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    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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    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

    Contour plot for 6% 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

    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

    Contour plot for 14% 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

    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

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    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

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    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

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    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.

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    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

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    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.

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    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=

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    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.

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    Condition of Continuous Bifurcation inCondition of Continuous Bifurcation in

    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

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    Condition of Continuous Bifurcation inCondition of Continuous Bifurcation in

    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:

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    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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    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