DEM Response of Sand With Moment Law

7/29/2019 DEM Response of Sand With Moment Law

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Analysis of the mechanical response of asand at grain-level with DEM-MTL

N. Belheine, F.V. Donz & F. DarveLaboratoire Sols, Solides, Structures et risques, Grenoble, France

[email protected]@hmg.inpg.fr

[email protected]

ABSTRACT

Particle rotations are known to have a dominant influence on the behavior of granular materials,

especially when the particles are circular or spherical. The definition of the relative displacement can

be used to decompose the motion into separate modes: incremental translational and rotationalmovements. The definition of the rotational motion allows us to identify four modes. Sliding and

rolling are the two principal local deformation mechanisms. This paper presents the results of a series

of numerical simulations conducted to examine the effect of the rolling and sliding mechanisms on the

macroscopic response. The finding results aggress well with experimental results. Comparisons are

presented between models with freely rolling (without Moment Transfer Law) and with incorporating

strongly rolling resistance (MTL). It was observed that the number of rolling contacts is more than the

sliding contacts, and rolling becomes more pronounced for the free rolling case. Thus the DEM with

MTL attenuate the particle rotation and has a strong effect upon the relative proportion ofsliding and

rolling between particles, and consequently upon the macroscopicstrength of the granular assembly.

INTRODUCTION

In granular soils, each particle can move against neighboring particles by sliding and/or rolling at

contact points. The classical microscopic theories of strength and dilatancy of granular media, were

emphasizing the influence of frictional sliding (Newland & Alley 1957, Rowe 1962, Horne 1965,

1969, Oda 1972, Shodja et al 2003), and neglected the effects of particle rolling. In fact, there exist

some experimental results contracting this. Skinner (1969) for example, observed that as the inter-

particle friction increases, rolling become dominant. This was later confirmed by Oda et al (1982) by

showing experimentally- during biaxial compression tests on photo-elastically sensitive oval cross-

sectional rods-, the effective presence of particle rolling as a dominant microscopic deformation

mechanism. They also concluded that rolling produces a softening effect and that particle rolling

partially negate the strengthening effect of inter-granular friction (Kuhn et al 2002, 2004). Oda et al(1982) observations also agree with the results obtained by Bardet and Proubet (Bardet et al 1994),

who numerically examined particle rotation within the shear band that forms during the biaxial

compression of two dimensional assemblies of circular cross-sectional grains. Because of the great

difficulty in measuring particle rotation in the geo-laboratory even with such advanced techniques as

the stereo-photogrammetric technique (Jiang et al 2005, Shodja et al 2003) and particle image

velocimetry, mechanisms of rolling and sliding contacts and their relation with dilatancy have not

been comprehensively investigated, and many aspects of these phenomena and the contributing

parameters remain unresolved. Since the pioneering work of Cundall and strack, the DEM has been

widely accepted to analyze the mechanical characteristics of granular material (Akke et al

2004).

The main objective of this paper is to study the contact mechanisms that include the particle rolling

and its effects on the macroscopic response of granular materials using Discrete Element Methodwith Moment Transfer Law ( DEM with MTL) (Plassiard et al 2007, Belheine et al 2008).


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Firstly, we develop a vocabulary and definitions for the various forms of inter-particle movements.

The simple term rolling admits numerous definitions, and the choice of one or another definitiondepends upon the contact mechanics (Kuhn et al 2002, 2004). The definition of rolling can be used to

decompose the motions of the two particles into separate modes: Rolling with sliding where the

rotation velocity has the same signs (2snde

mode), Pure rolling which the rotation velocity has the

opposite signs (3rd

mode, PR), Rolling with sliding where the rotation velocity has the opposite signs

(3rd mode, RS) and Sliding (4th mode) with one of the two spheres moves. The direction (sign) depends

on whether the path of the contact is clockwise or counterclock wise. Secondly, we present the results

of numerical simulations of triaxial loading with different cases. Several aspects of the effects of

rolling are examined. Particle rolling appears to be a dominant micro-mechanism between particle

pairs, and rolling is likely responsible for reducing the strength and stiffness of particle assemblies.

Particle rolling and contact couples control dilatancy, and the magnitude of the peak and residual shear

strength, that means that different deformation and collapse mechanisms observed at macro level can

be explained by related structural changes at micro level.

ROLLING AND SLIDING MECHANISMS AT CONTACTS

Particle rotations can have an important influence on the mechanical behaviour of granular materials,

especially in those composed with circular or spherical particles. The change in overall applied stress

involves creation and loss of contacts as well as restructuring by means of rolling and sliding contacts.

Figure 1 depicts two particles withA and B as their centres, which are initially in contact at pointC.The contact is assumed to be point-like, the contact area is negligibly small in comparison with thedisplacement. The particles undergo the incremental translational and rotational movements. Let

Ax and Bx denote the position vectors of the centres in the global set of axes, while their rotations are

given by A and B . Their rotation velocities are A and B and Ad and Bd denote the

incremental rotations (in radians) and their incremental translational are Adu and Bdu which take

place during the time increment dt. During a time step, the two spheres with radii Ar and Br , areassumed to remain in contact. Let n and 'n be two unit vectors at time increment dtwhich are normal

to the common surface at contact point C and 'C respectively. The new position of the centres

'A and 'B are defined by the vectors Ax' and B'x . These deformations are produced by the relative

motions of the two particles.

The relative position of the points A and B , compared to the new contact point 'C denoted by

MC' A and MC' B respectively, are given by:

=

=

+=+=

.dt.dn,drdu

ndrduMCCC'MC'

AA

AAA

AAAAAAA

=

=

+=+=

..dtd

,ndrdu

ndrduMCCC'MC'

BB

BBB

BBBBBBB

From Eqs. (1) and Eqs. (2), the relative displacement vector, denoted rdU , can be expressed

in terms of the relative position of points AM and BM , as follows:

(2)

(1)


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( ) ( )( ) ( )2

n.dtrn.dtrnnrnnr

2

MCMCdU BBAA

'

B

'

AB

'

A

'

r

+=

+=

The motion rdU is composed by the incremental translational and rotational movements.

According to the translational movement, the normal n and n exchanges their orientation (see

figure 2a), the translational velocity of the two points 1C and 2C is different to zero, but therotational velocity is null and the magnitude of rdU is defined by:

( ) ( )( )

=

=

0V

0V2

nnrnnrdU

rot

rel

'

B

'

Atrans_slid

r

The quantitytrans_slid

rdU is termed 1st

mode, pure sliding without rotations.

Generally, Ar Br and it is reasonable to consider that dUr is used to define the rolling part.

A second form of motion is based upon a rotational path of the contact points while the particles

move. The direction (sign) of an arc such as 1CC depends on whether the path of the contact from

C to 1C is clockwise or counterclockwise. Based on the relative position of the points C , 1C

and 2C , there are several cases to examine.

The first case is the rolling coupled with sliding which the rotation velocity has the same signs (see

figure 2b). In this case points 1C and 2C are located in the opposite sides of point C .The magnitude

of rdU is determined by the following expression:

( )

=

=

)()(

..__

BA

BBAAslidroll

r

BA

signsign

ndtrndtrdU

and

2

zerotodifferentare

2

The quantity2__slidroll

rdU is termed 2nd

mode, rolling with sliding which the rotation velocity has the

same signs.A second form of rotation is the rolling coupled with sliding which the rotation velocity has the

opposite signs and 1CC is greater than 2CC (see figure 2c). In this case points 1C and 2C are

located in the same side of point C .The magnitude of rdU is determined by the following

expression:

( )

=

+=

>

)(sign)(sign2

n.dtrn.dtrdU

zerotodifferentareand

CCCC

BA

BBAA3_slid_roll

r

BA

21

The motion3__slidroll

rdU can be separated into two motions. Firstly, the pure rolling which 1CC and

2CC are equal (see figure 2d),3__slidroll

rdU is quantified as:

(5)

(4)

(6)

(3)


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=

=

)(sign)(sign

n.dtrdU

equalareCCandCC

BA

AA

A3_slid_roll

r

'

2

'"

1

'

Secondly, we deduce the remainder between3__slidroll

rdU andAslidroll

rdU3__

. We obtain

Bslidroll

rdU3__

. This is the rolling with sliding which the rotation velocity has the opposite signs (see

figure 2e).

=

=

=

)(sign)(sign

dUdUdU

CCCCCC

BA

A3_slid_roll

r

3_slid_roll

r

B3_slid_roll

r

11

'

1

'

1

"

1

Also, it exist another type of movement, sliding with one of two spheres moves and the other is fixed

(see figure 2f). The motion is defined by:

=

=

=

0,02

n.dtrdU

0CC,0CC

BA

BB4_slid_roll

r

'

2

'

1

Figure 1. Kinematics of spheres in contact at time t and t+dt

G

A

B

C

B

A

dB

At time t +dt

dA

B

A

rBB

A

nG

C

At time t

rA

x

x

Figure 2. Different cases of motion.

x

z

o1

o2

C

n

C'1

C'2

b)

o1

o2

C

n

C'1

C'2

1

2

c

o1

o2

C

n

C"1

C'2

d

o1

o2

C

n

C'1 C"1

e

o1

o2

C

n

C'1

f

o

o2

C1

C2C

nn

n

a)

(7)

(8)

(9)


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MACROSCOPIC AND MICROSCOPIC NUMERICAL

RESPONSES IN TERM OF ROLLING AND SLIDING

CONTACTS

The particulate nature of soils is directly responsible for their complex overall behaviour. Sands

consist of particles with different sizes and shapes which interact with each other through contactforces (both normal and tangential) at the points of contact. Considering the particles essentially

incompressible, deformation of the granular assembly occurs as the particles translate; slip and/or roll

and either form or break contacts with neighbouring particles to define a new microstructure. The

result is an uneven distribution of contact forces that manifests in the form of complex macroscopic

material behaviour such as permanent deformation, anisotropy and localized instabilities. In this study

we propose to examine the effect of particle-to-particle rotation on the macroscopic response.

In contrast to (Manna and Hermann 1991), Oda et al (1982) in their experiments on plastic rods,

concluded that the particle rotations induced not only rolling but also sliding contacts with their

neighbours and that particle rolling indeed a dominant microscopic features, especially in the presence

of inter-particle friction. This section characterizes rolling and sliding in 3D granular material. In

section 2 the motion is decoupled into different cases. We are interested of the rotation mode 2snd

, 3rd

and 4th.

Figure 3. Microstructure of the assembly after deformation and macroscopic

Microscopic deformationsMacroscopic response evaluated from boundaryconditions

Initial state, H0, V0Deformed state, H, V

Macroscopic deformation


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On the basis of the lengths and signs of arcs 21 CCandCC which represent the displacement of the

contact pointC , the nature of the relative displacement of two contacting particles is quantitativelyexpressed by introducing the ratio SRof an individual contact as

( )21

21

z,y,xCC,CCmax

CCCCSR

+=

According to this definition, SR always falls in the range 0 to 2.

Moreover, for dominantly rolling contacts with sliding, where 1CC and 2CC are of same sign, SR

falls in the range 1 and 2 (2snd

mode, roll_slid_2). For pure rolling contacts where 1CC and 2CC are

of opposite sign and 21 CCCC = (3rd

mode, roll_slid_3A). Hence 0SR = . In dominantly rolling

contacts with sliding, 1CC and 2CC are of the opposite sign and 1SR0


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Figure 5.Frequency distribution for SRz ratio at %6and%1a = . a) DEM without MTL,

b) DEM with strong MTL. c) DEM with MTL for calibrated model.

b)

a

c

Figure 4. Influence of the Moment Transfer Law on the stress-strain behaviour.


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CONCLUSION

Based on the numerical simulations of the triaxial tests, analyses were made on the effect of the

particle rotations on the macroscopic response of granular material. The finding results are in

agreement with the experimental results of Oda et al (1982). We have identified four modes of

rotational motion: rolling with sliding where the rotation velocity has the same signs, Pure rolling

which the rotation velocity has the opposite signs, rolling with sliding where the rotation velocity has

the opposite signs and Sliding with one of the two spheres moves. These movements are quantified by

the ratio SR. The contacting particles is classified as predominantly rolling, predominantly sliding or

pure rolling. To reveal the degree of local rearrangement by sliding and rolling, two cases are studied.

Mechanisms for free rolling case (DEM without MTL) and for the incorporating rolling resistance

case (DEM with MTL). It was observed that the number of rolling contacts is more than the slidingcontacts, and rolling becomes more pronounced for the free rolling case. Thus the DEM with MTL

attenuate the particle rotation and has a strong effect upon the relative proportion ofsliding and rolling

between particles, and consequently upon the macroscopicstrength of the granular assembly.

REFERENCES

Akke, Suiker, S. J. Norman A. Fleck. Frictional Collapse of granular assemblies. Journal of Applied

Mechanics 2004; 71(3):350-358.

Bardet, JP. Observations on the effects of particles rotations on the failure of idealised granular

materials. Mechanics of Materials 1994; 18:159-182.

Belheine, N., J.P., Plassiard, F.V. Donze, F. Darve, A. Seridi, Numerical Simulation of Drained

Triaxial Test Using 3D Discrete Element Modeling, Computers and Geotechnics, DOI

10.1016/j.compgeo.2008.02.003, 2008.

Horne, M.R. (1965). The behaviour of an assembly of rotund, rigid, cohensionless particles (I and II),

Proc. R. soc. London A286, 62.

Horne, M.R. (1969). The behaviour of an assembly of rotund, rigid, cohesionless particles (III), Proc.R. Soc. London A 310, 21.

Figure 6.Variation ofs

r

N

Nratio versus axial strain. a) DEM without MTL, b) DEM with strong MTL.

c) DEM with MTL for calibrated model.

c)

a) b


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Jiang, MJ. Haris.D, H-S. A novel discrete model for granular material incorporating rolling resistance.

Comput Geotech 2005; 32(5):340-357.

Kuhn, M. R. Particle rotations in granular materials. 15th

ASCE eng. Mec. Conf. june 2002. Columbia

University, New York, NY.

Kuhn, M. R, Bagi, K. Contact rolling and deformation in granular media. Int. J. Sol. Struc 2004;

41:5793-5820.

Newland, P.L, Allely BH. Volume changes in drained triaxial tests on granular materials.

Gotechnique 1957; 7(1): 17-34.

Oda, M. (1972). The mechanism of fabric changes during compressional deformation of sand, soils

found. 12(2), 1.

Oda, M, Konishi J, Nemat-Nasser S. Experimental micromechanical evolution of strength of granular

materials: effects of particle rolling. Mechanics of materials 1982; 1(4):269-283.

Oda, M, Iwashita, K, and T. Kakiuchi. Importance oa particle roation in the mechanics of granular

materials. In R. P. Behringer and J. T. Jenkins, editors, Powders & Grains 97, pages 207-210.

A. A. Balkema, Roterdam, Ntherlands, 1997.

Plassiard, JP. Belheine, N. Donze, F-V. Calibration procedure for spherical discrete element using a

local moment Law. Discret Element Methods (DEM) 07, Brisbane, Australia, August 27-29,

2007.

Rowe, P.W. The stress-dilatancy relation for static equilibrium of an assembly of particles in contact

Proceeding of the Royale Society, London,Sries A 1962; 269:500-527.

Shodja, HM, Nezami EG. A micromechanical study of rolling and sliding contacts in assemblies of

oval granules. Int. J. Numer. Anal. Meth. Geomech., 2003; 27: 403 424.

Skinner, AE. A note on the influence of interparticle friction on the shearing strength of a random

assembly of spherical particles. Gotechnique 1969; 19:150-157.


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DEM Response of Sand With Moment Law