# Rapid Evaluation and Analysis of the Deformation of Filled Cylindrical Casing with Deforming Charge Width Based on Self-Compiled MATLAB Program

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Criterion α for Measuring the Deformable Surface Width

#### 2.1. Establishment of Criterion α

_{1}D = R

_{0}, the initial center is O

_{1}, the casing radius after deformation is O

_{2}A = R

_{1}, the corresponding center is O

_{2}, the chord length of the plane part is AE = 2h, the center angle of the chord length is $\angle $AO

_{2}E = 2δ, and the central angle corresponding to the initial deforming charge is $\angle $BO

_{1}D = 2θ, as shown in Figure 2b. In addition, the y-axis is parallel to the loading direction, and the x-axis is perpendicular to the loading direction. Criterion α is defined as the proportion of the deformed portion to the entire cylindrical casing. Thus, the expression of criterion α is as follows:

_{0}is known, and if any of the parameters δ or R

_{1}is known, another parameter can be calculated. Then, α can be calculated from the obtained parameter. Taking the radius of cylindrical casing R

_{0}= 50.5 mm as an example, combined with the above geometric analysis, the proportions of the straight segment and arc segment under five groups of different conditions (D-30°, D-60°, D-90°, D-120°, and D-150°) are shown in Table 1 and Figure 3.

#### 2.2. Optimum Range of Criterion α

## 3. Self-Compiled MATLAB Rapid Evaluation Program

#### 3.1. Parameter Initialization of the MATLAB Rapid Evaluation Program

_{0}, the thickness is t

_{0}, the width is L, and the explosion heat is Q

_{v}; the material density of Plane 1 is ρ

_{1}, the thickness is t

_{1}, the width is L, and the scattering velocity is v

_{1}under the loading of deforming charge; the material density of Plane 2 is ρ

_{2}, the thickness is t

_{2}, the width is L, and the scattering velocity is v

_{2}under the loading of deforming charge. Thus, the following relations can be obtained.

_{g}and the plate kinetic energy E

_{p}. Then, the following expression is:

_{g}≈ m

_{0}(v

_{1}

^{2}+ v

_{2}

^{2})/12. Thus, Equation (9) can be rewritten as:

_{1}and v

_{2}can be obtained, respectively.

#### 3.2. Verification of the Rapid Evaluation Program Based on LS-DYNA

_{1}= ρ

_{2}= 7.8 g/cm

^{3}, the Young’s modulus is E = 2.06 × 10

^{11}Pa, and the yield limit is σ

_{y}= 245 MPa. The thickness and diameter of the inner casing are t

_{1}= 4 mm and d

_{1}= 101 mm and the thickness of outer casing is t

_{2}= 2 mm. The material density of the deforming charge is ρ

_{0}= 1.67 g/cm

^{3}, the thickness of deforming charge is t

_{0}= 5 mm, the width of the deforming charge is φ = 90°, and the explosion heat of the deforming charge is Q

_{v}= 4520 kJ/kg. The material of the internal filling medium is soil, and its density is ρ

_{m}= 1.80 g/cm

^{3}. Substituting the above parameters into Equations (5) and (6), the specific values of v

_{1}and v

_{2}can be obtained.

#### 3.3. Verification of the Rapid Evaluation Program Based on the Experiments

## 4. Analysis and Discussion of the Results

#### 4.1. Effect of Deforming Charge Width on Filled Cylindrical Casing

_{1}. Based on the MATLAB rapid evaluation program, the spatial distributions of the casing elements of the filled cylindrical casing, under different loading widths at several typical moments, were quickly obtained, as shown in Figure 14. In addition, the final deformation surface of the casing corresponding to the four groups of working conditions was arranged as shown in Figure 15.

- (1)
- If the deforming charge width is too small, the resulting load will be distributed in a relatively narrow region. At this moment, fewer and relatively concentrated casing elements are driven by the lateral explosion, so that the difference in relative azimuth between the casing elements and the circle center is smaller. When velocity decomposition is performed, the velocity in the horizontal direction is larger, and the velocity of the casing elements at the loading region center is slightly larger than those at both ends. As a result, the deformed surface of the casing eventually results in an approximate “inner-concave” type.
- (2)
- If the deforming charge width is too large, the resulting load will be distributed in a relatively wider region, and more casing elements are driven by the lateral explosion. The wide loading region creates a large difference in the relative azimuths of the casing elements and the circle center. The difference in the azimuth angles will cause a greater difference in the velocity decomposition in the horizontal and vertical direction. The closer the casing element to the outside, the greater the velocity of its decomposition in the vertical direction, and the lower in the horizontal direction. Therefore, the final deformation surface is similar to the “outer-convex” type.
- (3)
- As the deforming charge width increases, the deformation surface of the casing gradually transfers from the inner-concave to outer-convex shape. Therefore, we inferred that a suitable width of the deforming charge must exist to make the corresponding deformation surface of the casing D-shaped.

#### 4.2. Relationship between Deforming Dharge Width φ and Criterion α

_{0}is assumed to be 30 to 150 mm. In addition, according to Equation (7), the relationship between h and R

_{0}can be rewritten as follows:

_{0}is the radius of the initial casing ($\overline{{O}_{1}D}$), as shown in Figure 2b. Thus, in combination with the range of α, the relationship curve between h and R

_{0}can be obtained as shown in Figure 16.

_{0}, combined with Equations (15) to (18), and using MATLAB for the iterative solution, the optimum deforming charge width corresponding to different α values under different initial casing sizes can be obtained as shown in Table 2.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram and action process of the deformable warhead. (

**A**) Initial projectile structure; (

**B**) Deforming charge detonation; (

**C**) D-shaped projectile surface; (

**D**) Fragments directional dispersion.

**Figure 2.**Schematic diagram of the casing element after deformation and the D-shaped profile. (

**a**) Casing element and (

**b**) D-shaped profile.

**Figure 5.**Relationships between the energy distribution and azimuth angle. (

**a**) Total energy of fragments varies with azimuth angle; (

**b**) Energy density of fragments varies with azimuth angle.

**Figure 6.**Schematic diagram of the unit length loading area. The mass of Plane 1 per unit length m

_{1}: m

_{1}= ρ

_{1}t

_{1}L; The mass of Plane 2 per unit length m

_{2}: m

_{2}= ρ

_{2}t

_{2}L; The mass of deforming charge per unit length m

_{0}: m

_{0}= ρ

_{0}t

_{0}L; The total energy released by unit length deforming charge E: E = m

_{0}Q

_{v}.

**Figure 8.**Deformed surface of the inner casing under several typical moments obtained from MATLAB: (

**a**) t = 50 μs; (

**b**) t = 100 μs; (

**c**) t = 150 μs.

**Figure 9.**Deformed surface of the inner casing under several typical moments obtained from LS-DYNA: (

**a**) t = 50 μs; (

**b**) t = 100 μs; (

**c**) t = 150 μs.

**Figure 10.**Various parts of the experimental device: (

**a**) upper end-plate; (

**b**) lower end-plate; (

**c**) filling medium; (

**d**) inner casing; (

**e**) outer casing.

**Figure 11.**Experiment preparation process: (

**a**) filling of inner medium; (

**b**) adhesion of deforming charge; (

**c**) installation of detonating cord; (

**d**) assembly of outer casing; (

**e**) installation of electric detonator.

**Figure 13.**Comparison of deformed casing profile based on the experiment and MATLAB results. (

**a**) φ = 60°; (

**b**) φ = 90°.

**Figure 14.**Spatial distribution of the casing elements of the filled cylindrical casing under different loading widths at several typical moments: (

**a**) φ = 30°; (

**b**) φ = 60°; (

**c**) φ = 90°; (

**d**) φ = 120°.

**Figure 15.**The final deformation surface of the casing corresponding to the four groups of working conditions.

**Figure 16.**Relationship curve between half of the length of the platform section after deformation h and radius of the initial casing R

_{0}.

**Figure 18.**Deformation surface of the casing under different deforming charge widths: (

**a**) φ = 30° (α = 17.8%); (

**b**) φ = 60° (α = 24.1%); (

**c**) φ = 90° (α = 31.2%); (

**d**) φ = 120° (α = 38.4%).

D-Shaped | Radius of Initial Casing $\overline{{\mathit{O}}_{1}\mathit{D}}$ | Length of Straight Segment $\overline{\mathit{A}\mathit{E}}$ | Length of Arc Segment $\stackrel{\u2322}{\mathit{A}\mathit{O}\mathit{E}}$ | Proportion of Straight Segment β |
---|---|---|---|---|

D-30° | 5.05 cm | 2.61 cm | 29.09 cm | 8.2% |

D-60° | 5.05 cm | 5.05 cm | 26.44 cm | 16.1% |

D-90° | 5.05 cm | 7.14 cm | 23.79 cm | 22.8% |

D-120° | 5.05 cm | 8.75 cm | 21.15 cm | 29.2% |

D-150° | 5.05 cm | 9.76 cm | 18.51 cm | 34.5% |

Initial Radius of Casing | Width of Deforming Charge φ | |||||
---|---|---|---|---|---|---|

R_{0} (mm) | α = 20% | α = 22% | α = 24% | α = 26% | α = 28% | α = 30% |

30 | 46° | 53° | 60° | 69° | 77° | 85° |

40 | 46° | 53° | 60° | 69° | 77° | 85° |

50 | 46° | 53° | 60° | 69° | 77° | 85° |

60 | 46° | 53° | 60° | 69° | 77° | 85° |

70 | 46° | 53° | 60° | 69° | 77° | 85° |

80 | 46° | 53° | 60° | 69° | 77° | 85° |

90 | 46° | 53° | 60° | 69° | 77° | 85° |

100 | 46° | 53° | 60° | 69° | 77° | 85° |

110 | 46° | 53° | 60° | 69° | 77° | 85° |

120 | 46° | 53° | 60° | 69° | 77° | 85° |

130 | 46° | 53° | 60° | 69° | 77° | 85° |

140 | 46° | 53° | 60° | 69° | 77° | 85° |

150 | 46° | 53° | 60° | 69° | 77° | 85° |

Width of Deforming Charge φ | α | |
---|---|---|

Simulation Result (LS-DYNA) | Calculation Result from Equation (19) | |

30° | 0.178 | 0.161 |

60° | 0.241 | 0.237 |

90° | 0.312 | 0.313 |

120° | 0.384 | 0.389 |

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**MDPI and ACS Style**

Ding, L.; Li, Z.; Lu, F.; Li, X.
Rapid Evaluation and Analysis of the Deformation of Filled Cylindrical Casing with Deforming Charge Width Based on Self-Compiled MATLAB Program. *Symmetry* **2018**, *10*, 310.
https://doi.org/10.3390/sym10080310

**AMA Style**

Ding L, Li Z, Lu F, Li X.
Rapid Evaluation and Analysis of the Deformation of Filled Cylindrical Casing with Deforming Charge Width Based on Self-Compiled MATLAB Program. *Symmetry*. 2018; 10(8):310.
https://doi.org/10.3390/sym10080310

**Chicago/Turabian Style**

Ding, Liangliang, Zhenduo Li, Fangyun Lu, and Xiangyu Li.
2018. "Rapid Evaluation and Analysis of the Deformation of Filled Cylindrical Casing with Deforming Charge Width Based on Self-Compiled MATLAB Program" *Symmetry* 10, no. 8: 310.
https://doi.org/10.3390/sym10080310