## 4 Numerical Analysis
### Distribution of Temperature Stress Field in Carbon Fiber-Heated Pavement  
Since the temperature mainly varied along the depth direction in the process of heating pavement with carbon fiber, the component of the principal stress along the depth direction (later referred to as the temperature stress) was chosen for the temperature stress studied in this paper to assess the safety of the heating process. Based on the above numerical model, this section investigates the temperature stress field distribution of concrete pavement heated by carbon fiber heating wire.  
Figure 10 depicts a cloud image of the temperature stress field fluctuation for various heating periods. The compressive stress was greater than the tensile stress. The stresses were mostly dispersed at the carbon fiber hairline at 0 h of heating, and the tensile and compressive stresses were quite minor. The maximum tensile and compressive stresses were 0.01 MPa and 0.06 MPa, respectively. Temperature stress was essentially non-existent in other areas of the pavement. The tensile and compressive stresses exhibited a tendency of initially rising and then stabilizing with an increase in heating time. The compressive tension around the carbon fiber heating wire steadily rose and expanded in all directions after 2 h of heating. The compressive tension decreased with increasing distance from the carbon fiber hairline and eventually turned into tensile stress. The aforementioned characteristics demonstrated that while the road surface closest to the carbon fiber hot line warmed up first, the road surface farther away did not warm up as quickly. This created a temperature gradient that caused the concrete road surface to heat unevenly, subjecting the concrete road surface's thermal expansion to the limitations of the surrounding concrete road surface. The result was compressive stresses on the road surface closest to the carbon fiber hot line and tensile stresses in the area between the two carbon fibers. Compressive stress, which was mostly spread close to the carbon fiber hairline, was the pavement's highest stress.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Depth (cm)**} & \multicolumn{2}{c}{**RMSE**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Temperature (\({}^{\circ}\)C)**} & \multicolumn{2}{c}{**Thermal Strain (\(\mathbf{\mu\epsilon}\))**} \\ \hline
2 & 0.53 & 23.82 \\
5 & 0.70 & 22.64 \\
10 & 0.29 & 20.74 \\
15 & 0.47 & 2.99 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Root mean square error analysis of experimental results and simulation results.  
Figure 9: Comparison of test results and simulation results of \(\text{TC}_{\text{A}}\) and \(\text{TS}_{\text{A}}\) at different depths.  
Figure 11 depicts the temperature stress trend over time in the presence of carbon fiber heating wires at various depths. The carbon fiber heating wires were buried in the concrete pavement at a depth of 5 cm. The thermal stress of the pavement at various depths increased and then smoothed out over time. The pavement, for example, was subjected to compressive stress at a depth of 5 cm, and temperature stress was greatest at this level. The compressive stress increased quickly from 0 MPa to 4 MPa with a rate of 4 MPa/h during heating from 0 h to 1 h. After 1 h of heating, the compressive stress was stabilized at 4 MPa when the heating duration was increased. According to the warming trend, 6 h of heating time did not stabilize the temperature of the pavement model, and the temperature continued to rise, although at a slower pace, so the compressive stress remained stable at around 4 MPa. The maximum tensile stress of 1 MPa and maximum pressure of 4 MPa were both less than the design value of compressive strength of C40 concrete pavement in the concrete pavement design specification, indicating that the concrete pavement was safe in the carbon fiber heating process.  
Figure 10: The numerical model of carbon fiber–heated pavement’s temperature stress field distribution at different times.