# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 3 Modeling
### Analytical Stress Calculation and Failure Prediction
#### 3.4.3 Failure Criteria  
To determine if the rotor will fail for aforementioned load cases, the computed stresses are compared to the material strengths. For this comparison, three mathematical failure criteria are considered in this study, i.e., Maximum Stress Failure Criterion, Maximum Strain Failure Criterion and Tsai-Wu Failure Criterion. The Maximum Stress Failure Criterion is defined as follows:  
\[R_{\rm 1t} = \frac{\sigma_{\rm\theta\Theta}}{X^{\rm T}},\ {\rm for}\ \sigma_{\rm\theta\Theta} \geq 0\] \[R_{\rm 1c} = \frac{|\sigma_{\rm\theta\Theta}|}{X^{\rm C}},\ {\rm for}\ \sigma_{\rm\theta\Theta} < 0\] \[R_{\rm 3t} = \frac{\sigma_{\rm rr}}{Y^{\rm T}},\ {\rm for}\ \sigma_{\rm rr} \geq 0\] \[R_{\rm 3c} = \frac{|\sigma_{\rm rr}|}{Y^{\rm C}},\ {\rm for}\ \sigma_{\rm rr} < 0,\]  
where \(X^{\rm T}\), \(X^{\rm C}\), \(Y^{\rm T}\), \(Y^{\rm C}\), are the tensile and compressive material strengths of the longitudinal and transversal direction, respectively. \(R_{\rm 1t}\), \(R_{\rm 1c}\), \(R_{\rm 3t}\), and \(R_{\rm 3c}\) denote the strength ratios which indicate failure if their value is greater than unity. For this reason, it is sufficient to calculate the maximum of the strength ratios \(R_{\rm Stress}=\max\big{(}R^{\rm 1t},R^{\rm 1c},R^{\rm 3t},R^{\rm 3c}\big{)}\). Similarly, the Maximum Strain Ratio can be written as:  
\[R_{\rm 1t}=\frac{\epsilon_{\rm\theta\Theta}E_{1}}{X^{\rm T}}, \ \ \ {\rm for}\ \epsilon_{\rm\theta\Theta}\geq 0\] \[R_{\rm 1c}=\frac{|\epsilon_{\rm\theta\Theta}|E_{1}}{X^{\rm C}}, \ \ \ {\rm for}\ \epsilon_{\rm\theta\Theta}<0\] \[R_{\rm 2t}=\frac{\epsilon_{\rm zz}E_{2}}{Y^{\rm T}}, \ \ \ {\rm for}\ \epsilon_{\rm zz}\geq 0\] \[R_{\rm 2c}=\frac{|\epsilon_{\rm zz}|E_{2}}{Y^{\rm C}}, \ \ \ {\rm for}\ \epsilon_{\rm zz}<0\] \[R_{\rm 3t}=\frac{\epsilon_{\rm rr}G_{\rm 12}}{Y^{\rm T}}, \ \ \ \ {\rm for}\ \epsilon_{\rm rr}\geq 0\] \[R_{\rm 3c}=\frac{|\epsilon_{\rm rr}|G_{\rm 12}}{Y^{\rm C}}, \ \ \ {\rm for}\ \epsilon_{\rm rr}<0.\]  
Again, the strength ratio can be expressed by a single value  
\[R_{\rm Strain}=\max\Big{(}R^{\rm 1t},R^{\rm 1c},R^{\rm 2t},R^{\rm 2c},R^{\rm 3t},R^{\rm 3c}\Big{)}. \tag{22}\]  
The Tsai-Wu Failure Criterion is defined as [22]:  
\[R_{\rm T}=\frac{\overline{\bf F}\sigma+\sqrt{\big{(}\overline{\bf F}\sigma \big{)}^{2}+4\Big{(}\sigma^{\rm T}\widetilde{\bf F}\sigma\Big{)}}}{2} \tag{23}\]  
where \(\overline{\bf F}\) and \(\widetilde{\bf F}\) denote strength parameters defined by material properties, i.e.,  
\[\widetilde{\bf F}=\left[\begin{array}{cc}\frac{1}{X^{\rm T}X^{\rm C}}&- \frac{1}{2}\sqrt{\frac{1}{X^{\rm T}X^{\rm C}}\frac{1}{Y^{\rm T}Y^{\rm C}}}\\ -\frac{1}{2}\sqrt{\frac{1}{X^{\rm T}X^{\rm C}}\frac{1}{Y^{\rm T}Y^{\rm C}}}& \frac{1}{Y^{\rm T}Y^{\rm C}}\end{array}\right],\ \widetilde{\bf F}=\left[\begin{array}{cc} \frac{1}{X^{\rm T}}-\frac{1}{X^{\rm C}}&\frac{1}{Y^{\rm T}}-\frac{1}{Y^{\rm C}} \end{array}\right]. \tag{24}\]None of these criteria is conservative. To develop an optimization algorithm which prefers the safest design, all criteria must be considered. Therefore, the maximum of these criteria is calculated:  
\[R=\max(R_{\text{Stress}},R_{\text{Strain}},R_{\text{T}}). \tag{25}\]