## 3 Dynamic Model for Low Aspect Ratio Keyholes or High Welding Speeds
### Results of the Generalized Piston Model for High Welding Speeds  
An example of the application of this model is discussed now, using experimental conditions close to the experiments on Ti-6 Al-4V material, published by [PERSON] et al. [21] where the results of experiments realized with a rather large set of operating parameters (incident laser power and welding speed) are available for discussion. In our model, we used three spot diameters (60, 120 and 180 \(\upmu\)m), with three welding speeds V\({}_{\mathrm{w}}\) (0.25, 0.5 and 1 m/s) and four different incident laser powers, in order to scan aspect ratios from 0.5 to 3. An example of the resulting aspect ratios R, defined by Equation (8), has been plotted on Figure 4 as a function of the incident power P, for three different welding speeds V\({}_{\mathrm{w}}\). For a given welding speed V\({}_{\mathrm{w}}\), one can first observe that the aspect ratio R is a linear function of the incident power, which can be written as:  
\[\mathrm{R}\approx\mathrm{r}_{0}\;(\mathrm{P}(\mathrm{W})-\mathrm{P}^{\prime}), \tag{10}\]  
where the slope \(\mathrm{r}_{0}\) of this linear function and P\({}^{\prime}\), the extrapolated threshold of laser power for the beginning of penetration, are dependent of the welding speed V\({}_{\mathrm{w}}\), and of the spot diameter d. It is also interesting to notice that the inverse of the slope \(\mathrm{r}_{0}\) appears to be a linear function of the welding speed, \(\mathrm{1/r}_{0}\approx\mathrm{r}_{1}+\mathrm{r}_{2}\mathrm{V}_{\mathrm{w}}\), where the parameters \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) are only dependent on the spot diameter d. In order to explain this dependence, one must analyze the evolution of the power threshold P\({}_{\mathrm{t}}\) for which R = 1. Figure 3: Scheme of the half-cone shape of the KH front used for applying the GPM that describes the dynamic welding regime. The transition of the KH front inclination around \(\alpha_{0}=\pi/4\), which occurs for a KH incident power threshold \(\mathrm{P_{t}}\) has significant consequences: For \(\mathrm{P<P_{t}}\), the reflected beam is directed upwards because \(\alpha\)\(>\pi/4\) (or R \(<\)1) and its power is lost for the process, and when \(\alpha<\pi/4\) (or R \(>\) 1), for \(\mathrm{P>P_{t}}\), the reflected beam is directed downwards and begins to add its contribution to the final penetration. Depending of d and V\({}_{\mathrm{w}}\), the corresponding incident power thresholds \(\mathrm{P_{t}(d,V_{w})}\), for which R = 1, inside the set of the previous operating conditions, have been determined. Knowing the dependence of \(\mathrm{P_{t}}\) defined in the case of macro-welding (see Equation (6)), it is tempting to plot the variation of the similar variable \(\mathrm{A_{F}(1)P_{t}(d,V_{w})/d}\), as a function of (V\({}_{\mathrm{w}}\)d): in fact, a linear dependence fits these data accurately (Figure 5), which is given by the relation:  
\[\frac{\mathrm{A_{F}(1)P_{t}(d,V_{w})}}{\mathrm{d}}=\mathrm{a_{1}}+\mathrm{b_{1 }}(V_{w}d), \tag{11}\]  
In Equation (11), \(\mathrm{A_{F}(1)}\approx 0.32\) is the Fresnel absorptivity for \(\alpha=\pi/4\), and the constants \(\mathrm{a_{1}}\approx 2.1\) x \(10^{5}\) W/m and \(\mathrm{b_{1}}\approx 1.2\) x \(10^{10}\) J/m\({}^{3}\) can then be derived. Moreover, if one considers that \(\mathrm{a_{1}}\approx\mathrm{n_{1}K(T_{v}-T_{0})}\) and \(\mathrm{b_{1}}\approx\mathrm{m_{1}K(T_{v}-T_{0})}/(2.\kappa)\) similarly to in Equations (4) and (5), one then finds that: \(\mathrm{m_{1}}\approx 2.1\) and \(\mathrm{n_{1}}\approx 2.2\). In fact, these values of \(\mathrm{m_{1}}\) and \(\mathrm{n_{1}}\) are rather close to the values of \(\mathrm{m}\) and \(\mathrm{n}\) observed in the case of a cylindrical KH. Indeed, they characterize the conductive losses of the geometry of this half-cone shape of the KH front used in this model.