Constitutive Modelling of Unsaturated Soils: Discussion of ... · Implementation of unsaturated soil models into FEM 7. ... Soils, Beijing, 1998, expansive soils) 4. Fredlund (3rd

Centre for Geotechnical and Materials Modellinghttp://livesite.newcastle.edu.au/cgmm/1/66

Daichao SHENG

The University of Newcastle, Australia

General Report5th International Conference on Unsaturated Soils

6-8 Sept, 2010, Barcelona, Spain

Constitutive Modelling of Unsaturated Soils: Discussion of Fundamental Principles

Centre for Geotechnical and Materials Modellinghttp://livesite.newcastle.edu.au/cgmm/2

Outline

1. Introduction

2. Volume change behaviour of unsaturated soils

3. Yield stress versus suction relationship

4. Shear strength of unsaturated soils

5. Hydro-mechanical coupling for unsaturated soils

6. Implementation of unsaturated soil models into FEM

7. Concluding remarks


(Li, 2010)

1. Introduction: Fundamental Issues

Volume change caused by suction change

(Gens, 2007 ← Viggiani)



Shear strength change caused by suction change

Zhou-Qu Landslide (8-8-2010, Gansu, China)

Thredbo Landslide (30-7-1997, NSW, Australia)



MINE WASTE

UNSATURATED SOIL LAYERS

Flow of moisture, oxygen, heat,…

Hydraulic properties of unsaturated soils



• Volume change behaviour associated with suction change.

• Shear strength behaviour associated with suction change.

• Flow characteristics (hydraulic behaviour) of unsaturated soils.

Saturated Soil Model

Complete Soil Model (saturated & unsaturated)

+


s

1. Introduction: BBM (Alonso, Gens & Josa, 1990)

Modified Cam Clay model (saturated soil) + shear strength vs suction

+ volume change vs suction

q

ap p u= −

Loading-collapse surface ↔ volume

Zero shear strength line


1. Introduction: BBM (Alonso, Gens & Josa, 1990)

BBM (Alonso et al, 1990):

1. It is the very first complete model that accommodates the key fundamental issues of unsaturated soils.

2. BBM’s seminal contributions: treating suction as an additional variable in the stress space and using the Loading-Collapse (LC) yield surface to model wetting-induced volume collapse.

3. It has inspired many researchers to unsaturated soil mechanics. Numerous other models have since been developed.


State-of-the-art reviews on constitutive modelling of unsaturated soils:

1. Gens (1st Int. Conf. Unsat. Soils, Paris, 1995)

2. Wheeler & Karube (1st Int. Conf. Unsat. Soils, Paris, 1995)

3. Alonso (2nd Int. Conf. Unsat. Soils, Beijing, 1998, expansive soils)

4. Fredlund (3rd Int. Conf. Unsat. Soils, Recife, 2002, hydraulic)

5. Vaunat (1st MUSE School, Barcelona, 2005)

6. Wheeler (4th Int. Conf. Unsat. Soils, Arizona, 2006)

7. Gens (1st European Conf. Unsat. Soils, Durham, 2008)

8. Gens (4th Asian-Pacific Conf. Unsat. Soils, Newcastle, 2009)

9. Gens (47th Rankine Lecture, Geotechnique, 2010)

This report focuses on a selected number of fundamental issues.

1. Introduction: State-of-Art Reports


Outline

1. Introduction







Alternative methods


2. Volume Change Behaviour: Saturated Soils

Saturated, normally consolidated soil:

N=3, λ=0.2,

sae>100kPa

Normally consolidated

1.5

2.0

2.5

3.0

1 10 100 1000

uw=0

uw=-10 kPa

uw=-100 kPa

v

lnp

( )wln lnv N p N p uλ λ′= − = − −

w

w w

d( )dd upvp u p u

λ λ −= − −

− −


2. Volume Change Behaviour: Unsaturated Soils

Saturated NC soils:

Approach A. Separate stress and suction approach (net stress and suction approach: Alonso et al. 1990; Wheeler & Sivakumar 1995; Cui & Delage 1996; ….)

Approach B. Combined stress-suction approach (effective stress approach: Kohgo et al 1993; Bolzen et al 1996; Loret & Khalili 2002; Sheng et al 2003, 2004; Sun et al. 2006; …)

Approach C. A more recent approach (SFG approach: Sheng, Fredlund & Gens, 2008)

Unsaturated soils?

w

w w

d( )dd upvp u p u

λ λ −= − −

− −

( )wln lnv N p N p uλ λ′= − = − −


Separate stress and suction (net stress and suction) approach:

Advantage: the compressibilities due to stress and suction changes are handled separately.

2. Volume Change Behaviour: Approach A

atvp vs

at

( ) ln ln s uv N s pu

λ λ⎛ ⎞+

= − − ⎜ ⎟⎝ ⎠

Toll (1990)

Toll & Ong (2003)

λ(Sr=1)

λvp

λvs


The normal compression line (NCL) is a straight line in the space:


atvp vs

at

( ) ln ln s uv N s pu

λ λ⎛ ⎞+

= − − ⎜ ⎟⎝ ⎠

s>0, ds=0

s=0

v

( )aln lnp p u= −

This has implications on yield surface.

lnv p−


The volume change is not well defined at the transition suction. For a stress change from to :

in unsaturated zone

in saturated zone

Implications on the zero shear strength (apparent tensile strength) surface.


ae

aevp

0 ae

Δ lns

p svp s

λ−

+= −

+

aevp

0

Δ lns

pvp

λ+ = −

0p p


Combined stress-suction (effective stress) approach:

2. Volume Change Behaviour: Approach B

s>0, ds=0

( )ln ( ) ln ( )v N p N s p f sλ λ′= − = − +

s=0

lnp

v

ln p

Advantage:

1. NCL is curved in space.

2. It recovers the saturated soil model.

lnv p−


Difficult to handle the different compressibilities due to stress and suction changes


Toll (1990)

Toll & Ong (2003)

λ(Sr=1)

λvp

λvs



( )ln ( ) ln ( )v N p N s p f sλ λ′= − = − +

1

Alternative forms of Approach B

NA

B

s=0

s>0

v

Drying path

ln p′

( ) 1(0)sλ

λ<


2. Volume Change Behaviour: Approach C

vp vsd dd

( ) ( )p sv

p f s p f sλ λ= − −

+ +

SFG Approach: A middle ground between Approach A and B:

( )f s s=Simplest form:

( )vp vs at atln ln ( ) /v N p s u uλ λ= − − +

( )ln ( )v N p f sλ= − +

Approach A:

Approach B:

Sheng, Fredlund & Gens (2008)


0.1

0.3

0.5

0.7

0.9

1.1

1 10 100 1000 10000

NCL: s=10 kPa

NCL: s=0 kPa

2.00

1.46

1.01

0.65

0.35

0.11

Potentially collapsible volume


Normal compression lines under constant suction

N=3.0, λvp=0.2, ssa=10kPa

e

: kPap

All curves are NCLs

vp vsd dd

( ) ( )p sv


+ +

NCL: s=1000 kPa



Volume change caused by suction change: vp vs

d dd =( ) ( )

p svp f s p f s

λ λ− −+ +

0.20

0.40

0.60

0.80

1.00

1.20

1 10 100 1000 10000s (kPa)

e

2.32

1.72

1.23

0.82

0.49

0.22

=100 kPap

=1 kPap

=10 kPap

=1000 kPap

VO

ID R

ATI

O

0.90

0.80

0.70

0.60

0.50

0.401 10 100 1000

SUCTION (kPa)

E1 NC 25 kPa

E2 NC 50 kPa

E7 NC 200 kPa

E11 NC 400 kPa

E5 OC 200 kPa

E6 OC 800 kPa

Vicol. (1990)



s (kPa)

Prediction of SFG model

1 10 100 1000 10000 100000

0.84

0.83

0.82

0.81

0.80

0.79

0.78

e

Air-dry silt: Data from Jennings and Burland (1962)

vp sa

vs savp sa

s ss s ss

λλ

λ

≤⎧⎪= ⎨

>⎪⎩



Compacted expansive clay: Sivakuma and Wheeler (2000)

(kPa)

Yield stress

NCL by Approach A

NCL by SFG model

v

△ s = 0○ s = 300 kPa

10 100 1000

2.25

2.20

2.15

2.10

2.05

2.00

1.95

Unloading-reloading line

p


2. Volume Change Behaviour

Comments on volume change modelling:

• Existing models all have advantages and disadvantages.

• There is no model that use one single stress variable to describe the volume change of unsaturated soils.

• The volume change model also underpins the yield stress – suction relation.

• Discussion: In the combined stress-suction approach (Approach B), it is perhaps worthwhile to explore:

In this case, Sr (instead of s) is used as an additional axis in the stress space.

( )r r( )ln ( ) ( )lnv N S p f s N S pλ λ ′= − + = −


Outline

1. Introduction








3. Yield Stress vs Suction: Approach A

Isotropic Compression Curves

v=1+e

ln p

s=0

s1

s2

s3

0<s1 < s2 < s3

pc1 < pc2 < pc3

c1p c2p c3p



p

s

s3

s2

s1

Loading-collapse yield surface (LC)

Zero shear strength

(Apparent tensile strength, ATS)

Suction Increase (SI)

pc1p c2p c3p


3. Yield Stress vs Suction: Soils reconstituted from slurry

Variation of yield stress with suction

Stress path B’D is elastoplastic, not purely elastic.

s

Initialelastic zone

p

D

45oA E F G

B'

B yield surface evolution



NCLs for s>0 are curved!

v=1+e

ln p

s=0

s1

s2

s3

All normal compression lines

?

A

B'

B'

B'

D D D


0

100

200

300

400

500

600

700

800

900

1000

-400 -300 -200 -100 0 100 200 300 400 500 600p (kPa)

A

B

45o

0p cBp

s

Initial elastic zone

3. Yield vs Suction: Approach C (SFG)

Evolution of yield surface during drying and compression

cDp

A

B

p

s

D

ATSSI

LC

D


3. Yield Stress vs Suction: Approach B (Effective Stress)

Evolution of yield surface in effective stress space

s

p′A

zero shear strength line: 0p′ =

45o

B

B' D

ABB' : Drying under 0p =

B'D: elastoplastic

SI LC

( ) ( ) lnv N s s pλ ′= −

?

ATS


3. Yield Stress vs Suction

Comments on yield stress – suction relationship:

• The yield stress – suction relationship is embedded in the volume change model.

• The zero shear strength (ATS) surface, the suction-increase (SI) surface and the loading-collapse (LC) surface are related to each other. In effective stress models and in the SFG model, the SI and LC surfaces are all evolved from the zero shear strength surface.


Outline

1. Introduction








4. Shear Strength of Unsaturated Soils

In a critical state constitutive model, the shear strength of the soil is fully defined by:- The slope of the critical state line M(s) (or the friction angle φ) and - The zero shear strength function, 0 ( )p s

0 ( )p s

p

s

45 o

q

CSL

M(s)

CSL

M(s)



Bishop & Blight (1963):

Fredlund et al (1978):

( )n tan tanc s cτ σ χ φ σ φ′= + + = +

bn tan tanc sτ σ φ φ= + +

0( )p s

Volume change


or

Pereira & Alonso (2009)

0

500

1000

1500

2000

0 200 400 600 800 1000 1200 1400

Suction (kPa)

Dev

iato

r stre

ss (k

Pa)


Various shear strength equations used to define χ or φb

Predictions

Test data (after Cunningham et al, 2003)

2

1

5

3

46 8

7 To capture the peak value:

( )rSχ χ=

( )reSχ χ=

( )b brSφ φ=



The slope of CSL (M) or the friction angle (φ):

Some experimental data support that M or φ does not depend on suction (Ng & Chiu, 2004; Thu et al 2007; Nuth & Laloui 2008)

Nuth & Laloui (2008)

Mean net stress (kPa)

Dev

iato

r stre

ss (k

Pa)

Suct

ion

(kPa

)

Thu et al (2007)



Some data support that the friction angle (φ) depends on suction or degree of saturation (Toll, 1990; Merchán et al 2008)

Merchán et al (2008)

Toll & Ong (2003)



Comment on shear strength of unsaturated soils:

• If the friction angle (φ) is independent of suction, all existing shear strength equations can be formulated in terms of a single effective stress. The real challenge is to find an effective stress when φdepends on suction.


Outline

1. Introduction








5. Hydro-Mechanical Coupling: SWCC & Its Hysteresis

Sr

lns

sI

sD

s0

1

Main drying curve

Main wetting curve

A

BC

D

B’ C’

D’

Soil water characteristic (or retention) curve, SWCC or SWRC

Scanning curves


5. Hydro-mechanical Coupling: VRJ Model

Vaunat, Romero & Jommi Model (2000)

lns

ELASTIC ZONE

SI Surface: Drying

SD Surface: Wetting

ln p

LC Surface: Loading


5. Hydro-mechanical Coupling: WSB Model

Wheeler, Sharma & Buisson Model (2003)

ns

ELASTIC ZONE

SI Surface: Drying

SD Surface: Wetting

p′

LC Surface: Loading

Fully coupled: movement of one surface will cause the movement of other surfaces.


5. Hydro-mechanical Coupling: SSG Model

Sheng, Sloan & Gens Model (2004)

s

ELASTIC ZONE

SI Surface: Drying

SD Surface: Wetting

p′

LC Surface: Loading

The movement of SI & SD surfaces are not coupled with the movement of LC surface.


5. Hydro-mechanical Coupling: Density Effect on SWCC

The density effect on SWCC (Sun et al, 2007b):

(1) The shift of SWCC as the initial void ratio of the soil changes,(2) The volume change along SWCC, or(3) The change of degree of saturation caused by loading/unloading

when the suction is kept constant.



In the literature, the change of degree of saturation is often attributed to the change of suction and the change of soil volume, in a form as:

……………………………(29)

This has been used in Sheng et al. (2004); Sun et al. (2007b); Nuth & Laloui (2008a); Mašín (2010); Nuth & Laloui (2008b); Khalili et al. (2008); ….

Note: SWCC is usually obtained under constant stress, not constant volume. Suction change (ds) will also cause volume change (dεv).

( ) ( )r vd d dS s ε= +



The embedded Sr – s relation is for constant volume and is different from the conventional SWCC. If this difference is neglected, we may run into inconsistency, as shown by Zhang & Lytton (2008):

( ) ( )r vd d dS s ε= +

Stress path (undrained: dw=0) Inconsistent change of Sr Consistent change of Sr

A

B

Main drying curve

Main wetting curve

s

Sr

C D

Scanning curveA AB

Main drying curve

Main wetting curve

s

Sr

CD

A

B

LC yield surface

SI surface

SD surface

s

CD

p


?


Alternative way to formulate the hydraulic equation:

where de0 is the change of void ratio purely due to stress change.

There are constraints on the Sr – e relationship:

Sheng & Zhou (2010)

( ) ( ) ( ) ( )r 0d d d d dS SWCC s p SWCC s e= + = +

r r r1S S Se e e

− ∂ −≤ ≤

∂

rr r0, when 1 or 0S S S

e∂

= = =∂

r r r

0 0

(1 )S S Se e

β∂ −= −

∂

rrd d , when d 0SS e w

e= − =



This approach is comparable with Gallipoli et al (2003b) where the van Genuchten equation is modified to account for void ratio effect:

This modified VG equation can also be written as:

r r r

0 0

(1 )S S Se e

β∂ −= −

∂

1/r r r

0 0

(1 )mS S Smne e

ψ∂ −= −

∂

( )r

0

1

1

m

nSe sψφ

⎡ ⎤⎢ ⎥=⎢ ⎥+⎣ ⎦

c.f.

= 1 Sheng & Zhou (2010)



Data of Sun et al (2007a, b)

r r r

0 0

(1 ) ,S S Se e

β∂ −= −

∂( )r1 ,

1

m

nSsα

⎡ ⎤= ⎢ ⎥

+⎢ ⎥⎣ ⎦

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

10 100 1000

Mean net stress, kPa

e

Test dataPredictionse0=1.73

e0=1.39

e0=1.28

e0=1.17

(a)

0.4

0.45

0.5

0.55

0.6

0.65

0.7

10 100 1000

Mean net stress, kPa

Sr

Test dataPredictions

(b)

and SFG model for volume change



Data of Jotisankasa (2005)r r r

0 0

(1 ) ,S S Se e

β∂ −= −

∂( )r1 ,

1

m

nSsα

⎡ ⎤= ⎢ ⎥

+⎢ ⎥⎣ ⎦

0.00

0.05

0.10

0.15

0.20

0.25

0.30

1.20 1.30 1.40 1.50 1.60 1.70 1.80

Sr

1+e

7-10-U

Calibration

7-10-V

7-10-P

7-10-W

5-10-K

5-10-L

5-10-I

Predictions



Data of Sharma (1998)r r r

0 0

(1 ) ,S S Se e

β∂ −= −

∂( )r1 ,

1

m

nSsα

⎡ ⎤= ⎢ ⎥

+⎢ ⎥⎣ ⎦

Sr

0.60

0.70

0.80

0.90

1.00

1.10

1.50 1.70 1.90 2.10 2.30 2.501+ e

s=100kPa

s=200kPas=300kPa

Calibration curve

Predictions


0.6

0.7

0.8

0.9

1

1.1

0.1 1 10 100 1000 10000


Data of Vanapalli (1999)r r r

0 0

(1 ) ,S S Se e

β∂ −= −

∂( )r1 ,

1

m

nSsα

⎡ ⎤= ⎢ ⎥

+⎢ ⎥⎣ ⎦

s (kPa)

e0=0.444e0=0.474e0=0.514e0=0.514Prediction

Sr



Data of Tarantino (2009)r r r

0 0

(1 ) ,S S Se e

β∂ −= −

∂( )r1 ,

1

m

nSsα

⎡ ⎤= ⎢ ⎥

+⎢ ⎥⎣ ⎦

0.50

0.60

0.70

0.80

0.90

1.00

1.10

10 100 1000 10000s (kPa)

e0=0.62

e0=0.54

e0=0.50

Prediction

Sr



Comment on hydro-mechanical coupling:

• The conventional soil-water characteristic curve (or soil-water retention curve) is obtained under constant stress, not constantvolume. This must be considered in the hydro-mechanical coupling.


Outline

1. Introduction








6. Implementation of Constitutive Model: Non-Convexity

s

p

Net stress – suction space: BBM

Elastic zone



s

p′

Effective stress – suction space

Elastic zone



Challenge:

An elastic trial stress path with both the starting and ending stress states inside the elastic zone can cause plastic yielding.

s

p

A

B


6. Implementation of Constitutive Model: Integration

Intersection between stress path and initial yield surface1. For a given strain increment, compute the elastic trial stress2. Find if the elastic trial stress path intersects the current yield surface: the

number (N) of roots of nonlinear function:3. If N > 1, divide the strain increment into two equal subincrements4. Find the number of roots within each subincrement5. Repeat the process until each subincrement contains at most one root

f

α

( ) ( , , ) 0kf f s zα αα σ= =

0 1

Pedroso et al (IJNME, 76: 2029-2062, 2008)



Number of Roots

The number (N) of roots of nonlinear function

f: yield functiong: first order derivative of fh: second order derivative of f

( ) ( , , ) 0kf f s zα αα σ= =

[ ]( )2

2 2 2 2

( ) ( ) ( ) ( )( ) ( ) ( ) 1d arctan( ) ( ) ( ) ( ) ( ) ( )

b

a

f a g b f b g af h gNf g f a f b g a g b

γγ α α α απ α γ α π γ

⎧ ⎫−− − ⎪ ⎪= + ⎨ ⎬+ +⎪ ⎪⎩ ⎭∫

Kronecker–Picard formula (Kavvadias et al, 1999)

It is computationally expensive to estimate N



Numerical Example: Integration of the SFG model

Sheng et al (Comput. Mech., 42: 685-694, 2008b)

p

s

0 20 40 60 80 100

020

4060

8010

0


7. CONCLUDING REMARKS

1. Partial saturation is a state of soil and any soil be unsaturated with water. Models for unsaturated soils should be able to deal with arbitrary suction and stress changes within possible pore pressures and stress ranges.

2. The volume change equation is one of the most fundamental properties for unsaturated soils. It underpins the yield stress – suction and shear strength – suction relationships.

3. When coupling the hydraulic component with the mechanical component in a constitutive model, it is recommended to take into account the volume change along soil-water characteristic curves.


7. CONCLUDING REMARKS

4. The volume change of unsaturated soils can not easily be described by one single stress variable.

5. On the other hand, there is little difference in formulating shear strength equations in one single stress variable or in two independent stress variables, if the friction angle is assumed to be independent of suction.

6. Unsaturated soil models are characterised by non-convex yield surfaces near the transition between saturated and unsaturated states. This non-convexity, if handled rigorously, can significantly complicate the implementation of these models into finite element codes.


Gens (2008, Durham)

+ + +BBM SFG


ACKNOWLEDGEMENT

A number of people have read the draft paper and provided valuable comments:YJ Cui, DG Fredlund, D Gallipoli, SL Houston, J Kodikara, DM Pedroso, JM Pereira, WT Solowski, DA Sun, C Yang, X Zhang, AN Zhou

The work presented in the paper involves important contributions from:DG Fredlund, A Gens, DM Pedroso, DA Sun and AN Zhou

Australian Research Council has provided financial support for research on unsaturated soils at Newcastle.

THANK YOU ALL


2. Volume Change Behaviour: Approach B vs C

vp vsd dd

( ) ( )p sv


+ +

Approach C (SFG):

Approach B (effective stress):

Approach C recovers Approach B only if:

Therefore, Approach B is more constraint than Approach C.

( )ln ( )v N p f sλ= − +

vs vpddfs

λ λ=



Isotropic compression curves for reconstituted soils

Jennings & Burland(1962)



Isotropic compression curves for compacted soils

Wheeler & Sivakumar(2000)


2. Volume Change Behaviour: Gallipoli et al (2003a)


Gallipoli et al(2003a), data of Sharma (1998)


2. Volume Change Behaviour: Romero & Jommi (2010)


Romero & Jommi (2010)


2. Volume Change: Approach B

( )( ) ( ) ln ( )v N s s p f sλ= − +

Why can’t we shift the NCLs upwards in the v – lnp' space?

N(s)

A

NCL(s=0)

NCL(s>0)

v

Drying path under constant p'?

N(0)

ln p′

v

p′

s

A

Drying under constant p':


Compression under constant s (elastic)

?

Where is the expansion of the elastic zone?


ln p′1

A

B

s>0

v

s=0

N Drying path under constant p

2. Volume Change: Approach B

Alternative forms of Approach B

( )r( )ln ( )v N S p f sλ= − +

Al-Badran & Schanz (2009)

Sr

p′

LC surface

Zero

she

ar

stre

ngth

line

( ) r

(1)( )

c c0Sp p

λ κλ κ

−−′ ′=


Questions

2. Why should the loading-collapse yield surface recover the apparent tensile strength surface when pc(s=0)=0?

s

pslurry (v=N)

s

p′slurry45o

• The zero shear strength line must be unique.• The LC surface is evolved from this slurry line.


s

p′

Questions

✘

3. Why do you state that the effective stress is not ‘effective’ in controlling soil volume, or why do we need suction as an additional axis in the stress space?

( )( ) ln ( ) ln ( )v N s p N s p f sλ λ′= − = − +

LC surface


Questions

Implications of Bi-modal vs mono-modal pore size distribution (PSD)

Unsaturated soils with a bi-modal PSD are usually collapsible. Therefore, wetting of such a soil (point A) can collapse the macro-pores and hence change the PSD to a mono-modal (point B). However, drying the mono-modal soil at point B to A and then compressing to C will regenerate the bi-modal PSD (soil at point C becomes collapsible again).

Bi-Modal: A

B: Mono-Modal

s

p

That means that the PSD can change with stress and hydraulic paths. Indeed, compressing a reconstituted soil (mono-modal) at unsaturated states should also be able to generate a collapsible soil (bi-modal).

C: Bi-Modal?


Questions

4. In a critical state model, should the critical state shear strength depend on hydraulic hysteresis? In other words, two samples of the same soil are sheared under exactly the same suction, the mean net stress and the same void ratio, but different degrees of saturation. Should its critical state strength be different?

o Implications: Is it necessary to have both Sr and s in the shear strength and volume change equations?

o Pros: Some data seem to show such differences in CS strength (e.g. Sun et al. 2010).

o Cons: CS shear strength is less dependent on the initial conditions of the sample (e.g. OCR, water content, void ratio, initial structure, cementation, …) for saturated soils. An unsaturated soil model can not be better than its base model for saturated states.


Questions

Implications of the maximum shear strength at intermediate suctions (SFG)

s

p45o


v

lns

vp sa

vs savp sa

SFG: s s

s s ss

λλ

λ

≤⎧⎪= ⎨

>⎪⎩ ?s

q


s

p

Stress Path Dependency in Elastic Zone (SFG Model)

Arbitrary yield surface

A (p0, s0)

D (p, s)B

C

evp vs

d dd p svp s p s

κ κ= − −+ +

evp vp vp

d( ) d dd p s p svp s p s p s

κ κ κ+= − = − −

+ + +

evpABD

Δ ln(36.9)v κ= −

evpACD

Δ ln(38.2)v κ= −

0 0

sa

11, 10 (kPa) 20, 20, 10 (kPa)

p sp s s

= = −= = =

vpe evp vpABD ACD

vp

ln1.0338.2Δ Δ ln( ) ln1.03, 0.008 0.8%36.9 ln 36.9

v vκ

κ κκ

−− = − = − = =

−

unsaturated

saturated


s

p

Stress Path Dependency in Elastic Zone (Model A)


A (p0, s0)

D (p, s)B

C

evp vs

d dd p svp s

κ κ= − −

evp vs

at

d dd p svp s u

κ κ= − −+

evp vp vp

d( ) d dd p s p svp s p s p s

κ κ κ+= − = − −

+ + +

evpABD

Δ ln(76.4)v κ= −

evpACD

Δ ln(60)v κ= −

0 0

sa

11, 10 (kPa) 20, 20, 10 (kPa)

p sp s s

= = −= = =

vpe evp vpABD ACD

vp

ln1.2776.4Δ Δ ln( ) ln1.27, 0.06 6%60 ln 60

v vκ

κ κκ

−− = − = − = =

−

unsaturated

saturated


Stress Path Dependency in Elastic Zone (Model B)

s

p


A (p0, s0)

D (p, s)B

C

unsaturated

saturated

s

p′

Corresponding paths in effective stress space

A

DB

C

D

Stress path in net stress space


Work-Conjugate Stress and Strain Variables

Work Input (Houlsby, 1997):

( ) ar w r a r r a

a(1 ) (1 )ij ij ij ijW S u S u n s S n S u ρσ δ δ ε

ρ= − − − + + −

( ) ( )

( )

( )

aa r w a r r a

a

aa r v r r a

a

aa r a

a

(1 )

(1 )

(1 )

ij ij ij ij ij

ij ij ij

ij ij ij

W u S u u nsS n S u

u S s nsS n S u

u s n S u

ρσ δ ε δ ερ

ρσ δ ε ερ

ρσ δ ε θρ

= − − − + + −

= − + + + −

= − + + −


Work-Conjugate Stress and Strain Variables

( ) ( )a a w w q 0 rd d d d dw p u p u q n s Sϕ ϕ ε= − + − + −

Work Input (Coussy, Pereira & Vaunat, 2010)

With two non-connected saturating fluids:

With connected fluids:

( )a v r v q rd d d d dw p u S s q n s Sε ε ε= − + + −


SFG Modelling Approach

Volumetric Model →

Hardening Law →

Yield Stress →

Shear Strength

SFG model (Sheng, Fredlund and Gens, Canadian Geotechnical Journal, 45(4), 511-523, 2008)


SFG MODEL: Hardening Law and Yield Surfaces

Volumetric model (BBM): ( ) vppv vp

ddd ( ) ln dd

ps p sp s

λε λ κ= − +

0(s)

c c0

r r

p pp p

λ κλ κ

−−⎛ ⎞

= ⎜ ⎟⎝ ⎠

Loading-Collapse YS:

( ) ( )pv vp vp vs vs

d dd( ) ( )

p sp f s p f s

ε λ κ λ κ= − + −+ +

Volumetric model (SFG):

Yield surface: ?


c0 sa

cc0 sa sa sa

saln

p s s sp sp s s s s

s

− <⎧⎪= ⎨ − − ≥⎪⎩

0

100

200

300

400

500

600

700

800

900

1000

-400 -300 -200 -100 0 100 200 300 400 500 600

0p

p

s

(kPa)

Initial elastic zone

cBp

ssa

B

c0p


Initial yield surface and its evolution during compaction

cDp

cn0 sa

cn cn0c0 sa sa sa

c0 saln

p s s sp p sp s s s s s s

p s

− <⎧⎪= ⎛ ⎞⎨ + − − − ≥⎜ ⎟⎪

⎝ ⎠⎩

cn0p

A

B

p

s

DATS SI

LC

D



Yield surface for soils reconstituted from slurry

p

sq

( )( )2 20 c 0f q M p p p p= − − − ≡



Yield surface for a compacted soil

p

q

s

( )( )2 20 c 0f q M p p p p= − − − ≡


0

100

200

300

400

500

600

700

800

900

1000

-400 -300 -200 -100 0 100 200 300 400 500 600

5. Hydro-mechanical Coupling: SFG Model

Sheng, Gens & Fredlund (2008)

0p

s

(kPa)

ELASTIC ZONE

SI: Drying

SD: Wetting

cp

p



s (kPa)1 10 100 1000 10000 100000

1.54

1.52

1.50

1.48

1.46

1.44

1.42

v

Reconstituted silty clay: Data from Cunningham et el. (2003)

Prediction of SFG model

vp sa

vs savp sa

s ss s ss

λλ

λ

≤⎧⎪= ⎨

>⎪⎩



v

s = 0 kPas = 400 kPas = 650 kPas =1000 kPa

Measured Predicted by SFG

1 10 100 1000 10000

1.54

1.52

1.50

1.48

1.46

1.44

1.42

1.40

1.38

Data from Cunningham et al.(2003)

Reconstituted silty clay: Data from Cunningham et el. (2003)

kPap


SFG MODEL: Shear Strength

Shear strength of reconstituted silty clay: Data from Cunningham et al. (2003)

Measured

SFG

(a) Unconfined

0 500 1000 1500

400

800

1200

1600

0

(σ1−

σ 3) (k

Pa)

s (kPa) 0 500 1000 1500s (kPa)

(b) Confining pressure 50kPa

Measured

SFG

400

800

1200

1600

0

(σ1−

σ 3) (k

Pa)

s (kPa)

(d) Confining pressure 200kPa

Measured

SFG

500

1000

1500

2000

0

0 500 1000 1500

(σ1−

σ 3) (k

Pa)

0 500 1000 1500

Measured SFG

500

1000

1500

2000

0

s (kPa)

(σ1−

σ 3) (k

Pa)

(c) Confining pressure 100kPa


SFG MODEL: Volume - Suction Behaviour

e

(a)Measured Predicted Series

(b)(c)

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.410 100 1000 10000 100000

s (kPa)

Compacted Brown London Clay: Data from Marinho et el. (1995)


SFG MODEL: Volume - Stress Behaviour

Compacted Kaolin: Thu et al. (2007)

1 10 100 1000 10000

v

2.20

2.15

2.10

2.05

2.00

1.95

p (kPa)

(a) s=0kPa

v

1 10 100 1000 10000

2.15

2.10

2.05

2.00

1.95

p (kPa)

(b) s=50kPa

v

1 10 100 1000 10000

2.15

2.10

2.05

2.00

1.95

p (kPa)

(c) s=100kPa

1 10 100 1000 10000

v

2.10

2.05

2.00

1.95

p (kPa)

(e) s=200kPa



Shear strength of air-dry sandy soil: Data from Röhm and Vilar (1995).

MeasuredSFG

(d) Suction 400 kPa0

100

200

300

400

500

600

700

0 500 1000 1500

(c) Suction 200 kPa

MeasuredSFG

0

100

200

300

400

500

600

700

0 500 1000 1500

(σ1+σ3) / 2 – ua (kPa)

(σ1−

σ 3) /

2 (k

Pa)

(a) Suction 20 kPa

MeasuredSFG

0

100

200

300

400

500

600

700

0 500 1000 1500

MeasuredSFG

(b) Suction 50 kPa0

100

200

300

400

500

600

700

0 500 1000 1500

(σ1−

σ 3) /

2 (k

Pa)

(σ1+σ3) / 2 – ua (kPa)

(σ1+σ3) / 2 – ua (kPa)

(σ1−

σ 3) /

2 (k

Pa)

(σ1+σ3) / 2 – ua (kPa)

(σ1−

σ 3) /

2 (k

Pa)



Measured Predicted

0 100 200 300 400 500 6000

50

100

150

200

τ(k

Pa)

s (kPa)

Shear strength of compacted glacial till: Data from Vanapilli (1996).

σn = 200 kPaσn = 100 kPaσn = 25 kPa


Measured Predicted

(kPa)

(σ1−

σ 3) (k

Pa)

0 50 100 150 200 250 3000

50

100

150

200

250

300

350

400

450


p

Shear strength of compacted kaolin: Data from Wheeler & Sivakuma (2000)

s = 300 kPa

s = 100 kPa

s = 0 kPa


0

100

200

300

400

500

-100 0 100 200 300 400

SFG MODEL PREDICTION

Example 1: Soil with low air-entry value – yield surface evolution

CA

B D

s

E

B' D'

p

ssa=10 kPa



Example 1: Soil with low air-entry value – volume change

0.40

0.45

0.50

0.55

0.60

0.65

0.70

1 10 100 1000

e

A

B

D

B'

E

NCL, s=0URL, s=0

p



Example 2: Soil with high air-entry value – yield surface evolution

0

500

1000

1500

2000

-1000 -500 0 500 1000 1500 2000

B D

A

s (kPa)

C

sae=500 kPa

kPap


0.1

0.3

0.5

0.7

0.9

1.1

1 10 100 1000 10000


Example 2: Soil with high air-entry value – volume change

p (kPa)

s=0 kPa

s=10 kPa

s=100 kPa

s=500 kPa

s=2000 kPa

e

s=1000 kPa

2.00

1.46

1.01

0.65

0.35

0.11


-50

0

50

100

150

200

-50 0 50 100 150 200 250 300 350 400


Example 3: Compacted soil: yield surface after compaction

p (kPa)

s (kPa)

Elastic zone after compaction

Measured suction levels where collapse starts

A

B E F G H

E' F' G' H'D'

D



Example 3: Compacted soil: volume collapse caused by wetting

D E F

D'

E'

F'

G'

H'I'

NCL (s=0)

AB

e

G HI

0.9

1.0

1.1

1.2

1.3

1.4

1 10 100 1000p (kPa)

Measured void ratio before and after collapse


5. Hydro-mechanical Coupling: Stress-Strain Relation

Hydro-mechanically coupled model (Sheng, Gens & Sloan, 2004)

( )7 1×

ep ep

r

dddd Ss G

⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠

D W εσR

( )7 1×( )7 7×

ep ep

r

d dd dS sG

⎛ ⎞⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

D Wσ εR

Unknown Known


Field Test Site

• 100 m2 polythene cover

• 300 mm wide and 500 mm deep trench

• Foundation movements were measured at 5 points

• Data for more than 8 years

trench at edge10 m

surface movement probe referred to in Figure 4


Field Test Site

6 m

flexible cover (impermeable)

10 m

constant head (6.0m)

impe

rmea

ble

impe

rmea

ble

wetting/drying boundary


Field Test Site

0

10

20

30

40

50

0 365 730 1095 1460 1825 2190 2555

Time (days)

Dis

plac

emen

ts (m

m)

measured

FEM prediction (x=0)

Predicted and measured heave under the cover


-10

0

10

20

30

40

0 365 730 1095 1460 1825 2190 2555

Time (days)

Dis

plac

emen

ts (m

m)

measured

FEM prediction (x=10m)

Predicted and measured heave outside the cover

Field Test Site


-0.05

0.00

0.05

0.10

0 2 4 6 8 10Distance from the cover centre (m)

Dis

plac

emen

ts (m

) Wet season

Dry season

Mound Shape during Dry and Wet Seasons

Dry → Cover → Cyclic Wetting and Drying

Documents

Constitutive Modelling of Unsaturated Soils: Discussion of ... · Implementation of unsaturated soil models into FEM 7. ... Soils, Beijing, 1998, expansive soils) 4. Fredlund (3rd