## 2 Theoretical Study
### Local Stiffness Coefficient  
Figure 3 is a schematic of the transverse crack. Figure 3a is the free body diagram, and \(P_{1}\), \(P_{1}^{\prime}\), \(P_{2}\), \(P_{2}^{\prime}\), \(P_{3}\), \(P_{3}^{\prime}\) are the external loads that the shaft suffering. Figure 3b is the section view of the crack, where, \(a\) is the crack depth, the width of the crack is \(2b\  
where, \(K_{n}\) (\(n\) = I, II, III) is the stress-intensity factor with three different crack modes. According to the definition, the local stiffness coefficients can be written as:  
\[c_{i}=\frac{\partial u_{i}^{c}}{\partial P_{i}}=\frac{\partial^{2}}{\partial P_{ i}\partial P_{i}}\int_{0}^{a}\int_{-b}^{b}J(A)d\xi d\eta \tag{6}\]  
where, \(i\) is the index of the external loads. Equation (6) is the equation for the local stiffness coefficients, substituting related loads into which, the coefficients can be obtained. For Mode I, only the loads \(P_{1}\) and \(P_{1}^{\prime}\) re needed to be considered, see Figure 3b. It can be deduced that the local stiffness coefficient along the crack direction is [31]  
\[c_{1}=2\frac{1-v^{2}}{E}\int_{0}^{b}\int_{0}^{\eta}\frac{32}{\pi R^{8}}\Big{(} R^{2}-\xi^{2}\Big{)}\eta F_{1}^{2}(\eta/h)d\eta d\xi \tag{7}\]  
where, \(h\) is the height of infinitesimal local area, and \(h=2\sqrt{R^{2}-\xi^{2}}\), \(F_{1}\big{(}\frac{\eta}{h}\big{)}=1.125+1.4\big{(}\frac{\eta}{h}\big{)}+7.33 \big{(}\frac{\eta}{h}\big{)}^{2}+13.08\big{(}\frac{\eta}{h}\big{)}^{3}+14 \big{(}\frac{\eta}{h}\big{)}^{4}\). For Mode II, only the loads \(P_{2}\) and \(P_{2}^{\prime}\) are needed to be considered. In addition, \(K_{II2}=\frac{kR_{2}^{2}}{\pi R^{2}}\sqrt{\pi\eta}F_{II}\big{(}\frac{\eta}{h} \big{)}\), \(k=6(1+v)/(7+6v)\), substituting all the related parameters into Equation (6), it can be deduced that  
\[c_{2}=\frac{\partial u_{i}^{c}}{\partial P_{i}}=\frac{\partial^{2}}{\partial P _{1}\partial P_{1}}\int_{-b}^{b}\int_{0}^{\eta}J(A)d\eta d\xi=2\frac{1-v^{2}} {E}\frac{36(1+v)^{2}}{(7+6v)^{2}}\int_{-b}^{b}\int_{0}^{\eta}\frac{\eta}{\pi R ^{4}}F_{II}^{2}\big{(}\frac{\eta}{h}\big{)}d\eta d\xi \tag{8}\]  
where, \(\eta=(a-R)+\sqrt{R^{2}-\xi^{2}}\), \(F_{II}\big{(}\frac{\eta}{h}\big{)}=\frac{1.122-0.561\big{(}\frac{\eta}{h} \big{)}+0.085\big{(}\frac{\eta}{h}\big{)}^{2}+0.18\big{(}\frac{\eta}{h}\big{)} ^{3}}{\sqrt{1-\big{(}\frac{\eta}{h}\big{)}}}\). Considering that Equation (8) is even symmetric, thus it can be further rewritten as  
\[c_{2}=4\frac{1-v^{2}}{E}\frac{36(1+v)^{2}}{(7+6v)^{2}}\int_{-b}^{b}\int_{0}^{ \eta}\frac{\eta}{\pi R^{4}}F_{II}^{2}\big{(}\frac{\eta}{h}\big{)}d\eta d\xi \tag{9}\]  
For Mode III, the strain energy density function and stress stiffness factor for the crack can be expressed as  
\[J(\text{A})=\frac{\big{(}1-v^{2}\big{)}(1+v)}{E}K_{III}^{2} \tag{10}\]  
\[K_{III}=\frac{B_{2}}{\pi R^{4}}\sqrt{R^{2}-\xi^{2}}\sqrt{\pi\eta}F_{III}\Big{(} \frac{\alpha}{h}\Big{)} \tag{11}\]  
Substituting Equations (10) and (11) into Equation (6), the local stiffness coefficient along the crack direction is then obtained  
\[c_{3}=\frac{\partial u_{i}^{c}}{\partial P_{i}}=\frac{\partial u_{2}^{c}}{ \partial P_{3}}=\frac{\partial^{2}}{\partial P_{3}^{2}}\int_{-b}^{b}\int_{0}^{ \eta}J(A)d\eta d\xi=2\frac{\big{(}1-v^{2}\big{)}(1+v)}{E}\int_{-b}^{b}\int_{0}^{ \eta}\frac{1}{\pi R^{8}}\Big{(}R^{2}-\xi^{2}\Big{)}d\eta d\xi \tag{12}\]  
where, \(F_{III}\big{(}\frac{\eta}{h}\big{)}=\sqrt{\Big{(}\frac{2h}{\pi\eta}\big{)}\tan \big{(}\frac{\pi\eta}{2h}\big{)}}\). Similarly, considering that the equation is even symmetric, thus, Equation (12) can be further rewritten as  
\[c_{3}=\frac{4(1+v)\big{(}1-v^{2}\big{)}}{E}\int_{0}^{b}\int_{0}^{\eta}\frac{ \big{(}R^{2}-\xi^{2}\big{)}}{\pi R^{8}}\eta F_{III}^{2}\Big{(}\frac{\eta}{h} \Big{)}d\eta d\xi \tag{13}\]  
Substituting Equations (7), (8) or (12) into Equation (3) separately, the transfer matrix of the elastic wave with different crack mode can then be deduced.