- Discovery Science: 10th International Conference, DS 2007 Sendai, Japan, October 1-4, 2007. Proceedings.
- High-Velocity Impact Phenomena by Ray Kinslow.
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- Nano, Quantum and Molecular Computing: Implications to High Level Design and Validation!
- High-Velocity Impact Phenomena.
In the presented model, the Lennard-Jones potential is modified slightly to refine the description of the physics of particle interactions: A cutoff distance, set to the potential minimum, is defined to remove the attractive component. Beyond this distance the potential is defined to be zero. Neighboring particles are linked together to form a crystalline lattice structure. The bonded particle pairs can experience both cohesive and repulsive forces. A quadratic spring potential.
Figure 3 displays the various potential contributions to the total interaction potential for a particle pair. The modified Lennard—Jones potential is shown in blue, with the cut-off tail shown as a dotted line. Likewise the quadratic potential is shown in red. The distance at which the spring elements fail, r c u t is marked by the vertical dotted black line. 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.
With the simplified model that we present in this paper, we deliberately exclude dissipation caused by friction or damping. However, some energy is removed from the system when failure of the material occurs.
When the distance between two bonded particles exceeds a certain distance r c u t , the bond is considered to be broken and is removed. The two particles however, may continue to interact with each other or any other particle via the contact potential, and with other particles to which they may still be bonded. For a simulation to be robust it must reproduce the same results regardless of the arbitrarily chosen number of particles used.
This requirement can be separated into two parts: geometric sizing which relates to the value of the model parameters, and numerical convergence which relates to the number of particles used in the simulation. We treat these requirements in turn with the following two case studies. In this way the simulation results are independent of the model sizes or particle numbers, respectively.
We prove this analytically and numerically by considering two models, A and B , representing the same macroscopic solid of length L but with a different number of particles N A and N B , scaled by factor a , as shown in Figure 2 , such that:. Similarly, for model B we obtain:. Replacing the B model size variables in Equation 13 with Equations 7 to 11 ,. Using the fact that the average energy of each particle pair can be found by dividing the total potential by the number of particles, we sum over N A and N B particles in system A to get the same average energy such that.
Equations 12 and 14 can then be substituted into Equation 15 :. Finally, Equation 7 can be substituted into Equation 17 :. For the kinetic energy of the system to be independent of size, the mass of the particles must also be scaled. This is done by solving. The same geometry is modeled with different numbers of particles. Figure 4 shows the scaled and unscaled effects on the energy in the plates with the number of particles spanning almost three orders of magnitude. The unscaled simulations are widely divergent while the geometric sizing renders the three solutions very similar.
Total potential and kinetic energies of colliding plates with different numbers of particles. Figure 5 shows a convergence study measuring the fragmentation of the debris cloud after HVI. In this paper, a fragment is defined as an interconnected collection of simulation particles; the smallest fragment consists of a single particle. Fragmentation is defined as the ratio of fragments over the total number of simulation particles. Therefore, a fragmentation equal to one represents a simulation where every single bond is broken, and therefore the number of fragments is equal to the number of simulation particles.
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In Figure 5 , each simulation performed was first sized geometrically, therefore representing the same physical system, only with a different number of particles. Fragmentation in debris clouds simulated with a varying number of particles shows convergence. The study shows that the simulations begin to converge at about a quarter of a million particles. Simulations for calibrating the model parameters, described in Section 3.
In this section we discuss the results of our simulation study. After a detailed analysis of our choice of model parameters, we validate our simulations by comparison with experiments. This is followed by a comprehensive parameter study of HVI simulations which we use to analyze the shape of the resulting debris cloud and the degree of material fragmentation after impact. Due to the challenges in performing HVI experiments, multiple experiments with the exact same parameters were unavailable.
The experiment shown in Figure 6 was taken from a series of experiments studying the scalability of HVI, all of which had the same cloud expansion properties. 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. The snapshots are full 3D simulations displayed from a side view. Parameter r c u t is set to 1.
This parameter models the cohesive force of the material. Impactor particles are shown in red and target particles are shown in blue. The overlapping allows the high-velocity impactor particles to more easily penetrate past the stationary target particles without transferring as much momentum. Only a fraction of the impactor particles fully collide with the stationary target particles and transfer their momentum. Because only some particles in the impactor directly impact the target particles, while others penetrate the target without transferring much momentum, more of the connecting bonds are broken leading to smaller fragment clusters in the debris cloud.
On the extreme top right corner of the chosen parameter space of our model displayed in Figure 7 , the material is too hard and tough to represent the experimental findings displayed in Figure 6. On the lower left extreme, the material modeled is too weak and brittle. For finally determining the simulation parameters that best fit to the experiment, we use a visual, qualitative comparison and two quantitative properties. A close examination of the experimental situation shown in Figure 6 reveals the abundance of small fragments of different sizes in the cloud. Small impact craters on a witness plate, as shown in Figure 8 , confirm that the debris cloud contains many small and medium fragments of approximately 1 mm, and that no larger pieces of the impactor remain intact.
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. As a second parameter, the dimensionless length ratio R of the cloud provides a quantitative measure of the cloud shape.
A ratio equal to one represents a circular shape of the debris cloud and values larger than one represents an elliptical shape of the cloud. The target value, as measured in the experiment presented in Figure 6 , is 1. Color code refers to the length ratio R shown on the right side of the figure.
See main text for details. In Figure 9 , we can see a plateau at a length ratio of approximately 1. Our third quantitative comparison criteria is the cloud expansion velocity v a , which we normalize with the initial impact velocity v 0.
Experimentally, the cloud expansion velocity is determined by measuring the distance traveled by the cloud front between two high-speed photographs and dividing by the time between each snapshot. This can be done because after impact, the cloud continues on its trajectory at constant velocity.
The red contour in Figure 10 represents the experimentally measured axial velocity. The simulation points closest to the red line which have not already been ruled out by the other first two comparison criteria remain as the best choices for a realistic representation of the experiment. The experimentally measured value is shown as a red contour line. Blue represents target plate particles and red represents impactor particles. Comparing this figure directly to the experimental photograph in Figure 6 , shows the similarity between experiment and simulation. Figure 12 shows a 3D series of snapshots of the developing debris cloud directly after impact.
Red particles represent the impactor and gray particles represent the target. A video sequence of this simulation can be found as Supplementary Material in Video S1. This simulation corresponds to the experiment shown in Figure 6. Impactor particles are in red and target particles in blue. 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.
One challenge is the limited quantifiable data which can be obtained from HVI experiments. The extremely short time scale and limited instrumentation mean that often high-speed photographs are the only data available from the experiments. This restricts the quantitative comparison possible between our proposed numerical model and experiments. Nevertheless, a comparison is performed from the data that are available.
The diameter of the impacting sphere was 9. The dotted line represents linearly extrapolated experimental data. The simulation results are compared to experiments by Piekutowski [ 50 ]. In Figure 13 , the simulation over-predicts the expansion velocities, but still captures the overall decreasing trend. This increases the total mass in the debris cloud, but reduces its velocity.
Physically speaking, the passing of a shock wave is a highly transient process during which some of the kinetic energy is converted into heat as the material behind a shock wave experiences a sudden jump in thermodynamic variables such as pressure, energy, and density. This jump between two points of the Hugoniot curve takes place along the Rayleigh line and is a highly nonisentropic process.
The rarefaction waves that bring the material back to ambient condition occur on an isentropic path.
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