## 4 Identification of Breathing Cracks in Beams
### Vibration Model of a Beam Containing Breathing Cracks  
Figure 4 shows a typical open cracked beam element with a rectangular cross-section, the element length is \(L\), the width is \(b\), the height is \(h\), the cross-sectional area is \(A\). The modulus of elasticity, Poisson's ratio, and density of the material are \(E\), \(\mu\), and \(\rho\), respectively, and the moment of inertia is \(I\). There are two nodes \(i\) and \(j\) on the element, \(i\) is fixed, and there is an open crack of depth \(a\) at the distance \(L_{c}\) from the node \(j\). According to the above conditions, the strain energy \(W^{(0)}\) of the beam element in the absence of cracks and the additional strain energy \(W^{(1)}\) caused by the cracks can be expressed as the following formula [34; 35], respectively. Figure 3: Third CWT mode shape of the FF beam: (**a**) no boundary extension; (**b**) boundary extension. Figure 2: Third mode shape of the FF beam: (**a**) no boundary extension; (**b**) boundary extension. \[W^{(0)}=\frac{1}{2 EI}\bigg{(}M^{2}L+MPL^{2}+\frac{P^{2}L^{3}}{3}\bigg{)}+\frac{F^{2 }L}{2 EA} \tag{22}\]  
\[W^{(1)}=\int_{A_{e}}\frac{1}{E^{\prime}}[(K_{\rm II}+K_{\rm II}+K_{\rm I3})^{2}+ K_{\rm II}^{2}]dA \tag{23}\]  
where \(E^{\prime}=E\) when it is a plane stress state and \(E^{\prime}=E/(1-\mu^{2})\) when it is a plane strain state; \(A_{e}\) is the effective area of the crack. \(K_{\rm II}\), \(K_{\rm I2}\), \(K_{\rm I3}\), and \(K_{\rm II2}\) are the stress intensity factors corresponding to the axial force \(F\), shear force \(P\), and bending moment \(M\). They can be expressed as:  
\[K_{\rm II}=\frac{F}{bh}\sqrt{\pi\xi}F_{\rm I}\bigg{(}\frac{\xi}{h}\bigg{)} \tag{24}\]  
\[K_{\rm II}=\frac{6 PL_{c}}{bh^{2}}\sqrt{\pi\xi}F_{\rm I}\bigg{(}\frac{\xi}{h} \bigg{)} \tag{25}\]  
\[K_{\rm II}=\frac{p}{bh}\sqrt{\pi\xi}F_{\rm I}\bigg{(}\frac{\xi}{h}\bigg{)} \tag{27}\]  
where \(\xi\) is the crack depth in the integration process from 0 to the final depth \(a\). \(F_{1}\), \(F_{2}\), and \(F_{\rm II}\) are the modified coefficients of the stress intensity factor, and if \(s=\xi/h\), then the modified coefficients can be written as the following formulae:  
\[F_{1}(s)=\sqrt{\frac{\tan(\pi s/2)}{(\pi s/2)}}\frac{0.752+2.02s+0.37(1-\sin( \pi s/2))^{3}}{\cos(\pi s/2)} \tag{28}\]  
\[F_{2}(s)=\sqrt{\frac{\tan(\pi s/2)}{(\pi s/2)}}\frac{0.923+0.199(1-\sin(\pi s /2))^{4}}{\cos(\pi s/2)} \tag{29}\]  
\[F_{\rm II}(s)=\frac{1.122-0.561s+0.085s^{2}+0.180s^{3}}{\sqrt{1-s}} \tag{30}\]  
According to [PERSON]'s theorem, the partial derivative transformation of Formulas (22) and (23) yields the flexibility matrix \(c_{ij}^{(0)}\) of the intact beam element and the additional flexibility matrix \(c_{ij}^{(1)}\) of the open cracked beam element. Figure 2:0  
Figure 2:1  
Figure 2:2 Open cracked beam element. The above analysis is for an open cracked beam element that is fixed at the left end and subjected to forces at the right end. For an open cracked beam element in the FEM coordinate system, in the case of both nodes \(i\) and \(j\) are subjected to the axial force \(F\), shear force \(P\), and bending moment \(M\) and the additional flexibility matrix \(c_{ij}^{\rm(add)}\) caused by the crack can be expressed as:  
\[c_{ij}^{\rm(add)}= \begin{bmatrix}c_{11}^{(1)}&-c_{12}^{(1)}&-c_{13}^{(1)}\\ -c_{21}^{(1)}&c_{22}^{(1)}&c_{23}^{(1)}\\ -c_{31}^{(1)}&c_{32}^{(1)}&c_{33}^{(1)}\end{bmatrix}  \tag{33}\]  
The relationship between the forces and displacements in the intact beam element is described by the flexibility matrix \(c_{ij}^{\rm(int)}\), where \(c_{ij}^{\rm(int)}=c_{ij}^{(0)}\), and the total flexibility matrix \(c_{ij}^{\rm(tot)}\) of the open cracked beam element can be expressed as:  
\[c_{ij}^{\rm(tot)}=c_{ij}^{\rm(int)}+c_{ij}^{\rm(add)}= \begin{bmatrix}\frac{L} {EA}+c_{11}^{(1)}&-c_{12}^{(1)}&-c_{13}^{(1)}\\ -c_{21}^{(1)}&\frac{L^{3}}{3 EI}+c_{22}^{(1)}&\frac{L^{2}}{2 EI}+c_{23}^{(1)}\\ -c_{31}^{(1)}&\frac{L^{2}}{2 EI}+c_{32}^{(1)}&\frac{L}{EI}+c_{33}^{(1)}\end{bmatrix}