Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks

This is the first paper of a series of three, in which we report on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. The influence of various assembling processes on packing microstructures is investigated. It is accurately checked that frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, higher solid fractions corresponding to more ordered configurations with traces of crystallization. Specific properties directly related to isostaticity of the force-carrying structure in the rigid limit are discussed. With frictional grains, different preparation procedures result in quite different inner structures that cannot be classified by the sole density. If partly or completely lubricated they will assemble like frictionless ones, approaching the RCP solid fraction ${\Phi }_{\mathrm{RCP}}\simeq 0.639$ with a high coordination number: $z^{*}\simeq 6$ on the force-carrying backbone. If compressed with a realistic coefficient of friction $\mu =0.3$ packings stabilize in a loose state with $\Phi \simeq 0.593$ and $z^{*}\simeq 4.5$. And, more surprisingly, an idealized “vibration” procedure, which maintains an agitated, collisional regime up to high densities results in equally small values of $z^{*}$ while $\Phi$ is close to the maximum value ${\Phi }_{\mathrm{RCP}}$. Low coordination packings have a large proportion $(>10\%)$ of rattlers—grains carrying no force—the effect of which should be accounted for on studying position correlations, and also contain a small proportion of localized “floppy modes” associated with divalent grains. Low-pressure states of frictional packings retain a finite level of force indeterminacy even when assembled with the slowest compression rates simulated, except in the case when the friction coefficient tends to infinity. Different microstructures are characterized in terms of near neighbor correlations on various scales, and some comparisons with available laboratory data are reported, although values of contact coordination numbers apparently remain experimentally inaccessible.

Figure 1

(Color online) Solid fraction $\Phi$ versus $n^{-1∕2}$. Blue dotted line: average value, with standard deviation indicated as error bars, according to OSLN’s results, extrapolated to $n\to \infty$. Black round dots with error bars: our $A$ samples for $n=4000$ and other, very similar results for $n=1372$. Square dot: $A^{\prime }$ samples with $n=4000$ (this point has a smaller error bar).

Figure 2

(Color online) Probability distribution function $P\left(f\right)$ of normalized contact forces $f=N∕⟨N⟩$ in $A$ and $A^{\prime }$ configurations at high $\kappa$. The dashed line is the fit proposed by DTS: $P\left(f\right)=[3.43f^2+1.45-1.18∕(1+4.71f\left)\right]\exp (-2.25f)$.

Figure 3

Pair correlation functions $g^{\mathrm{I}}\left(r\right)$ and $g^{\mathrm{II}}\left(r\right)$ versus $r∕a$ in $A$ samples at $\kappa =39000$. Both definitions coincide on this scale (only the peak for $r\to a^{+}$ is slightly different).

Figure 4

(Color online) Coordination number of near neighbors function of interstice $h$. The power-law regime extends to smaller and smaller $h$ values as $\kappa$ increases.

Figure 5

Pair correlation functions $g\left(r\right)$ [definitions $g^{\mathrm{I}}\left(r\right)$ and $g^{\mathrm{II}}\left(r\right)$ coincide on this scale] for configurations $B$, $C$, $D$.

Figure 6

(Color online) Coordination number for neighbors at distance $\le h$, $z^{\mathrm{II}}\left(h\right)$, for configurations $A$ (red, upper dashed line), $B$ (blue, middle dashed line), $C$ (black, solid line), and $D$ (green, bottom dashed line).

Figure 7

(Color online) $z^{\mathrm{I}}\left(h\right)-z$ (blue) and $z^{\mathrm{II}}\left(h\right)-z^{\mathrm{II}}\left(0\right)$ (black) versus $h$ on a doubly logarithmic plot for $D$ samples at lowest pressure. The slopes of the corresponding dotted straight lines (power law fits) are ${\beta }_{\mathrm{II}}=0.51$ and ${\beta }_{\mathrm{I}}=0.35$ [see Eq. (32)].

Figure 8

(Color online) Same as Fig. 7, for $B$ samples. Dotted lines have slopes ${\beta }_{\mathrm{II}}=0.56$ and ${\beta }_{\mathrm{I}}=0.45$.

Figure 9

(Color online) Same as Fig. 7, for $C$ samples. Slopes of dotted straight lines: ${\beta }_{\mathrm{II}}=0.6$ and ${\beta }_{\mathrm{I}}=0.37$.

Figure 10

Equilibrium and free motion (mechanism) of one sphere (marked 1) with two contacts (with particles marked 2 and 3). (a) In the plane of the three centers, normal and tangential components of the two contact forces balance along the line joining the contact points (dotted line). (b) Seen from above, along the direction of the line of the centers of spheres 2 and 3, sphere 1 can move and occupy the different positions depicted with dashed lines, its center describing the dotted circle around the 2-3 axis.

Figure 11

(Color online) Distribution of normal forces, normalized by its average, $f=\frac{N}{⟨N⟩}$, in states $A$ (red crosses), $B$ (blue asterisks), $C$ (black square dots), and $D$ (green open squares) at the lowest simulated pressure.

Figure 12

(Color online) Examples of trajectories of mobile divalent grains in the plane orthogonal to vector $\mathbf{e}$. All trajectories begin at $x=D$, $y=0$. Continuous lines depict the results obtained on ignoring contact creation, which end up, in two cases out of the four represented, in instabilities (black lines), while complete circles are observed in the other, stable ones (red). Dots mark the same trajectories, which are in practice arrested by other contacts when contact creation is taken into account.