A detailed model is constructed in order to determine the full 3D weld pool and keyhole geometry by setting the appropriate energy and pressure balances. The energy balance takes into account heat conduction, ablation losses and evaporation effects at the keyhole open surfaces, as well as the most relevant energy-absorption mechanisms, namely Fresnel and inverse Bremsstrahlung. The pressure balance ensures mechanical stability of the keyhole by including ablation pressure against surface tension pressure. The model provides a full description of the temperature field, electronic density, degree of ionization and absorption coefficient within the plasma, as well as setting the maximum penetration depth for a given set of laser parameters such as power, focusing radius and oscillation transversal mode. The keyhole boundary is initially taken to be an unknown free boundary and is obtained as a part of the solution of the problem. For low and medium welding speeds this boundary is successfully described with a family of ovoids. Good agreement with experimental results is achieved for a wide range of laser powers and plate thicknesses.

In the deep penetration laser welding of materials as well as in other laser material processing it is usual for the laser to generate a keyhole in the material. One of the principal mechanisms of energy absorption in the keyhole is the Fresnel absorption process on the keyhole wall. In order to take account of the Fresnel absorption process properly it is necessary to include the effect of multiple reflections in the keyhole. With this end in view an axisymmetrical numerical model is presented in which the keyhole geometry is not fixed at the outset, but instead the keyhole wall is considered to be a free boundary whose shape changes after each iteration in the numerical process. In this connection each complete set of reflections constitutes a step in the iteration process. Convergence is reached for a reasonably wide set of initial geometries and discretization steps. The model is used to analyse keyhole profiles and intensity distributions with depth for the cases of a Gaussian and of a uniform top-hat distribution in the laser beam. The model allows the effect of inverse bremsstrahlung absorption in the keyhole to be taken into account, this being the other principal mechanism for the absorption of laser light in the keyhole.

A mathematical model for the laser drilling of metals is given for the cases of constant and pulsed laser sources. Attenuation of the laser beam within the vapour is considered through an averaged absorption coefficient . The experimentally observed logarithmic dependence of the hole depth on the laser energy is predicted theoretically. A singular perturbation technique is used in order to find solutions valid for different regimes of time, namely pre-vaporization and post-vaporization times. Uniformly valid solutions are found for the one-dimensional analysis of the drilling-front position and speed by matching the inner and outer solutions. First-order approximations for the time-dependent hole profile for the various laser source profiles considered are also found. The model is compared with experimental data in the literature for the drilling speed of copper. An additional set of experiments is specifically carried out to allow comparison with the theoretical hole profiles for titanium. The predictions of the model are found to agree well with the experiments.

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