# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 3 The Free Vibration of a Loaded Single-Edge Cracked Rectangular Plate  
This section deals with the free vibration of a single-edge cracked rectangular plate under uniaxial in-plane tension using classical thin plate theory and the RRM. For a plate vibrating harmonically with amplitude \(\overline{w}\) and angular frequency \(\omega\), the maximum strain energy \(V_{max}\) and the maximum kinetic energy \(T_{max}\) can be expressed as follows:  
\[ \begin{array}{c}V_{max}=\frac{\mathcal{D}}{2}\iint\Big{\{}\big{(}\overline{ w}_{,xx}+\overline{w}_{,yy}\big{)}^{2}-2(1-v)\Big{[}\overline{w}_{,xx} \overline{w}_{,yy}-\big{(}\overline{w}_{,xy}\big{)}^{2}\Big{]}\Big{\}}dA\\ \qquad\qquad+\frac{1}{2}\iint\Big{(}N_{xx}^{(0)}\overline{w}_{,x}^{2}+2N_{xy}^ {(0)}\overline{w}_{,x}\overline{w}_{,y}+N_{yy}^{(0)}\overline{w}_{,y}^{2}\Big{)} dA\end{array}  \tag{1}\]  
\[ \begin{array}{c}T_{max}=\frac{\rho h\omega^{2}}{2}\iint\overline{w}^{2}dA \end{array}  \tag{2}\]where \(D\), \(v\), \(\rho\) and \(h\) are the flexural rigidity of the plate, Poisson's ratio, density and plate thickness, respectively; \(N_{xx}^{(0)}\), \(N_{xy}^{(0)}\) and \(N_{yy}^{(0)}\) are the in-plane stress resultants caused by in-plane tension, and the subscript comma denotes partial differentiation with respect to the coordinate after the comma. It should be noted that even when the cracked plate is subjected to uniform in-plane tension, \(N_{xx}^{(0)}\), \(N_{xy}^{(0)}\) and \(N_{yy}^{(0)}\) are functions of position. In this paper, \(N_{xx}^{(0)}\), \(N_{xy}^{(0)}\) and \(N_{yy}^{(0)}\) are found numerically using the XFEM. The natural frequencies of the cracked plate are obtained by minimizing the energy functional:  
\[\Pi=V_{max}-T_{max} \tag{3}\]  
For a single-edge cracked rectangular plate as shown in Figure 3, the admissible functions of the RRM should satisfy the boundary conditions of the cracked plate. Meanwhile, the stress singularities at the crack tip and the discontinuity of displacement and slope across the crack surfaces should be described. Consequently, the admissible functions are defined as follows:  
\[\overline{w}(x,y)=x\overline{J}y^{\overline{m}}(x-a)^{\overline{m}}(y-b)^{ \overline{q}}[\overline{w}_{p}(x,y)+\overline{w}_{c}(r,\theta)] \tag{4}\]  
where the function before the brackets is inserted to satisfy the geometric boundary conditions along \(x=0\), \(x=a\), \(y=0\) and \(y=b\); \(\overline{l}\) (or \(\overline{m}\), \(\overline{n}\), \(\overline{q}\)) = 0 for free (F) boundary, 1 for simply supported (S) boundary and 2 for clamped (C) boundary; \(\overline{w}_{p}(x,y)\) is a set of polynomials and can be expressed as the following:  
\[\overline{w}_{p}(x,y)=\sum_{i=1}^{I}\sum_{j=1}^{J}a_{ij}x^{i-1}y^{j-1} \tag{5}\]  
Equation (5) forms a mathematically complete set of functions if an infinite number of terms are used; \(\overline{w}_{c}(r,\theta)\) is a set of corner functions. In this paper, \(\overline{w}_{c}(r,\theta)\) is taken from the second set of functions in [30] and is expressed as follows:  
Figure 3: The divisions of the single-edge cracked plate. \[ \begin{split}\overline{w}_{c}(r,\theta)=\sum\limits_{n=1}^{N}\sum \limits_{l=0}^{n}B_{nl}r^{(2n+1)/2}cos\frac{2l+1}{2}\theta\\ +\sum\limits_{n=1}^{N}\sum\limits_{l=0}^{n}C_{nl}r^{(2n+1)/2}sin \frac{2l+1}{2}\theta\end{split}  \tag{6}\]  
where \(B_{nl}\) and \(C_{nl}\) are coefficients, and \(r\) and \(\theta\) are the polar coordinates as shown in Figure 3. The origin of the polar coordinate system is centred at the crack tip \(P(a-c,b/2)\). Therefore,  
\[r=\sqrt{\left(x-a+c\right)^{2}+\left(y-b/2\right)^{2}} \tag{7}\]  
The cracked plate can be divided into four subdomains separated by the extended line of the crack \(y=b/2\) and the vertical line \(x=a-c\). Then \(\theta\) in the corner functions for different subdomains is the following:  
\[\theta=\left\{ \begin{split} arct\frac{y-b/2}{x-a+c}\text{ for subdomains I and II}\\ arct\frac{y-b/2}{x-a+c}+\pi\text{ for subdomain III}\\ arct\frac{y-b/2}{x-a+c}-\pi\text{ for subdomain IV}\end{split} \right.