## 2 Composite Flywheel Rotor Modeling
### Tsai-Wu Failure Criterion  
The general Tsai-Wu failure criterion, described in [6; 7; 26; 27], can be reduced to nine terms for a transversely isotropic material and considering the absence of shear stresses. This criterion finds a relationship, \(F\), between the applied stress tensor and the material tensile strengths and predicts failure when \(F\geq 1\). At failure, the stress tensor represents the maximum allowable stress and \(F\) equals unity. Hence, the Tsai-Wu failure criterion can be written as  
\[\begin{split} F=F_{11}\big{(}\sigma_{1}^{\text{all}}\big{)}^{2}+F _{22}\big{(}\sigma_{2}^{\text{all}}\big{)}^{2}+F_{33}\big{(}\sigma_{3}^{\text {all}}\big{)}^{2}+2F_{12}\sigma_{1}^{\text{all}}\sigma_{2}^{\text{all}}+2F_{13 }\sigma_{1}^{\text{all}}\sigma_{3}^{\text{all}}+\\ 2F_{23}\sigma_{2}^{\text{all}}\sigma_{3}^{\text{all}}+F_{1} \sigma_{1}^{\text{all}}+F_{2}\sigma_{2}^{\text{all}}+F_{3}\sigma_{3}^{\text{ all}}=1,\end{split} \tag{10}\]  
where \(\big{(}\sigma_{i}^{\text{all}}\big{)}\) is the allowable stress in the \(i=1,2,\) or \(3\) directions at an arbitrary point in the rotor. Note as applied herein, the \(1\) and \(3\) directions refer to the circumferential and radial stress, respectively, while the \(2\) direction refers to the axial direction. While the plane strain condition eliminates axial strain, it allows for axial stress; therefore, it is included in the failure criterion. Then,  
\[\begin{split} F_{11}=\frac{1}{\sigma_{1}\sigma_{1}},\;F_{22}=F _{33}=\frac{1}{\sigma_{3}\sigma_{3}\sigma_{3}},\;F_{1}=\frac{1}{\sigma_{1}}- \frac{1}{\sigma_{1}},\\ F_{2}=F_{3}=\frac{1}{\sigma_{3}}-\frac{1}{\sigma_{3}},\;F_{12}=F _{13}=\frac{-1}{2\sqrt{\sigma_{1}\sigma_{1}\sigma_{1}\sigma_{3}\sigma_{3}}},\;F _{23}=F_{22}-\frac{1}{2\sigma_{23}^{2}},\end{split} \tag{11}\]  
where the subscripts \(t\) and \(c\) refer to the tensile and compressive ultimate strengths, respectively. The strength coefficients in the \(2\) and \(3\) directions are equal due to the transversely isotropic assumption, discussed further in Section 3.1.  
It is common, and more valuable, to define the relationship between maximum allowable stress and the applied stress as the failure ratio (\(R\)) [7]. This relationship is found by combining the maximum allowable stress tensor, \(\big{(}\sigma_{i}^{\text{all}}\big{)}\), with the applied stress tensor, \(\Big{(}\sigma_{i}^{\text{app}}\Big{)}\), multiplied with \(R\) such that  
\[\sigma_{1}^{\text{all}}=R\sigma_{1}^{\text{app}},\;\sigma_{2}^{\text{all}}=R \sigma_{2}^{\text{app}},\;\sigma_{3}^{\text{all}}=R\sigma_{3}^{\text{app}}. \tag{12}\]  
Then, substituting Equation (12) into Equation (10) yields a quadratic equation, i.e.,  
\[\begin{split} 0=\bigg{[}F_{11}\big{(}\sigma_{1}^{\text{app}} \big{)}^{2}+F_{22}\big{(}\sigma_{2}^{\text{app}}\big{)}^{2}+F_{33}\big{(} \sigma_{3}^{\text{app}}\big{)}^{2}+2F_{12}\sigma_{1}^{\text{app}}\sigma_{2}^{ \text{app}}+2F_{13}\sigma_{1}^{\text{app}}\sigma_{3}^{\text{app}}\\ +2F_{23}\sigma_{2}^{\text{app}}\sigma_{3}^{\text{app}}\bigg{]}R^{ 2}+\Big{[}F_{1}\sigma_{1}^{\text{app}}+F_{2}\sigma_{2}^{\text{app}}+F_{3} \sigma_{3}^{\text{app}}\Big{]}R-1.\end{split} \tag{13}\]  
Solving this quadratic equation for \(R\) defines the failure ratio. When \(\Big{(}\sigma_{i}^{\text{app}}\Big{)}\) equals \(\big{(}\sigma_{i}^{\text{all}}\big{)}\), then \(R\) equals unity, indicating failure, whereas \(R>1\) indicates \(\Big{(}\sigma_{i}^{\text{app}}\Big{)}\) is less than \(\big{(}\sigma_{i}^{\text{all}}\big{)}\) and no failure is predicted. It is convenient to define a strength ratio (\(SR\)) to be \(1/R\), as this is more intuitive conceptually and graphically [7]. Failure under this criterion is predicted when \(SR\geq 1\).