# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 5 Numerical Examples and Discussion
### Cracked Plates Made of Brittle Metals  
In this subsection, single-edge cracked plates made of AISI 52100 alloy steel and 2014-T651 aluminium alloy are investigated following the procedures shown in Figure 2. The cracked plates (i.e., the SENT specimens for fracture toughness test) are regarded as structural components carrying the service load \(P_{w}\). \(P_{w}\) is determined using yield strength \(\sigma_{0}\) and a series of safety factors \(\chi\). Referring to Equation (A4), when \(P_{c}=P_{w}\), substituting \(P_{c}\) and \(K_{c}\) into Equation (A4) yields the critical crack length \(c_{s}\). Here, \(c_{s}\) is the maximum crack length that a cracked plate can tolerate under the service load \(P_{w}\). When the crack length is beyond \(c_{s}\), the applied load has to be lower than \(P_{w}\) to avoid brittle fracture, which means the serviceability of the cracked plate has to be compromised. As multiple safety factors are considered, there is more than one critical crack length \(c_{s}\) and whether the ensuing \(\Delta\omega\) is measurable also partly depends on the safety factor being used. To make the final results comparable to each other, the size of all rectangular plates investigated in Section 5 is uniform, namely, the width \(a=0.8\) m, the height \(b=1.6\) m and the thickness \(h=0.008\) m. The boundary condition is CFCF as previously mentioned. The clamped boundaries represent the edges where the tensile load is applied when the plate is in service. For the cracked plates made of AISI 52100 alloy steel, Table 6 presents the critical crack length ratios and the corresponding natural frequency drops for different safety factors. For each critical crack length \(c_{s}\), the frequency drop of the loaded CFCF single-edge cracked plate is calculated. Only those frequency drops with a magnitude over 0.1 Hz are tabulated as 0.1 Hz is assumed to be the threshold for measurable frequency change in this study (the determination of the 0.1 Hz threshold has been clarified in Appendix D). It is worth mentioning that in Table 6 (also in Table 7 as shown later), when safety factor \(\chi=2.5\) and 3, the obtained critical crack length is smaller than \(2.5{(K_{Ic}/\sigma_{0})}^{2}\), which means in these two circumstances the crack length is too short to satisfy the plane strain condition. Therefore, the results when \(\chi=2.5\) and 3 are discarded. However, this does not cause any hindrance because the difficulty of measuring the frequency drop when \(\chi\leq 3.5\) is still obvious. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Safety Factor \(\chi\)**} & \multicolumn{6}{c}{**Natural Frequency Drop \(\Delta\omega\) (Hz)**} \\ \cline{2-9}  & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** & **Mode 5** & **Mode 6** \\ \hline
3.5 & 0.018\% & \textless{}0.1 & \textless{}0.1 & \textless{}0.1 & \textless{}0.1 & \textless{}0.1 & \textless{}0.1 \\
4 & 0.024\% & & & & & & & \\
4.5 & 0.031\% & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
10 & 0.151\% & \(-0.01\) (\textless{}0.1) & 0 (\textless{}0.1) & \(-0.06\) (\textless{}0.1) & \(-0.06\) (\textless{}0.1) & 0 (\textless{}0.1) & 0.06 (\textless{}0.1) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the loaded CFCF single-edge cracked plate made of AISI 52100 alloy steel. It can be observed from Table 6 that as the safety factor \(\chi\) ranges from 3.5 to 10, the service load \(P_{\varpi}\) decreases gradually and the critical crack length \(c_{s}\) increases progressively. The maximum critical crack length ratio is 0.151% (with \(c_{s}\) approximately 1.2 mm), which means only a tiny crack can be tolerated by the cracked plate and the resulting natural frequency drop is almost impossible to measure. Once the crack length reaches a critical value, either further crack growth or an increase in the working load may lead to brittle fracture.