A new constitutive model of unsaturated soils using bounding surface plasticity (BSP) and a non-associative flow rule

The mechanics of unsaturated soils started to develop from the 1960s. The first milestone for comprehensive unsaturated soil modeling can be attributed to [1], who have developed the Barcelona basic model (BBM) by extending the modified Cam-Clay model of saturated soils. Then, following the same direction, several works based on the classic Cam Clay model in a critical state framework have been developed (e.g. [2‐4]). However, the main drawback of these models is their difficulty to reproduce with accuracy a smooth stress–strain relation at pre-yield and post-yield, pre-peak and post-peak, as well as the volumetric behavior of soils (in particular the smooth transition from contractancy to dilatancy).

One option to improve the limits of classic models is the concept of Bounding surface Plasticity (BSP). This theory, firstly proposed by Dafalias [5] and Dafalias and Herrmann [6] to simulate metal behaviors, was applied by Bardet [7] on saturated sands and later on followed by Gajo and Muirwood [8] as well as by Yu and Khong [9]. The use of the BSP theory to model the behavior of unsaturated soils was originally introduced by Russell and Khalili [10]. These authors also took into account grain breakage, which occurs at high stresses. However, the determination of the model parameters is quite difficult, which prevents its practical application for ordinary loading conditions.

The aim of this paper is to develop a new constitutive model which can remove some short comings of classic elastic–plastic models and to reproduce complex volumetric behaviors by using the BSP theory with a non-associative plastic flow rule, but with a limited number of parameters which are easily measurable.

In this section, some general aspects of the bounding surface concept are represented. Contrary to classic plasticity, the BSP theory introduces a non-zero plastic strain rate even inside the bounding surface (analogous but not identical to the classic yield surface), with its amplitude increasing as the current stress point approaches the bounding surface. This is achieved by making the plastic modulus to depend on the distance δ between the current stress point σ′ and an arbitrarily defined image stress point, \( \bar{\varvec{\sigma }}^{\prime } \) located on the Bounding surface (see Fig. 1). This definition leads to a smoothing of the stress–strain curve from small to large strains, reproducing more realistically experimental observations. It exists different way to define the image stress, \( \bar{\varvec{\sigma }}^{\prime } \). Among them, the “radial mapping” method introduced by Dafalias [11] is the simplest and the most widely used. It consists of extrapolating the position vector linking the origin to the current stress until it intersects the image stress on the bounding surface as illustrated in Fig. 1.

The model developed in this study, called the CASMNS model, is based on the CASM-b model [12] for saturated soils: use of the effective stress, same form of the bounding surface, and non-associate law with the same form of plastic potential.

We consider the cylindrical symmetry corresponding to the classical triaxial test. Under this consideration, only two variables are required to define the strain state. In this study, we use the classic Cam Clay variables which are the volumetric strain, noted ε _v , and the equivalent deviatoric strain, noted ε _q :where \( \varvec{e} =\varvec{\varepsilon}- {\mathbf{1}} \times \varepsilon_{v} /3 \) is the deviatoric strain tensor, 1 is the second order identity tensor, and ε is the total strain tensor which is supposed to be the sum of an elastic and a plastic component:where elastic strain, noted \( \varvec{\varepsilon}^{e} \), and plastic strain, noted \( \varvec{\varepsilon}^{p} \).

Similarly, the stress state can be defined by the mean effective stress, p′ and the equivalent deviatoric stress q:where \( \varvec{s} =\varvec{\sigma}^{\prime } - {\mathbf{1}} \times {p}^{\prime } \) is the deviatoric stress tensor and \( \varvec{\sigma}^{\prime } \) is the effective stress tensor that can be expressed as Dangla and Coussy [13]:where σ is the total stress tensor, and π the equivalent pore pressure and 1 is the second order identity tensor. From the several definitions that have been proposed for π (e.g. [13‐15]), we chose to follow the expression proposed by Dangla and Coussy:where s is the suction and S_r is the saturation degree.

Note that this definition is not simply a weighted average of the bulk liquid and gaseous phases, but also accounts for the tangential forces at the liquid–gas interface arising from surface tension effects. Its computation requires information on the soil water retention curve (SWRC), which is the function linking suction s to the degree of water saturation S_r. For the sake of simplicity, the hydraulic hysteresis is neglected here and the following bijective relation proposed by Brooks and Corey [16] is assumed. Note that S_r remains constant equal to unity for suctions below the so-called air entry suction:where s_a is the air-entry suction and α is a material constant.

Altogether, 13 material constants are required to define completely this model. Note that 8 of these parameters can be determined by using experimental data. We used the knowledge of the water retention curve to obtain the parameters α and the air-entry suction s _a in Brooks and Corey’s relation. Furthermore, the deformability parameters κ, υ, λ₀ can be determined from classic oedometric and triaxial consolidation tests at full saturation. The oedometric test at different controlled suctions has been used to deduce the parameter k₁ defining the reduction of compressibility λ (s) with suction. The parameters M and υ can be determined using two triaxial test at different controlled suctions. The remaining parameters: h, m, n, r have to be indirectly determined by successive iteration using results from the triaxial and oedometric tests. Recall that the BBM model requires 12 parameters, taking into account the invariance of CSL in the (p′, q) plane as pointed out by Khalili and Khabbaz [19] and Nuth and Laloui [20]. Comparatively, this model has only one additional parameter while it uses a substantially more complicated plastic driver based on bounding surface plasticity, with a non-associative flow law to achieve a much better prediction on general stress–strain behaviour in general and a more precise description of volumetric behavior in particular.

This section presents a few numerical examples of simulation in order to show the applicability and the quality of the model. Figure 3 shows the flow chart for the implementation of this model in MatLab. In the following paragraphs, this computer code is applied to analyze the behavior of a soil sample subject to different loading paths.

A new bounding surface plasticity model is presented in this study. Of fundamental importance in practice, this model developed can be used to describe a wide range of unsaturated soils such as sand, silt, etc. Numerical results performed show that the model can reproduce the essential feature of unsaturated soil behavior including suction induced hardening, the possibility of wetting-collapse. They also underline its abilities to remove some short coming of classic elastic–plastic models, in particular its ability to simulate progressive transition from pre- to post-peak behavior, involving a smooth change of stress–strain variations from pre- to post-yield, as well as complex volumetric behavior.