Effect of confinement on dense packings of rigid frictionless spheres and polyhedra

We study numerically the influence of confinement on the solid fraction and on the structure of three-dimensional random close-packed granular materials subject to gravity. The effects of grain shape (spherical or polyhedral), material polydispersity, and confining wall friction on this dependence are investigated. In agreement with a simple geometrical model, the solid fraction is found to decrease linearly for increasing confinement no matter the grain shape. Furthermore, this decrease remains valid for bidisperse sphere packings, although the gradient seems to reduce significantly when the proportion of small particles reaches $40\%$ by volume. The confinement effect on the coordination number is also captured by an extension of the aforementioned model.

Figure 1

Pinacoid, a model polyhedra characterized by its length $L$, width $G$, height $E$, and angle $\alpha$.

Figure 2

Typical 3D snapshots of packings made of polysized spheres (a) and pinacoids (b). The confinement is characterized by the gap $W$ between sidewalls. The direction of gravity is $-z$.

Figure 3

Two polyhedra can experience simple contacts (a), double contacts (b), or triple contacts (c).

Figure 4

2D simplified representation of the four intersection situations between two pinacoids and a lattice cell: (a) solid volume $\ge$ cell volume and partial interpenetration in the cell; (b) solid volume $\ge$ cell volume and total interpenetration in the cell; (c) solid volume $<$ cell volume and interpenetration (partial as represented or total); and (d) solid volume $<$ cell volume and no interpenetration.

Figure 5

(a) Homogeneity of MSP ($\circ$) and MPP ($\times$) along the $z$ axis, expressed as the ratio of the number of grains to the total number of grains in $d_L$-thick layers. (b) Homogeneity of BSP along the $z$ axis, expressed as the ratio of the number of large (empty symbols) and small (solid symbols) grains to the total number of grains in $d_L$-thick layers for $x_s=10\%$ ($\square$), $x_s=25\%$ ($\diamond$), and $x_s=40\%$ ($▵$). (c) Homogeneity of BSP along the $y$ axis (same calculation method and same key). For (a)–(c), the gap between sidewalls is $W=5d_L$ and the data have been averaged over three simulations.

Figure 6

Plot of the average solid fraction versus $d_L/W$ for MSP, BSP, and MPP. The lines are fits from the geometrical model [Eq. (1)]. Error bars denote the standard deviation.

Figure 7

Solid fraction profiles as a function of distance $y/d_L$ from the confining wall for (a) MSP with $W=4d_L$, $W=5d_L$, $W=10d_L$, and $W=20d_L$; (b) BSP with $W=5d_L$ and $x_s=0,10,25,\text{and}40\%$; and (c) MPP with $W=5d_L$, $W=10d_L$, and $W=20d_L$. Fluctuations of the local solid fraction are due to the layering of particles in the vicinity of the sidewalls. The inset in Fig. 7a is a zoom over $3d_L$. The inset in Fig. 7b shows the solid fraction for BSP (here $x_s=40\%$) and the corresponding fit [Eq. (8)].

Figure 8

Characteristic length of confinement effect ${\lambda }_L$ versus fraction of small grains $x_s$ for MSP and BSP (circle) as well as for MPP (diamond) with $W=10d$. That length is defined in Eq. (8). The effect of confinement is found to decrease with increasing grain polydispersity or grain angularity. The inset reports the same length versus that of the fit parameter in Eq. (1): $C=2h({\varphi }_{\mathrm{bulk}}-{\Phi }_{\mathrm{BL}})$. The dashed line corresponds to a linear fit.

Figure 9

Pair correlation functions of MSP (a) and MPP (b) for several values of $W/d_L$. These exhibit local order that is stronger for MSP compared to MPP. When $r/d_L$ is large enough, $g\left(r\right)$ tends towards 1, indicating the absence of long-range translational order.

Figure 10

Contact normal orientation distributions for MSP inside the boundary layers (a) and inside the bulk region (b). The upper half of each chart (e.g., from $0^{\circ }$ to ${180}^{\circ }$) corresponds to the $xy$ plane, while lower half corresponds to the $yz$ plane. Several gap widths are considered.

Figure 11

Contact normal orientation distributions for MSP ($x_S=0$) and BSP ($x_S=40\%$) inside the boundary layers/closer than $1.5d_L$ to a sidewall (a) and inside the bulk region (b). The upper half of each chart (e.g., from 0 to ${180}^{\circ }$) corresponds to the $xy$ plane, while the lower half corresponds to the $yz$ plane. The gap width is $W=5d_L$ and the bulk region coincides with particles located at least $1.5d_L$ away from the sidewalls.

Figure 12

Contact normal orientation distributions for MPP inside the boundary layers (a) and inside the bulk region (b). The upper half of each chart (e.g., from $0^{\circ }$ to ${180}^{\circ }$) corresponds to the $xy$ plane, while the lower half corresponds to the $yz$ plane. Several gap widths are considered.

Figure 13

Distribution of the orientations of simple (a), double (b), and triple (c) contacts for MPP in the $yz$ plane. Periodic boundary conditions are used in the $x$ and $y$ directions.

Figure 14

Coordination number as a function of $d_L/W$ for MSP and MPP. Error bars on pinacoid packings results denote the standard deviation (not represented for sphere packings because errors are smaller than symbol size). The linear relationship between $Z$ and $d_L/W$ suggests that the geometrical model, initially derived for the packing fraction, is also valid for the coordination number.

Figure 15

Coordination number profiles (along $y$) for MSP (a) and for MPP (b) for several gap width. These profiles evidence a constant central zone and two drop zones in contact with the sidewalls.

Figure 16

Coordination number profiles (along $y$) of MPP for simple (a), double (b), and triple contacts (c) for confined ($W\in [5d_L,20d_L]$) and unconfined packings.

Figure 17

Coordination number profiles (with $y$) for BSP. The gap between sidewalls is $W=5d_L$.