Micromechanical description of the compaction of soft pentagon assemblies

We analyze the isotropic compaction of assemblies composed of soft pentagons interacting through classical Coulomb friction via numerical simulations. The effect of the initial particle shape is discussed by comparing packings of pentagons with packings of soft circular particles. We characterize the evolution of the packing fraction, the elastic modulus, and the microstructure (particle rearrangement, connectivity, contact force, and particle stress distributions) as a function of the applied stresses. Both systems behave similarly: the packing fraction increases and tends asymptotically to a maximum value ${\varphi }_{\max }$, where the bulk modulus diverges. At the microscopic scale we show that particle rearrangements occur even beyond the jammed state, the mean coordination increases as a square root of the packing fraction, and the force and stress distributions become more homogeneous as the packing fraction increases. Soft pentagons experience larger particle rearrangements than circular particles, and such behavior decreases proportionally to the friction. Interestingly, the friction between particles also contributes to a better homogenization of the contact force network in both systems. From the expression of the granular stress tensor we develop a model that describes the compaction behavior as a function of the applied pressure, the Young modulus, and the initial shape of the particles. This model, settled on the joint evolution of the particle connectivity and the contact stress, provides outstanding predictions from the jamming point up to very high densities.

Figure 1

Close-up views of the assembly of pentagons (a), (b) and disks (c), (d) at $\mu =0$ and for $P/E=0.025$ (a), (c) and $P/E=0.3$ (b), (d). The insets of (a) and (c) show the finite element mesh used for both pentagons and disks, respectively.

Figure 2

Packing fraction $\varphi$ as a function of the reduced pressure $P/E$ for assemblies of soft pentagons and different values of friction in (a) linear-linear scale and (b) log-linear scale. The insets show the same data for assemblies composed of soft disks. The dashed lines are the elastic approximations [Eq. (10)], and the continuous lines are the predictions given by Eq. (11) for $\mu =0$ (black) and $\mu =0.8$ (red).

Figure 3

Evolution of the bulk modulus $K$ normalized by $E$ as a function of the packing fraction $\varphi$ for soft assemblies of pentagons and disks (inset). The predictions given by Eq. (12) are shown in continuous lines.

Figure 4

Color map of the particle rearrangement parameter ${\widehat{\theta }}_i$ in assemblies of frictionless disks and pentagons for different levels of compaction.

Figure 5

${\widehat{\theta }}_{\varphi }$ as a function of the friction coefficient for assemblies of pentagons (continuous line) and disks (dashed line). The inset shows the evolution of $\widehat{\theta }$ as a function of the packing fraction $\varphi$ for two different values of friction.

Figure 6

(a) Evolution of $\widehat{R}/{\widehat{R}}_0$ as a function of $\varphi$ for different values of friction in assemblies of pentagons and disks (inset). (b) Group of particles extracted from the assemblies of pentagons and disks, respectively, undergoing the same cumulative packing fraction.

Figure 7

Evolution of the reduced coordination number $Z-Z_0$ as a function of the reduced packing fraction $\varphi -{\varphi }_0$ for assemblies of pentagons and disks (inset) and for different values of friction coefficient. The power-law relation $Z-Z_0=\xi {(\varphi -{\varphi }_0)}^{\alpha }$ with $\alpha =0.5$ and $\xi =5.1$ is shown in a continuous line.

Figure 8

Evolution of particle connectivity $P_c$ as a function of $\varphi$: for assemblies of (a) pentagons and (b) for disks.

Figure 9

Close-up views of the force chains in frictionless assemblies of pentagons (a), (b) and disks (c), (d) at the jammed state (a), (c) and for $\varphi \sim 1$ (c), (d). The magnitude of each normal force is represented by the thickness of the segment joining the centers of the particles in contact. The strong forces ($f_n⩾\langle f_n\rangle$) and weak forces ($f_n<\langle f_n\rangle$) are plotted in red and black, respectively.

Figure 10

Probability distribution function of the normal forces $f_n$ normalized by the average normal force $\langle f_n\rangle$ for frictionless (a) pentagon and (b) disk packings, and for different packing fractions.

Figure 11

Participation number $\mathrm{\Gamma }$ as a function of $\varphi$ for assemblies of pentagons and disks (inset) and different friction coefficients.

Figure 12

(a) Probability distribution function of the particle stress $P_i$, normalized by the mean $\langle P_i\rangle$, for frictionless assemblies of pentagons and increasing packing fraction $\varphi$. (b) Standard deviation of the distribution $P_p$ as a function of the packing fraction $\varphi$ for assemblies of pentagons and disks (inset) and for various values of the friction coefficient.

Figure 13

Macroscopic volumetric strain ${\epsilon }_v$ as a function of the local strain ${\varepsilon }_{\ell }$ at the small deformation domain in assemblies of pentagons and disks (inset) for different values of friction.