ball-bearing effect on shear behavior of binary granular...

BALL-BEARING EFFECT ON SHEAR BEHAVIOR

OF BINARY GRANULAR MIXTURE

Takao UEDA1, Takashi MATSUSHIMA 2 and Yasuo YAMADA3

1Member of JSCE, Graduate Student, Dept. of Eng. Mech. and Energy, University of Tsukuba (1-1-1, Tennodai, Tsukuba, Ibaraki 305-8573, Japan)

E-mail: [email protected] 2Member of JSCE, Associate Professor, Dept. of Eng. Mech. and Energy, University of Tsukuba

(1-1-1, Tennodai, Tsukuba, Ibaraki 305-8573, Japan) E-mail: [email protected]

3Member of JSCE, Professor, Dept. of Eng. Mech. and Energy, University of Tsukuba (1-1-1, Tennodai, Tsukuba, Ibaraki 305-8573, Japan)

E-mail: [email protected]

The shear behavior of binary granular mixtures was studied using the 2D discrete element method (DEM). Various specimens were prepared using different volume fractions and shapes i.e., disk and peanut, of small particles. The analyses showed that the average particle rotation of both small and large particles is equivalent to the continuum rotation field. However, when a specimen contains small disks of volume fractions in the range 5% - 15%, the average particle rotation is found to be much larger than the continuum rotation that accompanies the reduction in both shear resistance and thickness of the shear band. In such specimens, it was observed that many small particles were sandwiched by two large particles and under-went considerable rotation. In this paper, this irregular phenomenon is referred to as the “ball-bearing ef-fect.” Key Words: ball-bearing effect, shear band thickness, discrete element method (DEM), binary mixture

1. INTRODUCTION

It is well known that in the softening process under shear, the deformation of granular materials localizes and concentrates in an area of a specific thickness called shear band. The shear band is a unique de-formation characteristic of granular materials and its thickness has been the subject of extensive research. It is important to understand the actual phenomenon such as the effect of grain size on the shear behavior. For example, the bearing resistance of coarse soil is considered to be the same as that of fine soil ac-cording to classical soil mechanics, which does not consider the shear band thickness. However, it is well known that the bearing capacity of coarse soil is larger than that of fine soil. Experimental studies have indicated that the shear band thickness of poorly graded sand is 10–20 times larger than its mean di-ameter1)-3).

Binary mixtures, a mixture of granular materials of two sizes, have attracted attention in the geotechnical field for investigating the mechanical behavior of gravel–soil structures such as rockfill dams. Many

experimental studies have indicated that the shear strength of a gravel–soil mixture is mostly controlled by gravel when the gravel content is high, whereas it is controlled by soil when the gravel content is low4)-6). Furthermore, similar thresholds were found for other mechanical characteristics such as com-pressive strength and fall cone penetration resistance in gravel–clay mixtures7). However, despite many experimental achievements, only a few theoretical interpretations of the contribution of each sized par-ticles in binary mixtures have been presented8).

Lupini et al. presented a detailed interpretation of shear band development in soil–clay mixtures with varying clay content9). However, observation of the shear band is difficult and may associate with non-negligible uncertainty. A numerical photo-grammetry technique for measuring the movement of each grain including rotation was proposed, but this technique was only employed to observe the assem-blies of specially prepared cylinders10).

The discrete element method (DEM) is very useful for simulating the movement and rotation of each grain11). Previous research using DEM has revealed

Journal of Japan Society of Civil Engineers, Ser.A2 (Applied Mechanics (AM)), Vol. 68, No. 1, 1-9, 2012.

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Table 1 Simulation cases.

Case Packing state

(Relative density*)

Particle shape Particle size ratio, Content of small particles, WS (%)

Large Small

(Elongation ratio)

DD Dense (1.0) Disk Disk (1.0) 10 0, 5, 10, 15, 20, 30, 50, and 100

DPA Dense (1.0) Disk Type A (0.67) 10 0, 5, 10, 15, 20, 30, 50, and 100

LPA Loose (0.0–0.07) Disk Type A (0.67) 10 0, 5, 10, 15, 20, 30, 50, and 100

DPB Dense (1.0) Disk Type B (0.80) 10 5, 10, 15, and 100

* Relative density was calculated using the maximum and minimum densities obtained by the packing simulations with interpar-ticle friction angle of zero and 45 degrees, respectively18).

that grain rotation plays an important role in shear band development12) and the grain rotation of poorly graded granular material shows good agreement with the continuum rotation of the material13),14). For poorly graded granular materials, such as those de-scribed above, the shear band characteristics have been researched experimentally, numerically, and theoretically. However, we still do not have much knowledge about the shear band nature of binary mixtures.

We considered the simplest binary granular mix-ture, which is composed of mono-sized large and small particles, whose grain size distribution is characterized by only two parameters, i.e., the vol-ume fraction of the small particles WS and the particle size ratio . We selected these parameters by as-suming that the particle size effect due to gravity, viscosity of fines, and crushability of gravels is neg-ligible. WS and are derived from the following equations, where MS and MA are the gross mass of small particles and all particles, and DL and DS are the diameters of the large particles and small particles, respectively:

,100A

SS M

MW (1)

.S

L

D

D (2)

The ball-bearing effect in which the small particles are sandwiched by large particles and rotate and be-have like lubricating agents is well known in the field of concrete technology. Fine spherical coal called “fly ash” is mixed with concrete to increase its workability without increasing the water–cement ratioe.g.,15). We believe that the ball-bearing effect also affects the shear behavior of the binary mixture, de-pending on WS. In this study, a series of DEM shear simulations was conducted on three types of binary mixtures with different WS to investigate particle rotation, continuum rotation, and the shear band thickness. Furthermore, the influence of ball-bearing on these mechanical characteristics is discussed in this paper.

2. SIMULATION PROCEDURE Table 1 lists the simulation cases, where Types A and B are two types of peanut-shaped particles con-sisting of two clamped disks. Their elongation ratios (i.e., width(w) to length(l) ratios) are 0.67 and 0.80, respectively, as depicted in Fig. 1. Type A is more elongated and Type B is more round compared with well-known sands such as Toyoura and Monterey sand, whose elongation ratios are 0.716) and 0.7217), respectively. The particle size ratio was set as 10 in all cases and the content of small particles WS was varied from 0 to 100 %, as shown in Table 1. Note that of DPA and DPB are calculated using = DL / w. Table 2 lists the physical parameters of the parti-cles determined on the basis of previous studiese.g.,18). Note that we conducted additional shearing simula-tions of disk assemblies with an interparticle friction angle of 45 degrees to examine its influence on the shear deformation. No significant differences were found with respect to the shear band thickness and particle rotation, whereas shear resistance was larger.

Among the simulated cases, DD with 10% and 50% WS are referred to as DD10 and DD50, respec-tively, and DPA with 10% WS and LPA with 10% WS are called DPA10 and LPA10, respectively.

A series of shear simulations on binary granular mixtures was performed using a 2D DEM pro-gram19). The simulations were conducted using the following steps: (1) Approximately 4,000 large and small particles were generated according to and WS prescribed in Table 1.

Fig. 1 Peanut-shaped particle; (a) Type A, whose elongation ratio (w / l) is 0.67; (b) Type B, whose w / l is 0.80.


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(2) The assembled densely packed and loosely packed samples were isotropically compressed with interparticle friction angles of zero and 27 degrees, respectively. Note that the interparticle friction angle is an essential parameter that indicates grain surface roughness. In general, an assembly with a higher interparticle friction angle shows higher shear strength. Maximum and minimum densities were obtained for assemblies with interparticle friction angles of zero and 45 degrees, respectively18). (3) The particles in the top and bottom layers were unitized to form boundary plates and the right and left side of the assemblies were set as periodical boundaries. In particular, approximately 1/4 of the particles on the left side were kept as periodic parti-cles throughout the calculation. The particles on the right side were stressed by the particles on the left side and vice versa, as if the assembly was sand-wiched by the duplicated assemblies from the right and left sides. (4) The top plate was displaced in the right or left direction with a constant speed under a constant confining pressure (100 N/m) and the particle rota-tion was measured. The angle of shear resistance was calculated by the following equations:

j

iij l

f

V

1 , (3)

2211

22112

222111sin

. (4)

where (f1, f2) and (l1, l2) are the horizontal and vertical components of the contact force and the branch vector, respectively, (11, 12, 21, 22) are the stress components, and V is the total volume of the parti-

cles. The values of f and l were obtained by DEM. To measure the continuum rotation and the shear

band thickness, additional steps were conducted as follows: (5) The samples were horizontally divided into 10 layers, as shown in Fig. 2. This process is based on the assumption that the shear band develops in a horizontal direction under this condition. The effects of the number of layers were verified beforehand. For 3–20 layers, the accuracy of the local shear defor-mation increased as the number of layers increased. In contrast, the dispersion of the particle rotations increased because the number of particles in the layer decreased. Therefore, the number of layers was de-termined to be 10 to ensure balance between these two effects. (6) The continuum rotation n (where n = 1, 2, 3 … 10) of each layer was obtained by calculating the deformation of each layer (Fig. 2). (7) The particle rotation was measured layer by layer and compared with the continuum rotation. (8) The continuum rotation under the assumed uni-form deformation c was calculated (Fig. 2). (9) The shear band thickness was defined as the total thickness of layers sandwiched between the topmost and lowest layer, each having a value of n greater than c (Fig. 2). The shear band was observed to be continuous layers of n > c, except for DD50. The thickness of each layer was smaller than the large particle size in DD50, which might affect the calcu-lated value of n. However, the development of sep-arate shear bands was not observed in this study. 3. RESULTS

Figure 3 shows an example of the shear re-

sistance–strain curve of DD50, as an example, which shows the formation of a clear peak to reach the re-sidual state. The deformation modes of A, B, and C in Fig. 3, whose shear strains S are 0.038, 0.11, and 0.23, respectively, are shown in Fig. 4. After the assembly deforms uniformly (Fig. 4 A), a shear band begins to develop as the stress state of the assembly shifts toward the residual state (Fig. 4 B, C). The direction of the principal stress does not change in the residual state, and the influence of the change in the principal stress direction on shear band development is limited at this stage. (1) Particle rotation versus continuum rotation

Figure 5 schematically illustrates the relationship between the particle rotation and the continuum ro-tation on the basis of the finite deformation theory (Fig. 5(a)) and the ball-bearing effect (Fig. 5(b)). According to the finite deformation theory, the shear

Table 2 Simulation parameters.

Parameters Value

Density (kg/m2) 25.0 Normal spring coefficient (N/m) 1.0*108 Tangential spring coefficient (N/m) 2.5*107 Normal damping (N sec/m) 10.0 Tangential damping (N sec/m) 5.0 Inter particle friction angle during shear (degrees) 27

Fig. 2 Shear deformation and continuum rotation.


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deformation of an assembly is divided into two parts: rigid rotation and stretching (Fig. 5 (a)). Because of the rigid rotation, all particles rotate by some amount

(Fig. 5 (a) B). However, in the stretching phase, the average particle rotation becomes zero (Fig. 5 (a) C). As a result, the average particle rotation a is iden-tical to the continuum rotation (= rigid rotation ) of the assembly in the finite deformation theory. The particle rotation of the poorly graded granular mate-rials agrees with the continuum rotation14).

In contrast, in the ball-bearing condition in which the small particles are sandwiched by two large par-ticles, the shear deformation is caused by the small particle rotation (Fig. 5 (b)). The particle rotation b becomes larger than the continuum rotation when b > . In this paper, therefore, the rotation of the large particles L and the small particles S were compared with the continuum rotation to clarify the ball-bearing effect.

Figures 6 (a)–(d) show the shear deformations and Figs. 6 (e)–(f) compare the continuum rotation and the mean particle rotation of DD10, DD50, DPA10, and LPA10, respectively, in their residual state. Figure 6 (e) depicts that the large particle ro-tation shows good agreement with the continuum rotation, whereas the mean small particle rotation is

(a) Deformation based on the finite deformation theory: A)

itnitial state, B) rigid rotation and C) stretching.

(b) Deformation due to the ball-bearing effect: A) at initial

state and B) after deformation. Fig. 5 Deformation sketch of particle assembly (a) based on

the finite deformation theory and (b) due to the ball-bearing effect.

0.00 0.05 0.10 0.15 0.20 0.25

5

10

15

20

25

C : S = 0.23

B : S = 0.11

The

ang

le o

f sh

ear

resi

stan

ce,

(d

egre

es)

Shear strain, S

DD50A : S = 0.038

Fig. 3 Shear resistance-strain curve of DD50.

A: S = 0.038

B: S = 0.11

C: S = 0.23 Fig. 4 Deformation modes of DD50 whose shear strains S

are 0.038, 0.11 and 0.23, respectively.


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significantly different from the continuum rotations. This is because the small particles sandwiched by large particles rotate excessively like ball bearings, as shown in the close-up image of DD10 (Fig. 7). In Fig. 7, black denotes clockwise rotations greater than 180 degrees and gray denotes counterclockwise ro-

tations greater than 180 degree. Figure 6 (f) shows the particle rotations of both

large and small particles, which agree with the con-tinuum rotations of DD50. As shown in the close-up image of DD50 (Fig. 8), a few small particles rotate like ball bearings. This is mainly because small

Fig. 6 (a), (b), (c), (d) The images of particle assemblies in the residual state for DD10, DD50, DPA10, and LPA10, respectively.

(e), (f), (g), (h) The continuum rotations and the particle rotations for DD10, DD50, DPA10, and LPA10, respectively. The dashed lines show the continuum rotation under the assumed uniform deformation c.


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DD50 particles have less flexibility than DD10 ow-ing to more number of contact points.

Figure 6 (g) shows that the particle rotations agree well with the continuum rotations of DPA10 even though its WS is the same as that of DD10. Only a few small peanut-shaped particles rotate excessively, as shown in the close-up image of DPA10 (Fig. 9). The ball-bearing effect of DPA10 is considered to be less compared with that of DD10 because of the irregular shape of small particles.

Figure 6 (h) shows that the particle rotations generally agree with the continuum rotation. How-ever, the small particle rotation is somewhat larger than the continuum rotation. This is mainly because the number of contact points of small particles of LPA10 is less than that of DPA10 owing to its loose packing state.

For quantitatively evaluating the ball-bearing ef-fect, in Fig. 10 the average small particle rotations of

DD10 and DD50 are compared with respect to the coordination number and those of the ball-bearing particles are separately plotted as stars. Note that in this paper, the small particles that are in contact with only two large particles and maintain contact during each simulation term (corresponding to 0.15 s) are defined as the ball-bearing particles. In general, the rotations of the ball-bearing particles are considera-bly larger than those of other small particles. In DD10 and DD50, the number of ball-bearing parti-cles account for 4.7% and 0.13% of the total number of particles, respectively. The ball-bearing effect in DD50 is limited because the proportion of ball-bearing particles is very small compared with DD10 despite their average rotations are very large. Figure 11 compares the particle rotation and the continuum rotation for various values of WS. Small DD particles whose WS values range from 5% to 15% exhibit clear dominant ball-bearing effects compared with the other cases.

(2) Shear band thickness

A new parameter Dm, a type of mean diameter, was introduced to evaluate the shear band thickness of a binary mixture. Figure 12 is a schematic illustration

Fig. 7 The close-up image of DD10. Black denotes clock-

wise rotations greater than 180 degrees and gray denotes counterclockwise rotations greater than 180 degrees.

Fig. 8 The close-up image of DD50.

Fig. 9 The close-up image of DPA10.

1 2 3 4 5 6 7

0

20

40

60

80

100

120

140

160

1800.13%

Ave

rage

sm

all p

arti

cle

rota

tion

(de

gree

s)

Coordination number

Small particle in DD10 Ball-bearing particle in DD10 Small particle in DD50 Ball-bearing particle in DD50 Continuum rotation

4.7%

Ball-bearing particle rate

Fig. 10 Average small particle rotation vs. coordination number.

0 10 20 30 40 50 60 70 80 90 100

-5

0

5

10

15

20

25

Par

ticl

e ro

tati

on -

C

onti

nuum

rot

atio

n (d

egre

es)

Content of small particles, WS (%)

Large particle of DD/ Large / Small particle of DP

A

/ Large / Small particle of LPA

/ Large particle of DPB

Small particle of DD

Fig. 11 The difference between the absolute values of the particle rotation and the continuum rotation of DD, DPA, and LPA.


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defining Dm: Fig. 12(a) shows the particles in a bi-nary mixture color coded according to their size; Fig. 12(b) denotes the ratio of the areas of the small par-ticles to large particles WS:(100 − WS); and Fig. 12(c) shows that the large and small particles are stacked in a single column corresponding to the ratio WS:(100 − WS), where Nb

L and NbS denote the number of stacked

large and small particles, respectively; Dm is defined as a type of mean diameter, as follows:

bS

bL

bSS

bLL

m NN

NDNDD

, (5)

where DL and DS are the mean diameters of the large and small particles, respectively.

Figure 13 compares the shear band thickness of

DD, DPA, and DPB. The shear band becomes thinner with increasing WS and no significant difference is observed in the thickness in this simulation. Figure 14 shows the normalized shear band thickness cal-culated by dividing the shear band thickness by Dm. With the exception of the DD10 case, the normalized shear bands range from 8 to 16, i.e., the shear band thickness is 8–16 times larger than the mean diame-ter, which is consistent with results obtained in pre-vious studies1)-3). In contrast, the shear band thickness of DD10 is much smaller than the average thickness of DD.

Figure 15 shows the frequency distributions of the coordination numbers of DD10 in the initial and residual state, and Fig. 16 shows those of DD50 for comparison. The frequency distributions of the initial state of DD10 and DD50 are similar and those of the residual state are very different. The frequency of coordination numbers having a value of one or two in the residual state is more in DD10 than in DD50. In the former case, the small particles sandwiched by two large particles are believed to rotate like a ball bearing, and they enable large deformations associ-ated with the movement of the large particles in contact with them, as illustrated in Fig. 5 (b).

Figure 17 compares the number of small

Fig. 12 The overview of the proposed mean diameter.

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

160

DD DP

A

DPBDP

B

DPA

DD

Shea

r ba

nd th

ickn

ess

(cm

)


Fig. 13 The shear band thickness of DD and DPA.

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

DD DP

A

DPB

DPB

DPA

DD

Nor

mal

ized

she

ar b

and

thic

knes

s


Fig. 14 The normalized shear band thickness of DD and

DPA.

0 1 2 3 4 5 6 7 8 9 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Residual state

Initial state

Fre

quen

cy

Coordination number Fig. 15 Frequency distribution of the coordination number of

DD10.

0 1 2 3 4 5 6 7 8 9 100.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Residual state

Initial state

Fre

quen

cy

Coordination number Fig. 16 Frequency distribution of the coordination number of

DD50.


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ball-bearing particles per large particle of DD in the initial state with those in the residual state. In all cases, the number of small ball-bearing particles increases during shear compared with that in the initial state because of the positive dilatancy during shear. At 10% WS, the small ball-bearing particles ratio was considerably larger than in other cases. This proves that the ball-bearing effect of small particles was most significant in this situation. Consequently, the thinner shear band of DD10 is considered to be significantly related to these small ball-bearing par-ticles.

(3) Shear strength

Figure 18 shows the angles of shear resistance for DD, DPA, and DPB in the peak and residual states. The peak shear strength of DD is almost constant irrespective of WS. In contrast, the peak shear strengths of DPA and DPB increase with increasing WS, which may be attributed to the particle shape effect of small clumped particles. The residual shear strength of DD decreases to some extent when WS is 10% compared with other values of WS. This implies that the residual shear strength is affected by the

ball-bearing effect: the small ball-bearing particles reduce the chance that large particles will shear. However, contrary to our expectations, the effect of ball bearing on the shear strength is limited. It is mainly because the small particles sandwiched by large particles hardly keep rotating for a considerable term but were easily pushed out because the large particles were round in this simulation. 4. CONCLUSION

A series of DEM simulations was conducted on various types of binary mixtures to evaluate the shear band thickness, the continuum rotation, and the par-ticle rotation. The following observations were ob-tained. 1) The particle rotation showed good agreement with the continuum rotation with the exception of the case involving densely packed binary disks, whose pro-portion of small particles is 10% (DD10). 2) A representative diameter was proposed to vali-date the shear band thickness. Most shear band thicknesses were 8–16 times larger than the repre-sentative diameter. In contrast, the shear band thickness of DD10 was much smaller than that ob-tained in other cases. 3) The residual shear strength of DD10 was smaller than that of other cases to some extent. However, the effect of the ball bearing on the shear strength was limited under the simulation conditions used in this study. 4) Small particles sandwiched by two large particles may act like a ball bearing and cause significant de-formation that is associated with large particles. This mechanism is believed to be responsible for irregular particle rotation, thinner shear bands, and smaller residual shear strength of DD10. REFERENCES 1) Muhlhaus, H., B. and Vardoulakis., I.: The thickness of

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0 10 20 30 40 50

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Residual state

Initial state

The

num

ber

of th

e ba

ll-b

eari

ng

part

icle

s pe

r la

rge

part

icle


Fig. 17 The number of small ball-bearing particles per large particle.

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10

15

20

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30

35

40

45

res

DPB

res

DD

res

DPA

peak

DD

peak

DPB

/ Peak / residual shear resistance of DD/ Peak / residual shear resistance of DP

A

/ Peak / residual shear resistance of DPB

peak

DPA

The

ang

le o

f sh

ear

resi

stan

ce,

(de

gree

s)


Fig. 18 The peak shear strength and the residual shear

strength of DD, DPA, and DPB described using the angle of shear resistance.


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(Received January 31, 2011)


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ball-bearing effect on shear behavior of binary granular...

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