{\mathrm{d}(\mathrm{a_{1}}+\mathrm{b_{1}}( V_{w}d)}=\frac{\mathrm{A_{F}(R)}}{\mathrm{A_{F}(1)}}\frac{P}{\mathrm{P_{t}}}. \tag{13}\]  
Equation (13) shows that the inverse of the slope R/P is a linear function of the welding speed V\({}_{\mathrm{w}}\), whose parameters depend of the spot diameter d (in agreement with Equation (10)). Figure 4: Examples of the aspect ratio R dependence with the incident power P for different welding speeds V\({}_{\mathrm{w}}\) (a spot diameter d = 120 \(\mathrm{\SIUnitSymbolMicro m}\) is used). For the same set of operating parameters, the Figure 6 shows the evolution of the normalized inclination angle \(\alpha\)/(\(\pi\)/4) and the evolution of R as a function of the normalized incident power P/P\({}_{\rm t}\)(d, V\({}_{\rm w}\)), where P\({}_{\rm t}\)(d, V\({}_{\rm w}\)) is the determined incident power threshold for any given d and V\({}_{\rm w}\). These data follow unique curves, which can be easily understood because both \(\alpha\) = Arctg(1/R), given by Equation (8), and R, through Equation (13), depend on P/P\({}_{\rm t}\). The Fresnel absorptivity A\({}_{\rm F}\)(R) is also involved in these equations, but in the range of the aspect ratio used, the corresponding variation of A\({}_{\rm F}\)(R) is rather small: \(0.31<\) A\({}_{\rm F}\)(R) \(<0.36\). Finally, one must notice that Equations (13) and (7) are very similar. They only differ by the expressions of the absorptivity and the corresponding power thresholds involved for each geometry. Remark: For the derivation of Equation (13), it is supposed that there is no threshold for the process of melt pool surface depression, but this surface depression begins only if the velocity of the ejected melt V\({}_{\rm m}\)\(\geq\) 0. From Equation (A10), this is only possible if the temperature exceeds a threshold temperature T\({}_{\rm th}\), for which a threshold power P* is then determined by the model. One observes that P* increases with the spot diameter d. Therefore, taking into account this threshold power P*, Equation (13) should be modified as follows:  
\[{\rm R}=\frac{{\rm A_{F}(R)(P-P^{*})}}{{\rm A_{F}(1)P_{t}}}, \tag{14}\]  
Figure 5: Plot of A\({}_{\rm F}\)(1)P\({}_{\rm t}\)/d and A\({}_{\rm F}\)(R)P\({}_{\rm t}\)/dR as a function of (V\({}_{\rm w}\)d) and corresponding best linear fit. Figure 6: Variations of the normalized inclination angle of the KH front \(\omega\)/(\(\pi\)/4) and the aspect ratio R as a function of the normalized incident power P/P\({}_{\rm t}\). The speed \(\rm V_{m}\) of the ejected melt on the sides of the inclined KH front is another important parameter that characterizes this dynamic regime compared to the \"thermal\" one. On Figure 7, the parameter \(\rm\Delta V_{m}/V_{w}=(V_{m}/V_{w}-1)\), which represents the variation of the melt ejection speed compared to the welding speed, has been plotted as a function of the three welding speeds \(\rm V_{w}\) and the three spot diameters used, inside the range of the aspect ratios varying from 0.5 to 2.5. It can be seen that \(\rm\Delta V_{m}/V_{w}\) increases quite linearly with the spot diameter d and more rapidly with the welding speed \(\rm V_{w}\). Additionally, for a given d and \(\rm V_{w}\), \(\rm\Delta V_{m}/V_{w}\) increases with the aspect ratio R. This ejected melt velocity can be very high compared to the welding speed when this welding speed or the spot diameter becomes important.