## 2 Thermal Model for High Aspect Ratio Keyholes  
### Model for Deep Penetration Regime  
We recall here the model that allows determining the dependence of the KH depth in the case of keyholes with rather large aspect ratios R = e/d where e is defined as the KH depth and d is the spot diameter [16]. For this regime, one considers that the KH is a vertical cylinder with a diameter d given by the spot diameter, whose wall is at a uniform evaporation temperature T\({}_{\rm v}\), which is moving with a welding speed V\({}_{\rm w}\) inside a material at initial temperature T\({}_{0}\). The incident laser power P is homogeneously distributed inside the KH due to the multiple reflection process. Only the heat conduction loss process from the KH wall is taken into account. As the aspect ratio R is large for this deep penetration regime, the heat flux from the KH wall is considered mainly 2D, so the power loss per unit keyhole depth dP/dz (W/m) for this geometry can be rather easily determined for this cylindrical geometry [16; 24; 25] and is given by:  
\[\frac{\rm dP}{\rm dz}=\rm K\ (T_{\rm v}-T_{0})\ g(Pe) \tag{1}\]  
where K is heat conductivity of the solid material and g(Pe) is a function of the Peclet number Pe (Pe = V\({}_{\rm w}\)d/(2\(\kappa\)) where \(\kappa\) is the solid heat diffusivity). The function g(Pe) has to be computednumerically, but for a given range of Peclet numbers, one can use a simple and useful linear approximation g(Pe) \(\approx\) mPe + n. For example, many laser welding processes are characterized by a Peclet range 2 \(\leq\) Pe \(\leq\) 10; in that range, one determines that m \(\approx\) 2.4 and n \(\approx\) 3 [16]. As the absorbed laser power A(R)P is homogeneously distributed along the KH depth e, it is then given by (P--incident power, and A(R)--KH absorptivity which depends of its aspect ratio R = e/d):  
\[\mathrm{e}=\frac{\mathrm{A(R)P}}{\frac{\mathrm{dP}}{\mathrm{dz}}}=\frac{ \mathrm{A(R)P}}{\mathrm{K(T_{v}-T_{0})(mPe+n)}}, \tag{2}\]  
From Equation (2), the aspect ratio R = e/d is then written as:  
\[\mathrm{R}=\frac{\mathrm{R}_{0}}{\left(1+\frac{V_{w}}{V_{0}}\right)}, \tag{3}\]  
where R\({}_{0}\) = A(R)P/(ndK(T\({}_{v}-T_{0}\)) and V\({}_{0}\) = 2n\(\kappa\) /(md) have been defined. Equation (3) reproduces the KH depth variation that is usually observed for the macro-welding experiments, and its dependence with the characteristic thermo-physical properties of the material: at high welding speeds V\({}_{w}\), the KH depth is inversely proportional to the welding speed, whereas when the welding speed decreases, R increases and then saturates. Another way to write Equation (3) is the following:  
\[\frac{\mathrm{A(R)P}}{\mathrm{(dR)}}=\mathrm{a}+\mathrm{b(V_{w}d)}, \tag{4}\]  
where  
\[\mathrm{a}=\mathrm{nK(T_{v}-T_{0})}\mathrm{\ and\ }\mathrm{b}=\mathrm{mK(T_{v}-T_{0}) }\mathrm{/(2\kappa)}, \tag{5}\]  
Equation (4) is interesting because its shows that the variable A(R)P/(dR) is a linear function of the variable V\({}_{w}\)d--both depend only on the operating parameters--and the slope b and the ordinate at the origin a of this linear function depend only on the thermo-physical properties of the material and the conductive loss (through the dependence in m and n, which is defined by the shape of the KH used). The scaling law described by Equation (4) has been verified for different published data obtained for several macro-experiments with different materials, such as steel [26] and copper [27]. Figure 1 shows an example of this comparison for experiments realized on St35 steel and different laser powers and focal spot diameters [26]. The observed correct agreement between this rather simple model and the corresponding macro-experiments shows that this model contains the main parameters involved for the determination of the scaling law for the KH depth. Additionally, from Equation (4) it is possible to consider an important parameter, which is the incident power threshold P\({}_{t}\) necessary for obtaining an aspect ratio R = 1.