Structured Deformation Granular DEM

7/28/2019 Structured Deformation Granular DEM

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Structured deformation in granular materials

Matthew R. Kuhn 1

Department of Civil Engineering, University of Portland, 5000 N. Willamette Blvd., Portland, OR 97203, USA

Received 5 August 1998

Abstract

Microscale deformations are investigated in numerical DEM experiments of a large two dimensional assembly of

disks. The assembly was subjected to quasi-static biaxial loading at small to moderate strains. Deformations within

individual voids were computed from the relative motions of surrounding particles. Evolution of the local fabric was

measured in terms of void-based parameters, including eective void ratio, void cell valence, and shape-elongation of

the voids, all of which increased monotonically during loading. A direct correlation was measured between local void

shape and dilation, which accounts for the transition from compressive to dilatant behavior. Deformation was very

nonuniform at the microscale of individual voids. The predominant deformation structures were thin obliquely trending

bands of void cells within which slip deformation was most intense. These ``microbands'' appeared spontaneously

throughout the test, even at the start of loading. The microbands ranged in thickness between one and four particle

diameters. Unlike shear bands, the microbands were neither static nor persistent: they would emerge, move, and dis-

appear. Their orientation angle increased as deformation proceeded. Dilation was slightly larger within the microbands

than in the surrounding material. Void shapes within the microbands were somewhat elongated, with an elongationdirection that was related to the orientations of the microbands. Energy dissipation was concentrated within the

microbands, even at small strains. In a small cycle of loading and unloading, local uctuations in the elastic and plastic

slips occurred in opposite directions. No spatial relation was found between the deformation microbands and chains of

the most heavily loaded particles. Particle rotations were structured, with the most rapid rotations occurring within and

near microbands. The rotations tended to relieve sliding between most particles, but transferred the sliding to a few

contacts at which frictional slipping was most intense. 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Granular materials; Discrete element method; Localized deformation; Patterning; Shear bands; Voids; Microstructure;

Inhomogeneous material

1. Introduction

The purpose of this work is to experimentally

investigate deformation inhomogeneity in granular

materials at low to moderate strains. Both stress

and deformation within granular materials have

long been known to be nonuniform, particularly at

the microscale of particle groups. Nonhomogeneity

does not, however, imply randomness or disorder.

Early photoelastic and numerical experiments, for

example, revealed that stress is borne dispropor-

tionately by chains of particles (de Josselin de Jong

and Verruijt, 1969; Oda et al., 1980; Cundall et al.,

1982), and these force chains are the predominant

structure of internal force at a microscale. The

nonuniform but structured nature of deformation

is most vividly exhibited in the form of shear bands,

Mechanics of Materials 31 (1999) 407429

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which occur when a granular material has been

subjected to large shearing deformation.

We focus instead on nonuniform deformation

at low and moderate strains, well before the ap-

pearance of shear bands. The intent is to observe

and measure deformation during slow, quasi-static

loading at the microscale of, at most, a few particle

diameters. Two experimental approaches are pos-

sible at such a small scale:

1. Methods based upon particle chains, in which

deformation is viewed as a combination of

stretches and shears along the chains. This ap-

proach had its inception in early stress-dilatancy

studies (e.g., Horne, 1965) and has been recently

applied by Oda (1997) to the buckling of colum-

nar chains of particles within shear bands.2. Methods based on voids and the deformations

that occur within them. This approach is based

upon the graph theoretic work of Satake (see

Satake, 1992, 1993) and the methods that Bagi

(1996) developed for computing void deforma-

tions.

In this paper, the second method is applied to

numerical experiments on a large two-dimensional

assembly of circular disks. The experiments in-

volve the slow biaxial compression of the assembly

at pre-failure load levels. The methodology ofthese experiments is described in the next section.

In Section 3 we describe void cell geometrics and

their evolution during biaxial compression. We are

primarily interested, however, in the manner in

which local, nonhomogeneous deformation is dis-

tributed and organized. The dominant deforma-

tion structure appears to be thin microbands of slip

deformation, and these are described in Section 4.

We end with brief concluding observations.

The paper presents experimental observations,

and, as such, refrains from testing their confor-

mance with an encompassing constitutive theory.

The observations will, however, occasionally be

compared with the related experimental and the-

oretical work of other investigators.

1.1. Notation

Vectors and tensors will be represented by bold

letters, lower and upper case, respectively. Their

inner products are computed as

a b Y 1a

A B eqfqY 1b

with the associated norms

jaj a a1a2Y 2a

jAj A A1a2X 2b

A juxtaposed tensor and vector will represent

the conventional product

Ab eqq 3

and the cross and dyadic vector products

a b ijkjkY a b q 4

will occasionally be used.

2. Methods

2.1. Particle graphs and deformation measures

The topological association of particles within a

region e of a two-dimensional granular material

can be described by a planar graph, which parti-

tions e into a covering of polygonal micro-

domains ei

(Satake, 1992). In this paper, a particlegraph is employed in which the ei represent void

cells that are each surrounded by the branch vec-

tors of contacting particles (Fig. 1(a)). This parti-

tion is suited to measuring the average

deformation within individual voids. We modify

the particle graph to include only those particles

that share in the load-bearing framework of the

Fig. 1. Modied graph of particle arrangement.

408 M.R. Kuhn / Mechanics of Materials 31 (1999) 407429


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material, by disregarding all nonparticipating pen-

dant, island, peninsula, and isolated particles,

along with the branch vectors that would other-

wise connect them to the remaining particle graph

(Fig. 1(b)). Although such particles do not share in

supporting load, they are not entirely inert, and, as

deformation proceeds, may later be re-incorpo-

rated into the material's modied, load-bearing

particle graph.

The modied particle graph is represented by its

L void cells (faces), M contacts (edges or branch

vectors), and N particles (vertices). The overbars

designate the portion of the total M and N con-

tacts and particles that remain after the nonpartic-

ipating particles have been neglected. Superscripts i,

j, and k will be used as indices to represent voidcells, contacts, and particles, respectively.

The deformation rate within the material region

e will be expressed by the velocity gradient L, a

function of position x. In our experiments, L was

uniform on a macroscale but exhibited consider-

able variation at a microscale. We distinguish be-

tween the two scales by separately computing the

spatial average L within the entire particle assem-

bly and the average velocity gradients Li

within

each of the L void cells, such that

L 1

e

Li1

eiLiX 5

Bagi (1996) derived an exact expression for the

Li

of a triangular region in terms of the relative

velocities among its three nodes (vertices). The

principal assumption in her derivation is that ve-

locity varies linearly along the three edges of the

triangular region. By making a similar assump-

tion, the author extended Bagi's expression to a

polygonal void cell with mi edges (Kuhn, 1997):

Li

1

6ei

j1Yj2Pf0Y1YFFFYmi1g

mi

j1Yj2viYj1 biYj2 X 6

In this expression, vector viYj is the relative ve-

locity between two particles on edge j of the ith

void cell ei (Fig. 2(a)). That is, viYj is the relative

velocity viYk2 viYk1 between the two neighboringparticles k1 j and k2 j 1. The b vectorsgive the geometry of polygon i. Vector biYj is out-

wardly normal to edge j, with a magnitude equal

to the length of that edge (Fig. 2(c)). The num-

bering of nodes and edges in Eq. (6) begins with

zero to accommodate modulo arithmetic. This

expression is a linear combination of the m2 dyadic

products viYj1 biYj2 , which are weighted by the el-ements of an m m matrix m. We have chosen auniquely skew symmetric and circulant form of

m, which for a triangular region is given by

3

0 1 1

1 0 1

1 1 0PR QS

X 7

A recursive expression for the m matrices of a

general m-polygon is presented in Kuhn (1997),

and Table 1 gives the rst row of the m matrices

Fig. 2. Vectors associated with edges of the ith void cell.

M.R. Kuhn / Mechanics of Materials 31 (1999) 407429 409


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for triangles through hexagons. Because this form

ofm is circulant, the remaining rows are given as

mq m1Yq1 with the subscripts computed

modulo m. Eqs. (5) and (6) provide a means ofcomputing the average deformation of a planar

particle assembly from the relative movements of

neighboring particles. We use Eq. (6) to measure

deformation at the smallest scale that is physically

meaningful within individual void cells but

Eq. (6) will also aid in extracting the micro-de-

formation eects of various causative motions

(sliding, rolling, indentation, separation, etc.)

among neighboring particles. The two equations

are the compliment of similar expressions for av-

erage stress in terms of the contact forces among

particles (e.g., Christoersen et al., 1981).

Section 4 presents color plots of the local

deformations Li. These plots have a discontinuous,

patchwork appearance rather than continuous

gradations of color. The derivation of Eq. (6) as-

sumes that material velocities are continuous

within the void cells and vary linearly along their

edges. The velocity eld is therefore continuous,

although gradient L may be discontinuous. The

composite eld of local gradients Li

is less smooth,

however, as they represent spatial averages within

individual void cell subregions.

2.2. Biaxial testing

We tested a dense two-dimensional assembly of

4008 circular disks in biaxial compression by using

the numerical Discrete Element Method

(Fig. 3(a)). The method allows direct numerical

simulation of deformation processes in granular

materials. Each particle is an element that can

Table 1

First row of matrix mpq

Valence m q mod m

0 1 2 3 4 5

3 0 33

33

4 0 64

0 64

5 0 95

35

35

35

6 0 126

66

0 66

126

Fig. 3. Initial arrangement of 4008 particles (e22 0%).



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move and rotate to accommodate boundary dis-

placements, external forces, and the constraints

imposed by neighboring particles.

In our simulations, the collection of disk sizes

was randomly distributed over a fairly small range

of between 0X45h50 and 1X40h50, where h50 is themedian particle diameter. The material was ini-

tially compacted into a dense, random, and

isotropic arrangement, with an initial (two-

dimensional) void ratio of 0X179. The assemblywas initially square, with each side about 54h50wide. It was surrounded by two pairs of periodic

boundaries, which bestow a long-range transla-

tional (i.e., wallpaper group) symmetry (Cundall,

1988). As with rigid platens, these periodic

boundaries impose kinematic constraints on eachboundary particle. Periodic boundaries, however,

have the advantage of providing a more uniform

particle fabric throughout the assembly, which

would otherwise be disrupted by platens.

A simple force mechanism was employed be-

tween contacting particles. Linear normal and

tangential contact springs were assigned equal

stinesses, and slipping between particles would

occur whenever the contact friction coecient of

0X50 was attained. Unlike the recent model of

Iwashita and Oda (1998b), no resistance to rollingwas included at the contacts.

During the biaxial compression simulations,

the height of the assembly was reduced at a

constant rate, while the width was continually

adjusted to maintain a constant horizontal stress,

r11 (Fig. 4(a)). The vertical strain was advanced

in small increments of De22 1X0 106, and

several relaxation steps were performed within

each increment. These measures minimized the

transient inertial eects that would otherwise have

biased the results of a presumed quasi-static

loading. Fig. 4(b) shows the average mechanical

response for a compressive strain e22 of up to0X70%. The applied vertical stress is representedas Dr22 r22 r11, which is plotted in dimen-sionless form by dividing by the initial mean

stress o. Although the behavior is predominantly

plastic at strains above 0X2%, the stress has notyet peaked even at the nal 0X70% strain. More-

over, these strains are far less than the 2 12% to 6%strain at which shear bands develop (Cundall,

1989; Bardet and Proubet, 1991; Iwashita and

Oda, 1998b).

The assembly's particle graph was constructed

and stored with the algorithms and data structures

described by Kuhn (1997). In its initial state

(Fig. 3(b)), the particle graph contained 3950 void

cells (L), 7727 contacts (M), and 3777 load-

bearing particles (N). This fairly dense

arrangement was attained with relatively small

indentations at the particle contacts on average,less than 0X013% ofh50. During loading, the par-ticle graph was reconstructed after each strain in-

crement so that the current void cell geometrics

could be determined (Section 3).

Fig. 4. Biaxial compression test.



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The local velocity gradients Li

within individ-

ual void cells were computed with Eq. (6) by

measuring the changes in particle positions over

the course of a few increments of strain, which

would together produce a small compressive

strain De22 of just 0X001%. Although it is unusualto express quasi-static deformation with the

Eulerian rate measure L, the computed local de-

formations Li

will henceforth be presented in a

dimensionless, normalized form that will be de-

ned in Section 4.

3. Void fabric evolution

We consider geometric fabric changes within

individual void cells, including their valence

(number of edges per void cell), shape, and void

ratio. Our purpose is to explore the interrelation

between local void fabric and deformation. This

approach is distinct from the more commonly used

particle-based measures of fabric evolution (i.e.,

coordination number and orientations of particle

contacts).

3.1. Valence and coordination number

The void- and particle-based approaches are, of

course, closely related. The coordination numbers,

n and n, are the average numbers of contacts per

particle,

n 2MaN 8a

and

n 2MaNY 8b

with the eective coordination number n com-

puted from only load-bearing particles and con-

tacts. The average valence of the assembly m is

dened as

m 2MaLY 9

where only load-bearing contacts are included.

The connection between m and n is provided by the

Euler formula of the particle graph, which is given

by

LMN 0 10

for an assembly with periodic boundaries (since

the graph is homeomorphic with a torus). From

Eqs. (8a), (8b), (9) and (10), we have

m 2 4n 2

11

for a two-dimensional assembly with periodic

boundaries.

Fig. 5 shows measured changes in fabric during

our biaxial test. The eective coordination number

n decreased throughout deformation (Fig. 5(c)),

with a consequent increase in the average valence

m of the void cells (Fig. 5(d)). To compute these

results, the particle graph was continually updated

to account for newly established and newly sepa-

rated particle contacts. The measured changes invalence and coordination number will be discussed

in Section 3.4.

3.2. Void ratio

The void ratio, , of a granular material is a

conventional measure of its packing density. For a

two-dimensional material it is dened as the ratio

of void and solid areas, where the solid area esincludes the areas eks of all N particles:

e esaesY es Nk1

eks X 12

We also consider an eective void ratio, , that

includes only the area es of the N load-bearing

particles:

e esaesY es Nk1

eks X 13

This larger void ratio provides a more authentic

representation of the packing density. During

biaxial loading, the coordination number de-

creases, the average valence increases, and parti-

cles disengage from the assembly's load-bearing

framework. Such particles become temporarily

dormant within encompassing void cells. Fig. 5(b)

and (e) show these changes in the two void ratios

during biaxial compression. Prior to loading, only

231 of the 4008 particles were dormant, and this is

reected in the initial small dierence between

and : 0X179 versus 0X215. The initial small dier-



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ence increases greatly during deformation, with

changes in exceeding roughly ten-fold those in .

The eective void ratio reaches values of over

0X28, which is greater than that of an open squarepacking of equal size disks.

3.3. Fabric anisotropy

Other investigators have used a particle-based

approach to measure the stress-induced anisotro-

pic fabric of granular materials (e.g., Rothenburg

and Bathurst, 1989; Oda et al., 1980). Their

physical and numerical tests of disk assemblies

consistently reveal that particle contacts become

preferentially aligned in the direction of the ap-

plied major principal stress. Konishi and Naruse(1988) and Tsuchikura and Satake (1998) present

alternative, void-based measures of material fab-

ric. We use the void-based loop tensor Fi of Tsu-

chikura and Satake (1998), which is computed for

the ith void cell from its mi branch vectors liYj

(Fig. 2(b))

Fi 1

2

mi1j0

liYj liYjX 14

Tensor F

i

depends upon the size, shape, andorientation of the void cell. The elongation ratio

pi22api

11 is a measure of the average vertical elon-

gation of the ith void cell, and its height-to-width

ratio is given roughly by ai pi22ap

i11

p. Fig. 5(f)

show the average height-to-width ratio a of all

void cells during the numerical simulation. The

initial isotropic fabric (a % 1) becomes greatlyanisotropic, with a reaching a value of 1X21 at theend of loading. This anisotropy is visually appar-

ent in Plate 1(c). (Other features of this gure will

be discussed in Section 4.)

3.4. Discussion

Fig. 5 reveals a number of trends in fabric

change during biaxial compression.

(1) As biaxial compression proceeds, the num-

ber of void cells is reduced, and their average size

increases. Together, these changes suggest an in-

creasing sparsity in the load-bearing particle ar-

rangement, with fewer particles sharing an

Fig. 5. Fabric measures during biaxial compression.



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increasing vertical stress and fewer voids partici-

pating in the material's deformation.

(2) Except for the conventional void ratio , all

fabric measures in Fig. 5 change monotonically

throughout the range of strains that were inves-

tigated. The void cells become monotonically

larger and more elongated in the direction of

compression. The prevailing reason for void cell

growth and elongation is the predominant loss of

contacts between particles and their horizontally

oriented neighbors. This loss of contacts can be

attributed to the continual horizontal expansion

of the assembly, which occurs even when the

behavior is compressive at small strains. Fig. 6

shows the average orientation jhj of both dis-

continued and newly formed contacts. Only atstrains less than 0X1% does the vertical compres-sion consistently produce new contacts with an

average orientation greater than 45; but at larger

strains, the new contacts have average orienta-

tions of between 37 and 50. On the other hand,

horizontal expansion results in a continual loss of

contacts with an average orientation less than

45.

(3) Although compressive vertical strains and

expansive horizontal strains together produce void

anisotropy, a reciprocal eect also occurs. Themore vertically elongated void cells tend to dilate;

whereas, horizontally attened void cells tend to

compress. Statistical analysis shows that the dila-

tion rate of individual void cells, vi

11 vi

22, is re-

lated to their height-to-width ratio ai. The

coecient of correlation (i.e., the normalized co-

variance) between dilation and elongation varies

with compressive strain, but is consistently be-

tween 0X21 and 0X39. This statistical measure wascomputed for allL void cells samples of between

2300 and 3900 voids, depending on the compres-

sive strain. (A coecient of zero would indicate no

statistical correlation; values of 1 or 1 indicateperfect correlation.) The measured coecients of

0X21 to 0X39 indicate a moderate correlation be-tween void cell dilation and elongation, and their

positive values indicate a tendency toward in-

creased dilation with greater vertical elongation.

The coecients do not imply, of course, that allelongated void cells are dilatant, but only that

there is a moderate tendency for such behavior.

Fig. 7 shows the average dilation rates for void

cells within ten ranges of height-to-width ratio ai

at the strain e22 0X60%. Each bar representsone-tenth of 2370 voids. The vertical axis /

volYi is a

dimensionless and normalized measure of dilation

rate within void cells, which will be more precisely

dened in Section 4. The gure shows a consistent

tendency for greater dilation for voids that are

vertically elongated.This interrelation between dilation and void

fabric explains the reversal in volumetric behavior,

from compressive behavior at low strains to

Fig. 6. Average orientations jhj of discontinued and newly es-tablished contacts.

Fig. 7. Void cell elongation and dilation rate at strain

e22 0X60%.



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dilatant behavior at larger strains (Fig. 5(b)). The

slight reduction in void ratio that occurs at small

strains is typical of granular materials, even in

dense materials that later become vigorously

dilatant. The overall, macroscale compression oc-

curs even while the average void cell size and ef-

fective void ratio grow continually larger.

Because the average void cell shape is initially

isotropic (a % 1 at e22 0%, Fig. 5(f)), void cellelongation initially has no eect on volume

change. The early reduction in void ratio is a

consequence of an increasing mean stress

r11 r22a2, which pushes the particles moreclosely together. As the vertical stress increases

and the voids become vertically elongated, their

average behavior becomes increasingly moredilatant, eventually causing the entire assembly to

expand.

We have measured a microscale tendency for

individual voids to dilate when they are vertically

elongated. This result agrees with the macroscale

experimental and theoretical work of other in-

vestigators. Oda (1972) and Konishi et al. (1982)

conducted physical experiments on sands and

plastic disks to determine the eect of initial fabric

anisotropy on deformation behavior. Although

both tests involved slightly elongated particles,which could obscure the separate eect of fabric

anisotropy, dilation was most intense when the

initial fabric was more anisotropic, with the con-

tacts preferentially oriented in the direction of

applied compression. Chang et al. (1995) and

Maeda et al. (1995) developed theoretical elastic

moduli on the basis of idealized particle kinematics

and contact mechanics. Their moduli correspond

to greater dilation when the average fabric tensor

is anisotropic, such that particle contacts are

preferentially aligned in the direction of the com-

pressive strain.

(4) The void fabric continued to change signif-

icantly at strains greater than 0X3%, even thoughthe applied compressive stress r22 increased very

little. This progressive deformation-induced ani-

sotropy is an indication that signicant fabric

changes are occurring even while the stress is

nearly static. These fabric changes and their as-

sociated deformation structures are considered in

the next section.

4. Microbands of slip deformation

The previous section concerned aggregate

measures of void fabric and their changes during

biaxial compression. We now consider the spatial

distribution of local void cell deformations Li

within the two-dimensional material. Of particular

interest are local deformation patterns (or struc-

tures) that deviate from the mean deformation L

of the entire assembly. One diculty in studying

local velocity gradients Li

is presenting their four

Cartesian components in a meaningful way. We

address this problem by ``ltering'' a selected de-

formation mode, say U, by computing its inner

product with the local velocity gradient Li

within

each (ith) void cell. The result is a dimensionlessscalar measure /

i of the local deformation rate Li

that is aligned with mode U:

/i L

iUajLjjUjY 15

where the local rate is normalized with respect to

the norms of mean deformation L and mode U

(see Eqs. (1a), (1b), (2a), and (2b)).

We considered many dierent deformation

modesU, but two modes revealed especially strong

spatial organization on a microscale: left and rightslip deformations. In this section we consider the

two deformation modes Ub

which are shown in

Fig. 8 and correspond to simple isochoric shearing

along left and right slip planes b and b

. The

Cartesian components of the two lters Ub

and

Ub are

Fig. 8. Left and right slip deformation modes.



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Ub

cosbsinb cos2b sin2b cosbsinb

!16

and the corresponding void cell deformation rates

are designated /bYi and /b

Yi:

/bYi L

iUbajLjjUbjX 17

The spatial distribution of local void cell dila-

tions will also be considered in this section, and the

lter

Uvol

1 0

0 1

!18

is used with Eq. (15) to compute the dimensionless

dilation rate /volYi for each, ith void cell.

4.1. General observations

Plate 1 shows the spatial distributions of left

and right slip deformations /bYi and /b

Yi at three

compressive strains, 0%, 0X02%, and 0X60%. Eachdiagram shows between 2300 and 3900 polygonal

void cells which have been colored according to

the intensities of their slip deformation rates /bYi.

The local deformations are clearly nonhomoge-

neous, and the more intense local colors attest to

large deviations from the mean. Indeed, the

assembly's average deformation rate L has slip

values /b

,

/b

L Ub

ajLjjUb

j 19

of only 0X540X67, which are represented by thepale blue shades within boxes in the lower left

corners of the gures. Although slip deformations

are nonuniform, they are not randomly arranged,

but are organized into thin bands of intensely

colored void cells that trend obliquely through the

assembly. These microbands of slip deformation (or

simply ``microbands'') occur as conjugate systems

of left and right slip modes, trending downward to

the left and right, respectively (compare Plate 1(b)

and (d)). Microbands are faintly present even

when deformation is rst initiated (e22 0%), ascan be seen in Plate 1(a).

In producing these gures, the slip directions b

and b were both chosen as 50. The reason for

this choice and its eect on the results are discus-

sed at the end of Section 4.3.

The spatial arrangement of deformation non-

uniformity will determine the minimum size of the

mesodomain that is required to approximate

continuum behavior. If nonuniformity is randomly

arranged, a dozen or even fewer void cells may

closely represent the average behavior. The spatial

clustering of deformation that is seen in Plate 1,

however, indicates that a representative mesodo-

main must encompass a much larger region, of,

perhaps, 100 or more void cells (over 150 parti-

cles). Conversely, the limited size of our assembly

precludes studying any structured nonuniformity

that might otherwise be clustered within ``mega-

domains'' of, say, 1000 or more particles.

Evidence of slip deformation microbands at

small strains has appeared in the experimentalwork of others. In an early numerical biaxial test

on an assembly of 284 disks, Cundall et al. (1982)

observed a velocity discontinuity that trended

obliquely through the assembly when the strain

was 0X90%. In another numerical test, Koenders(1997) and his coworker found that most defor-

mation took place as slips between particles, which

lined up along complementary pairs of oblique

planes. Misra (1998) conducted physical experi-

ments on an assembly of 500 rods and observed

high shear strains at pre-failure stress levels withinnarrow regions that were 24 particles wide. His

statistical analysis revealed a spatial correlation

among particle motions that extended to a dis-

tance of 26 particle diameters. Hopper and trap-

door ow simulations by Langston et al. (1995)

and Murakami et al. (1997) have revealed the

presence of thin zones of intense shearing, al-

though the mean strain at which these zones de-

veloped was not given.

We should, of course, also mention the devel-

opment of shear bands, which are often observed

at the onset of failure. Of particular interest are

numerical simulations that have enabled the close

observation and measurement of deformation

within and adjacent to such shear bands (Cundall,

1989; Bardet and Proubet, 1991; Iwashita and

Oda, 1998a,b). Although we will often refer to

these results, shear bands and slip deformation

microbands are considered separate phenomena.

The primary dierences are in their thickness and

persistence and in the strains at which they appear.



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12/23

These and other aspects of slip deformation mic-

robands are discussed in the remainder of this

section.

4.2. Morphology, periodicity, and evolution

Microbands are thin obliquely trending chains

of void cells within which slip deformations are

most intense (Plate 1). These thin bands of intense

slip are separated by thicker zones within which

the slip deformations are less than the assembly

mean. The microbands range in thickness from

1 12h50 to 2

12h50 during the early stages of loading

(for example, at e22 0X02% in Plate 1(b) and(d)) and become slightly thicker as deformation

proceeds (between 1 12h50 and 4h50 at strains of0.400X60%). The microband slip zones are sepa-rated from each other by thicker zones that are 3

10 particle diameters wide.

The fact that these microbands are signicantly

thicker than a single particle diameter h50 suggests

that they are not just velocity discontinuities along

slip planes between particle groups. Rather, mic-

robands thicker than 2h50 are more properly

characterized as slip zones. The particle interac-

tions that produce microbands of slip deformation

are illustrated in Plate 2(a). Each line in the gureis a branch vector between two contacting parti-

cles, and its thickness indicates the tangential ve-

locity mtanYjrel between the two particle centers. Blue

() and red () lines correspond to counter-clockwise and clockwise velocities. Specically,

thickness and color for the jth contact are calcu-

lated as the cross product

mtanYjrel

lj

jljj vj Lljah50jLjY 20

where lj is the branch vector, jljj is its length, andvj is the relative velocity of the two particles

(Fig. 2(a) and (b)). To emphasize uctuations rel-

ative to the mean, we subtract the motion Llj

( vqljq) that corresponds to uniform deformation

and use the ``rel'' subscript. Quantity (20) is ren-

dered dimensionless by the quotient h50jLj.Plate 2(a) shows zones of intense clockwise

tangential motions (red lines) at strain

e22 0X02%. The zones trend downward andright (8), with their red branch vectors oriented

roughly perpendicular to the zones (a). Thesezones are, in fact, the microbands of right slip that

appear as dark blue bands in Plate 1(b). Likewise,

the conjugate zones of counterclockwise motions

(blue lines) in Plate 2(a) correspond to left slip

microbands (dark blue bands in Plate 1(d)). In

either case, the microbands, although thin, are

usually thick enough to encompass widths of

three, and sometimes four or more, branch vectors

that are all moving in the same tangential sense:

red widths of clockwise motions within the right

slip microbands, and blue widths of counter-

clockwise motions within the left slip microbands.

These patterns correspond to slip zones rather

than slip surfaces.

Plate 1 shows the evolution of local right slipdeformation. Although only faintly present at the

start of loading, the microbands become progres-

sively more intense, until, at strain e22 0X60%most void cells are engaged in either intense left or

right slip deformation. There are also several short

bands of red in Plate 1(c) which signify negative

slip occurring in a direction opposite the assem-

bly's average deformation. This observation of

negative slip suggests the hypothesis of Rice (1976)

in which neighboring regions of elastic unloading

and plastic ow can occur along a plane of dis-continuity in the velocity gradient eld.

The microbands that appear in these simula-

tions are neither static nor persistent features.

They emerge, move, and disappear as the assembly

is deformed. The microbands in Plate 1(b)

(e22 0X02%) are no longer present in Plate 1(c)(e22 0X60%), and they are almost entirely rear-ranged even before the strain reaches 0X10%. In-dividual microbands rarely persist for elapsed

strains of more than 0X20%.Microbands become steeper with increased de-

formation, although their orientation is likely in-

uenced by the boundary conditions that were

used in our simulation. With periodic boundaries,

a long microband must ``wrap around'' and join

with itself over an integer number of assembly

widths and heights (say, integers g1 and g2). (For a

discussion of admissible symmetry patterns for

deformation wallpaper groups, see Ikeda and

Murota, 1997.) This boundary eect can be seen in

Fig. 9(a) and (b), where nine assembly cells have



13/23



14/23

been stacked together, three wide and three tall.

The monochrome gures display only void cells

that are undergoing the most intense positive right

slip (/bYi b 0X8 or 1X0). At small strain, integers g1

and g2 are both 1 (Fig. 9(a)). For larger strains, g1and g2 are 2 and 3, so that the average orientation

of the microbands is about 56 from horizontal

(Fig. 9(b)).

4.3. Relative signicance of slip deformations

Local slip deformation is the predominant

source of nonhomogeneous deformation. To ver-

ify the relative signicance of local slip deforma-

tions, we consider their participation in the

statistical variance of the velocity gradient, varL,throughout the material:

varL 1

e

Li1

eijLi

Lj2ajLj

2X 21

In this form, varL is a dimensionless measureof the degree of inhomogeneity of deformation

(i.e., a squared coecient of variation ofL), and it

includes all forms of deformation uctuation, not

just slip deformation. Values of varL are quitelarge between 0X49 at the start of loading and2X94 at a compressive strain of 0X60% and arefurther evidence of the very nonuniform nature of

deformation in granular materials. To investigate

the combined participation of both left and right

slip deformations in this nonuniformity, we mustaccount for the fact that the two slip modes Ub

and Ub

are not orthogonal to each other. We

introduce the tensor subspace Sb spanned by

tensors Ub

and Ub

,

Sb Ub

Ub

X Y P RX 22

The contribution of local slip deformations

(both left and right) to the overall nonuniformity

varL is denoted as varbL and given by

Fig. 9. Stacked assembly showing zones of the most intense right slip deformation.



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varbL 1

e

Li1

eijLi

L projSbj2ajLj

2Y 23

where L

i

L projS

b is the orthogonal projec-tion of the tensor dierence Li

L onto subspaceSb.

Throughout the biaxial compression test,

varbL accounted for at least 65% of varL and,at times, for as much as 70% of varL. That is, thelocal nonuniform slip deformations could account

for the bulk of deformation nonuniformity. No

other combination of two deformation modes

could account for as much nonuniformity as could

the left and right slip deformations Ub

and Ub

.

By comparison, nonuniform dilation alone ac-

counted for less than 17% of varL; the subspaceof combined void cell dilations and rotations ac-

counted for less than 50% of varL.We chose the slip directions b 50 to pro-

duce Plate 1. A choice of between 45 and 50

maximizes the value of varbL, resulting in greatercolor contrasts in the gures. If b is chosen be-

tween 35 and 65, however, the change is only

subtly noticeable, with the blue coloration within

microbands only slightly less intense and the

banded patterning just slightly more diuse.

4.4. Relation to circulation patterns

Williams and Nabha (1997) and Murakami

et al. (1997) have reported the presence of circu-

lation cells or vortex structures in two-dimensional

DEM simulations. These clusters of circulating

particles can be seen in Fig. 10(a) and (b). The

movement arrows in these gures are the scaled

velocities vk of the 4008 particles relative to the

background velocities Lxk that correspond to

uniform deformation

vrelYk vk Lxkah50jLjX 24

The relative velocities are computed and displayed

in dimensionless form, by dividing by the velocity

norm h50jLj.The habitual presence of microbands demon-

strates that the motions of individual particles are

not random or unrelated, but instead collaborate

Fig. 10. Particle velocity vectors relative to uniform deformation.



16/23

with the motions of nearby particles to form large

long-range deformation structures. The circulation

cells in Fig. 10 are also manifestations of this

complex patterning of particle motions. Although

circulation cells and microbands are related, their

relationship is complex and not easily reduced to

simple rules. As might be expected, slip deforma-

tion microbands are often found between two

separate co-circulating cells (e.g., or ). In

some instances, however, slip deformation micro-

bands pass between counter-circulating cells (e.g.,

or ), and, in other instances, microbands

pass directly through the middle of small circula-

tion cells.

4.5. Relation to local fabric and dilation

The simulations reveal a number of connections

between local, void cell fabric and the intensity of

slip deformation. These relationships, elaborated

below, are by no means without exceptions, but

are merely trends and tendencies in the widely

varying fabric and behavior of between 2300 and

3900 void cells.

(1) Larger void cells tend to participate dis-

proportionately in both left slip and right slip de-

formations (the darker blue zones in Plate 1). Onthe other hand, regions of low left and right slip

deformations, the lighter zones in these gures,

tend to be occupied by smaller void cells. The

trend is strongest at low strain levels

(e22 ` 0X05%), at which the statistical coecientof correlation between local slip deformation /b

Yi

and void cell size is 0X25. At a larger strain of0X60%, the correlation is 0X10. These trends re-semble the observations of higher than average

void ratios within fully developed shear bands

(Oda, 1997; Iwashita and Oda, 1998b).

The correlation between void size and slip de-

formation is largely due to the relative lack of

triangular voids within slip deformation micro-

bands. Fig. 11 shows histograms of void cell va-

lence mi for two sets of void cells at strain

e22 0X60%. The light bars are for void cells withrelatively little right slip deformation the lower

quartile of/bYi. The dark bars are for the upper

quartile of/bYi, found primarily within right slip

microbands. The greatest dierence in the two

histograms is in the number of triangular voids,

with far fewer triangles among the microband void

cells.

(2) In Section 3, we noted that dilation is most

intense within void cells that are vertically elon-

gated (i.e., in the direction of the principal com-

pressive stress). Right and left slip deformations,

however, are most intense in obliquely elongated

void cells. In this regard, we consider the principaldirections of the fabric tensor Fi of each void cell

(Eq. (14)). The orientation angle fi of the tensor's

major principal axis can be computed from its

Cartesian components piq (Fig. 12, refer also to

Konishi and Naruse, 1988 and Tsuchikura and

Satake, 1998). The relationship between void ori-

entation fi and right slip deformation /b

Yi is il-

lustrated in Fig. 13 at a compressive strain of

0X60%. Each of the fourteen bars represents arange of orientations f

i, with each bar representing

one-fourteenth of the 2370 void cells. Right slip

deformation is least intense in those void cells that

are elongated in the direction of right slip

(fi % 90 b 40) and most intense at orien-

tations of between 5 and 30, which is some-what steeper than an orientation perpendicular to

the slip plane (fi 50). Our measurements of

void orientations within microbands are similar to

void orientations within fully developed shear

bands, as can be seen in gures presented by

Iwashita and Oda (1998b).

Fig. 11. Histograms of valence among two void cell groups:

voids in the lower quartile of right slip intensity (light bars) and

voids in the upper quartile of right slip intensity (dark bars), atstrain e22 0X60%.



17/23

(3) The development of slip deformation mic-

robands bears likeness to double-slip and multi-

slip continuum models, in which deformation at

every material point is presumed to occur as a

combination of shearing in two or more preferred

directions (e.g., Nemat-Nasser, 1983). Such mod-

els require a kinematic assumption of the dilation

that is produced by the slip deformations.

In this regard, we found a moderate correlation

between local slip deformation and dilation. At

lower strains, where the average material behavior

is compressive (Fig. 5(b)), local compressive de-

formation is slightly greater within microbands of

both left and right slip deformation than in the

neighboring material. At higher compressive

strains, however, where the behavior is dilatant,

the tendency is reversed: void cell dilation is

greater within the microbands than in the neigh-

boring material. The relationship between dilation

and slip deformation is illustrated in Fig. 14 forstrain e22 0X60%. Although the deformationsof all 2370 void cells are represented in this gure,

the widely scattered data have been greatly

smoothed. Each point represents a traveling av-

erage of 150 void cell deformations, which had

been arranged in order of ascending right slip de-

formation /bYi. The dilation rate within individual

void cells is expressed in dimensionless form as

/volYi, dened at the beginning of this section

(Eqs. (15) and (18)). The range of dilation rates in

Fig. 14, values of /volYi between 0.2 and 1.0, is

quite large in comparison with the mean dilation

rate /vol

of just 0X28. The gure shows a tendencyof increased dilation among void cells that are

engaged in more intense right slip deformation.

In some double-slip constitutive formulations, a

distinction is made between dilations (or stretches)

that occur perpendicular and parallel to the slip

planes (Nemat-Nasser, 1983). We found that

microband dilation occurs in both directions and

in roughly equal amounts.

Fig. 12. Void cell orientation fi.

Fig. 13. Relationship between right slip deformation and void

cell orientation at strain e22 0X60%.

Fig. 14. Relationship between void cell dilation and right slip

deformation at strain e22 0X60%.



18/23

4.6. Energy dissipation

In our quasi-static loading, work exerted by the

moving boundaries is either expended in frictional

contact slipping at the particle contacts or accu-

mulated within the elastic contact springs. Studies

have shown that at any stage of deformation, only

a small minority of contacts are slipping (Cundall

et al., 1982; Oda et al., 1982; Bardet, 1994). At a

compressive strain of 0X60%, only 7X8% of the 6127contacts are slipping, and over 90% of the energy

dissipation occurs within just 140 (about 2%) of

the contacts. A disproportionate number of slip-

ping contacts are located within the thin micro-

bands of left and right slip (the dark blue zones of

Plate 1) or along the edges that separate thesemicrobands from neighboring void cells.

The relationship between energy dissipation

and right slip deformation is shown in Fig. 15 for

a compressive strain of 0X60%. When comparingdissipation and deformation at the microscale, we

must resolve an inherent inconsistency: slipping

and energy dissipation occur at particle contacts,

which form the edges of a particle graph; whereas,

deformation occurs within the void cells that form

the faces of the graph (Fig. 1(a)). Our approach is

to assign half of the dissipation at a slipping con-tact to each of the two void cells that share the

contact. The dissipation rate li associated with the

ith void cell is then computed as half of the dissi-

pations in the mi particle contacts around its pe-

rimeter:

li 1

2

1

h250ojLj

mi1j0

fj vslipYjY 25

where fj vslipYj is the inner product of the contactforce and slipping velocity at the jth contact.

Quantity li is made dimensionless by the quotient

h250ojLj, where o is the initial mean stress.The wide scatter in the data of 2370 void cells is

resolved in Fig. 15 by using a traveling average of

150 void cell deformations and dissipations. The

results show that energy dissipation occurs dis-

proportionately among those void cells that are

undergoing large slip deformations (in this case,

right slip deformation). In contrast to this mod-

erate correlation between slip deformation anddissipation, we found little if any correlation be-

tween void cell dilation and energy dissipation.

4.7. Local elastic and plastic deformations

The frictional nature of particle contacts pro-

duces an inelastic material behavior. This behavior

was examined in a small cycle of loading, un-

loading, and reloading at a compressive strain of

0X035%. Fig. 16 shows this cycle and denes the

elastic and plastic increments of strain, Deel

22 andDe

pl22. At the start of biaxial loading, the behavior is

almost entirely elastic, but as loading proceeds,

plastic deformation becomes dominant. With a

compressive strain of 0X035%, the two contribu-tions Deel22 and De

pl22 are about equal.

Fig. 15. Relationship between frictional energy dissipation and

right slip deformation at strain e22 0X60%.Fig. 16. Cycle of loading, unloading, and reloading at

e22 0X035% (see also Fig. 4(b)).



19/23

At a microscale, the loadingunloading cycle did

not return particles to their original positions. We

could, therefore, separate particle movements

(treated as velocities) into elastic and plastic com-

ponents and compute their separate contributions

to the deformation of each, ith, void cell, LiYel

and

LiYpl

. The corresponding mean velocity gradients of

the entire assembly, Lel

and Lpl

, can be computed

as the spatial average of the local void cell defor-

mations LiYel

and LiYpl

(see Eq. (5)). Although the

mean velocity gradients Lel

and Lpl

are coaxial (i.e.,

share the same principal axes), they are far from

collinear. IfLel

and Lpl

are treated as vectors, their

angular separation, cos1Lel

LplajL

eljjL

plj, is a

signicant 52.We observed the following microscale behavior:

(1) At strain e22 0X035%, the mean elastic andplastic deformations, L

eland L

pl, were of roughly

the same magnitude, but there was far more local

variation in the local plastic deformation. The sta-

tistical spatial variance of plastic deformation,

varLpl 1

e

Li1

eijLiYpl

Lpl

j2ajL

plj

226

was more than ve times greater than the corre-

sponding variance in elastic deformation: 8X1versus 1X5.

(2) As has been mentioned, the mean elastic and

plastic deformations Lel and Lpl were only partially

aligned, with an angular separation of 52. The

local uctuations LiYel

Lel

and LiYpl

Lpl

,however, were consistently in opposite directions

within individual void cells, with an average an-

gular separation of 160. This interesting behavior

was revealed in plots of particle movement vectors

similar to those of Fig. 10. When elastic and

plastic velocities are separately plotted relative to

their background movements Lel

and Lpl

, the two

whorled circulation patterns occur in opposite di-

rections. In a similar fashion, the darker and

lighter zones of slip deformation (such as in

Plate 1) become interchanged when the separate

elastic and plastic slips are plotted. This phenom-

ena is also illustrated in Fig. 17, which shows

elastic and plastic right slip velocities at strain

e22 0X035%. The velocities are along a line ABthat is oriented perpendicular to the direction b

of right slip (Fig. 17(a)). These are ``smoothed''

particle velocities, which were computed by

Fig. 17. Elastic and plastic velocity uctuations in direction nb

along cross section AB at strain e22 0X035%.



20/23


21/23

distinguish between contact slipping, which pro-

duces frictional energy dissipation, and contact

sliding, which can produce slipping, elastic chan-

ges in the contact force, or both.

The eect of particle rotation is illustrated in a

small group of 85 particles located to the left of

and below the center of the entire assembly (box in

Plate 1(c)). A right slip microband of width 3h50passes through the center of the group, which also

includes the neighboring void cells that have low

and even negative right slip deformations. The

relative motions of the particles are represented in

Plate 2(c) and (d) at a vertical strain of 0X60%.Plate 2(c) shows the tangential relative velocities

mtanYj of the particle centers, presented in a similar

manner as Plate 2(a) of Section 4.2,

mtanYj lj

jljj vj

0h50jLjY 27

but in which the average motion Llj has not been

subtracted (cf. Eq. (20)). Red and blue lines rep-

resent the clockwise and counterclockwise tan-

gential motions of contacting particle pairs. The

microband void cells are lightly shaded in blue. As

was discussed in Section 4.2, a right slip micro-

band is produced by the relative clockwise motions

of pairs of contacting particles that lie along theband's length, and these red contacts appear as the

rungs of a ladder through the middle of the group

(Plate 2(c)). The tangential motions of particles

outside the band are much smaller in magnitude.

Plate 2(d) includes the additional inuence of

particle rotations by showing the sliding motions

mslideYj that occur at particle contacts within the

group. The sliding motion at a contact results from

a combination of the particles' relative tangential

velocity (mtanYj, Plate 2(c)) and the rotational ve-

locities of the two particles. Such sliding motionschange the contact force and can produce fric-

tional slipping between particles. Iwashita and

Oda (1998b) derived an expression for the sliding

velocity, which we adopt in the following dimen-

sionless form:

mslideYj lj

jljj

vj xk1rk1 xk2rk2

0h50jLjX

28

This expression is similar to that for the relative

tangential velocity mtanYj of Eq. (27), but also in-

cludes the counterclockwise rotational velocities x

and radii rof the two particles k1

and k2

at the jth

contact.

For most contacts within the microband, par-

ticle rotations reduce contact sliding, as is evident

in the reduced line thicknesses of Plate 2(d).

Contact sliding is increased, however, in a few

microband contacts six thick blue lines in Pla-

te 2(d). Of the 42 contacts that lie within the

microband zone, contact sliding was reduced by

more than half in 36 contacts and increased in only

the six contacts. Frictional slipping within the

microband occurs in just four contacts, and in

each case, particle rotations had greatly increasedthe sliding motions.

Particle rotations can reduce contact sliding

mslideYj in two ways:

(i) By the rolling between pairs of counter-ro-

tating particles ( or , see Bardet, 1994 and

Iwashita and Oda, 1998b). Rolling, which oc-

curred in 35 of the 42 contacts of Plate 2(d), was

particularly eective within the microband and

greatly reduced particle sliding. The relative lack

of triangular voids within microbands (Sec-

tion 4.5) likely promotes such rolling among par-ticles.

(ii) By co-rotating particles ( or ), that,

together with the tangential motions mtanYj, produce

a rigid rotation of particle pairs. Indeed, the ro-

tations xk1 and xk2 can cancel the sliding eect of

particle movements vj when xk1rk1 xk2rk2 lj vjajljj. Although its eect is smaller thanthat of rolling, co-rotation occured in 18 of the 36

contacts in which sliding was greatly reduced by

the particle rotations. In all 18 of these contacts,

the co-rotations contributed to a reduction. Co-

rotations, however, can also increase the sliding

between particles when they are excessive or

occur in a direction opposite the tangential motion

mtanYj. This occurs in all six of the contacts in which

sliding was increased by particle rotations.

In summary, particle rotations are organized in

a manner that relieves the sliding among most

microband particles, but sliding is transfered to a

few contacts in which sliding and frictional slip-

ping is most intense.



22/23

Outside the microband, particle rotations also

reduce the contact sliding mslideYj among particles,

but the reduction is much less pronounced. Rota-

tional velocities xk are smaller and are organized so

that the inuences of rolling and rigidly rotating

pairs are more subdued than within the microband.

5. Conclusions

Deformation can be measured at the microscale

of individual voids. At this small scale we nd

interrelations between the local void fabric and

deformation: vertically elongated voids tend to

dilate, obliquely elongated voids tend to undergo

slip deformation, and smaller voids are less likelyto participate in either type of deformation. This

interrelation is dynamic, with void fabric changing

signicantly as deformation proceeds and, in turn,

aecting the subsequent deformation.

Deformations are structured, with microbands

of slip deformation being the most prominent

feature at the mesoscale of several particle diam-

eters. These transient zones of considerable slip

deformation trend obliquely through the material.

They are evidence of a complex organization of

particle motions. Microbands result from largetangential particle movements within zones that

are a few particles wide and tens of particles long.

Because of particle rotations, these motions pro-

duce very little sliding between the particles. The

sliding that does occur is concentrated within a few

intensely sliding contacts.

We have discussed other aspects of slip defor-

mation microbands: their orientation and progres-

sion, their measured signicance relative to other

deformation modes, their relation to circulating

particle clusters and chains of heavily loaded par-

ticles, and the frictional energy dissipation thatoccurs within them. We have related the interesting

phenomena of oppositely directed elastic and plas-

tic deformation uctuations. Many aspects remain

unexplored, most notably the evolution of micro-

bands at larger strains and their relation to shear

bands. Our observations may, however, provide a

basis for developing, testing, and calibrating non-

local constitutive formulations that accomodate

deformation patterning at small scales.

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Structured Deformation Granular DEM

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