Friction force regimes and the conditions for endless penetration of an intruder into a granular medium

An intruder penetrating into a granular column experiences a depth-dependent friction force F(z). Different regimes of F(z) have been measured depending on the experimental design: a nearly linear dependence for shallow penetrations, total saturation at large depths, and an exponential increase when the intruder approaches the bottom of the granular bed. We report here an experiment that allows us to measure the different regimes in a single run during the quasistatic descent of a sphere in a light granular medium. From the analysis of the resistance in the saturation zone, it was found that F(z) follows a cube-power-law dependence on the intruder diameter and an exponential increase with the packing fraction of the bed. Moreover, we determine the critical mass mc required to observe infinite penetration and its dependence on the above parameters. Finally, we use our results to estimate the final penetration depth reached by intruders of masses m < m(c). The results indicate that an intruder of any density (larger than the density of the granular bed) can sink indefinitely into the granular medium if the bed packing fraction is smaller than a critical value.