# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 2 Methodology  
The methodology of the current study is shown in Figure 2. The process includes three stages (as indicated by different colours), i.e., the application of linear elastic fracture mechanics (LEFM), natural frequency calculation for single-edge cracked rectangular plates under uniaxial in-plane load, and natural frequency measurement. In the first stage, LEFM is employed to determine the critical loads causing brittle fracture or the initiation of crack propagation. For brittle metals, the material behaviour is dominantly elastic and the fracture response is brittle. Thus, LEFM applies. The plane strain fracture toughness \(K_{Ic}\) is a given material property (the subscript \(I\) in \(K_{Ic}\) represents Mode \(I\) of crack surface displacement). A given material can resist a crack without brittle fracture as long as the stress intensity factor \(K\) is below the critical value \(K_{Ic}\). For a SENT specimen, the solution of \(K\) has been formulated as a function of the applied load, the crack length, and the geometrical configuration of the specimen using LEFM and is available in the literature [25; 26]. By referring to the plane strain fracture toughness \(K_{Ic}\) of the material under investigation and utilizing the solution of \(K\) inversely, the critical load \(P_{c}\) applied to the SENT specimen (i.e., the cracked plate) for a specific crack length can be determined (as detailed in Appendix A). Thus, as the crack length increases, the development of the critical load \(P_{c}\) can be obtained. Assuming the cracked plate is a structural component under uniform uniaxial tension, the service load \(P_{w}\) of the structural component can be decided using yield strength \(\sigma_{0}\) and appropriate safety factor \(\chi\). Once the service load \(P_{w}\) is determined, it can be compared with the development of the critical load \(P_{c}\) as the crack grows. When \(P_{w}=P_{c}\), the corresponding crack length is the critical crack length \(c_{s}\) beyond which brittle fracture may occur. Figure 1: The SENT specimen for fracture toughness test. Figure 2: The methodology of the current study. For ductile metals, the fracture response is ductile and the material is capable of considerable plastic deformation. When excessive yielding occurs, the stress intensity factor \(K\) underestimates the crack severity and LEFM no longer applies. However, for the'small-scale yielding' termed by [PERSON] [27], the yielded zone is confined to a small region near the crack tip that is negligible in size in comparison to geometric dimensions such as crack length, uncracked specimen width, etc. In this situation, the crack length can be modified to accommodate the stress redistribution near the crack tip [28]. By substituting the modified crack length back into the solution of \(K\), the critical load \(P_{c}\) corresponding to a specific crack length can be determined. Thus, the development of the critical load \(P_{c}\) as the crack grows can be obtained. The subsequent procedures are the same as those for brittle metals. The critical load solution considering small-scale yielding is attached in Appendix B. It is worth mentioning that instead of causing brittle fracture, the critical load \(P_{c}\) for ductile metals only initiates crack growth. Once the critical crack length \(c_{s}\) is obtained, the natural frequency drop caused by a crack of length \(c_{s}\) can be computed using the RRM in the second stage. Then essentially the question is to calculate the natural frequency of a loaded single-edge cracked rectangular plate with CFCF boundaries (i.e., the SENT specimen). To ensure the accuracy of natural frequency results, some important measures are taken in this stage:  
Firstly, as the SENT specimen is assumed to be a structure carrying a service load \(P_{w}\), the in-plane load should be considered when calculating the natural frequency. To this end, the XFEM [29] is employed to calculate the initial stress of the cracked plate.