icces05 presentation

8/13/2019 ICCES05 Presentation

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1/ 20/ 2006 1

Experimental and Analytical Aspects of StrainExperimental and Analytical Aspects of Strain

Localization for Cohesive Frictional MaterialsLocalization for Cohesive Frictional Materials

Dayakar Penumadu: Professor, Department of Civil and Environ. Engineering,

University of Tennessee, Knoxville, TN 37996, USA

Ajanta Sachan: Former graduate student, Department of Civil and Environ.Engineering, University of Tennessee, Knoxville, TN 37996, USA

Amit Prashant:Assistant Professor, Department of Civil Engineering, IndianInstitute of Technology, Kanpur, UP 208016, India

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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1/ 20/ 2006 Penumadu, Sachan, and Prashant. ICCES-2005 2

Digital image analysis (DIA) for studying theDigital image analysis (DIA) for studying the

strain localizationstrain localization

Deformation of a soil element

Digital

Imaging

setup

Zone of measurement

Circumferential co-ordinate

10 mmVerticalCo-ordinate

Cast-acrylic

Cell

water

Clay

specimen

Triaxial Clay specimen

Shear

Band

(a) Before Loading (b) Uniform Deformation (c) Strain Localization

Shear

Band

Shear

Band

(a) Before Loading (b) Uniform Deformation (c) Strain Localization


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Strain contour plots for solid cylindricalStrain contour plots for solid cylindrical

specimens of Kaolin clayspecimens of Kaolin clay

Angle of orientation of shear band=Global strain at shear band formation

sb =

Maximum local strainm

=

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00

Contour plot for 11% global axial strain

2.00

3.00

4.00

5.00

6.00

7.00

8.00

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

Initiation of

shear band

Local strainValues

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00


2.00

3.00

4.00

5.00

6.00

7.00

8.00

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

Relatively Uniform

Deformation

Local strainValues

-5.00 -4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00


2.00

3.00

4.00

5.00

6.00

7.00

8.00

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

Local strain

Values

Zone A

Zone B

Formation of

shear band


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Shear banding in Kaolin clay specimens usingShear banding in Kaolin clay specimens using

Lubricated end triaxial setupLubricated end triaxial setup


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Observed Strain Localization within the CubicalObserved Strain Localization within the Cubical

Specimens during True Triaxial TestingSpecimens during True Triaxial Testing

Undeformed Specimen

Shear Banding at FailureDefused Localization


6/201/ 20/ 2006 Penumadu, Sachan, and Prashant. ICCES-2005 6

Onset of Localization:Onset of Localization:

Variation in Sum of Principal StrainsVariation in Sum of Principal Strains

-6

-3

0

3

6

9

12

0 3 6 9 12 15 18

MAJOR PRINCIPAL STRAIN,1(%)

SUMO

F

PRINCIPALST

RAINS,1+2+

3

(%)

b=0

b=1.0

b=0.75

b=0.5

b=0.25- Failure Location.

-

-

-

-

-1.2

-0.9

-0.6

-0.3

0.0

3 5 7 9

b=1.0 b=0.75 b=0.5



Deformation Under Flexible BoundaryDeformation Under Flexible Boundary

ConditionCondition

These marks represent the location for

Measurement of deformation.

Center area (Shaded) on each face

deformed slightly more than the corner

and edge area (Not shaded)

Pre-failure deformations were uniformon the center of the faces

Localization developed near failure

produces non-uniformity on the surfaces

Non-uniformity in deformationDeformation profile



Sudden Failure Response of Clay Specimens Due toSudden Failure Response of Clay Specimens Due to

Strain LocalizationStrain Localization

The pre-failure elasto-plastic deformation in mostconstitutive theories is modelled by considering its strong

relationship with failure stress state parameters. It isassumed that when a soil element is subjected to shearloading, it yields consistently following a hardening ruleand smoothly reaches a stress state where continuously

decreasing shear stiffness becomes zero, which theauthors define as a reference state.

A series of true triaxial experiments performed duringthis study suggest that the strain localization occursduring hardening of clay, which leads to a sudden failureresponse within the specimen. In the absence oflocalized deformation, the soil element may sustainhigher stress and eventually reach the reference state.



Strain Localization and Sudden Failure ResponseStrain Localization and Sudden Failure Response

During triaxial undrained shearing, the specimens ofKaolin clay experienced localized deformations in theform of thin shear bands and/or local bulging at the peakshear stress location.

Due to localized deformations, the specimenexperienced an abrupt loss of the shear stiffness at peakshear stress and showed a sudden failure.

Smooth Failure

Sudden Failure

Reference

StateSmooth Failure

Sudden Failure

Reference

State



Concept of Reference State and its SignificanceConcept of Reference State and its Significance

The strain localization may have a relationship with theconstitutive properties of soil; however, it is also largelyinfluenced by many other factors such as specimenboundary and material imperfections. Due to strainlocalizations, soil elements may show early failure andreach the critical state before the shear stiffness decreasesto zero i.e. before reaching the reference stress state.

These sudden failure conditions (caused by strainlocalization) might be independent of the soil propertiesdefining the pre-failure elasto-plastic yielding of clay.

A constitutive theory developed for such clay should defineits formulation of material-yielding independent of thefailure surface. The failure surface may be defined as alower bound for the Reference surface (especially in

deviatoric plane), and that would ensure the applicability ofthe definition of failure at peak deviatoric stress.



Failure Surface and Reference State in DeviatoricFailure Surface and Reference State in Deviatoric

PlanePlane

x y

z

OCR=1

x y

z

OCR=5

Reference state

Failure point

I3surfaceJ2'surface

Experimental

referencesurface

I3surface

J2'surface

Experimental

reference

surface

The failure envelop could be reasonably described using the I3 = constant (for a

deviatoric plane) surface for both the OCR values. The reference stress states

followed a different pattern than the failure points in deviatoric plane, and the

surface connecting these reference states was observed between the I3

and J2

surfaces.

3 1 2 3. .I =2

23

qJ=



Condition of Continuous Bifurcation inCondition of Continuous Bifurcation in

Elastoplastic MaterialElastoplastic MaterialThe 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.




ElastoplasticElastoplasticMaterialMaterial

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 tensor eB , 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 ellipticityHle is determined as

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

Loss of Ellipticity:Loss of Ellipticity:

elastoplastic acoustic tensor




ElastoplasticElastoplasticMaterialMaterial

For a nonassociative elastoplastic model (fg), the stiffness tensor Dacoustic 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 theare not symmetric, and the condition of bifurcation is defined

Loss of Strong Ellipticity:Loss of Strong Ellipticity:



A Nonassociative Elastoplastic ModelA Nonassociative Elastoplastic Model

The analysis presented in this paper is based on a nonassociative elastoplastic constitutive

model developed by the authors. A detailed description of the proposed model can be

found in Prashant and Penumadu [3]. Following are the key components of the model.

( ) ( )2 2

ln of q p L p p =

2 1o

g p

p p

=

1g

gn

q

=

Yield Surface:

Plastic Potential:

yq q=

( ) oy y oq C p p p

=

Mapping Variable:

Reference state shear stress:

Here, p( 1 3I= ) is mean effective stress, op is mean effective pre-consolidation stress,

23J= ) is deviatoric stress, L is a state variable, and , ng, Cy, and o are materialq (

constants.


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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 (%)

NormalizedHardein

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

Major principal Strain (%)

NormalizedHard

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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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

A series of triaxial shear tests using different specimenshapes and loading/boundary conditions were performedin this study, and they all indicated the occurrence of

strong strain localization zones within a deformingspecimen near the peak shear stress location.

New techniques have been developed for identifying theonset of strain localization, which is important to consider

for modeling the initiation of catastrophic failures ingeotechnical structures.

It was found that mere implementation of thin shear band

type strain localization theory for a single homogeneoussoil element does not provide sufficient condition formodeling the sudden failures triggered by localizeddeformations as observed in the experiments.


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1/ 20/ 2006 Penumadu Sachan andPrashant ICCES-2005 20

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 shearbehavior 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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