Constitutive modelling of partially saturated soils: Hydro-mechanical coupling in a generic thermodynamics-based formulation
Introduction
The majority of geotechnical structures are constructed in soils well above the groundwater table which are normally partially saturated over their entire service life (Wheeler and Sivakumar, 1995; Vanapalli et al., 1999). Thus, geohazards usually involve partially saturated soils and their responses under different loading and hydraulic conditions. A constitutive model being able to predict the soil behaviour under a wide range of both mechanical loading and saturation conditions is essentially needed for the prediction of geotechnical failure involving unsaturated soils. This is a challenge given the complex and coupled hydro-mechanical interactions at the grain scale that govern the macro responses. For example, suction and surface tension along with interfaces between phases at the grain scale have long been recognised as key factors governing the macro responses (Gallipoli et al., 2003; Likos, 2014). They produce the capillary forces exerted by water menisci and affect the grain-to-grain contact behaviour (Gallipoli et al., 2003; Hoxha et al., 2007; Xie and Shao, 2006). The macro behaviour is therefore governed by the coupled hydro-mechanical mechanism due to the interaction between the two grain scale phenomena, (i) grain sliding and rearrangement, (ii) ruptures of liquid bridge and their redistributions (Bianchi et al., 2016; Mani et al., 2013). These grain-scale phenomena result in observable hydro-mechanical coupling at the continuum level that has been extensively investigated in several suction-controlled (i.e. Alonso, 1987; Cui and Delage, 1996; Wheeler and Sivakumar, 1995; Chen, 2007; Macari et al., 2003) and constant water content triaxial tests (Thu et al., 2006; Marinho et al., 2016; Rahardjo et al., 2004; Maleki and Bayat, 2012; Li, 2015; Zhang, 2016). Particularly, when suction increases, the shear strength (Vaunat et al., 2007; Toyota et al., 2001), yield limit (Alonso et al., 1990; Wheeler and Sivakumar, 1995; Cui and Delage, 1996) and dilatancy (Ng and Zhou, 2005; Cui and Delage, 1996) increases. All of the features described above are accompanied by the volume change associated with irreversible change of saturation (Sharma, 1998; Wheeler et al., 2003), both of which are induced by coupled mechanical and wetting/drying processes. This volume change is considered as one of the most fundamental properties of partially saturated soils (Sheng et al., 2008).
The above key characteristics of partially saturated soil behaviour should be reflected in a constitutive model to capture the transition between partially and fully-saturated conditions (Zhou and Sheng, 2009; Sheng, 2011). For example, the effects of suction on the stress-strain relationships have been addressed in several papers (e.g. Alonso et al., 1990; Wheeler and Sivakumar, 1995; Cui and Delage, 1996; Sun et al., 2000; Stropeit et al., 2008; Blatz and Graham, 2003; Macari et al., 2003). In particular, suction was used in a loading-collapse (LC) yield function to capture the plastic compression due to the wetting-induced collapse behaviour. The normal compression line (NCL) shifts with suction where the compression index was found to decrease (Alonso, 1987; Alonso et al., 1990; Cui and Delage, 1996; Zhang and Lytton, 2009b, Zhang and Lytton, 2009a) or increase (Wheeler and Sivakumar, 1995; Matsuoka et al., 2002; Sun et al., 2000, 2004) with increasing suction. However, the link between suction and degree of saturation was missing in these models, making it hard to reproduce the dependence of model responses on different saturation regimes (capillary in fully saturated conditions, funicular at high saturations, pendular at low saturations). For example, cohesion induced by the distributions of liquid bridges between particles are very different in the three saturation regimes (Louati et al., 2015, 2017; Wang et al., 2017). To address this, Zhou et al., 2012a, Zhou et al., 2012b and Zhou and Sheng (2015), used an NCL with soil compression index varying with effective degree of saturation, while Alonso et al. (2013) proposed a NCL dependent on both suction and saturation degree. Wheeler et al. (2003), Tamagnini (2004), Xie and Shao (2006) and Buscarnera and Nova (2009) suggested different hardening constitutive laws governing the coexistence of strain and saturation rates. These studies demonstrate that both suction and saturation degree play an indispensable role in modelling the wetting/drying-induced collapses of partially saturated soils.
The use of a SWCC independent of the volumetric behaviour has been the focus in several models (Thu et al., 2007a; Zhou and Sheng, 2009; Russell and Khalili, 2006) to reflect the effects of degree of saturation and suction on the mechanical behaviour. This kind of SWCC can facilitate the development of the models, despite neglecting its non-uniqueness observed and addressed in several papers (Nuth and Laloui, 2008; Miller et al., 2008; Gallipoli et al., 2003; Vanapalli et al., 1999; Tarantino, 2009; Mašín, 2010; Mbonimpa et al., 2006). Other models employing a SWCC dependent on volume change to have stronger interactions between hydraulic and mechanical responses have also been proposed (i.e. Buscarnera and Nova, 2009; Chiu and Ng, 2003; Jommi, 2000; Bolzon et al., 1996). Despite their successes in taking into account the hydro-mechanical coupling in the constitutive behaviour of partially saturated soils, the difference in wetting and drying paths was not taken into account in these models.
More complete representations of water retention behaviour, taking into account the irreversibility between wetting and drying processes, has been successfully addressed in several fully coupled hydro-mechanical models (e.g. Gallipoli et al., 2003; Loret and Khalili, 2000, 2002; Wheeler et al., 2003; Hu et al., 2014; Khalili et al., 2008; Muraleetharan et al., 2009; Liu and Muraleetharan, 2011a, Liu and Muraleetharan, 2011b; Sheng et al., 2008; Sun et al., 2007, 2010; Sun and Sun, 2012; Zhou et al., 2012a, Zhou et al., 2012b; 2018; Zhou and Sheng, 2015; Lloret-Cabot et al., 2017; Ghorbani et al., 2018; Gholizadeh and Latifi, 2018; Bruno and Gallipoli, 2019; Kodikara et al., 2020). These fully coupled models can capture several important features of the coupled hydro-mechanical behaviour under different loading and saturation conditions, such as irreversible swelling/shrinkage upon wetting/drying, load/deformation-dependency of capillary hysteresis, together with effects of hydraulic hysteresis on shear strength, stiffness, and dilation. Nevertheless, the identification and calibration of several parameters in these models (e.g. Bruno and Gallipoli, 2019; Gholizadeh and Latifi, 2018; Liu and Muraleetharan, 2011a, Liu and Muraleetharan, 2011b; Zhou et al., 2012a, Zhou et al., 2012b; 2018) is a challenge for their applications. To capture the wetting-drying difference in the behaviour, several models (e.g. Khalili et al., 2008; Muraleetharan et al., 2009; Liu and Muraleetharan, 2011a, Liu and Muraleetharan, 2011b; Zhou et al., 2018; Sun and Sun, 2012; Kodikara et al., 2020) adopted a separate law for hydraulic hysteresis requiring different sets of parameters for drying and wetting paths. In addition, the hydro-mechanical coupling in some models (e.g. Wheeler et al., 2003; Sheng et al., 2004; Muraleetharan et al., 2009; Sun et al., 2010; Lloret-Cabot et al., 2017; Kodikara et al., 2020) requires the use of multiple yield surfaces, e.g. one mechanical (Loading Collapse; LC) and two hydraulic (Suction Increase/Decrease; SI/SD, usually as horizontal straight lines) with complicated treatments for the coupled evolutions of all yield functions (Wheeler et al., 2003; Sheng et al., 2004). Delage and Graham (1996) and Tang and Graham (2002) found that LC and SI/SD should merge into a single yield locus to capture the micromechanical nature of coupled hydro-mechanical yielding. This issue has been investigated through the combination of drained isotropic compression and drying tests of Sivakumars and Doran (2000) and the extended Barcelona Basic model of Pedroso and Farias (2011). On the other hand, net stress-controlled experiments by Thu et al. (2007b) and Sivakumars and Doran (2000) showed that these widely adopted horizontal straight lines for SI/SD are not reasonable despite their usefulness in constitutive modelling as discussed in several papers (e.g. Delage and Graham, 1996; Robles et al., 2002; Tang and Graham, 2002; Zhang et al., 2009).
The development of thermodynamic-based approaches to constitutive modelling of partially saturated soils have attracted considerable attention in the last 15 years (Sheng et al., 2004; Tamagnini and Pastor, 2005; Uchaipichat, 2005; Santagiuliana and Schrefler, 2006; Li, 2007a, Li, 2007b; Coussy et al., 2010; Buscarnera and Einav, 2012; Dangla and Pereira, 2014; Hu et al., 2015; Lei et al., 2016). The key advantage of such developments is that all essential behavioural characteristics of the considered material can be rigorously incorporated in a thermodynamics-based model, whilst the number of arbitrary assumptions and also model parameters can be reduced without compromising the model performance. The success of such approaches to constitutive modelling has been demonstrated for not only partially unsaturated soils, but a wide range of engineering materials (Buscarnera and Einav, 2012; Liu et al., 2018; Zhang, 2017; Balieu and Kringos, 2015; Nguyen et al., 2015; Lai et al., 2016; Al-Rub and Darabi, 2012; Darabi et al., 2018). Despite the attempts and some successes, full coupling between plasticity and hydraulic irreversibility and their associated hydro-mechanical dissipation properties are usually not adequate in previous thermodynamics-based approaches. In particular, the hydraulic dissipation attributed to the irrecoverable change of saturation degree is overlooked in Tamagnini and Pastor (2005), Uchaipichat (2005), Coussy et al. (2010), Buscarnera and Einav (2012), Dangla and Pereira (2014) and Lei et al. (2016). Consequently, these models cannot naturally capture different responses under wetting and drying paths. Additionally, despite bringing the usefulness, the use of multiple yield surfaces in several models (Sheng et al., 2004; Santagiuliana and Schrefler, 2006; Hu et al., 2015) does not reflect the inseparable nature of the hydro-mechanical interactions at the grain scale, given the dissipative stresses are not dependent on the rates of all internal variables.
In this paper, a new generic thermodynamics-based approach incorporating key characteristics of partially saturated soil behaviour is developed. We aim for the establishment of a rigorous thermodynamics-based approach that leads to a good balance between rigour, simplicity, number of parameters and performance in the derived constitutive models. From the above literature review, this balance has not always been the case in existing models. The proposed thermodynamic formulation is used as a versatile means to connect all essential behavioural characteristics of partially saturated soils, resulting in a single yield surface in stress-suction space and two evolution rules for plastic strains and irrecoverable saturation with a single “plastic” multiplier. This is thanks to the inter-dependence of mechanical and hydraulic dissipations in the proposed dissipation potential, providing a smooth transition between saturated and partially saturated conditions. This coupling naturally induces the effects of mechanical behaviour on the saturation-suction relationship. As a result, models derived from this approach possess an implicitly defined SWCC dependent on the volumetric behaviour, reflecting the hydro-mechanical interactions at the grain scale. This is consistent with suggestions in Wheeler et al. (2003), based on both micro-mechanical reasoning and experiments (e.g. Sharma, 1998; Gallipoli et al., 2003), that the irreversible part of the volumetric strain, arising from the mechanical energy lost due to slippage at inter-particle contacts, results in the translation of the water retention curve. The proposed thermodynamic formulation also helps minimise the number of parameters required, while not compromising the performance of the derived models. The obtained model, formulated under infinitesimal strain assumption, possesses a small number of identifiable parameters, which have clear physical meanings and can be calibrated from standard tests on partially saturated soils. The model performance is assessed and validated against a range of experiments on partially saturated soils.
The outline of this paper is as follows. In section 2, a critical state thermo-mechanical framework for partially saturated soils is described to provide a basis for the development of constitutive models. This is followed by the formulation of a model and its dissipation properties. Section 3 describes the numerical implementation algorithms and their verification. The parameter identification and determination are presented in Section 4, followed by the validation and demonstration of the capabilities of the proposed model in Section 5.
Section snippets
A generic thermodynamics-based framework
The experimentally observed behaviour of partially saturated soils requires coupling between internal variables representing the hydro-mechanical behaviour in constitutive modelling. In this section, a generic thermodynamic approach is described to serve as a basis for the interaction between the mechanical and hydraulic responses represented by plastic strain and irrecoverable saturation, respectively. The inseparable nature of this interaction will be reflected in the proposed approach
Numerical implementation
For implementation purpose, the model descriptions can be summarized and rewritten in tensorial form as follows.
The stress-strain and suction-saturation relationships:where is the plastic strain tensor and is the pressure-dependent elastic stiffness tensor of the following form:
The yield function and evolution rules:
Model parameters
The proposed model possesses 13 parameters categorised into five groups, namely Group 1 (), Group 2 (), Group 3 (), Group 4 () and Group 5 (). This section aims to provide details on the calibration of these model parameters, using a suction-controlled triaxial test on Bourke silt (Bourke silt-SCT, Uchaipichat, 2005; Uchaipichat and Khalili, 2009). A step by step approach is presented in which parameters in each group are calibrated using relevant sets of
Model behaviour and validation
This section is to present several examples to highlight the predictive capacity of the proposed model. The model performance is assessed against experimental results of both isotropic compression and triaxial shear tests under drained and undrained conditions.
Conclusions
In the proposed generic thermodynamics-based framework for partially saturated soils and a constitutive model derived from it, we strike a good balance between rigour, simplicity, number of parameters and performance. The rigour in the formulation of the proposed generic framework guarantees the thermodynamic admissibility of any models derived from it. The obtained model described in equations (56), (57), (58), (59), (60), (61) is simple in its structure given it possesses a single yield
CRediT authorship contribution statement
Dat G. Phan: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft. Giang D. Nguyen: Supervision, Conceptualization, Methodology, Formal analysis, Writing - review & editing. Ha H. Bui: Supervision, Conceptualization, Formal analysis, Writing - review & editing. Terry Bennett: Supervision, Conceptualization, Formal analysis, Writing - review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors gratefully acknowledge support from the Australian Research Council via Discovery Projects FT140100408(Nguyen), DP170103793 (Nguyen & Bui), and DP190102779 (Bui & Nguyen).
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