# Macroscale laser-surface interactions

High-power lasers are used in many manufacturing processes, including cutting, welding, brazing, surface texturing and surface cleaning. The thermal analysis of these processes requires treatment of the laser-surface radiative energy transfer coupled to a transient conduction analysis within the work piece. These interactions are treatable without near-field effects in many cases. For welding and brazing, the analysis is complicated by the presence of the moving boundary between the molten and solid material. In addition, a layer of metal vapors or ionized gases near the hot surface, called the Knudsen layer, may interfere with the incident laser energy, requiring analysis of radiative transfer through a participating medium. Rosenthal (1946) derived the first simplified analysis of the welding process, and reviews of recent modeling improvements are in Frewin and Scott (1999), who used finite element modeling with variable material properties for ultrafast laser welding. In these analyses, it is usually assumed that the laser beam for a continuous wave laser has a Gaussian shape for normal incidence of

${q_{absorbed}}(r) = C(1 - {\rho _\lambda }){q_{laser}}\exp ( - {r^2}/R_{laser}^2)\qquad \qquad(1)$

where C accounts for interaction with the the Knudsen layer, ρ λ is the reflectivity of the surface, r is the radial beam position, Rlaser is the laser beam radius, and qlaser is the incident laser power. The value of C depends on the particular material being welded, the weld velocity v, and the laser power q laser . For a pulsed laser, the shape in space and time is given by

${q_{absorbed}}(r,t) = C(1 - {\rho _\lambda })F(t){q_{laser}}\exp ( - {r^2}/R_{laser}^2)\qquad \qquad(2)$
Coordinate system for laser welding.

where F(t) is the temporal pulse shape. The laser is generally not positioned at normal incidence for these processes to avoid reflections into the beam optics, so the shape of the incident beam is generally ellipsoidal rather than circular (see figure).

Rosenthal (1946) used an analytical solution to show the three-dimensional temperature profile present around a point source moving at velocity v on the surface of a thick material with thermal conductivity k and thermal diffusivity α . The temperature profile approaches a steady solution with the respect to the source position. The result is

$T(\xi ,y,z) = {T_o} + \frac{q}{{4\pi kR}}\exp \left( { - \frac{{v\xi }}{{2\alpha }}} \right)\exp \left( { - \frac{{vR}}{{2\alpha }}} \right)\qquad \qquad(3)$

where

$R \equiv {\left( {{\xi ^2} + {y^2} + {z^2}} \right)^{1/2}}$

and ζ = vt.

This quasi-steady solution is invalid near the x-boundaries of the welded material. Additional information on numerical modeling is in Rohsenow et al. (1998) and Han et al. (2005).

## References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Frewin, M.R. and Scott, D.A., 1999, ‘‘Finite Element Model of Pulsed Laser Welding,” Welding Research Supplement 1999, 15s-22s.

Han, L., Liou, F.W., and Musti, S., 2005, ‘‘Thermal Behavior and Geometry Model of Melt Pool in Laser Material Process,” J. Heat Transfer, Vol. 127, pp. 1005-1014.

Rosenthal, D., 1946, ‘‘The Theory of Moving Sources of Heat and its Application to Metal Treatments,” Trans. ASME, Vol. 43, pp. 849–866.

Rohsenow, W.M., Hartnett, J.P., and Cho, Y.I., 1998, Handbook of Heat Transfer, 3rd ed., Chap. 18, Heat Transfer in Materials Processing, McGraw-Hill, New York, NY.