Binary mixtures of disks and elongated particles: Texture and mechanical properties

We analyze the shear strength and microstructure of binary granular mixtures consisting of disks and elongated particles by varying systematically both the mixture ratio and degree of homogeneity (from homogeneous to fully segregated). The contact dynamics method is used for numerical simulations with rigid particles interacting by frictional contacts. A counterintuitive finding of this work is that the shear strength, packing fraction, and, at the microscopic scale, the fabric, force, and friction anisotropies of the contact network are all nearly independent of the degree of homogeneity. In other words, homogeneous mixtures have the same strength properties as segregated packings of the two particle shapes. In contrast, the shear strength increases with the proportion of elongated particles correlatively with the increase of the corresponding force and fabric anisotropies. By a detailed analysis of the contact network topology, we show that various contact types contribute differently to force transmission and friction mobilization.

Figure 1

Geometry of rcr particles.

Figure 2

Representations of disk-disk (dd), disk-rcr (dr), and rcr-rcr (rr) contacts.

Figure 3

Examples of the generated packings at the initial state, for $\alpha =0.5$ (left column) and $\alpha =0.1$ (right column) and for $M^{*}=0.0$ (a),(b), $M^{*}=0.5$ (c),(d), and $M^{*}=1.0$ (e),(f). A zoom near the central zone is shown for $\alpha =0.5$ and $M^{*}=0.0$ (g) and $M^{*}=1.0$ (h).

Figure 4

Normalized shear stress $q/p$ (a) and packing fraction $\rho$ (b) as a function of cumulative shear strain ${\varepsilon }_q$ for five extreme combinations (see schematic representation in inset) of the mixture ratio $\alpha$ and mixture homogeneity $M^{*}$.

Figure 5

Macroscopic friction angle $sin{\phi }^{*}$ as a function of the ratio (a) and homogeneity (b) of the mixture for all values of $M^{*}$ (a) and $\alpha$ (b) averaged in the steady state. Error bars correspond to the standard deviation of the fluctuations in the steady state.

Figure 6

Packing fraction ${\rho }^{*}$ as a function of the proportion $\alpha$ of rcr particles for all values of $M^{*}$ averaged in the residual state. Inset: ${\rho }^{*}$ as a function of $M^{*}$ for all values of $\alpha$. Legend are the same as for Figs. 5 and 5 for the inset. Error bars correspond to the standard deviation of the fluctuations in the steady state.

Figure 7

Snapshots of force-bearing particles for (a) $(\alpha ,M^{*})=(0.50,0.00)$ and (b) $(\alpha ,M^{*})=(0.5,1.00)$ in the critical state at ${\varepsilon }_q\simeq 0.5$. 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 to the contact point. The diameters of light-gray (yellow) circles are proportional to the friction mobilization $I_c=\left|f_t^c\right|/\left(\mu f_n^c\right)$ at each contact $c$.

Figure 8

Polar representation of the functions $P\left(\theta \right)$ (a), $\langle {\ell }_n\rangle \left(\theta \right)$ (b), $\langle {\ell }_t\rangle \left(\theta \right)$ (c), $\langle f_n\rangle \left(\theta \right)$ (d), and $\langle f_t\rangle \left(\theta \right)$ (e) for different values of the $M^{*}$ and $\alpha$.

Figure 9

Evolution of contact anisotropies as a function of $\alpha$ (a) for all values of $M^{*}$ and as a function of $M^{*}$ for all values of $\alpha$ in the steady shear state. Error bars correspond to the standard deviation of the fluctuations in the steady state.

Figure 10

Evolution of normal (up) and tangential (down) branch anisotropies as a function of $\alpha$ for all values of $M^{*}$ and in the insets as a function of $M^{*}$ for all values of $\alpha$ in the steady shear state. Error bars correspond to the standard deviation of the fluctuations in the steady state. Legends are the same as in Figs. 9 and 9 for the insets.

Figure 11

Evolution of normal (up) and tangential (down) force anisotropies as a function of $\alpha$ for all values of $M^{*}$ and in the insets as a function of $M^{*}$ for all values of $\alpha$ in the steady shear state. Error bars correspond to the standard deviation of the fluctuations in the steady state. Legends are the same as for Figs. 9 and 9 for the insets.

Figure 12

Snapshots of force-bearing particles for $\alpha =0.5$ and $M^{*}=0.0$ (a), $M^{*}=0.5$ (b), and $M^{*}=1.0$ (c) in the critical state at ${\varepsilon }_q\simeq 0.5$. The floating particles (i.e., particles with one or no contacts) are shown in white, and the normal forces are represented by the thickness of the segments joining the particle centers to the contact points, in red for disk-disk contacts, yellow for disk-rcr contacts, and in black for rcr-rcr contacts.

Figure 13

Proportions of disk-disk ($dd$), disk-rcr ($dr$), and rcr-rcr ($rr$) contacts as functions of the homogeneity parameter $M^{*}$ and for $\alpha =0.5$. The error bars represent the standard deviation in the steady state.

Figure 14

Partial contact anisotropies $a_c^{dd}$, $a_c^{dr}$, and $a_c^{rr}$ (a), partial normal force anisotropies $a_{fn}^{dd}$, $a_{fn}^{dr}$, and $a_{fn}^{rr}$ (b), and partial tangential force anisotropies $a_{ft}^{dd}$, $a_{ft}^{dr}$, and $a_{ft}^{rr}$ (c) of $dd$, $dr$, and $rr$ contacts as functions of the homogeneity parameter $M^{*}$, together with total anisotropies $a_c$, $a_{fn}$, and $a_{ft}$, for $\alpha =0.5$. The error bars represent the standard deviation in the steady state.

Figure 15

Macroscopic friction angle $sin{\phi }^{*}$ and partial shear strengths for disk-disk ($dd$), disk-rcr ($dr$), and rcrc-rcrc ($rr$) contacts as functions of the homogeneity parameter $M^{*}$, together with the predicted values by Eq. (13) (empty symbols), for $\alpha =0.5$. The error bars represent the standard deviation in the steady state.