# Application of Dimension Extending Technique to Unified Hardening Model
## 2 Introduction to the UH Model and Transformed Stress
### Transformed Stress Method  
The transformed stress method [23] extends the two-dimensional constitutive model to three-dimensions. It converts the strength criterion in the ordinary stress space into the extended Mises strength criterion in the transformed stress space without changing the  
Figure 1: Current and reference yield surfaces.  
form of the yield function, as shown in Figure 2. [PERSON] et al. proposed the transformed stress method based on strength theories, such as SMP criterion [24], and Lade criterion [25].  
The transformed stress method based on the SMP strength criterion is used to extend the UH model to a three-dimensional model, and the corresponding transformation stress tensor is given by:  
\[\widetilde{\sigma}_{ij}=\left\{\begin{array}{cc}p\delta_{ij}+\frac{q_{c}}{q}( \sigma_{ij}-p\delta_{ij})&q\
eq 0\\ \sigma_{ij}&q=0\end{array}\right. \tag{8}\]  
\[q_{c}=\frac{2I_{1}}{3\sqrt{\left(I_{1}I_{2}-I_{3}\right)/\left(I_{1}I_{2}-9I_{ 3}\right)}-1} \tag{9}\]  
\(\widetilde{\sigma}_{ij}\) is the stress tensor in the transformed stress space, and the characters with '\(\sim\)' atop thereafter denote the variables in the transformed stress space. \(\delta_{ij}\) is the Kronecker's delta; \(I_{1}\), \(I_{2}\) and \(I_{3}\) are the first, second and third invariants, respectively.  
\[\left\{\begin{array}{c}I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3}\\ I_{2}=\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+\sigma_{3}\sigma_{1}\\ I_{3}=\sigma_{1}\sigma_{2}\sigma_{3}\end{array}\right. \tag{10}\]  
In the transformed stress space, the equations of the current yield surface and the reference yield surface are given by:  
\[\widetilde{F}=c_{p}\ln\frac{\widetilde{p}}{\widetilde{p}_{x0}}+c_{p}\ln(1+ \frac{\widetilde{q}^{2}}{M^{2}\widetilde{p}^{2}})-\widetilde{H}=0 \tag{11}\]  
\[\widetilde{\widetilde{F}}=c_{p}\ln\frac{\widetilde{\widetilde{p}}}{\widetilde {\widetilde{p}}_{x0}}+c_{p}\ln(1+\frac{\widetilde{q}^{2}}{M^{2}\widetilde{ \widetilde{p}}^{2}})-\epsilon_{v}^{p}=0 \tag{12}\]  
with  
\[\widetilde{H}=\int\frac{\widetilde{M}_{f}^{4}-\widetilde{\eta}^{4}}{M^{4}- \widetilde{\eta}^{4}}d\epsilon_{v}^{p} \tag{13}\]  
\(\widetilde{M}_{f}\) signifies the potential failure stress ratio, varying with the changes in the current stress point, and \(\widetilde{\eta}\) is the stress ratio under the transformed stress space.  
\[\widetilde{M}_{f}=6\left[\sqrt{\frac{k}{\widetilde{R}}(1+\frac{k}{\widetilde {R}})}-\frac{k}{\widetilde{R}}\right] \tag{14}\]  
\[\widetilde{R}=\frac{\widetilde{p}}{\widetilde{p}_{x0}}(1+\frac{\widetilde{ \eta}^{2}}{M^{2}})exp(-\frac{\epsilon_{v}^{p}}{c_{p}}) \tag{15}\]  
Figure 2: Transformed stress method based on SMP strength criterion. (**a**) \(\pi\) plane; (**b**) \(\widetilde{\sigma}_{i}\) space.  
In the transformed stress space, the associated flow rule is adopted:  
\[d\epsilon^{p}=\Lambda\frac{\partial\widetilde{F}}{\partial\widetilde{\sigma}} \tag{16}\]  
where \(de^{p}\) is the plastic strain increment and \(\Lambda\) is the plastic multiplier.  
From Equation (13) it can be observed that, when the stress approaches the limit state, it will cause the denominator in the equation to be zero and the calculation will fail. For the stability of numerical calculations, the hardening Equation (13) is now rewritten in incremental form and organized as:  
\[d\widetilde{H}=\frac{\widetilde{M}_{f}^{4}-\widetilde{\eta}^{4}}{M^{4}- \widetilde{\eta}^{4}}d\epsilon_{p}^{p}=\Lambda\frac{\widetilde{M}_{f}^{4}- \widetilde{\eta}^{4}}{M^{4}-\widetilde{\eta}^{4}}\frac{\partial\widetilde{F}}{ \partial\widetilde{p}}=\Lambda c_{p}\frac{1}{\widetilde{p}}\frac{\widetilde{M}_ {f}^{4}-\widetilde{\eta}^{4}}{\left(M^{2}+\widetilde{\eta}^{2}\right)^{2}} \tag{17}\]  
In terms of format, the transformed stress method adopts the form of the associated flow rule, but in fact it adopts the non-associated flow rule, which is more in line with the actual situation of the soil.