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4/29/2009
1
Constitutive Modeling
Total Stress Analysis vs. Effective Stress
Analysis
• Difference on how to approximate liquid face
of soil
• Effective Stress Analysis is more precise
• VELACS project
4/29/2009
2
Total Stress Analysis
• Modulus Reduction Curves
• Ramberg Osgood
Effective Stress
• Elastic (linear/ non-linear/ visco-elastic)
• Perfectly plastic
• Mohr Coulomb (Elastic/ Perfectly plastic)
• Cam Clay
• Multi Yield Surface Models
• Bounding Surface Plasticity
4/29/2009
3
Elasto-Plasticity: Useful Readings
• Computational Inelasticity, Simo and Hughes
• Computational Geomechanics notes, Boris
Jeremic
Perfectly Plastic Models
• No deformation Before Yielding
• Successfully utilized for Static Foundation
problems
• Obviously not good for Wave propagation
analyses
σ
ε
4/29/2009
4
Elasto Plasticity
Elastic-Friction Model
σ
ε
σ
Deformation ε
Yield Surface
• σ=Ε(ε-εp)
• Yield funtion:
– f=|σ|-σγ<0
• dεp=γsign(σ)
• Loading/ Unloading
• Consistency
– Upon loading, df=0
σ
ε
εp
4/29/2009
5
Hardening
σ
ε
σy
2σy
σ
ε
σyn
2σyn
Kinematic Hardening Isotropic Hardening
Yield Surfaces, 2D
p
q
Mohr Coulomb
4/29/2009
6
Yield Surfaces
p, εv
q,
εs
Mohr Coulomb
Associative vs non-associative plastic flow rule
Yield Surface
Dilation Surface
Mohr Coulomb
• Statically great for a variety of problems
– Foundations
– Slope Stability
– Gravity Walls
• Dynamically not good: excessive strains after
yielding
4/29/2009
7
Yield Surfaces
3D Principal Space
Matsuoka Nakai Yield Surface
Loading-Unloading
• Stress paths moving away from the yield
surface are in general defined as loading
4/29/2009
8
Neutral Loading
Rotation of principal stresses
Problems with Traditional Plasticity
• Neutral loading paths are not always observed
in nature
4/29/2009
9
Problems with Traditional Plasticity
• Elastic Region Very small
Drained Triaxial Test (Nevada Sand Dr=63.9%)
Problems with Traditional Plasticity
• Unloading introduced plasticity
Undrained Triaxial Test (Nevada Sand Dr=63.9%)
4/29/2009
10
“Ingredients” of Constitutive Soil
Models for Cyclic Behavior of Sands
• Monotonic Response
– Large Deformations post-seismic event
• Excess pore pressure accumulation (shear
induced contraction and dilation)
• Phase Transformation
“Ingredients” of Constitutive Soil
Models for Cyclic Behavior of Sands
• Dependence of Peak and Phase Tranformation
stress-ratios on the state parameter (void ratio
& stress)
• Dependence of Plastic Modulus and the
Dilatancy on the inherent anisotropy
• Dependence of Plastic Modulus and the
Dilatancy on the evolving anisotropy
4/29/2009
11
Bounding Surface Plasticity
• Originally formulated from Dafalias, 1987
Dafalias – Manzari, 2004
4/29/2009
12
Dafalias Manzari
Bounding Surface Plasticity
Success
• Excellent Simplicity
• Same parameters for various stress levels and
voids ratio
• Critical State Soil mechanics concepts
implemented
4/29/2009
13
Dafalias Manzari - Simulations
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 20 40 60 80 100
τ (k
Pa
)
σv (kPa)
Dr: 50%
-50
-40
-30
-20
-10
0
10
20
30
40
50
-0.04 -0.02 0 0.02 0.04
τ (k
Pa
)γ
Dr: 50%
-50
-40
-30
-20
-10
0
10
20
30
40
50
-0.04 -0.02 0 0.02 0.04
τ (k
Pa
)γ
Dr: 50%
Undrained Simple Shear Test (Toyoura Sand)
Dafalias Manzari - Simulations
0
0.05
0.1
0.15
0.2
0.25
1 10 100
τ/σ
v0
N
Dr=50%
γ=1%
γ=2.5%
Undrained Simple Shear Test (Toyoura Sand)
4/29/2009
14
Dafalias Manzari
Pitfalls
• The model does not predict cycle to cycle
evolving damage (changes in the stress strain
properties are mainly attributed to the
increase of excess pore pressure)
• Critical State is unique
Multi-Yield-Surface
Yield Surface
Yang et al
4/29/2009
15
Multi-Yield-Surface
Yield Surface
Yang et al
Model Hardening
Yang et al
4/29/2009
16
Multi-Yield-Surface:
Success
Undrained Simple Shear Test (Medium-Loose Nevada Sand)
Yang et al
Multi-Yield-Surface:
Success
• Relatively Simple
• Can model Excess Pore Pressure Accumulation
• Can model accumulative Damage Effects
• Can model stress level dependence
4/29/2009
17
Multi-Yield-Surface: Pitfalls
• Properties have to be calculated at every void
ratio
• Phase transformation angle is a property for a
specific voids ratio (we know that it depends
on stress level)
History
Mohr Coulomb (1776)
Von Mises (1913)
Perfect Plasticity
Cam Clay
MIT-E1
MIT-E3
MIT-S1
Multi Yield Surface
Plasticity(MYSP), Prevost
Bounding Surface
Plasticity (BSP), Dafalias
MYSP, Elgamal
MYSP, Yang
BSP, Manzari Dafalias
BSP, Papadimitriou et al
BSP, Dafalias Manzari
Hypoplasticity
Micromechanics
Statistical Mechanics,
Thermodynamics
4/29/2009
18
Future
• A lot of improvements on the current models.
• Shear banding, localization. Monte Carlo
simulations.
• Multi-axial Conditions