## 4 Identification of Breathing Cracks in Beams
### Vibration Model of a Beam Containing Breathing Cracks
 \tag{34}\]  
On the basis of the above analysis, the stiffness matrix \(K_{\infty}\) of the open cracked beam element can be written as:  
\[K_{\infty}=T(c_{ij}^{\rm(tot)})^{-1}T^{T} \tag{35}\]  
where \(T\) is the transformation matrix, which can be expressed as:  
\[T= \begin{bmatrix}-1&0&0&1&0&0\\ 0&-1&-L&0&1&0\\ 0&0&-1&0&0&1\end{bmatrix} ^{T} \tag{36}\]  
In fact, cracks under dynamic loading have a 'breathing effect'. A simple way to construct the stiffness matrix of the breathing cracked beam element is to use the ratio of the instantaneous curvature at the crack to the maximum curvature to determine the degree of opening and closing in the breathing crack [36], which is expressed as the following formula:  
\[K_{cc}=K_{e}+\frac{1}{2}(K_{\infty}-K_{e})\left[1+\frac{d^{\prime\prime}}{d^{ \prime\prime}_{max}}\right] \tag{37}\]  
\[d^{\prime\prime}=\frac{\partial^{2}y}{\partial x^{2}} \tag{38}\]  
where \(K_{cc}\) is the stiffness matrix of the breathing cracked beam element, \(K_{e}\) is the stiffness matrix of the intact beam element, and \(d^{\prime\prime}\) is the instantaneous curvature at the crack location. From Formula (37), it can be seen that when \(d^{\prime\prime}\) reaches the maximum instantaneous curvature \(d^{\prime\prime}_{max}\), \(K_{cc}=K_{\infty}\), the crack is completely opened. When \(d^{\prime\prime}\) equals to \(-d^{\prime\prime}_{max}\), \(K_{cc}=K_{e}\) and the crack is completely closed. When \(-d^{\prime\prime}_{max}<d^{\prime\prime}<d^{\prime\prime}_{max}\), the cracked beam is in the process of gradually opening or closing. It is worth noting that the value of \(d^{\prime\prime}_{max}\) is calculated in the case of the crack being fully open (\(K_{cc}=K_{\infty}\)). Using the assumption of Rayleigh damping, the displacement response can be solved using the Newmark-\(\beta\) method. The strain response can be obtained by the second-order numerical differentiation of the displacement response as follows:  
\[\varepsilon_{i}=\frac{x_{i-1}-2x_{i}+x_{i+1}}{(L)^{2}}\frac{h}{2} \tag{39}\]where \(\epsilon_{i}\) and \(x_{i}\) are the strain and vertical displacement of the \(i\)th measurement point, respectively. The cantilever beam model containing two breathing cracks is established according to the above method, and the relevant parameters of the model are shown in Table 1. The cantilever beam is discretized into 200 two-node beam elements with three degrees of freedom (horizontal translation, vertical translation, and rotation) considered at each node. Two breathing cracks with the same depth were established at \(L_{1}=4.2\) m (\(\zeta=0.35\)) and \(L_{2}=7.2\) m (\(\zeta=0.6\)) from the fixed end, respectively. In addition, two damage scenarios are set up according to different crack depths \(a\) (see Table 1 for details). The placement of 200 measurement points on all nodes except the fixed end and the reason for setting up a large number of dense measurement points is that the greater the number of measurement points, the greater the accuracy of the damage location. In practical engineering applications, the dynamic response of these dense measurement points can be obtained with the help of modern visual measurement techniques (such as digital image correlation [37] and gradient-based optical flow [38]). The cracked cantilever beam model is shown in Figure 5, and the standard Gaussian white noise excitation is applied at the free end, while the time history of the white noise is shown in Figure 6. The sampling time is set to 15 s, the sampling frequency is 200 Hz, and after extracting the displacement response in the \(y\)-direction of the cantilever beam, Formula (39) is used to solve the strain response. Figure 7 shows the strain response of measurement points 75 and 120 under the scenario I.