## 2 Numerical Methodology
### Model Validation  
In our previous work, we experimentally validated the accuracy and reliability of the electromagnetic heating of coal [29]. In order to demonstrate the reliability of thermal damage, the ETMD model was verified by the thermal damage experiment of [PERSON] [44]. In this experiment, a rock sample was heated to 800 \({}^{\circ}\)C and held for 2 h. The uniaxial compression strength of the rock specimen was compared before and after heat treatment. The settings of the simulations for heat and uniaxial compression (Figure 7) were as follows. The material parameters (Table 5) were as follows [44; 45]. Figure 6: Relationship between the number of elements and the NPA. Figure 7: Thermal damage model: (**a**) heating model and (**b**) uniaxial compression model. The stress-strain curves of rocks before and after thermal damage calculated from this model were largely consistent with the experiment. The peak strength and elasticity modulus of the rock were reduced after thermal damage (Figure 8). The overall relative deviation in the untreated group was 12.41%, with a maximum relative deviation of 14.68%. In the heating group, the overall relative deviation was 10.13%, with the maximum relative deviation reaching 68.03%. The maximum deviations were concentrated in the elastic deformation stage, mainly due to the stronger inhomogeneous distribution of the experimental coal samples compared to the simulated coal models. The stronger random heterogeneity of rocks led to more significant upward fluctuations in the stress curve during the elastic stage. In addition, there were differences in the strength reduction after thermal damage. This was due to the strength distributions of the rock samples used in the experimental fracturing being inherently different, while the strength distribution of the rock samples used in the simulation was consistent. The overall deviations were relatively small. These discrepancies were kept within reasonable bounds. This indicated that the present thermal damage model accurately calculated and derived the damage extent of the coal sample by the thermal damage process. Therefore, the thermal damage in the ETMD model was reasonable and reliable. In a previous work, a coupled electromagnetic-thermomechanical (ETM) model without damage judgment was used to study coal damage by microwave heating [24,46]. In the ETM model, Von Mises stresses were used to speculate on the rock damage caused by microwave radiation [24,47]. When the Von Mises stress was larger than its yield stress, the rock was considered to experience plastic failure. New cracks would form and increase with  
Figure 8: Stress–strain curves of numerical and experimental results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Homogeneity index, m & 10 & - \\ Elastic modulus, E & 37.6 & GPa \\ Uniaxial compressive strength, \(f_{c}\) & 183.2 & MPa \\ Uniaxial tensile strength, \(f_{t}\) & 22.9 & MPa \\ Density, \(\rho\) & 2760 & kg/m\({}^{3}\) \\ Poisson’s ratio, \(\
u\) & 0.25 & - \\ Thermal conductivity, \(k\) & 0.1 & W/(m\(\cdot\)k) \\ Specific heat capacity, \(C_{p}\) & 700 & J/(kg\(\cdot\)K) \\ Thermal expansion coefficient, \(\alpha_{T}\) & \(2.0\times 10^{-6}\) & 1/K \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mechanical and thermal parameters. the microwave heating time. The coal sample was heated in the 0\({}^{\circ}\) position for 15 s by the ETM and ETMD models. It was thus found that the Von Mises stresses were not uniformly distributed around each pyrite particle (Figure 9a) due to the non-uniform distribution of the electromagnetic field intensity (Figure 10 S\({}_{0}\)M). The peak value of Von Mises stresses occurred mainly near pyrite A\({}_{0}\) and D\({}_{0}\). Moreover, the Von Mises stresses of pyrite B\({}_{0}\) and E\({}_{0}\) were also noticeable. Therefore, it was predictable for the crack to first form near pyrite A\({}_{0}\) and D\({}_{0}\) and then propagate to pyrite B\({}_{0}\) and E\({}_{0}\).