The distribution of forces affects vibrational properties in hard sphere glasses

We study theoretically and numerically the elastic properties of hard sphere glasses, and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero-temperature, we argue that the presence of certain pairs of particles interacting with a small force $f$ soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting $P(f)\sim f^{\theta_e}$ the force distribution of such pairs and $\phi_c$ the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale $\omega^*$, rising up to it as $D(\omega) \sim \omega^{2+a}$, and decaying above $\omega^*$ as $D(\omega)\sim \omega^{-a}$ where $a=(1-\theta_e)/(3+\theta_e)$ and $\omega$ is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with $\langle \delta R^2\rangle\sim1/\mu\sim (\phi_c-\phi)^{\kappa}$ where $\kappa=2-2/(3+\theta_e)$, and (iii) continuum elasticity breaks down on a scale $\ell_c \sim1/\sqrt{\delta z}\sim (\phi_c-\phi)^{-b}$ where $b=(1+\theta_e)/(6+2\theta_e)$ and $\delta z=z-2d$, where $z$ is the coordination and $d$ the spatial dimension. We numerically test (i) and provide data supporting that $\theta_e\approx 0.41$ in our bi-disperse system, independently of system preparation in two and three dimensions, leading to $\kappa\approx1.41$, $a \approx 0.17$, and $b\approx 0.21$. Our results for the mean-square displacement are consistent with a recent exact replica computation for $d=\infty$, whereas some observations differ, as rationalized by the present approach.Comment: 5 pages + 4 pages supplementary informatio

On the dependence of the avalanche angle on the granular layer thickness

A layer of sand of thickness h flows down a rough surface if the inclination is larger than some threshold value theta which decreases with h. A tentative microscopic model for the dependence of theta with h is proposed for rigid frictional grains, based on the following hypothesis: (i) a horizontal layer of sand has some coordination z larger than a critical value z_c where mechanical stability is lost (ii) as the tilt angle is increased, the configurations visited present a growing proportion $_s of sliding contacts. Instability with respect to flow occurs when z-z_s=z_c. This criterion leads to a prediction for theta(h) in good agreement with empirical observations.Comment: 6 pages, 2 figure