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System size effects on the mechanical response of cohesive-frictional gran-
ular ensembles
Saurabh Singh1,,Ramesh Kannan Kandasami1,,Rupesh Kumar Mahendran2, , and Tejas Murthy1,
1Indian Institute of Science, Bangalore, India - 560 012
2Indian Institute of Technology Madras, Chennai, India - 600 036
Abstract. Shear resistance in granular ensembles is a result of interparticle interaction and friction. However,
even the presence of small amounts of cohesion between the particles changes the landscape of the mechanical
response considerably. Very often such cohesive frictional (c-φ) granular ensembles are encountered in nature
as well as while handling and storage of granular materials in the pharmaceutical, construction and mining
industries. Modeling of these c-φmaterials, especially in engineering applications have relied on the oft-made
assumption of a “continua” and have utilized the popular tenets of continuum plasticity theory. We present
an experimental investigation on the fundamental mechanics of c-φmaterials specifically; we investigate if
there exists a system size effect and any additional length scales beyond the continuum length scale on their
mechanical response. For this purpose, we conduct a series of 1-D compression (UC) tests on cylindrical
specimens reconstituted in the laboratory with a range of model particle–binder combinations such as sand-
cement, sand-epoxy, and glass ballotini-epoxy mixtures. Specimens are reconstituted to various diameters
ranging from 10 mm to 150 mm (with an aspect ratio of 2) to a predefined packing fraction. In addition to the
effect of the type of binder (cement, epoxy) and system size, the mean particle size is also varied from 0.5 to 2.5
mm. The peak strength of these materials is significant as it signals the initiation of the cohesive-bond breaking
and onset of mobilization of the inter particle frictional resistance. For these model systems, the peak strength is
a strong function of the system size of the ensemble as well as the mean particle size. This intriguing observation
is counter to the traditional notion of a continuum plastic typical granular ensemble. Microstructure studies in a
computed-tomograph have revealed the existence of a web patterned ‘entangled-chain’ like structure, we argue
that this ushers an additional length scale as well as presents a system size effect.
1 Introduction
In nature, granular materials when stored or under the
action of natural forces, develop inter particle cohesion.
They develop organic or inorganic bonds between the par-
ticles, which changes the mechanical behaviour at the
ensemble significantly. These materials acquire their
strength from cohesive bonds, in addition to the frictional
resistance due to interaction at contacts. For engineer-
ing of these materials, i.e. to understand the mechanical
behaviour, weak c-φgranular materials are treated as a
“continua" with no consideration of dependency on length
scales [1–4]. In case of brittle and quasi-brittle materials
such as glass, rock, concrete etc. scaling is well under-
stood and is considered in any numerical modeling which
is inspired from Weibull’s statistical model [5]. It is well
known that the fabric or structure of c-φgranular mate-
rials change with changes in amount of cohesion/binder
present [10]. This proportion of binder and the arrange-
ment of particles due to the presence of binders, brings
e-mail: saurabh@civil.iisc.ernet.in
e-mail: ramesh.k.kannan@gmail.com
e-mail: dmrupeshkumar@gmail.com
e-mail: tejas@civil.iisc.ernet.in
forth fabrics in c-φmaterials such as a matrix bound struc-
ture (asphalt - wherein individual particles are suspended
in a matrix of the binder), void bound structure (concrete,
mortar, rocks - wherein the voids in-between particles are
filled with the binder) and contact bound structure(weak
c-φmaterials, geological materials such as sandstones). It
has been widely accepted that materials with void bound
structure follow Weibull’s statistical scaling model, in that
with increase in the size of the system (or specimen) the
strength reduces and has been well adapted into the me-
chanics and design of such systems. In this study we seek
to understand if scaling other than the continuum scaling
exists in weak c-φmaterials (i.e. materials with a con-
tact bound structure) with varying system size and particle
size.
2 Background
In order to model weakly cohered frictional particulate en-
sembles (in short c-φmaterials), two broad approaches
have been practiced. In the first approach, models for fric-
tional granular materials were used and modified to incor-
porate additional features exhibited due to the presence of
DOI: 10.1051/
,08007 (2017 ) 714008007
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© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative
Commons Attribution
License 4.0 (http://creativecommons.org/licenses/by/4.0/).
a weak cohesion between the particles. In order to accom-
plish this, utilising the framework of the mathematical the-
ory of continuum plasticity, the features of a plastic model
are modified, i.e. the plastic potential function, yield func-
tion are modified to take care of cohesion by considering
this cohesion as an additional confinement on the ensem-
ble. In other words, the yield surface and plastic potential
surfaces were translated along hydrostatic axis to include
this cohesion or cementation in the material [3, 4]. While
another approach has been less phenomenological, by ob-
serving the governing mechanism and the evolving mecha-
nism in the inter particulate cohesion to reach failure [1, 2].
In a typical study of this kind, [1, 2] assumed that net load
taken by the ensemble can be thought as a combination of
resistance of particles at the contact and bonds which is
true at both micro (inter particle) and macro (ensemble or
continuum) levels. Separate elastic-plastic evolution laws
were considered for two phases and then combined using
the micro-mechanical equilibrium. The model parameters
for such constitutive models were obtained by perform-
ing “continuum" elemental tests on cylindrical specimens
of standard dimension (38x76mm). These elemental tests
can be relied upon to provide an appropriate response of
the material only if the response is that of a true or a typ-
ical "continuum material". In that, the overall mechani-
cal response along with failure strength is not affected by
changing the size of the system under consideration. This
has been clearly documented through simulations and ex-
periments for the case of a purely frictional granular ma-
terial [6]. However, thus far, no study has thrown light
if such continuum assumptions and scale independence
is valid for these weak c-φmaterials (or materials with
a contact bound structure). As stated earlier, in case of
brittle and quasi brittle materials such as (c-φmaterial
with void bound structure) concrete, rock show significant
scaling in the strength of the material. With increase in
specimen size, peak strength (failure strength) decreases.
This appears akin to weibull’s distribution which was ini-
tially coined for failure of a chain like structure with links.
Where failure of a link is considered as failure of the en-
tire chain. With increasing number of links probability of
failure increases which resembles the behaviour of these
quasi-brittle materials. That is, with increase in size of a
specimen, the structural strength decreases (or the proba-
bility of failure increases). Bazant et al.[7] has provided
an extension of this in a framework of fracture mechanics.
In recent times, discrete element simulations for weak
c-φmaterials have been carried out [8]. The results show
a systematic transition from elastic behaviour followed by
bond breakage which in turn leads to dilation, as a result of
the frictional rearrangement of the particulates. A change
in the stress distribution from web patterned force chains
to columnar structure of force chains was also shown from
these simulations. A computed tomographic scan image of
an initial contact bound structure is shown in the figure 1.
3 Experimental
In these experiments, we used two model granular mate-
rials - sand and glass ballotini and two materials for im-
Figure 1. Computed tomography image of a weakly cemented
specimen (glass ballotini +1% epoxy) showing a contact bound
structure. The inset shows the chain like cemented structure
Figure 2. Artificially reconstituted weakly cemented (sand +4%
cement) with specimen/system size varying from 150 to 10 mm
dia keeping the particle size, density and aspect ratio constant
parting inter grain cohesion - cement and epoxy, in effect,
we used three c-φsystems sand+cement (cemented sand),
sand+epoxy (SE) and glass beads+epoxy (GBE).
For the case of cemented sands, specimens were re-
constituted by mixing quartzitic sand (specific gravity 2.65
and mean grain size 0.45 mm) with ordinary Portland ce-
ment (OPC) at a density of 1.4 g/cc. Three types of ce-
mented sand specimens were prepared using 2%, 4%, and
8% OPC by weight of sand. For each specimen the ratio
of height to diameter (aspect ratio) was kept constant at
two. To understand the system size effect specimens with
diameter of 10 mm,20mm,38mm, 100 mm, 150 mm were
prepared. All the specimens were prepared to a desired
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Figure 3. Artificially reconstituted weakly cemented (glass bal-
lotini +1% epoxy) specimens with particle size varying from 0.5
to 2.5 mm dia while keeping the system size and density constant
packing fraction and were cured under moist conditions to
achieve a contact bound structure and a characteristic com-
pressive strength. Figure 2 shows the prepared specimen
at different diameters.
For GBE, specimens were prepared by thoroughly
mixing glass beads (specific gravity 2.5) with 1% epoxy
uniformly to create ensembles of density of 1.5 g/cc.
Three types of specimens with particle diameter 0.5 mm,
1.0 mm, 2.5 mm were chosen to understand the particle
size effect on the failure strength of weak c-φgranular ma-
terials. Specimens were cured by heating in oven at 50◦C
for 48 hours. Figure 3 shows the specimens prepared to
study particle size effect. Similar procedure was followed
to prepare SE specimen.
All the uniaxial compression tests were performed at a
constant strain rate of 0.5% per min.
Figure 4. Stress strain response for cemented sand sample with
specimen diameters of 10 mm,20mm,38mm
4 Results
A typical stress-strain plot is shown in figure 4 with vary-
ing specimen dimension (or with increasing number of
Figure 5. The peak strength increases with increase in the per-
centage of cementation as well as increase in the number of par-
ticles across the cross section or as the system size increases
particles) for SE specimen. The increase in specimen di-
mension not only reflects in the change of peak strength
but also other features of stress behaviour such as elastic
stiffness, failure stress, hardening, and softening are sig-
nificantly affected. Other parameters such as mean parti-
cle size (0.45 mm), density (1.4 g/cc), binder content (4%)
were kept constant for studying system size effect.
This effect of system size on cemented sand specimens
was studied for various amounts of cohesion of 2%, 4%,
8%, it should be noted that even with increase in the binder
content, a contact bound structure was ensured. A plot of
peak strength with number of particles (or specimen di-
mension) is presented in figure 5 along with a linear fit
for each binder content. With increase in binder content
normalized peak strength with number of particles (slope
of fit lines) increases. This linear fit can be interpreted
as the number of particles directly controlling the failure
strength.
Figure 6 shows a typical stress-strain curve for GBE
with 2% binder content. For understanding the effect of
particle size; the density of the ensemble, specimen di-
mension and binder content were kept constant. With in-
creasing particle diameter the failure strength correspond-
ingly decreases. We refer to this as the particle size effect.
We further studied this by varying binder content, how-
ever, ensuring a contact based structure. A plot of peak
strength is presented for 1% and 2% epoxy in figure 7. The
peak strength is shown to increase with increasing num-
ber of particles and binder content. The normalized peak
strength with number of particles appears to be constant.
Similar results were obtained for sand+epoxy specimen
with changing mean grain size.
5 Discussion
Brown et. al [9] present a set of experiments on granu-
lar chains, especially highlighting the shear stiffening of
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Figure 6. Stress strain response for GBE with changing particle
dia for specimen diameter of 38 mm
Figure 7. The peak strength increases with increase in the num-
ber of particles across the cross section or as the particle size
decreases
random packed granular chains. In order to explain the in-
teresting phenomenology of their experiments, the authors
suggest that the increase in the length of a granular chain
contributes to increased entanglement, in turn contribut-
ing to higher shear stiffening (or hardening). A similar
increase in failure strength with specimen size (or number
of particles) is observed in our experiments. We conjecture
that the structure of weak c-φmaterials can be construed
as an ensemble comprising of multiple granular chains.
Given that the structure of the ensemble is complex, we
further suggest that this c-φensemble can be modelled as a
series of knotted (entangled) granular chains. Brown et al.
[9] further defined a minimum loop circumference (mlc)
which was a function of particle diameter. With increase
in the particle diameter the minimum loop circumference
decreased for a given chain length whereas this minimum
loop diameter also increases as the particle size decreased.
The entanglement, or the minimum loop circumference
governed the overall mechanical behaviour of the ensem-
ble. If the chain length was smaller than mlc then material
behaviour resembled a typical granular ensemble. If chain
length was sufficiently larger than the mlc then significant
strain stiffening was observed within a test along with an
increase in peak strength as the chain length increases.
For weak c-φmaterials considered in our experiments,
system size or the particle size both affect the length of
a “hypothetical granular chain". As the system size in-
creases, we further conjecture that the lengths of these
granular chain structures also increases, in other words,
the possibility of obtaining more entangled structure in-
creases. Similarly with increase in particle size although
the chain length would remain the same, however, the
equivalent mlc decreases which further causes reduction
in entanglement measure.
Figure 5 and figure 7 represent the system size and
particle size effect, respectively. The results are plotted
between the peak strength and number of particles to sup-
port our argument of minimum loop circumference.
We present exploratory set of experimental results, and
the first evidence of the existence of a second length scale
beyond the traditional continuum length scale in weakly
cohered-granular ensembles. Ongoing work is implement-
ing these systems through extensive tomography for un-
derstanding the microstructure, and the manifestation of
this microstructure at the ensemble level.
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