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

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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

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5

Hardening

σ

ε

σy

2σy

σ

ε

σyn

2σyn

Kinematic Hardening Isotropic Hardening

Yield Surfaces, 2D

p

q

Mohr Coulomb

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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

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11

Bounding Surface Plasticity

• Originally formulated from Dafalias, 1987

Dafalias – Manzari, 2004

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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)

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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