# Discrete Particle Method for Simulating Hypervelocity Impact Phenomena

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## Abstract

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## 1. Introduction

## 2. Simulation Model

#### 2.1. Initial Setup

#### 2.2. Particle Potentials

#### 2.2.1. Contact Potentials

#### 2.2.2. Bonded Potentials

#### 2.3. Sizing and Convergence Properties of Our Particle Model

#### 2.3.1. Geometric Sizing Properties

#### 2.3.2. Convergence Properties

## 3. Results and Discussion

#### 3.1. Choice of Model Parameters

#### 3.2. Validation with Experiment

#### 3.2.1. Extension of Debris Cloud

#### 3.2.2. Shape and Degree of Fragmentation

- The well defined front end (left side of debris cloud) as seen in the experiment is missing in the simulation.
- The large central fragment in the simulation did not fracture into a distinctive debris bubble behind the dense cloud center (right side of debris cloud) as seen in the experiment.

#### 3.3. Analyzing Fragmentation

## 4. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**High-speed photograph sequence of HVI performed at Fraunhofer Ernst–Mach–Institut [8]. A cylindrical impactor approaches from the left, collides with the target plate, and the resulting debris cloud propagates to the right.

**Figure 2.**Particles are initiated into a regular cubic lattice. The model’s properties are independent of number of particles.

**Figure 3.**Repulsive and cohesive potentials used in the model. The solid blue line (left side) represents the Lennard-Jones potential and the solid red line (right side) represents the spring potential. The combined blue and red solid lines govern the forces acting on each particle pair; the dotted lines are excluded.

**Figure 4.**Total potential and kinetic energies of colliding plates with different numbers of particles. (

**top**) The unscaled simulation is strongly affected by the number of simulation particles used. (

**bottom**) Resizing the simulation renders the resulting energy independent of the number of particles used.

**Figure 5.**Fragmentation in debris clouds simulated with a varying number of particles shows convergence.

**Figure 6.**Experimental high-speed photograph of the debris cloud showing the cloud’s length ratio $R={L}_{a}/{L}_{r}$, and axial expansion velocity, ${v}_{a}$. The witness plate seen in the right side of the snapshot is shown in Figure 8. The image intensity has been inverted for better viewing. Figure courtesy of Fraunhofer Ernst–Mach–Institut.

**Figure 7.**Simulation snapshots of the debris cloud calculated with different values of $\epsilon $ and $\kappa $ taken at 32 $\mathsf{\mu}$s after impact. Impactor particles are shown in red and target particles are shown in blue.

**Figure 8.**The witness plate from the experiment shown in Figure 6 contains many small impact craters left by the debris cloud fragments. The average crater has a diameter of approximately 1 mm, but there is no crater left by a large central fragment. Figure courtesy of Fraunhofer Ernst–Mach–Institut.

**Figure 9.**Contour plot representing the debris cloud’s axial to radial extension ratio, $R={L}_{a}/{L}_{r}$, for each simulation in Figure 7. Color code refers to the length ratio R shown on the right side of the figure. See main text for details.

**Figure 10.**Contour plot representing the debris cloud’s axial expansion velocity normalized by the impact velocity. Color code refers to the velocity ${v}_{a}/{v}_{0}$ shown on the right side of the figure. The experimentally measured value is shown as a red contour line. See main text for details.

**Figure 11.**A snapshots of the resulting 3D simulation using the best fit parameters: $\epsilon =0.01$, $\kappa =4.56\times {10}^{6}$ and ${r}_{cut}=1.5\sigma $. This simulation corresponds to the experiment shown in Figure 6. Impactor particles are in red and target particles in blue.

**Figure 12.**A series of 3D simulation snapshots showing an aluminum sphere impacting an aluminum plate at hypervelocity. Red indicates impactor particles and gray indicates target plate particles. The simulation was run with $N=7.4\times {10}^{5}$ particles, with a timestep of $\mathsf{\Delta}t=1\times {10}^{-10}$ s, and took nine hours to complete on a single processor. Impact parameters are: ${v}_{0}=6.7\phantom{\rule{0.277778em}{0ex}}{\mathrm{kms}}^{-1}$ and $t/D=0.425$. A video sequence of this simulation can be found as Supplementary Material in Video S1.

**Figure 13.**Debris cloud axial expansion velocity with ${v}_{0}=6.7\phantom{\rule{0.277778em}{0ex}}{\mathrm{kms}}^{-1}$ at different t/D ratios. The simulation results are compared to experiments by Piekutowski [50]. The dotted line represents linearly extrapolated experimental data.

**Figure 14.**Simulations with high $t/D$ ratios match the experiment closely with complete fragmentation of the impactor and a similar cloud shape. (

**a**) High-speed photograph of experiment (intensity inverted); (

**b**) 3D simulation shown with ${v}_{0}=6.7\phantom{\rule{0.277778em}{0ex}}{\mathrm{kms}}^{-1}$ and $t/D=0.425$.

**Figure 15.**The shape of the debris cloud is highly affected by the $t/D$ ratio. (

**a**) High-speed radiograph of experiment; (

**b**) 3D simulation shown with ${v}_{0}=6.7\phantom{\rule{0.277778em}{0ex}}{\mathrm{kms}}^{-1}$ and $t/D=0.05$. The simulation performs poorly at extremely small $t/D$ ratios, but very well at large values, cf. Figure 14. Experimental snapshot from [50].

**Figure 16.**Fragmentation of the debris cloud at various $t/D$ ratios and impact velocities ${v}_{0}$. (

**a**) Three–dimensional representation of fragmentation level; (

**b**) Two–dimensional contour plot of fragmentation level. Black dots represent simulation points from which the contour surface was created and the red line is the fragmentation cutoff boundary.

**Figure 17.**Snapshots of the debris clouds produced under different impact conditions. The impact conditions leading to a valid model is shown in red; invalid regions are shown in blue.

**Table 1.**Approximate shock-melting properties of aluminum [51].

Shock | Impact | |
---|---|---|

Pressure [GPa] | Velocity [km/s] | |

Incipient Melting | 70 | 5.6 |

Complete Melting | 100 | 7.0 |

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**MDPI and ACS Style**

Watson, E.; Steinhauser, M.O.
Discrete Particle Method for Simulating Hypervelocity Impact Phenomena. *Materials* **2017**, *10*, 379.
https://doi.org/10.3390/ma10040379

**AMA Style**

Watson E, Steinhauser MO.
Discrete Particle Method for Simulating Hypervelocity Impact Phenomena. *Materials*. 2017; 10(4):379.
https://doi.org/10.3390/ma10040379

**Chicago/Turabian Style**

Watson, Erkai, and Martin O. Steinhauser.
2017. "Discrete Particle Method for Simulating Hypervelocity Impact Phenomena" *Materials* 10, no. 4: 379.
https://doi.org/10.3390/ma10040379