Fuentes, W. & Triantafyllidis, Th. (2013). Ge ´otechnique 63, No. 16, 1451–1455 [http://dx.doi.org/10.1680/geot.13.T.013] 1451 TECHNICAL NOTE On the effective stress for unsaturated soils with residual water W. FUENTES  and TH. TRIANTAFYLLIDIS  In the present work an expression describing the rate of internal energy density for unsaturated soils is derived. The main difference here, with respect to preceding works, lies in the fact that the free and residual water are treated as two different phases from the soil. The latter presents intrinsic velocity equal to that of the solid phase, and is able to exchange mass with the free water phase. Under these conditions, a Bishop-type effective stress is identified as the energy-conjugated variable with the solid deformation rate, and presents a Bishop parameter equal to the effective degree of saturation. Furthermore, the negative rate of effective degree of saturation is energy-conjugated with the suction times the effective porosity. Other contributors to the internal energy come from the constituent compressibilities, the relative flow and the mass exchange between the free and residual water. KEYWORDS: numerical modelling; partial saturation; stress analysis INTRODUCTION The simulation of the mechanical behaviour of unsaturated soils requires the definition of a constitutive stress  ó able to link the solid deformation å through constitutive models. The literature offers several constitutive stresses  ó , each with different definitions. Among them, the Bishop stress ó defined as (Bishop & Blight, 1963) ó ¼ ó tot þ p a 1  ÷s1 (1) where ó tot is the total stress, p a is the pore air pressure, s ¼ p a  p w is the suction, p w is the pore water pressure, 1 is the unit tensor and ÷ is the Bishop parameter, has become frequently used in constitutive modelling (Bolzon et al., 1996; Jommi, 2000; Gallipoli et al., 2003; Wheeler et al., 2003; Borja, 2004; Sheng et al., 2004; Tamagnini, 2004; Fuentes & Triantafyllidis, 2013). This is fairly attributed to some advantages obtained when selecting a proper function for the Bishop parameter ÷ (Nuth & Laloui, 2008; Gens, 2010). Actually, the definition of the Bishop parameter ÷ (equa- tion (1)) has alone emerged from different theories. Just to mention some examples, volume averaging was used by Hassanizadeh & Gray (1979) and Lewis & Schrefler (1998), entropy inequality by Hassanizadeh & Gray (1980) and Hutter et al. (1999), Lagrangian saturation concept by Coussy (2007) and theory of porous media with their respec- tive balance equations by Coussy (2004), Ehlers & Ammann (2004), Borja (2006) and Borja & Koliji (2009). Detailed reviews of these approaches can be found in papers by Coussy (2007) and Gens (2010). Particularly, if the soil is idealised as a triphasic material composed by air, water and solid (without considering the phases’ interfaces), and the intrinsic compressibilities of the constituents are ignored, then the Bishop stress ó can be recognised as an energy- conjugated stress with the solid deformation rate _ å and the corresponding Bishop parameter is equal to the degree of saturation ÷ ¼ S w (Houlsby, 1997; Hutter et al., 1999; Borja, 2006; Borja & Koliji, 2009). The choice of ÷ ¼ S w in order to fulfil some thermo- dynamical considerations has been the subject of debate in recent years (Tarantino & Tombolato, 2005; Pereira & Alon- so, 2009; Alonso et al., 2010; Pereira et al., 2010; Vlahinic et al., 2011). The arguments are convincing when analysing its limitations, among them the fact that for soils presenting a residual degree of saturation S w0 . 0, the product sS w diverges to infinite when the suction s tends to infinite as well. The direct consequence is the need to contrarest this effect with indefinite compressive mean Bishop stress p ¼ (1=3)tró !1 under constant total stress _ ó tot ¼ 0 (see equation (1)). Considering that none of these scenarios seems realistic, many undesired restrictions are usually adopted by the modellers, such as setting S w0 ¼ 0 in hydraulic models and choosing simultaneously appropriated hydraulic para- meters in order to guarantee that the product sS w never diverges. Others prefer to avoid large suction values in their simulations. At this point, the current authors note that the conventional formulation presents serious limitations for materials with a non-negligible amount of residual water. To overcome these issues, some authors have proposed setting the Bishop parameter equal to the effective degree of saturation ÷ ¼ S e w (Tarantino & Tombolato, 2005; Pereira & Alonso, 2009; Alonso et al., 2010; Pereira et al., 2010; Lu et al., 2010) for its use in constitutive modelling; for example, see the recent models from Zhou et al. (2012) and Alonso et al. (2010). By doing this, the advantages of the Bishop stress with ÷ ¼ S w remain, and the mentioned limit- ations are eliminated. Above this, it is suggested to add the fact that with the choice ÷ ¼ S e w the relations used at the saturated state to predict the critical state surface in the stress space and the hypo-elastic behaviour of the material can be used without further extensions for unsaturated states for the whole range of degree of saturation (Tarantino & Tombolato, 2005; Alonso et al., 2010; Pereira et al., 2010). It seems that the only controversial point is that choosing ÷ ¼ S e w is not consistent with the energy-conjugated stress already identified in many works, whereby the corresponding Bishop parameter is equal to the degree of saturation ÷ ¼ S w (Houlsby, 1997; Borja, 2006; Coussy, 2007). To the authors’ knowledge, none of these theories recognises the residual Manuscript received 29 March 2013; revised manuscript accepted 2 August 2013. Published online ahead of print 23 September 2013. Discussion on this paper closes on 1 May 2014, for further details see p. ii.  Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany. Downloaded by [ Fundacion Universidad del Norte] on [28/06/16]. Copyright © ICE Publishing, all rights reserved. Journal: Géotechnique Year: 2013