## 2 Formulation and Theoretical Derivation of the Elastoplastic Phase Field Fracture Model
### Elastic Strain Energy
\tag{11}\]  
In this formula, \(\tau\) is the shear strength and \(\lambda\) and \(\mu\) are Lame constants, related to Young's \(E\) modulus and [PERSON]'s ratio \(\
u\)  
\[\left\{ \begin{array}{l}\lambda=\frac{E\
u}{(1+\
u)(1-2\
u)}\\ \mu=\frac{E}{2(1+\
u)}\end{array} \right. \tag{12}\]  
Post compressive-shear failure, the shear plane orientation relative to the axial direction is well-described by the Mohr-Coulomb criterion. This criterion establishes that shear failure depends on cohesion and internal friction angle, which quantify interparticle bonding and frictional resistance. In this framework, shear strain energy regulates phase field damage evolution, bridging plasticity theory with continuum damage mechanics [28]. Under plane stress conditions, the Mohr-Coulomb shear strength \(\tau\) is expressed as  
\[\tau=\left\{\begin{array}{l}\frac{\sigma_{1e}-\sigma_{2e}}{2}\cos\varphi\ \left( \sigma_{n}<0\right)\\ 0\ \  
plastic deformation and damage evolution, thereby supporting more precise model refinement [29; 30]. The subsequent discussion addresses plastic damage definition, evolution equations, and coupling relationships.  
Under isotropic hardening assumptions, the plastic strain energy \(E_{\text{pla}}\) is generally expressed as  
\[E_{\text{pla}}=\int_{\Omega}\psi(\varepsilon^{p})\text{d}\Omega \tag{15}\]  
In this formula, \(\psi(\varepsilon^{p})\) represents the plastic strain energy density. Plastic damage originates from irreversible plastic deformation in loaded rock materials, causing internal structural degradation [31]. The model quantifies plastic damage severity through a plastic damage variable \(D_{p}\) (\(D_{p}\in[0,1]\)), where \(D_{p}=0\) indicates undamaged material and \(D_{p}=1\) signifies complete plastic failure. Distinct from elastic damage, \(D_{p}\) captures energy dissipation during plastic flow--particularly in shear bands where accumulated plastic slip reduces material stiffness and modifies crack propagation paths. Given that plastic deformation involves energy dissipation, the plastic damage evolution is defined via accumulated plastic strain energy. Assuming proportionality between damage evolution and plastic energy, the evolution rate is formulated as  
\[\frac{dD_{p}}{dt}=\frac{1}{G_{pD}}\frac{d\psi(\varepsilon^{p})}{dt} \tag{16}\]  
In this formula, \(G_{pD}\) denotes the plastic damage-related critical energy release rate; \(E_{\text{pla}}\) is the plastic strain energy. Following the plastic constitutive framework, the plastic strain energy \(E_{\text{pla}}\) is derived from plastic work concepts. Assuming isotropic hardening, the plastic strain energy density is expressed as  
\[\psi(\varepsilon^{p})=\frac{1}{2}K_{p}(tr(\varepsilon^{p}))^{2}+\mu_{p}tr( \varepsilon^{p})^{2} \tag{17}\]  
In this formula, \(K_{p}\) is the plastic bulk modulus; \(\mu_{p}\) is the plastic shear modulus; and \((\varepsilon^{p})^{2}\) denotes the squared plastic strain tensor. Differentiating Equation (17) temporally yields the plastic strain energy rate:  
\[\frac{\text{d}\psi(\varepsilon^{p})}{\text{d}t}=K_{p}\text{tr}(\varepsilon^{p })\text{tr}\bigg{(}\frac{\text{d}\varepsilon^{p}}{\text{d}t}\bigg{)}+2\mu_{p} \text{tr}\bigg{(}\varepsilon^{p}\frac{\text{d}\varepsilon^{p}}{\text{d}t} \bigg{)} \tag{18}\]  
Substituting Equation (18) into Equation (16) results in  
\[\frac{\text{d}D_{p}}{\text{d}t}=\frac{1}{G_{pD}}\bigg{\{}K_{p}\text{tr}( \varepsilon^{p})\text{tr}\bigg{(}\frac{\text{d}\varepsilon^{p}}{\text{d}t} \bigg{)}+2\mu_{p}\text{tr}\bigg{(}\varepsilon^{p}\frac{\text{d}\varepsilon^{p }}{\text{d}t}\bigg{)}\bigg{\}} \tag{19}\]  
In practical scenarios, plastic damage evolution depends on material yielding. A plastic yield function \(f(\sigma,\varepsilon^{p})\) is introduced to determine yielding states. When condition \(f(\sigma,\varepsilon^{p})\leq 0\) is satisfied, the material remains in the elastic state; when condition \(f(\sigma,\varepsilon^{p})>0\) is met, the material enters the plastic state, and plastic damage begins to evolve.