Introduction: After our initial simulations outlined in the first progress report, we determined that a shear cell with a bidisperse composition of particles would be a promising system to explore. However, before doing so, we ran a few more simulations to determine appropriate parameters for the bidisperse simulations. Some of these considerations included appropriate elastic constant values, angular velocity of plate rotation, angular amplitude of plate oscillation, and the frictional interaction between the particles and the cylindrical cell wall. Elastic constant: For our initial simulations, an elastic constant of k = 2*10^6 mg/d was most appropriate for our simulations. A k value one degree of magnitude lower resulted in particles leaving the system for the oscillatory motion shear cell. Before proceeding, we tested a k value one order of magnitude higher to see if there was any appreciable difference in the particle dynamics that would affect our results. Ideally, physically realistic k values are reduced in simulations to speed up computation time, but not reduced so much as to dramatically change the behavior of the system. Torque (z-direction) vs. time (continuous rotation): The plot above shows the torque exerted by the particles on the plate over time for the following four k values: 4*10^4 (low k), 2*10^5 (mid k), 2*10^6 (high k), and 2*10^7 mg/d (high k 2). The torque for highest k value of 2*10^7 mg/d is not surprising based on what we had seen for the other three k values tested. There are differences in the magnitude and shape of the transient response seen at the beginning, the torque value to which the system eventually equilibrates, and the variance of the torque. However, these differences do not concern us, as we are more interested in the general behavior of the way the torque evolves over time. The increased variance of the torque is likely due to the rearrangement of stiffer particles causing the plate to slip. Even if we were concerned about the specific value of the torque, rather than the general 1
behavior, the torque appears to approach a value asymptotically as the k value is increased. For instance, if the mean value of the torque is calculated after a certain timestep (200 τ), the low k, mid k, and high k, have a mean torque values that are 84%, 91%, and 98% of the next highest mean torque values. For example the low k mean torque is 84% of the mid k mean torque, and so on. From this general trend, we might assume that the torque values for physically realistic values are not much higher than those for the high k and high k 2 values. Our collaborators at Bethcare gave us physical parameters for possible granular materials to be used in the prosthetic socket membranes. The materials are zirconia, acrylic, and melamine. However, our calculations indicate that elastic constants for these materials are in the range of 10^10-10^11 mg/d. This means that any simulation we perform will not likely resolve any meaningful differences between using particles of these different materials. This is another reason for pursuing a bidisperse composition, which seems like a more promising area for interesting physics. Angular velocity: The angular velocity at which the plate was rotated was also considered. The default value we had previously used for testing was approximately 0.126 rad/s, or a period of 500 s for one rotation. Given that one of our goals is to identify how granular materials might behave in varying conditions in a prosthetic socket membrane, we considered how the speed of the shearing might affect the particles. Torque (z-direction) vs. time (continuous rotation): The plot above shows the torque exerted by the particles on the plate over time for three different angular velocities: the default 50 s rotational period, as well as half the speed and twice the 2
speed. The plot above seems to indicate that the torque is a function of the angular displacement rather than the velocity. Given this, we plotted the torque versus the angular displacement below. Torque (z-direction) vs. angular displacement (continuous rotation): There seems to be no difference in torque based on the angular velocity. Previous, similar studies have shown the same results. Speeding up the angular velocity of the shearing scales temporally; e.g. speeding up the rotation by a factor of two is the same as speeding up the simulation by a factor of two. Angular amplitude for oscillations: For simulations involving a plate shearing the cell in an oscillatory motion, we had previously considered motion subtending 10 degrees (angular amplitude of 5 degrees). Here we consider how the torque response changes for different amplitude oscillations. The motion of the plate is driven by a sine wave. Torque (z-direction) vs. time (oscillatory motion): 3
The plot above shows the torque response on the plate for oscillations of varying angular amplitude. In the legend, values are given as fractional values of the default angular amplitude (5 degrees): 1/4, 1/2, 3/4, 1, 5/4, 3/4, 7/4, and 2. We can see that for larger values of angular amplitudes the peak torque that is reached in each oscillation is reduced. The graphs below also show this with the torque plotted versus the angular displacement as well as the angular displacement normalized to the angular amplitude. Torque (z-direction) vs. angular amplitude (oscillatory motion): Torque (z-direction) vs. angular amplitude (oscillatory motion): 4
This could be a result of how we are moving the plate. We prescribe a set angular displacement, but move the plate in the z-direction in response to the particle forces. The following plot shows the z-displacement of the plate as it oscillates. z-displacement vs. time (oscillatory motion): The plate has its highest z-displacement when it is moving the fastest at the origin between the two angular extrema, and its z-displacement is lowest at the extrema. The plate is at a higher overall z-displacement for faster oscillations. Wall friction: To more accurately characterize the effect of shearing on granular material, we modified the frictional interaction of the particles with the cylindrical wall of the shear cell. Previously, the frictional interaction was the same as for two particles interacting. We compared this to making the walls frictionless--setting the tangential elastic and damping constants to zero. When we did this, the effect was quite noticeable. As the plate rotates, the particle layers are sheared and move in the direction of the shear. However, for our previous simulations, only the top several particle layers moved. The frictional interaction with the wall made it difficult to observe the movement of lower particle layers. When the friction with the walls was turned off, we observed a very clear velocity gradient where the particles moved at a rate that steadily decreased for lower layers. Because the wall interaction played such a strong role in altering the particle dynamics, we will be turning the wall friction off in all subsequent simulations to more accurately characterize shearing of bulk granular media. In addition to this, we also saw a slight decrease in the torque exerted on the plate by the particles when the wall friction was turned off. The plot below shows this. 5
Torque (z-direction) vs. time (continuous rotation): Bidisperse composition For our bidisperse simulations we will be using an elastic constant of k = 2*10^6 mg/d. For oscillations, the period is T = 50 τ = 0.05 s, and for continuous rotations the period of one rotation is T = 500 τ = 5 s. The oscillations have an angular amplitude of 5 degrees. The following bidisperse compositions will be considered, in addition to a monodisperse composition for comparison. diameter 1 (mm) diameter 2 (mm) diameter 1 / diameter 2 1 1 1 0.95 1.05 0.904761904762 0.9 1.1 0.818181818182 0.85 1.15 0.739130434783 0.8 1.2 0.666666666667 0.75 1.25 0.6 0.7 1.3 0.538461538462 0.65 1.35 0.481481481481 6
diameter 1 (mm) diameter 2 (mm) diameter 1 / diameter 2 0.6 1.4 0.428571428571 0.55 1.45 0.379310344828 0.5 1.5 0.333333333333 0.45 1.55 0.290322580645 0.4 1.6 0.25 The third column simply lists the ratio of the diameter of the smaller particles to the diameter of the larger particles. For our initial tests, we will be considering a 50-50 percent composition by number of particles, meaning the average particle diameter will still be 1 mm, but the mass of the system will be different. The system under consideration will still occupy the size geometry, so the number of particles in the system will decrease for the compositions with a lower d1/d2 ratio. For our range of bidisperse compositions, there could be anywhere from 4 to 6 layers of particles between the plates. In our previous tests, we found that the depth of the shear cell did not alter the torque response. Because of the different particle sizes, the mass of the plate as calculated previously (simply as the mass of all particles within the region z = 7-9) will be different for each composition. For these simulations, we have simply set the mass of the plate to a fixed value: the density of the particles multiplied by the volume of the cylindrical top plate region. This more accurately matches our experimental setup. We also remove any particles higher than z = 9 before the shearing motion begins. Also, for bidisperse compositions, there may be more gaps in the particle packing. Because our simulation takes place in microgravity, it is possible that particles will easily move up through these cracks and leave the system. To prevent this, we have inserted a horizontal wall at z = 9 to prevent particles from floating away. The particle interaction with this wall is the same as with the cylindrical walls. The particles are also poured into the system in an unbiased fashion, which means LAMMPS on average has to try more insertions for larger particles. As a result, the time for pouring takes longer for compositions with a lower d1/d2 ratio. Also, the insertion region for pouring must be slightly adjusted for larger particles so they are not inserted into the edge of the cylinder. For our simulations, the pouring and settling of particles is considered complete when the rotational kinetic energy of all of the particles is of the order of magnitude 10^-3. Since we will be considering a wide range of bidisperse compositions, we will use the Harvard computing cluster to run multiple jobs at once. Each run is submitted via a batch script using the sbatch system on the Odyssey cluster. Jobs are submitted to the general queue, which as a maximum runtime of 3 days. As a starting point, we will run these simulations for a time of 100,000,000 timesteps = 2500 τ = 25 seconds. Typically each job takes ~500 Mb of memory and 7
we are using 32 cores for each job. Dump files containing particle data are outputted only every 100,000 timesteps to avoid large data files. The dump file produced for each run is between 300 and 400 Mb. These jobs can take anywhere from 1 to 5 days each depending on cluster performance. It may help to use a --contiguous flag after the sbatch command to request contiguous memory for faster MPI operations. The image above shows bidisperse composition of particles. d1 = 0.8 mm and d2 = 1.2 mm. After running the simulations we will plot values of interest such as torque as before, but because these simulations are running for longer we would also like to observe any rearrangement or crystallization of the particles. To do this, we take the particle data from a dump file and perform Voronoi tesselation, which assigns a cell region around each particle. Each cell contains space which is closer to that particle than any other. These cells help us determine which particles are neighbors to each other (they share a cell wall). From this, we will use a script to analyze what neighbor particle arrangements are most common and how this structure evolves over time. Depending on our results, it may be necessary to run the simulation over a longer timescale or output dump files with more frequency near the beginning of the simulation. Additionally, instead of a 50-50 composition, we could have a bidisperse composition but keep the total mass the same as the monodisperse case. This may create some more interesting packing as there would be more smaller particles to fill the gaps between the larger particles. 8