The Use of Nonlinear Guided-Wave Features for Detection and Assessment of Intermetallic Compounds Within Dissimilar Joints—An Experimental Investigation

Ensuring the integrity of our structural systems requires the development of accurate and robust methods. The development of structural health monitoring (SHM) systems led to many interesting applications in the civil, mechanical, and aerospace industries [1]. SHM systems use sensing technologies to track and evaluate the symptoms of operational incidents, anomalies, and deterioration or damage indicators that may affect the operation, serviceability, safety, or reliability [1,2]. Over the past three decades, numerous SHM methods have been developed, encompassing global and local damage detection techniques. Global damage detection focuses on the overall structural response, while local damage detection aims to identify damages at small scales, such as cracks or corrosion [3].

Many welding techniques have been developed and are widely used processes for joining materials, typically metals [4]. While welding similar metals is efficient, joining dissimilar metals presents many challenges. Dissimilar-material joining can be described as combining materials or material combinations that are often more difficult to join than two pieces of the same material or alloys with minor differences in composition. Traditional welding methods, such as tungsten inert gas welding and metal inert gas welding, have limitations in terms of longer processing times and weaker joints, which can affect the efficiency and reliability of dissimilar-material joining. Modern welding techniques, such as electron beam welding, diffusion welding, laser welding, and friction stir welding (FSW), have been developed to address such limitations [5–9].

FSW offers several advantages such as reduced heat input, decreased distortion, absence of solidification cracking, and reliable dissimilar joint production [9–11]. Kundu et al. [12] have used FSW to join titanium to steel in a lap joint. The obtained microstructure at the joint interface was intensively examined, and the different phases and grain sizes were identified. The micro-hardness test across the joint interface indicated that the hardness at the joint interface is greater than the base materials due to the distribution of small intermetallic compounds/particles (IMCs) and the formation of β titanium at the interface. The development of IMCs at the joint level remains a significant challenge that may endanger the integrity of dissimilar welds [5–8].

Various nondestructive testing (NDT) techniques are employed to inspect and assess welds for potential defects. Thermal imaging, modal-based methods, optical fiber sensing, laser Doppler vibrometry (LDV), and X-ray imaging are commonly used [13–17]. On the other hand, Lamb waves (LWs) have gained attention in NDT and SHM due to their ability to inspect metallic and composite structures and travel long distances in relatively complex systems.

Ultrasonic Lamb-wave techniques are proven testing methods for inspecting components. Detection using these techniques entails detecting changes in the elastic properties of the propagating medium using linear features of the ultrasound wave such as its speed, attenuation, transmission, and reflection coefficients. Fakih et al. [18–21] examined the potential of LWs for inspecting FSW joints and scrutinized the assessment of flaws in AZ31B FSW magnesium plates both experimentally and numerically [18,19]. To detect wormholes of various severities, LWs were excited and received using piezoelectric transducers (PZTs) on both sides of the weld line, and the S₀ mode was separated from the complex signals, which were then classified using an attenuation-based damage index. In another study [20], LWs were excited using PZT wafers and received using a scanning LDV, and the frequency–wavenumber filtering technique was implemented to separate the wave reflected from the welded joint. The reflection of the A₀ mode from the material interface in an intact dissimilar AA6061-T6/AZ31B FSW was minimal, but it became significant in the presence of a wormhole within the welded joint. In Ref. [21], surrogate models based on artificial neural networks (ANNs) were trained using finite element simulations to predict Lamb-wave sensor measurements when specific damage is introduced. The ANNs were then employed to perform a probabilistic damage inference on simulated data with and without noise using approximate Bayesian computation, thus, providing posterior probability density functions of six damage parameters, including length, width, thickness, and x-, y-, and z-positions. It has been demonstrated that the algorithm has a high potential for detecting various damage scenarios that could result in the same sensor measurement, with a precision rate of over 95%. Despite successful or promising implementations, the standard linear features of LWs may not be effective in detecting microstructural changes or small-scale damage (e.g., fatigue cracks), as there would be no significant linear wave scattering observed in the ultrasound signals [22,23].

Nonlinear ultrasonic guided-wave features have a strong potential to be used for the detection and assessment of micro-size damage. Since the characteristics of their propagation are directly related to the properties of the material, this makes them a highly effective method for nondestructive material degradation assessment [24]. Examples of nonlinear features of ultrasonic waves are sub-harmonics [25–27], higher harmonics [28–30], nonlinear modulation [31–33], and nonlinear resonance [34,35].

Zhang et al. [36] presented a peri-ultrasound theory based on ordinary state-based peridynamics to model nonlinear elastic waves in 3D plate structures and their interaction with multiple cracks. The sideband peak count-index (SPC-I) technique was used to extract the nonlinear response and find any correlations related to the type (thin and thick) and size of the crack. The results show larger SPC-I values for thin cracks than for thick cracks and the case of no cracks. Zhang et al. [37] further examined the effects of various nonlinearities on the SPC-I technique. Three types of nonlinearity were investigated including material, structural, and contact. Numerical modeling and experimental analysis were performed on composite and aluminum plates. The results showed that the sideband peak values do not increase proportionally to input signal strength, indicating a nonlinear response, with different nonlinearities impacting SPC-I measurements differently, validating the technique's effectiveness in assessing material and structural nonlinearities. Furthermore, Jiang et al. [38] investigated the quasistatic pulses (QSPs) generation in fiber reinforced plastic thin plates. The authors investigated the dependency of QSPs on wave propagation direction and material properties. Comparisons show that QSP generation efficiency varies with mode types and primary wave propagation direction, with appropriate primary wave excitation achieving higher cumulative QSP generation efficiency than phase-matched second harmonics, despite divergence loss.

Zhu et al. [39] have used the static component of Lamb waves (SCLWs) and second harmonics to detect closed cracks in aluminum alloy samples. The authors observed that the acoustic nonlinearity parameter of SCLWs increases with crack length and orientation angle, but drops with the crack width. Also, Jiang et al. [40] examined the static component (SC) generation of guided waves (GWs) experimentally and numerically for assessing damage in carbon fiber reinforced plastic (CFRP) composites. The results showed a linear relationship between SC pulse magnitude and material nonlinearity, demonstrating effectiveness in detecting hygrothermal damage and low-velocity impact damage in CFRP plates.

Zhao et al. [28] examined the use of third-harmonic and third-order nonlinear parameters for detecting early fatigue damage in aluminum alloys. It was found that the amplitude of the wave's third-harmonic component increases when the fatigue life falls below about 80%. While above this, the third-harmonic component decreases, and the third-order nonlinear parameters progressively increase before showing a downward trend. Sampath and Sohn [30] employed a nonlinear ultrasonic three-wave mixing technique to detect and localize an incipient fatigue crack in aluminum 6069-T6 plate specimens, correlating the normalized γ′ nonlinear feature with fatigue cycles and fatigue crack size. Furthermore, Aslam et al. [41] investigated defect-localization techniques using the nonlinear interaction between primary Lamb-wave modes, with a focus on crack-wave interaction at the damage zone. Moreover, Lee and Lu [42] studied and identified fatigue cracks in steel joints under vibration using nonlinear guided waves based on second-harmonic generation (SHG); it was observed that the percentage of reduction in nonlinearity increased with the crack growth.

One of the most important challenges of dissimilar-metal welding is the development of intermetallic compounds at the joint level. IMCs are formed when there is a major difference in the physical quantities of the welded materials, including melting temperature, thermal conductivity, and thermal expansion. The compounds may also be formed due to the low solubility between the joined materials such as that of steel in aluminum [43]. IMCs reduce the bonding strength by causing high brittleness and low fracture toughness and accelerating the formation of cracks and voids [44]. First, IMCs tend to exhibit high brittleness and low fracture toughness, a characteristic attributed to their inclusion of covalent bonds; consequently, the interior of the IMC layer is a primary location for crack initiation and expansion [44]. Deng et al. [44] investigated the microstructure of IMCs at the interface of explosively welded metal laminates and discovered that the solidification contraction and the IMC formation in a localized melting region resulted in crack formation. Second, IMCs and metal matrices often differ significantly in terms of their physical properties and are susceptible to stress concentrations, thus accelerating the emergence of cracks. Tanaka et al. [45] showed that the joint strength increased with the reduction in the thickness of the intermetallic compound at the weld interface. Therefore, detecting and quantifying the presence of IMCs within the joint to evaluate its strength is exceedingly important.

This paper represents a continuation of previous work [46], which focused on the impact of intermetallic compounds on LWs in dissimilar friction stir welds. The previous research primarily involved numerical simulations, where Lamb waves were excited from only one side and using a high-frequency range. However, a notable challenge remained unaddressed: the variations in the trend of the relative acoustic nonlinear parameter across different sensing positions. In the current study, the assessment of IMC thickness in dissimilar FSW lap joints between AA5052-H32 aluminum and ASTM 516-70 steel was investigated using the low-frequency range. The nonlinear characteristics of the materials were experimentally studied. LWs were excited using PZT transducers with a central excitation frequency of 300 kHz when exciting from the aluminum side and 350 kHz when exciting from the steel side. This precise frequency selection is fundamental to meeting the synchronism and zero-flux conditions of SHG. Following this, the impact of the change in the IMC thickness on the LWs was explored, specifically focusing on the parameter known as the relative acoustic nonlinear parameter. The collected LW signals were analyzed in both the time and frequency domains to understand the behavior of β′ with the thickness of the intermetallic layer. The rest of the paper is organized as follows: Sec. 2 introduces the methodology, including a theoretical background about nonlinear ultrasound and descriptions of the experimental setups; Sec. 3 presents and discusses the obtained results; and the concluding remarks are presented in Sec. 4.

Second-order harmonics of LWs in particular have drawn the attention of many researchers in recent decades due to their demonstrated sensitivity to various microstructural changes and degradation in both composite and metallic structures. Dislocations [47], voids [47], and closed cracks have all been identified as probable sources of nonlinearities [48], which can be discovered or even evaluated by various nonlinear properties of ultrasonic waves. Furthermore, nonlinear ultrasonic techniques for measuring the acoustic nonlinearity parameter have received a lot of attention as a promising method for evaluating material degradation, such as thermal aging and fatigue damage [49].

Due to the low amplitudes of the second-harmonic modes, it is preferable to excite LWs at particular frequencies to generate a cumulative second-harmonic generation with a good signal-to-noise ratio (SNR). This happens when the following two conditions are met:

1. the phase velocity of the fundamental wave at the excitation frequency matches or comes close to the phase velocity of another or the same mode at double the excitation frequency and

2. a non-zero power flux between the two modes [54] (i.e., both modes should be either symmetric or antisymmetric).

When these two conditions are met, an internal resonance between the two modes will be generated, and energy can move efficiently from the fundamental mode to the second-harmonic mode during the wave propagation. This ensures that this particular second-harmonic mode gets stronger, while other modes at double the frequency fade away due to their decreasing amplitude over the propagation distance [51].

It is important to note that nonlinearity is not claimed to be solely generated in the weld region. The synchronism and non-zero power flux conditions were used to make sure that there is an accumulation of second-harmonic generation in the base material itself (from which the wave is being excited). This nonlinearity is being disrupted at the weld region which causes a change in the nonlinear parameter. It is expected also to have other sources of nonlinearity, possibly from the weld interface and its microstructural composition. However, analyzing such phenomena could be a real challenge with many uncertainties about the distribution, exact composition, and nonlinear material properties of such microstructures, with possibly yet ununderstood physics around that topic.

In this study, the use of RANP is explored on a 2-mm thick AA5052-H32 aluminum plate and a 7-mm thick ASTM 516-70 steel plate friction-stir-lap welded, toward the detection of microstructural features at the weld line. The dispersion calculator [55] was used to determine the theoretical phase- and group-velocity dispersion curves, shown in Figs. 1 and 2 for the samples under investigation. As shown in Fig. 1(a) and due to the almost flat shapes of the S₀ curve between 0 and 600 kHz, it can be concluded that the two conditions of internal resonance are satisfied between the S₀–S₀ modes in this frequency range. Similarly, in Fig. 2(a), the two conditions of internal resonance are satisfied between the S₀–S₀ or A₀–A₀ modes at fundamental excitation frequencies up to 120 kHz and between 350 and 750 kHz, respectively.

Friction stir diffusion cladding (FSDC) is a newly developed cladding technique, similar to FSW lap welding, in which the entire cladding surface is processed [56]. Figure 3 shows the top and cross-sectional views of a cut FSDC specimen with two cladding passes. The microstructural and mechanical characteristics of several FSDC samples, resulting from welding using different welding parameters, were thoroughly investigated in a previous study [56]. Four samples of one-pass FSDC joints between 2-mm AA5052-H32 aluminum and 7-mm ASTM 516-70 steel samples are used in the current study.

Table 1 lists the dimensions of the specimens' dimensions and their corresponding FSDC process parameters. Figure 4 shows the top and side views of one of the samples, all of which have the same geometrical shape.

Energy dispersive spectroscopy (EDS) line scan analysis was used to characterize the diffusion zone, which defines the interface between the two materials, and the composition variation technique was used to measure the interface thickness. As shown in Fig. 5, the different process parameters caused a variation in the interface thickness between the specimens. For instance, when the same tool rotational speed of 1000 rpm was applied to both A50 and A100 specimens but with varying tool feed speeds (50 m/s for one and 100 m/s for the other), this resulted in distinct interface layers of 6-μm and 3-μm thickness, respectively. The thickness of IMCs available within the weld region can be related to the interface thickness [57]. Insets of the zoomed-in weld region in Fig. 6 display the employed interface sublayers.

Two experiments with two different boundary conditions were conducted utilizing the laser Doppler vibrometer (LDV) technique. The first experiment provides support to the base of the specimen, whereas the second experiment is carried out in free-free boundary conditions. Through the two experimental setups and data collection process, a good understanding of how β′ accumulates within the specimens under investigation is obtained.

Two circular PZTs, each with a diameter of 7 mm and thickness of 0.5 mm, were placed on the upper surfaces of aluminum and steel, as shown in Fig. 7. These PZTs served as actuators for generating LWs, where only one actuator PZT was utilized at any given time. To improve the measurement quality, retroreflective tape was placed on the specimen's upper surface.

In the first setup, gummy tape was wrapped around the specimen's edges (to minimize boundary reflections), which was then attached to a robotic arm also using gummy tape. To generate the desired excitation waveforms, a Keysight 33500B arbitrary function generator was used. The wave excitation signals utilized were 15-cycle Hann-windowed sinusoidal signals, hereafter referred to as 15HC signals. The excitation signals were further amplified to 100 volts peak-to-peak using a Krohn-hite 7602M wideband power amplifier before being applied to the PZT transmitter. This configuration allowed the guided waves to propagate along the specimen, interact with the weld, and be captured by the LDV receiver. The sampling frequency, which denotes the number of measurements taken per second, was set to be at least ten times higher than the excitation frequency to avoid aliasing and allow an accurate time-domain analysis of the recorded data, and the same was applied in both experiments. To achieve comprehensive coverage and obtain meaningful data, the sensing points were positioned along the specimen at intervals of 0.9 mm, ranging from 8 to 80 mm from the left border of the model. Precise positioning was accomplished by giving G-code instructions to the robot arm to ensure accurate and repeatable measurements, while the LDV was kept fixed. The resulting sensing waveforms were then recorded using a Keysight DSO-X 3014T digital storage oscilloscope after adding some averaging for the signal. Figure 8 presents the experimental supported setup for the nonlinear ultrasonic tests.

In this setup, the samples were suspended freely from support stands using threads affixed to their sides. This has ensured consistent boundary conditions among the samples during measurement. To initiate the LW, an arbitrary waveform generator (specifically, the TTi TGA1241) was used, which produced the excitation signals. These signals were characterized by central excitation frequencies identical to those employed in the supported experiment. Before wave excitation, the 15HC excitation signals were amplified to a 100-volt peak-to-peak using a Krohn-Hite Model 7500 amplifier. The amplified signals were then fed into the actuator PZTs for wave generation. For measuring the waves propagating within the plates, a single scanning laser Doppler vibrometer laser head, the Polytec PSV-400-3D model, was used. The synchronization between laser measurements and wave excitation was achieved by utilizing the output of the waveform generator as a trigger. During the LDV line scan, an average of 50 measurements were collected at each sensing point to enhance the SNR. The performed line scans covered nearly the entire length of the specimen, with a nominal distance of 0.5 mm between the edge and the nearest measurement point on each side. The sensing points were spaced at regular intervals of 0.85 mm along the specimens. Figure 9 illustrates the experimental setup used in the support-free condition.

In this section, the experimental results are presented and discussed. The findings from both approaches are compared to evaluate the effectiveness of the nonlinear LW technique in detecting and assessing the thickness of IMCs within dissimilar joints.

To investigate the presence of second-harmonic generation and analyze the frequency content of the measured time-domain signals, the signals were transformed to the frequency domain using the fast Fourier transform (FFT). The FFT is used to identify the frequencies that are being excited and the amplitude at each frequency in a complex signal. In the first set of measurements, the excitation was initiated from the aluminum side of the sample, and multiple excitation frequencies were assessed between 0 and 300 kHz where the two conditions of internal resonance are satisfied, as discussed in Sec. 2.1. On this basis, the second-harmonic generation was well seen at a frequency of 300 kHz.

Figures 10–12 display sample measurements at sensing positions, 20 mm (aluminum), 34 mm (weld region), and 72 mm (steel), respectively, measured from the aluminum edge of the sample. These measurements were taken for both the supported and support-free samples of the B150 specimen. The presence or absence of second-harmonic generation was determined by observing the FFT of the signals. Due to the large amplitude difference between the fundamental and second-harmonic frequencies, the FFT plot was divided into two plots which display the excitation frequency and double-excitation frequency domains separately. The frequency-domain plots show a second-harmonic component with multiple peaks or side bands.

A similar experiment was conducted while exciting from the steel side of the sample, and multiple excitation frequencies were used within the low-frequency range up to 120 kHz and between 300 and 750 kHz. A clear second-harmonic generation was observed at an excitation frequency of 350 kHz. Figures 13–15 display sample measurements at different sensing positions in the steel (at 20 mm from the sample's steel edge), weld (at 34 mm), and aluminum regions (at 72 mm), respectively. The measurements were taken for both the supported and support-free for the B150 specimen.

In both supported and support-free samples, a strong second-harmonic peak was observed at a frequency of 350 kHz. However, when observing the signals' change over time in both experiments at the three different sensing positions, it is observed that both exhibit multiple modes, but the raw data in the supported experiment appears to be less complex. Second, when comparing the FFT plots, the presence of multiple peaks around the second harmonic in the support-free-sample experiment is noticed.

The multiple frequency peaks appearing sometimes in the main-lobe region (fundamental mode) could be due to a phenomenon called spectral leakage. This phenomenon could occur when the signal being analyzed by FFT is not perfectly periodic or symmetric within the used window. This can cause energy to “leak” into adjacent frequency bins which distorts the actual frequency spectrum of the signal and makes the main lobe appear as if it is split. This can also be a reason for the peak frequency in the second-order harmonic region not to be perfectly equal to two times the frequency of the fundamental mode. Interactions between different modes, boundary reflections, and other possible nonlinearities in the measurements can also cause this small shift in the peak frequency of the second-harmonic component.

The amplitudes associated with excitation frequency and double-excitation frequency, as determined through FFT analysis, were subsequently used as inputs for calculating the relative acoustic nonlinearity parameter presented in Sec. 3.3.

The frequency–wavenumber plot was created by plotting k as a function of f [60]. Figure 16 illustrates the results obtained when exciting the specimen from the aluminum side, while Fig. 17 presents the results obtained when exciting from the steel side. Both figures show the experimental results of the supported and support-free B150 sample. The black-colored solid curves represent the theoretical dispersion curves of the excitation material obtained using the dispersion calculator [55]. The dominant propagating wave mode was identified in each case by analyzing the obtained f–k plots. When the LW was excited from the aluminum side, the S₀ mode was found to be the most significant in the supported-sample experiment (Fig. 16(a)). However, in the free sample, several pronounced modes (A₀ and S₀) are observed (Fig. 16(b)). On the other hand, when the LW was excited from the steel side, multiple propagating modes were seen to co-exist in both the free (A₀, S₀, and A₁) and supported (A₀ and S₀) samples as shown in Figs. 17(a) and 17(b). This disparity can be attributed to variations in material properties and geometric characteristics between the two sides of the specimen.

When examining the second harmonic, it was observed that in the supported experiment, the second harmonic is clearly and distinctly visible. In contrast, in the free experiment, the second harmonic appears with lower intensity, indicating a weaker presence. In the supported setup, the A₀ mode might appear weaker due to the presence of different boundary conditions. Overall, the results underscore the influence of material properties, specimen geometry, and boundary conditions on the behavior of LWs, particularly concerning their propagation modes. This analysis provided valuable insights to understand Lamb-wave propagation in the examined specimens.

To visualize the cumulative second-harmonic generation along the propagation distance in the models, the parameter β′ was plotted against the sensing positions for each sample. These plots are shown in Fig. 18 for the supported-sample experiments while exciting the waves from the aluminum side. Figure 19 shows the same plots when exciting from the steel side in the supported-sample experiments. The plots reveal different patterns and variations in β′ when moving along the length of the specimen in the supported experiment. However, the support-free experiment does not exhibit a clear trend due to the presence of A₀ as the dominant mode.

Looking at the pattern of one of the specimens in Fig. 18, from the start of the measurement at 9 mm up to 24 mm (just before the beginning of the steel plate), β′ generally exhibits a slight linear increase. This indicates a gradual accumulation of SHG as the guided waves propagate through the aluminum section of the specimen. The linearity of this increase suggests a uniform interaction between the guided waves and the material properties of this region. Moving further along the specimen, between 25 mm and 56 mm (the lap-weld region consisting of both metals), β′ shows a higher increase at the beginning and then undergoes some sharp changes. This may be related to the sharp changes in the material and geometrical properties of the propagating medium within and on the borders of the combined steel-aluminum region, causing complex interactions and reflections in the measured waves. Beyond 56 mm, after reaching the end of the aluminum plate, where the waves were measured on the surface of the steel plate, another sharp variation is observed in β′. This sharp variation can be attributed to the transition from the aluminum plate to the steel plate, signifying a distinct change in the propagation behavior of the waves as they transition from one medium to another.

When comparing specimens with different intermetallic layer thicknesses, a clear trend of β′ emerged in the region between 30 mm and 54 mm as shown in Figs. 18 and 19. The data from the supported experiment consistently revealed a decrease in β′ as the intermetallic layer thickness increased. This trend is of significant importance, as it indicates a direct relationship between the thickness of the intermetallic layer and the amount of nonlinearity in the waves. However, it is important to note that in the case of the support-free sample, no such consistent trend was evident within this region. This discrepancy can be attributed to the presence of multiple propagating wave modes in the free sample, which led to the disappearance of such a trend. The interaction of these modes can lead to variations in β′ values, making it challenging to establish a linear relationship with the intermetallic layer thickness.

When exciting from the steel side (Fig. 19), a clear pattern in the β′ values was not seen within any specific region. This observation suggests that the steel plate's material properties and thickness (Fig. 4) play a crucial role in the wave propagation behavior and their nonlinear interaction with microstructural changes. When excited from the steel plate, multiple propagating wave modes can exist within the plate, as shown in Fig. 17, leading to interference and mode coupling. This complexity can make it difficult to establish a direct relationship between β′ and the intermetallic layer thickness.

It can be concluded that exciting waves from the aluminum side are more promising, especially in supported experiments where the presence of different boundary conditions contributes to the excitation of S₀ mode as dominant mode. Overall, these plots (Figs. 18 and 19) provide a comprehensive visual representation of the cumulative SHG along the propagation distance within the specimens. They demonstrate the gradual accumulation of SHG in the aluminum section, a consistent trend in the lap-weld region section, and a sudden variation at the transition point between the two measurement materials.

The next section builds upon these findings by focusing on the region between 30 and 54 mm (from the aluminum edge), where intriguing trends were observed in the β′ values while exciting the waves from the aluminum side. Specifically, the average values of β′ will be calculated within this region for various thicknesses of the IMC layers. By doing so, it is aimed to systematically analyze and establish a more precise relationship between β′ and the thickness of the IMC layer.

It is worth mentioning that the measurements are affected by boundary reflections due to the small size of the available sample. It was proven, in previous work [46], that the boundary reflections and the contributions from different modes, which are difficult to exclude in our test bed, can significantly disturb the monotonic accumulation of acoustic nonlinearity. For this reason, the acoustic nonlinearity is not expected to grow linearly with the propagation distance in the current experiment.

On the other hand, a clear distinct pattern was observed with the changes in the geometry and when the waves propagate from one material to another. This pattern was disrupted differently by the existence of different microstructural amounts of IMCs in the weld region, which was successfully measured in the current study. Again, this distinct pattern is affected if more modes and boundary reflections are included in the considered signals, which may affect the results and might risk the efficiency of the IMC assessment.

The influence of the IMC layer thickness on the value of the relative acoustic nonlinear parameter, represented by β′, is described in this section.

The average of β′ values (⁠$βa′$⁠) was analyzed as discussed in Sec. 3.3, based on the values where a trend was observed, specifically, in the weld region (from 30 to 54 mm from the aluminum edge) and when excited from the aluminum side. The variations in $βa′$ were plotted against the corresponding IMC layer thickness in Fig. 20 for the supported-sample experimental result. A clear linear correlation was observed between $βa′$ and the thickness of the IMC layer. Specifically, $βa′$ demonstrated a linear decrease with the increase in the thickness of the IMC layer. Supported-sample experimental result enhances the reliability of the proposed method for detecting and assessing IMCs in dissimilar welded structures.

It is crucial to note that this linear relationship was not observed in the case of the support-free boundary condition. This can be attributed to the presence of A₀ as the dominant mode. In a support-free sample, the multiple observed modes interact and interfere with each other, making it challenging to establish a straightforward correlation between β′ and IMC thickness.

These findings collectively demonstrate that β′ serves as a valuable metric for quantifying IMC layer characteristics in dissimilar lap-weld models. The complexity introduced by multiple wave modes in the free boundary condition highlights the importance of controlled boundary conditions or selective mode excitation when utilizing the proposed approach. These findings not only enhance our understanding of the field of nondestructive testing but also stress the importance of well-defined interfaces and boundary conditions in wave-based assessments of materials and structures. To conclude, it becomes evident that a closer consideration of mode selection is warranted. Analysis reveals that the S₀ mode yields particularly promising results.

This research aimed to explore the applicability of Lamb-wave nonlinear features, namely the second-harmonic generation, for quantitatively evaluating micro-scaled IMCs found within the interface region of dissimilar welded joints. Two experiments were conducted using the laser Doppler vibrometer technique. One of these experiments suppressed one of the propagating modes through a supported boundary condition, while the other was carried out under free boundary conditions. Lamb waves were excited from both aluminum and steel sides based on the synchronism and zero-flux conditions of second-harmonic generation. Waves were measured at different sensing positions and analyzed in the time, frequency, and frequency–wavenumber domains. The relative acoustic nonlinearity parameter (β′) was calculated using the fast Fourier transform. This calculation allowed for the interpretation of the cumulative second-harmonic generation of the propagating modes and its relation to the thickness of the IMC layer within the weld. β′ measured from sensing positions in the lap-weld region exhibited a consistent decreasing trend with the thickness of the IMC layer when the waves are excited from the aluminum side in the supported-sample experiments. However, this trend was not observed in the support-free sample, which may be attributed to the presence of A₀ mode as the dominant mode and the findings indicate that the S₀ mode produces notably favorable results. Conversely, when exiting from the steel side, no distinct trend was noticed. The attained results demonstrate the potential of nonlinear Lamb-wave characteristics, particularly the second-harmonic generation, for the precise assessment of small-scale damage or microstructural differences within dissimilar welded joints.

Future research will focus on refining experimental methods and signal-processing techniques to enhance the reliability of this approach. Additionally, exploring the impact of different welding conditions and material properties on Lamb-wave characteristics will contribute to improving the precision of assessing small-scale damage and microstructural differences in welded joints.

The authors acknowledge the support from the University Research Board at the American University of Beirut for their Award No. 104392.

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.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.