# Numerical and Experimental Studies of the ŁK Type Shaped Charge

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

_{p}is the penetration depth, r is the square root of target to jet density, and ρ

_{b}and ρ

_{j}are the partition and jet density.

_{0}means the virtual beginning of the jet, the time it takes the jet front to reach the partition is determined by the relation

_{0}means the distance from the beginning of the jet to the barrier, t

_{p}is the penetration time, and V

_{0}is the initial jet velocity.

_{p}(0) = 0:

_{p}can be estimated from Equation (4).

## 3. Experimental Method and Results

#### 3.1. Assumptions

#### 3.2. The Structure of Shaped Charge Tests

^{3}, and a detonation speed of 7500 m/s.

#### 3.3. Experimental Results

## 4. Numerical Analysis

#### 4.1. Assumptions for Modelling

#### 4.2. Description of Materials

_{T}means the TNT equivalent of the explosive in question, m is the mass of the explosive in question, m

_{T}is the mass of the reference explosive, E is the explosive energy of the charge in question, and E

_{T}is the explosive energy of the referenced explosive.

_{0}) of the original explosive, κ

_{pd}is the pre-detonation bulk modulus, and A*, B*, R

_{1}, R

_{2}and ω are the fitting coefficients obtained from experiments.

_{0}is the initial air pressure, γ is the Gruneisen coefficient, ρ

_{a}is the density of air, and E

_{0}is the initial internal energy.

_{0}is the room temperature, T

_{m}is the melting temperature, A, B, C, n and m are the J-C material behavior coefficients, ${\epsilon}^{f}$ is the plastic strain to fracture, ${\sigma}^{*}$ is the equivalent stress, and D

_{1}, D

_{2}, D

_{3}, D

_{4}and D

_{5}are the input constants determined empirically.

#### 4.3. ŁK Charge Options

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Cumulative charges prepared for testing: (

**a**) axial of cumulative charge; (

**b**) peripheral of cumulative charge.

**Figure 6.**Plate class 1 after firing with ŁK charge: (

**a**) holes after axial (1A, 2A and 3A) and peripheral charges (1P, 2P and 3P); (

**b**) no perforation for 1A and 2A shots.

**Figure 7.**Examples of results of the entry and the exit openings after a shot with a ŁK type shaped charge: (

**a**) hole of axial charge entrance; (

**b**) hole of axial charge exit; (

**c**) hole of peripheral charge entrance; (

**d**) hole of peripheral charge exit.

**Figure 9.**Numerical models of the ŁK charge components: (

**a**) plastic case; (

**b**) explosive; (

**c**) cumulative liner; (

**d**) armor plate.

**Figure 11.**Stages of formation jet of the axial charge for individual time steps: (

**a**) process of propagation of explosion detonation wave; (

**b**) stages of the jet’s formation in the liner.

**Figure 12.**Stages of jet formation of the peripheral charge for individual time steps: (

**a**) the process of propagation of an explosion detonation wave onto an additional insert in order to increase the angles of incidence on the liner; (

**b**) stages of the jet formation in main liner.

**Figure 14.**Time steps of the three types of ŁK charges analyzed: (

**a**–

**c**) time step for t = 2.5 × 10

^{−5}; (

**d**–

**f**) time step for t = 4.5 × 10

^{−5}; (

**g**–

**i**) time step for t = 6.5 × 10

^{−5}; (

**j**–

**l**) time step for t = 8.0 × 10

^{−5}.

**Figure 16.**Qualitative comparison of the effect of the collapsing liner material on the formation of the slug and the jet. (

**a**) the jet from detonation of an axial charge; (

**b**) the jet from detonation of an peripheral charge.

**Figure 17.**Simulations of cumulative charges with liners of different material structures: (

**a**) formation of the cumulative jet; (

**b**) jet velocities for different insert materials.

Specimens Thickness (mm) | Yield Strength R _{p02} (MPa) | Tensile Strength R _{m} (MPa) | Hardness (HBW) | Charpy-V (J) | Elongation A_{5}(min %) | Elongation A_{50}(min %) |
---|---|---|---|---|---|---|

plate 80 | 800 | 900–1100 | 280–330 | 60 J/−40 °C | 13 | 15 |

plate 100 | - | - | 300–350 | 40 J/−40 °C | - | - |

Specimens [mm] | Axial Charges | Peripheral Charges | |||||
---|---|---|---|---|---|---|---|

1A | 2A | 3A | 1P | 2P | 3P | ||

plate 100 | front | 21.2 | 22.7 | 23.0 | 12.9 | 14.1 | 13.2 |

back | None | None | 14.8 | 12.0 | 13.4 | 12.9 | |

plate 80 | front | 22.0 | 23.0 | 24.0 | 12.0 | 12.5 | 13.0 |

back | 15.3 | 16.0 | 15.0 | 11.7 | 12.3 | 12.8 |

Explosive | |||||||||||||||||

ρ* | D | P_{CJ} | A* | B* | R_{1} | R_{2} | ω | κ_{pd} | Sources | ||||||||

(kg/m^{3}) | (m/s) | (GPa) | (GPa) | (GPa) | (-) | (-) | (-) | (GPa) | |||||||||

TNT | 1730 | 8193 | 28.00 | 609.00 | 13.00 | 4.50 | 1.40 | 0.25 | 9.00 | [48,53] | |||||||

Liner Material | |||||||||||||||||

ρ | E | ν | A | B | C | n | m | Sources | |||||||||

(kg/m^{3}) | (GPa) | (-) | (MPa) | (MPa) | (-) | (-) | (-) | ||||||||||

Zn5Al | 7010 | 98 | 0.30 | 180 | 200 | 0.008 | 0.100 | 1.0 | [53,54] | ||||||||

Cooper | 8960 | 1.28 | 0.36 | 80 | 500 | - | 0.605 | 1.00 | [55,56] | ||||||||

AC-44200 | 2730 | 70 | 0.33 | 110 | 330 | 0.008 | 0.100 | 1.00 | [57] | ||||||||

Steel S355 | 7820 | 210 | 0.35 | 807 | 1660 | 0.008 | 0.100 | 1.00 | [26,53] | ||||||||

Armco | 7870 | 210 | 0.37 | 233 | 460 | 0.047 | 0.320 | 0.55 | [53] | ||||||||

Lead | 11,300 | 115 | 0.42 | 24 | 40 | 0.010 | 0.500 | 1.00 | [53] | ||||||||

Elastomer | 1.200 | 0.01 | 0.49 | 10 | 20 | - | - | - | [53,54] |

_{CJ}is the Chapman–Jouguet pressure; A*, B*, R

_{1}and R

_{2}are parameters; ω is the Gruneisen parameter; κ

_{pd}is the pre-detonation bulk modulus; ρ is the density of material insert; E is the Young’s modulus; ν is the Poisson’s ratio; A is the yield strength of the material; B is the strain hardening constant; C is the strengthening coefficient of strain rate; n is the hardening exponent; m is the thermal softening exponent.

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**MDPI and ACS Style**

Pyka, D.; Kurzawa, A.; Bocian, M.; Bajkowski, M.; Magier, M.; Sliwinski, J.; Jamroziak, K.
Numerical and Experimental Studies of the ŁK Type Shaped Charge. *Appl. Sci.* **2020**, *10*, 6742.
https://doi.org/10.3390/app10196742

**AMA Style**

Pyka D, Kurzawa A, Bocian M, Bajkowski M, Magier M, Sliwinski J, Jamroziak K.
Numerical and Experimental Studies of the ŁK Type Shaped Charge. *Applied Sciences*. 2020; 10(19):6742.
https://doi.org/10.3390/app10196742

**Chicago/Turabian Style**

Pyka, Dariusz, Adam Kurzawa, Miroslaw Bocian, Marcin Bajkowski, Mariusz Magier, Janusz Sliwinski, and Krzysztof Jamroziak.
2020. "Numerical and Experimental Studies of the ŁK Type Shaped Charge" *Applied Sciences* 10, no. 19: 6742.
https://doi.org/10.3390/app10196742