This letter reports designs of adaptive metasurfaces capable of modulating incoming wave fronts of elastic waves through electromechanical-tuning of their cells. The proposed elastic metasurfaces are composed of arrayed piezoelectric units with individually connected negative capacitance elements that are online tunable. By adjusting the negative capacitances properly, accurately formed, discontinuous phase profiles along the elastic metasurfaces can be achieved. Subsequently, anomalous refraction with various angles can be realized on the transmitted lowest asymmetric mode Lamb wave. Moreover, designs to facilitate planar focal lenses and source illusion devices can also be accomplished. The proposed flexible and versatile strategy to manipulate elastic waves has potential applications ranging from structural fault detection to vibration/noise control.

Manipulating or guiding elastic/acoustic waves have attracted a great deal of attention on various applications including medical imaging,1–3 structural health monitoring,4,5 and enhanced sensing,6,7 to mention a few. The recently proposed concept of metasurfaces,8–16 i.e., artificially engineered sheet-like materials, provides the great promise to effectively mold wave fronts through adjusting discrete phase shifts along the specific metasurfaces in different ways. Unlike the bulky metamaterials that direct waves by spatially varying the properties of the material through which the wave propagates,17–21 the metasurfaces are more compact but can still realize similar functionalities in wave manipulation. Different from that of metamaterials, the physical mechanism of a metasurface focuses on altering the phases as wave components exit it. Anomalous propagations of acoustic waves modulated by metasurfaces have been demonstrated.8–14 For instance, anomalous reflection, ultrathin planar lenses, and nonparaxial beams were investigated with acoustic metasurfaces constructed by labyrinthine units.9 The metasurfaces on elastic waves, involving more degrees of freedom than their acoustic counterparts, are expected to be more intriguing.15,16 A recent innovation is the elastic metasurfaces that are built upon geometric tapers, which accomplished the anomalous refraction of guided waves in solids both with and without mode conversion.15 Nevertheless, existing designs of metasurfaces have generally been facilitated through choosing or modifying the constituent materials or microstructures/cells, and one of the challenges is the capability of online tuning their performances without the modification of the physical microstructures/cells.

Piezoelectric transducers possess two-way electromechanical coupling. Integrating circuitry elements to piezoelectric transducers bonded to or embedded in a host structure may alter the dynamics of the integrated system. The integrated system can be further made adaptive, as tunable circuitry elements can be used. Adaptive metamaterials with integrated piezoelectric circuits have shown promising aspects in achieving wave attenuation or localization.22–27 A tunable prism based on piezoelectric metamaterial concept was developed, which could continuously steer acoustic waves.28 A piezo-lens was designed to focus flexural waves in thin plates with an array of piezoelectric patches where the focal locations could be adjusted by tuning the shunting negative capacitance (NC) values.29 To enhance the flexural wave sensing, a piezoelectric metamaterial-based system with gradient negative capacitance circuits was framed to compress and amplify the flexural waves in an adaptive manner.7 

In this research, we explore the design of elastic metasurfaces utilizing piezoelectric circuitry, aiming at realizing adaptive and tunable elastic metasurfaces capable of arbitrarily manipulating refracted waves without altering the host structure. The underlying mechanism is that, in such a system, the wavenumbers can be altered with circuitry tuning such that a proper gradient of phase discontinuity along the metasurface can be achieved as waves are stretched or shortened through the metasurface. With this physical mechanism, we illustrate that elastic metasurfaces can be designed to accomplish anomalous refraction and planar focal lenses using the same cell arrangement with different circuitry tuning. Fundamentally, we can accomplish a qualitatively similar outcome as geometric tapers do15 but with an electromechanical synthesis that can be online tuned. We further demonstrate that this physical mechanism can yield a metasurface design to regulate wave fronts from a point source into different prescribed profiles, known as the source illusion,30,31 which again features adaptivity owing to the circuitry tunability.

We start from examining the A0 mode Lamb wave across the basic unit-cells to be used for the proposed adaptive elastic metasurfaces. We randomly select the excitation frequency as 20 kHz. A basic unit-cell of the adaptive elastic metasurface is shown in Fig. 1(a). It consists of one square-shaped piezoelectric transducer bonded onto the host substrate. The host substrate is made of aluminum with a dimension of 7 × 7 × 1.6 mm3 and material properties, Young's modulus Eb=70GPa, density ρ=2700kg/m3, and Poisson's ratio v=0.33, while the piezoelectric material is the standard PZT-5H with a dimension of 6.4 × 6.4× 1 mm3. For the aluminum host structure, the wavelength of the A0 mode Lamb wave at this selected frequency is λ=27.6mm. In each basic unit-cell, a negative capacitance element is connected to the piezoelectric transducer. In practice, a semi-active negative capacitance (NC) realized by utilizing the op-amp circuit is used [Fig. 1(a)], which is tunable online.32 Finite element simulations using COMSOL Multiphysics® are conducted throughout this research for performance illustration. Specifically, the piezoelectric-device module is utilized to perform the eigenfrequency and harmonic analyses, and the piezoelectric shunting boundary condition is implemented in terms of the impedance values in a weak form expression.7 We first calculate the dispersion curves of the A0 mode Lamb wave propagating along the ΓX orientation for various non-dimensional NC values CN/CpT, where CpT is the capacitance of the piezoelectric transducer under constant stress. Here, the ΓX orientation represents the x-direction shown in Fig. 1(a). The Floquet periodicity condition is applied to the side boundary of the basic unit-cell as boundary conditions. By parametric sweeping of the wave vector kx from 0 to π/Lb, eigenfrequencies are determined at each wavenumber value, and the band structure is obtained. Figure 1(b) shows that the frequency band of the A0 mode shifts with the change in the NC value, causing the propagating wave to be stretched or shortened due to different wavenumbers involved at the designed frequency (20 kHz). Consequently, upon transmitting across the medium with properties changed with different NC values along the y-direction, the propagating wave will feature different phase shifts along the y-direction. This strong tunability of phases by using different NC values will be exploited in the design of the elastic metasurface to manipulate wave propagation.

FIG. 1.

(a) Schematic of the basic unit-cell with a piezoelectric transducer shunted with negative capacitance (NC). (b) Dispersion curves of the basic unit-cell with different NC values along the ΓX direction. (c) Out-of-plane displacement fields produced by different NC values illustrating the phase shifts covering the 2π range at frequency f=20kHz. (d) Phase shift and normalized transmission ratio plots of the frequency responses of the functional unit as a function of the NC value at f=20kHz. The six dots indicate the chosen NC values for the six discrete functional units shown in (c).

FIG. 1.

(a) Schematic of the basic unit-cell with a piezoelectric transducer shunted with negative capacitance (NC). (b) Dispersion curves of the basic unit-cell with different NC values along the ΓX direction. (c) Out-of-plane displacement fields produced by different NC values illustrating the phase shifts covering the 2π range at frequency f=20kHz. (d) Phase shift and normalized transmission ratio plots of the frequency responses of the functional unit as a function of the NC value at f=20kHz. The six dots indicate the chosen NC values for the six discrete functional units shown in (c).

Close modal

To realize the metasurface design that can effectively steer the transmitted guided wave, a functional unit of the metasurface must have the capability of shaping the phase over a full 2π range and also allow the wave energy to penetrate through the metasurface with a high transmission. Since the phase shift yielded by a single basic unit-cell mentioned above is limited within a small range (maximum 1.24 rad under the given frequency with high transmission), here similar to other investigations,10,13 we serially connect five identical basic unit-cells together to form a functional unit, which can span the phase shift over an entire 2π range with tunable NC values. In Fig. 1(c), each row in the metasurface is one functional unit. To calculate the phase shift and the transmitted amplitude of functional units with different NC values, the frequency response analysis for each functional unit is conducted, where a beam-like model is employed to encompass the functional unit and the host structure along the wave propagation direction. The side boundaries of the host substrate along the wave propagation direction are set as periodic boundary conditions as depicted in Fig. 1(c). Perfectly matched layers (PMLs) are used on both ends of the beam-like model and an incident A0 beam from the left side as the external excitation is applied. The phase shift and transmitted amplitude can be obtained by taking averaged displacement data along an extra line in the far field.8,15 The phase shift performance of the functional units under different NC values can be observed in Fig. 1(c), where the out-of-plane displacement distribution is plotted. It is evident that a cumulative phase shift covers the full 2π range. Figure 1(d) illustrates the phase shift and the normalized transmission with respect to the incident wave as a function of the NC value. The transmission ratio is large except in the region with the NC value (CN/CpT) ([0.88105,0.83680] [between two vertical dashed lines in Fig. 1(d)], where the transmission ratio is lower than 0.8, primarily because our designed frequency (20 kHz) is located inside the bandgap within this NC value range. While we will not choose the NC values in this aforementioned region in our metasurface design, the functional unit can still achieve a full coverage of 2π with high transmission. The six dots in Fig. 1(d) correspond to the chosen NC values for the six discrete functional units along the y-direction in Fig. 1(c) to span the entire phase range with an increased step of π/3. In the subsequent metasurface design, for a given phase shift requirement, we can calculate the required NC value.

We now demonstrate the anomalous refraction through the adaptive elastic metasurface design, in which the functional unit topology is kept unchanged, but the circuitry can be tuned to different NC values to yield different refraction angles. Under the guidance of generalized Snell's law (GSL), the refracted angle, θt, is related to the incident angle, θi, as follows:8–12,15

θt=arcsin[λt2πdϕdy+λtλisinθi],
(1)

where λt and λi denote the wavelengths in the refracted and incident domains, respectively, and dϕ/dy is the phase gradient on the metasurface. The above equation implies that the refracted angle can be framed arbitrarily via the proper design of the phase gradient (dϕ/dy). To facilitate anomalous refraction, the metasurface is designed to consist of an array (along the y-direction) functional cell. In order to increase the refraction angle and also to simplify the design, every two adjacent functional cells are grouped together with the same NC value. Therefore, dy =14 mm. Similar to previous studies, here we apply a normally incident A0 Gaussian beam from the left side as the external excitation, and PMLs are introduced to surround the simulation domain to suppress wave reflections.15 We investigate three phase gradient designs, dϕ=π/6, dϕ=π/4, and dϕ=π/3. For dϕ=π/6, we need to select 12 NC values, each corresponding to a phase shift required, which can be solved through the beam-like model frequency response mentioned above. Similarly, for dϕ=π/4 and dϕ=π/3, we can solve for 8 and 6 NC values, respectively. Figures 2(a)–2(c) display the transmitted A0 mode Lamb wave fields (out-of-plane displacements) under different phase gradients. For all three cases, the NC values solved are plotted as well. Based on the GSL, the theoretical refracted angles can be obtained as θt(π/6)=9.46°, θt(π/4)=14.27°, and θt(π/3)=19.18°, which are presented as arrows in Figs. 2(a)–2(c). The patterns of the simulation results match with the theoretical values very well. The proposed elastic metasurface is capable of inducing anomalous refraction, and more importantly, the refraction angles can be online adjusted easily by tuning the NC values differently.

FIG. 2.

The transmitted A0 mode wave field (out-of-plane displacement) for three metasurfaces with different gradient phase shifts facilitated by the corresponding NC values. (a) dϕ=π/6, (b) dϕ=π/4, and (c) dϕ=π/3. Insets show the NC values solved for each case.

FIG. 2.

The transmitted A0 mode wave field (out-of-plane displacement) for three metasurfaces with different gradient phase shifts facilitated by the corresponding NC values. (a) dϕ=π/6, (b) dϕ=π/4, and (c) dϕ=π/3. Insets show the NC values solved for each case.

Close modal

We then proceed to the synthesis of adaptive focal lenses. While a number of previous investigations have proposed various metasurface-enabled focal lenses, the designs were mostly based on mechanical tailoring or addition. Such fixed designs do not allow the tuning of focal locations once they are implemented.8,9,15 While a tunable, metamaterial-based piezo-lens was recently developed,29 the metamaterial synthesis was based on changing the refractive index inside the metamaterial through a homogenization formulation. With the homogenization under the long wavelength assumption, the metamaterial-based piezo-lens was mainly applicable to low frequency ranges. As the wave components were bent inside of it to the focal point, a significant thickness of the lens may be needed. Here, we demonstrate that a tunable focal lens can be designed based on the metasurface concept that creates phase shifts as wave components exit it. After exiting, the wave components will undergo either constructive or destructive interference due to the relative phase differences, resulting in the wave focusing effect. The design is realized by utilizing the same functional cell topology mentioned above but with different NC values [Fig. 3(a)]. The NC values can be further tuned to yield adaptively the focal locations. To construct an elastic flat lens, the phase distribution on the metasurface is modulated to be hyperbolic. The relationship between the phase profile ϕ(y) and the given focal length F is expressed as8,15

ϕ(y)=2πλ(y2+F2F),
(2)

where λ is the wavelength. Two flat lenses with focal lengths F =250 mm and F =300 mm, respectively, are designed to illustrate the capability and tunability of the adaptive elastic metasurface that serves as focal lens. Again, every two adjacent functional cells are grouped together with the same NC value. For 48 functional units employed in the case illustration, with the symmetry, 12 NC values are solved from the beam-like model frequency response aforementioned. Figures 3(b) and 3(c) illustrate the transmitted A0 wave patterns together with the respective NC values employed, where the black dashed arcs manifest the arc wave fronts and arrows indicate the propagation directions which confirm the energy concentration owing to the proposed adaptive elastic metasurface. Figures 3(d) and 3(e) indicate the squared absolute displacements (z-components) for the two cases of focal locations, showing that narrow focal points can be observed at x =235 mm and x =286 mm, respectively, which are very close to the target focal points. It is worth emphasizing that the focal lens performances showcased in Figs. 3(b) and 3(c) are based upon exactly the metasurface topology but with different tunings of NC values. In order words, the proposed adaptive metasurface design could provide significant flexibility in devising focal lens with varying focal location requirements.

FIG. 3.

(a) Schematic of the adaptive elastic metasurface for planar focal lenses with two different focal locations. Transmitted wave patterns with black dashed arcs indicating wave fronts and arrows denoting wave propagation directions when (b) F =250 mm and (c) F =300 mm are shown. Insets show the determined NC values. The squared absolute displacement (z-component) of transmitted A0 mode Lamb waves for the two lenses with focal points (d) 250 mm and (e) 300 mm is shown.

FIG. 3.

(a) Schematic of the adaptive elastic metasurface for planar focal lenses with two different focal locations. Transmitted wave patterns with black dashed arcs indicating wave fronts and arrows denoting wave propagation directions when (b) F =250 mm and (c) F =300 mm are shown. Insets show the determined NC values. The squared absolute displacement (z-component) of transmitted A0 mode Lamb waves for the two lenses with focal points (d) 250 mm and (e) 300 mm is shown.

Close modal

To further demonstrate the versatility of the piezoelectric circuitry-based metasurface design, we finally extend the straight metasurface design circular-shape, aiming at shifting and transforming the point source [Fig. 4(a)]. 60 functional units are evenly distributed circumferentially, where, as mentioned above, each functional unit consists of 5 serially connected unit-cells (along the radial direction). Correspondingly, R0 = 67 mm and R =102 mm. Based on the discontinuous phase modulations offered by the metasurface, the wave pattern originated from a point source can be turned into arbitrary target profiles after passing across the designed metasurface tailored by corresponding NC values. The point source is realized using a circular PZT-5H transducer with the diameter 12 mm bonded onto the host structure at the origin. Outside this circular metasurface, one may view the wave fields as if they were emanated from other types of sources, which results in the source illusion phenomena.30,31,33–35 We first construct a source shifting metasurface which shifts the point source located at (0,0) to the left with a horizontal offset Δd =25 mm. If the inspection on the wave fields is taken from outside, the virtual point source should be at (−Δd, 0). To design the metasurface to realize the shifting effect, the desired phase discontinuity can be expressed as16 

ϕ(θ)=2πλ(RmR),
(3)

where θ is the azimuthal angle and Rm=(Rcosθ+Δd)2+(Rsinθ)2. Owing to the symmetry, 31 NC values for the functional units located at the upper half-circle (θ from 0° to 180°) are determined based on frequency responses of the beam-like model. Figure 4(b) illustrates the A0 mode wave fields inside and outside the metasurface excited by a point source. The results illustrate that the circular wave fronts centered at (0, 0) mm are anticipated inside the metasurface. Outside the metasurface, similar circular wave fronts are also observed, and the solid lines represent the target wave fronts. The results show good agreement between the desired and the realized source shifting.

FIG. 4.

(a) Schematic of the circular-shape adaptive elastic metasurface composed of 60 functional units evenly distributed along the circle with a radius R. (b) Full wave fields modulated by the source shifting metasurface with the shifting effect of Δd =25 mm. (c) The Archimedean spiral wave fronts generated by the source transforming metasurface imposing an orbital angular momentum (m =4) to a point source. Insets illustrate the corresponding NC values determined from the beam-like model. The analytical predications are superimposed on the wave crests with different lines.

FIG. 4.

(a) Schematic of the circular-shape adaptive elastic metasurface composed of 60 functional units evenly distributed along the circle with a radius R. (b) Full wave fields modulated by the source shifting metasurface with the shifting effect of Δd =25 mm. (c) The Archimedean spiral wave fronts generated by the source transforming metasurface imposing an orbital angular momentum (m =4) to a point source. Insets illustrate the corresponding NC values determined from the beam-like model. The analytical predications are superimposed on the wave crests with different lines.

Close modal

Moreover, we can also utilize the circular metasurface to transform a point source into a vortex wave front by adding an orbital angular momentum m (an integer number).16,36 Here, we demonstrate the Archimedean spiral wave front fulfilled by the source transforming metasurface with the desired phase profile as

ϕ(θ)=mθ+ϕ0,
(4)

where θ represents again the azimuthal angle and ϕ0 denotes an arbitrary constant. It should be noted that the same metasurface topology as shown in Fig. 4(a) is employed but with different tunings of the NC values to yield the required phase shifts. Without loss of generality, we let m =4 in case demonstration. A point source is also applied at the center. Figure 4(c) shows the full wave fields of the A0 mode, where four equally distributed branches with the Archimedean spiral wave fronts are generated after the illuminations of the circular A0 mode waves into the circular-shaped metasurface. The number of the branches is expected to be four, equal to the orbital angular momentum imparted (m =4), since the wave experiences the 2π phase change when the azimuthal angle θ rotates every quarter of the whole circle. Therefore, with this rotational symmetry, 15 NC values are tuned accordingly in the design stage. The phases outside the metasurface can be evaluated based on the relation ϕ(x,y)=mθ+kx2+y2+ϕ0, where k is the wavenumber. The analytical wave fronts are represented with different types of lines in Fig. 4(c), showing good agreement with simulation results and demonstrating the generation of the vortex wave front by only adjusting the circuitry NC values.

In summary, a concept of the adaptive elastic metasurface facilitated through piezoelectric circuitry integration is demonstrated. Different phase profiles along metasurfaces can be accomplished through proper selection of the values of negative capacitances employed. Owing to the tunability of the circuitry elements, the metasurface can exhibit tunable and adaptive performance with the same physical structure and cell topology. Several metasurfaces are synthesized to realize anomalous refractions, planar flat lenses, and source illusions. The results confirm the strong wave manipulation capability of the metasurfaces in an adaptive manner. We believe that the adaptive elastic metasurfaces with shunted piezoelectric circuitry can open possibilities in elastic wave manipulation and lead to potential applications ranging from nondestructive testing to vibration/noise control.

This research was supported by NSF under Grant CMMI–1544707.

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