Frictionless bead packs have macroscopic friction, but no dilatancy

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 $\dot{\gamma }$ in the limit of small $\dot{\gamma }$ and static approaches, in which packings are equilibrated under growing deviator stresses. The internal friction angle $\phi$, equal to $5.76°\pm 0.22°$ 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 $\Phi$ remains equal to the random close packing value $\simeq 0.64$ 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 $I=\dot{\gamma }\sqrt{m/\left(aP\right)}$, with $m$ the grain mass and $a$ its diameter, both internal friction coefficient ${\mu }^{\ast }=\text{tan}\phi$ and volume $1/\Phi$ 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.

Figure 1

(Color online) $\left|{\sigma }_{12}\right|$ (left axis, in black) and ${\sigma }_{22}$ (right axis, in gray, red online) as functions of strain $\gamma$. Note that the left and right scales are different. Time series obtained with $I=3.2\times {10}^{-5}$, $\kappa ={\kappa }_1$, $\zeta =0.98$, and $N=4000$.

Figure 2

(Color online) Kinetic energy associated with velocity fluctuations, as defined in Eq. (3), normalized by $m{a}^2{\dot{\gamma }}^2$, versus $I$, in simulations with 4000 beads, for $\kappa ∊\left\{{\kappa }_2,{\kappa }_1\right\}$ and $\zeta =0.98$.

Figure 3

(Color online) Two velocity profiles at randomly chosen times, for $I=3.2\times {10}^{-2}$ (top), $I=3.2\times {10}^{-3}$ (middle), $I=3.2\times {10}^{-4}$ (bottom). $\kappa ={\kappa }_1$, $\zeta =0.98$ and $N=4000$.

Figure 4

(Color online) Velocity profile after shear strain intervals $\gamma$ equal to 0.004 (red dotted curve), 0.02 (blue dashed curve), and 0.1 (black solid curve), following the instant corresponding to the localized profile marked $\mathit{L}$ in Fig. 3, bottom graph.

Figure 5

(Color online) Macroscopic friction ${\mu }^{\ast }$ vs inertial number $I$ for stiffness parameter $\kappa ∊\left\{{\kappa }_2,{\kappa }_1\right\}$, damping parameter $\zeta =0.98$, and number of beads $N=4000$. The solid line is Eq. (6) with the parameters of Table (no visible difference on using best parameters with ${\kappa }_1$ or ${\kappa }_2$).

Figure 6

(Color online) $\Delta \mu /{\mu }^{\ast }$ as a function of $N$ for $I=3.2\times {10}^{-5}$, $\kappa ={\kappa }_1$ and $\zeta =0.98$. The solid line equation is relation (5).

Figure 7

(Color online) Volume fraction $\Phi$ as a function of inertial number $I$ (for $\zeta =0.98$, $N=4000$), for both stiffness levels $\kappa ={\kappa }_1$ (blue squares) and $\kappa ={\kappa }_2$ (red triangles) The solid lines are given by Eq. (11) with the parameters of Table .

Figure 8

(Color online) $\Delta \Phi /\Phi$ as a function of $N$ for the same time series as in Fig. 6, fitted with Eq. (10) (solid line).

Figure 9

(Color online) Size dependence of $⟨{\mu }^{\text{stat}}⟩$. $N$ denotes the number of particles in the system. The solid line is the fit of Eqs. (14, 15). Crosses are the top percentile values extracted from the time series of $\left|{\sigma }_{12}\right|/{\sigma }_{22}$ obtained in procedure $\mathit{D}$, as listed in Table . The hashed region represents the estimate, from $\mathit{D}$ simulations, of ${\mu }_0^{\ast }$ with its error bar (Table ). The (blue) dot with an error bar on the left axis is the static estimate ${\mu }_{\infty }^{\text{stat}}$.

Figure 10

(Color online) Variation of the volume fraction $\Phi$ with the static shear $\left|\tau \right|/P$ imposed to five different samples of 4000 beads with $\kappa ={\kappa }_2$. Each curve stops at a given value of $\left|\tau \right|/P=\left|{\tau }_{\text{max}}\right|/P$: this is the greatest value for which the packing has managed to reach mechanical equilibrium.

Figure 11

The model of the slider on a rough surface. (a) Case of a sinusoidal profile. Forces at point $\mathit{S}$ are drawn as vectors. (b) Profile for which ${\mu }_D={\mu }_S$.

Figure 12

(Color online) Coordination number $z$ as a function of inertial number $I$, for $\kappa ={\kappa }_1$ (red square dots joined by dotted line, bottom data points) and $\kappa ={\kappa }_2$ (blue crosses, top, dashed line).

Figure 13

(Color online) Probability distribution function of $f=N/⟨N⟩$ for $I={3.210}^{-1}$ (red triangles), $I={3.210}^{-3}$ (blue round dots), and $I={3.210}^{-5}$ (black square dots).

Figure 14

(Color online) $F_{12}$ as a function of inertial number $I$, with $\zeta =0.98$ and $N=4000$, for $\kappa ={\kappa }_1$ (red square dots connected by a dotted line) and for $\kappa ={\kappa }_2$ (blue crosses connected by a dashed line). Both lines are power law fits of $F_{12}$.

Figure 15

(Color online) Crystalline order induced by shear in one $\mathit{S}$ sample with $N=256$.