In this paper, the emphasis is on the modeling of the stress tensor of the solid particles (clay). However, for the closure of the governing equations, we also need constitutive relations for the stress tensor for the fluid, ${\mathit{\sigma}}^{f}$, and the interaction forces ${\mathit{\mathcal{M}}}^{i}$.

#### 3.1. Fluid Component and the Interaction Forces

The following assumptions, as elaborated in Martins-Costa et al. [

28] and Rajagopal [

27], would lead to the classical Darcy’s equation. The interaction forces designated by

${\mathit{\mathcal{M}}}^{i}$, within the context of mixture theory and many of the multiphase theories, are usually based on generalizing the interaction force for very special cases, such as the Stokes drag on a single spherical particle.

We assume that the frictional (viscous) forces within the fluid can be ignored and as a result the partial stress tensor for the fluid can be given by a Eulerian fluid model:

where

${p}^{f}$ is, in general, a function of density, and

$\mathit{I}$ is the identity tensor [note that compressive stresses are assumed to be negative in these theories, whereas in, geomechanics-related problems, the opposite convention is used]. We further assume, as is customary in geomechanics problems and basic flows through porous structures (see Oka and Kimoto [

29], p. 34):

The interaction force is assumed to be

where

$\alpha $ is a coefficient that can depend on porosity, viscosity, permeability, etc. This is basically a generalization of the Stokes’ drag on a single spherical particle. In general, the interaction forces could depend on other kinematical quantities such as the relative acceleration, velocity gradients, etc. (see Massoudi [

30,

31] for a review of this topic). To obtain the Darcy’s equation, we ignore the inertial effects, i.e., we ignore the left-hand side of Equation (18) (when written for the fluid component), and, by using Equations (13) and (23), we obtain the Darcy’s equation:

where

${\rho}_{f}$, as mentioned before, is the density of the fluid. Furthermore, by assuming (see Martins-Costa, et al. [

28], Williams [

32])

where

$\mathit{k}$ represents the specific permeability, which for anisotropic materials is generally a second-order tensor. Equation (26) can be re-written as

In soil mechanics literature, Darcy’s equation is sometimes expressed using the concept of hydraulic conductivity (

$\mathit{K}$) (see Bear and Bachmat (1990, p. 294)), defined by

Here,

${\rho}_{f}\mathit{g}$ represents the volumetric weight of the fluid,

${\gamma}_{f}$, which is assumed to act in the vertical direction (

${\mathit{b}}_{f}={\left[0,{\gamma}_{f},0\right]}^{T}$). Equation (28) is then re-written as:

We use this form of the equation in our finite element simulation.

#### 3.2. Solid Component

The partial stress for the solid component can be defined as (Lewis and Schrefler [

33]):

where

${\mathit{\sigma}}^{\prime}$ is the effective stress tensor and

${p}_{f}$ is the pore (fluid) pressure. We can relate the total stress tensor of the mixture

$\mathit{\sigma}$. [see Equation (19)] to the effective stress tensor

${\mathit{\sigma}}^{\prime}$ [by adding Equations (19), (24) and (31)], namely

We assume that the strain in clay can be decomposed into an elastic part and a viscoplastic part. In plasticity theory, we find it more convenient to assume this decomposition applies to for the strain rates [see Davis and Selvadurai [

34], p. 97]

We also assume that the elastic part of the strain can be represented by the “small-strain” or the linearized theory of elasticity, where, as customary in soil mechanics, the strain is assumed to depend on the effective stress (Terzaghi [

4], pp. 11–15), Schofield and Wroth ([

35], p. 9). For an isotropic material, using the index notation, the elastic strain is given by (Matsuoka and Sun [

36], p. 37)

Where in accordance with the critical state theory, we assume

$E=\frac{3\left(1-2\nu \right)\left(1+{e}_{0}\right)p}{\kappa}$ is the (modified form of the) Young’s modulus,

$\nu $ is the Poisson’s ratio,

$\kappa $ is the slope of the unloading and reloading path (see [Matsuoka and Sun [

36] (p. 35, Figure 2.8); Desai and Sriwardane [

9] (p. 289, Figure 11.7)),

$p=\frac{{\sigma}_{kk}}{3}$ is the initial mean pressure,

${e}_{0}$ is the initial void ratio, and

${\delta}_{ij}$ is the Kronecker delta (

${\delta}_{ij}=1\left(i=j\right),0\left(i\ne j\right)$). In a more compact form (using the index notation), Equation (34) can be written as

where

${S}_{ijkl}$ is the fourth-order compliance tensor, related to

${C}_{ijkl}$ the stiffness tensor. Since we have assumed that the material is isotropic, in short hand notation [recalling Hooke’s law

${\dot{\sigma}}_{kl}^{\prime}={C}_{ijkl}{\dot{\epsilon}}_{kl}^{e}$],

In Equation (35)

${\dot{\sigma}}_{kl}^{\prime}$ can be generalized to the case of an elasto-viscoplastic case (see Desai and Sriwardane [

9], p. 294, Equation 11.32), i.e.,

where

${D}_{ijkl}$ is a fourth-order tensor similar to the elastic moduli.

For the viscoplastic modeling of the strain rate, we start with Perzyna, who used the associated flow rule and assumed the plastic potential function coincides with the loading function [

20]. To apply this theory to geomaterials, the main challenge is to define the static loading function [

12]. By definition,

${f}_{s}$, represents the stress state where the strain rate is assumed to be zero. Here, we assume that the viscoplastic part of the strain rate in Equation (33) is based on Perzyna’s approach, where

In the above equations,

$\psi $ is the rate sensitivity function, and its functional form can be obtained either experimentally or theoretically; 〈 〉 is the Macaulay’s bracket (in Equation (38), the Macaulay’s bracket ensures that the function inside the bracket only has a value when it is positive, otherwise its magnitude is zero);

$F$ is the over-stress function; and

${f}_{p}$ is a new term, representing the new potential surface (surfaces of the proposed EVP model are obtained by extending the Modified Cam Clay surface [Roscoe and Burland [

21]; in our EVP model, we require a total of three surfaces (see

Figure 1): the loading surface

${f}_{l}$, the reference surface

${f}_{r}$, and the potential surface

${f}_{p}$) given by

Potential surface:

and

${f}_{l}\mathrm{and}\text{}{f}_{r}$ are given by the same expression as those in the Perzyna model. They are the dynamic loading function (the potential surface) and the static loading functions, respectively, given by

Loading surface:

where

$p={\sigma}_{oct}=\frac{{\sigma}_{kk}}{3}$,

$q=\frac{3}{\sqrt{2}}{\tau}_{oct}={\left[\frac{3}{2}{\left({\sigma}_{d}\right)}_{ij}{\left({\sigma}_{d}\right)}_{ij}\right]}^{\frac{1}{2}}$. Similar expressions for

$p$ and

$q$ (also known as the deviatoric stress) can also be found in the work by Borja and Kavazanjian [

19]. In the principal stress space,

${\sigma}_{oct}$ and

${\tau}_{oct}$ are the octahedral normal stress and the octahedral shear stress, respectively (see Matsuoka and Sun [

36], pp. 29–30). In the above equations, the suffixes

$r,\text{}l,\text{}\mathrm{and}\text{}p$ represent the reference surface, the loading surface, and the potential surface, respectively. The meaning of these surfaces is shown in

Figure 1. At any given time, the reference stress state and the reference surface are known from the laboratory test. The current stress state and the potential stress state are related to the reference stress state through the radial mapping rule, which was proposed by Phillips and Sierakowski [

37]. In this paper, we used two image parameters and the details are given in

Appendix A. It is worth mentioning that Islam and Gnanendran [

22] demonstrated the strengths and the limitations of the existing methods using the two-surface approach in the EVP models. They used the associated flow rule for their EVP model, while, in the present paper, we use the non-associated flow rule and the three-surface approach. In the previous paper, the surface shapes are two ellipses, while, in the present study, we assume all the surfaces are given by a single ellipse, which is close to the MCC surface. To formulate the new EVP model, we assume that the “projection center" is in the origin of the stress space, which is identical to the MCC model. However, to define the surfaces, the expression of the slope of the critical state line

$\left(M\right)$ is changed with respect to the

b-value

$\left(=\frac{{\sigma}_{2}-{\sigma}_{3}}{{\sigma}_{1}-{\sigma}_{3}}\right)$ [

38] as

where

$\varphi $ is the internal angle of friction at the failure for each

$b$-value test, ranging from 0 (triaxial compression) to 1 (triaxial extension).

We have introduced a new parameter

$M$ in order to obtain a more realistic surface shape in the

$\pi $-plane (see Islam and Gnanendran [

22]). It is observed that, in any stress state

$\left(0<b-value\le 1\right)$ other than the triaxial compression state

$\left(b-value=0\right)$, the MCC equivalent surface overestimates the stress. For the sake of completeness, in

Figure 2, we compare the EVP model based on the MCC surface with the new modified surface presented in this paper. It is observed that the newly extended MCC surface captures and compares well with the experimental results.

In

Figure 3, we can see that, at any arbitrary reference time

$\overline{t}$, the soil state is at “A”, where the corresponding void ratio is

$\overline{e}$. With time changing from

$\overline{t}$ to t, due to creep, the soil moves from “A” to “B” where the corresponding void ratio is

$e$. Then, the following expressions are obtained from Borja and Kavazanjian [

19]:

where

$\lambda $ and

$\kappa $ are the slope of the normal consolidation line (the

$\lambda $-line) and the unloading-reloading line (the

$\kappa $-line), respectively, and

${e}_{N}$ is the void ratio corresponding to the

$\lambda $-line when

$p=1$ kPa at

$\overline{t}$.

It should be mentioned that

$\lambda $,

$\kappa $ and

${e}_{N}$ are the necessary parameters in the EVP model; these can be obtained either from the oedometer test or the triaxial test. The meaning of these parameters is the same as those in the MCC model. As time increases, the

$\lambda $-line changes to the

$\dot{\lambda}$-line and

${e}_{N}$ is transformed to

${\dot{e}}_{N}$. The

$\dot{\lambda}$-line will generate the new bounding surface. The

$\lambda $-line and the

$\dot{\lambda}$-line are parallel, as shown by Bjerrum theory [

39]. Using Borja and Kavazanjian’s [

19] concept and the multisurface theory,

$\overline{t}$ is the arbitrary time, representing the state of stress prior to the surface evolving. In Equations (40) and (41),

${p}_{cl}$ and

${p}_{cr}$ are also known as the creep exclusive preconsolidation pressure and the creep inclusive preconsolidation pressure, respectively (see Islam and Gnanendran [

22]). The expression for

${p}_{cl}$ is similar to the one used in the MCC model:

After rearranging Equation (44), the expression for

${p}_{cr}$ can be obtained

From

Figure 3 for the definitions of

$\lambda $ and

$\kappa $, we can obtain an expression for

${p}_{cp}$:

The detailed derivation of

$\langle \psi \left(F\right)\rangle $ and

${\dot{\epsilon}}_{ij}^{vp}$ are presented in

Appendix B and

Appendix C, respectively. In closing this section, we need to mention that the overstressed EVP models are usually based on the Perzyna’s theory [

20] in combination with the critical state soil mechanics theory, e.g., the Modified Cam Clay (MCC) model [

21]. In these approaches, the viscous nature of the EVP model is introduced in the theory using a secondary compression index, a creep function and a relaxation function. However, in most cases, to reduce the complexity of the model and to minimize the number of parameters, the EVP models are developed considering the associated flow rule, where it is assumed that the yield surface is identical to the potential surface. To capture the behavior of geomaterials, using the non-associated flow rule is essential [

40]. Depending on the application of the Critical States Soil Model (CSSM) in the EVP models using the non-associated flow rule, there are two approaches we can consider: (i) those with critical state [

40]; and (ii) those without critical state [

41]. The required parameters for the EVP models, satisfying the non-associated flow rule, ranges from 7 parameters [

40] to 44 parameters [

41]. A summary of the EVP models with the non-associated flow rule for different geomaterials is presented in

Table 1.