An analytical solution for the three-dimensional temperature field in the liquid and heat-affected zones around a welding cavity produced by a moving distributed low- or high-power-density-beam is provided. The incident energy rate distribution is assumed to be Gaussian and the cavity is idealized by a paraboloid of revolution in workpieces of infinite, semi-infinite, or finite thicknesses. The present study finds that temperature fields can be described by the Laguerre and confluent hypergeometric functions. By satisfying a momentum balance at the cavity base and utilizing a consequence of the second law of thermodynamics, the depth of penetration is uniquely determined. The results show that the predicted depths and temperatures of the cavity agree with available experimental data. Some crucial factors affecting the transition from low- to high-power-density-beam welding are presented.

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