Ballistic resistance of mild steel plates of various thicknesses against 7.62 AP projectiles

Abstract

The ballistic resistance of mild steel plates has been studied against 7.62 AP projectiles through numerical simulations using ABAQUS/Explicit commercial finite element package. The projectiles were impacted on 4.7, 6, 10, 12, 16, 20 and 25 mm thick target plates at varying incidence angles. The material parameters proposed by authors for the Johnson–Cook model were used to predict the material behavior of target, while the material behavior of projectile was incorporated from the available literature. The numerical results thus obtained have been compared with the experiments available in the literature. The experimental and numerical results with respect to failure mechanism, residual projectile velocity, and maximum angle for perforation and the effect of configurations on spacing and critical angle of ricochet have been compared. A close correlation between the experimental findings and the predicted results has been found. In general, the resistance of target has been found to increase with an increase in target obliquity. The critical angle of the projectile ricochet has been found to decrease with an increase in target thickness. The ballistic limit for all given thicknesses of mild steel targets has also been obtained numerically. The ballistic limit thus obtained has been used to calibrate the Recht–Ipson empirical model for calculating the residual projectile velocity corresponding to a given incidence velocity. Simulations were also done for three-layered target of 4.7- and 6-mm-thick plate and spacing was varied to study its effect on their ballistic resistance. The variation of spacing at normal impact was found to have an influence as long as the spacing was smaller than the projectile length.

Keywords

Oblique impact ballistic limit layered target spaced target 7.62 AP projectiles

Introduction

In the 1990s, small-arms projectiles were the central weapon in 47 of the 49 world’s major conflicts (Borvik et al., 2009). Small-arms projectiles are relatively cheap and portable and are easy to use and maintain. Therefore, the need for protection against small arms and light weapons is large both from a civilian and a military point of view. For successful military operations, high-strength steel may be considered; however, conventional structural steels are used in common building. For instance, the tallest structures are constructed using structural steel due to its constructability. The ballistic investigations of mild steel plates against normal and oblique impact by 7.62 AP projectiles with the help of computer tool are yet interesting. The majority of ballistic studies is further concerned with the worst case, that is, the normal impact. However, in most real cases, the projectile strikes the target with some degree of obliquity. Earlier works on oblique impact have been found in the review papers (Awerbuch and Bodner, 1977; Borvik et al., 2011; Goldsmith and Finnegan, 1986; Iqbal et al., 2010; Johnson et al., 1982; Teng et al., 2005). It was observed that the velocity drop during perforation was almost unaffected by the oblique angle up to 30°. At higher oblique angles, the velocity drop was significant. However, numerical studies on metallic plates against oblique impact with AP projectiles are limited. Similarly, Senthil and Iqbal (2013) carried out numerical simulations on 1100-H12 aluminum of layered and monolithic targets against 15, 19 and 24 mm diameter ogival nose projectiles. It was concluded that the ballistic resistance has been found to increase with an increase in the projectile diameter. For a given diameter, the monolithic target has been found to offer better resistance (Iqbal et al., 2016; Johnson et al., 2011; Nia and Hoseini, 2011; Senthil et al., 2013). The studies reporting the response of mild steel targets against AP projectiles, however, are limited. The numerical reproduction of the experimental results is, however, limited due to the unavailability of the material parameters required for constitutive modeling. Therefore, it is concluded that the studies on numerical investigations of ballistic resistance at varying angles of incidence against armor-piercing projectile are limited.

In this article, the ballistic performance of mild steel targets of various thicknesses of 4.7, 6, 10, 12, 16, 20, and 25 mm has been studied against 7.62 AP projectiles at normal and oblique angles of incidence until the occurrence of projectile ricochet using ABAQUS/Explicit finite element code. The results thus obtained were compared with our earlier experimental results by Gupta and Madhu (1992, 1997) with same projectile and target configuration. The Johnson–Cook (JC) constitutive model has been used for predicting the material behavior of the projectile and mild steel targets. The ballistic limit of 4.7, 6, 10, 12, 16, 20 and 25 mm thick target has also been obtained numerically against 7.62 AP projectiles at normal impact. The spacing between three layers of 4.7 and 6 mm individual layer thickness was also varied as 10, 20, 30, 40, 50, and 60 mm to study its influence on ballistic resistance at normal incidence.

Constitutive modeling

The flow and fracture behavior of mild steel target and projectile was predicted using the JC (Johnson and Cook, 1983, 1985) elasto-viscoplastic material model available in ABAQUS finite element code. The material model includes the effect of linear thermoelasticity, yielding, plastic flow, isotropic strain hardening, strain rate hardening, softening due to adiabatic heating, and damage. The equivalent von Mises stress $\bar{σ}$ of the JC model is defined as

\bar{σ} ({\bar{ε}}^{pl}, {\dot{\bar{ε}}}^{pl}, \hat{T}) = [A + B {({\bar{ε}}^{pl})}^{n}] [1 + Cln (\frac{{\dot{\bar{ε}}}^{pl}}{{\dot{ε}}_{0}})] [1 - {\hat{T}}^{m}]

(1)

where A, B, n, C, and m are the material parameters determined from different mechanical tests. ${\bar{ε}}^{pl}$ is the equivalent plastic strain, ${\dot{\bar{ε}}}^{pl}$ is the equivalent plastic strain rate, ${\dot{ε}}_{0}$ is the reference strain rate, and $\hat{T}$ is the non-dimensional temperature defined as

\hat{T} = \frac{(T - T_{0})}{(T_{melt} - T_{0})} T_{0} ⩽ T ⩽ T_{melt}

(2)

where T is the current temperature, $T_{melt}$ is the melting point temperature, and $T_{0}$ is the room temperature. Johnson and Cook (1985) extended the failure criterion proposed by Hancock and Mackenzie (1976) by incorporating the effect of strain path, strain rate, and temperature on the fracture strain expression, in addition to stress triaxiality. The fracture criterion is based on the damage evolution wherein the damage of the material is assumed to occur when the damage parameter, $ω$ , exceeds unity

ω = \sum^{} (\frac{Δ {\bar{ε}}^{pl}}{{\bar{ε}}_{f}^{pl}})

(3)

where $Δ {\bar{ε}}^{pl}$ is an increment of the equivalent plastic strain, ${\bar{ε}}_{f}^{pl}$ is the strain at failure, and the summation is performed over all the increments throughout the analysis. The fracture model proposed by Johnson and Cook takes into account the effect of stress triaxiality, strain rate, and temperature on the equivalent failure strain. The equivalent fracture strain ${\bar{ε}}_{f}^{pl}$ is expressed as

{\bar{ε}}_{f}^{pl} (\frac{σ_{m}}{\bar{σ}}, {\dot{\bar{ε}}}^{pl}, \hat{T}) = [D_{1} + D_{2} \exp (- D_{3} \frac{σ_{m}}{\bar{σ}})] [1 + D_{4} \ln (\frac{{\dot{\bar{ε}}}^{pl}}{{\dot{ε}}_{0}})] [1 + D_{5} \hat{T}]

(4)

where $D_{1} - D_{5}$ are material parameters determined from different mechanical tests, $σ_{m} / \bar{σ}$ is the stress triaxiality ratio, and $σ_{m}$ is the mean stress (s⁻¹). The associated high strain rate and temperature further complicate the failure process in perforation problems. A number of metallurgical events such as nucleation, growth, and coalescence of holes precede failure, and none of these definitions of damage initiation could be considered definitive. However, the available (Borvik et al., 2002; Teng and Wierzbicki, 2004) studies suggest that stress triaxiality is a major factor responsible for material damage and therefore has a major influence on fracture strain. The effect of stress triaxiality on the material failure could be studied with the help of tension tests conducted on notched specimens with varying initial notch radii. A careful examination of these tests suggests that the effective cross-sectional area of the specimen is progressively reduced due to the formation of microscopic holes which grow throughout the tension test and coalesce to develop a large crack which causes a sharp reduction in the cross-section and a fall in the average stress. The first bracket, $[D_{1} + D_{2} {\exp (D}_{3} (σ_{m} / \bar{σ}))]$ , of the equivalent plastic failure strain expression incorporates the effect of stress triaxiality on the material fracture. The parameters D₁, D₂, and D₃ are obtained by performing the tension tests on the cylindrical notched specimens at quasi-static loading. The above expression of the JC model is fitted with the measured true fracture strain and stress triaxiality relationship for calibrating these parameters. The strain rate–dependent damage parameter D₄ could be obtained by fitting the JC expression, $[1 + D_{4} \ln ({\dot{\bar{ε}}}^{pl} / {\dot{ε}}_{0})]$ , with the experimentally obtained true fracture strain and strain rate relationship. The temperature-dependent fracture strain parameter D₅ is obtained by fitting the corresponding expression, $[1 + D_{5} \hat{T}]$ , with the experimentally obtained failure strain and temperature relationship.

When material damage occurs, the stress–strain relationship no longer accurately represents the material’s behavior (ABAQUS, 2016). Continuing to use the stress–strain relationship introduces a strong mesh dependency based on strain localization, such that the energy dissipated decreases as the mesh is refined. A different approach is required to follow the strain-softening branch of the stress–strain response curve. Hillerborg et al.’s (1976) fracture energy proposal is used to reduce mesh dependency by creating a stress–displacement response after damage is initiated. It is defined as the energy required to open a unit crack area, $G_{f}$

G_{f} = \frac{σ_{y 0} \times {\bar{u}}_{f}^{pl}}{2}

(5)

where $σ_{y 0}$ is the ultimate stress and ${\bar{u}}_{f}^{pl}$ is the equivalent plastic displacement at failure. With this approach, the softening response after damage initiation is characterized by a stress–displacement response rather than a stress–strain response. In this study, therefore, the fracture energy approach has been used as a damage evolution criterion in conjunction with JC damage initialization criteria. The damage evolution criterion assumes that the damage is characterized by the progressive degradation of material stiffness leading to its failure. It also takes into account the combined effect of different damage mechanisms acting simultaneously on the same material.

Numerical investigations

The numerical simulations were carried out using commercial finite element software ABAQUS/Explicit. The geometric model of the target and the projectile was made in ABAQUS/CAE (Figure 1(a) and (b)). The target plate and steel core of the 7.62 AP projectiles modeled as a deformable body have been made in accordance with the geometries used in Gupta and Madhu (1992, 1997). The shank diameter of the projectile was 6.06 and total length was 28.4 mm. The length of shank and ogival part was 20.75 and 7.65 mm, respectively. The target was modeled as square, 200 mm × 200 mm, and assigned fixed boundary conditions at the periphery. The projectile was assigned initial velocities equivalent to those obtained by Gupta and Madhu (1992, 1997). All the experimental results with respect to incident and residual projectile velocities are the average of the three tests considered (Gupta and Madhu, 1992, 1997). The jacket and lead cap of the AP projectile were not considered in this study. In this regard, there has been an extensive study relating to this subject in the literature. It was observed through numerical simulations by Borvik et al. (2009) that the brass jacket was just ripped off the hard steel core but not perforated into the steel target plate. Borvik et al. (2010) experimentally found that the ballistic limit velocity of AP bullet is 4%, 6%, and 12% less than that of its hard steel core when penetrating into 20, 40, and 60 mm 5083-H116 aluminum plates, respectively. Forrestal et al. (2010) experimentally found that the ballistic limit velocity of APM2 bullet was 1% and 8% less than that of its hard steel core when penetrating into 20 and 40 mm 7075-T651 aluminum plates, respectively. Chen et al. (2013) concluded that the rigid steel core of bullet dominates in the perforation, but the brass jacket, lead cap, and end cap have minor influence on the perforation process. Although the bullet deforms, the assumption of rigid projectile model is still applicable to predict the ballistic performance of the bullet. Therefore, it is concluded that the effect of brass jacket and lead cap has been neglected since it has little contribution to the perforation process. The size of the element for the projectile was considered to be 1.0 mm³. The interaction between the projectile and target was assigned using the kinematic contact algorithm considering the projectile as master surface and contact region of the target as slave surface. The meshing of the target and the projectile was done with C3D8R elements. For discretization, the target was divided into three different regions. The mesh sensitivity in the target was studied by varying the element size as 0.8, 0.6, 0.2, and 0.1 mm³ in the impact region corresponding to 15, 20, 60, and 120 elements at the target thickness (Iqbal et al., 2015). The projectile was impacted normally at an incidence velocity of 818 m/s on 12-mm-thick target and the residual velocity was found to be 669, 663, 658, and 657 m/s, respectively. In the central impact region equal to the diameter of projectile, the size of the element was considered to be 0.2 mm³. The JC material parameters proposed by the authors (Iqbal et al., 2015) were used to predict the material behavior of the target, while the material behavior of the projectile was incorporated from Niezgoda and Morka (2009). The complete failure coefficients of projectile materials (D₁–D₅) have not been described by Niezgoda and Morka (2009); however, the available material constants (only D₁) proposed by Niezgoda and Morka (2009) have been used in the present simulations considering the projectile as rigid in comparison with the mild steel targets. Gupta and Madhu (1997) also confirmed that no permanent deformation of projectile was observed in any of their experiments. The friction coefficient was assumed negligible between the projectile and 4.7–16 mm target thickness, considering the fact that the global structural deformation, such as combination of global bending and membrane action, and localized bulging take place during penetration on a typically thin target plate (<16 mm). The coefficient of friction was considered as 0.045 for 20- and 25-mm-thick targets since most of the projectile kinetic energy has to be absorbed in the highly localized shear zones surrounding the projectile due to ductile hole enlargement during penetration on a typically thick target plate (>16 mm). The material model of the target and the projectile considered in this study is shown in Table 1.

Figure 1.

Finite element model of (a) isometric view and (b) impact zone of 12.0-mm-thick target.

Table 1.

Material parameter for target and projectile materials.

Description	Notations	Target (22)	Projectile (23)
Modulus of elasticity	E (N/m²)	203 × 10⁹	202 × 10⁹
Poisson’s ratio	ν	0.33	0.32
Density	ρ (kg/m³)	7850	7850
Yield stress constant	A (N/m²)	304.330 × 10⁶	2700 × 10⁶
Strain hardening constant	B (N/m²)	422.007 × 10⁶	211 × 10⁶
Strain hardening constant	n	0.345	0.065
Viscous effect	C	0.0156	0.005
Thermal softening constant	m	0.87	1.17
Reference strain rate	${\dot{ε}}_{0}$	0.0001 s⁻¹	0.0001 s⁻¹
Melting temperature	$θ_{melt}$ (K)	1800	1800
Transition temperature	$θ_{transition}$ (K)	293	293
Fracture strain constant	D₁	0.1152	0.4
	D₂	1.0116	0
	D₃	−1.7684	0
	D₄	−0.05279	0
	D₅	0.5262	0

Results and discussion

The ballistic resistance of mild steel targets has been studied against 7.62 AP projectiles by carrying out the finite element simulations on ABAQUS/Explicit. The simulations were performed on 4.7, 6, 10, 12, 16, 20 and 25 mm thick targets at varying obliquity and until the occurrence of critical angle of ricochet. It should be noted that for 4.7- and 6-mm-thick targets, the experiments were performed only at the normal impact (Gupta and Madhu, 1997); hence, the finite element simulations have also been carried out at the normal incidence for these targets.

Effect of varying thicknesses of target on ballistic resistance

The experimental and numerical results of 4.7, 6, 10, 12, 16, 20 and 25 mm thick targets against normal incidence are presented in Table 2. The velocity drop was found to be almost linear against target thickness. The resistance of the target has been found to increase almost linearly with an increase in target thickness (see Figure 2). The residual projectile velocity for 4.7 and 6 mm thick target has been predicted with 2% and 1% deviation, respectively. The predicted residual projectile velocity for 10- and 12-mm-thick target was found to be exactly the same as experimental residual velocity. At 16 and 20 mm thick targets, a maximum deviation of 6% was found between the actual and predicted residual velocities. At 25-mm-thick target, a maximum deviation of 50% was found between the actual and predicted residual velocities. It may be due to the fact that the experimental residual velocity might not be measured accurately.

Table 2.

Comparison of experimental and numerical results against normal impact.

Thickness of target (mm)	Impact velocity (m/s)	Residual velocity (m/s)
Thickness of target (mm)	Impact velocity (m/s)	Experimental results	Numerical results
4.7	821.0	758.6	757.20
6.0	866.3	792.2	799.79
10.0	827.5	702.2	701.40
12.0	818.0	661.5	662.42
16.0	819.7	562.0	594.09
20.0	820.6	404.8	429.15
25.0	842.3	107.6	272.62

Figure 2.

Residual velocity versus target thickness against normal impact.

The 7.62 AP projectile failed the target through hole enlargement. The formation of petals at the front surface and bulge at the rear surface has also been witnessed through experiments (Figures 3 and 4). The numerical simulations accurately predicted the failure mode, including the size of hole, petalling at the front, and the bulge at the rear surface. The hole actually formed in the target was slightly bigger in size at the front than at the rear surface. The diameter of the hole in 12-mm-thick target was 9.6 and 9.0 mm at the front and rear surface, respectively. A similar variation in the size of hole was predicted through the numerical simulations. The predicted diameter of the hole was 9.32 and 7.66 mm at the front and rear surface, respectively. The actual and predicted failure modes of 20-mm-thick target are compared in Figure 5 at normal impact, and these have been found to be in close agreement. The failure of the target occurred through ductile hole enlargement, making a clear circular hole in the target and a bulge at the front surface. The size of the hole was larger at the front and smaller at the rear surface of the target.

Figure 3.

Front surface failure mode of 12-mm-thick target.

Figure 4.

Rear surface failure mode of 12-mm-thick target.

Figure 5.

Front side failure surface of 20-mm-thick target.

Effect of obliquity on various thicknesses of target

The experimental and numerical results for 10, 12, 16, 20 and 25 mm thick targets are presented in Table 3 corresponding to varying obliquity. The experimental and numerical results with respect to failure mechanism, residual projectile velocity, maximum angle of perforation, effect of layer, and critical angle of ricochet have been compared.

Table 3.

Experimental and numerical results of monolithic target impacted with varying angle of incidence.

Thickness of target (mm)	Oblique angle (°)	Impact velocity (m/s)	Residual velocity (m/s)
Thickness of target (mm)	Oblique angle (°)	Impact velocity (m/s)	Experimental results	Numerical results
10	00	827.5	702.2	701.40
	15	815.0	690.4	679.48
	30	825.7	654.0	674.24
	45	790.0	500.0	556.18
	52	819.9	–	517.31
	54	819.9	–	452.09
	56	819.9	–	366.62
	60	819.9	493.7	Ricochet
	61.5	827.9	293.6
	62	821.4	0
12	00	818.0	661.5	662.42
	15	842.7	671.6	677.73
	30	801.8	598.0	603.97
	45	808.0	555.3	515.82
	50	808.0	–	409.59
	53	815.3	–	318.76
	57	809.0	368.9	0.0
	59	815.3	Ricochet	Ricochet
16	00	819.7	562.3	594.09
	15	817.3	544.4	577.83
	30	817.7	496.3	526.09
	45	806.1	0.0	314.88
	47	819.2	–	266.01
	49	819.2	–	133.18
	51	819.2	Ricochet	0.0
	58	819.2		Ricochet
20	00	820.6	404.8	429.15
	15	825.6	295.6	408.72
	30	817.5	151.9	312.77
	35	817.5	–	232.35
	38	817.5	–	106.62
	45	797.3	0.0	0.0
	51	812.8	Ricochet	0.0
	57	817.5		Ricochet
25	00	842.3	107.6	272.62
	15	799.5	0.0	14.0
	50	819.0	Ricochet	0.0
	57	819.0		Ricochet

The experimental and numerical results for 10-mm-thick target are presented in Table 3 corresponding to varying incidence angles. The target was impacted experimentally at 0°, 30°, 45°, 60°, 61.5°, and 62° at incidence velocities of 790–827 m/s; however, it experienced perforation up to 61.5° obliquity (Figure 6(a)). The simulations, however, predicted the perforation up to 56° obliquity. At 58° obliquity, the simulations predicted embedment, while at 60° obliquity, the simulations predicted critical projectile ricochet. The velocity drop at 0°, 15°, and 30° obliquity was found to be almost identical, 15%, and this finding was also witnessed through finite element simulations. At 45° obliquity, a maximum deviation of 10% has been found between the actual and predicted residual velocities. The actual and predicted failure mechanisms of 10-mm-thick target at 59.7° obliquity and 58° obliquity, respectively, are compared in Figure 7(a) and (b). Close correlation between the two has been found. However, it should be noted that experimentally the projectile perforated the target; however, the simulations predicted the embedment of the projectile. The residual projectile velocity could not be obtained through experimental results for this test. However, the actual residual velocity at 45° obliquity was 500 m/s, which has been reproduced, 556 m/s, within 10% deviation. However, at 60° obliquity, the actual residual velocity was 493.7 m/s, while at 61.5° obliquity, it was 293.6 m/s. It should be noted here that with an increase in target obliquity from 45° to 60°, the residual velocity remained almost the same, while with a further increase of 1.5° obliquity, it suddenly dropped almost 200 m/s. Thus, the actual residual velocities at 60° and 61.5° obliquity seem to have been overestimated.

Figure 6.

Residual velocity of projectile function of obliquity for (a) 10 mm, (b) 12 mm, (c) 16 mm, (d) 20 mm, and (e) 25 mm thick target.

Figure 7.

(a) Perforation of 10-mm-thick target at 59.7° and (b) embedment of the projectile at 58° obliquity.

In case of 12-mm-thick target, the angle of incidence during experimentation was varied as 0°, 15°, 30°, 45°, 57°, and 59° obliquity (see Table 3). At 0°, 15°, and 30° obliquity, the predicted residual velocities are in close agreement with their actual values (see Figure 6(b)). The maximum difference between the actual and predicted residual velocity was found to be 7.6% at angle of incidence 45°. At 57° obliquity, the experimental results suggested perforation of target with a residual velocity of 368.9 m/s. At the same angle of obliquity, however, the numerical results predicted that the projectile embedded in the target. The residual velocity, 368.9 m/s, measured during experiments through the optical measurement system could be the velocity of the fragments ejected out of the projectile or target material. At 59° obliquity, both actual and predicted results showed ricochet of the projectile (Figure 8). The projectile has registered an elliptical deformation pattern and erosion of material while sliding over the target surface. An exact pattern of deformation and material erosion has been predicted through the numerical simulations.

Figure 8.

Deformation of 12-mm-thick target at 59° obliquity of (a) experiments and (b) simulations.

The experimental and numerical results of 16-mm-thick target are presented in Table 3 corresponding to 0°, 15°, 30°, 45°, 47°, 49°, 51°, and 58° obliquity (see Figure 6(c)). At normal incidence, the velocity drop of the projectile was found to be 31%. A maximum deviation of 6% was found between the actual and predicted residual velocities. However, at 45° obliquity, the experiments revealed the embedment of the projectile, while the numerical simulations predicted perforation. The embedment of the projectile, however, was predicted at 51° obliquity. The critical ricochet of the projectile was found to occur at 51° obliquity through experiments and at 58° obliquity through finite element simulations (see Table 3). The actual and predicted deformation of the target as a result of projectile ricochet has been compared in Figure 9. The projectile has registered an elliptical deformation pattern and caused erosion of material while sliding over the target surface. An exact pattern of deformation and material erosion has been predicted through the finite element simulations.

Figure 9.

Deformation of 16-mm-thick target as a result of projectile ricochet (a) observed at 54.6° obliquity and (b) predicted at 58° obliquity.

The experimental and numerical results of 20-mm-thick target are presented in Table 3 corresponding to varying angles of obliquity. The target was impacted at 0°, 15°, 30°, and 45° obliquity at incidence velocity of 812–825 m/s (see Figure 6(d)). At normal incidence, the velocity drop of the projectile was found to be 50%. However, at 15° obliquity, it was almost 64%, and at 30° obliquity, 81%. The actual and predicted residual projectile velocities at the normal impact have been found in close agreement. At 15° and 30°obliquity, a larger difference between the experimental and numerical residual velocities was found. It may be due to the fact that the experimental residual velocity might not be measured accurately. This is also evident from the experimental results at normal impact, wherein the residual velocity was measured to be 404.8 m/s. However, at 15° obliquity, there is a sudden drop in the measured residual velocity which seems to be unrealistic. At 45° obliquity, the projectile embedded after hitting the target. The numerical simulations also witnessed embedment of projectile at 45° obliquity. The actual critical angle of ricochet was found to be 51° and it was predicted to be 57° obliquity.

The experimental and numerical results of 25-mm-thick target are presented in Table 3 for different angles of obliquity. The target was impacted at 0°, 15°, and 50° obliquity at incidence velocity of 799–842 m/s (see Figure 6(e)). However, it experienced perforation only at the normal impact. At normal incidence, the velocity drop of the projectile was found to be 87%. At 15° obliquity, the projectile lost all the kinetic energy, while the finite element simulations predicted the perforation with a very low residual velocity, 14 m/s. At 50° obliquity, the experiments showed critical ricochet of the projectile, while the same has been predicted to be at 57° obliquity.

The predicted critical angle of projectile ricochet has been compared with the experiments and found to be reasonably in good agreement. The maximum difference between the actual and predicted critical projectile ricochet was found to be 12%, at a thickness of 25 mm. It may be concluded that the actual critical angle of ricochet decreased with an increase in target thickness (see Figure 10(a)). The actual angle of critical ricochet for 10, 12, 16, 20 and 25 mm thick target was found to be 62°, 59°, 51°, 51°, and 50°, respectively, and the same has been reproduced to be 60°, 59°, 58°, 57° and 57°, respectively (see Table 4). It may also be concluded that the maximum angle for perforation also decreased with an increase in target thickness (see Figure 10(b)). Up to a thickness of 20 mm, the simulations are found to be in good agreement with available experiments. The maximum difference between the actual and predicted critical projectile ricochet was found to be 17%, at 25-mm-thick target. It may be due to the fact that the amount of local work continues to increase up to a thickness of 20 mm, while the amount of global work may decreases thereafter with an increase in thickness. Therefore, it is concluded that as the thickness is increased, more and more energy is absorbed in combination with localized shearing and hole enlargement. The simulations are not able to predict the damages due to the problem involving localized shearing actually not been accounted in the simulations. The maximum angle of perforation for 10-, 12-, 16-, 20- and 25-mm-thick target was 61°, 59°, 45°, 45°, and 0°, respectively, and the same was reproduced to be 58°, 56°, 51°, 40°, and 17°, respectively (see Table 4).

Figure 10.

Plots for (a) critical angle of ricochet and (b) maximum angle of perforation.

Table 4.

Critical angle of ricochet and maximum angle for perforation of mild steel target.

Thickness of target (mm)	Experimental results	Numerical results
	Critical angle of ricochet (°)
10	62°	60°
12	59°	59°
16	51°	58°
20	51°	57°
25	50°	57°
	Maximum angle of perforation (°)
10	61°	58°
12	59°	56°
16	45°	51°
20	45°	40°
25	0°	17°

Ballistic limit evaluation

The ballistic limit of 4.7, 6, 10, 12, 16, 20 and 25 mm thick mild steel target has also been obtained numerically against 7.62 AP projectile at normal impact. The numerical simulations were carried out at assumed incidence velocities to obtain the ballistic limit. The ballistic limit velocity was calculated as the average of the highest projectile velocity not giving perforation and the lowest projectile velocity giving complete perforation of the target. After obtaining the ballistic limit velocity, the residual projectile velocity corresponding to a given incidence velocity was also calculated using the Recht–Ipson model. The residual velocities were calculated based on the following model originally proposed by Recht and Ipson (1963)

v_{r} = a {(v_{i}^{p} - v_{bl}^{p})}^{\frac{1}{p}}

(6)

where $v_{i}$ , $v_{r}$ , and $v_{bl}$ are initial, residual, and ballistic limit velocity, and “a” and “p” are the model constants. These calculated residual projectile velocities for 4.7, 6, 10, 12, 16, 20 and 25 mm thick target are shown in Figure 11. The least square method was used to obtain a best fit to the numerical data and thus calibrate the parameters “a” and “p” of the model. For 4.7, 6, 10, 12, 16, 20 and 25 mm thick target, the ballistic limit velocity, V₅₀, was found to be 274, 304.5, 400.5, 447.5, 533, 682.5, and 791 m/s, and the corresponding model parameters “a” and “p” were calibrated to be 1 and 1.9, respectively (see Table 5). The value of Recht and Ipson model parameters “a” and “p” is 1 and 1.9, respectively, which is same for all given target thicknesses. Figure 11 shows the impact and residual velocity data points and the corresponding Recht–Ipson fit through solid lines for 4.7, 6, 10, 12, 16, 20 and 25 mm thick target. The ballistic limit of 25-mm-thick target was found to be 13%, 32%, 43%, 49%, 61%, and 65% higher than 4.7, 6, 10, 12, 16, 20 and 25 mm thick target respectively. Moreover, the ballistic limit was found to increase almost linearly with an increase in target thickness (see Figure 12).

Figure 11.

Impact and residual velocity of projectiles with function of thickness of targets.

Table 5.

Ballistic limit velocity corresponding target thickness.

Thickness of target (mm)	Ballistic limit velocity (m/s)	Recht–Ipson constants
Thickness of target (mm)	Ballistic limit velocity (m/s)	a	p
4.7	274.0	1	1.9
6.0	304.5
10.0	400.5
12.0	447.5
16.0	533.0
20.0	682.5
25.0	791.0

Figure 12.

Ballistic resistance of varying thicknesses of mild steel targets.

Experimental and numerical results of layered targets

Numerical simulations were carried out to evaluate the response of layered plates of 4.7, 2 × 4.7, 3 × 4.7, 4 × 4.7, 5 × 4.7, and 6 × 4.7 mm thick mild steel under the normal impact against AP projectile (see Table 6). The projectiles perforated all the layered configurations and its impact, and residual velocities were obtained in each case and compared with the experimental results. The experimental and numerical results were found to be reasonably in good agreement for monolithic, two-layer, three-layer, and six-layer configurations. For four-layer and five-layer configurations, the numerical results are underpredicted to be almost 9% and 15% compared with experimental results. Gupta and Madhu (1997) observed small petals on the impacted side of the front plate, and extrusion of material was seen toward the distal side of the rear layers and the same results were predicted in this study (see Figure 13). It was observed that the perforation surface in the first and last layer of chosen target configuration is almost the same (straight or outward) as that of experiments except the second layer (inward). It may be due to the fact that the extrusion of material from the rear side of first layer (accumulation of fragments and debris) is more during the travel of projectile up to second layer, and this effect may be diminished after the second layer. After the projectile travels into the second layer, the target experiences hole enlargement, and therefore, perforation surface seems straight. The numerical simulations were also carried out in layered in-contact plates of 1 × 6, 2 × 6, 3 × 6, 4 × 6, 5 × 6 and 6 × 6 mm thickness under normal impact (see Table 6). The experiment and numerical results were found to be in close agreement for two- and three-layer configurations, as well as for six-layer configuration. For four-layer and five-layer configurations, the numerical results are underpredicted to be almost 18% and 44% compared with experimental results. This may be due to the fact that the experimental residual velocity might not be measured accurately. The deformed profiles of layered targets against AP projectile are shown in Figure 14(a) to (e). In addition to that, numerical simulations were also performed on layered in-contact plates of 10 + 10, 10 + 16, 16 + 10, 10 + 20, 20 + 10, 16 + 20, 20 + 16 and 25 + 12 mm thickness under normal impact (Table 6). The experimental and numerical results were found to be in close agreement except for 10 + 16 and 16 + 10 layered configurations.

Table 6.

Comparison of experimental and numerical results of layered targets.

Layer configuration	Impact velocity (m/s)	Residual velocity (m/s)
Layer configuration	Impact velocity (m/s)	Experimental results	Numerical results
1 × 4.7	821.0	758.6	757.20
2 × 4.7	818.3	711.3	698.28
3 × 4.7	827.1	636.6	635.36
4 × 4.7	826.3	607.3	553.70
5 × 4.7	820.3	510.0	442.55
6 × 4.7	811.7	303.0	307.68
1 × 6	866.3	792.2	799.79
2 × 6	869.6	724.6	720.14
3 × 6	863.9	612.8	612.16
4 × 6	874.7	488.8	397.51
5 × 6	866.8	378.0	210.94
6 × 6	862.6	0.0	0.0
10 + 10	826.8	410.0	437.41
10 + 16	820.1	113.2	177.75
16 + 10	833.1	117.3	198.45
10 + 20	821.0	Embed	Embed
20 + 10	811.9	Rebound	Rebound
16 + 20	812.0	Embed	Embed
20 + 16	805.7	Rebound	Rebound
25 + 12	810.3	Rebound	Rebound

Figure 13.

Comparison of experimental and numerical results of 4.7-mm-thick layered in-contact target: (a) monolithic, (b) two-layer configuration, (c) three-layer configuration, (d) four-layer configuration, and (e) five-layer configuration.

Figure 14.

Deformation of (a) 6, (b) 2 × 6, (c) 3 × 6, (d) 4 × 6, and (e) 5 × 6 mm thick layered in-contact targets.

Effect of number of layer on layered target

The numerical investigations were carried out in monolithic and layered targets with a constant initial velocity of 821 m/s. The performance of layered targets of 1 × 4.7, 2 × 4.7, 3 × 4.7, 4 × 4.7, 5 × 4.7, and 6 × 4.7 mm are compared with the equivalent thick monolithic target of 4.7, 9.4, 14.1, 18.8, 23.5, and 28.2 mm, which is shown in Figure 15. The resistance of layered in-contact and monolithic targets was found to be same up to four layers, that is, 4 × 4.7 and 18.8 mm thickness, respectively. The residual velocity of projectile increased almost 50% at 5 × 4.7 mm thickness compared to equivalent thick monolithic target of 23.5 mm. The 28.2-mm-thick monolithic target stopped projectile velocity of 821 m/s, while at 6 × 4.7 configuration, the residual velocity was found to be 307.68 m/s (see Table 7). Therefore, it is concluded that the resistance of equivalent monolithic targets was found to be the same compared to first, second, third, and fourth layers of layered targets; thereafter, the monolithic target offered better ballistic resistance than the layered target, that is, the fifth and sixth layers.

Figure 15.

Resistance of 4.7-mm-thick monolithic and layered target.

Table 7.

Resistance of monolithic and layered target.

Monolithic target (mm)	Layered targets	Impact velocity (m/s)	Residual velocity (m/s)
Monolithic target (mm)	Layered targets	Impact velocity (m/s)	Monolithic target	Layered targets
4.7	1 × 4.7	821.0	772.36	757.20
9.4	2 × 4.7		695.82	698.28
14.1	3 × 4.7		630.64	635.36
18.8	4 × 4.7		532.73	553.70
23.5	5 × 4.7		301.53	442.55
28.2	6 × 4.7		0.0	307.68

Effect of spacing in between the targets

Numerical simulations were carried out in three layers of 4.7 and 6 mm target with spacing of 10, 20, 30, 40, 50, and 60 mm against normal impact with an incidence velocity of 824.7 m/s (see Table 8). For 4.7-mm-thick target, the residual velocity was found to increase almost linearly with an increase in spacing between the targets of 10, 20, 30, 40, 50, and 60 mm. For 6-mm-thick target, however, the residual velocity of projectile increased initially when the spacing increased from 0 to 20 mm, and subsequently, the residual velocity became almost constant (Iqbal et al., under revision). The variation in spacing at normal impact was found to have influence as long as the spacing was smaller than the projectile length.

Table 8.

Effect of configuration on spaced targets.

Layer configuration	Spacing between targets (mm)	Impact velocity (m/s)	Residual velocity (m/s)
3 × 4.7 mm	0	824.7	635.36
	10		641.36
	20		644.40
	30		651.17
	40		660.33
	50		661.98
	60		663.72
3 × 6 mm	0		577.85
	10		598.00
	20		604.70
	30		605.93
	40		607.36
	50		609.09
	60		610.27

Conclusion

The ballistic performance of mild steel targets has been studied under varying obliquity and configurations against 7.62 AP projectiles. The numerical simulations have been carried out wherein mild steel plates of thicknesses 4.7, 6, 10, 12, 16, 20 and 25 mm were impacted at varying oblique angles and the results obtained have been compared with the experiments performed earlier by the authors. The experimental and numerical results with respect to failure mechanism, residual projectile velocity, maximum angle of perforation, and critical angle of ricochet have been compared. The effect of spacing between the layers has also been studied numerically and the following conclusions are drawn.

In general, a close correlation between the experimental findings and the predicted results has been found. The resistance of the target has been found to increase with an increase in target obliquity. For 10, 12 and 16 mm thick targets, however, the resistance of the target remained almost same up to 30° obliquity and thereafter increased with a further increase in angle of obliquity. For 20-mm-thick target, the residual velocity of the projectile decreased linearly with an increase in target obliquity. For 25-mm-thick target, the perforation occurred only at the normal incidence. The critical angle of projectile ricochet has been found to decrease with an increase in target thickness. The ballistic limit for all the mild steel targets has also been obtained numerically. The ballistic limit of 25 mm thickness increased by 13%, 32%, 43%, 49%, 61%, and 65% in comparison with 20, 16, 12, 10, 6 and 4.7 mm thickness, respectively, at normal incidence. The ballistic limit has been found to increase almost linearly with an increase in target thickness.

Numerical simulations were also carried out to evaluate the response of layered and equivalent thick monolithic plates of 4.7 and 6 mm under the normal impact, and the layers are varied as two, three, four, five, and six. The experimental and numerical results were found to be reasonably in good agreement.

The resistance of target decreased linearly with an increase in spacing between the layers when the layer thickness was 4.7 mm. When the layer thickness increased to 6 mm, the residual velocity increased initially when spacing increased from 0 to 20 mm, and subsequently, it became almost constant. The variation in spacing at normal impact was found to have influence as long as the spacing was smaller than the projectile length.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

ABAQUS (2016) Explicit User’s Manual Version 6.14.

Awerbuch

Bodner

(1977) An investigation of oblique perforation of metallic plates by projectiles. Experimental Mechanics 17: 147–153.

Borvik

Dey

Clausen

(2009) Perforation resistance of five different high-strength steel plates subjected to small-arms projectiles. International Journal of Impact Engineering 36: 948–964.

Borvik

Forrestal

Warren

(2010) Perforation of 5083-H116 aluminum armor plates with ogive-nose rods and 7.62 mm APM2 bullets. Experimental Mechanics 50(7): 969–978.

Borvik

Hopperstad

Berstad

. (2002) Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses, part II: numerical simulations. International Journal of Impact Engineering 27: 37–64.

Borvik

Olovsson

Dey

. (2011) Normal and oblique impact of small arms bullets on AA6082-T4 aluminium protective plates. International Journal of Impact Engineering 38: 577–589.

Chen

Gao

(2013) Analysis on the perforation of ductile metallic plates by APM2 bullets. International Journal of Protective Structures 4: 65–78.

Forrestal

Borvik

Warren

(2010) Perforation of 7075-T651 aluminum armor plates with 7.62 mm APM2 bullets. Experimental Mechanics 50(8): 1245–1251.

Goldsmith

Finnegan

(1986) Normal and cylindrical and oblique projectiles impact of cylindro-conical on metallic plates. International Journal of Impact Engineering 4: 83–105.

10.

Gupta

Madhu

(1992) Normal and oblique impact of a kinetic energy projectile on mild steel plates. International Journal of Impact Engineering 12: 333–343.

11.

Gupta

Madhu

(1997) An experimental study of normal and oblique impact of hard-core projectile on single and layered plates. International Journal of Impact Engineering 19: 395–414.

12.

Hancock

Mackenzie

(1976) On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. Journal of the Mechanics and Physics of Solids 24: 147–169.

13.

Hillerborg

Modeer

Petersson

(1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6: 773–782.

14.

Iqbal

Gupta

(2010) 3D numerical simulations of ductile targets subjected to oblique impact by sharp nosed projectiles. International Journal of Solids and Structures 47: 224–237.

15.

Iqbal

Senthil

Bhargava

. (2015) The characterization and ballistic evaluation of mild steel. International Journal of Impact Engineering 78: 98–113.

16.

Iqbal

Senthil

Madhu

. (Under revision) Oblique impact on single, layered and spaced mild steel targets by 7.62 AP projectiles. International Journal of Impact Engineering.

17.

Iqbal

Senthil

Sharma

. (2016) An investigation of constitutive behavior of Armox 500T steel and armour piercing incendiary projectile material. International Journal of Impact Engineering 96: 146–164.

18.

Johnson

EAF

Salesh

Edwards

(2011) Ballistic performance of multi-layered metallic plates impacted by a 7.62-mm APM2 projectile. International Journal of Impact Engineering 38: 1022–1032.

19.

Johnson

Cook

(1983) A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the 7th international symposium on ballistics, The Hague, 19–21 April, pp. 541–547. Netherlands: American Defense Preparedness Association, Koninklijk Instituut van Ingenieurs.

20.

Johnson

Cook

(1985) Fracture characteristics of three metals subjected to various strains, strain rates, temperatures, and pressures. Engineering Fracture Mechanics 21: 31–48.

21.

Johnson

Sengupta

Ghosh

(1982) High velocity oblique impact and ricochet mainly of long rod projectiles: an overview. International Journal of Impact Engineering 24: 425–436.

22.

Nia

Hoseini

(2011) Experimental study of perforation of multi-layered targets by hemispherical-nosed projectiles. Materials & Design 32: 1057–1065.

23.

Niezgoda

Morka

(2009) On the numerical methods and physics of perforation in the high-velocity impact mechanics. World Journal of Engineering 6: 414–416.

24.

Recht

Ipson

(1963) Ballistic perforation dynamics. Journal of Applied Mechanics 30: 384–390.

25.

Senthil

Iqbal

(2013) Effect of projectile diameter on ballistic resistance and failure mechanism of single and layered aluminum plates. Theoretical and Applied Fracture Mechanics 67–68: 53–64.

26.

Senthil

Tiwari

Iqbal

. (2013) Impact response of single and layered thin plates. Proceedings of the Indian National Science Academy 79(4): 705–716.

27.

Teng

Chu

Chang

. (2005) Numerical analysis of oblique impact on reinforced concrete. Cement and Concrete Composites 27: 481–492.

28.

Teng

Wierzbicki

(2004) Effect of fracture criteria on high velocity perforation of thin beams. International Journal of Computational Methods 1: 171–200.