The following application — based on a misfit strain type formalized over the course of my doctoral studies — computes the residual stress field associated with the Rosenthal solution through a quasi-analytical method that exploits its symmetry under the linearity assumption. The results are represented on a z ≤ 0 half-plane perpendicular to the heat source trajectory y = z = 0, as the steady state assumption makes them independent of the position along said trajectory at rest. I.e., the inelastic strain induced by the gradient of thermal expansion at solidification.
Mattia Moda
Please note that the above method implies the assumptions of isothermal solidification and homogeneous, isotropic, linear elastic material, in addition to those underlying the Rosenthal solution. Although this may limit accuracy in case of significant creep, plastic deformations, and metallurgical transformations, the ensuing simplifications allow for first- or higher-order approximations with minimal computational costs. For a full theoretical explanation and further details, you can refer to my doctoral thesis.