# Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils
## 3 Implicit Algorithm of SBSP-R Model
### 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. \begin{table}
\begin{tabular}{l} \hline (i) Initialize: \\ Set the values of material parameters: \(M\), \(\lambda\), \(\kappa\), \(\mu\), \(c_{b}\), \(\epsilon_{r}\), and \(r\) \\ Set the initial value of variables : \(\alpha_{0}=\left\{\rho^{n},0,0,p^{trg},q^{trg},p_{c}^{n+1},p_{s}^{n},0,k_{p}^{n}\right\}\) \\ Set a residual error vector: \(\hat{f}^{0}=\left\{f_{1}{}^{0},f_{2}{}^{0},f_{3}{}^{0},f_{4}{}^{0},f_{5}{}^{0}, f_{6}{}^{0},f_{7}{}^{0},f_{8}{}^{0},f_{9}{}^{0}\right\}\) \\ Set an error value: tol \\ (ii) Iteration: \\  Calculate \(\left|\left|\hat{f}^{(i)}\right|\right|\), \(\left|f^{(i)}\right|=\left\{f_{1}{}^{(i)},f_{2}{}^{(i)},f_{3}{}^{(i)},f_{4}{}^{ (i)},f_{5}{}^{(i)},f_{6}{}^{(i)},f_{7}{}^{(i)},f_{8}{}^{(i)},f_{9}{}^{(i)}\right\}\) \\  Jacobian determinant \(\boldsymbol{f}^{(i)}\) \\  Calculate \(\boldsymbol{x}^{(i+1)}\) in each step : \(\boldsymbol{x}^{(i+1)}=\boldsymbol{x}^{(i)}-\left(\boldsymbol{f}^{(i)}\right)^ {-1}\boldsymbol{f}^{(i)}\) \\  If \(\left|\left|\hat{f}^{(i+1)}\right|\right|\geq\text{ tol}\), \(i=i+1\), Continue (ii) \\  If \(\left|\left|\hat{f}^{(i+1)}\right|\right|<\text{tol}\), Goto (iii) \\ (iii) Stress and variables update : \(\boldsymbol{x}^{(i+1)}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Procedure of the algorithm. \[\left\{\begin{array}{l}\text{(i)}\quad\text{Plastic correction using a Newton-Simpson scheme:}\\ \\ \left\{ \begin{array}{l}f_{1}=\left(\rho^{n+1}q^{n+1}\right)^{2}+M^{2}\big{(} \rho^{n+1}p^{n+1}+p_{s}^{n+1}\big{)}\Big{(}\rho^{n+1}p^{n+1}-\overline{p}_{0}^ {n+1}\Big{)}\bigg{(}\theta+(1-\theta)\frac{\rho^{n+1}p^{n+1}+p_{s}^{n+1}}{ \overline{p}_{0}^{n+1}}\bigg{)}^{2}\\ f_{2}=\left(\Delta\varepsilon_{q}^{p}\right)^{n+1}-M^{2}Ln^{n+1}\rho^{n+1}p^{n+1 }+L^{n+1}\rho^{n+1}\big{(}q^{n+1}\big{)}^{2}\big{(}p^{n+1}\big{)}^{-1}\\ f_{3}=\left(\Delta\varepsilon_{q}^{p}\right)^{n+1}-2L^{n+1}\rho^{n+1}q^{n+1}\\ f_{4}=p^{n+1}-p^{n}-K^{n+1}\Big{(}\left(\Delta\varepsilon_{q}\right)^{n+1}- \left(\Delta\varepsilon_{q}^{p}\right)^{n+1}\Big{)}\\ f_{5}=q^{n+1}-q^{n}-3G^{n+1}\Big{(}\left(\Delta\varepsilon_{q}\right)^{n+1}- \left(\Delta\varepsilon_{q}^{p}\right)^{n+1}\Big{)}\\ f_{6}=p_{s}^{n+1}-p_{s}^{n}+\varepsilon_{r}^{n}p_{s}^{n+1}\left(\Delta \varepsilon_{q}^{p}\right)^{n+1}\\ f_{7}=\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}\\ f_{8}=\rho^{n+1}-\rho^{n}-X\gamma^{n+1}\Big{(}1-\rho^{n+1}/k_{r}^{n+1}\Big{)} \big{(}p^{n+1}X^{n+1}+q^{n+1}Y^{n+1}\big{)}^{-1}\\ f_{9}=k_{p}^{n+1}-\frac{-C_{p}^{n+1}\overline{p}_{0}^{n+1}Z^{n+1}+ \varepsilon_{r}^{-1}P_{s}^{n+1}W^{n+1}}{-C_{p}^{n+1}\overline{p}_{0}^{n+1}Z^{ n+1}+\varepsilon_{r}^{-1}P_{s}^{n+1}W^{n+1}+10H_{n}^{n+1}{\rm COR}^{-2}\big{(}\rho^{n+1}-1 \big{)}^{2}}\\ \text{(ii)}\quad\text{In the formula set, }\Big{[}\rho^{n+1},\left(\Delta \varepsilon_{v}^{p}\right)^{n+1},\left(\Delta\varepsilon_{q}^{p}\right)^{n+1},p^{n+1},q^{n+1},p_{0}^{n+1},p_{s}^{n+1},L^{n+1},k_{p}^{n+1}\Big{]}\text{ is considered as unknown, and its Jacobian determinant \(\mathbf{J}\) is expressed as follows:}\\ \\ J=\left[\frac{\partial f_{i}}{\partial x_{j}}\right]\end{array} \right. \tag{40}\]  
where its elements are shown in Appendix B. \[ \begin{array}{l}\mathbf{x}^{0}\quad=\Big{\{}\rho^{n+1},\left(\Delta \varepsilon_{v}^{p}\right)^{n+1},\left(\Delta\varepsilon_{q}^{p}\right)^{n+1},p^{n+1},q^{n+1},p_{c}^{n+1},p_{s}^{n+1},L^{n+1},k_{p}^{n+1}\Big{\}}\\ \quad=\Big{\{}\rho^{n},0,0,p^{trg},q^{trg},p_{c}^{n+1},p_{s}^{n},0,k_{p}^{n}\Big{\}} \end{array}