a numerical examination of the hollow cylindrical torsional shear test using dem

RESEARCH PAPER

A numerical examination of the hollow cylindrical torsionalshear test using DEM

Bo Li • Fengshou Zhang • Marte Gutierrez

Received: 10 October 2013 / Accepted: 20 April 2014

� Springer-Verlag Berlin Heidelberg 2014

Abstract This paper presents results of three-dimen-

sional simulations of the hollow cylindrical torsional shear

test using the discrete element method. Three typical stress

states that can be applied in the hollow cylindrical appa-

ratus (HCA), i.e. triaxial, torsional compression and pure

torsional, are examined in terms of the distributions of

stresses and strains in the HCA sample. The initiation and

propagation of the shear bands in the sample were char-

acterized by porosity and shear strain rate distributions in

the sample. The results show that the shear strain rate

contour is a better indicator for shear band development

than the porosity contours. It is demonstrated that the

stresses and strains measured in the shear zone are signif-

icantly different from the boundary measurements and the

average values used in HCA testing. Initially, the peak

strength measured from the boundary forces was found to

be slightly lower than that measured in the shear band.

Subsequently, due to the formation of shear band, the stress

ratio from boundary forces decreased significantly

especially when the major principal stress is oriented 30�and 45� from the vertical. The evolutions of porosity,

coordination number and particle rotation at different

locations in the sample were also monitored. Finally, the

appropriateness of the HCA is evaluated in comparison

with previously published data.

Keywords Discrete element method � Granular soils �Hollow cylinder apparatus � Shear bands � Strain

localization

1 Introduction

The hollow cylinder apparatus (HCA) is a versatile testing

device widely employed to investigate the constitutive

behavior of soils under generalized stress conditions,

including principal stress rotation, anisotropy and non-

coaxiality (e.g., [5, 21, 28, 29]). Saada et al. [23, 24] pio-

neered the use of HCA for investigating the effects of

principal stress rotation in sands and clays. Later, Saada

et al. [25, 26] investigated the localization of deformation

of homogeneous sand specimens by using a digital imaging

system in conjunction with HCA testing. Observations

backed by photographic records indicated that for sands, a

dominant shear band is initiated in the vicinity of the peak

strength and fully develops as the loading moves toward

the critical state. The inclination of the shear band appears

to depend on both friction angle and the dilation angle. The

non-coaxial behavior of sand has been extensively studied

using the HCA (e.g., [5, 6, 9, 11, 20, 38]). Lade et al. [11,

12] also investigated the shear banding and cross-aniso-

tropic behavior of sands observed in the HCA with prin-

cipal stress rotation. The friction and dilation angles of the

specimen with different principal stress directions were

B. Li (&)

Department of Civil Engineering, Wenzhou University,

Wenzhou 325000, People’s Republic of China

e-mail: [email protected]

F. Zhang

Itasca Houston, Inc., Itasca, TX, USA


M. Gutierrez

Department of Civil, Infrastructure and Environmental

Engineering, Khalifa University, Abu Dhabi, UAE


M. Gutierrez

Department of Civil and Environmental Engineering,

Colorado School of Mines, Golden, CO 80401, USA

123

Acta Geotechnica

DOI 10.1007/s11440-014-0329-9

studied. Also, Lade et al. [11] stated that shear banding was

initiated before the smooth peak failure, similar to the

observation of Saada et al. [26].

Even though the HCA offers highly sophisticated

capabilities for the study of soil behavior, including the

ability to apply true triaxial loading and rotate the principal

stresses, its use has been subject to criticisms. The issues

are primarily focused on the non-uniform distribution of

stresses and strains within the specimen, which are induced

by the specimen geometry, end restraints, the applied tor-

que and the difference between internal and external con-

fining pressures (e.g., [7, 13, 19, 24, 27, 32]). Hight et al.

[7] proposed equations for the average stress and strain

calculations for the HCA. In addition, error indices were

defined to characterize stress non-uniformities in the sam-

ple. These indices were dependent on the stress state,

specimen geometry and constitutive law of the test mate-

rial. Vaid et al. [32] also analyzed non-uniformity in hol-

low cylinder specimens by using linear elastic model.

Through optimization, two criteria based on elasticity

theory and finite element methods are widely accepted for

dimensioning HCAs that will yield minimal non-uniform

loading (e.g., [24, 27]).

Similar with other element tests, traditional interpreta-

tions of the stress and strain states in the HCA have been

based on continuum mechanics. However, a continuum

approach might not be suited for treating complex stress

paths especially when the exact state of the specimen

during loading is non-uniform. Hence, doubt is cast upon

the reliability of HCA test results for their use in practical

engineering designs. From the microscopic point of view,

the investigation of strain localization has been primarily

based on the observations of sample deformations at the

boundary, and thus could not capture the evolution of the

microstructure of the shear band. Although various imag-

ing methods including stereo-photogrammetry, digital

image correlation, X-ray CT and micro-focus X-ray CT

have been used for local measurements on sand specimens,

and significant progress has been made in understanding

the relationship between local phenomena in shear bands

and global behavior, these technological and practical

difficulties make these methods unable or too expensive to

continuously track physical quantities inside and outside

the shear bands during the course of loading. (e.g., [18, 34])

In recent years, there has been a strong interest in the use of

discrete element modeling (DEM) to improve the under-

standing of the behavior of granular soils particularly from

the micro-mechanical perspective. DEM also has enabled

continuous and accurate quantification of fabric evolution

as well as strain localization by local measurements in the

virtual experiments.

Many investigators have used DEM to examine the

effectiveness of the physical data from lab tests and explore

the internal structure of test specimens. For instance, Ni

et al. [15] reported results of three-dimensional DEM

simulations of the direct shear test in which non-spherical

particles were generated by bonding pairs of unequal-sized

spheres. They showed that the dilation and bulk friction

angles increased with inter-particle friction and with

composite particle shape factor. Ni et al. [15] also exam-

ined the effect of the number of particles used to fill the

same sized box and showed that, as the number of particles

increased from 5,000 to 50,000, the dilation angle reduced

exponentially. There was no significant change in the peak

bulk friction angle, but the residual friction angle was

reduced from 37� to 28�. It was also shown that deforma-

tion was restricted to a narrow shear zone near the mid-

height of the specimen. The thickness of the shear zone

was approximately 11�D50, where D50 is the grain size at

Fig. 1 Forces acting on the hollow cylinder specimen

Fig. 2 Three-dimensional DEM model of hollow cylinder sample

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123

50 % passing. Within this zone was a narrower zone of

intense particle rotations. Thornton and Zhang [31, 39] also

examined the effectiveness of the direct shear test using

DEM. It was shown that the evolution of the stress ratio

(s/r) inside the shear band was very similar to the inferred

values from boundary force calculation. It was demon-

strated that dilation in the shear band was much greater

than the one obtained from boundary observations. DEM

simulation results reported by Wang and Gutierrez [35]

showed that the maximum shear strength measured at the

model boundary increased with decreasing specimen

length scale and increasing specimen height scale. More-

over, micromechanics-based analysis indicated that the

local and global aspects of fabric change and failure were

the major mechanisms responsible for the specimen scale

effect (e.g., [36, 37]).

Despite wide use of the HCA, there are no published

literatures on the comprehensive DEM examination of the

HCA under complex stress paths. In this regard, this paper

presents a comprehensive DEM study of the hollow cyl-

inder specimen under complex stress states. Three-dimen-

sional DEM simulation of the HCA test with rigid spherical

particles in loose and dense states are performed under

three typical loadings, namely, triaxial, torsional com-

pression and pure torsional conditions. The DEM simula-

tions were carried out using PFC3D, developed by Itasca

Consulting Group, Inc. Macroscopic behavior of the HCA

specimen is interpreted using stresses, strains and volu-

metric change measured at the boundary. At the same time,

to represent the microscopic behavior and strain localiza-

tion of the specimen at different stress states, the ‘‘mea-

suring sphere scheme’’ is used to illustrate the mechanical

response at specified sampling locations in the specimen.

The appropriateness of the HCA is discussed in details.

Shear banding in terms of spatial distribution of porosity,

shear strain rate and particle rotations are studied to illus-

trate the microstructure evolution. Micromechanics-based

shear banding analysis is used to provide fundamental

explanations of the effects that the test scale has on the

macro-scale behavior of granular soils inside the HCA.

2 Simulation details

2.1 Model setup

The HCA allows independent control of the magnitudes of

the three principal stresses and direction a of one of the

principal stresses. Figure 1 illustrates the idealized stress

condition in a hollow cylindrical element subject to axial

load W, torque Mt, internal pressure Pi and external cell

pressure Po. During shearing, the torque Mt applies shear

Table 1 Equations used for calculating stresses and strains in the HCA

Stress Strain

Vertical rz ¼ Wpðr2

o�r2iÞ þ

por2o�pir

2i

ðr2o�r2

iÞ

ez ¼ zH

Circumferential rh ¼ poro�piri

ro�rieh ¼ � uoþui

roþri

Radial rr ¼ poroþpiri

roþrier ¼ � uo�ui

ro�ri

Shear szh ¼ T2

32pðr3

o�r3iÞ þ

4ðr3o�r3

iÞ

3pðr2o�r2

iÞðr4

o�r4iÞ

h iczh ¼

hðr3o�r3

iÞ

3Hðr2o�r2

iÞ

Major principalr1 ¼ rzþrh

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirz�rh

2

� �2þs2zh

qe1 ¼ ezþeh

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiez�eh

2

� �2þc2zh

q

Intermediate principal r2 ¼ rr e2 ¼ er

Minor principal r3 ¼ rzþrh2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrz�rh

2Þ2 þ s2

zh

qe1 ¼ ezþeh

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðez�eh

2Þ2 þ c2

zh

q

Fig. 3 Locations of measurement spheres

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123

stresses shz (= szh) in the horizontal plane of the sample.

The axial load W applies the vertical stress rz, and Pi and

Po control the radial and circumferential stresses rr and rh.

Three-dimensional simulations of the HCA that mimic the

physical torsional shear test using the discrete element

method (DEM) are performed on an assemblage of 18,000

rigid spheres. In the DEM technique, the rigid particles do

not deform. The inter-particle forces are calculated from

the particle overlap at the contact point. The actual sizes of

the randomly generated particles are in the range of

0.8–1.2 mm. The specified mechanical properties of the

particle contacts are: normal contact stiffness

kn = 29105 N/m, shear contact stiffness ks = 29105 N/m

and inter-particle friction coefficient l = 0.5. Li et al. [10]

have extensively investigated the parameters in terms of

loading velocity, normal and shear stiffness of particle,

friction coefficient and wall stiffness. The conclusions

drawn from combination of compression and torsion tests

are opposite from that in triaxial stress state: the particle

contact shear stiffness also significantly affects the

mechanical response of the particle assemblage due to

introduction of torque.

Ng [14] has investigated the effects particle shape on

the macro–micro-behavior of granular materials. He

found that particle shape affects the volume change

behavior but not the stress–strain response. Hence,

spherical particles are used in this study for easy inter-

pretation of the results. A hollow cylinder assembly of

spheres with an internal diameter of 60 mm, external

diameter of 100 mm and height of 200 mm is shown

schematically in Fig. 2, and these dimensions are the

same as the physical test. The DEM specimens were

created to a void ratio e = 0.68. The membrane is

simulated by ‘‘20-stacked wall’’ technique (Fig. 2),

Fig. 4 Relationship between the length on the image and the length

on the specimen’s surface

0 10 20 30 40 50 600

500

1000

1500

2000

2500

Dev

iato

r st

ress

, q=

σ 1−σ3 (

kPa)

Principal stress rotation, α (o)

45ο

30ο

pure torsion state

torsional compression stateTriaxial state

α=0ο

Fig. 5 Stress paths followed in monotonic loading test: theoretical

versus results from simulation

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

400

800

1200

1600

2000

2400

2800

3200

Str

ess

com

pone

nt (

kPa)

Devitoric strain ε13= ε1−ε 3 (%)

τθz

σθ

σr

σz

(a)

0 500 1000 1500 2000 2500-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Deviatoric stress q=σ1−σ 3 (kPa)

Str

ain

com

pone

nt (

%)

εz

εr

εe

γθz

(b)

Fig. 6 Macroscopic response of a hollow cylinder specimen with a = 0�: a stress components versus deviatoric strain, b strain components

versus deviatoric stress

Acta Geotechnica

123

which can replicate the real condition of the specimen

and appropriately show the microstructure evolution of

the specimen. Li et al. [10] have suggested the optimum

ratio of H/D50 (H is the height of specimen and D50 is

mean diameter of the particles), which indicate how

many walls it needs to be related to mean particle

diameter. The superiority of this technique has been

demonstrated by Zhao and Evans [40]. The details for

the chosen model parameters can be obtained from Li

et al. [10].

In order to simulate the torque applied on the specimen

for the compression torsional state, the angular velocity xis applied to the particles near the top boundary at a certain

value (x = 0.2 rad/s) marked by red ball shown in Fig. 2.

The specimens are first consolidated to equilibrium under a

specified confining pressure applied on boundary through a

servo-control mechanism. Here, the ‘‘consolidated to

equilibrium’’ simply refers to the process of sample crea-

tion and application of the initial confinement before the

specimens were sheared. All displacement (but not forces,

and hence stresses) were zeroed out after ‘‘consolidation’’

before shearing was applied. It is well recognized that the

initial porosity and packing geometry will have consider-

able effects on the ensuing shear behavior. Therefore, a

uniform inter-particle friction coefficient of 0.0 is used in

the initial consolidation stage for all the simulations to

minimize the variation of initial porosity owing to particle

interlocking. Isotropically compressed specimens were

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

200

400

600

800

1000

1200

1400

1600

1800

2000S

tres

s co

mpo

nent

(kP

a)

Deviatoric strain ε13= ε1−ε3 (%)

σθ

σr

σz

τθz

0 500 1000 1500 2000

0

10

20

30

40 εz

εr

εe

γθz

Str

ain

com

pone

nt (

%)

Deviatoric stress q=σ1−σ 3 (kPa)



0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.20

200

400

600

800

1000

1200

Deviatoric strain ε13=ε1−ε3 (%)

Str

ess

com

pone

nt (

kPa)

τθz

σθ

σr

σz

(a) (b)

0 200 400 600 800

0

10

20

30

40ε

z

εr

εe

γθz

Str

ain

com

pone

nt (

%)

Deviatoric stress σ1−σ 3 (kPa)



Acta Geotechnica

123

prepared for subsequent shear deformation at a constant

mean stress of p’ = 1,000 kPa. The values of the assigned

model parameters, other than those directly measured, were

shown to provide realistic macro- and micromechanical

model behavior. Readers are referred to Li et al. [10] for

detailed information on validation of the DEM model.

2.2 Data analysis

The global stress and strain response were determined by

monitoring the loads applied on the boundaries similar to

the real test. Table 1 lists the equations to calculate the four

components of average stresses and strains obtainable from

the measured boundary forces in the HCA test. To

characterize the variability of the stress–strain response in

the sample, ‘‘measuring spheres’’ are placed in different

sampling locations in the specimen. Figure 3 shows the

‘‘measuring spheres’’ located near the top, middle and

bottom to record the stress–strain response, porosity and

coordination number evolution. To visualize the shear band

initiation and formation, a slice of the hollow cylinder with

3 mm thickness is unwrapped as shown in Fig. 4. The

relationship between the length S measured on the speci-

men and the distance L measured on the viewing plane is:

S ¼ Rsin�1 L

R

The shear band indicated by porosity, shear rate contour

and particle rotation are presented below.

Fig. 9 Maximum shear stress c = (r1 - r3)/2 distribution in the

hollow cylinder specimen at failure states a a = 0�; b a = 30�;

c a = 45�

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5α=0o

α=30o

α=45o

Str

ess

ratio

R=

σ 1/σ3


Fig. 10 Comparisons of stress–strain responses in three typical stress

states. Note The solid lines are measurements in the shear band

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Vol

umet

ric s

trai

n, ε v (

%)

Deviatoric strain ε13=ε1−ε 3 (%)

0o

30o

45o

Fig. 11 Volumetric behavior of specimen in the different stress states

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3 Results and discussion

DEM simulation excels in yielding both micro- and mac-

roscopic data on granular materials, which facilitates the

understanding of granular material behavior from a micro-

mechanical point of view. The following sections focus on

addressing the issues mentioned in the introduction section.

3.1 Macro-response for different stress states

Figure 5 presents comparison between theoretical and

simulated results as an indication of the accuracy of the

control of the principal stress direction a and subsequently

the accuracy of the applied stress paths. It is observed that

a was controlled sufficiently well so as to be consistent

with its prescribed variation during loading. Figures 6, 7,

8a illustrate the variations of the stress components,

including axial stress rz, radial stress rr, circumferential

stress rh and shear stress shz during the tests while shearing

with fixed principal stress directions of a = 0�, 30� and

45�. These stress components are measured from the

boundary loads as in the laboratory test. It is observed that

for a = 0�, the axial stress rz increased with the develop-

ment of strain, while the confining pressure was kept

constant without the torque, which is the loading condition

for triaxial compression test. For a = 30�, torque is intro-

duced. It is observed that the shear stress shz and axial

stress were increasing with the increase in the deviatoric

strain e13 = e1 - e3. The principal stress direction a = 30�is determined by the combination of axial and torsional

loads by using the relationship tan 2a ¼ shz=ðrz � rhÞ. For

a = 45�, the specimen was subjected to changes in cell

pressures and torque only, and no change in axial loading

rz was imposed, which meant the specimen is in pure

torsional shear state. It is observed the shear stress

increased monotonically as the development of strain, and

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2500

1000

1500

2000

2500

3000

3500

4000

Prin

cipa

l str

ess,

kP

a


σ3-MS2σ1-MS2σ3-MS5σ1-MS5σ3-MS8σ1-MS8σ3-MS11σ1-MS11

α=0ο

0.0 0.5 1.0 1.5 2.0 2.5 3.0200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400α=45ο


Prin

cipa

l str

ess,

kP

a

σ3-MS11σ1-MS11σ3-MS8σ1-MS8σ3-MS5σ1-MS5σ3-MS2σ1-MS2

(a) (b)

Fig. 12 Stress components from the measuring sphere under different loading directions

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2-6

-4

-2

0

2

4

6

8

α=0ο

Str

ain

com

pone

nt a

t diff

eren

t dire

ctio

n (%

)


ε3-MS2

ε1-MS2

ε3-MS5

ε1-MS5

ε3-MS8

ε1-MS8

ε3-MS11

ε1-MS11

0.0 0.5 1.0 1.5 2.0 2.5 3.0-50

-40

-30

-20

-10

0

10

20

30

40

Deviatoric strain ε13= ε1-ε3 (%)

α=45ο

Prin

cipa

l str

ain

(%)

ε3-MS11ε1-MS11ε3-MS8ε1-MS8ε3-MS5ε1-MS5ε3-MS2ε1-MS2

(a) (b)

Fig. 13 Strain components from the measuring sphere under different loading directions

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123

the rest of the stresses were kept constant (i.e.,

rr = rh = rz). These three typical stress states are the

most commonly used in HCA testing.

The relationship between strain components and devia-

tor stress q = r1 - r3 are presented in Figs. 6, 7, 8b. The

strain evolution is significantly dependent on the inclina-

tion of principal stress axes during the shearing. The axial

strain ez, circumferential strain eh, radial strain er and shear

strain shz varied along with the deviatoric stress. The radial

strain er and circumferential strain eh were found to be

equal in this series of tests due to the equal inner and outer

cell pressures and horizontal isotropic behavior of the

sample. It is clearly observed that the axial strain ez

decreased with the increase in the principal stress rotation

angle, and the radial strain and circumferential strain

evolved to opposite trend. From a = 0� to a = 30�, the

specimens were compressed along the vertical axis and

expand along the radial direction. When a = 45�, eh and er

were almost equal to zero, and only a small amount of axial

strain was produced until failure. Figure 9 shows the shear

stress of the three different stress states of the hollow

cylinder specimen at failure state. The onset, orientation

and thickness of the shear band will be discussed in next

sections.

Figure 10 illustrates the deviatoric stress versus devia-

toric strain for different principal stress directions mea-

sured inside and outside the shear band. It can be seen that

the mobilized stress ratios are significantly different for the

three typical stress states, with maximum values of 4.0, 3.2

and 2.3. The specimen with a = 0� has the largest shear

strength, while the specimen with a = 45� has the lowest.

This is the result from the combination of inherent and

stress induced anisotropy which is widely discussed in the

literature (e.g., [28, 30]). Firstly, although spherical parti-

cles were used, the initial fabric of the sample was not

necessarily isotropic because of the manner by which the

samples were created and consolidated. Specifically,

gravity was active during the consolidation phase, and

several studies have shown that the effect of gravity during

sample consolidation is to create particle contact normals

that are oriented more in the vertical direction. The samples

were thus inherently anisotropic due to the initially non-

Fig. 14 Porosity evolution versus deviatoric strain for a = 30�

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uniform arrangements of particle contacts, and the

observed direction-dependent response was due to this

inherent anisotropy. At the same time, the inside and out-

side cell pressures were maintained at the same value in all

tests presented. For this condition, the value of b = (r2 -

r3)/(r1 - r3) is related to the angle a between the vertical

and the direction of major principal stress r1, that is

b = sin(2b). Hence, the observed differences between the

stress–strain relationships of the three cases are partly due

to the variation in b-value as well.

For comparison, the results for a = 0� inside the spec-

imen, where no obvious shear band was observed, are very

similar to the data measured from the boundary, while for

a = 30� and a = 45�, the peak stress ratio in the shear

band is slightly higher than those measured from the

boundary and decrease significantly afterward due to the

shear band initiation and formation. For the volumetric

response (Fig. 11), for the medium dense state, the speci-

men with a = 0� experienced some amount of volumetric

contraction of ev = 0.75 % at the beginning of the shear-

ing, then the sample dilated afterward. For the specimen

with a = 30�, only a very small amount of volumetric

contraction was obtained at the beginning stage of shear-

ing. The specimen with a = 45� dilated immediately

without contraction behavior. It is concluded that the

contraction decreased as the principal stress rotation angle

increased. This is in good agreement with the laboratory

test results [26].

3.2 Strain localization

To explore the strain localization in the hollow cylinder

specimen, Figs. 12 and 13 show the evolutions of the stress

and strain components against deviatoric strain measured

in the measuring spheres MS2, MS5, MS8 and MS11,

which are located in the middle height of the specimen. For

the case a = 0�, although there are some fluctuations for

the axial stress component, the average stress is consistent

with the value measured from the boundary. The evolution

of the strain components for the same measuring spheres

exhibit similarity. As a consequence, it is shown that the

localized response in the measuring sphere is similar to the

global response measured from the boundary condition,

which prove that the mechanical response of specimen in

Fig. 15 Porosity evolution versus deviatoric strain for a = 45�

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the triaxial stress state is homogenous. The results also

indicate that stress and strain components in the HCA can

be reliably determined from the boundary loads and dis-

placements for triaxial loading.

To investigate the effects of introducing torque in the

HCA specimen, pure torsional shear (a = 45�) test is

chosen as an example. The stress and strain components

from the measuring sphere are illustrated in Figs. 12, 13b.

It is observed that stress and strain components signifi-

cantly deviate from the average boundary values in MS5

and decrease when deviatoric strain exceeds 0.47 %. This

could be explained that the particle in the MS5 is under-

going shear banding. The evolution of strain components

showed that the localized phenomenon was significant. The

principal strain in MS5 deviates considerably significantly

from the principal stress for deviatoric strain e13 above

1.1 %, which indicates non-coaxiality plastic flow. A

companion paper will discuss this non-coaxiality in-depth.

As a is increased, the stress and strain components deviated

remarkably compared to the results from boundary. This is

mainly due to the localization of the specimen.

To further visualize the spatial distributions of porosity

and shear strain rate in the samples, which are potential

indicators of shear banding initiation and formation,

Figs. 14 and 15 show the contours of porosity at the dif-

ferent levels of deviatoric strain e13. For the case a = 45�in the pure torsional stress state, the porosity contour at

e1–2 = 0.12, 0.30, 0.45, 2.5 % are presented. Porosity

started to localize at the shear strain of e13 = 0.30 % and

localized further afterward. Finally, one dominant then two

minor shear bands were observed. The dominant shear

band from the porosity contour was flatter compared to the

case of a = 30� as shown in Fig. 14. However, the porosity

distribution as a shear band indicator is not sensitive to the

shear band initiation and formation. Saada et al. [25]

indicated that the shear strain rate is a more precise mea-

sure to characterize the shear banding initiation and for-

mation than porosity.

Fig. 16 Shear strain rate versus deviatoric strain for a = 30�

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For the sake of a clear representation, Figs. 16 and 17

show the contours of shear strain rate _c ¼ ð _e1 � _e3Þ=2 in

the specimen from the sliced and unwrapped projections of

the HCA. For a = 30� (Fig. 16), the shear band is clearly

observed at the strain of e13 = 0.8 %, while it cannot be

observed in the porosity contour. For the pure rotation case,

the shear band observed from contour of shear rate is

flatter. Hence, it can be concluded that the shear strain rate

is a reliable indicator to characterize shear band initiation

and evolution.

3.3 Microstructure evolution

3.3.1 Porosity evolution

Figure 18 shows the porosity evolution at specified loca-

tions in the sample at different stress states. For a = 0�, it is

shown that the porosity at different locations evolved sim-

ilarly to each other. The evolution of sample microstructure

in the different measuring spheres also proves that the DEM

results are realistic. It is observed that the porosity firstly

decreased due to sample contraction until around ea = 6 %

and increased afterward due to dilation. Finally, failure

occurred accompanied by bulging of the specimen. For

a = 30�, due to the application of torque, it is observed that

the porosity measured from the sampling spheres contracted

initially and then dilated. Specially, for locations MS5 and

MS9, the porosity significantly increased which indicated

these two locations were shear band. For pure torsional

shear, it is shown that the porosity fluctuated except the

porosity of the spheres in the shear band. The porosity of

MS9 increased significantly until a shear strain of

e13 = 1.5 % and thereafter entered a constant value. It is

probable that the particle in this location rotated without

significant displacement. Meanwhile, the porosity of MS5

increased similarly to that observed in the case of a = 30�.

At e13 = 2.0 %, the porosity of MS2 increased signifi-

cantly. This proves that the inclination of shear band for

a = 45� is flatter and twisted since the sudden change of

porosity occurred in the top, middle and bottom zone.

Fig. 17 Shear strain rate versus deviatoric strain for a = 45�

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123

3.3.2 Coordination number

The coordination number cn is an important indicator of

the stability potential of a granular assembly with overall

stability increasing with larger cn [16]. It is equal to the

number of contacts each particle has with other particles,

and it can be monitored at selected measuring spheres in

different loading stages. To further evaluate the micro-

structure evolution of the HCA samples, cn versus axial

strain curves are presented in Fig. 19. Prior to shearing, cn

values of all the measured spheres are close to 6.8. Once

shearing started, cn for different loading directions behave

in their own characteristics as described below.

For the case a = 0�, cn is increasing slightly from 6.8 to

7.0 and subsequently decreasing drastically especially for

MS2 and MS8. It can be recalled from Figs. 11 and 19 that

the specimen contracted first and then dilated. Initially, the

specimen became denser so that cn increased, then at the

certain value of strain, the specimen dilated and cn

decreased. These results are consistent with observations

from the lab tests. For the case a = 30�, cn decreased from

6.4. It is observed that the cn in the sampling volume MS5

inside the shear band experienced drastic decrease from

6.6 to 4.8. For the pure torsional test, since the component

rz was zero, cn decreased not as significantly as for

a = 30�. It should be pointed out that the cn for MS5 and

MS9, both of which were located in the shear band,

decreased from 6.8 to 5.2 and 4.8, respectively. Observed

from the loading at a = 30� and 45�, although the incli-

nation of the shear band is different, the shear bands were

initiated at the same point. Overall, after introduction of

the torque, the shear band initiated and formed. The cn

curves in the shear band decrease significantly and showed

larger fluctuations. Considering that void space inside the

shear band was significantly larger than that inside the

shear block, the particle had more freedom to move inside

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.300

0.305

0.310

0.315

0.320

0.325

0.330

0.335

0.340

0.345

0.350

0.355

Deviatoric strain ε13 = ε1−ε3 (%)

Por

osity

var

iatio

nMS1MS2MS3MS4MS5MS6MS7MS8MS9MS10MS11MS12

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.34

0.35

0.36

0.37

0.38

0.39

0.40

0.41

0.42


Por

osity

var

iatio

n

MS1MS2MS3MS4MS5MS6MS7MS8MS9MS10MS11MS12

(a) (b)

0.0 0.5 1.0 1.5 2.0 2.5 3.00.33

0.34

0.35

0.36

0.37

0.38

0.39

0.40

0.41

0.42

0.43

0.44


Pro

sity

var

iatio

n


(c)

Fig. 18 Porosity evolution for: a a = 0�, b a = 30�, and c a = 45�

Acta Geotechnica

123

the shear band, and thus, it caused the cn curves to fluc-

tuate substantially.

The shear strength of a granular assemblage is governed

by the stability of the force chains. A decrease in coordi-

nation number reduces the number of force chains, which

may reduce the stability of the force chains and, corre-

spondingly, the shear strength. It could be observed that a

high concentration of the force chains occurs right outside

the shear band but fewer inside, which means the transfer

of load-carrying capacity to fewer force chains inside the

band during shear bands formation. The reduction in

number of force chains is accompanied by the formation

and buckling of column of particles in the shear band.

Several researchers have proved that formation and buck-

ling of such column-like structures. The reduced number of

force chains and the buckling of column of grains lead to

strain softening and the onset of strain localization and

shear band formation [e.g., 1, 2, 14, 20].

3.3.3 Particle rotation

Oda et al. [17] observed that particle rotation is a major

microscopic mechanism which cannot be neglected in the

micromechanics of granular soils. Desrues [4] indicated

that particle rotation should be considered to reasonably

replicate the shear banding formation in lab tests. Bardet

and Proubet [1, 2] showed that particle rotations tend to

concentrate inside the shear band. They found the effects of

particle rotation on the shear band thickness were signifi-

cant. To replicate the microstructure evolution indicated by

particle rotation, this section will address how particle

rotation varies during a different loading conditions and

how it relates to the shear band formation. Particle rotation

is an extra degree of freedom that enhances particle

mobility and has a significant influence on the localized

failure mechanism of idealized granular media as shown by

Oda and Kazama [18], and Iwashita and Oda [8].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.56.0

6.2

6.4

6.6

6.8

7.0

7.2

7.4


Coo

rdin

atio

n N

o.,c

n


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.54.0

4.4

4.8

5.2

5.6

6.0

6.4

6.8


Coo

rdin

atio

n N

o.,c

n


(a) α=0o (b) α=30o

0.0 0.5 1.0 1.5 2.0 2.5 3.0

4.4

4.8

5.2

5.6

6.0

6.4

6.8

7.2


Coo

rdin

atio

n N

o.


(c) α=45o

Fig. 19 Evolution of coordination number cn versus representative strain in different measurement spheres for: a a = 0�, b a = 30�, and

c a = 45�

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123

Figure 20 illustrates particle rotation h in the specimen

with a = 0� at e13 = 0.5, 1.07, 2.5 and 3.0 %. The

amplitude of the particle rotation is evaluated from the

initial particle positions and represented by the color scale.

As shown in Fig. 20, the particle rotations are random near

the boundary at the strain e13 = 0.2 %. As the axial strain

increased, the amplitude of particle rotation also increased.

However, there is no obvious localized particle rotation

observed during the loading, and all of the particle rota-

tions were uniform. The magnitude of particle rotation

during loading is not very large. In the physical test, the

bulging of the specimen is observed in the failure condition

in the triaxial stress state. For a = 30� shown in Fig. 21,

which illustrates the particle rotation h in the specimen at

different shear strain levels, the particle rotation is more

significant due to the torque. With increasing strain, the

particle rotation exhibits some directional tendency and

becomes banded. Afterward, once the band extends all

around the specimen and failure is incipient, the thickness

of the band becomes uniform at around 10 times mean

particle diameter. The evolution of shear band indicated by

the particle rotation distribution is clearly visible. Specifi-

cally, at deviatoric strain of e13 = 0.57 %, the localized

particle rotation is initiated. However, no band was formed.

With increase in the shear strain, the shear band is visually

initiated and evolved afterward. Up to e13 = 2.49 %, there

is one dominant shear band in the specimen without the

restrain of the lateral boundary. This demonstrates that,

during shearing, the particles inside the central shear zone

rotate more significantly (at about 15�) than the particle

outside the band (\2�). Bardet [3] stated that the effects of

particle rotations on the failure of idealized granular

material are significant, which could be considered as a

good indicator of shear band evolution. Recalling the shear

rate contours shown in the Fig. 17, the shear band from

particle rotation is almost coincident with that observed

from shear rate contour.

For the case of pure torsional test shown in Fig. 22,

compared to the compression torsional state, the rotations

of particles were more pronounced. At the deviatoric strain

of e13 = 0.12 %, the particle rotation occurred near the

boundary. At the deviatoric strain of e13 = 0.25 %, the

shear band indicated from the particle rotations is formed

and evolved as characterized by two bands marked by red

in the figure. This demonstrates that during shearing, the

particles inside the shear band rotate more significantly

(about 15�) than the particle outside the band (\2�). Out-

side the shear band, the particle rotation was small. At the

shear strain of e13 = 0.55 %, three dominant bands are

formed. It is observed that the shape of the specimen at the

13=0.5% 13=1.07%

13=2.5% 13=3.0%

Fig. 20 Evolution of particle rotation for a = 0�

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123

end of test is twisted similar to the observation (e.g., [11,

26]). High particle rotations are indicative of strain local-

ization leading to the formation of shear bands.

Li et al. [10] showed the experimental results for the

specimen in the different levels of principal stress rotation.

It is shown that at the failure state of the specimen, the

gridlines marked on the membrane exhibit the twist for the

pure torsion (a = 45�). Compared to a = 30�, the shear

band is flatter with inclination angle dob = 14�. The failure

state of specimen at three typical stress conditions shown in

Li et al. [10] can further prove that the simulations are

reasonable. Table 2 lists the summary of numerical test

results at failure for torsional test. The friction angle,

dilation angle and inclination of shear band are reasonably

in agreement with those from physical tests. The compar-

ison of results shown in Table 2 shows that DEM is an

effective tool to evaluate the macro- and micro- behavior

of sand specimen under complex stress states.

3.4 Thickness and shear band orientation

Ample experimental evidence has demonstrated that the

width of shear bands is related to mean grain size. Roscoe

[22] has shown that the width of the shear band is

approximately 10 times the average grain diameter. Later,

Vardoulakis et al. [33], using X-ray, suggested a shear band

thickness that is approximately 16 times the mean grain

diameter. Oda and Kazama [18] visualized the shape,

thickness and inclination of the shear band. In the present

study, some findings in the numerical simulation reason-

ably support previous published conclusions: a high gra-

dient of particle rotation can be developed within a

relatively narrow zone (Figs. 21, 22), which can be con-

sidered as a criterion to judge the shear band formation.

The thickness of shear band was about 8–10 times the

mean particle size depending on the principal stress rota-

tion. However, the shear band is not uniform and smooth

and extremely large voids were produced in the shear band

from the measuring sphere. From the numerical test, the

thickness of the shear band from high gradient of particle

rotation is around 10 times the mean particle size and

coincident with the observation from the physical test.

Regarding to the inclination of the shear band, Saada

et al. [26] investigated the localization of deformation of

homogeneous sand specimen under complex stress condi-

tions. The initiation and propagation of the shear bands that

developed in the HCA under different principal stress

rotation angles was recorded by digital imaging process.

With variations of a, one or several parallel groups of

surfaces coalesced and resulted in what was referred to as a

dominant shear band. Similarly, Lade et al. [11] discussed

the inclination of shear band theoretically and experimen-

tally. The photographic records of the complete pattern of

shear bands in which the shear band angle with the hori-

zontal was measured directly on the specimen were made

at the end of the shearing test. For a = 30�, one major

shear band and one minor one with inclination angles rel-

ative to horizontal of 24� were clearly observed in the

physical test [26]. This supports the result from numerical

results of dob = 21� shown in Fig. 14. Table 2 summarizes

the theoretical and numerical results for other a-values,

which show that the inclination of shear band from

numerical tests are reasonably in agreement with those

Table 2 Summary of numerical tests results at failure for torsional tests on medium dense assemblage with void ratio 0.68

Test no. Inclination of r1

with vertical, a (�)

b Friction angle,

u (�)

Dilation angle,

w (�)

Inclination of shear band

with horizontal, x (�)

Remarks at

end of tests

1 0 0 35 (35.6) 13 (15.6) – Bulging

2 30 0.25 34 (34.4) 10 (8.8) 24 (26) Shear band

3 45 0.5 33 (31) 9 (9.36) 14 (15.8) Twist

4 60 0.75 32.5 (32) 8.5 (9.2) 0.4 (0.7) Shear band

5 90 1 32 (31.5) 3.0 (4.4) – Necking

Values in the bracket are obtained from physical tests

Fig. 23 Particle rotation distribution for D50 = 0.6 mm (number of

particles = 52,487)

Acta Geotechnica

123

from physical tests. However, the particles used in the

DEM models are coarse relative to the size of the speci-

men. To assess the effect of the particle size on the

thickness of the shear band, a specimen with

D50 = 0.7 mm consisting of 52,784 particles is tested to

observe the effect of particle size on the shear band incli-

nation and thickness. From Fig. 23, it is observed that the

inclination of shear band for both of these cases is almost

the same at around 14�. The thickness of the shear band

observed from the particle rotation distribution is 0.015 m,

which is about 15�D50. For the specimen with smaller

particle sizes, the thickness of the shear band is 0.01 m,

which is also around 15�D50. It is concluded that the effects

of particle size on the shear band inclination and thickness

(as function of D50) almost could be ignored in the

analysis.

4 Conclusions

The behavior of soils under generalized stress conditions

has been a challenging topic.

Constitutive theories for granular media are needed to

model complex stress loading path involving general

changes in the principal stresses and their directions.

Meanwhile, modern testing devices, such as hollow cyl-

inder apparatus, are extremely complicated and the data

obtained are still not reliable. A major problem, with lab-

oratory experiments is the inability to prepare exact repli-

cates of the physical system, a situation that does not occur

in numerical simulations. The results reported in this paper

illustrate that, at least in a qualitative sense, the mechanical

behavior of hollow cylinder specimen in three typical stress

states can be generated by discrete element modeling.

Numerical simulations with DEM can not only monitor the

global mechanical response from boundaries but also be

able to examine the localized response using measuring

scheme at sampling locations in the test specimen, which

may improve the understanding the physics of particle

systems.

The results presented in the paper show that for HCA

specimen in triaxial stress state, the global stress strain

behavior can be considered as the real response observed

from the uniform localized response measured from dif-

ferent locations in the sample despite certain minor devi-

ations. However, after the introduction of the torque, as

principal stress rotation is increased, the response measured

from the boundary is not that inconsistent with the local-

ized response, especially after the strain at which the shear

banding is initiated. Shear band formation was observed

from the shear strain rate, porosity and particle rotation.

Further, the effects of particle size on the thickness and

orientation is examined. It is shown that the particle size

did not affect the orientation. The thickness of the shear

band is related to the D50 of the assembly.

This paper has shown that it is possible to capture using

DEM the essential features of the mechanical behavior of

granular materials under complex stress state. However,

there are still some limitations in this study. The most

important limitation is that spherical particles were adopted

for simplicity which could not consider the effects of

particle shape. One fast way of accounting for non-spher-

ical particles is by the use of rolling resistance. In two-

dimensional DEM, rolling resistance has been shown to

have important effects on the stress–strain and strain

localization behavior of granular assemblies. In three-

dimension, introduction of rolling resistance is more

challenging as this will require the addition of three more

degrees of freedom per particle. Three-dimensional DEM

modeling accounting for rolling resistance will be the

subject of future studies.

Acknowledgments The work reported here is supported by the

National Natural Science Foundation of China (No. 41202186,

11372228), the Zhejiang Natural Science Foundation (No.

LQ12E08007) and partially Ministry of Housing and Urban- Rural

Development of People’s Republic of China (2013-K3-28). The

authors would like to express their appreciation to the financial

assistance.

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a numerical examination of the hollow cylindrical torsional shear test using dem

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