Journal of Applied Mechanics Vol.11, pp.369-376
An elasto-viscoplastic
(August 2008)
JSCE
numerical analysis of the swelling process of unsaturated
bentonite
Fusao Oka1, Huaiping Feng2, Sayuri Kimoto3 and Yosuke Higo4
1FellowMemberDr. of Engng.Prof Dept.of Civiland EarthResourcesEngng.KyotoUniversity(Cl Bd. KatsuraCampus,
Kyotodaigaku-katsura
4 Nishikyo-ku,Kyoto,615-8540)
2Dr. of Engng.Inst.of CivilEng. ShijiazhuangRailwayInstitute(Shijiazhuang,
China,050043),
FormerDr. coursestudentof KyotoUniversity,
3MemberDr. of Engng.AssociateProf.Dept of Civiland EarthResourcesEngng.KyotoUniversity(C1Bd. KatsuraCampus,
Kyotodaigaku-katsura
4 Nishikyo-ku,Kyoto,615-8540)
4MemberDr . of Engng.Research Associate Dept of Civiland Earth ResourcesEngng.KyotoUniversity(C1Bd. Katsura
Campus,Kyotodaigaku-katsura
4 Nishikyo-ku,Kyoto,615-8540)
A numerical
analysis
elasto-viscoplastic
of the swelling
theory.
behavior
of bentonite
is presented
using
an
It is an extension of an elasto-viscoplastic model for unsaturated
soil, which can describe the behavior of macrostructures,
such as change of suction, pore
pressure and degree of saturation. The volume increase of montmorillonite minerals due to
water absorption into the interlayers, is assumed to be a part of viscoplastic volumetric strain.
An internal variable H which controls an increase in water absorption into clay interlayers is
adopted to describe
compaction
overconsolidation
model,
the swelling behavior
boundary
In addition, the internal
surface and the static yield surface. Based on the proposed
a fully coupled soil-water-air
development
of microstructure.
effect caused by swelling of clay unit is described by the expansion of the
finite element analysis is conducted
of swelling pressure. Comparing
the experimental
to study the
results and the simulated
results, it is found that the proposed model can reproduce the effects of dry density and the
initial water content on the swelling behavior.
Key Words: bentonite, elasto-viscoplastic model, swellingpressure,
1. Introduction
Highly expansive soil, such as bcntonitc,is currently
consideredto be suitablebather materialfor the isolationof
waste, e.g., nuclear,industrial,and mining wastes, from the
surroundingenvironmentbecauseof its low permeability.Due
to the swelling property, cracks that may exist in the
surrounding soil and rock can be filled with bentonite.
According to conventional design practices,however, it is
acceptableto assumethat eventuallygroundwaterwill saturate
these bentonitebathers. Therefore,it is importantto evaluate
the swellingpressurethat bentoniteimposeson containersand
surroundingsoil and rock due to groundwaterseepage,as well
as the long-termstabilityof the barrierstructureitself In fact,
determiningthe swellingpressureis an importantaspectof all
high-levelradioactivewaste disposalprojects (e.g., Tripathy,
Sridharanand Schanz2004).
unsaturation
Many attemptshave been made in the past to understand
the swelling mechanism of expansive soils. The
Gouy-Chapmandiffusedoublelayer theory (Gouy 1910)has
been the most widely used approach to relate clay
compressibility to basic particle-water-cationinteraction
(Marcailet al. 2002). Duringthe last few years, a number of
experimentaland theoreticalresearchworks have been carried
out on bentoniteand bentonite-soilmixtures.Swellingpressure
tests on compactedbentonitehave been conductedby some
researchers(Push 1982;Kanno and Wakamatsu 1992).The
relationshipbetween swellingdeformationand the distance
between two montmorillonitelayers was proposed (Komine
and Ogata 1996).Sridharanand Choudhury(2002) proposeda
swelling pressure equation for Na-montmorillonitewhile
analyzing the compression data of slurred samples of
montmorillonitereported by Bolt (1956). However, some
researchers(Mitchell 1993; Tripathy, Sridharanand Schanz
― 369―
2004)have shownthat verylittleinformationis availableon the
use of the diffusedoublelayer theoryfor the determinationof
the swellingpressureof compactedbentonite.
Numericalmodelshave also been proposed to simulate
expansivesoil (Alonsoet al. 1991and 1999;Gens and Aloso
1992) based on the elastoplastictheory. Accordingto their
theory, two levels are distinguishedfor the structure,(1) a
microstructural
levelthatcorrespondsto the activeclayminerals
and their vicinityand (2) a macrostructurallevelthat accounts
for the larger structuralsoil arrangements.The microstructure,
namely,the swellingdomain that expandswhen hydrated,is
thoughtto be water-saturatedeven at high levelsof suction.In
contrast,the macrostructureis assumedto be unsaturatedwhen
subjectedto suction,and its behaviormay be describedby a
conventionalframeworksfor unsaturatedsoils. In addition,a
theoreticalmodel has been proposed by Shuai and Fredlund
(1998) to describe the volume changes during various
oedometerswellingtests.
In the present paper, an elasto-viscoplastic
for
unsaturated
bentonite
elasto-viscoplastic
model
is
developed
for unsaturated
swelling model
based
on
the
soil. An internal
variable H, which controls the growth of the absorption
of
water into the clay interlayer, is introduced to describe the large
volumetric
expansive
behavior
of the microstructure.
namely, solid (S), water (W), and air (G), which are
continuouslydistributedthroughout space. Volume fraction
na is definedas the local ratio of the volume element with
respecttothe totalvolume,namely,
(1)
The volume fraction of the void, n , is written as
(2)
Finally, the material density p,
for a phase is given by
(3)
where
M a is the mass
The
effectives
saturated
soil;
of
stress
for
compressible.
In present
as the
stress
tensor „qij
the
defined
stress
soil
where
the
stress
skeleton
skeleton
variables.
stress"
is obtained
by
effective
study,
as the
"average
values,„qaij,
been
unsaturated
are adopted
same
has
however,
reconsidered
suction
a phase.
from
needs
to
fluids
are
tensor
and
stress
is tha
(2000).
Total
of the partial
stress
namely,
This
model includes the effects of suction and the swelling effect into
the hardening
parameter.
Using the proposed
model,
be
the
Jommi
sum
for
stress
Skeleton
by
the
Terzaghi
(4)
the
(5)
swelling behavior of bentonite is simulated by the multi-phase
finite element method.
(6)
2. Elasto-viscoplasticconstitutive model for unsaturated
swellingsoil
where
gas
Adoptingskeletonstress and suctionas the stress variables,
an elasto-viscoplasticmodel for unsaturatedsoil has been
proposed (Oka et al. 2006; Feng et al. 2006; Oka et al.
2008). Based on this model, three-dimensionalmultiphase
numericalanalysishas been carriedout. The simulationresults
showthat the behaviorof unsaturatedsoil,such as the changes
in poreair pressure,pore waterpressure,and volumetricstrain,
can be simulatedwell with this model (Oka et al. 2008).This
model also can describe the viscoplasticvolumetriccollapse
phenomenon due to decrease of suction. To reproduce the
swellingphenomenon caused by the clay particles, such as
montmorilloniteparticles,the elasto-viscoplasticconstitutive
model for unsaturatedsoil is extendedto be able to reproduce
the volumetric swelling. In the present model, a swelling
equation is proposed to describe the viscoplasticvolumetric
swelling.
2.1 Skeleton stress
and
nG are the
phase,„qij is
average
the
pore
pressure,
volume
skeleton
PF
fraction
stress
is defined
of liquid
phase
and
present
study,
the
in the
as,
(7)
where
Sr is the degree of saturation.
22 Stretching
It is assumed that the strain rate tensor consists of elastic
stretching tensor Deij, the viscoplastic stretching tensor Dvpij,and
an additional viscoplastic
microstructural
stretching tensor
Dyvp(s)ij
due to the
swelling. Total stretching tensor Dij
is defined
in the following equation:
(8)
The elastic stretching tensor
Deij is given by a generalized
Hooke type of law, namely,
(9)
where
The material to be modeled is composed of three phases,
nw
skeleton
•\ 370•\
Sij
is
stress,
the
G
deviatoric
is
the
stress
elastic
tensor,„qm
shear
modulus,
is
the
e0
mean
is
the
initial void ratio, K is the swelling index, and the
superimposeddotdenotesthe timedifferentiation.
2.5 Overconsolidation
BoundarySurface
In the model, the overconsolidationboundary surface is
defined to delineate the normal consolidation(NC) region,
2.3
fb>0, and the overconsolidation(OC) region, fb<0,
follows:
Swellingdue to absorptionof water into the interlayers
of clay particles
From the experimentalresults on bentonite (Komine and
Ogata 1996;Push 1982),it has been shown that the swelling
phaseis followedby an asymptotictendencytowardsa constant
final value. In the model, an evolutionalequation is used to
describethe viscoplasticvolumetricswellingof theparticlesas
(10)
(12)
(13)
where nij* is
denotes
(11)
as
the
the
the state
initial
state
stress
ratio
at the end
before
=•@
tensor
of the consolidation,
deformation
at
the
(nij*=Sij/„qm),
occurs.
critical
and
in other
(0)
words,
M.*m is the value
state
Mf. „qmb
of
is
the
n*
where H is an intemal variable that describes the growth in the
absorption of water into the clay particles, and A and B are
strain-hardening
material parameters.
size of the boundary
Figure
1 shows the swelling equation
and
softening
parameter,
which
controls
the
surface.
curves at various values for parameters A and B. It can be seen
that A is a parameter for the potential of the absorption of water,
2.6
and B is a parameter which controls the evolution rate of H.
Static yield function
To describe the mechanical
behavior
of clay at its static
equilibrium state, a Cam-clay type of static yield function is
2.4 Internalcompactionphenomenon
With the absorption of water into the interlayers,the
distancebetweenthe two plateletsincreasesfrom 15A to 20 A.
Figure2 illustratesthe swellingprocessin the case where the
swelling deformationis restricted.With wetting, the macro
voids graduallybecomespackedwith swollenmontmorillonite
particles.As shownin Figure3, the SEM imageof the wetting
process of the bentonitehas been reported by Komine and
Ogata(2004),in whichthe bentonitecontentis 100%.It can be
seen that the macro voids are finallyfilled up by the volume
increasein bentonite.
As a result,the sample becomes stiffer and has a higher
strength,which is similarto the soil being highly compacted.
Hereafter,the phenomenonis calledthe "internal compaction
effect";it is differentfromtraditionalcompactioninthe changes
inwatercontent
assumed to be
(14)
where „q(s)my
is
which
controls
2.7
Hardening
According
soil
(Oka
the
strain-hardening
rule
to the
in
of viscoplastic
considering
internal
compaction
model
hardening
Eqs.
et al. 2006),
(11)
strain evpij
and
(13)
and
parameter
function.
Feng
are
suction
for
effect
unsaturated
parameters:
assumed
„qm
Pc
to
be
as follows:
(15)
(b)
Swelling equations with different parameters for A and B
•\ 371•\
yield
softening
elasto-viscoplastic
(a)
Fig. 1
and
size of the static
et al. 2006;
b and „q(s)my
function
the
During water uptake
Before water uptake
Platelets
Water and cation
Bentoniteparticles
(beforewetting)
particles
swollen
by absorbing
water into interlayers
Platelets
Fig.2
Benotonite
(Kunigel-V1Void
Bentonite
Process of swelling with restricted deformation
Voids are filled up completely by the
volume increase of bentonite absorbing
water
Process of filling the voids by the volume
increase of bentonite can be observed
Before water supply
After water supply
Under water supply
Fig.3 Swellingbehaviorof bentonite(bentoniteA, Kunigel-V1)(Komine
and Ogata2004)
soil when
material
parameter
decrease
in „qmb
(16)
where Evpkk
are
the
e0
is
viscoplastic
compression
is the
used
the
and
initial
void
to describe
the
assumed
to
volumetric
the
swelling
ratio. „qma
effect
decrease
index,
and
structural
an
increase
In
the
expressed
and
surface,
and
to reference
which
with
present
as the
equals
changes
study,
expansion
the static
controls
yield
suction
the
rate
Pic
of
. Sd
is a
increase
or
in suction.
the
internal
of the
surface
compaction
overconsolidation
effect
is
boundary
as follows:
parameter
degradation,
in
K
respectively,
is a strain-softening
of
with
strain, ă
the suction
which
viscoplastic
(19)
is
strain,
namely,
(20)
(17)
where Evp*kk is the viscoplastic volumetric strain including the
(18)
in which, „qmai
for „qma,
respectively,
controls
strain
and „qmaf
the
is
rate
equal
consolidation
The
the
of
last
terms
suction
effect
and
zero
at
the
the
initial
on
is equal
(15)
and
unsaturated
and
the
a material
changes.
stress „qmbi
in Eqs.
initial
,B is
structural
to
yield
are
final
the
state,
we
(21)
values
parameter
Since
where, r
that
viscoplastic
obtain
are
soil,
in
is adopted to reflect the percentage of the swelling
viscoplastic strain considered, which varies from 0 to 1. r
being equal to 0 means that the swelling viscoplastic strain does
the
not affect the expansion or shrinkage of the overconsolidation
to „qmai.
(16)
special swelling effect taken into account which is defined as
adopted
which
boundary surface and the static yield boundary surface, while
to reflect
Pc
is
and
SI
being equal to 1 means that the viscoplastic
the
swelling
r
volumetric strain has an effect on expansion of the boundary
present
a material
suction
values,
parameter
Pci
denoting
is the reference
the
strength
suction,
ratio
to the saturated
is
surface.
•\ 372•\
2.8
3.1
Viscoplastic potential function
Conservation of momentum
The momentum balance for each phase is obtained with the
The viscoplastic potential function is given by
following equations when the acceleration is disregarded,
(22)
where M*
(27)
is assumed to be constant in the NC region and
varies with the current stress in the OC region as
NC
region
OC
region
(28)
(23)
(29)
where „qmc
intersection
axis.
denotes
of the
the
mean
overconsolidation
skeleton
stress
at
the
boundary
surface
and
the
in which
„qm
water and gas phases
Fi is the body force and the interaction between
DWG and DGW is assumed to be zero.
the interaction between solid phase and other two phases are
2.9
Viscoplasticflow rule
The viscoplasticstretching tensor is expressed by the
following equation which is based on Perzyna's type of
viscoplastictheory(Perzyna,1963;Kimoto& Oka 2005)as
(24)
in which <> are Macaulay'sbrackets;
if fy>0 and
indicatesstrain
rate sensitivity.Based on the experimentaldata fromthe strain
rate constanttriaxialtests,the materialfunctionis givenas
(25)
given as
(30)
in which, g is the acceleration of gravity, kw and k G are the
permeability coefficients for water and air phase, respectively.
The sum of Eqs. (27)-(29) leads to
(31)
Finally, we used the rate type of conservation
of momentum
neglecting body force for updated Lagrangian scheme, which is
given by
(32)
(26)
3. Multiphase finite element formulation for the analysisof
unsaturated bentonite
It is foundthatthe waterabsorbedinto theseaggregatesdoes
not flow,which can be consideredas a part of solid phase. In
contrast, the micro pores between the clay aggregatesare
assumedto be occupiedby free water and air, which can be
describedwithin the frameworkof a macroscopiccontinuum
mechanicalapproachthroughthe use of the theory of porous
media.
Proceeding from the general geometrically non-linear
formulation,the governing balance relations for multiphase
materialcan be obtained(e.g., Ehlers 2004; Loret and Khalili
2000).Using the skeletonstressand suctionas stressvariables,
Oka et al. (2006) and Kimoto et al. (2007) proposed an
air-water-soil coupled finite element model. And a
three-dimensionalnumericalanalysishas been carriedout to
investigatethe triaxialbehaviorof unsaturatedsilt (Feng et al,
2006).
where
32
Sij is the total nominal stress rate tensor.
Conservation of mass
The conservation
laws of mass for liquid and gas phase are
given in the following equations:
(33)
(34)
where
Vwiand VGi are the apparent velocity for water and air,
n is the porosity,
3.3
Di, is the stretching tensor.
Soil water characteristic curve
The relation between suction and saturation is given in the
following equation proposed by van Genuchten (1980)
(35)
where a , m, and n are material parameters and the relation
m=1-1/n
saturation.
•\ 373•\
is assumed.Sre
is an effective
degree
of
4. Simulation results
water contentvalues (SW5, SW6),a larger r value (0.3) is
used to representthe initialhardeningeffect.
In the finite element analysis,an eight-nodequadrilateral
element is used for the displacementand the four nodded
elementis used for the pore pressure.The finiteelementmesh
and boundaryconditionsfor the simulationare shownin Figure
4. The bottomboundaryis assumedto be the wettingboundary
with water pressure of 10 kPa. The other boundaries are
assumed to be impermeablefor water and air. The main
Table 1 Material parameters
parametersand initialconditionsare listed in Table 1. Elastic
shear modulus, viscoplastic parameter C, SI, Sd, air
permeability,and shape factor a and b are assumed and the
other parameters are determined by the existing data
(Horikoshiet al. 2007).For the initialporewater pressure,i.e.,
initialsuctionis assumedto be 100kpawhich is smallerthan the
expectedone. The reasonis thatthe initialsuctionis onlyfor the
initialmatrixsuctiondue to the meniscusbetweenparticles.
Fig.4
Table2 Experimentalcases(Ono,2006)
Finite element mesh and boundary conditions
Figure5 showsthe changesin the degreeof saturationfor
each elementwithwetting.In this analysis,it is assumedthatthe
element starts swellingwhen the saturationreaches a given
value. From Figure5, it can be seen that swellingstartsfrom
bottomelementand movesupwards,elementby element.
To investigate the swelling behavior of bentonite, as
shown in Table 2, swelling pressure tests of bentonite
(KunigeruGX)have been carriedoutby Ono et al.(2006).The
experimentalresultsare shown in Figure 6(a). It is confirmed
that dry densitycontrolsthe finialswellingpressure.Meanwhile,
initialwatercontentaffectsthe typeof swellingcurve.
In the model, parameter A is adopted to describe the
swellingpotential,namely, thy density effect.The parameter
in Eq.(21) is a parameterused to describe ther "intemal
compactioneffect".Consideringthat the initialwater content
also can lead to some degree of swelling.It is reasonableto
investigatethe swellingbehaviorwith differentdry densityand
initial water content by adoptingproper value of A and r .
Table3 liststhe value of A and y used in this simulation.For
casesSW3 and SW6, withhigherdry densities,higherA values
(0.18, and 0.16) are adopted. For cases with higher initial
Fig. 5 Simulation
results
of degree
saturation-time
relation with wetting
of
The predictedswellingpressuresduringwettingprocessare
shown in Figure 6(b). Simulationresult shows that large
parameterA givesthe largemagnitudeof swellingpressure.It is
•\ 374•\
Table 3 Parameter for simulation
seenthatthe effectof dry densitycanbe describedby parameter
A. Accordingto experimentalresults, for samples with high
initialwater content,such as SW5 and SW6, swellingpressure
increasesmonotonically,while a time-softeningbehaviourcan
be observedin the sampleswith low initialwater content,such
as SW2, SW3. As mentionedbefore,parameter y is adopted
to describethe "internalcompactioneffect"in this model.The
softeningcomes from the structuraldegradationeffectby the
parameter B (Kimotoand Oka,2005) . When largervalueof
is
adopted the hardenging behavior ris
significant(SW5,SW6).On the other hand, in the case of
smallervalue of y , the softeningeffect is significantdue to
degradationof materialsis.Thesecasescorrespondto the lower
initial water content cases (SW2,SW3). Simulationalresults
show that by adjusting the value of parameter y we can
reproduce the hardeing effect associated with initial water
contents.For the maximumvalue of the swellingpressure,it is
seen that the higher magnitudeof the swelling pressure is
obtainedby the largevalue ofA.
5. Conclusion
In this paper,an elasto-viscoplastic
model for unsaturated
expansivesoil is proposed.An internalvariableH that reflects
the growth of the absorptionof water into the interclayis
adoptedto describethe expansivebehaviorof microstructures.
An internal compactioneffect affects the expansion of the
overconsolidation
boundarysurfaceand the staticyield surface.
Using the proposedmodel, FEM analyseswere conductedto
simulate the swelling pressure. Parameter A and y are
adopted to describe the swelling potentialand the internal
compactioneffectof bentonite(KunigelGX). Comparedwith
the experimentalresults,it has been found that the proposed
(a)
Fig.6
(b)
Swellingpressure-timeprofile (a) experimentalresults(afterOnoet al.,2006);
(b) simulationresults
•\ 375•\
model can well reproduce the effect of dry density and initial
water content on the swelling pressure.
Acknowledgements
The authors express sincere thanks to Dr. H. Komine of Ibaraki
University and Mr.M.Chichimatsu
of Hazama cooperation for
giving us permission to use valuable data.
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(Received:April 14,2008)
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