Strong interlocking of nonconvex particles in random packings

PHYSICAL REVIEW E 85, 051307 (2012)

Strong interlocking of nonconvex particles in random packings F. Ludewig and N. Vandewalle GRASP, Physics Department, University of Li`ege, B-4000 Li`ege, Belgium (Received 20 March 2012; published 15 May 2012) We present a numerical study of random packings made of nonconvex grains. These particles are built by the agglomeration of overlapping spheres in order to control their sphericity φ. The contact number C is found to be much larger than the coordination number Z, providing a significant difference with convex grains. The packing properties are found to be highly dependent on the morphological parameters of the grains : packing fractions as low as 0.3 have been reached. More importantly, the way nonconvex grains develop multiple contacts, i.e., interlocking, is found to be a relevant effect in such packings. Interlocking provides more stability to loose packings. DOI: 10.1103/PhysRevE.85.051307

PACS number(s): 81.05.Rm, 45.70.Cc, 05.45.Pq

I. INTRODUCTION

How a large number of particles can fill a volume is one of the most persistent problems in mathematics and science. For identical spheres, the packing fraction η, defined as the volume of all particles divided by the apparent√volume of the assembly, has a maximum value of ηfcc = π/3 2  0.74, corresponding to the face-centered-cubic (fcc) lattice. When spheres are randomly packed, a broad range of values for the packing fraction are encountered : 0.60 < η < 0.64 = ηRCP , far below ηfcc (Random Close Packing). By playing with cohesive forces, loose assemblies could be generated with low η values [1]. Compaction experiments and simulations [2–4] have emphasized the slow dynamics that granular assemblies follow to approach jamming states from loose configurations. Numerous recent attempts have aimed to find a relationship between microscopic configurations and the global properties of a random packing [5,6]. When nonspherical particles are considered, such as anisotropic objects, higher packing fraction values can be reached [7]. The local ordering of grains leads to rotational and translational time scales in the dynamics of the system [8]. The question of nonconvex particles has been recently addressed in this context [9]. This case is of crucial importance since multiple contacts can be found for two neighboring grains in opposition to a single contact for convex bodies [10]. Rotational motions should be strongly reduced for nonconvex grains. How the grain morphology will affect the physical properties of the packing is the fundamental question addressed in the present paper. We propose to study the impact of grain morphological parameters on the packing structures in numerical simulations. In the present work, the nonconvex particles are built by agglomerating spheres [11,12] in order to obtain a specific shape. The grain size is adjusted in order to keep a constant volume (mass) for all considered particles. The structure of a grain is described by a central sphere of radius R with 14 overlapping spheres of radius R/2 placed around the central one. The shape of the grains is tuned by changing the relative distance between the central sphere and the overlapping smaller spheres. Table I gives the pictures of four typical nonconvex grains considered in our work. The resulting aspherical particles are described by morphological parameters such as the sphericity and the Feret diameters. The sphericity φ is defined by the ratio between the equivalent sphere area that has the same volume and the 1539-3755/2012/85(5)/051307(4)

real area of the particle. This dimensionless parameter φ could reach values between 0 (fractal dust) and 1 (sphere). In this work, we consider values between φ = 0.494 and 1 (see Table I for typical examples). The Feret diameter F is the distance between two parallel planes that are tangent to the particle. The Feret lengths are expressed in term of the sphere unit, i.e., the single characteristic length for φ = 1. For nonspherical particles, the minimum Fmin and maximum Fmax values are obtained over all plane orientations. They correspond to sieve mesh sizes to be used for selecting the considered grains. The ratio Fmax /Fmin is a shape factor which characterizes the particle anisotropy. Table I presents the values of the morphological parameters of four selected particles. II. NUMERICAL MODEL

Our granular systems have been numerically simulated using the molecular dynamics (MD) approach [13] where friction and angular momentum are taken into account. Normal rep forces Fijn are composed by a repulsive (Fij ) and a dissipative (Fijdis ) component. The repulsive component follows a simple Hooke’s Law, rep

Fij = −kn δij ,

(1)

where δij = dij − 2R. Here, dij is the distance between the centers of the solids i and j . The constant kn is the normal stiffness. Its value is calculated in order to control the maximum deformation of the particles. (In our simulations, kn = 106 N/m). The dissipative component is taken into account by viscous forces according to the following law: ∂δij , (2) ∂t where the viscous constant γn is a nontrivial function of the normal stiffness kn and the restitution coefficient ε [14]. The same restitution coefficient is used for both grain-grain and grain-wall collisions. Tangent forces Fijt are bounded and depend on the relative tangent velocities vijt between the colliding solids i and j . One has   F t = −kt v t and F t   μF n , (3) Fijdis = −γn (kn ,ε)

ij

ij

ij

ij

where μ is a friction coefficient and kt is a purely numerical constant. Friction coefficient μ (0.4) as well as restitution

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©2012 American Physical Society

F. LUDEWIG AND N. VANDEWALLE

PHYSICAL REVIEW E 85, 051307 (2012)

TABLE I. Morphological parameters (see definition in the main text) of four typical investigated particles. Values of the relevant parameters obtained in our simulations are also given. Grain

φ

Fmin

Fmax

η0

η500

Z500

C500

0.959

0.965

1.080

0.532

0.588

4.57

5.49

0.798

1.010

1.150

0.471

0.535

4.86

7.07

0.621

1.065

1.225

0.412

0.474

4.72

7.32

0.494

1.160

1.350

0.314

0.388

4.84

8.11

coefficient ε (0.25) can be adapted to fit the intrinsic properties of the system. A complete description of MD simulations is given by Taberlet [14]. The granular piles have been initialized by a gravity free fall in a cylindrical container. They are composed of N = 1000 particles, each of them being composed of 15 spheres. The compaction process consists of a repeated cycle during which mechanical energy is injected (one tap) and is thereafter relaxed. During each tap, a vertical sinusoidal motion z = A sin ωt of the container is applied during one period. In this work, the tap intensity de2 fined by the reduced acceleration = Aω is fixed to 4 g with a frequency of f = ω/2π = 25 Hz, which corre2 2 sponds to a reduced energy of  = 2d1 ( A gω + ωg2 + 2A) = 0.497 [1,4]. The relaxation phase corresponds to a gravity fall until a stable position is reached by the pile. The cycle tap relaxation has been realized 500 times in our simulations.

FIG. 2. (Color online) The left plot presents the semilog evolution of the packing fraction η during the compaction process for four different particle shapes (those illustrated in Table I). The curves presented in the right plot correspond to the packing fraction at different times t as a function of the sphericity φ.

trapped in the structures for low sphericity values as a function of φ. Let us analyze the volume fraction of our packings. The left semilog plot of Fig. 2 presents the compaction dynamics for the particles presented in Table I. As a function of the tap number t, the packing fraction ηt presents a slow evolution towards an asymptotic saturation η∞ . The major part of the compaction process is realized during the 100 first taps. Thereafter, the process becomes much slower. The compaction curves are well fitted with the inverse logarithmic law,

III. RESULTS

Figure 1 illustrates four typical configurations for a central grain and its neighborhood for the different selected shapes of Table I. Contacts are emphasized. As discussed above, multiple contacts can be seen between neighboring nonconvex grains. The average number of contacts will be analyzed below. From the pictures of Fig. 1, one understands that more voids are

FIG. 1. (Color online) Four local configurations for nonconvex shapes. A central grain, in light gray (green), is in contact with its neighbors, in dark gray. Multiple contacts, in black (red), could be observed for pairs of touching grains.

ηt = η∞ −

η∞ − η0 , 1 + ln(1 + t/τ )

(4)

proposed in earlier experimental, numerical, and theoretical works [2,15,16]. Two important parameters can be extracted from the fits: the asymptotic packing fraction η∞ and the characteristic relaxation time τ . The right plot of Fig. 2 presents, respectively, η0 , η500 , and η∞ values as a function of φ. While η0 and η500 are given by the simulations, η∞ is obtained by fitting. As expected, complex morphologies lead to extremely low packing fraction values. Indeed, the initial random assembly, at t = 0, presents a large spectrum of packing fraction values from 0.30 to 0.55. The asymptotic values η∞ are between 0.41 and 0.62. The latter value for spheres is a little bit smaller than ηRCP due to the finiteness of our system. The difference between the various packing fraction curves, i.e., η∞ − η0 , seems to be independent of φ, meaning that the characteristic compaction time τ does not depend on the particle morphology. This is confirmed by the fits of the compaction curves giving τ ≈ 5 ± 2 for all investigated φ values. Although the grain morphology affects deeply the packing fraction of a random assembly, the compaction dynamics remains unchanged. This compaction dynamics is, however, tuned by the tap characteristics through parameters and , as shown for spheres in [4]. The effect of the grain morphology on the packing structure is probed by looking at the coordination number Z. This parameter corresponds to the number of neighbors

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STRONG INTERLOCKING OF NONCONVEX PARTICLES IN . . . 8.5

φ = 0.959 φ = 0.798 φ = 0.621 φ = 0.494

4.8

PHYSICAL REVIEW E 85, 051307 (2012)

7.5

Z

C

4.6

6.5 4.4

5.5

4.2 1

10

100

500

10

t

100

500

t

FIG. 3. (Color online) Evolution of the coordination number for the neighbor Z (left graphic) and contact C (right graphic) as a function of the number of taps t.

with which at least one contact exists. This number should be compared with the total number of contacts C. Both parameters are averaged over the packing. Typical evolutions of those parameters in a semilog plot are shown in Fig. 3. The mean coordination number Z increases as the assembly becomes denser. However, the amplitude of this phenomenon is extremely limited when φ ≈ 1 and remains smaller than 1 for φ < 1. The morphological parameters have little effect on the coordination number. Figure 4 exhibits Z as a function of φ for t = 0 and t = 500. Nevertheless, the number of contacts C is strongly affected by the parameter φ, as seen in both Figs. 3 and 4. The number of contacts increases abruptly when the shape differs from a sphere.

FIG. 5. Rotational mobility μr at t = 500 as a function of the grain sphericity φ. Error bars are indicated.

studied from the positions of the grains (through η or Z). However, it could be evidenced by looking at the rotational mobility of the grains. This mobility μr is defined as the average angular displacement of the particles between two stable positions, i.e., before and after one tap. One has

μr (t) =

N 1  αi (t), N i=1

(5)

IV. DISCUSSIONS

It should be noticed that a remarkable effect takes place when the shape of the grains is modified: the packing fraction decreases while the number of contacts increases. Since the neighborhood (Z) remains unchanged, our numerical results imply that the interlocking of grains becomes more and more present when φ is reduced. Granular interlocking cannot be

30

t=0 t = 100 t = 500

C/λ

25

20

15 0.5

0.6

0.7

0.8

0.9

1

φ

FIG. 4. (Color online) Evolution of the coordination number for the neighbor Z and contact C as a function of the sphericity at two different times t = 0 and 500.

FIG. 6. (Color online) The top graphic presents the probability distribution functions (PDF) of the contact force normalized by the grain weight mg at the initial state for four different particles. The bottom plot shows the values of the mean C/λ as a function of the sphericity φ for the initial and final states.

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F. LUDEWIG AND N. VANDEWALLE

where

⎡ αi (t) = cos−1 ⎣

PHYSICAL REVIEW E 85, 051307 (2012)

⎤

1  1 ( vt−1 · vt ) − ⎦ . 2 2

(6)

v∈{ n,t,s }

In this formula, n, t, and s are the vectors of the local base for the particle i. After 500 taps, the grain mobility μr reaches nearly a steady state. Figure 5 presents this residual mobility as a function of φ. The rotational mobility is found to vanish when φ is reduced. This is a clear signature of grain interlocking. The relevant fundamental question is whether or not the interlocking of neighboring grains confers more stability to the random assembly. One would believe that more contacts would divide contact forces into components, inducing some fragility of the assembly. The probability distribution functions (PDF) of the normal contact forces are presented in the top graphic of Fig. 6. As observed in the spherical case [17], each contact force distribution follows an exponential law f (x) = λ exp(−xλ),

(7)

its neighbors. This average contact force is seen to increase when the morphology deviates from the sphere. The difference between both states at t = 0 and t = 500 reveals the evolution of the force network during the compaction process. Large C/λ ratios indicate that nonconvex grains are more “pressed” than spheres, therefore reducing the possibility to move into structure. V. CONCLUSIONS

In summary, the compaction process for aspherical grains has been studied. The nonconvex shape of the particles has a limited influence on the characteristic time of the compaction. However, grain interlocking and multiple contact are developing in the assembly as the grain shape is modified towards low sphericities. It has been shown that the force network is deeply affected by the grain shape and the occurrence of multiple contacts. Nonconvex grains lead to low density packings, which exhibit more stability than sphere packing.

where x is the dimensionless contact force N/mg. Both the mean and the standard deviation of f are given by λ−1 . The evolution of the ratio C/λ as a function of the sphericity at the initial and final states is presented in the bottom plot of Fig. 6. This ratio represents the total force exerted onto a grain from

We thank E. Opsomer for fruitful discussions. This work has been supported by the T-REX Morecar project (Feder, Wallonia).

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ACKNOWLEDGMENTS

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