## 2 Theoretical Background
### Governing Equation for PFM  
The initial elastic strain energy density is expressed as:  
\[\psi_{\epsilon_{c}}(\epsilon_{c})=\frac{\lambda}{2}\mathrm{tr}^{2}(\epsilon_{ c})+\mu\mathrm{tr}\Big{(}\epsilon_{c}^{2}\Big{)} \tag{6}\]  
where \(\lambda\) and \(\mu\) denote the Lame constant.  
\[\left\{\begin{array}{l}\lambda=\frac{E_{V}}{(1+\
u)(1-2\
u)}\\ \mu=\frac{E}{2(1+\
u)}\end{array}\right. \tag{7}\]  
The damage function \(g(d)=(1-d)^{2}+\delta_{0}\) is adopted to characterize the deterioration of material stiffness with damage. Additionally, a small dimensionless parameter \(\delta_{0}\) (\(0<\delta_{0}<<1\)) is introduced into the degradation function to prevent numerical singularity arising from complete loss of material stiffness when \(d=1\). The elastic strain energy stored in a solid can be written as:  
\[\psi_{\epsilon_{c}}=\int_{\Omega}g(d)[\psi_{\epsilon_{c}}(\epsilon_{c})]\, \mathrm{d}\Omega \tag{8}\]  
This thermal component \(\psi_{T}\) is defined as:  
\[\psi_{T}=\int_{\Omega}\big{(}\rho C_{p}\dot{T}+\
abla\cdot\mathbf{J}-Q\big{)}\, \mathrm{d}\Omega+\int_{\Omega_{f}}\mathbf{J}\cdot\mathbf{n}\,\mathrm{d}S \tag{9}\]  
where \(\rho\) denotes the density, \(C_{p}\) represents the specific heat capacity, and \(k\) is the thermal conductivity. \(\mathbf{J}=-k\
abla T\) is obtained from [PERSON]'s law. Furthermore, thermal conductivity undergoes degradation proportional to material damage, ensuring zero heat flux across fully developed cracks. This is implemented through the conductivity degradation function [35; 40].  
\[k=g(d)k_{0} \tag{10}\]where \(k_{0}\) is the initial thermal conductivity.  
Therefore, the Lagrange energy functional \(\Pi(\mathbf{u},T,d,\
abla d)\) can be rewritten as:  
\[\begin{array}{l}\Pi(\mathbf{u},T,d,\
abla d)=\frac{1}{2}\int_{\Omega}\rho\dot{\mathbf{ u}}\cdot\mathbf{u}\mathrm{d}\Omega-\int_{\Omega}g(d)[\psi_{\epsilon_{\epsilon}}( \epsilon_{\epsilon})]\mathrm{d}\Omega\\ -\int_{\Omega}G_{\epsilon}\bigg{(}\frac{d^{2}}{2l_{0}}+\frac{l_{0}}{2}|\
abla d |^{2}\bigg{)}\mathrm{d}\Omega+\int_{\Omega}\mathbf{b}\cdot\mathbf{u}\mathrm{d}\Omega+ \int_{\Omega_{t}}\bar{\mathbf{t}}\cdot\mathbf{u}\mathrm{d}S\\ +\int_{\Omega}\big{(}\rho C_{p}\bar{T}+\
abla\cdot\mathbf{J}-Q\big{)}\mathrm{d} \Omega+\int_{\Omega_{t}}\bar{\mathbf{J}}\cdot\mathbf{\mathrm{n}}\mathrm{d}S\end{array} \tag{11}\]  
Applying the variational principle (\(\delta\Pi(\mathbf{u},T,d,\
abla d)=0\)) to the total energy functional yields the following governing equations for the TM-PFM:  
\[\
abla\cdot\mathbf{\sigma}(\epsilon_{\epsilon},d)+\mathbf{b}=\rho\ddot{\mathbf{u}} \tag{12}\]  
\[\rho C_{p}\dot{T}+\
abla\cdot\mathbf{J}=Q \tag{13}\]  
\[\frac{G_{\epsilon}}{l_{0}}\Big{(}d-l_{0}^{2}\
abla^{2}d\Big{)}-2(1-d)H( \epsilon_{\epsilon})=0 \tag{14}\]  
where Equation (12) is the displacement field governing equation, Equation (13) is the temperature field governing equation, and Equation (14) is the phase-field governing equation. \(\mathbf{\sigma}(\epsilon_{\epsilon},d)=\partial\psi_{\epsilon_{\epsilon}}( \epsilon_{\epsilon},d)/\partial\epsilon_{\epsilon}\). \(H(\epsilon_{\epsilon})=\max\limits_{s\in[0,l]}\{\psi_{\epsilon_{\epsilon}}( \epsilon_{\epsilon}(\mathbf{u},s))\}\) is the historical strain field providing the driving force for PFM.  
The Dirichlet boundary conditions and Newman boundary conditions for the corresponding field variables are as follows:  
\[\mathbf{\sigma}(\epsilon_{\epsilon},d)\cdot\mathbf{n}=\bar{\mathbf{I}}\qquad\text{ on }\partial\Omega_{t} \tag{15}\]  
\[\mathbf{J}\cdot\mathbf{n}=\bar{\mathbf{J}}\qquad\qquad\text{ on }\partial\Omega_{j} \tag{16}\]  
\[\
abla d\cdot\mathbf{n}=0\qquad\qquad\text{ on }\partial\Omega\cup\Gamma \tag{17}\]