Finite Element Simulation of Strain Localization in Unsaturated Soils Xiaoyu Song, Gregor Idinger, Ronaldo I. Borja, and Wei Wu
Abstract. We investigate the impact of spatially varying saturation on the localization properties of unsaturated soils. The approach is based on mesoscale modeling in which continuum state variables including the suction stress are allowed to vary both spatially and temporally within a soil specimen. Continuum hydromechanical simulations are conducted on a rectangular specimen of unsaturated soil using stabilized low-order mixed finite elements in plane strain condition. The results reveal important patterns of shear band development triggered by the spatially varying saturation. Keywords: coupled analysis, critical state model, mesoscale modeling, strain localization, unsaturated soil.
1 Introduction Heterogeneity in the material properties is known to enhance localized deformation in soils (Borja and Andrade 2006; Andrade and Borja 2006). It is generally known that a heterogeneous soil sample sheared at a given mean density is expected to localize into a shear band sooner than an equivalent homogeneous sample having the Xiaoyu Song Stanford University, California, USA e-mail:
[email protected] Gregor Idinger Universit¨at f¨ur Bodenkultur, Vienna, Austria e-mail:
[email protected] Ronaldo I. Borja Stanford University, California, USA e-mail:
[email protected] Wei Wu Universit¨at f¨ur Bodenkultur, Vienna, Austria e-mail:
[email protected]
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same mean density. Apart from dry density, soil heterogeneity could also manifest in the form of nonuniform grain size distribution and spatially varying degree of saturation. In this paper, we focus on the effect of spatially varying degree of saturation on the localization properties of a rectangular soil specimen loaded in plane strain compression. In order to study the impact of nonuniform saturation on the localization properties of soils, we conduct numerical simulations of boundary-value problems using a constitutive model for unsaturated soils formulated by Borja (2004). The simulations are based on mesoscale modeling of a partially saturated soil sample with imposed heterogeneity in the form of nonuniform degree of saturation. The sample is loaded in plane strain compression under a locally-drained/globally-undrained condition. The constitutive model takes into account the increase in the preconsolidation pressure with increasing suction stress (Gallipoli et al. 2003). We use a fully coupled hydromechanical model employing stabilized mixed finite elements with equal-order interpolation (bilinear) of the displacement and pore pressure fields (White and Borja, 2008). Heterogeneity in the degree of saturation is introduced into the soil sample at the mesoscopic scale, defined as a scale that is smaller than the specimen scale but larger than the particle scale. We then show that this form of heterogeneity could trigger a shear band and alter not only the deformation pattern but also the local flow of fluids inside the soil sample.
2 Hydromechanical Model The hydromechanical model is based on a mixed finite element formulation satisfying the momentum and mass conservation equations. It is presented extensively in Borja and White (2010) and will not be repeated here. It suffices to note that the model uses a u/p-formulation, where u and p are the solid displacement and pore water pressure fields, respectively. The model is an extension of Richards’ equation to accommodate the solid deformation. It is assumed that the pore air pressure remains atmospheric; however, because the soil is unsaturated, the degree of saturation Sr enters into the formulation through the effective stress equation (Bishop and Blight 1963; Borja 2006) σij = σi j + BSr pδi j , (1) where σij and σi j are the effective and total Cauchy stress tensors, respectively, B is the Biot coefficient (B = 1 is valid for soils), and δi j is the Kronecker delta. In the above equation, continuum mechanics convention is used (tension is positive) and deformation is assumed to be infinitesimal (geometrically linear). The formulation for the constitutive model, described by Borja (2004), has been implemented into a nonlinear finite element code for the analysis of boundaryvalue problems under unsaturated soil condition. Figure 1 summarizes the plasticity model. It is a two-invariant elasto-plastic constitutive model based on modified Cam-Clay plasticity, but the preconsolidation pressure pc is now a function of the suction stress through the bonding variable ξ introduced by Gallipoli et al. (2003). When the soil is fully saturated the bonding variable is zero; in this case the
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preconsolidation stress varies according to the plastic volumetric strain as in standard Cam-Clay models. However, when the soil is partially saturated the bonding variable is greater than zero, and the preconsolidation stress varies not only with the plastic volumetric strain but also with the degree of saturation through the bonding variable ξ . Mathematically, we have pc = − exp[a(ξ )](−pc )b(ξ ) ,
(2)
where a(ξ ) and b(ξ ) are real scalar functions given by Borja (2004), pc is the preconsolidation pressure at full saturation, and pc is the preconsolidation pressure in the unsaturated state. For comparison, yield surfaces in the saturated and unsaturated states are shown in Figure 1. Note from Figure 1 that both yield surfaces pass through the origin of the plane defined by the effective mean normal stress p and von Mises stress q axes. When the degree of saturation approaches 100% from the unsaturated state the two yield surfaces coincide, resulting in a mechanism called “load-collapse.”
Fig. 1. Yield surfaces for the modified Cam-Clay plasticity model enhanced to accommodate suction stress effects. CSL = critical state line. After Borja (2004).
The constitutive model has been implemented into a nonlinear finite element code using a fully implicit algorithm in both the plasticity model and the soil-water characteristic curve. The finite element code uses stabilized low-order mixed finite elements advocated by White and Borja (2008). For the 2D example discussed in this paper, four-node quadrilateral finite elements with bilinear interpolation in both the solid displacement and pore water pressure degrees of freedom have been used (equal-order pair). These stabilized elements do not suffer from mesh locking and pore pressure oscillations in the incompressible and nearly incompressible limits, and generally have the same desirable properties as the higher-order biquadraticdisplacement/bilinear-pressure mixed elements.
3 Shear Band in Unsaturated Soil We consider the finite element mesh for a plane strain soil sample shown in Fig. 2a on which the initial degree of saturation is superimposed. The mesh is 5 cm wide
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and 10 cm tall and is modeled with 200 bilinear-pair mixed finite elements. The top and bottom ends of the sample are supported on rollers with the bottom left corner fixed with a pin for stability. The bottom face is constrained from vertical movement whereas the top face is given a vertical displacement. The vertical faces are subjected to an initial pressure of 100 kPa. As for the flow equations, all faces of the sample are no-flow boundaries to simulate a globally undrained condition. The initial degree of saturation in the sample ranges from 71% to 89%, whereas the initial specific volume is uniform at 1.60. The saturation profile shown in Fig. 2a may be interpreted as resulting from sample preparation in which the soil is deposited into a mold in thin horizontal layers and sprayed with water to induce a layered degree of saturation profile. The constitutive model parameters are [see Borja (2004) for the physical meanings of the parameters]: recompression index κ˜ = 0.03, e = 0, reference pressure p = −100 kPa, conreference elastic volumetric strain εv0 0 stant shear modulus μ0 = 10, 000 kPa, slope of critical state line (CSL) M = 1.3, compressibility parameter λ˜ = 0.10, and evolution parameter N = 2.80 for the preconsolidation pressure. The isotropic hydraulic conductivity at complete saturation K = 10−7 m/s; the van Genuchten parameters are: S1 = 0.0, S2 = 1.0, sa = 10.0 kPa, n = 2.0; the fitting parameter values in the Gallipoli et al. curve are c1 = 0.185 and c2 = 1.42 (Borja 2004). The time step is Δ t = 5 s, and 100 steps were used to compress the sample to a 5% nominal axial strain.
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Fig. 2. Contours of degree of saturation in a rectangular soil sample: (a) initial saturation; (b) saturation at 5% axial strain. Note that as the air pores as squeezed the overall degree of saturation increases even as the water content remains constant.
With the degree of saturation serving as the only imperfection in the soil specimen, the sample is then compressed vertically. Figure 2b shows that after a nominal vertical strain of 5%, the two initial high-saturation zones in Fig. 2a merged to form a high-saturation band. Note that the initial irregular saturation distribution shown in Fig. 2a becomes a more regular distribution except for the high-saturation band that
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Fig. 3. Volumetric deformation at 5% axial strain: (a) specific volume; (b) volumetric strain.
separates the top and bottom portions of the soil sample. A question arises as to why such high-saturation band formed from the initial irregular saturation distribution. Figure 3 sheds some light onto the above question. The two high-saturation zones in Fig. 2a are weaker zones in the sense that their preconsolidation pressures are lower than those in the lower-saturation zones, so they yielded first to form what appears to be a developing compactive shear band (Borja and Aydin 2004), as shown in Fig. 3. The mechanical compaction along this band reduces the volume of the pores, resulting in a localized increase of the degree of saturation. As the highsaturation band is a zone of decreased preconsolidation pressure, the material inside the band weakens further, enhancing strain localization. Figure 4 shows the deformed sample at 5% nominal axial strain, along with contours of deviatoric plastic strain and localization function(Borja 2006). The figure corroborates the early statement that the high-saturation zone is a developing compactive shear band. The high saturation induces a pressure gradient that expels water from the band, as suggested by the fluid flow vectors. Note that the localization function is close to zero in the neighborhood of the band, but never becomes negative. The specimen can exhibit a shear band deformation and still not bifurcate. This simply means that the problem remains well posed and the FE solution does not suffer from mesh sensitivity even if deformation was inhomogeneous. To capture bifurcation, a constitutive model such as those presented by Borja and Andrade (2006) and Andrade and Borja (2006) may be used.
4 Closure We have presented a mesoscale modeling approach to investigate strain localization in a partially saturated soil with inhomogeneous degree of saturation. Our studies showed that a non-uniform degree of saturation could serve as an imperfection to trigger strain localization. A fully coupled hydromechanical model is required to
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LOC_FUNC X 10^7 30
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Fig. 4. Deformed sample and fluid flow vectors at 5% axial strain: (a) contour of deviatoric plastic strain; (b) contour of localization function. Arrows are fluid flow vectors.
capture the relevant multiphysics processes: the fluid flow processes drive the mechanical model, while the mechanical model triggers more fluid flow processes. Only by accounting for both processes simultaneously can one obtain a full understanding of the important mechanisms leading to strain localization in partially saturated soils. Acknowledgement. Support for this work was provided by the US National Science Foundation (NSF) under Contract Numbers CMMI-0824440 and CMMI-0936421 to Stanford University, and by Fond zur F¨orderung der wissenschaftlichen Forschung (FWF) of Austria under Project Number L656-N22 to Universit¨at f¨ur Bodenkultur.
References Andrade, J.E., Borja, R.I.: Capturing strain localization in dense sands with random density. Int. J. Numer. Methods Eng. 67, 1531–1564 (2006) Borja, R.I.: On the mechanical energy and effective stress in saturated and unsaturated porous continua. Int. J. Solids Struct. 43, 1764–1786 (2006) Borja, R.I.: Cam-Clay plasticity, Part V: A mathematical framework for three-phase deformation and strain localization analyses of partially saturated porous media. Comput. Methods Appl. Mech. Engrg. 193, 5301–5338 (2004) Borja, R.I., Andrade, J.E.: Critical strate plasticity, Part VI: Meso-scale finite element simulation of strain localization in discrete granular materials. Comput. Methods Appl. Mech. Engrg. 195, 5115–5140 (2006); Comput. Methods Appl. Mech. Engrg., 78, 49–72 Borja, R.I., Aydin, A.: Computational modeling of deformation bands in granular media, I: Geological and mathematical framework. Comput. Methods Appl. Mech. Engrg. 193, 2667–2698 (2004)
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Borja, R.I., White, J.A.: Conservation laws for coupled hydromechanical processes in unsaturated porous media: Theory and implementation. In: Laloui, L. (ed.) Mechanics of Unsaturated Geomaterials, ch. 8, pp. 185–208. ISTE Ltd. and John Wiley & Sons, Hoboken, NJ (2010) Bishop, A.W., Blight, G.E.: Some aspects of effective stress in saturated and partly saturated soils. G´eotechnique 13, 177–197 (1963) Gallipoli, D., Gens, A., Sharma, R., Vaunat, J.: An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour. G´eotechnique 53, 123–135 (2003) White, J.A., Borja, R.I.: Stabilized low-order finite elements for coupled solid-deformation/ fluid-diffusion and their application to fault zone transients. Comput. Methods Appl. Mech. Engrg. 197, 4353–4366 (2008)
