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Article

Experimental Investigation of One-Way Lamb and SH Mixing Method in Composite Laminates

by
Siyang Xie
1,
Youxuan Zhao
1,* and
Yuzi Liu
2
1
College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
2
China Aircraft Intensity Research Institute, Xi’an 710065, China
*
Author to whom correspondence should be addressed.
Sensors 2025, 25(24), 7631; https://doi.org/10.3390/s25247631
Submission received: 16 November 2025 / Revised: 12 December 2025 / Accepted: 15 December 2025 / Published: 16 December 2025
(This article belongs to the Section Fault Diagnosis & Sensors)
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Abstract

This paper experimentally investigates the resonant behavior of the one-way Lamb and SH (shear horizontal) mixing method in composite laminates with inherent quadratic nonlinearity, delamination damage and impact damage. When the fundamental S0-mode Lamb waves and SH0 waves mix in the damage regions of composite laminates, experimental results demonstrate the generation of the resonant SH0 waves with the resonance condition. Meanwhile, the damage localization method in composite laminates is experimentally verified by the time-domain signal of resonant waves. Furthermore, it is found that the one-way Lamb and SH mixing method is sensitive to inherent quadratic nonlinearity and impact damage but insensitive to delamination damage.

1. Introduction

The application of composite laminates has grown significantly in aeronautical and other industries due to their excellent mechanical properties, lightweight characteristics and corrosion resistance. However, different types of damage could gradually lead to the performance degradation of composite laminates in service, which are mainly fiber fracture, matrix cracking, matrix–fiber interface debonding and delamination [1]. The development of performance degradation could result in a final failure. Therefore, it is essential to detect and evaluate performance degradation for ensuring the integrity and durability of composite structures.
Non-destructive testing (NDT) technology is widely used in composite material testing applications [2,3], including the ultrasonic method [4,5,6,7,8], X-ray detection method [9,10], infrared thermal imaging method [11,12] and acoustic emission (AE) method [13,14]. Among them, the ultrasonic method has attracted wide attention due to its high sensitivity and strong penetration even in composite structures, which includes the linear ultrasonic method [15,16] and nonlinear ultrasonic method [17,18,19]. Furthermore, nonlinear guided wave techniques have great potential to evaluate early material degradation in composite laminates with their long propagation and high sensitivity to material microstructures. Considering different nonlinear effects, the higher harmonics technique [20,21], nonlinear wave mixing technique [22,23,24] and quasi-static component (QSC) technique [25,26] have been widely studied in recent years.
Many efforts have been devoted to the application of the higher harmonics technique in composites. Shan et al. [27] have proposed two damage indices to evaluate delamination in composites based on the amplitude of second harmonic Lamb waves. Rauter et al. [28] have applied a second harmonic guided wave to detect fatigue damage in composites. Alnuaimi et al. [29] have proposed the modified Sideband Peak Count Index (SPC-I) technique to detect and monitor impact damage in composite plates. During the propagation of nonlinear ultrasonic waves, the quasi-static component can also be generated with the generation of higher harmonics. Wang et al. [30] have employed QSC to evaluate the thickness of materials with high attenuation. Li et al. [25] have experimentally evaluated thermal degradation by detecting QSC pulses. Jiang et al. [31] have investigated static component (SC) generation of guided waves (GWs) under group velocity matching condition to assess hygrothermal damage and low-velocity impact damage in composite plates.
Although the higher harmonics technique and quasi-static component technique are able to evaluate nonlinear damage, the nonlinear wave mixing technique has received more attention for its abilities of locating damage positions and avoiding interference from noise signals. The nonlinear wave mixing method with bulk wave has been employed to detect plasticity, fatigue, corrosion [32] and thermal aging [33] in materials. Sun et al. [34] have analytically derived and numerically investigated the resonance conditions for one-way resonant mixing of nonlinear Lamb waves. Lissenden et al. [35] have proposed internal resonance criteria for the guided wave-mixing method: phase–velocity matching and the nonzero power flux. Li et al. [36] have proposed the one-way S0-A0 mixing method with the resonance condition ω S 0 / ω A 0 = 2 κ / ( κ + 1 ) to maximize the energy of the resonant wave (where κ = c S 0 / c A 0 , c S 0 and c A 0 are the phase velocities of the S0 and A0 mode Lamb waves) and experimentally investigated the corrosion damage evaluation [37]. Jiao et al. [38] have experimentally and numerically investigated residual stress measurement in metal plates based on the collinear Lamb wave mixing method. Li et al. [39] have experimentally investigated zero-group velocity (ZGV)-combined harmonic generation induced by the mixing of counter-directional Lamb waves. Zhu et al. [40] have explored the low-frequency Lamb wave mixing method for fatigue damage evaluation. Liu et al. [41] have proposed and verified the one-way Lamb and SH mixing method for evaluating and locating the damage region in thin plates. Shan et al. [42] have investigated the generation of cumulative secondary S0 Lamb waves at the sum frequency when two primary codirectional shear-horizontal SH0 waves converge. Based on the progress in guided wave mixing technique research, recent work has begun exploring its applicability to composite materials. Li et al. [43] have employed the guided wave mixing technique combined with the mixing frequency peak counting approach to evaluate impact damage in composites. Pineda et al. [44] have applied the A0-A0 Lamb mixing method to detect delamination damage in composite laminates. Lan et al. [45] have employed the chirp guided wave mixing method (CGWM) to detect delamination in composite laminates. Aslam et al. [46] have investigated the nonlinear acoustic response of feature-guided waves (FGWs) in welded joints by one-way and two-way mixing methods.
Compared with other guided wave mixing methods, the S0-SH0 mixing method could have great potential for damage evaluation due to the approximate non-dispersion characteristic of SH0 waves in the low-frequency range [47], which could reduce the complexity of ultrasonic signal processing. Recently, the S0-SH0 mixing method has been numerically and experimentally investigated on a 2 mm aluminum plate [41]. Considering the anisotropy of composite laminates, the feasibility of the S0-SH0 mixing method needs to be discovered through numerical simulations [48] and experimental measurements. Therefore, this paper aims to experimentally investigate the feasibility of the one-way Lamb and SH mixing method for the damage detection of composite laminates with inherent quadratic nonlinearity, delamination damage and impact damage. Meanwhile, the method to locate the impact damage region in composite laminates is investigated, and the result is compared with a C-scan result. The novelty of this paper could be the experimental demonstration of the feasibility of the one-way S0-SH0 mixing method to evaluate damage in composite laminates.

2. Lamb and SH Mixing Method in Composite Laminates

A transverse wave pulse and a longitudinal wave pulse are successively emitted at the same side and propagate in the same direction with the resonance condition ω L / ω T = 2 κ / ( κ + 1 ) ; this process is called the one-way mixing technique [49], where κ = c L / c T , c L and c T are the longitudinal and transverse velocities, and ω L and ω T are the circular frequencies of the longitudinal and transverse wave pulses, respectively. Then, when two fundamental wave pulses mix in the region with quadratic nonlinearity, a transverse resonant wave with difference frequency of these wave pulses can be generated and propagate in the opposite direction of the fundamental transverse wave pulse, where quadratic nonlinearity derives from the quadratic nonlinear elastic constitutive model, including second-order and third-order constants [41]. Similarly, the one-way S0-A0 mixing method with the resonance condition ω S 0 / ω A 0 = 2 κ 1 / ( κ 1 + 1 ) and the one-way S0-SH0 mixing method with the resonance condition ω S 0 / ω S H 0 = 2 κ 2 / ( κ 2 + 1 ) are successively proposed and verified by Zhao et al. [37,41], where κ 1 = c S 0 / c A 0 and κ 2 = c S 0 / c S H 0 , c S 0 , c A 0 and c S H 0 are the phase velocities of the S0- and A0-mode Lamb waves and SH0 waves, respectively.
Considering composite laminates with quadratic nonlinearity, the one-way Lamb and SH mixing method is also effective. The schematic of the one-way S0-SH0 mixing method in composite laminates with quadratic nonlinearity is described in Figure 1. To locate the damage region with the length L, SH0 wave pulses and S0-mode Lamb wave pulses are sequentially excited in a fundamental waves excitation position. Then, they interact with each other in the mixing zone with the length Ld. When the resonance condition ω S 0 / ω S H 0 = 2 κ 2 / ( κ 2 + 1 ) is satisfied and the quadratic nonlinearity of composite laminates exists, a series of nonlinear effects could be triggered, which leads to the generation of the resonant SH0 wave. The frequency of the resonant SH0 wave is ω R = ω S 0 − ω S H 0 , and its propagation direction is opposite to that of fundamental SH0 waves, where ω S 0 and ω S H 0 are the circular frequencies of S0-mode Lamb waves and SH0 waves, respectively. The resonant SH0 wave is finally received at the signal-receiving position, which is Dd away from the fundamental waves excitation position.
Then, the damage localization formula based on the heterodyne detection technique can be expressed by Equations (1)–(3) [41]:
P S = ( T s t a r t − Δ T + D d / V g S H R ) V g S V g S H R / ( V g S + V g S H R )
P E = ( T e n d − T S − Δ T + D d / V g S H R ) V g S V g S H R / ( V g S + V g S H R )
L = P E − P S
where P S is the distance between the left boundary of the damage region and fundamental waves excitation position in Figure 1, P E is the distance between the right boundary of the damage region and fundamental waves excitation position, L is the length of the damage region, Dd is the distance between fundamental waves excitation position and receiving position, T S is the time of exciting fundamental S0-mode Lamb wave pulses, Δ T is the time interval between exciting S0-mode Lamb wave pulses and SH0 wave pulses, V g S H R is the group velocity of the resonant wave, V g S is the group velocity of the fundamental S0-mode Lamb waves, and T s t a r t and T e n d are the start and the end time of the resonant wave, respectively.

3. Experimental Measurement

In this section, the feasibility of the one-way S0-SH0 mixing method to detect damage regions in composite laminates is explored through experimental measurements. Inherent quadratic nonlinearity, delamination damage and impact damage are considered, respectively. Meanwhile, the effectiveness of one-way S0-SH0 and S0-A0 mixing methods for delamination damage detection is also investigated. In addition, the reliability of the damage localization method for impact damage in composite laminates is verified.

3.1. Experimental Setup

3.1.1. Preparation of Composite Samples with Damage

In this study, experimental measurements are conducted on T300 carbon-fiber-reinforced composite laminates (Blue Label Composite Materials Inc., Dezhou, China), which have dimensions of 1000 mm × 400 mm × 3.2 mm with the stacking sequence [0/90]4s.
Three cases are prepared to investigate the feasibility of the one-way S0-SH0 mixing method for detecting damage regions in three individual composite laminates.
  • For case A, inherent quadratic nonlinearity of composite laminates is introduced as one type of early material degradations, which is naturally caused by the manufacturing process [50]. Thus, case A, with inherent quadratic nonlinearity, is employed to verify the generation mechanism and the resonance condition of the one-way S0-SH0 mixing method in composite laminates.
  • For case B, delamination damage in composite laminates is introduced by prefabricated delamination, which is arranged at the center of the composite sample and between the 8th and 9th layers through a 100 × 100 mm polytetrafluoroethylene (also known as Teflon [51]) film, which is effective in separating two mating surfaces without significantly altering the properties of the composite laminates [52]. The schematic of delamination damage in composite laminates is shown in Figure 2. Meanwhile, the one-way S0-SH0 mixing method and the one-way S0-A0 mixing method are both employed to detect delamination damage, and the comparison of their effects are discussed in detail.
  • For case C, impact damage with an impact energy of 10 J [53,54] is introduced in composite laminates by dropping a 1 kg weight and 62.5 mm diameter iron ball from a 1 m height position, which is employed to verify the capability of the one-way S0-SH0 mixing method for detecting and locating impact damage.

3.1.2. Ultrasonic Measurement

The experimental setup of the S0-SH0 mixing method in composite laminates is shown in Figure 3. The fundamental signals are generated by two channels, respectively, from the RAM-5000 SNAP system (RITEC Inc., Warwick, RI, USA). Channel 1 is applied to excite SH0 waves by the PZT-5H piezoelectric ceramic(d15) [55] (Baoding Hongsheng Ceramic Inc., Baoding, China). According to the theoretical dispersion curve, the excited frequencies of SH0 waves in the low-frequency range (100 kHz–250 kHz) are selectively employed to ensure only zero-mode SH waves in experiments. Then, channel 2 is applied to excite S0-mode Lamb waves by Olympus transducer (Model: A413S-SB, Olympus Inc., Tokyo, Japan) through the wedge block (Model: ABWX-2001, Olympus Inc., Tokyo, Japan) with an oblique angle of 26.2° [41]. The PZT-5H piezoelectric ceramic and the wedge block are located at excitation positions with zero distance. Meanwhile, the transducer, wedge, and composite laminates are coupled with an ultrasonic coupling agent. The PZT-5H piezoelectric ceramic is bonded to the surface of the composite laminate using conductive silver adhesive. When SH0 waves and S0-mode Lamb waves interact with each other in the damage region, the resonant waves can be generated and propagate in the opposite direction of two fundamental waves, which can be received and converted into a voltage signal by another PZT-5H piezoelectric ceramic at the receiving position. The distance between the wave excitation position and receiving position is fixed at 100 mm. Finally, to reduce uncertainty, the voltage signals are received by the DPO 3014 digital oscilloscope (Tektronix Inc., Beaverton, OR, USA) with 512 times sampling average operation, which is saved into the computer. Note that the size and location of the damage region in three cases are different. Therefore, the damage region in Figure 3 is shown for illustrative purposes only.
The frequencies of two fundamental waves are selected according to dispersion curves (discussed in Section 3.2.1 in detail), and both fundamental wave pulses have 10 cycles with a continuous sinusoidal signal [37]. In addition, the synthesizer amplitude for SH0 waves is set as 50%; the output level of gated amplitude is set as 50. For S0-mode Lamb waves, the synthesizer amplitude and the output level of gated amplitude are the same as for SH0 waves.
The signal processing method in Ding’s experiment [37] is employed to effectively obtain the resonant wave signals. The received signal α is acquired by only emitting SH0-mode fundamental waves, then the received signal β is acquired by only emitting S0-mode fundamental waves; finally, the received signal γ is acquired by emitting both S0- and SH0-mode fundamental waves. As a result, the resonant wave signal η = signal γ − signal α − signal β.

3.2. Experimental Results

3.2.1. The Selection of Fundamental Waves

The frequencies of S0-mode Lamb waves and SH0 waves are vital for the generation of resonant waves, which need to be carefully selected. The frequency pairs of two fundamental waves need to satisfy the resonance condition ω S 0 / ω S H 0 = 2 κ / ( κ + 1 ) and only generate S0-mode Lamb waves and SH0 waves. Considering the material properties of T300 carbon-fiber-reinforced composite listed in Table 1 (from the software Dispersion Calculator V2.0 [56]), the dispersion curves of Lamb waves and SH waves in a 3.2 mm thick T300 composite laminates are shown in Figure 4. Therefore, the representative frequency pairs are carefully selected and listed in Table 2, where f S 0 and f S H 0 are, respectively, the frequencies of S0-mode Lamb waves and SH0 waves; f R = f S 0 − f S H 0 is the theoretical frequency of the resonant waves.
More importantly, the frequencies and group velocities of two fundamental waves need to be prudently obtained to verify the generation of SH0 waves and S0-mode Lamb waves in experiments. Figure 5 shows the typical time-domain and frequency-domain signals of fundamental SH0 waves and S0-mode Lamb waves ( f S 0 = 250 kHz and f S H 0 = 160 kHz). The frequency-domain signals are obtained by performing FFT on time-domain signals. Then, it can be found that the experimental frequencies agree well with excited frequencies of fundamental waves. The experimental results of other frequency pairs listed in Table 2 are also as expected. Meanwhile, the group velocities of fundamental waves with different frequencies are listed in Table 3 and Table 4. The theoretical group velocities can be determined from dispersion curves, and the experimental group velocities can be calculated by Time-of-Fly of wave packets between two fixed points [57]. Then, the approximate non-dispersion characteristic of SH0 waves and low-level dispersion characteristic of S0-mode Lamb waves in low-frequency range can be verified in quasi-isotropic composite laminates. Moreover, it can be seen that the error of SH0 waves’ group velocities between experiment and theory decreases with the increase in the frequency, while the error of group velocity of S0-mode Lamb waves between experiment and theory changes within a small range. The errors could be caused by the deviation between actual material properties in experimental composite laminates and theoretical material properties listed in Table 1. The experimental group velocities are also in good agreement with the theoretical group velocities from the dispersion curves in Figure 4. Therefore, it can be confirmed that S0-mode Lamb waves and SH0 waves in composite laminates are successfully excited by the Olympus transducer and the PZT-5H piezoelectric ceramic, respectively.

3.2.2. Results of Case A

In this section, a series of experiments based on the one-way S0-SH0 mixing method with different frequency pairs are conducted in composite laminates with inherent quadratic nonlinearity, which aim to investigate the feasibility of the generation of resonant waves. The frequencies and waveforms of resonant waves are obtained to verify the generation of SH0 resonant waves. Figure 6, Figure 7 and Figure 8 show the time-domain and frequency-domain signals of resonant waves with different frequencies. Note that, considering the frequency bandwidth from ( f R − 10) kHz to ( f R + 10) kHz, a Butterworth filter is applied to clearly obtain resonant wave signals. Distinct diamond shapes could be found in the waveforms of resonant waves, which is consistent with the results of numerical simulation from Liu [48]. Then, the frequencies of different resonant waves can be obtained, as listed in Table 5. It is found that the errors of resonant frequencies between experiment and theory change within a small range. Meanwhile, resonant waves collected at receiving position can also prove the opposite propagation direction of fundamental waves, which is located between the excitation position and the damage region. Therefore, it is experimentally confirmed that resonant waves can be successfully generated in composite laminates with inherent quadratic nonlinearity when the resonance condition ω S 0 / ω S H 0 = 2 κ / ( κ + 1 ) is satisfied.
Additionally, to further validate the damage localization method in Section 2, the frequency pair of f S 0 = 250 kHz and f S H 0 = 160 kHz is adopted to investigate the start and end time of resonant waves. Based on the one-way S0-SH0 mixing method, the mixing zone of resonant waves can be controlled by changing the delay time. Thus, the intervals between S0-mode Lamb wave and SH0 wave excitation time are set to 120 μs, 200 μs and 260 μs, respectively. Figure 9 shows the time-domain signals of resonant waves with different intervals, clearly indicating the start and end time. Note that Tstart in experiments shows the start time of resonant waves when the variation magnitude of the amplitude of the time-domain signal suddenly increases, and Tend shows the end time of resonant waves when the variation magnitude of the amplitude of the time-domain signal suddenly decreases. The localization results with the comparison between experiment and theory are listed in Table 6. Small errors between experimental values and theoretical values can be found. Then, the position and the length of the damage region in composite laminates can be accurately and experimentally obtained through time-domain signals of resonant waves. Therefore, based on the one-way S0-SH0 mixing method, the damage localization in composite laminates with inherent quadratic nonlinearity can be achieved with remarkable potential.

3.2.3. Results of Case B

Different from case A, composite laminates for case B with delamination damage aim to investigate the feasibility of delamination damage detection in composite laminates based on the S0-SH0 mixing method. It should be noted that, due to appropriate sensitivity and intensity, the frequency pair ( f S 0 = 250 kHz and f S H 0 = 160 kHz) is employed in the following experiments.
The resonant wave signals in composite laminates with/without delamination are shown in Figure 10. Due to delamination damage, the amplitude of resonant waves decreases remarkably with distinct degradation of the diamond-shaped waveform. Meanwhile, the ANP (acoustic nonlinearity parameter) [41] is defined as β = A R / ( A S 0 · A S H 0 ) , where A R , A S 0 and A S H 0 are the amplitudes of resonant waves, S0-mode Lamb waves and SH0 waves, respectively. In this section, the ANPs without delamination and with delamination are denoted as β i and β d , respectively. Delamination damage in composite laminates is cumulatively generated on the basis of inherent quadratic nonlinearity. Thereby, the ANP focused on delamination damage is denoted as β d ′ = β d − β i . The experimental values of β i and β d ′ are 3.24 and −0.68, respectively, which means that ANP observably decreases with delamination damage. It could be inferred that the one-way S0-SH0 mixing method is not sensitive to delamination damage in composite laminates. The vibration direction of SH0 waves is perpendicular to the direction of delamination type I, which could lead to ineffective energy transmission of fundamental SH0 waves to the delamination damage. Thus, lower-level resonant waves could be generated in the process, which could be the possible reason.
To verify the above inference, a similar experiment based on the one-way S0-A0 mixing method is employed for comparison. Similarly to the experiments of the one-way S0-SH0 mixing method, the selection of fundamental frequency pairs of the one-way S0-A0 mixing method is also based on the dispersion curves and the resonance condition [36]. Thus, 250 kHz of the S0-mode Lamb wave and 150 kHz of the A0-mode Lamb wave are appropriately selected for the S0-A0 mixing experiment. As a result, a 100 kHz A0 resonant wave could be generated in composite laminates. Note that the A0-mode wave with the frequency of 150 kHz is excited via another Olympus transducer (Model: A413S-SB) at the excitation position [37]. Figure 11 shows the time-domain and frequency-domain signals of 150 kHz A0-mode fundamental Lamb waves with 1 MHz filtering processing, which could reduce the oscillations. The experimental frequencies agree well with excited frequencies of fundamental waves. Compared with the theoretical group velocity of A0-mode Lamb waves at 1572.63 m/s, the experimental value is 1654.56 m/s, which shows a small error of 5.21%. Therefore, it can be confirmed that A0-mode Lamb waves in composite laminates are successfully excited by the Olympus transducer.
Next, the one-way S0-A0 mixing experiments are conducted in the composite laminates without/with delamination, respectively. The signals of corresponding resonant waves are shown in Figure 12. The amplitude of A0 resonant waves increases significantly with the distinct diamond-shaped waveform. Meanwhile, the ANP for the one-way S0-A0 mixing method is defined as β = A R / ( A S 0 · A A 0 ) [36], where A A 0 is the amplitude of A0-mode Lamb waves. And the similar ANPs β i , β d and β d ′ are also adopted in the S0-A0 mixing experiments. The comparison of ANPs between S0-SH0 and S0-A0 mixing methods is shown in Table 7. It can be found that the ANP β d ′ of the S0-A0 mixing method increases significantly compared with that of the S0-SH0 mixing method. The vibration direction of the A0 resonant wave is parallel to the direction of delamination type I, which leads more sensitivity to the delamination damage area in composite laminates of the one-way S0-A0 mixing method. Thus, considering delamination damage in composite laminates, it is suggested that the S0-A0 mixing method could be more appropriate than the S0-SH0 mixing method.

3.2.4. Results of Case C

Different from cases A and B, impact damage is introduced to investigate the feasibility of the S0-SH0 mixing method in detecting impact damage of composite laminates. Based on the research results from References [53,54], the energy limit of barely visible impact damage (BVID) is approximately 5–15 J, and the approximate circular shape of the impact damage region is obtained by the C-scan result. Thus, impact energy of 10 J is adopted to introduce impact damage in case C.
Similarly to case B, the signals of resonant waves and the corresponding ANPs are obtained to investigate the sensitivity of the S0-SH0 mixing method to impact damage in composite laminates. The signals of resonant waves with/without impact damage are shown in Figure 13. Compared with the case without impact damage, a more distinct diamond-shaped waveform of resonant waves could be discovered in the case with impact damage. Meanwhile, the ANPs without/with impact damage are denoted as β i and β p , respectively. Impact damage in composite laminates is also cumulatively generated on the basis of inherent quadratic nonlinearity. In addition, the ANP focused on impact damage is denoted as β p ′ = β p − β i . The experimental values of β i and β p ′ listed in Table 8 are 2.93 and 21.38, respectively, which means that ANP is significantly increased with impact damage. Therefore, it could be concluded that the one-way S0-SH0 mixing method has a remarkable effect on the detection of impact damage in composite laminates.
In addition, the damage localization method mentioned in Section 2 is employed to detect the impact damage region in composite laminates. Based on the theoretical arrival time of resonant waves and the distinct diamond-shaped waveform, the signals of resonant waves generated in the impact damage region are shown in Figure 14 (in the blue rectangular box). Then, the start and end time of resonant waves can be obtained, which are 295 μs and 407 μs, respectively. Based on the damage localization formula, the damage localization results could be calculated, as shown in Table 9.
Moreover, the wheel-mounted ultrasonic fast C-scan system is employed to verify the reliability of the damage localization method [58]. The system consists of a 128-element linear array wheel-mounted composite material probe, with an excitation frequency of 5 MHz ultrasonic waves. The handheld probe moves at a speed of 50 mm/s along the surface of composite laminates to acquire ultrasonic data for full coverage. The gate range is set to extract the reflected wave signal occurring between the initial pulse and the first full-thickness reflection pulse. Then, these data are processed by the C-scan system to generate C-scan images, which visually characterize the size, location, and depth of any damage. As a result, a two-dimensional cloud map is generated, as shown in Figure 15. The damage localization results from the C-scan method are also listed in Table 9.
Considering the damage localization results in Table 9, the positions of the damage center from the S0-SH0 mixing method and C-scan method are strongly consistent, being 513.15 mm and 512.31 mm, respectively. However, the size of the impact damage region from the S0-SH0 mixing method (118.10 mm) is obviously larger than that from the C-scan method (35.42 mm). It could be caused by the differences in fundamental principles between the S0-SH0 mixing method and the C-scan method. Due to impact damage, the transition region with material degradation may exist between a large-sized defect region and an undamaged region. Based on the linear reflected signals, only a large-sized defect region could be imaged by the C-scan method [59]. Nevertheless, both the transition region and large-sized defect region could lead to nonlinear ultrasonic signals based on the S0-SH0 mixing method. Thus, the larger size of the impact damage region could be obtained by the S0-SH0 mixing method, with high sensitivity to the transition region with material degradation. In conclusion, the one-way S0-SH0 mixing method could be a potential and effective technique for locating the impact damage region.

4. Conclusions

In this paper, the one-way S0-SH0 mixing method in composite laminates with inherent quadratic nonlinearity, delamination damage and impact damage, respectively, are investigated by a series of experiments. The damage localization method based on the one-way S0-SH0 mixing method is also experimentally verified. The following conclusions are drawn accordingly:
Firstly, in composite laminates with inherent quadratic nonlinearity, the resonant SH0 wave can be generated by a pair of S0-mode Lamb waves and SH0 waves when the resonance condition ω S 0 / ω S H 0 = 2 κ / ( κ + 1 ) is satisfied. The resonant wave propagates in the opposite direction of fundamental waves. The frequency of the resonant wave is equal to the difference between two fundamental wave frequencies.
Secondly, because the vibration direction of the SH0 resonant wave is perpendicular to the direction of delamination type I, the one-way S0-SH0 mixing method is not sensitive to the delamination damage in composite laminates. However, the one-way S0-A0 mixing method could be more suitable for the detection of delamination damage in composite laminates than the one-way S0-SH0 mixing method.
Finally, the method to evaluate and locate the impact damage region is experimentally demonstrated based on the one-way S0-SH0 mixing method. As a contrast, the C-scan method is also employed to verify the feasibility of the damage localization method in this paper. Therefore, this study provides the experimental foundation for the one-way Lamb and SH mixing method to evaluate and locate the impact damage region in composite laminates.

Author Contributions

Experimentation, S.X. and Y.L.; Resources, Project Administration, Methodology, Funding Acquisition, Conceptualization, Y.Z.; Writing—Original Draft, S.X.; Writing—Review and Editing, Y.Z. and Y.L.; Data Curation, S.X.; Validation, Investigation, S.X. and Y.L. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the National Science Foundation of China (No. 12474449 and No. 11604033).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data will be made available on request.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. Schematic of one-way mixing of S0-mode Lamb waves and SH0 waves in composite laminates with quadratic nonlinearity. Red box—mixing zone, green box—damage region.
Figure 1. Schematic of one-way mixing of S0-mode Lamb waves and SH0 waves in composite laminates with quadratic nonlinearity. Red box—mixing zone, green box—damage region.
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Figure 2. Schematic of delamination damage in composite laminates ((Left): layout diagram; (Right): the position of polytetrafluoroethylene film in the thickness direction).
Figure 2. Schematic of delamination damage in composite laminates ((Left): layout diagram; (Right): the position of polytetrafluoroethylene film in the thickness direction).
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Figure 3. Ultrasonic experimental setup. (a) Nonlinear ultrasonic experimental system: RAM-5000, signal generator; Oscilloscope, signal receiver; Computer, signal processor; (b) schematic diagram of ultrasonic experimental setup, corresponding to the ultrasonic experimental system; (c) layout of composite laminate in ultrasonic experiments.
Figure 3. Ultrasonic experimental setup. (a) Nonlinear ultrasonic experimental system: RAM-5000, signal generator; Oscilloscope, signal receiver; Computer, signal processor; (b) schematic diagram of ultrasonic experimental setup, corresponding to the ultrasonic experimental system; (c) layout of composite laminate in ultrasonic experiments.
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Figure 4. The dispersion curves of T300 composite laminates (only phase velocity): (a) Lamb waves; (b) SH waves.
Figure 4. The dispersion curves of T300 composite laminates (only phase velocity): (a) Lamb waves; (b) SH waves.
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Figure 5. Time-domain and frequency-domain signals of two fundamental waves collected in the experiment: (a1,a2) 250 kHz S0-mode Lamb waves; (b1,b2) 160 kHz SH0 waves.
Figure 5. Time-domain and frequency-domain signals of two fundamental waves collected in the experiment: (a1,a2) 250 kHz S0-mode Lamb waves; (b1,b2) 160 kHz SH0 waves.
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Figure 6. Resonant wave signals with the frequency 80 kHz, generated from frequency pair of 140 kHz SH0 fundamental wave and 220 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
Figure 6. Resonant wave signals with the frequency 80 kHz, generated from frequency pair of 140 kHz SH0 fundamental wave and 220 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
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Figure 7. Resonant wave signals with the frequency 90 kHz, generated from frequency pair of 160 kHz SH0 fundamental wave and 250 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
Figure 7. Resonant wave signals with the frequency 90 kHz, generated from frequency pair of 160 kHz SH0 fundamental wave and 250 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
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Figure 8. Resonant wave signals with the frequency 120 kHz, generated from frequency pair of 210 kHz SH0 fundamental wave and 330 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
Figure 8. Resonant wave signals with the frequency 120 kHz, generated from frequency pair of 210 kHz SH0 fundamental wave and 330 kHz S0-mode fundamental wave: (a) time domain; (b) frequency domain.
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Figure 9. Resonant wave signals with the frequency 90 kHz with different delay times: (a) 120 μs, (b) 200 μs, (c) 260 μs. The left and right ends of the blue dashed box indicate the start and end times of resonant wave signals.
Figure 9. Resonant wave signals with the frequency 90 kHz with different delay times: (a) 120 μs, (b) 200 μs, (c) 260 μs. The left and right ends of the blue dashed box indicate the start and end times of resonant wave signals.
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Figure 10. Resonant wave signals with the frequency 90 kHz generated from S0-SH0 mixing experiments in composite laminates with/without delamination: (a) time domain; (b) frequency domain.
Figure 10. Resonant wave signals with the frequency 90 kHz generated from S0-SH0 mixing experiments in composite laminates with/without delamination: (a) time domain; (b) frequency domain.
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Figure 11. A0-mode fundamental wave signals with the frequency 150 kHz collected in the experiment: (a) time domain (with 1 MHz Butterworth filtering processing); (b) frequency domain.
Figure 11. A0-mode fundamental wave signals with the frequency 150 kHz collected in the experiment: (a) time domain (with 1 MHz Butterworth filtering processing); (b) frequency domain.
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Figure 12. Resonant wave signals with the frequency 100 kHz generated from S0-A0 mixing experiments in composite laminates with/without delamination: (a) time domain; (b) frequency domain.
Figure 12. Resonant wave signals with the frequency 100 kHz generated from S0-A0 mixing experiments in composite laminates with/without delamination: (a) time domain; (b) frequency domain.
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Figure 13. Resonant wave signals with the frequency 90 kHz generated from S0-SH0 mixing experiments in composite laminates with/without impact damage: (a) time domain; (b) frequency domain.
Figure 13. Resonant wave signals with the frequency 90 kHz generated from S0-SH0 mixing experiments in composite laminates with/without impact damage: (a) time domain; (b) frequency domain.
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Figure 14. Resonant wave signal with the frequency 90 kHz from the impact damage region. The left and right ends of the blue dashed box indicate the start and end times of the resonant wave signal.
Figure 14. Resonant wave signal with the frequency 90 kHz from the impact damage region. The left and right ends of the blue dashed box indicate the start and end times of the resonant wave signal.
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Figure 15. C-scan result of impact damage in T300 composite laminates.
Figure 15. C-scan result of impact damage in T300 composite laminates.
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Table 1. Material parameters of T300 composite laminates.
Table 1. Material parameters of T300 composite laminates.
Materialρ (kg/m3)E1
(GPa)		E2
(GPa)		E3
(GPa)		ν12ν13ν23G12
(GPa)		G13
(GPa)		G23
(GPa)
T3001760139.9210.0510.050.310.310.485.703.403.40
Table 2. The representative frequency pairs of two fundamental waves in T300 composite laminates for experiments.
Table 2. The representative frequency pairs of two fundamental waves in T300 composite laminates for experiments.
f S 0 (kHz) f S H 0 (kHz) f R (kHz)
22014080
25016090
330210120
Table 3. Theoretical and experimental group velocities of SH0 fundamental waves with different frequencies.
Table 3. Theoretical and experimental group velocities of SH0 fundamental waves with different frequencies.
Frequency (kHz)Group Velocity (m/s)
(Experiment)		Group Velocity (m/s)
(Theory)		Errors
1401688.231799.626.19%
1601720.541799.624.93%
2101763.871799.621.98%
Table 4. Theoretical and experimental group velocities of S0-mode fundamental waves with different frequencies.
Table 4. Theoretical and experimental group velocities of S0-mode fundamental waves with different frequencies.
Frequency (kHz)Group Velocity (m/s)
(Experiment)		Group Velocity (m/s)
(Theory)		Errors
2206198.826267.481.09%
2506098.646125.090.43%
3305134.285062.671.41%
Table 5. The errors between experimental and theoretical frequencies of three resonant waves, corresponding to three frequency pairs of fundamental waves.
Table 5. The errors between experimental and theoretical frequencies of three resonant waves, corresponding to three frequency pairs of fundamental waves.
f S 0 (kHz) f S H 0 (kHz) f R (kHz) (Experiment) f R (kHz) (Theory)Errors
22014077.5803.13%
25016090900%
3302101151204.17%
Table 6. Damage localization results with different delay times. PS—start position, PE—end position.
Table 6. Damage localization results with different delay times. PS—start position, PE—end position.
∆T (μs)PS (mm)
(Theory)		PS (mm)
(Experiment)		ErrorsPE (mm)
(Theory)		PE (mm)
(Experiment)		Errors
120155.2170.69.92%361382.65.98%
200345.7334.33.30%536.3514.14.14%
260496527.56.35%744.9702.05.75%
Table 7. Comparison of ANPs obtained from S0-SH0 and S0-A0 mixing methods.
Table 7. Comparison of ANPs obtained from S0-SH0 and S0-A0 mixing methods.
Method β d β i β d ′
S0-SH02.563.24−0.68
S0-A04.392.792.6
Table 8. Comparison of ANPs obtained from S0-SH0 mixing experiments in composite laminates without/with impact damage.
Table 8. Comparison of ANPs obtained from S0-SH0 mixing experiments in composite laminates without/with impact damage.
β i β p ′
2.9321.38
Table 9. Damage localization results of S0-SH0 mixing method and C-scan method.
Table 9. Damage localization results of S0-SH0 mixing method and C-scan method.
PositionS0-SH0 Mixing MethodC-Scan MethodErrors
PS (mm)454.10494.608.59%
PE (mm)572.20530.027.95%
L (mm)118.1035.42233.43%
Damage center (mm)513.15512.310.16%
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Xie, S.; Zhao, Y.; Liu, Y. Experimental Investigation of One-Way Lamb and SH Mixing Method in Composite Laminates. Sensors 2025, 25, 7631. https://doi.org/10.3390/s25247631

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Xie S, Zhao Y, Liu Y. Experimental Investigation of One-Way Lamb and SH Mixing Method in Composite Laminates. Sensors. 2025; 25(24):7631. https://doi.org/10.3390/s25247631

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Xie, Siyang, Youxuan Zhao, and Yuzi Liu. 2025. "Experimental Investigation of One-Way Lamb and SH Mixing Method in Composite Laminates" Sensors 25, no. 24: 7631. https://doi.org/10.3390/s25247631

APA Style

Xie, S., Zhao, Y., & Liu, Y. (2025). Experimental Investigation of One-Way Lamb and SH Mixing Method in Composite Laminates. Sensors, 25(24), 7631. https://doi.org/10.3390/s25247631

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