PSI - Issue 45

Xianwen Hu et al. / Procedia Structural Integrity 45 (2023) 20–27 Hu, X., Liang, P., Ng, C.T., and Kotousov, A. / Structural Integrity Procedia 00 (2019) 000 – 000

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Fig. 1. Schematic diagram of the plate edge

Fig. 2. Dispersion curves for the ES 0 , S 0 , and SH 0 wave modes

The equation of motion can be written as follows (Wilde, Golub & Eremin 2019): (1) where λ=(2 μυ ) / (1-2 ν ); v , and μ are the Poisson’s ratio, and the shear modulus of the material . ρ denotes the plate density. Considering that the loading of the PZT has a symmetrical pattern, stress tensor components ( i, j =1, 2, 3) can be expressed in terms of displacement as follows (Liang & Ng 2021): (2) where = { 0 ≠ 1 = is the Kronecker delta. The vector of transformed displacements is (Liang & Ng 2021): (3) where ω and ξ are the Laplace transform parameter and the Fourier transform parameter, respectively. For the homogeneous boundary conditions of the target edge of the plate, the solution of the generated edge wave can be expressed as a superposition of two wave modes. One is shear wave mode (su perscript ‘ H ’) and the other is symmetric Lamb wave mode (superscript ‘ L ’). The solution form can be rewritten as follows (Liang & Ng 2021): (4) where and denote the coefficients to be determined based on the boundary conditions at the edge (x 3 = 0). and represent eigenfunctions of wavenumbers. Figure 2 illustrates the dispersion curves of ES 0 , Symmetric (S 0 ) and Shear-horizontal (SH 0 ) Lamb wave modes. k represents the wavenumber and FTV = (2 Ω·f· H ) / C 2 , where C 2 is the shear wave velocity, which is 3120 m/s for aluminium (Hughes, Kotousov & Ng 2022). f is the central frequency of the excitation signal. It can be observed that ES 0 couples with SH 0 and S 0 when the FTV is lower than 2 or higher than 6. This phenomenon can significantly increase the difficulties of signal processing. Thus, the present study focuses on the FTV range between 2 and 6 (Hughes, Kotousov & Ng 2022). In order to compare the degradation levels of the intact and fatigued materials, the relative nonlinearity parameter β’ = A 2 / A 1 2 is adopted in this study (Jhang et al. 2020). A 1 is the signal amplitude of the primary wave at the excitation frequency, and A 2 is the signal amplitude of the second harmonic at twice the excitation frequency. + 1 1 + 2 2 + 3 3 + 2 1 2 + 2 22 + 2 32 + = 2 2 =1,2,3 = 1 1 + 2 2 + 3 3 + , + , 2 , 3 , , = , − − 1 1 ∞ −∞ ∞ −∞ 2 , 3 , , = 2 − 3 + 2 − 3 ∞ =0 ∞ =0