Numerical Simulations of Impact Cratering in Porous Materials

Asteroid Mathilde's large craters apparently formed in close proximity without noticeable damage to their neighbors either by seismic jolts or by ejecta deposition [1]. The lack of ejecta blankets is especially puzzling in light of scaling relationships and laboratory experiments that predict kilometer-deep deposits of ejecta around MathildeOs large craters [2]. The unusual appearance of these craters has been suggested to result from MathildeOs high porosity of ~50% [1, 3, 4]. Recent impact experiments in a porous, crushable soil provide one explanation for MathildeOs lack of ejecta [5]. Craters in this porous material form primarily by compaction of the soil due to the outgoing pressure shock, whereas craters in typical soils such as sand form by shearing and excavation. Thus the ejection velocities in the porous material were so low that nearly all of the ejecta from a large crater on Mathilde would fall back into the crater bowl. The ejecta does not refill the crater because it is a small part of the volume created by compaction. The end result is a bowl-shaped crater with no appreciable ejecta blanket or raised rim [5]. This unusual mechanism introduces a new regime of crater formation, distinct from the well-known strength and gravity regimes. Whereas gravity-dominated cratering has been proposed for impacts on even small asteroids [6], it now seems likely that craters on porous asteroids never form in the gravity regime: these craters are likely dominated by compaction. To augment our experimental studies of compaction cratering, we use the finite-difference hydrocode, CTH, to study the mechanics of crater formation in porous materials. Dry sand was selected as the first material to model, because it is a relatively simple porous material, and because there is an abundance of detailed data about the kinematics of the crater formation available from the soil mechanics literature and from laboratory cratering experiments. We find it is crucial to compare the code results to this detailed data in order to construct an accurate physical model for the code calculations. The numerical simulations use a dissipative P-alpha crush-up model [8] with an underlying Mie-Gruneisen equation of state for the fully crushed material. The parameters of the model were determined from fits to Hugoniot dynamic data [7] and quasistatic compression data for sand [9]. The modeled sand is cohesionless with shear strength given by a Mohr-Coulomb model with a constant friction angle. All of the simulations are 2D axially symmetric, with ~20 mesh cells across the diameter of the projectile or explosive source. Test calculations with finer meshes produced essentially identical results. Figure 1 compares the crater profiles calculated with CTH to that measured for shot N10, a tangent-below explosion in dry Ottawa sand [0.4 gm PETN, gravity=1G, ref. 10]. The profiles are compared at a time of 2.5 ms, at which point the experimental crater is nearly at its final depth, and the radius is at 46% of its final value. A series of calculations show that the angle of internal friction (φ) is the material property that most strongly controls crater size, shape and ejection angles. Satisfactory agreement between the calculated and measured crater profiles is obtained only with φ ~ 35 deg. This value agrees well the range of 30-35 deg. typically measured in quasistatic tests [9]. These values should also be applicable to the dynamic case of impact because dynamic compression and shear tests have shown that sand does not have a significant strain-rate dependence [11,12]. Figure 2 compares the calculated ejection angles and velocities to those measured for shot N10. As with crater size, the ejection angles match the experimental values only for a friction angle of ~35 deg. However, the magnitude of the ejecta velocities is about a factor of 1.5 to 2 greater than the measured values. The cause of this discrepancy is currently being investigated. Figure 3 shows the calculated profile (φ=35 deg.) at time=70 ms. The CTH crater has reached its final profile, but the ejecta curtain is still in flight. Comparison to the experimental crater profile shows that, while there are some differences, the overall agreement is quite good. Figure 4 shows the result of a CTH simulation of shot 35-X, a 1.9 km/s impact of a polyethylene cylinder into dry sand on a centrifuge at an acceleration of 500G [13]. The calculation, which used a friction angle of 35 deg., produced a deeper final crater than observed in the experiment. Interestingly, calculations with lower values of φ produce better agreement with the experiment, because late-time slumping from the crater walls reduces the depth of the crater. While significant progress has been made in modeling crater formation in porous materials, there are notable discrepancies between the ejection velocities and crater profiles for the simulations and experiments. We have found that the details of the numerical models are very important in reproducing the correct physical results. Additional calculations are being performed to understand the sources of discrepancies and to provide comparisons with additional experiments.

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