Start-up shear of spherocylinder packings: effect of friction

We study the response to shear deformations of packings of long spherocylindrical particles that interact via frictional forces with friction coefficient $\mu$. The packings are produced and deformed with the help of molecular dynamics simulations combined with minimization techniques performed on a GPU. We calculate the linear shear modulus $g_\infty$, which is orders of magnitude larger than the modulus $g_0$ in the corresponding frictionless system. The motion of the particles responsible for these large frictional forces is governed by and increases with the length $\ell$ of the spherocylinders. One consequence of this motion is that the shear modulus $g_\infty$ approaches a finite value in the limit $\ell\to\infty$, even though the density of the packings vanishes, $\rho\propto\ell^{-2}$. By way of contrast, the frictionless modulus decreases to zero, $g_0\sim\ell^{-2}$, in accordance with the behavior of density. Increasing the strain beyond a value $\gamma_c\sim \mu$, the packing undergoes a "shear-thinning" transition from the large frictional to the smaller frictionless modulus when contacts saturate at the Coulomb inequality and start to slide. In this regime, sliding friction contributes a "yield stress" $\sigma_y=g_\infty\gamma_c$ and the stress behaves as $\sigma=\sigma_y+g_0\gamma$. The interplay between static and sliding friction gives rise to hysteresis in oscillatory shear simulations.