## 2 Governing Equations
### Incremental Stress-Strain Relation  
Following the conventional plasticity theory, the total incremental strain \(d\mathbf{\epsilon}\) considering the temperature effect is given as:  
\[d\mathbf{\epsilon}=d\mathbf{\epsilon}_{\sigma}^{e}+d\mathbf{\epsilon}_{T}^{e}+d\mathbf{\epsilon }^{p} \tag{18}\]  
where \(d\mathbf{\epsilon}_{\sigma}^{e}\) and \(d\mathbf{\epsilon}_{T}^{e}\) are the incremental strains induced by stress and thermo-loading, respectively. The incremental stress-strain relation is given as:  
\[d\sigma=\left(\mathbf{\mathrm{C}}^{e}-\mathbf{\mathrm{C}}_{\sigma}^{p}\right):\mathbf{ \epsilon}+\left(\alpha_{t}1:\mathbf{\mathrm{C}}^{e}-\mathbf{\mathrm{C}}_{T}^{p}\right)dT \tag{19}\]  
where \(\mathbf{\mathrm{C}}^{e}\) is the elastic stress-strain tensor that corresponds to the elastic bulk modulus \(K\) and the shear modulus \(G\), and is expressed as:  
\[\mathbf{\mathrm{C}}^{e}=K\mathbf{\mathrm{1}}\otimes\mathbf{\mathrm{1}}+2G\left(\mathbf{ \mathrm{I}}-\frac{1}{3}\mathbf{\mathrm{1}}\otimes\mathbf{\mathrm{1}}\right) \tag{20}\]  
where \(\mathbf{\mathrm{I}}\) is the fourth-order unity tensor and the symbol \"\(\otimes\)\" denotes a tensor product. Similar to the modified Cam-clay model, \(K\) and \(G\) depend on the initial void ratio and the current mean effective stress, which are defined as  
\[K=(1+e_{0})p/\kappa \tag{21}\]  
\[G=3(1-2\
u)/(2+2\
u) \tag{22}\]  
where \(\
u\) is the [PERSON]'s ratio. The plastic strain tensors \(\mathbf{\mathrm{C}}_{\sigma}^{p}\) and \(\mathbf{\mathrm{C}}_{T}^{p}\) are, respectively, calculated by:  
\[C_{\mathrm{P}}^{e}=\frac{\mathbf{\mathrm{C}}_{\mathrm{e}}:\partial f^{a}/\partial \sigma\otimes\partial^{a}f/\partial\sigma^{a}:\mathbf{\mathrm{C}}_{\mathrm{e}}}{K_ {\mathrm{P}}+\partial f/\partial\sigma:\mathbf{\mathrm{C}}_{\mathrm{e}}:\partial ^{a}f/\partial\sigma^{a}} \tag{23}\]  
\[C_{\mathrm{T}}^{p}=\frac{\mathbf{\mathrm{C}}^{e}:\partial f^{a}/\partial\sigma^{a} [\alpha_{T}\mathbf{\mathrm{1}}:\partial/\partial\sigma:\mathbf{\mathrm{C}}^{e}+ \partial f/\partial T]}{K_{\mathrm{P}}+\partial f/\partial\sigma:\mathbf{\mathrm{C }}^{e}:\partial^{a}f/\partial\sigma^{a}} \tag{24}\]  
where \(K_{p}\) is the plastic modulus that can be expressed as:  
\[K_{p}=\left(q^{2}+M_{cs}^{2}p^{2}\right)\left[\frac{1+e_{0}}{\Lambda-\kappa} \left(1-\frac{\ln R}{RS}\right)\frac{\partial^{a}f}{\partial p^{a}}\right] \tag{25}\]  
To simulate the arbitrary laboratory tests with homogeneous stress and strain fields applicable to the proposed model, the method by [PERSON] and [PERSON] [25] is adopted here by solving the following vector-valued function:  
\[f\left(\Delta\mathbf{\epsilon}_{g}\right)=S_{\mathcal{S}\Delta\sigma}\left(\Delta \mathbf{\epsilon}_{g},\mathbf{\sigma}_{g},q\right)+E\Delta\mathbf{\epsilon}_{g}-\Delta \mathbf{\mathrm{y}}=0 \tag{26}\]  
where the stress tensor \(\mathbf{\sigma}_{g}\) and strain tensor \(\mathbf{\mathrm{e}}_{g}\) are denoted as \(\mathbf{\sigma}_{g}=\{\sigma_{11},\sigma_{22},\sigma_{33},\sigma_{12},\sigma_{13},\sigma_{23}\}^{T}\) and \(\mathbf{\epsilon}_{g}=\{\mathbf{\epsilon}_{11},\Figure 6 shows the measured stress-strain curves and volumetric deformation of sedimentary rock under the drained compression condition with various confining pressures. It can be found that the volume of rock samples is first reduced and then expanded during shearing. Once the peak strength state is reached, continuous loading will lead to the strain-softening phenomenon, and the rock sample can still resist extra deformation at the critical state. The proposed model can accurately describe the abovementioned features. In addition, to consider the temperature effect on the behaviour of soft rock, the experimental data of Ohya rock ([PERSON] et al. [10]) is adopted here and predicted by the proposed model. The cylindrical specimens with 50 mm in diameter and 100 mm in height were obtained from Ohya Village of Tochigi Prefecture.