## 3 Formulation of the Plastic Damage Model
### Free Energy and State Equations  
For micro-crack quasi-brittle rocks, the REV (Representative Element Volume) is taken to be a matrix-cracks system, it is assumed composed of an isotropic solid matrix and randomly distributed microcracks. Each micro-crack is geometrically approximated by an oblate spheroid called a penny-shaped crack. When the cracked materials subjected to the a local compressive stress, the friction sliding acting on the surface of closed micro-cracks which induced macroscopic inelastic deformation and micro-crack growth result in the material damage, are two main dissipation processes [39; 40].  
With the assumption of small deformations, the total macroscopic strain over REV can be divided into two parts: an elastic part \(\epsilon^{e}\) attributed to the deformation of isotropic matrix and a plastic part \(\epsilon^{p}\) due to the frictional sliding along the surfaces of closed microcracks:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \(q\) (MPa) & \(\epsilon_{0}\)(\(10^{-3}\)) & \(\sigma_{t}\) (MPa) & \(\lambda_{\text{12h}}\) & \(\lambda\) & \(\lambda_{\text{12h}}/\lambda\) & \(u_{A}\)(MPa-\(h^{-1}\)) \\ \hline
171 MPa & 3.984 & 158.1 & 0.061 & 0.075 & 80.4\% & 0.013 \\ \hline
199 MPa & 4.584 & 180 & 0.067 & 0.076 & 88.4\% & 0.053 \\ \hline
221 MPa & 5.381 & 202.6 & 0.079 & 0.083 & 94.6\% & 0.051 \\ \hline \end{tabular}
\end{table}
Table 2: Comparison of single-stage relaxation test results between Beishan granite.  
Figure 4: The single relaxation curves of Beishan granite under different levels of deviatoric stress with the confining pressure \(p_{c}=15\) MPa. (**a**) \(q=171\) MPa; (**b**) \(q=199\) MPa; (**c**) \(q=221\) MPa.  
\[\epsilon=\epsilon^{p}+\epsilon^{p} \tag{2}\]  
Apart from the friction-induced plastic strain, another internal damage variable is studied to describe the density parameter of micro-crack which is denoted by \(d\)[41]. From the above analysis, one can obtain the free energy for the case of closed cracks [39]:  
\[W=\frac{1}{2}(\epsilon-\epsilon^{p}):\mathbb{C}^{m}:(\epsilon-\epsilon^{p})+ \frac{1}{2d}\epsilon^{p}:\mathbb{C}^{b}:\epsilon^{p} \tag{3}\]  
Above, the first term on the right-hand side represents the elastic strain energy and the second one represents plastic the strain energy which related to friction sliding along the surface of micro-crack. The fourth tensor \(\mathbb{C}^{b}\) can be interpreted as the stiffness modification of damaged matrix-crack system, and it takes the form \(\mathbb{C}^{b}=\frac{3\epsilon^{m}}{\pi_{1}}\mathbb{J}+\frac{2\mu^{m}}{ \pi_{2}}\mathbb{K}\), Where \(\kappa^{m}\) and \(\mu^{m}\) denote the bulk modulus and shear modulus of solid matrix phase. According to the Mori-Tanaka homogenization method [42; 43], \(\alpha_{1}=\frac{16}{9}\frac{1-(\
u^{m})^{2}}{1-2\
u^{m}}\) and \(\alpha_{2}=\frac{32}{45}\frac{(1-\
u^{m})(5-\
u^{m})}{2-\
u^{m}}\) are two constants only dependent on the [PERSON]'s ratio \(\
u^{m}\) of the solid matrix.  
Under isotropic assumption, \(\mathbb{C}^{m}\) is elasticity stiffness tensor of the solid matrix which can be expressed in the following form:  
\[\mathbb{C}^{m}=3\kappa^{m}\mathbb{J}+2\mu^{m}\mathbb{K} \tag{4}\]  
The macroscopic stress-strain relation can be derived according to the free energy Equation (3):  
\[\sigma=\mathbb{C}^{m}:(\epsilon-\epsilon^{p}) \tag{5}\]  
One then obtains the damage thermodynamic force associated with the internal variables \(d\):  
\[Y_{d}=-\frac{\partial W}{\partial d}=\frac{1}{2d^{2}}\epsilon^{p}:\mathbb{C}^ {b}:\epsilon^{p} \tag{6}\]  
The driving force for plastic strain \(\epsilon^{p}\) is also derived:  
\[\sigma^{p}=-\frac{\partial W}{\partial\epsilon^{p}}=\sigma-\frac{1}{d} \mathbb{C}^{b}:\epsilon^{p} \tag{7}\]