Force transmission in a packing of pentagonal particles

We perform a detailed analysis of the contact force network in a dense confined packing of pentagonal particles simulated by means of the contact dynamics method. The effect of particle shape is evidenced by comparing the data from pentagon packing and from a packing with identical characteristics, except for the circular shape of the particles. A counterintuitive finding of this work is that, under steady shearing, the pentagon packing develops a lower structural anisotropy than the disk packing. We show that this weakness is compensated by a higher force anisotropy, leading to enhanced shear strength of the pentagon packing. We revisit “strong” and “weak” force networks in the pentagon packing, but our simulation data also provide evidence for a large class of “very weak” forces carried mainly by vertex-to-edge contacts. The strong force chains are mostly composed of edge-to-edge contacts with a marked zigzag aspect and a decreasing exponential probability distribution as in a disk packing.

Figure 1

(Color online) Representation of simple (vertex-to-edge) and double (edge-to-edge) contacts between two pentagons.

Figure 2

(Color online) Snapshots of a portion of the samples $S2$ (a) and $S1$ (b) composed of circular and pentagonal particles, respectively.

Figure 3

(Color online) Normalized shear stress $q∕p$ as a function of cumulative shear strain ${\epsilon }_q$ for the samples $S1$ and $S2$.

Figure 4

(Color online) Cumulative volumetric strain ${\epsilon }_p$ as a function of cumulative shear strain ${\epsilon }_q$ for the samples $S1$ and $S2$.

Figure 5

(Color online) Stress-dilatancy relation between dilatancy angle $\psi$ and internal angle of friction $\phi$ for the samples $S1$ and $S2$.

Figure 6

(Color online) The coordination number $z$ (a) and the fraction $\gamma$ of persistent contacts (b) as a function of cumulative shear strain ${\epsilon }_q$ for the samples $S1$ and $S2$.

Figure 7

(Color online) Connectivity diagram for the samples $S1$ and $S2$ expressing the fraction $P\left(c\right)$ of particles with exactly $c$ contacts in the residual state.

Figure 8

Polar representation of the probability density function $P_{\theta }$ of the contact normal directions $\theta$ for the samples $S1$ and $S2$ in the residual state.

Figure 9

(Color online) Evolution of the anisotropy $a^{\prime }$ with cumulative shear strain ${\epsilon }_q$ for the samples $S1$ and $S2$.

Figure 10

(Color online) Polar representation of the angle-averaged normal (a) and tangential (b) forces $⟨f_n⟩\left(\theta \right)$ and $⟨f_t⟩\left(\theta \right)$ for the samples $S1$ and $S2$ in the residual state.

Figure 11

(Color online) Evolution of force anisotropies $a_n$ (a) and $a_t$ (b) as a function of cumulative shear strain ${\epsilon }_q$ in samples $S1$ and $S2$.

Figure 12

(Color online) Evolution of the normalized shear stress $q∕p$ for the samples $S1$ and $S2$ with ${\epsilon }_q$ together with the corresponding predictions from its expression as a function of fabric and force anisotropies, Eq. (19).

Figure 13

(Color online) Snapshots of normal forces in samples $S2$ (a) and $S1$ (b). Line thickness is proportional to the normal force.

Figure 14

(Color online) Probability density functions of normal forces in samples $S1$ and $S2$ in log-linear (a) and log-log scales (b).

Figure 15

(Color online) Partial shear stress $q\left(\xi \right)∕p$ as a function of force cutoff $\xi$ (normalized by the mean force) for the samples $S1$ and $S2$ in the residual state.

Figure 16

(Color online) Partial fabric anisotropy $a^{\prime }\left(\xi \right)$ as a function of force cutoff $\xi$ (normalized by the mean force) in the samples $S1$ and $S2$.

Figure 17

(Color online) A map of the contact network composed of very weak (black), intermediate (light gray) and strong (dark gray) contacts in the pentagon packing.

Figure 18

(Color online) Normalized shear stress $q∕p$ for simple $\left(s\right)$ and double $\left(d\right)$ contacts, as well as for all contacts $(s+d)$, as a function of cumulative shear strain ${\epsilon }_q$ in the pentagon packing.

Figure 19

(Color online) Proportions $k_s$ and $k_d$ of simple and double contacts, and the corresponding relative force averages $f_s$ and $f_d$, as a function of cumulative shear strain ${\epsilon }_q$.

Figure 20

(Color online) Cumulative proportion $\mathrm{\Delta }\gamma$ of simple contacts turning to double $(s\to d)$ and vice versa $(d\to s)$.

Figure 21

Gray level map of the connectivity function $P(m_s,m_d)$ of the pentagon packing in the residual state.

Figure 22

(Color online) The anisotropy $a^{\prime }$ of simple $\left(s\right)$ and double $\left(d\right)$ contacts as a function of cumulative shear strain ${\epsilon }_q$ in the pentagon packing.

Figure 23

(Color online) Probability density function of normal forces for simple $\left(s\right)$ and double $\left(d\right)$ contacts in log-linear $\left(a\right)$ and log-log scales $\left(b\right)$.

Figure 24

(Color online) A map of the normal force network in the residual state with simple contacts $\left(s\right)$ in blue (black) and double contacts $\left(d\right)$ in red (gray). Line thickness is proportional to the normal force.

Figure 25

(Color online) Proportions $k_s^S$ and $k_s^W$ of strong $\left(S\right)$ and weak $\left(W\right)$ simple $\left(s\right)$ contacts, respectively, as well as the proportions $k_d^S$ and $k_d^W$ of strong and weak double $\left(d\right)$ contacts as a function of cumulative shear strain ${\epsilon }_q$ in the pentagon packing.