Post on 04-Jun-2018
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Constitutive Modeling of ClaysUsing Cam Clay and Modified
Cam Clay Models
A PRESENTATIONBY
AMIT PRASHANTGRADUATE STUDENT
CLARKSON UNIVERSITY
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Discussion on -1. Important definitions
2. Original Cam-clay model parameters3. Stable State Boundary Surface
4. Cam:clay Flow Rule5. Plastic potentials and Normalitycondition
6. Modified Cam-clay Model7. Stress:Strain relationship
8. Examples
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State of Sample During Triaxial test
Mean effective stress, p= ( 1+ 2+ 3)/3 = ( 1+ 2+ 3)/3 uShear stress, q= [{( 1- 2)2+( 2- 3)2+( 3- 1)2}/2] 1/2
Specific volume, v=1+e
Strains Corresponding to p and q:- Volumetric strain, v = 1+ 2+ 3Shear strain, = 1/3[2{( 1-2)2+( 2-3)2+( 3-1)2}]1/2
p'
v p' p'
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DEFINITIONS Yield point:- This is the state of stress at which the
sample starts to deform with plastic deformations.
Failure point:- This is state of stress at which theapplied shear stress is maximum while shearing the sample.
Critical state:- This isthe state of stress (for monotonicloading) at which further sheardeformation can occur without
further change in effective stressand void ratio.
Yield pointFailure point
Critical statefor both the cases
1
q
Yield point may besomewhere in this range
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Soil Strength ParametersSoil constants:-
Slope of Iso-NCL in v-ln p plane, Slope of URL in v-ln p plane, Specific volume on CSL at unit pressure in v-ln p
plane, Slope of CSL in p-q plane, MShear modulus, G
Two tests required to determine parameters:-Triaxial Compression testIsotropic Consolidation test
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Soil Strength Parameters
CSL
Iso-NCL
URL
p'=1
p'
v
vv
Modified Cam-clay
Original Cam-clay
p'
q
CSLYield Surface
1
M
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Volume and Pressure Relationships
NCL :- v=v - ln(p)URL :- v=v - ln(p)CSL :- v= - ln(p)
q=Mp
NOTE:-Specific volume and mean effective stress at critical state, found inthe triaxial compression test may be used to determine the valueas the CSL has the same slope as NCL.
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Stable State Boundary Surface
q
p'
v
CSL
SSBS
q
v
p'CSL
q
pc' pc'/2.72
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Stable State Boundary Surface
Equation of SSBS :-
q=Mp( + v ln(p))/( )OR
V= +( )(1 /M)
Where = Current value of q/p
Equation of Yield Surface in q-p plane:-q=M.p.ln(p c/p)
Where pc = Isotropic pre-consolidation pressure
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Cam-clay Flow ruleEnergy dissipated on yielding:-
p vp+q
p=Mp
p
Flow Rule:- (rearranging above equation)
vp / p = M M > gives positive volumetric strain, i.e. the sample iscompressive in nature. The stress state is called on the dry side ofcritical state line.
M < gives negative volumetric strain, i.e. the sample isdilatative in nature. Stress state is called on the wet side of criticalstate line.
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Plastic Potential Surface
Plastic Strain Increment Vector:- Plastic volumetricstrain and plastic shear strain take the same direction as p and qrespectively. The vector sum of increments of these strains at anystress state is called as plastic strain increment vector.
Plastic Potentials:-Plastic potentials form the familyof curves to which the plastic
strain increment vectors areorthogonal.
p'
q p
p p
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Normality Condition
When the yield loci and plastic potentials coincide each
other, normality condition or associative flow rule exists.This means the plastic strain vectors are orthogonal to theyield loci itself.Cam-clay model follows this normality condition, finding itto be reasonably good assumption from the simplificationpoint of view.In original cam-clay model, theyield curve shows a kink at theIsotropic pre-consolidationpressure and that shows twoplastic strain vectors at one point.
p' p c'
q Strain vectors at pre-consolidation pressure
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Modified Cam-clay ModelThis was proposed with elliptical yield surface, having nocontradiction to normality condition for strain vectors atIsotropic pre-consolidation pressure.Equation of SSBS :-
V= +( ){ln(2) ln(1+( /M) 2)}
Where = Current value of q/p
Equation of Yield locus in q-p plane:-q2+M2p 2=M2.p.p c
Where p c = Isotropic pre-consolidation pressure
pc'
q
p'
CSL YieldSurace
1M M.p'/2
p'/2
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Modified Cam-clay Flow Rule
Energy dissipated on yielding:-
p vp
+q p
=p[ vp2
+(M p
)2
]1/2
Flow Rule:- (rearranging above equation)vp / p = (M 2 2)/2
This again explains deformation the same way as original Cam-clayModel.
M > gives positive volumetric strain, i.e. the sample iscompressive in nature.
M < gives negative volumetric strain, i.e. the sample isdilatative in nature.
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Elastic and Plastic stress-strain response
Elastic stress:strain response:-
Plastic stress:strain response:-
Pre-yield (elastic) andPost-yield (elastic-plastic)deformations.
p p
q p = q
p'
2
()
vp' (M + )2 2(M - )22 2
/(M - )2 242 q
p' p'cA cB p'
= p0q
e1/3G q
/vp'e
0 p'
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Undrained Test Normally Consolidated Sample
B
A p'
u = p - p'
G
H
A
CSL
v NCL
q
cH p'
A p'
cA
H
cG p'
u p'
G
CSLB
ESP
TSPq
H
G
B
cB p'
AH
GB
uAH
uAG
uf
f u
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Undrained Test Lightly Over-consolidated Sample
CSL
p' p'
v
p'
CSL NCL u
u = p - p'
ESP
TSP
cA p'cC cG p'A
B
G
C
A
B
GC
A
B
G
C
CG B A
uAB
uAGuf
uf
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Undrained Test Highly Over-consolidated Sample
B
TSP
ESP
CSL
A
A
p'
NCLvCSL
u
A
DB C
u = p - p'
q q
p'
C D
B
C
D
uBD
uAB
ABu
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Drained Test Normally Consolidated or Lightly Over-consolidated Sample
A
A
v NCLCSL
p'
e
TSP & ESP H
A
B
CCSL
p'
GB
H
G
A BGH
C
BG
H
C
C
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Drained Test Highly Over-consolidated Sample
CC
p'
A
A
TSP & ESP
v
CSL NCL
q
p'
AB
A
v
B
C
CSL q
B
B
C
e
ACt
e
ABe
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Comments-This model explains most of the trends observed instress:strain behavior of clay, Hardening; Softening
dilatation etc.This model uses Extended von-misses criteria as its failureenvelop and that doesnt include the variation of shear
strength due to changing intermediate principal stress.This is probably due to the unavailability of threedimensional test data during the development of thesemodels.Predictions of undrained tests always show no variation inmean effective stress before yielding, which is not alwaystrue.
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Comments-
The p-q plane superimposed
on triaxial plane shows thatthe strength of highly over-consolidated clay is
over predicted as the claymay not bear this much oftension in principal directions
before failure.
p'
q
2. '3
1 '
Tension