Internal friction and absence of dilatancy of packings of frictionless polygons

By means of numerical simulations, we show that assemblies of frictionless rigid pentagons in slow shear flow possess an internal friction coefficient (equal to $0.183\pm 0.008$ with our choice of moderately polydisperse grains) but no macroscopic dilatancy. In other words, despite side-side contacts tending to hinder relative particle rotations, the solid fraction under quasistatic shear coincides with that of isotropic random close packings of pentagonal particles. Properties of polygonal grains are thus similar to those of disks in that respect. We argue that continuous reshuffling of the force-bearing network leads to frequent collapsing events at the microscale, thereby causing the macroscopic dilatancy to vanish. Despite such rearrangements, the shear flow favors an anisotropic structure that is at the origin of the ability of the system to sustain shear stress.

Figure 1

A classical representation of dilatancy mechanism in quasistatic simple shear test: Some expansion in direction $y$ is necessary for adjacent layers to flow past one another.

Figure 2

(a) A polygon rolling on another polygon in side-to-side contact, whence (b) an effective dilatancy angle ${\psi }_{\text{loc}}$.

Figure 3

A simple (vertex-side) contact and a double (side-side) one between polygonal grains, with corresponding normal forces.

Figure 4

Steady shear flow results. (a) $sin\phi =q/p$ versus $I$, with(red) squares for pentagons and (black) dots for disks. (b) $\nu$ versus $I$ (same symbols and colors). The horizontal straight line corresponds to isotropic equilibrium (RCP) values, hashed regions indicating the error bar.

Figure 5

Each polygon is colored according to the sign of the local area change: orange (or darker gray) for a local dilation; light blue (or gray) for a local contraction. The inset shows a smaller scale detail.

Figure 6

Polar diagrams of $P\left(\theta \right)$ (black dots) and $\langle f_n\rangle \left(\theta \right)$ (red squares).

Figure 7

An improved schematic representation of dilatancy. As grains of the upper row, on average, flow past the bottom ones, some displacements (left; color code as in Fig. 5) increase the volume; others (right) decrease it. Friction favors the effects of the former.