## 2 Analytical Solution of Multiple Contact Pairs for the MLTWC
### Analytical Solution of Bi-layer Thick-walled Cylinders  
The left subgraph in Figure 2 describes the interference problem of bi-layer thick-walled cylinders. Within small elastic deformation, such problems satisfy superposition principle, and can be decomposed into two independent space axisymmetric problems of thick-walled cylinder, as shown in right side of the equal sign in Figure 2. The overall calculation process of interference algorithm for BTC is shown in the flow diagram, Figure 3.  
Figure 3: Calculation flow diagram for interference algorithm for BTC. Contact pressure is the key of solving interference problem of bi-layer thick-walled cylinders. With the contact pressure meeting all the constraints, the interior stress of the contact can be derived. Figure 2: Superposition principle for interference problem of bi-layer thick-walled cylinders (BTC). The inner and outer cylinders are represented by two different colors. \(E\), \(v\), and \(\alpha\) represent Young modulus, Poisson’s ratio, and coefficient of thermal expansion, respectively, and the index 1 and 2 represent outer and inner cylinders, respectively. The pressure on the contact surface is \(\mathrm{p_{i}}\). Before the contacting, the inside radius of the outer cylinder is \(r_{2,out}\), and the outside radius of the inner cylinder is \(r_{2,in}\). After the contacting, the radius of the contact surface is \(r_{2}\), and the inside radius of the inner cylinder is \(r_{3}\), and the outside radius of the outer cylinder is \(r_{1}\). The thick-walled cylinder with inner and outer radius of \(r_{2,out}\) and \(r_{1}\) respectively as shown in Figure 2 is under uniform internal pressure \(P_{i}\), and there is no external pressure. The Young modulus of this cylinder is \(E_{1}\); its Poisson's ratio is \(v_{1}\), and the thermal linear expansion coefficient is \(\alpha_{1}\). According to the theoretic solution of space axisymmetric problems of thick-walled cylinder, the radial displacement at the inner surface of this cylinder can be expressed as:  
\[u_{r,\;out|r=r_{2}}=\frac{r_{2,out}^{3}P_{i}\left(1-v_{1}\right)}{E_{1}\left( r_{1}^{2}-r_{2,out}^{2}\right)}+\frac{r_{1}^{2}r_{2,out}P_{i}\left(1+v_{1} \right)}{E_{1}\left(r_{1}^{2}-r_{2,out}^{2}\right)} \tag{4}\]  
where the subscript \(r\) in \(u_{r,\;out|r=r_{2}}\) represents radial direction, and the subscript out indicates outside of the contact surface. The thick-walled cylinder with inner and outer radius of \(r_{3}\) and \(r_{2,in}\) respectively as shown in Figure 2 is under uniform external pressure \(P_{i}\), and there is no internal pressure. The Young modulus of this cylinder is \(E_{2}\); its Poisson's ratio is \(v_{2}\), and the thermal linear expansion coefficient is \(\alpha_{2}\). According to the theoretic solution of space axisymmetric problems of thick-walled cylinder, the radial displacement at the outer surface of this cylinder can be expressed as:  
\[u_{r,\;in|r=r_{2}}=-\frac{r_{2,in}^{3}P_{i}\left(1-v_{2}\right)}{E_{2}\left( r_{2,in}^{2}-r_{3}^{2}\right)}-\frac{r_{2,in}r_{3}^{2}P_{i}\left(1+v_{2} \right)}{E_{2}\left(r_{2,in}^{2}-r_{3}^{2}\right)} \tag{5}\]  
where the subscript \(r\) in \(u_{r,\;in|r=r_{2}}\) represents radial direction, and the subscript \(in\) indicates inside of the contact surface. Assembling two concentric thick-walled cylinders together, \(r_{2,out}\) is slightly smaller than \(r_{2,in}\), and the difference between them is the interference \(\delta_{i}\) as shown in Equation (6), which determines the displacement coordination between the inner and outer cylinders on the contact surface. \[\delta_{i}=u_{r,out|r=r_{2}}-u_{r,in|r=r_{2}} \tag{6}\]  
The displacement along the radial direction is set to be positive, otherwise it is negative.