Effects of shape and size polydispersity on strength properties of granular materials

By means of extensive contact dynamics simulations, we analyze the combined effects of polydispersity both in particle size and in particle shape, defined as the degree of shape irregularity, on the shear strength and microstructure of sheared granular materials composed of pentagonal particles. We find that the shear strength is independent of the size span, but unexpectedly, it declines with increasing shape polydispersity. At the same time, the solid fraction is an increasing function of both the size span and the shape polydispersity. Hence, the densest and loosest packings have the same shear strength. At the scale of the particles and their contacts, we analyze the connectivity of particles, force transmission, and friction mobilization as well as their anisotropies. We show that stronger forces are carried by larger particles and propped by an increasing number of small particles. The independence of shear strength with regard to size span is shown to be a consequence of contact network self-organization, with the falloff of contact anisotropy compensated by increasing force anisotropy.

Figure 1

Regular pentagon ($\delta =0$) deformed into irregular pentagons for two values of parameter $\delta$.

Figure 2

Snapshots of dense packings obtained at the end of compression, here for (a) $(s,\delta )=(0.01,0)$, (b) $(s,\delta )=(0.9,0)$, (c) $(s,\delta )=(0.01,1)$, and (d) $(s,\delta )=(0.9,1)$.

Figure 3

Normalized shear stress $q/p$ (a) and solid fraction $\rho$ (b) as a function of cumulative shear strain ${\varepsilon }_q$ for different values of the polydispersity parameters $s$ and $\delta$.

Figure 4

(a) Normalized friction angle $sin{\phi }^{*}$ as a function of $s$ for all values of $\delta$ averaged in the steady state. Inset: $q/p$ as a function of $\delta$ for all values of $s$. (b) Solid fraction ${\rho }^{*}$ as a function of $s$ for all values of $\delta$ averaged in the residual state. Inset: ${\rho }^{*}$ as a function of $\delta$ for all values of $s$.

Figure 5

Snapshots of the contact network for (a) $(s,\delta )=(0.01,0)$, (b) $(s,\delta )=(0.9,0)$, (c) $(s,\delta )=(0.01,1)$, and (d) $(s,\delta )=(0.9,1)$. Floating particles (i.e., particles with one or no contacts) are shown in white, and contacts are represented by line segments joining the centers of mass of the particles to the contact points.

Figure 6

(a) Proportion $X_f$ of floating particles as a function of size span for all values of shape polydispersity and (b) mean coordination number $Z$ as a function of reduced diameter $d_r$ for all values of size polydispersity.

Figure 7

Snapshots of force-bearing particles for (a) $(s,\delta )=(0.01,0)$, (b) $(s,\delta )=(0.9,0)$, (c) $(s,\delta )=(0.01,1)$, and (d) $(s,\delta )=(0.9,1)$. Floating particles (i.e., particles with one or no contacts) are shown in white and normal forces are represented by the thickness of the segments joining the particle centers. The diameters of light-gray (red) circles are proportional to the friction mobilization at each contact.

Figure 8

Local geometry.

Figure 9

(a) Normal forces normalized by the mean $\langle f_n\rangle$ and (b) friction mobilization index $I_{\mu }$ normalized by the mean $\langle I_{\mu }\rangle$ as a function of the normalized intercenter length $\ell /R_{\max }$.

Figure 10

Polar representation of the functions (a) $P\left(\theta \right)$, (b) $\langle {\ell }_n\rangle \left(\theta \right)$, (c) $\langle {\ell }_t\rangle \left(\theta \right)$, (d) $\langle f_n\rangle \left(\theta \right)$, and (e) $\langle f_t\rangle \left(\theta \right)$ (symbols) and with truncated Fourier approximations (solid lines) for different values of $s$ and $\delta$.

Figure 11

Evolution of state anisotropies (a) as a function of $s$ for all values of $\delta$ and (b) as a function of $\delta$ for all values of $s$ in the steady shear state. Error bars correspond to the standard deviation of the fluctuations in the steady state.

Figure 12

Normalized shear stress as a function of $s$ for all values of $\delta$ (symbols and solid line) together with harmonic approximations given by Eqs. (11) (dashed line). Error bars correspond to the standard deviation of the fluctuations in the steady state.