## 4 Simulation Results and Discussion  
A summary of the flight characteristics predicted using the velocity simulations is provided in Table 4. Figure 5 shows the predicted velocity-time histories for the first 10 us from the velocity simulation approach and the ball bearing simulation approach. The results are separated by charge diameter for ease of interpretation. The different times after detonation at which the velocity sharply increased are due to the height of the explosive charge. Increased charge heights mean the detonation wave must travel further to reach the ball bearing, delaying the transfer of momentum to the bearing as expected.  
Figure 5a shows that, for the 18 mm charge diameter, there was a negligible (less than 1%) difference in the velocity histories in each case, except in the 3 mm ball bearing simulation where the peak velocity was approximately 3% higher in the velocity approach. For the 30 mm diameter charges, shown in Figure 5b, the velocity simulation approach provided similar predictions to the ball bearing approach but with slightly (1-3%) lower peak velocities.  
The results show that including damage modelling in the ball bearing has a negligible effect on its flight characteristics. This means the velocity simulations can be used with reasonable confidence based on the good agreement here and the satisfactory correlation with experiments demonstrated in reference [17]. Throughout the following sections of the paper, the velocity simulation results are used when describing the bearing flight characteristics, while the bearing simulation approach is used to elucidate the failure and damage within the ball bearings.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Charge Mass** & **Charge** & **Aspect** & **Ball Bearing** & **Ball Bearing** & **Impact** & **Mean Plateau** & **Peak** \\
**(g)** & **Diameter (mm)** & **Ratio** & **Diameter (mm)** & **Mass (g)** & **Velocity** & **Velocity** & **Velocity** \\ \hline
5.13 & 18 & 0.7 & 3 & 0.11 & 711 & 720 & 795 \\
5.13 & 18 & 0.7 & 5 & 0.51 & 498 & 505 & 540 \\
11.0 & 18 & 1.5 & 5 & 0.51 & 616 & 622 & 664 \\
11.0 & 18 & 1.5 & 8 & 2.09 & 426 & 428 & 452 \\
20.37 & 30 & 0.6 & 5 & 0.51 & 572 & 573 & 701 \\
27.16 & 30 & 0.8 & 5 & 0.51 & 623 & 625 & 781 \\
20.37 & 30 & 0.6 & 8 & 2.09 & 448 & 449 & 507 \\
27.16 & 30 & 0.8 & 8 & 2.09 & 497 & 498 & 570 \\
20.37 & 30 & 0.6 & 10 & 4.08 & 384 & 385 & 426 \\
27.16 & 30 & 0.8 & 10 & 4.08 & 436 & 437 & 482 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Summary of flight characteristics obtained from the velocity simulations using different diameter ball bearings and charge masses.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Johnson–Cook Strength Model**} & \multicolumn{3}{c}{**Principal Stress Damage Model**} \\ \hline
**Shear Modulus** & **Yield Stress** & **Hardening Constant** & **Hardening Constant** & **Strain Rate** & **Tensile Failure** & **Shear Failure** \\
**(GPa)** & **(MPa)** & **(MPa)** & **Exponent** & **Constant** & **Stress** & **Stress** \\
80 & 2173 & 1785 & 0.6 & 0.023 [29] & 21 & 390 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Properties of the ball bearing material used in simulations.  
Figure 5: Ball bearing velocity–time histories for the first 10 \(\upmu\)s, comparing the velocity simulation approach and the ball bearing simulation approach. (**a**) 18 mm diameter charge detonations, (**b**) 30 mm diameter charge detonations.