Geometry-dependent constitutive law for granular slow frictional drag

Frictional constitutive law for very slow vertical withdrawing of a thin rod from a granular bed is experimentally studied. Using a very precise creep meter, geometry-dependent granular frictional constitutive law is particularly examined. In some previous works, a dimensionless number $I=\dot{\gamma}D_g/\sqrt{p/\rho_g}$ has been used to characterize granular frictional constitutive laws, where $\dot{\gamma}$, $D_g$, $p$, and $\rho_g$ are the shear strain rate, grain diameter, confining pressure, and bulk density of granular bed, respectively. It has been considered that granular frictional constitutive law expressed by $I$ is universal (almost geometry-independent) in dense flow regime. In this study, however, we find that the geometry of the system is much more crucial to characterize granular friction in a very slow withdrawing regime. Specifically, the ratio between rod and grain diameters must be an essential parameter to describe the granular frictional constitutive law. Physical meaning of the geometry-dependent constitutive law is discussed on the basis of grains-contact-number dependence of granular behavior.