Sage Journals: Discover world-class research

Abstract

A method for predicting the residual velocity and deceleration of a projectile during normal low-velocity impact on a 2024-O thin aluminium plate is developed based on the similarity theory. Geometric scaling, the dissimilar materials of the projectile and different target thicknesses are considered. By a similitude analysis, the simulation criteria between the modelling and prototype experiments are obtained. The dimensionless velocity and deceleration of a projectile can be predicted by the relationship equations with the dimensionless dynamic pressure, projectile density and target thickness. On the basis of experimental data, the dimensionless residual velocity relationship is obtained and verified. In the range of normalised target thicknesses of $0.5 \leq H / D_{p} \leq 1$ (where H is target thickness and $D_{p}$ is projectile diameter), the deceleration–time data during penetration is simplified as a triangular wave. Moreover, it can be characterised using the maximum deceleration, the time to the maximum deceleration and the period of the triangular wave. Through a simulation analysis, three dimensionless deceleration characteristics of the projectile are developed to replicate a prototype-like deceleration–time process in a scaled model.

Keywords

Introduction

In dimensional analysis, a replica of a prototype is derived by scaling one or more parameters of the prototype. Scaled replicas enable the economical and easy set-up of the experiments, especially in the engineering field. In terminal ballistics experiments, a scaling law is widely used in researching penetration and perforation,¹ and the dimensionless parameters make the relationship among the parameters intuitive.

In terminal ballistics experiments, geometric scaling is a convenient approach to test prototypes. Jones et al.^2–4 conducted geometric-scale (geometric-scale factor = 0.25, 0.5, 0.75) and full-scale experiments to compare the plate perforation and reported that the dimensionless perforation energy and the dimensionless transverse displacement of the ductile target conform to the scaling law. Noam et al.⁵ simulated the scaling of dynamic failure using two types of failure criteria, namely, the maximum stress criterion and strain energy density criterion; the obtained results could adequately describe the separation and adiabatic shear failure, respectively.

In hypervelocity impact experiments, impact velocities higher than 7 km/s are difficult to achieve using two-stage light-gas guns; thus, velocity scaling is used in order to reduce the impact velocity. Mullin et al.⁶ used a dissimilar-material model and lower velocities through velocity scaling; and this approach yielded high correlations for the debris cloud structure, materials, and velocities. In a structure impact, the approach VSG-D⁷ (initial Velocity, V₀; dynamic Stress, σ_d and impact mass, G; with density factor) makes the target response (including the scaled force, displacement and final plate profile) in scaled replicas with different materials highly consistent with a prototype.

Rosenberg and Dekel⁸ determined the relationship between the effective resisting stress $σ_{r}$ and the thickness of the target H in perforation for a wide range of plate thicknesses; they found that $σ_{r}$ and H are normalised by the target flow stress $Y_{tf}$ and the diameter of the projectile D, respectively, and presented a dimensionless piecewise equation for three $H / D$ values, from which the perforation energy can be calculated.

For terminal ballistics research, the geometric scaling, velocity scaling and normalised target thickness have been used, as mentioned above. The present work extends previous work by coupling these scaling methods for perforation research. In perforation experiments, the deceleration–time data and the residual velocity of the projectile are valuable output response parameters. The residual velocity of the prototype can be estimated through geometric-scaling experiments. However, in geometric scaling, a smaller model results in a higher projectile deceleration, which means that a downscaled experiment cannot adequately reflect the real-world (prototype) environmental and deceleration characteristics. Thus, in engineering experiments, the anti-overload capabilities of the projectile components (e.g. the fuse and warhead) of a downscaled geometric-scaling model may be difficult to assess.

Therefore, in this study, the geometry (i.e. the geometric scaling and target thickness), dissimilar projectile materials and impact velocity are considered for the prediction of the residual velocity and deceleration (specifically, for rigid projectile perforation of a thin aluminium plate at low velocities). The target thickness range in this study ranges from half of the projectile diameter to the projectile diameter. The rest of this article is organised as follows: first, the experiments and results are introduced; then, an equation for predicting the projectile residual velocity through a dimensional analysis is obtained; after describing and validating the simulation process, the dimensionless prediction equations for deceleration of the projectile are obtained by simulation; finally, using the obtained equations, the deceleration–time data of projectiles in the prototype model can be appropriately represented using a scaled model.

Experiments

Experimental setup

A gas-gun apparatus (Figure 1) with an inner diameter of 16 mm is used to launch the projectiles. The projectile impact velocity is less than 500 m/s. A steel chamber with a metal target is fixed in front of the gun barrel; this chamber has a circular hole in line with the barrel, through which the projectile could pass into and penetrate the target. The target is sandwiched between a C-shaped clamping rig and a C-shaped support plate using 10 bolts, and the support plate is welded with steel plates to ensure its rigidity (Figure 2). Rubber plates are placed behind the target as a buffer to stop and recover the projectile after perforation. On one side of the target chamber is a transparent acrylic plate window through which a high-speed camera equipped with a flash unit can record the projectile action. The camera is placed in such a way that it can photograph both the front and back of the target. The flash unit is placed between the window and the camera and can provide an exposure time of approximately 21 ms. A copper wire connected to a trigger mechanism is connected to the gas-gun muzzle. The wire is cut when the projectile passes through the muzzle, which in turn triggers the flash; simultaneously, photographs are manually captured using the camera at a frame rate of 44,000 fps.

Figure 1.

Gas-gun experimental system.

Figure 2.

Metal target, C-shaped support and clamping rig.

Similitude analysis and experimental design

For a projectile perforating a plate, the three characteristic parameters are the residual velocity $v_{r}$ , the deceleration of the projectile a and the penetration time $T_{pene}$ , which is the time taken to completely perforate the plate. These parameters can be determined using the main parameters as follows

v_{r} = f_{r} (ρ_{p}, L_{e f f}, D_{p}, N, ρ_{t}, H, v_{i}, E_{p y}, E_{t y}, Y_{p}, Y_{t}, E_{p t}, E_{t t}, Y_{p s}, Y_{t s}, ε_{p f}, ε_{t f}, μ)

(1)

a = f_{a} (ρ_{p}, L_{e f f}, D_{p}, N, ρ_{t}, H, v_{i}, E_{p y}, E_{t y}, Y_{p}, Y_{t}, E_{p t}, E_{t t}, Y_{p s}, Y_{t s}, ε_{p f}, ε_{t f}, μ)

(2)

T_{p e n e} = f_{p e n e} (ρ_{p}, L_{e f f}, D_{p}, N, ρ_{t}, H, v_{i}, E_{p y}, E_{t y}, Y_{p}, Y_{t}, E_{p t}, E_{t t}, Y_{p s}, Y_{t s}, ε_{p f}, ε_{t f}, μ)

(3)

In this study, all the targets were composed of 2024-O aluminium alloy. Due to it is a ductile metal with unchanged elastic modulus $E_{ty}$ , tangent modulus $E_{tt}$ , yield strength $Y_{t}$ , ultimate strength $Y_{ts}$ and failure strain $ε_{tf}$ during the experiments. Hence, the only measurable constant, the yield strength $Y_{t}$ , is used to express the behaviour of the target material. The ultimate tensile strength of the 2024-O aluminium alloy is 220 MPa.⁹ We used projectiles fabricated from dissimilar high-strength materials, AISI 1045 steel and 7055-t77 aluminium, the ultimate tensile strength of which are 625 and 645 MPa, respectively.^9,10 These projectiles can be considered rigid, as their ultimate tensile strengths are higher than that of the target materials. Because of this assumption, the effects of the elastic modulus $E_{py}$ , tangent modulus $E_{pt}$ , yield strength $Y_{p}$ , ultimate strength $Y_{ps}$ and failure strain $ε_{pf}$ , which are constant for the projectile materials, can be ignored. Because the densities of AISI 1045 steel and 7055-t77 aluminium differ greatly, inertial effects should be considered $ρ_{p}$ . To simplify the analysis, the coefficient of dynamic friction between the projectile and the target is also assumed to be a constant irrespective of the projectile material. In addition, because the size ratio of the projectiles remains unchanged, the projectile’s nose-shaped factor N can be ignored. Regarding the effective length $L_{eff}$ and the diameter of the projectile $D_{p}$ , a constant projectile size ratio is maintained in the dimensional analysis; this means that $L_{eff} / D_{p}$ is a constant. Therefore, $D_{p}$ is investigated in this study. According to the foregoing assumptions, equations (1)–(3) can be simplified to

v_{r} = f_{r} (ρ_{p}, D_{p}, ρ_{t}, H, v_{i}, Y_{t})

(4)

a = f_{a} (ρ_{p}, D_{p}, ρ_{t}, H, v_{i}, Y_{t})

(5)

T_{p e n e} = f_{p e n e} (ρ_{p}, D_{p}, ρ_{t}, H, v_{i}, Y_{t})

(6)

Equations (4)–(6) are the dimensionless forms of equations (1)–(3)

\frac{v_{r}}{v_{i}} = f_{r} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y_{t}})

(7)

\frac{a D_{p}}{v_{i}^{2}} = f_{a} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y_{t}})

(8)

\frac{T_{pene} v_{i}}{D_{p}} = f_{pene} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y_{t}})

(9)

Thus, three dimensionless pi terms, namely, $H / D_{p}, ρ_{p} / ρ_{t} and ρ_{p} v_{i}^{2} / Y_{t}$ , are obtained. On the basis of equations (7)–(9), the following conditions should be satisfied to ensure that the phenomena or responses in the replicas are similar to the prototype

{\begin{matrix} {(\frac{H}{D_{p}})}_{rep} = {(\frac{H}{D_{p}})}_{prot} \\ {(\frac{ρ_{p}}{ρ_{t}})}_{rep} = {(\frac{ρ_{p}}{ρ_{t}})}_{prot} \\ {(\frac{ρ_{p} v_{i}^{2}}{Y_{t}})}_{rep} = {(\frac{ρ_{p} v_{i}^{2}}{Y_{t}})}_{prot} \end{matrix}

(10)

where $H / D_{p}$ represents the dimensionless target thickness, $ρ_{p} / ρ_{t}$ is the ratio of the inertial effect and $ρ_{p} v_{i}^{2} / Y_{t}$ is the dynamic pressure relative to the target strength.

The perforation experiments are designed considering these three pi terms and the scaling factor λ. In the geometric-scaling experiments, $ρ_{p} / ρ_{t}$ is maintained constant and AISI 1045 projectiles are used. Two types of projectiles are used in the experiments (Figure 3): for the 16- and 10-mm-diameter projectiles, the dimensions of the targets are 140 mm × 140 mm × 8 mm and 140 mm × 140 mm × 5 mm, respectively. The 16-mm projectile perforating the 8-mm-thick plate is considered the prototype, and the latter projectile–target combination is considered its geometric-scale model (hereafter, replica-G).

Figure 3.

Dimensions of the two types of projectiles: (a) λ = 0.625 and (b) λ = 1.

In the dissimilar-materials experiments (hereafter, replica-D), 7055-t77 aluminium projectiles perforate 2024-O aluminium plates; the projectile and target dimensions are the same as those of the prototype in the geometric-scaling experiment. The normalised projectile density in the prototype is $ρ_{p} / ρ_{t} = 2.84 (7.85 / 2.78)$ and that in replica-D is $ρ_{p} / ρ_{t} = 1.02 (2.84 / 2.78)$ .

In experiments where the target thickness is varied (hereafter, replica-T), we use 9.5- and 12-mm-thick targets. However, the projectile material and dimensions are the same as those of the prototype. The normalised target thicknesses in the prototype and in replica-T are $H / D_{p} = 0.5 (8 / 16)$ and $H / D_{p} = 0.59 (9.5 / 16), 0.75 (12 / 16)$ , respectively.

Experimental results and analysis

The experimentally obtained projectile residual velocities are presented in Table 1 and Figure 4, and illustrative high-speed camera images are depicted in Figure 5. After the perforation experiments, the steel projectiles did not exhibit any substantial deformation (Figure 6). The deformation of the nose tip of the steel projectile used in experiment S16-20 was not caused by penetration but by impact with the steel box after perforation; the aluminium projectiles exhibited only slight nose-tip deformation.

Table 1.

Experimental data for the perforation of the 2024-O target.

Experiment	Test ID	$H_{t}$ (mm)	$D_{p}$ (mm)	$M_{p}$ (g)	$v_{i}$ (m/s)	$v_{r}$ (m/s)
Prototype	S16-1	8.0	16	45.57	242.76	177.17
	S16-3	8.0	16	45.55	222.68	147.08
	S16-5	8.0	16	45.57	190.71	84.23
Replica-G	S10-6	5.0	10	11.27	199.17	93.55
	S10-10	5.0	10	11.16	243.78	169.45
	S10-9	5.0	10	11.17	245.07	172.88
	S10-8	5.0	10	11.12	262.83	196.12
	S10-7	5.0	10	11.15	273.86	214.03
	S10-5	5.0	10	11.16	283.21	226.10
	S10-1	5.0	10	11.13	298.10	243.97
Replica-D	A16-1	8.0	16	16.13	345.46	191.24
	A16-4	8.0	16	16.13	333.89	168.60
	A16-5	8.0	16	16.15	319.12	134.28
	A16-8	8.0	16	16.16	308.89	105.90
Replica-T	S16-15	9.5	16	45.55	230.30	141.89
Replica-T	S16-20	12.0	16	45.55	239.00	134.32

Figure 4.

Residual velocity as a function of the impact velocity.

Figure 5.

Illustrative images captured using the high-speed camera.

Figure 6.

Dimensions of example projectiles after the perforation experiments, as well as the standard projectile.

Figure 4 illustrates the impact velocity as a function of the residual velocity using equation (11)

v_{bl} = (\frac{M_{p}}{M_{p} + M_{s}}) {(v_{i}^{2} - v_{r}^{2})}^{0.5}

(11a)

This equation was proposed by Recht and Ipson.¹¹ Here, $M_{s}$ is the mass of the plug formed on the perforation, $v_{i}$ is the impact velocity of the projectile and $v_{r}$ is the residual velocity of the projectile and the plug. The plug was very small, as shown in Figure 5, and its mass was therefore negligible; hence, equation (11a) becomes

v_{bl} = {(v_{i}^{2} - v_{r}^{2})}^{0.5}

(11b)

Accordingly, the perforation energy¹² can be estimated as the loss of projectile’s kinetic energy

W_{p} = \frac{M_{p} v_{i}^{2}}{2} - \frac{M_{p} v_{r}^{2}}{2}

(12)

As shown in Figure 4, the results of replica-G experiments have a good agreement with results of prototype experiments (S16-1, S16-3 and S16-5). According to the fitted curves (red line and green line), the ballistic limit velocities are 169.47 and 174.29 m/s for prototype and replica-G, respectively. But the results of replica-T (S16-15 and S16-20) are offset from the fitted curve (red line) of prototype experiments, which means that the perforation of thicker target costs higher kinetic energy. In replica-D experiments (A16-1, A16-4, A16-5 and A16-8), the ballistic limit velocity is 289.31 m/s and the average perforation energy is 673.61 J. The average perforation energy for prototype (S16-1, S16-3 and S16-5) is 643.75 J. 7055-t77 aluminium projectiles cost more kinetic energy than AISI 1045 steel projectiles to perforate the 2024-O aluminium plate of the same thickness. Different dynamic frictional coefficients of different projectile materials may cause the deviation, but the deviation is lower than 5%.

Figure 7(a) shows the normalised impact and residual velocities obtained through the experiments. The $v_{r} / v_{i}$ curve overlaps that of $ρ_{p} v_{i}^{2} / Y_{t}$ for the prototype, replica-G and replica-D experiments for $H / D_{p} = 0.5$ . Hence, $v_{r} / v_{i}$ of the rigid projectiles with dissimilar materials and different geometric-scaling models can be predicted using

\frac{v_{r}}{v_{i}} = {(1 - 1.750 {(\frac{ρ_{p} v_{i}^{2}}{Y_{t}})}^{- 0.987})}^{0.458}

(13)

Figure 7.

Normalised residual velocity as a function of (a) $ρ_{p} v_{i}^{2} / Y_{t}$ and (b) $(ρ_{p} v_{i}^{2} / Y_{t}) / (H / D_{p})$ .

Figure 7(a) shows that when $ρ_{p} v_{i}^{2} / Y_{t}$ for the 16-mm steel and aluminium projectiles are the same, the normalised projectile density ( $ρ_{p} / ρ_{t}$ ) does not substantially affect $v_{r} / v_{i}$ . For replica-H, where the values of $H / D_{p}$ of the projectiles are different (0.59 and 0.75), the actual curve is offset from the fitted curve. If the abscissa in Figure 7(a) is changed from $ρ_{p} v_{i}^{2} / Y_{t}$ to $(ρ_{p} v_{i}^{2} / Y_{t}) / (H / D_{p})$ , the $H / D_{p}$ curve overlaps the fitted curve (Figure 7(b)). Thus, the new prediction equation can be expressed as

\frac{v_{r}}{v_{i}} = {(1 - 2.753 {(\frac{(ρ_{p} v_{i}^{2} / Y_{t})}{(H / D_{p})})}^{- 0.775})}^{0.341}

(14)

From Figure 7(b), the residual velocity prediction for rigid projectiles for different target thicknesses, dissimilar materials and different geometric-scale factors can be achieved using a single dimensionless relationship equation.

As explained in section ‘Introduction’, the effective resisting stress ( $σ_{r}$ )⁸ is used to estimate perforation energy

W_{p} = \frac{π D_{p}^{2} H σ_{r}}{4}

(15)

According to Rosenberg and Dekel,⁸ the relationship between the normalised effective resisting stress ( $σ_{r} / Y_{tf}$ ) and the normalised target thickness ( $H / D$ ) for the perforation of ductile plates by sharp-nosed rigid projectiles is

\frac{σ_{r}}{Y_{tf}} = {\begin{matrix} \frac{2}{3} + 4 \cdot (\frac{H}{D}) for : \frac{H}{D} \leq \frac{1}{3} \\ 2.0 for : \frac{1}{3} \leq \frac{H}{D} \leq 1.0 \\ 2.0 + 0.8 \cdot \ln (\frac{H}{D}) for : \frac{H}{D} \geq 1.0 \end{matrix}

(16)

Combining equations (12) and (16), the fit curve of equation (17) is also listed in Figure 7(b). This method of residual velocity prediction⁸ has demonstrated that equation (15) is valid

\frac{M_{p} v_{i}^{2}}{2} - \frac{M_{p} v_{r}^{2}}{2} = \frac{π D_{p}^{2} H σ_{r}}{4}

(17)

On the basis of the research of residual velocity prediction, for a rigid projectile perforating a 2024-O target, replicas with different geometries (geometric scaling and target thickness), dissimilar projectile materials and impact velocities can be similar to the prototype. The effect of the normalised projectile density $ρ_{p} / ρ_{t}$ on a short projectile (spheres and $L / D = 1$ cylinders, where L is the height of the cylinder and D is diameter of the cylinder.) should be considered.¹² In our experiments, the projectiles are not short, and the nose shape of the projectiles is conical. This is why it does not substantially influence the normalised residual velocity $v_{r} / v_{i}$ .

Numerical simulation

Numerical model and validation

Numerical model

A two-dimensional axisymmetric model is used to simulate normal penetration using the finite element code LS-DYNA (Figure 8). The projectiles can be assumed to be rigid; the constants used in the rigid material model for the steel and aluminium projectiles obtained from the literature are listed in Table 2.^9,10 Similar to the target dimensions used in the experiments, the radius of the targets in the simulation models is set as 50 mm. The target is fully clamped at the boundary. *Contact_2D_Automatic_Single_Surface is used for the friction setting, and the targets are assigned the *Hourglass setting with the Flanagan–Belytschko stiffness. For reducing the computing time, refined shell elements are used as shown in Figure 8.

Figure 8.

Finite element model showing a conical projectile and Zones I and II of the target plate.

Table 2.

Material constants for rigid projectiles.

Materials	ρ (g/cm³)	E (GPa)	υ
AISI 1045⁹	7.85	203.00	0.290
7055-T77¹⁰	2.84	71.08	0.324

The dynamic frictional force influences the perforation behaviour of the conical-nosed projectiles but is difficult to determine, as it varies with factors such as the stress state, temperature, material(s) between the contact surfaces, contact direction, projectile velocity and melting point of the material. This coefficient has been assumed to be 0–0.5 in simulations and theoretical analyses of aluminium-plate perforation in the literature.^13–20 To simplify the model, ignoring the difference in the dynamic frictional coefficients of dissimilar materials, this coefficient is assumed to be 0.05 in all simulations in this study.

Using a bilinear elastic–plastic model (*MAT_PLASTIC_KINEMATIC), the yield stress and tangent modulus of the target material are obtained from its dynamic stress–stain curves (Figure 9), which were measured using a split-Hopkinson pressure bar (SHPB) test. However, the Young’s modulus from the fitted curve substantially deviates from that of aluminium; hence, in Table 3, the Young’s modulus is assumed to be $E = 73.1 GPa$ according to MatWeb data.⁹ The 2024-O aluminium alloy has no obvious strain-rate effect;^21,22 hence, this effect is not considered in the simulation. In practice, the element size influences the effective plastic strain for erosion; therefore, the effective plastic strain for erosion in the simulation was adjusted on the basis of earlier ballistic experiments.^23,24 According to the experimental data, element erosion occurs at Zone I during perforation; thus, the effective plastic strain for erosion is set as $ε_{f} = 1.68$ .

Figure 9.

Stress as a function of the strain in 2024-O aluminium, obtained through an SHPB test.

Table 3.

Material constants for the 2024-O target.

ρ⁹ (g/cm³)	E⁹ (GPa)	υ ⁹	σ₀ (MPa)	E_t (MPa)	β	FS
2.78	73.10	0.33	138.09	108.91	1	1.68

Validation of the simulation method

The simulation can be validated by comparing the experimental and simulated impact velocity $v_{i}$ and penetration time $T_{pene}$ , as shown in Figure 10(a) and (b). The experimental and simulated perforation processes are depicted in Figure 10(c); as is evident, the penetration position and time of the projectile in the simulation are similar to those in the experiment. Moreover, these validations of the velocity and time validate the projectile deceleration data indirectly.

Figure 10.

Validation of the simulated (a) residual velocity and (b) penetration time; (c) images of the perforation process in the S16-1 experiment (captured using the high-speed camera) and the corresponding simulation.

Deceleration filtering

In the simulations, the deceleration–time history often contains high-frequency components, one of which is an elastic wave oscillating along the projectile axis.²⁵ For the A16-1 simulation, high-frequency components can be substantially reduced using a fast Fourier transform and low-pass filtering (57,000 Hz) of the deceleration–time history (Figure 11). Using the authentication method of Fasanella and Jackson,²⁶ we then integrated the filtered deceleration curve (red solid curve) to obtain the velocity curve (green dashed–dotted line), which fits well with the velocity–time history curve (black solid line) derived from the simulation (Figure 11).

Figure 11.

Projectile velocity and deceleration as a function of time. Black solid line: velocity–time curve from the simulation; blue solid line: deceleration–time curve from the simulation; red solid line: filtered deceleration–time curve; green dashed–dotted line: velocity–time curve integrated filtered deceleration–time curve.

After filtering, the profile of the deceleration–time curve can be simplified as a triangular wave, which can be characterised by the maximum deceleration during penetration ( $a_{\max}$ ), the time to $a_{\max}$ ( $T_{\max}$ ) and the time to complete perforation by the projectile nose ( $T_{all}$ ). After complete perforation by the projectile nose, only the frictional force acts on the cylindrical surface of the projectile, and this frictional force is very low. Therefore, $T_{pene}$ is not influential in filtering the deceleration–time curve. a and $T_{pene}$ in equations (8) and (9) can be substituted with $a_{\max}$ , $T_{\max}$ and $T_{all}$

\frac{a_{\max} D_{p}}{v_{i}^{2}} = f_{a} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y})

(18)

\frac{T_{\max} v_{i}}{D_{p}} = f_{Tm} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y_{t}})

(19)

\frac{T_{all} v_{i}}{D_{p}} = f_{Ta} (\frac{H}{D_{p}}, \frac{ρ_{p}}{ρ_{t}}, \frac{ρ_{p} v_{i}^{2}}{Y_{t}})

(20)

In the next section, the deceleration–time curve is filtered by the same manner. For all simulations, before analyses, the (integrated) velocity–time curve is matched with the (simulated) velocity–time curve. And for simplicity, the effects of the Young’s modulus and Poisson’s ratio of the rigid projectile have been ignored. In all subsequent simulations, the Young’s modulus and Poisson’s ratio of the projectiles are assumed to be 203 GPa and 0.29, respectively (the same as those of AISI 1045 steel in Table 2). In the following simulations of dissimilar-material projectiles, only the densities are assumed to be different. In addition, in the next section, the target diameter is determined using the equation $D_{p} / D_{t} = 0.16$ (16/100), where $D_{t}$ is the target diameter.

Deceleration

Similarity analysis of the projectile deceleration

Considering three deceleration characteristics ( $a_{\max}$ , $T_{\max}$ and $T_{all}$ ), dimensional analysis is used to obtain the deceleration–time data for downscaled models and to make it similar to those in the prototype. Let simulation 1 in Table 4 be the prototype simulation. The 17 listed simulations were performed as follows: when changing one pi term, the other two pi terms were unchanged. When varying $ρ_{p} / ρ_{t}$ , the projectile impact velocity was adjusted so that $ρ_{p} v_{i}^{2} / Y_{t}$ remained unchanged.

Table 4.

Simulation results.

No.	$ρ_{p}$ (g/cm³)	$v_{i}$ (m/s)	H (mm)	$ρ_{p} / ρ_{t}$	$ρ_{p} v_{i}^{2} / Y_{t}$	$H / D_{p}$	$a_{\max}$ (m/s²)	$T_{\max}$ (µs)	$T_{all}$ (µs)
1	7.85	300.00	8	2.824	5.116	0.500	720,414.9	82.1	150.2
2	3.14	474.34	8	1.130	5.116	0.500	1,884,765.9	49.3	92.9
3	4.71	387.30	8	1.694	5.116	0.500	1,234,074.3	61.8	114.9
4	6.28	335.41	8	2.259	5.116	0.500	908,359.5	72.9	133.5
5	9.42	273.86	8	3.388	5.116	0.500	587,976.9	89.2	165.3
6	10.99	253.55	8	3.953	5.116	0.500	496,999.5	98.4	179.6
7	12.56	237.17	8	4.518	5.116	0.500	434,408.9	104.6	192.3
8	7.85	180.00	8	2.824	1.842	0.500	639,700.2	161.3	387.9
9	7.85	200.00	8	2.824	2.274	0.500	660,802.7	137.4	285.5
10	7.85	250.00	8	2.824	3.553	0.500	694,939.3	102.4	193.5
11	7.85	350.00	8	2.824	6.964	0.500	735,077.8	68.6	123.6
12	7.85	400.00	8	2.824	9.096	0.500	748,308.0	58.4	105.5
13	7.85	500.00	8	2.824	14.212	0.500	781,543.4	45.9	82.0
14	7.85	300.00	10	2.824	5.116	0.625	870,891.4	80.1	161.2
15	7.85	300.00	12	2.824	5.116	0.750	987,898.2	82.6	174.7
16	7.85	300.00	14	2.824	5.116	0.875	1,155,259.6	80.1	190.2
17	7.85	300.00	16	2.824	5.116	1.000	1,274,979.2	80.6	209.7

According to the results of simulations, the dimensionless relationship for $a_{\max}$ , $T_{\max}$ and $T_{all}$ is as follows

\frac{a_{\max} D_{p}}{v_{i}^{2}} = 63.882 \cdot f_{1} (\frac{ρ_{p}}{ρ_{t}}) \cdot f_{2} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) \cdot f_{3} (\frac{H}{D_{p}})

(21)

where

f_{1} (\frac{ρ_{p}}{ρ_{t}}) = (0.136 - 0.025 e^{- 3.178 (\frac{ρ_{t}}{ρ_{p}})})

f_{2} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) = (0.632 e^{- 0.981 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}})} + 0.259 e^{- 0.229 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}})} + 0.038)

f_{3} (\frac{H}{D_{p}}) = (\frac{0.198 H}{D_{p}} + 0.029)

\frac{T_{\max} v_{i}}{D_{p}} = 0.425 \cdot f_{4} (\frac{ρ_{p}}{ρ_{t}}) \cdot f_{5} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) \cdot f_{6} (\frac{H}{D_{p}})

(22)

where

f_{4} (\frac{ρ_{p}}{ρ_{t}}) = (1.556 - 0.264 e^{- 0.903 (\frac{ρ_{p}}{ρ_{t}})})

f_{5} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) = (4.060 e^{- 1.870 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}})} + 0.420 e^{- 0.240 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}})} + 1.420)

f_{6} (\frac{H}{D_{p}}) = (1.553 - 0.044 (\frac{H}{D_{p}}))

\frac{T_{all} v_{i}}{D_{p}} = 0.126 \cdot f_{7} (\frac{ρ_{p}}{ρ_{t}}) \cdot f_{8} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) \cdot f_{9} (\frac{H}{D_{p}})

(23)

where

f_{7} (\frac{ρ_{p}}{ρ_{t}}) = (2.896 - 0.208 e^{- 0.343 (\frac{ρ_{p}}{ρ_{t}})})

f_{8} (\frac{ρ_{p} v_{i}^{2}}{Y_{t}}) = (1.035 e^{- 2.627 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}} - 1.835)} + 0.794 e^{- 0.327 (\frac{ρ_{p} v_{i}^{2}}{Y_{t}} - 1.835)} + 2.554)

f_{9} (\frac{H}{D_{p}}) = (0.482 e^{1.483 (\frac{H}{D_{p}})} + 1.803)

Discussion

To verify the validity of equations (21)–(23), we modelled two simulations as prototypes: a 64-mm steel projectile penetrating a 32-mm-thick plate and a 64-mm steel projectile penetrating a 64-mm-thick plate. The projectile is assumed to be rigid, the initial impact velocity is 500 m/s, and the plate material is 2024-O aluminium. $a_{\max}$ , $T_{\max}$ and $T_{all}$ of the prototypes were obtained through simulations and the dimensionless relationship equations (Groups A and B in Table 5). Substituting the prototype results ( $a_{\max}$ , $T_{\max}$ and $T_{all}$ ) obtained using equations (21)–(23) into equations (21)–(23) again but with different projectile diameters, $ρ_{p}$ , $v_{i}$ and H were calculated for the projectile diameters $D_{p} = 32, 16 and 8 mm$ . According to the obtained values, model perforation simulations were performed for the downscaled models. The theoretical and simulation results are listed in Tables 6 and 7 for Groups A and B, respectively. The theoretical and simulated results deviate slightly from each other, as shown in Tables 6 and 7; nevertheless, the deceleration–time data in most of the downscaled models are the same as those in the prototype models, as shown in Figures 12 and 13. However, when the projectile diameter $D_{p} = 8 mm$ , the deceleration–time data of the downscaled models do not agree well with those of the prototype. In Group B, the 8-mm-diameter projectile could not perforate the target plate.

Table 5.

Dimensions and parameters used in the simulations of the prototypes and replicas.

Group	Parameters	Prototype	Replica 1	Replica 2	Replica 3
A	$ρ_{p} (g / c m^{3})$	7.85	14.21	26.08	48.31
	$D_{p} (mm)$	64	32	16	8
	$H (mm)$	32.0	16.0	8.0	4.0
	$v_{i} (m / s)$	500.00	265.74	146.71	84.47
B	$ρ_{p} (g / c m^{3})$	7.85	14.01	24.45	34.10
	$D_{p} (mm)$	64	32	16	8
	$H (mm)$	64.0	31.5	15.0	5.5
	$v_{i} (m / s)$	500.00	263.39	142.82	86.60

Table 6.

Theoretical and simulated results for Group A.

Group A	Group A	Prototype	Replica 1	Replica 2	Replica 3
Equation	$a_{\max} (m / s^{2})$	204,736.3	204,736.3	204,736.3	204,736.3
	$T_{\max} (μ s)$	183.3	181.5	174.1	161.6
	$T_{all} (μ s)$	327.5	327.5	327.5	327.5
Simulation	$a_{\max} (m / s^{2})$	185,120.8	190,028.8	194,850.1	188,800.6
	$T_{\max} (μ s)$	178.7	180.8	181.3	170.3
	$T_{all} (μ s)$	327.0	329.5	328.0	331.0

Table 7.

Theoretical and simulated results for Group B.

Group B	Group B	Prototype	Replica 1	Replica 2	Replica 3
Equation	$a_{\max} (m / s^{2})$	362,542.5	349,757.0	351,594.9	363,657.3
	$T_{\max} (μ s)$	180.7	177.1	177.1	173.4
	$T_{all} (μ s)$	457.0	455.5	454.3	446.7
Simulation	$a_{\max} (m / s^{2})$	337,632.8	351,690.9	378,008.1	420,696.6
	$T_{\max} (μ s)$	176.9	177.4	169.5	162.9
	$T_{all} (μ s)$	409.5	427.1	477.0	357.2

Figure 12.

Deceleration–time curves of Group A: (a) before and (b) after filtering.

Figure 13.

Deceleration–time curves of Group B: (a) before and (b) after filtering.

For deceleration–time curve similarity research, a dimensional analysis with the geometry (i.e. geometric scaling and target thickness), dissimilar projectile materials and impact velocity is still a valid way to find a prototype-like deceleration–time process in a downscaled model.

Conclusion

This study focused on the similarity of the residual velocity and deceleration–time data in dissimilar projectile materials and geometric-scaling models with $H / D_{p} = 0.5 - 1$ under low-velocity impact loading. The conclusions are as follows:

The similarity of rigid projectiles normally impacting 2024-O aluminium plates at low velocity was evaluated in this study. On the basis of the similarity criteria of the residual velocity and deceleration–time data, replicas with geometric scaling, different target thickness and dissimilar projectile materials can be similar to the prototype.

Through experiments and simulations, the residual velocity can be predicted by the single relationship equation with the main dimensionless parameters. Only the dimensionless pi term $(ρ_{p} v_{i}^{2} / Y_{t}) / (H / D_{p})$ has an effect on the dimensionless residual velocity $v_{r} / v_{i}$ in the residual velocity equation.

According to the simulation results of a rigid projectile perforating a 2024-O aluminium plate in the range of $0.5 \leq H / D_{p} \leq 1$ , the deceleration–time data are simplified as a triangular wave and characterised using $a_{\max}$ , $T_{\max}$ and $T_{all}$ . The dimensionless equations for the three characteristic parameters $a_{\max}$ , $T_{\max}$ and $T_{all}$ were developed by numerical and dimensional analyses. A similar deceleration–time process can be replicated by these three dimensionless equations.

On the basis of the experimental and numerical results, the dimensionless term ( $ρ_{p} / ρ_{t}$ ) has a negligible effect on the dimensionless residual velocity $v_{r} / v_{i}$ for predicting the residual velocity. However, the term $ρ_{p} / ρ_{t}$ should be considered for predicting the deceleration.

Footnotes

Academic Editor: Rahmi Guclu

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the National Natural Science Foundation of China (no. 11472053).

References

Corbett

Reid

Johnson

Impact loading of plates and shells by free-flying projectiles: a review. Int J Impact Eng 1996; 18: 141–230.

Jones

Birch

Duan

Low-velocity perforation of mild steel rectangular plates with projectiles having different shaped impact faces. J Press Vessel Technol 2008; 130: 31206.

Jones

Birch

RS.

Low velocity perforation of mild steel circular plates with projectiles having different shaped impact faces. J Press Vessel Technol 2008; 130: 31205.

Jones

Birch

RS.

On the scaling of low-velocity perforation of mild steel plates. J Press Vessel Technol 2008; 130: 31207.

Noam

Dolinski

Rittel

Scaling dynamic failure: a numerical study. Int J Impact Eng 2014; 69: 69–79.

Mullin

Anderson

Wilbeck

JS.

Dissimilar material velocity scaling for hypervelocity impact. Int J Impact Eng 2003; 29: 469–485.

Mazzariol

Oshiro

Alves

A method to represent impacted structures using scaled models made of different materials. Int J Impact Eng 2016; 90: 81–94.

Rosenberg

Dekel

Revisiting the perforation of ductile plates by sharp-nosed rigid projectiles. Int J Solid Struct 2010; 47: 3022–3033.

MatWeb: online materials information resource, http://www.matweb.com/ (accessed 21 July 2016).

10.

Mondal

Mishra

Jena

. Effect of heat treatment on the behavior of an AA7055 aluminum alloy during ballistic impact. Int J Impact Eng 2011; 38: 745–754.

11.

Recht

Ipson

TW.

Ballistic perforation dynamics. J Appl Mech: T ASME 1963; 30: 384–390.

12.

Rosenberg

Dekel

Terminal ballistics. Berlin, Heidelberg: Springer Science & Business Media, 2012.

13.

Iqbal

Chakrabarti

Beniwal

. 3D numerical simulations of sharp nosed projectile impact on ductile targets. Int J Impact Eng 2010; 37: 185–195.

14.

Børvik

Clausen

Eriksson

. Experimental and numerical study on the perforation of AA6005-T6 panels. Int J Impact Eng 2005; 32: 35–64.

15.

Hub

Numerical simulations of the 12.7-PZ32 bullet penetration of the aluminium alloy sheet metal. Technology 2009; 8: 339–344.

16.

Ibrahim

Siswanto

Zaidi

AMA

. Numerical study on failure process of AL2024 T3 plate under normal impact by ogival projectiles. Int J Min Metall Mech Eng 2014; 2: 74–78.

17.

Forrestal

Okajima

Luk

VK.

Penetration of 6061-T651 aluminum targets with rigid long rods. J Appl Mech: T ASME 1988; 55: 755–760.

18.

Pirvu

Deleanu

Badea

Using simulation for comparing the response of materials in terminal ballistics. Mech Test Diagn 2015; 5: 32.

19.

Swaddiwudhipong

Islamb

Liu

ZS.

High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method. Int J Prot Struct 2010; 1: 489–506.

20.

Wang

Zhang

. Energy absorption mechanisms of modified double-aluminum layers under low-velocity impact. Int J Appl Mech 2015; 7: 1550086.

21.

Karagiozova

Langdon

Nurick

. Simulation of the response of fibre–metal laminates to localised blast loading. Int J Impact Eng 2010; 37: 766–782.

22.

Zhu

Zhao

. A numerical simulation of the blast impact of square metallic sandwich panels. Int J Impact Eng 2009; 36: 687–699.

23.

LS-DYNA Aerospace Working Group. Modeling guidelines document (version 13-1), 2013.

24.

Zhang

Chen

Guan

. Study on ballistic penetration resistance of titanium alloy TC4, Part II: numerical analysis. Chin J Aeronaut 2013; 26: 606–613.

25.

Mingjie

Xiaofeng

. Research on composition and formation mechanism of penetration acceleration signal. In: Proceedings of the 2010 international conference on intelligent computation technology and automation (ICICTA), Changsha, China, 11–12 May 2010, pp.1118–1121. New York: IEEE.

26.

Fasanella

Jackson

KE.

Best practices for crash modeling and simulation. NASA/ARL report, October 2002, https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20020085101.pdf

A similarity method for predicting the residual velocity and deceleration of projectiles during impact with dissimilar materials

Abstract

Keywords

Introduction

Experiments

Experimental setup

Similitude analysis and experimental design

Experimental results and analysis

Numerical simulation

Numerical model and validation

Numerical model

Validation of the simulation method

Deceleration filtering

Deceleration

Similarity analysis of the projectile deceleration

Discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References