## 2 Formulation and Theoretical Derivation of the Elastoplastic Phase Field Fracture Model
### Derivation of Phase Field Governing Equations and Equilibrium Equations  
Existing models account for the effects of phase field variable \(\phi\) and plastic damage \(D_{p}\) on crack evolution. However, their coupling interaction affects material properties and crack propagation. To address this coupling effect, the damage degradation function \(g\) (\(\phi\), \(D_{p}\)) is redefined as  
\[g(\phi,D_{p})=(1-\delta)(1-\phi-D_{p})^{2}+\delta \tag{22}\]  
In this formula, \(\delta\) is a regularization parameter to prevent numerical singularities. Thus, the total strain energy is then expressed as  
\[E_{\mathrm{str}} =E_{\mathrm{ela}}+E_{\mathrm{pla}} \tag{23}\] \[=\int_{\Omega}g(\phi,D_{p})\Big{(}\psi_{+}^{\mathrm{d}1}+\psi_{ +}^{\mathrm{e}\mathrm{I}}\Big{)}\mathrm{d}\Omega+\int_{\Omega}\{g(\phi,D_{p}) +\eta[1-H(\mathrm{tr}(\varepsilon))][1-g(\phi,D_{p})]\}\psi_{-}^{\mathrm{e} \mathrm{I}}\mathrm{d}\Omega+\int_{\Omega}\Big{\{}\frac{1}{2}K_{p}(\mathrm{tr} (\varepsilon^{p}))^{2}+\mu_{p}\mathrm{tr}(\varepsilon^{p})^{2}\Big{\}}\mathrm{ d}\Omega\]  
Equation (22) reveals that tensile elastic energy dissipates as \(\phi\)\(\rightarrow\)\(1\), the tensile elastic energy gradually disappears. To ensure the irreversibility of crack damage evolution, the energy takes the maximum value of the historical field variable, and we can set  
\[\left\{\begin{array}{l}\vspace{0.2 cm}\vspace{0.2 cm}\vspace{0.2 cm}\vspace{0.2 cm} \vspace{0.2 cm}\vspace{0.  
Of these, the influence of plastic deformation needs to be considered for the stress tensor \(\sigma\). The stress tensor is obtained by taking the partial derivative of the total strain energy \(\psi(\varepsilon,\varepsilon^{p})\) with respect to the strain tensor \(\varepsilon\)  
\[\sigma=\frac{\partial\psi}{\partial\varepsilon}=g(\phi)\sigma_{+}^{eff}+g(\phi )\sigma_{-}^{eff}+\eta[1-H(\text{tr}(\varepsilon))][1-g(\phi)]\sigma_{-}^{ eff}+\frac{\partial\psi(\varepsilon^{p})}{\partial\varepsilon^{p}} \tag{28}\]  
In this formula, \(\sigma_{+}^{eff}\) and \(\sigma_{-}^{eff}\) denote effective tensile and shear stresses, respectively. As can be seen from Equation (17), by taking the partial derivative of the plastic strain energy with respect to the strain tensor, we can obtain  
\[\frac{\partial\psi(\varepsilon^{p})}{\partial\varepsilon^{p}}=K_{p}\text{tr }(\varepsilon^{p})I+2\mu_{p}\varepsilon^{p} \tag{29}\]  
In this formula, \(I\) is the second-order identity tensor.  
Plastic damage modifies material stiffness, thereby reducing crack propagation resistance. At crack tips, localized softening from \(D_{p}\) accumulation decreases fracture energy. To account for the effect of plastic damage on crack propagation, a correction factor \(\beta(D_{p})=1-D_{p}\) is introduced to modify the critical energy release rates \(G_{c\Pi}^{+}\) and \(G_{c}^{-}\) into \(G_{c\Gamma}^{+}\beta(D_{p})\), \(G_{c\Pi}^{+}\beta(D_{p})\), and \(G_{c}^{-}\beta(D_{p})\).  
Applying variational minimization, the phase field governing equation with plastic damage coupling is derived as  
\[\frac{\partial g(\phi,D_{p})}{\partial\phi}\left\{\frac{\hbar_{+}^{\text{L}}} {G_{c\Gamma}^{+}\beta(D_{p})}+\frac{\hbar_{+}^{\text{L}}}{G_{c\Gamma}^{+}\beta (D_{p})}+\left\{1-\eta[1-H(\text{tr}(\varepsilon))]\right\}\times\frac{\hbar_ {-}}{G_{c}^{-}\beta(D_{p})}+\frac{\psi^{p}(\varepsilon^{p})}{G_{p}}\right\}- \ell_{0}\Delta\phi+\frac{1}{\ell_{0}}\phi=0 \tag{30}\]  
In this formula, \(G_{c\Gamma}^{+}\) is the tensile critical energy release rate of the material; \(G_{c\Gamma}^{+}\) is the tensile-shear critical energy release rate of the material; and \(G_{c}^{-}\) is the compressive-shear critical energy release rate of the material.