numog ix presentation

8/13/2019 NUMOG IX Presentation

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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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9/7/2004 Prashant and Penumadu. NUMOG-IX 2

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


3/33


= 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.


4/33


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.


5/33


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


6/33


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


7/33


Yield Surface in 3-D stress space


8/33


( )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)


9/33


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


10/33


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.


11/33


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


12/33


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



po'=300 kPa

L = 0.8

o= 0.7

Yield Surface

Failure Surface:

C = Cf= 0.75

Reference Surface:

C = Cy= 0.8


13/33


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


14/33


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


15/33


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


16/33


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


17/33


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)


18/33



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)


19/33



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)


20/33



(effective stress paths)

0

50

100

150

200

0 50 100 150 200 250 300


DeviatorStress,q(kPa)

Predicted

Measured


21/33


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.


22/33


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=


23/33


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


24/33


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


25/33


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


26/33


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


27/33


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:


28/33


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.


29/33


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


30/33

P di ti i Si l H d i


31/33


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.



32/33




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



33/33




0

40

80

120

160

200

0 0.03 0.06 0.09 0.12 0.15 0.18Shear Strain, q


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

numog ix presentation

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