NUMERICAL SIMULATIONS OF GRANULAR DYNAMICS IN VARIOUS CONDITIONS APPLICABLE TO REGOLITH MOTION ON SMALL BODY SURFACES. N. Murdoch1,2, P. Michel1, D .C. Richardson3, K. J. Walsh4, W. Losert3, C. Berardi3, and S. F. Green2. 1University of Nice-Sophia Antipolis, Observatoire de la Côte d’Azur (Laboratoire Cassiopée, BP 4229, 06304 Nice Cedex 4, France) 2The Open University, PSSRI (Walton Hall, Milton Keynes, MK7 6AA, UK) 3Institute for Physical Science and Technology, and Department of Physics (Uni-versity of Maryland, College Park, Maryland 20742, USA) 4Department of Space Studies, Southwest Research Insti-tute (1050 Walnut St. Suite 300, Boulder, CO 80302, USA)

Introduction: Surfaces of planets and small bodies in our Solar System are often covered by a layer of gran-ular material that can range from a fine regolith to a gravel-like structure of varying depths. This presence of a regolith layer plays a particularly important role in the surface geology of asteroids. The same can be stated, although to a lesser extent, for bodies like Mars and the Moon, whose surface gravities are also smaller than that of Earth.

The presence or relative absence of gravitational acceleration on granular flow is of importance for un-derstanding the geology of small bodies and planets, and to clarify the environments that may be en-countered during planetary exploration. Bodies with low surface gravity can be very sensitive to processes that appear irrelevant in the case of larger planetary bodies. For instance, seismic vibration is expected to occur when small projectiles impact the surface of small bodies. Due to the low gravity this seismic vibra-tion may potentially lead to various kinds of surface motion, such as down-slope migration and degrada-tion, or erasure of small craters. Regolith motion res-ulting from seismic shaking of asteroids has been ad-dressed in several papers, but without simulating expli-citly the dynamics of the granular material [1] [2].

Improving our understanding of the dynamics of granular materials under a wide variety of conditions requires that both experimental and numerical works be performed and compared. Once numerical ap-proaches have been validated by successful comparis-on with experiments, then they can cover a parameter space that is too wide for or unreachable by laboratory experiments. We have therefore performed a series of numerical simulations that currently focus on scenarios for which we have experimental data thus allowing dir-ect comparisons to be made with laboratory experi-ments.

Numerical Code: The code used for our numerical simulations is a modified version of the N-body code pkdgrav [3], that has been adapted to handle hard-body collisions [4][5]. The granular dynamics modifications consist primarily of providing wall “primitives” to sim-ulate the boundaries of the experimental apparatus [6]. Currently four wall primitives are supported that can be combined in arbitrary ways: infinite plane, finite disk, infinite cylinder, and finite cylinder (the finite primitives consist of a surface combined with one or

two thin rings). Certain primitives can have limited translational or rotational motion.

Advantages of pkdgrav over other many discrete element approaches include full support for parallel computation, the use of hierarchical tree methods to rapidly compute long-range interparticle forces (namely gravity, when included) and to locate nearest neighbors (for short-range Hooke’s-law type forces) and potential colliders, and options for particle bond-ing to make irregular shapes that are subjet to Euler’s laws of rigid-body rotation with non-central impacts (cf. [5]). In addition, collisions are determined prior to advancing particle positions, insuring that no collisions are missed and that collision circumstances are com-puted exactly (in general, to within the accuracy of the integration), which is a particular advantage when particles are moving rapidly.

Shaken Granular Material: Recent experiments investigating the nature of particle motion in a vibrated layer of densely packed hard spheres have demon-strated that the hard spheres spontaneously form crys-tallized regions known as grains. These grains are sep-arated by relatively disordered grain boundary (GB) re-gions. Within these GB regions, cooperatively moving clusters with a string like appearance have been ob-served [7]. The addition of small particles to the en-semble further increases the scale of the collective mo-tion.

We simulate the experiment of Berardi et al. (2010) [7] that involved shaking a densely packed layer of bidisperse glass beads at very high frequency (125 Hz) and low amplitude (0.07 mm). See an example simula-tion set-up in Figure 1. As a stringent test of the nu-merical code we investigate the complex collective motion of granular material by quantitative comparison with laboratory experiments.

FIGURE 1: The simulation set-up. Large (3mm) particles are red, small (2mm) particles are yellow and walls are invisible. Particles are shaken in the vertical

direction whilst being confined with walls.

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As found experimentally, in the simulations we find crystalised grains and less packed GB regions (see Fig 2). Additionally, inside the GB regions there are mo-bile particles that exhibit collective motion (granular strings). We are also capable of modulating the scale of the collective motion by altering the small particle con-centration. Full simulation results will be reported in Murdoch et al. (2011) [8].

FIGURE 2: Simulation results showing the grains and grain boundaries. Colours show the measure of lo-

cal order; Dark blue implies near hexagonal particle packing (grains) and red implies more disordered

packing (i.e. grain boundary regions)

Avalanches: Hofmeister et al. (2009) [9] used the Bremen drop tower to test the flow of glass beads at re-duced gravity levels of 0.01-0.3 g0 (where g0 is 9.81 ms-2). A 4x4x15cm3 box of material was housed in a centrifuge, such that terrestrial gravity vanished at drop, allowing 4.7s of micro-gravity. Different mor-phologies are observed at the end of drop-tower flight; the avalanches are shorter with decreasing gravity. As the final angle of repose should be independent of gravity this implies there are some cohesive forces act-ing on the glass beads. Currently we are simulating the case of an avalanche without cohesion (see Fig 3) but by adding cohesive forces we hope to help explain the experimental results.

FIGURE 3: An example avalanche simulation where the residual gravity is 0.1 g0

Tumbler Behaviour: We have performed a direct comparison of our simulations on granular flow in a tumbler to the lab based experiments of Brucks et al. (2007) [10]. The Froude number (Fr) is ratio of centri-fuging force to gravitational force. As Fr is increased the tumbling behaviour changes (see Fig 4). Our simu-lations correctly model the tumbling behaviour, the centrifuging beginning at Fr > 1.0 and also the correct trend of dynamical angle of repose with Fr.

FIGURE 4: Results of tumbler simulations as re-ported in [6]

Conclusion: The new implementation of pkdgrav al-lows convenient modeling of granular dynamics in a variety of conditions. We are capable of modeling ava-lanching granular material under varying gravitational conditions and tumbler behaviour for varying Froude numbers. Additionally we have shown that even though we are using a hard sphere method that resolves only two-body interactions, we are capable of reprodu-cing complex long-term and large-scale collective particle dynamics in a shaken granular material.

Studies of granular material dynamics, both with experiments and numerical simulations, will continue over a wide range of conditions adapted to the regolith covering the uppermost layer of solid planetary bodies.

References: [1] Michel, P. et al., (2009) Icarus 200, 503-513. [2] Richardson, J.E. et al., (2005) Icarus 179, 325-349. [3] Stadel (2001) PhDT. [4] Richardson, D. C. et al. (2000) Icarus 143, 45–59. [5] Richardson, D. C. et al. (2009) Plan. & Space Sci. 57, 183–192. [6] Richardson, D. C. et al. (2010), Icarus (in press). [7] Berardi, C. R. et al., (2010) Phys. Rev. E 81, 041301. [8] Murdoch, N. et al. (2011) Icarus (in preparation). [9] Hofmeister et al. (2009) AIP Conference Proceed-ings Volume 1145, 71-74. [10] Brucks, A. et al. (2007) Phys. Rev. E 75, 032301.

Acknowledgments: We would like to thank the Open University, Thales Alenia Space and the french Nation-al Program of Planetology for financial support. DCR acknowledges support from NASA (Grant No. NNX08AM39G). WL was supported by NSF-DM-R0907146.

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