A rigid projectile perforation model for metallic targets with the free-surface and fracture effects

Abstract

The metallic targets are regarded as the compressible power-law strain-hardening materials, and the resistance function coefficients for three kinds of steel and 11 kinds of aluminum targets are obtained based on the spherical cavity-expansion theory. Then, an explicit expression for projectile impact resistance is proposed by curve-fitting, which is simple to use without performing the complex numerical solution. For the projectile perforating on the finite thickness plate, the free-surface effect is realized by multiplying a decay function with the above explicit resistance. Furthermore, a new phenomenological perforation model considering the projectile entry and exit phases, compressibility, rear free-surface and fracture effects of target is established, which is sufficiently verified by comparing with the existing 19 sets of projectile perforation tests and two typical theoretical models. Finally, the influence of the impact velocity on the rear free-surface effect is discussed.

Keywords

Projectile perforation free-surface effect compressibility strain-hardening material

Introduction

Terminal ballistic parameters of kinetic energy projectiles perforating on the monolithic or multi-layered metallic targets are concerned by both weapon and armor designers. The energy consumption of a rigid projectile perforation is dominated by both perforation process and the target failure mode (ductile hole enlargement, shear plugging, petaling, etc.). Ductile hole enlargement (Figure 1) is a common failure mode for the sharp-nosed projectile perforating on the ductile metallic target, which is mainly discussed in this article.

Figure 1.

Ductile hole enlargement failure mode of the sharp-nosed projectile perforating metallic plate: (a) experimental picture (Børvik et al., 2010) and (b) sketch.

Cavity-expansion theory (CET) is widely used to predict the target resistance of the metallic target against projectile impact. Bishop et al. (1945) first presented CET for the quasi-static expansions of spherical and cylindrical cavities in elastic–plastic solids, which was then used to estimate the resistance of the target. Hill (1948) derived equations for the dynamic CET of an incompressible, elastic–plastic material; and Goodier (1965) further applied the equations in Hill (1948) in predicting the depths of penetration for rigid spheres into semi-infinite metallic targets. Furthermore, Forrestal and Luk (1988) and Luk et al. (1991) introduced similarity transformation to obtain the numerical solution of the dynamic spherical CET, in which the metallic target was considered as the compressible elastic–plastic material and compressible power-law strain-hardening material, respectively. Both of the calculation procedures in Forrestal and Luk (1988) and Luk et al. (1991) are complex, and there is no explicit formula for the resistance of the compressible target.

For the rigid projectile penetration/perforation models established on CET, Forrestal et al. (1990) proposed a perforation model for the conic-nosed rigid projectile striking into aluminum target, where the target is regarded as the incompressible power-law strain-hardening material. Then, Forrestal and Warren (2009) extended the above perforation model and obtained the closed-form and accurate perforation equations for the conic and ogive-nosed projectile. Chen and Li (2003) presented a model for the rigid projectile perforation on the relatively thick plate, in which the target was treated as the incompressible elastic–plastic and strain-hardening material. Both of the two models in Forrestal and Warren (2009) and Chen and Li (2003) assumed that the reduced energy consumption in the exit phase is basically equal with the increased energy in the entry phase and the projectile nose is completely embedded in the target medium, which are applicable for thick plate perforation analyses and are essentially the same in the formulae of ballistic limit and residual velocity (Chen et al., 2011). Furthermore, Li et al. (2004) defined a generalized geometry function of projectile with considering the projectile entry phase, and established a perforation model for rigid projectile striking into moderately thick plate. The target is regarded as compressible elastic–plastic material, and the exit phase as well as the rear free-surface effect is not considered in their model. It should be pointed out that the resistance function determined by the CET is based on the infinite thickness target assumption, which should be modified when applied in the finite thickness target due to the rear free-surface effect. Warren and Poormon (2001) constructed a decay function for the free-surface effect during the oblique penetration. The resistance function for the finite thickness target was obtained by multiplying the resistance for infinite target obtained by the CET with the decay function. Then, Jiang et al. (2007) established a three-stage perforation model with the consideration of the free-surface and fracture effects, in which the resistance function was determined based on the incompressible elastic–plastic assumption.

The impact resistance function for the incompressible material could be easily obtained based on the CET (e.g., Forrestal and Luk, 1988; Luk et al., 1991). However, it could only be determined numerically for the compressible material. Therefore, most of the existing models treated the target as the incompressible material, and the rear free-surface effect of target was always ignored. Both of the above two simplifications will overestimate the projectile impact resistance and are non-conservative for the protective design of armor materials.

In this article, the resistance function coefficients for 14 sets of armor materials are first determined based on the dynamic spherical CET proposed in Luk et al. (1991), where the target is considered as the compressible power-law strain-hardening material. Then, an explicit projectile impact resistance expression dependent on the target material properties is proposed by curve-fitting. Considering the rear free-surface effect, the resistance function for the finite thickness plate is obtained by adopting a decay function proposed by Warren and Poormon (2001). Furthermore, a new phenomenological perforation model considering the entry and exit phases, the compressibility and the rear free surface as well as the fracture effects of the target is established. The predictions of the present and two existing models are compared with a series of perforation test data. Finally, the influence of the impact velocity on the rear free-surface effect is discussed.

Resistance function of the compressible power-law strain-hardening targets

In this section, based on the dynamic spherical CET, the coefficients of the resistance functions for 14 kinds of armor materials are derived and the analytical expression of the resistance function of the compressible power-law strain-hardening metallic target is deduced.

During the projectile penetration, it is assumed that a spherical cavity is formed in the target, which expands from the projectile surface outwards at a constant radial speed v_c. For the metallic target, the cavity-expansion usually produces plastic and elastic regions, as shown in Figure 2, where r_c, r_p, and r_d are the radii of the cavity, plastic region, and elastic region, respectively. v_c, c, and v_d denote the radial velocities at cavity surface, elastic–plastic interface, and elastic interface, respectively. r and v are the radial distance and radial velocity, respectively.

Figure 2.

Response regions for the spherical cavity-expansion.

Most of the armor materials can be described by the power-law strain-hardening constitutive relationship, which is formulated as

{\begin{cases} σ = E ε, σ ⩽ Y \\ σ = Y {(E ε / Y)}^{n}, σ > Y \end{cases}

(1)

where σ and ε are the true stress and true strain, respectively. E, Y, and n are the Young’s modulus, yield stress, and strain-hardening exponent, respectively. For instance, Figure 3 shows the true strain–stress data of 6061-T651 aluminum (Luk et al., 1991), Weldox 460E steel (Dey et al., 2004), and 7075-T651 aluminum (Forrestal et al., 1992) as well as the power-law data-fitting curves, in which the blue lines denote the yield stresses.

Figure 3.

Power-law strain-hardening constitutive curves of three metals.

Based on the conservation equations of the mass and momentum of the target material, by using the constitutive relationship of the target material and the similarity transformation, the radial stress at the cavity surface can be obtained. The detailed numerical procedures are given in Luk et al. (1991), which will not be repeated here.

Following the numerical procedures suggested in Luk et al. (1991), the relationship between the radial stress at the cavity surface and the cavity-expansion velocity can be determined as follows

\frac{σ_{r}}{Y} = (A + B w^{2} v_{c}^{2})

(2)

where $σ_{r}$ is the radial stress, measured positive in compression. $w = \sqrt{ρ_{t} / Y}$ and ρ_t are the density of the target. $v_{c} = V \sin φ$ and V are the instantaneous velocity of projectile. $ϕ$ is the angle between the projectile axial with the tangent of an arbitrary point in the projectile nose surface. A and B are the dimensionless coefficients. For the incompressible power-law strain-hardening material, A and B have explicit expressions as (Luk et al., 1991)

A = \frac{2}{3} [1 + {(\frac{2 E}{3 Y})}^{n} \int_{0}^{1 - (3 Y / 2 E)} \frac{{(- \ln x)}^{n}}{1 - x} d x], B = 1.5

(3)

While for the compressible power-law strain-hardening material, the implicit differential conversation equations (Equation (32) in Luk et al. (1991)) have to be solved numerically to obtain the radial stress σ_r at the cavity surface and the corresponding cavity-expansion velocity v_c. Then, the coefficients A and B can be obtained by fitting the σ_r − v_c data. Table 1 lists the typical armor materials, including three kinds of steels and 11 kinds of aluminum. Taking 6061-T6 aluminum in Table 1 as an example, following the solution procedures suggested in Luk et al. (1991), Figure 4(a) and (b) show the curves of c − v_c and σ_r − v_c, respectively, in which ρ₀ is the initial density of target. By data-fitting, A = 4.403 and B = 1.132 can be obtained, which are consistent with the results given by Luk et al. (1991). Therefore, the solution procedure is validated.

Table 1.

Coefficients of impact resistance function for typical armor materials.

Ref.	Target						Resistance coefficients
	Materials		E (GPa)	ρ_t (g/cm³)	Y (MPa)	n	A	B	A ₁	B ₁
Dey et al. (2004)	Steel	Weldox 460E	210	7.85	499	0.0893	5.4294	1.1315	5.6199	1.5
Dey et al. (2004)		Weldox 700E	210	7.85	859	0.0529	4.4122	1.1277	4.5966
Børvik et al. (2002)		Weldox 460E	200	7.85	490	0.1366	6.2078	1.1736	6.4168
Børvik et al. (2010)	Aluminum	5083-H116	71	2.66	240	0.108	5.3044	1.1442	5.5049
Forrestal et al. (1990)		5083-H131	70.3	2.66	276	0.084	4.8202	1.1315	5.0068
Forrestal et al. (1992)		7075-T651	73.1	2.71	448	0.089	4.4131	1.0814	4.6090
Børvik et al. (2004)		5083-H116	70	2.7	157	0.205	7.9089	1.1520	8.1589
Rosenberg and Forrestal (1988)		6061-T6	68.9	2.71	276	0.051	4.4029	1.1322	4.5936
Piekutowski et al. (1996)		6061-T651	69	2.71	262	0.085	4.8665	1.1330	5.0543
Forrestal et al. (2014)		6082-T651	69	2.71	265	0.06	4.5463	1.1324	4.7341
Holmen et al. (2013)		6070-T0	70	2.7	51	0.213	11.0193	1.2522	11.275
Holmen et al. (2013)		6070-T4	70	2.7	187	0.166	6.6065	1.1364	6.8535
Holmen et al. (2013)		6070-T6	70	2.7	373	0.05	4.1058	1.0750	4.3259
Holmen et al. (2013)		6070-T7	70	2.7	341	0.036	4.0519	1.0852	4.2632

Figure 4.

Relationships of the cavity-expansion velocity with (a) elastic–plastic interface velocity and (b) radial stress.

The resistance coefficients A and B of the metallic targets from Børvik et al. (2010), Dey et al. (2004), Forrestal et al. (1992), Børvik et al. (2002), Børvik et al. (2004), Rosenberg and Forrestal (1988), Piekutowski et al. (1996), Forrestal et al. (2014), and Holmen et al. (2013) are obtained and listed in Table 1, in which A₁ and B₁ are the corresponding resistance coefficients of the incompressible power-law strain-hardening material given by Equation (3). It should be pointed out that the strain-hardening exponent n for 5083-H116 aluminum, Weldox 460E steel, and Weldox 700E steel from Børvik et al. (2010), Dey et al. (2004), and Børvik et al. (2002) are obtained through fitting the true stress–strain curves by Equation (1), while for other materials, it is given in the corresponding references.

From Table 1, it can be found that the coefficient B for different materials has slight difference, and consequently, the average value B = 1.1346 is suggested. Shown in Figure 5, the coefficient A increases almost linearly with the increase of A₁. Therefore, the formula for the coefficients of resistance function is proposed as follows

A = 0.9896 A_{1} - 0.1482 = 0.9896 [\frac{2}{3} + \frac{2}{3} {(\frac{2 E}{3 Y})}^{n} \int_{0}^{1 - (3 Y / 2 E)} \frac{{(- \ln x)}^{n}}{1 - x} d x] - 0.1482, B = 1.1346

(4)

Equation (4) greatly simplifies the determinations of the resistance coefficients for the compressible power-law strain-hardening metallic target. Therefore, Equations (2) and (4) can be utilized to quickly predict the resistance function during the projectile penetrating into the compressible power-law strain-hardening material without performing the complex numerical solution.

Figure 5.

Linear fitting of resistance coefficient A.

A new model for projectile perforating metallic targets

In this section, based on the proposed resistance function given in Equations (2) and (4), a new perforation model for rigid sharp-nosed projectile striking into metallic target is established with apprehensive considerations of the target compressibility, and the rear free-surface and fracture effects.

Resistance function with the free-surface effect

For a finite thickness metallic plate, the rear free surface has a decay effect on the target resistance. Aiming to predict the depth of penetration and projectile trajectory during the oblique penetration, Warren and Poormon (2001) constructed a decay function for considering the frontal free-surface effect, which is expressed as

f (R, r_{c}, {\dot{r}}_{c}) = {\begin{cases} \frac{(2 / 3) Y [\ln T + 1 - T {(r_{c} / R)}^{3}] + (1 / 2) ρ {\dot{r}}_{c}^{2} [3 + {(r_{c} / R)}^{4} - 4 (r_{c} / R)]}{(2 / 3) Y (\ln T + 1) + (3 / 2) ρ {\dot{r}}_{c}^{2}}, R ⩾ r_{p} \\ \frac{2 Y \ln (R / r_{c}) + 0.5 ρ {\dot{r}}_{c}^{2} [3 + {(r_{c} / R)}^{4} - 4 (r_{c} / R)]}{(2 / 3) Y (\ln T + 1) + (3 / 2) ρ {\dot{r}}_{c}^{2}}, R < r_{p} \end{cases}

(5)

where T = 2E/3Y, and r_p = T^1/3r_c is the radius of the plastic region. R is the radial distance of an arbitrary point on the projectile axis to the free surface of the target, as shown in Figure 6. The elastic and plastic response regions of the target are also illustrated in Figure 6, where V₀ is the striking velocity.

Figure 6.

Sketch of the elastic–plastic responses during projectile impact.

Since Equation (5) is too complex to derive the explicit expression of the projectile motion equation, only the quasi-static decay ( ${\dot{r}}_{c} = 0$ ) is considered; correspondingly, Equation (5) is simplified as

f (R, r_{c}) = {\begin{cases} 1 - \frac{T {(r_{c} / R)}^{3}}{\ln T + 1}, R ⩾ r_{p} \\ \frac{3 \ln (R / r_{c})}{\ln T + 1}, R < r_{p} \end{cases}

(6a, b)

The validity of the above simplification will be proved in section “Comparisons.” Equation (6) indicates that when R ⩾ r_p, the boundary of plastic region does not reach the rear free surface of the plate; correspondingly, the resistance is decayed elastically. While if R < r_p, the radial stress at the cavity surface is decayed plastically. A typical relationship of decay function with dimensionless distance to rear free surface is shown in Figure 7, where the target is 6061-T6 aluminum in Rosenberg and Forrestal (1988) (see Table 1).

Figure 7.

Decay function versus dimensionless distance to rear free surface.

Based on above discussions, the dimensionless radial stress with the consideration of the free-surface effect can be obtained as follows

\frac{σ_{r}}{Y} = (A + B w^{2} v_{c}^{2}) f (R, r_{c})

(7)

Projectile perforation model

The orientation and position of the projectile at an arbitrary time are shown in Figure 8, where [y, z] represent the coordinate in a local reference frame located at the current projectile nose tip. There are three regions associated with the projectile perforation, that is, the elastic-decay region, the plastic-decay region, and the fracture region. Shown in Figure 8, r_c₁ and r_c₂ are the cavity radii of an arbitrary point on the projectile nose surface, corresponding to the elastic decay–plastic decay interface and plastic decay–fracture interface. The corresponding distances to the rear free surface of the target denoted as R₁ and R₂, respectively, can be determined as follows (Jiang et al., 2007)

\frac{R_{1}}{r_{c 1}} = {(\frac{2 E}{3 Y})}^{1 / 3}

(8a)

\frac{R_{2}}{r_{c 2}} = {[1 - \exp (- 1.5 ε_{f})]}^{- (1 / 3)}

(8b)

where $R_{1} = (H - L_{p} + z + y \tan ϕ) / \sin ϕ$ , with y = y(z) being the nose shape function, and $R_{2}^{3} = R_{1}^{3} + r_{c_{2}}^{3}$ . H and L_p denote the target thickness and instantaneous depth of penetration, respectively. z₁ and z₂ are the corresponding values of r_c₁ and r_c₂, respectively (see Figure 9), which are expressed as where Equation (8) is used.

z_{1} = L_{p} - H + (T^{1 / 3} - 1) y \tan ϕ, z_{2} = L_{p} - H - {[1 - \frac{\exp (- 1.5 ε_{f})}{1 - \exp (- 1.5 ε_{f})}]}^{1 / 3} y \tan ϕ

(9)

when R > R₁ and R₁ ⩾ R > R₂, the resistance function is decayed elastically (Equation (6a)) and plastically (Equation (6b)), respectively. While R ⩽ R₂, the tensile strain of the target reaches the fracture strain ε_f; consequently, the fracture region appears and the resistance function should be set as zero. Therefore, by using Equations (7)–(9), the normal stress acting on the projectile nose with the considerations of the rear free-surface and fracture effects is obtained as

σ_{r} = {\begin{cases} \frac{Y (A + B w^{2} V^{2} \sin^{2} ϕ) [\ln T + 1 - T {(R / r_{c})}^{3}]}{(\ln T + 1)}, z_{1} ⩽ z < \min (L_{p}, l) \\ \frac{3 Y (A + B w^{2} V^{2} \sin^{2} ϕ) \ln (R / r_{c})}{(\ln T + 1)}, z_{2} ⩽ z < z_{1} \\ 0, 0 ⩽ z < z_{2} \end{cases}

(10)

where l is the length of the projectile nose.

The axial resistance of the projectile can be obtained by integrating the radial and tangential stresses on the nose surface

F = \int_{0}^{\min (L_{p}, l)} 2 π y σ_{r} \tan ϕ (1 + μ \cos ϕ) d z

(11)

where µ is the coefficient of sliding friction.

Figure 8.

Three regions in the target.

Figure 9.

Comparisons of the residual velocities between the test data and predictions by the present model.

Numerical study by Camacho and Ortiz (1997) concluded that there is a very thin melted layer in the target next to the projectile that provides a nearly frictionless interface. Thus, neglecting the friction (µ = 0), substituting Equation (10) into Equation (11) gives

F_{i} = 2 π Y (A p_{i} + B w^{2} V^{2} q_{i})

(12)

where p_i and q_i are

p_{i} = α_{i} - \frac{T}{1 + \ln T} β_{i} + \frac{3}{1 + \ln T} γ_{i}, q_{i} = a_{i} - \frac{T}{1 + \ln T} b_{i} + \frac{3}{1 + \ln T} c_{i}

(13)

where i = 1, 2, 3 represent the three stages during the perforation process, including the only elastic-decay region, the two regions of elastic-decay and plastic-decay, as well as the three regions of elastic-decay, plastic-decay, and fracture, respectively. The expressions of α_i, β_i, γ_i, A_i, B_i, and C_i are

α_{1} = \int_{0}^{h_{0}} y \tan ϕ d z, α_{2} = \int_{h_{1}}^{h_{0}} y \tan ϕ d z, α_{3} = α_{2}

β_{1} = \int_{0}^{h_{0}} {(\frac{r_{c}}{R_{1}})}^{3} y \tan ϕ d z, β_{2} = \int_{h_{1}}^{h_{0}} {(\frac{r_{c}}{R_{1}})}^{3} y \tan ϕ d z, β_{3} = β_{2}

γ_{1} = 0, γ_{2} = \int_{0}^{h_{1}} y \tan ϕ \ln \frac{R_{2}}{r_{c}} d z, γ_{3} = \int_{h_{2}}^{h_{1}} y \tan ϕ \ln \frac{R_{2}}{r_{c}} d z

a_{1} = \int_{0}^{h_{0}} y \tan ϕ \sin^{2} ϕ d z, a_{2} = \int_{h_{1}}^{h_{0}} y \tan ϕ \sin^{2} ϕ d z, a_{3} = a_{2}

b_{1} = \int_{0}^{h_{0}} {(\frac{r_{c}}{R_{1}})}^{3} y \tan ϕ \sin^{2} ϕ d z, b_{2} = \int_{h_{1}}^{h_{0}} {(\frac{r_{c}}{R_{1}})}^{3} y \tan ϕ \sin^{2} ϕ d z, b_{3} = b_{2}

c_{1} = 0, c_{2} = \int_{0}^{h_{1}} y \tan ϕ \sin^{2} ϕ \ln \frac{R_{2}}{r_{c}} d z, c_{3} = \int_{h_{2}}^{h_{1}} y \tan ϕ \sin^{2} ϕ \ln \frac{R_{2}}{r_{c}} d z

(14)

where $h_{0} = \min (L_{p}, l)$ , $h_{1} = \min (z_{1}, l)$ , and $h_{2} = \min (z_{2}, l)$ . For the conic-nosed projectile, $ϕ$ is a constant value, and $q_{i} = p_{i} \sin^{2} ϕ$ .

Using the Newton’s second law, it gives

F = - m \frac{d V}{d t} = - m \frac{d V}{d L_{p}} \frac{d L_{p}}{d t} = - m V \frac{d V}{d L_{p}}

(15)

where m is the projectile mass, and $d / d t$ is the derivative of time. Substituting Equation (12) into Equation (15) and denoting V^* = V², it gives

\frac{d V^{*}}{d L_{p}} = - \frac{4 π Y (A p_{i} + B q_{i} w^{2} V^{*})}{m}, i = 1, 2, 3

(16)

Equation (16) can be solved numerically using the fourth-order Runge-Kutta method given the initial conditions L_p = 0 and $V^{*} (0) = V_{0}^{2}$ . The numerical solution starts with L_p = 0 to z₂ = l where the perforation ends.

The proposed model can be applied for arbitrary axisymmetric sharp-nosed projectile, including the conical, ogival, and so on. It should be pointed out that when the projectile entry and exit phases and the free surface as well as the fracture effects are neglected, the proposed model reduces to the model proposed by Chen and Li (2003).

Comparisons

In this section, 19 sets of rigid projectile perforation tests on monolithic/multi-layered metallic plates are used to validate the proposed model. Furthermore, based on three typical test data, the comparisons with the existing models are made to demonstrate the advantages of the proposed model.

Comparisons with the available test data

Table 2 presents the available reduce-scaled armor piercing projectiles perforation test data, in which the tensile fracture strain ε_f is obtained from the true stress–strain curves in the corresponding references, and d is the projectile diameter. For the target configuration, “2 × 20” denotes the two contacted plates with each having a thickness of 20 mm, while “2 × 6+24” denotes the two spaced plates with each having a thickness of 6 mm and space distance of 24 mm.

Table 2.

Parameters of reduce-scaled armor piercing projectiles perforation tests.

Test No.	Configuration	Ref.	Projectile parameters				Target parameters
Test No.	Configuration		Nose shape	d/mm	m/g	l/mm	Materials		H/mm	ε_f	A	B
1	Single	Dey et al. (2004)	Conic	20	197	30	Steel	Weldox 460E	12.2	1.36	5.4133	1.1346
2		Dey et al. (2004)		20	197	30		Weldox 700E	12.1	1.26	4.4006
3		Børvik et al. (2004)		20	197	30	Aluminum	5083-H116	15	0.31	7.9258
4		Børvik et al. (2004)		20	197	30		5083-H116	20	0.31	7.9258
5		Forrestal et al. (1990)		8.31	26	14.83		5083-H131	12.7	0.13	4.8066
6		Forrestal et al. (1990)		8.31	26	14.83		5083-H131	50.8	0.13	4.8066
7		Forrestal et al. (1990)		8.31	26	14.83		5083-H131	76.2	0.13	4.8066
8		Rosenberg and Forrestal (1988)		7.1	25	10.7		6061-T6	25.4	0.17	4.3977
9		Børvik et al. (2010)	Ogive	20	197	33		5083-H116	20	0.31	5.2995
10		Piekutowski et al. (1996)		12.9	81	21.4		6061-T651	26.3	0.17	4.8536
11		Forrestal et al. (2014)		6.17	5.25	10.2		6082-T651	20	0.8	4.5367
12	Single/multi-layered	Børvik et al. (2010)		6.17	5.25	10.2		5083-H116	20	0.31	5.3000
13	Single/multi-layered	Børvik et al. (2010)		6.17	5.25	10.2		5083-H116	2 × 20	0.31	5.3000
14		Børvik et al. (2010)		6.17	5.25	10.2		5083-H116	3 × 20	0.31	5.3000
15		Forrestal et al. (2010)		6.17	5.25	10.2		7075-T651	20	0.75	4.4129
16		Forrestal et al. (2010)		6.17	5.25	10.2		7075-T651	2 × 20	0.75	4.4129
17		Dey et al. (2007)		20	197	33	Steel	Weldox 700E	6	1.26	4.4129
18		Dey et al. (2007)		20	197	33		Weldox 700E	12	1.26	4.4129
19		Dey et al. (2007)		20	197	33		Weldox 700E	2 × 6+24	1.26	4.4129

Figure 9 shows the predicted residual velocities of projectiles (V_r) and the corresponding test data. It should be pointed out that the contacted target plates (Tests 13, 14, and 16) are treated as monolithic targets with the equal total thickness, following the suggestion by Børvik et al. (2010), while the spaced target plates (Test 19) are treated as two independent plates.

It is observed that the predictions agree well with the total 19 sets of test data. Figure 9(i) further shows that the protective performance of two spaced-layered target is weaker than the monolithic target with the equal total thickness, which is consistent with the conclusion of Iqbal et al. (2012) drawn by numerical simulations.

Comparison with the existing models

Two typical projectile perforation models, that is, Forrestal and Warren (2009) (F-W) model and Chen and Li (2003) (C-L) model, are used to compare with the present model in this section. Taking three sets of test data with various H/d in Table 2 (Tests 6, 9, and 10) as examples, Figure 10 shows the test data, and the predicted results by the present model and above two models.

Figure 10.

Comparisons of the residual velocities of projectile predicted by the present and existing models.

It can be observed that (1) the proposed model agrees well with all the test data for a large range of H/d, while the existing two models overestimate the target resistance due to the rear free-surface effect, and consequently underestimate the residual velocities; (2) when the impact velocity is high, the predictions of three models all agree well with the test data. The reasons lie in that the influence of the free-surface effect on the residual velocity decreases with the increase of the impact velocity, which will be further discussed in the following section.

Influence of impact velocity on the rear free-surface effect

Taking the Test 8 in Table 2 as an example, the influence of the impact velocity on the rear free-surface effect is discussed by keeping the target thickness constant. When the projectile velocity is lower than the ballistic limit V_BL = 301.9 m/s, that is, penetration, the predicted curves of the instantaneous velocity (V) and the deceleration (a) under four different striking velocities are shown in Figure 11(a) and (b). When the velocity of the projectile is higher than the ballistic limit V_BL = 301.9 m/s, that is, perforation, the corresponding predicted curves under four different striking velocities are shown in Figure 11(c) and (d), in which the expressions of z₁ and z₂ are given in Equation (9). z₁ = 0 indicates that the projectile nose tip starts to enter the plastic-decay region, and z₂ = l indicates that the perforation process ends.

Figure 11.

Instantaneous velocity and deceleration of projectile versus instantaneous depth of penetration under different striking velocities.

It can be seen that the rear free-surface effect has a great influence on the terminal ballistic parameters of projectile penetrating and perforating into the finite thickness target:

Figure 11(a) and (b) show that when the impact velocity is less than the ballistic limit, the free-surface effect gradually weakens with the decrease of the impact velocity. It is mainly due to the fact that for the finite thickness target, when the impact velocity is lower, the distance of the projectile to the rear surface of target is relatively far and the decay effect of the free surface is weak. For example, for a relatively low velocity of 270 m/s, the depth of penetration considering the free-surface effect (0.0261 m) is only 0.38% higher than the depth without considering the free-surface effect (0.026 m), thus the free-surface effect can be neglected.

Figure 11(c) and (d) show that when the impact velocity is higher than the ballistic limit, the free-surface effect gradually weakens with the increase of the impact velocity. The reason lies in that for the finite thickness target, when the impact velocity is high, the time of the perforation process is short and the cumulative decay effect of rear free surface is small. For instance, for a relatively high velocity of 750 m/s, the residual velocity with the free-surface effect (678.9 m/s) is only 2.3% higher than that without the free-surface effect (663.1 m/s).

Figure 11(b) and (d) show that when the free-surface effect is considered, the so-called “deceleration drag” phenomenon is well reproduced, which is consistent with the measured curves of Forrestal et al. (2003) on the concrete target.

Conclusion

For the armor design under projectile impact, based on the spherical CET, a new projectile perforation model is established, which is compared with the available test data and the existing models. The main works are as follows: (1) considering the target as compressible power-law strain-hardening material, the coefficients of projectile impact resistance functions for 14 sets of metallic targets are obtained and an analytical expression of resistance function is further obtained, which relies on the elastic modulus, yield strength, and hardening exponent of the targets; (2) a new perforation model with the consideration of projectile entry and exit phases, compressibility, and rear free-surface and fracture effects of target is established and sufficiently verified by comparing with 19 sets of projectile perforation tests as well as two existing theoretical models; (3) for finite thickness plate, with the increase of the impact velocity, the rear free-surface effect weakens in the projectile perforation and is enhanced in the projectile penetration.

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

This study was supported by the National Natural Science Foundations of China (51522813, 51378015).

References

Bishop

Hill

Mott

(1945) The theory of indentation and hardness. Proceedings of the Physical Society 57(3): 147–159.

Børvik

Clausen

Hopperstad

. (2004) Perforation of AA5083-H116 aluminum plates with conical-nose steel projectiles-experimental study. International Journal of Impact Engineering 30(4): 367–384.

Børvik

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.

Børvik

Langseth

Hopperstad

. (2002) Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses: part I: experimental study. International Journal of Impact Engineering 27(1): 19–35.

Camacho

Ortiz

(1997) Adaptive Lagrangian modeling of ballistic penetration of metallic targets. Computer Methods in Applied Mechanics and Engineering 142: 269–301.

Chen

(2003) Perforation of a thick plate by rigid projectiles. International Journal of Impact Engineering 28(7): 743–759.

Chen

Huang

Liang

(2011) Comparative analysis of perforation models of metallic plates by rigid sharp-nosed projectiles. International Journal of Impact Engineering 38(5): 613–621.

Dey

Børvik

Hopperstad

. (2004) The effect of target strength on the perforation of steel plates using three different projectile nose shapes. International Journal of Impact Engineering 30(8): 1005–1038.

Dey

Børvik

Teng

. (2007) On the ballistic resistance of double-layered steel plates: an experimental and numerical investigation. International Journal of Solids and Structures 44(20): 6701–6723.

10.

Forrestal

Luk

(1988) Dynamic spherical cavity-expansion in a compressible elastic-plastic solid. Journal of Applied Mechanics 55(2): 275–279.

11.

Forrestal

Warren

(2009) Perforation equations for conical and ogival nose rigid projectiles into aluminum target plates. International Journal of Impact Engineering 36(2): 220–225.

12.

Forrestal

Børvik

Warren

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

13.

Forrestal

Børvik

Warren

. (2014) Perforation of 6082-T651 aluminum plates with 7.62 mm APM2 bullets at normal and oblique impacts. Experimental Mechanics 54(3): 471–481.

14.

Forrestal

Frew

Hickerson

. (2003) Penetration of concrete targets with deceleration-time measurements. International Journal of Impact Engineering 28(5): 479–497.

15.

Forrestal

Luk

Brar

(1990) Perforation of aluminum armor plates with conical-nose projectiles. Mechanics of Materials 10(1–2): 97–105.

16.

Forrestal

Luk

Rosenberg

. (1992) Penetration of 7075-T651 aluminum targets with ogival-nose rods. International Journal of Solids and Structures 29(14): 1729–1736.

17.

Goodier

(1965) On the mechanics of indentation and cratering in solid targets of strain hardening metal by impact of hard and soft spheres. In: Proceedings of the 7th Symposium on Hypervelocity Impact III, Tampa, FL, 17–19 November 1964, pp. 215–259. New York: AIAA.

18.

Hill

(1948) A Theory of Earth Movement Near a Deep Underground Explosion. Memo No. 21-48. Kent: Armament Research Establishment.

19.

Holmen

Johnsen

Jupp

. (2013) Effects of heat treatment on the ballistic properties of AA6070 aluminum alloy. International Journal of Impact Engineering 57: 119–133.

20.

Iqbal

Gupta

Deore

. (2012) Effect of target span and configuration on the ballistic limit. International Journal of Impact Engineering 42: 11–24.

21.

Jiang

Zeng

Zhou

(2007) A three-stage model for the perforation of moderately thick metallic plates. Acta Armamentarii 28(9): 1046–1052 (in Chinese).

22.

Weng

Chen

(2004) A Modified model for the penetration into moderately-thick plates by a rigid, sharp-nosed projectile. International Journal of Impact Engineering 30(2): 193–204.

23.

Luk

Forrestal

Amos

(1991) Dynamic spherical cavity expansion of strain-hardening materials. Journal of Applied Mechanics 58(1): 1–6.

24.

Piekutowski

Forrestal

Poormon

. (1996) Perforation of aluminum plates with ogive-nose steel rods at normal and oblique impacts. International Journal of Impact Engineering 18(7): 877–887.

25.

Rosenberg

Forrestal

(1988) Perforation of aluminum plates with conical-nosed rods—additional data and discussion. Journal of Applied Mechanics 55(1): 236–238.

26.

Warren

Poormon

(2001) Penetration of 6061-T6511 aluminum targets by ogive-nosed VAR 4340 steel projectiles at oblique angles: experiments and simulations. International Journal of Impact Engineering 25: 993–1022.