Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils

Abstract

The non-unique critical state represents the distance between the critical state line (CSL) and the isotropic consolidation line (ICL) that significantly varies with stress paths and particle size distribution of soils. A structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was implemented using an explicit algorithm. However, the explicit algorithm did not well capture the non-unique critical state of soils with a large spacing ratio r, which prevented the soil mechanics research on non-unique critical state via finite element analysis. To overcome the limitation, the implicit algorithm of the SBSP-R model is formulated, and it mainly includes elastic prediction and plastic correction. The plastic correction is realized using the Newton–Simpson scheme with a controlling equation set related to consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. Case studies indicate that the implicit algorithm of the SBSP-R model is right and stable in predicting non-unique critical states. Comparisons between predicted and tested results indicate that the implicit algorithm of the SBSP-R model not only captures the critical state, stress-strain, and stress paths of various soils but also shows higher computational accuracy and efficiency compared with the previous explicit algorithm. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm.

1. Introduction

The mechanism and prediction of critical state is an important research topic in soil mechanics [1,2,3,4], especially the non-unique critical state of gap-graded soils that attracts many researchers’ attention [5,6]. The non-unique critical state is related to the distance between the critical state line (CSL) and the isotropic consolidation line (ICL), and the distance varies with stress paths or particle size distributions. So, predicting the non-unique critical state of soils is important for solving soil mechanics problems.

Over the past decades, considerable efforts have been devoted to the constitutive modeling of soils. A unified hardening model was proposed and improved to reflect the overconsolidation, anisotropy, and thermo-elasto-plastic behavior of clays [7,8,9]. A double-yield surface model was proposed and modified to investigate dilatancy, fabric anisotropy, and consolidation conditions of sands [10,11,12,13]. Several other types of constitutive models were formulated for predicting the effect of soil structure [14,15,16] and unsaturated characteristics [17,18,19]. Recently, the framework of the bounding surface plasticity model proposed by Dafalias et al. [20] has been widely developed, and it shows a robust capacity in predicting the overconsolidated soils under monotonic and cyclic loading, such as saturated clays [21,22,23], unsaturated clays [24,25], sands [26], and frozen soils [27]. The bounding surface plasticity model was also applied to solve multi-axial wave propagation problems [28] and boundary value problems [29].

However, predicting the non-unique critical state of soils is still a challenge, because the predicted distance between CSL and ICL using classical constitutive models does not vary with soil types. Lists of experimental results indicate that the distance varies with soil types, especially for gap-graded soils [5,6]. To solve the problem, a material parameter “spacing ratio r” was improved to reflect the distance between CSL and ACL (see Figure 1) [30], and a structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was formulated [31]. The SBSP-R model well captures the non-unique critical state of soils because of the spacing ratio r (see Figure 1).

Furthermore, the algorithm of the SBSP-R model is essential in capturing the non-unique critical state of soils. The SBSP-R model was once implemented using an explicit algorithm to predict the non-unique critical state and strain-softening of clays, and the limitations of the explicit algorithm are obvious: (a) it even does not capture the non-unique critical state of soils with large spacing ratio r; (b) it consumes much computation time in predicting critical state at the intersection of the distorted bounding surface and CSL (see Figure 1b). So, it is necessary to formulate an algorithm for the SBSP-R model to predict the non-unique critical state of soils. Fortunately, the implicit algorithm has been proven to be an efficient method to eliminate the limitations [32]. The implicit algorithm of the SBSP-R model may be an alternative approach to the previous explicit algorithm.

This paper aims to formulate the implicit algorithm of the SBSP-R model to efficiently predict the non-unique critical state of soils with larger spacing ratios. The implicit algorithm mainly undergoes elastic prediction and plastic correction. The plastic correction is conducted using a Newton–Simpson scheme with a list of controlling equations related to consistency condition, plastic flow, hardening parameter, bounding surface equation, plastic modulus, and mapping rule. The implicit algorithm is analyzed via case studies and is further verified by comparisons between calculated results and experimental results of triaxial compression tests on soils including Sandy clay, Kaolin clay, and Intact clayey loess.

2. SBSP-R Model

The structural bounding surface plasticity model with spacing ratio (SBSP-R model) was formulated for both soils and soft rocks [30]. It mainly contains a structural bounding surface, loading surface, mapping rule, plastic potential, and hardening modulus.

2.1. Structural Bounding Surface

The bounding surface derived by Kang et al. (2020) [31] is adopted:

$$\overline{f}={\overline{q}}^2-{\overline{M}}^2({\overline{p}}^{\prime }+{p^{\prime }}_s)({{\overline{p}}^{\prime }}_0-{\overline{p}}^{\prime })=0$$

(1)

where (${\overline{p}}^{\prime }$, $\overline{q}$) is an image stress that is the stress state on the bounding surface, ${p^{\prime }}_s$ is a structure strength that reflects the bonding among soil particles, ${{\overline{p}}^{\prime }}_0$ is a hardening parameter, and $\overline{M}$ is a stress ratio related to the spacing ratio, structure strength, and hydrostatic pressure (see Figure 2).

The original expression of $\overline{M}$ shows corners in the deviatoric plane because the expression of unified strength criterion was adopted, and it will induce singularity at the non-derivable points if Lode’s angle $\theta$ equals a specific angle ${\theta }_b$ (see Appendix A). To overcome the limitation, the original expression of $\overline{M}$ is modified as follows:

$$\overline{M}=\left(\vartheta +(1-\vartheta )({\overline{p}}^{\prime }+{p^{\prime }}_s){{\overline{p}}^{\prime }}_0{}^{-1}\right)M$$

(2)

where $M$ is the slope of the critical state line that is determined by $6\sin \phi /(3-\sin \phi )$, $\vartheta$ is a parameter directly determined by the spacing ratio $r$. The value of $\vartheta$ ($0\le \vartheta \le 1$) is obtained by:

$$\vartheta =\frac{r-\sqrt{r-1}}{(r-1)\sqrt{r-1}}$$

(3)

where the spacing ratio $r$ is related to the distance between CSL and ICL in the e-lnp′ plane. It is noteworthy that different soils show various spacing ratios [5], and one kind of soil shows various spacing ratios influenced by stress paths and particle size distribution [6,30].

2.2. Loading Surface and Mapping Rule

According to the bounding surface plasticity theory, the loading surface should be similar to the bounding surface [31]:

$$f=q^2-M^2(p^{\prime }+{p^{\prime }}_s)({p^{\prime }}_0-p^{\prime })=0$$

(4)

where ($p^{\prime }$, $q$) is a current stress that is the stress state on the loading surface; ${p^{\prime }}_0$ charges the size of the loading surface. The current stress and the image stress follow a radial mapping rule (Figure 3).

The mapping rule is essential for bounding surface plasticity models [22,23]. A radial mapping rule is expressed as follows:

$$\left\{\begin{array}{l}{\overline{p}}^{\prime }=\rho p^{\prime }\\ \overline{q}=\rho q\end{array}\right.$$

(5)

where $\rho$ ($\rho$ ≥ 1) is a scaler factor. If $\rho$ = 1, the loading surface coincides with the structural bounding surface; if $\rho$ > 1, the loading surface locates inside the structural bounding surface.

2.3. Plastic Potential

Plastic potential function adopts the following expression [31]:

$$\overline{g}={\overline{q}}^2-M^2{\overline{p}}^{\prime }({p^{\prime }}_g-{\overline{p}}^{\prime })=0$$

(6)

The incremental plastic strain is obtained by the non-associated flow rule:

$${\dot{\epsilon }}_{ij}^p=L\frac{\partial \overline{g}}{\partial {\overline{\sigma }}_{ij}}$$

(7)

The plastic multiplier $L$ is expressed as follows (Dafalias 1986 [20]):

$$L=\frac{1}{H}\frac{\partial \overline{f}}{\partial {{\overline{\sigma }}^{\prime }}_{ij}}{{\dot{\sigma }}^{\prime }}_{ij}=\frac{1}{H_b}\frac{\partial \overline{f}}{\partial {{\overline{\sigma }}^{\prime }}_{ij}}{\dot{{\overline{\sigma }}^{\prime }}}_{ij}$$

(8)

2.4. Hardening Law

The structure strength decay is obtained by [31]:

$${{\dot{p}}^{\prime }}_s=-{\dot{\epsilon }}_q^p{\epsilon }_r^{-1}{p^{\prime }}_s$$

(9)

where ${\epsilon }_r^{}$ is a reference strain, ${\dot{\epsilon }}_q^p$ is the incremental plastic deviatoric strain.

The isotropic hardening law reflecting the structural decay is obtained:

$${{\dot{\overline{p}}}^{\prime }}_0=C_p{{\overline{p}}^{\prime }}_0{\dot{\epsilon }}_p^p-{\epsilon }_r^{-1}{p^{\prime }}_s{\dot{\epsilon }}_q^p$$

(10)

where ${\dot{\epsilon }}_p^p$ is the incremental plastic volumetric strain, $C_p$ = $(1+e)/(\lambda -\kappa )$, $\lambda$ and $\kappa$ is the slope of the isotropic consolidation line and the unloading-reloading line, respectively.

2.5. Hardening Modulus and Strain Increment

The hardening modulus $H$ equals to $H_b$ + $H_l$, where $H_b$ relates to the image stress ${{\overline{\sigma }}^{\prime }}_{ij}$:

$$H_b=-C_p{{\overline{p}}^{\prime }}_0\frac{\partial \overline{g}}{\partial {\overline{p}}^{\prime }}\frac{\partial \overline{f}}{\partial {{\overline{p}}^{\prime }}_0}+{p^{\prime }}_s{\epsilon }_r^{-1}\frac{\partial \overline{g}}{\partial \overline{q}}(\frac{\partial \overline{f}}{\partial {{\overline{p}}^{\prime }}_0}+\frac{\partial \overline{f}}{\partial {p^{\prime }}_s})$$

(11)

The plastic modulus $H_l$ relates to the mapping rule in Equation (5):

$$H_l=10H_{bi}{\mathrm{OCR}}^{-2}{(\rho -1)}^2$$

(12)

where $H_{bi}$ is the initial value of $H_b$, $\mathrm{OCR}$ is the over-consolidation ratio.

The elastic strain increment is given by:

$${\dot{\epsilon }}_{ij}^e=C_{ijkl}^e{\dot{\sigma }}_{kl}$$

(13)

where $C_{ij}^e$ is the tensor of the flexibility matrix. The elastic bulk modulus and elastic shear modulus are respectively calculated by:

$$K=\frac{\nu }{\kappa }{p}^{\prime }$$

(14)

$$G=\frac{3(1-2\mu )}{2(1+\mu )}K$$

(15)

where $\nu$ is specific volume, and μ is Poisson’s ratio.

The plastic strain increment is calculated by:

$${\dot{\epsilon }}_{ij}^p=\frac{1}{H}\frac{\partial \overline{g}}{\partial {\overline{\sigma }}_{ij}}\frac{\partial \overline{f}}{\partial {\overline{\sigma }}_{ij}}{\dot{\sigma }}_{ij}$$

(16)

3. Implicit Algorithm of SBSP-R Model

Based on the implicit integration method [32], the implicit algorithm of the SBSP-R model is formulated through three steps: elastic prediction, state judgment, and plastic correction.

3.1. Elastic Prediction

Elastic prediction derived from generalized Hooke’s law is expressed as follows [32]:

$$p^{n+1}=p^n+K^{n+1}\Delta {\epsilon }_v$$

(17)

$$q^{n+1}=q^n+3G^{n+1}\Delta {\epsilon }_q$$

(18)

where n and n + 1 represent the current time and the next time increment, $\Delta {\epsilon }_v$ and $\Delta {\epsilon }_q$ are the incremental volumetric strain and incremental deviatoric strain, respectively. $K^{n+1}$ and $G^{n+1}$ are current volumetric modulus and bulk modulus, respectively.

$$K^{n+1}=v^n{p}^n/\kappa$$

(19)

$$G^{n+1}=(3/2)K^{n+1}(1-2\mu )/(1+\mu )$$

(20)

3.2. State Judgment

To judge the image stress state inside the bounding surface or not, it is necessary to calculate the $\overline{f}$. Submitting the radial mapping rule (Equation (5)) into Equation (1), the values of $\overline{f}({\rho }^n{p}^{n+1},{\rho }^n{q}^{n+1})$ are obtained. If $\overline{f}({\rho }^n{p}^{n+1},{\rho }^n{q}^{n+1})$ is smaller than 0.0, the structural bounding surface will shrink; If $\overline{f}({\rho }^n{p}^{n+1},{\rho }^n{q}^{n+1})$ is larger than 0.0, the estimated stress must be corrected via plastic correction. In such a way, the image stress should locate on the bounding surface.

3.3. Plastic Correction

(1)

Controlling equation of plastic strain.

According to Equations (6) and (7), the current incremental plastic strain is obtained by:

$${(\Delta {\epsilon }_v^p)}^{n+1}=L^{n+1}{M}^2{\rho }^{n+1}{p}^{n+1}-L^{n+1}{\rho }^{n+1}{(q^{n+1})}^2{(p^{n+1})}^{-1}$$

(21)

$${(\Delta {\epsilon }_q^p)}^{n+1}=2L^{n+1}{\rho }^{n+1}{q}^{n+1}$$

(22)

(2)

Controlling equation of hardening parameter.

According to Equations (9) and (10), the current hardening parameters are obtained by:

$$p_s^{n+1}=p_s^n-{\epsilon }_r^{-1}{p}_s^{n+1}{(\Delta {\epsilon }_q^p)}^{n+1}$$

(23)

$${\overline{p}}_0^{n+1}={\overline{p}}_0^n+C_p^{n+1}{\overline{p}}_0^{n+1}{(\Delta {\epsilon }_v^p)}^{n+1}-{\epsilon }_r^{-1}{p}_s^{n+1}{(\Delta {\epsilon }_q^p)}^{n+1}$$

(24)

(3)

Controlling equation of the bounding surface.

According to the consistency condition, the corrected image stress should locate on the bounding surface. Submitting Equation (5) into Equation (1), the following controlling equation is obtained:

$${({\rho }^{n+1}{q}^{n+1})}^2+M^2({\rho }^{n+1}{p}^{n+1}+p_s^{n+1})({\rho }^{n+1}{p}^{n+1}-{\overline{p}}_0^{n+1}){\left(\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)}^2=0$$

(25)

(4)

Controlling equation of stress correction.

Based on Equations (17), (18), (21) and (22), Current stress can be corrected by:

$$p^{n+1}=p^n+K^{n+1}({(\Delta {\epsilon }_v)}^{n+1}-{(\Delta {\epsilon }_v^p)}^{n+1})$$

(26)

$$q^{n+1}=q^n+3G^{n+1}({(\Delta {\epsilon }_q)}^{n+1}-{(\Delta {\epsilon }_q^p)}^{n+1})$$

(27)

.

(5)

Controlling equation for ${\rho }^{n+1}$.

Submitting $d\overline{p}=pd\rho +\rho dp$ and $d\overline{q}=qd\rho +\rho dq$ into Equation (1):

$${\rho }^{n+1}={\rho }^n+X{Y}^{n+1}\frac{1-{\rho }^{n+1}/k_p^{n+1}}{p^{n+1}{X}^{n+1}+q^{n+1}{Y}^{n+1}}$$

(28)

where $X^{n+1}$, $Y^{n+1}$, and $X{Y}^{n+1}$ are expressed as follows:

$$\begin{array}{l}{X}^{n+1}=M^2(2{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}-{\overline{p}}_0^{n+1}){\left[\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right]}^2+\\ \begin{array}{cc}& \end{array}2M^2(1-\vartheta )({\rho }^{n+1}{p}^{n+1}+p_s^{n+1})({\rho }^{n+1}{p}^{n+1}-{\overline{p}}_0^{n+1})\frac{1}{{\overline{p}}_0^{n+1}}\left[\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right]\end{array}$$

(29)

$$Y^{n+1}=2{\rho }^{n+1}{q}^{n+1}$$

(30)

$$X{Y}^{n+1}=L^{n+1}{H}_b^{n+1}=-\left((p_s^{n+1}-p_s^n){\left(\frac{\partial \overline{f}}{\partial p_s}\right)}^{n+1}+({\overline{p}}_0^{n+1}-{\overline{p}}_0^n){\left(\frac{\partial \overline{f}}{\partial {\overline{p}}_0}\right)}^{n+1}\right)$$

(31)

(6)

Controlling equation for $k_p^{n+1}$.

In Equation (28), $k_p^{n+1}$ is expressed as follows:

$$k_p^{n+1}=\frac{H_b^{n+1}}{H_{}^{n+1}}=\frac{-C_p^{n+1}{\overline{p}}_0^n{Z}^{n+1}+{\epsilon }_r^{-1}{p}_s^n{W}^{n+1}}{-C_p^{n+1}{\overline{p}}_0^n{Z}^{n+1}+{\epsilon }_r^{-1}{p}_s^n{W}^{n+1}+10H_{bi}^n{\mathrm{OCR}}^{-2}{\left({\rho }^{n+1}-1\right)}^2}$$

(32)

$$Z^{n+1}={\left(\frac{\partial \overline{g}}{\partial \overline{p}}\right)}^{n+1}{\left(\frac{\partial \overline{f}}{\partial {\overline{p}}_0}\right)}^{n+1}$$

(33)

$$W^{n+1}={\left(\frac{\partial \overline{g}}{\partial \overline{q}}\right)}^{n+1}\left({\left(\frac{\partial \overline{f}}{\partial {\overline{p}}_0}\right)}^{n+1}+{\left(\frac{\partial \overline{f}}{\partial p_s}\right)}^{n+1}\right)$$

(34)

where the partial derivative in Equations (33) and (34) are shown as follows:

$$\begin{array}{l}{\left(\frac{\partial \overline{f}}{\partial {\overline{p}}_0}\right)}^{n+1}=-M^2({\rho }^{n+1}{p}^{n+1}+p_s^{n+1}){\left(\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+{\overline{p}}_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)}^2+\\ \begin{array}{cc}& \end{array}-2M^2(1-\vartheta )({\rho }^{n+1}{p}^{n+1}-{\overline{p}}_0^{n+1}){({\rho }^{n+1}{p}^{n+1}+p_s^{n+1})}^2\frac{1}{{({\overline{p}}_0^{n+1})}^2}\left(\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)\end{array}$$

(35)

$$\begin{array}{l}{\left(\frac{\partial \overline{f}}{\partial p_s}\right)}^{n+1}=M^2({\rho }^{n+1}{p}^{n+1}-{\overline{p}}_0^{n+1}){\left(\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)}^2+\\ \begin{array}{cc}& \end{array}2M^2(1-\vartheta )({\rho }^{n+1}{p}^{n+1}-p_0^{n+1})({\rho }^{n+1}{p}^{n+1}+p_s^{n+1})\frac{1}{{\overline{p}}_0^{n+1}}\left(\vartheta +(1-\vartheta )\frac{{\rho }^{n+1}{p}^{n+1}+{\overline{p}}_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)\end{array}$$

(36)

$${\left(\frac{\partial \overline{g}}{\partial \overline{p}}\right)}^{n+1}=M^2({\rho }^{n+1}{p}^{n+1}+p_s^{n+1})-{\rho }^{n+1}\frac{{(q^{n+1})}^2}{p^{n+1}}$$

(37)

$${\left(\frac{\partial \overline{g}}{\partial \overline{q}}\right)}^{n+1}=2{\rho }^{n+1}{q}^{n+1}$$

(38)

It is noteworthy that the controlling Equation (32) is essential for plastic correction because calculated stress paths will not pass the critical state line if the controlling equation is not suitable. If the plastic hardening modulus related to current stress $H_l$ is replaced with a function A multiplies $H_b$, the expression of $k_p^{n+1}$ in Equation (32) is $1/(1+A)$. This should be avoided because it will cause the calculated deviatoric stress increment to be zero if the stress state comes to the CSL.

3.4. Detailed Algorithm

The detailed algorithm of the SBSP-R model is further derived and rearranged, and the procedure of the algorithm is shown in

Table 1. Procedure of the algorithm.

(i) Initialize:

Set the values of material parameters: M, λ, κ, μ, cb, εr, and r

$\mathrm{Set} \mathrm{the} \mathrm{initial} \mathrm{value} \mathrm{of} \mathrm{variables}: {\mathit{x}}_0=\left\{{\rho }^n,0,0,p^{try},q^{try},p_c^{n+1},p_s^n,0,k_p^n\right\}$

Set a residual error vector: ${\mathit{f}}^0=\left\{f_1{}^0,f_2{}^0,f_3{}^0,f_4{}^0,f_5{}^0,f_6{}^0,f_7{}^0,f_8{}^0,f_9{}^0\right\}$

Set an error value: tol

(ii) Iteration:

$\mathrm{Calculate} \left|\right|{\mathit{f}}^{(i)}\left|\right|$$, {\mathit{f}}^{(i)}=\left\{f_1{}^{(i)},f_2{}^{(i)},f_3{}^{(i)},f_4{}^{(i)},f_5{}^{(i)},f_6{}^{(i)},f_7{}^{(i)},f_8{}^{(i)},f_9{}^{(i)}\right\}$

$\mathrm{Jacobian} \mathrm{determinant} {\mathit{J}}^{(i)}$

$\mathrm{Calculate} {\mathit{x}}^{(i+1)}$$\mathrm{in} \mathrm{each} \mathrm{step}: {\mathit{x}}^{(i+1)}={\mathit{x}}^{(i)}-{({\mathit{J}}^{(i)})}^{-1}{\mathit{f}}^{(i)}$

$\mathrm{If} \left|\right|{\mathit{f}}^{(i+1)}\left|\right|$$\ge \mathrm{tol}, i=i+1$, Continue (ii)

$\mathrm{If} \left|\right|{\mathit{f}}^{(i+1)}\left|\right|$ < tol, Goto (iii)

$\left(\mathrm{iii}\right) \mathrm{Stress} \mathrm{and} \mathrm{variables} \mathrm{update}: {\mathit{x}}^{(i+1)}$

.

(i)

Plastic correction using a Newton–Simpson scheme:

$$\left\{\begin{array}{c}{f}_1={({\rho }^{n+1}{q}^{n+1})}^2+M^2\left({\rho }^{n+1}{p}^{n+1}+p_s^{n+1}\right)\left({\rho }^{n+1}{p}^{n+1}-{\overline{p}}_0^{n+1}\right){\left(\vartheta +\left(1-\vartheta \right)\frac{{\rho }^{n+1}{p}^{n+1}+p_s^{n+1}}{{\overline{p}}_0^{n+1}}\right)}^2\hfill \\ f_2={(\Delta {\epsilon }_v^p)}^{n+1}-M^2{L}^{n+1}{\rho }^{n+1}{p}^{n+1}+L^{n+1}{\rho }^{n+1}{(q^{n+1})}^2{(p^{n+1})}^{-1}\hfill \\ f_3={(\Delta {\epsilon }_q^p)}^{n+1}-2L^{n+1}{\rho }^{n+1}{q}^{n+1}\hfill \\ f_4=p^{n+1}-p^n-K^{n+1}\left({(\Delta {\epsilon }_v)}^{n+1}-{(\Delta {\epsilon }_v^p)}^{n+1}\right)\hfill \\ f_5=q^{n+1}-q^n-3G^{n+1}\left({(\Delta {\epsilon }_q)}^{n+1}-{(\Delta {\epsilon }_q^p)}^{n+1}\right)\hfill \\ f_6=p_s^{n+1}-p_s^n+{\epsilon }_r^{-1}{p}_s^{n+1}{(\Delta {\epsilon }_q^p)}^{n+1}\hfill \\ f_7={\overline{p}}_0^{n+1}-{\overline{p}}_0^n-C_p^{n+1}{\overline{p}}_0^{n+1}{(\Delta {\epsilon }_v^p)}^{n+1}+{\epsilon }_r^{-1}{p}_s^{n+1}{(\Delta {\epsilon }_q^p)}^{n+1}\hfill \\ f_8={\rho }^{n+1}-{\rho }^n-X{Y}^{n+1}\left(1-{\rho }^{n+1}/k_p^{n+1}\right){\left(p^{n+1}{X}^{n+1}+q^{n+1}{Y}^{n+1}\right)}^{-1}\hfill \\ f_9=k_p^{n+1}-\frac{-C_p^{n+1}{\overline{p}}_0^{n+1}{Z}^{n+1}+{\epsilon }_r^{-1}{p}_s^{n+1}{W}^{n+1}}{-C_p^{n+1}{\overline{p}}_0^{n+1}{Z}^{n+1}+{\epsilon }_r^{-1}{p}_s^{n+1}{W}^{n+1}+10H_{\mathit{bi}}^{n+1}{\mathrm{OCR}}^{-2}{\left({\rho }^{n+1}-1\right)}^2}\hfill \end{array}\right.$$

(39)

(ii)

In the formula set, $\left[{\rho }^{n+1},{(\Delta {\epsilon }_v^p)}^{n+1},{(\Delta {\epsilon }_q^p)}^{n+1},p^{n+1},q^{n+1},p_0^{n+1},p_s^{n+1},L^{n+1},k_p^{n+1}\right]$ is considered as unknown, and its Jacobian determinant J is expressed as follows:

$$J=\left[\frac{\partial f_i}{\partial x_j}\right]$$

(40)

where its elements are shown in Appendix B.

(iii)

Initial vector

$$\begin{array}{ll}{\mathit{x}}^0& =\left\{{\rho }^{n+1},{(\Delta {\epsilon }_v^p)}^{n+1},{(\Delta {\epsilon }_q^p)}^{n+1},p^{n+1},q^{n+1},p_c^{n+1},p_s^{n+1},L^{n+1},k_p^{n+1}\right\}\hfill \\ & =\left\{{\rho }^n,0,0,p^{try},q^{try},p_c^{n+1},p_s^n,0,k_p^n\right\}\hfill \end{array}$$

(41)

Set a residual error vector

$${\mathit{f}}_0=\left\{f_1{}^0,f_2{}^0,f_3{}^0,f_4{}^0,f_5{}^0,f_6{}^0,f_7{}^0,f_8{}^0,f_9{}^0\right\}$$

(42)

(iv)

Calculate $\left|\right|\mathit{f}\left|\right|$

$$\mathit{x}={\mathit{x}}^0-{\mathit{J}}^{-1}\mathit{f}$$

(43)

where, if $\left|\right|\mathit{f}\left|\right|\le \mathrm{tol}$, exit iteration; if $\left|\right|\mathit{f}\left|\right|>\mathrm{tol}$, continue iteration until $\left|\right|\mathit{f}\left|\right|\le \mathrm{tol}$.

(v)

Parameters and stress update

$$\left\{\begin{array}{l}{v}^{n+1}=v^n(1-\Delta {\epsilon }_v^{n+1})\\ K^{n+1}=v^{n+1}{p}^{n+1}/\kappa \\ G^{n+1}=(3/2)((1-2\mu )/(1+\mu ))K^{n+1}\end{array}\right.$$

(44)

4. Influence of Material Parameters

4.1. Model Parameters

Seven material parameters need to be determined in the SBSP-R model including M, λ, κ, μ, c_b, ε_r, and r. M, λ, κ, μ, and c are the same as in classical constitutive models [18]. M is the slope of CSL in the p′-q plane, λ is the slope of the ICL in the e-lnp′ plane, κ is the slope of the unloading-reloading line in the e-lnp′ plane, μ is Poisson’s ratio, c_b is the cohesive force. r is the spacing ratio, and ε_r is a reference strain [31]. The value of M is determined by a drained or undrained triaxial shearing test. The values of λ and κ are determined via an isotropic consolidation test with a loading-unloading-reloading path. μ generally takes 0.3. The cohesive force c_b is obtained via a direct shearing or triaxial shearing test. The reference strain ε_r is determined according to the evolution of structural strength. The spacing ratio r is determined via triaxial compression tests of samples under different confining pressures.

4.2. Case Studies

A list of case studies is conducted to show the influence of the parameters (see

Table 2. Material parameters for case studies.

Case	M	λ	κ	μ	εr	cb/kPa	r	Results
Case1	1	0.15	0.01	0.28	0.01	0, 10, 20	2.0	Figure 4
Case2	1	0.15	0.01	0.28	0.02, 0.01, 0.001	20	2.0	Figure 5 and Figure 6
Case3	1	0.15	0.01	0.28	0.01	0	1.5, 2.0, 2.5	Figure 7
Case4	1	0.15	0.01	0.28	0.01	10	2.0	Figure 8

), and the accuracy of the implicit algorithm of the SBSP-R model is analyzed.

(a)

Influence of structure strength ${p^{\prime }}_s$.

The implicit algorithm of the SBSP-R model reflects the influences of the structure strength ${p^{\prime }}_s$ on stress-strain and strength of normally consolidated samples (see Figure 4). The residual strength decreases as ${p^{\prime }}_s$ increases when OCR = 1, while the peak strength is not obviously influenced by ${p^{\prime }}_s$. The sample with a larger ${p^{\prime }}_s$ (c_b = 20 kPa) shows strain-softening behavior, and the sample with a smaller ${p^{\prime }}_s$ (c_b = 0 kPa) shows strain-hardening behavior. This observation is in accordance with experimental tests on intact soil with bonding [33]. Reconstituted soils without bonding prefer to show strain-hardening [6]. Note, that the strain increment in the implicit algorithm does not influence the calculated results while the results from the explicit algorithm are affected by strain increment.

(b)

Influence of reference strain ε_r.

The reference strain ε_r does not influence the residual strength but the peak strength (see Figure 5). For normally consolidated samples, the peak strength increases with increasing the reference strain ε_r. The peak strength and the residual strength of overconsolidated samples are merely influenced by the reference strain ε_r. This is because the reference strain ε_r controls the decay rate of the structure strength ${p^{\prime }}_s$, according to Equation (9). A larger ε_r leads to a smaller structure decay rate, otherwise, a smaller ε_rcauses a faster decay rate (see Figure 6). These simulations indicate that it is necessary to consider the structure strength decay in the constitutive model of a structured soil.

(c)

Influence of spacing ratio r.

The values of r reflect the “distance” between CSL and ICL. That is, the critical state (point A1) is close to the initial state (point A0) if the spacing ratio r = 1.5, while it is far away from the initial state (point A3 and A0) if the spacing ratio r = 2.5 (see Figure 7). When the spacing ratio r > 2.5, the implicit algorithm of the SBSP-R model has advantages in computation time and accuracy as it shows higher accuracy and consumes less computation time. The consumed computation time decreases with the increase in strain increment (see Figure 8a), and the stress-strain and stress paths are stable (see Figure 8b). The previous explicit algorithm is obviously influenced by the strain increment. The previous explicit algorithm will consume much computation time if the strain increment is small, and its results are not right if the strain increment is 1.0 × 10⁻⁵, such as point A in Figure 8a and point B in Figure 8b. Compared with the previous explicit algorithm, the implicit algorithm runs well, even the strain increment is as large as 2.0 × 10⁻³.

5. Validation of the Implicit Algorithm of the SBSP-R Model

5.1. Sandy Clay

Triaxial compression tests on the sandy clay are adopted to validate the accuracy and efficiency of the implicit algorithm of the SBSP-R model with the parameters shown in

Table 3. Material parameters for Sandy clay.

M	λ	κ	μ	εr	cb/kPa	r
1.18	0.063	0.009	0.25	0.03	0	2.3

. The values of M, λ, κ, and μ are taken from the thesis by Gens [34]. The values of ε_r, c_b, and r are determined according to the reference [31].

As shown in Figure 9, the critical state, stress-strain and strength of the sandy clay are not captured if r = 2.0 (MCCM) or r = 2.7 (CCM). These behaviors are well captured by the implicit algorithm of the SBSP-R model with r = 2.3, which reflects the real “distance” between CSL and ICL (see Figure 1). Besides, the implicit algorithm requires less computation time and shows higher accuracy than the explicit algorithm.

5.2. Kaolin Clay

Triaxial compression tests on Kaolin Clay are adopted for verifying the implicit algorithm of the SBSP-R model with parameters shown in

Table 4. Material parameters for Kaolin clay.

M	λ	κ	μ	εr	cb/kPa	r
1.05	0.14	0.05	0.3	0.03	0	2.9

, the values of M, λ, κ, and μ are set according to the thesis [35], and the values of ε_r, c_b, and r are determined according to Section 4.1 and the test results from the thesis.

As shown in Figure 10, the critical state of Kaolin Clay is well captured by the implicit algorithm of the SBSP-R model with r = 2.9, compared with r = 2.0 (MCCM) or r = 2.7 (CCM). That is, point A′ corresponding to r = 2.9 is close to the final stress state of Kaolin clay, while point A″ corresponding to r =2.0 is far away from the final stress state (see Figure 10f). This is because r = 2.9 is determined by the triaxial compression test reflecting the real “distance” between CSL and ICL. While r = 2.0 and r = 2.7 are, respectively, taken from MCCM and CCM they cannot reflect the real “distance” between CSL and ICL of Kaolin Clay. Besides, the implicit algorithm requires less time and shows high accuracy in capturing the critical state of Kaolin clay, especially when the spacing ratio r is large than 2.5.

5.3. Intact Clayey Loess

Intact clayey loess is hard to be predicted because of the bonding and particle size distribution [33]. Particle size distribution influences the non-unique critical state, and bonding damage affects the strain-softening of intact clayey loess. The critical state and strain-softening of intact clayey loess are predicted using the implicit algorithm of the SBSP-R model (see

Table 5. Material parameters for Intact clayey loess.

M	λ	κ	μ	εr	cb/kPa	r
1.3	0.14	0.0134	0.3	0.03	30	3.43

).

The critical state and stress-strain are well captured by the present model with r = 3.43 and ${p^{\prime }}_s$ = 47.5 kPa (see Figure 11), because the spacing ratio r = 3.43 is obtained from the triaxial compression test. r = 3.43 reflects the real location of the critical state in the v-lnp’ plane. The behaviors are not captured in the case of r = 2.0 and ${p^{\prime }}_s$ = 47.5 kPa, because r = 2.0 is just suitable for Cam-clay soils.

When r = 2.0 and ${p^{\prime }}_s$ = 0 kPa, the present model degenerates to the modified Cam-clay model. When r = 2.7 and ${p^{\prime }}_s$ = 0 kPa, the present model degenerates into the Cam-clay model. They are further compared in Figure 12. The experimental data are better predicted by the present model as the real spacing ratio of intact loess (r = 3.43). This indicates that the structural strength and spacing ratio should be both considered in a constitutive model to capture the critical state, stress-strain, and stress paths of soils with a bonding structure. Besides, the implicit algorithm requires less time and shows high accuracy in predicting the mechanical behaviors of Intact clayey loess, especially when the spacing ratio r is larger than 3.0.

6. Summary and Conclusions

To overcome the limitation of the explicit algorithm in predicting the critical state of soils, the implicit algorithm of the bounding surface plasticity model with the spacing ratio (SBSP-R model) is formulated and verified. The main conclusions are shown as follows:

The implicit algorithm of the SBSP-R model is formulated through three steps: elastic prediction, state judgment, and plastic correction which is the most important step. The plastic correction is conducted using the Newton–Simpson scheme with controlling equations derived from consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. The implicit algorithm of the SBSP-R model is right and stable in predicting the non-unique critical state of soils. In the implicit algorithm, the controlling equation of the plastic hardening modulus is essential for plastic correction in overconsolidated conditions, since the calculated stress paths will not pass the critical state line if it is not suitable. Especially, the plastic hardening modulus $H_l$ related to the current stress cannot be a homogeneous linear function with an independent variable $H_b$ that is the plastic hardening modulus related to image stress.

The implicit algorithm of the SBSP-R model is further verified by comparing simulated results with experimental data on soils with different spacing ratios. On one hand, the implicit algorithm of the SBSP-R model is not obviously influenced by strain increment in predicting non-unique critical states, especially when the spacing ratio is large. On the other hand, the implicit algorithm can reproduce the non-unique critical state and stress-strain of various soils, as well as show higher computational accuracy and efficiency. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm.