Active control on topological immunity of elastic wave metamaterials


The topology concept in the condensed physics and acoustics is introduced into the elastic wave metamaterial plate, which can show the topological property of the flexural wave. The elastic wave metamaterial plate consists of the hexagonal array which is connected by the piezoelectric shunting circuits. The Dirac point is found by adjusting the size of the unit cell and numerical simulations are illustrated to show the topological immunity. Then the closing and breaking of the Dirac point can be generated by the negative capacitance circuits. These investigations denote that the topological immunity can be achieved for flexural wave in mechanical metamaterial plate. The experiments with the active control action are finally carried out to support the numerical design.


Phononic crystals and elastic wave metamaterials are artificial structures which are arranged periodically and have received lots of attention1,2,3,4,5,6,7,8. These new kinds of structures have many extraordinary properties, e.g. the wave band gaps9,10,11,12, negative refraction13,14, acoustic/elastic wave cloaks15,16,17, etc. These interesting behaviors can be used to tune the wave propagation properties and have a wide variety of potential engineering applications. Acoustic couplers18, sensors19 and waveguides20 are several typical applications of phononic crystals and elastic wave metamaterials.

In recent years, a lot of attention has been focused on the extraordinary transmission phenomena of acoustic/elastic waves near the Dirac point. The periodic structures with the Dirac point can present many interesting behaviors, e.g. carrier mobility21, zero refractive index22,23,24,25, Hall effect26 and topological edge states27,28,29,30, etc. It should be mentioned that in the photonic crystal, the spatial phase can be reconstructed by the degenerate Bloch modes at the Dirac point31. Moreover, the topological phase can also transform because of the band inversion32.

On the other hand, some investigations have been reported on the phononic crystals and elastic wave metamaterials with the active control action33,34,35,36,37. To present the controllable flexural waves with the broadband characteristics by the active control, an experiment using the shunted piezoelectric patches was reported38. Furthermore, another experiment was performed to show the tunable waveguide of the phononic plate39. Although the active control action has been applied on the waveguide structures, its tunable effects on the topological properties of elastic wave metamaterials have not been considered. In this work, we propose a tunable topological state of the flexural wave with the active control. Based on the electrical control action, the Dirac point and its corresponding topological immunity can be achieved.


In this work, we focus on elastic wave metamaterials with double Dirac points locating at the center of the Brillouin zone. A hexagonal unit cell in the elastic wave metamaterial plate is showed in Fig. 1(a). The cell is made of the resin and its six corners are attached with the piezoelectric material P-4 on both sides. The parameters are illustrated in Table 1, where a is the lattice constant, R is the radius of the piezoelectric patch, h1h3 are the thicknesses of the middle layer, groove and piezoelectric sheet, respectively. Each piezoelectric patch is connected by a negative capacitance circuit to behave as the active control load in Fig. 1(b). As shown in Fig. 1(c), the negative capacitance circuit is used in the active control system to change the elastic modules, in which the capacitance Cp, compensation resistance R0, operational amplifier, fixed resistances R1 and sliding rheostat R2 are considered. The derivation of the negative capacitance circuit is presented in Supplementary Material. Both the dispersion curves and propagation properties are calculated by the finite element software COMSOL. Our attention is focused on the topological immunity of the elastic wave in which the energy bands of the flexural modes are presented.

Figure 1

(a) A unit cell bonded by piezoelectric patches, (b) the piezoelectric patches attached by the negative capacitance circuits and (c) the negative capacitance circuit.

Table 1 The parameters of the unit cell.

Results and discussions

When the negative capacitance circuits are not connected, the band structure calculated by the finite element method is shown in Fig. 2(a) and we can see that the band gap width is about 35 Hz. In order to generate the Dirac point, the negative capacitance circuits are connected and the parameter of the resistance α = 0.9 in Eq.(A.4) is applied. The band structure with the Dirac point is shown in Fig. 2(b), in which a quadruple degeneracy at 2256 Hz can be observed. The displacement field distributions for the four degenerate states at the Dirac point are presented in Fig. 2(c).

Figure 2

(a) Band gap, (b) the Dirac point induced by the electric circuit and (c) its displacement fields of the four degenerate states at the Dirac point.

An interesting behavior is the extraordinary transmission at the Dirac cone. In this work, it is designed by the configuration and electrical tuning to show the immune property of the elastic wave to the object. Then the concept “topological immunity” is applied to present the propagation of the flexural wave around the circular defect. This phenomenon makes the active controllable structure can immunize to the object, which means that the metamaterial plate exhibits a robust property at a certain frequency. Then, we can design an elastic wave metamaterial plate as a new function to achieve the topological immunity of the flexural wave. This work demonstrates the similar physics to ref. 40 for elastic wave and an additional tuning approach. Due to both the P-symmetry and T-symmetry can be found, the Berry curvature and Chern number are zero. The periodic structure is excited by a shaker at the right boundary and the frequency response is considered.

Here we define the transmission coefficient as the ratio of the left to the right flexural displacements. From the transmission coefficient curve for the elastic wave metamaterial plate with a defect in Fig. 3(a), a peak with 0.91 at the frequency of 2377.9 Hz is found. At this frequency, the flexural waves are almost perfectly transmitted through the whole system. Furthermore, when the frequency of the incident wave is 2377.9 Hz which is close to the Dirac point, the phase change becomes almost invisible in Fig. 3(b). Although there is a weak reflection on the right side, the defect is undetectable from the left field. Generally speaking, 2377 Hz does not shift obviously compared to 2256 Hz. It is mainly because of the problems during the mesh generation and boundary condition setting. An important influence is that the band structure is calculated with the infinite periodic boundary condition but the transmission spectrum is calculated with finite periodic structures.

Figure 3

(a) Transmission coefficient, (b) wave propagation at 2377.9Hz with electrical circuits and (c) wave propagation without electrical circuits at 2377.9 Hz.

In order to support these numerical results, the transmission response of the metamaterial plate without external circuits is calculated and shown in Fig. 3(c). We can see that the transmission becomes low and the waveform is observably distorted, which means that the topological immunity is achieved by the active control. The tunable effects of the active control on the topological immunity are considered, in which the negative capacitance circuits are applied to generate the Dirac point. When the Dirac point appears in the band structure, the topological immunity in the elastic wave metamaterial plate can be found. At the same time, there is only one peak in the transmission coefficient curve, which corresponds to the Dirac point.

Then, the active control experiments are performed on the elastic wave metamaterial plate in which the P-4 piezoelectric patches are boned at the corners. The elastic wave metamaterial plate is fabricated by the 3D printing technology with 8000 synthetic resin, in which the Young’s modulus E = 2.5 GPa, density ρ = 1300 kg/m3 and Poisson’s ratio ν = 0.41. The material parameters of the P-4 piezoelectric patch are shown in Table2 and the operational amplifier LM324N is applied.

Table 2 The material parameters of the piezoelectric patch.

The experimental setup is illustrated in Fig. 4(a), in which the elastic wave metamaterial plate is connected by the negative capacitance circuits. The exciter and the four received points e–h are illustrated in Fig. 4(b). The plate and exciter are connected by a fixture with a high rigidity to generate the flexural wave. During the experiment, 4 × 9 unit cells are applied to the elastic wave metamaterial plate and the sample size in Fig. 4(b) is same as that in Fig. 3(b). On the other hand, the power supply depends on the operational amplifier which is shown in Fig. 4(c). The description of the circuit is listed in Table 3. Due to the limitation of experimental conditions, only the finite periodic structures can be used for to support the numerical simulation. Although there is a little difference between the experimental and the numerical results, similar topological immune phenomenon can also be found during the experiment.

Figure 4

Experimental setup: (a) The testing system, (b) the whole structure with the active control system and (c) the negative capacitance circuit.

Table 3 The parameters of the negative capacitance circuit.

Figures 5 and 6 show the experimental results, in which the solid lines correspond to the points a–d (wave source) and dotted lines represent points e–h (response). With the connecting circuits on the elastic wave metamaterial plate in Fig. 5, we can see that although there is a little difference between the exciting and receiving points, their responses are quite similar at 2247 Hz. According to both numerical and experimental results, the waveform keeps the same from the right to the left sides. It means that the topological immunity is realized by the active control action. Then, the experiment is performed for the periodic structure without active control systems at the same frequency. As shown in Fig. 6, we can obviously see that the responses of the points e–h are quite smaller than those of the points a–d. It denotes that the flexural wave signal cannot be received on the left side for the structure without the active control.

Figure 5

Responses of points a-h at 2247 Hz when the elastic wave metamaterial plate is attached by the active control system.

Figure 6

Responses of points a-h at frequency at 2247Hz when the elastic wave metamaterial plate is not attached by the active control system.


In this work, an elastic wave metamaterial plate with the active control system is design to illustrate the topological immune property. The band structure is calculated by the numerical simulation and the double Dirac point is achieved by the external negative capacitance circuits. Around the Dirac point, the transmission responses of the metamaterial plate with a defect are discussed. Experiments are performed with the 3D printing metamaterial plate being bonded by the external electrical circuits. The topological immunity is observed by the responses behind the defect and achieved by the control action.


  1. 1.

    Krodel, S. & Delpero, T. 3D auxetic microlattices with independently controllable acoustic band gaps and quasi-static elastic moduli. Advanced Engineering Materials 4, 357–363 (2014).

    Article  Google Scholar 

  2. 2.

    Qureshi, A., Li, B. & Tan, K. T. Numerical investigation of band gaps in 3D printed cantilever-in-mass metamaterials. Scientific Reports 6, 28314 (2016).

    ADS  CAS  Article  Google Scholar 

  3. 3.

    Li, B., Alamri, S. & Tan, K. T. A diatomic elastic metamaterial for tunable asymmetric wave transmission in multiple frequency bands. Scientific Reports 7, 6226 (2017).

    ADS  Article  Google Scholar 

  4. 4.

    Li, X., Chen, Y., Hu, G. & Huang, G. A self-adaptive metamaterial beam with digitally controlled resonators for subwavelength broadband flexural wave attenuation. Smart Materials and Structures 27, 045015 (2018).

    Article  Google Scholar 

  5. 5.

    Zhang, Y. F., Li, B., Zheng, Q. S., Genin, G. M. & Chen, C. Q. Programmable and robust static topological solitons in mechanical metamaterials. Nature Communications 10, 5605 (2019).

    ADS  CAS  Article  Google Scholar 

  6. 6.

    Park, H. W. & Oh, J. H. Study of abnormal group velocities in flexural metamaterials. Scientific Reports 9, 1–13 (2019).

    Article  Google Scholar 

  7. 7.

    Elmadih, W. et al. Three-dimensional resonating metamaterials for low-frequency vibration attenuation. Scientific Reports 9, 11503 (2019).

    ADS  CAS  Article  Google Scholar 

  8. 8.

    Li, S. B. et al. Designing a broad locally-resonant bandgap in a phononic crystals. Physics Letters A 382, 1371–1377 (2019).

    ADS  Article  Google Scholar 

  9. 9.

    Wu, Z. J., Li, F. M. & Zhang, C. Z. Vibration band-gap properties of three-dimensional Kagome lattices using the spectral element method. Journal of Sound and Vibration 341, 162–173 (2015).

    ADS  Article  Google Scholar 

  10. 10.

    Dong, H. W., Wang, Y. S. & Zhang, C. Z. Topology optimization of chiral phoxonic crystals with simultaneously large phononic and photonic bandgaps. IEEE Photonics Journal 9, 19430655 (2017).

    Google Scholar 

  11. 11.

    He, J. H. & Huang, H. H. Complete vibrational bandgap in thin elastic metamaterial plates with periodically slot-embedded local resonators. Archive of Applied Mechanics 88, 1263–1274 (2018).

    ADS  Article  Google Scholar 

  12. 12.

    Fu, C. Y., Xu, J. X., Zhao, T. F. & Chen, C. Q. A mechanical wave switch with tunable frequency output. Applied Physics Letters 115, 191902 (2019).

    ADS  Article  Google Scholar 

  13. 13.

    Croenne, C., Morvan, B., Vasseur, J., Dubus, B. & Hladky-Hennion, A. Analysis of elastic waves transmitted through a 2-D phononic crystal exhibiting negative refraction. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 58, 178–186 (2011).

    Article  Google Scholar 

  14. 14.

    Zhu, R., Liu, X. N., Hu, G. K., Sun, C. T. & Huang, G. L. Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nature Communications 5, 5510 (2014).

    ADS  CAS  Article  Google Scholar 

  15. 15.

    Torrent, D. & Sánchez-Dehesa, J. Acoustic cloaking in two dimensions: a feasible approach. New Journal of Physics 10, 063015 (2008).

    ADS  Article  Google Scholar 

  16. 16.

    Zhang, S., Xia, C. & Fang, N. Broadband acoustic cloak for ultrasound waves. Physical Review Letters 106, 024301 (2011).

    ADS  Article  Google Scholar 

  17. 17.

    Farhat, M. et al. A homogenization route towards square cylindrical acoustic cloaks. New Journal of Physics 10, 115030 (2008).

    ADS  Article  Google Scholar 

  18. 18.

    Herrmann, H., Schaefer, K. & Sohler, W. Polarization independent, integrated optical, acoustically tunable wavelength filters/switches with tapered acoustical directional coupler. IEEE Photonics Technology Letters 6, 1335–1337 (1994).

    ADS  Article  Google Scholar 

  19. 19.

    Lucklum, R., Zubtsov, M. & Ke, M. Liquid sensor utilizing a regular phononic crystal with normal incidence of sound. IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 59, 463–471 (2012).

    Article  Google Scholar 

  20. 20.

    Nishizawa, J. et al. THz generation from gap rod-type waveguides. IEEE Photonics Technology Letters 19, 143–145 (2007).

    ADS  CAS  Article  Google Scholar 

  21. 21.

    Li, F. et al. Benzene-like N6 rings in a Be2N6 monolayer: a stable 2D semiconductor with high carrier mobility. Journal of Materials Chemistry C 5, 11515 (2017).

    CAS  Article  Google Scholar 

  22. 22.

    Edwards, B., Alu, A., Young, M. E., Silveriinha, M. & Engheta, N. Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide. Physical Review Letters 100, 033903 (2008).

    ADS  Article  Google Scholar 

  23. 23.

    Moitra, P. et al. Realization of an all- dielectric zero-index optical metamaterial. Nature Photonics 7, 791–795 (2013).

    ADS  CAS  Article  Google Scholar 

  24. 24.

    Li, Y. et al. On-chip zero-index metamaterials. Nature Photonics 9, 738–742 (2015).

    ADS  CAS  Article  Google Scholar 

  25. 25.

    Dubois, M., Shi, C., Zhu, X., Wang, Y. & Zhang, X. Observation of acoustic Dirac-like cone and double zero refractive index. Nature Communications 8, 14871 (2017).

    ADS  CAS  Article  Google Scholar 

  26. 26.

    Hatsugai, Y., Fukui, T. & Aoki, H. Topological analysis of the quantum hall effect in graphene: Dirac-Fermi transition across van hove singularities and edge versus bulk quantum numbers. Physical Review B 74, 205414 (2006).

    ADS  Article  Google Scholar 

  27. 27.

    Huo, S. Y., Chen, J. J., Huang, H. B. & Huang, G. L. Simultaneous multi-band valley-protected topological edge states of shear vertical wave in two-dimensional phononic crystals with veins. Scientific Reports 7, 10335 (2017).

    ADS  Article  Google Scholar 

  28. 28.

    Yang, Y. H. et al. Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal. Nature Communications 10, 5185 (2019).

    ADS  Article  Google Scholar 

  29. 29.

    Jia, D. et al. Acoustic topological insulator by honeycomb sonic crystals with direct and indirect band gaps. New Journal of Physics 20, 093027 (2018).

    ADS  Article  Google Scholar 

  30. 30.

    Xia, J. P. et al. Programmable coding acoustic topological insulator. Advanced Materials 30, 1805002 (2018).

    Article  Google Scholar 

  31. 31.

    Huang, X., Lai, Y., Hang, Z. H., Zheng, H. & Chan, C. T. Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials. Nature Materials 10, 582–6 (2011).

    ADS  CAS  Article  Google Scholar 

  32. 32.

    Dai, H. Q., Xia, B. Z. & Yu, D. J. Dirac cones in two-dimensional acoustic metamaterials. Journal of Applied Physics 122, 065103 (2017).

    ADS  Article  Google Scholar 

  33. 33.

    Chen, S. B., Wen, J. H., Wang, G., Yu, D. L. & Wen, X. S. Directionality of wave propagation and attenuation in plates with resonant shunting arrays. Journal of Intelligent Material Systems and Structures 27, 28–38 (2016).

    Article  Google Scholar 

  34. 34.

    Wang, Y. Z., Li, F. M. & Wang, Y. S. Active feedback control of elastic wave metamaterials. Journal of Intelligent Material Systems and Structures 28, 2110–2116 (2017).

    CAS  Article  Google Scholar 

  35. 35.

    Cardella, D., Celli, P. & Gonella, S. Manipulating waves by distilling frequencies: a tunable shunt-enabled rainbow trap. Smart Materials and Structures 25, 085017 (2016).

    Article  Google Scholar 

  36. 36.

    Lossouarn, B., Deu, J. F. & Aucejo, M. Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network. Smart Materials and Structures 24, 115037 (2015).

    Article  Google Scholar 

  37. 37.

    Manuel, C., Morvan, O. & Mohamed, N. I. Structural energy flow optimization through adaptive shunted piezoelectric metacomposites. Journal of Intelligent Material Systems and Structures 23, 1661–1677 (2012).

    Article  Google Scholar 

  38. 38.

    Tateo, F., Collet, M., Quissen, M., Cunefare, K. & Abbe, P. Experimental characterization of a bi-dimensional array of negative capacitance piezo-patches for vibroacoustic control. Journal of Intelligent Material Systems and Structures 26, 952–964 (2015).

    Article  Google Scholar 

  39. 39.

    Casadei, F., Beck, B., Cunefare, K. A. & Ruzzene, M. Vibration control of plates through hybrid configurations of periodic piezoelectric shunts. Journal of Intelligent Material Systems and Structures 23, 1169–1177 (2012).

    Article  Google Scholar 

  40. 40.

    Chen, Z. G. et al. Accidental degeneracy of double Dirac cones in a phononic crystal. Scientific Reports 4, 4613 (2014).

    Article  Google Scholar 

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The authors acknowledge the supports by the National Natural Science Foundation of China (Grant Nos. 11772039 and 11991031), the Joint Sino-German Research Project (Grant No. GZ 1355) and the German Research Foundation (DFG, Grant No. ZH 15/27-1) for this research work.

Author information




Guan-Hua Li performed the numerical simulation and experiment with the cooperation of Tian-Xue Ma. Yi-Ze Wang discussed the results and developed the experimental model. Yue-Sheng Wang presented the helpful suggestions about the underlying mechanism and participated in the revision of the manuscript. All authors contributed to the writing and editing of the manuscript.

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Correspondence to Yi-Ze Wang.

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Li, G., Ma, T., Wang, Y. et al. Active control on topological immunity of elastic wave metamaterials. Sci Rep 10, 9376 (2020).

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