Nuth Presentationkkk

8/18/2019 Nuth Presentationkkk

http://slidepdf.com/reader/full/nuth-presentationkkk 1/20

USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 1

Implications of a GeneralisedEffective Stress

on the Constitutive Modellingof UnsaturatedSoils

Mathieu NUTH

Lyesse LALOUI

Soil Mechanics Laboratory (LMS)

Ecole Polytechnique Fédérale de

Lausanne, EPFL - Switzerland

Mathieu NUTH

Lyesse LALOUI




Mechanics of Unsaturated Soils, Weimar, 8th March 2007

Funds framework: European Project COST C15 -

OFES funds C03.0021


OFES funds C03.0021

Ai r, u a

Water, uw

Solid

grains

v a

vw

v s




OutlineOutline

1. Introduction2. Unified stress framework for unsaturated soils

3. Critical state analysis

4. Unsaturated mechanical compression

5. Constitutive modelling framework

6. Conclusion




Introduction (1a)Introduction (1a)

Context and aims of the study

- Long time debate on stress frameworks for unsaturated soils

- Need a better understanding of the effective stresses concept

Contribution to the clarification of notions of effective

stress in unsaturated soils

Investigations on a particular unif ied stress framework , on

the basis of reinterpretation of experimental results




Introduction (1b)Introduction (1b)

e e

ij ij klkld C d ε σ ′=

( )e pd d d ε ε ε = +

Effective stress principle in porous medium saturated with n immiscible fluids

Effective stressMain rheological

characteristicsStrains

2) Effective stress is a function of external stress, internal pressures and

fluids repartition

Reference

medium

- Multiphase

- Multistress

Mechanically

equivalent

- Single phase

- Single stress

1) Stress-strain relationship in mechanically equivalent continuum :

σ1

σ3σ3

σ1

u1

u2

u3un

σ'1

σ'3σ'3

σ'1




Stress framework for unsaturatedsoils (2a)Stress framework for unsaturatedsoils (2a)

w a β = or

Continuum

solid

Three-phase descriptionSingle-phasedescription

Effective stress

2

1

'ij ij ijd d u β β

β

σ σ α δ =

= − ∑

A possible formulation is that of generalised Bishop-Schrefler :

( ) ( )ij ij a ij a w ijr u S u uσ σ δ δ ′ = − + −

S r , accounting for volumetric fractions, is the ‘effective stress parameter’

Scaling factorfor phase

+ inner

hydraulic

behaviour*

Ai r, u a

Water, uw

Solid

grains

v a

vw

v s

wr

w a

vS

v v=

+Degree of

saturation




Stress framework for unsaturatedsoils (2b)

. r ds A dS =

*Stress-strain relation for the

hydraulic part

Stress-strain relation for themechanical part

skeleton strain variations are

associated to a single

generalised effective stress

The hydraulic behaviour is described

via the work conjugate suction and

variation in degree of saturation

Hydraulic properties

Double-way

H-M coupling

ε

σ ′

Mechanical

behaviour

Hydraulic

behaviour r S

s

e e

ij ijkl kld C d ε σ ′=

( ) ( )ij ij a ij a w ijr u u uS σ σ δ δ ′ = − + −




Bishop generalisedeffective stress

s

Other conceptual analyses leading to generalised form of effective

stress

- Houlsby (1997): The rate of input work (per unit volume) to the soil, is

expressed as the sum of the products of the stresses with their corresponding

strain rates:

- Thermodynamic mixture theory: Hutter et al. (1999), Laloui et al. (2003)

Examples of constitutive frameworks using the generalised effective

stress:- François et al. (2006)

- Sheng et al. (2004)

- Gallipoli et al. (2003)

- Wheeler et al. (2003)

hk hk ( ) σ ( (1(1 ) ) )[ ]δa

a r r hk

a

a w r w r aW u n S nS u u S u S u

ρ ε

ρ = − − +− − + −

&&& &

hk σ ′




Advantages and shortcomings vs other contexts (a)

0( , , )

m ext a w

u uσ µ σ = %

Following Gens (1995) classification

( )2 ,net r

s S σ σ µ ′ = + %

Independent stress variables :

Form of the mechanical constitutive equation

Effective stress variable – Category 1:Form of the mechanical constitutive equation

Effective stress variable – Category 2:

Form of the mechanical constitutive equation

1 ( )net sσ σ µ ′ = + %

( ),r s S ξ ξ = %

( ), r s S ξ ξ = %

( )sξ ξ = %e e s

ij ijhk hk hk d C d C d ε σ ξδ = +

e e


e e





Advantages and shortcomings vs other contexts (b)

- Choice of stress variables is mostly a matter of convenience

- It is proposed to verify that the shortcomings of the use of the generalised effective

stress (∈C2) do not overcome the simplifications inherent to the stress context

Equations

+++-Category 2

effective

stresses

+-+-

Category 1

effectivestresses

---+Independent

stress

variables

Direct

accounting of

increase in

strength

Hydraulic

Hysteresis

effects

Saturated

Unsaturated

transition

Representation

ComplexityCategory

0 ( , , )m ext a w

u uσ µ σ = %

( )2 ,net r s S σ σ µ ′ = + %

1 ( )net sσ σ µ ′ = + %

( ),r

s S ξ ξ = %

( ), r s S ξ ξ = %

( )sξ ξ = %




OutlineOutline

3. Critical state analysis

4. Unsaturated mechanical compression

5. Further constitutive aspects (within context of elasto-plasticity)

net r S sσ σ ′ = +

r S

1. Introduction

2. Unified stress for unsaturated soils- The generalised effective stress belongs to

category 2 of effective stresses

- The effective stress is either incrementedby changes in matric suction or net

stress

- The mechanical stress state is uniquely

described by the means of the effective

stress, sufficiency of a unique constitutivematrix

e e


Reinterpretation of experimental results within the

generalised effective stress concept:




- CSL are uniformly shifted by amount of mean stress

- Unification of unsaturated critical state lines with saturated (s=0) failure criterion

Critical state analysis (3a)Critical state analysis (3a)

Conventional (q-p net ) plane vs effective (q-p’) plane

0

50

100

150

200

250

300

350

400

50 100 150 200 250 300

s = 300 kPas = 200 kPas = 100 kPas = 0 kPa

D e v

i a t o r i c s t r e s s q ( k P a

)

Mean net stress pnet

(kPa)

(a)

Saturated

CSL

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600

s = 300 kPas = 200 kPas = 100kPas = 0 kPa

D e v i a t o r i c s t r e s s q ( k P a

)

Mean effective stress p' (kPa)

(b)

Saturated

CSL

Experimental data from Sivakumar (1993), Kaolin

( . )r p S s∆ = ∆

MM




Critical state analysis (3b)Critical state analysis (3b)

0

500

1000

1500

2000

0 400 800 1200 1600

s > 0 kPa

s = 100 kPa

s = 0 kPa

D e v i a t o r i c s t r e s s q ( k P a )


(kPa)

(a)

40

200

50280

75

5560

114

Suction level (kPa)

SaturatedCSL

0

500

1000

1500

2000

0 400 800 1200 1600

s > 0 kPa

s = 100 kPa

s = 0 kPa

D e v i a t o r i c s t r e s s q ( k P a )


(b)

40200

50

280

75

5560

114

Suction level (kPa)

SaturatedCSL

Experimental data from Geiser, Laloui, Vulliet (2006), Sion silt

Practical implications and limitations

- Parameter determination reduced to saturated shear resistance parameters

- No suction-dependent apparent cohesion

- Unification observable in silty, clayey materials




Unsaturatedmechanical compression (4a)Unsaturatedmechanical compression (4a)

( )lnv pε ′−

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

20 40 60 80100 300 500

s = 0 kPa

s = 100 kPas = 200 kPas = 300 kPa

V

o l u m e t r i c s t r a i n ε v ( - )


(kPa)

(a)

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

20 40 60 80100 300 500

s = 0 kPa

s = 100 kPas = 200 kPas = 300 kPa

V o l u m e t r i c s t r a i n ε v (

- )


(b)

- Increase in preconsolidation pressure* with s is amplified in plane

- Higher magnitude for compressibility variations*

- Linear fitting of curves with Log x-scale remains appropriate

Experimental data from Sivakumar (1993), Kaolin

Complexity and non linearity of consolidation lines in plane ?( )lnv pε ′−




Unsaturatedmechanical compression (4b)Unsaturatedmechanical compression (4b)

0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300

C o m p r e s s i b i l i t y

c o e f f i c i e n t λ

Matric suction s (kPa)

Net stressinterpretation

Effective stressinterpretation

0

1 105

2 105

3 105

4 105

0 1 105

2 105

3 105

M a t r i c s u c t i o n s ( P a )

Preconsolidation stress pc-net

or p'c (Pa)

Net stress

interpretation

Effective stress

interpretation

Pc0

se

Compressibility

- Similar trends for evolution of

compressibility with suction in both

conceptions

- Principally, coefficients are higher in

effective representation

‘Loading collapse’ yield curve

- Similar shape for the traces of yield locii- Continuous LC along the full range of

suction: straightforward transition between

saturated and unsaturated domains

- Particular shape for es s<




Unifiedconstitutive modelling framework (5a)Unifiedconstitutive modelling framework (5a)

Unicity of mechanical yield surface :

Particular case of the hydraulic loading

Suction increase under constant net stress

-2 105

0

2 105

4 105

6 105

8 105

1 106

0 4 104

8 104

1.2 105

M a t r i c s u c t i o n s

( P a )

Mean stress pnet

or p' (Pa)

net stressinterpretation

effective stressinterpretation

possibleyield limits - Drying path in net stress

representation

- A specific plastic yield limit isrequired

- Drying path in effective

stress representation

- Unified yield limit alreadydefined

net r sS σ σ ′ = +




Influence of hydraulichysteresis (5b)Influence of hydraulichysteresis (5b)

0

0.2

0.4

0.6

0.8

1

1 100 104

106

D e g r e e o f s a t u r a t i o n S r ( - )

Matric suction s (Pa)

wetting

drying

(a)

1

10

100

1000

104

105

106

1 100 104

M a t r i c s u c t i o n s

( P a )

Mean effective stress p' (Pa)

wetting

drying

(b)

- The complete soil water retention curve evidences a dissipation: the main drying curve

is distinct from the main wetting curve. For the same level of suction, two saturation

states are possible

- The hysteresis is naturally repercuted in the stress plane (s-p’)

Y i e l d

l o c u s

Experimental data from Fleureau et al. (1993), Jossigny loam




ConclusionsConclusions

Generalised effective stress owns

a number of advantages:

- Unification of critical state line

- Natural saturated-unsaturated

transition

- Simplified stress-strain relationship

- Inclusion of Hydraulic hysteresis

- Simplicity of stress formulation

This study is the pretext to the formulation of an Advanced

Constitutive Model for Environmental Geomechanics ACMEG-s

ThanksThanks forfor your your attentionattention

Contribution to the clarification of the effective stress use forunsaturated soils modelling

Example of ACMEG-s yield surface

(Laloui and Nuth, 2005)




Implications of a GeneralisedEffective Stress

on the Constitutive Modellingof UnsaturatedSoils

Mathieu NUTH

Lyesse LALOUI




Mathieu NUTH

Lyesse LALOUI




Mechanics of Unsaturated Soils, Weimar, 8th March 2007


OFES funds C03.0021


OFES funds C03.0021

Ai r, u a

Water, uw

Solid

grains

v a

vw

v s




s

p

εv

s

Wetting collapse

For given soils, a decrease in suction

can induce a collapse.

A necessary condition to obtain plastic

compression on wetting is a preliminary

mechanical consolidation.

• AB: drying (p=const.)

• BC: mechanical consolidation

(s=const.)

• CC’: wetting – elastic swelling

• C’D: wetting – plastic collapse

A

A

B

B

C

C

C’

C’

D

D

CollapseCollapse

LC curve Elastic domain

SwellingSwelling




YieldsurfaceYieldsurface

Example of ACMEG-s yield surface

(Laloui and Nuth, 2005)

Nuth Presentationkkk

Documents

Transcript of Nuth Presentationkkk