## 3 Formulation of the Plastic Damage Model
### Plasticity Criterion and Damage Evolution  
A generalized Coulomb-type plastic yield criterion is formulated in terms of the local stress \(\sigma^{p}\) acting on crack surfaces.  
\[f=\|\mathbf{s}^{p}\|+\eta\sigma^{p} \tag{8}\]  
\(\mathbf{s}^{p}=\mathbf{\sigma}^{p}:\mathbb{K}\) and \(\sigma^{p}=\frac{1}{3}\sigma^{p}:\delta\) which represent the deviatoric and mean parts of the local stress respectively. \(\eta\) denotes the friction coefficient of crack's surfaces.  
The evolution of plastic strain can be derived by applying the normality rule:  
\[\dot{\epsilon}^{p}=\lambda^{p}\frac{\partial f}{\partial\sigma^{p}}=\lambda^ {p}\mathbf{D} \tag{9}\]  
with \(\lambda^{p}\) is a non-negative plastic multiplier and \(\mathbf{D}=\frac{\mathbf{s}^{p}}{\|\mathbf{s}^{p}\|}+\frac{1}{3}\eta\delta\) denotes the plastic flow direction.  
The damage criterion can be expressed as a function of the damage thermodynamic force:  
\[f_{d}=Y_{d}-R_{d} \tag{10}\]  
where \(R_{d}\) represents the damage resistance function which usually depends on the damage state [44]. The following classic form is adopted for \(R_{d}\) in this paper:\[R_{d}=r_{c}\frac{2\xi}{1+\xi^{2}} \tag{11}\]  
with \(\xi=\frac{d}{d_{c}}\) which is a dimensionless parameter. Physically, \(d_{c}\) represents the critical value of damage density variable that corresponds to the peak strength. Moreover, \(r_{c}\) is maximal value of damage resistance function \(R_{d}\). The damage increment can be determined by applying the normality rule:  
\[\dot{d}=\lambda^{d}\frac{\partial f_{d}}{\partial Y_{d}}=\lambda^{d} \tag{12}\]  
For the case of time-dependent part, the total damage can be divided into two parts: the instantaneous damage \(\omega\) induced by external loads and time-dependent damage related to subcritical cracking:  
\[d=\omega+\varsigma \tag{13}\]  
where \(\omega\) and \(\varsigma\) denote the instantaneous damage due to the external mechanical loading, and time-dependent damage which is relative to subcritical cracking due to stress corrosion. The damage criterion (10) is used for instantaneous damage evolution while the time-dependent damage rate can be described in the terms of the distance between the current damage and its equilibrium state \(\varsigma\). [PERSON] et al. [30] gave the detailed description of the parameter \(\varsigma\). The simple linear form is adopted here to describe the time-dependent damage:  
\[\xi=\Gamma(\varsigma-\varsigma) \tag{14}\]  
where \(\Gamma\) is the model parameter which controls the rate of accelerated creep and steady relaxation of the presented time-dependent model which will be discussed in Section 4.4.  
By setting the initial condition \(\varsigma(0)=0\), the time-dependent damage criterion can be expressed using the Laplacian transformation and convolution operation:  
\[\varsigma(t)=\int_{0}^{t}\xi(\tau)\Gamma e^{-\Gamma(t-\tau)}d\tau \tag{15}\]  
A relevant explicit integration method is taken here [45]:  
\[\varsigma_{i}=\varsigma_{i-1}e^{-\Gamma t}+\frac{1}{2}\Gamma(\varsigma_{i}+ \varsigma_{i-1})e^{-\frac{1}{2}\Gamma t}\dot{t} \tag{16}\]  
From above analysis, the parameter \(\xi\) represents a stationary state of microstructure equilibrium associated with the current damage \(d\). Due to the three-stage feature of creep and relaxation tests on Beishan granite, the following form is adopted:  
\[\varsigma=\frac{d^{2}}{C+d}\exp\biggl{(}\frac{\langle d-d_{c}\rangle}{d_{c}} \biggr{)} \tag{17}\]  
where \(\langle x\rangle=\frac{\tau+|x|}{2}\) is the Macauley bracket and \(C\) is a time-dependent parameter whose calibration will de addressed in Section 4.2.