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International Journal of Fracture

, Volume 180, Issue 1, pp 53–70 | Cite as

Total Lagrangian SPH modelling of necking and fracture in electromagnetically driven rings

  • T. De Vuyst
  • R. Vignjevic
Original Paper

Abstract

This paper describes research on the prediction of necking and failure in metals at very high strain rates. The model developed in this paper uses a total Lagrangian SPH formulation with a normalised kernel. The detailed data from electromagnetically driven ring experiments by Zhang and Ravi-Chandar (Int J Fract 142:183–217, 2006) is used to evaluate the accuracy of the model predictions. In order to correctly model fracture in the total Lagrangian SPH formulation a visibility criterion based on a truncated cone has been implemented to remove particles obscured by a failed particle. A Johnson-Cook plasticity model is used in combination with a Lemaitredamage model to describe the plastic deformation and fracture of the rings. The effect of Joule heating due to the current induced in the ring is taken into account in the constitutive model. The acceleration due to the ring currents was implemented in the SPH code as a body force. The results demonstrate that this type of model is capable of predicting the number of fragments as well as the time of fracture. In agreement with experimental data, the model also predicts arrested necks and bending in the fragments.

Keywords

Smoothed particle hydrodynamics Meshless Modelling Fracture Expanding ring Necking Strain-rate 

References

  1. Becker R (2002) Ring fragmentation predictions using the Gurson model with material stability conditions as failure criteria. Int J Solids Struct 39:3555–3580CrossRefGoogle Scholar
  2. Belytschko T, Xiao SP (2000) Stability analysis of particle methods with corrected derivatives. Comput Math Appl 43: 329–350Google Scholar
  3. Corbett BM (2006) Numerical simulations of target hole diameters for hypervelocity impacts into elevated and room temperature bumpers. Int J Impact Eng 33:431–440CrossRefGoogle Scholar
  4. Dong Q, Thomas E, Shaofan L, Wing Kam L (2008) Meshfree simulation of failure modes in thin cylinders subjected to combined loads of internal pressure and localised heat. Int J Numer Methods Eng 76:1159–1184Google Scholar
  5. Fyfe JM, Rajendran AM (1979) Dynamic pre-strain and inertia effects on the fracture of metals. J Mech Phys Solids 28:17–26CrossRefGoogle Scholar
  6. Grady DE, Benson DA (1983) Fragmentation of metal rings be electromagnetic loading. Exp Mech 23:393–400CrossRefGoogle Scholar
  7. Guduru PR, Freund LB (2002) The dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials. Int J Solids Struct 39:5615–5632CrossRefGoogle Scholar
  8. Hiroe T, Fujiwara K, Hata H, Takahashi H (2008) Deformation and fragmentation behaviour of exploded metal cylinders and the effects of wall materials configuration, explosive energy and initiated locations. Int J Impact Eng 35:1578–1586CrossRefGoogle Scholar
  9. Hopson MV, Scott CM, Patel R (2011) Computational comparisons of homogeneous and statistical descriptions of AerMEt100 steel subjected to high strain rate loading. Int J Impact Eng 38:451–455CrossRefGoogle Scholar
  10. Hu X, Daehn GS (1996) Effect of velocity on flow localisation on tension. Acta Mater 44(3):1021–1033 Google Scholar
  11. Johnson JN (1981) Dynamic fracture and spallation in ductile solids. J Appl Phys 52(4):2812–2824CrossRefGoogle Scholar
  12. Landen D, Wetz D, Satapathy S, Levinson S (2009) Electromagnetically driven expanding ring with preheating. IEEE Trans magn 45:598–603CrossRefGoogle Scholar
  13. Lemaitre J, Chaboche JL (1990) Mechanics of solid materials. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Lemaitre J, Sermage JP (1997) One damage law for different mechanisms. Comput Mech 20:84–88CrossRefGoogle Scholar
  15. LSTC (2007) LS-Dyna keyword user’s manual, version 971, volume I and II. LSTC, Livermore, CaliforniaGoogle Scholar
  16. Meulbroek JP, Ramesh KT, Swaminathan PK, Lennon AM (2008) CTH simulations of an expanding ring to study fragmentation. Int J Impact Eng 35:1661–1665CrossRefGoogle Scholar
  17. Meyer HW Jr, Brannon RM (2012) A model for statistical variation of fracture properties in a continuum. Int J Impact Eng 42:48–58CrossRefGoogle Scholar
  18. Meyers MA (1994) Dynamic behavior of materials. Wiley, New YorkCrossRefGoogle Scholar
  19. Mercier S, Molinari A (2004) Analysis of multiple necking in rings under rapid radial expansion. Int J Impact Eng 30: 403–419Google Scholar
  20. Mercier S, Granier N, Molinari A, Llorca F, Buy F (2010) Multiple necking during the dynamic expansion of hemispherical metallic shells, from experiments to modelling. J Mech Phys Solids 58:955–982CrossRefGoogle Scholar
  21. Pandolfi A, Krysl P, Ortiz M (1999) Finite Element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. Int J Fract 95:279–297CrossRefGoogle Scholar
  22. Rabczuk T, Belytschko T, Xiao SP (2004) Stable particle methods based on Lagrangian kernels. Comput Methods Appl Mech Eng 193:1035–1063CrossRefGoogle Scholar
  23. Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61:2316–2343Google Scholar
  24. Randles PW, Libersky LD (1996) Smoothed particle hydrodynamics: some recent improvements and applications. Comput Methods Appl Mech Eng 139:375–408CrossRefGoogle Scholar
  25. Rusinek A, Zaera R (2007) Finite element simulation of steel ring fragmentation under radial expansion. Int J Impact Eng 34:799–822CrossRefGoogle Scholar
  26. Satapathy S, Landen D (2006) Expanding ring experiments to measure high-temperature adiabatic properties. Int J Impact Eng 33:735–744CrossRefGoogle Scholar
  27. Shenoy VB, Freund LB (1999) Necking bifurcations during high strain rate extension. J Mech Phys Solids 47:2209–2233CrossRefGoogle Scholar
  28. Sørensen NJ, Freund LB (2000) Unstable neck formation in a ductile ring subjected to impulsive radial loading. Int J Solids Struct 37:2265–2283Google Scholar
  29. Swegle JW, Hicks DL, Attaway SW (1995) Smoothed particle hydrodynamics stability analysis. J Comp Phys 116:123–134CrossRefGoogle Scholar
  30. Triantafyllidis N, Waldenmyer JR (2004) Onset of necking in electro-magnetically formed rings. J Mech Phys Solids 52:2127–2148Google Scholar
  31. Vadillo G, Rodriguez Martinez JA, Fernandez Saez J (2012) On the interplay between strain rate and strain rate sensitivity on flow localisation in the dynamic expansion of ductile rings. Int J Solids Struct 49:481–491Google Scholar
  32. Vignjevic R, Reveles JR, Campbell J (2006) SPH in a total lagran- gian formalism. CMES Comput Model Eng Sci 14(3):181–198Google Scholar
  33. Vignjevic R, Campbell J, Jaric J, Powell S (2009) Derivation of SPH equations in a moving referential coordinate system. Comp Methods Appl Mech Eng 198(30–32):2403–2411Google Scholar
  34. Zhang H, Ravi-Chandar K (2006) On the dynamics of necking and fragmentation—I. Real-time and post-mortem observations in Al 6061-O. Int J Fract 142:183–217CrossRefGoogle Scholar
  35. Zhang H, Ravi-Chandar K (2008) On the dynamics of necking and fragmentation—II. Effect of material properties, geometrical constraints and absolute size. Int J Fract 150:3–36CrossRefGoogle Scholar
  36. Zhang H, Ravi-Chandar K (2009) On the dynamics of necking and fragmentation—III. Effect of cladding with a polymer. Int J Fract 155:101–118 Google Scholar
  37. Zhang H, Ravi-Chandar K (2009) On the dynamics of necking and fragmentation—IV., Expansion of Al 6061-O tubes. Int J Fract 163:41–65CrossRefGoogle Scholar
  38. Zhang H, Ravi-Chandar K (2009) Dynamic fragmentation of ductile materials. J Phys D: Appl Phys 42(21):1–16Google Scholar
  39. Zhou F, Molinari JF, Ramesh KT (2006) Analysis of the brittle fragmentation of an expanding ring. Comput Mater Sci 37: 74–85Google Scholar
  40. Zhou F, Molinari JF, Ramesh KT (2006) An elastic-visco-plastic analysis of ductile expanding ring. Int J Impact Eng 33: 880–891Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Crashworthiness, Impact and Structural Mechanics Group, School of EngineeringCranfield UniversityCranfieldUK

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