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

doi:10.1007/s10483-026-3367-9

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

2010 MSC Number:

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

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Figures/Tables 16

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Table 1

Table 2

Experimental results of abrasion of literature projectile[19]"

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

Table 2

Table 3

Material parameters"

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

Table 3

Table 4

Concrete material parameters"

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

Table 4

Table 5

Table 6

Indentation depth between the theoretical and simulated hm under low and high speed impact loads"

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

Table 6

Fig. 8

Fig. 9

Table 7

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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