Acta Mechanica Sinica

, Volume 34, Issue 2, pp 334–348 | Cite as

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

  • W. J. Jiao
  • X. W. Chen
Research Paper


The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod (tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically. By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretical solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical (exact) solution, and the first-order perturbation solution obtained by Walters et al. (Int. J. Impact Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity, approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical (exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical (exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.


Long-rod penetration Alekseevskii–Tate model Theoretical solution Approximate solution Perturbation solution 



The project was supported by the National Outstanding Young Scientist Foundation of China (Grant 11225213) and the Key Subject “Computational Solid Mechanics” of China Academy of Engineering Physics.


  1. 1.
    Allen, W.A., Rogers, J.W.: Penetration of a rod into a semi-infinite target. J. Franklin Inst. 272, 275–284 (1961)CrossRefGoogle Scholar
  2. 2.
    Alekseevskii, V.P.: Penetration of a rod into a target at high velocity. Combust. Explos. Shock Waves 2, 63–66 (1966)CrossRefGoogle Scholar
  3. 3.
    Tate, A.: A theory for the deceleration of long rods after impact. J. Mech. Phys. Solids 15, 387–399 (1967)CrossRefGoogle Scholar
  4. 4.
    Tate, A.: Further results in the theory of long rods penetration. J. Mech. Phys. Solids 17, 141–150 (1969)CrossRefGoogle Scholar
  5. 5.
    Christman, D.R., Gehring, J.W.: Analysis of high-velocity projectile penetration mechanics. J. Appl. Phys. 37, 1579–1587 (1966)CrossRefGoogle Scholar
  6. 6.
    Hohler, V., Stilp, A.J.: Hypervelocity impact of rod projectiles with L/D from 1 to 32. Int. J. Impact Eng. 5, 323–331 (1987)CrossRefGoogle Scholar
  7. 7.
    Rosenberg, Z., Dezel, E.: The relation between the penetration capability of long rods and their length to diameter ratio. Int. J. Impact Eng. 15, 125–129 (1994)CrossRefGoogle Scholar
  8. 8.
    Anderson, C.E., Walker, J.D., Bless, S.P., et al.: On the L/D effect for long-rod penetrators. Int. J. Impact Eng. 18, 247–264 (1996)CrossRefGoogle Scholar
  9. 9.
    Anderson, C.E., Walker, J.D.: An examination of long-rod penetration. Int. J. Impact Eng. 11, 481–501 (1991)CrossRefGoogle Scholar
  10. 10.
    Walker, J.D., Anderson, C.E.: A time-dependent model for long-rod penetration. Int. J. Impact Eng. 16, 19–48 (1995)CrossRefGoogle Scholar
  11. 11.
    Rosenberg, Z., Marmor, E., Mayseless, M.: On the hydrodynamic theory of long-rod penetration. Int. J. Impact Eng. 10, 483–486 (1990)CrossRefGoogle Scholar
  12. 12.
    Zhang, L.S., Huang, F.L.: Model for long-rod penetration into semi-infinite targets. J. Beijing Inst. Technol. 13, 285–289 (2004)Google Scholar
  13. 13.
    Rosenberg, Z., Dezel, E.: Further examination of long-rod penetration: the role of penetrator strength at hypervelocity impacts. Int. J. Impact Eng. 24, 85–102 (2000)CrossRefGoogle Scholar
  14. 14.
    Walters, W.P., Segletes, S.B.: An exact solution of the long-rod penetration equations. Int. J. Impact Eng. 11, 225–231 (1991)CrossRefGoogle Scholar
  15. 15.
    Segletes, S.B., Walters, W.P.: Extensions to the exact solution of the long-rod penetration/erosion equations. Int. J. Impact Eng. 28, 363–376 (2003)CrossRefGoogle Scholar
  16. 16.
    Forrestal, M.J., Piekutowski, A.J., Luk, V.K.: Long-rod penetration into simulated geological targets at an impact velocity of 3.0 km/s. In: 11th International symposium on ballistics, vol. 2. Brussels, Belgium (1989)Google Scholar
  17. 17.
    Walters, W., Williams, C., Normandia, M.: An explicit solution of the Alekseevskii–Tate penetration equations. Int. J. Impact Eng. 33, 837–846 (2006)CrossRefGoogle Scholar
  18. 18.
    Orphal, D.L., Anderson, C.E.: The dependence of penetration velocity on impact velocity. Int. J. Impact Eng. 33, 546–554 (2006)CrossRefGoogle Scholar
  19. 19.
    Orphal, D.L., Franzen, R.R.: Penetration of confined silicon carbide targets by tungsten long rods at impact velocities from 1.5 to 4.6 km/s. Int. J. Impact Eng. 19, 1–13 (1997)CrossRefGoogle Scholar
  20. 20.
    Orphal, D.L., Franzen, R.R., Charters, A.C., et al.: Penetration of confined boron carbide targets by tungsten long rods at impact velocities from 1.5 to 5.0 km/s. Int. J. Impact Eng. 19, 15–29 (1997)CrossRefGoogle Scholar
  21. 21.
    Sternberg, J., Orphal, D.L.: A note on the high velocity penetration of aluminum nitride. Int. J. Impact Eng. 19, 647–651 (1997)CrossRefGoogle Scholar
  22. 22.
    Anderson, C.E., Riegel, J.P.: A penetration model for metallic targets based on experimental data. Int. J. Impact Eng. 80, 24–35 (2015)CrossRefGoogle Scholar
  23. 23.
    Rosenberg, Z., Dezel, E.: Terminal Ballistics. Springer, Berlin (2012)CrossRefGoogle Scholar
  24. 24.
    Anderson, C.E., Littlefield, D.L., Walker, J.D.: Long-rod penetration, target resistance, and hypervelocity impact. Int. J. Impact Eng. 14, 1–12 (1993)CrossRefGoogle Scholar
  25. 25.
    Anderson, C.E., Orphal, D.L.: Analysis of the terminal phase of penetration. Int. J. Impact Eng. 29, 69–80 (2003)CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of Modern MechanicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Institute of Systems EngineeringChina Academy of Engineering PhysicsMianyangChina
  3. 3.Advanced Research Institute for Multidisciplinary ScienceBeijing Institute of TechnologyBeijingChina
  4. 4.The State Key Lab of Explosion Science and TechnologyBeijing Institute of TechnologyBeijingChina

Personalised recommendations