## 2 Theoretical Background  
### Regularization and Variational Formulation for Brittle Fracture  
An arbitrary thermoelastic continuum \(\Omega\subset R^{\text{dim}}(\text{dim}\in\{2,3\})\) serves as the physical domain for this investigation, with an outer boundary \(\partial\Omega\) and an inner sharp crack \(\Gamma\), as shown in Figure 1. Herein, \(\partial\Omega_{u}\) and \(\partial\Omega_{T}\) denote Dirichlet boundaries, while \(\partial\Omega_{t}\) and \(\partial\Omega_{f}\) represent Neumann boundaries. Within the standard PFM formulation, sharp cracks are regularized via a continuous scalar field \(d(\mathbf{x},t)\in[0,1]\) introduction, where \(d=0\) indicates the solid is intact (not damaged), and \(d=1\) represents the solid is completely broken. \(\mathbf{x}\) is the position vector, and \(t\) is the time. Following variational regularization theory, the sharp crack geometry is asymptotically approximated by a diffuse fracture representation governed by the phase-field functional.  
\[\underbrace{A=\int_{\Gamma}\text{d}S}_{\text{Sharp crack}}\approx\underbrace{ \int_{\Gamma_{d}}\gamma(d,\
abla d)\,\text{d}\Omega=A_{d}}_{\text{Diffuse crack}} \tag{1}\]  
where \(\gamma(d,\
abla d)\) denotes the crack surface density per unit volume, defined as [21]:  
\[\gamma(d,\
abla d)=\frac{d^{2}}{2l_{0}}+\frac{l_{0}}{2}|\
abla d|^{2} \tag{2}\]  
where \(l_{0}\) denotes the characteristic length scale, representing the regularization parameter in the phase-field formulation. The smaller \(l_{0}\) means the narrower the crack. When \(l_{0}\) tends to zero, the crack is closer to the real crack. Therefore, the crack fracture energy \(\psi_{f}\) can be approximately written as:  
\[\psi_{f}=\int_{\Gamma}G_{c}\text{d}S\approx\int_{\Gamma}G_{c}\bigg{(}\frac{d^{ 2}}{2l_{0}}+\frac{l_{0}}{2}|\
abla d|^{2}\bigg{)}\,\text{d}\Omega \tag{3}\]  
Figure 1: Schematic representation of the PFM.  
Consistent with [PERSON]'s fracture theory [38], crack propagation arises from the competition between external work expenditure for new crack formation and the strain energy released during fracture advancement. Consequently, the total energy functional \(\Pi(\mathbf{u},T,d,\
abla d)\) of the PFM is defined as [25; 39]:  
\[\begin{split}&\Pi(\mathbf{u},T,d,\
abla d)=\frac{\frac{1}{2}\int_{ \Omega}\rho\dot{\mathbf{u}}\cdot\dot{\mathbf{u}}\,\mathrm{d}\Omega}{\psi_{kin}}-\underbrace {\int_{\Omega}\psi_{\epsilon_{c}}(\epsilon_{c})\,\mathrm{d}\Omega}_{\dot{ \psi}_{\epsilon_{c}}}-\underbrace{\int_{\Gamma}G_{c}\,\mathrm{d}S}_{\dot{\psi} _{f}}\\ &+\underbrace{\int_{\Omega}\psi_{T}(T)\,\mathrm{d}\Omega+\int_{ \Omega_{f}}\bar{\mathbf{J}}\cdot\mathbf{u}\,\mathrm{d}S}_{\dot{\psi}_{T}}+\underbrace {\int_{\Omega}\mathbf{b}\cdot\mathbf{u}\,\mathrm{d}\Omega+\int_{\Omega_{t}}\bar{\mathbf{I }}\cdot\mathbf{u}\,\mathrm{d}S}_{\dot{W}_{ext}}\end{split} \tag{4}\]  
where \(\psi_{kin}\) is the kinetic energy, \(\psi_{\epsilon_{c}}\) represents the elastic strain energy, \(G_{c}\) denotes the critical energy release rate, \(\psi_{T}\) is the thermal energy, \(W_{ext}\) is the external force work, \(T\) is the temperature, \(\mathbf{b}\) is the body force, \(\bar{\mathbf{I}}\) is the traction, \(\mathbf{u}\) is the displacement, \(\dot{\mathbf{u}}=\partial u/\partial t\), and \(\epsilon_{c}\) represents the elastic strain tensor. Under thermo-mechanical coupling, materials undergo thermal expansion. The thermal elastic strain \(\epsilon_{th}\) of the material is defined as:  
\[\left\{\begin{array}{l}\epsilon_{c}=\frac{1}{2}\Big{[}\
abla\mathbf{u}+(\
abla \mathbf{u})^{\mathrm{T}}\Big{]}-\epsilon_{th}\\ \epsilon_{th}=\mathbf{\alpha}_{T}(T-T_{0})\mathbf{I}\end{array}\right. \tag{5}\]  
where \(\alpha_{T}\) represents thermal expansion tensor.