Journée d'hommage au Prof. Biarez MODELLING OF LANDSLIDES AS A BIFURCATION PROBLEM F. DARVE, F. PRUNIER, S. LIGNON, L. SCHOLTES, J. DURIEZ Laboratoire Sols, Solides, Structures, INPG-UJF-CNRS, Grenoble, France –ALERT, RNVO, DIGA, LESSLOSS, SIGMA, STABROCK F. NICOT CEMAGREF Grenoble, ALERT, RNVO F. LAOUAFA INERIS, Paris, France
I Introduction 1 Failure : the classical view 2 An example of diffuse failure strictly inside Mohr-Coulomb limit surface 3 Hill’s sufficient condition of stability and bifurcation criteria : phenomenological analyses for axisymmetric and plane strain conditions 4 A multiscale analysis of instabilities with a micromechanical model 5 A direct numerical analysis of instabilities with a discrete element method II Landslides modelling : Trévoux and Petacciato examples III Rockfalls modelling : Acropolis example IV Loading paths with constant q
I Introduction FAILURE : THE CLASSICAL VIEW Rate independent materials :
d σ = F (d ε ) ∀ λ ≥ 0 : F ( λ d ε ) ≡ λ F ( d ε ) ⇒ F Homogeneous of degree 1 Euler’s identity :
d σ = F ( d ε )≡ ∂ F d ε ∂ (d ε)
d σ α = M αβ d ε β
⎛ dε dσ=Mh(v)dε ⎜⎜ v= ⎝ dε
⎞ ⎟⎟ ⎠
Perfect plasticity :
dσ=0 ⇒
⎧ det M h = 0 ⎨ M hdε=0 ⎩
and
dε ≠0 Plastic limit condition Plastic flow rule
(Limit stress state)
UNDRAINED LOADING ON LOOSE SANDS Typical behaviour of a loose sand : undrained (isochoric) triaxial compression Δu σ 3
σ1 σ3 Δu σ 3
σ1 σ3
→ Experimental evidence : ε1
q goes through a maximum before Mohr-Coulomb limit
q q
p
ε1
UNDRAINED LOADING ON LOOSE SANDS Experimental observations (I. Georgopoulos, J. Desrues – Grenoble, DIGA project):
Partial diffuse failure
Total diffuse failure
UNDRAINED LOADING ON LOOSE SANDS A typical example of a diffuse mode of failure : Δq = ΔF S : « small » additional force (stress controlled loading) q peak is unstable according to LYAPUNOV definition Non-controllability after R. NOVA definition Second order work criterion : d ²W = dσ 1dε1 + 2dσ 3 dε 3
= dqdε1
(1)
(1)
with the isochoric condition :
dε v = 0
After HILL condition of stability, q peak is unstable For axisymmetric conditions : ⎡ dq ⎤ = N ⎡ dε1 ⎤
⎢dε ⎥ ⎣ v⎦
• Undrained loading : dε v = 0 • At q peak : dq = 0
⎢ dσ ⎥ ⎣ 3⎦
• Bifurcation criterion : det N = 0 ⎡ dε 1 ⎤ ⎡ 0 ⎤ N • Failure rule : ⎢dσ ⎥ = ⎢0⎥ ⎣ 3⎦ ⎣ ⎦
Conclusions : q peak is a proper failure state, strictly inside the plastic limit condition, without localization pattern, properly described by Hill’s condition and a bifurcation criterion
HILL’S CONDITION OF STABILITY DRUCKER’s postulate ∀ dσ, dε p : d²W p=dσ : dε p > 0 always satisfied in associated elasto-plasticity dε p = dλ (df/dσ) ⇒ d²W p= dλ (dσ : df/dσ) >0 >0 HILL’s condition of stability ∀ dσ, dε : d²W =dσ : dε > 0
DRUCKER ⇒ HILL
Incrementally linear constitutive relations dσ = M dε
d²W = t dε M dε = t dε Ms dε
d²W > 0 ⇔ det Ms > 0
• Associated elasto-plasticity : M≡Ms plasticity limit condition : det M=0 stability condition : det Ms=0
IDENTICAL !
• Non-associated elasto-plasticity : det Ms is always vanishing before det M and det tnLn=0, ⇒ det Ms=0 is satisfied stricly inside the plastic limit condition and, inside the localisation condition
BIFURCATION ANALYSIS IN AXISYMMETRIC CONDITIONS σ1- σ3/R ε1+ 2R ε3=0 ; R=cst.
(σ1-σ3/R) peak is unstable according to LYAPUNOV d²W = dσ1dε1+ 2 dσ3dε3 = (dσ1-dσ3/R) dε1 ≤ 0 from (σ1-σ3/R) peak
ε1
dε1
dσ1-dσ3/R =Q dε1+ 2R dε3
dσ3/R
dε1
At (σ1-σ3/R) peak : 0 =Q 0
dσ3/R
BIFURCATION CRITERION : det Q=0 ⇔ 2 E1-/E3-(1-ν33-)R²-2(ν13-+ E1- ν31-/E3-)R+1=0 Δ=-(2/(E3-)²) det Ms ≥ 0 Loss of controllability after NOVA (1994) First unstable direction : dσ1 √2 dσ3
=
E1-
(1-ν33-)(E1- ν31--E3- ν13-)
√2 E3- E1-(1-ν33-)-ν13-(E1- ν31--E3- ν13-)
Rupture rule : E1- dε1+ (2 ν31- E1-/E3--1/R) dσ3=0
BIFURCATION ANALYSIS IN PLANE STRAIN CONDITIONS (H.D.V. KHOA) •
Second order work: d 2W = d σ 1 d ε 1 + d σ 3 d ε 3
= (dσ 1 − dσ 3 R ) dε 1 + dσ 3 (dε 1 + R dε 3 ) R
peak
= (dσ 1 − dσ 3 R ) dε 1 ≤ 0
⎧dε1 = positive constant ⎪ ⎨dε 2 = 0 ⎪dε + Rdε = 0 (R=cst.) 3 ⎩ 1
det Q =0
fro m
(σ 1
− σ 3 R ) pe a k
⇒ The peak is potentially unstable in Hill’s sense
•
⎡dσ 1 − dσ 3 R ⎤ ⎢ ⎥ =Q d ε + Rd ε 3 ⎦ ⎣ 1
Octo-linear model:
Bifurcation criterion
(
)
(
)
(
)
⇔ E 1− 1 −V 23−V 32 − R 2 − ⎡E 3− V13− +V12 −V 23− + E 1− V 31− +V 32−V 21− ⎤ R + E 3− ⎣ ⎦ ⎛ ⎡dε1 ⎤ 2 2 ⎡ dσ 1 ⎤ Δ = −4 E 1− E 3− det P S ≥ 0 = P ⎜⎜ ⎢ ⎥ ⎢ ⎥ d ε d σ ⎣ 3⎦ ⎝ ⎣ 3⎦
( )( )
det P s =0
⎡ dε1 ⎤ ⎢ ⎥ ⎣dσ 3 R ⎦
First bifurcation point
(1 −V
V
2 − 1− 1 2
)=0
⎞ ⎟⎟ ⎠
First bifurcation direction E 1− (1 −V 23−V 32 − ) ⎡⎣E 1− (V 31− +V 32−V 21− ) − E 3− (V13− +V12−V 23− ) ⎤⎦ ⎛ dσ ⎞ 1 ⎜ ⎟ = ⎝ dσ 3 ⎠c E 3− ⎡E 1− D + 1 −V12 −V 21− 1 −V 23−V 32 − ⎣⎢
( (
)(
))
(
− E 3− V13− +V12−V 23−
) ⎤⎦⎥ 2
AXISYMMETRIC PROPORTIONAL STRAIN PATHS – INL2 model, DENSE SAND R ∈ (0.3, 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0)
Volumetric strain versus axial strain on the left and deviatoric stress versus mean pressure on the right
Deviatoric stress versus axial strain on the left and (σ1-σ3/R) versus axial strain on the right
PLANE STRAIN PROPORTIONAL STRAIN PATHS (H.D.V. KHOA) Loose Hostun sand ⇒ liquefaction for all R values
Dense Hostun sand
octo-linear model
⇒ liquefaction for only low R values
Proportional strain paths simulated by the incrementally octo-linear model for different R values (R=[0.1,0.2,0.3,0.35,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0])
BIFURCATION DOMAIN IN AXISYMMETRY- INL2 and octolinear models, DENSE SAND
Instability domain for the dense sand in axisymmetric conditions.
σ1- 2σ3 plane in the cases of the octo-linear model (+) and the non linear one (•).
First stress directions giving a nil d²W for the dense sand in the σ1- 2 σ3 plane, with the octo-linear model (+) and the non linear one (•).
BIFURCATIONS IN AXISYMMETRIC CONDITIONS Cones of unstable stress directions : Dense sand ; non-linear model
BIFURCATIONS DOMAINS IN PLANE STRAIN (H.D.V. KHOA)
octo-linear model Bifurcation domains and first bifurcation directions corresponding to: • vanishing d2W
• vanishing d2W (red) or detPs (blue)
d e tP
Loose Hostun sand
Dense Hostun sand
Bifurcation domains are larger for the case of loose Hostun sand than for the dense one.
Loose Hostun sand
s
Dense Hostun sand
The above results of two bifurcation indicators completely numerically coincide.
BIFURCATIONS IN PLANE STRAINs (H.D.V. KHOA)
Loose Hostun sand
Dense Hostun sand
Boundary of the bifurcation domain and the first incremental stress directions of vanishing d2W, according to Hill’s condition, for dense Hostun sand (left) and loose Hostun sand (right). Diagrams in the (σ1; σ3) plane (ε2= 0) in the case of the octo-linear constitutive model.
BIFURCATION DOMAIN ( 3D )
Instability surface for the dense sand (tr σ = 300 kPa)
Stress paths in the deviatoric plane
Instability surface for the loose sand (tr σ = 300 kPa)
THE MICRO-DIRECTIONAL MODEL ( F. NICOT) Nicot F. and Darve F. (2005) : A multiscale approach to granular materials. Mech of Mat., 37,980-1006
RVE
uˆt ,
rˆ r F , uˆ
uˆn ,
σ,ε
Contact direction
r n (θ , ϕ ) l ≈ 2rg
rˆ dF
rˆ du
dε Strain localisation operator
dσ Stress averaging
Local behaviour
duˆi (θ ,ϕ ) ≈ 2rg dε ij n j (θ ,ϕ )
σ ij =
dFˆn = kn duˆn
2rg ve
∑ Fˆ n i
c
rˆ rˆ rˆ rˆ r r F + k d u ˆ ⎧ ⎫ t t t ˆ ˆ ˆ dFt = min ⎨ Ft + kt dut , tan ϕ g Fn + kn dun ⎬ r − Ft ⎩ ⎭ Fˆ + k durˆ t t tt
(
)
(Chang, 1992; Cambou, 1993, etc)
j
Probing tests and normalised second order work
ε1
dε 1
M4
r d σ dσ 1
r dε
M3
αε
M2
ασ 2 dε 2
M1
2 dσ 2
2 ε2
General expression of the second order work
d 2W = d σ d ε
Expression of the second order work in axisymmetric conditions
d 2W = dσ 1 dε 1 + 2 dσ 2 dε 2
Normalised second order work
d w=
d 2W
2
dσ 1 + 2 dσ 2 2
2
dε 1 + 2 dε 2 2
2
Polar diagrams of the normalised second order work Axisymmetric case 1
1
0.8
0.8
0.6
0.6
0.4
0.4
q
0.2
p 0
= 0.725
q p
0.2
q
= 0.701
q
-0.2
-0.2
q p
-0.4
= 0.875
p
0
p
= 0.439 -0.4
= 0.800 q p
-0.6
-0.6
-0.8
-0.8
= 0.439
-1
-1 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Micro-directional model
0.6
0.8
1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Incremental octolinear model
After Nicot, F., and Darve, F. (2005) : Micro-mechanical investigation of material instability in granular assemblies. Int. J. of Solids and Structures, 43, 3569-3595
0.6
0.8
1
Micro-mechanical interpretation
d 2W = d 2Wl + d 2Wv + d 2W f
Local second order work
d 2Wl =
ρo 3 2 2π N g rg ρ g
Change in volume
Change in fabric
dFˆi (θ , ϕ ) duˆi (θ , ϕ ) ω e (θ , ϕ ) sin ϕ dθ dϕ ∫∫ 2 − 1 ε kk [o ;π ]
d 2W f =
d 2Wv =
3 ρo 2 2π N g rg ρ g
∫∫
[o ;π ]2
ρo 3 2 2π N g rg ρ g
∫∫
[o ;π ]2
Fˆi (θ , ϕ ) duˆi (θ , ϕ ) dω e (θ , ϕ ) sin ϕ dθ dϕ 1 − ε kk
Fˆi (θ , ϕ ) duˆi (θ , ϕ ) ω e (θ , ϕ ) dε kk sin ϕ dθ dϕ (1 − ε kk )2
Micro-mechanical interpretation r n
r t1 r t2
rt Fc r r Fct + dFct =
tan ϕ g (F + kn du n c
n c
α
Quadratic form of the second order work
r k t duct
rˆ r d 2Wˆ = dF ⋅ duˆ = dFˆn duˆ n + dFˆt1 duˆt1 + dFˆt2 duˆ t2
)
Plastic case
In the plastic case 2 2 d 2Wˆ = k n duˆ n + tan ϕ g cos α k n duˆ n duˆ t + k t sin 2 α duˆt
duˆ n ≤ 0
d Wˆ ≤ 0 2
tan α ≤
tan ϕ g 2
kn kt
duˆ t ∈ [U 1 ; U 2 ]
⎡ π π⎤ ; ⎣ 2 2 ⎥⎦
α ∈ ⎢−
2 ⎛ k tan α t − tan ϕ g k n duˆ n cos α ⎜1 + ξ i 1 − 4 ⎜ k n tan 2 ϕ g ⎝ Ui = 2 k t sin 2 α
ξ 1 = −1
ξ2 = 1
⎞ ⎟ ⎟ ⎠
Discrete analysis of stability from D.E.M. computations (L. Sibille) The Discrete Element Model : SDEC software (Donzé & Magnier 1997): molecular dynamics approach such as Cundall’s one (1979).
Contact interaction defined by 3 mechanical parameters: 105 ≤ kn ≤ 106 N / m
kt = kn / 0.42 Φ cont = 35°
Cubic form of the specimen; 10 000 spheres; continuous size distribution; dense specimen: 2 ≤ Dsphere ≤ 9 mm
For p = 100 kPa:
n = 0.38
All paths in principal stress (σ1, σ2, σ3) or strain (ε1, ε2, ε3) spaces are allowed. In this lecture: axi-symmetric conditions only.
kt kn
φcont
Discrete analysis of stability from D.E.M. computations ( L. SIBILLE) Stress probes :
on
The initial axi-symmetric stress states: 1. isotropic compression (100, 200 or 300 kPa), 2. triaxial drained compression (σ2 = σ3 = cst.) characterised by: q σ1 − σ 3 = p (σ 1 + σ 2 + σ 3 ) 3
The loading program: • defined in the Rendulic plane of stress increments,
• dσ =
( dσ 1 ) + 2
(
2 dσ 3
)
2
= cte = 1kPa
• 0 ≤ αdσ < 360° → response vectors dε defined in dual plane ( 2dε3, dε1). 2 2 → d W = d W (α dσ )
Discrete analysis of stability from D.E.M. computations Unstable directions : Diagram of f (α dσ ) = d Wnorm. + c 2
with:
d 2Wnorm. =
dσ ⋅ d ε dσ d ε
c such as: f (α dσ ) > 0 σ3 = 100 kPa
σ3 = 200 kPa
→ Cones of "unstable" stress directions observed with the D.E.M.
Discrete analysis of stability from D.E.M. computations Cones of unstable stress directions : D.E.M. and Incremental Non-Linear model Discrete Element Model d²W > 0 Cones of unstable directions (L. Sibille)
Macroscopic phenomenological relation (I.N.L.2 model) calibrated on the drained triaxial responses of D.E.M. (L.Scholtes).
2nd step – from the bifurcation point to the failure • Imposimato and Nova (1998) shown that the full controllability of a loading programme defined by its control parameters can be lost before reaching the plastic limit condition. • The control parameters can be linear combinations of stresses or strains (e.g. volumetric strains).
Loose specimen, η = 0.46, α = 215.3 deg (q = cst ) • Stress probe is fully stress controlled → no failure observed. • Can we choose others control parameters ? d²W = dσ1 dε1 + 2 dσ3 dε3 = dq dε1 + dσ3 dεv • Can we control the loading programme defined by:
α = 215.3°
dq = 0 and dεv = - 0.002 % (dilatancy) ? f (α ) = d 2Wnorm. + c
v
0) (α = 200°, d²W > 0) (α = 210°, d²W < 0) (α = 215.3°, d²W < 0) (α = 220°, d²W < 0) (α = 230°, d²W < 0) (α = 240°, d²W > 0) (α = 250°, d²W > 0) f (α ) = d 2Wnorm. + c
• Small perturbation of the numerical specimen External input of kinetic energy (1 10-5 J) to the specimen by excitements of some floating grains (simulation without gravity).
2nd step – from the bifurcation point to the failure
• For R = 4.01; 1.94; 0.408; 0.257 (d²W > 0) a new stable state is reached. • Total collapse for R = 1.00; 0.843; 0.593 (d²W < 0).
Kinetic energy (J)
• For R = 1.22 (d²W < 0 but stress direction close to the border of the cone ) the collapse is partial.
Simulation time (s)
Simulation time (s)
CONCLUSIONS 1.
Phenomenological analysis : for non associated materials like geomaterials, there is not a single plastic limit surface where failure occurs, but rather a whole domain in the stress space where bifurcations, losses of uniqueness, instabilities … i.e. FAILURES can appear, according to : the stress-strain history the current direction of loading the loading mode
In this bifurcation domain, various failure modes can develop (material instabilities leading to diffuse or localized failures, geometric instabilities, … ) Second order work criterion seems to detect diffuse failure 2.
Micromechanical analysis : it confirms these analyses. Moreover a new micro-mechanical understanding of these material instabilities is proposed by considering the local, discrete, second order work at the grain level.
3.
Discrete element analysis : bifurcation domains and cones of unstable stress-strain directions also exhibited in good qualitative agreement. Diffuse failure was simulated exactly for the conditions predicted by the theory.
These 3, basically different, methods give similar results
Some recent contributions
Book : Degradations and Instabilities in Geomaterials, F. Darve and I. Vardoulakis eds, Springer publ., 367 pages, 2005 Papers : - Darve, F., Servant G., Laouafa F., Khoa H.D.V. (2004): Failure in geomaterials, continuous and discrete analyses, Comp. Meth. In Applied Mech. and Eng., vol. 193, 27-29, 3057-3085. - Nicot, F., and Darve, F. (2005) : A multiscale approach to granular materials. Mechanics of Materials, 37, 980-1006 - Darve, F., and Nicot, F. (2005): On incremental non linearity in granular media, phenomenological and multi-scale views.(I) Int. J. Numerical Analytical Methods in Geomech., vol.29, n°14, 1387-1410 - Darve, F., and Nicot, F. (2005): On flow rule in granular media, phenomenological and multiscale views (II). Int. J. Numerical Analytical Methods in Geomech. ,vol. 29, n°14, 1411-1432 - Nicot, F., and Darve, F. (2005) : Micro-mechanical investigation of material instability in granular assemblies. Int. J. of Solids and Structures, 43, 3569-3595 - H.D.V. Khoa, I.O. Georgopoulos, F. Darve, F. Laouafa (2006): Diffuse failure in geomaterials , experiments and modelling, Comp. Geotechn., vol.33, n°1, 1-14. - Nicot, F., and Darve, F. (2006) : On the elastic and plastic strain decomposition in granular materials., Granular Matter, vol.8, n°3, 221-237 - Sibille L., Nicot F., Donze F.V., Darve F. (2007) : Material instability in granular assemblies from fundamentally different models, Int. J. Num. Anal. Methods in Geomechanics, vol. 31, n°3, 457482 - Nicot F., Sibille L., Donze F., Darve F. ( 2007) : From microscopic to macroscopic second-order work in granular assemblies, Mechanics of Materials, vol.39, n°7, 664-684
II Landslides modelling : Trévoux and Petacciato examples
FINITE ELEMENT MODELLING OF THE TREVOUX AND PETACCIATO LANDSLIDES BY TAKING INTO ACCOUNT UNSATURATED HYDRO-MECHANICAL COUPLING
Application to Trévoux and Petacciato landslides
Trévoux
Petacciato
deep crack (square in front of the parish church).
“Vaccareggia”, crack on the road near the slide boundary.
The Trevoux site
Geographic location of the studied area
The Trevoux site
Typical cross section of the Trevoux landslide
Finite element mesh of the Trevoux landslide
The Petacciato site
Geographic location of the studied area
The Petacciato site
~ 6°
Typical cross-section of the Petacciato landslide
Finite element mesh of the Petacciato coastal slope
Description of the Plasol constitutive model (Liège university)
• Van Eekelen yield criterion : f = I 2σ
⎛ 3c ⎞ ⎜ ⎟⎟ = 0 + m⎜ I σ − tan ϕ c ⎠ ⎝
– With : 1
⎛ rc ⎜⎜ re ⎝ b=
⎞n ⎟⎟ − 1 ⎠
⎛ rc ⎜⎜ ⎝ re
⎞ ⎟⎟ + 1 ⎠
rc =
m = a (1+ b sin 3ϕ )
n
a=
1 n
2 sin ϕ c
3 (3 − sin ϕ c )
re =
n = −0.229
rc (1 + b )n
2 sin ϕ e 3 (3 + sin ϕ e )
Description of the Plasol constitutive model
• evolution of the internal variables : – hyperbolic function of : 2 ε = ∫ ε& dt = ∫ tr (e&) dt 3 t
p eq
t
p eq
0
2
0
tr (ε& ) e& = ε& − 1 3 p
p
– Internal variables : ϕc = ϕc0
(ϕ +
ϕe = ϕe0
(ϕ +
c = c0
(c +
cf
ef
− ϕ c 0 )ε eqp
B p + ε eqp − ϕ e 0 )ε eqp
B p + ε eqp
f
− c0 )ε eqp
Bc + ε eqp
(ϕ
0
= 30°, ϕ f = 35°)
Trévoux soil parameters Comp act marl (6)
Soil parameters
Unit
Upper fill (1)
Low fill (2)
Sand clay (3)
Gravel Gravel ly sand ly marl (5) (4)
Grain specific weight
kN/ m3
26.
26.
29.00
28.00
27.
28.5
Young modulus
MP a
38.6
38.6
30.0
20.0
46.3
100.0
Poisson’s ratio
-
0.29
0.29
0.29
0.29
0.29
0.29
Porosity
-
0.3
0.3
0.5
0.5
0.3
0.3
Intrinsic permeability
m2
10-10
10-10
10-10
10-10
10-14
10-14
Initial friction angle
°
10.0
10.0
10.0
10.0
35.0
35.0
Final friction angle
°
35.0
35.0
35.0
40.0
35.0
35.0
B p coefficient
-
0.008
0.008
0.008
0.008
0.008
0.008
kPa
10.0
15.0
1.0
1.0
100.0
100.0
°
0
0
3.2
3.2
2.0
2.0
Cohesion Dilatancy angle
Petacciato soil parameters Soil parameters
Symbols
Unit
Blue-gray clay
Grain specific weight
ρs
kN/m3
27.
Young modulus
E
MPa
95.0
Poisson’s ratio
ν
-
0.21
Porosity
n
-
0.3
Intrinsic permeability
kw
M2
10-17
Initial friction angle
ϕ0
°
1.0
Final friction angle
ϕf
°
19.0
B p coefficient
Bp
-
0.01
Initial cohesion
c0
kPa
10
Final cohesion
cf
kPa
171
Bc coefficient
Bc
-
0.02
ψ 0 =ψ f
°
0
Dilatancy angle
Hydro-mechanical coupling for the unsaturated soil
• Time dependent model of water transfer : – Pressure head :
hw =
pw
γw
+y
– Generalised Darcy’s law : ν w = − K w ( pc )∇hw – Richard’s equation : water mass balance) With
θ w = nS ew
and
⎛V ⎞ S ew = ⎜⎜ w ⎟⎟ ⎝ Vv ⎠ current
∂θ w T = ∇ (K w ( pc )∇hw ) ∂t
(volume water content)
(Darcy +
Hydro-mechanical coupling for the unsaturated soil
• Richards equation depends on 2 hydrodynamic characteristics : – Water retention curve of Van-Genuchten : S ew = S w +
S w − S rw
(1 + (αp ) )
1 β 1− β
c
– Permeability :
K w = S ew k w
• Effective stress : With :
χ = S ew
⎛V ⎞ S rw = ⎜⎜ w ⎟⎟ ⎝ Vv ⎠ minimal
σ ' = σ − pa 1 + χ ( pa − p w )1
⎛ Vw ⎞ S w = ⎜⎜ ⎟⎟ ⎝ Vv ⎠ maximall
Water retention curve for Trévoux Parameters Maximal degree of saturation Residual degree of saturation
Symbols
S S
w
rw
Unit
Sands
Marls
-
1.0
1.0
-
0.1
0.1
First retention parameter
α
Pa–1
6.8 10–5
2.5 10–5
Second retention parameter
β
-
4.8
2.0
Water retention curve for Petacciato Parameters
Symbols
Maximal degree of saturation
S
w
Residual degree of saturation
S
rw
First retention parameter Second retention parameter
α β
Unit
Clay
-
1.0
-
0.1
Pa–1
1.0 10–5
-
1.35
Implementation of d2W in finite element code Lagamine
• Expression of Hill’s stability criterion : d2 Wpi = dσ'pi : dε pi
– Local second order work : normalised : d Wnorm. = 2
d2 Wpi dσ' pi dε pi
– Global second order work :
Npi
D2 W = ∑ dσ'pi : dε pi .ωpi Jpi pi=1
D2 W
2
normalised, weighted :
D Wnorm. =
Npi
Npi
∑ ω J ∑ dσ'
pi=1
pi pi
pi=1
pi
dε pi
with :
Npi : total number of integration points Jpi : determinant of Jacobian transformation matrix for point pi ωpi: weight factor for point pi
Application to landslides
• Loading program in the simulation : – Initial state: unsaturated soil (dry) – Progressive saturation by increasing the water table – Stability analysis thanks to second order work criterion
• Results : – Iso values of the second order work – Global second order work of the problem vs loading parameter
Application to Trévoux landslide
• Evolution of local second order work – Water rising modelling
Application to Trévoux landslide
• Evolution of local second order work – Water rising modelling
Swelling level level of flood
Application to Trévoux landslide
• Evolution of global second order work :
Downhill water level (m)
Application to Petacciato landslide
• Evolution of local second order work – Water rising modelling
Application to Petacciato landslide
• Evolution of global second order work : 60 STEP 25
50
D2Wnorm
40 STEP 75
30 20
STEP 85
10 0 0
STEP 96
100
200
Upside water
300 level(m)
400
CONCLUSIONS
¾ Locally, Hill’s bifurcation criterion comes before the other criteria (i.e. Mohr-Coulomb plastic limit condition, Rice’s localisation condition, etc.). ¾ Application of Hill’s criterion to stability analyses of non-linear boundary problems : – Material scale : local second order work criterion • detection of different failure modes (localized, diffuse), • description of the propagation of the potentially unstable zones. – Global scale : global criterion by integrating of local second order work into considered volume • description of global stability of the whole body, • highlight of the influence of the parameters of the constitutive behaviour (saturation, hydraulic conductivity, etc.) and events of hydraulic nature (raining, water flow, earthquakes, etc.) or anthropic (constructions, excavations, etc.) to the global stability of body.
