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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
Soil Mechanics Laboratory (LMS)
Ecole Polytechnique Fédérale de
Lausanne, EPFL - Switzerland
Mechanics of Unsaturated Soils, Weimar, 8th March 2007
Funds framework: European Project COST C15 -
OFES funds C03.0021
Funds framework: European Project COST C15 -
OFES funds C03.0021
Ai r, u a
Water, uw
Solid
grains
v a
vw
v s
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 2
OutlineOutline
1. Introduction2. Unified stress framework for unsaturated soils
3. Critical state analysis
4. Unsaturated mechanical compression
5. Constitutive modelling framework
6. Conclusion
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 3
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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 4
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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 5
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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 6
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 ′ = − + −
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 7
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 ′
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 8
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
ij ijkl kld C d ε σ ′=
e e
ij ijkl kld C d ε σ ′=
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 9
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 = %
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 10
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
ij ijkl kld C d ε σ ′=
Reinterpretation of experimental results within the
generalised effective stress concept:
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 11
- 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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 12
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 )
Mean net stress pnet
(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 )
Mean effective stress p' (kPa)
(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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 13
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 ( - )
Mean net stress pnet
(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 (
- )
Mean effective stress p' (kPa)
(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 ′−
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 14
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<
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 15
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 ′ = +
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 16
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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 17
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)
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 18
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
Soil Mechanics Laboratory (LMS)
Ecole Polytechnique Fédérale de
Lausanne, EPFL - Switzerland
Mechanics of Unsaturated Soils, Weimar, 8th March 2007
Funds framework: European Project COST C15 -
OFES funds C03.0021
Funds framework: European Project COST C15 -
OFES funds C03.0021
Ai r, u a
Water, uw
Solid
grains
v a
vw
v s
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 19
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
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USS07 08/03/07 Nuth and Laloui – Implications of Generalized Effective Stress 20
YieldsurfaceYieldsurface
Example of ACMEG-s yield surface
(Laloui and Nuth, 2005)