Ultrasonics 119 (2022) 106627

Available online 19 October 20210041-624X/© 2021 Elsevier B.V. All rights reserved.

A metamaterial ultrasound mode convertor for complete transformation of Lamb waves into shear horizontal waves

Yiran Tian a, Yihao Song a, Yanfeng Shen a,b,*, Zhengyue Yu c

a University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China b Shanghai Key Laboratory for Digital Maintenance of Buildings and Infrastructure, Shanghai 200240, China c Experimental Center of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

A B S T R A C T

This article reports a new mechanism involving a non-perforated resonant elastic metamaterial to achieve the complete conversion of Lamb waves (A0 and S0) into the fundamental shear horizontal (SH0) wave. The proposed metamaterial ultrasound mode convertor is studied via the observation of the special resonant shear motion of its unit cells, initiating with a conventional additive stub design. Thereafter, such a stubbed structure is further modified to fully couple the Lamb modes with the shear horizontal stub motion. By investigating the band structure of the metamaterial unit cell through modal analysis and tuning the shear resonant motions, a complete SH0 mode generation band within the simultaneous Lamb modes bandgap can be established in a wide frequency range. Such a special bandgap situation enables the complete mode conversion from Lamb waves into shear horizontal waves. The transformation capability of the proposed ultrasound mode convertor is further substantiated via the harmonic analysis of metamaterial chain model, showcasing the frequency spectrum of the transmitted wave modes. The optimal configuration is determined by conducting a parametric study to identify the most effective mode conversion performance. Finally, a coupled-field transient finite element simulation is carried out to acquire the dynamic response of the structure. The frequency-wavenumber analysis of the transmitted wave field illu-minates the successful realization of the mode conversion behavior. Experimental demonstrations are presented to validate the numerical predictions. The proposed complete mode conversion capability may possess great potential for wave control and manipulation.

1. Introduction

As a powerful nondestructive testing tool, ultrasonic guided waves have been widely investigated for structural damage detection, owing to their preferred features such as long propagation distance and superb sensitivity to a variety of damage types [1,2]. However, challenges may also arise from their dispersive and multi-modal nature. In the past decade, shear horizontal (SH) ultrasonic guided waves have received increasing attention within the Nondestructive Evaluation (NDE) and Structural Health Monitoring (SHM) communities. The advantages of SH waves over other wave types are their abilities to propagate without mode conversion, dispersion, and propagation around curved surfaces with little energy loss [3]. Fundamental shear horizontal mode (SH0) is non-dispersive, contributing to simplifying the interpretation of signals and improving the reliability of structural interrogation. Thus, it has been widely recognized as a competitive candidate for SHM systems [4–6]. Unfortunately, compared with other wave modes which can be easily generated by traditional thickness-polarized piezoelectric trans-ducers (PZT), pure SH waves are hard to obtain in most cases. Recent advancements in shear-type piezoelectric wafer transducers have attracted much attention, as an effective approach to obtain

interrogative SH waves [7–9]. Zhou et al. developed a new piezoelectric wafer made from a single crystal with dominate piezoelectric coefficient d36 to generate SH waves along one direction as well as Lamb modes in the orthogonal direction [9]. Miao et al. utilized a newly defined face- shear d24 PZT wafer, achieving the successful generation and recep-tion of pure SH0 waves [8]. Huan et al. further reported an omni- directional SH wave transducer based on two thickness-direction- poled piezoelectric half-rings, improving the performance of wave dy-namic directionality [7]. Recent achievements in electromagnetic acoustic transducers (EMATs) have also received increasing attention, as another approach to exciting SH waves [10–13]. Nurmalia et al. explored the mode conversion of shear horizontal guided waves when impinging on smooth defects in plates. The fundamental SH0 mode and the first higher SH1 mode could be selectively generated by an EMAT as demonstrated in Ref. [14]. Herdovics et al. presented a non-contact EMAT based on the Lorentz force for generating torsional guided waves for pipeline monitoring [15]. Wen et al. conducted a systematic investigation on the SH0 wave generation in an magnetostrictive transducers (MsTs)-activated SHM system containing adhesive layers [16]. However, when using EMAT for damage detection with different guided wave modes, different types of coils or magnets need to be

* Corresponding author at: University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail address: yanfeng.shen@sjtu.edu.cn (Y. Shen).

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https://doi.org/10.1016/j.ultras.2021.106627 Received 16 July 2021; Received in revised form 10 September 2021; Accepted 14 October 2021

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replaced, which is inconvenient. More recently, Liu et al. developed a mode-tuning magnetic-concentrator-type EMAT (MT-MC-EMAT), con-trolling the mode of the generated signal through adjusting the center distance of the static magnetic field provided by the magnets [17]. Unfortunately, the proposed MT-MC-EMAT can only generate pure A0 mode or S0 mode for the detection of multiple types of defects. There-fore, it is hard for the aforementioned piezoelectric transducers and EMATs to switch between the Lamb and shear horizontal excitation modes. In addition, each guided wave type generally demonstrates distinctive sensitivity to different kinds of structural damage. Thus, it is desirable to establish an effective approach to selectively transmit Lamb and shear horizontal interrogating wave modes into the monitoring area.

In 2000, Liu et al. realized the idea of employing localized resonant structures to form spectral gaps within certain tunable sonic frequency ranges [18]. Thereafter, the unique and bizarre characteristics of met-amaterials have been continuously excavated, such as bandgap [19,20], negative mass density, negative bulk modulus, negative shear modulus [21,22], and gradient-index bending stiffness [23]. Among all these metamaterial kinds, elastic pillared metamaterials, comprised of cylin-drical stubs on a thin homogeneous plate, have proved their effective-ness in manipulating the propagation of Lamb waves and SH waves, through coupling the resonant modes with Lamb or SH waves in the plate [24–29]. Wu et al. systematically demonstrated the existence of complete bandgaps and resonances in a plate with a periodic stubbed surface [27]. Wang et al. put forward a new mechanism involving the torsional resonance of stubs to achieve the negative effective shear modulus of an elastic metamaterial plate. Furthermore, they proposed a chiral pillar to efficiently couple the torsional resonance with an inci-dent A0 Lamb wave [25]. Assouar and Oudich reported a double-sided stubbed plate, achieving the enlargement of locally resonant bandgap [26]. Wang et al. developed a single-phase double-sided pillared meta-material, and by investigating different resonances of both pillars, the negative effective mass density and negative effective elastic modulus can be achieved [24]. Furthermore, metamaterials have been investi-gated for manipulating wave fields to achieve focusing, reflection, refraction, and mode conversion [30–37]. Li et al. utilized stubs with spatially graded height to control the propagation of SH0 waves in elastic plates and realized two novel phenomena, wave focusing and low-pass wave filtering [34]. Qiu et al. developed a general method to manipulate the SH0 waves in plates with arbitrary wave fronts by using metasurfaces consisting of multiple parallel strips [31]. To realize longitudinal-to-transverse mode conversion, Yang et al. designed a single-phase anisotropic metamaterial [32]. Besides, through a nonresonance-based singly polarized solid with deep-subwavelength scale microstructures, Zheng et al. experimentally demonstrated that elastic polarization can be tailored in a broad frequency range [36]. The authors’ previous investigations have demonstrated the great potential of metamaterials for NDE and SHM applications [38,39], and realized the complete conversion of Lamb modes into SH modes through a kind of perforated elastic metamaterial design [40]. However, the afore-mentioned perforated design will introduce cuttings to the monitored plate structure, thus weakening the stiffness and strength of the entire structure system. As a result, a non-perforated design is the key that enables such a mode convertor metamaterial to function in SHM and NDE systems.

This paper focuses on developing a completely different and new non-perforated resonance-based elastic metamaterial for achieving the complete mode conversion from Lamb waves into SH waves, by the means of forming a frequency band, within which, shear horizontal particle motion couples with Lamb motion, as the A0 and S0 modes simultaneously fall into a complete bandgap. In addition to the new metamaterial design, another novelty resides in the systematic presen-tation of the mode converting mechanism, which has not been detailed before.

It should be noted that the performance of metamaterials is very

sensitive to the target frequency, while most of the ultrasonic-based NDE/SHM techniques utilize short temporal tone bursts with a wide frequency spectrum. However, nonlinear ultrasonic methods usually adopt long burst signals with a large number of cycles for engaging the nonlinear interactions between the ultrasonic wave field and the struc-tural damage. The authors believe that the proposed metamaterial- manipulated interrogative waves may find potential applications in nonlinear ultrasonic techniques. For example, the authors’ previous investigation has successfully demonstrated such an aspect for improving the accuracy and identifiability of the superharmonic fea-tures from fatigue cracks by eliminating the inherent nonlinear com-ponents (refer to Ref. [38]). The advantage of our new design is to achieve the complete mode conversion from Lamb waves to SH waves by means of forming a frequency band, i.e., only one transmitter is needed to switch the interrogative wave types, allowing to take full advantage of the sensitivity from each wave mode in SHM and NDE procedures. For example, a potential application of the proposed metamaterial design is to detect the extension direction of fatigue cracks. If the extension di-rection of the fatigue crack is parallel to the particle motion of A0 and S0 modes (such as longitudinal cracks in pipelines), the nonlinear ultra-sonic technique employing S0 and A0 modes (compressive and flexural waves for pipelines) becomes less sensitive to such kind of cracks. At this point, the mode conversion behavior may become exceptionally important. The interrogative wave mode can be easily switched to SH (tortional modes for pipes) waves, with the particle motion perpendic-ular to the crack orientation, which may considerably improve the identifiability of the damage.

The current investigation starts with a careful scrutinization into the dispersion curve features of the conventional pillared/stubbed meta-material. The unique, standing-alone shear horizontal propagative wave band is identified from the observation. Then, the stub unit cell is further modified to couple Lamb motion with shear horizontal polarized movement. Subsequently, harmonic analysis of the unit cell substructure is carried out for illustrating the complete mode conversion frequency band. Numerical results further demonstrate that the extremely wide mode conversion band of the SH0 mode wave can be opened by the local resonance (LR) mechanism. Then the frequency spectrum of a chain model is obtained and analyzed to substantiate the mode conversion phenomenon. Finally, the frequency-wavenumber analysis validates the successful achievement of the complete mode conversion behavior via both transient finite element (FE) simulations and experimental demonstrations.

2. Band features of conventional stubbed metamaterials

This section presents the analysis for the conventional stubbed metamaterials. Through investigating the band structures of a unit cell and carefully observing the resonant motions on the potential mode conversion dispersion branch, an in-depth understanding of such a mechanism behind the formation of a complete conversion frequency band is demonstrated.

2.1. Conventional stubbed metamaterial and its band structure

As schematically shown in Fig. 1(a), a single-sided pillared meta-material unit model, comprised of stub I and stub II with the same diameter, but different heights and materials, is proposed to represent the conventional stubbed metamaterial widely investigated in the existing literatures. Two stubs are connected to each other, constructing a composite cylinder, deposited on a 1-mm thick aluminum host sub-strate. The composite cylinder is configured by its height h = h1 + h2, where h1 and h2 are the heights of stub I and stub II (1 mm and 3 mm, respectively), and by the diameter d of its cross-section, which is 4 mm. The lattice constant a of the unit cell is 10 mm. The periodicity extends in plane along x-y directions, and z-axis is chosen perpendicular to the plate. For the showcase, stub II is set as aluminum invariably, same as

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the host plate, promising the strong coupling between the Lamb and stub modes, leading to a wide resonant frequency band. And several common materials easy to obtain and fabricate are chosen as candidates for the stub I, such as aluminum, lead, steel, and polylactic acid (PLA). There-fore, it is necessary to explore the influence of different material com-binations on the evolution of dispersion curves. The elastic properties of the materials used for the unit cells are listed in Table 1.

Fig. 1(b) shows the finite element model (FEM) for calculating the dispersion relation by treating a unit cell with Bloch-Floquet boundary conditions. This study employed standard element code ANSYS 15.0 for the computation of wave dispersion curves [41]. Fig. 1(c) displays the irreducible Brillouin zone with primitive vectors of the square lattice.

The band structures along ΓX direction for the pristine plate and the metamaterials constructed with different stub I combinations were computed, as depicted in Fig. 2. It can be observed that the dispersion curves of the metamaterials have considerably complex characteristics due to the strong coupling and interaction between the wave motions in the plate and the stubbed structure.

Fig. 2(a) exhibits the dispersion curves of a pristine plate. When the excitation frequency stays within 80 kHz, only three fundamental wave modes A0, SH0, and S0 are present. In comparison with Fig. 2(a), Fig. 2 (b)-2(e) designate a single branch of the dispersion curves, within which, only one wave mode may transmit through the metamaterial region [39]. Such a single-mode band within the light-yellow scope potentiates to serve for SH0-mode conversion purpose, and this hy-pothesis from the observation needs to be further investigated through a scrutiny into the vibrational motions of the metamaterial, which will be discussed in Section 2.2 and Section 3.1. Analyzing Fig. 2(b)-2(e), it can be concluded that the density of stub I larger than that of aluminum (stub II) will result in a decrease and narrowing of the potential mode conversion frequency band; whereas, when the material of stub I is PLA, with a density smaller than aluminum, the potential mode conversion frequency band will move to a higher range and become wider. Subse-quently, the structure shown in Fig. 2(e) is tentatively designated as the proposed metamaterial unit cell, showing a potential mode conversion band ranging from 44 kHz to 52 kHz.

2.2. Resonant motions of the metamaterial system for selective shear type mode guiding

The selective bandgap behavior of the guided wave modes as well as the coupling between the metamaterial stub motions with the plate guided SH0 mode place a foundation for the establishment of the com-plete mode-converting band. To further understand such a mechanism behind, the resonant motions of the stubbed metamaterial unit at α-, β-, γ-, and δ-position (indicated by the blue circles) along the single dispersion curve branch are scrutinized in Fig. 3(b)-3(e). As is well- known, under the current Cartesian coordinate system, at the low fre-quencies of interest, ux is mainly contributed by the motion of the S0 mode; for the A0 mode, the dominant surface motion is uz with slightly small ux; the SH0-mode waves propagate along the x-direction, however, the particle vibration is polarized along the y-direction and follow a symmetric motion with respect to the midplane of the plate. From Fig. 3 (b)-3(e), it can be noticed that the composite stubs consistently vibrate along y-direction, representing a coupling between the stub modeSH and SH0 mode in the host substrate, contributing to the only allowed transmitted SH0-mode wave. On the other hand, the antisymmetric Lamb mode A0 and symmetric Lamb mode S0 are totally decoupled with such a kind of stub motion, forming a Lamb modes bandgap. It should be noted that as the single branch dispersion curve approaches to location δ, contemporaneous LR phenomenon may exert an influence on the effectiveness of the SH0 mode transmission. By an in-depth observation of the displacement distribution, the wave energy is concentrated in the stubs, whereas it is almost zero in the host plate, indicating that the transmission of SH0-mode wave into the plate may also become forbidden.

Fig. 4(a) presents the setup of a conventional metamaterial chain model for harmonic analysis to better understand the bandgap and wave motion behaviors. A 50-N distributed external line force acting along x- direction was applied on the top surface of the metallic strip to generate an incident wave containing both A0 and S0 modes into the structure. The Lamb waves would interact with the metamaterial unit cells and further propagate into the host plate. Two non-reflective boundaries (NRB) were implemented on both ends of the structure to eliminate the boundary reflections [42]. Periodic boundary conditions were applied along the strip edges to simulate the wave propagation in a large plate. Fig. 4(b) exhibits the spectral response of A0, SH0, and S0 mode decomposed from the displacement responses of two sensing points located at the top and bottom surfaces of the metamaterial strip corre-sponding to their mode shapes. The directional complete bandgap covers a frequency range from 44 kHz to 55 kHz, totally overspreading the scope of the single-branch propagative shear horizontal band. From Fig. 4(c), it can be seen that when incident waves were excited at 50 kHz, the waves propagating through the metamaterial structures were pro-hibited, i.e., all wave modes were forbidden with negligible magnitudes. In conclusion, although the conventional stubbed metamaterial system could sustain the transmission of shear type waves carried by the single-

Fig. 1. (a) Schematic diagram of the proposed single-sided stubbed metamaterial unit cell. (b) Finite element model for obtaining the dispersion curves of the metamaterial system. (c) The irreducible Brillouin zone with primitive vectors.

Table 1 The elastic properties of the materials used for the metamaterial unit cells.

Materials Elastic modulus, E (GPa)

Poisson’s ratio, v

Density, ρ (kg/ m3)

Aluminum (Al)

70.0 0.33 2700

Steel (Fe) 220.0 0.30 7760 Lead (Pb) 16.0 0.42 11,340 PLA 3.5 0.36 1250

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branch propagative shear horizontal stub mode, but there is no mech-anism to trigger and achieve the mode conversion from Lamb waves into shear horizontal waves [43].

3. Skewed-stubbed metamaterials for shear horizontal mode conversion

This section presents a new concept of the stubbed metamaterial system by adding a “trigger” that can couple the Lamb-motion-related stub modes with shear-horizontal-motion-related stub modes, i.e., the resonant motions of the new metamaterial system on the single mode- converting branch display a strong coupling between the two target modes.

3.1. Band structure and spectral response of the skewed-stubbed metamaterial system

Fig. 5(a) and 5(b) display the three-dimensional (3-D) configuration and top view of the new metamaterial system configuration. To trigger the coupling between the Lamb-related and shear-horizontal-related stub modes and ensure the transmission efficiency of wave energy from the stubs into the plate, a 1-mm thick rectangular aluminum wafer

with a skewed angle of θ, was added between the stub II and the aluminum substrate. The length and width of the aluminum wafer are 8 mm and 5 mm, respectively. The deflection angle θ is determined by conducting a parametric study. Fig. 5(c) shows the FEM for calculating the dispersion relation, and Fig. 5(d) demonstrates a 10 × 1 unit-cell- chain model for the harmonic analysis. The detailed model setup took the same fashion as that in Section 2.2 and can be safely omitted here.

Fig. 6(a) and 6(b) compare the dispersion curves of the proposed metamaterial unit cell and the frequency spectra of the wave compo-nents after the metamaterial manipulation. Remarkable agreement can be found with only very little deviation caused by the limited number of the unit cells arranged along x-direction in the harmonic analysis. And the new prominent SH0 mode conversion band covering the range from 48 kHz to 66 kHz is wider than that of the conventional metamaterial system. Fig. 6(c)-6(f) display the shear horizontal motions at α’-, β’-, γ’-, and δ’-location (indicated by the blue circles) on the single mode- converting branch. As excepted, the stubs mainly vibrate along y-di-rection, illustrating the coupling between the stub mode and the SH0 mode. However, unlike the conventional metamaterial system, the skewed-stubbed design couples the Lamb-related and shear-horizontal- related stub modes together, which could trigger the mode conversion and sustain the propagation of shear horizontal waves simultaneously.

Fig. 2. The corresponding dispersion curves of (a) a smooth plate and metamaterial unit cells with different stub I components: (b) aluminum; (c) lead; (d) steel; (e) polylactic acid (PLA).

Fig. 3. (a) Band structure of the conventional metamaterial system with stub I of PLA. (b)-(e) Shear resonant motions of the metamaterial unit cell at α-, β-, γ-, and δ-position along the propagative shear horizontal stub mode: Uy contour plot.

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3.2. Parametric study for the optimal mode conversion performance

To investigate the optimal SH0 mode-converting performance, a

parametric study of the proposed metamaterial system was conducted. The skew angle θ shown in Fig. 5(b), pertaining to the stub mode coupling effect, gradually increases from 0 degree to 90 degree with a

Fig. 4. (a) Setup of a conventional metamaterial chain model for harmonic analysis. (b) Frequency spectra of each guided wave mode decomposed by the displacement responses from the metamaterial chain model. (c) Displacement wave fields of the metamaterial strip along x-, y-, and z-direction at 50 kHz (within the scope of the single-branch propagative shear horizontal band).

Fig. 5. (a) Three-dimensional configuration and (b) top view of the proposed skewed-stubbed metamaterial unit cell. (c) Finite element model for obtaining the dispersion curves of the metamaterial system and the irreducible Brillouin zone with primitive vectors. (d) Numerical model of a 10 × 1 unit-cell-chain structure for the acquisition of the spectral response.

Fig. 6. (a) Dispersion curves of the proposed skewed-stubbed metamaterial system along ΓX direction. (b) Frequency spectra of transmitted Lamb and SH0 modes. (c)-(f) Shear horizontal motions of the metamaterial unit cell at α’-, β’-, γ’-, and δ’-position: Uy contour plot.

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step of 15 degree. Both modal and harmonic analyses were performed to confirm the efficacy of the SH0 mode conversion band and evaluate the magnitudes of the converted SH0 mode and the remnant Lamb modes.

Fig. 7(a) and 7(b) exhibit the variation of SH0 mode conversion bandwidth versus the skew angle θ. From Fig. 7(a), the potential mode conversion band can open for all θ in modal analysis. However, from Fig. 7(b) and 7(c), when θ refers to 0 or 90 degrees, forming a sym-metrical structure along the wave propagation direction, the magnitudes of the transmitted Lamb modes and converted SH0 mode all equal to zero, i.e., symmetrical setup could not trigger any mode conversion between Lamb modes and shear horizontal modes. Ulteriorly, as the symmetry is broken, it is remarkable that the magnitude of the con-verted SH0 mode monotonically increases with the augment of θ, reaching a peak at 60 degree. After that, it starts to decrease until reaching zero again. By further referring to the ratio of magnitudes between mode-converted SH0 wave and transmitted Lamb waves dis-played in Fig. 7(d), 30 and 75 degrees are two best candidates for the new metamaterial design. For each case, the width of SH0 mode con-version band is large; in addition, the converted SH0-mode is much stronger than the transmitted Lamb modes. As a result, 30 degree is selected as the optimal choice for the proposed unit cell.

3.3. Frequency-wavenumber analysis of the transmitted wave field

Fig. 8(a) presents the numerical model setup of a 10 × 1 unit-cell- chain structure with periodic boundary conditions for transient dy-namic simulation and frequency-wavenumber analysis. Such a case study of wave propagation is utilized to further substantiate the mode conversion capability. A 10 mm × 10 mm × 0.2 mm square piezoelectric wafer active sensor (PWAS) was bonded on the left side 50 mm away from the left edge of the metamaterial region. The data acquisition points cover a total length of 150 mm. The in-plane and out-of-plane displacements of these points along both x-, y-, and z-directions on the top surface along the wave propagation direction were extracted to conduct the frequency-wavenumber analysis, allowing for the identifi-cation of Lamb and SH0 modes in the transmission region. A 100 V-peak- to-peak (vpp) 50-count Hanning-window modulated sine tone burst

signal was applied on the transmitter PWAS (T-PWAS). The test fre-quency used in this case study was 56 kHz, within the SH0 mode con-version band.

Consequently, only strong SH0 mode showed up in the frequency- wavenumber analysis result, as depicted in Fig. 8(b), agreeing well with the dispersion curves, i.e., Lamb modes can be converted into SH0 mode through the proposed metamaterial unit cells. From Fig. 8(c), it can be seen that when the incident waves were excited at 56 kHz, the waves propagating through the metamaterial structures only supported the SH0 mode, both A0 and S0 modes were forbidden with negligible magnitudes.

4. Experimental demonstration of the SH0 mode conversion behavior

This section reveals the frequency spectra of the converted SH0-mode waves from the experimental data through the frequency sweeping method and the frequency-wavenumber analysis of the metamaterial plate.

4.1. Experimental setup

In order to validate the numerical predictions and demonstrate the SH0 mode conversion behavior of the metamaterial system, experiments using the Scanning Laser Doppler Vibrometry (SLDV) were carried out on a metamaterial plate. Fig. 9(a) shows the experimental setup for the SLDV test. A Keysight 33500B arbitrary function generator was used to generate excitation waveforms of 50-count Hanning window modulated sine tone bursts. The excitation signal was further amplified to 100 vpp by a Krohn-hite 7602 M wideband power amplifier and was applied on the T-PWAS. The fundamental frequencies of the excitation signals were chosen as 54 kHz and 56 kHz within the SH0 mode conversion band.

To capture all possible modes in the wave field, the 1-D SLDV pro-jected with an oblique angle was used to measure the in-plane particle motion and the out-of-plane particle motion normal to the wave prop-agation direction [39]. Fig. 9(b) presents the experimental specimen as well as the zoom-in details of the metamaterial unit cells. A total of 10 ×

Fig. 7. (a) Variation of SH0 mode conversion bandwidth versus the skew angle θ obtained from modal analysis. (b) Variation of SH0 mode conversion bandwidth versus the skew angle θ acquired from harmonic analysis. (c) Variation of magnitudes of the transmitted modes versus the skew angle θ from harmonic analysis. (d) Ratio of the magnitudes between mode-converted SH0 wave and transmitted Lamb waves from harmonic analysis.

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5 metamaterial unit cells were attached on the host aluminum plate. Damping clay was wrapped surrounding the plate to absorb the boundary reflections. Two T-PWASs were bonded on the carefully pol-ished aluminum plate to generate Lamb waves into the plate structure. For measuring both Lamb and shear horizontal waves, the plate spec-imen will be tested twice. In the first round, the laser head pointed

perpendicularly towards the plate surface to record the out-of-plane velocity component. In the second round, the specimen was tilted to introduce an inclination angle between the laser beam and the upper surface of the plate structure, as shown in Fig. 9(c). Then, the in-plane horizontal velocity component at the scanning points can be calcu-lated by solving two linear algebraic equations, following the vector

Fig. 8. (a) Numerical model setup of a plate implemented with the new metamaterial unit cells for the transient analysis. (b) Frequency-wavenumber analysis of wave fields in a metamaterial plate structure at 56 kHz. (c) Displacement fields of the metamaterial strip along x-, y-, and z-direction at 56 kHz (within the mode conversion band).

Fig. 9. (a) Experimental setup using 1D-SLDV for measuring both A0 and SH0 modes. (b) Experimental specimen for SLDV tests and zoom-in details of the unit cells. (c) Schematic diagram for the second test with an oblique incident angle. (d) Spectral response for the decomposed A0 and SH0 modes.

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operation rules depicted in Fig. 9(d): {

V1 = Vy⋅cosθ1 + Vz⋅sinθ1V2 = Vy⋅cosθ2 + Vz⋅sinθ2

(1)

where V1 and V2 are the velocity data collected from the first and second tests; Vy represents the in-plane velocity component (the target mea-surement quantity); Vz stands for the out-of-plane velocity component; θ1 and θ2 are the angles between the incident laser beam and the plate surface at each scanning point for the two test rounds, respectively, considering laser head pointing toward the plate surface with a rela-tively large distance to record the velocity component in the tests. For this research, θ1 = 90

◦

and θ2 = 30◦

were adopted. Fig. 9(e) shows the frequency spectra of A0 and SH0 modes from the

experimental data. It can be noticed that within the SH0 mode conver-sion band, the A0 mode component gradually reached the minimum magnitude. However, the SH0 mode component still maintained at a high amplitude. This may substantiate the elimination of the Lamb modes and the synchronous formation of the SH0 mode, contributing the complete conversion from Lamb waves into shear horizontal waves. It should be noted that the 1-D SLDV can only pick up the particle velocity along the laser beam direction. As mentioned above, the equipment projected with an oblique angle was used to measure the in-plane par-ticle motion and the out-of-plane particle motion normal to the wave propagation direction. Although the out-of-plane particle motion was successfully recorded, from the mode shapes of A0, S0, and SH0 mode at 54 kHz and 56 kHz within the mode conversion band, which can be calculated using the semi-analytical finite element (SAFE) method [44], the out-of-plane displacement component (Uz) of S0 mode is negligibly small compared to that of A0 mode at such low frequencies. Thus, it is very hard to capture the S0 mode simultaneously.

4.2. Frequency-wavenumber analysis of the experimental wave field

For further illustration, the frequency-wavenumber analysis was adopted to analyze the wave mode components propagating through the

metamaterial region in the aluminum plate. The frequency-wavenumber analysis for 1-D wave propagation, by its nature, is a 2-D Fourier transform signal processing technique on the time–space wavefield data. The comprehensive descriptions and examples of such a technique can be found in Ref. [45]. In this study, the frequency-wavenumber analysis was carried out by utilizing the 2-D Fast Fourier Transform (FFT) function in MATLAB.

Fig. 10(a) and 10(b) display the representative time traces of the sensing signals at 56 kHz for the metamaterial plate structure at the starting and ending locations along the scanning line, x1 and x2, as shown in Fig. 9(c). Fig. 10(c) and 10(d) display the frequency- wavenumber analysis results of the experimental data at different fre-quencies: 54 kHz and 56 kHz, within the mode conversion band. It can be noticed that the experimental results are consistent with the results depicted in Fig. 8(b). The Lamb modes were successfully prohibited by the proposed metamaterial substructure, while SH0-mode waves were generated. This may further substantiate the SH0 mode conversion capability of the proposed metamaterial system.

5. Concluding remarks and future work

This article presented a new non-perforated resonance-based elastic metamaterial for achieving complete mode conversion from Lamb waves (A0 and S0) into the fundamental shear horizontal (SH0) wave. The research started with a careful observation and in-depth analysis of the dispersion curve features of the conventional stubbed metamaterial system. Although a single shear-mode-carrying branch can be clearly identified, the symmetric unit cell layout could not trigger the conver-sion from Lamb type into shear horizontal fashion. Thereafter, the stubbed metamaterial unit cell was further modified by adding a skewed trigger to strengthen the stub mode coupling effect. Subsequently, the frequency spectra of the decomposed wave components were further obtained by harmonic analysis to substantiate the mode conversion phenomenon. Ultimately, the frequency-wavenumber analyses from the numerical simulation and experiments ulteriorly demonstrated the

Fig. 10. Experimental sensing signals of the 10 × 5 unit cells plate structure at 56 kHz excitation: (a) time trace of the sensing signal at x1 location; (b) time trace of the sensing signal at x2 location (x1 and x2 are defined in Fig. 9(c)). Experimental frequency-wavenumber analysis testing results at (c) 54 kHz and (d) 56 kHz within the mode conversion band. (Note: only SH0 wave mode exists in the transmission wave field).

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successful achievement of the SH0 mode conversion behavior. The proposed method possesses great potential in future SHM and NDE ap-plications, allowing the selective emission of various interrogative wave modes into the host structure from a single wave source.

For future work, active materials such as Shape Memory Alloys (SMA) and electrical-magnetic active materials should be incorporated into the design for the tunable performance of the mode conversion.

Declaration of Competing Interest

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

Acknowledgments

The support from the National Natural Science Foundation of China (contract number 51975357 and 51605284) is thankfully acknowl-edged. This work was also sponsored by the Shanghai Rising-Star Pro-gram (contract number 21QA1405100).

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