# Application of Dimension Extending Technique to Unified Hardening Model
## 3 Incremental Constitutive Integration
### The Dimension-Extending Technique and Constitutive Integration
#### 3.3.2 The MiCP with Exact Elastic Properties (EEP)  
An alternative to deal with the rate form of the constitutive equation is to precisely integrate the equation. If the plastic strain increment is frozen, considering that the strain increment only causes pure elastic behavior, the elastic trail stress can be obtained as:  
\[p^{n+1,tr}=p^{n}\exp(\overline{K}_{0}\Delta\varepsilon_{kk}) \tag{53}\]  
\[s_{ij}^{n+1,tr}=s_{ij}^{n}+2G_{ave}\Delta\varepsilon_{ij} \tag{54}\]  
\[G_{ave}=\frac{\overline{G_{0}}p^{n}(\exp(\overline{K}_{0}\Delta\varepsilon_{kk })-1)}{\overline{K}_{0}\Delta\varepsilon_{kk}} \tag{55}\]  
\(\overline{K}_{0}\) and \(\overline{G}_{0}\) are defined in Equation (27). \(s_{ij}^{n}\) and \(s_{ij}^{n+1,tr}\) are the deviatoric stress at time \(n\) and the trial deviatoric stress at time \(n+1\), respectively. \(\Delta\varepsilon_{kk}\) and \(\Delta\varepsilon_{ij}\) are the incremental volumetric and deviatoric strains, respectively.  
Combining Equations (53) and (54), the stress tensor, which is in the original \(\sigma\)-space, is derived by Equation (56):  
\[\sigma_{ij}^{n+1,tr}=s_{ij}^{n+1,tr}+p^{n+1,tr}\delta_{ij} \tag{56}\]  
According to the unloading/loading criterion in Section 3.3.1, determine whether GSPC needs to be invoked.  
\[\Delta\varepsilon_{ij}^{p}=\Lambda\widetilde{r}_{ij}^{n+1} \tag{57}\]  
\[\widetilde{r}_{ij}=\frac{\partial f}{\partial\widetilde{\sigma}} \tag{58}\]  
By combining the precise integral, we can obtain:  
\[p^{n+1}=p^{n}\exp[\overline{K}_{0}(\Delta\varepsilon_{kk}-\Lambda\widetilde{r }_{kk}^{n+1})] \tag{59}\]  
\[s_{ij}^{n+1}=s_{ij}^{n}+2\overline{G}_{ave}[\Delta\varepsilon_{ij}-\Lambda \widetilde{d}_{ij}^{n+1}] \tag{60}\]  
\[\overline{G}_{ave}=\frac{\overline{G}_{0}p^{n}[\varepsilon^{\overline{K}_{0}( \Delta\varepsilon_{kk}-\Lambda\widetilde{r}_{kk}^{n+1})}-1]}{\overline{K}_{0}( \Delta\varepsilon_{kk}-\Lambda\widetilde{r}_{kk}^{n+1})} \tag{61}\]  
\(\widetilde{d}_{ij}^{n+1}\) is the deviatoric part of \(\widetilde{r}_{ij}^{n+1}\) defined as:  
\[\widetilde{d}_{ij}^{n+1}=\widetilde{r}_{ij}^{n+1}-\frac{1}{3}\widetilde{r}_{ kk}^{n+1}\delta_{ij} \tag{62}\]Then, the stress tensor and hardening parameter are obtained as:  
\[\sigma_{ij}^{n+1}=s_{ij}^{n+1}+p^{n+1}\delta_{ij} \tag{63}\]  
\[\widetilde{H}^{n+1}=\widetilde{H}^{n}+\Delta\widetilde{H} \tag{64}\]  
The calculation formula \(\Delta\widetilde{H}\) refers to Equation (40).  
For simplicity of notation, let us define  
\[\overline{\rho}=\overline{K}_{0}(\Delta\epsilon_{kk}-\Lambda\widetilde{r}_{ kk}^{n+1}) \tag{65}\]  
\[\overline{\theta}=\Delta\epsilon_{ij}-\Lambda\widetilde{d}_{ij}^{n+1} \tag{66}\]  
Then we have  
\[\sigma_{ij}^{n+1}=s_{ij}^{n}+2\overline{G}_{ave}\overline{\theta}+p^{n} \overline{\theta}\delta_{ij} \tag{67}\]  
with  
\[\overline{G}_{ave}=\frac{\overline{G}_{0}p^{n}[e^{\overline{\rho}}-1]}{ \overline{\rho}} \tag{68}\]  
Then \(f_{1}\) and \(f_{E}\) can be obtained from Equations (41) and (42).  
The above process provides the derivation of incremental constitutive integration in the transformed stress space. Of course, if the three-dimensional situation is not considered, the UH model can be directly applied to the two-dimensional model through the dimension-extending technique.