Experimental demonstration of topological waveguiding in elastic plates with local resonators

Our system consists of a thin plate and bolts that are tightly fastened to form a periodic pattern. As a starting point, we take a hexagonal lattice pattern shown in figure 1(A). We focus on a large unit-cell representation, i.e. a rhombus-shaped unit cell (lattice constant a = 45 mm) that contains six bolts to facilitate zone-folding. We then perturb this unit cell by varying the circumferential radius R of the mounted bolts around R = a/3 (figure 1(B)). This is to obtain two topologically distinct unit-cell configurations of the bolted plate. Note that, in the process, we keep the C₆ symmetry intact in the unit cell, so that we obtain degenerate modes in the dispersion to be shown later. These two types of unit cells, specifically $R=0.8a/3$ and $1.1a/3$ in this study, are then tessellated to form a periodic pattern in the plate (figure 1(C)). We construct a domain wall consisting of three linear segments (one horizontal and two inclined at 60^◦) by joining these two types of periodic patterns (figure 1(D)).

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The substrate is an aluminum 6061-T6 plate (91.4 × 61.0 cm and thickness t = 2 mm). We use a CNC milling machine to drill 1320 holes, followed by manual tapping and fastening M4-0.7 mm black-oxide alloy steel bolts (91290A180, McMaster-Carr). To facilitate the firm contact between the bolts and the threads of the plate, we add a drop of glue (Henkel Loctite adhesive) in the process of mounting bolts using a screwdriver (movie 1 is available online at stacks.iop.org/NJP/20/113036/mmedia). The partial thread on the screws comes handy, as it allows tightening them firmly on the plate without using any additional nuts.

To excite the system, we bond a piezoelectric ceramic disc (STEMiNC, diameter 10 mm, and thickness 1 mm) with silver epoxy adhesive on the plate as shown in figure 1(D). We mount the Polytec OFV 534 laser Doppler vibrometer (LDV) on the two-axis linear stage and control its motion using MATLAB to measure vibrations on the plate. The LDV takes point-by-point measurements by moving in a square grid of 7.5 × 7.5 mm. We measure 114 points horizontally and 60 points vertically, i.e. a total of 6840 points. All measurements are synchronized with respect to the onset of the input voltage signal from the function generator, which sends a signal to the piezoelectric actuator via an amplifier. To get the steady-state transmission profiles, we send a 100 ms wide frequency sweep signal from 2 to 40 kHz. Then, we employ fast Fourier transformation on the time-history obtained by each measurement and construct a 2D transmission profile for a frequency slice. To get the transient wavefields, we send a Gauss-modulated sine pulse with 80 cycles at a given center frequency. We apply a bandpass filter around the center frequencies to eliminate ground noise on the plate. Then, we use a cubic interpolation on the measured grid data for a better visualization of wavefield on the plate.

We use commercial finite element software (COMSOL Multiphysics) to perform numerical simulations. We take nominal material properties of the aluminum plate (Young's modulus E = 68.90 GPa, density ρ = 2700 kg m⁻³, Poisson ratio ν = 0.33) and steel bolts (Young's modulus E = 210.60 GPa, density ρ = 7800 kg m⁻³, Poisson ratio ν = 0.30). See appendix A for the geometric details of the bolt. The threaded contact between the bolt and plate is modeled using a simple approach, which captures the realistic contact stiffness by modeling an effecting contact area (see appendix B). We mesh the model with tetrahedral elements. To obtain the dispersion diagrams, we take a unit-cell and apply Bloch–Floquet boundary conditions and solve for eigenfrequencies along the irreducible Brillouin zone.

In figure 2, we numerically show dispersion diagrams for the unit-cell configurations with radii varying across R = a/3. The colormap indicates the polarization index defined as ${P}_{z}=\displaystyle \frac{{\int }_{{V}_{u}}| {u}_{z}{| }^{2}{\rm{d}}{V}}{{\int }_{{V}_{u}}(| {u}_{x}{| }^{2}+| {u}_{y}{| }^{2}+| {u}_{z}{| }^{2}){\rm{d}}{V}}$, where V_u is the plate volume in the unit cell, and u_x, u_y, and u_z are the displacement components in x, y, and z axis, respectively. P_z would be close to 1 for out-of-plane plate modes and near 0 for in-plane plate modes. For R = 0.8a/3, we observe two distinct bandgaps in the range of 6.2–8.5 kHz in figure 2(A). Clearly, the polarization index confirms that the bandgaps are for out-of-plane modes (blue markers). The lower bandgap is a partial bandgap (PBG) because it supports nearly in-plane modes (S₀ and SH₀) shown with red and yellow markers. Notably, the upper one is a CBG. This is because the bending-dominated local resonance of the bolts occurs near 6.9 kHz (see the horizontal blue markers in figure 2(A)), and it is coupled with in-plane modes of the plate (further explanations to follow next). This results in a locally resonant bandgap (see the two-sided arrow) for 'undesired' in-plane modes (red and yellow markers) in the range of 6.9–9.4 kHz. Consequently, we observe a CBG between 6.9 and 8.5 kHz for all plate modes.

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This is an example of judicious engineering of overlapping two types of bandgaps: locally resonant bandgap for in-plane modes and Bragg bandgap for out-of-plane modes in the same system. We call the later Bragg bandgaps because those emerge solely when the translational symmetry in the system is changed by varying the radius across R = a/3. This point can be further justified if we plot the dispersion diagram for the case of the perfectly hexagonal system, i.e. with R = a/3, in figure 2(B). We observe that both bandgaps (denoted by PBG and CBG above) close. However, the bandgap for in-plane modes (red markers) remains intact, because it is a locally resonant bandgap caused by the local bending of the bolts, which we do not change in the process (see the two-sided arrow). Remarkably, near the resonant frequency of the bolts, we observe the formation of a pair of distinct double Dirac cones (see the panel below figure 2(B) for the zoomed-in view of the dispersion curves along with the mode shape of the resonant bolts). To the best of the authors' knowledge, the creation of such multiple double Dirac cones at different frequencies in an elastic system has not been reported in the literature.

In figure 2(C), we plot the dispersion diagram for the case when we further increase the unit-cell radius to R = 1.1a/3. We choose this radius to avoid the bolts approaching too close to the unit cell boundary, and at the same time to ensure an overlap with the bandgaps established in $R=0.8a/3$ (compare the locations of bandgaps between figures 2(A) and (C)). We observe that the two double Dirac cones open again and form two bandgaps. However, these are different from the ones observed in figure 2(A) in terms of topology. For further investigation, we revisit figure 2(A), examine the out-of-plane mode shapes, and plot them in figure 2(D). For the lower bandgap, i.e. PBG, we extract the mode shapes of its edges at the Γ point. We observe that two p-type modes are degenerate at the lower edge, i.e. Γ(DOWN), and two d-type modes are degenerate at the upper edge, i.e. Γ(UP). Naturally, the d-type modes show prominent bending of the bolts, approaching the vicinity of local resonance region around 6.9 kHz. A similar pattern is observed for the upper bandgap (CBG) as well, wherein d-type modes are at the upper edge. Sandwiched between these two regions are the locally resonant bending modes of the bolts.

In figure 2(E), we plot the mode shapes at the aforementioned band edges at the Γ point, but for the configuration with $R=1.1a/3$ shown in figure 2(C). A similar degeneracy of modes is observed. However, p-type modes are at higher frequencies than d-type modes for both bandgaps. This so-called band-inversion is a crucial ingredient of the topological phenomena at work and makes this configuration topologically non-trivial. This mechanism is reminiscent of the pseudo-spin Hall effect in photonics [33], where the degeneracy of p- and d-type modes is used for creating two counter-propagating pseudo-spin modes at the domain wall between two topologically distinct unit cells. Here we realize this effect for a flexural wave at a low-frequency regime (i.e. in the vicinity of the bending mode of the bolts) in a continuum plate.

Now moving to the multicell configuration, we first take a supercell consisting of eight unit cells of each type and join them to form a domain wall as shown in figure 3(A). We apply a periodic boundary condition in parallel to the domain wall to investigate the wave dynamics of this tessellated system. We keep the other boundaries free. In figure 3(B), we show the eigenfrequencies of the supercell as a function of wavevector k_//. The colormap indicates the localization index for out-of-plane wave modes. This is defined as ${L}_{z}=\tfrac{{\int }_{{V}_{s}^{* }}| {u}_{z}{| }^{2}{\rm{d}}{V}}{{\int }_{{V}_{s}}| {u}_{z}{| }^{2}{\rm{d}}{V}}$, where V_s is the entire plate volume of the supercell and V_s* is the plate volume of two unit cells touching the domain wall (see the enclosed cells in figure 3(A) with a darker solid line). The high value of the localization index in the colormap clearly highlights the presence of localized modes at the domain wall (red markers) inside both bandgaps (PBG and CBG). The modes in the upper region are desolate as expected in the complete bandgap. However, the localized modes in the lower region are populated with other modes in this partial bandgap.

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To verify the polarization of these localized modes, we replot figure 3(B) in figure 3(C) but with a different colormap indicating the wave polarization, as was used in figure 2. We clearly observe that the localized modes at the domain wall are out-of-plane modes (blue markers). Again, the upper region shows clean out-of-plane modes. The additional branch in this bandgap corresponds to the modes at the other ends of the supercell, and therefore is not of any important considerations here. The lower region, however, shows weaker out-of-plane localized modes, and those are intermixed with other in-plane modes (red and yellow markers). This fact leads us to understand the difference of topological wave localization between PBG and CBG. The advantage of having the CBG is evident as it facilitates a clean wave localization at the domain wall and eliminates the possibility of wave leakage into other modes. Note in passing that there is a small gap at the k_// = 0 point in both regions, where there is no localization at the domain wall. As highlighted in the previous studies [26, 33], this is due to the breakage of the C₆ symmetry at the domain wall. By employing engineering tricks, such as a slowly-varying (i.e. gradient) domain wall, one can also reduce this small gap at the k_// = 0 point and impart greater protection to the topological modes.

We proceed to investigate the mode shapes of the aforementioned localized modes. We plot out-of-plane displacements at selected frequencies marked as circles in figure 3(B). Figures 3(D), (E) correspond to the frequencies inside PBG, and figures 3(F), (G) correspond to the frequencies inside CBG, all arranged in the order of increasing frequencies. It is clear that these modes are localized at the domain wall. Insets show zoomed-in views of the mode shapes on which the superimposed arrows indicate in-plane time-averaged mechanical energy flux (${I}_{j}=-{\sigma }_{{ij}}{v}_{j}$, where σ_ij and v_j are stress tensor and velocity vector, respectively) over a harmonic cycle. It is evident that these modes have spin characteristics. More specifically, in figures 3(D), (F) that correspond to the cyan markers in figure 3(B), we observe clockwise spins. In figures 3(E), (G) (corresponding to the green markers in figure 3(B)), we witness counterclockwise spins (see the lattices with smaller R for clearer visualization). As the time-reversal symmetry is intact in our system, it is obvious that we will have exactly the opposite spins at these frequencies for a negative wavevector k_//. Therefore, judging from the group velocity, i.e. the slope of the dispersion branch at the green circles in figure 3(B), we expect that the counterclockwise spin propagates along the domain wall towards positive wavevector, while the clockwise spin propagates in the opposite direction. We will look for these spins in the experimental results later.

Building from this numerical result, we experimentally verify the existence of topological modes at the domain wall for both PBG and CBG regions. In figure 4, we show the evolution of steady-state transmission plots on the plate (see movie 2 for more plots is available online at stacks.iop.org/NJP/20/113036/mmedia), as we inject vibrational energy through a piezoelectric actuator attached at the center of the domain wall and increase its input frequency via a sweep signal. At low excitation frequencies (figures 4(A)–(C)), we observe that energy is primarily localized at the domain wall. This waveguiding pattern detected in this low-frequency range corresponds to the PBG (see the dispersion relationship in figures 3(B), (C)). We observe some traces of elastic energy far away from the domain wall. This can be attributed to the nature of this PBG as it allows other wave modes to co-exist in the system. In addition, the smaller size of this bandgap would generally mean higher localization length of the modes at the domain wall, and hence, the flexural wave would penetrate more into the bulk. Beyond this PBG, the wave localization is lost, marking the presence of global modes (extended in the plate as shown in figure 4(D)). At 7.33 kHz (figure 4(E)), the wave is highly attenuated around the point of excitation.

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Remarkably, for the frequencies above this point, we again observe wave localization along the domain wall (figures 4(F)–(H)). This corresponds to the localization region in the CBG as indicated in figures 3(B), (C). We clearly observe a persistent wave localization along the horizontal domain wall for a wide range of frequencies. Among these, the transmission at 7.70 kHz (figure 4(G)) is robust, in which the wave follows the entire domain wall without any significant back-scattering around the sharp bends.

It is worth pointing out that at 7.51 and 7.98 kHz (figures 4(F) and (H)), the wave back-scatters around the sharp bends. A similar behavior was also observed recently in experiments on a mechanical counterpart of quantum valley Hall effect [35], where guided topological waves are back-scattered around a sharp bend for some frequencies inside the topological bandgap. However, here we observe that the wave is able to pass through the left bend at a lower frequency (figure 4(F)) and through the right bend at a higher frequency (figure 4(H)). This implies the preference of the topological wave propagation around one bend over the other, depending on the excitation frequency in the CBG. We explain this mechanism by the asymmetry of the two bends experienced by the spin waves. That is, as marked by the circular arrows in figure 4(H), the bending angle experienced by the counterclockwise spin that propagates towards the right is different from that by the clockwise spin towards the left. Note that in this symmetry-protected topological system, we have intentionally broken the C₆ symmetry at the domain wall, including the sharp bends. This makes the modes less 'protected' (i.e. less robust), thereby resulting in the spin-dependent transmission efficiency around the bends (see the transient wave propagation in movie 3 for its further verification is available online at stacks.iop.org/NJP/20/113036/mmedia).

Now we demonstrate robust waveguiding capabilities of our system by capturing the transient wavefield when excited at the frequencies inside PBG and CBG. Navigating the steady-state profiles observed in the previous section, we choose two central frequencies of excitation, 6.3 kHz (in PBG) and 7.7 kHz (in CBG), and send Gaussian pulses from the piezoelectric actuator. In figures 5(A)–(C), we show transient wave propagation on the plate at three different time steps inside PBG. We observe that the wave, emanating from the point of excitation at the middle of the domain wall, is effectively guided through the domain wall. Here, we verify that a counterclockwise spin propagates towards the right and a clockwise spin propagates towards the left (see movie 4 is available online at stacks.iop.org/NJP/20/113036/mmedia). Note in figure 5(C) that there is some visible leakage into the bulk, complying well with the result from the steady-state response (figure 4(B)). In figures 5(D)–(F), we show transient wave propagation inside CBG. It is evident that this wave is also being guided along the domain wall without any significant back-scattering at the sharp bends (see movie 5 is available online at stacks.iop.org/NJP/20/113036/mmedia). However, the key difference is that there is no evident leakage into the bulk due to the presence of CBG (figure 5(F)). This result is consistent with the supercell analysis (as shown in figure 3(B)) and the steady state response (figure 4(G)). In appendix C, we discuss a way to quantify the wave transmission efficiency in these two cases.

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