Smoothed Particle Hydrodynamics for Ductile Solid Continua

  • Peter Eberhard
  • Fabian Spreng
Living reference work entry


In this chapter, a numerical simulation model for ductile solid continua is presented. It is based on the Smoothed Particle Hydrodynamics (SPH) method, which serves to spatially discretize and, thus, solve the governing equations of continuum mechanics. Due to the meshless, Lagrangian character of the SPH spatial discretization technique, the introduced model is naturally well-suited for the simulation of continua featuring large deformations, major changes in topology, material failure including structure disintegration, and/or a large number of contacts with the environment occurring at the same time. For this reason, it has the potential to become a beneficial complement to the well-established numerical solid models, which mainly make use of mesh-based methods. To that end, however, the original SPH discretization scheme is to be variously extended and modified as discussed in detail in the course of this chapter. Besides, also its efficient implementation, i.e. the efficient numerical solution of the SPH-discretized governing equations of continuum mechanics, is addressed. The quality of the developed SPH formulation for ductile solids including its versatility and accuracy is demonstrated on the basis of two exemplary applications, namely, the industrial processes of friction stir welding and orthogonal metal cutting. It is shown as part of this contribution that, in either case, the proposed SPH model for ductile solid continua is capable of reproducing both the mechanical and the thermal macroscopic behavior of the real processed material in the simulation.


  1. 1.
    Gurtin ME. An introduction to continuum mechanics. San Diego: Academic Press; 1981.zbMATHGoogle Scholar
  2. 2.
    Gingold RA, Monaghan JJ. Smoothed Particle Hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc. 1977;181(3):375–89.zbMATHCrossRefGoogle Scholar
  3. 3.
    Lucy LB. A numerical approach to the testing of the fission hypothesis. Astron J. 1977;82(12):1013–24.CrossRefGoogle Scholar
  4. 4.
    Pasimodo [Internet]. Stuttgart: Institute of Engineering and Computational Mechanics, University of Stuttgart; 2009 [updated 7 Aug 2015; cited 1 Aug 2016]. Available from:
  5. 5.
    Liu MB, Liu GR. Smoothed Particle Hydrodynamics (SPH): an overview and recent developments. Arch Comput Method E. 2010;17(1):25–76.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Monaghan JJ. Smoothed Particle Hydrodynamics. Rep Prog Phys. 2005;68(8):1703–59.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Violeau D. Fluid mechanics and the SPH method: theory and applications. Oxford: Oxford University Press; 2012.zbMATHCrossRefGoogle Scholar
  8. 8.
    Dehnen W, Aly H. Improving convergence in Smoothed Particle Hydrodynamics simulations without pairing instability. Mon Not R Astron Soc. 2012;425(2):1068–82.CrossRefGoogle Scholar
  9. 9.
    Wendland H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math. 1995;4(1):389–96.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Amicarelli A, Marongiu JC, Leboeuf F, Leduc J, Neuhauser M, Fang L, Caro J. SPH truncation error in estimating a 3D derivative. Int J Numer Meth Eng. 2011;87(7):677–700.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Belytschko T, Krongauz Y, Dolbow J, Gerlach C. On the completeness of meshfree particle methods. Int J Numer Meth Eng. 1998;43(5):785–819.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Haupt P. Continuum mechanics and theory of materials. 2nd ed. Berlin: Springer; 2002.zbMATHCrossRefGoogle Scholar
  13. 13.
    Batchelor GK. An introduction to fluid dynamics. Cambridge: Cambridge University Press; 2000.zbMATHCrossRefGoogle Scholar
  14. 14.
    Eringen AC. Mechanics of continua. New York: Wiley; 1967.zbMATHGoogle Scholar
  15. 15.
    Acheson DJ. Elementary fluid dynamics. Oxford: Clarendon; 1990.zbMATHGoogle Scholar
  16. 16.
    Dill EH. Continuum mechanics: elasticity, plasticity, viscoelasticity. Boca Raton: CRC; 2006.Google Scholar
  17. 17.
    Sod GA. Survey of several Finite Difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys. 1978;27(1):1–31.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Monaghan JJ, Gingold RA. Shock simulation by the particle method SPH. J Comput Phys. 1983;52(2):374–89.zbMATHCrossRefGoogle Scholar
  19. 19.
    Monaghan JJ. Smoothed Particle Hydrodynamics. Annu Rev Astron Astr. 1992;30:543–74.CrossRefGoogle Scholar
  20. 20.
    Glatzmaier GA. Introduction to modeling convection in planets and stars: magnetic field, density stratification, rotation. Princeton: Princeton University Press; 2014.zbMATHGoogle Scholar
  21. 21.
    Chakrabarty J. Applied plasticity. New York: Springer; 2000.zbMATHCrossRefGoogle Scholar
  22. 22.
    Kaviany M. Principles of heat transfer. New York: Wiley; 2002.zbMATHGoogle Scholar
  23. 23.
    Cleary PW, Monaghan JJ. Conduction modelling using Smoothed Particle Hydrodynamics. J Comput Phys. 1999;148(1):227–64.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Burshtein AI. Introduction to thermodynamics and kinetic theory of matter. 2nd ed. Berlin: Wiley; 2005.CrossRefGoogle Scholar
  25. 25.
    Doghri I. Mechanics of deformable solids: linear, nonlinear, analytical and computational aspects. Berlin: Springer; 2000.zbMATHCrossRefGoogle Scholar
  26. 26.
    Simo JC, Hughes TJR. Computational inelasticity. 2nd ed. New York: Springer; 1998.zbMATHGoogle Scholar
  27. 27.
    Dieter GE. Mechanical metallurgy. New York: McGraw-Hill; 1961.CrossRefGoogle Scholar
  28. 28.
    Paterson MS, Wong TF. Experimental rock deformation – the brittle field. 2nd ed. Berlin: Springer; 2005.Google Scholar
  29. 29.
    Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: American Defense Preparedness Association, Royal Netherlands Society of Engineers. Proceedings of the 7th international symposium on ballistics, The Hague; 19–21 Apr 1983. pp. 541–7.Google Scholar
  30. 30.
    Spreng F. Smoothed Particle Hydrodynamics for ductile solids [dissertation]. Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, Band 48. Aachen: Shaker Verlag; 2017.Google Scholar
  31. 31.
    Farren WS, Taylor GI. The heat developed during plastic extension of metals. P Roy Soc Lond A Mat. 1925;107(743):422–51.CrossRefGoogle Scholar
  32. 32.
    Johnson GR, Cook WH. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech. 1985;21(1):31–48.CrossRefGoogle Scholar
  33. 33.
    Grady DE, Kipp ME. Continuum modelling of explosive fracture in oil shale. Int J Rock Mech Min. 1980;17(3):147–57.CrossRefGoogle Scholar
  34. 34.
    Monaghan JJ. SPH without a tensile instability. J Comput Phys. 2000;159(2):290–311.zbMATHCrossRefGoogle Scholar
  35. 35.
    Gray JP, Monaghan JJ, Swift RP. SPH elastic dynamics. Comput Method Appl Mech. 2001;190(49–50):6641–62.zbMATHCrossRefGoogle Scholar
  36. 36.
    Müller A. 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 Verlag; 2017.Google Scholar
  37. 37.
    Randles PW, Libersky LD. Smoothed Particle Hydrodynamics: some recent improvements and applications. Comput Method Appl Mech. 1996;139(1–4):375–408.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Shapiro PR, Martel H, Villumsen JV, Owen JM. Adaptive Smoothed Particle Hydrodynamics, with application to cosmology: methodology. Astrophys J Suppl Ser. 1996;103(2):269–330.CrossRefGoogle Scholar
  39. 39.
    Owen JM, Villumsen JV, Shapiro PR, Martel H. Adaptive Smoothed Particle Hydrodynamics: methodology. II. Astrophys J Suppl Ser. 1998;116(2):155–209.CrossRefGoogle Scholar
  40. 40.
    Thacker RJ, Tittley ER, Pearce FR, Couchman HMP, Thomas PA. Smoothed Particle Hydrodynamics in cosmology: a comparative study of implementations. Mon Not R Astron Soc. 2000;319(2):619–48.CrossRefGoogle Scholar
  41. 41.
    Springel V, Hernquist L. Cosmological Smoothed Particle Hydrodynamics simulations: the entropy equation. Mon Not R Astron Soc. 2002;333(3):649–64.CrossRefGoogle Scholar
  42. 42.
    Allen MP, Tildesley DJ. Computer simulation of liquids. Oxford: Clarendon Press; 1989.zbMATHGoogle Scholar
  43. 43.
    Schinner A. Fast algorithms for the simulation of polygonal particles. Granul Matter. 1999;2(1):35–43.CrossRefGoogle Scholar
  44. 44.
    Müller M, Schirm S, Teschner M, Heidelberger B, Gross M. Interaction of fluids with deformable solids. Comput Anim Virtual Worlds. 2004;15(3–4):159–71.CrossRefGoogle Scholar
  45. 45.
    Monaghan JJ, Kos A, Issa N. Fluid motion generated by impact. J Waterw Port C-ASCE. 2003;129(6):250–9.CrossRefGoogle Scholar
  46. 46.
    Hut P, Makino J, McMillan S. Building a better leapfrog. Astrophys J. 1995;443(2):L93–6.CrossRefGoogle Scholar
  47. 47.
    Mishra RS, Ma ZY. Friction stir welding and processing. Mat Sci Eng R. 2005;50(1–2):1–78.CrossRefGoogle Scholar
  48. 48.
    Tang W, Guo X, McClure JC, Murr LE, Nunes A. Heat input and temperature distribution in friction stir welding. J Mater Process Manu. 1998;7(2):163–72.CrossRefGoogle Scholar
  49. 49.
    Armarego EJA, Brown RH. The machining of metals. Englewood Cliffs: Prentice-Hall; 1969.Google Scholar
  50. 50.
    Boothroyd G, Knight WA. Fundamentals of machining and machine tools. 3rd ed. Boca Raton: CRC; 2006.Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

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

Personalised recommendations