A Theoretical Model for the Cross-sectional Area of a Projectile Considering Deformation and Erosion Coupling

doi:10.12382/bgxb.2024.0547

Abstract

Abstract:

When a projectile penetrates a concrete target at medium-highspeed, the high pressure between projectile and target induces severe deformation of the projectile. Concurrently, the intense friction results in the erosion of the projectile’s mass, further resulting in a significant decrease in penetration performance. To delve into the penetration behavior of projectile under medium-highspeed conditions, the critical velocities corresponding to the transition in penetration modes are determined using the cavity expansion theory and the Alekseevskii-Tate model. Subsequently, considering the coupled effects of deformation and mass erosion during medium-highspeed penetration, a computational model for the cross-sectional area of projectileunder these conditions is constructed. To verify the rationality and reliability of the theoretical model, the calculated results are compared with the experimental data in Ref.[6, 11]. The results indicate that the theoretical model’s predicted results of critical penetration velocities are in good agreement with the experimental data. Under the combined effects of deformation and erosion during medium-highspeed penetration, the cross-sectional area of the recovered projectile increases approximately exponentially with the initial impact velocity.The deforming effect causes theradial upsetting of projectile, while the erosion effect gradually strips away the surface material of projectile.

Key words: medium-highspeed penetration, deformation, erosion, critical penetration velocity, cross-sectional area

CLC Number:

O 385

XU Hengwei, LU Yonggang, LI Junrun, FENG Xiaowei, LU Zhengcao. A Theoretical Model for the Cross-sectional Area of a Projectile Considering Deformation and Erosion Coupling[J]. Acta Armamentarii, 2024, 45(S1): 97-104.

Figures/Tables 8

Fig.1 Penetration modes at different velocities

Fig.2 Deformation/erosion coupling state of projectile

Fig.3 Stress state of the head of ogival-nosed projectile

Table 1 Material parameters of experimental projectiles

案例	弹体
案例	M₀/g	L₀/mm	$E p t$ /MPa	Y_p/GPa	ρ_p/(kg·m^-3)	f_c/MPa
1^[6]	6.58	30	700	1.1	7800	50
2^[11]	38.8	80	700	0.9	7800	45.4
3^[22]	68.5	96	1.2	1.45	7800	42.8
4^[22]	322.5	90	1.2	1.95	7800	42.8

Table 1 Material parameters of experimental projectiles

案例	弹体
案例	M₀/g	L₀/mm	$E p t$ /MPa	Y_p/GPa	ρ_p/(kg·m^-3)	f_c/MPa
1^[6]	6.58	30	700	1.1	7800	50
2^[11]	38.8	80	700	0.9	7800	45.4
3^[22]	68.5	96	1.2	1.45	7800	42.8
4^[22]	322.5	90	1.2	1.95	7800	42.8

Table 2 Material parameters of experimental targets

案例	靶体
案例	ρ_t/(kg·m^-3)	f_c/MPa	a₀/(10⁸Pa)	a₁/(10⁶kg·m^-2·s^-1)	a₂/(10³kg·m^-3)
1^[6]	6.58	30	700	1.1	50
2^[11]	38.8	80	700	0.9	45.4
3^[22]	68.5	96	1.2	1.45	42.8
4^[22]	322.5	90	1.2	1.95	42.8

Fig.4 Comparison among the calculated results and experimental results of critical velocity under different conditions

Fig.5 Comparison of predictions of dimensionless cross-sectional area by different models with experimental data

Fig.6 The effects of different projectile strengths and plastic wave velocities on non-dimensional cross-sectional area K

References 27

[3]	高飞, 张国凯, 纪玉国, 等. 卵形弹体超高速侵彻砂浆靶的响应特性[J]. 兵工学报, 2020, 41(10): 1979-1987.
	GAO F, ZHANG G K, JI Y G, et al. Response characteristics of hypervelocity ogive-nose projectile penetrating into mortar target[J]. Acta Armamentarii, 2020, 41(10): 1979-1987. (in Chinese) doi: 10.3969/j.issn.1000-1093.2020.10.007
[4]	ZHANG Y D, LU Z C, WEN H M. On The penetration of semi-infinite concrete targets by ogival-nosed projectiles at different velocities[J]. International Journal of Impact Engineering, 2019, 129: 128-140.
[5]	WEN H M, XIAN Y X. A unified approach for concrete impact[J]. International Journal of Impact Engineering, 2015, 77: 84-96.
[6]	KONG X Z, WU H, FANG Q, et al. Projectile penetration into mortar targets with a broad range of striking velocities: test and analyses[J]. International Journal of Impact Engineering, 2017, 106:18-29.
[7]	ALEKSEEVSKII V P. Penetration of a rod into a target at high velocity[J]. Combustion, Explosion, and Shock Waves, 1996, 2(2):63-66.
[8]	TATE A. A theory for the deceleration of long rods after impact[J]. Journal of The Mechanicsand Physics of Solids, 1967, 15(6):387-399.
[9]	高光发. 长杆弹侵彻半无限靶板流体动力学理论[J]. 科技导报, 2016, 34(2): 287-298. doi: 10.3981/j.issn.1000-7857.2016.2.049
	GAO G F. Hydro-dynamics theory for long-rod projectile penetrating semi-infinite target: reviews and studies[J]. Science and Technology Review, 2016, 34(2): 287-298. (in Chinese)
[10]	刘闯, 张先锋, 黄长强, 等. 半球头长杆弹高速侵彻半无限厚靶临界速度理论模型[J]. 振动与冲击, 2019, 38(9):8-14.
	LIU C, ZHANG X F, HUANG C Q, et al. Critical velocity theoretical model for hemispherical long rod projectiles’penetrating semi-infinite thick target at high velocity[J]. Journal of Vibration and Shock, 2019, 38(9):8-14. (in Chinese)
[11]	LIU C, ZHANG X F, CHEN H, et al. Experimental and theoretical study on steel long-rod projectile penetration into concrete targets with elevated impact velocities[J]. International Journal of Impact Engineering, 2019, 138:103482.
[12]	卢正操, 张元迪, 文鹤鸣, 等. 长杆弹侵彻半无限混凝土靶的理论研究[J]. 现代应用物理, 2018, 9(4):17-28.
	LU Z C, ZHANG Y D, WEN H M, et al. Theoretical study on the penetration of long rods into semi-infinite concrete targets[J]. Modern Applied Physics, 2018, 9(4):17-28. (in Chinese)
[13]	欧阳昊, 陈小伟. 混凝土骨料对高速侵彻弹体质量侵蚀的影响分析[J]. 爆炸与冲击, 2019, 39(7): 64-73.
	OUYANG H, CHEN X W. Analysis of mass abrasion of high-speed penetrator influenced by aggregate in concrete target[J]. Explosion and Shock Waves, 2019, 39(7): 64-73. (in Chinese)
[14]	刘均伟, 张先锋, 刘闯, 等. 考虑摩擦因数变化的弹体高速侵彻混凝土质量侵蚀模型研究[J]. 爆炸与冲击, 2021, 41(8): 114-124.
	LIU J W, ZHANG X F, LIU C, et al. Study on mass erosion model of projectile penetrating concrete at high speed considering variation of friction coefficient[J]. Explosion and Shock Waves, 2021, 41(8): 114-124. (in Chinese)
[15]	CHEN X W, LI Q M. Deep penetration of a non-deformable projectile with different geometrical characteristics[J]. International Journal of Impact Engineering, 2002, 27:619-637.
[16]	TATE A. A possible explanation for the hydrodynamic transition in high-speed impact[J]. International journal of mechanical sciences, 1997, 19(2):121-123.
[17]	ALEKSEEVSKII V P. Penetration of arod into atarget at high velocity[J]. Combustion, Explosion, and Shock Waves, 1966, 2(2): 63-66.
[18]	ZHANG L S, HUANG F L. Model for long-rod penetration into semi-infinite targets[J]. Journal of Beijing Institute of Technology, 2004, 13(3): 285-289.
[1]	姚志彦, 李金柱, 齐凯丽, 等. 长杆弹超高速侵彻砂浆靶临界速度的实验和计算[J]. 兵工学报, 2022, 43(7): 1578-1588.
	YAO Z Y, LI J Z, QI K L, et al. Experiment and calculation of critical velocity of long-rod projectile penetrating mortar target at hypervelocity[J]. Acta Armamentarii, 2022, 43(7): 1578-1588. (in Chinese) doi: 10.12382/bgxb.2021.0403
[2]	宋春明, 李干, 王明洋, 等. 不同速度段弹体侵彻岩石靶体的理论分析[J]. 爆炸与冲击, 2018, 38(2):250-257.
	SONG C M, LI G, WANG M Y, et al. Theoretical analysis of projectiles penetrating into rock targets at different velocities[J]. Explosion and Shock Waves, 2018, 38(2):250-257. (in Chinese)
[19]	KONG X Z, WU H, FANG Q, et al. Rigid and eroding projectile penetration into concrete targets based on an extended dynamic cavity expansion model[J]. International Journal of Impact Engineering, 2017, 100: 13-22.
[20]	TATE A. Long rod penetration models-Part Ⅱ. Extensions to the hydrodynamic theory of penetration[J]. International Journal of Mechanical Sciences, 1986, 28(9): 599-612.
[21]	WEN H M, LAN B. Analytical models for the penetration of semi-infinite targets by rigid, deformable and erosive long rods[J]. Acta Mechanica Sinica, 2010, 26(4): 573-583.
[22]	郭磊, 何勇, 潘绪超, 等. 高速侵彻混凝土弹体侵蚀效应试验研究[J]. 实验力学, 2020, 35(1): 82-90.
	GUO L, H E Y, PAN X C, et al. Experimental study on mass loss of projectile subjected to high-velocity penetration into concrete target[J]. Journal of Experimental Mechanics, 2020, 35(1): 82-90. (in Chinese)
[23]	MU Z C, ZHANG W. An investigation on mass loss of ogival projectiles penetrating concrete targets[J]. International Journal of Impact Engineering, 2011, 38(8/9): 770-778.
[24]	SILLING S A, FORRESTAL M J. Mass loss from abrasion on ogive-nose steel projectiles that penetrate concrete targets[J]. International Journal of Impact Engineering, 2007, 34:1814-1820
[25]	WU H J, HUANG F L, WANG Y N, et al. Mass loss and nose shape change on ogive-nose steel projectiles during concrete penetration[J]. International Journal of Nonlinear Sciences and Numerical Simulation, 2012, 13(3/4): 273-280.
[26]	李干, 宋春明, 邱艳宇, 等. 超高速弹对花岗岩侵彻深度逆减现象的理论与实验研究[J]. 岩石力学与工程学报, 2018, 37(1):60-66.
	LI G, SONG C M, QIU Y Y, et al. Theoretical and experimental studies on the phenomenon of reduction in penetration depth of hyper-velocity projectiles into granite[J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(1):60-66. (in Chinese)
[27]	JIAO W J, CHEN X W. Approximate solutions of the Alekseevskii-Tate model of long-rod penetration. Acta Mechanica Sinica, 2018, 34(2): 334-348.