Shock Waves

pp 1–13 | Cite as

A mesoscopic reaction rate model for shock-to-detonation of PBX explosives having different mean particle sizes

  • Y. R. Liu
  • X. M. HuEmail author
  • Z. P. Duan
  • Z. Y. Zhang
Original Article


In order to well predict mean explosive particle size effects on the shock-to-detonation transition (SDT) process of a plastic bonded explosive (PBX), some improvements to a previous three-term mesoscopic reaction rate model that consists of a hot-spot ignition term, a hot-spot growth term, and an overall reaction term are made: A set of new switch conditions, which depend on mean explosive particle size, is proposed for the operations of the three terms; a new expression is obtained for the hot-spot growth term by combining an ignition efficiency factor which depends on mean explosive particle size and replacing the original burning topology geometric factor, which merely describes the characteristic of the outward pore surface burning reaction, by a more reasonable one, which combines the characteristic of inward particle surface burning reaction and that of outward pore surface burning reaction. Furthermore, for verification, the improved reaction rate model is incorporated into the DYNA2D code to simulate numerically the SDT process of three formulations of PBXC03 having the same density but different mean particle sizes, and the numerical results of pressure histories at different Lagrangian locations in the explosive are found to be in good agreement with previous experimental data.


Mesoscopic reaction rate model Shock-to-detonation transition Particle size effects Hot-spot ignition PBX 



The authors gratefully acknowledge the financial support for the present study by the National Natural Science Foundation of China under Grant U1630113. The authors also thank the anonymous reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially. Finally, the authors would like to thank the Managing Editor and Xia Jin for their linguistic assistance during the preparation of this manuscript.


  1. 1.
    Forest, C.A.: Burning and Detonation. Los Alamos Scientific Laboratory Report LA-UR-81-839 (1981).
  2. 2.
    Tarver, C.M.: Ignition and growth modeling of LX-17 hockey puck experiments. Propellants Explos. Pyrotech. 30(2), 109–117 (2010). CrossRefGoogle Scholar
  3. 3.
    Tang, P.K., Johnson, J.N., Forest, C.A.: Modeling heterogeneous high explosive burn with an explicit hot-spot process. Proceedings of the 8th Symposium (International) on Detonation, Albuquerque, NM, pp. 52–61 (1985)Google Scholar
  4. 4.
    Kaneswaran, M.A., Curtis, J.P., Reaugh, J.E.: Modeling the shock to detonation transition in PETN using CREST. Proceedings of the 15th International Detonation Symposium, San Francisco, CA, pp. 1042–1051 (2014)Google Scholar
  5. 5.
    Starkenberg, J.: Shock-pressure and pseudo-entropic approaches to explosive initiation modeling. Proceedings of the 15th International Detonation Symposium, San Francisco, CA, pp. 908–916 (2014)Google Scholar
  6. 6.
    Wescott, B.L., Stewart, D.S., Davis, W.C.: Equation of state and reaction rate for condensed-phase explosives. J. Appl. Phys. 98(5), 053514 (2005). CrossRefGoogle Scholar
  7. 7.
    Massoni, J., Saurel, R., Baudin, G., Demol, G.: A mechanistic model for shock initiation of solid explosives. Phys. Fluids 11(3), 710–736 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hamate, Y., Horie, Y.: Ignition and detonation of solid explosives: a micromechanical burn model. Shock Waves 16(2), 125–147 (2006). CrossRefzbMATHGoogle Scholar
  9. 9.
    Kim, K., Sohn, C.H.: Modeling of reaction buildup processes in shock porous explosive. Proceedings of the 8th Symposium (International) on Detonation, Albuquerque, NM, pp. 926–933 (1985)Google Scholar
  10. 10.
    Kim, K.: Development of a model of reaction rates in shocked multicomponent explosives. Proceedings of the 9th Symposium (International) on Detonation, Portland, OR, pp. 593–603 (1989)Google Scholar
  11. 11.
    Duan, Z.P., Wen, L.J., Liu, Y., Ou, Z.C., Huang, F.L.: A pore collapse model for hot-spot ignition in shocked multi-component explosive. Int. J. Nonlinear Sci. Numer. Simul. 11(Supplement), 19–24 (2011). CrossRefzbMATHGoogle Scholar
  12. 12.
    Zhang, Z.Y., Lu, F.Y., Wang, Z.B., Huan, S.: Studies on high-pressure reaction rate of PBX-9404. Explos. Shock Waves 19(4), 360–364 (1999) (in Chinese) Google Scholar
  13. 13.
    Wen, L.J.: Research on mesoscopic reaction rate model of shock initiation of PBX. Ph.D. Thesis, Beijing Institute of Technology (2011) (in Chinese) Google Scholar
  14. 14.
    Duan, Z.P., Liu, Y.R., Zhang, Z.Y., Ou, Z.C., Huang, F.L.: Prediction of initial temperature effects on shock initiation of solid explosives by using mesoscopic reaction rate model. Int. J. Nonlinear Sci. Numer. Simul. 15(5), 299–305 (2014). CrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, Y.R., Duan, Z.P., Zhang, Z.Y., Ou, Z.C., Huang, F.L.: A mesoscopic reaction rate model for shock initiation of multi-component PBX explosives. J. Hazard. Mater. 317, 44–51 (2016). CrossRefGoogle Scholar
  16. 16.
    Wen, L.J., Duan, Z.P., Zhang, L.S., Zhang, Z.Y., Ou, Z.C., Huang, F.L.: Effects of HMX particle size on the shock initiation of PBXC03 explosive. Int. J. Nonlinear Sci. Numer. Simul. 13(2), 189–194 (2012). CrossRefzbMATHGoogle Scholar
  17. 17.
    Moulard, H.: Particular aspect of the explosive particle size effect on shock sensitivity of cast PBX formulations. Proceedings of the 9th Symposium (International) on Detonation, Portland, OR, pp. 18–24 (1989)Google Scholar
  18. 18.
    Price, D.: Effect of particle size on the shock sensitivity of porous HE. J. Energ. Mater. 6(3–4), 283–317 (1988). CrossRefGoogle Scholar
  19. 19.
    Zhang, J., Jackson, T.L.: Effect of microstructure on the detonation initiation in energetic materials. Shock Waves (2017). CrossRefGoogle Scholar
  20. 20.
    Sutherland, G.T., Zakraysek, A.J.: Modeling of the effect of crystal quality and particle size on the shock reactivity and detonation properties of simple nitramine based explosives. Proceedings of the 13th Symposium (International) on Detonation, Norfolk, VA, pp. 23–28 (2006)Google Scholar
  21. 21.
    Garcia, F., Vandersall, K.S., Tarver, C.M.: Shock initiation experiments with ignition and growth modeling on low density HMX. J. Phys. Conf. Ser. 500(5), 052048 (2014). CrossRefGoogle Scholar
  22. 22.
    Howe, P., Frey, R., Taylor, B., Boyle, V.: Shock initiation and the critical energy concept. Proceedings of the 6th Symposium (International) on Detonation, Arlington VA, pp. 11–19 (1976)Google Scholar
  23. 23.
    Tarver, C.M., Chidester, S.K., Nichols, A.L.: Critical conditions for impact- and shock-induced hot spots in solid explosives. J. Phys. Chem. 100(14), 5794–5799 (1996). CrossRefGoogle Scholar
  24. 24.
    Taylor, B.C., Ervin, L.W.: Separation of ignition and buildup to detonation in pressed TNT. Proceedings of the 6th Symposium (International) on Detonation, Arlington VA, pp. 3–10 (1976)Google Scholar
  25. 25.
    Khasainov, B.A., Ermolaev, B.S., Presles, H.N., Vidal, P.: On the effect of grain size on shock sensitivity of heterogeneous high explosives. Shock Waves 7(2), 89–105 (1997). CrossRefGoogle Scholar
  26. 26.
    Carroll, M.M., Holt, A.C.: Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. Phys. 43(4), 1626–1636 (1972). CrossRefGoogle Scholar
  27. 27.
    Greenaway, M.W., Gifford, M.J., Proud, W.G., Field, J.E., Goveas, S.G.: An investigation into the initiation of hexanitrostilbene by laser-driven flyer plates. AIP Conf. Proc. 620, 1035–1038 (2002). CrossRefGoogle Scholar
  28. 28.
    Starkenberg, J.: Modeling detonation propagation and failure using explosive initiation models in a conventional hydrocodes. Proceedings of 12th Symposium (International) on Detonation, San Diego, CA (2002)Google Scholar
  29. 29.
    Hussain, T., Liu, Y., Huang, F., Duan, Z.P.: Ignition and growth modeling of shock initiation of different particle size formulations of PBXC03 explosive. J. Energ. Mater. 34(1), 38–48 (2016). CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina
  2. 2.State Key Laboratory of Explosion Science and TechnologyBeijing Institute of TechnologyBeijingChina
  3. 3.Institute of Technical Physics, College of ScienceNational University of Defense TechnologyChangshaChina

Personalised recommendations