## 2 Tests and Modeling
### Heat Transfer  
To obtain thermal fields in the disc during a stop braking, finite element analysis was carried out. When a car is decelerating from \(V_{0}\), the braking power \(P\), is given by  
\[P=Q\omega=Q(V_{0}-\alpha t)/r_{\omega} \tag{1}\]  
where \(Q\) is the torque applied on the disc surfaces by the brake pads, \(\omega\) is the angular velocity of the axle, \(V_{0}\) is the initial braking speed of the vehicle, \(\alpha\) is the deceleration of the vehicle, and \(r_{w}\) is the wheel diameter of the vehicle. The torque \(Q\), on one side of the disc is  
\[Q=\int_{r_{i}}^{r_{o}}\mu p(r)2\pi rdr \tag{2}\]  
where \(\mu\) is the friction coefficient between the disc and pads, and \(p(r)\) is the contact pressure between the disc and pads. It is assumed that the contact pressure is uniform in the circumferential direction. This assumption is acceptable because infrared images show similar temperatures in the circumferential direction. \(r_{i}\) = 195 mm, and \(r_{o}\) = 330 mm are the inner and outer radii of the disc, respectively. During braking, some of the frictional heat is transferred to the disc, and the rest to the brake pads. The proportion of braking energy transferred to the disc, and pads can be calculated from a theoretical  
Figure 1: Brake dynamometer. 1-D.C. motor, 2-bearing, 3-Inertial wheel, 4-Tread brake, 5-air cylinder, 6-brake disc, 7-pad, 8-control box. (**a**) Schematic diagram; (**b**) Photograph.  
Figure 2: Ventilated cast-iron brake disc and composite brake pad used in this study. (**a**) Brake disc mounted to the axle; (**b**) Assembled brake disc; (**c**) Composite brake pad.  
solution of temperature fields for semi-infinite solids [16]. In this study, it was assumed that 90% of the frictional heat was given to the disc, and radiation heat exchange was negligible.  
In this study, an isotropic and homogeneous disc was considered. And an uncoupled thermo-elastic problem was solved. The heat conduction equation and boundary conditions take the forms of Cartesian coordinates.  
\[\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}+ \frac{\partial^{2}T}{\partial z^{2}}+\frac{\Omega}{k}=\frac{\rho c}{k}\frac{ \partial T}{\partial t}\text{ in D, t }\geq 0 \tag{3}\]  
\[-k\frac{\partial T}{\partial n}=q\text{ on }\Gamma_{1}\text{, t }\geq 0 \tag{4}\]  
\[-k\frac{\partial T}{\partial n}=h(T-T_{\infty})\text{ on }\Gamma_{1}\text{, }\Gamma_{2}\text{, }\Gamma_{3}\text{, }\Gamma_{5}\text{, }\Gamma_{7}\text{, t }\geq 0 \tag{5}\]  
\[-k\frac{\partial T}{\partial n}=0\text{ on }\Gamma_{4}\text{, }\Gamma_{6}\text{, t }\geq 0 \tag{6}\]  
\[T(x,y,z,0)=T_{0}\text{ in D, t }=0 \tag{7}\]  
where \(n\): normal to the boundary; \(T_{\infty}\): the ambient temperature; \(T_{0}\): initial temperature of the disc. The convective heat transfer coefficient, \(h\) was calculated through fluid flow analysis for a ventilated railway brake disc [9]. It can be expressed in a function of the Reynolds number, Re:  
\[h=0.6698\frac{k_{air}}{r_{0}}\text{Re}^{0.6164}=15.9219\
u_{0}^{0.2328}w^{0.6164} \tag{8}\]  
where \(k_{air}=0.025\) (W/m\({}^{\circ}\)C) is the thermal conductivity of the air. All numerical analyses were performed using Abaqus (Ver. 6.16) software (Company, City, State, Country, Dassault Systemes HQ, Waltham, MA, USA). Figure 3 shows the finite element model. The disc consisted of 407,775 quadratic tetrahedron elements. Considering the symmetry of the disc, one-fourth of the disc was included in the model. The values of the specific heat, density, thermal conductivity and thermal expansion coefficient used in the simulation were given in the function of temperatures in reference [17]. Heat flux of 40 s was given to a stop braking as shown in Figure 3. The two seconds in the beginning is the air pressure build-up time of the brake cylinder.