# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 3 Modeling
### Analytical Stress Calculation and Failure Prediction
#### 3.4.2 Analytical Stress Calculation  
An analytical stress analysis which also considers interference and residual stresses can be found in previous publications [9; 14] and is therefore only summarized. Introducing the radial and circumferential stresses \(\sigma_{\mathrm{r}}\) and \(\sigma_{\Theta}\), respectively, the following equation can be written  
\[\frac{\partial\sigma_{\mathrm{rr}}}{\partial r}+\frac{\sigma_{\mathrm{rr}}- \sigma_{\Theta\Theta}}{r}+\rho r\omega^{2}=0. \tag{14}\]  
The stress-strain-temperature relation is defined as  
\[\mathbf{\sigma}=\mathbf{Q}(\mathbf{\epsilon}-\mathbf{\alpha}\Delta T). \tag{15}\]  
where \(\mathbf{Q}\) is the stiffness matrix expressed in cylindrical coordinates, \(\mathbf{\alpha}\) is the thermal expansion coefficient vector, \(\Delta T\) is the temperature change, \(\mathbf{\epsilon}\) and \(\mathbf{\sigma}\) are the vector of strains and vector of stresses, respectively. The temperature change is based on the temperature difference between the stress-free curing temperature and the operating temperature. Hence, a higher curing temperature will lead to increased residual stresses. The stiffness matrix \(\mathbf{Q}\) is defined by the material properties [22] and can be expressed as:  
\[\left[\begin{array}{ccc}\frac{1}{E_{\Theta}}&-\frac{\
u_{\Theta\tau}}{E_{ \mathrm{r}}}&-\frac{\
u_{\Theta\tau}}{E_{\mathrm{z}}}\\ &\frac{1}{E_{\mathrm{r}}}&-\frac{\
u_{\mathrm{rz}}}{E_{\mathrm{z}}}\\ \mathrm{sym.}&&\frac{1}{E_{\mathrm{z}}}\end{array}\right]^{-1}. \tag{16}\]  
For flywheels with a low height to diameter aspect ratio, a plane stress state can be assumed. Therefore, the stresses \(\sigma_{\mathrm{zz}}\) and \(\sigma_{\Theta\tau}\) can be neglected. Each of the equations discussed in this subsection can be applied to any rim of the rotor. To calculate stresses and displacements for the entire rotor \(2(N_{\mathrm{rim}}-1)\), continuity conditions are introduced. The radial stresses are continuous at the rim surfaces, while the radial displacements may differ by an interference \(\delta^{(i)}\) due to press fitting.  
\[\sigma_{r_{i}}^{(i+1)}=\sigma_{r_{o}}^{(i)}, \tag{17}\]  
\[u_{r_{i}}^{(i+1)}=u_{r_{o}}^{(i)}+\delta^{(i)}, \tag{18}\]  
for \(i=1\) to \(N_{\mathrm{rim}}-1\). The continuity of radial stresses is only valid as long as the rims are attached to each other. In case of interference fits the attachment vanishes when the tensile stresses induced by rotation exceed the radial stresses due to the interference fit. Hence, a computed tensile stress would lead to a detachment failure.  
Since there are \(2N_{\rm rim}\) constraint equations and therefore \(2N_{\rm rim}\) unknown constants, two additional equations are required for the solution. They can be obtained by predefining the radial stresses at the innermost and outermost rotor radius.  
\[\sigma_{\rm r_{1}}^{(1)}=p_{\rm in},\ \sigma_{\rm r_{o}}^{(N_{\rm rim})}=-p_{\rm out}, \tag{19}\]  
with the inner and outer pressure \(p_{\rm in}\) and \(p_{\rm out}\), respectively. In this study, \(p_{\rm in}\) and \(p_{\rm out}\) are set to zero.