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1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
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Peer-review under responsibility of the organizing committee of Implast 2016
doi: 10.1016/j.proeng.2016.12.100
ScienceDirect
11th International Symposium on Plasticity and Impact Mechanics, Implast 2016
Prediction of mechanical response of geomaterials using an
advanced elasto-plastic constitutive model
S. Singha, R.K. Kandasamia, T.G. Murthya,∗
aIndian Institute of Science, Bangalore - 560012, INDIA
Abstract
Phenomenological models using plasticity theory are constantly evolving with the increased computational advancements. This
evolution is inevitable especially for modeling heterogeneous, anisotropic materials systems like granular or cemented granular
materials as newer aspects of their physics come to light. In this study, we present a brief compendium of the evolution of
constitutive models for granular materials. The essence of this research study lies in selecting an appropriate advanced third
generation elasto-plastic constitutive model and checking its efficacy in predicting the various aspects of granular and cemented
granular response. The Lade‘s isotropic single hardening elasto-plastic constitutive model is selected for this study. The model
description is laid out for granular system and the modifications that are made to incorporate the effect of cohesion are also clearly
established. Further, the material/model parameters are obtained from appropriate experimental procedures and a comparison
between the single point integrated model predictions and experimental results are made.
c
2016 The Authors. Published by Elsevier B.V.
Peer-review under responsibility of the organizing committee of Implast 2016.
Keywords: cemented granular materials; ductile-brittle transition; constitutive modelling
1. Introduction and history of constitutive models for soils
The goal of providing a competent infrastructure requires an extensive understanding of underlying components.
With increasing demand on space to build structures, the number of high rise buildings are increasing day by day.
Due to these structures a great amount of load is transferred to the soil through substructure which introduces a
possibility of failure of soil. Therefore a complete understanding of the mechanics of soil, structure, and its interaction
is inevitable. With the advent of high speed computing it is possible to ensure safety and serviceability requirement of
a structure by analyzing and designing it through numerical modeling. These numerical models should be validated,
using a laboratory experiment or in-situ test results, before using them for analysis and design purpose.
Soil is a complex material (non-homogeneous, anisotropic, multi-phase) to predict its response mechanically due to
varied composition (unknown structure, non uniform grain shape and size), loading history, drainage condition, etc. A
satisfactory design of geotechnical structure should take care of safety and serviceability criteria. Safety requirement is
related to failure or bearing capacity while serviceability requirement is related to settlement (mostly elastic). Prior to
∗Corresponding author. Tel.: +0-802-293-3125.
E-mail address: tejas@civil.iisc.ernet.in
© 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the organizing committee of Implast 2016
794 S. Singh et al. / Procedia Engineering 173 ( 2017 ) 793 – 799
advent of high speed computing tools, settlement and bearing capacity were obtained using analytical solutions using
linear elasticity and limit equilibrium or limit analysis, respectively. Hooke’s law, with material parameters - elastic
modulus (E) and Poison’s ratio (ν), was used to calculate settlement assuming soil as linear elastic and isotropic.
Limit equilibrium methods assumed soil as perfectly plastic with failure identified with Mohr-Coulomb criteria. Such
simplifying assumptions often fails to capture the important characteristics of the soil behavior. The next stage of
advancement came through utilizing a non-linear elastic hyperbolic model [1–3] with two parameters. Even though
such a non-linear model was able to capture the stress strain relations for specific loading paths in small strain domain,
it was incapable of predicting the real soil behavior under general conditions (large strain, varied stress paths, drainage
conditions, cyclic loading, etc). This hyperbolic model had no strong theoretical basis like linear elastic isotropic
model. These two aspects of constitutive models (elastic and plastic) were clubbed into elastic perfectly plastic
models with elastic stress-strain relation, failure criteria, associated flow rule. In elastic-perfectly plastic models,
history effect as observed in compression was not taken into account. Therefore it failed to differentiate between
loading, unloading and reloading response. With associated flow rule, model predicted excessive dilation during
plastic shearing. Additionally, model predicted infinite elastic volumetric strain under isotropic compression since
there was no cap along hydrostatic axis.
To overcome drawbacks of elastic-perfectly plastic models, [4] introduced a movable cap along with fixed yied
cone, the movement of cap was due to the volumetric plastic strain hardening of the cap. These models were called
cone-cap models which assumed associated flow rule for the yield cap. The models were capable of capturing
volumetric (p
v) and deviatoric (p
q) plastic strains. Perhaps one of the important developments in soil constitutive
models came in 1958 when [5] introduced a framework called “critical state soil mechanics”. A modified cam clay
model given by [6] based on these critical state concepts assuming an associated flow rule and isotropic hardening/
softening has become one of the most frequently used constitutive models in prediction and design. The important
feature that distinguishes this model from other plasticity models is the dependency on history of the material which
evolves as a function of accumulated plastic strains and is capable of simulating both consolidation and shearing
of soils. These models were able to reproduce a number of facets of real soil behavior including stress-strain
response, volumetric or pore pressure behaviour under drained and undrained conditions. However predictions of
super-critical region, large deformations and cyclic response were not captured accurately. To refine this model
further [7] introduced Hvorslev’s failure surface in place of Cam clay surface in super-critical region. This generation
models used associative plasticity which satisfies stability and uniqueness postulates but predicts excessive dilation.
The next generation of constitutive models were marked with the introduction of true triaxial and hollow cylinder
experiments. These apparatus were equipped to provide a mapping of yield surface, failure surface along with plastic
strain increments and its direction with stress increments. Results from these experiments enabled to introduce
combined hardening of yield cap and cone with p
vand p
q[8–12] whereas previous generation models only considered
hardening of yield cap with p
valong with a fixed yield cone [4,7]. These hardening concepts were extended from
combined isotropic hardening to kinematic hardening. The models of this generation incorporates non-associated flow
rule by introducing plastic potential function to avoid excessive dilatancy. The isotropic hardening models have the
capability to predict the material response under the monotonic load history with irreversible plastic deformation.
The material response is elastic as predicted by previously discussed models with stress state lying within the
current yield surface, which implies models inability to predict the response under radial loading or cyclic loading
history for soils. Significant amount of plastic deformation can be observed in case of cyclic loading even under
elastic limits (of monotonic stress history) which causes the evolution of center along with changes in the shape and
size of the yield surface. Such a response can be modeled using anisotropic hardening i.e. mix hardening models.
To incorporate this, in the recent past the ideas of bounding surface and multi-surface plasticity were introduced
[13,14]. The yield surface of previous models is identified as bounding surface evolving with the internal variable
according to isotropic hardening rules. These models allowed irreversible deformation inside the bounding surface to
simulate the response of soils under cyclic loading. To do so, an inner yield surface was conceived inside bounding
surface which evolves with the mix hardening rule. This inner yield surface demarcates the region of elastic and
plastic deformations (of slightly lesser magnitude in comparison to plastic deformations due to bounding surface). To
evaluate the model response a translation rule and interpolation rules are employed for yield surface and hardening
variables, respectively. Further development in this field came with the introduction of bubble models [15]. In regular
loading unloading experiments it can be observed that the point of unload does not match with the point of onset of
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plastic deformation during reloading. The bubble model tries to address this issue. To this end it has been realised
that addressing one issue affects the other and complexity of the model also increases.
The above discussed models belong to elasto-plastic framework, in which, underlying components are obtained
from curve fitting or phenomenological in nature. Several researchers have tried other frameworks such as rational
continuum mechanics and thermodynamics, etc. The models from these belongs to hypo-plastic models [16,17],
endochronic theory [18], multi-laminate [19], disturbed state model [20] and breakage mechanics models [21].
In this study we examine one of the advanced elasto-plastic constitutive models, Lade’s model [12,22,23] to predict
mechanical response of weakly cemented granular material. Model was established phenomenologically for the
frictional granular materials (such as sand) and extended later to incorporate effect of cohesion or bond between
the cohesionless grains. This model is equipped with failure criteria, yield criteria, non-associated flow rule and
hardening/softening rule. Model has good prediction ability with purely frictional granular materials [22]. The model
surfaces are translated in stress space to adapt bond strength between the grains. We provide a brief description of
model components and model material used along with experiments performed to calibrate and validate the model. A
detail discussion on the performance of model for predicting the response of cemented granular material is provided.
2. Lade’s constitutive model
A single hardening elasto-plastic constitutive model given by [12,22,24,25] is utilized in this study. Components
of the model are described below.
2.1. Elastic stress-strain relation
A non-linear elastic stress-strain relation is employed in this model. Structure of elastic stress strain relation is
similar to linear elastic isotropic solid
dσ=Ced(1)
Where Ceis fourth order elasticity tensor. The young modulus as derived by Lade and Nelson (1988) is given below
E(I1,J2)=MP
a⎡
⎢
⎢
⎢
⎢
⎢
⎣I1
Pa2
+6(1+ν)
(1−2ν)
J2
Pa⎤
⎥
⎥
⎥
⎥
⎥
⎦
λ
(2)
Where I1is the first invariant of stress tensor and J2is the second invariant of deviatoric stress tensor. Pais the
atmospheric pressure and νis the Poisson’s ratio. Parameters Mand λare elastic model parameters.
2.2. Failure criteria
In this criteria, peak stress state is used as failure point which is also used to differentiate between hardening and
softening. The locus of these points (failure criteria F(I1,I3)=0) is given below
F(σ)=fn(I1,I3)−η=I13
I3
−27I1
Pam
=0 (3)
Where I3is the third invariant of stress tensor and m,ηare the failure parameters which controls geometry of the
failure surface.
2.3. Flow rule
Flow used in this model is non-associative with the plastic potential function given by
gp(I1,I2,I3)=ψ1
I13
I3
−I12
I2
+ψ2I1
Paμ
(4)
Where ψ1,ψ2, and μare the plastic potential parameters.
796 S. Singh et al. / Procedia Engineering 173 ( 2017 ) 793 – 799
2.4. Yield criteria and work hardening/softening function
Yield criteria is used to address hardening and softening with state variable as plastic work (Wp).
f(σ,Wp)=f1(σ)−f2(Wp)=ψ1
I13
I3
−I12
I2I1
Pah
exp(q)−f2(Wp) (5)
Where q=αS
1−(1−α)S,q∈(0,1)and Sis the stress level (S=fn
η). h,αare the yield parameters.
The function f2(Wp) is defined as follows
f2(Wp)=Wp
DP
a
1
ρhardening regime (6)
f2(Wp)=Aexp −BWp
Pasoftening regime (7)
Where D=C
(27 ψ1+3)ρ,ρ=p
hand C,pare hardening parameters.
The model was originally coined for cohesionless soils which was modified later to accommodate cohesion between
the grains. To allow the cohesion, original stress space is translated by bond strength (σt=aP
a) along the hydrostatic
axis since cohesion will act as extra confinement. This exercise is fairly common for models of elasto-plastic
framework where cohesion between the grains is treated as extra confinement. Current study focuses on critically
assessing the efficacy of such a treatment.
3. Model material and experimental details
Natural soils are susceptible to additional cohesion due to moisture, silicates, carbonates and presence of organic
matter. The cohesion imparted by these can change the mechanical behaviour of cohesionless soils significantly. Many
a times to provide additional strength and stiffness, cohesion is artificially added to cohesionless grains. The effect of
this cohesion is studied experimentally by choosing an appropriate model material whose characteristics are known
a priori and a uniformity can be maintained among the experiments performed throughout the study. For the current
study, a weakly cemented sand with angular sand and ordinary Portland cement (53 grade) is artificially reconstituted
in the laboratory. Mean grain size of the angular sand (specific gravity =2.65) is 0.45 mm with optimum moisture
content and maximum dry density of 18% and 1.6g/cc, respectively. A hollow cylinder apparatus was employed
to perform tests on specimen with 200 mm height, 20 mm thickness and 100 mm outer diameter. This apparatus is
capable of performing test along different radial stress path unlike triaxial apparatus which can only perform triaxial
compression and extension. In this study, isotropic compression test, unload-reload test, triaxial compression tests are
performed to calibrate the Lade’s constitutive model. Then model is validated using test performed at constant mean
effective stress of 300 kPa and with varying intermediate principal stress ratio (b=σ2−σ3
σ1−σ3). Table 1 presents the model
parameter obtained by calibration exercise. An extensive description of extraction of material parameter is given in
[22,24,25].
Table 1. Model parameters determined from triaxial compression test
Elastic parameters Failure parameters Plastic potential Hardening Yield
νMλamηψ
1ψ2μCphα
0.230 456.886 0.265 1.125 0.105 27.92 0.027 -3.619 2.552 0.000352 1.6 1.0562 0.065
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S. Singh et al. / Procedia Engineering 173 ( 2017 ) 793 – 799
Fig. 1: a: Comparison of predicted and experimentally obtained response of stress strain behaviour at b=0 and mean effective stress (p)of300
kPa b: Comparison of predicted and experimentally obtained response of volumetric behaviour at b=0 and mean effective stress (p) of 300 kPa
Fig. 2: a: Comparison of predicted and experimentally obtained response of stress strain behaviour at b=0.2 and mean effective stress (p)of300
kPa b: Comparison of predicted and experimentally obtained response of volumetric behaviour at b =0.2 and mean effective stress (p) of 300 kPa
4. Result and discussion
Current study focuses on constitutive modeling of weakly cemented granular materials and its prediction ability.
In the calibration exercise, we have utilised experiments on the triaxial plane (σ2=σ3) for extraction of model
parameters. Validation of the model is performed by comparison of experimentally obtained results on octahedral
plane with predicted response. The results from this comparison exercise indicates the accuracy of failure surface,
798 S. Singh et al. / Procedia Engineering 173 ( 2017 ) 793 – 799
plastic potential function, yield surface and its evolution in stress space. Results are plotted between deviatoric
effective strain (qin %) vs deviatoric effective stress (qinkPa), deviatoric effective strain vs volumetric strain (vin %)
as shown in fig 1, fig 2 and fig 3 for b values 0.0, 0.2 and 0.4.
Fig. 3: a: Comparison of predicted and experimentally obtained response of stress strain behaviour at b =0.4 and mean effective stress (p)of300
kPa b: Comparison of predicted and experimentally obtained response of volumetric behaviour at b =0.4 and mean effective stress (p) of 300 kPa
Parameter bor intermediate principal stress ratio takes values from 0 to 1 where 0 reflects compression test
and 1 is for tension test. Figure 1a shows that the stress strain behaviour is predictd quite well but as we move
from compression to tension regime the degree of mismatch increases particularly during softening (fig 2a, fig 3a).
Prediction of volumetric response is poor since the predicted response shows only contraction whereas material shows
dilation after contraction (fig 1b, fig 2b, fig 3b). And other contradiction between predicted and experimentally
obtained response arises from observing that with increasing b material shows more contractive behaviour but the
predicted response becomes more dilative.
This exercise implies that failure criteria is fairly accurate whereas softening rule and plastic potential function do
not stand correct for cemented granular materials. Other cause of mismatch can be the sensitivity of model material
parameters to the extraction process and choice of experiments. In a good constitutive model, material parameters
should be independent of choice of experiments to calibrate the model.
5. Conclusion
Constitutive models are essential components for analysis and design of a structure subjected to body forces,
Neumann and Dirichlet boundary condition. Due to complexity of accurate physical and mechanical characterization
of granular materials in geomechanics, constitutive models are keep on evolving to give better prediction. This study
presents a brief description on evolution of soil models from elastic-perfectly plastic to multisurface plasticity or
bubble models. An advance third genration constitutive model (Lade’s model) with 13 material parameters is chosen
for calibration and validation exercises for cemented granular materials. Model was originally devised for granular
materials without cohesion which has been extended to cemented granular materials by accounting cohesion or bond
strength as additional confinement and translating the stress space along hydrostatic axis. This study comprises of
several triaxial compression tests and constant mean effective stress tests. Model is calibrated using data obtained
from triaxial compression tests, isotropic compression test and unload reload tests. Then stress-strain behaviour
and volumetric response, as obtained from prediction exercise, is validated using experimental results for constant
mean effective test result with b values of 0.0, 0.2, 0.4. From the validation exercise, we conclude that predicted
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stress-strain response is satisfactory in comparison to volumetric behaviour which is not properly replicated through
model integration.
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