**5. Conclusions**

A physics-based analytical model is proposed to predict the residual stress in laser powder bed fusion process. The proposed model was validated by measuring the residual stress of the fabricated IN718 samples via X-ray diffraction. The proposed thermo-mechanical analytical model predicts the temperature field of the additively manufactured part using moving point heat source approach by considering the effects of scan strategies, heat loss due to convection and radiation, and energy needed for solid-state phase change. The build part may experience high thermal stress due to the high-temperature gradient. The thermal stress is obtained using Green's function of stresses due to the point body load. The thermal stress may exceed the yield strength of the material. Thus, the Johnson–Cook flow stress model is used to determine the yield surface. Due to the cyclic heating and cooling, the material experiences high residual stress. The residual stress is predicted using incremental plasticity and kinematic hardening behavior of the metal according to the property of volume invariance in plastic deformation in coupling with the equilibrium and compatibility conditions.

The residual stress is predicted along the scan direction and build direction under three di fferent process conditions. The results show that the residual stress is highly tensile in both directions since, during the cooling cycle, the heat-a ffected zone begins to cool down and the shrinkage of material in this zone tends to occur; thus, the tensile stress state builds up.

Moreover, a comparison of the residual stress along the scan direction and build direction showed that the predicted residual stress along the build direction is higher than that along the scan direction. Di fferent heat transfer mechanisms along the scan direction and build direction could contribute to this di fference.

Furthermore, the residual stress is predicted under two di fferent boundary conditions: (1) the effect of heat loss due to the convection and radiation is considered; and (2) the e ffect of heat loss from boundaries is ignored. The results show that, when the heat loss is ignored, the predicted residual stress has higher value compared to the results where the e ffect of heat loss is considered. Although in most of the points through thickness, the results for both conditions are within the range of experimental measurements, the predicted residual stress without the e ffect of heat loss boundary conditions is an upper bound, and the predicted residual stress with the e ffect of heat loss boundary conditions is a lower bound.

Results from the proposed analytical residual stress model showed good agreemen<sup>t</sup> with X-ray di ffraction measurements used to determine the residual stresses in the IN718 specimens built via L-PBF. Thus, the proposed model is a valuable tool for the rapid and accurate prediction of the residual stress build-up in the parts built via L-PBF. Due to the high computational e fficiency of the proposed model, this model can also be used for the real-time monitoring and control of the build process, as well as optimization of the process parameters in achieving a high-quality part.

**Author Contributions:** Conceptualization, E.M.; methodology, E.M.; software, E.M.; validation, E.M., H.-C.T., Y.-L.L., Y.-C.C., H.-Y.L.; formal analysis, E.M.; investigation, E.M.; writing—original draft preparation, E.M.; writing—review and editing, S.Y.L.; visualization, E.M.; supervision, S.Y.L.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The experimental data were financially supported by the "Intelligent Manufacturing Research Center" (iMRC) from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.

**Conflicts of Interest:** The authors declare no conflict of interest.
