On the shape of impact craters and the collapse of an unstable transient cavity in granular media

We study low energy impact craters formed in a granular target, and the processes which determine their shape. Projectiles are dropped vertically into a granular target. Upon impact they excavate an unstable transient crater which then collapses, producing the final crater. A laser profilometer is used to digitize the crater surfaces in three dimensions. The craters are found to be hyperbolic in profile. Using the digitized profiles, we measure a number of characteristic crater dimensions. The dependence of each crater dimension on the radius, density and kinetic energy of the projectile is determined. We find that less than 1% of the projectile's kinetic energy is required to excavate the resultant crater, and that the excavation efficiency is dependent on the projectile density alone. In the second portion of this work, we investigate how the collapse of an unstable well in a granular medium determines the shape of the metastable final profile. The collapse of a rectangular well in two dimensions is studied experimentally in a Hele-Shaw cell, and numerically with a model based on the Saint-Venant equations. If the well is sufficiently narrow, then the collapsing walls collide at the midpoint and the resultant profile shape is hyperbolic. The dependence of both the late time profile shape and the collapse dynamics on the geometry of the initial well is investigated. The collapse of a cylindrical well in three dimensions is studied both experimentally and using the Saint-Venant model. Collapse craters resemble impact craters in shape. By comparing collapse craters to impact craters, we infer that the impact of a low density projectile will excavate a wide, shallow transient crater, while a high density projectile excavates a narrow, deep transient crater. This result is consistent with our calculations of the excavation efficiency.