## 2 Coupled Thermo-Mechanical Phase-Field Modeling
### Considering the Temperature-Dependent Phase-Field Fracture Model  
As seen in Figure 1, the external boundary of a cracked solid \(\Omega\) is represented by \(\partial\Omega\), and the crack zone is marked by \(\Gamma\). As per the [PERSON] theory of fracture mechanics, the following equation expresses the total fracture energy.  
\[\Psi_{c}=\int_{\Gamma}G_{c}(T)\mathrm{d}A=\int_{\Omega}G_{c}(T)\gamma(d,\
abla d )dV \tag{19}\]  
[PERSON] et al. [27] and [PERSON] et al. [29] used the volumetric integral rather than the surface integral to estimate the fracture energy. The crack surface density function \(\gamma\) is written as follows.  
\[\gamma(d,\
abla d)=\frac{1}{c_{0}}(\frac{1}{\ell_{0}}\omega(d)+\ell_{0}|\
abla d |^{2}),\ c_{0}=4\int_{0}^{1}\sqrt{\omega(\beta)}d\beta \tag{20}\]  
The above equation characterizes the evolution of phase-field cracks using a geometric function with the following properties:  
\[\omega(d)\in[0,1],\omega(0)=0,\omega(1)=1;\omega^{\prime}(d)>0 \tag{21}\]  
where \(\ell_{0}\) is the length scale parameter of a regularized sharp crack, which is related to the width of the diffusion crack And \(c_{0}>0\) represents the scaling parameter [27; 29].  
The momentum balance equation of the micro force is illustrated below for quasi-static fracture, assuming negligible micro-inertia.  
\[\
abla\cdot\mathbf{H}=\mathbf{K} \tag{22}\]  
The governing equation of the phase field can be yielded by combining the micro-force balance Equation (22) with the fracture energy Equation (19).  
\[\frac{G_{c}}{c_{0}\ell_{0}}\omega^{\prime}(d)-Y-\frac{2G_{c}\ell_{0}}{c_{0}} \
abla^{2}d=0 \tag{23}\]  
where \(Y\) is the driving force of the phase field, which is expressed as follows.  
\[Y=\frac{\partial\psi_{e}}{\partial d}=f^{\prime}(d)\mathcal{H} \tag{24}\]  
Consequently, the governing equation of the phase field can be further simplified as shown below.  
\[\frac{G_{c}}{\pi\ell_{0}}(2-2d)+f^{\prime}(d)\mathcal{H}=0 \tag{25}\]  
In thermo-mechanical coupling problems, the temperature variation is significant. Consequently, the influence of temperature close to the crack tip cannot be overlooked when it comes to crack propagation. Therefore, the temperature dependence of the critical  
Figure 1: A cracking solid with cracks.  
energy release rate \(G_{c}\) needs to be considered. According to the research of [PERSON] et al. [43] and [PERSON] et al. [44], the temperature dependence of the critical energy release rate \(G_{c}\) can be described as follows for brittle and quasi-brittle materials:  
\[G_{c}=G_{c0}[1-b_{1}\frac{T-T_{ref}}{T_{max}}+b_{2}(\frac{T-T_{ref}}{T_{max}}) ^{2}] \tag{26}\]  
where \(T_{ref}\) and \(T_{max}\) denote the reference temperature and the maximum temperature, respectively; \(b_{1}\) and \(b_{2}\) represent constant model parameters; and \(G_{c0}\) represents the value of \(G_{c}\) at the reference temperature \(T_{ref}\).  
The temperature dependence of material properties affects not only the elastic performance, but also the fracture behavior. Hence, the elastic modulus E, the critical energy release rate \(G_{c}\), and material strength are all thought to be temperature-dependent [45].  
\[E(\theta)=\phi_{1}(\theta)E_{0},\ G_{c}(\theta)=\phi_{2}(\theta)G_{c0},\ \sigma_{u}( \theta)=\phi_{3}(\theta)\sigma_{u0} \tag{27}\]  
In the above equation, \(\phi_{1}\), \(\phi_{2}\), and \(\phi_{3}\) are related to the temperature degradation function. Then, \(l_{ch}\) is as follows.  
\[l_{ch}:=\frac{EG_{c}}{\sigma_{u}^{2}}=\beta_{1}(\theta)l_{ch0}\ \text{with}\ \beta_{1}(\theta)=\frac{\phi_{1}(\theta)\phi_{2}(\theta)}{\phi_{3}^{2}(\theta)} \tag{28}\]  
Consequently, \(a_{1}\) in Equation (16) is presented as below.  
\[a_{1}(\theta)=\beta_{1}(\theta)\frac{4l_{ch0}}{\pi\ell_{0}} \tag{29}\]