A new method to estimate the residual stresses in additive manufacturing characterized by point heat source

  • Li Sun
  • Xiaobo Ren
  • Jianying He
  • Jim Stian Olsen
  • Sakari Pallaspuro
  • Zhiliang ZhangEmail author


Residual stress in additive manufacturing (AM) is one of the key challenges in terms of structural integrity and the finish quality of printed components. Estimating the distribution of residual stresses in additively manufactured components is complex and computationally expensive with full-scale thermo-mechanical FE analysis. In this study, a point heat source is utilized to predict the thermal field and residual stress distribution during the manufacturing processes. Numerical results show that the residual stress at a single material point can be expressed as a function of its spatial position and the peak nodal temperature it has experienced during thermal cycles. The distribution of residual stress can be divided into three segments according to the peak nodal temperature. The peak nodal temperature only depends on the heat flux and the distance to the point heat source center. A semi-analytical approach to predict the peak nodal temperature and residual stresses, once the heat flux is known, is proposed. The proposed approach is further validated by a numerical case study, and a very good agreement has been achieved. Compared with traditional thermo-mechanical FE analysis of additive manufacturing, the proposed method significantly improves the computational efficiency, showing great potential for prediction of residual stresses and distortion.


Point heat source Residual stress Peak nodal temperature Additive manufacturing 



Distance to point heat source center


Young’s modulus


Heat flux


Radius of point heat source


Radius of axisymmetric model


Height of axisymmetric model


Peak temperature the node has experienced during thermal cycles


Maximum temperature the model has experienced during thermal cycles


Room temperature


Melting temperature

Te, 1

First critical temperature in three-segment equivalent residual stress model

Te, 2

Second critical temperature in three-segment equivalent residual stress model

T1, 1

First critical temperature in three-segment maximum principal residual stress model

T1, 2

Second critical temperature in three-segment maximum principal residual stress model


The angle to heat surface


The coefficient of thermal expansion


Radiation coefficient


Convection coefficient


Inherent strain


Plastic strain


Thermal plastic strain


Phase transformation strain


Yield stress


Residual stress

\( {\sigma}_e^{res} \)

Von Mises equivalent residual stress

\( {\sigma}_1^{res} \)

Maximum principal residual stress

\( {\sigma}_{e,1}^{res} \)

First critical equivalent residual stress

\( {\sigma}_{e,2}^{res} \)

Second critical equivalent residual stress

\( {\sigma}_{1,1}^{res} \)

First critical maximum principal residual stress

\( {\sigma}_{1,2}^{res} \)

Second critical maximum principal residual stress


Funding information

The Chinese Scholarship Council is greatly acknowledged for financial support. The authors wish to thank the Research Council of Norway for funding through the BIA Program, Contract No. 269558/O20.


  1. 1.
    Kranz J, Herzog D, Emmelmann C (2015) Design guidelines for laser additive manufacturing of lightweight structures in TiAl6V4. J Laser Appl 27(S1):S14001. CrossRefGoogle Scholar
  2. 2.
    Carroll BE, Palmer TA, Beese AM (2015) Anisotropic tensile behavior of Ti–6Al–4V components fabricated with directed energy deposition additive manufacturing. Acta Mater 87:309–320. CrossRefGoogle Scholar
  3. 3.
    Brandl E, Palm F, Michailov V, Viehweger B, Leyens C (2011) Mechanical properties of additive manufactured titanium (Ti–6Al–4V) blocks deposited by a solid-state laser and wire. Mater Des 32(10):4665–4675. CrossRefGoogle Scholar
  4. 4.
    Mercelis PKJ (2006) Residual stresses in selective laser sintering and selective laser melting. Rapid Prototyp 12(5):254–265CrossRefGoogle Scholar
  5. 5.
    Gary K, Lewis ES (2000) Practical considerations and capabilities for laser assisted direct metal deposition. Mater Des 21:417–423CrossRefGoogle Scholar
  6. 6.
    Deng D (2009) FEM prediction of welding residual stress and distortion in carbon steel considering phase transformation effects. Mater Des 30(2):359–366. CrossRefGoogle Scholar
  7. 7.
    Labudovic MHD, Kovacevic R (2003) A three dimensional model for direct laser metal powder deposition and rapid prototyping. J Mater Sci 38(1):35–49CrossRefGoogle Scholar
  8. 8.
    Smith J, Xiong W, Yan W, Lin S, Cheng P, Kafka OL, Wagner GJ, Cao J, Liu WK (2016) Linking process, structure, property, and performance for metal-based additive manufacturing: computational approaches with experimental support. Comput Mech 57(4):583–610. zbMATHCrossRefGoogle Scholar
  9. 9.
    Songa JL, Dowson AL, Jacobsa MH, Brooksb J, Bedenc I (2002) Coupled thermo-mechanical finite-element modelling of hot ring rolling process. J Mater Process Technol 121:332–340CrossRefGoogle Scholar
  10. 10.
    Zhang Z, Zhang HW (2007) A fully coupled thermo-mechanical model of friction stir welding. Int J Adv Manuf Technol 37(3-4):279–293. CrossRefGoogle Scholar
  11. 11.
    Yuan MG, Ueda Y Prediction of residual stresses in welded T- and l-joints using inherent strains. J Eng Mater Technol 118(2):229–234CrossRefGoogle Scholar
  12. 12.
    Li C, Fu CH, Guo YB, Fang FZ (2016) A multiscale modeling approach for fast prediction of part distortion in selective laser melting. J Mater Process Technol 229:703–712. CrossRefGoogle Scholar
  13. 13.
    Yang Y, Ayas C (2017) Computationally efficient thermal-mechanical modelling of selective laser melting. 1892:040005. doi:
  14. 14.
    Mukherjee T, Manvatkar V, De A, DebRoy T (2017) Mitigation of thermal distortion during additive manufacturing. Scr Mater 127:79–83. CrossRefGoogle Scholar
  15. 15.
    Mukherjee T, Zuback JS, De A, DebRoy T (2016) Printability of alloys for additive manufacturing. Sci Rep 6:19717. CrossRefGoogle Scholar
  16. 16.
    Cheng W (2005) In-plane shrinkage strains and their effects on welding distortion in thin-wall structures. The Ohio State University, OhioGoogle Scholar
  17. 17.
    Camilleri D, Comlekci T, Gray TGF (2005) Computational prediction of out-of-plane welding distortion and experimental investigation. J Strain Anal Eng Des 40(2):161–176. CrossRefGoogle Scholar
  18. 18.
    Ding J, Colegrove P, Mehnen J, Williams S, Wang F, Almeida PS (2013) A computationally efficient finite element model of wire and arc additive manufacture. Int J Adv Manuf Technol 70(1-4):227–236. CrossRefGoogle Scholar
  19. 19.
    Radaj D (1992) Heat effects of welding: temperature field, residual stress, distortion. Springer Science & Business Media, BerlinCrossRefGoogle Scholar
  20. 20.
    Chao CK, Chen FM, Shen MH (2006) Green's functions for a point heat source in circularly cylindrical layered media. J Therm Stresses 29:809–847. MathSciNetCrossRefGoogle Scholar
  21. 21.
    Pratik P, Shukla PTS, Colin J (2014) Laser shock peening and mechanical shot peening processes applicable for the surface treatment of technical grade ceramics: a review. J Eng Manuf 228(5):639–652. CrossRefGoogle Scholar
  22. 22.
    Hasebe KYN (1999) Green's function for a heat source in an infinite region with an arbitrary shaped hole. J Appl Mech 66:204–210MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nazemi N (2015) Identification of the mechanical properties in the heat-affected zone of aluminum welded structures. University of Windsor, WindsorGoogle Scholar
  24. 24.
    Meco SAM (2016) Joining of steel to aluminium alloys for advanced structural applications. Cranfield University, CranfieldGoogle Scholar
  25. 25.
    Ding J (2012) Thermo-mechanical analysis of wire and arc additive manufacturing process. Cranfield University, CranfieldGoogle Scholar
  26. 26.
    Michaleris P, Feng Z, Campbell G (1997) Evaluation of 2D and 3D FEA models for predicting residual stress and distortion. ASME PVP- Approximate Methods in the Design and Analysis of Pressure Vessels and Piping Components 347:91–102Google Scholar
  27. 27.
    Martukanitz R, Michaleris P, Palmer T, DebRoy T, Liu Z-K, Otis R, Heo TW, Chen L-Q (2014) Toward an integrated computational system for describing the additive manufacturing process for metallic materials. Addit Manuf 1-4:52–63. CrossRefGoogle Scholar
  28. 28.
    DebRoy T, Wei HL, Zuback JS, Mukherjee T, Elmer JW, Milewski JO, Beese AM, Wilson-Heid A, De A, Zhang W (2018) Additive manufacturing of metallic components – process, structure and properties. Prog Mater Sci 92:112–224. CrossRefGoogle Scholar
  29. 29.
    Ueda Y, Fukuda K, Nakacho K, Endo S (1975) A new measuring method of residual stresses with the aid of finite element method and reliability of estimated values. Soc Nav Archit:499-507CrossRefGoogle Scholar
  30. 30.
    Okerblom NO (1958) The calculations of deformations of welded metal structures. Her Majesty’s Stationery Office, LondonGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Li Sun
    • 1
  • Xiaobo Ren
    • 2
  • Jianying He
    • 1
  • Jim Stian Olsen
    • 1
  • Sakari Pallaspuro
    • 3
  • Zhiliang Zhang
    • 1
    Email author
  1. 1.Department of Structural EngineeringNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.SINTEF IndustryTrondheimNorway
  3. 3.Materials and Mechanical Engineering, Centre for Advanced Steels ResearchUniversity of OuluOuluFinland

Personalised recommendations