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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
www.elsevier.com/locate/mechmat
1 E-mail: [email protected].
0167-6636/99/$ - see front matter
1999 Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 6 3 6 ( 9 9 ) 0 0 0 1 0 - 1
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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.
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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%).
410 M.R. Kuhn / Mechanics of Materials 31 (1999) 407429
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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-
412 M.R. Kuhn / Mechanics of Materials 31 (1999) 407429
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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.
M.R. Kuhn / Mechanics of Materials 31 (1999) 407429 413
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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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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
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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.
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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%.
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(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%.
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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)).
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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%.
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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.
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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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