## 4 Numerical Validations
### Validation of the Characteristics of the Torsional Wave-Damage Interaction  
The characteristics of the interaction between torsional wave and damage are evaluated. The excitation is a 5-cycle tone-burst signal with the center frequency at 50 kHz. A typical wavefield at 200 us in terms of the deformation is captured in Figure 4b before the incident wave reaches the defect, showing the generated torsional wave. After the wave interaction with damage, the transmitted and reflected torsional waves are observed in the wavefield at 350 us in Figure 4c. No mode conversion takes place, which is conducive to the extraction of damage information in the response signals in the subsequent analyses. The influence of the defect width on the reflected waves is first evaluated. By fixing the defect depth at 0.7 mm (\(r_{d}=0.2r\)), the widths vary from 0.05\(\delta\) to 1.2\(\delta\) with an increment of 0.05\(\delta\), where \(\delta\) is the wavelength of the torsional wave at 50 kHz (\(\delta=62.85\) mm). Two representative response signals are discussed. Specifically, Figure 5a shows the time-domain response at the receiving point for the 0.25\(\delta\)-wide defect case. According to the torsional wave velocity and time of flight, the wave packets are identified. The reflections from the defect and end are denoted as damage reflection and end reflection, respectively, in the figure for simplicity. With the complex Morlet wavelet transform [35], the amplitudes of the wave packets in Figure 5a are extracted as the peaks in Figure 5b. When the defect width is 0.5\(\delta\), the response is captured in Figure 5c where the reflection from the defect is relatively weak and a split in the wave packet is observed. This is associated with the anti-resonance phenomenon in a dynamic system, which is demonstrated in the literature [36; 37]. After the complex Morlet wavelet transform, the valley amplitude is regarded as the actual wave amplitude as illustrated in Figure 5d to provide an accurate characterization. Figure 5: Typical time-domain responses and their corresponding wavelet coefficient modulus for: (**a**,**b**) the 0.25\(\delta\)-wide defect; (**c**,**d**) the 0.5\(\delta\)-wide defect. Through extracting the amplitudes of incident and reflection waves in all the cases with different defect widths, the reflection coefficients are calculated according to Equation (17). Results are plotted in Figure 6 compared to the theoretical results. The variation pattern in the finite element results matches well with the theoretical predictions. The variations of the reflections from the defect and end exhibit the same period of 0.5\(\delta\) but show an opposite trend. Specifically, when the defect width is \((2n+1)\delta/4\) with \(n\) being the integers, the reflection from the defect reaches the maximum while that from the end tends to the minimum. The discrepancy between the numerical and theoretical results is observed in terms of the amplitude. This can be attributed to the use of the tone-burst excitation with finite duration in the finite element simulations. As indicated by the theory, the received reflection wave results from the superposition of multiple reflections from the two interfaces of the defect. When the tone-burst excitations are used, the phase lag associated with wave propagation distance will affect the wave amplitude after the superposition of the reflections. The influence of defect depth on the wave-damage interaction is then assessed. The depths of the defects vary from 0.1\(r\) to 0.9\(r\) with an increment of 0.1\(r\) while the defect width is set as 0.25\(\delta\) in these cases. Typical response signals are shown in Figure 7a where the reflections from the defects and end are obvious. With the extracted wave amplitudes from the complex Morlet wavelet transform, the reflection coefficients from the defect and end are calculated and compared to the theoretic predictions in Figure 7b,c. A good agreement between the numerical and theoretical results is observed.