TY - JOUR

T1 - Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

AU - Jiao, W. J.

AU - Chen, X. W.

N1 - Publisher Copyright:
© 2017, The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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.

AB - 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.

KW - Alekseevskii–Tate model

KW - Approximate solution

KW - Long-rod penetration

KW - Perturbation solution

KW - Theoretical solution

UR - http://www.scopus.com/inward/record.url?scp=85019674192&partnerID=8YFLogxK

U2 - 10.1007/s10409-017-0672-9

DO - 10.1007/s10409-017-0672-9

M3 - Article

AN - SCOPUS:85019674192

SN - 0567-7718

VL - 34

SP - 334

EP - 348

JO - Acta Mechanica Sinica/Lixue Xuebao

JF - Acta Mechanica Sinica/Lixue Xuebao

IS - 2

ER -