## 4 Analytical Prediction of Short- and Long-Term Strengths under Conventional Triaxial Compression
### Long-Term Strength  
In this paper, the crack damage stress \(\sigma_{cd}\) is used as the long-term strength of Beishan granite which can be defined as the compressibility-dilatancy (C/D) transition stress [18; 37; 46; 47; 48]. In our proposed micromechanics-based analytical model, one assumes that there exists a crack critical damage value \(d_{cd}\) corresponding to the transition stress \(\sigma_{cd}\). Then the damage resistance value \(R_{cd}\) in terms of the crack critical damage \(d_{cd}\) can be derived:  
\[R_{cd}=r_{c}\frac{2\xi_{cd}}{1+\sigma_{cd}^{2}},\ \ \text{with}\ \ \ \xi_{cd}=\frac{d_{cd}}{d_{c}} \tag{26}\]Finally, the crack damage stress \(\sigma_{cd}\) can be expressed in terms of the damage resistance value \(R_{cd}\). \[\sigma_{cd}=\frac{\sqrt{6}+2\eta}{\sqrt{6}-\eta}\sigma_{3}+\frac{3\sqrt{2R_{cd} \chi}}{\sqrt{6}-\eta} \tag{27}\]  
The short-term peak strength, the crack damage stress and its linear fitting functions under different confining pressures are presented in Figure 5.  
It is highlighted that the proposed micromechanics-based analytical time-dependent model contains seven parameters which can be determined from a series of conventional compression tests on Beishan granite. The [PERSON]'s modulus \(E^{m}\) and the Poisson's ratio \(\
u^{m}\) are determined from the linear part of the stress-strain curves as shown in Figure 2a. According to Equations (11) and (23), the friction coefficient \(\eta\), the critical damage resistance \(r_{c}\) and the crack critical damage \(d_{cd}\) can be calibrated by using the linear fitting function of peak stress and transition stress of Beishan granite under different confining pressure, respectively, as shown in Figure 5. The critical damage \(d_{c}\) is related to plastic strain in its peak stress state. In this context, the time-dependent parameter \(C\) can be expressed as following form [48]:  
\[C=\frac{d_{c}^{2}}{d_{c}-d_{cd}}-d_{c} \tag{28}\]  
The time-dependent parameter \(\Gamma\) can be calibrated by the creep deformation or relaxation stress rate of third stage in triaxial time-dependent test of Beishan granite. The relevant values of model parameters are presented in Table 3.  
A damage-controlled time-dependent model is proposed for the simultaneous account of creep and relaxation mechanical behaviors. The instantaneous damage can be defined by damage evolution \(f_{d}\) while the time-dependent damage can be updated by time-dependent damage criterion \(\varsigma(t)\). Figure 5: Linear fitting of the peak strength and transition stress extracted from triaxial compression tests on Beishan granite under different confining pressures. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & \(E^{m}\) **(MPa)** & \(\
u^{m}\) & \(d_{c}\) & \(\eta\) & \(r_{c}\) & \(C\) \\ \hline Values & 45,000 & 0.15 & 8 & 1.71 & 0.012 & 2.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter’s values of Beishan granite. For the case of creep condition, the axial stress \(\sigma_{1}\) keeps unchanged. Assuming that during time-dependent stage plastic yield criterion still work, the accumulated multiplier \(\Lambda^{p}\) can be expressed by:  
\[\Lambda^{p}=\frac{\|\mathbf{s}\|+\eta\sigma}{\chi} \tag{29}\]  
According to Equation (25) and plastic yield criterion \(f\), the macroscopic strain-stress relation can be rewritten for both instantaneous and time-dependent conditions:  
\[ \begin{array}{l}\varepsilon_{1}=\frac{1}{E^{m}}\sigma_{1}-\frac{2\
u^{m}}{E^ {m}}\sigma_{3}\ +\ \mathbf{D}_{11}\frac{\sigma_{1}\mathbf{D}_{11}+2\sigma_{2}\mathbf{D}_{1 1}}{\frac{\chi}{\chi}}d\\ \varepsilon_{2}=\varepsilon_{3}=\frac{1-\
u^{m}}{E^{m}}\sigma_{3}-\frac{\
u^{ m}}{E^{m}}\sigma_{1}+\mathbf{D}_{33}\frac{\sigma_{1}\mathbf{D}_{11}+2\sigma_{2}\mathbf{D}_{ 11}}{\chi}d\end{array}  \tag{30}\]  
The second term on the right hand of the equation is plastic strain including instantaneous part and time-dependent part, due to that the damage composed of its instantaneous damage and creep damage.