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Effect of particle shape on the mechanical response of a granular ensemble

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The mechanical behaviour of a granular ensemble at a macro scale is an integration of the inter-particle interactions. Even though manifold parameters at multiple length scales govern the behaviour of granular materials, particle shape and size are considered paramount in governing the ensemble level mechanical response such as the strength, stiffness and packing fraction (e max , e min), friction angle etc. In this research programme, an experimental study using hollow cylinder apparatus is undertaken in order to study the effect of particle shape on the ensemble material behaviour. Two model materials are chosen for these series of experiments-angular sand and spherical glass ballotini. The mean grain size of both sand and glass ballotini used in this study is 0.5 mm. Also the specimens are prepared at a relative density of 30% using the same preparation technique, ensuring almost similar initial fabric. These specimens are subjected to a gamut of stresses by varying the intermediate principal stress ratio (b) such that the critical state surface (or yield locus) could be traced on an octahedral plane (in the principal stress space). The angular particles showed increased critical state strength when compared to the glass ballotini at different intermediate principal stress ratio. The volume change in the angular sand particles are seen to be mostly contractive, while at the same relative density, the glass ballotini showed a dilative response. Finally, the particle shape controls the size of the critical state locus on the octahedral plane, in that, the angular particles show an increased size when compared to the spherical particles.
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Effect of particle shape on the mechanical response of a granular
ensemble
Ramesh K Kandasami & Tejas G Murthy
Department of Civil Engineering
Indian Institute of Science, Bangalore, India
ABSTRACT: The mechanical behaviour of a granular ensemble at a macro scale is an integration of the
inter-particle interactions. Even though manifold parameters at multiple length scales govern the behaviour of
granular materials, particle shape and size are considered paramount in governing the ensemble level mechanical
response such as the strength, stiffness and packing fraction (emax,emin), friction angle etc. In this research
programme, an experimental study using hollow cylinder apparatus is undertaken in order to study the effect of
particle shape on the ensemble material behaviour.
Two model materials are chosen for these series of experiments - angular sand and spherical glass ballotini.
The mean grain size of both sand and glass ballotini used in this study is 0.5 mm. Also the specimens are
prepared at a relative density of 30% using the same preparation technique, ensuring almost similar initial
fabric. These specimens are subjected to a gamut of stresses by varying the intermediate principal stress ratio
(b) such that the critical state surface (or yield locus) could be traced on an octahedral plane (in the principal
stress space). The angular particles showed increased critical state strength when compared to the glass ballotini
at different intermediate principal stress ratio. The volume change in the angular sand particles are seen to be
mostly contractive, while at the same relative density, the glass ballotini showed a dilative response. Finally, the
particle shape controls the size of the critical state locus on the octahedral plane, in that, the angular particles
show an increased size when compared to the spherical particles.
1 INTRODUCTION
The mechanical behaviour of granular materials is
very complex due to some intriguing properties such
as dilatancy, shear banding, mean stress dependence
etc. This response of a granular ensemble is also
strongly influenced by the individual particle mor-
phology (such as shape, size, inter-grain interaction,
surface characteristics etc.). Granular materials re-
sist ensemble level loads due to the frictional inter-
actions between the individual particles, and these in-
teractions depend on the particle morphology, which
in turn is reflected in the macro/continuum level re-
sponse. The particle shape also strongly affects the
arrangement/fabric; this fabric also presents an inher-
ent anisotropy to the granular ensemble. The effect of
grain characteristics such as roundness, sphericity and
smoothness have been investigated through experi-
ments and is known to affect the packing (void ratio)
(Cho et al. 2006), internal friction angle (Holtz and
Kovacs 1981), liquefaction potential (Kramer 1996),
compressibility potential (Santamarina & Cho 2004),
small strain stiffness (Cho et al. 2006) etc. Simu-
lations using discrete element method (DEM) have
also contributed to an understanding of individual
particle effects to the ensemble mechanical response
(O‘Sullivan 2010), however, most DEM studies have
used spherical or elliptical particle shape in the simu-
lations.
In order to understand the continuum response of
granular materials, conventional elemental tests such
as triaxial compression, biaxial compression, direct
shear, simple shear etc. are used to delineate var-
ious aspects of the mechanical behaviour. The ef-
fects of particle morphology on the overall mechan-
ical response has been investigated by using various
shapes and roughness of particles in specimens for
these elemental tests. However, the effect of inter-
mediate principal stress and principal stress inclina-
tion, which also strongly influence the mechanical re-
sponse of sands with different particle morphologies
are generally not considered because of requirements
of specialised laboratory equipment. The HCT (Hol-
low Cylinder Torsion) apparatus can independently
control both the magnitude and direction of the prin-
cipal stresses. Bishop (1966) quantified the effect of
intermediate principal stress using a dimensionless
parameter ‘b’ where b= (σ2σ3)/(σ1σ3)which
varies from 0 to 1. Also ‘b’ can be associated with
the Lode angle; due to which variation of ‘b’ can be
thought of as traversing different planes on an octa-
hedral plane in the principal stress space. Lade et al.
(2014) performed a series of tests on sub-rounded
Nevada sand using HCT tests to understand the ef-
fect of ‘b’ on the critical state friction angle. Symes
et al. (1988) have used sub angular sand particles to
obtain a state boundary surface using hollow cylinder
apparatus.
DEM simulations investigating the effect of inter-
mediate principal stresses on peak strength were con-
ducted for particulate systems with different rough-
ness, the particles with increased roughness showed
an increase in peak strength at different intermediate
principal stress ratios (O‘Sullivan et al. 2013), how-
ever comprehensive experimental validation of such
simulations were still lacking.
This paper presents results of HCT tests on two par-
ticulate systems. These two particles are considered
from two extreme corners based on the shape classi-
fication (‘R’ classification system, Powers (1953)). A
series of hollow cylinder tests were conducted under
monotonic drained conditions in order to understand
the effect of intermediate principal stress ratio on the
strength, friction angle at critical state and eventually
to understand the effect of individual particle mor-
phology on the shape of the critical state loci on an
octahedral plane.
2 EXPERIMENTAL
In this research programme a series of drained tests
were performed using HCT apparatus to investigate
the effect of intermediate principal stress ratio. A hol-
low cylinder torsion testing system with fully auto-
mated controls (GDS Instruments Co.,UK) was used
in this study. This HCT apparatus allowed indepen-
dent control over the axial load (W), torque (Mt), in-
ternal pressure (Pi) and external pressure (Po), which
was in-turn used to control the magnitude and direc-
tion of the three principal stresses. By solving the bal-
ance equations (Hight et al. 1983) the average ax-
ial, radial, tangential and shear stresses on an ele-
ment was obtained. The HCT apparatus used in these
experiments was equipped with high precision digi-
tal encoders to measure the average axial displace-
ments, and rotation angles, while the digital pressure
volume controllers (DPVC) was used to measure the
change in the volume of the specimen. The interme-
diate principal stress ratio was related to the principal
stress inclination (b=sin2α) under specific condi-
tions when the specimen was subjected to an internal
pressure equal to the external pressure (Po=Pi). In a
hollow cylinder torsional apparatus, the intermediate
principal stress was always equal to the radial stress
(σ2=σr) (Hight et al. 1983).
3 MODEL MATERIAL
Two model materials with significantly different par-
ticle morphologies were chosen for the study i.e.
spherical glass ballotini and very angular quartzitic
sand with ‘R’ (Roundness) value of 0.99 (well
rounded) and 0.17 (very angular) and sphericity (S)
value of 0.99 (circular) and 0.42 (semi elongated) re-
spectively. Figure 1 shows the scanning electron mi-
crographs of the model materials used in this study.
Glass ballotini and sand have a specific gravity of
about 2.5 and 2.65 respectively and both have a mean
grain size of about 0.5 mm. Both the particulate sys-
tems chosen here were poorly graded with the grain
size distribution (based on ASTM D6913) as shown
in the Fig. 2. The maximum and minimum void ratios
for the very angular sand (emax = 0.97 and emin =
0.53) and spherical glass ballotini (emax = 0.73 and
emin = 0.6) were estimated using ASTM D 4254 &
4253 protocols respectively. From the value of max-
imum and minimum void ratio obtained for the two
material system, the void ratio interval Ie=emax
emin increased with decrease in the sphericity and/or
roundness of the materials akin to the observations
made by Cho et al. (2006).
Figure 1: Particle shape of the model materials (Glass ballotini,
Sand)
Specimens with glass ballotini and sand were pre-
pared such that the relative densities were equal at the
end of isotropic consolidation, i.e. prior to the shear-
ing stage. Water pluviation technique (Vaid and Ne-
gussey 1984, Cresswell et al. 1999) was used to target
a particular packing/density. The dimensions of the
specimen used in this study were 200 mm height, ex-
ternal diameter of 100 mm and internal diameter of
60 mm such that the non-uniformities were kept at
a minimum (Sayao and Vaid 1991). Additionally, the
internal and external pressures during the test were
maintained such that the stress ratio (ratio of external
pressure ‘Po’ and internal pressure ‘Pi’) lies between
0.9 and 1.2 (Hight et al. 1983).
3.1 Tests
The HCT specimens were saturated at a back pressure
of about 300 kPa. Both glass ballotini and sand were
saturated with an effective stress of 100 kPa until the
B-value reached a value of 0.95 or more. Further the
specimens were isotropically consolidated to an ef-
fective stress of about 300 kPa in several stages. At
the end of consolidation, the relative density of the
two material systems was maintained at 34% before
shearing the specimen. Stress controlled monotonic
drained tests were conducted at different values of ‘b
keeping the major principal stress direction to be ver-
tical. As stated earlier, the non-dimensional parame-
ter ‘b’ depicts a particular plane in the principal stress
space. The value of b = 0 - the triaxial compression
plane (σ1> σ2=σ3) and b = 1 - the triaxial extension
plane (σ1=σ2> σ3).
Figure 2: Particle size distribution of the poorly graded material
system with the mean grain size of about 0.5 mm
4 RESULTS & DISCUSSION
A series of 12 HCT tests were performed and the
behaviour of the two model systems under different
loading conditions was presented here. A comparative
study of the two particulate response on its octahedral
shear stress and volumetric strain response due to the
change in ‘b’ was also performed. In addition to the
stress strain response, the specimen were sheared to
severe plastic deformations, the critical state stresses
and the friction angle due to the variation of interme-
diate principal stress were investigated here .
4.1 Stress state
The three normal component stresses (axial, radial
and tangential stress) were changed independently so
as to control the magnitude of the principal stresses
as desired, i.e., allowing a control of the dimension-
less parameter ‘b’. Shear stresses or torques were not
applied in this series of tests thereby keeping the prin-
cipal stress direction vis-a-vis the vertical (α) at zero
throughout the testing process. Figure 3 shows varia-
tion of the three principal stresses at critical state for
different values of intermediate principal stress ratio.
It was found that the major and the minor principal
stresses (σ0
1and σ0
3) decreased, while the intermediate
principal stress increased with increase in ‘b’ as also
evident from the experiments by Zdravkovic and Jar-
dine (1997) and the DEM simulations of O‘Sullivan
et al. (2013). The condition α= 0and b= 0 is sim-
ilar to triaxial compression test where the intermedi-
ate and minor principal stresses are equal throughout
the test and major principal stress increases. Since the
mean effective normal stress is kept constant through-
out the test, σ0
2=σ0
3decreased while σ0
1increased in
this series of tests. Similarly the condition α= 0and
b= 1 is akin to triaxial extension test where the inter-
mediate and major principal stresses are equal and in-
creases throughout the test and minor principal stress
decreases.
Critical state: Critical state was identified when the
deviatoric stress or the octahedral shear stress, and
the volumetric strain reached a constant value un-
der large deformation. In certain cases, the specimens
continuously contract or dilate and also collapse be-
fore reaching a constant volumetric strain. In such
cases, a stress-dilatancy plot as outlined in Been and
Jefferies (2004) was used to identify if the specimens
reached a critical state. When the dilatancy values
were almost 0, the specimen was considered to have
reached a critical state.
Figure 3: The values of σ0
1, σ0
2, σ0
3at critical state for ‘b’ values
from 0 to 1 at 0.2 intervals
4.2 Effect of particle shape on stress - strain
response
The individual particle shape and its effect on the
overall mechanical response of the granular ensemble
was measured through these tests. It was observed
that as the value of ‘b’ increased, the critical state
stresses decreased and reached a minimum value
at b= 1 immaterial of the particle morphology.
Figure 4 and 5 shows the variation of octahedral
shear stress and volumetric strain for sand and glass
ballotini on the plane b= 0 and b= 1 respectively.
Even though the sand particles presents a greater
strength due to its angularity compared to spherical
particles, the percentage reduction (about 16 to 17 %)
in the critical state strength from b= 0 to b= 1 was
observed to be similar in both these systems when
sheared at the same initial relative density and mean
effective stress. At b= 0, the specimens reached a
clear critical state for both sand and glass ballotini
at large strains. The volumetric strain response on
the compression plane (b= 0) showed the angular
sand to be ‘contractive’, however, the spherical glass
ballotini showed an initial contractive response at
small values of octahedral shear strain after which the
response was dilative until it reached critical state.
While, on the extension plane (b= 1), the octahedral
shear stress increased continuously for the angular
sand ensemble and did not show a clear critical state,
the volumetric strains were contractive (it should be
noted that the volumetric strain was greater on the
extension plane than on the compression plane) and
reached a constant value. In case of glass ballotini, the
volumetric strain showed a purely dilative response
and did not show a clear critical state.
Figure 4: A comparative plot showing the variation of octahedral
shear stress and volumetric strain with octahedral shear strain for
the two material systems on the compression plane b = 0 at a
constant mean effective stress (300 kPa)
Figure 5: A comparative plot showing the variation of octahedral
shear stress and volumetric strain with octahedral shear strain
for the two material systems on the extension plane b = 1 at
a constant mean effective stress (300 kPa). Specimens on this
plane did not show a clear critical state.
Specimens were sheared with different values of
intermediate principal stress ratio by varying the
internal and external pressures on a hollow cylinder
specimen in addition to application of axial load. The
unequal pressures (internal and external) applied on
the specimen creates stress non-uniformities across it.
In the tests carried out on the extension plane (b= 1),
pressure ratio (external to internal) exceeds the ac-
ceptable range suggested by Hight et al. (1983). The
test conducted at b= 1 on glass ballotini; violated the
suggested pressure ratio, additionally, the dilatancy
value (Dp) at the end of the test was about 0.15,
after which the specimen collapsed. Hence, a clear
critical state could not be established on this plane for
both the model systems. The stress non-uniformity
(0.9Po/Pi1.2) and dilatancy (Dp<0.05) were
in the acceptable range in case of specimens tested
between b= 0 to b= 0.8and showed a clear critical
state. Even though the specimens on the extension
plane did not show a clear critical state, the stresses at
the end of the test are used in estimating the friction
angle and the critical state locus on the octahedral
plane.
4.3 Effect of particle shape on critical state friction
angle
The friction angle at the critical state (φc) which has
traditionally been used as one of the indicators of soil
properties, was also used to portray the results of this
testing programme. The ‘φc’ is known to primarily
depend on the mean effective stress, principal stress
inclination and packing fraction. At the critical
state, both these model systems showed a lower
value of friction angle (φc) on the extension plane
(b= 1) when compared to the compression plane
(b= 0). In case of the angular sand, the critical state
friction angle increased upto a 0b0value of 0.4 after
which it decreased to a minimum on the extension
plane (shown in Fig. 6). In case of spherical glass
ballotini, the critical state friction angle decreased
with increase in ‘b’.
Figure 6: Variation of critical state friction angle with intermedi-
ate principal stress ratio for the two model material system
4.4 Effect of particle shape on the critical state loci
The shape of the critical state locus was investigated
in this research programme by changing the value
of ‘b’, i.e. exploring the octahedral plane. The vari-
ation of intermediate principal stress ratio (akin to
Lode angle) was used to traverse one sector (30to
30) on the octahedral plane. Figure 7 shows the two
dimensional representation of the three dimensional
stress space where the critical states (the stress state
at the end of the test) were plotted for both the partic-
ulate systems. This representation of the octahedral
plane was obtained by rotating the intermediate prin-
cipal stress so as to coincide with the hydrostatic axis
(σ0
1=σ0
2=σ0
3), while the other two principal stresses
lie on the octahedral plane (Rao and Nott 2008). The
variables a1and a3are defined as shown in the equa-
tions 1 to 2. The parameters a1and a3were normal-
ized by ‘S’ as shown in the equation 3.
a1=((2σ0
1)σ0
2σ0
3)
6,(1)
a3=(σ0
2σ0
3)
2,(2)
S= (σ0
1+σ0
2+σ0
3),(3)
Figure 7 shows the critical state loci obtained
for these two particulate ensembles. From the ex-
periments, only one sector of the octahedral plane
was explored. However using six fold symmetry
the entire critical state loci was constructed (Davis
and Selvadurai 2002). The critical state locus was
benchmarked with Lade0s failure criteria to fit the
experimental results. Lade0s dimensionless constant
η’ were considered as 37.36 and 14.72 to benchmark
the experimentally obtained critical state points of
sand and glass ballotini respectively. The perfectly
rounded particles have a critical state loci akin to the
very angular sand in the shape of a curved triangle.
The experimentally obtained locus does not change
in characteristic shape or form when different mor-
phologies of particulate assemblies were used in the
experiment. As suggested by (Kandasami and Murthy
2014), the shape of the critical state loci was predom-
inantly controlled by the critical state stresses under
different intermediate principal stress ratio, while the
size of the critical state surface was controlled by
the fabric (which has been studied by understanding
the effect of inclination of the principal stresses
α’). The critical state locus of the sand encapsulates
the locus of the glass ballotini when the tests were
conducted under similar conditions of mean stress
(p0) and density. Thus the particle morphology/shape
plays an important role in controlling the size of
the critical state loci and not the shape. The results
of DEM simulations on spherical particles can be
equally well applicable to other particle shapes,
with allowances made for increasing angularity and
roughness, tending to increase the size of the critical
state surface, in other words enhancing the strength
of the granular ensemble.
Figure 7: Reduction in the size of the critical state yield loci due
to the variation of particle morphology
5 CONCLUSION
A study using HCT apparatus was made on two ex-
treme particle morphologies, with an angular sand
and spherical glass ballotini. This study examined the
effect of particle shape on the octahedral shear stress,
volumetric response, critical state friction angle and
characteristics of the critical state loci. The octahedral
shear stress at critical state significantly depend on the
particle morphology, with increase in particle angu-
larity, interlocking of the grains also increases which
elevates the stiffness levels in the ensemble. The crit-
ical state stress ratio (and octahedral shear stress) was
clearly affected by the intermediate principal stress
ratio for both these model systems. In case of vol-
umetric response, the angular sand shows predomi-
nantly contractive response while spherical glass bal-
lotini shows dilation. The effect of ‘b’ on the volumet-
ric strain behaviour was greatly influenced by the an-
gularity of the particles, predictably, its effect (i.e. the
effect of ‘b’) on spherical glass ballotini is almost neg-
ligible. The critical state friction angle varies with ‘b
for both sand and glass ballotini. However the initial
increase in critical state friction angle upto a value of
b= 0.5which was generally observed in sand was un-
seen in glass ballotini. When the stresses at the critical
state were plotted on the octahedral plane, the shape
of the critical state loci obtained was a curved triangle
akin to the Lade0s failure model and was unaffected
by the particle shape. The particle morphology plays
an important role in controlling the size of the criti-
cal state locus. These experiments provide insight into
the manifestation of particle morphology on the me-
chanical response of granular ensembles. DEM sim-
ulations which predominantly use spherical particles
can be extended to granular materials of various mor-
phologies by appropriately accounting for changes in
stresses, as can be quantified through these experi-
ments.
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... Angular and elongated particles have many sharp protrusions (or corners), which interlock with each other to form a larger coordination number and a stronger load-bearing skeleton. Therefore, experimental and numerical studies have shown that angular and elongated soils exhibit larger values of index void ratios, angle of internal friction, dilatancy, constant volume friction angle, and small-strain modulus than rounded and spherical soils [1][2][3][4][5]. ...
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... Particle shape profoundly affects the engineering behavior of coarse-grained soils. Experimental studies have shown that angular and elongated particles exhibit larger values of index void ratios, internal friction, dilatancy, constant volume friction angle, compressibility, and small-strain modulus than rounded and spherical soils [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. With the development of computer modeling in geotechnical engineering, the discrete element method (DEM) has become a popular method to investigate the relationship between micro-level particle morphology and macro-mechanical behavior of granular soils. ...
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