Geometric cohesion in two-dimensional systems composed of star-shaped particles

Using a discrete element method, we investigate the phenomenon of geometric cohesion in granular systems composed of star-shaped particles with 3 to 13 arms. This was done by analyzing the stability of columns built with these particles and by studying the microstructure of these columns in terms of density and connectivity. We find that systems composed of star-shaped particles can exhibit geometric cohesion (i.e., a solidlike behavior, in the absence of adhesive forces between the grains), depending on the shape of the particles and the friction between them. This phenomenon is observed up to a given critical size of the system, from which a transition to a metastable behavior takes place. We also have evidence that geometric cohesion is closely linked to the systems' connectivity and especially to the capability of forming interlocked interactions (i.e., multicontact interactions that hinder the relative rotation of the grains). Our results contribute to the understanding of the interesting and potentially useful phenomenon of geometric cohesion. In addition, our work supplements an important set of experimental observations and sheds light on the complex behavior of real, three-dimensional, granular systems.

Figure 1

Particles geometry: (a) Schematic representation of an arm, made from a rectangle and a disk. (b) Schematic representation of a five-arm particle with a diameter (i.e., of the circumscribing circle) $D$. (c) Star-shaped particles with numbers of arms $N_a$ from 3 to 13. (d) Cap-cap contact ($cc$) considered as a disk-disk contact. (e) Cap-side contact ($cs$) considered as a disk-rectangle contact. (f) Side-side contact ($ss$) considered as two $cs$-contacts.

Figure 2

Sample preparation: (a) Initial state of the sample construction, where $N_p$ randomly oriented particles were initially laid on a triangular lattice (here for particles with three arms). (b) Final state, at the end of the gravity deposition phase; this pile served as an initial state for the stability test.

Figure 3

The initial (dark blue) and final (magenta) states of columns made of star-shaped particles with (a) $N_a=3$ and (b) $N_a=8$ are shown, for increasing values of the scaled initial height $H_0/D$. For $N_a=3$, the columns collapse towards a more or less triangular heap regardless of $H_0$. For $N_a=8$, the integrity of the granular column is maintained, even in the absence of the side walls, for $H_0<150D$. Note that both heights are normalized by the diameter of the particles $D$.

Figure 4

(a) Scaled final height $H_f/D$ as a function of the normalized initial height $H_0/D$, for columns composed of grains with different numbers of arms $N_a$. Error bars represent the standard error between several repetitions for each $H_0$. (b) Schematic representation of $H_0$ and $H_f$. Particles in the initial state are shown in gray color. In the final state, the particles that remain inside the initial area are shown in blue, while that particles that fall outside this area are shown in red.

Figure 5

(a) Evolution of the collapse ratio $r$ of the columns as a function of the scaled initial height $H_0/D$, for columns composed of star-shaped particles with different numbers of arms $N_a$. Error bars represent the standard error between several repetitions for each $H_0$. The lines show a fit of Eq. (3). (b) Stability phase diagram as a function of $N_a$ and the scaled initial height $H_0/D$. The color scale is proportional to $r$. The evolution of the critical height $H_c$ and the crossover height $H_{50}$ (see text for a definition) is also shown using dashed lines.

Figure 6

Collapse ratio $r$ as a function of the scaled initial height $H_0/D$, for columns composed of grains with $N_a=6$ arms and for contact friction, $\mu$, from 0.01 to 0.7. Error bars represent the standard error of the collapse ratio. The inset shows the evolution of the critical height $H_c/D$ and the crossover height $H_{50}/D$ as functions of $\mu$.

Figure 7

Averaged packing fraction, $\langle \nu \rangle$, as a function of the number of arms $N_a$. Data points and bars (too small to be seen) represent the mean and standard error over all repetitions and for all column heights $H_0$.

Figure 8

Schematic representation of the types of contacts that can be observed between the strongly nonconvex grains explored in this study. Without any loss of generality, here examples are given for particles with $N_a=8$ arms. Red lines represent $cc$ or $cs$ contacts, and green lines represent $ss$ contacts (counted twice since a $ss$ contact represents two geometrical constraints).

Figure 9

(a) Coordination number $Z$ and connectivity number $Z_c$ as functions of the number of arms $N_a$. (b) Evolution of the partial connectivity numbers $Z_{i1}$ (dark red circle), $Z_{i2}$ (clear red square), $Z_{i3}$ (dark green lozenge), $Z_{i4}$ (clear green cross) as functions of $N_a$. (c) The connectivity numbers distinguishing $Z_{i1}+Z_{i2}$ and $Z_{i3}+Z_{i4}$. The bars represent the standard error of the plotted data.

Figure 10

Collapse sequence for a column with (a) $N_a=3$ arms and an initial height $H_0/D=66$ and (b) for $N_a=8$ and $H_0/D=240$. $t_f$ corresponds to the final time from which the heap is at rest. The color scale represents the particles' velocity.

Figure 11

Connectivity profiles $Z_c\left(y\right)$ as functions of the vertical position $y$, scaled by the diameter $D$, for columns with (a) $N_a=3$ arms and (b) $N_a=8$. On the left side of the figure, the average of $Z_c$ is shown using a full line, and the limit $z_{\text{iso}}$ is shown using a purple dashed line. On the right-hand side, we show the partial connectivity number profiles $Z_{i1}$ to $Z_{i4}$ as functions of $y/D$; the data for all initial heights $H_0$ are shown using the same color for the sake of clarity.

Figure 12

(a) Evolution of the collapse ratio $r$ for $N_a=8$ arms, for both uncompressed and compressed columns. Error bars represent the standard error. (b) Profiles of the connectivity number $Z_c\left(y\right)$ (black) and partial connectivity numbers $Z_{i1}+Z_{i2}$ (red) and $Z_{i3}+Z_{i4}$ (green), as functions of the vertical position $y/D$ for initially uncompressed (full lines) and compressed (dashed lines) columns with $H_0/D=240$.