Hemispherical nosed steel projectile high-speed penetration into aluminum target

Materials and Design 133 (2017) 237–254

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Hemispherical nosed steel projectile high-speed penetration into aluminum target Y.K. Xiao a, H. Wu a,⁎, Q. Fang a, W. Zhang b, X.Z. Kong a a b

State Key Laboratory for Disaster Prevention & Mitigation of Explosion & Impact, PLA University of Science and Technology, Nanjing 210007, China Hypervelocity Impact Research Center, Harbin Institute of Technology, Harbin 150080, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Non-monotonic dependence of penetration depth on the impact velocity is observed. • Alekseevskii-Tate model is better in predicting the non-monotonic eroding penetration depth. • Judgement criterion for the occurrence of the non-monotonic dependence is proposed.

a r t i c l e

i n f o

Article history: Received 13 June 2017 Received in revised form 30 July 2017 Accepted 1 August 2017 Available online 02 August 2017 Keywords: Projectile Penetration Depth of penetration Alekseevskii-Tate model Johnson-Cook model

a b s t r a c t Terminal ballistic performance of high-strength projectiles penetrating into metallic targets is mostly concerned by both weapon and armor designers. Most existing works are concentrated on the rigid-eroding penetration regime, and limited studies have addressed the rigid-deforming-eroding penetration regime. In this paper, nineteen shots of hemispherical nosed D6A steel projectiles penetration test on 5A06-H112 aluminum targets is conducted with a wide range of velocities (696 m/s–1870 m/s). The non-monotonic dependence of depth of penetration (DOP) on the impact velocity is observed, which successively corresponds to the three penetration stages, i.e., rigid projectile penetration, deforming projectile penetration without eroding and eroding projectile penetration. Then, for the non-monotonic rigid-deforming-eroding projectile penetration regime, the applicability of the existing six classical theoretical models for both rigid and eroding projectile penetrations is evaluated. Furthermore, the transition velocities (the upper limit of rigid penetration and the lower limit of eroding penetration) are discussed and an empirical judgement criterion for the occurrence of non-monotonic dependence is proposed. Finally, by conducting the dynamic compression test, quasi-static tension test under varying temperature, etc., the Johnson-Cook model parameters for the present target and projectile are calibrated and validated by numerically simulating the present test. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Aluminum alloy is widely used in the armored vehicles which are designed to withstand the intentional or accidental impact caused by high-speed projectiles and blast induced fragments. Terminal ballistic performance of the high strength projectiles penetrating into aluminum targets is mostly concerned by both armor-piercing projectile and ⁎ Corresponding author. E-mail address: [email protected] (H. Wu).

http://dx.doi.org/10.1016/j.matdes.2017.08.002 0264-1275/© 2017 Elsevier Ltd. All rights reserved.

armor designers. For the combinations of projectiles and targets with various strengths, densities, etc., two dependences of DOP (depth of penetration) on the impact velocity of projectile V0 are observed in the existing experiments. They are the rigid-eroding projectile penetration regime in which DOP increases monotonically with V0, and the rigid-deforming-eroding projectile penetration regime in which DOP increases non-monotonically with V0. Based on the review of the existing experimental, theoretical and numerical works in Section 2, the following limitations can be drawn: (i) most of the existing reduce-scaled projectile high-speed penetration

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tests are focused on the tungsten or steel projectiles and targets, which mainly leads to the monotonous dependence of DOP on the striking velocity. While for the non-monotonic dependence, e.g., steel projectile penetrating into aluminum target with a wide velocities range, the corresponding test data are very limited; (ii) the validities of the existing theoretical models, especially the applicability of which for the nonmonotonic dependence regime, as well as the adoptions of the projectile strength and target resistance in the existing models should be further assessed and clarified; (iii) little attention is paid on the determination of the transition velocities for the non-monotonic rigiddeforming-eroding projectile penetration regime, as well as the judgement criterion for the occurrence of the non-monotonic penetration regime, which are important for both weapon and armor designers; (iv) for the complex penetration mechanism and the few test data, limited numerical simulation works are conducted for the non-monotonic penetration regime. In the present paper, nineteen shots of hemispherical nosed D6A steel projectiles penetrating into 5A06-H112 aluminum targets are firstly conducted with the striking velocities ranged from 696 m/s to 1870 m/s. Three penetration regimes, i.e., rigid projectile penetration, deforming projectile penetration without eroding and eroding projectile penetration, are occurred successively with the increase of the initial striking velocities of projectiles. It further validates that the nonmonotonic dependence of DOP ~ V0 exists for aluminum targets under high-speed steel projectile penetration. Then, for the above nonmonotonic rigid-deforming-eroding projectile penetration regime, the applicability of the existing six models for predicting the rigid and eroding DOPs are evaluated. Furthermore, the transition velocities and the judgement criterion for the occurrence of non-monotonic dependence of DOP ~ V0 are discussed. Finally, the majority of Johnson-Cook model parameters for the present target and projectile are obtained experimentally, and the corresponding numerical simulations for the present penetration test are conducted. 2. Review of the existing works For the projectile penetration tests, Anderson et al. [1] summarized about 2300 shots data of projectiles striking into semi-infinite and finite thickness metallic targets, in which most of the projectile and target materials are tungsten or steel and almost all the DOPs are monotonically increasing with the impact velocity. Afterwards, Piekutowski et al. [2] and Forrestal and Piekutowski [3] conducted the ogival and hemispherical nosed AerMet100 and 4340 steel projectiles penetrating into 6061T6511 aluminum targets with the velocities ranged from 0.5 km/s to 3 km/s, the non-monotonic dependence of DOP on the impact velocity is found by the X-ray technology. Matthias et al. [4] further performed the tungsten sinter alloy projectiles with the truncated conical nose penetration test on the 7072 aluminum targets at the velocity of 0.25– 1.9 km/s, and the non-monotonic variation is also observed. For the rigid projectile penetration analyses, based on the cavity expansion theory (CET), Forrestal et al. [5] proposed an engineering model (denoted as Forrestal model) for rigid hemispherical nosed projectile penetrating into metallic targets, which was validated by amounts of test data for rigid steel projectiles penetrating into aluminum targets. Afterwards, Forrestal and Warren [6] further extended the Forrestal model for the ogive nosed steel projectile striking into aluminum target (F-W model). Besides, by introducing the impact and geometry functions of projectile into the target resistance equation drawn from CET, and considering the different projectile nose shapes and projectile/target interfacial frictions, Chen and Li [7] further presented a nondimensional model for predicting the DOP of rigid projectile (C-L model). From the view of CET [5–7], the resistance of target acted on the projectile is dependent on the nosed geometry and striking velocity of projectile as well as the material properties (density, yield strength and elastic modulus) of target. However, based on the 2D numerical simulations, Rosenberg and Dekel [8] observed that the resistance

force acted on the rigid projectile is a constant and independent on the projectile instantaneous velocity during the penetration. They further derived the resistance force formula through fitting the existing experimental data and established an empirical rigid projectile penetration model for predicting the DOP (R-D model). For the eroding projectile penetration analyses, several theoretical models were established. Alekseevskii [9] and Tate [10,11] independently proposed a 1D semi-hydrodynamic model (A-T model) by introducing both the projectile strength and target resistance into the Bernoulli equation, which established the theoretical foundation for analyzing the eroding projectile penetrations. After, by assuming the particle velocity fields of both the target and projectile, Walker and Anderson [12] proposed a time-dependent model for the eroding projectile penetration (W-A model), which derived good agreements for the eroding projectile penetrations. Furthermore, by introducing the flow region into the dynamic CET model and assuming the relationship of the penetration velocity with the particle velocity on the interface of flow/plastic regions, Lan and Wen [13] proposed an extended CET model (L-W model) for the eroding projectile penetration. Recently, through regression analysis of large amounts of penetration test data, Anderson and Riegel [14] proposed a simple empirical formula for predicting the DOP (A-R model), which was validated by the penetration data collected in Anderson et al. [1]. For the numerical projectile penetration analyses, Rosenberg and Dekel [15] conducted the numerical simulation for the 4340 steel projectile penetration test on 6061-T6511 aluminum targets [2] with the PISCES 2DELK Euler code. Afterwards, by using the Steinberg material model [16], Lan and Wen [17] also performed the numerical simulations for the identical test by using the finite element program LS-DYNA with the Arbitrary Lagrangian-Eulerian (ALE) algorithm. Generally, both of the predicted results in Refs [15,17] basically show the trend of the non-monotonic dependence, while the DOP results are unsatisfactory. Additionally, by introducing rigid material model for the rigid penetration stage and the Johnson-Cook model [18] for the eroding penetration stage, Lou et al. [19] conducted the simulation of tungsten alloy projectile penetrating into 7020 aluminum targets with the 2D Lagrange algorithm, and the reasonable agreement of the predicted DOP with the test data were derived. 3. Test 3.1. Projectile, sabot and obturator The cylindrical hemispherical nosed projectiles were machined from D6A (45CrNiMoV) steel rods and heat treated to a Rockwell hardness C value (Rc) of 41.2. Fig. 1 shows the photographs and sketch of the projectile. The length, diameter, average mass and density of the projectiles are 33.11 mm, 6.02 mm, 7.18 g and 7852 kg/m3, respectively. As shown in Fig. 2, the quasi-static tension test on three D6A steel specimens (the strain rate is 0.00045 s− 1) is conducted on the MTS810 testing machine conforming to Chinese Standard GB/T 2282010 with the loading speed of 0.168 mm/s. The power-law strainhardening constitutive model has been validated for representing the stress-strain relationship of most metallic materials, which has the form of


σ ¼ Eε σ ¼ Y ðEε=Y Þn

σ ≤Y σ NY

ð1Þ

where σ and ε are the true stress and true strain, respectively. E, Y and n are the Young's modulus, yield strength and strain-hardening exponent, respectively. From Fig. 2(c), E = 209GPa, Y = 1210 MPa and n = 0.0607 are derived by fitting the true stress-strain curves with the power-law strain-hardening model. Since the projectile diameter is smaller than that of the gun barrel, the sub-caliber technology was used. The projectiles were placed in a

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Fig. 1. D6A steel projectile (a) photograph (b) sketch.

set of special designed polylactide sabots with the density of 820 kg/m3 and polycarbonate obturator with the density of 1220 kg/m3, which fit snugly into the gun barrel. A 2 mm-thick steel disk was stuck to the obturator in order to prevent the projectile penetrating inversely into the obturator during launching. Fig. 3(a) shows the package of the separated projectile, obturator and the sabots, while Fig. 3(b) gives the detailed dimensions of the sabot and obturator. A cut angle at the front end of sabot was designed in order to ensure the sabots can be separated from the projectile aerodynamically after propelling out of the muzzle and before impacting on the target. 3.2. Target Shown in Fig. 4, the cylindrical target was manufactured by 5A06H112 aluminum with the density of 2703 kg/m3 and heat treated to a Rockwell hardness A value (Ra) of 41.8, which is widely used in the armored vehicles. The ingredients of the 5A06-H112 aluminum are shown in Table 1. The diameter and thickness of the target are both 200 mm, which are large enough to avoid the rear surface and boundary effects. Similarly, the quasi-static tensile stress-strain curves (the strain rate is 0.00056 s−1) of 5A06-H112 material as well as the power-law strain-hardening fitting curve are shown in Fig. 5, in which E = 73.5GPa, Y = 280 MPa and n = 0.0825 are derived. 3.3. Setup As shown in Fig. 6, the projectiles are launched by the two-stage light-gas gun, which mainly consists of six parts, i.e., air vessel, pump tube, conical section, gun barrel, separation chamber and target chamber. The diameters of the pump tube (first-stage) and the gun barrel (second-stage) are 50 mm and 20 mm, respectively. The shooting procedures are as follows:

Firstly, the air is pushed into the air vessel by an air compressor until a designed pressure. Simultaneously, the pump tube is filled with highpressure hydrogen gas and the gun barrel as well as the separation and target champers are filled with nitrogen gas. A counterweight with a polycarbonate piston and the assembled projectile are placed at the beginnings of the first-stage pump tube and second-stage gun barrel, respectively. Then, the valve of air vessel is opened and the counterweight with polycarbonate piston is accelerated in the pump tube until the polycarbonate piston entering the conical section. In order to increase the second-stage air pressure in the pump tube, a pre-nicked disk is placed after the conical section with a “+” pattern scored onto the middle surface (Fig. 6). The high-pressure hydrogen gas is compressed into the conical section until the pressure reaches the critical value to bursting the disk. Furthermore, the hydrogen flows through the nicked hole and accelerates the projectiles with sabots and obturators located at the beginnings of the second-stage gun barrel. After the projectiles are pulled out of the gun barrel, the sabots will be separated from the projectiles aerodynamically in the separation chamber. Besides, a mushroomed 45# steel block is designed and fixed at the end of the separation chamber in order to further prevent the sabot impacting on the target (Fig. 6). Afterwards, a pair of laser beam is located between the separation chamber and the target chamber to measure the impact velocity V0. The cylindrical aluminum targets are placed in the target chamber with their impacted surfaces perpendicular to the gun barrel. The impact process of the projectile after entering the target chamber is recorded by the high-speed camera system. Fig. 7 shows the typical photographs of the projectile before impacting on the target, which indicates that the projectile impacts the target nearly perpendicularly. Some small pitch angles (the angle between the axis and velocity of the projectile) existed in some shots, which are given in Table 2.

Fig. 2. Quasi-static tension test of D6A steel (a) specimen (b) MTS810 testing machine (c) true stress-strain curves.

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Fig. 3. Sabot and obturator of projectile (a) photograph (b) sketch.

3.4. Results Among the total twenty-four shots, large deviations of the impacting pitch angles occurred in five of them. In this section, the experimental targets and projectiles damages, as well as the DOPs of the other nineteen shots are given.

3.4.1. Target damages The front damages of the targets are shown in Fig. 8 with the Shots No. written on the labels. The penetration of projectiles formed a relatively deep borehole. Particularly in some shots, e.g., Shots 5–11, due to the high impact velocity, the largely deformed 2 mm-thick steel disk (shown in Fig. 3) or the debris of the obturator also passed through the mushroomed steel block (shown in Fig. 6) and impacted on the target following the projectile, which leads to another shallow holes or tiny pits on the target surface. By using the milling machine, the targets were carefully cut along the penetration trajectory after the penetration tests. Fig. 9 shows the sectional views of the targets including the residual projectiles, in which the residual projectile in Shot 16 is missed due to the mistake.

From Figs. 8 and 9, it indicates that: (i) the DOPs of 2 mm-thick steel disks are negligible due to the smaller masses; (ii) for the relatively low impact velocity, the diameter of the penetration borehole is almost the same with (e.g., Shots 2 and 15) or a little larger (e.g., Shots 1 and 20) than the projectile diameter. With the impact velocity further increasing, the penetration borehole diameter is becoming much larger than the projectile diameter (e.g., Shots 16, 18 and 22); (iii) with the enlarging of the penetration borehole, obvious radial cracks appear on the impacted surfaces around the borehole, e.g., Shots 16 and 22. 3.4.2. Projectiles damages The residual projectiles were carefully recovered by sectioning the targets. Fig. 10 shows the photographs of the unfired (far left one) and recovered projectiles, in which the projectiles are arranged with the increase of the impact velocity from left to right. 3.4.3. DOP The projectile mass (M0), striking velocity (V0), the impacting pitch angle (β) and the DOP of the nineteen shots are listed in Table 2 with the increase of impact velocity. Due to the difficulties in accurately cutting the aluminum targets and taking out of the residual projectiles, partial projectiles are also peeled slightly. Thus the residual masses of the projectiles after the test are not given. From the Figs. 9 and 10 as well as Table 2, for the present projectile/ target combination, it can be seen that, with the impact velocity increasing, three projectile penetration regimes occurred successively, i.e., the rigid projectile penetration, the deforming projectile penetration without eroding, and then the eroding projectile penetration. In details, when the impact velocities of projectiles are lower than 951 m/s, the lengths of the residual projectiles have no obvious variation and minor deformations of the projectiles are observed (Shots 14, 2, 5), the projectile can be approximately regarded as rigid body. While the impact velocities of projectiles are larger than 1104 m/s, the residual lengths of the projectiles reduce significantly and the mass losses increase rapidly (Shots 7–13, 17, 18, 22), which can be regarded as the eroding projectile penetration. Besides, for the projectiles with the initial velocities ranging from 951 m/s to 1104 m/s, obvious deformations of the projectiles can be observed without significant mass losses (Shots 1, 6, 20, 23, 24), which can be considered as deforming projectile penetration without eroding. Corresponding to the above three penetration regimes, a nonmonotonic dependence of DOP ~ V0 is observed. That is, the DOPs Table 1 Ingredients of the 5A06-H112 aluminum (mass fraction (%)).

Fig. 4. Cylindrical 5A06-H112 aluminum target.

Al

Mg

Mn

Si

Fe

Zn

Cu

Ti

Other

91.53

6.5

0.7

0.4

0.4

0.2

0.1

0.07

0.1

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4.1. Rigid projectile penetration 4.1.1. Existing models As introduced in Section 2, four theoretical models (Forrestal, F-W, C-L and R-D models) are widely used in predicting the DOP of rigid projectile. Actually, based on the CET, Forrestal, F-W and C-L models are essentially the identical in predicting the DOP. Hence, only the F-W and RD models are adopted at follows, in which the sliding frictions between the projectile and target are neglected. (1) F-W model [6]

Forrestal and Warren [6] regarded the aluminum target as the incompressible power-law strain-hardening material and gave the nondimension DOP of projectile as Fig. 5. Quasi-static true stress-strain curves of the 5A06-H112 aluminum.

increase gradually with the impact velocities of projectile increasing in the rigid and eroding projectile penetration stages, while the DOPs decrease gradually with the impact velocities of projectile increasing in the deforming projectile penetration stage. The above non-monotonic dependence is also observed in the existing experimental studies of hemispherical and ogive nosed steel projectiles penetrating into aluminum targets [2,3]. For example, in Ref. [3], the velocity range of the deforming projectile penetration stage for the hemispherical nosed 4340 steel projectiles penetrating into 6061-T6511 aluminum targets is approximate 932 m/s–1193 m/s, which is basically consistent with the present data of about 951 m/s–1104 m/s. Considering the limited test data except for Refs [2–4] are available for the non-monotonic rigid-deforming-eroding projectile penetration regime, the present test is a good complement for the existing database.

4. Discussion In this section, for the present test data of non-monotonic rigiddeforming-eroding projectile penetration regime, the applicability of the existing two models for rigid projectile penetration (F-W and R-D models), as well as the four models for eroding projectile penetration (A-T, W-A, L-W and A-R models) are firstly evaluated. In particular, the adoption of the projectile strength and target resistance in these models are clarified. Then, the two deceleration equations of the A-T model proposed by Tate [10,11] and Tate [20] successively are further assessed based on the present and existing monotonous and nonmonotonic test data. Furthermore, the transition velocities for the non-monotonic dependence, e.g., the upper limit velocity of rigid penetration and lower limit velocity of eroding penetration, are discussed. Finally, the judgement criterion for the occurrence of non-monotonic dependences is given.

    P 1 ρp 3N  ρt 2 ln 1 þ V0 ¼  2σ yt L 3N ρt N ¼

8ψ−1 24ψ2

ð2aÞ

ð2bÞ

where P, L, ρp and ρt are the DOP, effective length of projectile, densities of projectile and targets, respectively. σyt is the quasi-static yield strength of target. N⁎ is the projectile nose coefficient. ψ is the caliberradius-head (CRH) of the projectile and ψ = 0.5 for the hemispherical nosed projectile. (2) R-D model [8]

Considering that the resistance acted on the projectile is constant during the rigid projectile penetration, Rosenberg and Dekel [8] proposed an empirical formulae for predicting the DOP of rigid projectile as ρp V 20 P     ¼ L 2σ ft 1:1 ln Et =σ ft −ϕ

ð3Þ

where ϕ is the projectile nose parameter and ϕ = 1.15, 0.93 and 0.2 for the 3CRH ogival, the conical and the hemispherical nosed projectiles, respectively. Et and σft are the elastic modulus and flow stress of the target. 4.1.2. Comparisons Fig. 11 shows the non-dimensional experimental DOPs of rigid projectile from present test and Ref. [2,3], as well as the predicted curves by qffiffiffiffiffiffiffiffiffiffiffiffi above two models, in which μ ¼ ρp =ρt . It can be seen that: (i) the predictions of F-W model basically agree well with the present test data, while they are slightly larger for tests from Refs [2,3].; (ii) the predictions of R-D model show good agreement

Fig. 6. Sketch of two-stage light-gas gun.

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Fig. 7. Typical photographs of projectiles from high-speed camera.

with the test data in Refs [2,3]., but they are quite small for the present test data, which may attributed to the adopted value of the target flow stress σft in Eq. (3). It should be pointed out that, the flow stress of the target σft in R-D model is critical for the predictions. Here, Et = 73.5GPa, σyt = 280 MPa, n = 0.0825 and σft = 490 MPa (ultimate strength) are obtained from Sections 3.2 and 4.2.2 for the 5A06-H112 aluminum target. And Et = 68.9GPa, σyt = 276 MPa, n = 0.051 and σft = 420 MPa are adopted for the 6061-T6511 aluminum targets from Refs. [5,8]. 4.2. Eroding projectile penetration

Tate [20] further proposed that when the projectile completely eroded for the high-speed penetration, there is still kinetic energy residing in the elements of target material, which give rise to an after-flow penetration stage. The total DOP is the sum of the DOP in primary stage by solving Eq. (4a) and the DOP in the after-flow stage given as h i 1=3 P after−flow ¼ Dc 1 þ 1:5ρt u20 =Rs −1 =4

where u0 is the initial penetration velocity when the erosion of projectile occurs. Rs = 2σyt/3[1 + ln(2Et/3σyt)] is the energy per unit volume. Dc is the impact crater diameter and has the form of  2 2ρp ðV 0 −u0 Þ2 Dc h

i ¼1þ D σ yt 2=3 þ ln 2Et =3σ d

4.2.1. Existing models (1) A-T model [9–11]

ð4fÞ

ð4gÞ

yt

By assuming that an infinitesimal fluid region exists around the projectile-target interface during the steady state eroding penetration, Alekseevskii [9] and Tate [10,11] independently proposed the modified Bernoulli equation with considering both the projectile strength and target resistance as 2

2

ρp ðv−uÞ =2 þ Y p ¼ ρt u =2 þ Rt

ð4aÞ

The erosion rate, deceleration and penetration equations of the eroding projectile are dl=dt ¼ u−v

ð4bÞ

dv=dt ¼ −Y p =ρp l

ð4cÞ

dPi =dt ¼ u

ð4dÞ

where v, u, l, t and Pi are the projectile tail velocity, penetration velocity, instantaneous projectile length, time and instantaneous DOP, respectively. Yp = 1.7σdyp is the projectile strength and σdyp = 4.2BHN (Brinell hardness of material) is the dynamic yield strength of projectile [20]. Rt is the target resistance which has the form of [20] Rt ¼ σ dyt

" !# 2 2Et þ ln 3 σ dyt ð4−e−0:7 Þ

ð4eÞ

(2) W-A model [12]

By assuming the velocity profile along the centerline of both the projectile and target as well as the shear behavior of the target material, Walker and Anderson [12] proposed an eroding projectile penetration model based on the momentum equation along the centerline of the penetration as ρp ðv−uÞ2 =2 þ σ dyp ¼ ρt u2 =2 þ 7 ln ðα Þσ ft =3

ð5aÞ

The dimensionless extent of the plastic zone characterized by the parameter β, which is determined by 

1 þ ρt u2 =σ ft



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K t −ρt β2 u2 ¼ 1 þ ρt β2 u2 =2Gt K t −ρt u2

ð5bÞ

where Kt and Gt are the bulk modulus and shear modulus of the target material, respectively. The erosion and penetration equations are similar with the A-T model and the deceleration equation of projectile is substituted by dv=dt ¼ −σ dyp =ρp l

where σdyt is the dynamic yield strength of the target.

ð5cÞ

(3) L-W model [13]

Table 2 Test data. Shot no.

M0 (g)

V0 (m/s)

β (°)

DOP (mm)

Shot no.

M0 (g)

V0 (m/s)

β (°)

DOP (mm)

14 2 5 20 1 23 24 6 9 13

7.14 7.19 7.18 7.13 7.14 7.19 7.19 7.16 7.14 7.16

696 750 787 951 955 986 1066 1104 1193 1266

2.2 1.97 5.1 5.13 0 3.49 4.82 4.1 0.71 3.2

31.72 37.05 38.87 46.50 43.15 44.30 38.40 31.57 32.31 36.22

11 7 10 8 17 12 18 22 16

7.18 7.19 7.2 7.21 7.2 7.2 7.17 7.18 7.17

1356 1413 1470 1553 1656 1670 1684 1808 1870

0 0 0 1.3 0.73 2.35 1.45 3.62 0.7

43.50 40.60 45.99 46.00 49.06 45.86 48.10 52.80 54.37

By assuming the different response regions in the deformed targets (i.e., flow, plastic and elastic regions when the penetration velocity u is larger than a critical velocity UF0), as well as the relationship between the penetration velocity and the particle velocity on the interface of flow/plastic regions, the pressure balance equation on the interface of target and projectile is (

2

ρp ðv−uÞ2 =2 þ Y p ¼ ρt ½u−f ðuÞ =2 þ S þ Cρt f ðuÞ2 ; uNU F0 ρp ðv−uÞ2 =2 þ Y p ¼ S þ C t ρt f ðuÞ2 ; 0bu≤U F0

n o f ðuÞ ¼ U F0 exp −½ðu−U F0 Þ=Nt U F0 2

ð6aÞ

ð6bÞ

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243

Fig. 8. Front damages of the targets.

where S is the static target resitance and determined by spherical CET. Ct is the coefficient of the dynamic resistance of target and Ct = 1.5 for inpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi compressible material. U F0 ¼ HELt =ρt is the critical penetration velocity and HELt is the Hugoniot Elastic Limit of the target material. Here, Yp = σyp(1− ν)/(1− 2ν), σyp is the quasi-static yield strength of projectile in the bilinear constitutive model and ν is the Poisson's ratio. Nt is a constant and Nt = 2.45 for incompressible material. The after-flow stage is also considered following Tate [20] with Eq. (4f). It should be noted that, in L-W model, the parameters S, σyp and σyt are determined by regarding both the projectile and target as the bilinear constitutive materials, and the governing equations are identical with which of the A-T model (Eqs. (4b–4d)). (4) A-R model [14]

By regressing amounts of eroding projectile penetration test data, Anderson and Riegel [14] gives the dimensionless empirical formula of DOP as P 1:16 ¼ −0:0567 þ 1:96ð1:86−V Þ μL 1 þ 10

0:2 V ¼ ρp V 0 2 =σ ft μ −0:14

ð7aÞ

ð7bÞ

The above equations are based on the penetration data with L/D = 10. Furthermore, the dimensionless DOP for other L/D of projectiles is P=L L=D¼ξ ¼ P=L L=D¼10 −0:194 ln ðξ=10Þ

ð7cÞ

where ξ = L/D is the ratio of projectile length to diameter. 4.2.2. Comparisons Aiming to determine the dynamic mechanical properties of the target in the eroding projectile penetration models, e.g., the dynamic yield strength and the flow stress, the corresponding split Hopkinson pressure bar (SHPB) test was conducted (introduced in Section 5.2.1). The dynamic true stress-strain curve of 5A06-H112 aluminum under the strain rate of 2600 s− 1 is shown in Fig. 12, in which σdyt = 400 MPa and σft = 490 MPa are derived. Furthermore, in L-W model, ν = 0.3 is adopted for the projectile. σyp = 1260 MPa and σyt = 300 MPa are obtained for the present test by the bilinear-fitting of the quasi-static stress-strain curves of D6A steel projectile and 5A06-H112 aluminum target materials. σyp = 3.45BHN and σyt = 317 MPa are used for the 4340 steel and 6061T6511 aluminum target from Ref. [21]. In W-A model, σdyp = 4.2BHN is adopted for the projectile. σ dyt = 335 MPa and σft = 420 MPa are adopted for 6061-T6511 aluminum target from Refs. [8,22].

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Fig. 9. Sectional views of the targets.

Fig. 13 shows the comparisons between the present and existing test data from Refs. [2,3], as well as the predicted curves by above four models in Section 4.2.1. It can be seen that, the predictions of A-T model basically agree well with the four sets of eroding projectile penetration data of hemispherical nosed projectiles for the non-monotonic penetrations (Fig. 13(a–d)). While the predictions

of the above four models for two sets of ogival nosed projectile penetration data are quite smaller than test data (Fig. 13(e, f)). The reason lies in that the influence of the projectile nose shape is not considered in above four models. Thus, the semi-hydrodynamic model with the consideration of the projectile noses shapes needs further studies.

Fig. 10. Recovered projectiles.

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Fig. 11. Predicted and experimental dimensionless DOPs for the (a) present test (b) test in Ref. [2] (c) test in Ref. [3].

4.2.3. Discussions on the A-T model There are two different deceleration equations in A-T model proposed successively by Tate [10,11] and Tate [20], which will be discussed in this section. For the original A-T model from Tate [10, 11], Y p in the projectile deceleration equation (Eq. (4c)) is the Hugoniot Elastic Limit of the projectile. Afterwards, Tate [20] further modified the deceleration equation of the projectile and Yp was substituted by the dynamic yield strength σdyp of projectile and Yp = 1.7σ dyp . Therefore, the modified deceleration equation is the same as Eq. (5c).

In this section, the above two deceleration equations (Eqs. (4c) and (5c)) in A-T model are assessed by the present and the existing five sets of penetration data listed in Table 3. In which the Tests 1–2 and the Tests 3–6 are corresponding to the monotonous and nonmonotonic dependences of DOP ~ V0, respectively. In Table 3, the hardness is represented by the Brinell hardness (BHN), Rockwell hardness (Rc) or the dynamic yield strength. The relationship of the Brinell hardness and Rockwell hardness is [1] BHN ¼ 7:595Rc þ 63:86 ðMPaÞ

ð8Þ

Fig. 14 shows the comparisons of test data as well as the predicted DOPs by the A-T model with above two deceleration equations, respectively. It can be seen that, for the monotonous dependence (Tests 1–2), the predicted results according to the two deceleration equations are almost identical and basically agree well with the test data. However, the two deceleration equations lead to obvious differences for the nonmonotonic dependence, i.e., steel projectiles penetrating into aluminum targets (Tests 3–6), Eq. (4c) is more preferable. 4.3. Transition velocity

Fig. 12. Dynamic true stress-strain curve of the 5A06-H112 aluminum target.

There are two transition velocities for the non-monotonic rigiddeforming-eroding projectile penetration regime, i.e., the upper limit of rigid projectile penetration stage and the lower limit of the eroding projectile penetration stage. In this section, the above two transition velocities of the present non-monotonic penetration data are discussed by the existing models.

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Fig. 13. Predicted and experimental dimensionless DOPs.

For the upper limit velocity of the rigid projectile penetration VR, Rosenberg and Dekel [27] proposed a one-dimensional stress balanced equation on the projectile/target interface, which is expressed as ð9Þ

is the effective target resistance. b is the nose shape factor of projectile and b = 0.15, 0.24 and 0.5 for 3CRH ogive, conical and hemispherical nosed projectiles, respectively. From Eq. (9), for the hemispherical nosed projectile, VR can be derived as

   0:5 V R ¼ 2σ fp þ 3:2σ ft −1:1σ ft ln Et =σ ft =ρt ð10Þ

where σfp is the flow stress of the projectile. pc = 3σft is the pressure needed to open a cylindrical cavity, and H = [1.1ln(Et/σft) − ϕ + 3]σft/2

For the lower limit velocity of the eroding projectile penetration VE, Tate [28] assumed that, the projectile starts to erode when the eroding

σ fp þ pc ¼ H þ bρt V 2R

Table 3 Parameters of projectile penetration tests. Test no.

1 2 3 4 5 6

Ref.

[23–26] [3] [3] [3] Present test

Projectile

Target

Material

ρp (kg/m3)

Hardness

Material

ρt (kg/m3)

Hardness

C110W1 steel D17.6 tungsten 4340 steel 4340 steel 4340 steel D6A steel

7850 17,600 7830 7830 7830 7852

BHN 230 BHN 406 Rc 36.6 Rc 39.5 Rc 46.2 Rc 41.2

Armor Steel HzB, A steel 6061-T6511 aluminum

7850 7850 2710

BHN 295 BHN 255 335 MPa [22]

5A06-H112 aluminum

2703

400 MPa

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Fig. 14. Comparisons of experimental and predicted DOPs by A-T model with two deceleration equations.

rate exceeds the deforming rate of the projectile (plastic wave velocity), thus it has V E −u ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffi Eh =ρp

ð11Þ

where Eh is the dynamic hardening module of the projectile material, i.e., the slope of the plastic hardening curve in the dynamic bilinear constitutive model. Substituting Eq. (11) into Eq. (4a), it gives [29] VE ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffin     ffio Eh =ρp 1 þ ρp 1−2 Rt −Y p =Eh =ρt

ð12Þ

Therefore, the upper limit velocity of the rigid penetration VR and the lower limit velocity of eroding penetration VE can be obtained by using Eqs. (10) and (12). In the present test, due to the high hardness and strength of the D6A steel, there is no available dynamic stress-strain

data for the D6A steel at high strain-rate. Hence, the true stress-strain curve of D6AC steel under 2300 s−1 strain rate from Ref. [30] is fitted by the bilinear constitutive model and Eh = 808 MPa is obtained and adopted for the present D6A steel. Therefore, VR = 867 m/s and VE = 1166 m/s are obtained by using Eqs. (10), (12). Fig. 15 shows the predicted DOP for the whole range velocities of the present test as well as the predicted transition velocities VR and VE, and satisfied agreements are derived.

4.4. Judgement criterion for the occurrence of non-monotonic dependence The criterion for the occurrence of the non-monotonic dependence of DOP ~ V0 is mostly concerned by the designers. Several projectile highspeed penetration tests from Refs [1–3]. are listed in Table 4 and classified into two categories according to the dependence of DOP ~ V0, in which HELp and HELt represent the Hugoniot Elastic Limit of projectile and target

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5. Numerical simulation In this section, by utilizing the Johnson-Cook (J-C) constitutive model, the present penetration test is numerically simulated with the finite element (FE) program LS-DYNA, and the 2D Lagrange algorithm is adopted due to the high computational efficiency. Firstly, the mechanical performance tests for the 5A06-H112 aluminum including dynamic SHPB compression and quasi-static tension under varying temperatures are conducted. Then, the strain hardening, strain rate hardening and thermal softening effects for the 5A06-H112 aluminum are studied, the parameters in the J-C model for the 5A06H112 aluminum and D6A steel are calibrated by the present test and the existing literature. Furthermore, the numerical simulation for the rigid and eroding projectile penetration stages in the above test is conducted. 5.1. Constitutive model Fig. 15. Comparisons of the transition velocities.

materials. The relationships of HEL with the dynamic yield strength σdy and the Brinell hardness BHN are [11]

σ dy ¼ 4:2  BHN ðMPaÞ; HEL ¼ σ dy

1−ν 1−2ν

ð13Þ

where the Poisson's ratio ν = 0.3, 0.3 and 0.33 for tungsten, steel and aluminum materials, respectively. From Table 4, it can be found that, the monotonous and nonmonotonic dependences are corresponding to the steel and aluminum targets, and the inequalities HELp N 3.5HELt and HELp b 3.5HELt always holds true for the two dependences, respectively. Thus we can take it as a simple judgement criterion for the occurrence of non-monotonic dependence. Actually, HELp and 3.5HELt represent the projectile and target strengths during semi-hydrodynamic penetration proposed by Tate [11], also more in-depth theoretical studies need to be further conducted. It should be noted that, Matthias et al. [4] also observed the non-monotonic dependence of DOP ~ V0 by conducting the tungsten sinter alloy projectiles penetration test into 7072 aluminum targets at the velocity of 0.25–1.9 km/s. Since the yield strengths as well as the hardness of the above projectile and target are not given, the corresponding test data are not discussed at present.

5.1.1. Johnson-Cook model J-C constitutive model [18] given in Eq. (14) has been widely used for the materials subjected to large strains, high strain rates and high temperatures, which considers the coupled influences of strain hardening, strain rate hardening and thermal softening. The stress-strain relationship has the form of

    1−T M σ ¼ A þ Bεp N 1 þ C ln ε_

ð14Þ

where A is the quasi-static yield strength under the reference temperature T0 and reference strain rate ε_ 0. B and N are the strain hardening coefficients. C and M are the parameters corresponding to the strain rate hardening and thermal softening. T∗ = (T − T0)/(Tm − T0) is the dimensionless temperature, in which T and Tm are the current temperature and melting temperature of the material. εp is the effective plastic strain.  _ ε_ 0 is the dimensionless plastic strain rate and ε_ is the current ε_ ¼ ε= plastic strain rate. Johnson and Cook [38] further proposed a damage equation for the above J-C constitutive model, which has the form of   ε f ¼ ½D1 þ D2 expðD3 σ  Þ 1 þ D4 ln ε_ ð1 þ D5 T  Þ

ð15Þ

where εf is the failure strain. D1-D5 are the material parameters. σ∗ = σm/σeff is the ratio of the mean pressure σm and the equivalent von-Mises stress σeff.

Table 4 Parameters of projectile penetration tests. Dependence

Ref.

Projectile

Target

Material

σdyp (MPa)

HELp (MPa)

Material

σdyt (MPa)

HELt (MPa)

3.5HELt (MPa)

992/1323 2024 1323/2852 1874 1985 1977 1977 2867 1940 662

3473/4631 7083 4631/9981 6560 6946 6920 6920 10,033 6791 2317

6061-T6511 aluminum

336

662

2317

D6A steel

1691 1691 2161 1595/2984 2587 3212 2727/2800 3006/2800 2536–2690 2607 3445 2513 2674 3048 2769

567/756 1156 756/1630 1071 1134 1130 1130 1638 1109 336

Present test

966 966 1235 911/1705 1478 1835 1558/1600 1718/1600 1449–1537 1490 1968 1436 1528 1742 1582

Steel 37/52 Armor Steel Steel 52/W8 tungsten HzB, A steel RHA steel RHA steel RHA steel RHA steel RHA steel 6061-T6511 aluminum

[3]

C110W1 steel C110W1 steel D17 tungsten C110W2 steel/D17.6 tungsten W10 tungsten U-3/4Ti uranium U-3/4Ti uranium/93W tungsten W92.5/W10 tungsten X27C/X27/X21C/W-2 tungsten 4340 steel AerMet100 steel 4340 steel

5A06-H112 aluminum

400

788

2758

Monotonous

[23–26]

Non-monotonic

[31] [32] [33] [34] [35–37] [2]

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Fig. 16. Typical compression specimen after SHPB test.

5.1.2. Grüneisen equation of state Grüneisen equation of state (EOS) is employed corresponding to the J-C constitutive model, in which the pressure p for the compressed materials has the form of [39] 8 > > > > > <"

h

γ  α i ρ0 C 1 2 λ 1 þ 1− 0 λ− λ2 2 2 # þ ðγ0 þ αλÞE1 ; λ2 λ3 p¼ −S3 1−ðS1 −1Þλ−S2 > > λþ1 > ðλ þ 1Þ2 > > : 2 ρ0 C 1 λ þ ðγ0 þ αλÞE1 ;

λ ≥0

λb0

are taken as the reference strain rate and temperature. Thus the J-C model (Eq. (14)) can be simplified as Y ¼ A þ Bε p

ð17Þ

Through fitting the stress-strain curves before necking in Fig. 5 by using the Eq. (17), the strain hardening parameters A = 280 MPa, B = 298.64 and N = 0.5079 for the 5A06-H112 aluminum are obtained. The fitting curve of J-C constitutive model is also shown in Fig. 5. (2) Strain rate effect

ð16Þ where λ = ρ/ρ0–1 is the volume strain, ρ and ρ0 are the current and initial density. C1 is the intercept of the us-up curve, us and up denote the shock wave velocity and particle velocity. S1, S2, S3 are the coefficients of the us-up curve. γ0 is the dimensionless Grüneisen parameter and α is the first order volume correction to γ0. E1 is the internal energy of the material. 5.2. Calibration of Johnson-Cook model 5.2.1. 5A06-H112 aluminum target Based on the single variable method, by conducting the quasi-static tension test under room temperature, SHPB compression test under room temperature as well as the quasi-static tension test under varying temperatures (50–350 °C), the strain hardening effect, strain rate effect and temperature effect contained in the J-C constitutive model are experimentally discussed, and the J-C model parameters for the 5A06H112 aluminum are determined in this section.

Under the room temperature, the SHPB compression test on nine cylindrical 5A06-H112 aluminum specimens is conducted with the strain rate from 423 s − 1 to 2600 s− 1. Fig. 16 shows the intact (far left) and typical recovered specimens after the SHPB test, in which the initial height of the cylindrical specimen is 5.5 mm and the diameter is 11 mm. It can be seen that, with the strain rate increasing, the specimens expanded radially and the corresponding heights were reduced gradually. Fig. 17 shows the dynamic stress-strain curves under different strain rates. For the dynamic yield strength under the reference temperature (εp = 0 and T⁎ = 1), the J-C constitutive model (Eq. (14)) can be simplified as   Y ¼ A 1 þ C ln ε_

ð18Þ

Fig. 18 further shows the experimental dynamic yield strength data under different strain rates and the fitting curve by Eq. (18), C = 0.017 is obtained through fitting the test data. (3) Temperature effect

(1) Strain hardening effect

As shown in Section 3.2, the quasi-static tension test for the 5A06H112 aluminum is conducted on the MTS810 machine, the corresponding strain rate is 0.00056 s−1 and the room temperature is 25 °C, which

The quasi-static tension test for eighteen standard specimens under 50–350 °C temperatures is conducted on the INSTRON 5965 testing machine, and the detailed dimensions of the specimen are shown in Fig. 19.

Fig. 17. Dynamic stress-strain curves of 5A06-H112 aluminum under different strain rates.

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Eq. (19), the thermal softening parameter M = 0.9028 is obtained through fitting the yield strength data under different temperatures. It should be pointed out that, three specimens were tested for each temperature, while only two of them are valid and shown in Fig. 22 for the temperatures of 150 °C, 200 °C and 350 °C. Up to now, the J-C constitutive model parameters of 5A06-H112 aluminum target are all determined and listed in Table 5, in which G and Cv are the shear modulus and specific heat capacity of the material. Besides, EOS parameters proposed by Lan and Wen [17] are adopted.

Fig. 18. Dynamic yield strength data and the fitting curve.

Fig. 20 shows the testing machine and the typical fractured specimens in the quasi-static tension test under different temperatures. Fig. 21 further illustrates the typical stress-strain curves of the specimens under 50 ~ 350 °C temperatures, in which the strain rate is 0.0021. The yield strength under different temperatures (εp =0) can be simplified from Eq. (15) as  "   # T−T 0 M ε_ 1− Y ¼ A 1 þ C ln T m −T 0 ε0

ð19Þ

As shown in Fig. 22, substituting A = 280 MPa, C = 0.0158, ε_ = 0.0021 s− 1, ε0 = 0.00056, T0 = 298 K (25 ̊C) and Tm = 853 K into

5.2.2. D6A steel projectile The J-C constitutive model parameters of D6A steel are determined by conducting the quasi-static tension test and referencing the existing literatures. As shown in Section 3.1, the quasi-static tension test for the D6A steel is conducted with the strain rate of 0.00045 s−1 and the room temperature of 25 °C, which are taken as the reference strain rate and reference temperature in the J-C model. Correspondingly, A = 1210 MPa, B = 1071.6 MPa and N = 0.6 are obtained, and the fitted curve of J-C model is also shown in Fig. 2(c). The J-C model parameters of the D6A steel are listed in Table 6. It should be noted that, as described in Section 4.3, due to the limited loading capability of the testing machine, the dynamic compression and high temperature tension tests for D6A steel are not conducted at present. The strain rate hardening coefficient C = 0.014 and the thermal softening coefficient M = 1 are referred from Ref. [40], and the EOS parameters are referred from [17]. 5.3. Simulation In this section, the numerical simulations are conducted by the commercial FE program LS-DYNA with the above calibrated constitutive model parameters, in which the rigid penetration stage and the eroding

Fig. 19. Dimensions of 5A06-H112 aluminum tensile specimen under different temperatures.

Fig. 20. Quasi-static tension test for 5A06-H112 aluminum under different temperatures (a) INSTRON 5965 testing machine (b) typical fractured specimens.

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Table 6 J-C model parameters of D6A steel. Basic parameter ρ (kg/m3) E (GPa) G (GPa) Tm (K) T0 (K) Cv (J/(g ∗ K))

Fig. 21. Typical quasi-static stress-strain curves under different temperatures.

Fig. 22. Fitting curve of the yield strength under different temperatures.

penetration stage under the non-monotonic variation regime are discussed, respectively. 5.3.1. Rigid projectile penetration As shown in Section 3.4, the projectile is considered as the rigid body when the penetration velocity below 951 m/s. Thus the material models of MAT_RIGID (020#) and MAT_JOHNSON_COOK (015#) embedded in LS_DYNA [39] are employed for the projectile and target, respectively. The calibrated parameters for the J-C model and Gruneisen EOS in Section 5.2.1 are adopted for the target. 2D axisymmetric Lagrange algorithm is employed to improve the computational efficiency. Fig. 23 shows the FE model, in which the geometry of the projectile is identical with the experiment. The radius and height of the target are 30.1 mm and 90.3 mm, and the boundary of the target is set as non-reflect. Based on the mesh sensitivity analyses, the element size for the projectile and target is selected to be 0.5 mm, and there are 387 elements in the projectile and 10,800 elements in the target. The 2D_AUTOMATIC_SINGLE_SURFACE contact is employed

7852 209 81.78 1793 298 469

Constitutive parameter

EOS parameter [17]

A (MPa) B (MPa) N C M ε_ 0

C1 (m/s) S1 S2 S3 γ0 α

1210 1071.6 0.6 0.014 [40] 1.00 [40] 0.00045

to define the contact behavior between the projectile and target without friction. Furthermore, limited by the testing capability, the corresponding test for determining the damage parameters in Eq. (15) is not conducted. At present, only D1 in the J-C damage equation (Eq. (15)) is used following Rosenberg and Dekel [15] and Li et al. [41], and D1 = 1.8 is adopted for the 5A06-H112 aluminum target. Fig. 24 shows the typical penetration process for the rigid projectile penetration at the striking velocity of 696 m/s. The comparisons between the experimental and simulated DOPs are further shown in Fig. 25, and good agreements are obtained. 5.3.2. Eroding projectile penetration As shown in Section 3.4, when the penetration velocity is larger than 1104 m/s in the present experiment, the projectile begins to erode with obvious mass loss. Therefore, for the eroding projectile penetration stage, the material model MAT_JOHNSON_COOK (015#) in the LSDYNA [39] is employed for both the projectile and target, and the Gruneisen EOS is used. The FE model for the eroding penetration simulation is identical with the rigid penetration model in Section 5.3.1. For the target and projectile, the J-C constitutive model parameters are referred from Tables 5 and 6, and the corresponding damage parameters D1 in the J-C damage equation (Eq. (15)) are adopted as 1.8 and 0.9, respectively. Fig. 26 shows the typical eroding steel projectile penetrating process into the aluminum target at the striking velocity of 1870 m/s, and Fig. 27 shows the final cross-sectional comparisons. Fig. 28 further shows experimental and simulated DOPs of projectile with the striking velocity ranged up to 5 km/s. Based on the discussions in Section 4.2, the A-T model give better predictions for the high-speed eroding projectile penetrations. Therefore, aiming to verify the validations of the present simulated results, the predicted curve by the A-T model is also given in Fig. 28. It can be seen that: (i) the numerical simulation expanded the striking velocity up to 5 km/s, and rather good agreements with the theoretical predictions are obtained; (ii) the simulated results including the DOP and final penetration cross-sectional images agree reasonably

Table 5 J-C model parameters of 5A06-H112 aluminum. Basic parameter ρ0 (kg/m3) E (GPa) G (GPa) Tm0 (K) T0 (K) Cv (J/(g*K))

2703 73.5 27.632 853 298 921

Constitutive parameter

EOS parameter [17]

A (MPa) B (MPa) N C M ε_ 0

C1 (m/s) S1 S2 S3 γ0 α

280 298.64 0.5079 0.017 0.9028 0.00056

5376 1.55 0.00 0.00 2.19 0

4580 1.34 0.00 0.00 1.97 0

Fig. 23. 2D axisymmetric FE model.

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Fig. 24. Typical penetration process for the rigid projectile penetration (696 m/s).

numerical simulation for the DOP in the deforming projectile penetration stage is still a challenge and not performed at present, more indepth works are needed. 6. Conclusion In the present paper, addressing the rigid-deforming-eroding projectile penetration regime, in which the DOP increases nonmonotonically with the projectile impact velocity rising, the experiments and numerical simulations for the steel projectiles penetrating into aluminum targets as well as the evaluation of the existing theoretical models are performed. The main works and conclusions are as follows:

Fig. 25. Comparisons between the experimental and simulated DOPs.

with the present eroding projectile penetration test; (iii) the present calibrated J-C model parameters for the D6A steel projectile and 5A06H112 aluminum target are validated. It should be pointed out that, as indicated from the previous works of Rosenberg and Dekel [15] and Lan et al. [17] in simulating the 4340 steel projectiles penetrating into 6065-T6511 aluminum targets, the

(1) The hemispherical nosed D6A steel projectiles penetration test on 5A06-H112 aluminum targets is conducted with a wide range of impact velocities from 696 m/s to 1870 m/s, in which the non-monotonic dependence of DOP ~ V0 is observed. (2) The existing six models for predicting the DOPs of rigid and eroding projectiles are evaluated by comparing with the present and existing non-monotonic penetration test data. Both F-W model and R-D model give reasonable agreements with the rigid projectile penetration data, and A-T model is preferable for eroding projectile penetrations in the non-monotonic penetration regime. (3) Two deceleration equations of projectile in A-T model are discussed, both of them give almost identical predictions and basically agree well with the monotonous dependence test data,

Fig. 26. Penetration process for the eroding projectile penetration (1870 m/s).

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Fig. 27. Cross-sectional comparisons of experimental and simulated results (a) 1553 m/s (b) 1684 m/s (c) 1870 m/s.

Fig. 28. Comparisons between the experimental, simulated and theoretical predicted DOPs.

while the original expression is much better for the nonmonotonic penetration regime. The existing models in predicting the transition velocities for the present non-monotonic penetration data are validated. A simple empirical judgement criterion for the occurrence of the non-monotonic dependence of DOP ~ V0 is further given. (4) By conducting the SHPB compression test, quasi-static tension test under varying temperatures, etc., the majority of J-C model parameters for 5A06-H112 aluminum targets and D6A steel projectile are determined. The corresponding numerical simulations with the Lagrange algorithm are conducted based on the FE program LS-DYNA, and good agreements are obtained.

Acknowledgement This study is supported by the National Natural Science Foundation of China (51522813). References [1] C.E. Anderson Jr., B.L. Morris, D.L. Littlefield, A Penetration Mechanics Database. No. SwRI 3593/001, Southwest Research Inst, San Antonio TX, 1992. [2] A.J. Piekutowski, M.J. Forrestal, K.L. Poormon, T.L. Warren, Penetration of 6061T6511 aluminum target by ogive-nosed steel projectiles with striking velocities between 0.5 and 3.0 km/s, Int. J. Impact Eng. 23 (1) (1999) 723–734.

[3] M.J. Forrestal, A.J. Piekutowski, Penetration experiments with 6061-T6511 aluminum targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3.0 km/s, Int. J. Impact Eng. 24 (1) (2000) 57–67. [4] W. Matthias, Penetration data for a medium caliber tungsten sinter alloy penetrator into aluminum alloy 7020 in the velocity regime from 250 m/s to 1900 m/s, The 23rd International Symposium on Ballistics, Tarragona, Spain, 2007. [5] M.J. Forrestal, N.S. Brar, V.K. Luk, Penetration of strain-hardening targets with rigid spherical-nose rods, J. Appl. Mech. 58 (1) (1991) 7–10. [6] M.J. Forrestal, T.L. Warren, Penetration equations for ogive-nose rods into aluminum targets, Int. J. Impact Eng. 35 (8) (2008) 727–730. [7] X.W. Chen, Q.M. Li, Deep penetration of a non-deformable projectile with different geometrical characteristics, Int. J. Impact Eng. 27 (6) (2002) 619–637. [8] Z. Rosenberg, E. Dekel, Terminal Ballistics, Springer, Berlin, Germany, 2012. [9] V.P. Alekseevskii, Penetration of a rod into a target at high velocity, Combust. Explos. Shock Waves 2 (2) (1966) 63–66. [10] A. Tate, A theory for the deceleration of long rods after impact, J. Mech. Phys. Solids 15 (6) (1967) 387–399. [11] A. Tate, Further results in the theory of long rod penetration, J. Mech. Phys. Solids 17 (3) (1969) 141–150. [12] J.D. Walker, C.E. Anderson Jr., A time-dependent model for long-rod penetration, Int. J. Impact Eng. 16 (1) (1995) 19–48. [13] B. Lan, H.M. Wen, Alekseevskii-Tate revisited: an extension to the modified hydrodynamic theory of long rod penetration, Sci. China Technol. Sci. 53 (5) (2010) 1364–1373. [14] C.E. Anderson, J.P. Riegel, A penetration model for metallic targets based on experimental data, Int. J. Impact Eng. 80 (2015) 24–35. [15] Z. Rosenberg, E. Dekel, Numerical study of the transition from rigid to eroding-rod penetration, J. Phys. IV 110 (2003) 681–686. [16] D.J. Steinberg, S.G. Cochran, M.W. Guinan, A constitutive model for metals applicable at high-strain rate, J. Appl. Phys. 51 (3) (1980) 1498–1504. [17] B. Lan, H.M. Wen, Numerical simulation of the penetration of a spherical nosed 4340 steel long rod into semi-infinite 6061-t6511 aluminum targets, Eng. Mech. 10 (26) (2009) 183–190 (In Chinese). [18] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperature, Proceedings of the 7th International Symposium on Ballistics, 541–47, 1983. [19] J.F. Lou, Z. Wang, T. Hong, et al., Numerical study on penetration of semi-infinite aluminum-alloy targets by tungsten-alloy rod, Chin. J. High Pressure Phys. 1 (2) (2009) 65–70 (In Chinese). [20] A. Tate, Long rod penetration models-part II. Extensions to the hydrodynamic theory of penetration, Int. J. Mech. Sci. 28 (9) (1986) 599–612. [21] H.M. Wen, B. Lan, Analytical models for the penetration of semi-infinite targets by rigid, deformable and erosive long rods, Acta Mech. Sinica 26 (4) (2010) 573–583. [22] T.L. Warren, M.J. Forrestal, Effects of strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods, Int. J. Solids Struct. 35 (35) (1998) 3737–3753. [23] V. Hohler, E. Schneider, A.J. Stilp, R. Tham, Length- and velocity reduction of high density rods perforating mild steel and armor steel plates, Proceedings of the 4 th International Symposium on Ballistics, Monterey, CA, 17–19 October 1978. [24] V. Hohler, A.J. Stilp, Penetration of steel and high density rods in semi-infinite steel targets, Proceedings of the 3rd International Symposium on Ballistics, Karlsruhe, FRG, 23–25 March 1977. [25] V. Hohler, A. Stilp, Influence of the length-to-diameter ratio in the range from 1 to 32 on the penetration performance of rod projectiles, Proceedings of the 8th International Symposium on Ballistics, Orlando, FL, 23–25 October 1984. [26] V. Hohler, A. Stilp, Hypervelocity impact of rod projectiles with L/D from 1 to 32, Int. J. Impact Eng. 5 (4) (1987) 323–331. [27] Z. Rosenberg, E. Dekel, On the deep penetration of deforming long rods, Int. J. Solids Struct. 47 (2) (2010) 238–250.

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[28] A. Tate, A possible explanation for the hydrodynamic transition in high speed impact, Int. J. Mech. Sci. 19 (2) (1977) 121–123. [29] J.F. Lou, Y. Zhang, Z. Wang, et al., Long-rod penetration: the transition zone between rigid and hydrodynamic penetration modes, Def. Tech. 10 (2) (2014) 239–244. [30] C.M. Hu, H.L. He, S.S. Hu, Research of dynamic mechanical behavior of D6AC steel, Ordinance Mater. Sci. Eng. 26 (2) (2003) 26–29 (In Chinese). [31] G.F. Silsby, Penetration of semi-infinite steel targets by tungsten rods at 1.3 to 4.5 km/s, Proceedings of the 8th International Symposium on Ballistics, TB/31-35, Orlando, Florida, 1984. [32] M. Keele, E.J. Rapacki, W.J. Bruchey, High velocity ballistic performance of a uranium alloy long rod penetrator, Proceedings of the 12th International Symposium on Ballistics, San Antonio, TX, 30 Oct.–1 Nov. 1990. [33] L.S. Magness, T.G. Farrand, Deformation behavior and its relationship to the penetration performance of high-density KE penetrator materialsProceedings of the 1990 Army Science Conference, Durham, NC, 1990. [34] M. Wilkins, J. Gibbons, V. Hohler, A.J. Stilp, M. Cozzi, Ballistic performance of AIN, SiC, and A1203 ceramic tiles impacted by tungsten alloy long rod projectiles, Proceedings of the U. S. Army TACOM Combat Survivability Symposium, Vol. II, 15–17 April 1991, pp. 75–95.

[35] P. Woolsey, S. Mariano, D. Kokidko, Alternative test methodology for ballistic performance ranking of armor ceramicsProceedings of the Fifth Annual TACOM Armor Coordinating Conference, 7 March 1989. [36] P. Woolsey, D. Kokidko, S. Mariano, Alternative Test Methodology for Ballistic Performance Ranking of Armor Ceramics. Report No. MTL TR 89–43, U. S. Army Materials Technology Laboratory, Watertown, Massachusetts, 1989. [37] P. Woolsey, D. Kokidko, S. Mariano, Progress report on ballistic test methodology for armor ceramics, Proceedings of TACOM Combat Vehicle Survivability Symposium, Gaithersburg, MD, March 1990. [38] G.R. Johnson, W.H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Fract. Mech. 21 (1) (1985) 31–48. [39] LS-DYNA Keywords User's Manual, Livermore: Livermore Software Technology Corporation, 2001. [40] M. Qu, X.W. Chen, G. Chen, Numerical study on dynamic plastic buckling of deep penetrating projectile, Expl. Shock Wave 28 (2) (2008) 116–123 (In Chinese). [41] J.C. Li, X.W. Chen, F.L. Huang, FEM analysis on the “self-sharpening” behavior of tungsten fiber/metallic glass matrix composite long rod, Int. J. Impact Eng. 86 (2015) 67–83.