Applied Physics A

, 125:797 | Cite as

Analytical modeling of lack-of-fusion porosity in metal additive manufacturing

  • Jinqiang NingEmail author
  • Wenjia Wang
  • Bruno Zamorano
  • Steven Y. LiangEmail author


This work presents a physics-based analytical modeling methodology for the prediction of the lack-of-fusion porosity in powder bed metal additive manufacturing (PBMAM) considering the molten pool geometry, powder size variation, and packing. The presented model has promising short computational time without resorting to the finite element method or any iteration-based simulations. The temperature profiles were calculated using a closed-form temperature solution. Multiple transverse sectional areas of the molten pool geometry were plotted on a cross-sectional area of the part based on hatch space and layer thickness to calculate the lack-of-fusion area. The powder bed porosity was calculated using advancing front approach with consideration of powder statistical distribution and powder packing. The part porosity was converted from the calculated lack-of-fusion area by multiplying the calculated powder bed porosity. Acceptable agreements were observed upon validation against experimental measurements under various process conditions in PBMAM of Ti6Al4V. The computational time was recorded less than 26 s for the porosity calculation of five consecutive layers. The presented model has high prediction accuracy and high computational efficiency, which allow the porosity calculation for large-scale parts and process parameters planning through inverse analysis, and thus improves the usefulness of analytical modeling in real applications.


Lack-of-fusion porosity Powder bed metal additive manufacturing Closed-form temperature solution Statistical powder size variation and packing High computational efficiency 



The authors would like to acknowledge the funding support from The Boeing Company.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    D.S. Thomas, S.W. Gilbert, Costs and cost effectiveness of additive manufacturing. NIST Spec. Publ. 1176, 12 (2014). CrossRefGoogle Scholar
  2. 2.
    J.J. Lewandowski, M. Seifi, Metal additive manufacturing: a review of mechanical properties. Annu. Rev. Mater. Res. 46, 151–186 (2016). ADSCrossRefGoogle Scholar
  3. 3.
    B. Van Hooreweder, Y. Apers, K. Lietaert, J.P. Kruth, Improving the fatigue performance of porous metallic biomaterials produced by selective laser melting. Acta Biomater. 47, 193–202 (2017). CrossRefGoogle Scholar
  4. 4.
    R. Wauthle, J. Van Der Stok, S.A. Yavari, J. Van Humbeeck, J.P. Kruth, A.A. Zadpoor, H. Weinans, M. Mulier, J. Schrooten, Additively manufactured porous tantalum implants. Acta Biomater. 14, 217–225 (2015). CrossRefGoogle Scholar
  5. 5.
    H. Yi, L. Qi, J. Luo, D. Zhang, N. Li, Direct fabrication of metal tubes with high-quality inner surfaces via droplet deposition over soluble cores. J. Mater. Process. Technol. 264, 145–154 (2019). CrossRefGoogle Scholar
  6. 6.
    H. Yi, L. Qi, J. Luo, D. Zhang, H. Li, X. Hou, Effect of the surface morphology of solidified droplet on remelting between neighboring aluminum droplets. Int. J. Mach. Tools Manuf. 130, 1–11 (2018). CrossRefGoogle Scholar
  7. 7.
    H. Yi, L. Qi, J. Luo, N. Li, Hole-defects in soluble core assisted aluminum droplet printing: Metallurgical mechanisms and elimination methods. Appl. Therm. Eng. 148, 1183–1193 (2019). CrossRefGoogle Scholar
  8. 8.
    H. Yi, L.H. Qi, J. Luo, Y. Jiang, W. Deng, Pinhole formation from liquid metal microdroplets impact on solid surfaces. Appl. Phys. Lett. 108(4), 041601 (2016). ADSCrossRefGoogle Scholar
  9. 9.
    N.T. Aboulkhair, N.M. Everitt, I. Ashcroft, C. Tuck, Reducing porosity in AlSi10Mg parts processed by selective laser melting. Addit. Manuf. 1, 77–86 (2014). CrossRefGoogle Scholar
  10. 10.
    R. Li, J. Liu, Y. Shi, L. Wang, W. Jiang, Balling behavior of stainless steel and nickel powder during selective laser melting process. Int. J. Adv. Manuf. Technol. 59(9–12), 1025–1035 (2012). CrossRefGoogle Scholar
  11. 11.
    G. Vastola, Q.X. Pei, Y.W. Zhang, Predictive model for porosity in powder-bed fusion additive manufacturing at high beam energy regime. Addit. Manuf. 22, 817–822 (2018). CrossRefGoogle Scholar
  12. 12.
    C. Teng, H. Gong, A. Szabo, J.J.S. Dilip, K. Ashby, S. Zhang, N. Patil, D. Pal, B. Stucker, Simulating melt pool shape and lack of fusion porosity for selective laser melting of cobalt chromium components. J. Manuf. Sci. Eng. 139(1), 011009 (2017). CrossRefGoogle Scholar
  13. 13.
    G. Kasperovich, J. Haubrich, J. Gussone, G. Requena, Correlation between porosity and processing parameters in TiAl6V4 produced by selective laser melting. Mater. Des. 105, 160–170 (2016). CrossRefGoogle Scholar
  14. 14.
    J.A. Slotwinski, E.J. Garboczi, K.M. Hebenstreit, Porosity measurements and analysis for metal additive manufacturing process control. J. Res. Nat. Inst. Stand. Technol. 119, 494 (2014). CrossRefGoogle Scholar
  15. 15.
    A.B. Spierings, M. Schneider, R. Eggenberger, Comparison of density measurement techniques for additive manufactured metallic parts. Rapid Prototyp. J. 17(5), 380–386 (2011). CrossRefGoogle Scholar
  16. 16.
    C. Bruna-Rosso, A.G. Demir, B. Previtali, Selective laser melting finite element modeling: validation with high-speed imaging and lack of fusion defects prediction. Mater. Des. 156, 143–153 (2018). CrossRefGoogle Scholar
  17. 17.
    T. Mukherjee, T. DebRoy, Mitigation of lack of fusion defects in powder bed fusion additive manufacturing. J. Manuf. Process. 36, 442–449 (2018). CrossRefGoogle Scholar
  18. 18.
    P. Wei, Z. Wei, Z. Chen, Y. He, J. Du, Thermal behavior in single track during selective laser melting of AlSi10Mg powder. Appl. Phys. A 123(9), 604 (2017). ADSCrossRefGoogle Scholar
  19. 19.
    Z. Chen, Y. Xiang, Z. Wei, P. Wei, B. Lu, L. Zhang, J. Du, Thermal dynamic behavior during selective laser melting of K418 superalloy: numerical simulation and experimental verification. Appl. Phys. A 124(4), 313 (2018). ADSCrossRefGoogle Scholar
  20. 20.
    C. Körner, E. Attar, P. Heinl, Mesoscopic simulation of selective beam melting processes. J Mater. Process. Technol. 211(6), 978–987 (2011). CrossRefGoogle Scholar
  21. 21.
    Y. Xiang, S. Zhang, Z. Wei, J. Li, P. Wei, Z. Chen, L. Yang, L. Jiang, Forming and defect analysis for single track scanning in selective laser melting of Ti6Al4V. Appl. Phys. A 124(10), 685 (2018). ADSCrossRefGoogle Scholar
  22. 22.
    J.L. Tan, C. Tang, C.H. Wong, A computational study on porosity evolution in parts produced by selective laser melting. Metall. Mater. Trans. A 49(8), 3663–3673 (2018). CrossRefGoogle Scholar
  23. 23.
    M. Bayat, S. Mohanty, J.H. Hattel, Multiphysics modelling of lack-of-fusion voids formation and evolution in IN718 made by multi-track/multi-layer L-PBF. Int. J. Heat Mass Transf. 139, 95–114 (2019). CrossRefGoogle Scholar
  24. 24.
    A. AlFaify, J. Hughes, K. Ridgway, Controlling the porosity of 316L stainless steel parts manufactured via the powder bed fusion process. Rapid Prototyp. J. 25(1), 162–175 (2019). CrossRefGoogle Scholar
  25. 25.
    G. Tapia, A.H. Elwany, H. Sang, Prediction of porosity in metal-based additive manufacturing using spatial Gaussian process models. Addit. Manuf. 12, 282–290 (2016). CrossRefGoogle Scholar
  26. 26.
    M. Khanzadeh, S. Chowdhury, M.A. Tschopp, H.R. Doude, M. Marufuzzaman, L. Bian, In-situ monitoring of melt pool images for porosity prediction in directed energy deposition processes. IISE Trans. 51(5), 437–455 (2019). CrossRefGoogle Scholar
  27. 27.
    A. Garg, J.S.L. Lam, M.M. Savalani, A new computational intelligence approach in formulation of functional relationship of open porosity of the additive manufacturing process. Int. J. Adv. Manuf. Technol. 80(1–4), 555–565 (2015). CrossRefGoogle Scholar
  28. 28.
    J. Ning, S.Y. Liang, A comparative study of analytical thermal models to predict the orthogonal cutting temperature of AISI 1045 steel. Int. J. Adv. Manuf. Technol. 102(9–12), 3109–3119 (2019). CrossRefGoogle Scholar
  29. 29.
    J. Ning, V. Nguyen, Y. Huang, K.T. Hartwig, S.Y. Liang, Constitutive modeling of ultra-fine-grained titanium flow stress for machining temperature prediction. Bio-Des. Manuf. 2(3), 153–160 (2019). CrossRefGoogle Scholar
  30. 30.
    M. Van Elsen, M. Baelmans, P. Mercelis, J.P. Kruth, Solutions for modelling moving heat sources in a semi-infinite medium and applications to laser material processing. Int. J. Heat Mass Transf. 50(23–24), 4872–4882 (2007). CrossRefzbMATHGoogle Scholar
  31. 31.
    D. Rosenthal, The theory of moving sources of heat and its application of metal treatments. Trans. ASME 68, 849–866 (1946)Google Scholar
  32. 32.
    J. Ning, D.E. Sievers, H. Garmestani, S.Y. Liang, Analytical modeling of in-process temperature in powder bed additive manufacturing considering laser power absorption, latent heat, scanning strategy, and powder packing. Materials 12(5), 808 (2019). ADSCrossRefGoogle Scholar
  33. 33.
    J. Ning, D.E. Sievers, H. Garmestani, S.Y. Liang, Analytical thermal modeling of metal additive manufacturing by heat sink solution. Materials 12(16), 2568 (2019). ADSCrossRefGoogle Scholar
  34. 34.
    J. Ning, E. Mirkoohi, Y. Dong, D.E. Sievers, H. Garmestani, S.Y. Liang, Analytical modeling of 3D temperature distribution in selective laser melting of Ti-6Al-4V considering part boundary conditions. J. Manuf. Process. 44, 319–326 (2019). CrossRefGoogle Scholar
  35. 35.
    J. Ning, D.E. Sievers, H. Garmestani, S.Y. Liang, Analytical modeling of in-process temperature in powder feed metal additive manufacturing considering heat transfer boundary condition. J. Precis. Eng. Manuf. Green Technol. Int. (2019). CrossRefGoogle Scholar
  36. 36.
    J. Ning, D.E. Sievers, H. Garmestani, S.Y. Liang, Analytical modeling of transient temperature in powder feed metal additive manufacturing during heating and cooling stages. Appl. Phys. A 125(8), 496 (2019). ADSCrossRefGoogle Scholar
  37. 37.
    M. Tang, P.C. Pistorius, J.L. Beuth, Prediction of lack-of-fusion porosity for powder bed fusion. Addit. Manuf. 14, 39–48 (2017). CrossRefGoogle Scholar
  38. 38.
    Y.T. Feng, K. Han, D.R.J. Owen, Filling domains with disks: an advancing front approach. Int. J. Numer. Methods Eng. 56(5), 699–713 (2003). CrossRefzbMATHGoogle Scholar
  39. 39.
    A.V. Gusarov, T. Laoui, L. Froyen, V.I. Titov, Contact thermal conductivity of a powder bed in selective laser sintering. Int. J. Heat Mass Transf. 46(6), 1103–1109 (2003). CrossRefzbMATHGoogle Scholar
  40. 40.
    J. Ning, V. Nguyen, Y. Huang, K.T. Hartwig, S.Y. Liang, Inverse determination of Johnson-Cook model constants of ultra-fine-grained titanium based on chip formation model and iterative gradient search. Int. J. Adv. Manuf. Technol. 99(5–8), 1131–1140 (2018). CrossRefGoogle Scholar
  41. 41.
    J. Ning, S.Y. Liang, Model-driven determination of Johnson-Cook material constants using temperature and force measurements. Int. J. Adv. Manuf. Technol. 97(1–4), 1053–1060 (2018). CrossRefGoogle Scholar
  42. 42.
    Y. Yang, M.F. Knol, F. Van Keulen, C. Ayas, A semi-analytical thermal modelling approach for selective laser melting. Addit. Manuf. 21, 284–297 (2018). CrossRefGoogle Scholar
  43. 43.
    J.J.S. Dilip, S. Zhang, C. Teng, K. Zeng, C. Robinson, D. Pal, B. Stucker, Influence of processing parameters on the evolution of melt pool, porosity, and microstructures in Ti-6Al-4V alloy parts fabricated by selective laser melting. Prog. Addit. Manuf. 2(3), 157–167 (2017). CrossRefGoogle Scholar
  44. 44.
    M.D. Abramoff, P.J. Magalhaes, S.J. Ram, Image processing with ImageJ. Biophoton. Int. 11(7), 36–42 (2004).
  45. 45.
    H.W. Mindt, O. Desmaison, M. Megahed, A. Peralta, J. Neumann, Modeling of powder bed manufacturing defects. J. Mater. Eng. Perform. 27(1), 32–43 (2018). CrossRefGoogle Scholar
  46. 46.
    I.A. Roberts, C.J. Wang, R. Esterlein, M. Stanford, D.J. Mynors, A three-dimensional finite element analysis of the temperature field during laser melting of metal powders in additive layer manufacturing. Int. J. Mach. Tools Manuf. 49(12–13), 916–923 (2009). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.The Boeing CompanyHuntsvilleUSA

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