## 2 Coupled Thermo-Mechanical Phase-Field Modeling
### Heat Conduction  
The following is the energy balance equation, which accounts for the thermal diffusion and the power generated by the microscopic and macroscopic forces [46].  
\[\rho\dot{e}=\sigma:\dot{\epsilon}+K\dot{d}+H\cdot\
abla\dot{d}-\
abla\cdot \mathbf{q}+Q \tag{30}\]  
We can the following based on the given relation \(e=\psi+T\eta\).  
\[\dot{e}=\dot{\psi}+\dot{T}\eta+T\dot{\eta} \tag{31}\]  
According to the Helmholtz free energy \(\psi=\psi(\epsilon,d,\
abla d,T)\) and the constitutive relationship given by Equation (7), we can obtain the following expression.  
\[\dot{\psi}=\frac{1}{\rho}\Big{(}\sigma:\dot{\epsilon}+K\dot{d}+H\cdot\
abla \dot{d}-\rho\dot{T}\eta\Big{)} \tag{32}\]  
Combining Equations (31) and (32) yields the following.  
\[\rho\dot{e}=\sigma:\dot{\epsilon}+K\dot{d}+H\cdot\
abla\dot{d}+\rho T\dot{\eta} \tag{33}\]  
We can obtain the following conclusion by comparing Equations (30) and (33).  
\[\rho T\dot{\eta}=-\
abla\cdot\mathbf{q}+Q \tag{34}\]  
Based on Equation (7), \(\eta(\epsilon,d,\
abla d,T)=-\frac{\partial\psi}{\partial T}\), we obtain the following equation for a specific entropy rate.  
\[\begin{array}{l}\dot{\eta}=-\Big{(}\frac{\partial^{2}\dot{\varphi}}{\partial T \dot{\alpha}}:\dot{\epsilon}+\frac{\partial^{2}\dot{\varphi}}{\partial T \dot{\alpha}d}\dot{d}+\frac{\partial^{2}\dot{\varphi}}{\partial T\dot{\
abla} \dot{d}}\
abla\dot{d}+\frac{\partial^{2}\dot{\varphi}}{\partial T^{2}}\dot{T }\Big{)}\\ =-\frac{1}{\rho}\Big{(}\frac{\partial\psi}{\partial T}:\dot{\epsilon}+\frac{ \partial\psi}{\partial T}\dot{d}+\frac{\partial H}{\partial T}\
abla\dot{d}- \rho\frac{\partial\eta}{\partial T}\dot{T}\Big{)}\end{array} \tag{35}\]We consider that [PERSON]'s law through the inequality relation in Equation (8) automatically satisfies the following equation:  
\[q=-k(d)\
abla T \tag{36}\]  
where \(k(d)\) represents the phase-field-influenced degraded thermal conductivity and is indicated as follows.  
\[k(d)=g(d)k_{0} \tag{37}\]  
In the above equation, \(k_{0}\) represents the thermal conductivity of the undamaged material, and \(g(d)\) denotes the thermal degradation function that causes the absence of heat flux on the crack. We may construct the following equation by combining Equations (34) and (35).  
\[-T\bigg{(}\frac{\partial\mathbf{\sigma}}{\partial T}:\dot{\varepsilon}+\frac{ \partial\mathbf{K}}{\partial T}\dot{d}+\frac{\partial\mathbf{H}}{\partial T} \
abla\dot{d}-\rho\frac{\partial\eta}{\partial T}\dot{T}\bigg{)}=k(d)\
abla^{ 2}T+\mathbf{Q} \tag{38}\]  
The specific heat is defined as follows.  
\[c=T\frac{\partial\eta}{\partial T} \tag{39}\]  
Consequently, the following is the complete form of the heat equation.  
\[\rho c\dot{T}=k(d)\
abla^{2}T+T\bigg{(}\frac{\partial\mathbf{\sigma}}{\partial T} :\dot{\varepsilon}+\frac{\partial\mathbf{K}}{\partial T}\dot{d}+\frac{ \partial\mathbf{H}}{\partial T}\
abla\dot{d}\bigg{)}+\mathbf{Q} \tag{40}\]  
Equation (40) can degenerate into the standard heat conduction equation for thermal conduction with an internal heat source. In this case, we apply the formula for quasi-static crack extension, where the transient coupling terms \(\dot{\varepsilon}\) and \(\dot{d}\) vanish. Hence, the heat equation takes the following form.  
\[\rho c\dot{T}=k(d)\
abla^{2}T+\mathbf{Q} \tag{41}\]