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Mechanics and modeling of cohesive frictional
granular materials
S. Singh1, R. K. Kandasami2, T. G. Murthy3
1PhD student, 2Research Associate, 3Assistant Professor, Department of Civil Engineering,
Indian Institute of Science Bangalore, 560012, INDIA
In nature, weakly cemented granular materials are encountered in the
form of soft rocks such as limestone, sandstone, mudstone, shale, etc. The
mechanical behaviour of these materials is quite different from the purely
frictional granular materials. The presence of cementation between the grains
causes a significant variation in mechanical response under complex
boundary conditions. In order to understand the manifestation of this
interparticle cohesion at the ensemble level, we have used a hollow cylinder
torsional testing apparatus which is capable of independently controlling the
magnitude and the direction of the three principal stresses. From this
experimental programme, the small strain response, peak strength and post
peak behaviour with changing intermediate principle stress ratio (b) and
initial mean effective stress (I1) is studied. In addition to the analysis of stress
strain behaviour at different b and I1, stress-dilatancy characteristics of these
cohesive frictional material are also discussed. This experimental study is
followed by calibration and validation of a single hardening constitutive
model which considers cementation as additional confinement. Observations
from validation exercises suggest that this consideration works well for stress-
strain response whereas it fails to predict the volumetric behaviour.
Introduction
Natural sands subjected to saline environment over a long period of time de-
velop cementatious bonds between the grains due to precipitation of organic and
inorganic matter (Clough et al., 1981; O'Rourke and Crespo, 1988; Santamarina et
al., 2001; Mitchell and Soga, 1976). Other instances of cohesive-frictional materi-
als are seen in many other geo-engineering applications, for example, geomaterials
are stabilized using cement in high way application for strengthening subgrade, for
canal lining, in earthen dams and for other earthen fills. Cement is also used to
improve the liquefaction resistance of contractive sands. Presence of these cohe-
sive bonds changes the mechanical behaviour of sands under different loading
conditions. When sheared, these cemented sands shows a more rigid and contract-
ant behaviour in comparison with reconstituted uncemented sands under similar
test conditions. At small strain, these cemented sands show a pseudo-equilibrium
2
state or metastable state (since at larger strains, inter particulate cohesive bonds
are destroyed and a reduction in the ensemble stress is seen with further shearing).
Obtaining undisturbed cemented sand specimens for testing in the laboratory is
extremely challenging, because of which most laboratory studies of soft rocks and
cohesive frictional materials have involved artificially reconstituted “weakly ce-
mented” sands (Clough et al., 1981; Coop and Atkinson, 1993; Huang and Airey,
1993). To study the mechanical behaviour of these weakly cemented sands, direct
shear, triaxial and simple shear test were carried out with varying initial density,
confining pressure, amount and type of cementation etc. (Coop and Atkinson,
1993; Leroueil and Vaughan, 1990; Huang and Airey, 1998; Lade and Overton,
1989; Airey, 1993; Cuccovillo and Coop, 1997; Menendez et al., 1996; Abdulla
and Kiousis, 1997; Ismail et al., 2002; Schnaid et al., 2001). These studies suggest
that the presence of cement or cohesion between particles enhances elastic stiff-
ness and peak strength with reduction in ductility. The volumetric response shows
more dilative behaviour in comparison to pure sand under similar conditions. With
further increase in the cementation, it is observed that the peak strength increases
and the strain required to mobilize this peak strength decreases. In the post peak
behaviour a transition from brittle to ductile mode of failure is observed with i n-
creasing confining pressure and decreasing density.
We present results of conventional triaxial compression tests and hollow cylin-
der tests performed on a reconstituted weakly cemented sand ensemble at different
confining pressures and intermediate principal stress ratios, respectively. The
stress-dilatancy plots obtained from these experiment is discussed next which is
followed by a study on the performance of a single hardening elasto-plastic consti-
tutive model to predict the stress strain and volumetric response of these weakly
cemented materials.
Experimental
A series of HCT experiments on reconstituted weakly cemented sand specimens
are performed in this programme. These specimens are prepared by mixing sand
with 4% of ordinary Portland cement (53-grade) at optimum moisture content of
18% to a density of 1.5 g/cc (through static compaction). A hollow cylinder mould
(inner diameter - 60 mm, outer diameter - 100 mm, height – 200 mm) was used to
cast the specimens. After preparation, specimens were cured under moist condi-
tion for 14 days.
This experimental study is performed using hollow cylinder torsional shear ap-
paratus (GDS, UK) shown in the figure 1. Unlike conventional triaxial shear appa-
ratus, hollow cylinder apparatus is capable of performing tests along different
stress paths, not only on the triaxial plane but also on the octahedral plane, identi-
fied by intermediate principal stress ratio
. This study comprises of
conventional triaxial experiments (with constant confining pressure of 50, 300,
450 kPa) and test performed at constant mean effective stress of 300 kPa with b
3
ranging from 0.0 to 1.0 at an interval of 0.2. In all the experiments, specimens
were saturated with an effective stress of 20 kPa and isotropically consolidated to
desired effective stress before shearing under drained condition at a strain rate of
0.5% per minute.
Fig. 1: An image of HCT apparatus with specimen inside HCT cell
The peak stress state and the critical state were identified in all the tests. The
results are analysed in the critical state soil mechanics framework. We further uti-
lize these experimental results for calibrating a third generation constitutive mod-
el, details of which are present in the ensuing.
Description of constitutive model
In this study, Lade’s single hardening constitutive model (Kim and Lade, 1988;
Lade and Kim, 1988a; Lade and Kim, 1988b) is calibrated with the results of con-
ventional triaxial experiments and further validated with tests performed at a con-
stant mean effective stress of 300 kPa with b ranging from 0.0 to 1.0 at an interval
of 0.2. The details about the model and extraction of model parameters are provid-
ed in Kim and Lade, 1988; Lade and Kim, 1988a; Lade and Kim, 1988b. Model
parameters for artificially weakly cemented sand used in this study are provided in
Table 1.
Table 1: Lade’s model parameters for artificially weakly cemented sand
Elastic
Parameters
Failure
Parameters
Plastic Potential
Parameters
Hardening
Parameters
Yield
Parameters
M
a
m
C
p
h
0.23
456
0.27
1.14
0.19
27
0.013
-3.3
2.59
8.9E-5
2.24
0.069
1.42
4
Results and discussion
In this study, two sets of tests were performed as discussed in experimental
section. A series of conventional triaxial compression tests with varying confining
pressures are performed. The mechanical behaviour of these cohesive frictional
granular materials obtained from these experiments is presented in figure 2 with
deviatoric effective stress (q) vs. deviatoric effective strain () and volumetric
strain () vs. deviatoric effective strain plots. Deviatoric effective stress (q), devi-
atoric effective strain (), volumetric strain () and mean effective stress (p) are
defined as follows:
,
, ,
Where, are first invariants of stress and strain tensor, respectively. ,
are second invariants of the deviatoric stress and strain tensor.
Fig. 2: Stress strain and volumetric strain plots for confining pressure () of 50 kPa, 300
kPa, and 450 kPa
Fig. 3: Stress strain and volumetric strain plots for intermediate principal stress ratio (b) of
0.2, 0.4, 0.6, 0.8 at a constant mean effective stress of 300 kPa (mean effective stress of 300
kPa was kept constant throughout the test.)
5
From figure 2, it is observed that elastic stiffness along with peak strength of
cemented sand specimen increases with increase in confining pressure whereas
volumetric response becomes increasingly contractive. Ductility of the specimen
also increases with increasing confining pressure.
Second set consists of experiments performed at a constant mean effective
stress of 300 kPa with b of 0.2, 0.4, 0.6, and 0.8. The results of stress strain and
volumetric response are plotted in figure 3. For test preformed at constant mean
effective stress, elastic stiffness is independent of b, which implies that specimen
is isotropic, although peak strength varies with changing b. With change in ‘b’
from the compression zone (b = 0–0.2) to tension side (b = 0.8–1.0), peak strength
decreases and material response changes from ductile to brittle behaviour. How-
ever, the volumetric response is not significantly affected with intermediate prin-
cipal stress ratio. As a general trend, the specimen initially contracts following
which it dilates with progress of shear.
Fig. 4: Stress dilatancy plots for b of 0.2, 0.4, 0.6, 0.8 and confining pressure () of 50 kPa,
300 kPa, and 450 kPa
Figure 4 shows stress ratio
vs. plastic dilatancy
plots for
constant mean effective stress tests with different intermediate principal stress ra-
tio and varying confining pressure. Typically, for granular materials, the point of
maximum dilatancy coincides with point of peak stress ratio. However, in case of
weakly cemented granular material there is a lag between these two states. It is
apparent that material initially contracts and then dilates before reaching a brittle
failure or zero dilatancy state (critical state). Maximum plastic dilatancy remains
unaffected with changing intermediate principal stress ratio whereas peak stress
ratio decreases with increasing b. Mode of failure for specimen tested at lower b
and higher confining pressure is ductile since final plastic dilatancy reaches a near
zero value. However for higher b and lower confining pressure failure mode is
brittle, in which, failure is accompanied with strain localization or shear banding.
6
Fig. 5: Comparison of experimental and model response at intermediate principal stress ra-
tio of 0.2, 0.4, and 0.6
The behaviour of weakly cemented sand is intriguing as it has facets of behav-
iour that resemble both a cohesive material such as concrete/rock and a typical
frictional ensemble such as sand. At low strains the network of soil and cementa-
tious bonds resist the load with initial mobilization of strain occurring due to the
breakage/damage in the bonds since the sand grains are significantly stiffer and
stronger than cohesive/ cementatious bonds. For traditional cohesive materials like
soft rocks or concrete wherein the cohesive matrix dominates the behaviour
(Moavenzadeh and Kuguel, 1969; Van Mier, 1984), the fractures lead to a drastic
brittle failure, while in case of weakly cemented sands resistance beyond the peak
brittle failure is provided from particle rearrangement (dilation). With further
shearing material reaches a state where resistance comes from pure friction.
The volumetric response at low strain level remains contractive till the start of
bond breakage, following which the sample dilates (and records a drop in
strength). Further increase in strain level brings forth dilation due to particle rear-
rarrangement (and increase in the strength). When these cemented sand samples
are sheared, at the microscale, a slight densification of bonds, breakage of bonds,
particle rearrangement, and eventually localization mobilized only through friction
occur simultaneously, in that no clear demarcation of strain magnitudes or thresh-
old stress values can be obtained here. This simultaneous occurrence of bond
breakage and dilation at the interparticle level manifests itself as a peak in stress-
strain response as has been observed through some post deformation studies
through microscopy and tomography (Kandasami et al., 2016).
These experimental results are used to benchmark Lade’s model for weakly
cemented sand. This constitutive model was originally suggested for purely fric-
tional granular materials, which was subsequently extended to cohesive frictional
granular materials by translating the stress space along the hydrostatic axis to pro-
vide extra confinement offered by the cohesive bonds. In this study we have used
single point integration to validate the model for elemental test response rendered
7
by hollow cylinder testing. Results from the validation exercise are plotted in fig-
ure 5 for different intermediate principal stress ratio. It can be seen that the model
prediction response matches well with the experimental results for stress-strain
behaviour whereas prediction of volumetric response is not so satisfactory. We be-
lieve that the reason for this mismatch is because of addition of extra confinement
for weakly cemented granular materials in the elasto-plastic constitutive model.
This addition of confinement to a purely frictional material shows higher peak
strength while at the same time becomes more contractive. In the suite of experi-
ments conducted, the weakly cemented granular material shows higher peak
strength but relatively dilative response due to presence of cementation bonds.
Conclusions
The mechanical behaviour of weakly cemented sand is studied using conven-
tional triaxial compression test at different confining pressure. Further a set of ex-
periments were performed on octahedral plane of stress space by keeping the
mean effective stress constant and with changing intermediate principal stress ra-
tio to traverse the behaviour along different stress paths other than just compres-
sion or tension. For this purpose a hollow cylinder torsional shear apparatus was
used which is capable of independently controlling the 4 components of stress ten-
sor (). The results of these experiments are presented on stress-strain
and volumetric strain plots. The initial stiffness of the weakly cemented sand was
found to be invariant of intermediate principal stress ratio. The peak strength de-
creases from compression (b = 0) to tension (b = 1) path zone which mobilizes at
lower strain levels. A transition of ductile to brittle failure mode is observed with
increase in b. Further these results are analyzed using stress-dilatancy plots.
Next, an advanced single hardening constitutive model is calibrated using con-
ventional triaxial compression tests. Further, this is used for model validation ex-
ercise using experiments performed on octahedral plane. Lade’s model, used in
this study, was originally suggested for purely frictional granular materials. For
cemented granular materials, a translation of stress space is performed to allow ex-
tra confinement offered by cemented sand in modeling. The result of this valida-
tion exercise shows that model is capable of predicting stress response satisfactori-
ly but does not perform well in capturing the volumetric behaviour.
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