Timoshenko–Ehrenfest Beam‐Based Reconfigurable Elastic Metasurfaces for Multifunctional Wave Manipulation

Abstract Herein, a Timoshenko–Ehrenfest beam‐based reconfigurable elastic metasurface is introduced that can perform multifunctional wave phenomena within a single substrate, featuring high transmission in the ultrabroadband frequency range. Conventional elastic metasurfaces are typically limited to specific purposes and frequencies, thereby imposing significant constraints on their practical application. The approach involves assembly‐components with various geometries on a substrate for reconfigurability, enabling to easily control and implement multifunctional wave phenomena, including anomalous‐refraction, focusing, self‐acceleration, and total‐reflection. This is the first study on elastic metasurfaces to theoretically analyze the dispersion relation based on the Timoshenko–Ehrenfest beam theory, which considers shear deformations and rotational inertia. The analytical model is validated by demonstrating an excellent agreement with numerical and experimental results. The findings include full‐wave harmonic simulations and experimentally visualized fields for measuring various wave modulations. Furthermore, the practicality of the system is verified by significantly enhancing the piezoelectric energy harvesting performance within the focusing configuration. It is believed that the reconfigurable elastic metasurface and analytical model based on the Timoshenko–Ehrenfest beam theory have vast applications such as structural health monitoring, wireless sensing, and Internet of Things.

To demonstrate the superiority of our proposed Timoshenko-Ehrenfest (TE) beam theorybased analytic approach, we compared it with the Euler-Bernoulli (EB) beam theory, which does not consider shear deformation and rotational inertia.For linear elastic, isotropic, and homogeneous EB beams, the equation of motion is derived as follows: 4   (,)  4 +      2   (,)  2 = 0. (S1) Then, we considered the harmonic traveling wave as: (, ) =    (  −  ) .(S2) By substituting Equations S1 and S2, we obtain the frequency dispersion relation of the EB beam as follows: To connect each adjacent beam section of the unit cell, we applied the compatibility conditions: (  − ) =  +1 (  + ), (S4a) ′′ (  − ) =  +1  +1  +1 ′′ (  + ), ′′′ (  − ) =  +1  +1  +1 ′′′ (  + ), where each equation represents the compatibility conditions for the displacement, slope, bending moment, and shear deformation between the adjacent beam sections.Therefore, we derived a transfer matrix   that connects the adjacent  ℎ and ( + 1) ℎ beam sections as follows: where , (S6) Next, to investigate the wave physics of the infinitely arrayed unit cells, we applied the Floquet-Bloch boundary conditions at the ends of the unit cells as follows: Therefore, we derived boundary matrix  that connects the adjacent  ℎ and ( + 2) ℎ periodic unit beam sections as follows: where By integrating Equation S5 and S9, For obtaining a non-trivial solution, Consequently, we analytically derived the relationship between  and  , i.e., the frequency dispersion relation based on the EB beam theory.The TE beam theory is considered equivalent to the EB beam theory, which is a valid approximation when the condition (3) ( 2 ) ⁄ ≪ 1, and the EB beam theory is recovered in the limit as  → ∞ and  → 0. To calculate the wavelength of the propagating waves on the thin plate, we employed the Kirchhoff-Love plate theory, which describes the transverse displacement behavior, whose governing equation is defined as follows: By assuming a monochromatic time dependence of the angular frequency , we derived a quadratic dispersion relation for the thin plate for planar flexural wave propagation with  for various thicknesses of the plate: By expressing the phase velocity of the propagating flexural wave under the linear relation: 6 we can obtain the wavelength  under the relation as: We calculated the dispersion curves for a high slenderness ratio, where the length/width is ten.
The unit cell described in the main manuscript for designing the Timoshenko-Ehrenfest beambased reconfigurable elastic metasurface (TREM) has a low slenderness ratio.In this case, the results based on the EB beam theory tended to deviate from the trends observed in the TE beam theory and FEM results.However, for a high slenderness ratio, it can be assumed that the influence of rotational inertia and shear deformation is negligible, generally aligning well with the trends.Nevertheless, even in this scenario, as the wavenumber increases, the cutoff frequency introduces a larger error compared to the TE beam theory.Consequently, for both low and high slenderness ratio, we have demonstrated that our proposed analytic model based on the TE beam theory accurately represents the dispersion relation.To determine the number of unit cells composing the single meta-slab, we numerically calculated the transmission by varying the ℎ m with respect to the number of unit cells.In general, regardless of the number of unit cells, the maximum transmission exceeded 80%, but when there were 8-unit cells, the maximum transmission reached its highest value at 81.5%.
Consequently, we adopted a meta-slab composed of 8-unit cells.However, even with a reduced number of unit cells to compose the TREM, the proposed TREM can achieve a high transmission ratio.It is also possible to design the total length of functional units to be smaller than the working wavelength.For example, considering a working wavelength of 62.3 mm at 5 kHz, a TREM composed of fewer than 6-unit cells has a total length smaller than the working wavelength with high transmission ratio.To validate the reconfigurability of TREM and explore multifunctional wave phenomena, experiments were conducted using the following setup.Initially, it is divided into a mechanical domain for generating and measuring elastic waves and an electrical domain for measuring piezoelectric energy harvesting performance.Firstly, to generate an incident wave, a continuous sinusoidal wave signal is generated from a function generator.Subsequently, to enhance the signal-to-noise ratio, the wave signal is refined through a power amplifier, and this electrical signal is then transmitted to a transducer array consisting of 13 piezoelectric disc.To generate a plane wave from the 13-piezoelectric-transducer array, each transducer with a radius of 20 mm and height of 0.5 mm was aligned with one-quarter of the wavelength on the substrate plate.
The transducer array generates planar waves through the piezoelectric effect and propagates them through the TREM's domain based on the desired functionality.Consequently, to visualize the behavior of the transmitted wave beyond the TREM domain, a scanning laser Doppler vibrometer was employed.Secondly, the electrical domain for measuring piezoelectric energy harvesting performance, conducted to validate the practicality of the TREM, is built upon the mechanical domain described earlier.The harvesting performance was measured at three positions: before the TREM to assess performance in the incident wave, after the TREM to evaluate performance in the transmitted wave, and at the focal to demonstrate how the confined elastic wave energy enhances electrical power in the configuration designed for wave focusing.Consequently, to determine the maximum output power and corresponding optimal load resistance, the load resistance was varied using a resistance substituter, and the electrical voltage signal extracted through the reverse piezoelectric effect was measured by an oscilloscope.To verify the successful formation of a planar wave from the transducer array, we examined the transverse displacement field from the transducer array to the TREM.Additionally, we confirmed the maintained planar wave through the TREM without configuring the assemblycomponents.Consequently, we confirmed that the transmission of the transmitted wave beyond the TREM is 87.7%, validating the high transmission ratio even as it propagates through the TREM.

Figure S8.
Transverse displacement fields demonstrating the variation in the anomalous refraction angle of the transmitted wave with respect to the number of meta-slabs.
To examine the variation in refractive angle with the number of meta-slabs, we numerically calculated the transmitted wave field by arranging meta-slabs from 4 to 8 at 5 kHz.The refractive angle, calculated by the generalized Snell's law, decreased as the number of metaslabs increased, and the transverse displacement field exhibited corresponding results.
Consequently, we verified the ability to freely manipulate the refractive angle by varying the number of meta-slabs.For piezoelectric energy harvesting, we described the piezoelectric element and its corresponding electrical circuit.To attach the piezoelectric disc with electrodes on the upper and lower surfaces to a flat aluminum thin plate, the electrode on the lower surface was extended upward.Subsequently, the two electrodes were connected by a resistance substituter, and the electrical voltage signal passing through the load resistance was measured by an oscilloscope.To investigate the self-acceleration phenomenon in which waves propagate along the trajectory we defined, we calculated the phase-shift distribution based on the generalized Snell's law according to the desired trajectory function.Subsequently, we reconfigured the TREM based on this information and conducted numerical simulations of the wave field.Consequently, we observed that the degree of wave bending varied depending on the values of the constants  and  for each function, representing the self-acceleration level.We confirmed the propagation of waves along the trajectories of each function, depicted in white-line.

Figure S1 .
Figure S1.Schematic of the comparison between the Timoshenko-Ehrenfest beam theory and

Figure S2 .
Figure S2.Frequency-dependent wavelength variations of flexural waves based on the

)Figure S3 .
Figure S3.Dispersion curves for the high slenderness ratio beam based on the (a) Euler-

Figure S4 .
Figure S4.Boundary conditions for numerically analyzing the meta-slab.

Figure S5 .
Figure S5.(a) Transmission spectrum based on the variation of ℎ m per number of unit cells,

Figure S6 .
Figure S6.Picture of the experimental setup used for verifying the multifunctional wave

Figure S7 .
Figure S7.Experimentally scanned transverse displacement field of the plane wave generation

Figure S9 .
Figure S9.Transverse displacement fields indicating the variation in the anomalous refraction

Figure S10 .
Figure S10.Electrical circuit of the piezoelectric energy harvesting system.

Figure S11 .
Figure S11.Intensity fields in the ultrabroadband frequency range for the elastic metasurfaces

Figure S12 .
Figure S12.Intensity fields in the reconfigurable elastic metasurface at ultrabroadband

Figure S13 .
Figure S13.Intensity fields in the reconfigurable elastic metasurface with varying focal

Figure S14 .
Figure S14.Intensity fields in the reconfigurable elastic metasurface with varying focal

Figure S15 .
Figure S15.Transverse displacement fields for the self-accelerating phenomena propagating

Figure S16 .
Figure S16.Transverse displacement fields for the self-accelerating phenomena propagating