## Physical model of failure
### Analytical Model for Temperature Loads  
Motor components generate various losses in the operation process, mainly including stator and rotor core losses, eddy current losses of permanent magnets, copper losses of windings, and mechanical losses. Losses are the source of heat that causes temperature variations in various parts of the motor. Therefore, when solving for the temperature distribution of the motor, the heating rate of each part of the motor must be obtained. The core losses of the stator and rotor as well as the eddy current losses of the permanent magnets can be obtained by an electromagnetic simulation. Assuming the speed of the motor is 1000 rpm and since eddy current losses are proportional to the square of the frequency, Equation (2) can be used to derive the losses for other operating conditions at 1000 rpm, thus covering the entire operating condition at the current speed [9].  
\[P(n,T)=\Big{(}\frac{n}{1000}\Big{)}^{2}P(1000,T) \tag{2}\]  
where \(P\) (\(n\), \(T\)) is the eddy current loss when the speed is \(n\) rpm, and the torque is \(T\). The winding copper consumption and mechanical losses can be calculated using the empirical formulae in the literature [15]. The calculation of the motor thermal field is based on the basic theory of thermodynamics. In cartesian coordinates, the boundary conditions and calculation equations of the three-dimensional heat conduction problem are as follows [16]:  
\[\left.\begin{array}{l}\frac{\partial}{\partial x}\Big{(}\lambda_{x}\frac{ \partial T}{\partial x}\Big{)}+\frac{\partial}{\partial y}\Big{(}\lambda_{y} \frac{\partial T}{\partial y}\Big{)}+\frac{\partial}{\partial z}\Big{(} \lambda_{z}\frac{\partial T}{\partial z}\Big{)}+qv=\rho c\frac{\partial T}{ \partial t},\\ -\lambda\frac{\partial T}{\partial n}\Big{|}_{s1}=q_{0},\\ \lambda\frac{\partial T}{\partial n}\Big{|}_{s2}=-\alpha(T-T_{w}).\end{array}\right\} \tag{3}\]  
where \(\lambda_{x}\), \(\lambda_{y}\), and \(\lambda_{z}\) are the thermal conductivity of the object in the \(x\), \(y\), and \(z\) directions in the thermal field; \(T\) is the motor temperature; \(qv\) is the heat source density of each part inside the motor; \(\rho\) is the density; \(c\) is the specific heat capacity; \(t\) is the time; \(s_{1},s_{2}\) is the edge surface of the thermal field; \(q_{0}\) is the heat flux density across the boundary surface \(s_{1}\); \(n\) is the boundary vector of the object; \(T_{w}\) is the temperature of the surrounding objects at the boundary \(s_{2}\); \(\alpha\) is the heat dissipation coefficient of the edge surface. The stator surface dissipation coefficient \(\alpha_{s}\) and the rotor surface heat dissipation coefficient \(\alpha_{r}\) can be calculated using Equation (4) [17].  
\[\left.\begin{array}{l}\alpha_{s}=\frac{1+0.04v}{0.045}\\ \alpha_{r}=28(1+\sqrt{0.45v})\end{array}\right\} \tag{4}\]  
where \(v\) is the circumferential linear speed of the rotor. When the internal temperature of the motor changes, a temperature gradient is formed within the rotor due to the different heat generation rates in the structure, and thermal stresses are generated as a result [18]. Based on the theory of thermo-elasticity, the equation for calculating the thermal stress on the structure is as follows [19].  
\[\left.\begin{array}{l}\sigma_{ij,j}=0,\\ \epsilon_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,i}\big{)},\\ \epsilon_{ij}=\frac{1+\varphi}{E}\sigma_{ij}-\frac{v}{E}\alpha_{ik}\delta_{ij }+\beta\Delta T.\end{array}\right\} \tag{5}\]  
where \(\beta\) is the coefficient of thermal expansion of the rotor and \(\Delta T\) is the temperature variation.