Micro-mechanical modelling of unsaturated granular media

Transcription

1 Micro-mechanical modelling of unsaturated granular media L. Scholtès, B. Chareyre, F. Darve Laboratoire Sols, Solides, Structures, Grenoble, France ABSTRACT A discrete three-dimensional (3-D) poly disperse model for unsaturated granular soils has been developed. In unsaturated granular materials, presence of inter particular water introduces new interaction forces between grains, which can be quantified through capillary theory. Capillary forces depend both on physical parameters and on grain/fluid phase configuration as it is described by the Laplace-Young equation. In order to deal with water transfer mechanisms, the simplified model is suction-controlled, and consequently, reacts just as granular soil samples behave in laboratory tests. At every suction level, capillary forces are directly linked to the water content. Thus, the model is able to provide new insights into unsaturated soils behaviour under combined changes of stress and suction, such as wetting-induced collapse or increase of the soil strength, and to correlate this behaviour to the water-retention curve. INTRODUCTION Macroscopic properties of granular materials such as soils depend on interparticle contact properties. For dry materials, interparticle forces are related to the applied external stresses. In unsaturated soils, new additional elements should be defined in order to understand properly their behavior. When a soil is unsaturated (above the water-table level), presence of water between particles leads to the formation of liquid bridges (meniscus), introducing new interparticle forces which can be quantified through capillary theory and the Laplace equation. Naturally, the amplitude of these capillary forces depends upon the degree of saturation and morphology at particle level, but they usually provide a capillary suction involving a visible cohesion inside the material which can reach some hundreds of kilo Pascals. For example, this cohesion allows sand castles edification, and its particularity is that it can completely disappear during a wetting of the material (like in case of diluvian rains), which involves a cancellation of capillary forces, and hence, induces its collapse. The multi scale approach, based upon the mechanics of granular materials, or micromechanics [Cambou et Jean, 001], is an alternative tool to the phenomenological approach F 1

2 usually used to model solid materials behavior; and discrete element methods (DEM), [Cundall and Strack, 1979], allow to introduce capillary phenomenon at the local scale in order to analyze its implications at the macroscopic level. This approach consequently permits to develop macroscopic constitutive relations better funded on the micro structure, and it clearly appears that discrete simulation of granular materials constitutes a pertinent tool for studying non saturation of soils. Along these lines, we present a 3-Dimensional micro mechanical model for unsaturated granular media made of spherical particles which has the particularity to be suctioncontrolled, in order to deal with water phase changes, and simulates some laboratory s experiments on unsaturated soils. This model can be an acceptable approximation to granular soils such as medium to fine sands and silts, where suction is mostly owed by capillarity (no osmotic suction like in clays). WATER INFLUENCE: CAPILLARY COHESION Origin of the phenomenon Cohesion in granular materials has several origins, physical or chemical ones, but this paper focuses more particularly on cohesion caused by capillary forces appearing inside the material, bringing about by inter particle liquid bridges. This choice is well-funded by the prominence of the phenomenon in a lot of domains where granular media interfere, but also by the fact that it is only due to the presence of fluids (water and air in this case). This cohesion explains why some sands embankments can stay vertically. It is to be noticed that when sand is submerged, capillary forces vanish with no more cohesion; mechanism which is at the origin of landslides caused by great precipitations. Capillarity can be explained by superficial tension phenomenons which develop at the interface between two non miscible liquids or between a liquid and a gas. Liquid-gas interface behave like a stretched membrane defined by a surface tension σ, opposing the deformations. The origin of surface tension is a difference in the number and the nature of attractive interactions between molecules of a same phase. At the liquid-gas interface, molecules are subjected to an ensemble of unbalanced interaction forces, contrary to molecules situated in the liquid womb. A molecule localized inside a mass of liquid is subjected to interactions of the same nature (figure 1(a)), whereas one on the interface is under the influence of different actions: some due to gas, and others due to liquid (figure 1(b)). Molecules of the interface are consequently attracted to the liquid mass, and liquid surface is subjected to a perpendicular force which bends it : this attraction is at the origin of the surface tension σ [De Gennes et al., 00; Delage et Cui, 000]. Figure 1- acting forces on a water molecule : (a) inside a liquid mass, (b) at the liquid-gas interface. In a liquid bridge between two grains, actions of surface tension provoke the liquid-gas interface to behave like a stretched membrane, which as a consequence, maintains solid F

3 particles together. This surface tension also generates a depression inside the liquid bridge, which creates a pressure jump between the gas phase and the liquid phase. Laplace theorem (1805) expresses this pressure difference Δp = p gaz pliquide like the product of the surface tension by the mean curvature C of the liquid bridge surface: Δ p = σ C This pressure difference also contributes to the attractive action between grains. To sum up, surface tension is at the origin of the cohesion between two particles by, on one hand, the liquid-gas interface behaving like a stretched membrane, and on the other hand, the pressure difference making the two elements attracting together, this two contributions being completely linked. Description of the phenomenon at the grain scale Because of the complexity and diversity of possible shapes for grains, the granular media is assumed as an idealized media to facilitate its study. Description of the media is therefore based upon some assumptions: grains are supposed spherical by shaped and perfectly smooth (no roughness), and constituted of a unique material (glass beads). In fact, real grains surface always presents a roughness which can modified capillary forces [Bocquet et al., 00], and, related on this roughness, several variation scheme of the capillary phenomenon can be distinguish, but, as a consequence of the phenomenon complexity, this aspect is not taken into account in this study. Capillary phenomenon is considered at the scale of a capillary doublet, made of a pair of grains, linked by a liquid bridge which shape and profile are definite by the Laplace equation. The geometrical description of a capillary doublet is presented on figure. Grains of radius R1 and R are separated by a length called inter granular distance D, angles δ1 and δ represent the wetting angles corresponding to the wetting part of the solid grain. The way the grains are wet by the liquid is blatant by a contact angle marks θ. The x axis is definite like the line reaching centers of the two spheres, and constitutes a symmetrical axis for the doublet, the liquid bridge being as an axi symmetric volume which shape is described by the y(x) profile. The gorge radius y0 is definite like the smaller value of the profile. Figure Liquid bridge between two grains of different sizes (poly disperse case): (a) global view of the capillary doublet, (b) geometry of the liquid bridge. F 3

4 Obviously, in a granular assembly, particularly for high water content, several particles can be included in the meniscus and capillary interactions are then more complexes than this doublet representation. This model is, therefore, not available at every saturation degree. Laplace equation entirely describes the liquid bridge geometry (volume V, intergranular distance D), and also the inter particular force produced (F cap ), by the way of the definition of the profile y(x), through the analytical relationships (1), (1.1), (1.1.1), (1.), and (1.3): 1+ y' (x) y(x)y'' (x) Δpy(x) + σ / ( 1+ y' (x)) 3 = 0 (1) D = x Vi = 3 x c V = π y (x) dx V1 V x c1 where 1 3 πr i i ( 1 cosδ ) ( + cosδ ) i (1.1) (1.1.1) R( 1 cosδ ) xc1 R 1( 1 cosδ ) (1.) c 1 Fcap = πyoσ + πyo Δp (1.3) In this study, it appears reasonable to neglect gravitational effects considering the small volume of water introduced. By a general way, these effects can be neglected for small values of the Bond number [Pitois, 1999], definite as: L Δρg B 0 = << 1 σ where Δρ is the volumetric mass difference between the two phases constituting the interface, and L a characteristic length of the system (typically, grains diameters). In the case of soil particles linked by a liquid bridge constituted of water, surface tension σ is of the order of N/m for a temperature of 0 C, and density difference is equal to 1000 kg/m 3. If we consider grains of about 1 mm radius, then B 0 = 0., which well verifies precedent inequality. Gravity effects can then be neglected, both in term of bridge distortion than in term of applied force. Moreover, liquid bridge is studied in a quasi-static configuration, so, viscosity effects are also neglected: actually, liquid viscosity and movements between grains are supposed to be sufficiently small to neglect the dynamical force of the liquid bridge. This assumption can be verified comparing contributions of the viscosity and of the surface tension on the applied forces by the dimensionless capillary number [Mikami et al., 1998]: μν Ca = σ By definition, viscosity effects dominate when Ca > 1, and, to the contrary, surface tension effects dominate when Ca < In the case of water at 0 C, viscosity is equal to 1, Pa.s; estimating that relative speed between grains never exceeds 0,01 m/s, then Ca < 1, The assumption is already verified and, consequently, grains are considered to F 4

5 move tangentially without any viscous resistance: liquid bridges only exert a normal force between grains. DEM USED IN THIS STUDY Contact interaction A 3D code called YADE (Yet Another Dynamic Engine) has been developed based upon the work of Cundall and Strack [Cundall and Strack, 1979]. Each particle of the material (soil) mass is a sphere that is identified independently with its own mass, m, and radius, R. The deformability of the particles is taken into account via contact laws as described below. Figure 3 interaction geometry. Contact interactions are described by a linear elastic law, which defines the normal force Fn in relation with the inter granular distance (see figure 3 for the definition of the local coordinates system): Kn D si D 0 Fn = 0 si D > 0 where Kn is the contact stiffness. Friction between grains produces a shear force Ft in the tangential plane of contact, opposite to the incremental displacement du, and obeying to Coulomb friction law: dft = K tdu Ft max = μf Kt is the shear stiffness and μ the friction coefficient. At contact, energy dissipations appear due to friction forces. A damping coefficient is therefore introduced to take it into account. Interparticle force due to capillary water To consider capillarity as it interferes in granular media, it is essential to be able to link the 3 elements included in the Laplace equation: capillary pressure (equal to suction in the case of granular media), water volume and capillary force. The originality of this work is that a numerical resolution of the Laplace equation has been developed in order to have a suction-controlled model. At every time of the simulations, for a given capillary pressure, micro structure of the sample (inter granular distances) n F 5

6 automatically defines the engendered capillary forces (F cap ) and water volumes (Vw) as it is describes by equations (1) to (1.3). Figure 4 shows the capability of the model comparing its results to the exact numerical solution of Laplace equation for two grains of radius R1 = 1 mm and R = mm subjected to suctions of 50 and 100 kpa, in terms of capillary forces, distances and meniscus volume. Figure 4 Capillary Law for a doublet (R1 = 1 mm, R = mm) a) capillary force b) water volume. It can be shown that Laplace equation provides non unique solutions for a given inter granular distance. Previous works [Lian et al., 1993; Molenkamp et al., 003], have shown that, through energy considerations, only solutions with the greater water volume have to be considered, the other corresponding to an unstable state of the liquid bridge, with water-phase changes and non existing interface. F 6

7 The authors made the choice to define the appearance of a meniscus between grains strictly in contact (D creation = 0), considering perfectly smooth grains without any roughness. Besides, it is to be noted that this formulation, as a matter of course, intrinsically defines the distance from which meniscus breaks off (D rupture ), depending on the given pressure and interacting geometry. A scheme of the capillary law implemented is shown in figure 5. This hysteresis will certainly have an influence in suction variation simulations. Figure 5 Capillary law: definition of D creation and D rupture. Figure 6 displays the influence of radius ratio (r = R1 / R) on dimensionless capillary force for a constant suction. Figure 6 Influence of radius ratio on capillary forces PERSPECTIVES The current aim of this work is to study the results of this capillary when numerical samples (figure 7) are subjected to loading programs such as triaxial and suction variation F 7

8 tests (drying-wetting), in order to provide insights in the study of unsaturated materials behaviour. Actually, the model enables to directly link suction to capillary forces and water content inside the material at the micro-level. Development of homogenisation techniques will allow obtaining informations on capillary stress induced, and on suction-saturation degree curves. Figure 7 Yade numerical sample ( elements). REFERENCES L. Bocquet, E. Charlaix, F. Restagno. Physics of humid granular media. Compte rendu de physique de l Académie des Sciences, 3:07-15, 00. B. Cambou, M. Jean. Micromécanique des milieux granulaires. Hermes Sciences, Mars 001. P. A. Cundall, O. D. L Strack. A discrete numerical model for granular assemblies. Géotechnique,9,pp.47-65, P.-G. De Gennes, F. Brochard-Wyart, D. Quéré. Gouttes, bulles, perles et ondes. Collection Echelles, Belin, Paris, 00. J. A. Gili, E. E. Alonso. Microstructural deformation mechanisms of unsaturated granular soils. International Journal for Numerical and Analytical Methods in Geomechanics, 6: , 00. G. Lian, C. Thorton, M.J. Adams. A theoretical study of the liquid bridge forces between two rigid spherical bodies. Journal of Colloid an Interface Science, 161: , T. Mikami, H. Kamiya, M.Horio. Numerical simulation of cohesive powder behaviour in a fluidized bed. Chemical Engineering Science, 53(10): , F. Molenkamp, A.H. Nazemi. Interaction between two rough spheres, water bridge and water vapour. Géotechnique, 53, n, 55-64, 003. O. Pitois. Assemblées de grain lubrifiés : élaboration d un système modèle expèrimental et étude de la loi de contact. Thèse de doctorat, Ecole Nationale des Ponts et Chaussées, F. Soulié, F. Cherblanc, M. S. El Youssoufi. Influence of liquid bridges on the mechanical behaviour of polydisperse granular materials, International Journal for Numerical Methods in Geomechanics, 30:13-8, 006. F 8