Skip to main content
SpringerLink
Log in

Projectile oblique impact on granular media: penetration depth and dynamic process

  • Original Paper
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

The motion of a projectile oblique impacting sand bed with deep penetration was captured in experiment. Penetration depth, velocity and acceleration of the projectile were analyzed in terms of kinematic data. Horizontal and vertical penetration depths were measured under a series of impact angles, impact velocities and projectile diameters, and the collapsed curves were obtained. Specifically, an impact drag force model was assumed, which consists of an inertial term, a linear viscosity term, a linear static term, and a constant term. Then, we confirmed the terms related with velocities for both horizontal and vertical direction at the fixed depth-dependent term and obtained the linear depth-dependent static force at certain velocities. It demonstrated that viscous drag force is significant in deep penetration cases (vertical penetration depths are larger than the diameter of projectiles), but it can be neglected in shallow penetration cases. The generalized phenomenological resistance force model were proposed, which can predict the motion of projectile and resistance force exerted on the projectile by the granular medium at oblique impact cases.

Graphic abstract

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. van der Meer, D.: Impact on granular beds. Annu. Rev. Fluid Mech. 49, 463–484 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  2. Tiwari, M., Mohan, T.K., Sen, S.: Drag-force regimes in granular impact. Phys. Rev. E 90(6), 062202 (2014)

    Article  ADS  Google Scholar 

  3. de Bruyn, J.R., Walsh, A.M.: Penetration of spheres into loose granular media. Can. J. Phys. 82, 439 (2004)

    Article  ADS  Google Scholar 

  4. Goldman, D.I., Umbanhowar, P.: Scaling and dynamics of sphere and disk impact into granular media. Phys. Rev. E 77, 021308 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ambroso, M.A., Kamien, R.D., Durian, D.J.: Dynamics of shallow impact cratering. Phys. Rev. E 72, 041305 (2005)

    Article  ADS  Google Scholar 

  6. Katsuragi, H., Durian, D.J.: Unified force law for granular impact cratering. Nat. Phys. 3, 420 (2007)

    Article  Google Scholar 

  7. Katsuragi, H., Durian, D.J.: Drag force scaling for penetration into granular media. Phys. Rev. E 87(5), 052208 (2013)

    Article  ADS  Google Scholar 

  8. Nordstrom, K.N., Lim, E., Harrington, M., Losert, W.: Granular dynamics during impact. Phys. Rev. Lett. 112(22), 228002 (2014)

    Article  ADS  Google Scholar 

  9. Joubaud, S., Homan, T., Gasteuil, Y., Lohse, D., van der Meer, D.: Forces encountered by a sphere during impact into sand. Phys. Rev. E 90(6), 060201 (2014)

    Article  ADS  Google Scholar 

  10. Clark, A.H., Petersen, A.J., Behringer, R.P.: Collisional model for granular impact dynamics. Phys. Rev. E 89(1), 012201 (2014)

    Article  ADS  Google Scholar 

  11. Uehara, J.S., Ambroso, M.A., Ojha, R.P., Durian, D.J.: Low-speed impact craters in loose granular media. Phys. Rev. Lett. 90, 194301 (2003)

    Article  ADS  Google Scholar 

  12. Walsh, A.M., Holloway, K.E., Habdas, P., de Bruyn, J.R.: Morphology and scaling of impact craters in granular media. Phys. Rev. Lett. 91, 104301 (2003)

    Article  ADS  Google Scholar 

  13. Newhall, K.A., Durian, D.J.: Projectile-shape dependence of impact craters in loose granular media. Phys. Rev. E 68, 060301 (2003)

    Article  ADS  Google Scholar 

  14. Boudet, J.F., Amarouchene, Y., Hellay, H.: Dynamics of impact cratering in shallow sand layers. Phys. Rev. Lett 96, 158001 (2006)

    Article  ADS  Google Scholar 

  15. de Vet, S.J., de Bruyn, J.R.: Shape of impact craters in granular media. Phys. Rev. E 76, 041306 (2007)

    Article  ADS  Google Scholar 

  16. Tsimring, L.S., Volfson, D.: Modeling of impact cratering in granular media. Powders Grains 2, 1215–1223 (2005)

    Google Scholar 

  17. Clark, A.H., Kondic, L., Behringer, R.P.: Particle scale dynamics in granular impact. Phys. Rev. Lett. 109(23), 238302 (2012)

    Article  ADS  Google Scholar 

  18. Hinch, J.: Particles impacting on a granular bed. J. Eng. Math. 84(1), 41–48 (2014)

    Article  MathSciNet  Google Scholar 

  19. Clark, A.H., Behringer, R.P.: Granular impact model as an energy-depth relation. Europhys. Lett. 101(6), 64001 (2013)

    Article  ADS  Google Scholar 

  20. Seguin, A., Bertho, Y., Gondet, P.: Influence of confinement on granular penetration by impact. Phys. Rev. E 78, 010301 (2008)

    Article  ADS  Google Scholar 

  21. Kang, W., Feng, Y., Liu, C., Blumenfeld, R.: Archimedes’ law explains penetration of solids into granular media. Nat. Commun. 9, 1101 (2018)

    Article  ADS  Google Scholar 

  22. Hou, M., Peng, Z., Liu, R., Lu, K., Chan, C.K.: Dynamics of a projectile penetrating in granular systems. Phys. Rev. E 72, 062301 (2005)

    Article  ADS  Google Scholar 

  23. Ruiz-Suárez, J.C.: Penetration of projectiles into granular targets. Rep. Prog. Phys. 76(6), 066601 (2013)

    Article  ADS  Google Scholar 

  24. Zheng, X.J.: Mechanics of wind-blown sand movements. Springer Science & Business Media (2009)

    Book  Google Scholar 

  25. Nishida, M., Okumura, M., Tanaka, K.: Effects of density ratio and diameter ratio on critical incident angles of projectiles impacting granular media. Granul. Matter 12, 337–344 (2010)

    Article  Google Scholar 

  26. Nishida, M., Nagamatsu, J., Tanaka, K.: Discrete element method analysis of ejection and penetration of projectile impacting granular media. J. Solid Mech. Mater. Eng. 5, 164–178 (2011)

    Article  Google Scholar 

  27. Ye, X.Y., Wang, D.M., Zheng, X.J.: Criticality of post-impact motions of a projectile obliquely impacting a granular medium. Powder Technol. 301, 1044–1053 (2016)

    Article  Google Scholar 

  28. Zheng, X.J., Wang, Z.T., Qiu, Z.G.: Impact craters in loose granular media. Eur. Phys. J. E 13, 321–324 (2004)

    Article  Google Scholar 

  29. Wang, D.M., Zheng, X.J.: Experimental study of morphology scaling of a projectile obliquely impacting into loose granular media. Granul. Matter 15(6), 725–734 (2013)

    Article  Google Scholar 

  30. Takizawa, S., Katsuragi, H.: Scaling laws for the oblique impact cratering on an inclined granular surface. Icarus 335, 113409 (2020)

    Article  Google Scholar 

  31. Ye, X.Y., Wang, D.M., Zheng, X.J.: Influence of particle rotation on the oblique penetration in granular media. Phys. Rev. E 86, 061304 (2012)

    Article  ADS  Google Scholar 

  32. Wang, D.M., Ye, X.Y., Zheng, X.J.: The scaling and dynamics of a projectile obliquely impacting a granular medium. Eur. Phys. J. E 35, 7–19 (2012)

    Article  Google Scholar 

  33. Lee, S., Marghitu, D.B.: Analysis of a rigid body obliquely impacting granular matter. Nonlinear Dyn. 57, 289–301 (2009)

    Article  Google Scholar 

  34. Holsapple, K.A.: The scaling of impact processes in planetary sciences. Annu. Rev. Earth Planet. Sci. 21, 333 (1993)

    Article  ADS  Google Scholar 

  35. Umbanhowar, P., Goldman, D.I.: Granular impact and the critical packing state. Phys. Rev. E 82, 010301(R) (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support for this work from National Natural Science Foundation of China (Nos. 11490553, 11572144, 11502104, 11872028, 11872029), Ministry of Education, the Fundamental Research Funds for the Central Universities (lzujbky-2015-177, lzujbky-2017-168, lzujbky-2018-124), the Natural Science Foundation of Gansu Province (No. 17JR5RA206).

Author information

Authors and Affiliations

  1. Key Laboratory of Mechanics On Disaster and Environment in Western China Attached To the Ministry of Education of China, and Department of Mechanics and Engineering Science, School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Xiaoyan Ye, Dengming Wang, Xingyi Zhang, Chaofeng Zhang, Wei Du & Xiyang Su

  2. Science and Technology On Vacuum Technology and Physics Laboratory, Lanzhou Institute of Physics, Lanzhou, 730000, Gansu, People’s Republic of China

    Gang Li

Authors
  1. Xiaoyan Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dengming Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xingyi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chaofeng Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wei Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Xiyang Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Gang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaoyan Ye.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

  1. (a)

    \(\beta_{j1} {, }\, \beta_{j2} , \, \beta_{j3}\) are fitting parameters are functions of \(\theta\), \(D_{b}\) and \(V_{0}\)

    $$\beta_{x1} = 1724D_{b}^{0.55} \theta^{ - 1.3} V_{0}^{0.65}$$
    $$\beta_{x2} = 1816D_{b}^{0.55} \theta^{ - 1.3} V_{0}^{0.65}$$
    $$\beta_{x3} = 2.76D_{b}^{0.4} \theta^{ - 0.73} V_{0}^{ - 0.6}$$
    $$\beta_{y1} = 8.5D_{b}^{0.4} \theta^{ - 0.1} V_{0}^{0.65}$$
    $$\beta_{y2} = 8.85D_{b}^{0.4} \theta^{ - 0.1} V_{0}^{0.65}$$
    $$\beta_{y3} = 4.2D_{b}^{0.6} \theta^{ - 0.8} V_{0}^{ - 0.5}$$
  2. (b)

    The fitted parameters \(\lambda_{x2}\), \(\lambda_{y2}\), \(F_{0x}\) and \(F_{0y}\) that are functions of \(\theta\), \(D_{b}\).

    $$\lambda_{x2} /m = - 0.00016\theta^{2.7} D_{b}^{ - 1/3}$$
    $$F_{0x} = - \theta^{13} D_{b}^{13}$$
    $$\lambda_{y2} /m = - 0.05\theta^{1.6} D_{b}^{ - 1/33}$$
    $$F_{0y} = - \theta^{ - 2.4} D_{b}^{ - 3}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ye, X., Wang, D., Zhang, X. et al. Projectile oblique impact on granular media: penetration depth and dynamic process. Granular Matter 23, 48 (2021). https://doi.org/10.1007/s10035-021-01108-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-021-01108-3

Keywords

Search

Navigation