Abstract
In this study, the penetration behavior of a cone-shaped projectile into granular particles was analyzed using simulations based on the discrete element method (DEM). The rate-independent friction force and inertial drag force proportional to the squared projectile velocity are the principal force terms that interact between the projectile and the particles. Simulation results show that the friction force and inertial drag force follow the power law with respect to penetration depth and have changing tendencies before and after the complete penetration of the projectile into particles. Based on the results, a mathematical model is proposed to simplify the force terms using the penetration depth, projectile tip angle, and projectile length. The simplified force terms are physically explained using changes in the projectile–particle contact area and the fluidization of particles during dynamic collisions. Experiments were conducted using steel projectiles and ABS plastic beads to verify the accuracy of the mathematical model for real-life cases. The results of this study validate the proposed mathematical model of the rate-independent friction force and inertial drag force regarding the cone-shaped projectile behavior during penetration into granular particles.
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References
Horabik J, Molenda M (2016) Parameters and contact models for DEM simulations of agricultural granular materials: A review. Biosys Eng 147:206–225. https://doi.org/10.1016/j.biosystemseng.2016.02.017
Lim NH, Kim KJ, Bae HU, Kim S (2020) DEM analysis of track ballast for track ballast-wheel interaction simulation. Appl Sci 10(8):2717. https://doi.org/10.3390/app10082717
Zhou L, Gao J, Cheng P, Hu C (2020) Study on track-soil traction using discrete element method simulation and soil bin test. AIP Adv 10(7):075307. https://doi.org/10.1063/5.0016448
Mudarisov S, Farkhutdinov I, Khamaletdinov R, Khasanov E, Mukhametdinov A (2022) Evaluation of the significance of the contact model particle parameters in the modelling of wet soils by the discrete element method. Soil and Tillage Res 215:105228. https://doi.org/10.1016/j.still.2021.105228
Nordstrom KN, Lim E, Harrington M, Losert W (2014) Granular dynamics during impact. Phys Rev Lett 112(22):228002. https://doi.org/10.1103/PhysRevLett.112.228002
Xu Y, Padding JT, Kuipers JAM (2014) Numerical investigation of the vertical plunging force of a spherical intruder into a prefluidized granular bed. Phys Rev E 90(6):062203. https://doi.org/10.1103/PhysRevE.90.062203
Zaidi AA, Müller C (2017) Vertical drag force acting on intruders of different shapes in granular media. In: EPJ Web of Conferences. Vol.140. EDP Sciences. pp 02011. https://doi.org/10.1051/epjconf/201714002011
Roth LK (2021) Constant speed penetration into granular materials: drag forces from the quasistatic to inertial regime. Granular Matter 23(3):1–17. https://doi.org/10.1007/s10035-021-01106-5
Feng Y, Blumenfeld R, Liu C (2019) Support of modified Archimedes’ law theory in granular media. Soft Matter 15(14):3008–3017. https://doi.org/10.1039/C8SM02480D
Cheng B, Yu Y, Baoyin H (2018) Collision-based understanding of the force law in granular impact dynamics. Phys Rev E 98(1):012901. https://doi.org/10.1103/PhysRevE.98.012901
Hou M, Peng Z, Liu R, Lu K, Chan CK (2005) Dynamics of a projectile penetrating in granular systems. Phys Rev E 72(6):062301. https://doi.org/10.1103/PhysRevE.72.062301
Ogawa K, Takeda S, Kobayashi H (2015) Dynamic simulations of projectile penetration into granular medium. Mech Eng J 2(1):14–00427. https://doi.org/10.1299/mej.14-00427
Zaidi AA (2018) Study of particle inertia effects on drag force of finite sized particles in settling process. Chem Eng Res Des 132:714–728. https://doi.org/10.1016/j.cherd.2018.02.013
Chian SC, Tan BCV, Sarma A (2017) Reprint of: Projectile penetration into sand: Relative density of sand and projectile nose shape and mass. Int J Impact Eng 105:80–88. https://doi.org/10.1016/j.ijimpeng.2017.03.026
Cheng B, Yu Y, Baoyin H (2017) Asteroid surface impact sampling: dependence of the cavity morphology and collected mass on projectile shape. Sci Rep 7(1):1–10. https://doi.org/10.1038/s41598-017-10681-8
Newhall KA, Durian DJ (2003) Projectile-shape dependence of impact craters in loose granular media. Phys Rev E 68(6):060301. https://doi.org/10.1103/PhysRevE.68.060301
Zaidi AA (2020) Granular drag force during immersion in dry quicksand. Powder Technol 364:986–993. https://doi.org/10.1016/j.powtec.2019.10.048
Nouguier-Lehon C, Vincens E, Cambou B (2005) Structural changes in granular materials: the case of irregular polygonal particles. Int J Solids Struct 42(24–25):6356–6375. https://doi.org/10.1016/j.ijsolstr.2005.04.021
Uehara JS, Ambroso MA, Ojha RP, Durian DJ (2003) Low-speed impact craters in loose granular media. Phys Rev Lett 90(19):194301. https://doi.org/10.1103/PhysRevLett.90.194301
Seguin A, Bertho Y, Gondret P (2008) Influence of confinement on granular penetration by impact. Phys Rev E 78(1):010301. https://doi.org/10.1103/PhysRevE.78.010301
Goldman DI, Umbanhowar P (2008) Scaling and dynamics of sphere and disk impact into granular media. Phys Rev E 77(2):021308. https://doi.org/10.1103/PhysRevE.77.021308
Hou M, Peng Z, Liu R, Lu K, Chan CK (2005) Dynamics of a projectile penetrating in granular systems. Phys Rev E 72(6):062301. https://doi.org/10.1103/PhysRevE.72.062301
Ambroso MA, Kamien RD, Durian DJ (2005) Dynamics of shallow impact cratering. Phys Rev E 72(4):041305. https://doi.org/10.1103/PhysRevE.72.041305
Katsuragi H, Durian DJ (2013) Drag force scaling for penetration into granular media. Phys Rev E 87(5):052208. https://doi.org/10.1103/PhysRevE.87.052208
Katsuragi H, Durian DJ (2007) Unified force law for granular impact cratering. Nat Phys 3(6):420–423. https://doi.org/10.1038/nphys583
Shen W, Zhao T, Crosta GB, Dai F, Dattola G (2022) Influence of inter-particle friction and damping on the dynamics of spherical projectile impacting onto a soil bed. Front Earth Sci 10:835271. https://doi.org/10.3389/feart.2022.835271
Guo J (2018) Exact solution for depth of impact crater into granular bed. J Eng Mech 144(1):06017018. https://doi.org/10.1061
Tiwari M, Mohan TK, Sen S (2014) Drag-force regimes in granular impact. Phys Rev E 90(6):062202. https://doi.org/10.1103/PhysRevE.90.062202
Tsimring LS, Volfson D (2005) Modeling of impact cratering in granular media. Powders and grains 2:1215–1223
Brzinski TA III, Mayor P, Durian DJ (2013) Depth-dependent resistance of granular media to vertical penetration. Phys Rev Lett 111(16):168002. https://doi.org/10.1103/PhysRevLett.111.168002
Hill G, Yeung S, Koehler SA (2005) Scaling vertical drag forces in granular media. EPL (Europhysics Letters) 72(1):137. https://doi.org/10.1209/epl/i2005-10203-3
Clark AH, Behringer RP (2013) Granular impact model as an energy-depth relation. EPL (Europhysics Letters) 101(6):64001. https://doi.org/10.1209/0295-5075/101/64001
Bester CS, Behringer RP (2017) Collisional model of energy dissipation in three-dimensional granular impact. Phys Rev E 95(3):032906. https://doi.org/10.1103/PhysRevE.95.032906
Mishra BK, Rajamani RK (1992) The discrete element method for the simulation of ball mills. Appl Math Model 16(11):598–604. https://doi.org/10.1016/0307-904X(92)90035-2
Wu CY, Cocks AC, Gillia OT, Thompson DA (2003) Experimental and numerical investigations of powder transfer. Powder Technol 138(2–3):216–228. https://doi.org/10.1016/j.powtec.2003.09.011
Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65. https://doi.org/10.1680/geot.1979.29.1.47
Ghaboussi J, Barbosa R (1990) Three-dimensional discrete element method for granular materials. Int J Numer Anal Meth Geomech 14(7):451–472. https://doi.org/10.1002/nag.1610140702
Albert R, Pfeifer MA, Barabási AL, Schiffer P (1999) Slow drag in a granular medium. Phys Rev Lett 82(1):205. https://doi.org/10.1103/PhysRevLett.82.205
Van Der Meer D (2017) Impact on granular beds. Annu Rev Fluid Mech 49:463–484. https://doi.org/10.1146/annurev-fluid-010816-060213
Kang W, Feng Y, Liu C, Blumenfeld R (2018) Archimedes’ law explains penetration of solids into granular media. Nat Commun 9(1):1–9. https://doi.org/10.1038/s41467-018-03344-3
Moghisi M, Squire PT (1981) An experimental investigation of the initial force of impact on a sphere striking a liquid surface. J Fluid Mech 108:133–146. https://doi.org/10.1017/S0022112081002036
Acknowledgements
This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No.2018R1A5A7025522).
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Lee, H.M., Kim, T.H. & Yoon, G.H. Analysis of cone-shaped projectile behavior during penetration into granular particles using the discrete element method. Comp. Part. Mech. 11, 689–703 (2024). https://doi.org/10.1007/s40571-023-00647-1
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DOI: https://doi.org/10.1007/s40571-023-00647-1