[29] The development of constitutive modelling of unsaturated soils in the framework of elastoplasticity originated from the first generalised hardening model proposed by *Alonso et al.* [1990]. The term generalised is adopted here to highlight that hardening of unsaturated soils depends not only on plastic strains, but on variables describing water content too. Following *Alonso et al.* [1990], suction is adopted most frequently for the description of the hardening mechanisms, but other choices may be equally acceptable [*Gens*, 1996; *Gens et al.*, 2006]. Moreover, suction itself is not sufficient to describe an unsaturated state for a given soil. To complete the description, a measure of the soil water content must be provided, at least in between the state parameters of the soil [*Wheeler*, 1996]. Besides, the role of hysteresis of the water retention mechanisms on the mechanical behavior of unsaturated soils should be considered [*Tamagnini*, 2004; *Sheng et al.*, 2004; *Nuth and Laloui*, 2008].

[30] Notwithstanding the many developments in constitutive modelling of unsaturated soils, only few efforts were devoted to the study of the shape of the yield function and of the plastic potential of anisotropic compacted soils. To this respect, the work by *Cui and Delage* [1996] must be mentioned. To model the initial anisotropic fabric of a compacted silt, they proposed an analytical expression for the yield function, in the mean net stress – deviator stress plane, describing an elastic limit centred on an axis rotated with respect to the hydrostatic axis. Suggestion of a non-associated flow rule was motivated by the elaboration of the experimental data in the superposed net stress – plastic strain increment planes. The mechanical behavior of the compacted soil was then modelled only by assuming a change in the dimensions of the yield locus as a function of volumetric plastic strains and suction, with no further modification of the inclination of the initial rotated yield surface.

[31] An elastic-plastic model with generalised isotropic and rotational hardening depending on plastic strains and on degree of saturation was proposed by *Jommi and di Prisco* [1994]. Rotational hardening was a basic capability of the original model previously conceived by di Prisco [*di Prisco*, 1992; *di Prisco et al.*, 1993] for saturated sands, but *Jommi and di Prisco* did not exploit this feature of the model, as at that time no experimental data were available to appreciate the relevance of anisotropy on the strain response. Anyway, to the authors' knowledge, no other model allowing for rotational hardening has been published so far for unsaturated soils.

#### 5.1. Conceptual Basis of the Coupled Elastic-Plastic Model

[32] In the model presented in the following, the analytical expressions of the yield function and of the plastic potential, the hardening laws and the number of parameters were kept as simple as possible. Its conceptual bases stem from the proposal by *Jommi and di Prisco* [1994]. A modified Cam Clay model, with associated flow rule, was adopted instead of the original non-associated *di Prisco*'s [1992] model, to simplify its calibration for Boom clay. Although simple, modified Cam Clay usually proves to be an effective choice as reference model for compacted clayey soils in saturated conditions, and only four parameters, coming from conventional saturated tests, are needed for its calibration.

[33] As in the original proposal by *Jommi and di Prisco* [1994], the yield function and the plastic potential for unsaturated conditions are analytical functions of the skeleton stress. This choice allows for a few advantages in formulating the constitutive model, as discussed by *Jommi* [2000] and most recently by *Nuth and Laloui* [2008]. Among them, the most important one is that the effects of hysteretic water retention mechanisms may be built naturally into the constitutive formulation. To the latter aim, the hysteretic retention curve proposed by *Romero and Vaunat* [2000] is introduced in the coupled hydromechanical model, in order to reproduce cyclic wetting and drying paths correctly [*Tamagnini*, 2004]. A hysteretic retention curve will prove to be quite important in modelling the mechanical response of the soil undergoing drying and wetting paths, as the following numerical simulations will demonstrate with reference to the experimental data discussed.

[34] A mixed isotropic-rotational hardening law depending on plastic strain and degree of saturation is introduced to model the soil structure evolution. Again, simple choices were made for the analytical functions adopted, to limit to a minimum the number of new parameters introduced to describe the behavior of the compacted soil in unsaturated conditions.

#### 5.2. Hydraulic Model: The Water Retention Domain

[35] Recent trends in the interpretation and modelling the coupled hydromechanical behavior of unsaturated soils suggest that knowledge of the water retention properties of a soil is of vital importance not only to describe properly the changes in its water content, but also to complete the description of its actual stress state in unsaturated conditions [*Jommi*, 2000; *Sheng et al.*, 2004; *Nuth and Laloui*, 2008].

[36] Both literature data [*Ng and Pang*, 2000; *Kawai et al.*, 2000; *Gallipoli et al.*, 2003b; *Tarantino and Tombolato*, 2005] and the experimental data presented in the previous section show that irreversible strains undergone by a soil along different hydromechanical stress paths affect its water retention properties. Nevertheless, as a first approximation, in the following it will be assumed that the water storage capacity of the soil investigated could be described by a unique hysteretic retention curve in the suction-degree of saturation plane.

[37] As the dependence of the water retention characteristics on irreversible strains is disregarded, the retention properties of the soil samples may be described by means of data coming from different data sets. Figure 10 shows the main wetting and drying branches of the water retention domain of the soil based on experimental data obtained for a constant void ratio of *e*_{0} = 0.93, hence with reference to the as-compacted condition. Data points at *s* > 3 MPa were obtained from different samples using vapour equilibrium technique. Data points at *s* ≤ 0.45 MPa were determined using axis translation technique under constant volume conditions [*Romero*, 1999].

[38] Water retention data may be fitted to a modified form of *van Genuchten* [1980] expression for the degree of saturation, *S*_{r}, as a function of suction, *s*, [*Romero and Vaunat*, 2000]:

Parameters *n*, *m* and *α* describe the shape of the retention functions, as in the original van Genuchten's expression. Hysteresis is introduced by defining two different sets of parameters for the main drying branch and the main wetting branch of the retention domain, respectively. The inverse of parameter *α*_{D} (for main drying) is related to the air-entry value of the soil in a drying path, while the inverse of *α*_{W} (for main wetting) is related to the air occlusion pressure in a wetting path. The original van Genuchten's expression is not adequate to fit retention data for clayey soils at high values of suction. For this reason, the expression was modified by a correction function *C*(*s*), which makes the curve tend to a linear relationship between the logarithm of *s* and *S*_{r} in the high-suction zone, with intersection parameter *a* at *S*_{r} = 0.

[39] Inside the domain delimited by the main wetting and the main drying branches, the hydraulic behavior is assumed to be reversible, with a simple linear relationship between suction and degree of saturation. The slope of the scanning curves, ∂*S*_{r}/∂*s* = *l*, is given a constant value. Fitted parameters for the description of the water retention domain of Boom clay are listed in Table 1.

Table 1. Parameters of the Retention Curves According to Equation (1)Parameter | Wetting Branch | Drying Branch | Scanning Curves |
---|

*a* (MPa) | 300 | 400 | — |

*α* (MPa^{−1}) | 19.3 | 1.40 | — |

*n* | 1.12 | 0.95 | — |

*m* | 0.20 | 0.41 | — |

*l* (MPa^{−1}) | — | — | 0.02 |

#### 5.3. Mechanical Model: Plastic Potential, Yield Function and Hardening Laws

[40] The experimental program was mainly devoted to the study of the strain response of Boom clay under drying and wetting paths. No attempts were made to determine the actual shape of the current yield locus. Hence modelling was concentrated primarily on the evolution of plastic potential. In the development of models with rotational hardening for saturated clays some authors adopted associated flow rules [see, e.g., *Dafalias*, 1986; *Korhonen and Lojander*, 1987; *Wheeler et al.*, 2003], while others suggested different analytical functions for the plastic potential and the yield locus [see, e.g., *Newson and Davis*, 1996; *Dafalias et al.*, 2003, 2006]. In the absence of specific experimental data on compacted Boom clay, and following a common choice in unsaturated soil modelling, the yield function was assumed to coincide with the plastic potential. This choice may pose some limitations to the capabilities of the present model. Nevertheless, normality could be easily removed conceptually if the need for distinguishing between yield function and plastic potential should arise.

[41] Although some degree of anisotropy is expected for the elastic behavior too, the adoption of an isotropic elastic law is rather common in constitutive modelling of anisotropic soils [see, e.g., *di Prisco et al.*, 1993; *Wheeler et al.*, 2003; *Dafalias et al.*, 2006]. This choice is usually justified by the assumption that elastic strains are small with respect to plastic strains, and do not affect the overall strain response in a relevant way. Here the classical non-linear elastic law adopted in critical state models, described by the logarithmic compliance, κ, and a constant shear modulus, *G*, is introduced.

[42] By just substituting the soil skeleton stress for saturated effective stress in the original form of modified Cam Clay [*Roscoe and Burland*, 1968], a convenient isotropic yield function for an unsaturated clayey soil may be written in the form:

In the previous equation, _{0} is the preconsolidation pressure in unsaturated conditions, and parameter *M*, describing the slope of the critical state line, is assumed to be independent of suction. Although the latter assumption is still a matter of debate, many experimental results suggest that it can hold true (see, e.g., the data collection presented by *Nuth and Laloui* [2008]).

[43] To allow for the description of an anisotropic response, a rotation of the axis of the yield surface and plastic potential must be introduced. The proposal by *Dafalias* [1986] was followed here and the analytical expression of the yield surface and of the plastic potential adopted finally reads:

In equation (3), *M*_{α} represents the inclination of the current rotated yield surface with respect to the axis.

[44] The two internal variables, _{0} and *M*_{α}, govern the isotropic and rotational hardening, respectively, provided convenient hardening laws are defined. Figure 11 shows the shape of the rotated yield surface in the triaxial skeleton stress plane together with the geometrical interpretation of the two internal variables. In the figure, the plastic strain increment vector associated with an isotropic stress state at yield is drawn. The direction of the vector clearly shows that even adopting an associated flow rule, due to the rotation of the axis of the yield surface, a shear strain increment will be predicted for a purely isotropic stress increase at null deviator stress. Hence the previous choices, although rather simple, allow for modelling the distortional strain observed along an isotropic loading as a consequence of the anisotropic soil behavior.

[45] Taking advantage of the suggestion by *Gallipoli et al.* [2003a], the preconsolidation pressure in unsaturated conditions, _{0}, is defined as the sum of the preconsolidation pressure in saturated conditions, *p**_{0}, depending on volumetric plastic strains, plus a term also depending on the degree of saturation:

In the previous expression, *b*_{1} and *b*_{2} are model parameters ruling the dependence of preconsolidation pressure on the degree of saturation. For *p**_{0}, the classical critical state evolution law is adopted:

where *λ* and κ are the elastic-plastic and elastic logarithmic volumetric compliances, respectively, and *e* is the void ratio.

[46] Rotational hardening is assumed to be governed by the angle between the current obliquity, = *q*/, and the current inclination *M*_{α} of the yield surface:

as originally proposed by *Dafalias* [1986]. The dimensionless constant, *c*, governs the rate of evolution of *M*_{α}, while *ξ* controls the target value of *M*_{α} for a given obliquity, i.e., d*M*_{α} = 0 when *ξ* = /*M*_{α}.

[47] Although more advanced evolution laws have been proposed in the last years – an updated review is provided by *Dafalias et al.* [2006] –, the assumed rotational law has the advantage to allow for a direct calibration of the constants on the basis of relatively simple laboratory tests.

#### 5.4. Mechanical Model: Calibration of the Parameters

[48] The model is defined in terms of eight material parameters. Four of them, *M*, *λ*, κ and *G*, describe the behavior of the isotropic soil under saturated conditions, and may be calibrated on the basis of conventional laboratory tests performed on saturated samples.

[49] The parameters *M* and *G* could be obtained from standard triaxial tests on saturated samples. A collection of values for the two parameters for various samples of Boom clay is presented in the report by *Horseman et al.* [1987]. The elastic shear modulus does not play a prominent role in the overall behavior. A value of *G* = 40 MPa was adopted based on experimental data from *Coop et al.* [1995]. As for the slope of the critical state line, a collection of consolidated-undrained triaxial data coming both from *Horseman et al.* [1987] and from tests run at Universitat Politècnica de Catalunya, on undisturbed samples of Boom clay from Mol, is presented in Figure 12. Two limit envelopes are shown in the figure for the data set. The lowest one, with a slope of *M* = 0.87, was adopted, as it is thought to be more representative of the behavior of Boom clay at critical state.

[50] The logarithmic elastic and elastic-plastic compliances, κ and *λ*, were determined from experimental data coming from a wider experimental investigation on compacted Boom clay [*Romero*, 1999]. Values for the two compliances as a function of suction are presented in Figure 13. As in the present model the values for saturated conditions are needed, *λ* = 0.125 and κ = 0.014 were chosen.

[51] As for the parameter *ξ*, the experimental data associated with static compaction under oedometer conditions (path *AB* in Figure 2) could be exploited, following a similar way of reasoning as in the study of *Wheeler et al.* [2003]. Knowing the initial suction, *s*_{0}, the initial degree of saturation, *S*_{r0}, and the net stress coefficient of earth pressure at rest, *K*_{0}, from the experimental data, the obliquity _{0} at the end of compaction in the (, *q*) plane could be easily calculated and was found to be _{0} = 0.509. Assuming that, at least at the end of compaction, plastic strain increment could be assimilated to total strain increment, the initial value of the rotational parameter, *M*_{α0}, could be determined on the basis of the stress dilatancy law obtained by derivation of the plastic potential (equation (3)), and observing that under oedometer conditions the ratio between volumetric and deviatoric strain increments must be equal to 3/2:

A value of *M*_{α0} = 0.343 was determined. Finally, by assuming that at the end of compaction, the direction of plastic strain increments were not changing any more, hence *dM*_{α0} = 0, from equation (6), the value of *ξ* = _{0}/*M*_{α0} = 1.484 could be determined.

[52] Only the two parameters *b*_{1} and *b*_{2}, strictly related to the evolution of the preconsolidation pressure with degree of saturation, need experimental data coming from tests run on samples in unsaturated conditions for calibration. To this aim, the experimental data presented in the study of *Romero* [1999] were exploited to determine possible ranges for the two parameters. Afterwards, they were optimised by numerical simulation of the isotropic experimental test (Figure 8) on the basis of the evolution of volumetric plastic strain along the first wetting path alone.

[53] In principle, parameter *c*, ruling the velocity of rotational hardening, could be determined with reference to saturated samples too. As no experimental data from saturated samples were available for calibration, the material parameter *c* was calibrated on the basis of the data presented in Figure 8, by minimising the difference between observed and simulated deviatoric strains (as shown in Figure 15).

[54] The complete set of parameters adopted in the simulations presented in the next paragraphs is listed in Table 2.

Table 2. Material Parameters Adopted in the Numerical Simulations | *G* (MPa) | *λ* | *M* | *ξ* | *b*_{1} | *b*_{2} | *c* |
---|

0.014 | 40 | 0.125 | 0.87 | 1.484 | 0.11 | 8.2 | 136 |