effective stress

MUSE Mechanics of Unsaturated Soils for Engineering

1st MUSE School: Fundamentals of Unsaturated SoilsBarcelona, 1-2 June, 2005

EFFECTIVE STRESSES

Antonio GensUniversitat Politècnica de Catalunya, Barcelona

OUTLINEIntroductionEffective stress for saturated geomaterialsEffective stress for unsaturated soils

The meaning of suctionProposals for a single effective stressTwo independent stress variablesWork-conjugate stress and strain variablesExamples of stress variables for constitutive models

Conclusions

Saturated soils

wu′σ = σ −

Two variables:

Total stresses: σWater pressure: uw

Saturated soils

Karl Terzaghi (1883 - 1963)

All the measurable effects of a change of stress, such as compression, distortion and a change in the shearing resistance are exclusively due to changes in effective stress…every investigation of the stability of a saturated body of earth requires the knowledge of both the total and the neutral stresses

(Terzaghi, 1936)

Saturated soils

wu′σ = σ −

Saturated soils

Joseph V. Boussinesq(1842-1929)

One can ignore the atmospheric pressure...for this acts in all directions within the sand and all around each grain…It therefore has no influence on their mutual action and consequently will not modify the supplementary pressure that cats at the contact between the grains… Only this supplementary pressure has to be considered.

Essai théorique sur l’équilibre d’élasticité des massifs pulvérulents (Boussinesq, 1876)

Effective stress

Skempton (1960a). Terzaghi’s discovery of effective stress. From theory to practice in soil mechanics, Wiley, 42-53Skempton (1960 b). Effective stress in soils, concrete and rocks. Pore pressure and suction in soils, Butterworths, 4-16

Saturated soils, concrete and rocks

Unsaturated soils

Effective stress for unsaturated soils

Skempton on effective stress for unsaturated soil

• Bishop’s expression

' ( )a a wu u uσ = σ − +χ −

χ coefficient may be different for shear strength or consolidation

Effective stress for saturated geomaterials

Skempton on effective stress for saturated soil, rock & concrete

• Theory I: Intergranular stress' (1 )a uσ = σ − −

' uσ = σ −• Theory II: Terzaghi’s expression

• Theory III: Micromechanical variables

shear strength:

volume change:

tan' (1 )tan '

a uψσ = σ − −φ

' (1 )sC uC

σ = σ − −

Effective stress for for saturated geomaterials

Sand Granite

Effective stress for for saturated geomaterials

(Lade & De Boer, 1997)

' u= −σ σ B' u= −βσ σ I


Poroelasticity ' (1 )sC uC

σ = σ − −

(Nur & Byerlee, 1971)

(1 )sC uC

σ − −uσ −σ

Tests on Weber sandstone


Poroplasticity (Coussy, 1995)Tensor B is constant in perfect plasticity but not in hardening plasticityPlastic ‘effective stress’ does not necessarily coincide with elastic ‘effective stress’ It is not required that some aspects of the plastic model (e.g. yield surface) depends on plastic ‘effective stress’

Theory of porous media (De Boer & Ehlers, 1990, De Boer, 1996)Classical mixture theory + volume fractionsAn ‘effective’ or ‘extra’ stress arises naturally Unfortunately, the effective stresses depends on the constitutive hypothesis


β Reference Comment 1 Terzaghi (1923) n Hoffman (1928) (n: porosity) Biot (1955) Pietruszczak & Pande

(1995) Saturated cemented material

1-a Skempton & Bishop (1954) (a: contact area ratio) De Buhan & Dormieux

(1999) Porous rock

1- Cs/C Biot & Willis (1957) Geertsma (1957) Nur & Byerlee (1971) Bishop (1973)

Cs: compressibility of grains C: compressibility of skeleton

1-(1-n) Cs/C Suklje (1969) 1-(1-n) Cgu/Csks

Lade & De Boer (1997) Granular material

1- Cgu/Csks Lade & De Boer (1997) Solid rock with interconnected pores Cgu: compressibility of grains due to a pore pressure change Csks: compressibility of skeleton due to a confining pressure change


Some conclusions:The effective stress definition depends on the hypothesis made:

Constitutive lawMicromechanical model

Material parameters appear in the effective stress definition

It is highly unlikely that an universal effective stress expression will be ever found for the full range of porous geomaterials

wu′σ = σ −

However, Terzaghi´s expression is always recovered when grain compressibility can be neglected (soils)

Solid

Air

Water

Effective stresses for unsaturated soils

Unsaturated soils

Two variables:

Total stresses: σWater pressure: uwAir pressure: ua

Suction: ua - uw

WATER POTENTIAL: Measure of free energy/unit change of mass

zgoc Ψ+Ψ+Ψ+Ψ=Ψ

:Ψ Total potential

Matric (capillary) potentialOsmotic potential:RTcmo =Ψ:)( awc uu −=Ψ

:)( atmag uu −=Ψ Air pressure potential

:zwz γ=Ψ Gravitational potential

SOIL SUCTION

Sand Clay

WATER POTENTIAL: Measure of free energy/unit change of mass

zgoc Ψ+Ψ+Ψ+Ψ=Ψ

:Ψ Total potential

Matric (capillary) potentialOsmotic potential:RTcmo =Ψ:)( awc uu −=Ψ

:)( atmag uu −=Ψ Air pressure potential

:zwz γ=Ψ Gravitational potential

SOIL SUCTION

π+=π+−=Ψ− suu wat )(:tΨ− Total suction

:)( wac uus −=Ψ−= Matric suctionOsmotic suction:oΨ−=π

Total water potential controls water flowSuction affects mechanical behaviour. Not all suction components have the same effectMechanical behaviour is mainly affected by matric suctionOsmotic effects only in clay-rich materials

Solid

Air

Water


Unsaturated soils

Two variables:

Total stresses: σWater pressure: uwAir pressure: ua

Suction: ua - uw

Bishop’s (1959) expression

' ( )a a wu u uσ = σ − +χ −


Bishop’s (1959) expression' ( )a a wu u uσ = σ − +χ −

χ = Sr


' ( ) (1 )a r a w r a r wu S u u S u S uσ = σ − + − = σ − − −Average skeleton stress

Entropy inequality + Coleman.Noll procedure (Hassanizadeh & Gray, 1980)

Volume averaging (Lewis & Schrefler, 1987)

Mixture theory (Hutter et al., 1999)

Average skeleton stress + additional termsEnergy approach to extend Biot’s theory of poroelasticity (Dangla & Coussy, 1998; Cousssy & Dangla, 1992)

Macroscale thermodynamic approach, solid phase surface in contact with water as weighing parameter (Gray & Schrefler, 2001)


Suction term, should it be a scalar quantity? (Li, 2003)Fabric dependent (tensorial quantity?)

' ( )ij ij a ij ij a wu F u uσ = σ − δ + −

ij r ij ij ïjF S ′ ′′= δ +ξ + ξ

ij′ξ

ij′′ξ : effect of the contractile skin

: distribution of pore fluid on particle surfaces


Effective stress from water menisci forces (Fleureau et al., 2003)

( )2

2

3 3 9 8 ( )' ; 4

2 ( ) ( )a w

ij ij u ij ua w

R u up p R

g e R u u

γ − γ + γ −πγ ′ ′σ = σ − δ = + −


Effective stress from water menisci forces (Fleureau et al., 1995)

(b)

(d)

(a) (Escario & Saez, 1986)

(c)

(Wheeler & Sivakumar, 1992)


Effective stress from shear strength data (Khalili & Khabbaz, 1998)

( )( )

m

a w

a w b

u uu u

− −χ = −

' ( )a a wu u uσ = σ − +χ − compacted kaolin

sand.-clay mixture


Effective stress from shear strength data (Khalili & Khabbaz, 1998)

0.55( )( )

a w

a w b

u uu u

− −χ = −

' ( )a a wu u uσ = σ − +χ −


A single effective stress?

' ( )a a wu u uσ = σ− +χ −

(Jennings and Burland 1962)


Collapse behaviour upon wetting

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 90.1 1.0 10.0

Vertical stress (MPa)

-1.0

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

Vert

ical

str

ain

(%)

Series D-D

(a)

Barcelona silt

(Suriol et al., 2002)


Collapse behaviour upon wetting

Lixhe chalk

(De Gennaro et al., 2004)

Effect of intergranular forces due to external stresses and suction

Stress variables for unsaturated soils

(Coleman, 1962; Bishop & Blight, 1963; Matyas & Radhakrishna, 1968)

Two sets of stress variables are required :au−σ Net stress :)( wa uus −= Matric suction

Any two of the following stress variables (Fredlund & Morgenstern, 1977):

auσ− ( )a ws u u= − wuσ−


(Tarantino et al., 2000)

Null tests auσ− ( )a ws u u= −


The full description of the behaviour of unsaturated soils require the use of two independent stress variables

Which ones?

( )(1 ) / ( ) (1 )a r a a a w r ij r w r a ij ijW u n S u u nS S u S u ≡ − ρ ρ − − + σ − + − δ ε (Houlsby,1997)

On average, contractile skin will move with the soil skeleton

Work dissipated by flow of fluids is not included in the expression

Work input rate to an unsaturated granular material




Neglecting air compressibility term

( )(1 ) ( )ij r w r a ij ij a w rW S u S u u u nS ≡ σ − + − δ ε − −

( )a wn u u−

Bishop’s stress: ( )(1 )ij r w r a ijS u S uσ − + − δ

rS−

ijε

Modified suction:




Neglecting air compressibility term

( )

( ) ( )ij ij a ij r v r

ij ij a ij w w r

W u sS snS

W u s nS

≡ σ −δ ε + ε +

≡ σ −δ ε + ε ε =

( )auσ−Suction: ( )a wu u−Net stress:

( )w r r vnS Sε = − + εijε


The full description of the behaviour of unsaturated soils require the use of two independent stress variables

( )(1 ) / ( ) (1 )a r a a a w r ij r w r a ij ijW u n S u u nS S u S u ≡ − ρ ρ − − + σ − + − δ ε

( )auσ−Suction: ( )a wu u−Net stress:

( )a wn u u−

Bishop´s stress: ( )(1 )ij r w r a ijS u S uσ − + − δ

rS−

( )w r r vnS Sε = − + εijε

ijε

Modified suction:

Other stress combinations are possible…

(Houlsby,1997)

(Gens,1995)

1( , )a ru s Sσ− +µ

2( , )rs Sµ


Class I


1( , )a ru s Sσ − + µ

2 ( , )rs Sµ

1 ( 0)auσ− µ = Alonso et al.(1990); Josa et al.(1992), Wheeler andSivakumar (1995), Cui et al (1995)

Easy representation of conventional stress pathsDifficulties in the transition saturated-unsaturatedHysteresis and hydraulic effects difficult to incorporateIndependent function required to model the increase of strength with suction

Class II


1( , )a ru s Sσ − + µ

2 ( , )rs Sµ

Kohgo et al. (1993), Modaressi and Abou Bekr (1994), Pakzad (1995), Geiser et al. (2000), Loret and Khalili(2002)

Representation of conventional stress paths not straightforwardDifficulties in the transition saturated-unsaturated (even when incorporating desaturation suction)Hysteresis and hydraulic effects difficult to incorporateThe increase of strength with suction results from stress variable definition

1( )au sσ− +µ

Class III


1( , )a ru s Sσ − + µ

2 ( , )rs Sµ

Jommi and de Prisco (1994), Bolzon et al. (1996), Jommi (2000) , Wheeler et al, (2003), Gallipoli et al. (2003), Sheng et al. (2004)

Representation of conventional stress paths not straightforward, sometimes impossibleNo difficulties in the transition saturated-unsaturatedHysteresis and hydraulic effects can be naturally incorporatedThe increase of strength with suction results from stress variable definition

1( , )a ru s Sσ− +µ

An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviourD. Gallipoli, A. Gens, R. Sharma, J. Vaunat (2003) Géotechnique, 53, 123-135

(Buisson & Wheeler, 2000)

Menisci domain

Saturated domain

Stress variables for unsaturated soils: an example

• Two stress variables

' ( )a r a wu S u uσ = σ− + −Average skeleton stress:

( )(1 )a w rf u u Sξ = − −Bonding stress:

: variation of interparticle stress with suction( )a wf u u−

Stress variables for unsaturated soils : an example

( )(1 )a w rf u u Sξ = − −

0 1000 2000 3000 4000

s (kPa)

1

1.1

1.2

1.3

1.4

1.5

f(s)

(Fisher, 1926)

Bonding stress:


(d)

(a) (b)

(c)

(Sharma, 1998) (Sivakumar, 1993)

( )( )ξbaee

s

⋅⋅= exp-1-1


(Wheeler & Sivakumar, 2000)

( )( )ξbaee

s

⋅⋅= exp-1-1

0.8 0.9 1 1.1 1.2

Experimental e

0.8

0.9

1

1.1

1.2

Pred

icte

d e

Identity functions=100 kPa (series I)s=200 kPa (series I)s=300 kPa (series I)s=100 kPa (series II)s=300 kPa (series II)s=100 kPa (series III)s=300 kPa (series III)

Critical state


( )( )ξbaee

s

⋅⋅= exp-1-1

0 0.1 0.2 0.3 0.4

ξ

1

1.1

1.2

1.3

1.4

e/e s

ExperimentalModel equation

0.950 0.2 0.4 0.6

ξ

1

1.1

1.2

1.3

1.4

1.5

e/e s

ExperimentalModel equation

0.95

(Toll, 1990)

(w/c= 24.9% – 27.7%)

(w/c= 19.6% – 21.9%)

Critical state



Some conclusions:The description of the behaviour of unsaturatedsoils require the use of two independent stress variables

“in fact, no single stress variable has ever been found which, substituted for effective stress, allows for a description of all the aspects of the mechanical behaviour of a given soil in the unsaturated range”.

(Jommi, 2000)


Géotechnique editorial (Houlsby, 2004):

“The above somewhat over-simplifies the picture. as it is now widely recognized that the mechanics of unsaturated soil is (almost certainly) not explicable in terms of a single ‘effective stress’ but that a further variable (e.g. the difference between the pore air pressure and pore water pressure) is needed too. Even so, the unequivocal identification of the best choice of two variables on which to base the hypothesis has not, I believe yet been achieved.

It is a challenge to our readers to achieve the same breakthrough for unsaturated soils that Terzaghi did for saturated materials. We need clear definitions, empirical proof taht the mechanical behaviour of unsaturated soils does indeed depend on the chosen variables, and preferably a satisfying ‘explanation’ in terms of well articulated principles. It is not an easy task.”

So far, the selection of stress variables for unsaturated soils is a matter for convenience (within reason)

Effective stress

Some final conclusions:

It is highly unlikely that an universal effective stress expression will be ever found for the full range of porous geomaterials

wu′σ = σ −

However, Terzaghi´s expression is always recovered when the material is saturated and grain compressibility can be neglected

effective stress

Documents