# Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils
## 3 Implicit Algorithm of SBSP-R Model
### Plastic Correction  
(1) Controlling equation of plastic strain. According to Equations (6) and (7), the current incremental plastic strain is obtained by:  
\[\left(\Delta\varepsilon_{v}^{p}\right)^{n+1}=L^{n+1}M^{2}\rho^{n+1}p^{n+1}-L^{n+ 1}\rho^{n+1}{\left(q^{n+1}\right)^{2}}{\left(p^{n+1}\right)^{-1}} \tag{21}\]  
\[\left(\Delta\varepsilon_{q}^{p}\right)^{n+1}=2L^{n+1}\rho^{n+1}q^{n+1} \tag{22}\]  
(2) Controlling equation of hardening parameter. According to Equations (9) and (10), the current hardening parameters are obtained by:  
\[p_{s}^{n+1}=p_{s}^{n}-\varepsilon_{r}^{-1}p_{s}^{n+1}{\left(\Delta\varepsilon_ {q}^{p}\right)^{n+1}} \tag{23}\]  
\[\overline{p}_{0}^{n+1}=\overline{p}_{0}^{n}+C_{p}^{n+1}\overline{p}_{0}^{n+1} \left(\Delta\varepsilon_{v}^{p}\right)^{n+1}-\varepsilon_{r}^{-1}p_{s}^{n+1}{ \left(\Delta\varepsilon_{q}^{p}\right)^{n+1}} \tag{24}\]  
(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:  
\[\left(\rho^{n+1}q^{n+1}\right)^{2}+M^{2}(\rho^{n+1}p^{n+1}+p_{s}^{n+1})(\rho^{ n+1}p^{n+1}-\overline{p}_{0}^{n+1})\Bigg{(}\theta+(1-\theta)\frac{\rho^{n+1}p^{n+1 }+p_{s}^{n+1}}{\overline{p}_{0}^{n+1}}\Bigg{)}^{2}=0 \tag{25}\]  
(4) Controlling equation of stress correction. Based on Equations (17), (18), (21) and (22), Current stress can be corrected by:  
\[p^{n+1}=p^{n}+K^{n+1}(\left(\Delta\varepsilon_{v}\right)^{n+1}-\left(\Delta \varepsilon_{v}^{p}\right)^{n+1}) \tag{26}\]  
\[q^{n+1}=q^{n}+3G^{n+1}(\left(\Delta\varepsilon_{q}\right)^{n+1}-\left(\Delta \varepsilon_{q}^{p}\right)^{n+1}) \tag{27}\]  
(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):  
\[\rho^{n+1}=\rho^{n}+XY^{n+1}\frac{1-\rho^{n+1}/k_{p}^{n+1}}{p^{n+1}X^{n+1}+q^{ n+1}Y^{n+1}} \tag{28}\]  
where \(X^{n+1}\), \(Y^{n+1}\), and \(XY^{n+1}\) are expressed as follows:  
\[ \begin{split} X^{n+1}=M^{2}(2\rho^{n+1}p^{n+1}+p_{s}^{n+1}- \overline{p}_{0}^{n+1})\bigg{[}\phi+(1-\theta)\frac{\rho^{n+1}p^{n+1}+p_{s}^{n +1}}{\overline{p}_{0}^{n+1}}\bigg{]}^{2}+\\ 2M^{2}(1-\theta)(\rho^{n+1}p^{n+1}+p_{s}^{n+1})(\rho^{n+1}p^{n+ 1}-\overline{p}_{0}^{n+1})\frac{1}{\overline{p}_{0}^{n+1}}\bigg{[}\phi+(1- \theta)\frac{\rho^{n+1}p^{n+1}+p_{s}^{n+1}}{\overline{p}_{0}^{n+1}}\bigg{]} \end{split}  \tag{29}\]  
\[Y^{n+1}=2\rho^{n+1}q^{n+1} \tag{30}\]  
\[XY^{n+1}=L^{n+1}H_{b}^{n+1}=-\Bigg{(}(p_{s}^{n+1}-p_{s}^{n})\Bigg{(}\frac{ \partial\overline{f}}{\partial p_{s}}\Bigg{)}^{n+1}+(\overline{p}_{0}^{n+1}- \overline{p}_{0}^{n})\Bigg{(}\frac{\partial\overline{f}}{\partial\overline{p} _{0}}\Bigg{)}^{n+1}\Bigg{)} \tag{31}\]  
(6) Controlling equation for \(k_{p}^{n+1}\). In Equation (28), \(k_{p}^{n+1}\) is expressed as follows:  
\[k_{p}^{n+1}=\frac{H_{b}^{n+1}}{H^{n+1}}=\frac{-C_{p}^{n+1}\overline{p}_{0}^{n}Z^{ n+1}+\epsilon_{r}^{-1}p_{s}^{n}W^{n+1}}{-C_{p}^{n+1}\overline{p}_{0}^{n}Z^{n+1}+ \epsilon_{r}^{-1}p_{s}^{n}W^{n+1}+10H_{bi}^{n}\text{OCR}^{-2}{\left(\rho^{n+1}- 1\right)}^{2}} \tag{32}\]  
\[Z^{n+1}=\left(\frac{\partial\overline{\delta}}{\partial\overline{p}}\right)^{ n+1}\left(\frac{\partial\overline{f}}{\partial\overline{p}_{0}}\right)^{n+1} \tag{33}\]  
\[W^{n+1}=\left(\frac{\partial\overline{\delta}}{\partial\overline{q}}\right)^{ n+1}\left(\left(\frac{\partial\overline{f}}{\partial\overline{p}_{0}}\right)^{n+1}+ \left(\frac{\partial\overline{f}}{\partial p_{s}}\right)^{n+1}\right) \tag{34}\]  
where the partial derivative in Equations (33) and (34) are shown as follows:  
\[