# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 5 Numerical Examples and Discussion  
In this section, numerical solutions for cracked plates made of different metals, namely, AISI 52100 alloy steel, 2014-T651 aluminium alloy and 6061-T6 aluminium alloy, are presented. AISI 52100 alloy steel is taken as the brittle metal due to its high strength and relatively low fracture toughness, whereas 6061-T6 aluminium alloy is selected as the ductile material. A 'less brittle' material, i.e., 2014-T651 aluminium alloy, is also studied for the convenience of direct comparisons between brittle materials and between two aluminium alloys. AISI 52100 alloy steel is known as a high-carbon, chromium-containing low alloy steel that is mainly used for the manufacture of highly stressed parts such as anti-friction bearings, mill rolls, fasteners, etc. 2014-T651 aluminium alloy is one of the strongest available aluminium alloys. It is widely used for manufacturing high-strength components, especially in the aerospace and defence industries. It is not as brittle as AISI 52100 alloy steel but its ductility is relatively weak among aluminium alloys. Thus, it is also taken as a brittle material in this study, although it should be borne in mind that the results may be conservative. 6061-T6 aluminium alloy is one of the most common alloys of aluminium for general-purpose use. It has good mechanical properties, exhibits good weldability, and is commonly extruded. Typical material properties of these metals are listed in Tables 3-5. As shown in the first stage in Figure 2, by using the solution of the stress intensity factor \(K\) inversely, a relationship describing the critical state between the applied load and crack length is established. From the relationship, each applied load maps to a unique crack length, and either an increase in the applied load or a growth in the crack length would cause unstable crack propagation or failure. Therefore, the applied load should be chosen judiciously. For this reason, it is necessary to introduce the safety factor \(\chi\). When the load causing yielding fully across the uncracked section of the plate is a definite figure (i.e., the limit load \(P_{0}\)), each safety factor \(\chi\) corresponds to a unique service load \(P_{w}\), and hence maps to a unique critical crack length \(c_{s}\). Knowing the critical crack length, the natural \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \begin{tabular}{c} **Young’s** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Bulk** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Shear** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Yield** \\ **Strength** \\ **(MPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Tensile** \\ **Strength** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Fracture** \\ **Toughness** \\ **(MPa\(\sqrt{\text{m}}\))** \\ \end{tabular} \\ \hline
203.33 & 157.81 & 79.10 & 0.285 & 7346 & 2070 & 2241 & 14.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Typical material properties of AISI 52100 alloy steel [42; 43]. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \begin{tabular}{c} **Young’s** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Bulk** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Shear** \\ **Modulus** \\ **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Yield** \\ **Strength** \\ **(MPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Tensile** \\ **Strength** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Fracture** \\ **Toughness** \\ **(MPa\(\sqrt{\text{m}}\))** \\ \end{tabular} \\ \hline
203.33 & 157.81 & 79.10 & 0.285 & 7346 & 2070 & 2241 & 14.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Typical material properties of 2014-T651 aluminium alloy [43; 44].