Journée d'hommage au Prof. Biarez MODELLING OF LANDSLIDES AS A BIFURCATION PROBLEM

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1 Journée d'hommage au Prof. Biarez MODELLING OF LANDSLIDES AS A BIFURCATION PROBLEM I Introduction 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 1 Failure : the classical view 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

2 FAILURE : THE CLASSICAL VIEW Rate independent materials : Euler s identity : I Introduction d σ = F ( d ε ) λ 0 : F ( λdε) λf ( dε) F Homogeneous of degree 1 dσ = F ( dε) F dε ( dε) dσ = M v ε v = d dε ε h ( )d d σ α =M αβ dε β Perfect plasticity : d σ = 0 and d ε 0 (Limit stress state) det M M h d ε = = 0 0 h Plastic limit condition Plastic flow rule

3 UNDRAINED LOADING ON LOOSE SANDS Typical behaviour of a loose sand : undrained (isochoric) triaxial compression σ 1 σ 3 3 Δ u σ Δ u σ 3 σ 1 σ 3 Experimental evidence : ε 1 q goes through a maximum before Mohr-Coulomb limit q q p ε 1

4 UNDRAINED LOADING ON LOOSE SANDS Experimental observations (I. Georgopoulos, J. Desrues Grenoble, DIGA project): Partial diffuse failure Total diffuse failure

5 Δq = ΔF S UNDRAINED LOADING ON LOOSE SANDS A typical example of a diffuse mode of failure : : «small» additional force (stress controlled loading) q peak is unstable according to LYAPUNOV definition Non-controllability after R. NOVA definition Second order work criterion : d² W = dσ = dqdε 1dε1 + 1 (1) dσ dε 3 3 (1) with the isochoric condition : dε v = 0 After HILL condition of stability, q peak is unstable For axisymmetric conditions : dq dε v dε1 = N dσ 3 Undrained loading : dε v = 0 Bifurcation criterion : det N = 0 dε1 0 At q peak : dq = 0 Failure rule : N = dσ 3 0 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

6 DRUCKER s postulate HILL S CONDITION OF STABILITY 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 M s dε d²w > 0 det M s > 0 Associated elasto-plasticity : M M s plasticity limit condition : det M=0 stability condition : det M s =0 IDENTICAL! Non-associated elasto-plasticity : det M s is always vanishing before det M and det t nln=0, det M s =0 is satisfied stricly inside the plastic limit condition and, inside the localisation condition

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8 BIFURCATION ANALYSIS IN AXISYMMETRIC CONDITIONS σ 1- σ 3 /R ε 1 + R ε 3 =0 ; R=cst. (σ 1 -σ 3 /R) peak is unstable according to LYAPUNOV d²w = dσ 1 dε 1 + dσ 3 dε 3 = (dσ 1 -dσ 3 /R) dε 1 0 from (σ 1 -σ 3 /R) peak ε 1 dσ 1 -dσ 3 /R dε 1 = Q dε 1 + R dε 3 dσ 3 /R At (σ 1 -σ 3 /R) peak : 0 dε 1 = Q 0 dσ 3 /R BIFURCATION CRITERION : det Q=0 E 1- /E 3- (1-ν 3 3- )R²-(ν E 1- ν 3 1- /E 3- )R+1=0 Δ=-(/(E 3- )²) det M s 0 Loss of controllability after NOVA (1994) First unstable direction : dσ 1 E - 1 (1-ν 3-3 )(E 1- ν 1-3 -E 3- ν 3-1 ) = dσ E E 1- (1-ν 3-3 )-ν 3-1 (E 1- ν 1-3 -E 3- ν 3-1 ) Rupture rule : E 1- dε 1 +( ν 3 1- E 1- /E 3- -1/R) dσ 3 =0

9 BIFURCATION ANALYSIS IN PLANE STRAIN CONDITIONS (H.D.V. KHOA) Second order work: peak d W = dσ dε + dσ dε ( dσ 1 dσ 3 R) dε1 dσ 3 ( dε1 Rdε 3) R ( dσ dσ R) dε 0 fro m ( σ σ R ) = + + = peak The peak is potentially unstable in Hill s sense d ε1 = positive constant d ε = 0 dσ1 dσ3 R dε1 Octo-linear model: = Q dε1 + Rdε3 = 0 (R=cst.) dε1 + Rdε3 dσ3 R det Q=0 Bifurcation criterion 3 ( ) ( ) ( ) 1 ( ) dε S 1 dσ 1 Δ= 4( E1 ) ( E 3 ) de t P 0 = P E 1 V V R E V + V V + E V + V V R + E 1 V V = 0 s det P = dε3 dσ 3 First bifurcation point First bifurcation direction dσ E1 ( 1 V V3 ) E1 ( V3 + V3 V ) E 3 ( V1 + V1 V ) 1 = dσ c E 3 E1 ( D+ ( 1 V1 V )( 1 V V3 )) E 3 ( V1 + V1 V )

10 AXISYMMETRIC PROPORTIONAL STRAIN PATHS INL 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 (σ 1 -σ 3 /R) versus axial strain on the right

11 PLANE STRAIN PROPORTIONAL STRAIN PATHS (H.D.V. KHOA) Loose Hostun sand liquefaction for all R values octo-linear model Dense Hostun sand liquefaction for only low R values Proportional strain paths simulated by the incrementally octo-linear model for different R values (R=[0.1,0.,0.3,0.35,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0])

12 BIFURCATION DOMAIN IN AXISYMMETRY- INL and octolinear models, DENSE SAND Instability domain for the dense sand in axisymmetric conditions. σ 1 - σ 3 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 σ 1 - σ 3 plane, with the octo-linear model (+) and the non linear one ( ).

13 BIFURCATIONS IN AXISYMMETRIC CONDITIONS Cones of unstable stress directions : Dense sand ; non-linear model

14 BIFURCATIONS DOMAINS IN PLANE STRAIN (H.D.V. KHOA) octo-linear model Bifurcation domains and first bifurcation directions corresponding to: vanishing d W vanishing d W (red) or detps (blue) detp s Loose Hostun sand Dense Hostun sand Loose Hostun sand Dense Hostun sand Bifurcation domains are larger for the case of loose Hostun sand than for the dense one. The above results of two bifurcation indicators completely numerically coincide.

15 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 d W, according to Hill s condition, for dense Hostun sand (left) and loose Hostun sand (right). Diagrams in the (σ 1 ; σ 3 ) plane (ε = 0) in the case of the octo-linear constitutive model.

16 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)

17 THE MICRO-DIRECTIONAL MODEL ( F. NICOT) Nicot F. and Darve F. (005) : A multiscale approach to granular materials. Mech of Mat., 37, RVE rˆ r F, u ˆ σ, ε uˆt, uˆn, Contact direction n r ( θ,ϕ ) dε duˆ i Strain localisation operator ( θ, ϕ ) r dε n ( θ, ϕ ) g rˆ df t ij j du ˆr Local behaviour dfˆ = k rˆ r = min Ft + kt duˆ t, tanϕ g n n duˆ n df ˆr rˆ σ ij Stress averaging = r v e g Ft kt dut ( Fˆ + rˆ n + kn duˆ n ) Ft rˆ F t l r g + k t rˆ r duˆ tt c Fˆ i n j dσ (Chang, 199; Cambou, 1993, etc)

18 Probing tests and normalised second order work ε 1 M 3 M 4 dε 1 dε r σ r d 1 dσ M α ε α σ dε dσ M 1 ε General expression of the second order work d W = dσ dε Expression of the second order work in axisymmetric conditions d W = dσ 1 dε1 + dσ dε Normalised second order work d w = dσ 1 d W 1 + dσ dε + dε

19 Polar diagrams of the normalised second order work Axisymmetric case q = 0.75 p q = p q = p q = p q = p q = p Micro-directional model Incremental octolinear model After Nicot, F., and Darve, F. (005) : Micro-mechanical investigation of material instability in granular assemblies. Int. J. of Solids and Structures, 43,

20 f v l W d W d W d W d + + = Micro-mechanical interpretation ( ) ( ) ( ) [ ] = ; sin 1,, ˆ, ˆ 3 π ϕ θ ϕ ε ϕ θ ω ϕ θ ϕ θ ρ ρ π o kk e i i g o g g l d d du df r N W d ( ) ( ) ( ) ( ) [ ] = ; sin 1,, ˆ, ˆ 3 π ϕ θ ϕ ε ε ϕ θ ω ϕ θ ϕ θ ρ ρ π o kk kk e i i g o g g v d d d du F r N W d ( ) ( ) ( ) [ ] = ; sin 1,, ˆ, ˆ 3 π ϕ θ ϕ ε ϕ θ ω ϕ θ ϕ θ ρ ρ π o kk e i i g o g g f d d d du F r N W d Change in volume Change in fabric Local second order work

21 Micro-mechanical interpretation n r tr 1 t F r c α Quadratic form of the second order work r r r t t t t r F kt du c + dfc = c r n n tanϕ g ( Fc + kn duc ) ˆ ˆ r d W = df duˆ = dfˆ ˆ ˆ ˆ ˆ n dun + dft du ˆ 1 t + df 1 t du t Plastic case In the plastic case d Wˆ = k n duˆ n + tanϕ cosα k duˆ duˆ + k g n n t t sin α duˆ t ˆ 0 du n d Wˆ 0 tanα duˆ t tanϕ g [ U U ] 1 ; k k n t π π α ; U i = tanϕ k g n duˆ n cosα 1 + ξi 1 4 k sin α t k k n t tan tan α ϕ g ξ1 = 1 ξ = 1

22 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: kn 10 / k t = k n Φ = 35 cont N m /0.4 k n k t φ cont Cubic form of the specimen; spheres; continuous size distribution; dense specimen: D 9mm sphere For p = 100 kpa: n = 0.38 All paths in principal stress (σ 1, σ, σ 3 ) or strain (ε 1, ε, ε 3 ) spaces are allowed. In this lecture: axi-symmetric conditions only.

23 Discrete analysis of stability from D.E.M. computations ( L. SIBILLE) Stress probes : on The initial axi-symmetric stress states: 1. isotropic compression (100, 00 or 300 kpa),. triaxial drained compression (σ = σ 3 = cst.) characterised by: q p = σ σ 1 3 ( σ + σ + σ ) The loading program: defined in the Rendulic plane of stress increments, dσ = dσ1 + dσ3 = cte= 1kPa 0 α dσ < 360 response vectors dε defined in dual plane ( dε, dε ). dw= dwαd σ ( ) ( ) ( ) 3 1

24 Discrete analysis of stability from D.E.M. computations Unstable directions : Diagram of f ( α dσ ) = dwnorm. + c with: dwnorm. = dσ dε dσ dε c such as: f ( α ) > 0 dσ σ 3 = 100 kpa σ 3 = 00 kpa Cones of "unstable" stress directions observed with the D.E.M.

25 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. model) calibrated on the drained triaxial responses of D.E.M. (L.Scholtes).

26 nd 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 = 15.3 deg (q = cst ) Stress probe is fully stress controlled no failure observed. Can we choose others control parameters? d²w = dσ 1 dε 1 + dσ 3 dε 3 = dq dε 1 + dσ 3 dε v Can we control the loading programme defined by: dq = 0anddε v = % (dilatancy)? α = 15.3 f α = dw + c ( ) norm.

27 Comparison of the response of the numerical specimen controlled by: dq = 0 and dσ 3 < 0 or by dq = 0 and dε v < 0 Loss of controllability for dq = 0 and dε v < 0, the specimen totally collapses. Kinetic energy (J) dq = 0, dε v < 0 dq = 0, dσ 3 < 0 Simulation time (s) Stresses: σ 1 σ 3 (kpa) dq = 0, dε v < 0 dq = 0, dε v < 0 dq = 0, dσ 3 < 0 dq = 0, dσ 3 < 0 Strains: ε 1 ε 3 (%) dq = 0, dε v < 0 dq = 0, dε v < 0 dq = 0, dσ 3 < 0 Simulation time (s) Simulation time (s)

28 nd step from the bifurcation point to the failure Generalization to proportional stress paths dw= dσ1dε1+ dσ3dε3 dσ 3 dσ 3 dw= dε1 dσ1 + dε1+ Rdε3 R R with R = cst. for a given loading path. ( ) Can we control the loading programme defined by: dσ 1 -dσ 3 / R =0 and dε 1 + R dε 3 = %? with: R = 1.94 (α = 00, d²w > 0) R = 1. (α = 10, d²w < 0) R = 1.00 (α = 15.3, d²w < 0) R = (α = 0, d²w < 0) R = (α = 30, d²w < 0) R = (α = 40, d²w > 0) f α = dw + c ( ) norm.

29 nd step from the bifurcation point to the failure The loading programme is controllable for R = 1.94 and (d²w > 0). Loss of controllability for R = 1.00; 0.843; (d²w < 0). For R = 1. (d²w < 0 but stress direction close to the border of the cone ) the loss of controllability is not total. Simulation time (s) Simulation time (s)

30 nd step from the bifurcation point to the failure For R = 1.94 and (d²w > 0) a new stable state close to the initial one is reached. Total collapse for R = 1.00; 0.843; (d²w < 0). For R = 1. (d²w < 0 but stress direction close to the border of the cone ) the collapse is partial. Simulation time (s) Simulation time (s) Loading parameters exist such as the specimen collapses from a bifurcation point detected by the sign of d²w.

31 nd step from the bifurcation point to the failure In their tests Chu et al. 003 do not apply any loading path. They verify if a mechanical state governed by specific control parameters can be sustained (Notion of loss of sustainability, Nicot and Darve 006). Control of the mechanical state by: dσ 1 -dσ 3 / R =0 and dε 1 + R dε 3 = 0 with: R = 4.01 (α = 190, d²w > 0) R = 1.94 (α = 00, d²w > 0) R = 1. (α = 10, d²w < 0) R = 1.00 (α = 15.3, d²w < 0) R = (α = 0, d²w < 0) R = (α = 30, d²w < 0) R = (α = 40, d²w > 0) R = 0.57 (α = 50, d²w > 0) Small perturbation of the numerical specimen f α = dw + c ( ) norm. External input of kinetic energy ( J) to the specimen by excitements of some floating grains (simulation without gravity).

32 nd step from the bifurcation point to the failure For R = 4.01; 1.94; 0.408; 0.57 (d²w > 0) a new stable state is reached. Total collapse for R = 1.00; 0.843; (d²w < 0). For R = 1. (d²w < 0 but stress direction close to the border of the cone ) the collapse is partial. Kinetic energy (J) Simulation time (s) Simulation time (s)

33 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. 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

34 Some recent contributions Book : Degradations and Instabilities in Geomaterials, F. Darve and I. Vardoulakis eds, Springer publ., 367 pages, 005 Papers : - Darve, F., Servant G., Laouafa F., Khoa H.D.V. (004): Failure in geomaterials, continuous and discrete analyses, Comp. Meth. In Applied Mech. and Eng., vol. 193, 7-9, Nicot, F., and Darve, F. (005) : A multiscale approach to granular materials. Mechanics of Materials, 37, Darve, F., and Nicot, F. (005): On incremental non linearity in granular media, phenomenological and multi-scale views.(i) Int. J. Numerical Analytical Methods in Geomech., vol.9, n 14, Darve, F., and Nicot, F. (005): On flow rule in granular media, phenomenological and multiscale views (II). Int. J. Numerical Analytical Methods in Geomech.,vol. 9, n 14, Nicot, F., and Darve, F. (005) : Micro-mechanical investigation of material instability in granular assemblies. Int. J. of Solids and Structures, 43, H.D.V. Khoa, I.O. Georgopoulos, F. Darve, F. Laouafa (006): Diffuse failure in geomaterials, experiments and modelling, Comp. Geotechn., vol.33, n 1, Nicot, F., and Darve, F. (006) : On the elastic and plastic strain decomposition in granular materials., Granular Matter, vol.8, n 3, Sibille L., Nicot F., Donze F.V., Darve F. (007) : Material instability in granular assemblies from fundamentally different models, Int. J. Num. Anal. Methods in Geomechanics, vol. 31, n 3, Nicot F., Sibille L., Donze F., Darve F. ( 007) : From microscopic to macroscopic second-order work in granular assemblies, Mechanics of Materials, vol.39, n 7,

35 II Landslides modelling : Trévoux and Petacciato examples FINITE ELEMENT MODELLING OF THE TREVOUX AND PETACCIATO LANDSLIDES BY TAKING INTO ACCOUNT UNSATURATED HYDRO-MECHANICAL COUPLING

36 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.

37 The Trevoux site Geographic location of the studied area

38 The Trevoux site Typical cross section of the Trevoux landslide Finite element mesh of the Trevoux landslide

39 The Petacciato site Geographic location of the studied area

40 The Petacciato site ~ 6 Typical cross-section of the Petacciato landslide Finite element mesh of the Petacciato coastal slope

41 Description of the Plasol constitutive model (Liège university) Van Eekelen yield criterion : With : ( ) n b a m ϕ sin 3 1+ = = n e c n e c r r r r b ( ) n c b r a = 1+ ( ) c c r c ϕ ϕ sin 3 3 sin = ( ) e e r e ϕ ϕ sin 3 3 sin + = n = tan 3 = + = c c I m I f ϕ σ σ

42 Description of the Plasol constitutive model evolution of the internal variables : hyperbolic function of : ε p eq t p = &ε dt = 0 e& = & ε p eq tr( e& ) 3 dt Internal variables : ϕ = ϕ c c0 + t 0 p tr( & ε ) 1 3 ( ϕ ϕ ) cf B p + ε c0 p ε eq p eq ϕ = ϕ e e0 c = c ( ϕ ϕ ) ef ( c c ) f B B c p + + e0 p ε eq ε ε p 0 eq p ε eq p eq ( ϕ = 30, ϕ = 35 ) 0 f

43 Trévoux soil parameters Soil parameters Unit Upper fill (1) Grain specific weight Young modulus B p coefficient Low fill () Sand clay (3) Gravel ly sand (4) Gravel ly marl (5) Comp act marl (6) kn/ m 3 MP a Poisson s ratio Porosity Intrinsic permeability Initial friction angle Final friction angle m Cohesion kpa Dilatancy angle

44 Petacciato soil parameters Soil parameters Symbols Unit Blue-gray clay ρ s Grain specific weight kn/m 3 7. Young modulus E MPa 95.0 Poisson s ratio ν Porosity n k w Intrinsic permeability M Initial friction angle 1.0 ϕ 0 Final friction angle 19.0 ϕ f B p coefficient B p Initial cohesion kpa 10 c 0 Final cohesion kpa 171 c f B c coefficient B c Dilatancy angle ψ 0 =ψ f 0

45 Hydro-mechanical coupling for the unsaturated soil Time dependent model of water transfer : Pressure head : h p = γ w w + w y Generalised Darcy s law : ν w w ( pc ) hw = K θ w T Richard s equation : = ( K w( pc ) hw ) (Darcy + t water mass balance) With θ = w ns ew (volume water content) and S ew V = V w v current

46 Hydro-mechanical coupling for the unsaturated soil Richards equation depends on hydrodynamic characteristics : Water retention curve of Van-Genuchten : S ew = S w + S w S β ( ( α ) ) β p c rw S rw V = V w v minimal S w V = V w v maximall Permeability : K = S w ew k w Effective stress : σ ' = σ p 1+ χ( p a a p w )1 With : χ = S ew

47 Water retention curve for Trévoux Parameters Symbols Unit Sands Marls Maximal degree of saturation Residual degree of saturation First retention parameter Second retention parameter S S rw α β w Pa

48 Water retention curve for Petacciato Parameters Symbols Unit Clay Maximal degree of saturation Residual degree of saturation First retention parameter Second retention parameter S S α w rw β Pa

49 Implementation of d W in finite element code Lagamine Expression of Hill s stability criterion : Local second order work : d Wpi = dσ' pi : dε pi normalised : d Wnorm. = d dσ' W pi pi dε pi Global second order work : D W N pi = dσ' : dε. ω pi= 1 pi pi pi J pi normalised, weighted : D W norm. = N pi= 1 pi N pi ω pi J D pi pi= 1 W dσ' pi dε pi with : N pi J pi ω pi : total number of integration points : determinant of Jacobian transformation matrix for point pi : weight factor for point pi

50 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

51 Application to Trévoux landslide Evolution of local second order work Water rising modelling

52 Application to Trévoux landslide Evolution of local second order work Water rising modelling level Swelling of flood level

53 Application to Trévoux landslide Evolution of global second order work : Downhill water level (m)

54 Application to Petacciato landslide Evolution of local second order work Water rising modelling

55 Application to Petacciato landslide Evolution of global second order work : STEP 5 D W norm STEP 85 STEP Upside water level(m) STEP 96

56 CONCLUSIONS Locally, Hill s bifurcation criterion comes before the other criteria (i.e. Mohr-Coulomb plastic limit condition, Rice s localisation condition, etc.). 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.