Study on indentation formation mechanism and mass loss under impact loading in concrete penetration

doi:10.1007/s10483-026-3367-9

摘要/Abstract

Abstract:

Current erosion models for penetrating projectiles are almost exclusively developed for either rigid-body assumptions or hyper-velocity impacts. At such velocities, projectile deformation is primarily rheologically dominated. These models rarely capture the medium-velocity regime, where the impact speed drives plastic deformation, and the surface indentation governs progressive wear. The gap is addressed in this study by presenting a multi-scale framework that couples indentation mechanics, cavity-expansion theory, and impact-wave dynamics, and quantitatively correlates the indentation depth with the material properties and impact speed for the first time. Dimensional analysis identifies the dimensionless governing group, i.e., the dimensionless projectile elastic-inertial ratio T_p, dimensionless penetration resistance-inertia ratio T_t, dimensionless projectile ductility parameter λ₁, and dimensionless target strength parameter λ₂ as the trigger for erosion initiation. Then, a physically based wear model is derived analytically and calibrated against high-resolution simulations using adaptive mesh refinement to capture the projectile indentation. The resulting empirical formula is validated against independent experimental data, yielding errors of less than 7%. The proposed model provides armor-piercing projectile designers with an accurate and ready-to-use predictive tool for erosion in the medium-velocity regime.

Key words: projectile erosion, finite element simulation, indentation mechanics, cavity expansion theory, concrete, plastic strain, penetration

中图分类号:

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 791-814.

Yulong ZHENG, Yao TANG, Duo ZHANG, Xianwen RAN. Study on indentation formation mechanism and mass loss under impact loading in concrete penetration[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 791-814.

图/表 16

Warhead pattern projectile	Velocity/ $(m ⋅ s − 1)$	Volumetric abrasion/%
Q235 ogive-nosed	576	2.57
HPb63-3 ogive-nosed	498	4.16
HPb63-3 ogive-nosed	602	5.94

Material	Density $ρ / (kg ⋅ m − 3)$	Young’s modulus E/GPa	Poisson’s ratio μ	Yield strength Y/MPa
Q235	7 910	210	0.3	235
HPb63-3	8 500	105	0	140

Parameter	Value	Parameter	Value
$ρ / (kg ⋅ m − 3)$	2 440	A₀/MPa	$− 48$
μ	0.2	α	394
$F c$ /MPa	4	β	145

Material	Impact load application rate/ $(m ⋅ s − 1)$	Theoretical $h m$ /cm	Simulated $h m$ /cm	Error/%
Q235	300	0.000 910 8	0.001 580 0	42.35
Q235	200	0.000 403 5	0.001 270 0	68.23
Q235	100	0.000 100 6	0.000 842 8	88.06
Q235	1 000	0.010 790 0	0.004 300 0	150.93
Q235	1 200	0.015 880 0	0.005 100 0	211.37
Q235	2 000	0.031 980 0	0.008 210 0	289.52

参考文献 43

[1]	CHEN, X. W. Mechanics of structural design of EPW (I): the penetration/perforation theory and the analysis on the cartridge of projectile. Expolosion and Shock Waves, 25(6), 499–505 (2005)
[2]	FELDGUN, V., YANKELEVSKY, D., and KARINSKI, Y. A new simplified analytical model for soil penetration analysis of rigid projectiles using the Riemann problem solution. International Journal of Impact Engineering, 101, 49–65 (2017)
[3]	SONG, D. Y., LIU, F., and JIANG, Z. G. An analytical model for penetration into cylindrical metallic thick target by rigid sharp-nosed projectiles. Engineering Mechanics, 30(1), 31–36 (2013)
[4]	LUK, V. K. and FORRESTAL, M. J. Penetration into semi-infinite reinforced-concrete targets with spherical and ogival nose projectiles. International Journal of Impact Engineering, 6(4), 291–301 (1987)
[5]	JONES, S. E. and RULE, W. K. On the optimal nose geometry for a rigid penetrator, including the effects of pressure-dependent friction. International Journal of Impact Engineering, 24(4), 403–415 (2000)
[6]	DENG, Y. J., CHEN, X. W., ZHONG, W. Z., and HE, L. L. Experimental and numerical study on normal penetration of a projectile into a reinforced concrete target. Expolosion and Shock Waves, 40(2), 023101 (2020)
[7]	WANG, Z. L., LI, Y. C., SHEN, R. F., and WANG, J. G. Numerical study on craters and penetration of concrete slab by ogive-nose steel projectile. Computers and Geotechnics, 34(1), 1–9 (2007)
[8]	FORRESTAL, M. J., FREW, D. J., HANCHAK, S. J., and BRAR, N. S. Penetration of grout and concrete targets with ogive-nose steel projectiles. International Journal of Impact Engineering, 18(5), 465–476 (1996)
[9]	HUANG, M. and GU, G. H. Experiment and simplified analytical model for penetration of rigid projectile in a reinforced concrete target. Journal of Experimental Mechanics, 24(4), 283–290 (2009)
[10]	CHENG, X. W., ZHANG, F. J., YANG, S. Q., XIE, R. Z., GAO, H. Y., XU, A. M., JIN, J. M., and QU, M. Mechanics of structural design of EPW (III): investigations on the reduced scale tests. Expolosion and Shock Waves, 26(2), 105–214 (2006)
[11]	TABOR, D. The physical meaning of indentation and scratch hardness. British Journal of Applied Physics, 7(5), 159 (1956)
[12]	SILLING, S. A. and FORRESTAL, M. J. Mass loss from abrasion on ogive-nose steel projectiles that penetrate concrete targets. International Journal of Impact Engineering, 34(11), 1814–1820 (2007)
[13]	WEN, H. M., YANG, Y., and HE, T. Effects of abrasion on the penetration of ogival-nosed projectiles into concrete targets. Latin American Journal of Solids and Structures, 7, 413–422 (2010)
[14]	WU, H., QIAN, F., HUANG, F., and WANG, Y. Projectile nose mass abrasion of high-speed penetration into concrete. Advances in Mechanical Engineering, 4, 296503 (2012)
[15]	CHEN, X. W., HE, L. L., and YANG, S. Q. Modeling on mass abrasion of kinetic energy penetrator. European Journal of Mechanics-A/Solids, 29(1), 7–17 (2010)
[16]	HE, L. L. and CHEN, X. W. Analyses of the penetration process considering mass loss. European Journal of Mechanics-A/Solids, 30(2), 145–157 (2011)
[17]	HE, L. L. and CHEN, X. W. Simulation of variation of projectile nose during high-speed penetration into concrete. Chinese Journal of Theoretical and Applied Mechanics, 43(4), 707–715 (2011)
[18]	FORRESTAL, M. J., ALTMAN, B. S., CARGILE, J. D., and HANCHAK, S. J. An empirical equation for penetration depth of ogive-nose projectiles into concrete targets. International Journal of Impact Engineering, 15(4), 395–405 (1994)
[19]	WU, H. J., HUANG, F. L., WANG, Y. N., DUAN, Z. P., and PI, A. G. Experimental investigation on projectile nose eroding effect of high-velocity penetration into concrete. Acta Armamentarii, 33(1), 48–55 (2012)
[20]	LU, F. Y., LI, X. Y., TIAN, Z. D., and JIANG, B. Weapon Damage and Assessment, Science Press, Beijing, 263 (2021)
[21]	LIU, Q., YAO, Y., and ZHOU, L. Theoretical analysis of indentation size effect. Journal of Test and Measurement Technology, 23, 1–6 (2009)
[22]	KONG, X. Z., WU, H., FANG, Q., and PENG, Y. Rigid and eroding projectile penetration into concrete targets based on an extended dynamic cavity expansion model. International Journal of Impact Engineering, 100, 13–22 (2009)
[23]	LANKFORD, J., ANDERSON, C. E., ROYAL, S. A., and RIEGEL, J. P. Penetration erosion phenomenology. International Journal of Impact Engineering, 18, 565–578 (1996)
[24]	BISHOP, R. F., HILL, R., and MOTT, N. F. The theory of indentation and hardness tests. Proceedings of the Physical Society, 57(3), 147 (1945)
[25]	DUAN, J., WANG, K. H., ZHOU, G. G., and XUE, B. J. Critical ricochet performance of penetrator impacting concrete targets. Expolosion and Shock Waves, 36(6), 797–802 (2016)
[26]	CHEN, X., LI, X., CHEN, Y., WU, H. J., and HUANG, F. L. The third dimensionless parameter in the penetration dynamics of rigid projectiles. Chinese Journal of Theoretical and Applied Mechanics, 23(1), 77–84 (2007)
[27]	GAO, S. Q., LIU, H. P., and JIN, L. A fuzzy model of the penetration resistance of concrete targets. International Journal of Impact Engineering, 36(4), 644–649 (2009)
[28]	LI, P. C., ZHANG, X. F., and WANG, G. J. Dynamic cratering process during penetration of rigid projectile into concrete target. Expolosion and Shock Waves, 43(9), 091402 (2023)
[29]	FREW, D. J., HANCHAK, S. J., GREEN, M. L., and FORRESTAL, M. J. Penetration of concrete targets with ogive-nose steel rods. International Journal of Impact Engineering, 21(6), 489–497 (1998)
[30]	DENG, Y. J., SONG, W. J., CHEN, X. W., and YAO, Y. A dynamic cavity-expansion penetration model of compressible elastic-plastic response for reinforced concrete targets. Expolosion and Shock Waves, 38(5), 1023–1030 (2018)
[31]	KURTARAN, H. H., BUYUK, M., and ESKANDARIAN, A. Ballistic impact simulation of GT model vehicle door using finite element method. Theoretical and Applied Fracture Mechanics, 113–121 (2003)
[32]	BROITMAN, E. Indentation hardness measurements at macro-, micro-, and nanoscale: a critical overview. Tribology Letters, 65(1), 23 (2017)
[33]	XIE, W. Q., LIU, X. L., and ZHANG, X. P. A review of rock macro-indentation: theories, experiments, simulations, and applications. Journal of Rock Mechanics and Geotechnical Engineering, 16(6), 2351–2374 (2024)
[34]	FENG, S. X., ZHANG, Y. X., ZHAO, Y. F., and YUN, M. C. Rock indentation behavior: effects of penetration rates and indenter types. Applied Sciences (Switzerland), 15(4), 1785 (2025)
[35]	LIU, X. K., CAI, L. X., and CHEN, H. Residual stress indentation model based on material equivalence. Chinese Journal of Aeronautics, 35(8), 304–313 (2022)
[36]	HAUŠILD, P., ČECH, J., MERLE, B., and NOHAVA, J. Effect of temperature and strain rate on indentation size effect at shallow indentation depths. Materials and Design, 255, 114–196 (2025)
[37]	CLATON, J. D., LLOYD, J. T., and CASEM, D. T. Simulation and dimensional analysis of instrumented dynamic spherical indentation of ductile metals. International Journal of Mechanical Sciences, 251, 108333 (2023)
[38]	HE, Z. H., JIN, D., SHI, J. Y., HAN, X. D., and JAMAL, A. S. Multi-scale creep analysis of SCM-modified concrete, indentation test and multiscale homogenization method. Archives of Civil and Mechanical Engineering, 24(2), 1–14 (2024)
[39]	LI, K. Theoretical Studies on the Characterization of Uniaxial Mechanical Properties of Metallic Materials by Indentation Method, M. Sc. dissertation, Xiangtan University, 1–70 (2021)
[40]	WITTMANN, F. H., ROELFSTRA, P. E., and SADOUKI, H. Simulation and analysis of composite structures. Materials Science and Engineering, 68(2), 239–248 (1985)
[41]	MIRSHAMS, R. A. and POTHAPRAGADA, R. M. Correlation of nanoindentation measurements of nickel made using geometrically different indenter tips. Acta Materialia, 54(4), 1123–1134 (2006)
[42]	YANG, B. and VEHOFF, H. Dependence of nanohardness upon indentation size and grain size. Acta Materialia, 55(3), 849–856 (2007)
[43]	WANG, H. M., DENG, M., HUANG, Z. Y., LU, K. L., SHEN, G. B., QIAN, H. J., HUANG, Z. Y., QI, J. Q., and WANG, Q. Y. Extraction of the dynamic plastic behavior of AlON single crystals by nanoimpact. Journal of Advanced Ceramics, 13(10), 1566–1577 (2024)

Operating condition	Mesh size
Operating condition	0.2 mm	0.1 mm	0.05 mm
HPb63-3 (400 m/s)	0.012 792 20	0.011 478 700	0.011 461 100
HPb63-3 (500 m/s)	0.019 144 70	0.018 918 300	0.018 868 900
HPb63-3 (602 m/s)	0.020 279 20	0.020 137 500	0.020 127 600
HPb63-3 (700 m/s)	0.020 400 50	0.020 277 600	0.020 259 300
Q235 (400 m/s)	0.003 317 28	0.003 249 930	0.003 298 380
Q235 (576 m/s)	0.004 317 60	0.004 107 683	0.004 100 275
Q235 (600 m/s)	0.004 602 17	0.004 407 683	0.004 350 130
Q235 (700 m/s)	0.005 480 17	0.005 334 110	0.005 317 090

Operating condition	H	Experimental γ/%	Calculated γ/%	Error/%
Q235	0.004 107 683	2.57	2.396	6.77
Q235	0.002 044 388	2.57	2.396	6.77
HPb63-3 (602 m/s)	0.020 187 500	5.94	5.88	1.01
HPb63-3 (602 m/s)	0.006 219 890	5.94	5.88	1.01
HPb63-3 (498 m/s)	0.018 724 900	4.16	4.27	2.64
HPb63-3 (498 m/s)	0.004 384 180	4.16	4.27	2.64

Abrasion rate of projectile under experimental conditions/%	Abrasion rate under finite element simulation conditions/%	Abrasion rate calculated by the empirical formula/%
5.94	5.88	5.453
4.16	4.27	4.076