Frictionless bead packs have macroscopic friction, but no dilatancy

Abstract

The statement of the title is shown by numerical simulation of homogeneously sheared assemblies of frictionless, nearly rigid beads in the quasistatic limit. Results coincide for steady flows at constant shear rate $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ in the limit of small $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}$ and static approaches, in which packings are equilibrated under growing deviator stresses. The internal friction angle $\ensuremath{\varphi}$, equal to $5.76\ifmmode^\circ\else\textdegree\fi{}\ifmmode\pm\else\textpm\fi{}0.22\ifmmode^\circ\else\textdegree\fi{}$ in simple shear, is independent of average pressure $P$ in the rigid limit and stems from the ability of stable frictionless contact networks to form stress-induced anisotropic fabrics. No enduring strain localization is observed. Dissipation at the macroscopic level results from repeated network rearrangements, similar to the effective friction of a frictionless slider on a bumpy surface. Solid fraction $\ensuremath{\Phi}$ remains equal to the random close packing value $\ensuremath{\simeq}0.64$ in slowly or statically sheared systems. Fluctuations of stresses and volume are observed to regress in the large system limit. Defining the inertial number as $I=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\gamma}}\sqrt{m/(aP)}$, with $m$ the grain mass and $a$ its diameter, both internal friction coefficient ${\ensuremath{\mu}}^{\ensuremath{\ast}}=\text{tan}\text{ }\ensuremath{\varphi}$ and volume $1/\ensuremath{\Phi}$ increase as powers of $I$ in the quasistatic limit of vanishing $I$, in which all mechanical properties are determined by contact network geometry. The microstructure of the sheared material is characterized with a suitable parametrization of the fabric tensor and measurements of coordination numbers.