# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## Appendix B Critical Load Solution from LEFM: For Small-Scale Yielding  
As shown in Figure 1, for a load-controlled system where the load is monotonically increasing, the crack growth undergoes four stages during the ductile fracture process, namely, blunting of the initially sharp crack, crack initiation, slow (stable) crack growth and unstable crack growth. When small-scale yielding near the crack tip is considered, it is possible to extend LEFM beyond linear elasticity by adjusting the crack length to account for the yielding. In this case, the solution of \(P_{c}\) has the same formula as Equation (4),although the crack length \(c\) should be replaced by [PERSON]'s effective crack length \(c_{eff}\)[28]. Therefore, the critical load solution can be expressed as follows:  
\[P_{c}=K_{c}at/\left[F\left(c_{eff}/a\right)\sqrt{\pi c_{eff}}\right] \tag{10}\]  
where \(F\left(c_{eff}/a\right)\) can be obtained from Equation (10) by substituting \(c_{eff}\) for \(c\). Instead of leading to brittle fracture, now \(P_{c}\) indicates the second stage in Figure 11, i.e., the initiation of crack growth.  
In Equation (10), \(c_{eff}\) is defined as the sum of the actual crack length and a plastic zone correction:  
\[c_{eff}=c+\psi\tau_{y} \tag{11}\]  
where  
\[\psi=\frac{1}{1+\left(P/P_{0}\right)^{2}} \tag{12}\]  
\[r_{y}=\frac{1}{\beta\pi}\left[\frac{n-1}{n+1}\right]\left(\frac{K_{eff}}{ \sigma_{0}}\right)^{2} \tag{13}\]  
\[K_{eff}=F\left(\frac{c_{eff}}{a}\right)\frac{P}{at}\sqrt{\pi c_{eff}} \tag{14}\]  
In Equation (12), \(P_{0}\) is the limit load based on yield strength \(\sigma_{0}\) and is given as follows:  
\[P_{0}=1.455\eta(a-c)t\sigma_{0}\ \ \ \ \ for\ plane\ strain \tag{15}\]  
\[P_{0}=1.072\eta(a-c)t\sigma_{0}\ \ \ \ \ \ for\ plane\ stress \tag{16}\]  
where \(\eta\) is defined as the following:  
\[\eta=\sqrt{1+\left[c/(a-c)\right]^{2}}-c/(a-c) \tag{17}\]  
In Equation (13), \(\beta=2\) for plane stress and \(\beta=6\) for plane strain; \(n\) is the strain hardening exponent of the material. For a given load \(P\), an iterative calculation is involved in the computation of \(c_{eff}\). That is, \(K_{eff}\) is first determined in the absence of a plastic zone correction; \(c_{eff}\) is then obtained using Equations (A7)-(A9) which in turn is used to estimate \(K_{eff}\); thus, a new \(c_{eff}\) can be computed and the process is repeated until successive \(c_{eff}\) estimates converge.  
Therefore, for a given ductile metal, by referring to its plane strain fracture toughness \(K_{lc}\) in the literature, converting \(K_{lc}\) to \(K_{c}\) using Equation (A3), and equating \(K_{eff}\) in Equation (A10) with \(K_{c}\), the solution of \(P_{c}\) (i.e., Equation (A6)) is obtained. Here, it is worth mentioning that unlike the critical load solution for brittle metals, the solution of \(P_{c}\) considering small-scale yielding is conservative, which means the cracked plate may not fail immediately when \(P=P_{c}\) or the actual load causing ductile fracture may exceed \(P_{c}\). As the critical load \(P_{c}\) only indicates the initiation of crack growth, after \(P\) exceeds \(P_{c}\), the excessive load still can be accommodated by the ensuing phase of stable crack growth until the unstable crack growth starts. That is to say, for the ductile metal, the load-carrying capability remains when the crack growth begins, and the load causing final failure may exceed the critical load causing the onset of crack growth. This is quite different from the brittle metal for which the initiation of crack propagation and brittle fracture happen almost simultaneously.