Journal of Applied Mechanics Vol.11, pp.389-398
Modeling
of Microstructural
(August 2008)
Evolution
Variation
JSCE
to Simulate
of Kaolin
Undrained
Shear
Strength
Clay
N.H. Minh *,M. Oda**,K. Suzuki*** and L.C. Kurukulasuriya****
* Doctoralstudent
, Dept of Civil & Environ.Eng., SaitamaUniversity(255 Shimo-okubo,Sakura-ku,Saitama338-8570)
**Professor, Dept of Civil& Environ.Eng., SaitamaUniversity(255 Shimo-okubo,Sakura-ku,Saitama338-8570)
*** Assoc. Professor,Dept.of Civil& Environ.Eng., SaitamaUniversity(255 Shimo-okubo,Sakura-ku,Saitama338-8570)
**** SeniorLecturer
, Dept of CivilEng. UniversityofPeradeniya (Peradeniya,20400, Sri Lanka)
The
microstructure
described
deformation.
angle Į
on constitutive
classical
plasticity
shear
strength
of an
evolution
variation
Kaolin
is, in general,
and Carillo1),
The undrained
the inclination
study
of soils
by Casagrande
shear
modeling,
of Kaolin
rule
The effects
shear
tensor.
strength
soils,
with
quantity
of inherent
the "inherent"
responses
changes
the
fabric
tensor
of microstructure
and
induced
that
in plane
Words:
inherent
anisotropy,
induced
anisotropy,
1.Introduction
experiments
fabric
tensor,
can
for
which
was
soils is commmonlyclassified into two categories, i.e., inherent and
soils one-dimensionally
inducedanisotropies
(Casagrande
and Carillo1)).
The formeris
such
concerned with the anisotropy developed during the sedimentation
consolidated
process under gravity, while the latter is mainly concerned with the
anisotropy arising from the evolution of microstructure associated
interesting
observation?
yet
presented.
with plastic deformation after sedimentation. From a micro-structural
subsequent
point of view, however, both are formed by preferred orientation of
constituent elements, such as particles, voids, and contact surfaces
such
(Oda4)).
Furthermore,
SatakeandOdaet al.6)haveshownthatthe
et al.9)), the
preferred
terms
deformation
which
changes
of
plays
these
quantity
anisotropy
soils.
a dominant
considerably
For
the
elements
called
the fabric
responses
example,
the
on
be
tensor.
quantified
undrained
analyses
the
inclination
by
The structural
in both the strength
role in stability
depending
can
shear
strength
cu,
of soil foundations,
angle Į
of
―389―
all soils
condition.
However,
might
know,
the
be
were
How
on
of
natural
1). Importantly,
one-dimensionally
can
no valid
initial
closely
and
each
tests
(Fig.
the
a relation
Duncan
patterns,
triaxial
in the field
though
such
soils.
different
in undrained
As far as we
study
we
explain
this
explanation
has
microstructure
related
on the modeling
microstructure
which
with Į,
approximately
modeling
even
that
to
Kazama3);
to the
and
its
formation
of
patterns.
of a constant
plotted
the
three
(normal
Seee7),8);
for different
consolidated
a similar
plane
and
importance
that
found
evolution
In a previous
strength
showed
exist
different
consolidation
Duncan
be different
individually
under
anisotropy,
and plastic
example,
differences
strength
It is of particular
Seed8),
been
the
(e.g.,
microstructure of the soil skeleton becomes anisotropy. Anisotropy in
yields
well
consolidated
shear
against
their faces perpendicular to the direction of consolidation so that the
anisotropy
the
in terms
simulate
of normally
axis)
Studieson the microstructure
of clays (e.g., Kazama3); Kurukulasuriya2)).
Kurukulasuriya2))
haveshownthatplatyclayminerals
tendto align between cu and Į
a tensorial
can
undrained
direction
consolidation
introducing
into
of undrained
are considered
model
on
In the present
is incorporated
anisotropies
strain
plastic
depending
plane.
on the variation
the proposed
loading
of
greatly
senses
and
clay by Kurukulasuriya2).
Key
orientation
strength
to the consolidation
called
It is shown
and "induced"
for both
for example,
respect
the effects
observed
in both
anisotropic
of clayey
a tensorial
fabric
yield
direction
to simulate
clay.
of the
of undrained
strength
of the loading
framework
anisotropic
which
of Kaolin
fabric
leads
is
bilinear
tensor.
clay
relationship
microstructural
to
for Kaolin
is
(Minh
account
clay.
shear
successfully
Cu and Į
extended
in
of inherent
of undrained
simulate
between
anisotropy
into
the modeling
increase
sufficient
anisotropy
is taken
However,
to a monotonic
not
in Fig. 2 by Kurukulasuriya2)
of
of inherent
as
the
being
In this study,
such
that
the
the
effects of both inherent and induced anisotropies can be taken into
The sample was next placed in a plane strain test apparatus and
account with the introduction of an evolution rule of the fabric tensor
consolidatedunder an isotropicconfiningpressurepo. After isotropi
during the shearing process as well as its stationary values at the initial
consolidation,the sample was sheared under the undrained plane
and ultimate conditions. The simulated results using the proposed
strain condition to determine the undrained shear strength (=
constitutive model agree well with the experimental
Kurukulasuriya2) for normally consolidated Kaolin clay. Furthermore,
maximum shear stressesat failure).Thereare two points worthnoting
here. 1) Experimental evidence suggests that the anisotropic
the
microstructuredeveloped during the K0-consolidation process is
study
provides
micromechanics
an
approach
research
to
field with
connect
data by
results
conventional
from
continuu
preserved,to a certain extent,throughout the isotropicconsolidation.
2) The isotropic consolidationexcludesthe effectsof the anisotropic
initial stress condition on the variation of undrained shear strength
modeling.
(seeDuncanandSeed).Accordingly,
theundrained
shearstrength
anisotropy,
if
microstructure
Figure
between
Fig. 1
loading
direction
and consolidation
plane
(ƒÆ•‹)
can
clay
2 describes
Kaolin
clay
strain
testing
similar
Angle
it exists,
of Kaolin
as
the
variation
a function
with
The
values.
3. Constitutive
Modeling
3.1 Fabric
Tensor
and
from
the
anisotropic
duringK0-consolidation.
undrained
inclination
undrained
a minimum
OCR(=150/p0)
develop
of the
of the
condition.
manner
only
developed
shear
angle Į
shear
for the
strength
around ƒÆ=30•‹,
the Evolution
strength
plane
varies
in a
irrespective
of Fabric
of
of the
Tensor
Variations of strength using UU tests
Particles
(Duncan
andSeed8))
in
soils
consequently
by
particles.
so that
considering
major
seldom
produced
non-spherical
platy
are
the
the
In the case
microstructures
the spatial
planes.
spherical
The fabric
tensor
shape,
anisotropy
preferred
orientation
of Kaolin
minerals,
can
distribution
in
conveniently
of unit
Fij can be given
of
these
particles
are
defined
by
be
vectors
is
n normal
to their
as:
(1)
where Ħ
1,2,3)
function
such
oriented
within
satisfy
E(n
Fii being
soils
is
Strain
order
anisotropy
Tests
to
of clay,
equal
experiments,
of Kaolin
analyze
in
detail
we frequently
the
refer
undrained
to a series
shear
Kaolin
clay,
a
By
the
that
symmetry
vertical
(i=
E(n)
rate
is a
of unit
definition,
trace
of
E(n)
the
fabric
the microsbucture
axis
parallel
direction).
of
to
That
is,
the
natural
clay
fabric
tensor
was
prepared
under
a K0-consolidation
of 150 kPa.
After
parallelepiped
strain
from
samples
the material
used
process
K0-consolidation
(5•~10•~12.5
such
that
cm)
the
angles Į
to the consolidation
plane,
the
it is isotropic
consolidation
the microscopic
and
on the horizontal
direction.
observations
of sand
including
This
the
plane
assumption
is
on the microstructure
of
of Eq. (1) can further
by Oda4).
be simplified.
If this
is the case,
the
Let xƒ¿ (ƒ¿=1,2,3)
be
coordinate
system
such
that
x1 is the consolidation
direction,
in the
with
x2
and
x3 are
on
the
plane
perpendicular
to
x1.
Since
the
a
axes
is axially
(horizontal)
with
symmetric
the major
with
principal
axis
a symmetry
of fabric
axis
tensor
parallel
to
Fƒ¿ƒÀ.(Note
strain
the subscripts ƒ¿
and ƒÀ refer to the principle
axes
xƒ¿ (ƒ¿=1,2,3),
were
whereas
at different
vertical
was
for plane
sample
the
the
tests
that
made
whereas
by Kazame
microstructure
pressure
on
strength
of plane
the material
to
with
x1, it agrees
inclined
(or
anisotropic
direction,
perpendicular
Clay
by Kurukulasuriya2).
vertical
were
us assume
with
direction
to
sphere, ni
n, and
to the
angle dĦ.
leading
to 1. Let
is
and
tests
of a unit
vector
corresponds
solid
)dĦ=1,
microstructure
a local
out
completed,
E(n)dĦ
a small
consolidation
consistent
maximum
that
axial-symmetric
consolidation
carried
to a surface
Variations of undrained shear strength
ofKaolin
clay(Kurukulasuriya2))
In
equal
of a unit normal
vectors
tensor
2. Plane
angle
density
must
Fig. 2
is a solid
are Xi-components
the subscripts
i and j
in Eq.
(1) refer
to global
coordination
plane.
axes
―390―
Xi, (i=1,2,3)
defined
later.)
In this
case,
F11, F22 = F33 are the
principal
values.
Let ƒÁ be
a ratio
ofF11
to
F22 (=F33),
then
we
apparent
have:
(2)
anisotropy
stress-induced
isotropic
where ƒÁ is hereinafter
the
trace
of the
referred
fabric
tensor
to as the degree
is equal
to
of anisotropy.
1, we
have
the
Since
expression:
their
vary
must
be also
between
aforementioned
If the microstructure
is isotropic,
1 with the following
isotropic
the degree
of anisotropy ƒÁ
is set to
fabric
soil
properties
the
mechanical
the
hand,
critical
parameter
at the initial
fabric
extremes
and
tensor
could
be
to be
soil
or
be calculated
ultimate
plastic
at any
conditions
other
detemined
rule of the fabric
at step n could
M
The
value.
tensor
evolution
tensor
mechanical
so as the
two
state".
other
the
e.g.
quantified
these
stress
On
on Į
of the fabric
initial
requires
model,
depending
The states
(3)
nature.
in this
modulus,
by the anisotropic
anisotropy
in
properties
following
caused
tensor.
For
state
in
using
the
example,
the
as follows:
tensor:
(5)
(4)
where F inoij, dFkij are
the initial fabrictensor and the increment
of fabrictensorat step k,respectively.It is noted that in Eq. (5),all the
components of the newly updated fabrictensor F
with
=1
the trace
from
Oda1),
the
of itself
the definition
increment
Principal
axes xƒ¿(ƒ¿=1,2,3)
and Xi(i=1,2,3)
of the fabric
axes of the global
value
tensor
coordination
coordination
X2 is parallel
plane
to the vertical
(Fig.
3). In
compression
defined
and
system
the
plane,
direction
is
definition
principal
of Į
components
is
axis
(3)
and
parallel
between
such
simulations,
to X2. In
the global
x1 of the
fabric
to that
in Fig.
equivalent
of Eqs.
is introduced
numerical
always
angle
tensor
axes
(4) were
with
xƒ¿ of the fabric
respect
the
If only the effects
fabric
tensor
process.
is
addition, Į
In this
case,
since
The
conditions
are the same,
conditions
measured
with
different
anisotropy,
values
anisotropy
be
the
fabric
its value
(see
the distribution
during
the shearing
process.
the
invariant
of
a certain
normals.
saturated
Based
on this
tensor,
including
value
for the
observation,
of the
fabric
the
initial
the
fabric
Minh
into account,
out
the
at the initial
be calculated
condition
from
the
and
be defined
in terms
of•@
. The valu
of•@
at
ultimate
condition,
(•@) ult,
and
consequently,
(Fij) ult could
be calculated
from
the measured
stress
values
state.
et al9). for
Applying
the calculation
the testing
obtain
the
from
two
condition,
data
procedure
of Kaolin
degree
which
consequently,
clay
described
by Minh
by Kurukulasuriya2),
of
anisotropy
rult=1.06
at the
we
ultimate
Consequently,
then
could
be
used
to
calculate
(F
ij)ult
(•@)ult=0.01132.
Details
on the calculation
of
the stress
rult are
described
tensor
at the
later
in this
paper.
On
the
other
hand,
the
fabric
for
induced
beginning
of the
shearing
process,
(Fij)ini,
which
which reflects the
the effects
of the inherent
anisotropy
of Kaolin
clay, could
with the
by conducting
the simulation
of the K0-consolidation
it is necessary to
Note that the induced
anisotropy
phenomenon.
stress-induced
and
experiments
the evolution of the contact normals
described
at the
shearing
and ultimate
et al.9)). However,
anisotropy
could
is
process.
relation
For
on
simplicity,
a tentative
however,
basis;
i.e.,
we
assume
here
the
following
(•@)ini=0.8(•@)ult.
This
is
Ohta and
equivalent to the assumption that when K0-consolidation
Nishihara10), for example,
it is
rule of the tensor.
through
tensor
could
of the loading increments.
different with the stress-induced
deviatoric
for
if necessary,
changes its value in accordance
define an equation to formularize
state
conditions,
be determined
application
limitation
to the
of the
could,
of the contact normals,
of materials,
represents
any
that
represents
microstructure
second
(where
this
that
respect
components
are taken
constant
at the ultimate
of Į
is the
a
on
is
that
also
with
transformation
of inherent
to
which
assumed
ultimate
axis X1
(Note
1.) Note
axes Xi, (i=1,2,3)
the coordinate
assumed
tensor,dsij
exists
to
to depend
that
major
horizontal
tensor.
calculated
tensor.
to the global
using
tensor),
of the contact
could
be calculated
there
(where JF2D
concentration
critical
principal
is found
stress
Furthermore,
of Fii
in Eq . (1). According
normals
deviatoric
the condition
and X1 and X3 are on the horizontal
following
as the inclination
the major
Xi(i=1,2,3)
tensor
of the contact
of
of.•@
to maintain
system
the fabric
A global
in order
of the fabric
the concentration
sij=ƒÐ'ij-pƒÂij).
Fig. 3
FL
un are normalized
anisotropy
as: "an
•\ 391•\
is
completed,
as the
end
the
microstructural
result
of the
the
shearing
of
normals
fabric
anisotropy
evolution, ƒÁ
process.
is related
to
Since
dsij, dFij, can
is given
by
consequently
the
rult=1.048,
reaches
concentration
beformulated
of
in the
and
1.06
at the
the
3.2 Modified
following
In order
form:
Tobita12)
(6)
where k is a constant of proportionality expressed as:
shows
(7)
tensor
dSij. It is reasonable
when
As
to think
the compressive
the minimum
the
result,
k should
of ƒÆ=•‹
deformation
than
fabric
axes
three
change
of Fij and
particles
to occur
located
is likely
is applied
parallel
to the direction
normal
at the current
be
that
principal
of the contact
That
at least
the following
large
loading
at ƒµ=90•‹.
the major
that
concentration
maximum
case
between
an
increasing
is, the fabric
in the
Based
case
on
function
change
of ƒÆ=90•‹
the above
of ĵ
occurs
with
faster
in the early
consideration,
of
a
of
show
force
areas
More
importantly,
another
a constant
parameter
decreasing ĵ.
values
of
serving
to control
With
as
how
the introduction
(Fij)
and
a scaling facctor,
rapidly
k
of Eqs.
(6) and
(Fij)ult' the
and
lowers
evolution
b
to deal
each
with
planes
of the fabric
the
beginning
completely
Oda11)
behavior.
Kaolin
that
based
on
In this
study,
clay.
and
test
Since
Eq.
microscopic
we
have
for
Kaolin
clay
an alternative
modeling
solution.
tensor
prediction
phenomenon,
conventional plasticity
and
clear
of Kaolin
we accept
This
ultimate
observation
no
However,
good
the
Eq. (7) were
it is an attempt
evolution
(7)
Eq. (6) and
uncertainties.
of fabric
until
condition
is
fabric
of
to apply
clay, the direct
case
As it is turn
for Kaolin
could
constitutive
models.
are actually
is then
enlarged
application
of Eq.
conventional
involve
is
integration
some
to seek
for
acting
of
hollow
which
are
enclosed
by
defined
areas
and then
as
through
such
stress ƒÐƒ¿ƒÀ
a
over
stress
To
the
over
can be
are associated
with
summed
that
in
of
can
through
which
stress
that
the
equal
to 3
reflect
the
the applied
The
modified
Tƒ¿ƒÀacting
on
4b). The modified
that
the
xƒ¿-planes
integration
must
the xƒ¿-surfaces.
the
be clarified
such
(ƒ¿=1,2,3)
(Fig.
with
the xƒ¿-
to obtain
will
the assembly.
condition
stress
Cƒ¿ (Cƒ¿=acƒ¿)is
the imaginary
Cƒ¿ (ƒ¿=1,2,3)
by
that
the unit area
enlarged
areas
satisfied
associated
For reasons
that Cƒ¿
transmitted
than
(plastic
stress.
areas
areas
groups.
think
soils,
rather
contact
contact
two
to
of granular
proportionally
Note
of modified
the following
many
through
particles
to conditions
surfaces
of the contact
for
to these
reasonable
The contact
are
surfaces
be transferred
of conventional
enlarged
equal
the
stress
of
the
This leads to
relations:
clay
variation
not
be
actually
leads
of Kaolin
simulated
to
(9)
clay.
using
(b)
(a)
Fig. 4 (a) Orthogonal
to the
and
cube
4a
referred
are parallel
of
according
cƒ¿ (ƒ¿=1,2,3).
introduced
contact
tensor
in order
hereinafter,
centers
rolling
on the xƒ¿-planes
areas
distribution
on the
may
the
of
Figure
out later, the application
strength
otherwise,
for
Ty in terms
hatched
belonging
behavior
that
C1+C2+C3=3).
stress
material
results
The
it appears
unit plane.
of three
evidence
experimental
in this
shear
same
area,
would
and
to the contact
are so small
surface
forces
particles
the plastic
cƒ¿ (ƒ¿=1,2,3)
(i.e.,
introduced
granular
the
this assumption
change
of undrained
firstly
tensor ƒÐij
of a unit
planes
at contacts
are projected
anisotropic
be noted
microstructural
(6)
the
defined.
It should
by
of
the
in the definition
summation
from
tensor.
of particles,
to these
respect
orthogonal
later,
tensor
tensor
of which
respectively,
sliding
with
with
contact
as the
stress
stress
unit
directions
Accordingly,
the xƒ¿-planes
is an
its value
(8) as well
groups
occurs
Particles
is
having
fabric
outside,
between
forces.
defined
where ƒ¿
the
applied
contact
order
(8)
planes
materials,
sets of xƒ¿-planes.
contact
function:
two
and
of granular
a modified
the normal
xƒ¿ of
inside
deformation)
assume
the microstructure
the conventional
orthogonal
axes
Any
in the
stage
we
three
state.
for
Fij and
to as the xƒ¿-plane,
principal
is the angle
to account
and Oda11) introduced
the fabric
where ĵ
Stress
contact
planes
on a unit plane
cutting
and modified
through
stress
a particle
assembly
tensor,
Tƒ¿ƒÀ,
acting
•\ 392•\
(b) conventional
on enlarged
contact
stress
tensor, ƒÐƒ¿ƒÀ,
surfaces
Equation
=1
(9)
is given
with
,2,3) of the microstructure.
we can easily
generalize
respect
to
Referring
Eq. (9)
the
principal
to the global
axes
xƒ¿ (ƒ¿
(14)
axes Xi (i=1,2,3),
as:
where
q=•@
is
proportional
to
the
second
(10)
invariant
where
C-1ik
is
the
inverse
tensor
of
Cik
and
satisfies
of
deviatoric
C-1ikCkJ=„qiJ,
where „qij
is Kronecker's
delta.
Based
considerations,
relation
between
Oda11)
the tensors
derived
the
following
Cij and Fij for a spherical
sij(=„qij-p„qij)
is the
mean
effective
stress, „qij=„qij-uw„qij
on
is the effective
statistical
tensor
the
p=(1/3)„qij„qij
relation
stress
proportional
and
assembly:
change
end
Kaolin clay consistsof platy particles.For simplicity,however,we
adoptthis relationas the first step of the presentapproachand observe
the results.Equation(10) is then rewrittenas:
tensor, ă
indices,
representing
(11)
stress
swelling
the
effect
of clay,
K correspond
D
of the
ratio
stress
e0 and p0 are the void
of consolidation,
respectively.
behaviors,
all of the stress
calculated
from the modified
Eq. (14)
and
respectively,
can then
and
coefficient
on
mean
the
volume
pressure
to account
at the
for anisotropic
by their
tensor Tij
compression
dilatancy
increment
ratio
are replaced
stress
to the
the
In order
terms
be rewritten
is
equivalences
The yielding
as
function
as:
(12)
The
scalar
1/3 is chosen
such
that
the
modified
stress
(15)
tensor
where
should
reduce
isotropic
to the conventional
with
the isotropic
stress
tensor
tensor „qijc so long
and
conventional
of Eq.
stress
(4). Note,
are not symmetric,
which
means
the characteristics
of the
Consserat
tensors
furthermore,
that
resulting
from
Tij•‚Tji and „qij•‚„qji.
p,
the stress
tensor
of which
is related
to the
q
are the equivalences
in temis
of modified
that
This
i.e., q=•@
is
proportional
to
the
second
stress
tensor
Sij(=T'ij-p„qij),
leads to
the nonsymmetTy
spinning
and
Eq. (12)
invariant
continuum,
p0,
as the soil is
stress;
the modified
of
of
deviatoric
modified
of
of the unit
cells
=(1/3)T'ij„qij is
the
mean
effective
effective
modified
modified
p
stress,
and
(grain)of the materials(seeSchaeffer13).
In the presentstudy,
however,we considerthe case of symmetricstress tensors.Thus, the
T'ij=Tij-uw„qij
stresstensorscalculatedbyEq. (12) are symmetrizedas follows:
is the
stress
tensor.
Moreover,
similar to the definition by Schofield and Wroth15, the critical
T'ij=Tij-uw„qij
state
(13)
line can be also given in terms ofthe modified stress as:
(16)
3.3
Failure
Modified
Criterion
Stress
and
Yield
Function
in
terms
of
the
where
present
Tensor
M
approach,
anisotropic
The
modified
granular
soils,
surfaces
and
of particles
rather
for
function
Sekiguchi14)
the
effects
behavior
stress
should
be
of
an
of the
condition,
yield
function
Wroth15).
In this
original Cam-clay
anisotropic
of the
case,
initial
and
the
in terms
stress
in the
Sekiguchi
original
Ohta
stress
yield
such
take
condition
on
the
of isotropic
coincides
model
(Schofield
formulation
given
critical
fabric
we
principal
axes
calculated
and
soils.
The
present
a fabric
axis of anisotropy
that
ikTkj
the
Fƒ¿ƒÀ with
for
to be
degree
respect
to the
tensor
Fij is
r and the inclination
the
conventional
(Eq. (12)).
Hence,
stress
even
of
to the
Referring
the fabric
the
criterions
is said
(called
earlier,
of anisotropy
X1. Note
by „qij=3F
approach
(i.e., xa(a=1,2,3)).
that
tensors,
and failure
r
tensor
as discussed
of the degree
Note
stress
yielding
a parameter
can determine
x1 and
parameter.
modified
for any
Using
Xi(i=1,2,3),
in terms
between
isotropic
sense.
state
angle
tensorĮ is
if the modified
and
stress
into account
function
Sekiguchi
of the
a yield
Ohta
for the
global
Yield
in terms
that
could
Cam-clay
and
tensor.
for „qij.
case
major
function
using
in this
anisotropy),
of sliding
modified
soils can be applied
proposed
general
of the modified
formulated
which
for
contact
the yield
substituting Tij
function
that,
the
as a result
it is assumed
However,
the Ohta
to
words,
conventional
by
a yield
materials.
defined
Here,
generalized
introduces
be
the idea
referring
In other
are usually
tensor „qij.
on
of soils occur
the
soils
based
defined
at contacts.
than
stress
can
be
behaviors
isotropic
conventional
introduced
should
criterion
tensor,
functions
was
plastic
the failure
stress
the
stress
because
rolling
and
stress
is the
depending
plastic
Tij is identical,
the
on Į. As a result,
a failure
line in (q,p)
with
function
of the inclination
and
constant
in the
initial
of
tensor
the parameter
- space,
modified
conventional
is not
angle Į.
stress
stress
M(=q/p),
constant
This
space
but
which
rather
is true even
(q,p,).
tensor „qij
differs
represents
is given
though
Similarly,
as a
M
we
is
could
the
obtain
yield function can be given as:
yielding
―393―
the same
condition.
result
for the analysis
The same
yielding
of the stress
point
in terms
points
satisfying
of Tij could
give
different
equivalent
. As
the
yielding
result,
experimental
this
observation
differently
as being
stress
points
modeling
by
sheared
in terms
of „qij depending
approach
could
Kurukulasuriye2)
at different
that
inclination
onĮ
simulate
samples
assumed
behave
Since
updated
Equations
the
plasticity
normality
written
theory,
we
incremental
analysis.
at the end of each
yield
function
conventional
rule
use
which
particular
angles Į.
The
3.4 Constitutive
to be constant,
plays
a dominant
it in the present
pole
study.
in the
The normality
tensor,
of
parameter
df=0
of proportionality.
along
with
the
Using
and ă
normality
the consistency
rule,
we
constitutive
a function
but also of the fabric
not
tensor.
rule (Eq. (17)),
axes
plastic
strain
increment
dƒÃpij are
of
of the stress
axes
of ƒÐij and Fij.
tensor
tensor
but rather
That
is not guaranteed
are
only
manner.
only
of the
If such
a yield
the principal
not
coaxial
with
pointed
is, the
depend
coaxiality
in the present
out
that
on both
the principal
in the classical
model.
noncoaxiality
plasticity
Gutierrez
is
one
et al.17), for
of
the
most
is a
derive
aspects
evidence
supports
condition
(e.g.,
of granular
soils.
the noncoaxiality
Lin
and
In fact,
for
recent
clays
Prashant18).
experimental
sheared
The
in undrained
modification
of
the
the
conventional
following
is now
with the normality
those
condition
can
(15)
Fij
linear
along
fundamental
scalar
of
in a piecewise
is used
example,
increment
tensor
components
step
of Eq.
stress
The
loading
function
model
strain
a
rule is
(17)
dƒÃpij is the plastic
during
classical
as:
where
means •ÝFij/•ÝƒÐmn=0,
the
relationship:
plasticity
theory
of the term•Ýf/•ÝƒÐij,
an isotropic
elastic
in Eq. (20)
also
means
Nij•‚Nji.
which
constitutive
matrix
leads to non-symmetry
defined
However,
withii
as:
(22)
(18)
The product Aijof Cei
jkl and Nkbas calculated as below, retains its
symmetry, which means Aij=Aji:
where
Nij=•Ýf/•ÝƒÐij,Cepijkl
and
Ceijkl
are
the
elastoplastic
(23)
and
elastic
matrixes,
respectively
and, ƒÃpv,ƒÃpij,ƒÃij,
are
the
plastic
where
K and G in Eqs.
modulus,
volumetric
strain,
plastic
strain,
and
total
strain
tensors.
Here,
as the
defined
hardening
parameter
in this
model
with
the
hardening
(19)
Desai
and
function
the
symmetric
calculation
for the derivation
Siriwardanen.
f in the present
derivative
of Eq. (18)
of
Note,
model
however,
is given
of •Ýf/•ÝƒÐij in
can be found
Eq.
that
in terms
(18)
elsewhere
since
of modified
must
be
the
(e.g.,
yield
stress
calculated
Tij,
conventional
4.
Effects
bulk
into Eq. (18),
and
shear
results
in a
matrix
Cepijkl. Thus,
all
subsequent
Strength
of
as those
in the case of the
theory.
Induced
Anisotropy
on
the
Undrained
Shear
Anisotropy
of Model
Parameters
as
follows:
(20)
how
by which
model
to determine
addition,
stress,
can also be written
are the elastic
Eq. (23)
can be kept the same
plasticity
4.1 Determination
constitutive
Eq. (12), •ÝTij/•ÝƒÐmn,
constitutive
procedures
The method
From
and (23)
rule
as:
Details
(22)
Substituting
p0
final
acts
respectively.
to determine
is critical.
the parameters ă,
in order
to define
we newly
introduce
degree
of anisotropy
the parameters
Barka
and
Ohta19)
involved
discussed
K and D appearing
the critical
state
in the present
in terms
study
two
in the
in detail
in Eq.
14. In
of the modified
parameters
which
as:
are the
(21)
The constitutiveequation for anisotropicsoils is now completed
state
parameter
value
of rult, we
with the inclusionof the fabrictensor. Theeffect ofthe microstructure
initial
condition)
on yieldingbehavior appears explicitlyin the terms of F-1ij in Eq.
previous
and
(21). Note that in the calculationof Eq. (21), the fabric tensor Fij is
•\ 394•\
M
M
in
could
part of the paper.
referring
(q,p)
also
using
at ultimate
-stmss
obtain
the
space.
rini (the
simple
Here,
condition
we
to the experimental
data
the critical
Furthermore,
degree
relationship
focus
rult and
from
of anisotropy
described
on the determination
of normally
the
at the
in
the
of rult
consolidated
Kaolinclay.Let us choosea modifiedstresspoint (q,p) located
value
of
ruly can
conesponding
onthecriticalstateline,namely,q/p=M.
Assumingfurtherthat
T11,T22and T33(=T11)are the principalstresseswith a maximum
values
be determined
are determined
of MtĮ
at the
intersection
to ƒÆ=30•‹and ƒÆ=90•‹in
as
Fig.
M=0.935
and rut=
at ƒÆ=30•‹and ƒÆ=90•‹were
of the
two
lines
5. Consequently,
1.06.
specially
(The
chosen
their
two
values
to determine
principalvalueequalto T22,
we canexpressthecriticalparameterM
as:
rult and
M
because
the effects
of the modified
stress
on the ultimate
(24)
If the minimum stress T33(=T11)is set to unity,the modifiedstress
tensor TFij on the critical state line can be written in terms of M
parameter
increase
in this
with
90•‹. The
in Fig.
(25)
It is of particular
representative
importance
for
many
to know
conventional
to various
given
in terms
of rult,and ƒÆ, the stress „qFij
TFij
can
calculated
changing
and
the value
the
as qF/pF
entire
set of
data
the relation
of
TFij
(Eq.
determined
using
using
M
among
and ƒÆ=90•‹
are
the fabric
at failure
we
of qF and
can
rult, M and
On
one
q and
hand,
data
by
data
can
obtain
be noted
values,
the
corresponding
to
M)
are separately
values
case
of (rult,
(ruly, M)
uniquely
point
anisotropy
using
is
set
of
(rult,
different
a straight
M,
M)
of Kaolin
clay.).
the measured
stress
obtained
by
is used
which
defined
these
yield
two
the
irrespective
lines
represent
same
different
value
of Į, its value
of
values
of
volume
A=(1-K/ă))
change,
is a parameter
as shown
index,
which
related
by the last equation
is equal
to 51.2%
to the ineversible
of Eq. (26),
for Kaolin
to fit the data
and
the
Since
clay.
and PL the
Generally
In
combinations
MtĮ.
a
Į
of
for
clay might
simplicity,
be anisoiropic
Young's
based
on
in the elastic
modulus
E and
the assumption
the
response.
Poisson
of isotropy,
M
of
is
corresponding
•\ 395•\
1 for normally
consolidated
Kaolin
However,
ratio
v were
as summarized
M).
on
using
in this study.
for
combinations
=90•‹. These
line
of
(26)
Table
Any
deformation
r is calculated
0•‹to 30•‹
effects
according to the method suggested by Iizuka and Ohta19),as follows:
determined
points
the
to
at ƒÆ=30•‹
in Fig. 5 for ƒÆ=30•‹and ƒÆ=90•‹.
of ƒÆ=30•‹and ƒÆ=90•‹,
of
method
firm
only
Other parameters used in the yield function were calculated
of Į by
Kurukulastwiya2).
speaking,
each
on the plastic
since
with Į
is
30•‹to
MtƒÆ=90•‹, respectively.
MtƒÆ=30•‹and MtƒÆ
plotted
degree
is due
increase
from
Fig.5 Parameter
determination
(Kurukulasuriya2)
plasticity
(rult,
that
strength
later,
an
of Į
an
where
(rult,M)
described
range
is then
obtain
experimentally
determined
as MtƒÆ =30•‹and
we
M
we
be
selected
shear
be
et al,9)). Such
limited
MtĮ must
By
of the choice
for any
of undrained
that
various „qFij,
failure.
also
in the
anisotropy
It should
Minh
only
2, as will
microstructural
(see
characteristics
to
is independent
can
decrease
to the
is
angle Į. Note
conducted
calculated
tensor Tij
observed
phenomenological
failure
inclination
of M experimentally
denoted
=30•‹and •@ƒÆ=90•‹,
M
p at failure
values
previously
M
is
corresponding
obtain
pF at
for a given
experimental
two
the
values
at
equation „qFij=3FikTFkj.
in Eq. (25),
(rult,M,M)
using
Hereinafter,
the
of Eq. (25)
for any set of rult and Į. In this way,
(25)).
the
From
sets of rult and Į. Since
corresponding
calculated
that
of
the stress
stresses „qFij
corresponding
be
that
lead
increasing Į
experimentally
as:
model
clay.
Table 1 Input parameters for numerical modeling
in
Fig. 6 Flow chart ofthe program used for numerical modeling of undrained shear strength anisotropy
4.2 Modeling
Results
Undrained
Shear
on the
Effects
Strength
of Induced
Anisotropy
on the
that
the
values
(Fij)ult
Based
on
calculate
(18)
the
plasticity
numerically
so as we
samples
could
incorporation
obtain
a program
the
the
fabric
plasticity
and strain
undrained
and
theory,
code
was
of the constitutive
the stress
under
of
conventional
theory,
the integration
sheared
equation
behaviors
Figure
the undrained
6 shows
numerical
a flow
analysis
increments.
The
implemented
such
shear
chart
was
stress
target
strength
of the
carried
out
requirement
that
in Eq.
With
tensors
anisotropy
main
by
of
steps
means
assumption
fabric
the
of Kaolin
of the
was
Ciejki and
at the current
solved
modified
to
obtain
stress
was
6, namely,
and
A
tensor
stress
corresponding
calculated
twice
strain
matrix
point,
Note,
yielding
condition
however,
that
conventional
stress
constitutive
modeling.
M
and
was
then
(Fij)uly
of Eq. (15)
the
in the
failure
same
Here,
at the
and
was
The
running
Cepijkl, and to check
satisfied
condition
manner
M
beginning
was
was
loop
of the
first
throughout
the simulation
process.
based
on
the
in Eq. (2). On the other
step,
(Fij)F,
may
hand,
the
not be transverse
depends
on
both
the
initial
fabric
tensor
the
stress
path,
which
controls
the
term
dsij
the
in
Eq.
(6).
As
the
result,
(Fij)F
should
not
be
the
same
as
(Fij)ult
however,
according
to
the
checked
from
rule
in Eq. (8), the
second
deviatoric
invariant
(•@
)ult,calculated
of
(Fij)F
not
exceed
the value
of
other
words,
at the
ultimate
approach
calculation
It should
(Fij)ult.
of
In
components
of the
second
fabric
deviatoric
condition
tensors
invariants
need
of the
not
fabric
be uniquely
are uniquely
defined
tensor,
the
defined
but
as being
equal
value
step
(•@
)ult
irrespective
of the stress
path leading
to failure.
state.
in terms
the
horn
whether
at the current
as the common
computed
data
Eq. (18)
increments.
in a complete
was
experimental
anisotropy
value
rule
necessarily
Figure
of
7 shows
of
undrained
of
clay,
which
from
Minh
shear
also
the
strength
experimental
anisotropy
includes
et al.9) for
and
simulated
of normally
the simulated
comparison.
result
It is
results
consolidated
of inherent
shown
of the
Kaolin
anisotropy
in Fig.
7 that
the
and
modeling
used
and
Cepijkl were
and then
stress
the matrixes
the
at the last loading
Its
evolution
to
the
i,
dƒÃ33=0.
the elastoplastic
modified
the
to calculate
(Fij)in
strain
their
Fig.
define
clay.
program.
plane
from
of transverse
anisotropic.
will
first calculated
also
program
of controlling
undrained
dƒÃ11+dƒÃ=0
medrix
ruly, which
the
into
of the written
are calculated
evolution
The elastic
rini and
to
of the soil
condition.
modified
the main
written
(Fij)ini
is to simulate
of
Anisotropy
increase
be noted
with
―396―
the
of
inherent
of undrained
experimental
anisotropy
shear
data
could
strength
for Į
only
with Į
ranging
simulate
a monotonic
and, hence,
from
0•‹to
fail to agree
30•‹.The
modeling of induced anisotropy, on the other hand, could capture
well the full range of undrained shear strength anisotropic behavior of
(a,b)=(6.5•~10-6,17.0)
Kaolin clay observed by Kurukulasuriya2).
from
plane
strain
stress-strain
strength
The
constitutive
well
as
Figure
9,
9, there
and
simulation
exists
results
move
along
leads
to
stress
values
of
a=6.5•~10-6
parameters
changes
and
in Eq. (8). With
very
rapidly
we
could
b=17.0
the following
for
the parameter
within
obtain
the
evolution
b is set as much
60•‹•…ƒÕ•…90•‹,
so that
the
path
develops
direction
X2 (see
stage
of
initial
if Į
a typical
fabric
principal
axis
the
rotated
so quickly
the
case
undrained
until
that
shear
. Initially,
k
induced
the
axis
In other
to the
(90-Į)
us consider
model
path
can be produced
axis
since
the
strength
the shear
quickly
soil
parallel
case
of
major
sample
equals
(induced)
present
is
90•‹so that
dilatancy
This
stage
continuum
doing
because
it is
the induced
for
in the case
theoretical
tensor
the
fabric
looks
that
constitutive
model
of this
concentrated
microstructural
with
a
the
study
at the
on setting
up a
evolution.
In
rather
simple
and
original
Cam
clay
the
drastically
improved
of ƒÆ=90•‹.
the
stress-strain
curves
Experimental
values
curves calculated
defined
numerically
using
•\ 397•\
pore
shear
becomes
gaining
pressure
strength
in terms
higher
leads
of the induced
dominant
is closely
paths
related
in the undrained
obtained
of excess
values
anisotropy
at ƒÆ=0•‹
pore
from
to slower
in this
stress
values
deformation
water
in Eq. (8)
less
and modeling
different ƒÆ(•‹)
the plastic
anisotropy
As Į
effects
In Fig. 8, the stress-strain
of
a more
the
like
set irrespective
by employing
for the
framework
stress
characteristics
the scope
first
Cam
unique
similar
observation
undrained
different ƒÆ(•‹)
data
of
is
of excess
with
one
is firstly
start
stiffness
as a merit
of X2. As a
fabric
Since
and modeling
to
the
model.
with
Experimental
study
attempt
in
beyond
approach
an
well-established
Fig. 9
Fig. 8
the
higher
deformation
can be improved
relationship
which
with
if the original
only
experimental
of the
is, however,
modeling
so,
hand,
then
paths
to the axis X2 (not the axis
in the case of ƒÆ=0 (•‹is
strength
framework.
X1. The
in the direction
x1 of the
explains
complicated
changes
for the same
on the other
stress
process,
the
as, for example,
prediction
in the case
simulated
with
data
as well as the
can be considered
in the simulation
The
paths
in comparison
is consistent
model
is applied
value.
in an early
the
This
in Fig. 8 and
experimental
of shearing
strengths
and
regarding
For example,
and
as
simulated
results
the
values.
characteristic
constitutive
clay
the sample,
Į
axis x1 of the
principal
global
to X2
words,
This
compression
microstructure
the major
parallel
it becomes
of ƒÆ=90•‹.
it reaches
equals
hence
0•‹ to 30•‹. Let
develops
principal
X1 at the beginning).
to
is
to
to the X2 axis. Accordingly, ƒÕ
anisotropy
major
that ƒÕ
is parallel
dsij
parallel
least,
is within
parallel
induced
result,
at
example
of
quickly
3). Note
tensor
compressed
the
Fig.
deformation
significantly
=0•‹as
very
of ƒÆ=30•‹.
strain
the experimental
the
of the stress
line in the beginning
shear
higher
are discrepancies
the experimental
The
of ƒÆ=30•‹.
failure
with
between
9
clay.
case
smaller
and simulated
tendency
higher
the
shows
there
different
in Fig. 8. This
of the Į
anisotropy
a vertical
the proposed
as 17.0,
with
significant
observed
hand,
for the shifting
Fig. 9, both
Experimental and simulated results on the undrained
other
Kaolin
with
compared
Although
similar
curves
shear strength anisotropy of NC Kaolin clay
studies,
the
obtained
and ƒÆ=90•‹possess
predicts
as
of the experimental
Fig.
of ƒÆ=0•‹in
parametric
however,
paths.
the results
consolidated
of ƒÆ=0•‹
stiffness,
on
stress
with
in comparison
model,
initial
the magnitude
By conducting
stiffness
smaller
stress-strain
of normally
in the cases
and
experimental
Fig. 7
tests
curves
shear
results.
are compared
evolution
on the
range.
condition,
could
water
30•‹to
to the development
the higher
be mainly
pressure
90•‹, the
plastic
to
development.
evolution
of the fabric
the
rule
tensor,
deformation
Consequently,
due
the
hence
effects
of
induced
anisotropy
controlling
from
30•‹to
with Į,
on the shear
factor.
is
"inherent"
and
and
Carillo1).
and
the
now
become
tendency
increases
relationship
that,
the
with Į
monotonically
between
the
critical
present
Remarks
senses
direction
the framework
the microstructure
introduction
induced
respect
process
as
ultimate
conditions.
well
The
by
are
strength
the
classical
results
of the
of
is
to
fabric
values
numerical
plane.
Į In
theory
can be taken
stationary
angle
plasticity
clay. Furthermore,
rule
of clayey
tensor)
of Kaolin
its
both
fabric
anisotropies
as
and
for
consolidation
(called
the
Casagrande
on the inclination
to the
of the
in both
obtained
shear
depending
quantity
of an evolution
shearing
described
Undrained
with
into
anisotropic
responses
greatly
a tensorial
and
, as
deformation.
changes
study,
inherent
is, in general,
anisotropic
plastic
incorporated
both
parameter
for the
strength
the
of soils
Therefore,
for example,
simulate
to
Concluding
"induced"
of the loading
the
shear
similar
The microstructure
soils,
strength
explain
M and Į.
5. Discussion
strength
could
90•‹, undrained
which
parameter
This
the effects
into account
of
with
tensor
during
the
at the
initial
and
simulations
can
be
deformationof sand,Soilsand Foundations,12 (2), pp.1-18,
1972.
5) Satake,M., Fabrictensorin granularmaterials,Proc. IUTAM
Corferenceon Deformation
and Failureof GranularMaterials,
Balkema,Rotterdam,
pp.63-68,1982.
6) Oda,M., Nemat-Nasser,
S. and Mehrabadi, M.M,
A statistical
studyof fabricin a randomassemblyof sphericalgranules,Int.
for Num.&Ana.Meth inGeomech,6,pp.77-94,1982.
7) Duncan,J.M.and Seed, H.B.,
Anisotropyandstressreorientation
inclay,J ofSMFEDiv.,ASCE,92(SM5),pp.21-50,1966.
8) Duncan,J.M.and Seed,H.B., Strengthvariationalongfailure
surfacesin clay, J. ofSMFEDiv.,ASCE,92 (SM6),pp.81-104,
1966.
9) Minh,N.H., Oda, M., Suzuki,K. and Kurukulasuriya,
L.C.,
Modelingof inherentanisotropyto simulateundrainedshear
strengthvariation of Kaolin clay, Submittedto the J. of
Engineering
Mechanics,
ASCE,2008.
10)Ohta, H. and Nishrlara,A., Anisotropyof undrainedshear
strengthof claysunderaxi-symmetric
loadingconditions,Soils
and Foundations,
25(2),pp.73-86,1985.
11)Oda,M., Inherentand inducedanisotropyin plasticitytheoryof
granularsoils,MechanicsofMaterials,16,pp.35-45,1993.
12)
Tobita,Y., Contacttensor in constitutivemodelfor granular
strength
of Kaolin
clay is caused by both the anisotropy
in terms of
materials,Proc. US-JapanSeminar on Micromechanksof
shear strength
parameters
and the anisotropy
plastic response,
which
Granular Materials, Jenkins and Satake, eds., Elsevier,
leads to anisotropy
in excess pore water pressure
development.
2) By
Amsterdam,
pp.263-270,1988.
taking
the evolution
of the fabric tensor
(induced
anisotropy)
into
13)Schaeffer, H, Das Cosserat-Kontinuum,
ZAMM, 47, pp.
account,
the proposed
model
can simulate
well the variation
of
485-498,
1967.
undrained
shear
strength
with Į observed
from
the plane
strain
14)Ohta,H. and Sekiguchi,H., Constitutive
equationsconsidering
experiments
by Kurukulasuriya2)
for normally
consolidated
Kaolin
anisotropy
and
stress
orientation
in
clay,
Proc.
3rdInt coif on
clay. 3) The incorporation
of the fabric tensor into the yield function
NumericalMethodin Geomechanics,Aachen,pp.475-484,
in this model
naturally
leads to noncoaxiality
between
the plastic
1979.
strain
increment
tensor
and
the stress
tensor,which
has
been
15)
Schofield,
A. and Wroth, P., Critical State Soil Mechanics,
confirmed
experimentally
(e.g., Lin and Prashant18)
for Kaolin
clay
McCiraw-Hill,
London,1968.
sheared
under
the undrained
condition.
4) The proposed
approach
16)Desai, C.S. and Siriwardane,H.J., Constitutivelaws for
may be used to incorporate
the results
of discrete
studies
into the
engineeringmaterialswith emphasison geologicmaterials,
continuum
constitutive
modeling
of soils.
Prentice-Hall,
Inc.,NewJersey,1984.
17)
Gutierrez,
M.,
Ishrirara,
K. and Towhata,L, Noncoaxiality
and
References
stressdilatancy
relationsforgranularmaterials,Computermethod
and
advanced
ingeomechanics,
G. Beer,J.R.Bookeerand J.P.
1) Casagrande,A. and Carillo,N., Shear failureof anisotropic
Carter,eds.,Balkema,
Rotterdam,pp.625-630,1991.
materials,
Proc.BostonSoc.Civ.Engrs.,31,pp.74-87,1944.
18)Lin,H. and Prashant,A., Singlehardeningelasto-plasticmodel
2) Kurukulasuriya,L.C., Undrained shear strength of an
for Kaolinclaywithloading-history-dependent
plasticpotential
overconsolidated
clay and its effectson the bearingcapacityof
function., int.
J. ofGeomechanks,
ASCE,6 (1),pp.55-63,2006.
stripfooting DoctoralDissertation,SaitamaUniversity,Japan,
19)
Iizuka,
A.
and
Ohta,
H.,
A
determination
procedureof input
1998.
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parameters
in elasto-viscoplastic
finiteelementanalysis,Soilsand
Foundations,
27
(3),
pp.71-87,
1987.
effectson their consolidationand shear behaviors,Doctoral
Dissertation,
SaitamaUniversity,
Japan,1996.
(Received:
April14,2008)
4) Oda,M., Themechanismof fabricchangesduringcompressional
summarized
as
follows.
1) The
anisotropy
in the
undrained
shear
•\ 398•\