3MIT Presentation

8/13/2019 3MIT Presentation

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8/24/2005 1

Localization Analysis for OverconsolidatedLocalization Analysis for Overconsolidated

Kaolin Clay BehaviorKaolin Clay Behavior

Amit Prashant: Department of Civil Engineering,Indian Institute of Technology, Kanpur, UP 208016, India

Dayakar Penumadu: Department of Civil and Environ. 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.

Session 101 - Computational plastic ity, Part VIa

Room: 4-370, Chairpersons: F.J. Montans and J.C. Glvez


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8/24/2005 Prashant and Penumadu. Third MIT Conference-2005 2

Condition of Continuous Bifurcation inCondition of Continuous Bifurcation in

Elastoplastic MaterialElastoplastic Material

The classical elastoplasticity theory defines the following tangential constitutive relationships.

: = E Elasticity,

Elastic stiffness,

Elastoplasticity,

Elastoplastic stiffness,

where, and

( )ijkl ij kl ik jl il jk = + +E

: = D

: :

: :H

=

+

E P Q ED E

Q E P

f f =

Q

g g

= P

Here, f is the yield function, g is the plastic potential, and is the Euclidian norm of the tensor.


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ElastoplasticElastoplastic MaterialMaterial

The theory of localization defines the condition of continuous bifurcation forelastoplastic deformations across the shear band based on the vanishing of the

determinant of acoustic tensor, which is derived from the constitutive stiffness tensor.

For a unit vector n normal to the shear band, the elastic acoustic tensore

B , and

B is defined using the equations below.

n ne = B E and n n= B D

The hardening modulus corresponding to the loss of ellipticity Hle is determined as

( ) ( ) ( )1 : n n : : :leH = Q E E E P Q E P

Loss of Ellipticity:Loss of Ellipticity:

elastoplastic acoustic tensor


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ElastoplasticElastoplastic MaterialMaterial

For a nonassociative elastoplastic model (f g), the stiffness tensor D

acoustic tensors B

by the vanishing of the symmetric part of the acoustic tensor (loss of strong ellipticity),

det 0sym=B . In such condition, the hardening modulus Hlse is determined as

( ) ( )

( ) ( ){ }

( ) ( ){ }

( )

1

1/ 2 1/ 21 1

: n n :1

: :2 : n n : : n n :

sym

lse

sym sym

H

+ =

Q E E E PQ E P

P E E E P Q E E E Q

The occurrence of shear banding and its orientation is obtained by searching the largest

critical hardening modulus and the corresponding unit vector .n

and the

are not symmetric, and the condition of bifurcation is defined

Loss of Strong Ellipticity:Loss of Strong Ellipticity:


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= Elastic volumetric strain

= Elastic shear strain

= slope of unload-reload curve in e-ln(p')space

Specific volume, v = 1 + e

= Poissons ratio

A Nonassociative Elastoplastic ModelA Nonassociative Elastoplastic Model

Isotropic elasticityIsotropic elasticity

e

p

( )( )

v ' 0

'10

3 1-2 v '

e

pe

q

p

pq

p

+=

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-straincompliance 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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Yield surfaceYield surface2

2ln

opqf Lp p

=

0

50

100

150

200

0 50 100 150 200 250 300 350

Mean Effective Stress, p' (kPa)

Deviatoric

Stress,

q(

k

Pa)

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


D

eviatoric

Stress,

q(

k

Pa)

po'=100 kPa

po'=200 kPa

po'=300 kPa

L=1


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Yield Surface in 3Yield Surface in 3--D stress spaceD stress space


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Normalized failure surface in qNormalized failure surface in q--p' spacep' space

0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

Mean Effectiv e 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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Reference SurfaceReference 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


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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Mean Effect ive Stress, p'

Deviat

oric

Stress,

q

Yield Surface

ReferenceSurface

q

qy


Hardening RuleHardening Ruleop

( )v o o

p

p

p p

= 0o

p

q

p

= 0

p

p

L

= ( ) 1Lp

q

L n

=

y

q

q

=

Mapping Function:

Two hardening variables, 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 PotentialPlastic Potential

-1

-0.5

0

0.5

1

0 0.5 1

p'/po'

g/p'

= 0.8

= 1.2

more dilative

2 1

o

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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Parameters for Kaolin ClayParameters 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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Model Predictions forModel Predictions forTriaxialTriaxial Compression TestsCompression 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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Model Predictions forModel Predictions forTriaxialTriaxial Compression TestsCompression 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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Loss of Ellipticity vs. Loss of Strong EllipticityLoss of Ellipticity vs. Loss of Strong Ellipticity

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 5 10 15

Major principal Strain (%)

Normaliz

edHardein

gModulus

Hlse/GHle/G

OCR = 5

OCR = 1


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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 5 10 15


Norma

lizedHard

eingModulus

H/G

Hlse/G

OCR = 1

= 40

= 35

= 42 = 42

Strain Localization Analysis using Concept ofStrain Localization Analysis using Concept of

the Loss of Strong Ellipticitythe Loss of Strong Ellipticity

H = Hardening modulus of the clay

Hlse = Critical Hardening modulus

3'

1'

Plane of possible

shear banding


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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 5 10 15


Norma

lizedHard

eingModulus

H/G

Hlse/G

OCR = 1.5

= 42 = 41

= 43 = 43




Hlse = Critical Hardening modulus3'

1'

Plane of possible

shear banding


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-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 5 10 15Major principal Strain (%)

NormalizedHardeingModulu

sH/G

Hlse/G

OCR = 10

= 48 = 49

= 47 = 47


Hlse = Critical Hardening modulus

3'

1'

Plane of possible

shear banding


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Comments on Strain LocalizationComments on Strain Localization

According to the theory of the vanishing of the acoustic tensor, the

onset of shear band type strain localization in a soil element occurs

when H = Hlse. This condition was never achieved during hardening

regime for the model to predict the onset of localization, which is

consistent with the findings of Rudnicki and Rice [5] using a

generalized and simple constitutive law for soils and rocks.

The angle calculations may not match with the experimentally

observed as the theory did not predict shear banding at all. Incompression mode it was difficult to observe shear banding visually;

however, indirect methods suggested some kind of strain localization

at the peak shear stress location.

It should also be noted that some other modes of instabilities might

occur before shear banding, such as the growing nonuniformities

due to undrained instability under globally undrained but locally

drained conditions, Rice [4].


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ConclusionsConclusions

The concept of instability based on the vanishing of the acoustic

tensor was evaluated using the experimental data obtained from a

series of undrained triaxial compression test on normally to highly

overconsolidated Kaolin clay.

It was shown by a numerical evaluation that the loss of strong

ellipticity would occur before the loss of ellipticity for non-associative

materials.

For all OCR cases, it was observed that a soil element subjected to

undrained triaxial compression loading would not experience

instability of the acoustic tensor before the experimentally observed

failure location.

The inclination of the possible weak planes was observed to be a

function of the OCR value.


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ReferencesReferences

Bardet, JP. A comprehensive review of strain localization in elastoplastic soils. Computers

and Geotechniques 1990; 10:163-188.

Neilsen, MK, Schreyer, HL. Bifurcations in elastic-plastic materials. Int. J. Solids Struct 1993;

30:521-544. Prashant, A, Penumadu, D. Modeling the effect of overconsolidation on shear behavior of

cohesive soils. In: Proc. 9th Symp. Num. Models in Geomech., Ottawa, Canada, 2004; 131-

137.

Rice, RJ. On the stability of dilatant hardening of structured rock masses. J. Geophys.

Research 1975; 80(11):1531-1536.

Rudnicki, JW, Rice, JR. Conditions for the localization of deformation in pressure-sensitive

dilatant materials. J. Mech. Phys. Solids 1975; 23:371-394.

Szab, L. Comments on loss of strong ellipticity in elastoplasticity. Int. J. Solids and

Structures 2000; 37:3775-3806

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