## 3 Dynamic Model for Low Aspect Ratio Keyholes or High Welding Speeds
### Example of Application of This Model to the Analysis of Experiments on Ti-6 Al-4V  
[PERSON] et al. [21] have recently published a rather large set of experiments on Ti-6 Al-4V material for welding speeds varying from 0.4 to 1.2 m/s and incident laser powers from 100 to 500 W. They used a Gaussian distribution of laser intensity, characterized by a laser spot diameter at 1/e\({}^{2}\) maximum intensity of 95 \(\upmu\)m. This intensity distribution is different from the uniform one used in the present GPM. Therefore, a smaller laser spot diameter d\({}_{0}\), typically defined at about half maximum intensity, should be used instead, which for this Gaussian distribution would correspond to approximately 56 \(\upmu\)m.  
Their measured penetration depth variation e, as a function of the incident power P, appears to have a linear scaling with incident power P above a threshold P' of about 90 W:  
\[\text{e}(\text{m})\approx\text{S}(\text{P}-\text{P}^{\prime})\approx\left( \frac{\text{k}}{\text{V}_{\text{w}}}\right)(\text{P}-\text{P}^{\prime}), \tag{16}\]  
The slope S decreases with the welding speed V\({}_{\text{w}}\) and by analyzing their published data, one finds that the slope variation S with the welding speed can be fitted by the hyperbolic relation: S \(\approx\) k/V\({}_{\text{w}}\), with k \(\approx\) 0.6-0.7 10\({}^{-6}\) m\({}^{2}\)/J. Using Equation (13), this would mean that these experiments are realized in the range of high V\({}_{\text{w}}\) of this dynamic regime, for which b\({}_{1}\)(V\({}_{\text{w}}\)d) \(>\) a\({}_{1}\). Indeed, this is the case for their operating parameters, so Equation (14) becomes:  
\[\text{e}\approx\frac{\text{A}_{\text{F}}(\text{R})(\text{P}-\text{P}^{\ast})} {(\text{b}_{1}\text{V}_{\text{w}}\text{d})}, \tag{17}\]  
By comparing Equations (16) and (17), one has the same slope if k \(\approx\) A\({}_{\text{F}}\)(R)/b\({}_{1}\)d. This implies that a spot diameter d \(\approx\) 40 \(\upmu\)m (by using A\({}_{\text{F}}\)(R) \(\approx\) 0.31, k \(\approx\) 6.5 x10\({}^{-7}\) m\({}^{2}\)/J and b\({}_{1}\)\(\approx\) 1.2 x 10\({}^{10}\) J/m\({}^{3}\)). Considering the approximation made on b\({}_{1}\) (through the uncertainty on m\({}_{1}\), K and \(\kappa\)), this derived value of d seems in a quite reasonable agreement with the expected one of 56 \(\upmu\)m defined at half maximum intensity of the Gaussian intensity distribution. (In fact, [PERSON] et al. [21] used a similar spot diameter (\(\approx\) 50 \(\upmu\)m) for the discussion of their results).  
As a conclusion of this section, it is shown that Equations (14) and (11), which define the aspect ratio R and the incident power threshold P\({}_{\text{t}}\) respectively, as a function of the operating parameters and mean thermophysical properties of the used material, are representative solutions of this GPM which describes this dynamic welding regime where the inclined KH front is mainly irradiated, and KHs with rather low aspect ratios result. Despite the differences in the various physical mechanisms involved in these two models describing these two very distinct welding regimes, in particular, the importance of evaporation processes in the second model, it appears that the resulting scaling laws are almost identical, thereby explaining our previous finding that the penetration depths for the additive manufacturing processes followed the same scaling laws [19].