Tracking of material orientation in updated Lagrangian SPH

  • Elizaveta Shishova
  • Fabian Spreng
  • Dominik Hamann
  • Peter EberhardEmail author


This contribution demonstrates a continuum-mechanics-based method which yields the possibility of material orientation tracking within the smoothed particle hydrodynamics framework. The functionality provides the information on local orientation of particles, which is necessary for anisotropic material models, e.g., the kinematic hardening rule proposed by Prager. Such a model is expected to provide a possibility of a more precise simulation of complex industrial processes such as friction stir welding with three-dimensional material flow. The derivation of the method is presented for the updated Lagrangian formulation, in which the increment of the quaternion for the material point can be extracted from the gradient of velocity, following the continuum mechanics description. The examples of different complexity are demonstrated in order to verify the method: the rotating cylinder, torsion test, and industrial turning process. Additionally, the importance of kernel gradient correction adoption in order to preserve angular momentum in the presence of rotations is showcased.


Smoothed particle hydrodynamics (SPH) Material orientation tracking Quaternion Kernel gradient correction 



This research has received funding from the German Research Foundation (DFG) in the projects EB 195/30-1 “Simulation of Friction Stir Welding” and SP 1526/1, SP 1526/2 “Development and Validation of Novel, Discretization-Error-Based Adaptivity Criteria for Smoothed Particle Hydrodynamics.” This support is highly appreciated.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Gingold R, Monaghan J (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181(3):375–389CrossRefzbMATHGoogle Scholar
  2. 2.
    Lucy L (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82(12):1013–1024CrossRefGoogle Scholar
  3. 3.
    Libersky L, Petschek A (1991) Smooth particle hydrodynamics with strength of materials. In: Trease H, Fritts M, Crowley W (eds) Advances in the free-lagrange method including contributions on adaptive gridding and the smooth particle hydrodynamics method, vol 395. Lecture notes in physics. Springer, Berlin, pp 248–257CrossRefGoogle Scholar
  4. 4.
    Randles P, Libersky L (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139(1–4):375–408MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Madaj M, Pís̆ka M (2013) On the SPH orthogonal cutting simulation of A2024-T351 alloy. In: 14th CIRP conference on modeling of machining operations (CIRP CMMO), procedia CIRP, vol 8, pp 152–157Google Scholar
  6. 6.
    Vignjevic R, Reveles J, Campbell J (2006) SPH in a total Lagrangian formalism. Comput Model Eng Sci 14(3):181–198MathSciNetzbMATHGoogle Scholar
  7. 7.
    Thomas W, Nicholas E, Needham J, Murch M, Temple-Smith P, Dawes C (1991) Friction-stir butt welding. Great Britain Patent application no. GB9125978.8Google Scholar
  8. 8.
    Pan W, Li D, Tartakovsky AM, Ahzi S, Khraisheh M, Khaleel M (2013) A new smoothed particle hydrodynamics non-Newtonian model for friction stir welding: process modeling and simulation of microstructure evolution in a magnesium alloy. Int J Plast 48:189–204CrossRefGoogle Scholar
  9. 9.
    Fraser K, St-Georges L, Kiss LI (2016) A mesh-free solid-mechanics approach for simulating the friction stir-welding process. In: Ishak M (ed) Joining technologies. IntechOpen, RijekaGoogle Scholar
  10. 10.
    Hossfeld M, Roos E (2013) A new approach to modelling friction stir welding using the CEL method. In: Advanced manufacturing engineering and technologies (NEWTECH 2013), Stockholm, Sweden, p 179Google Scholar
  11. 11.
    Kim J, Lee W, Chung KH, Kim D, Kim C, Okamoto K, Wagoner RH, Chung K (2011) Springback evaluation of friction stir welded TWB automotive sheets. Met Mater Int 17(1):83–98CrossRefzbMATHGoogle Scholar
  12. 12.
    Bastier A, Maitournam M, Roger F, Van KD (2008) Modelling of the residual state of friction stir welded plates. J Mater Process Technol 200(1):25–37CrossRefGoogle Scholar
  13. 13.
    Shield RT, Ziegler H (1958) On Prager’s hardening rule. Z Angew Math Phys ZAMP 9(3):260–276MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chaboche J (1986) Time-independent constitutive theories for cyclic plasticity. Int J Plast 2(2):149–188CrossRefzbMATHGoogle Scholar
  15. 15.
  16. 16.
    Dehnen W, Aly H (2012) Improving convergence in smoothed particle hydrodynamics simulations without pairing instability. Mon Not R Astron Soc 425(2):1068–1082CrossRefGoogle Scholar
  17. 17.
    Bonet J, Lok TS (1999) Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. Comput Methods Appl Mech Eng 180(1–2):97–115MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu M, Liu G (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Archiv Comput Methods Eng 17(1):25–76MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Monaghan J (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68(8):1703–1759MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rabczuk T, Belytschko T, Xiao S (2004) Stable particle methods based on Lagrangian Kernels. Comput Methods Appl Mech Eng 193(12):1035–1063MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Monaghan J, Gingold R (1983) Shock simulation by the particle method SPH. J Comput Phys 52(2):374–389CrossRefzbMATHGoogle Scholar
  22. 22.
    Monaghan J (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30:543–574CrossRefGoogle Scholar
  23. 23.
    Gray J, Monaghan J, Swift R (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190(49–50):6641–6662CrossRefzbMATHGoogle Scholar
  24. 24.
    Müller A (2017) Dynamic refinement and coarsening for the smoothed particle hydrodynamics method. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, Band 46. Aachen: Shaker VerlagGoogle Scholar
  25. 25.
    Cole R (1948) Underwater explosions. Princeton University Press, PrincetonCrossRefGoogle Scholar
  26. 26.
    Chakrabarty J (2000) Applied plasticity. Springer, New YorkCrossRefzbMATHGoogle Scholar
  27. 27.
    Spreng F (2017) Smoothed particle hydrodynamics for ductile solids. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol 48, Aachen, Shaker VerlagGoogle Scholar
  28. 28.
    Yamaji A (2007) An introduction to tectonophysics: theoretical aspects of structural geology. Terrapub, TokyoGoogle Scholar
  29. 29.
    Bathe KJ (2014) Finite element procedures. Prentice-Hall, Upper Saddle RiverzbMATHGoogle Scholar
  30. 30.
    Lai W, Rubin D, Krempl E (1993) Introduction to continuum mechanics. Pergamon Press, OxfordzbMATHGoogle Scholar
  31. 31.
    Gürlebeck K, Sprössig W (1998) Quaternionic and Clifford calculus for physicists and engineers. Wiley, ChichesterzbMATHGoogle Scholar
  32. 32.
    Fleissner F (2010) Parallel object oriented simulation with Lagrangian particle methods. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, Band 16. Aachen, Shaker VerlagGoogle Scholar
  33. 33.
    Zhao F, van Wachem B (2013) A novel Quaternion integration approach for describing the behaviour of non-spherical particles. Acta Mech 224(12):3091–3109MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Monaghan J, Kos A, Issa N (2003) Fluid motion generated by impact. J Waterw Port Coast Ocean Eng 129(6):250–259CrossRefGoogle Scholar
  35. 35.
    Meyers M, Chawla K (2009) Mechanical behavior of materials. Cambridge University Press, New YorkzbMATHGoogle Scholar
  36. 36.
    Courant R, Friedrichs K, Lewy H (1967) On the partial difference equations of mathematical physics. IBM J Res Dev 11(2):215–234MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    SAE J403 (2014) Chemical compositions of SAE carbon steels. Warrendale: SAE InternationalGoogle Scholar
  38. 38.
    Johnson G, Cook W (1983) A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the 7th international symposium on ballistics, The Hague, pp 541–547Google Scholar
  39. 39.
    Dobrowolski P (2015) Swing-twist decomposition in Clifford algebra. CoRR. arXiv:1506.05481
  40. 40.
    Gross D, Hauger W, Schröder J, Wall W, Bonet J (2018) Engineering mechanics: mechanics of materials, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  41. 41.
    Boothroyd G, Knight W (2006) Fundamentals of machining and machine tools. CRC, Boca RatonGoogle Scholar
  42. 42.
    Thomas W, Nicholas E, Needham J, Murch M, Temple-Smith P, Dawes C (1993) Improvements relating to friction welding. International Pattent publication no. WO/1993/010935Google Scholar
  43. 43.
    Johnson G, Cook W (1985) Fracture characteristics of three metals subjected to various strains, strain rates. Temp Press Eng Fract Mech 21(1):31–48CrossRefGoogle Scholar
  44. 44.
    Simkins DC, Li S (2006) Meshfree simulations of thermo-mechanical ductile fracture. Comput Mech 38(3):235–249CrossRefzbMATHGoogle Scholar
  45. 45.
    Gaugele T (2011) Application of the discrete element method to model ductile, heat conductive materials. Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol 20, Aachen, Shaker VerlagGoogle Scholar

Copyright information

© OWZ 2019

Authors and Affiliations

  1. 1.Institute of Engineering and Computational MechanicsUniversity of StuttgartStuttgartGermany

Personalised recommendations