10 th Jubilee National Congress on Theoretical and Applied Mechanics, Varna September 2005

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1 th Jubilee National Congress on Theoretical and Alied Mechanics, Varna 6 Setember 25 FEM APPLIED IN HYDRO-MECHANICAL COUPLED ANALYSIS OF A SLOPE A. Yanakieva, M. Datcheva, R. Iankov, F. Collin 2, A. Baltov Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev, bl.4, BG Sofia 2 University of Liege, Belgium aniyanakieva@imbm.bas.bg. Introduction Landslides can be attributed to a number of factors, such as geologic features, toograhy, vegetation, eather, or combinations of these factors. Rock and soil mass is involved in tectonic or other natural rocesses that result in aearance and develoment of different kind of discontinuities such as faults, cracks and breaks filled ith ater or softer/harder minerals. The existence of such discontinuities is one of the essential failure sources in landslides. When ater is resent in or near the discontinuities, as ell as hen the ater is in the area of contact beteen to or more different soil layers, decrease of soil internal resistance is accounted for by means of shear strength decrease on the otential failure surface. The resent study assumes that the cause of the fracture rocesses is the develoment of stress and/or strain localization bands. Recently, the Finite Element Method (FEM) rovides a more oerful alternative to traditional methods in assessing stability of non-reinforced sloes and embankments. This aer resents an examle of alication of a couled hydro-mechanical finite element model to simulate the stress and strain state in a sloe and to investigate the influence of the existence of a fault on the sloe behavior. A secial finite element is used to model the influence of ater ressure that is alied into the fault. To elucidate the imact of the fault ater ressure variation on the otential failure hereafter to numerical simulations are erformed. One of them considers the alication of hydrostatic ressure on art of the sloe surface hich is near the fault, and the other one does not consider the alication of hydrostatic ressure. The change of the sloe mechanical behavior considering to or more soil layers and more than one faults, ill be an object of further studies. 2. Couled hydro-mechanical model A large strain finite element formulation of the mechanical roblem is used to describe the couled hydro-mechanical behavior of the sloe. In articular, the kinematics of solid continuum is described using Lagrangian formulation. To exress the equilibrium udate Lagrangian formulation is alied in the current configuration. In terms of the FEM aroach, this means that the reference configuration evolves at each time ste. The Cauchy stress tensor is chosen as a stress measure. The general relation for the static equilibrium of an elementary volume is given by: ( i,,2, ), j + ρb i, j, () here ρ is the unit volume mass and ρ b i is the mass force. Terzaghi effective stress hyothesis is assumed and the effective tress δ if and if < here is the ater ressure. The strain rate tensor is a sum of elastic strain rate, & & + & e 42 is defined as: (2) e & and lastic strain rate,, () The elastic art of the strain rate is related to an objective rate of the stress (i.e. Jaumann derivative) through a linear elastic la and the lastic art obeys a lastic flo rule defined for each of the lastic mechanisms in the folloing ay, [], [4]: g & λ & (4) & :

2 hile & λ is the lastic multilier; g is a otential function hich may be different for different lastic mechanisms f and is defined by so called discontinuous lastic flo function. The corresonding lastic flo rule reads in the stress sace, []. We assume in the resent ork that. Then, the lastic flo function is given as: f II m I c tan φc ˆ + here ˆ, (5) I, is the first stress invariant and I δ II ˆ ˆ 2 ˆ, is the second stress invariant,, is a stress deviator, c is cohesion, φ c is the angle of friction and m is a factor deending on the model. In the case of Drucker-Prager model m ( β ) n a + bsin, here a and b are functions of c c m, and for Van-Eeckelen model 2sinφ ( sinφ ) φ, φ e and n is a Van-Eeckelen arameter, β is the Lode angle hich deends on the third invariant of the stress deviator. The non-associate lastic flo rule is introduced using dilatancy angleψ, and the lastic otential exression reads: + m I II ˆ c tanψ g. (6) The equivalent lastic strain rate is given by 2 ˆ& ˆ& & λ 2 g g g g ll &, (7) eq kk here &ˆ is the lastic strain rate deviator. Material hardening/softening is described based on the evolution of cohesion and friction angle as functions of the equivalent lastic strain Fluid flo is described by the generalized Darcy la kint kr ( ) i [, i + gρ y i ] f, µ t & dt. (8) here f is the Darcy fluid flux; k int is the intrinsic ermeability; µ is the dynamic viscosity coefficient; g is the gravity coefficient; ρ is the ater density; k r is the relative ermeability, and y is the u don coordinate.. FEM model and numerical examles To examles for a sloe ith a fault are studied. In the first one it is considered a hydrostatic ressure alied near the fault (model CON4), hile the second one is ithout such loading condition (model VER2). The geometry of the boundary value roblem and the sace discretization are shon in Fig. a. The couled hydromechanical model described in the Section 2 is used for the sloe material descrition. It is assumed that the material is isotroic. The mechanical and hydraulic boundary conditions alied are deicted in the Fig. b and are that follos: fixed horizontal dislacements (u x ) of the left and right model boundaries; fixed vertical dislacements (u у ) on the bottom boundary; for model CON4 only hydrostatic ressure is alied on art of the sloe surface attaining a value of Е5 kpa for a rescribed eriod of time. Dead eight load acts ithin the hole volume. Numerical solution of the roblem is obtained by the FE code Lagamine ULg, secially its soil version, a tool designed to handle couled roblems in geological media, [4]. Material and model arameters used in code Lagamine eq c eq 4

3 are given in the Tables and 2; arameters of the mechanical model are given in Table ; and hydraulic roerties are given in Table 2. L injection of ater L 2 A H 2 H L Fig. a Sloe geometry dimensions: L 64 m; L 2 2 m; L 2 m; H 4m H 2 2м; A8,8m. Fig. b Boundary conditions arameter Yong elastic modulus E (MPa) Poisson ratio ν Dilatancy angle (for comressive aths) ψ / ( ) c ψ e Table Secific mass ρ (kg/m ) Initial/Final Coulomb s angle (for comressive aths) ϕ c / ϕ cf ( ) Initial/Final value of cohesion c (MPa) Only if there is hardening/softening B / value 2.25 / 265 5/5./. / B c arameter Soil ermeability k (m 2 ) int Secific mass of the fluid ρ (kg/m - ) Table 2 Soil orosity n Fluid comressibility coefficient (Pa - ) χ fluid dynamic viscosity µ (Pa.s) value Е-.5.E- Е- E4 CSR SRES SRFIELD CKW CKW2 arameter S S res value. 4 4 rfield The fault is numerically modeled by means of a secial finite element, hich allos introducing ater ressure into the fault as a third unknon in the element nodes. This tye of FE is incororated into the FE code LAGAMINE and is described in [2]. The alicability of this finite element is verified in []. In both examle cases CON4 and VER2 it is assumed that the ater-level reaches height of H 2. Since there is a fixed ater level, the sloe art that is above that level is considered to be in unsaturated condition, and the sloe art belo, and a function determining CSR (Pa) that level in ater saturated condition. Hence, a function of soil-ater interaction, S ( s) relative ermeability, k ( ), have to be defined. For the model ith key ISRW8, the exression for ( s ) [2]: r S CSR2 (Pa) S reads, CSR s + CSR2 CSR S ( s) arctan +, (9) π CSR 2 s a is ore air ressure. In our case the system is oened to the air to flo out and therefore it can here, a be taken that form [2]:. For the model ith key IKW4, the exression for the relative ermeability ( ) a ( S S ) CKW res k r if S S ; k k if S < CKW2 res r r, min res ( S S ) r, field res S k has the r S, () 44

4 The body of the sloe is modeled by means of a 2D solid hydro-mechanical couled FE. The initial stresses in the sloe are obtained using an excavation technique. The simulation starts for a model ith rectangular geometry. After several stes, the requested number of finite elements is removed that finally the model geometry attains the sloe shae. Then, the calculation roceeds ith alying loading conditions on the sloe boundary that is in our case rescribing hydraulic head on the art of the sloe next to the fault. The alied ore ater ressure as a function of time is shon in Fig E- *E5 (kpa) 6.E-.E-.E+.E+ 4.E+4 8.E+4.2E+5 t (sec) Fig. 2 Alied ore ater ressure as a function of time The influence of ore ater ressure on the evolution of equivalent strain has been further studied. The evolution of the equivalent strain for case model CON4 is resented in Fig..,.2,.,.4. The results are given at four different times, namely t Е4, t 2 7Е4, t 9Е4 and t 4 Е4 seconds. The evolution corresonds to the different ater ressure alied due to the ater injection along the sloe boundary, just above the fault. The results sho an increase of the maximal value of the equivalent strain at to characteristic oints of the area under consideration the sloe toe and the fault ti. For a certain value of the ressure alied, the line connecting those oints is ossibly the most robable one along hich a strain localization line is exected to occur. This is an area here fracture can develo. For a comarison in Fig..5,.6,.7,.8 at the same time stes the evolution of equivalent strain is deicted for the model examle VER2 hen there is no alication of hydrostatic ressure. Comaring figures.-.4 and.5-.8 it is evident that the injection of ater is the reason for the develoment of a zone of high deformation near the fault ti. Fig.. Evolution of equivalent strain at t 9Е4 s,con4 Fig..4 Evolution of equivalent strain at t 4 Е4 s,con4 45

5 Fig..5 Evolution of equivalent strain at t Е4 s,ver2 Fig..6 Evolution of equivalent strain at t 2 7Е4 s, VER2 Fig..7 Evolution of equivalent strain at t 9Е4 s, VER2 Fig..8 Evolution of equivalent strain at t 4 Е4 s, VER2 4. Conclusions In both CON4 and VER2 cases the attern of equivalent strain distribution is qualitatively different. Plots in Fig. 4 of the equivalent strain ( t eq 2 & ˆ ˆ& dt ith &ˆ - strain rate deviator) occurring ithin some elements along the line beteen the sloe toe and the fault ti sho an imortant influence of the eventual increase of ater ressure along a art of the sloe surface on the strain magnitude. The increase of ore ater ressure may result from heavy rainfall and/or human activity. The fault existence increases the hazard of catastrohic destruction and subsequent landslide due to the occurrence of strain localization. ELEMENT 264 ELEMENT 268,2E-,7E- ith ater injection ithout ater injection 2,E- 2,E- ith ater injection ithout ater injection,4e-,7e- eq,e- eq,4e-,8e-,e-,5e- 8,E-4,2E ste number 5,E ste number 46

6 ELEMENT 28 ELEMENT 287,E-,2E- ith ater injection ithout ater injection,2e-,5e- ith ater injection ithout ater injection,e-,e-,e-,5e- eq eq,e- 9,E-4 9,5E-4 8,E-4 9,E-4 7,E-4 8,5E-4 6,E ste number Fig. 4 8,E ste number Acknoledgement This ork as suorted by the Bulgarian National Science Fund, Grants TN-4/ and MM- /. Partly the ork on this roblem has been done during the visit of the first author in the Deartment GeoMac of the University of Liege in the frame of the bilateral roject (BAS-CGRI) Couled THM modeling for multihase materials and robust methods for model validation and verification. References. A. Baltov, R. Charlier, M. Datcheva, F. Collin, Modelling of dynamic lastic strain, ater ressure and temerature localizations in unsaturated soils, Journal of Theoretical and Alied Mechanics, Sofia, 24, vol. 4, No. 4, FE code LAGAMINE, Manuel, Technical Notes, University of Liege, 2.. C. Guiducci, A. Pellegrino, J.P. Radu, F. Collin, R. Charlier. Numerical modeling of hydro-mechanical fracture behavior. Numerical Models in Geomechanics Eds G.N. Pande & S. Pietruszczak (Balkema-Rotterdam), Proc. NUMOG VIII Numerical Models in Geomechanics, Roma, F. Collin, Coulage thermo-hydro-mécanique dans les sols et les roches, PhD Thesis, University of Liege, 2. 47