# Application of Dimension Extending Technique to Unified Hardening Model
## 3 Incremental Constitutive Integration
### The Constitutive Integral of the UH Model under Different Paths  
Based on the above analysis, we can achieve the calculation of constitutive integrals for 2D and 3D UH models under different paths. Taking the calculation of unit with equal \(p\) and \(b\) paths in a three-dimensional principal stress space as an example, the derivation process is detailed below.  
In the three-dimensional principal stress space, the unknown quantities are:  
\[\boldsymbol{s}^{T}\triangleq(\sigma_{1},\sigma_{2},\sigma_{3},\epsilon_{1}, \epsilon_{2},\epsilon_{3};\widetilde{H}) \tag{76}\]  
and plastic multiplier \(\Lambda\), eight unknown quantities in total.  
The Equation (47) only has five independent equations, and the solution is not closed, so three additional equations need to be added. \(\dot{\epsilon}_{1}\) as an undiminished quantity of a process can always be specified, and then the unknown quantity to be solved is:  
\[\boldsymbol{s}^{\prime T}\triangleq(\sigma_{1},\sigma_{2},\sigma_{3}, \epsilon_{2},\epsilon_{3};\widetilde{H}) \tag{77}\]  
For the path conditions of equal \(p\) and \(b\), the equation that needs to be supplemented is:  
\[\left\{\begin{array}{l}(\dot{\sigma}_{2}-\dot{\sigma}_{3})-b(\dot{\sigma}_{ 1}-\dot{\sigma}_{3})=0\\ (\dot{\sigma}_{1}+\dot{\sigma}_{2}+\dot{\sigma}_{3})=0\end{array}\right. \tag{78}\]  
Then, the backward Euler integration is applied to Equation (78).  
\[\left\{\begin{array}{l}(\Delta\sigma_{2}-\Delta\sigma_{3})-b(\Delta\sigma_{ 1}-\Delta\sigma_{3})=0\\ \Delta\sigma_{1}+\Delta\sigma_{2}+\Delta\sigma_{3}=0\end{array}\right. \tag{79}\]  
\(\epsilon_{2},\epsilon_{3}\) and \(\widetilde{H}\) in \(\boldsymbol{s}^{\prime}\) are used as initial variables along with stress and solved simultaneously. A better convergence with dimensional processing is required, and then the initial vector is:  
\[\boldsymbol{s}^{\prime T}\triangleq(\sigma_{1},\sigma_{2},\sigma_{3},\xi_{2}, \xi_{3};\widetilde{\Omega}) \tag{80}\]  
with  
\[\xi_{2}=K_{n}\Delta\epsilon_{2} \tag{81}\]  
\[\xi_{3}=K_{n}\Delta\epsilon_{3} \tag{82}\]  
\(\widetilde{\Omega}=K_{n}\widetilde{H}\) (83)  
\(K_{n}\) is the bulk modulus at the initial moment of this time step.  
Then \(\text{MiCP}(\boldsymbol{f}_{E},f_{1})\) is rewritten as:  
\[f_{E}(\Lambda;\boldsymbol{S}^{\prime})=\left\{\begin{array}{l}\boldsymbol{ \sigma}+\boldsymbol{\sigma}_{p}-\boldsymbol{\sigma}_{c}\\ \widetilde{\Omega}_{n+1}-\widetilde{\Omega}_{n}-\Delta\widetilde{\Omega}\\ (\Delta\sigma_{2}-\Delta\sigma_{3})-b(\Delta\sigma_{1}-\Delta\sigma_{3})\\ \Delta\sigma_{1}+\Delta\sigma_{2}+\Delta\sigma_{3}\end{array}\right. \tag{84}\]\[f_{1}(\Lambda,\vec{\textbf{{S}}}^{\prime})=-\widetilde{F}(\widetilde{\textbf{{ \sigma}}}) \tag{85}\]  
with \(\widetilde{\textbf{{S}}}^{\prime}\):  
\[\widetilde{\textbf{{S}}}^{\prime T}\triangleq(\widetilde{\textbf{{\sigma}}}_{1}, \widetilde{\textbf{{\sigma}}}_{2},\widetilde{\textbf{{\sigma}}}_{3},\xi_{2}, \xi_{3};\widetilde{\Omega}) \tag{86}\]  
Equations (84) and (85) are \(\boldsymbol{f}_{E}\) and \(f_{I}\) in the MiCP.  
For other different paths, corresponding equations can also be supplemented based on the path conditions, such as equal \(b\) with undrained path.  
If it is an equal \(b\) with an undrained path, then the equations are:  
\[\left\{\begin{array}{l}(\dot{\sigma}_{2}-\dot{\sigma}_{3})-b(\dot{\sigma}_{ 1}-\dot{\sigma}_{3})=0\\ (\dot{\epsilon}_{1}+\dot{\epsilon}_{2}+\dot{\epsilon}_{3})=0\end{array}\right. \tag{87}\]  
If it is a confining pressure path with equal \(b\), the equations are:  
\[\left\{\begin{array}{l}(\dot{\sigma}_{2}-\dot{\sigma}_{3})-b(\dot{\sigma}_{ 1}-\dot{\sigma}_{3})=0\\ \dot{\sigma}_{3}=0\end{array}\right. \tag{88}\]  
The above provides several common control equations for different paths in three-dimensional principal stress space.  
Alternatively, we can also embed the algorithm into the finite element calculations, and then the boundary value problems can be solved directly.