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9/7/ 2004 Prashant and Penumadu. NUMOG-IX 1
Modeling the Effect of Overconsolidation
on Shear behavior of Cohesive Soils
Amit Prashant and Dayakar PenumaduDepartment of Civil and Environmental Engineering,University of Tennessee, Knoxville, TN 37996, USA
Acknowledgements: Financial support from the National Science Foundation
(NSF) through grants CMS-9872618 and CMS-0296111 is gratefully acknowledged.
Any opinions, findings, and conclusions or recommendations expressed in this
presentation are those of authors and do not necessarily reflect the views of NSF.
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Presentation outline
Isotropic elasticity.
Experimentally evaluated yield surface.
Failure surface for normally and overconsolidated clay.
Introducing reference surface based on the failuremodes.
Plastic potential and hardening rule.
Predictions for Kaolin clay
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9/7/2004 Prashant and Penumadu. NUMOG-IX 3
= Elastic volumetric strain
= Elastic shear strain
= slope of unload-reload curve in e-ln(p')
spaceSpecific volume, v = 1 + e
= Poissons ratio
Isotropic elasticity
e
p
( )
( )
v ' 0'
1
0 3 1-2 v '
e
p
e
q
pp
qp
+=
The present model uses q-p-e space for defining the model surfaces and material
hardening, and the material response is assumed to be a function of pre-consolidation
stress. Therefore, as an obvious choice in this case, the elasticity model is defined based
on the Cam clay elasticity (Schofield & Wroth, 1968)*. The elastic stress-strain
compliance matrix:
*Schofield, A.N. & Wroth, C.P. (1968). Critical State Soil Mechanics. Maidenhead, McGraw-Hill.
e
p
Many clays show a non-linear e-log p relationship during isotropic unloading and the
value varieswith the mean effective stress; however, the model assumes a constant average value of consideringthat a small variation of will have less significant influence on the overall stress-strain responseduring monotonic shear loading.
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9/7/2004 Prashant and Penumadu. NUMOG-IX 4
Determination of yield location from
experimental data
0
0.4
0.8
1.2
1.6
0 20 40 60 80 100
W(
kPa)
YP
OCR = 1.5
0
0.4
0.8
1.2
1.6
0 20 40 60 80 100
YP
OCR = 2
0
0.3
0.6
0.9
1.2
0 20 40 60 80
LSSV (kPa)
W(
kPa)
YP
OCR = 5
0
0.3
0.6
0.9
1.2
0 20 40 60 80
LSSV (kPa)
YP
OCR = 10
0
30
60
90
120
0.0 0.5 1.0 1.5Shear Strain, eq(%)
DeviatoricSt
ress,q(kPa)
OCR=1.5
OCR=2
OCR=5
OCR=10
YP (Yield Point)
Length of stress
space vector:
Yield point determination using
Stress-strain relationship*
Yield point determination using
strain energy approach*
( )3
2
1
i
i
LSSV norm =
= = 3
1
i i
i
dW =
= Strain energyIncrement :
*Graham, J., Noonan, M. L., and Lew, K. V. (1983). Yield
States and Stress-Strain Relationships in a Natural Plastic
Clay. Canadian Geotechnical Journal, Vol. 20, pp. 502-516.
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9/7/2004 Prashant and Penumadu. NUMOG-IX 5
Experimental yield surface in q-p' space
Reference: Prashant, A. & Penumadu,
D. Three-Dimensional Mechanical
Behavior of Kaolin Clay. Soils and
Foundations, Personal communication.
3835353110
614061405
14163142682
17576175781.5
p
(kPa)
q
(kPa)
p
(kPa)
q
(kPa)
Strain Energy
Approach
Bi-linear Elasto-
plasticity
ApproachOCR
Point of initial yield during shearing
0
50
100
150
200
0 50 100 150 200 250 300
Mean Effective Stress, p' (kPa)
DeviatorStress,
q(kPa)
OCR=1OCR=10
OCR=5
OCR=2
OCR=1.5
Elastic
Zone
Failure Point
Initial Yield Point
Yield
Surface
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9/7/2004 Prashant and Penumadu. NUMOG-IX 6
Yield surface
2
2 ln opq
f Lp p
=
0
50
100
150
200
0 50 100 150 200 250 300 350
Mean Effect ive Stress, p' (kPa)
DeviatoricStress,q(kPa)
L=0.4
L=0.8
L=1.2
po' = 300 kPa
L = Another state variable
po = Pre-consolidation pressure
0
50
100
150
200
0 50 100 150 200 250 300 350
Mean Effect ive Stress, p' (kPa)
D
eviatoricStress,q(k
Pa)
po'=100 kPa
po'=200 kPa
po'=300 kPa
L=1
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9/7/2004 Prashant and Penumadu. NUMOG-IX 7
Yield Surface in 3-D stress space
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9/7/2004 Prashant and Penumadu. NUMOG-IX 8
( )o
f OC f ef f e initial f i o initial iq C p C p C p p p
= = =
Failure surface
( ) oe op p p p
=
VoidRatio
Natural Log of Mean Effective Stress
(po, eo)
(pe, e)(p, e)
NCL
URL
1
10
1 10
Overconsoli dation Ratio, OCR
x
y=x0.9
o= 0.90
1o s c
C C = ( )logo C e oe e C p p =
( )logo S oe e C p p =
( )0
of OC f NC
i
q qOCR
p p=
f f e
q C p=
o
of f
pq C p
p
=
Ladd, and Foott (1974)
Mayne and Swanson (1981) ( ) ( ) ou vo u voOC NC S S OCR
=
True triaxial data
Failure Points
NCL = Isotropic normalconsolidation line
URL = Isotropic unload-
reload line
f NC f ef f e initial f o initialq C p C p C p = = =
( )ep f e=
(Undrained shearing)
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Normalized failure surface in q-p' space
0
0.2
0.4
0.6
0.0 0.2 0.4 0.6 0.8 1.0
Mean Effect ive Stress, p'/po '
FailureS
hearStress,qf/p
o'
o= 0.5
o= 0.7
o= 0.9o= 1.0
Cf= 0.51
M
o
of f
pq C p
p
=
0
, Cf
= Failure surface parameter in proposed model
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9/7/2004 Prashant and Penumadu. NUMOG-IX 10
Failure modes in various stress-strain
relationships Localized deformations may occur during hardening of material that lead to a sudden
loss of material stiffness and eventually failure.
The sudden failure conditions (caused by strain localization) may be independent ofthe soil properties defining the pre-failure elasto-plastic yielding of clay.
The surface defining the ultimate growth of yield surface is separated form the failure
surface to model both sudden and smooth failure modes. This surface is named as a
reference surface. The failure surface defines lower
bound of the reference surface, and
these surfaces will be identical for
smooth failure conditions.
In triaxial compression plane (q-p
space), the reference surface can
reasonably be assumed to have a
similar shape as the failure surface.
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0
20
40
60
80
100
120
140
160
180
0 0.03 0.06 0.09 0.12 0.15
Shear Strain, eq
DeviatorStress,q(kPa)
b = 0.5
b=1
OCR = 1
OCR = 5
Sudden Failure Conditions at high
intermediate principal stress ratio
2 3
1 3
b
=
Intermediate principalstress ratio
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9/7/2004 Prashant and Penumadu. NUMOG-IX 12
Reference Surface
o
op
q C pp
=
Cy = Reference surface parameter in proposed model
Cf= Reference surface parameter in proposed model
0
50
100
150
200
250
0 50 100 150 200 250 300 350
Mean Effective Stress, p' (kPa)
DeviatoricStress,q(kPa)
po'=300 kPa
L = 0.8
o= 0.7
Yield Surface
Failure Surface:
C = Cf= 0.75
Reference Surface:
C = Cy= 0.8
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9/7/2004 Prashant and Penumadu. NUMOG-IX 13
Mean Effective Stress, p'
Devia
toricStress,q
Yield Surface
ReferenceSurface
q
qy
Hardening Rule
op
( )
v
o o
p
p
p p
= 0o
p
q
p
= 0pp
L
= ( ) 1Lpq
Ln
=
y
q
q
=
Mapping Function:
Two hardening variable, and define the shape and size of the yield
surface at current stress state
L
nL = Hardening parameter in
proposed model
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Plastic Potential
-1
-0.5
0
0.5
1
0 0.5 1
p'/po'
g/p'
= 0.8
= 1.2
more dilative
2 1o
g p
p p
=
1
g
gn
q
=
0
0.5
1
0 5 10 15 20
g/ q
ng= 1
ng= 2
Asymptotic
, ng = Plastic potential parameter in proposed model
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Incremental Stress-Strain Compliance
( )( )
1 13 1 2 '
ij ij kk e pij ij ij mn
mn ij
d d f gd d d d vp H
+ = + = +
( )
2 222 ln 2 1o
L g
o
p vL p pH n n L p
p p
= +
Hardening Function, H
Stress-Strain Compliance
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9/7/2004 Prashant and Penumadu. NUMOG-IX 16
Parameters for Kaolin Clay
= 0.016, is determinedfrom o and
oLop and have to be
determined from state of soil
Proposed Model Parameters Value
0.016Elastic Behavior
0.28
Cf 0.63Failure Surface
o 0.9
Reference Surface Cy 0.66
0.16Hardening Parameter
nL 22
0.92Plastic potential
ng 3.5
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9/7/2004 Prashant and Penumadu. NUMOG-IX 17
Predictions for triaxial compression tests
0
40
80
120
160
200
0 0.06 0.12 0.18
Shear Strain, q
Predicted
Measured
OCR = 1.5
q (kPa)
0
40
80
120
160
200
0 0.06 0.12 0.18
Shear Strain, q
Predicted
Measured
OCR = 1
q (kPa)
0
40
80
120
160
200
0 0.06 0.12 0.18
Shear Strain, q
OCR = 1
Predicted
Measuredu (kPa)
0
40
80
120
0 0.06 0.12 0.18
Shear Strain, q
OCR = 1.5
Predicted
Measuredu (kPa)
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9/7/2004 Prashant and Penumadu. NUMOG-IX 18
Predictions for triaxial compression tests
0
40
80
120
160
200
0 0.06 0.12 0.18
Shear Strain, q
OCR = 2
Predicted
Measuredq (kPa)
0
40
80
120
160
200
0 0.06 0.12 0.18
Shear Strain, q
OCR = 5
Predicted
Measuredq (kPa)
0
40
80
120
0 0.06 0.12 0.18
Shear Strain, q
OCR = 2
Predicted
Measured
u kPa
-40
0
40
80
0 0.06 0.12 0.18
Shear Strain, q
OCR = 5
Predicted
Measured
u (kPa)
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Predictions for triaxial compression tests
0
40
80
120
160
0 0.06 0.12 0.18
Shear Strain, q
OCR = 10
Predicted
Measuredq (kPa)
-40
0
40
80
0 0.06 0.12 0.18
Shear Strain, q
OCR = 10
Predicted
Measured
u (kPa)
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9/7/2004 Prashant and Penumadu. NUMOG-IX 20
Predictions for triaxial compression tests
(effective stress paths)
0
50
100
150
200
0 50 100 150 200 250 300
Mean Effective Stress, p' (kPa)
DeviatorStress,q(kPa)
Predicted
Measured
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Summary
Proposed a new constitutive approach:
Isotropic elasticity
Yield surface with teardrop shape
Failure surface as a function of loading history Introducing reference surface as a control for yielding to
capture brittle response
Plastic potential was different from the yield surface, hence
non-associative flow rule.
Uncoupled hardening for volumetric and shear deformation
The model parameters can be determined using two standard
laboratory tests, isotropic consolidation and undrained triaxialcompression test on normally consolidated clay.
New model predicted the kaolin clay behavior with reasonable
accuracy.
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9/7/2004 Prashant and Penumadu. NUMOG-IX 22
Failure and Reference Surface Parameters
f NC f o initialq C p=
y NC y o initialq C p=
1o = Plasticity Ratio or the
exponent in failure surface
o
of f
pq C p
p
=
o
oy y
pq C p
p
= Reference Surface:
For undrained shearing of
normally consolidated clay
Failure Surface:
( )ln 10c
C=
From Consolidation data From Consolidation data or Plasticity ratio
( )ln 10s
C=
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9/7/2004 Prashant and Penumadu. NUMOG-IX 23
Hardening Parameters
0.00
0.40
0.80
1.20
1.60
0 0.01 0.02 0.03 0.04
( L/nL) from hardening rule
(L)fromY
ieldFunction
1
nL= 22
Lo= 0.75
( )1 pqL
L
n
=
Hardening Parameter nL
( )vo o
p
p
p p
=
( )1Lpq
Ln
=
Volumetric Hardening
ParameterOCR = 1
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Plastic Potential Parameters
0.000
0.003
0.006
0.009
0.012
0.015
0.018
0 0.001 0.002 0.003 0.004 0.005
pp
(Rgq
p)
1
ng= 3.5
pq
p
p
d g g
q pd
=
p p
g q g pR n =
( ) ( ){ }1 2 1og
p pR
=
0g
p =
Stress state at
reference surface2 1
o
g p
p p
=
1g
gn
q
=
Where,
OCR = 1
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9/7/2004 Prashant and Penumadu. NUMOG-IX 25
Typical shape of the plastic potential
Typical plastic potential
surface for = 0.8
p'/2
q
p'
Typical plastic potential
surface for = 1.2
2
1
3
Hydrostatic Line
Typical plastic potentialsurface for > 1
Ellipsoidal plastic
potential surface for = 1
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1
Plastic Stress-Strain Compliance
0oo
f f f ff dp dq dp dL
p q p M
= + + + =
( )p p po o o
o p q pp p
p q
p p vpdp d d d
= + =
( )1p p pp q L qp pp q
L LdL d d n d
= + =
( )
( )1p po p L qo
vpf f f fd n d dp dq
p M p q
+ = +
Consistency
condition requires:
From hardening rule:
2
3
4
Using Eq. 2 and 3, Eq. 1 can be rearranged in the following form
p
p
gd d
p
=
p
q
gd d
q
=
Flow rule: 5 6
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Plastic Stress-Strain Compliance
cont.
22 ln opf
LpL p
=
22
o o
f pL
p p
=
( )
2 222 ln 2 1o
L g
o
p vL p pH n n L p
p p
= +
( )
( )
1
1 oLo
f fd dp dq
vpf g f g p qn
M q p p
= + +
( )( )
1 oLo
vpf g f gH n
L q p p
= +
Using Eq. 4 to 6, the
loading function can
be derived as:
Therefore, material
hardening:
Derivatives of yield
surface:
Material hardening:
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9/7/2004 Prashant and Penumadu. NUMOG-IX 28
Predictions using Modified Cam-clay
Model: NC Kaolin clay
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18
Shear Strain, q
DeviatoricStress,q
(kPa)
Predicted
Measured
Comparison of experimental data with modified cam-clay predictions for
undrained triaxial compression test on normally consolidated Kaolin clay
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
ExcessPorePressure
,u
(kPa) Predicted
Measured
Reference: Schofield, A.N. & Wroth, C.P. (1968). Critical State Soil Mechanics.
Maidenhead, McGraw-Hill.
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Predictions using Modified Cam-clay
Model: Shear Stress-Strain Relationship
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18
Shear Strain, q
DeviatorStress,q(kPa
)
OCR=1
OCR=5OCR=10
Experimental
Predictions
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P di ti i Si l H d i
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9/7/2004 Prashant and Penumadu. NUMOG-IX 31
Predictions using a Single Hardening
Model proposed by Lade (1990)
Comparison of experimental data with single hardening model predictions for
undrained triaxial compression test on normally consolidated Kaolin clay
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18
Shear Strain, q
DeviatoricStress,q
(kPa)
Predicted
Measured
0
40
80
120
160
200
240
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
ExcessPorePressure,u
(kPa)
Predicted
Measured
Reference: Lade, P. V. (1990). Single Hardening Model with Application to NC Clay.
Journal of Geotechnical Engineering, 116(3), 394-415.
P di ti i Si l H d i
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9/7/2004 Prashant and Penumadu. NUMOG-IX 32
Predictions using a Single Hardening
Model proposed by Lade (1990)
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
Deviatoric
Stress,q(kPa)
Predicted
Measured
OCR = 1.5
0
40
80
120
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
ExcessPorePressure,u
(kP
a)
OCR = 1.5
Predicted
Measured
P di ti i Si l H d i
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9/7/2004 Prashant and Penumadu. NUMOG-IX 33
Predictions using a Single Hardening
Model proposed by Lade (1990)
0
40
80
120
160
200
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
DeviatoricStress,q(kPa)
OCR = 2
Predicted
Measured
0
40
80
120
0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q
ExcessPore
Pressure,u
(kPa)
OCR = 2
Predicted
Measured
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