# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 5 Numerical Examples and Discussion
### Cracked Plates Made of Ductile Metals  
For single-edge cracked plates made of ductile metals such as 6061-T6 aluminium alloy, the analysis follows similar procedures as in the previous subsection. The only difference is that the crack length should be modified to accommodate the stress redistribution near the crack tip (i.e., small-scale yielding). This is achieved by using [PERSON]'s effective crack length \(c_{eff}\) (as detailed in Appendix B). The critical crack length ratios and the corresponding natural frequency drops for various safety factors are listed in Table 8.  
Table 8 shows that as the safety factor \(\chi\) increases from 2.5 to 5, the critical crack length ratio increases from 7.538% to 21.036%. Unlike the cracked plates made of AISI 52100 alloy steel and 2014-T651 aluminium alloy, a large crack can be contained by the cracked plates made of 6061-T6 aluminium alloy. This is also shown in Figure 8 where the development of critical crack length ratio versus safety factor is plotted. From AISI 52100 alloy steel to 6061-T6 aluminium alloy, as the strength weakens and the ductility advances, the maximum crack length that can be accommodated by the structural component under service loads grows larger and larger. In other words, the relative sensitivity to cracks varies with metals. AISI 52100 alloy steel, known as a high-strength brittle metal, is sensitive to rather small cracks. By comparison, 6061-T6 aluminium alloy is less sensitive to cracks due to its high fracture toughness and good ductility. This observation coincides with the description in [26] regarding the effects of cracks on brittle versus ductile behaviour. Thus, for the structural component made of 6061-T6 aluminium alloy, it is possible to get measurable natural frequency drops before the crack reaches the critical length. This is favourable to natural frequency-based early crack detection because there is a larger 'window period' to measure the natural frequency drop before fracture happens. Besides, as a large crack can be contained, it is possible to measure the natural frequency drop at low modes. For example, when \(\chi=2.5\), only the natural frequency drop of the first mode is below 0.1 Hz. When \(\chi>2.5\), all the natural frequency drops are measurable. This is quite different from the former two metals for which most of the frequency drops are below 0.1 Hz. As the brittleness of the employed metal weakens, it is likely to get measurable frequency drops using the first few modes before the crack threatens the serviceability of the structural component. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Safety Factor \(\chi\)**} & \multicolumn{5}{c}{**Natural Frequency Drop \(\Delta\omega\) (Hz)**} \\ \cline{2-7}  & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** & **Mode 5** & **Mode 6** \\ \hline
2.5 & 7.538\% & 0.09 & **0.24** & **\(\mathbf{-0.75}\)** & **0.13** & **1.73** & **2.76** \\
3 & 10.284\% & **0.18** & **0.43** & **\(\mathbf{-1.06}\)** & **0.38** & **2.98** & **4.35** \\
3.5 & 13.102\% & **0.31** & **0.66** & **\(\mathbf{-1.35}\)** & **0.87** & **4.44** & **6.13** \\
4 & 15.874\% & **0.48** & **0.93** & **\(\mathbf{-1.68}\)** & **1.94** & **5.91** & **8.13** \\
4.5 & 18.530\% & **0.67** & **1.22** & **\(\mathbf{-1.77}\)** & **3.90** & **6.91** & **10.13** \\
5 & 21.036\% & **0.86** & **1.51** & **\(\mathbf{-1.65}\)** & **5.84** & **7.66** & **12.12** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of the loaded CFCF single-edge cracked plate made of 6061-T6 aluminium alloy. As mentioned in Appendix B, the critical crack length results for ductile metal are conservative because only small-scale yielding is considered. The critical crack length ratios presented in Table 8 indicate the maximum crack length beyond which the crack growth begins. After the initiation of crack growth, the ensuing phases of stable crack growth and unstable crack growth during which large-scale yielding occurs cannot be depicted by the above results.