ICCES05 Presentation
Transcript of ICCES05 Presentation
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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