## Appendix A  
Figure 11a shows the crack in the starting position. From this position, the non-cracked area can be given by [12]:  
\[A_{1}=R^{2}[\pi-cos^{-1}(1-\mu)-(1-\mu)\gamma] \tag{1}\]also the radial distance of the centroid of area \(A_{1}\) from the origin \(O\) is  
\[e=\frac{2R^{3}}{3A_{1}}\gamma^{3}. \tag{10}\]  
The area moment of inertia of the crack segment about the \(X\) and \(Y\) axes are  
\[I_{X}^{c}=\frac{\pi R^{4}}{8}-\frac{R^{4}}{4}\Big{[}(1-\mu)\Big{(}2\mu^{2}-4\mu+ 1\Big{)}\gamma+sin^{-1}(1-\mu)\Big{]}, \tag{11}\]  
\[I_{Y}^{c}=\frac{R^{4}}{12}\Big{[}(1-\mu)\Big{(}2\mu^{2}-4\mu-3\Big{)}\gamma+3 sin^{-1}(\gamma)\Big{]}. \tag{12}\]  
The area moment of inertia of non-cracked segment \(A_{1}\) about the \(X\) and \(Y\) axes, respectively, are  
\[I_{1}=I-I_{X}^{c}, \tag{13}\]  
\[I_{2}=I-I_{Y}^{c} \tag{14}\]  
where, \(I=\pi R^{4}/4\). Using the parallel axis theorem, the area moment of inertia of \(A_{1}\) about the \(\overline{X}\) and \(\overline{Y}\) axes yields  
\[\overline{I}_{1}=I_{1}-A_{1}e^{2}=\frac{\pi R^{4}}{8}+\frac{R^{4}}{4}\Big{[}( 1-\mu)\Big{(}2\mu^{2}-4\mu+1\Big{)}\gamma+sin^{-1}(1-\mu)\Big{]}-A_{1}e^{2}, \tag{15}\]  
\[\overline{I}_{2}=I_{2}=\frac{\pi R^{4}}{4}-\frac{R^{4}}{12}\Big{[}(1-\mu) \Big{(}2\mu^{2}-4\mu-3\Big{)}\gamma+3 sin^{-1}(\gamma)\Big{]} \tag{16}\]  
_Appendix A.1 Fully Open Crack Equations_  
These equations are only valid for \(0\leq\Omega t\leq\theta_{1}\) and \(2\pi-\theta_{1}\leq\Omega t\leq 2\pi\).  
Using Figure 10, the distance between the \(Y\) and \(\overline{Y}\) axes is denoted \(X_{\alpha}\) and the distance between the \(X\) and \(\overline{X}\) axis is \(Y_{\alpha}\) and these values are calculated by  
\[X_{\alpha}=-e\,sin(\Omega t) \tag{17}\]  
\[Y_{\alpha}=e\,cos(\Omega t). \tag{18}\]  
Additionally, for a fully open crack the total non-cracked area is equivalent to \(A_{1}\)  
\[A_{\alpha}=R^{2}[\pi-cos^{-1}(1-\mu)-(1-\mu)\gamma], \tag{19}\]  
the angle between the first principal axis \(U\) and the bending moment vector \(\theta^{*}\) and is given as  
\[\theta^{*}(t)=\frac{\pi}{2}-\Omega t. \tag{20}\]  
Furthermore, in [24], the angle between the first principal axis \(U\) and the neutral axis is given by  
\[\phi^{*}(t)=tan^{-1}\Big{(}\frac{I_{U}}{I_{V}}tan\theta^{*}\Big{)}. \tag{21}\]  
where \(I_{U}\) and \(I_{V}\) are the area moments of inertia of \(A_{1}\) about the principal centroidal axes. The angle of inclination of the neutral axis to the bending moment vector \(\xi\) is determined by  
\[\xi(t)=\Omega t+\phi^{*}-\frac{\pi}{2}. \tag{22}\]For a fully open crack the \(\overline{X}\)-\(\overline{Y}\) plane is rotated by \(\Omega t\) clockwise from the principal plane. As such, the product and area moment of inertia of the uncracked area about the \(\overline{X}\) and \(\overline{Y}\) axes are  
\[I_{\overline{X}}(t)=\frac{\overline{I}_{1}+\overline{I}_{2}}{2}+\frac{ \overline{I}_{1}-\overline{I}_{2}}{2}cos(2\Omega t), \tag{115}\]  
\[I_{\overline{Y}}(t)=\frac{\overline{I}_{1}+\overline{I}_{2}}{2}-\frac{ \overline{I}_{1}-\overline{I}_{2}}{2}cos(2\Omega t), \tag{116}\]  
\[I_{\overline{X}\overline{Y}}(t)=-\frac{\overline{I}_{1}-\overline{I}_{2}}{2} sin(2\Omega t). \tag{117}\]  
At the instant the crack begins closing (\(\Omega t=\theta_{1}\)), the leading vertex of the crack becomes coincident with the neutral axis, creating a right-angled triangular region bounded by half of the crack front and the \(U\)-axis, as seen in Figure 11b; this region has been isolated in Figure 12, where  
\[tan(\phi^{*})=\frac{s}{e+b}. \tag{118}\]  
and \(s=R\gamma\) and \(b=R(1-\mu)\) (these identities are provided in [12]).  
Substituting Equation (113) into Equation (118), knowing that \(I_{V}=\overline{I}_{1}\) and \(I_{U}=\overline{I}_{2}\), results in  
\[\frac{\overline{I}_{1}}{\overline{I}_{2}}tan^{*}=\frac{R\gamma}{e+R(1-\mu)}. \tag{119}\]  
Substituting Equation (112) into Equation (119) and \(\theta_{1}\) for \(\Omega t\) gives  
\[\theta_{1}=tan^{-1}\left(\frac{\overline{I}_{2}}{\overline{I}_{1}}\cdot\frac{ e+R(1-\mu)}{R\gamma}\right). \tag{120}\]