Analysis of the size effect on the penetration depth of earth-penetrating projectiles and practical calculating formula

HE Yong; XU Tianhan; ZHANG Xiaohan; SUI Yaguang; XING Haozhe

doi:10.11883/bzycj-2024-0248

Article Contents

Article Navigation > Explosion And Shock Waves > 2024 >

HE Yong, XU Tianhan, ZHANG Xiaohan, SUI Yaguang, XING Haozhe. Analysis of the size effect on the penetration depth of earth-penetrating projectiles and practical calculating formula[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2024-0248

Citation:

HE Yong, XU Tianhan, ZHANG Xiaohan, SUI Yaguang, XING Haozhe. Analysis of the size effect on the penetration depth of earth-penetrating projectiles and practical calculating formula[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2024-0248

Citation:

HE Yong, XU Tianhan, ZHANG Xiaohan, SUI Yaguang, XING Haozhe. Analysis of the size effect on the penetration depth of earth-penetrating projectiles and practical calculating formula[J]. Explosion And Shock Waves. doi: 10.11883/bzycj-2024-0248

PDF( 1327 KB)

Analysis of the size effect on the penetration depth of earth-penetrating projectiles and practical calculating formula

doi: 10.11883/bzycj-2024-0248

1.
Institute of Engineering Safety Protection Technology, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
2.
State Key Laboratory of Explosion and Impact and Disaster Prevention and Mitigation, Army Engineering University of PLA, Nanjing 210007, Jiangsu, China
3.
Northwest Institute of Nuclear Technology, Xi’an 710024, Shaanxi, China

Received Date: 2024-07-22
Rev Recd Date: 2024-10-15

Available Online: 2024-11-05

Abstract

Abstract

The penetration depth of the earth-penetrating projectile is a basic problem in the design of protection engineering. Scaled testing is an important method to study the penetration law. The size effect between the model test results and the prototype is a problem that must be solved to establish the calculation method of penetration using scaled tests. In this study, the stress and strain state evolution of the rock-like target medium subjected to the penetration of earth-penetrating projectiles and the penetration resistance function of the projectiles were derived using cavity expansion theory. The formula for the caliber coefficient characterizing the size effect was obtained, and a simplified analysis of the nose shape coefficient and caliber coefficient was conducted using curve fitting and Taylor expansion within the penetration velocity range of the conventional earth-penetrating weapons. A practical calculation formula for the penetration depth of conventional earth-penetrating weapons into rock-like media was proposed, whose coefficients can be directly determined by parameters of target and projectiles. The results show that the main influencing factor of the projectile’s penetration resistance is the impedance of the target. The source of the size effect is originated from the fact that the ranges of the target damage zones do not satisfy the geometric similarity law. The nose shape coefficient can be simplified into a linear function of the projectile’s aspect ratio, and the nose shape coefficient of a flat-nosed projectile is 0.57. The caliber coefficient of the projectile is determined by the ratio of the cavity radius of the penetration to the radius of the fracture zone and can be taken as 1.2−1.4 for conventional earth-penetrating weapons. The theoretical calculation formula of penetration depth is in good agreement with experimental results, and thus, has high reliability.
- penetration depth,
- rigid projectile,
- size effect,
- caliber coefficient,
- nose shape coefficient

FullText(HTML)

References(52)

References

[1]	KENNEDY R P. A review of procedures for the analysis and design of concrete structures to resist missile impact effects [J]. Nuclear Engineering and Design, 1976, 37(2): 183–203. DOI: 10.1016/0029-5493(76)90015-7.
[2]	HUGHES G. Hard missile impact on reinforced concrete [J]. Nuclear Engineering and Design, 1984, 77(1): 23–35. DOI: 10.1016/0029-5493(84)90058-X.
[3]	KAR A K. Local effects of tornado-generated missiles [J]. Journal of the Structural Division, 1978, 104(5): 809–816. DOI: 10.1061/JSDEAG.0004915.
[4]	ACE. Fundamentals of protective design: AT1207821 [R]. Pennsylvania, USA: Office of the Chief of Engineers, 1946.
[5]	NDRC. Effects of impact and explosion: summary technical report of division 2, vol. 1 [R]. Washington: National Defense Research Committee, 1946.
[6]	BERNARD R S. Depth and motion prediction for earth penetrators: ADA 056701 [R]. Vicksburg: U. S. Army Engineer Waterways Experiment Station, 1978.
[7]	BERNARD R S. Empirical analysis of projectile penetration in rock: ADA 047989 [R]. Vicksburg: U. S. Army Engineer Waterways Experiment Station, 1977.
[8]	YOUNG C W. Penetration equations: AC04-94AL85000 [R]. Albuquerque: Sandia National Labs. , 1997.
[9]	FORRESTAL M J, OKAJIMA K, LUK V K. Penetration of 6061-T651 aluminum targets with rigid long rods [J]. Journal of Applied Mechanics, 1988, 55(4): 755–760. DOI: 10.1115/1.3173718.
[10]	FORRESTAL M J, WARREN T L. Penetration equations for ogive-nose rods into aluminum targets [J]. International Journal of Impact Engineering, 2008, 35(8): 727–730. DOI: 10.1016/j.ijimpeng.2007.11.002.
[11]	FORRESTAL M J, LUK V K. Dynamic spherical cavity-expansion in a compressible elastic-plastic solid [J]. Journal of Applied Mechanics, 1988, 55(2): 275–279. DOI: 10.1115/1.3173672.
[12]	FORRESTAL M J. Penetration into dry porous rock [J]. International Journal of Solids and Structures, 1986, 22(12): 1485–1500. DOI: 10.1016/0020-7683(86)90057-0.
[13]	WANG M Y, QIU Y Y, LI J, et al. Theoretical and experimental study on penetration in rock and ground impact effects of long rod projectiles of hyper speed [J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(3): 564–572. DOI: 10.13722/j.cnki.jrme.2017.1348.
[14]	LI J, WANG M Y, CHENG Y H, et al. Analytical model of hypervelocity penetration into rock [J]. International Journal of Impact Engineering, 2018, 122: 384–394. DOI: 10.1016/j.ijimpeng.2018.08.008.
[15]	LUK V K, FORRESTAL M J. Penetration into semi-infinite reinforced-concrete targets with spherical and ogival nose projectiles [J]. International Journal of Impact Engineering, 1987, 6(4): 291–301. DOI: 10.1016/0734-743X(87)90096-0.
[16]	FORRESTAL M J, TZOU D Y. A spherical cavity-expansion penetration model for concrete targets [J]. International Journal of Solids and Structures, 1997, 34(31/32): 4127–4146. DOI: 10.1016/S0020-7683(97)00017-6.
[17]	HE T, WEN H M, GUO X J. A spherical cavity expansion model for penetration of ogival-nosed projectiles into concrete targets with shear-dilatancy [J]. Acta Mechanica Sinica, 2011, 27(6): 1001–1012. DOI: 10.1007/s10409-011-0505-1.
[18]	FORRESTAL M J, ALTMAN B S, CARGILE J D, et al. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets [J]. International Journal of Impact Engineering, 1994, 15(4): 395–405. DOI: 10.1016/0734-743X(94)80024-4.
[19]	FREW D J, HANCHAK S J, GREEN M L, et al. Penetration of concrete targets with ogive-nose steel rods [J]. International Journal of Impact Engineering, 1998, 21(6): 489–497. DOI: 10.1016/S0734-743X(98)00008-6.
[20]	FORRESTAL M J, FREW D J, HICKERSON J P, et al. Penetration of concrete targets with deceleration-time measurements [J]. International Journal of Impact Engineering, 2003, 28(5): 479–497. DOI: 10.1016/S0734-743X(02)00108-2.
[21]	GOMEZ J T, SHUKLA A. Multiple impact penetration of semi-infinite concrete [J]. International Journal of Impact Engineering, 2001, 25(10): 965–979. DOI: 10.1016/S0734-743X(01)00029-X.
[22]	吴飚, 任辉启, 陈力, 等. 弹体侵彻混凝土尺度效应试验研究与理论分析 [J]. 防护工程, 2020, 42(2): 1–10. DOI: 10.3969/j.issn.1674-1854.2020.02.001. WU B, REN H Q, CHEN L, et al. Experimental study and theoretical analysis of size effect on projectile penetrating concrete [J]. Protective Engineering, 2020, 42(2): 1–10. DOI: 10.3969/j.issn.1674-1854.2020.02.001.
[23]	ROSENBERG Z, KREIF R, DEKEL E. A note on the geometric scaling of long-rod penetration [J]. International Journal of Impact Engineering, 1997, 19(3): 277–283. DOI: 10.1016/S0734-743X(96)00023-1.
[24]	FENG J, SUN W W, LI B M. Numerical study of size effect in concrete penetration with LDPM [J]. Defence Technology, 2018, 14(5): 560–569. DOI: 10.1016/j.dt.2018.07.006.
[25]	PENG Y, WU H, FANG Q, et al. Geometrical scaling effect for penetration depth of hard projectiles into concrete targets [J]. International Journal of Impact Engineering, 2018, 120: 46–59. DOI: 10.1016/j.ijimpeng.2018.05.010.
[26]	CANFIELD J A, CLATOR I G. Development of a scaling law and techniques to investigate penetration in concrete: NWL report No. 2057[R]. Dahlgren: U. S. Naval Weapons Laboratory, 1966.
[27]	ME-BAR Y. A method for scaling ballistic penetration phenomena [J]. International Journal of Impact Engineering, 1997, 19(9/10): 821–829. DOI: 10.1016/S0734-743X(97)00020-1.
[28]	ANDERSON C E, MULLIN S A, KUHLMAN C J. Computer simulation of strain-rate effects in replica scale model penetration experiments [J]. International Journal of Impact Engineering, 1993, 13(1): 35–52. DOI: 10.1016/0734-743X(93)90107-I.
[29]	LI Q M, REID S R, WEN H M, et al. Local impact effects of hard missiles on concrete targets [J]. International Journal of Impact Engineering, 2005, 32(1/2/3/4): 224–284. DOI: 10.1016/j.ijimpeng.2005.04.005.
[30]	王安宝, 邓国强, 杨秀敏, 等. 一个新的通用型侵彻深度计算公式 [J]. 土木工程学报, 2021, 54(10): 36–46. DOI: 10.15951/j.tmgcxb.2021.10.004. WANG A B, DENG G Q, YANG X M, et al. A new general formula for calculating penetration depth [J]. China Civil Engineering Journal, 2021, 54(10): 36–46. DOI: 10.15951/j.tmgcxb.2021.10.004.
[31]	彭永, 卢芳云, 方秦, 等. 弹体侵彻混凝土靶体的尺寸效应分析 [J]. 爆炸与冲击, 2019, 39(11): 113301. DOI: 10.11883/bzycj-2018-0402. PENG Y, LU F Y, FANG Q, et al. Analyses of the size effect for projectile penetrations into concrete targets [J]. Explosion and Shock Waves, 2019, 39(11): 113301. DOI: 10.11883/bzycj-2018-0402.
[32]	程月华, 姜鹏飞, 吴昊, 等. 考虑尺寸效应的典型钻地弹侵彻混凝土深度分析 [J]. 爆炸与冲击, 2022, 42(6): 063302. DOI: 10.11883/bzycj-2021-0373. CHENG Y H, JIANG P F, WU H, et al. On penetration depth of typical earth-penetrating projectiles into concrete targets considering the scaling effect [J]. Explosion and Shock Waves, 2022, 42(6): 063302. DOI: 10.11883/bzycj-2021-0373.
[33]	何勇, 戎晓力, 吴威涛, 等. 数物驱动智能仿真在重要目标毁伤效应评估中的应用 [J]. 防护工程, 2023, 45(6): 19–30. DOI: 10.3969/j.issn.1674-1854.2023.06.005. HE Y, RONG X L, WU W T, et al. Application of data-and-physics-driven intelligent simulation in damage effects assessment of critical targets [J]. Protective Engineering, 2023, 45(6): 19–30. DOI: 10.3969/j.issn.1674-1854.2023.06.005.
[34]	李峰, 石全, 孙正. 目标毁伤效果评估技术研究综述 [J]. 兵器装备工程学报, 2018, 39(9): 69–72. DOI: 10.11809/bqzbgcxb2018.09.015. LI F, SHI Q, SUN Z. Summary of technical research on the evaluation of target mutilation [J]. Journal of Ordnance Equipment Engineering, 2018, 39(9): 69–72. DOI: 10.11809/bqzbgcxb2018.09.015.
[35]	BERNARD R S, CREIGHTON D C. Projectile penetration in soil and rock: analysis for non-normal impact: SL-79-15 [R]. Vicksburg: U. S. Army Engineer Waterways Experiment Station, 1979.
[36]	任辉启, 穆朝民, 刘瑞朝. 精确制导武器侵彻效应与工程防护[M]. 北京: 科学出版社, 2016.
[37]	周健南, 金丰年, 王斌. 别列赞公式中弹体参数取值的探讨 [J]. 弹道学报, 2008, 20(2): 20–23. ZHOU J N, JIN F N, WANG B. Discussion on projectile parameters in Березаиъ formula [J]. Journal of Ballistics, 2008, 20(2): 20–23.
[38]	LONGCOPE D B, FORRESTAL M J. Penetration of targets described by a Mohr-Coulomb failure criterion with a tension cutoff [J]. Journal of Applied Mechanics, 1983, 50(2): 327–333. DOI: 10.1115/1.3167040.
[39]	FORRESTAL M J, LONGCOPE D B. Closed-form solutions for forces on conical-nosed penetrators into geological targets with constant shear strength [J]. Mechanics of Materials, 1982, 1(4): 285–295. DOI: 10.1016/0167-6636(82)90028-X.
[40]	FREW D J, FORRESTAL M J, HANCHAK S J. Penetration experiments with limestone targets and ogive-nose steel projectiles [J]. Journal of Applied Mechanics, 2000, 67(4): 841–845. DOI: 10.1115/1.1331283.
[41]	蒋东, 史文卿, 黄瑞源, 等. 高速/超高速侵彻的尺度效应及相似规律 [J]. 中国科学: 物理学力学天文学, 2021, 51(10): 104710. DOI: 10.1360/SSPMA-2021-0070. JIANG D, SHI W Q, HUANG R Y, et al. Scale effects and similarity laws on high/hypervelocity impact penetration [J]. SCIENTIA SINICA Physica, Mechanica & Astronomica, 2021, 51(10): 104710. DOI: 10.1360/SSPMA-2021-0070.
[42]	钱七虎, 王明洋. 岩土中的冲击爆炸效应 [M]. 北京: 国防工业出版社, 2010. QIAN Q H, WANG M Y. Impact and explosion effects in rock and soil [M]. Beijing: National Defense Industry Press, 2010.
[43]	杨卫. 宏微观断裂力学 [M]. 北京: 国防工业出版社, 1995.
[44]	杨占军. 岩石抗剪强度的检测与计算 [J]. 内蒙古煤炭经济, 2022(21): 148–150. DOI: 10.3969/j.issn.1008-0155.2022.21.049. YANG Z J. Detection and calculation of rock shear strength [J]. Inner Mongolia Coal Economy, 2022(21): 148–150. DOI: 10.3969/j.issn.1008-0155.2022.21.049.
[45]	徐志英. 岩石力学 [M]. 北京: 水利电力出版社, 1986.
[46]	CHANG S H, LEE C I, JEON S. Measurement of rock fracture toughness under modes i and ii and mixed-mode conditions by using disc-type specimens [J]. Engineering Geology, 2002, 66(1/2): 79–97. DOI: 10.1016/S0013-7952(02)00033-9.
[47]	ZHANG Z X. An empirical relation between mode Ⅰ fracture toughness and the tensile strength of rock [J]. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(3): 401–406. DOI: 10.1016/S1365-1609(02)00032-1.
[48]	王启智, 吴礼舟. 用平台巴西圆盘试样确定脆性岩石的弹性模量、拉伸强度和断裂韧度——第二部分: 试验结果 [J]. 岩石力学与工程学报, 2004, 23(2): 199–204. DOI: 10.3321/j.issn:1000-6915.2004.02.004. WANG Q Z, WU L Z. Determination of elastic modulus, tensile strength and fracture toughness of britle rocks by using flattened Brazilian disk specimen-part Ⅱ: experimental results [J]. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(2): 199–204. DOI: 10.3321/j.issn:1000-6915.2004.02.004.
[49]	何唐甫. 美国钻地武器研究发展状况及钻地计算分析 [C]//第三届全国工程结构防护学术会议论文集. 北京: 中国力学学会, 1999: 350–358.
[50]	徐天涵, 谢方, 何勇. 刚性弹侵彻缩比实验尺寸效应分析 [J]. 南京理工大学学报, 2024, 48(2): 141–147. DOI: 10.14177/j.cnki.32-1397n.2024.48.02.003. XU T H, XIE F, HE Y. Analysis of size effect for scaled penetration test of rigid projectiles [J]. Journal of Nanjing University of Science and Technology, 2024, 48(2): 141–147. DOI: 10.14177/j.cnki.32-1397n.2024.48.02.003.
[51]	张德志, 张向荣, 林俊德, 等. 高强钢弹对花岗岩正侵彻的实验研究 [J]. 岩石力学与工程学报, 2005, 24(9): 1612–1618. DOI: 10.3321/j.issn:1000-6915.2005.09.024. ZHANG D Z, ZHANG X R, LIN J D, et al. Penetration experiments for normal impact into granite targets with high-strength steel projectile [J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(9): 1612–1618. DOI: 10.3321/j.issn:1000-6915.2005.09.024.
[52]	MARKESET G, LANGBERG H. High performance concrete-penetration resistance and material development [C]//Proceedings of the 9th International Symposium on Interaction of the Effects of Munitions with Structures. Berlin-Strausberg: Bundesrepublik Deutschland, 1999.