Skip to main content
SpringerLink
Log in

A level set approach for shock-induced \(\upalpha \)–\(\upgamma \) phase transition of RDX

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

We present a thermodynamically consistent level sets approach based on regularization energy functional which can be directly incorporated into a Galerkin finite element framework to model interface motion. The regularization energy leads to a diffusive form of flux that is embedded within the level sets evolution equation which maintains the signed distance property of the level set function. The scheme is shown to compare well with the velocity extension method in capturing the interface position. The proposed level sets approach is employed to study the \(\upalpha \)–\(\upgamma \) phase transformation in RDX single crystal shocked along the (100) plane. Example problems in one and three dimensions are presented. We observe smooth evolution of the phase interface along the shock direction in both models. There is no diffusion of the interface during the zero level set evolution in the three dimensional model. The level sets approach is shown to capture the characteristics of the shock-induced \(\upalpha \)–\(\upgamma \) phase transformation such as stress relaxation behind the phase interface and the finite time required for the phase transformation to complete. The regularization energy based level sets approach is efficient, robust, and easy to implement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Adalsteinsson D, Sethian JA (1995) A fast level set method for propagating interfaces. J Comput Phys 118(2):269–277

    Article  MathSciNet  MATH  Google Scholar 

  2. Adalsteinsson D, Sethian JA (1999) The fast construction of extension velocities in level set methods. J Comput Phys 148(1):2–22

    Article  MathSciNet  MATH  Google Scholar 

  3. Allen SM, Cahn JW (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall 27(6):1085–1095

    Article  Google Scholar 

  4. Angheluta L, Jettestuen E, Mathiesen J, Renard F, Jamtveit B (2008) Stress-driven phase transformation and the roughening of solid–solid interfaces. Phys Rev Lett 100(9):096105

    Article  Google Scholar 

  5. Barth TJ, Sethian JA (1998) Numerical schemes for the Hamilton–Jacobi and level set equations on triangulated domains. J Comput Phys 145(1):1–40

    Article  MathSciNet  MATH  Google Scholar 

  6. Barton NR, Winter NW, Reaugh JE (2009) Defect evolution and pore collapse in crystalline energetic materials. Model Simul Mater Sci Eng 17(3):035003

    Article  Google Scholar 

  7. Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  8. Becker R (2004) Effects of crystal plasticity on materials loaded at high pressures and strain rates. Int J Plast 20(11):1983–2006

    Article  MATH  Google Scholar 

  9. Birch F (1947) Finite elastic strain of cubic crystals. Phys Rev 71(11):809–824

    Article  MATH  Google Scholar 

  10. Brooks AN, Hughes TJ (1982) Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259

    Article  MathSciNet  MATH  Google Scholar 

  11. Caselles V, Catté F, Coll T, Dibos F (1993) A geometric model for active contours in image processing. Numer Math 66(1):1–31

    Article  MathSciNet  MATH  Google Scholar 

  12. Cawkwell MJ, Ramos KJ, Hooks DE, Sewell TD (2010) Homogeneous dislocation nucleation in cyclotrimethylene trinitramine under shock loading. J Appl Phys 107(6):063512

    Article  Google Scholar 

  13. Cawkwell MJ, Sewell TD, Zheng L, Thompson DL (2008) Shock-induced shear bands in an energetic molecular crystal: application of shock-front absorbing boundary conditions to molecular dynamics simulations. Phys Rev B 78(1):014107

    Article  Google Scholar 

  14. Choi CS, Prince E (1972) The crystal structure of cyclotrimethylenetrinitramine. Acta Crystallogr Sect B Struct Crystallogr Cryst Chem 28(9):2857–2862

    Article  Google Scholar 

  15. Chopp DL (1993) Computing minimal surfaces via level set curvature flow. J Comput Phys 106(1):77–91

    Article  MathSciNet  MATH  Google Scholar 

  16. Connick W, May FGJ (1969) Dislocation etching of cyclotrimethylene trinitramine crystals. J Crys Growth 5(1):65–69

    Article  Google Scholar 

  17. Davidson AJ, Oswald ID, Francis DJ, Lennie AR, Marshall WG, Millar DI, Pulham CR, Warrenc JE, Cumming AS (2008) Explosives under pressure—the crystal structure of \(\upgamma \)-RDX as determined by high-pressure x-ray and neutron diffraction. CrystEngComm 10(2):162–165

    Article  Google Scholar 

  18. De S, Zamiri AR, Rahul (2014) A fully anisotropic single crystal model for high strain rate loading conditions with an application to \(\upalpha \)-RDX. J Mech Phys Solids 64:287–301

    Article  MathSciNet  Google Scholar 

  19. Dreger ZA, Gupta YM (2010) Phase diagram of hexahydro-1, 3, 5-trinitro-1, 3, 5-triazine crystals at high pressures and temperatures. J Phys Chem A 114(31):8099–8105

    Article  Google Scholar 

  20. Dreger ZA, Gupta YM (2012) Decomposition of \(\upgamma \)-cyclotrimethylene trinitramine (\(\upgamma \)-RDX): relevance for shock wave initiation. J Phys Chem A 116(34):8713–8717

    Article  Google Scholar 

  21. Gallagher HG, Halfpenny PJ, Miller JC, Sherwood JN, Tabor D (1992) Dislocation slip systems in pentaerythritol tetranitrate (PETN) and cyclotrimethylene trinitramine (RDX) [and Discussion]. Philos Trans R Soc Lond Ser A Phys Eng Sci 339(1654):293–303

    Article  Google Scholar 

  22. Gurtin ME (1981) An introduction to continuum mechanics. Academic, New York

    MATH  Google Scholar 

  23. Hallquist JO (1993) LS-DYNA3D theoretical manual. Livermore Software Technology Corporation, 18.1-7

  24. Hooks DE, Ramos KJ, Martinez AR (2006) Elastic-plastic shock wave profiles in oriented single crystals of cyclotrimethylene trinitramine (RDX) at 2.25 GPa. J Appl Phys 100(2):024908

    Article  Google Scholar 

  25. Ji H, Chopp D, Dolbow JE (2002) A hybrid extended finite element/level set method for modeling phase transformations. Int J Numer Methods Eng 54(8):1209–1233

    Article  MathSciNet  MATH  Google Scholar 

  26. Josyula K, Rahul, De S (2014) Thermomechanical properties and equation of state for the \(\upgamma \)-polymorph of hexahydro-1, 3, 5-trinitro-1, 3, 5-triazine. RSC Adv 4(78):41491–41499

    Article  Google Scholar 

  27. Juric D, Tryggvason G (1996) A front-tracking method for dendritic solidification. J Comput Phys 123(1):127–148

    Article  MathSciNet  MATH  Google Scholar 

  28. Landshoff R (1955) A numerical method for treating fluid flow in the presence of shocks. Los Alamos National Laboratory, Rept. LA-1930

  29. Levitas VI, Javanbakht M (2015) Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach. J Mech Phys Solids 82:287–319

    Article  MathSciNet  Google Scholar 

  30. Li C, Xu C, Gui C, Fox MD (2005) Level set evolution without re-initialization: a new variational formulation. In: 2005 IEEE computer society conference on computer vision and pattern recognition (CVPR’05), vol 1, pp 430–436

  31. Li C, Xu C, Gui C, Fox MD (2010) Distance regularized level set evolution and its application to image segmentation. IEEE Trans Image Process 19(12):3243–3254

    Article  MathSciNet  MATH  Google Scholar 

  32. Lloyd JT, Clayton JD, Becker R, McDowell DL (2014) Simulation of shock wave propagation in single crystal and polycrystalline aluminum. Int J Plast 60:118–144

    Article  Google Scholar 

  33. Lukyanov AA (2008) Constitutive behaviour of anisotropic materials under shock loading. Int J Plast 24(1):140–167

    Article  MATH  Google Scholar 

  34. Luscher DJ, Addessio FL, Cawkwell MJ, Ramos KJ (2017) A dislocation density-based continuum model of the anisotropic shock response of single crystal \(\upalpha \)-cyclotrimethylene trinitramine. J Mech Phys Solids 98:63–86

    Article  MathSciNet  Google Scholar 

  35. Lynch DR, O’Neill K (1981) Continuously deforming finite elements for the solution of parabolic problems, with and without phase change. Int J Numer Methods Eng 17(1):81–96

    Article  MathSciNet  MATH  Google Scholar 

  36. Malladi R, Sethian JA, Vemuri BC (1995) Shape modeling with front propagation: a level set approach. IEEE Trans Pattern Anal Mach Intell 17(2):158–175

    Article  Google Scholar 

  37. McDermott IT, Phakey PP (1971) A method of correlating dislocations and etch pits: application to cyclotrimethylene trinitramine. J Appl Crystallogr 4(6):479–481

    Article  Google Scholar 

  38. Menikoff R, Sewell TD (2002) Constituent properties of HMX needed for mesoscale simulations. Combust Theory Model 6(1):103–125

    Article  MATH  Google Scholar 

  39. Meyers MA (1994) Dynamic behavior of materials. Wiley, New York

    Book  MATH  Google Scholar 

  40. Munday LB, Chung PW, Rice BM, Solares SD (2011) Simulations of high-pressure phases in RDX. J Phys Chem B 115(15):4378–4386

    Article  Google Scholar 

  41. Olinger B, Roof B, Cady H (1978) The linear and volume compression of \(\upbeta \)-HMX and RDX to 9 GPa (90KiloBar). In: Symposium International Sur Le comportement Des Milieux Denses Sous Hautes Pressions Dynamiques, pp 1–7

  42. Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2):463–502

  43. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1):12–49

    Article  MathSciNet  MATH  Google Scholar 

  44. Patterson JE, Dreger ZA, Gupta YM (2007) Shock wave-induced phase transition in RDX single crystals. J Phys Chem B 111(37):10897–10904

    Article  Google Scholar 

  45. Peng D, Merriman B, Osher S, Zhao H, Kang M (1999) A PDE-based fast local level set method. J Comput Phys 155(2):410–438

    Article  MathSciNet  MATH  Google Scholar 

  46. Ramos KJ, Hooks DE, Bahr DF (2009) Direct observation of plasticity and quantitative hardness measurements in single crystal cyclotrimethylene trinitramine by nanoindentation. Philos Mag 89(27):2381–2402

    Article  Google Scholar 

  47. Rao VS, Hughes TJ, Garikipati K (2000) On modelling thermal oxidation of silicon II: numerical aspects. Int J Numer Methods Eng 47(1–3):359–377

    Article  MATH  Google Scholar 

  48. Rider WJ, Kothe DB (1998) Reconstructing volume tracking. J Comput Phys 141(2):112–152

    Article  MathSciNet  MATH  Google Scholar 

  49. Schreiber E, Anderson OL, Soga N (1973) Elastic constants and their measurement. McGraw-Hill, New York

    Google Scholar 

  50. Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci 93(4):1591–1595

    Article  MathSciNet  MATH  Google Scholar 

  51. Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  52. Sewell TD, Bennett CM (2000) Monte Carlo calculations of the elastic moduli and pressure–volume–temperature equation of state for hexahydro-1, 3, 5-trinitro-1, 3, 5-triazine. J Appl Phys 88(1):88–95

    Article  Google Scholar 

  53. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  54. Stolarska M, Chopp DL, Moës N, Belytschko T (2001) Modelling crack growth by level sets in the extended finite element method. Int J Numer Methods Eng 51(8):943–960

    Article  MATH  Google Scholar 

  55. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1):146–159

    Article  MATH  Google Scholar 

  56. Truesdell C (1984) Rational thermodynamics. Springer, Berlin

    Book  MATH  Google Scholar 

  57. Udaykumar HS, Mittal R, Shyy W (1999) Computation of solid–liquid phase fronts in the sharp interface limit on fixed grids. J Comput Phys 153(2):535–574

    Article  MATH  Google Scholar 

  58. Von Neumann J, Richtyer RD (1950) A method for the numerical calculation of hydrodynamic shocks. J Appl Phys 21(3):232–237

    Article  MathSciNet  Google Scholar 

  59. Winey JM, Gupta YM (2006) Nonlinear anisotropic description for the thermomechanical response of shocked single crystals: inelastic deformation. J Appl Phys 99(2):023510

    Article  Google Scholar 

  60. Yamasaki S, Nishiwaki S, Yamada T, Izui K, Yoshimura M (2010) A structural optimization method based on the level set method using a new geometry-based re-initialization scheme. Int J Numer Methods Eng 83(12):1580–1624

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhao HK, Chan T, Merriman B, Osher S (1996) A variational level set approach to multiphase motion. J Comput Phys 127(1):179–195

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been supported in part by the Office of Naval Research (ONR) Grants N000140810462 and N000141210527 with Dr. Cliff Bedford as the cognizant program manager.

Author information

Authors and Affiliations

  1. Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Kartik Josyula,  Rahul & Suvranu De

Authors
  1. Kartik Josyula
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Suvranu De
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Suvranu De.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Josyula, K., Rahul & De, S. A level set approach for shock-induced \(\upalpha \)–\(\upgamma \) phase transition of RDX. Comput Mech 61, 19–32 (2018). https://doi.org/10.1007/s00466-017-1493-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1493-1

Keywords

Search

Navigation