## 4 Integration Algorithm
 \tag{43}\]  
The relationship between \(\partial\text{\it{in}}^{\text{\it trial}}_{n+1}/\partial\left(\epsilon^{\epsilon }_{n+1}\text{\it trial}-\epsilon_{n+1}^{T}\right)\) and \(\partial\Delta\lambda_{n+1}/\partial\left(\epsilon^{\epsilon}_{n+1}\text{\it trial }-\epsilon_{n+1}^{T}\right)\) is derived through differentiation of Equation (39) with respect to \(\left(\epsilon^{\epsilon}_{n+1}\text{\it trial}-\epsilon_{n+1}^{T}\right)\) as follows:  
\[\frac{\partial}{\partial\left(\epsilon^{\epsilon}_{n+1}\text{\it trial}- \epsilon_{n+1}^{T}\right)}\left(\text{\it{in}}^{\text{\it trial}}_{n+1}-\Delta \lambda_{n+1}(3G+\left(\sum\limits_{ \begin{subarray}{c}n\
u=1\\ n+1\end{subarray} }^{3}C^{\text{\tiny{pr}}}/1+\Delta\lambda_{n+1}\sum\limits_{ \begin{subarray}{c}n\
u=1\\ n+1\end{subarray} }^{3}\gamma^{\text{\tiny{pr}}}\right)\right)-\sigma_{y}\right) = 0 \tag{44}\]  
Through algebraic manipulation and rearranging the Equation (44), the following result is yielded:  
\[\frac{\partial\text{\it{in}}^{\text{\it trial}}_{n+1}}{\partial\left(\epsilon^{ \epsilon}_{n+1}\text{\it trial}-\epsilon_{n+1}^{T}\right)} = \left(3G+\sum\limits_{ \begin{subarray}{c}n\
u=1\\ n+1\end{subarray} }^{3}C^{\text{\tiny{pr}}}/\left(1+\sum\limits_{ \begin{subarray}{ c}n\
u=1\\ n+1\end{subarray} }^{3}\gamma^{\text{\tiny{pr}}}\Delta\lambda_{n+1}\right)^{2}\right) \frac{\partial\Delta\lambda_{n+1}}{\partial\left(\epsilon^{\epsilon}_{n+1} \text{\it trial}-\epsilon_{n+1}^{T}\right)} \tag{45}\]  
where:  
\[ \begin{array}{l}\frac{\partial\Delta\lambda_{n+1}}{\partial\left(\epsilon^{ \epsilon}_{n+1}\text{\it trial}-\epsilon_{n+1}^{T}\right)}=\sqrt{3/2}\mathbf{N} _{n+1}(2G)/3G+\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}C^{\text{\tiny{pr}}}/\left(1+\sum\limits_{\begin{subarray}{ c}n\
u=1\\ n+1\end{subarray}}^{3}\gamma^{\text{\tiny{pr}}}\Delta\lambda_{n+1}\right)^{2}-( \sqrt{3/2}\mathbf{N}_{n+1}\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}\mathbf{X}_{n\prime}\\ \left(\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}\gamma^{\text{\tiny{pr}}}\right)^{2}\right)=\sqrt{3/2} \mathbf{N}_{n+1}\mathbf{X}\end{array}  \tag{46}\]  
Substituting Equation (45) into Equation (46) gives:  
\[ \begin{array}{l}\frac{\partial\text{\it{in}}^{\text{\it trial}}_{n+1} \text{\it trial}}{\partial\left(\epsilon^{\epsilon}_{n+1}\text{\it trial}- \epsilon_{n+1}^{T}\right)}=\sqrt{3/2}\mathbf{N}_{n+1}(2G)\left(3G+\sum\limits_{ \begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}C^{\text{\tiny{pr}}}/\left(1+\sum\limits_{\begin{subarray}{c}n \
u=1\\ n+1\end{subarray}}^{3}\gamma^{\text{\tiny{pr}}}\Delta\lambda_{n+1}\right)^{2} \right)/3G+\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}C^{\text{\tiny{pr}}}\\ \left(1+\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}\gamma^{\text{\tiny{pr}}}\Delta\lambda_{n+1}\right)^{2}- \left(\sqrt{3/2}\mathbf{N}_{n+1}\sum\limits_{\begin{subarray}{c}n\
u=1\\ n+1\end{subarray}}^{3}\mathbf{X}_{n\prime}\left(\sum\limits_{\begin{subarray}{c}n \
u=1\\ n+1\end{subarray}}^{3}\gamma^{\text{\tiny{pr}}}\right)^{2}\right)\right)=\ 2G\sqrt{3/2}\mathbf{N}_{n+1}\end{array}  \tag{47}\]  
Using Equations (43), (46) and (47) with simple manipulation, the following expression for the applied constitutive model is derived as:  
\[\mathbf{W}^{\text{\tiny{pr}}}=2G\left(1-\frac{3G\Delta\lambda_{n+1}}{\text{\it{ in}}_{n+1}^{\text{\it trial}}}\right)\mathbf{I}_{d}+(K)\mathbf{I}\otimes\mathbf{I}-(3G \mathbf{\chi})\mathbf{N}\otimes\mathbf{N}+\left(\frac{6G^{2}\Delta\lambda_{n +1}}{\text{\it{i}}_{n+1}^{\text{\it trial}}}\mathbf{v}\right)\tilde{\mathbf{N}} \otimes\mathbf{N} \tag{48}\]  
where \(\mathbf{j}_{n+1}^{\text{\it trial}}\),\(K\), \(\mathbf{\chi}\),\(\mathbf{N}\) and \(\mathbf{v}\) were defined in Equations (37), (28), (46), (32) and (47). The stress integration method applied for the present elastic-plastic constitutive equations through the user-defined material subroutine (UMAT) is summarized and presented in Figure 3.