# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 3 Analytical Method of Second Harmonics Under Stress Change
### Analysis of Second Harmonics in Plate  
Suppose the model of infinite width free plate is as shown in Figure 1; its cross-section is the y-z plane, and the wave propagation direction is the x direction. A one-dimensional three-nodes element is applied to discretize the cross-section of the plate. Each element has three nodes, and each node hasthree degrees of freedom, corresponding to the displacement in the coordinate axis direction, as shown in Figure 2.  
The thickness of the plate model is 15 mm, the material is defined as steel, and the relevant density, Lame coefficients, and third-order elastic constants of the model are tabulated in Table 1[39]. According to Equation (25), the dispersion curves of phase velocity and group velocity of Lamb waves in stress-free steel plate can be obtained by MATLAB software. The dispersion curves are shown in Figures 3 and 4. In Figures 3 and 4, \(C_{p}\) (\(C_{p}=\omega/k\)) is the phase velocity of ultrasonic guided wave mode, \(C_{g}\) (\(C_{g}=d\omega/dk\)) is the group velocity of ultrasonic guided wave mode, and \(fd\) is the frequency-thickness product. It can be seen from Figures 3 and 4 that there are many modes at the same frequency, and the number of modes increases with the increase of frequency, which reflects the multi-modal characteristics of ultrasonic guided waves. Each point in the dispersion curve represents a mode at the certain frequency. Through the dispersion curve, the phase velocity, group velocity, and mode shape of any mode can be obtained. Without loss of generality, the excitation signal frequency of 2.5 MHz is used in the following study, to facilitate experimental verification of the proposed method with the 2.5 and 5 MHz transducers in the laboratory. According to the phase matching and cumulative second harmonic mode that should be a symmetric mode, all the mode pairs that may generate cumulative second harmonic are selected, the mode shapes of each mode pairs are obtained, and the value of \(f_{n}{}^{surf}+f_{n}{}^{vol}\) is calculated. The relevant \begin{table}
\begin{tabular}{c c c c c c} \hline \(\rho(kg\mathbf{m}^{3})\) & \(\Lambda(GP\mathbf{a})\) & \(\mu(GP\mathbf{a})\) & \(A(GP\mathbf{a})\) & \(B(GP\mathbf{a})\) & \(C(GP\mathbf{a})\) \\ \hline
7932 & 116 & 83 & \(-\)340 & \(-\)647 & \(-\)17 \\ \hline \end{tabular}
\end{table} 
Table 1: Material properties of the plate. Figure 1: Semi-analytical finite element (SAFE) model of wave propagating in plate. Figure 2: Discretization of a plate with one-dimensional three-node elements. values are shown in Table 2. In Table 2, A represents antisymmetric mode, and S represents symmetric mode. In order to form a comparison, three mode pairs that do not meet the phase matching principle are also introduced in Table 2.  
In Table 2, mode pairs 1-15 meet the phase matching principle, and mode pairs 16-18 do not meet. In regard to the mode pairs that meet the phase matching principle, only the mode pairs 7-15 meet the condition of \(f_{n}{}^{surf}+f_{n}{}^{vol}\
eq 0\), so these mode pairs can generate cumulative second harmonic, and these mode pairs are sorted by the value of \(f_{n}{}^{surf}+f_{n}{}^{vol}\) in Table 2, from small to large. Substitute the mode shapes of mode pairs 7-15 and 16-18 into Equation (18), and the variations of amplitude coefficients, \(\overline{A}_{n}(x)\), with the propagation distance are shown in Figures 5 and 6. Figure 5 shows that the amplitudes of second harmonic modes generated by mode pairs 7-15 rise with the increase of propagation distance, indicating that mode pairs 7-15 can generate the cumulative second harmonics. However, the magnitudes of second harmonic modes generated by different mode pairs are not the same under the same distance.