 Original Article
 Open Access
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Application of Nonlinear Lamb Wave Mixing Method for Residual Stress Measurement in Metal Plate
Chinese Journal of Mechanical Engineering volume 36, Article number: 12 (2023)
Abstract
Harmonic nonlinear ultrasound can offer high sensitivity for residual stress measurements; however, it cannot be used for local stress measurements at a point in space and exhibits nonlinear distortions in the experimental system. This paper presents a feasibility study on the measurement of residual stress in a metal plate using a nonlinear Lamb wavemixing technique. The resonant conditions for two Lamb waves to generate a mixing frequency wave are obtained via theoretical analysis. Finite element simulations are performed to investigate the nonlinear interactions between the two Lamb waves. Results show that two incident A0 waves interact in regions of material nonlinearity and generate a rightward S0 wave at the sum frequency. Residual stress measurement experiments are conducted on steel plate specimens using the collinear Lamb wavemixing technique. By setting different delays for two transmitters, the generated sumfrequency component at different spatial locations is measured. Experimental results show that the spatial distribution of the amplitude of the sumfrequency component agrees well with the spatial distribution of the residual stress measured using Xrays. The proposed collinear Lamb wavemixing method is effective for measuring the distribution of residual stress in metal plates.
1 Introduction
Residual stress is the internal stress within a structure that remains stationary and maintains equilibrium with its surroundings [1]. Residual stress is generated owing to nonuniform elastoplastic deformation after external forces or inhomogeneous temperature fields are removed. Residual stresses significantly affect the properties of mechanical components, such as their deformation, fatigue strength, dimensional stability, corrosion resistance, and brittleness, thereby affecting their performance [2, 3]. Many engineering accidents and disasters are associated with residual stress; therefore, residual stress measurements are critical for structural performance assessment and failure prevention [4].
Residual stress measurement techniques can be classified into two categories: destructive and nondestructive methods [5, 6]. Destructive methods, such as hole drilling, slicing, or ring coring, facilitate stress release but can cause irreversible damage to the material [7, 8]. Recently, nondestructive testing methods, such as Xray diffraction, digital image correlation, magnetic methods, and ultrasonic methods, have been used widely for detecting residual stress [9, 10]. Xray diffraction can estimate the residual stress on surface of components by measuring the lattice spacing of materials; however, its sensitivity is significantly affected by surface conditions [11]. Digital image correlation allows noncontact estimation of the stress field by calculating the correlation of images obtained under stressfree and stressed conditions [12]. However, this technique requires a speckle pattern on the surface of the target structure and is not suitable for continuous monitoring. Magnetic methods, such as Barkhausen [13], can estimate the residual stress of materials based on the relationship between the stress and magnetization curve during ferromagnetic saturation; however, they are only applicable to ferromagnetic materials.
Ultrasonic testing is a more convenient and efficient technique for residualstress measurement compared with other nondestructive testing (NDT) methods. Stress measurement based on ultrasonic waves primarily involves the acoustoelastic effect, wherein the velocity of the ultrasonic wave is linearly related to the stress of the material as it propagates through the material [14,15,16]. Studies pertaining to stress measurements using the acoustoelastic effect involves multiple modes of ultrasonic waves, such as conventional bulk, guided, Rayleigh, Lamb, and longitudinal critically refracted waves [17,18,19,20]. In particular, the longitudinal critically refracted wave is widely adopted for stress measurement because of its sensitivity to inherent stress, as it propagates beneath the surface to a depth equivalent to the wavelength. In most materials, the variation in wave velocity caused by internal stress is insignificant; therefore, the acoustoelastic effect is extremely weak [21]. For example, a stress variation of 100 MPa can only change the wave velocity in steel by 0.01% [22]. Consequently, residual stress measurement methods based on the acoustoelastic effect exhibit low sensitivity and require high accuracy for measuring the arrival time of the ultrasonic wave packet.
Compared with conventional linear ultrasonic technologies, nonlinear ultrasonic testing technology is more sensitive to microstructural changes in a material [23,24,25]. Studies indicate that the stress of mechanical components can affect the nonlinear properties of ultrasonic waves [26, 27]. Stressinduced microstructural evolution of mechanical components can result in ultrasonic distortion, and higher harmonics associated with stress are generated. In addition, the theoretical relationship between the nonlinear coefficients and stress can be represented by a dislocation model. Nonlinear ultrasonic testing has been successfully performed to assess residual stress in mechanical components. Experiments were conducted to confirm the relationship between the amplitude of the second harmonics and residual stress [28]. For example, Yan et al. [29] experimentally investigated the variation of the surface wave nonlinear coefficient with increasing tensile stress; Liu et al. [30] experimentally investigated the propagation of nonlinear surface waves in aluminum alloy (AA 7075) samples shotpeened at different peening intensities and demonstrated the feasibility of nonlinear surface waves for residual stress measurements; Mao et al. [31] applied nonlinear critically refracted longitudinal waves to evaluate the stress state of metallic materials under steady state. The experimental results showed that the second and thirdorder relative nonlinear coefficients increased monotonically with stress, and that the normalized relationship was consistent with the simplified dislocation model.
Although the aforementioned nonlinear ultrasonic methods based on the effects of harmonic generation can offer higher sensitivity for residualstress measurements [32], these methods cannot be used for localstress measurements at a point in space. The nonlinear coefficients measured by these methods are the cumulative nonlinear responses in the region between the transmitter and receiver; therefore, the measured stresses are also the average stresses in the corresponding region.
This limitation can be overcome using the wavemixing technique, which is based on the fact that the nonlinear interaction of two incident waves with nonlinear sources (such as material nonlinearity or microdamage) can generate a third wave (known as the mixing frequency wave), whose frequency and wave vector are the sum of the incident frequencies and wave vectors, respectively [23, 24, 32]. Notably, the measured nonlinearity is related only to the acoustic nonlinearity within the mixing zone [33]. Thus, the spatial distribution of the nonlinear sources in the structure can be obtained by adjusting the intersection positions of the two incident waves.
The main objective of this study is to conduct a feasibility study on the measurement of residual stress in metal plates using a nonlinear Lamb wave mixing technique. First, a theoretical analysis was performed to determine the resonant conditions for generating the mixing frequency effect of two Lamb waves. Subsequently, finite element simulations were performed to investigate the nonlinear interaction of the two Lamb waves in regions of material nonlinearity. Finally, residual stress measurement experiments were performed on the steel plate specimens using the collinear Lamb wave mixing technique.
2 Fundamental Theories of Nonlinear Analysis
Under certain resonant conditions, two incident waves (\(\omega_{1}\) and \(\omega_{2}\)) interact with the nonlinear source to generate a mixing frequency wave (\(\omega_{g}\)). The resonant conditions for its generation are as follows:
where \({\varvec{k}}_{1}\), \({\varvec{k}}_{2}\), and \({\varvec{k}}_{g}\) are the wave vectors of two incident waves and the mixing frequency wave, respectively; “+” indicates sum frequency; “−” indicates the difference frequency.
Furthermore, the relationship between the two incident waves and mixing frequency waves can be represented by the geometry shown in Figure 1, where \(\alpha\) is the interaction angle of the two incident waves, and \(\psi\) is the scattering angle.
For Lamb wave mixing, in addition to the aforementioned resonance conditions, the dispersion relationship of the Lamb wave must be satisfied.
Considering the dispersion characteristics of Lamb waves, the resonance conditions for two incident Lamb waves for generating the mixing frequency waves can be expressed as follows:
where, \(K_{1}\) and \(K_{2}\) are the wave numbers of two incident waves, whereas \(K_{{{\text{sum}}}}\) and \(K_{{{\text{dif}}}}\) are the wave numbers of the sumfrequency and differencefrequency waves, respectively. The wave numbers in the expressions above can be expressed by the dispersion characteristics of Lamb waves.
where ν_{p(f1)} and ν_{p(f2)} are the phase velocities of two incident Lamb waves; ν_{p(f1+f2)} and ν_{p(f1f2)} are the phase velocities of the sumfrequency and differencefrequency waves, respectively.
Furthermore, the interaction angle α and scattering angle ψ for sumfrequency and differencefrequency Lamb waves can be expressed as follows:
As shown in Eqs. (5) and (6), the incident angle and scattering angle are related to the frequencies and phase velocities of the incident and mixing frequency waves. As an example, Figure 2 shows the resonant conditions for the interaction of two A0 waves resulting in the generation of a sumfrequency S_{0} wave in an aluminum plate with a thickness of 1 mm when the incident excitation frequency f_{1} = 1.26 MHz.
When the incident frequency f_{2} is 0.32 MHz and incident angle is 180°, the two incident A0 waves will interact, resulting in the generation of a sumfrequency S0 wave. Moreover, the scattering angle is 0°, which implies that the sumfrequency S0 wave propagates in the same direction as one of the incident waves.
3 Finite Element Simulation
The relationship between the thirdorder elastic constants of materials and their stresses has been clarified in numerous studies [34, 35]. To validate the Lamb wave mixing technique for local stress measurements, numerical simulations of Lamb wave mixing were performed. In the simulation model, the local stresses of the structure were simulated by varying the local thirdorder elastic constants.
3.1 Finite Element Model
A twodimensional finite element model of thin plates with local material anomalies was constructed using the commercial finite element software ABAQUS, as shown in Figure 3.
An aluminum sample with crosssection of 500 mm × 1 mm is modeled. To simulate localized stress concentrations, the material properties of a localized area with 40 mm ×1 mm at the center of the model were set to nonlinear, and the material properties of the rest of the model were set to linear elasticity. The nonlinear elastic parameters are listed in Table 1, and the elastic parameters associated with the linear elasticity were consistent with those of the nonlinear elasticity. Two excitation signals with different frequencies (f_{1} and f_{2}) were applied to each side of the model in an antisymmetric loading manner. Three monitoring points located at the upper surface, i.e., 330, 270, and 230 mm away from the left end, were used to receive the displacement in the U2 (offplane) direction.
In addition to satisfying the resonant conditions, another necessary condition for generating wavemixing effect is that the two incident waves must coincide at the nonlinear source. This implies that two incident waves must reach the nonlinear region simultaneously. Based on the difference in the group velocity of the two excitation signals (as shown in Figure 4), the time delay of the two excitations was determined to be 1.4 × 10^{−5} and 0 s. Table 2 lists the primary parameters of the two excitation signals. The excitation signal in the simulations was a sinusoidal signal modulated by a Hanning window.
For comparison, a finite element simulation of the complete linear elasticity model was performed under the same conditions. The finite element model showed no local material anomalies. These two models are hereinafter referred to as the nonlinear and linear models, respectively.
3.2 Simulation Results and Discussion
Figure 5 shows the waveforms and spectra received by monitoring point 1 in the linear and nonlinear models. The signals received from the two models were different; however, it was difficult to determine the mode and frequency of each echo from waveform. Two new frequency components appeared in the spectrum of the signal received by the nonlinear model: the second harmonics (0.64 MHz) and sumfrequency component (1.58 MHz).
Figure 6 shows the time–frequency distribution of the two signals processed by the continuous wavelet transform. The time–frequency results show two distinct peaks in the time–frequency distribution of the signal obtained from the linear model, which corresponded to the two A0 waves at the fundamental frequency.
The signal obtained from the nonlinear model contained five separated echoes in the waveform; however, the time–frequency distribution of the second and third echoes contained two frequency components, which resulted in seven distinct peaks in the time–frequency distribution.
Using the travel time, central frequency, and possible propagation paths of each echo (as shown in Figures 5 and 6), their velocities can be calculated; consequently, the mode of each echo can be determined. Table 3 summarizes the analysis process and the results for each echo. Echo “1” is the first arriving packet, and it is the second harmonics of the S0 wave (0.64 MHz) generated by the excitation signal at f_{2} = 0.32 MHz. Echo “2” includes two components, i.e., the fundamental component (0.32 MHz) and the second harmonics (0.64 MHz) of the A0 wave generated by the excitation signal at f_{2} = 0.32 MHz. Notably, the geometric nonlinearity enabled in the simulations allowed the generation of the second harmonics in the linear material region. Echo “3” includes two components, i.e., the fundamental component (1.26 MHz) of the A0 wave generated by the excitation signal at f_{1} = 1.26 MHz, and the sumfrequency component (1.58 MHz) of the S0 wave generated by the interaction of two incidents. Echo “4” is the echo reflected by echo “1” at the left boundary. Echo “5” is the echo reflected by the sumfrequency (1.58 MHz) S0 wave at the right boundary.
To further illustrate that the sumfrequency component was generated by the interaction of two incident A0 waves, numerical simulations were performed on the nonlinear model under the individual and synchronous excitation of two incident waves.
Figures 7 and 8 present the waveforms and spectra of the simulation signals received at monitoring points 2 and 3, respectively, including the signals obtained under synchronous excitation, sum of signals obtained under individual excitations, and difference between signals under individual and synchronous excitations.
As indicate by the signals detected at monitoring point 2 in Figure 7, the signals under individual and synchronous excitations differed significantly.
Specifically, a distinct echo was observed in the waveform of the differential signal with a center frequency exactly at a sum frequency of 1.58 MHz. However, no sumfrequency component was observed in the spectrum of the signals under individual excitations. Therefore, the sumfrequency component was generated by the interaction of two incident A0 waves under synchronous excitations, and no sumfrequency component was generated under individual excitations of the two A0 waves.
A similar analysis can be performed for the signals received at monitoring point 3, as shown in Figure 8.
Similarly, a discernible echo was observed in the differential signal detected at monitoring point 3. However, the echo at monitoring point 3 differed significantly from that at monitoring point 2, i.e., it featured a smaller amplitude, an earlier arrival time, and a center frequency of 0.64 MHz. The echo was identified as the secondharmonic component at 0.64 MHz, which was generated under both individual and synchronous excitations; furthermore, the secondharmonic components obtained from the two monitoring points were similar. Notably, no discernible sumfrequency component was observed in the signals detected at monitoring point 3. Therefore, it can be inferred that the sumfrequency wave was generated at the intersection of two incident A0 waves and propagated to the right. This is consistent with the propagation direction determined based on the theoretical resonance conditions.
To show the nonlinear interaction of two incident A0 waves more clearly, the displacement distribution of the points on the horizontal plane in the middle of the plate at typical moments is shown in Figure 9. The displacements in the U_{1} and U_{2} directions are plotted in red and blue, respectively. The wave propagation process can be observed clearly. The two incident A0 waves propagated in opposite directions, and a rightward propagating S0 wave was generated at the intersection.
4 Experiments for Stress Distribution Measurement
Collinear Lamb wave mixing experiments were conducted to investigate the effectiveness of the mixing technique for measuring the residual stress distribution in metal plates.
4.1 Finite Element Model
Figure 10 shows a schematic illustration of the collinear wavemixing measurement system. The SNAP RAM5000 is the core component of the nonlinear measurement system. Two highvoltage signals from the output of “RF burst” channels were applied to the two piezoelectric transducers (A and B).
The central frequencies of the two transmitters were 0.304 and 0.77 MHz. Lamb waves were received by a piezoelectric transducer located at the midpoint of the connection between the two transmitters. The center frequency of the receiver was 1 MHz. The frequency responses of the three transducers above are shown in Figure 11.
Considering the frequency response of the transmitters and receiver (shown in Figure 11(a)), the center frequency f_{1} of the excitation signal applied to transmitter A was 0.304 MHz. To obtain a strong mixingfrequency nonlinear response, the effect of the driving frequency f_{2} on the amplitude of the sumfrequency component was investigated via frequencysweep experiments. The driving frequency f_{2} applied to transmitter B was varied from 0.2 to 2 MHz at intervals of 0.05 MHz. Figure 11(b) shows the frequency dependence of the sumfrequency component when f_{1} was 0.304 MHz. The amplitudes of the sumfrequency component were particularly sensitive to the driving frequency, and four local resonant points were observed. Specifically, three resonant points (0.304, 0.636, and 0.95 Hz) corresponded to the fundamental and harmonic components of f_{1}. At these three resonance points, the sumfrequency components were the higher harmonics of the primary waves that were susceptible to system nonlinearity. To avoid the effect of nonlinearity on harmonic generation, the center frequency f_{2} of the excitation signal applied to transmitter B was set to 0.77 MHz. Table 5 lists the specific parameters of the two excitation signals.
The experimental measurements were conducted on two 65 Mn steel sheets with dimensions of 300 mm × 200 mm × 3 mm, as shown in Figure 12.
Both specimens were sampled from an industrial component with a local residual stress. A rectangular area of 200 mm × 75 mm in the middle of specimen 2# was partially machined, as shown in Figure 12(b). Residual stresses were measured using Xrays at 15 typical positions on three straight lines in the specimen; the measured residual stresses are listed in Table 4.
Figure 13 presents the experimental scheme for the stress measurement via collinear Lamb wave mixing. In the experiment, one transmitter was placed at each of the two ends of three straight lines, i.e., A–A, B–B, and C–C, and the receiver was placed in the center of the connection between the two transmitters. During the stress measurement, the positions of the transmitters and receiver were fixed. Different delays were set for the two transmitters such that the intersection position of two Lamb waves can be controlled. The receiver was used to receive the generated Lamb waves at the sumfrequency. Thus, the spatial sweep of the ultrasonic beam intersection can be regarded as a measurement of nonlinearity at different spatial locations in the metal plate. If the attenuation is disregarded, then the sumfrequency component in the measured signal will be related to the stress level at the intersection point.
4.2 Experimental Results and Discussion
Compared with the linear response, the nonlinear response was relatively weak. Therefore, collinear Lamb wave mixing experiments were conducted to investigate the extraction of mixedfrequency nonlinear components from the detected signals. The receiver picked up the signals when the two transmitters were excited simultaneously and respectively. Specifically, the superposition of the received signals obtained by applying the excitations respectively is known as the “sum of signals received through separate excitations.” The difference between the “sum of signals received through separate excitations” and the “sum of signals received through simultaneous excitations” was used to analyze the nonlinear effects.
Figure 14 shows the typical signals obtained from the collinear Lamb wave mixing experiments. The waveforms of the sum of signals received through separate excitations were similar to those received through simultaneous excitations, i.e., they contained the direct wave and reflected echo of the A0 mode at 50 and 150 µs. However, the waveform of the difference signal contained only direct waves. The corresponding spectrum presented in Figure 14(b) shows that each signal contained multiple frequency components. In particular, no mixedfrequency component was observed in the spectrum of the sum of signals received through separate excitations.
Under the same experimental setup, collinear Lamb wave mixing experiments were conducted by reversing the polarities of the two excitations, i.e., the phases of the two excitations were (0°, 0°), (0°, 180°), (180°, 0°), and (180°, 180°). The superposition of the detection signals obtained at four different polarity combinations is known as the “polarity reversal signal.”
Figure 15 shows a comparison of the polarity reversal results and the difference signals. In the time domain, the amplitude of the difference signal exceeded that of the polarity reversal signal.
Furthermore, the spectrum presented in Figure 15(b) shows that the difference signal contained more components than the polarity reversal signal. Twelve components were observed in the spectrum of the difference signal: 0.15, 0.304, 0.46, 0.608, 0.77, 0.92, 1.074, 1.22, 1.46, 1.53, 1.716, and 1.824 MHz. However, four components were observed in the spectrum of the polarity reversal signal: 0.165, 0.46, 1.074, and 1.716 MHz. The generation of each component can be analyzed specifically based on the center frequency of the two excitation signals (f_{1} = 0.304 MHz and f_{2} = 0.77 MHz). In detail, the components at 0.15, 0.304 0.608, 0.77, 0.92, 1.22, 1.46, 1.53, and 1.716 were regarded as the subharmonic or higher harmonic response of the two excitations. The components at 1.074 and 0.46 MHz were regarded as the sumfrequency and differencefrequency responses of the two excitations, respectively. The component at 1.716 MHz was regarded as the sumfrequency response generated by the subharmonic of f_{1} (0.152 MHz) and second harmonics of f_{2} (1.54 MHz). The component at 0.165 MHz was regarded as the differencefrequency response generated by the fundamental f_{2} (0.77 MHz) and second harmonics of f_{1} (0.608 MHz). Multiple harmonic components were observed in the difference signal; however, no harmonic components were observed in the polarity reversal signal. To avoid the effects of harmonics generated by system nonlinearity, the polarity reversal signals should be used to extract the mixedfrequency nonlinear component. In this study, the sumfrequency component at 1.074 MHz was extracted from both the difference and polarity reversal signals; subsequently, it was used to characterize the residential stress.
Figure 16 presents the amplitude of the sumfrequency component (measured via collinear Lamb wave mixing experiments) and stress (measured by Xrays) at different spatial locations. In general, the spatial distribution of the amplitudes of the sum frequency components measured via the two methods was similar to that of the residual stresses measured using Xrays. Therefore, the proposed amplitude of the sumfrequency component can be used to evaluate the residential stress in a metal structure. However, the amplitude of the sumfrequency component fluctuated significantly in the local spatial range (as shown in Figure 16(c)) and deviated considerably from the stress distribution.
This is attributable to the average effect and insufficient spatial resolution due to the probe size, delay errors, attenuation of ultrasonic waves, etc. In future investigations, immersionfocusing probes will be used to improve the spatial resolution and repeatability of the measurements.
5 Conclusions
In this study, collinear Lamb wave mixing was adopted to measure the residual stress in metal plates.

(1)
Based on theoretical analysis, the resonant conditions required for two Lamb waves to generate a mixing frequency wave were obtained. Finite element simulations were performed to investigate the nonlinear interactions between the two Lamb waves. Results showed that two incident A0 waves interacted in regions of material nonlinearity and generated an S0 wave at the sumfrequency. Moreover, the generated S0 wave propagated to the right, which was consistent with the propagation direction determined based on the theoretical resonance conditions.

(2)
The excitation frequencies for the Lamb wave mixing experiments were determined by considering both the theoretical resonance conditions and the experimental amplitudefrequency dependence of the sumfrequency component.

(3)
Residual stress measurement experiments were conducted on steel plate specimens using the collinear Lamb wave mixing technique. By setting different delays for the two transmitters, the generated sumfrequency component at different spatial locations was measured. The experimental results showed that the spatial distribution of the amplitude of the sumfrequency component agreed well with the spatial distribution of the residual stress measured using Xrays.
Therefore, the proposed collinear Lamb wave mixing method is effective for measuring the distribution of residual stress in metal plates.
Data availability statement
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy.
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Acknowledgements
Supported by National Natural Science Foundation of China (Grant Nos. 11972053, 12274012).
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JJ, LL were in charge of the whole trial; JJ and LL wrote the manuscript; QC and XG assisted with experiments; QC assisted with FE modeling; CH and BW assisted with framework design of manuscript. All authors read and approved the final manuscript.
Authors’ Information
Jingpin Jiao, born in 1973, is currently a professor at Faculty of Materials and Manufacturing, Beijing University of Technology, China. Her research interests include nondestructive testing, experimental mechanics, and signal process.
Li Li, born in 1992, is currently a PhD candidate at Faculty of Materials and Manufacturing, Beijing University of Technology, China. He received his Bachelor degree from Anhui Jianzhu University, China, in 2013.
Xiang Gao, born in 1991, is currently a lecture at Faculty of Materials and Manufacturing, Beijing University of Technology, China. His research interests include nondestructive testing, ultrasonic imaging.
Quan Cheng, born in 1994. She received master degree from Beijing University of Technology, China, in 2019.
Cunfu He, born in 1958, is currently a professor at Faculty of Materials and Manufacturing, Beijing University of Technology, China. His research interests include nondestructive testing, experimental mechanics.
Bin Wu, born in 1962, is currently a professor at Faculty of Materials and Manufacturing, Beijing University of Technology, China. His research interests include nondestructive testing, experimental mechanics.
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Jiao, J., Li, L., Gao, X. et al. Application of Nonlinear Lamb Wave Mixing Method for Residual Stress Measurement in Metal Plate. Chin. J. Mech. Eng. 36, 12 (2023). https://doi.org/10.1186/s10033023008326
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DOI: https://doi.org/10.1186/s10033023008326
Keywords
 Residual stress
 Lamb wave
 Collinear wave mixing
 Nondestructive testing
 Resonant conditions