† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11834008, 11704410, 11632004, 11474361, and U1930202).
On the basis of second-order perturbation approximate and modal expansion approach, we investigate the enhancement effect of cumulative second-harmonic generation (SHG) of circumferential guided waves (CGWs) in a circular tube, which is inherently induced by the closed propagation feature of CGWs. An appropriate mode pair of primary- and double-frequency CGWs satisfying the phase velocity matching and nonzero energy flux is selected to ensure that the second harmonic generated by primary CGW propagation can accumulate along the circumference. Using a coherent superposition of multi-waves, a model of unidirectional CGW propagation is established for analyzing the enhancement effect of cumulative SHG of primary CGW mode selected. The theoretical analyses and numerical simulations performed directly demonstrate that the second harmonic generated does have a cumulative effect along the circumferential direction and the closed propagation feature of CGWs does enhance the magnitude of cumulative second harmonic generated. Potential applications of the enhancement effect of cumulative SHG of CGWs are considered and discussed. The theoretical analysis and numerical simulation perspective presented here yield an insight previously unavailable into the physical mechanism of the enhancement effect of cumulative SHG by closed propagation feature of CGWs in a circular tube.
Circular tubes widely serve in chemical, pharmaceutical, food and power engineering industries due to their excellent mechanical properties.[1,2] As an effective means, ultrasonic guided waves have been effectively used for nondestructive assessment of circular tubes.[3–6] It is known that the guided wave modes propagating in circular tubes can be classified into two types: axial mode propagating along the axial direction (including longitudinal,[7] torsional,[8] and flexural mode[9]) and circumferential mode propagating along the circumferential direction.[10,11] The circumferential mode of guide wave propagation in a circular tube is referred to as circumferential guided wave (CGW), where its propagation path is closed and the corresponding standing wave is formed in the radial direction.
Due to the closed propagation feature of CGW mode (namely, its propagation path is closed), it is especially suitable for detecting the radial and axial cracks in the circular tube with a larger diameter.[10,11] Qu et al. laid a groundwork for investigations of dispersion relations of CGWs in a hollow cylinder.[12] Then some analytical and experimental investigations of propagation feature of CGWs have been performed.[13–16] Compared with the axial guided wave modes, the CGWs have some unique advantages in nondestructive assessment of circular tube structures. Firstly, when the axial length of a circular tube is relatively short, the axial guided waves can readily be affected by the end-face reflections, while the CGWs can completely avoid the influence of the end-face reflections because they propagate only along the circumference of the given circular tube. Secondly, due to the closed propagation feature of CGWs, the finite-duration CGW signal can propagate periodically along the circumference of the circular tube (or it can be repeatedly detected at the same position of the circular tube), which may make CGWs show more abundant nonlinear wave phenomena compared with the axial guided waves, Lamb waves and SH plate waves investigated previously.[17–20]
Considering the fact that the nonlinear ultrasonic technique is much more sensitive to early damage in materials than that based on the linear feature of ultrasonic wave propagation,[21–25] and that the CGWs have the unique advantages in nondestructive assessment of circular tube structures, it is expected that nonlinear CGWs can be used to effectively assess the early damage in circular tubes. Recently, Gao et al. established a theoretical model to analyze the effect of second-harmonic generation (SHG) of primary CGW propagation.[26] Then, Deng et al. experimentally demonstrated that the second harmonic generated by propagation of primary CGW mode satisfying the phase velocity matching and nonzero power flux could grow along the circumference of the tube.[27] On this basis, it has been experimentally validated that the level of the accumulated damage in a circular tube can effectively be revealed by the acoustic nonlinearity parameter of primary CGW propagation through one full circumference.[28] Furthermore, Li et al. analyzed the nonlinear feature of CGW propagation in a composite circular tube.[29–32] Although it is known that the nonlinear CGWs have the potential for nondestructive assessment of circular tube structures, the physical mechanism how the closed propagation feature of CGWs influences the effect of SHG of primary CGW still remains unrevealed. Specifically, it is necessary to investigate the enhancement effect of cumulative SHG, which is inherently induced by the closed propagation feature of CGWs.
In this paper, modeling and simulation of the enhancement effect of cumulative SHG by the closed propagation feature of CGWs will be conducted. The corresponding theoretical predictions will be examined by the finite element (FE) simulation. Moreover, the potential for applications of the enhancement effect of cumulative SHG of CGWs will be considered and discussed. The results obtained yield a previously unavailable insight into the physical mechanism of the enhancement effect of cumulative SHG by closed propagation feature of CGWs in the given circular tube.
The schematic diagram of a circular tube with inner radius R1 and outer one R2 is shown in Fig.
![]() | Fig. 1. Schematic diagram of the circular tube used for analyzing CGW propagation, where ![]() ![]() |
According to the stress-free boundary conditions on the inner and outer surfaces of the circular tube, both the dispersion relations for CGW propagation and the corresponding displacement fields can readily be determined.[26] When the primary CGW with the driving frequency f and order index l propagates clockwise along the circumference of the circular tube shown in Fig.
![]() |
![](cpb_29_2_024301/cpb_29_2_024301_ieqn3.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn4.gif)
Due to the geometric nonlinearity and the inherent elastic nonlinearity of solid, within a second-order perturbation, accompanying propagation of the lth primary CGW along the tube circumference, there are the traction stress tensors of double the fundamental frequency, denoted by
Based on the modal expansion approach for waveguide excitation,[26,33] both of
![]() |
![]() |
![](cpb_29_2_024301/cpb_29_2_024301_ieqn5.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn6.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn7.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn8.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn9.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn10.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn11.gif)
![]() |
Obviously, the magnitude of second-harmonic field will grow linearly with the circumferential angle θ. Until now, the solution of cumulative second harmonic of primary CGW propagation in a circular tube has been exactly determined using a second-order perturbation and modal expansion analysis approach.
Within a second-order perturbation, the amplitude of primary CGW signal can be regarded to be unchanged when the acoustic attenuation of solid is neglected, while the amplitude of cumulative second harmonic generated will grow linearly with the circumferential angle θ (see Eq. (
![]() |
Clearly, at the same position of the circular tube, the relative nonlinear acoustic parameter βR measured at θ + 2nπ will be (θ + 2nπ)/θ times that measured at θ. Namely, we can get a larger βR just letting the CGW signal propagate n cycles around the circumference.
Next some numerical and analytical considerations will be conducted to understand the foregoing theoretical analyses. The given circular tube shown in Fig.
![]() | Table 1. Material parameters of the circular tube.[26] . |
According to the equations of stress-free boundary conditions at the inner and outer surfaces of the circular tube, the dispersion relations for CGW propagation can readily be calculated and shown in Fig.
![]() | Fig. 2. Dispersion curves for the fundamental and double frequency CGWs in the given circular tube: (a) phase and (b) group velocities. |
In Fig.
Figure
It is known that the displacement field of the lth CGW mode (i.e.,
Generally, the CGWs generated by an excitation source exerted on the given circular tube propagate both clockwise and counterclockwise around the circumference. In the previous investigations,[29–31] only the CGWs propagating clockwise are taken into account. To avoid the interference of CGWs that propagate counterclockwise, in the previous FE simulation model, a section cut is made on the circular tube. For investigation of enhancement effect of cumulative SHG by closed propagation feature of CGWs, it is required that the CGW signal can unidirectionally propagate (e.g., clockwise) around the full circumference. Here a model is proposed based on the coherent superposition of multi-waves to ensure the unidirectional propagation of primary CGW around the full circumference. As shown in Fig.
![]() |
![](cpb_29_2_024301/cpb_29_2_024301_ieqn31.gif)
![](cpb_29_2_024301/cpb_29_2_024301_ieqn32.gif)
Here, FE simulations on enhancement effect of the cumulative SHG by closed propagation feature of CGWs are conducted using a commercial software COMSOL 5.3®. The model used for FE simulations is also shown in Fig.
In the model shown in Fig.
![]() | Fig. 4. Temporal signals received at points P and Q in Fig. |
Next, the enhancement effect of SHG by the closed propagation feature of unidirectional CGW propagation will be examined by FE simulations, which has been analyzed in Subsection
The probe points are placed around the outer surface of the given circular tube (see Fig.
The results of aforementioned theoretical considerations and FE simulations have verified that the closed propagation feature of CGWs can enhance the effect of cumulative second harmonic generated. What we are more interested in is how to apply this closed propagation feature of CGWs in practical applications. Generally, the potential is mainly reflected from the following aspects.
Firstly, the source of nonlinearity can be effectively distinguished using the enhancement effect of cumulative SHG by closed propagation feature of CGWs. It is known that in the measurement of second harmonics generated by primary guided wave propagation, the nonlinearity of the measurement system composed of signal generator, power amplifier, ultrasonic transducer, couplant and so on will lead to the appearance of an extra second-harmonic signal. This extra second-harmonic signal may be even larger than that induced by defects in material. Generally, it is difficult to distinguish the measured second-harmonic signal whether from the measurement system, material itself or both. It is worth noting that the material nonlinearity of interest can be distinguished from the measurement system using the enhancement effect of cumulative SHG by closed propagation feature of CGWs. Take our previous experimental investigation as an example,[28] when the wedge transmitting transducer for generation of the desired primary CGW is driven by the specified tone-burst voltage, the receiving wedge transducer successively receives two clear time-domain signals similar to that in Fig.
Secondly, the detection sensitivity of early damage stage can be improved using the enhancement effect of cumulative SHG by closed propagation feature of CGW. Previous investigates indicate that in the early damage stage of material, change in the second-order elastic constants (SOECs) of material is inconsiderable, and variation in the relative nonlinear acoustic parameter should be mainly attributed to change in the three-order elastic constants (TOECs) of material.[21] To facilitate the investigation of assessing early damage in material (characterized by change in TOECs) by using the relative nonlinear acoustic parameter, the TOECs are assumed to change from its initial values (A, B, C) to rc × (A, B, C),[31] where rc is the damage status coefficient used to describe the degree of early damage, while the remaining material properties and geometric parameters of the circular tube are kept unchanged. Base on Eq. (
Theoretically, increasing the propagation cycle number n can improve the detection sensitivity to the early damage. However, it should be noted that increasing the propagation cycle number n is equivalent to increasing the propagation path. Practically, the propagation attenuation of CGWs will inevitably increase with the propagation cycle number n, which will reduce the signal-to-noise ratio (SNR). Thus, it is necessary to find a balance between keeping an appropriate SNR and improving the detection sensitivity of early damage in practical measurements.
A theoretical model is established for analyzing the enhancement effect of cumulative SHG by the closed propagation feature of CGWs in a circular tube. Using a second-order perturbation and a modal expansion analysis, the mathematical expression of the second-harmonic fields by primary CGW propagation in a circular tube has been derived. In order to realize the unidirectional propagation of CGWs, a model is proposed based on coherent superposition of multi-waves to ensure the unidirectional propagation of primary CGWs around the full circumference. Regarding the enhancement effect of cumulative SHG by the closed propagation feature of CGWs, some numerical and analytical considerations are performed, and then FE simulations are conducted to further validate the said effect. A close agreement between the theoretical predictions and FE simulations verifies the effectiveness of using the model based on the coherent superposition of multi-waves to realize the unidirectional propagation of primary CGWs, and the enhancement effect of cumulative SHG by closed propagation feature of CGWs. On this basis, the potential for applications of the closed propagation feature of CGWs are considered and discussed. It is convinced that, in the SHG based nonlinear measurements, the enhancement effect of cumulative SHG by closed propagation feature of CGWs can be used to effectively remove the unwanted nonlinearity coming from the measurement system. In addition, with the enhancement effect of cumulative SHG by closed propagation feature of CGWs, the detection sensitivity of early damage of circular tube can be improved, especially when the SHG effect of CGW propagation is not enough to accurately characterize the early damage of material. The potential mentioned here is expected to play an important role in the nonlinear CGW detection technology. The results obtained here yield a previously unavailable insight into the physical mechanism of the enhancement effect of cumulative SHG by closed propagation feature of CGWs in a circular tube.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] |