## 3 Development of Analytical Crack Breathing Model
\[\bar{I}_{1}=\frac{\pi R^{4}}{8}+\frac{R^{4}}{4}\Big{[}(1-\mu)\Big{(}2\mu^{2}-4 \mu+1\Big{)}\gamma+sin^{-1}(1-\mu)\Big{]}-A_{1}e^{2}, \tag{6}\]  
\[\bar{I}_{2}=\frac{\pi R^{4}}{4}-\frac{R^{4}}{12}\Big{[}(1-\mu)\Big{(}2\mu^{2}- 4\mu-3\Big{)}\gamma+3 sin^{-1}(\gamma)\Big{]}. \tag{7}\]  
Figure 7: Milestones in the breathing of a fatigue crack; (**a**) initial position, (**b**) crack begins closing and (**c**) crack is fully closed. The angle of inclination of the neutral axis to the bending moment vector \(\xi\) is determined by  
\[\xi=\Omega t+\phi^{*}-\frac{\pi}{2}. \tag{8}\]  
Now, the crack closing angle \(\theta_{1}\) for this study can be numerically determined using Equation (9). Full derivation of this equation can be found in Appendix A.  
\[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]0  
For the horizontal neutral axis (HNA) model [12], the crack closing angle was determined as  
\[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]1  
The comparison between Equations (9) and (10) for crack depths \(\mu\) = 0 to \(\mu\) = 1 is shown in \[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]2 There is a clear divergence between the two models with the proposed model showing that the crack closes later when compared to the HNA model. The difference in the predicted crack closing angle becomes more severe at deeper cracks, where at \(\mu\) = 1, a 33.6\({}^{\circ}\) difference occurs (56.6\({}^{\circ}\) versus 23.0\({}^{\circ}\)). The comparison suggests that deeper cracks result in a more pronounced difference between the two models; hence, further investigations into the proposed model are valuable for deeper cracks. Figure 7c depicts the instant the crack is completely in compression and thus becomes fully closed. This occurs when the shaft rotation angle is equal to \(\theta_{2}\), where \(\theta_{2}\) can be calculated by  
\[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]3  
It is worth noting that in the second half of the crack's rotation, the crack remains closed until \(\Omega t\) = \(2\pi-\theta_{2}\) and then gradually opens until \(\Omega t\) = \(2\pi-\theta_{1}\), where it becomes fully opened again. Furthermore, the crack is partially open between these two angles. Table 2 summarises the crack status over one period of the shaft rotation. \[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]4 Comparison of the crack closing angle for the proposed model and horizontal neutral axis model. One method for illustrating crack breathing behaviour is to determine how much of the crack has closed as the shaft rotates, such as in \[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]5 To gauge the crack closing amount \(\lambda\), the following formula was used  
\[A_{1}=R^{2}(\pi-cos^{-1}(1-\mu)+(1-\mu)\gamma) \tag{1}\]6  
where \(\tilde{A}=\pi R^{2}\) (area of a full circle) and \(A_{cc}\) is the total non-cracked area. For the presented model, the value of \(A_{cc}\) was determined using the contents in Appendix A. Additionally, for the FEA model, the contact area of the crack was probed directly from the non-linear contact surface. At the crack depths chosen, \(\mu=0.5\), \(\mu=0.7\) and \(\mu=1.0\), the proposed model begins closing later, i.e., \(\lambda\) increases from 0% at a higher shaft rotation angle. Moreover, the proposed model becomes fully open sooner, i.e., \(\lambda\) reaches 0% again sooner when compared to the HNA model; this is merely a consequence of the \(\theta_{1}\) values being larger for the proposed model. Furthermore, the two crack models become fully closed at the same time, i.e., both models reach a \(\lambda\) value of 100%, and also begin opening again at the same time as the \(\theta_{2}\) values are identical. As such, when the crack is partially open (\(\lambda\) is not 0% or 100%) the HNA model overestimates the closed area of the crack which results in stiffer cracked rotor models when predicting vibration. Furthermore, the deeper the crack the more pronounced the overestimation of \(\theta_{1}\) becomes.