# Application of Dimension Extending Technique to Unified Hardening Model
## 3 Incremental Constitutive Integration
### The Dimension-Extending Technique and Constitutive Integration
#### 3.3.1 Description of MiCP with Approximate Elastic Properties (AEP)  
The rate form constitutive equations of the UH model in the \"increasing dimensional stress\" space can be expressed as:  
\[\left\{ \begin{array}{l}\dot{\mathbf{\sigma}}=\dot{\mathbf{\sigma}}_{e}-\mathbf{\sigma}_{ p}\\ \dot{\widetilde{H}}=\Lambda c_{p}\frac{1}{p}\frac{\widetilde{M}_{+}^{4}- \widetilde{y}^{4}}{(M^{2}+\widetilde{y}^{2})^{2}}\end{array} \right. \tag{28}\]  
\[\dot{\mathbf{\sigma}}_{e}=\mathbf{D}\dot{\mathbf{\varepsilon}} \tag{29}\]  
\[\mathbf{\sigma}_{p}\triangleq\mathbf{D}\dot{\mathbf{\varepsilon}}_{p} \tag{30}\]  
\[\dot{\mathbf{\varepsilon}}_{p}\triangleq\Lambda\
abla_{\widetilde{\sigma}}\widetilde {F} \tag{31}\]  
where \(\
abla_{\widetilde{\sigma}}\widetilde{\text{F}}\) represents the partial derivative of the yield function corresponding to the stress in the transformed stress space. \(\mathbf{D}\) is the symmetric positive elastic stiffness matrix:  
\[\mathbf{D}\triangleq\frac{3K(1-v)}{1+v} \begin{bmatrix}1&\frac{v}{1-v}&\frac{v}{1- v}\\ \frac{v}{1-v}&1&\frac{v}{1-v}\\ \frac{v}{1-v}&\frac{v}{1-v}&1\end{bmatrix}  \tag{32}\]  
\(K\) is defined as in Equation (26). \(v\) is the specific volume. \[\frac{\partial\widetilde{F}}{\partial\widetilde{\sigma}_{i}}=\frac{\partial \widetilde{F}}{\partial\widetilde{p}}\frac{\partial\widetilde{p}}{\partial \widetilde{\sigma}_{i}}+\frac{\partial\widetilde{F}}{\partial\widetilde{q}} \frac{\partial\widetilde{q}}{\partial\widetilde{\sigma}_{i}}\quad\quad i=1,2,3 \tag{33}\]  
\[\frac{\partial\widetilde{F}}{\partial\widetilde{p}}=c_{p}\frac{1}{\widetilde{ p}}\frac{M^{2}\widetilde{p}^{2}-\widetilde{q}^{2}}{M^{2}\widetilde{p}^{2}+ \widetilde{q}^{2}} \tag{34}\]  
\[\frac{\partial\widetilde{F}}{\partial\widetilde{q}}=c_{p}\frac{2\widetilde{q} }{M^{2}\widetilde{p}^{2}+\widetilde{q}^{2}} \tag{35}\]  
\[\frac{\partial\widetilde{p}}{\partial\widetilde{\sigma}_{i}}=\frac{1}{3}\quad \quad i=1,2,3 \tag{36}\]  
\[\frac{\partial\widetilde{q}}{\partial\widetilde{\sigma}_{i}}=\frac{3( \widetilde{\sigma}_{i}-\widetilde{p})}{2\widetilde{q}}\quad\quad i=1,2,3 \tag{37}\]  
Using the backward Euler integration on interval \((n,n+1)\) for Equation (28), then:  
\[\left\{ \begin{array}{l}\mathbf{\sigma}_{n+1}=\mathbf{\sigma}_{\varepsilon}-\mathbf{ \sigma}_{p}\\ \widetilde{H}_{n+1}=\widetilde{H}_{n}+\Delta\widetilde{H}\end{array} \right. \tag{38}\]\[\sigma_{e}\triangleq\sigma_{n}+D\Delta\epsilon \tag{39}\]  
\[\Delta\widetilde{H}=\Lambda c_{p}\frac{1}{\widetilde{p}_{n+1}}\frac{\widetilde{ M}_{f,n+1}^{4}-\widetilde{\eta}_{n+1}^{4}}{\left(M^{2}+\widetilde{\eta}_{n+1}^{2} \right)^{2}} \tag{40}\]  
\(\sigma_{p}\) is still calculated by Equation (30). \(f_{1}:R_{+}\times R^{4}\to R\)  
\[f_{1}(\Lambda,\widetilde{\sigma})\triangleq-\widetilde{F}(\widetilde{\sigma}) \tag{41}\]  
\[f_{E}(\Lambda,\hat{\sigma})\triangleq\left\{ \begin{array}{l}\sigma_{n+1}+ \sigma_{p}-\sigma_{e}\\ \widetilde{H}_{n+1}-\widetilde{H}_{n}-\Delta\widetilde{H}\end{array} \right. \tag{42}\]  
with  
\[\hat{\sigma}^{T}=(\sigma_{1},\sigma_{2},\sigma_{3},\widetilde{H}) \tag{43}\]  
\[\widetilde{\sigma}^{T}=(\widetilde{\sigma}_{1},\widetilde{\sigma}_{2}, \widetilde{\sigma}_{3},\widetilde{H}) \tag{44}\]  
Equations (41) and (42) form \(f_{1}\) and \(f_{E}\) in MiCP. The consistency condition is:  
\[\Lambda>0,\quad-\widetilde{F}=0,\quad\Lambda(-\widetilde{F})=0 \tag{45}\]  
The yield function of the UH model corresponds one-to-one to the current stress. In the \"increasing dimensional stress\" space, the stress point can only be located on the surface:  
\[\widetilde{E}\triangleq\{(\widetilde{\sigma}_{1},\widetilde{\sigma}_{2}, \widetilde{\sigma}_{3};\widetilde{H})\in R^{4}|\widetilde{F}(\widetilde{ \sigma}_{1},\widetilde{\sigma}_{2},\widetilde{\sigma}_{3};\widetilde{H})=0\} \tag{46}\]  
Therefore, the MiCP is adjusted to the following form:  
\[\left\{ \begin{array}{l}f_{E}(\Lambda,\hat{\sigma})=0;\hat{\sigma}\;free\\ \Lambda f_{1}(\Lambda,\widetilde{\sigma})=0,\;if\;\;\;f_{1}(\Lambda,\widetilde {\sigma})=0\\ \Lambda>0\end{array} \right.