# Nonlinear Vibration Analysis of a Beam with a Breathing Crack
## 3 Stiffness Matrix of a Beam Element with a Breathing Crack  
Figure 4 shows a beam element with a breathing crack. The element is defined by two nodes. Each node has three degrees of freedom (DOFs), where \(u_{1}\) and \(u_{4}\) are the longitudinal displacements of the node \(i\) and the node \(i+1\), respectively; \(u_{2}\) and \(u_{5}\) are the lateral displacement of the node \(i\) and the node \(i+1\), respectively; and \(u_{3}\) and \(u_{6}\) are the rotation angle of the node \(i\) and the node \(i+1\), respectively. The generalized coordinate of a beam element with a breathing crack \(\mathbf{u}\) is expressed as  
\[\mathbf{u}=\{u_{1}\ u_{2}\ u_{3}\ u_{4}\ u_{5}\ u_{6}\}^{\mathrm{T}}.\]  
Figure 3: Bending stiffness variation of the cracked beam. According to the finite element theory, the transverse deflection \(W(x)\) and longitudinal deflection \(V(x)\) at an arbitrary neutral axis point of the cracked bam element can be expressed as [42]  
\[\left\{ \begin{array}{ll}W(x)=\sum\limits_{i}B_{i}(x)u_{i},&i=2,3,5,6.\\ V(x)=\sum\limits_{j}B_{j}(x)u_{j},&j=1,4.\end{array} \right. \tag{6}\]  
where \(B_{1}(x)\), \(B_{2}(x)\), \(B_{3}(x)\), \(B_{4}(x)\), \(B_{5}(x)\), \(B_{6}(x)\) are the Hamiltonian interpolation functions, respectively, and  
\[\left\{ \begin{array}{ll}B_{1}=1-\frac{x}{l},\\ B_{2}=1-\frac{3}{l}x^{2}+\frac{2}{l^{2}}x^{3},\\ B_{3}=x-\frac{2}{l}x^{2}+\frac{1}{l^{2}}x^{3},\\ B_{4}=\frac{x}{l},\\ B_{5}=\frac{3}{l^{2}}x^{2}-\frac{2}{l^{2}}x^{3},\\ B_{6}=-\frac{1}{l}x^{2}+\frac{2}{l^{2}}x^{3}.\end{array} \right.\]  
where \(l\) is the length of the beam element with a breathing crack. According to the mechanics of materials, the strain at an arbitrary neutral axis \(\varepsilon_{x}\) is composed of two parts, namely the bending strain \(\varepsilon_{x}^{b}\) and the longitudinal strain \(\varepsilon_{x}^{l}\). Thus, the strain at \(r\) shown in Figure 4 can be written as [42]  
\[\varepsilon_{r}=\varepsilon_{r}^{l}+\varepsilon_{r}^{b}=\frac{\partial V(x)} {\partial x}-r\frac{\partial^{2}W(x)}{\partial x^{2}}. \tag{7}\]  
Applying Equation (6) to Equation (7) yields  
\[\varepsilon_{r}=-\frac{1}{l}u_{i}+\frac{1}{l}u_{i+1}-r[(\frac{12x}{l^{3}}- \frac{6}{l^{2}})v_{i}+(\frac{6x}{l^{2}}-\frac{4}{l})v_{i+1}\ +(-\frac{12x}{l^{3}}+\frac{6}{l^{2}})\theta_{i}+(\frac{6x}{l^{2}}- \frac{2}{l})\theta_{i+1}]. \tag{8}\]  
For the sake of calculation and experimental measurement, the strain at \(r\) is used here to distinguish the state of the breathing crack. When the strain at \(r\) is in compression (\(\varepsilon_{r}\leq 0\)), the crack is fully open. When the strain is transferred from compressive to tensile, the crack surfaces begin a preliminary contact. The crack surface acts as the partial crack closure state until the strain reaches the boundary value \(\varepsilon_{k}\). The boundary strain value can be found from the intersection of the curve of the partially closed crack with the curve of the fully closed crack. When the strain \(\varepsilon_{r}\) is tensile and greater than  
Figure 4: Model of beam element with a breathing crack. (\(\varepsilon_{r}\geq\varepsilon_{k}\)), the crack fully closes. Therefore, the stiffness matrix of the beam element with a breathing crack \(\mathbf{k}_{b}\) can be expressed as  
\[\mathbf{k}_{b}=\left\{ \begin{array}{ll}\mathbf{k}_{o}&\varepsilon_{r}\leq 0,\\ \mathbf{k}_{o}\ +\frac{\varepsilon_{r}}{\varepsilon_{k}}(\mathbf{k}_{c}- \mathbf{k}_{o})&\varepsilon_{k}>\varepsilon_{r}>0,\\ \mathbf{k}_{c}&\varepsilon_{k}\leq\ \varepsilon_{r}.\end{array} \right.