Shear strength and microstructure of polydisperse packings: The effect of size span and shape of particle size distribution

By means of extensive contact dynamics simulations, we analyzed the effect of particle size distribution (PSD) on the strength and microstructure of sheared granular materials composed of frictional disks. The PSDs are built by means of a normalized $\beta$ function, which allows the systematic investigation of the effects of both, the size span (from almost monodisperse to highly polydisperse) and the shape of the PSD (from linear to pronouncedly curved). We show that the shear strength is independent of the size span, which substantiates previous results obtained for uniform distributions by packing fraction. Notably, the shear strength is also independent of the shape of the PSD, as shown previously for systems composed of frictionless disks. In contrast, the packing fraction increases with the size span, but decreases with more pronounced PSD curvature. At the microscale, we analyzed the connectivity and anisotropies of the contacts and forces networks. We show that the invariance of the shear strength with the PSD is due to a compensation mechanism which involves both geometrical sources of anisotropy. In particular, contact orientation anisotropy decreases with the size span and increases with PSD curvature, while the branch length anisotropy behaves inversely.

Figure 1

Theoretical (solid lines) and generated (symbols) particle size distributions for different combinations of parameters $s$ and $b$. $s=0.2$ (black), 0.6 (red), and 0.95 (green); and $b=1$ (circles), 3 (squares), and 5 (triangles). The same distributions are shown in linear (a) and log-linear (b) scale.

Figure 2

Boundary conditions for isotropic compression (a) and biaxial shear (b).

Figure 3

Particle-scale view at the end of the isotropic compression phase for different combinations $(s,b)$. Floating particles (i.e., particles with one or no contacts) are shown in white.

Figure 4

Stress ratio $q/p$ (a) and volumetric strain ${\varepsilon }_p$ (b) as functions of the deviatoric strain ${\varepsilon }_q$ for different combinations of the size span $s$ and the shape $b$ of the PSD (see schematic representation in the inset).

Figure 5

Sine of the internal friction angle in the steady state $sin{\phi }^{*}$ as a function of $s$ for all values of $b$. Error bars represent the standard deviation in the steady state.

Figure 6

Packing fraction in the steady state ${\rho }^{*}$ as a function of $s$ for all values of $b$ (solid lines), and as a function of $b$ for all values of $s$ (inset, dashed lines). Error bars represent the standard deviation in the steady state.

Figure 7

Contact network (black lines) and particle connectivity (color scale) for $(s,b)=(0.9,1)$ (a) and $(s,b)=(0.9,5)$ (b). Color intensity is proportional to the grains' number of contacts. The floating particles (i.e., particles with zero or one contact) are shown in white.

Figure 8

(a) Proportion $P_{0-1}$ of floating particles and (b) Coordination number $z$, as a functions of $s$ for all values of $b$. The insets show $P_{0-1}$ and $z$ as a function of $b$ for all values of $s$. The error bars represent the standard deviation in the steady state.

Figure 9

Particle connectivity, defined as the proportion $P_c$ of particles with exactly $c$ contacts, as a function of $s$ for all values of $b$. The error bars represent the standard deviation in the steady state.

Figure 10

Snapshots of the force network for (a) $(s,b)=(0.2,2)$ and (b) $(s,b)=(0.9,5)$. Floating particles are shown in white and normal forces are represented by the thickness of the segments joining the particle centers. The red circles in the contact points represent friction mobilization, their diameter being proportional to the friction mobilization index (see text for definition).

Figure 11

Polar representation of the functions $P\left(\theta \right)$ (a), $\langle {\ell }_n\rangle \left(\theta \right)$ (b), $\langle f_n\rangle \left(\theta \right)$ (c), and $\langle f_t\rangle \left(\theta \right)$ (d), for $(s,b)=(0.2,1)$ (black circles) and $(s,b)=(0.9,1)$ (red squares). The corresponding harmonic approximations are drawn as solid lines.

Figure 12

Contact orientation $a_c$ (line) and branch length $a_{\ell }$ (dashed line) anisotropies as functions of $s$ for all values of $b$. Error bars represent the standard deviation in the steady state.

Figure 13

Contact anisotropies $a_c$ (a) and branch length anisotropies $a_{\ell }$ (b) as functions of $b$ for all values of $s$. Error bars represent the standard deviation in the steady state.

Figure 14

Normal $a_n$ (line) and tangential $a_t$ (dashed line) forces anisotropies as functions of $s$ for all values of $b$. Error bars represent the standard deviation in the steady state.