# Theoretical Model of the Axial Residual Velocity of PELE Projectiles Penetrating Thin Metal Targets

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of the Theoretical Model of Axial Residual Velocity

#### 2.1. Plane Shock Wave Assumption of PELE Projectile Penetrating Against Target Plate

_{0}represents the moment when the projectile has just touched the target plate; t

_{1}represents the moment when the rarefaction waves from the back side of the target plate reach the contact interface; t

_{2}represents the moment when the PELE projectile just perforates the target plate. The time range t

_{0}–t

_{1}is defined as the impact stage, and the time range t

_{1}–t

_{2}is defined as the plug shear stage. Thus, the process of the PELE projectile penetrating the target plate can be divided into two stages: projectile impact stage and plug shear stage.

#### 2.2. Internal Energy and Kinetic Energy Increment of the Target Plate Plug after Shock Waves

_{0}, the initial velocity and pressure of the target plate and the initial pressure of the projectile are both zero. Here, the following definitions are made. P stands for the pressure per unit area, ρ is the material density, D represents the shock wave velocity, u is the particle velocity, E means the energy, c, and λ represents the Hugoniot constant of the material. In addition, the subscript j is an abbreviation for “jacket,” and it is used to denote the parameters of the projectile outer casing material; the subscript f is an abbreviation for “filling,” and it is used to denote the parameters of the inner core material; the subscript t represents the parameters of the target plate, and it is an abbreviation for “target.” The subscript tj represents the post-wave state of target plate after the collision between the outer casing and target plate, and the subscript tf represents the post-wave state of the target plate after the collision between the inner core and target plate. The subscript 0 indicates the wave-front state of the material, and the subscript 1 indicates the post-wave state of the material. According to the interaction of the shock wave, the following relationships can be obtained.

_{0j}, ρ

_{0j}, u

_{0j}, c

_{0j}, and a left shock wave is generated in the outer casing after the impact, and the post-wave state is P

_{1j}, ρ

_{1j}, u

_{1j}, c

_{1j}. Thus, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.

_{0f}, ρ

_{0f}, u

_{0f}, c

_{0f}, and a left shock wave is generated in the inner core after the impact, and the post-wave state is P

_{1f}, ρ

_{1f}, u

_{1f}, c

_{1f}. Similarly, the following relationships can be obtained by the shock wave discontinuous equation and the linear shock equation.

_{0t}, ρ

_{0t}, u

_{0t}, c

_{0t}, and a right shock wave is generated in the target plate after the impact. The post-wave state is divided into two regions, as follows.

_{tj}, ρ

_{tj}, u

_{tj}, c

_{tj}. Similarly, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.

_{tf}, ρ

_{tf}, u

_{tf}, c

_{tf}. Similarly, the following relationships can be obtained through the shock wave discontinuous equation and the linear shock equation.

_{tj}= P

_{1j}, u

_{tj}= u

_{1j}. When Equations (1), (2), (5), and (6) are combined together, the expression of the particle velocity of the target plate after the shock wave is as follows.

_{tf}= P

_{1f}, u

_{tf}= u

_{1f}. When Equations (3), (4), (7), and (8) are combined together, the particle velocity after the impact between the inner core and the target plate can be obtained as follows.

_{0}. The relation between the material pressure and the specific volume under the impact adiabatic condition is shown in Figure 5. The relationship P >> P

_{0}is reflected in Figure 5 where point A is very close to point M, which means that the AB line can be considered to be the diagonal of the rectangular MNBC. Under this state, the influence of the initial pressure can be neglected, and it can be assumed that the total work of the shock compression is evenly distributed between the internal energy and kinetic energy.

#### 2.3. Internal Energy Increment of Projectile after Shock Waves

_{tj}, D

_{tf}, D

_{j}, and D

_{f}can be obtained. Where D

_{tj}represents the shock wave velocity of target plate under the impact between outer casing and target plate, D

_{tf}represents the impact wave velocity of target plate under the impact between inner core and target plate, and D

_{j}and D

_{f}represent the shock wave velocity of outer casing and inner core, respectively. The duration of the shock wave between the projectile and target plate is determined by the time from the projectile impacting the target plate to the rarefaction wave reaching the impact interface. It is assumed that the shock wave intensity unloading of the impact interface is completed once, and then the shock wave pressure duration t

_{j}and t

_{f}of the outer casing and the inner core can be expressed as follows.

_{j}and t

_{f}can be expressed by the kinetic energy change.

#### 2.4. Shear Energy Dissipation of the Target Plate Plug

_{τj}and that of the inner edge of outer casing on the target plate E

_{τf}can be expressed as follows.

#### 2.5. Axial Residual Velocity of PELE Projectile after Perforating Against the Target Plate

## 3. Verification and Analysis of the Theoretical Model

#### 3.1. Literature Verification and Analysis of the Theoretical Model

_{ref}represents the calculation result corresponding to the model established in Reference [3], u

_{cal}represents the calculation result corresponding to the model established in this paper, and u

_{exp}represents the experimental result in Reference [3].

#### 3.2. Experiment Verification and Analysis of the Theoretical Model

_{0}, the axial residual velocity of the projectile can be expressed as follows.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Longitudinal section of the simplified PELE (penetration with enhanced lateral efficiency) projectile.

**Figure 2.**Different penetration states of the PELE projectile. (

**a**) Initial penetration state; (

**b**) completely perforating state.

**Figure 3.**Propagating schematic diagram of shock waves and rarefaction waves upon PELE projectile impacting the target plate. (

**a**) Shock waves propagate in the projectile and target plate; (

**b**) rarefaction waves propagate in the projectile and target plate.

**Figure 4.**Simplified schematic diagram of the interaction between projectile and target plate. (

**a**) Interaction between outer casing and target plate; (

**b**) interaction between inner core and target plate.

**Figure 6.**Comparison of residual velocity between reference results and calculation results. (

**a**) A-G3 core, A-U4G target plate (3 mm); (

**b**) A-G3 core, A-U4G target plate (8 mm); (

**c**) A-G3 core, XC48 target plate (3 mm); (

**d**) PE core, A-U4G target plate (3 mm); (

**e**) PE core, A-U4G target plate (8 mm); (

**f**) PE core, XC48 target plate (3 mm).

**Figure 7.**Different parts of the experimental apparatus. (

**a**) Experimental projectile; (

**b**) experimental gun; (

**c**) light curtain target; (

**d**) copper foil target.

**Figure 9.**The fragments of PELE projectile outer casing collected from the water tank. (

**a**) Al-1#; (

**b**) PTFE-1#.

**Figure 10.**The residual velocity and relative error of experiment and calculation under different inner core material conditions. (

**a**) Residual velocity; (

**b**) relative error.

Material Properties | Outer Casing Material | Inner Core Material | Target Plate Material | ||
---|---|---|---|---|---|

D180K | A-G3 | PE | A-U4G | XC48 | |

Density ρ_{0} (g/cm^{3}) | 18.0 | 2.65 | 0.92 | 2.8 | 7.82 |

Sound velocity c_{0} (m/s) | 4029 | 5176 | 2187 | 5106 | 4797 |

Hugoniot constant λ | 1.24 | 1.35 | 1.48 | 1.35 | 1.49 |

Dynamic yield stress σ_{y}^{D} (Gpa) | -- | -- | -- | 1.16 | 2.16 |

Target Plate | Experimental Initial Velocity (m/s) | Experimental Residual Velocity (m/s) | Calculated Residual Velocity in Reference [3] (m/s) | Calculated Residual Velocity in this Paper (m/s) | |
---|---|---|---|---|---|

Material | Thickness (mm) | ||||

A-U4G | 3 | 929 | 914 | 916 | 912 |

1275 | 1261 | 1257 | 1264 | ||

2457 | 2444 | 2424 | 2436 | ||

8 | 937 | 900 | 898 | 902 | |

1254 | 1208 | 1206 | 1210 | ||

2472 | 2434 | 2384 | 2416 | ||

2984 | 2945 | -- | 2907 | ||

XC48 | 3 | 925 | 895 | 890 | 900 |

1261 | 1231 | 1214 | 1237 | ||

2441 | 2423 | 2352 | 2403 |

Target Plate | Experimental Initial Velocity (m/s) | Experimental Residual Velocity (m/s) | Calculated Residual Velocity in Reference [3] (m/s) | Calculated Residual Velocity in this Paper (m/s) | |
---|---|---|---|---|---|

Material | Thickness (mm) | ||||

A-U4G | 3 | 924 | 918 | 910 | 914 |

1279 | 1263 | 1260 | 1266 | ||

2420 | 2394 | 2386 | 2401 | ||

8 | 939 | 887 | 898 | 894 | |

1258 | 1203 | 1207 | 1200 | ||

2445 | 2393 | 2353 | 2389 | ||

2977 | 2952 | -- | 2934 | ||

XC48 | 3 | 936 | 889 | 899 | 897 |

1262 | 1206 | 1213 | 1211 | ||

2475 | 2462 | 2380 | 2436 |

Material Properties | Outer Casing Material | Inner Core Material | Target Plate Material | |
---|---|---|---|---|

Tungsten | 6061Al | PTFE | 2024Al | |

Density ρ_{0} (g/cm^{3}) | 18.0 | 2.7 | 2.15 | 2.8 |

Sound velocity c_{0} (m/s) | 4029 | 5350 | 1682 | 5370 |

Hugoniot constant λ | 1.24 | 1.34 | 1.82 | 1.3 |

Dynamic yield stress σ_{y}^{D} (Gpa) | -- | -- | -- | 1.2 |

Inner Core Material | Test Number | Quantity of Recycled Fragments | Total Mass of Recycled Fragments | Percentage of Recovery Mass of Outer Casing (%) |
---|---|---|---|---|

Al | 1# | 45 | 34.0 | 90.2 |

2# | 42 | 34.4 | 91.2 | |

PTFE | 1# | 34 | 31.7 | 84.1 |

2# | 29 | 31.8 | 84.4 |

Inner Core Material | Test Number | Experimental Initial Velocity (m/s) | Experimental Residual Velocity (m/s) | Calculated Residual Velocity (m/s) |
---|---|---|---|---|

Al | 1# | 806 | 788 | 779 |

2# | 798 | 780 | 772 | |

PTFE | 1# | 809 | 790 | 784 |

2# | 802 | 787 | 776 |

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**MDPI and ACS Style**

Ding, L.; Tang, W.; Ran, X.; Fan, Z.; Chen, W.
Theoretical Model of the Axial Residual Velocity of PELE Projectiles Penetrating Thin Metal Targets. *Symmetry* **2019**, *11*, 776.
https://doi.org/10.3390/sym11060776

**AMA Style**

Ding L, Tang W, Ran X, Fan Z, Chen W.
Theoretical Model of the Axial Residual Velocity of PELE Projectiles Penetrating Thin Metal Targets. *Symmetry*. 2019; 11(6):776.
https://doi.org/10.3390/sym11060776

**Chicago/Turabian Style**

Ding, Liangliang, Wenhui Tang, Xianwen Ran, Zijian Fan, and Weike Chen.
2019. "Theoretical Model of the Axial Residual Velocity of PELE Projectiles Penetrating Thin Metal Targets" *Symmetry* 11, no. 6: 776.
https://doi.org/10.3390/sym11060776