## 3 Results and Discussion
### Finite Element Analysis  
To investigate the relation between heat flux and temperature distribution in the radial direction, four types of heat flux (Figure 5) were applied on the frictional surface of the brake disc: (1) uniform distribution in the radial direction; (2) linearly increasing heat flux, which means uniform pressure in the radial direction, if the friction coefficient, \(\mu\) is constant at every instant; (3) quadratic distribution; (4) quartic distribution. The total amount of heat input on the contact surface was the same for all cases. Figure 6 shows the thermal and stress fields for the above-mentioned four types of heat fluxes at t = 16.26 s after brake application. Figure 7 represents the heat flux, temperature and stress distributions in the function of the radius on the surface of the disc. r = 0 and r = 135 are the inner and outer radii of the frictional surface of the disc. The ranges of maximum temperatures and von Mises stresses were 198-278 \({}^{\circ}\)C, and 165-378 MPa, respectively. The maximum temperatures and stresses along with positions on the surface are summarized in Table 1. The maximum temperature increased in the order of uniform, linear, quadratic, and quartic heat flux. In the case of the linear heat flux, the von Mises and absolute principal stress were minimum. The maximum von Mises stress and the absolute principal stress occurred in the same position except in the case of the linear heat flux. In the case of the uniform heat flux, the maximum stress occurred near the inner radius (r = 4). In the case of the linear heat flux, the variation in stress magnitude over the radius was not great. The results for the quadratic and quartic heat fluxes showed that the maximum stresses were generated in the middle of the heat flux bands. And the maximum stresses were greater than those for the uniform and linear heat fluxes. Therefore, in order to lower maximum thermal stress due to frictional heat, conformal contact between the disc and pads is desirable. Figure 4: Infrared images of stop braking and a temperature profile along a radial line. (**a**) 4.805 s after brake application, (**b**) 6.940 s, (**c**) 7.341 s, (**d**) 8.809 s, (**e**) 33.634 s, (**f**) 34.835 s, (**g**) 35.502 s, (**h**) 36.303 s.  
Figure 5: Four types of heat flux input in the radial direction on the brake disc surface. Figure 6: Thermal and stress fields at t = 16.26 s after brake application. (**a**) Uniform, (**b**) Linear, (**c**) Quadratic, (**d**) Quartic. Figure 8 shows the effect of boundary conditions on the stress along the symmetric line on the surface of the disc. r = 0 and r = 135 are the inner and outer radii of the frictional surface of the disc, respectively. Compared to the fixed bolt hole and fixed \(u_{y}\), the \(u_{y}\) free condition on the sides of the disc lowered the maximum von Mises stress by 27.5 MPa (10.3%), and the bolt hole contact condition gave a 15.8 MPa (5.9%) lower von Mises stress. Applying conditions-where bolt hole contact and \(u_{y}(u_{2})\) were free, lowered the maximum von Mises stress by 31.8 MPa (11.9%). Therefore, in terms of thermal stress, a disc composed of two pieces seems to be more thermoresistant. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type of** & \multirow{2}{*}{**T\({}_{\text{max}}\) (\({}^{\circ}\)C)**} & \multirow{2}{*}{**r (mm)**} & \multicolumn{2}{c}{**[PERSON]**} & \multirow{2}{*}{**r (mm)**} & \multirow{2}{*}{**Abs. Max Principal**} & \multirow{2}{*}{**r (mm)**} \\
**Heat Flux** & & & & \(\sigma\) (MPa) & & \(\sigma\) (MPa) \\ \hline Uniform & 180.4 & 82.1 & 233.4 & 4.0 & \(-255.4\) & 4.0 \\ Linear & 201.5 & 109.3 & 188.8 & 65.1 & \(-207.1\) & 91.0 \\ Quadratic & 242.1 & 65.1 & 310.8 & 58.2 & \(-355.3\) & 58.2 \\ Quartic & 262.9 & 10.26 & 380.2 & 16.9 & \(-424.1\) & 16.9 \\ \hline \hline \end{tabular}
\end{table}