# The Effects of Stress on Second Harmonics in Plate-Like Structures
## 1 Introduction  
When the ultrasonic waves propagate in the solid medium, because of the anharmonicity of the crystal structure and the existence of the internal micro defects (micro cracks, stress concentration, etc. ), the ultrasonic waves will have \"distortion\" in the process of propagation, showing a strong nonlinear effect. The nonlinear phenomena are specifically manifested in that the received signal will generate high-order harmonics in addition to the original fundamental frequency signal. The high-order harmonics contain the information that is difficult to be detected by traditional ultrasonic technologies, such as the micro defects in the structure and the material characteristics. Nonlinear ultrasonic technology applies acoustic means to extract the high-order harmonics in the received signal and calculates the unique ultrasonic nonlinear parameter. Nonlinear parameter can be used to quantify the change of material properties and the degree of damage inside the structure, so as to realize the health monitoring and evaluation of the structure [1; 2; 3; 4; 5; 6; 7]. In recent years, nonlinear ultrasonic guided waves have drawn increasing attention in the field of structural stress detection, as a result of their advantages of traditional ultrasonic guided waves [8; 9; 10; 11; 12; 13], high sensitivity, and unique characteristics of determining the internal state of structure. [PERSON] [14; 15] studied the application of nonlinear guided waves to the stress detection of pre-stressed steel barsin concrete structures and found that the contact stress between any strand and adjacent strand was closely related to the nonlinear effect of guided waves. [PERSON] (2016); [PERSON] (2016) proved that the nonlinear surface waves can be used to detect the residual stress on the surface of aluminum alloy. [PERSON] (2018); [PERSON] (2018) built a residual stress prediction model by exploring the nonlinear characteristics of the metal material in the process of tension and compression. In addition to the progress of applications, there is also a good development in theoretical researches. [PERSON] (2018) introduced an efficient technique based on the mode expansion method, to obtain second harmonic modal expression. [PERSON] (2018); [PERSON] (2018); [PERSON] (2018) investigated the nonlinear phenomenon of ultrasonic Lamb waves in the elastic plate by using second-order perturbation approximation. [PERSON] concluded that the cumulative second harmonic exists under some conditions. The cumulative second harmonic means that second harmonic modal amplitude increases with the increase of propagation distance. [PERSON] and [PERSON] (2018); [PERSON] (2018) arrived at the conclusion that the cumulative effect of second harmonic takes place only when the phase-velocity matching between fundamental mode and second harmonic mode is satisfied and the power flow from fundamental mode to second harmonic mode is not zero. [PERSON] and [PERSON] (2018) further analyzed the types of cumulative second harmonic modes in the plate and proposed that if a second harmonic mode whose amplitude can be accumulated in the plate is to be generated, the second harmonic mode must be a symmetric mode. [PERSON] and [PERSON] (2018) theoretically studied the symmetry characteristics of nonlinear Rayleigh-Lamb waves in plate and explained the reason behind the absence of antisymmetric modes at the cumulative second harmonic. [PERSON] (2018); [PERSON] (2018) proposed another condition for the generation of the cumulative second harmonic, that is, the group velocity of the fundamental mode should be equal to the group velocity of second harmonic mode. However, [PERSON] (2018) further proved both analytically and experimentally that, as long as the two aforementioned conditions (phase-matching and non-zero power flux) are satisfied, the cumulative second harmonic can be generated, even though the group velocity matching condition is not satisfied.