## 2 Numerical Methodology
### Governing Equations  
A coupled electromagnetic-thermomechanical model with damage judgment (ETMD) was established in this study. The electromagnetic field was determined by solving the Helmholtz equation, expressed as follows [29]:  
\[\
abla\times\mu_{r}^{-1}(\
abla\times\mathbf{E})-k_{0}^{2}\bigg{(}\varepsilon _{r}-\frac{j\sigma}{\omega\varepsilon_{0}}\bigg{)}\mathbf{E}=0 \tag{1}\]  
where \(\mathbf{E}\) is the electric field (V/m); \(\mu_{r}\) is the relative magnetic permeability; \(\omega\) is the angular frequency; \(k_{0}\) is the wave number in vacuum; \(j\) is the imaginary unit; \(\sigma\) is the electrical conductivity (S/m); and \(\varepsilon_{r}\) is the relative permittivity. The expression of the latter was given as follows (F/m):  
\[\varepsilon_{r}=\epsilon^{\prime}-\frac{ie^{\prime\prime}}{t_{0}} \tag{2}\]  
where \(\epsilon^{\prime}\) is the real part of the permittivity (F/m); \(\varepsilon^{\prime\prime}\) is the imaginary part of the permittivity (F/m); and \(\varepsilon_{0}\) is the vacuum permittivity (F/m). The waveguide and cavity boundaries were assumed to be ideal conductors. The expression for the boundary condition was as follows:  
\[\mathbf{E}_{tangential}=0 \tag{3}\]  
The volumetric heating term for coal (\(Q_{e}(t)\)) was the dielectric loss, expressed as follows [30; 31]:  
\[Q_{e}(t)=\tfrac{1}{2}\omega\varepsilon_{0}\varepsilon^{\prime\prime}|\mathbf{ E}|^{2} \tag{4}\]  
The temperature distribution was obtained by solving the heat transfer equation, expressed as follows [32; 33]:  
\[\rho C_{p}\frac{\partial T}{\partial t}-\
abla\cdot(k\
abla T)=Q_{e}(t) \tag{5}\]  
where \(t\) is the time; \(T\) is the temperature (\({}^{\circ}\)C); \(\rho\) is the density (kg/m\({}^{3}\)); \(C_{p}\) is the specific heat capacity (J/kg/K); and \(k\) is the heat transfer coefficient (W/(m\({}^{\circ}\)K)). The boundary between air and the material was set as an adiabatic boundary, expressed as follows:  
\[-n\cdot q=0 \tag{6}\]  
The above temperature distribution results were used to calculate the mechanical response of the coal body under thermal loading, and the elastic strain tensor expression was expressed as follows [24]:  
\[E_{th}=\alpha_{T}\Big{(}T-T_{ref}\Big{)} \tag{7}\]  
where \(\alpha_{T}\) is the thermal expansion coefficient (K/1); and \(T_{ref}\) is the reference temperature for strain (zero-strain temperature) (\({}^{\circ}\)C). Considering the thermal stress constitutive relationship, the elemental deformation control equation was formulated by combining the equations of mechanical equilibrium, geometric deformation, and constitutive behavior [34; 35]. According to the law of the conservation of momentum in a homogeneous isotropic medium, the static equilibrium equation was expressed as follows:  
\[\sigma_{ij,j}+F_{i}=0 \tag{8}\]  
where \(\sigma_{ij}\) and \(F_{i}\) are the stress tensor (N/m\({}^{2}\)) and volume force, respectively (N/m\({}^{2}\)). According to the theory of elasticity, the relationship between strain and displacement was as follows:  
\[\epsilon_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,i}\big{)} \tag{9}\]  
where \(\epsilon_{ij}\) is the strain tensor; and \(u_{i,j}\) and \(u_{j,i}\) are the displacement components (\(i,j=1,2,3\)). The constitutive equation of coal deformation was expressed as follows:  
\[\sigma_{ij}\ =2G\epsilon_{ij}+2G\frac{v}{1-2v}\epsilon_{kk}\delta_{ij}-K\alpha _{T}T\delta_{ij} \tag{10}\]  
\[\varepsilon_{r}=\epsilon^{\prime}-\frac{ie^{\prime\prime}}{t_{0}} \tag{2}\]0  
\[\varepsilon_{r}=\epsilon^{\prime}-\frac{ie^{\prime\prime}}{t_{0}} \tag{2}\]1  
\[\varepsilon_{r}=\epsilon^{\prime}-\frac{ie^{\prime\prime}}{t_{0}} \tag{2}\]2  
where \(G\) is the shear modulus of coal (Pa); \(E\) is the elastic modulus of coal (Pa); \(v\) is [PERSON]'s ratio; \(\epsilon_{kk}\) is the volumetric strain; \(\delta_{ij}\) is the Kronecker delta function, defined as 1 for \(i=j\) and 0 for \(i\
eq j\); and \(K\) is the bulk modulus of coal.