Support of modified Archimedes' law theory in granular media

Abstract

We study the resistance force of cylindrical objects penetrating quasi-statically into granular media experimentally and numerically. Simulations are validated against experiments. In contrast to previous studies, we find in both experiments and simulations that the force–depth relation consists of three regimes, rather than just two: transient and steady-state. The three regimes are driven by different dynamics: an initial matter compression, a developing stagnant zone, and an increase in steady-state force with a fully developed stagnant zone. By simulations, we explored the effects of a wide range of parameters on the penetration dynamics. We find that the initial packing fraction, the inter-granular sliding friction coefficient, and the grain shape (aspect ratio) have a significant effect on the gradient K_ϕ of the force–depth relation in the steady-state regime, while the rolling friction coefficient noticeably affects only the initial compression regime. Conversely, K_ϕ is not sensitive to the following grain properties: size, size distribution, shear modulus, density, and coefficient of restitution. From the stress fields observed in the simulations, we determine the internal friction angles ϕ, using the Mohr–Coulomb yield criterion, and use these results to test the recently-proposed modified Archimedes' law theory. We find excellent agreement, with the results of all the simulations falling very close to the predicted curve of ϕ vs. K_ϕ. We also examine the extreme case of frictionless spheres and find that, although no stagnant zone develops during penetration into such media, the value of their internal friction angle, ϕ = 9° ± 1°, also falls squarely on the theoretical curve. Finally, we use the modified Archimedes' law theory and an expression for the time-dependent growth of the stagnant zone to propose an explicit constitutive relation that fits excellently the force–depth curve throughout the entire penetration process.