Numerical Far-Field Investigation into Guided Waves Interaction at Weak Interfaces in Hybrid Composites

Abstract

Modern aerospace engineering places increasing emphasis on materials that combine low weight with high mechanical performance. Fiber metal laminates (FMLs), which merge metal layers with fiber-reinforced composites, meet this demand by delivering improved fatigue resistance, impact tolerance, and environmental durability, often surpassing the performance of their constituents in demanding applications. Despite these advantages, inspecting such thin, layered structures remains a significant challenge, particularly when they are difficult or impossible to access. As with any new invention, they always come with challenges. This study examines the effectiveness of the fundamental anti-symmetric Lamb wave mode (A0) in detecting weak interfacial defects within Carall laminates, a type of hybrid fiber metal laminate (FML). Delamination detectability is analyzed in terms of strong wave dispersion observed downstream of the delaminated sublayer, within a region characterized by acoustic distortion. A three-dimensional finite element (FE) model is developed to simulate mode trapping and full-wavefield local displacement. The approach is validated by reproducing experimental results reported in prior studies, including the author’s own work. Results demonstrate that the A0 mode is sensitive to delamination; however, its lateral resolution depends on local position, ply orientation, and dispersion characteristics. Accurately resolving the depth and extent of delamination remains challenging due to the redistribution of peak amplitude in the frequency domain, likely caused by interference effects in the acoustically sensitive delaminated zone. Additionally, angular scattering analysis reveals a complex wave behavior, with most of the energy concentrated along the centerline, despite transmission losses at the metal-composite interfaces in the Carall laminate. The wave interaction with the leading and trailing edges of the delaminations is strongly influenced by the complex wave interference phenomenon and acoustic mismatched regions, leading to an increase in dispersion at the sublayers. Analytical dispersion calculations clarify how wave behavior influences the detectability and resolution of delaminations, though this resolution is constrained, being most effective for weak interfaces located closer to the surface. This study offers critical insights into how the fundamental anti-symmetric Lamb wave mode (A0) interacts with delaminations in highly attenuative, multilayered environments. It also highlights the challenges in resolving the spatial extent of damage in the long-wavelength limit. The findings support the practical application of A0 Lamb waves for structural health assessment of hybrid composites, enabling defect detection at inaccessible depths.

1. Introduction

Modern industry, and particularly the aerospace sector, continues to seek materials that deliver higher performance while reducing weight and extending service life. This demand has led to the growing adoption of fiber metal laminates (FMLs), or hybrid composites, in structural applications [1,2,3]. FMLs are engineered by layering metal sheets with fiber-reinforced epoxy prepreg, creating a material that leverages the complementary properties of both constituents. By combining the toughness of aluminum alloys with the strength and fatigue resistance of fiber-reinforced laminates, FMLs offer a unique balance of mechanical and environmental performance. Common types of FMLs include GLARE (glass-fiber-reinforced aluminum laminate), ARALL (aramid/Kevlar-fiber-reinforced aluminum laminate), and CARALL (carbon-fiber-reinforced aluminum laminate). These materials are characterized by high impact resistance, excellent corrosion and burn resistance, superior residual strength around notches, and outstanding fatigue resistance, significantly impeding fatigue crack growth [4,5,6,7,8,9,10,11].

In recent years, significant research efforts have been devoted to understanding delamination and its inspection in CARALL laminates through various approaches [12,13,14,15,16]. Several studies have reported the initiation of internal delamination between layers, primarily due to tensile failure of the fibers. Bridging stresses play a critical role in transferring load between the metal and composite layers in these hybrid laminates; however, they are often accompanied by interfacial shear stresses that contribute to delamination between plies. Understanding the growth of delamination driven by shear is essential, as it directly impacts the fatigue and damage tolerance of the material [13].

The extent of delamination is particularly important, as it leads to localized regions of high stress concentration, typically at the onset or downstream end of the delamination zone. These regions, identified in this study as acoustically sensitive zones are key contributors to further delamination propagation and the initiation of cracks along the metal-composite interface. Under dynamic loading, plastic strain accumulation in these regions has been shown to correlate with interlaminar delamination length [14], making it a primary factor in CARALL failure. This behavior has been examined in previous work using full-field strain measurements obtained through optical and digital image correlation (DIC) techniques [16].

Given the limitations of conventional inspection methods in maintaining structural integrity, reaching inaccessible areas, and enabling on-site evaluation, a variety of nondestructive testing (NDT) techniques, such as X-ray radiography, thermography, computed tomography, and ultrasonics, have been explored for detecting hidden interlaminar flaws in hybrid composites under adverse conditions [17,18,19,20,21,22,23]. However, each technique has specific strengths and constraints, and many remain challenging to implement in the field, particularly when accessing far-field regions from a single-sided inspection point.

Among various inspection techniques, acoustic methods, such as guided wave inspection, have proven effective for characterizing multilayer structures. Ultrasonic guided waves (Lamb waves) propagate along structural boundaries and arise from multiple reflections within layered materials. These interactions generate distinct wave modes that travel as wave packets, with their characteristics governed by excitation frequency and layer thickness [24].

Each guided wave mode carries energy according to its mode shape, and the suitability of a mode for ultrasonic inspection depends on this shape and its interaction with the material system. The global transfer matrix method (GTMM) is often used to compute guided modes for multilayer structures. Numerous studies [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45] have investigated delamination detection in laminated composites and fiber metal laminates (FMLs), including CARALLs, using guided ultrasonic waves with single- and multi-sensor configurations.

Over more than a decade of research, it has become evident that accurately detecting delaminations or characterizing their interaction with specific guided wave modes remains challenging [25,26,27,28,29,30,31,32,33,34]. Many studies utilize low-frequency excitation to minimize the generation of higher-order modes and simplify signal processing, thereby improving the signal-to-noise ratio (SNR) [25,26,27,28,29,32,33,34]. This approach is particularly effective in acoustically anisotropic materials, where dispersion significantly influences the resolution of closely spaced defects and far-field detectability [24].

The presence of Lamb wave modes in FMLs can be confirmed using techniques such as Semi-Analytical Finite Element (SAFE) analysis or GTMM. Their existence is often verified by examining phase relationships near surface boundaries and analyzing time-frequency spectra [35]. Several studies have reported mode conversion phenomena, such as A1 converting to S0 at delamination sites, which is useful for detecting changes in dispersion behavior [36]. Other investigations have used A2 mode excitation to detect delamination by observing peak amplitude reductions in the affected region [37]. Capturing full wavefield and frequency spectra in layered laminates is particularly challenging due to strong dispersion effects [46,47,48]. For instance, dispersion in FMLs has been analyzed in the high-frequency domain using non-uniform discrete Fourier transforms [41].

Understanding the angular propagation characteristics of a given mode is critical for mapping its directivity and identifying scattering sources [24,28,32,48]. This can be achieved by launching multi-frequency excitations toward a feature of interest and identifying frequency components that are most affected, thereby improving spatial resolution and defect detectability [43]. However, exciting multiple modes complicates signal interpretation due to overlapping responses, reduced mode energy, and the need for extensive signal processing. This necessitates a trade-off between selecting an efficient wave mode and an appropriate frequency range.

In the present study, a low fixed-frequency band was selected to excite the fundamental anti-symmetric (A0) Lamb mode, based on GTMM calculations of the supported modes in the chosen CARALL configuration [46,47,48,49,50]. Section 2 outlines the problem statement and introduces the dispersion analysis and mode selection methodology. Section 3 presents the three-dimensional finite element (FE) model used to simulate wave propagation, followed by validation against published experimental and numerical results. Section 5 provides detailed results of the 3D wavefield analysis, with emphasis on the centerline through the delaminated region and its immediate surroundings. The dominant wave interactions are examined on the reflection side, focusing on estimating delamination extent. Section 6 further elaborates on this analysis, providing theoretical and analytical insights into resolving delamination boundaries. Finally, Section 7 concludes the study and outlines directions for future research.

2. Problem of Investigation

2.1. Material Domain Under Investigation

The three-dimensional finite domain is selected to record the wave field and attenuation data in situ conditions, and to validate the efficacy of the approach used in Section 3. The chances of interfering side wall reflections with the reflected wavefield from a faulty region are always challenging for a practical finite domain. However, the study focuses on investigating through-thickness delamination detectability at the accessible target sensing location along the centerline of the plate. Plates used to conduct this study were modeled as 9-layer Aluminum (Al)—Carbon fiber reinforced polymers (CFRPs), referred to as Carall–Carbon-fiber reinforced aluminum laminate, in an alternating pattern with Al being the topmost as well as bottommost layer, approximating perfect intralaminar bonding as shown in Figure 1. A similar layout has already been used in the study by Gupta et al. [16].

Two cases were considered: Carall1, where the fiber orientations of CFRP layers are [0/0]s, the unidirectional case, and Carall2, where fiber orientations are [0/90]s, the cross-ply case, to gain insight into the influence of transverse ply on delamination detectability. Delamination is considered a square that measures 50 mm, located centrally in the first case of consideration using the three-dimensional FE approach. The delamination extent is used by its left or right end/edge. $DL-l$ and $DL-r$. Delamination detectability in-between layers 1 and 2, 2 and 3, 3 and 4, 4 and 5, 5 and 6, 6 and 7, 7 and 8, and 8 and 9, which are denoted further in the paper as DL12, DL23, DL34, DL45, DL56, DL67, DL78, and DL89, respectively, has been studied first. The detectability resolution refers to the measurement of the delamination extent in terms of detecting delamination ends (i.e., the delamination entry point and the exit point, which corresponds to the downstream covered region of the delamination) based on the amplitude of the A0 Lamb mode calculated in the frequency domain. Delamination depth measures from the top surface are shown in

Table 1. This table shows the delamination positions and their depths relative to the top surface of the laminate.

Delamination Positions	Depth of Delamination (from Top Surface) [mm]
DL12	0.3
DL23	0.55
DL34	0.85
DL45	1.1
DL56	1.4
DL67	1.65
DL78	1.95
DL89	2.2
NDL	undefined

.

2.2. Dispersion

In solids, infinite modes are possible, and their presence is highly dependent upon the medium thickness and the operating frequency [45] in the case of guided waves. Investigating modes excited at the chosen operating frequency in the present study becomes essential. In this context, dispersion curves for A0 Lamb mode propagation in Carall1 and 2, shown in Figure 2, have been calculated using DISPERSE (version 2.0.11) [46,47]. At these lower frequencies (100 kHz), only A0 and S0 Lamb modes are present and can also be identified in the total displacement contour for the guided wave propagation obtained from the 3D-FE simulation (Section 5). The A0 Lamb mode has a significantly lower group velocity than S0 at 100 kHz, making it distinguishable and suitable for delamination detection. In addition to that, it is less dispersive [51], making analysis even more accessible. The A0 Lamb wave mode sensitivity towards delamination in these hybrid laminates (class of FML-Fiber metal laminate) is explored in the view of local wave dispersion and its effect on A0 Lamb mode sensitivity. Disperse uses the global transfer matrix method (GTMM) to estimate frequency-dependent phase and energy velocity vectors in the sub-layered regions in limits of infinite media in the direction of wave propagation.

2.3. Launching of A0 Lamb Mode

Launching of the A0 Lamb mode is attempted using a 5-cycle sinusoidal Hanning window tone burst using Equation (1), exerted in the negative Y direction, at the source node, which is located at the far-left edge of the plate (for further identification, named ‘front wall’ or ‘FW’, and the far-right edge of the plate will be referred to as ‘back wall’ or ‘BW’). Therefore, the wave propagates in the +X direction whereas the medium particle oscillates in the direction of Y. The schematic of the problem is shown in Figure 3.

$$u\left(t\right)=\left[{\displaystyle \frac{1}{2}}\left\{\begin{array}{c}1-\cos \left({\displaystyle \frac{{2\pi f}_{oc}}{n_c}}\right).\sin \left({2\pi f}_{oc}t\right),(0\le t\le {\displaystyle \frac{n_c}{f_{oc}}})\\ 0, (t>{\displaystyle \frac{n_c}{f_{oc}}})\end{array}\right\}\right]$$

(1)

where $u\left(t\right)$ is the input Hanning windowed tone burst impulse, in the time domain, $f_{oc}$ is the operating central frequency, and $n_c$ is the number of cycles. As the wave propagates and undergoes reflections, its coherence diminishes differently in the scattered and along the center line, resulting in more dispersion. To limit losses and dispersion, a low-frequency range has been chosen. The low frequency and number of cycles are chosen to ensure the existence of only fundamental modes. In addition, the number of cycles can be further increased or reduced as per the narrowing down/or broadband of the operating signal to limit the wave dispersion. Pitch-catch configuration is used, as the reflected waveforms are of interest at the selective nodal sensors. Two groups of sensor nodes were placed and used in this study. The first group (referred to in the future as Set 1) consists of sensor nodes placed in a 180-degree semi-circle, at a radius of 100 mm around the source, with an angular shift of 15 degrees, used to obtain in-plane wave energy distribution discussed further in Section 5. The second group of sensor nodes (in the future referred to as Set 2) was placed along the centerline from 50 mm to 200 mm, with the distance between nodes measuring 5 mm, counting 31 nodes in total. The sensor node layout can be seen in Figure 3.

3. 3D FE Methodology

The commercially available Abaqus/CAE 2024 [52] software was used to conduct the 3D FE simulations for A0 Lamb wave mode interaction with delamination. The dimensions of the Carall1 and 2 plates are 500 mm × 500 mm × 2.5 mm, where each aluminum layer is 0.3 mm in thickness and each CFRP layer is 0.25 mm. The dimensions of the plates were chosen so that side wall reflections do not interfere with the signals of interest. Laminate layers were modeled separately, and while aluminum is isotropic, for CFRP, fiber orientation was considered while modeling. Material mechanical properties chosen for Al7075 are Young’s modulus E = 70.3 GPa, Poisson’s ratio ν = 0.345, and density ρ = 2700 kg/m³, as for CFRP, the mechanical properties used are given in

Table 2. This table shows the material properties of CFRP used in Carall1 and Carall2.

CFRP Mechanical Properties
$E_{11}$ (GPa)	$E_{22}$ (GPa)	$E_{33}$ (GPa)	$G_{12}$ (GPa)	$G_{13}$ (GPa)	$G_{23}$ (GPa)	${Nu}_{12}$	${Nu}_{13}$	${Nu}_{23}$	ρ (kg/m3)
128.75	8.35	8.35	4.47	4.47	2.90	0.33	0.33	0.44	1517

.

Fiber orientations were modeled for CFRP layers by assigning a local coordinate system to each lamina individually (by rotating the assigned coordinate system around the vertical Y axis, we can model different fiber orientations). The delamination was located in the center of the plate, so the distance from the source to the left edge of the delamination is 225 mm. Delamination was modeled as an unbounded/disconnected surface between adjacent layers and is considered to be the equivalent to zero volume delamination, while the adhesion between layers was achieved using tie constraints. The delamination splits the laminate into the top and bottom sub-laminates. Once the wave propagating through the plate reaches the delaminated region, it propagates independently through the top and bottom sub-laminates. In the present study, the sensor nodes of interest were taken from Set 2 and were at 100 mm and 125 mm from the left edge of the delamination. They are chosen such that near-field effects could be avoided, and the side wall reflections could not interfere with the signals of interest. The theoretically calculated wavelength was λ = 13.5 mm, meaning that the approximated delamination length was 3.7 λ. The finite element (FE) mesh shape is a hexahedron sized 1 mm × 1 mm × 0.3 mm for Al layers and 1 mm × 1 mm × 0.25 mm for CFRP layers, with each node having 3 degrees of freedom (3 translations). The mesh elements are shown in Figure 4. There were 13.5 elements per wavelength, which satisfies the stability criteria established in previous studies [29,37,39]. The total time period for which the waveforms are captured was 800 $\mathsf{\mu }\mathrm{s}$ and the time increment was 0.01 $\mathsf{\mu }\mathrm{s}$. The number of elements per layer was 250,000, which, when calculated for all 9 layers, is 2.25 million elements per laminate. The computation time needed to complete one simulation was a little over 7 h for one model.

The time-of-flight method (TOF) was used to calculate the group velocity of the excited waveforms using Equation (2).

$$Vg or V_{ph}={\displaystyle \frac{∆x}{∆t}}$$

(2)

$V_{g/ph}$—represents the group/phase velocity, ∆x—the distance between nodes/sensors where the signals were received, calculated at the same phase, and ∆t—the time span between signal acquisitions. Further in the study, reflection coefficients were calculated to determine proportionately how much of the incident wave amplitude (energy) that reaches the delamination edge (left/right) is being reflected back. The calculation of reflection coefficients was completed according to Equation (3).

$${RC}_{A0,DL}={\displaystyle \frac{R_{\left(\omega \right)}}{I_{\left(\omega \right)}}}$$

(3)

where ${RC}_{A0,DL}$—implies that the reflection coefficient (RC) is calculated for the A0 Lamb mode and the waveforms reflected from the delaminated location are used. $R_{\left(\omega \right)}$ and $I_{\left(\omega \right)}$ represent the frequency spectra of the reflected and incident wave packet, respectively. A Fast Fourier Transformation (FFT) method was used to obtain the frequency spectra at the different TWs (time windows). When calculating RCs, the corresponding wave packet to the delamination entry and exit end has only been windowed for a selected TW, and RC ($\omega$) was calculated.

4. Validation

To validate the effectiveness of our approach, the results are compared with previous studies by Hayashi and Kawashima [32] and by Gupta and Rajagopal [33]. For the recreation of their work, we have used the same simulation strategies and methods (mesh used, excitation method, RC calculation method) as used in our current work and compared with their findings.

A group of authors has investigated the effect of delamination through-thickness position on the reflected waveforms of the fundamental modes. All the A0 Lamb mode cases are reproduced and compared in similar time frames. In this context, the time traces that show the interaction of A0 Lamb wave mode with delamination are compared and are shown in Figure 5. The incident-guided wave mode was observed at almost identical time ranges. For each wave packet received within the interested time window, an FFT analysis has been performed to verify the efficacy of the results about the launching frequency. The plate dimensions vary slightly from those previously used, resulting in a different time trace for multiple reflected waveforms. Considering the previous statement, it may have satisfied the dimension, excitation, and acquisition criteria given in the study.

Table 3. This table shows the reflectivity of delamination for different through-thickness positions, probed with A0 Lamb mode.

Incident Wave	Location of Delamination	Reflected Wave
A0	1–2 (interface)	Multiple reflections > 3-wave packets (highly dispersive)
	2–3	Multiple reflections > 2-wave packets (moderate dispersion)
	3–4	Multiple reflections > 3-wave packets
	4–5 (mid-plane)	Small (weak dispersion)

summarizes the results and insights on A0 Lamb wave interaction with delamination in the context of the degree of dispersion in the reflected wave field.

To further validate the approach, the author’s previous experimental data was compared [33] using the present FE methodology. The study has investigated the effect of through-thickness position on delamination detectability using the S0 Lamb mode. We have compared the previously obtained FE simulation and experimental results with the FE simulation results of our simulations, which are shown in Figure 6. Regarding the symmetry of delamination detectability in the given case, we have selected to recreate the results up to the middle layer. An agreement can be observed between the results we have produced and the previous ones, indicating that the detectability calculated using the present approach is not only close to the FE but also very close to the experimental findings.

To validate the model and approach effectively, results were generated for the S0 Lamb wave mode velocity and compared with the S0 group velocity of 2808 m/s, obtained numerically and experimentally by S. Gupta and P. Rajagopal. The S0 group velocity of 2793 m/s, acquired using the present approach, is found to be very close. Therefore, the efficacy of the approach has been proven and validated. In this sense, the current approach is deployed to explore various complex phases of the A0 Lamb wave mode interaction with weak interfaces where the detectability and resolution of the delamination extent are probed in the context of dispersion present in a sub-layered regime.

5. Results

Firstly, an attempt is made to observe the amplitude scattering plot (wavefront) for the intact laminate to confirm the maximum amplitude distribution at various angles. A0 Lamb mode amplitude (energy = square of the amplitude) is being dispersed along the centerline and angularly, as shown in Figure 7. However, most of the energy is observed to concentrate along the centerline (0 degrees) of the metal laminate (Figure 8).

Indeed, significant amplitude attenuation is noticed in this direction of wave propagation, as expected (Figure 9). The attenuation calculation is based on the arrival amplitudes at spatially located nodes ranging from 50 mm to 200 mm, with a frequency spectrum and increasing steps of 5 mm. However, to confirm this further, the arrival amplitudes of A0 Lamb modes are calculated at Set 1 of the sensor nodes (located angularly at every 15 degrees in a semi-circle around the source, as stated in Section 2). We have observed a substantial effect on the directivity of wave propagation as the distance from the source increases; however, we have confirmed that the A0 Lamb mode is highly dominant in the 0-degree direction compared to other directions and can be utilized to understand the scattered wavefield in the fiber-metal laminated composite plate structures.

Time traces were extracted for Carall1 and 2 at sensor nodes 100 mm and 125 mm (from Set 2) from the delamination left edge for the various delamination positions (Figure 8) as stated in Section 2 and Table 1. It should be emphasized that the node positions were chosen to avoid nearfield effects from the source as well as delamination. The sidewall reflections are avoided to an extent by selecting sensor nodes such that the capturing time trace window is out of it.

First, to confirm the correct excitation of the A0 Lamb mode, we have observed time traces for the intact laminates shown in Figure 10. All the time traces were extracted for the selected time range of 380–400 μs to avoid multiple reflections from the sidewall, backwall, and delamination interfaces. It is seen that for the intact laminates, only two waveforms can be observed, the incident A0 mode ($I_{A0}$) and the reflections from the sidewalls ($R_{sw}$). We can also observe that no waveforms are observed in the time window from 100 to 300 μs, making this time window, our window of interest, suitable for delamination extent wave signal acquisition.

As soon as the A0 mode excites, it leaks into the medium in all possible directions unless other physical constraints limit the excitability of specific modes. Therefore, it propagates in the +X direction and interacts with the sensor nodes at 100 mm and 125 mm (Figure 8), as the incident wave ${(I}_{A0}$) as shown in Figure 11.

In this view, the wave first reaches the left edge of the delamination and undergoes reflection ($R_{A0,DL23\_l}$) and transmission through the delaminated area. To abbreviate the notation of reflected waves, R represents here the first reflected wave, A0 represents the A0 Lamb mode at its fundamental tone [46,47], DL23 means the delamination is between layers 2 and 3 and r/l means that the reflected wave is from the right/left edge, respectively. The following reflection is observed from the right end of the delamination ($R_{A0,DL23\_r}$) where part of the energy is reflected, and part is transmitted. In this study, we have not considered the transmitted waves as a part of the present investigation. A schematic depicting how wave energy interacts with the delaminated area is shown in Figure 12.

Rsw, represents the reflections from the sidewalls. These sequences of the wave forms with scattering side lobes were observed for all the cases studied, as shown in Figure 13.

The time traces are shown in Figure 14 and Figure 15, at different depths from the top stress-free surface. It can be noticed that the reflected waveforms from the delamination are captured in the time range from 200 to 300 μs (for sensor nodes 100 mm from the delamination left edge), confirming the time window of interest. It should be noted that in addition to part of the incident wave energy being reflected and part being transmitted, the delamination acts as a secondary source and energy leaks through its sides due to its boundary effect phenomenon and participates in complex wave mode interference.

The trapping of energy in this context can be easily observed, as shown in Figure 13, in the near-field region, where mode trapping along the delamination boundary wall and within it occurs as a consequence of the non-propagating wave mode nature. Indeed, it easily distinguishes the delamination region from the intact region, and the same mechanism makes assessment complicated in the far-field, as it is one of the most accessible regions for the industrial on-site applications. Multiple mode conversions at the delaminated region perhaps could be present as a part of a non-propagating wave vector; calculation of these components is not the prime focus of this study. However, at the delaminated site, a significant distortion of waveforms was observed at near field as well as in far-field, mainly at the outer lobes that are radially visible in the total displacement contour extracted at the delamination proximity, as shown in Figure 13. This distortion makes the signal wider, and energy shifts from the operating central frequency. This phenomenon is observed to be more prominent at the wave-front lobes leaving the delamination site at an angle compared to the centerline, validating comparatively reduced energy lost along the centerline (Figure 7). Indeed, this is strongly linked with the dispersion in sub-layers, which has been explored later in the section and shown how the detectability is the strong function of dispersion in sub-layers. A very similar occurrence can be observed at the delamination proximity for Carall2, where distorted waveforms are visible in the extracted time traces, as shown in Figure 15. The incident time window and reflection time window are captured as being delayed due to a 90-degree fiber orientation in layers; however, this orientation makes side wall reflections faster, which are captured at 300 microseconds compared to 330 microseconds in Carall1, as shown in Figure 14 and Figure 15 for Carall2. It complicates the recognition of the delamination extents at its ends. The group velocity of the A0 mode is calculated using the ToF method, explained in Section 3, and is noted to be 1352.35 m/s in the case of Carall1 and 1272.26 m/s for Carall2. The results agree with the dispersion studies and are quite evident, as shown in Figure 2. The slower wave speed in the case of Carall2 can be attributed to the orientation of the second and third layers oriented at 90 degrees, exhibiting greater stiffness. Expanding on the differences present with different fiber orientations, sidewall reflections are observed earlier for Carall2 than for Carall1, making fiber orientation more favorable in this direction. Wave lengths λ1 = 13.5 mm and λ2 = 12.7 mm are calculated for the excitation frequency of 100 kHz for Carall1 and Carall2, respectively.

The reflection coefficients (${RC}_{A0,DL\_l} \mathrm{a}\mathrm{n}\mathrm{d} {RC}_{A0,DL\_r}$) were derived to gain further insight into the detectability for the Caralls studied in the frequency domain explained in Section 3. The (${RC}_{A0,DL\_l}$) represents the reflection coefficients from the left end, and (RC, DLA0-r) represents the reflection coefficients from the right end of the delamination; similar abbreviations will be followed throughout the investigation. Figure 16 and Figure 17 show the reflection coefficients (RC) with delamination through-thickness location for both Carall1 and Carall2 at 100 mm from delamination.

Additionally, for the observed cases, a linear least square fitting has been attempted to analyze the detectability pattern with the change of delamination position, which becomes crucial when the detectability of the delamination extent (left end-l and right end-r) must be calculated separately rather than just approximating the delamination detectability in the far-field. For both Caralls, the closest detectability behavior of ${RC}_{A0,DL\_r}$ was described by the fourth-degree polynomial, as shown in Figure 16b and Figure 17b. The calculated residual norms are 0.1453 and 0.0217 for Carall1 and Carall2, respectively, revealing the accuracy of the detectability pattern in the limits of minimal possible residual for the captured UT data. The detectability pattern calculated for (${RC}_{A0,DL\_l}$) is observed to be quadratic. It is very close to the detectability calculated numerically, as shown in Figure 16c and Figure 17c. The lowest norms of the residual are 0.02187 for Carall1 and 0.007477 for Carall2, predicting the detectability pattern of the delamination left edge very close to the detectability calculated numerically, as expected due to the involvement of less wave interference and dispersion in the captured signal. However, the larger residuals for the right edge of the delamination are due to multiple interferences between the delamination faces and dispersion at the sub-layered regions. The 90-degree ply at Carall2 also changes the local and energy velocity vector and affects the reflected wave energy. It increases the least square polynomial fit degree with the detectability of delamination on the right edge. Although structural nonlinearity is not considered here, it causes higher-degree-based polynomial behavior. However, in view of capturing out-of-plane displacement (u3), which is the only dominating mode with a magnitude range of 0.5 nm to 5 nm, analysis is suitable for capturing detectability behavior. Indeed, the losses and some structural nonlinearity can be expected at this point, which can be further explored based on the local dispersion between the layers and explained in the next section. This detectability shape is spatially consistent because, as Figure 18 illustrates, the pattern of A0 Lamb mode detection sensitivity is approximately equivalent at 125 mm to the detection recorded at 100 mm.

In both cases, the detectability of delamination for ${RC}_{A0,DL\_l}$, increases with proximity to the middle layers and deteriorates as it moves towards the outer layers. In this context, Carall1 detection for the delamination right edge (${RC}_{A0,DL\_r}$) increases monotonically from DL12 to DL34, with a significant decline at DL45, which is observed to be symmetric between DL56 and DL89. For Carall2, this behavior is interestingly reversed at almost all locations. It is indeed due to the reflections getting canceled out due to the unique destructive interference at the front and side faces of the delamination at those locations. It is likely due to the immediate lower sub-ply having a high degree of dispersion concerning the upper sub-ply, explained in detail in the next section. The amplitude spikes at these locations for Carall1 make the delamination sensitivity promising, except for the near-surface delamination DL12. We notice that the left edge (${RC}_{A0,DL\_l}$) delamination detectability increases from DL12 to DL45 and is followed to be symmetric from DL56 to DL89 for both the Caralls studied. The pattern is very similar following, except for a little more sensitivity towards middle-layered delamination DL45.

6. Discussion

6.1. Dispersion Analysis

Dispersion analysis is vital for understanding how guided wave modes propagate in plates and waveguides. Since guided waves are dispersive, their phase and group velocities depend on the frequency and thickness of the structure being studied. The dispersion behavior in hybrid laminates is even more complex than in isotropic materials due to mode coupling, layered anisotropy, and the frequency-dependent behavior of the propagating modes. In the present case of Carall1 and Carall2, as the A0 Lamb mode interacts with the delamination, it splits into two carrier waves and merges into a single wave at the acoustically prone distorted regions l and r. One propagates through the top sublayer and the second through the bottom sublayer, with different group and phase velocities, and exits the delamination as a single wave packet in a forward-propagation manner within the limits of the near-field criteria for the exit point r, serving as a secondary source. The sublayer thickness changes as the delamination position changes, as described in

Table 4. This table shows the sub-layered regions for different through-thickness delamination positions, probed with A0 Lamb mode.

Materials	Sub-Layered Domains
Carall1	Al–DL12, TSL	0/Al/0/Al/0/Al/0/Al–DL12, BSL
	Al/0–DL23, TSL	Al/0/Al/0/Al/0/AL–DL23, BSL
	Al/0/Al–DL34, TSL	0/Al/0/Al/0/Al–DL34, BSL
	Al/0/Al/0–DL45, TSL	Al/0/Al/0/AL–DL45, BSL
	Al/0/Al/0/AL–DL56, TSL	0/Al/0/Al–DL56, BSL
	Al/0/Al/0/Al/0–DL67, TSL	Al/0/Al–DL67, BSL
	Al/0/Al/0/Al/0/AL–DL78, TSL	0/Al–DL23, BSL
	Al/0/Al/0/Al/0/Al/0–DL89, TSL	Al–DL89, BSL
Carall2	Al–DL12, TSL	0/Al/90/Al/90/Al/0/Al–DL12, BSL
	Al/0–DL23, TSL	Al/90/Al/90/Al/0/AL–DL23, BSL
	Al/0/Al–DL34, TSL	90/Al/90/Al/0/Al–DL34, BSL
	Al/0/Al/90–DL45, TSL	Al/90/Al/0/AL–DL45, BSL
	Al/0/Al/90/AL–DL56, TSL	90/AL/0/AL–DL56, BSL
	Al/0/Al/90/Al/90–DL67, TSL	AL/0/AL–DL67, BSL
	Al/0/Al/90/Al/90/AL–DL78, TSL	0/AL–DL78, BSL
	Al/0/Al/90/Al/90/Al/0–DL89, TSL	AL–DL89, BSL

.

In this cycle of the wave propagation, part of the wave energy is reflected at the entrance into ML (Z1) and transmitted into the TSL (Z2) and BSL (Z3). Z1, Z2, and Z3 are the acoustic impedances for the main laminate, top-sub laminate, and bottom-sub laminate, respectively. The wave energy for the dispersive medium is carried with the group velocity of the wave [48,53]. It is calculated using GTMM by applying stable boundary conditions of stress and displacement at each interface for each TSL and BSL, considering the infinite domain in the direction of wave propagation, as shown in Figure 19.

The velocity vectors are calculated as shown in Figure 20 and Figure 21 for a specific TSL and BSL based upon corresponding delamination positions. It is evident by noticing the significant difference in group and phase velocity for delamination positioned at the outer layer compared to the middle layers, which are then symmetric towards the outer layer. This difference shall increase/decrease the degree of interference across the delamination extent and affect the detectability at the sensor nodes.

However, a favorable wave interference toward detecting l or r ends strongly and ties in with the degree of local dispersion, governed by a change in acoustic impedances. Indeed, it varies across the delimitation and the plane boundary passing through l and r, recognized by the acoustically prone to distorted regions (square colored area) as shown in Figure 19.

An attempt has been made to calculate the A0 Lamb mode reflection coefficient of amplitude for one of the Carall, using Equation (4) by considering a plane acoustic wave travelling in the +X direction and meeting a boundary separating two media by Z1 and Z2, Z1, and Z3 separately [33].

$$RC={\displaystyle \frac{Z1-Z2}{Z1+Z2}}+{\displaystyle \frac{Z1-Z3}{Z1+Z3}}$$

(4)

This approximate analytical procedure assumes a continuous particle velocity vector and excess acoustic pressure (local displacement vector in the present case) of the A0 Lamb mode at the separating boundaries Z1 (ML), Z2 (TSL) and Z1 (ML), Z3 (BSL) [53]. Physically, this ensures that the two media are in complete contact everywhere across Z1, Z2 and Z1, Z3. However, this assumption considers an infinite medium separated by the plane boundary in the direction of wave propagation. Therefore, this procedure cannot calculate the delamination extents l and r detectability separately. Only approximate detectability can be predicted by considering ML, TSL and ML, BSL separately for each case. The acoustic impedances can be calculated as $Z= \rho V_g$ where Z is the acoustic impedance, $V_g$ is the group velocity in the respective media, and ρ is the mass density. Substituting it into Equation (4), we can reach Equation (5) as:

$$RC={\displaystyle \frac{{\rho }_1{V_g}_1-{\rho }_2{V_g}_2}{{\rho }_1{V_g}_1+{\rho }_2{V_g}_2}}+{\displaystyle \frac{{\rho }_1{V_g}_1-{\rho }_3{V_g}_3}{{\rho }_1{V_g}_1+{\rho }_3{V_g}_3}}$$

(5)

The group/phase velocity of the A0 Lamb mode changes as the wave propagates throughout Carall1 and Carall2, TSL, and BSL. The theoretical detectability of the A0 Lamb mode, based on the rebounding acoustic wave spectrum for Z1 > Z2 or Z2 > Z1 (considering ML, TSL, at a time for each delamination position), is calculated as shown in Figure 22 for Carall1, using Equation (5). The same can be calculated considering ML, BSL, at a time for each delamination position. Interestingly, detectability closely follows the detectability calculated for l-end, except for DL12, DL23, and DL89 (due to symmetry). The significant disparity in detectability at this location is perhaps due to the substantial change in specific acoustic impedances and the noted assumption of infinite media [53]. It ensures that the two media are in complete contact everywhere across Z1, Z2 or Z1, Z3. However, this assumption considers an infinite medium separated by the plane boundary in the direction of wave propagation. Therefore, this procedure cannot calculate the delamination extents l and r detectability separately. Thus, the detectability pattern is between ${RC}_{A0, DL-l}$ and ${RC}_{A0, DL-r}$. That means it carries a higher rebounded total amplitude from TSL and BSL, which is sensed by the monitoring sensors at ML in the far-field. The significant changes in phase and group velocities, as shown in Figure 20 and Figure 21, capture attention towards the cause of this abrupt change in detectability for DL12. Although these changes are already incorporated into the calculation of the theoretical RC, the changes in the pattern of the velocity vectors at these sublayers are remarkable (Figure 20 and Figure 21) to the A0 wavefield sensing and the detectability resolution of the delamination extent, as explained in the next section. Starting at DL12, with a noticeable velocity difference between TSL and BSL, it converges at a minimum to the middle layer delamination location. Then this pattern diverges again to meet the symmetry of the Caralls studied.

6.2. Detectability Resolution of Delamination Extent

The detection thresholds were calculated for a specific TW (time window) compared to the non-detection (for the intact case). For example, the FFTs for detection cases were calculated and shown in Figure 23 and Figure 24.

However, for the non-detection case, that is, for the intact case, no signals were recorded in the detected TW (189–310 μs for Carall1, and 206–285 μs for Carall2). These observations are crucial because a similar phenomenon was reported [38] for fiber laminates for delamination positions. However, it will be completely different and more complex regarding the detectability resolution of delamination extent in the hybrid/FML laminates as reported in the present investigation. As the present investigation deals with hybrid laminate, exploring the sensitivity of delamination detection to detect the extent of l and r ends by providing detailed insights into local dispersion in the sub-layered region is essential to deliver optimized sensor locations in situ. The detection resolution sensitivity of delamination extents l and r is investigated in this section for A0 Lamb mode, given its strong bonding with a wave interaction corresponding to a sub-layered region. One can note here that for Carall2, because of the faster sidewall reflections, an interference to the right edge (r-end) delamination is identified at DL12, resulting in improved detection resolution at DL12. However, it also results in distortion and significant amplitude losses in all the wave packets recorded. This further reduces the detectability of the delamination extent l and r compared to Carall1, but on the other hand, it increases the detectability resolution of l and r.

The total spectral energy distribution corresponding to the different frequency bands can be estimated at the delamination site for the selected time window (TW) to gain insights into the detection resolution of the delamination extents l and r. This estimation can be essential in predicting mode conversions when dealing with guided waves. The TW is chosen to capture the delamination frequency spectrum based on the intact laminate’s baseline reference signal, as shown in Figure 10. The same methodology has been implemented for Carall1 and Carall2. There is no substantial evidence of mode conversion for the TW that captures the delamination from the entry of the A0 mode to the exit point of the delamination, as shown in Figure 23 and Figure 24. It validates our numerical predictions based on group velocity for individual wave packets at closely located receiver nodes. However, a shift in the frequency band from the central operating frequency is noticed for all the delamination positions, validating strong dispersion in sublayers by capturing velocity changes in sublayer regions, as shown in Figure 20, Figure 21 and Figure 25. It results in the redistribution of the energy in the frequency domain as delamination moves from top to bottom and impacts detectability and its resolution in terms of identifying extents separately, as shown in Figure 23d,e.

The detectability resolution of delamination extent increases from DL12 to DL34 (Figure 16, rectangular yellow color boxes recognized by i, ii, and iii), and the maximum resolution is achieved near the DL34 location, followed by a slight drop near the central zone (DL45-DL56) of Carall1. Multiple FFT peaks, resulting from the redistribution of the frequency band, are the reason for this abrupt drop in resolution and detectability. Figure 23 shows the detectability resolution in the frequency domain for the delamination extent l and r in the yellow-colored window. The higher amplitude peak corresponds to the delamination extent r-end, where the lower frequency band spectrum corresponds to the l-end. Indeed, this effect is more pronounced at DL34, and both extents are separated by corresponding amplitude peaks, demonstrating good detectability and resolution. The corresponding TW for the delamination extent at DL45 and DL56 shows the energy distribution populated by more than two FFT peaks. It is due to the complex wave interference in TSL and BSL regions, resulting in lower detectability and resolution.

However, a slight improvement can be seen at DL12 for Carall2, but it is insignificant on the same scale as Carall1. The detectability and delamination extents have shown a significant drop in the resolution of delamination ends l and r and their detectability at all the delamination depths for Carall2. This is due to the destructive interference of the energy velocity vector at TSL with the BSL regions and the increased energy leakage on the transmission side. The transverse ply effect in Carall2 cannot be ignored, though; higher transmission arises due to the complex interaction of the A0 Lamb mode in the TSL and BSL regions.

As stated in Section 2, two different configurations were used to determine the influence of fiber orientations on the detectability of delamination. Having observed the time traces and velocity curves at the TSL and BSL regions, fiber orientations may also significantly influence the group and phase velocity of the waves [28]. In this sense, for Carall1, it is easier for the A0 Lamb wave mode to pass through directly with little scattering and attenuation than the side-wall reflections, resulting in a delayed arrival time. However, Carall2 shows a reverse effect at those local plies, due to which side wall reflection reached the respective TW quickly. In the case of Carall2, non-favorable orientations influence the incident and scattered wave locally, causing a change in the degree of dispersion and local interference in the TSL and BSL regions. It reversely affects the detection and resolution of the delamination extent at all the delamination depths compared to Carall1. This effect is more dominant in resolving delamination extents (l and r), where the resolution of delamination extent for end-l improves significantly for Carall2 at most delamination locations, making this investigation more interesting to gain insights on A0 wave mode interaction with hybrid composite laminates. The scattered wavefield on the deformed shape has been extracted at different time instances, as shown in Figure 25, to represent the reflected wavefield and the effect of wave interference within the delamination-covered region.

The vertical wave-like lobes are arrows aligned in the direction of +y/−y. The dominance of the outward arrows in TW176-TW232 microseconds is evident, indicating the existence of the A0 Lamb mode, both in weak and strong forms, tied to the delamination extent, as discussed in Section 6.2. The redistribution of energy occurs in the form of side lobes skewed at a specific angle at certain depths, which can be observed in the scattered field and causes a reduction in the detection and resolution of the delamination extent along the centerline. It is interesting to notice the evidence of strong interferences within the delamination-covered region (Figure 25).

Indeed, the constructive and destructive interference within the delamination-covered region initiates strong dispersion at the sub-layered thicknesses and splits the A0 wave mode differently. It makes the dispersion as a monitoring tool to implement it for probing the delamination and its extent. This effect has essentially been observed for DL45, as shown by regions 1, 2, 3, 4, and 5 in Figure 26, and the local orientation effect cannot be ignored, as shown in Figure 26b, for Carall2, which makes the redistribution of the energy along the center line different. The dominance of the A0 Lamb wave mode is consistently observed at all time instances captured, and the delamination-covered region, indicated by outward-directed arrows, confirms the efficacy of the results presented in Section 6.1.

7. Conclusions and Future Directions

3D and 2D FE simulations were conducted for a hybrid laminated composite plate structure, presented as Caralls, with intentionally modeled square-shaped delamination at specific depths. The study focuses on the interaction between the A0 Lamb wave mode and delamination of a fixed size in a hybrid laminated environment. The novelty of this investigation lies in addressing the sensitivity of the A0 Lamb mode to detection and in resolving the extent of delamination at various depths in complex and highly attenuating environments. However, the significantly increased dispersion at TSL and BSL leads to additional challenges in terms of resolution and detectability, which had not previously been encountered, except as a scientific curiosity. It limits our inspection and imposes a minimum size constraint for resolving the delamination extents. Currently, examining the extent of delamination separately is more difficult near the surface compared to the in-depth location used for Caralls. Hybrid composite laminates exhibit a higher degree of local dispersion due to their specific layup configuration. It is observed to be highly dominant at the acoustically prone to distortion regions, the l and r ends, which limits the recognition of these ends with depth. It (resolving ends) is strongly tied to dispersion at the TSL and BSL and the corresponding thicknesses of these regions. Each step of the wave propagation within the TSL and BSL may disrupt the leakage wavefield across the delamination, limiting its detection and resolving extents. It further leads to the formation of a zone of constructive and destructive local interference, recorded in terms of an increased amplitude of delamination at the end of the r in the long-wavelength limit, as evident from Figure 25 and Figure 26. It increases the resolution for Carall1 at all the selected DL locations. It is perhaps due to the mode energy focusing from the lower to the higher stiffer region corresponding to the delamination entry and exit ends (l-r). However, these regions have not been explored in the present investigation and will be part of the future investigation. Indeed, the reverse happens in the Carall2 case. However, the quantitative wave field leakage across the delamination has not been explored and is part of a future investigation. A very engaging example of this phenomenon is observed, where more than two peaks in the frequency spectrum are captured at specific depths. It reduces sensitivity towards detecting and resolving delamination ends. The high-speed side wall reflections in the case of Carall2 cannot be ignored due to the embedded local transverse plies. That adverse effect on its detection and resolving delamination ends. It could be favorable or unfavorable, depending on the orientation of the locally oriented piles, which can be further explored to gain more insight into detection and delamination resolution for a particular operating mode.

Mode trapping occurs between the plies as one of the consequences, revealing local dispersion and capturing the complex interference pattern in the near field, as shown in the present investigation. However, it can reveal the geometry of the delamination in the limits of the local dispersion at the TSL and BSL regions. However, resolving delamination extents at various angles can be challenging in the hybrid laminated case, particularly in the proximity of the total wave field energy present at these locations. The results presented in this investigation are within the limits of the quasistatic far-field criteria, specifically those related to long-wavelength limits. The resolution of the delamination ends/extent reported here is restricted to its approximate length, size, depth, and the local stiffness variations. Variations of any parameter can strongly influence resolution and detectability. To further overcome this challenge, scattering analysis can be performed to probe the mode-trapped inclusion feature. The present investigation contributes to the NDE community as an added advantage for understanding the on-site inspection of hybrid composite laminated structures using the fundamental axis-symmetric mode.