# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 2 The Breathing Crack Model  
A data set of load \(P\) and crack opening displacement (COD) \(u\) of a beam with a fatigue crack was experimentally measured using four-point bend testing by [PERSON] et al. [32]. The COD gauge was fixed on the fatigue crack starter mouth of the specimen, which was used to measure the relative amount of the COD \(u\), as shown in Figure 1. The strain gauge was attached to the uncracked side of the beam specimen, which was used to monitor the applied load. Based on the mechanics of materials, the relation between the amount of the COD \(u\) at \(y_{1}\) and the strain at \(y_{2}\) can be written as  
\[\varepsilon_{y}=-\frac{y_{2}u}{y_{1}u_{0}}=-\frac{u}{u_{0}}. \tag{1}\]  
where \(y_{1}\) is the distance from the location of COD gauge to the neutral axis, \(y_{2}\) is the distance from the location of the strain gauge to the neutral axis, and \(u_{0}\) is the span of the COD gauge. Equation (1) indicates that the strain \(\varepsilon_{y}\) is proportional to the COD \(u\). In other words, the curve \(P/u\) and \(P/\varepsilon_{y}\) will be similar. For convenience, a set of load \(P\) and strain \(\varepsilon_{y}\) data points is used to determine the breathing crack function. As shown in Figure 2, the experimental data points between the load \(P\) and COD \(u\) are expressed by a \"scatter\" plot. These data are used to fit a three-section curve. The low section and upper section are a first-order polynomial (straight line), while the middle section is a second-order polynomial (quadratic) indicated by the red line in Figure 2. It is clear that the state of crack closing can be divided  
Figure 1: The geometric deformation of the cracked beam section: (**a**) undeformed, (**b**) deformed. into three regions: fully closed, partially closed and fully open. This observation was also made in [PERSON] et al. [15] and [PERSON] and [PERSON] [33]. From the fitting curves shown in Figure 2, it can be seen that the fully open and fully closed regions (states) are a straight line; i.e., the characteristics of the load \(P\) versus the COD \(u\) obey [PERSON]'s law in these two regions. However, the state of the partially closed region does not exhibit such a linear characteristic, and a transition occurs if the crack begins to close up. When the COD is transferred from tensile state to compressive state, the crack surfaces experience a preliminary contact. As the COD reaches a certain compression value, the crack becomes fully closed. The boundary between the partially closed state and the fully closed state is named the \"knot\", and the corresponding load \(P\) and COD \(u\) are denoted by \(P_{k}\) and \(u_{k}\), respectively. The stiffness of the cracked beam can be determined by finding the slope of the \(P\)-\(u\) curve at any displacement \(u\)[15; 33]. Thus, the bending stiffness of a beam with a breathing crack \(k_{t}\) can be described in the form:  
\[k_{t}=\left\{ \begin{array}{ll}k_{c}&u<u_{k},\\ c_{0}u+k_{o}&u_{k}\leq u<0,\\ k_{o}&u\geq 0.\end{array} \right. \tag{2}\]  
where \(k_{c}\) and \(k_{o}\) are the bending stiffness of the cracked beam with a fully closed crack and fully open crack, respectively; \(c_{0}\) is the unknown coefficient. If the crack repetitively opens and closes during the vibration, the bending stiffness \(k_{t}\) will vary periodically, as shown in Figure 3. Therefore, the bending stiffness of a beam with a breathing crack can be written as the complex Fourier series in the form  
\[k_{t}=k_{o}+f(t)(k_{c}-k_{o}). \tag{3}\]  
where \(t\) is the time,\(f(t)\) is the breathing crack function which describes the process of crack opening and closing, and  
\[f(t)=\frac{a_{0}}{2}+\sum_{n=1}^{N}{(a_{n}\cos n\omega t+b_{n}\cos n\omega t)}. \tag{4}\]  
Figure 2: Load versus crack open displacement.