# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 3 Analytical Method of Second Harmonics Under Stress Change
### Response of Second Harmonics under Stress  
In this paper, it is assumed that the plate is subjected to axial stress and other directions are in free state. After determining the mode pairs, assume that the axial stress on the steel plate gradually increases from the compressive stress of -200 MPa to the tensile stress of 200 MPa, and the stress changes by 40 MPa each time. By substituting the stress value into Equation (26), the dispersion curves of phase velocity and group velocity of Lamb waves in the steel plate under definite stress are acquired. The mode shapes of the corresponding mode pair can be extracted from the dispersion curves. Substitute the corresponding mode shapes into Equation (18), and the amplitude coefficient \(\overline{A}_{n}(x)\) of second harmonic modes generated by this mode pair under definite stress are obtained. After getting the magnitude of the amplitude coefficient, \(\overline{A}_{n}(x)\), of the second harmonic mode under different stress, the change trends of \(\overline{A}_{n}(x)\) are estimated by using the Least Squares method, as shown in Figure 7.  
Figure 7 presents that during the stress change process from -200 MPa compressive stress to 200 MPa tensile stress, the amplitudes of the second harmonic modes generated by mode pairs 7-9 and 12-15 decrease gradually, and the amplitudes of the second harmonic modes generated by mode pairs 10 and 11 increase gradually. In Figure 7, the amplitude coefficient, \(\overline{A}_{n}(x)\), shows a monotonic change trend, but the change slope is not always constant, indicating that the change trend of the second harmonic amplitude with the change of stress is not linear. The change of \(\overline{A}_{n}(x)\) is mainly due to the change of the mode shapes about the mode pair, and the change of the mode shapes come from the influence of the axial stress on the plate. Therefore, the variation trend of \(\overline{A}_{n}(x)\) can be used to express the change of the amplitude of the second harmonic mode with the stress. The simulation results also show that the change trends of the second harmonic modal amplitudes with the stress are not the same when different excitation and reception modes are selected. The method can be applied to provide the anticipation for the variation of the amplitude of second harmonic generated by mode pair in the nonlinear ultrasonic guided waves experiments.  
It should be noted that the numerical changes of some data points in Figure 7 are not consistent with the overall change trend. The error data points are caused by the small deviation of the dispersion curve obtained by Equation (26) after the stress is applied, which leads to the inaccuracy of some mode shapes extracted. The small change of mode shapes eventually leads to the appearance of wrong data points. Since the method proposed in this paper is used to predict the change trend of cumulative second harmonic mode in plate under stress, the existence of error points will not affect the overall trend identification. By increasing the number of simulation data points, we make the change trends of cumulative second harmonic modes obtained by this method accurate.  
Figure 7: _Cont._Figure 7: _Cont._