Cochlear outer hair cell bio-inspired metamaterial with negative effective parameters

Transcription

1 Appl. Phys. A (2016)122:525 DOI /s Cochlear outer hair cell bio-inspired metamaterial with negative effective parameters Fuyin Ma 1 Jiu Hui Wu 1 Meng Huang 2 Siwen Zhang 1 Received: 31 May 2015 / Accepted: 26 January 2016 Springer-Verlag Berlin Heidelberg 2016 Abstract Inspired by periodical outer hair cells (OHCs) and stereocilia clusters of mammalian cochlear, a type of bio-inspired metamaterial with negative effective parameters based on the OHC structure is proposed. With the structural parameters modified and some common engineering materials adopted, the bio-inspired structure design with length scales of millimeter and lightweight is presented, and then, a bending wave bandgap in a favorable low-frequency with width of 55 Hz during the interval or 116 Hz during the interval Hz is obtained, i.e., the excellent low-frequency acoustic performance turns up. Compared with the local resonance unit in previous literatures, both the size and weight are greatly reduced in our bio-inspired structure. In addition, the lower edge of low-frequency bandgap is reduced by an order of magnitude, almost to the lower limit frequency of the hearing threshold as well, which achieves an important breakthrough on the aspect of low-frequency and great significance on the noise and vibration reduction in lowfrequency range. & Jiu Hui Wu ejhwu@mail.xjtu.edu.cn Fuyin Ma mafuyin@163.com 1 2 School of Mechanical Engineering and State Key Laboratory for Strength and Vibration of Mechanical Structure, Xi an Jiaotong University, Xi an 71009, China State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing , China 1 Introduction Natural life has evolved through billions of years to complete all the process of intelligent manipulation, learning from the nature, concluding bionic and bio-inspire, which have become the forever theme for material design [1]. Through a bionic and bio-inspired design, we could design the advanced materials with both smartness as the biological tissues and mechanical performance exceeding the natural biological materials [2]. At present, bionic material design has become the hot topic for advanced material design [3 5]. In recent years, bionic material design concept was used in mechanical and optical material design [6 8]. In addition, a cochlear bionic acoustic metamaterial design concept was introduced in previous work [9]. Because the acoustic metamaterials have important applications in low-frequency noise and vibration reduction, numerous researches have been conducted by scholars and discovered a number of special physical phenomena, such as the negative parameters [10 17]. The acoustic metamaterials are developed from the phononic crystal. Because of the existence of mechanical bandgaps, the phononic crystals have potential applications in vibration and noise control [18]. The main mechanisms responsible for the mechanical bandgaps creation are based on Bragg scattering and local resonance [18 22], wherein the local resonance mechanism was first introduced by Liu et al. [22]. Such behaviors have also been studied by other researchers [19, 20, 23 28]. Recently, a two-dimensional thin plate structure with local resonance was studied by Hsu and Wu [19] and Xiao et al. [21]. Wu et al. [29] and Pennec et al. [30] have reported independently on the elastic behavior of a stubbed plate. And a membrane-type acoustic metamaterial with dynamic negative mass [10] has been proposed to realize favorable sound absorption ability

2 525 Page 2 of 8 F. Ma et al. [31], which is of the low-frequency broad sound black hole ranging between 100 and 1000 Hz. An important application of acoustic metamaterials is to realize the noise and vibration reduction at the deep subwavelength low-frequency. Previous elastic membranetype acoustic metamaterial provides very excellent solution for low-frequency sound insulation and absorption [10, 17, 31]. However, since the film thickness of these structures is small, the sound transmission loss (STL) or absorption coefficient is sensitive to the film tension, which is very difficult to be controlled for the reason that very slight tension differences could result in a large frequency deviation. In our previous studies, we have reduced the tension dependence by increasing the film thickness [25, 26]. In addition, we also proposed a type of one-dimensional (1D) rigid film-type acoustic metamaterials to completely eliminate the tension dependence [32]. In order to achieve the sound attenuation in lower frequency and realize more stable acoustic performance and higher STL amplitudes, this study employs a bio-inspired material design method to obtain the advanced acoustic metamaterials for lowfrequency noise and vibration reduction in the range of below 200 Hz. 2 OHC structure and function In the cochlear, there are OHCs periodically distributed on the basilar membrane (BM), and periodic stereocilia clusters on these OHCs. The active motility of stereocilia clusters enables the cochlea to yield a large gain by interacting with pressure waves. And simultaneously a periodic functional structure to modulate mechanical waves is constituted by the periodic OHCs and the attached stereocilia clusters. In fact, previous literature experimentally has indicated that the OHC manifests one metamaterial constitutive characteristic, i.e., the negative stiffness [33]. Since the tiny cochlea biologic structure has some amazing abilities to modulate the mechanical waves, it is expected that through a bio-inspired design, this type of structure with special mechanical and acoustic characteristics can be introduced into the low-frequency noise and vibration reduction. This paper puts forward the bio-inspired acoustic metamaterial design concept based on OHC structure. The structures with cochlea and BM are shown in Fig. 1a. The BM and the complicatedly structured Corti organ attached onto the membrane in scala media are the major structures to fulfill the auditory function, wherein details of the latter are shown in Fig. 1b, principally including hair cells (HCs), supporting cell, reticular lamina (RL), tectorial membrane (TM) and pillar cells (PCs) [9, 34]. In detail, the HCs consist of inner hair cells (IHCs) and OHCs, and the supporting cells are several types surrounding the hair cells, such as the inner phalangeal cells (IPCs), the outer phalangeal cells (OPCs), Hensen s cells and Claudius cells. Moreover, there are three rows of V-shaped clustered stereocilia distributed on each OHC, in which each row is different in the setting angles and lengths, and the structures are detailed in Fig. 1c, d. Inspired by the periodic OHCs mentioned above and the attached stereocilia clusters with negative stiffness, a bioinspired OHC structure is proposed, which is of millimetric length scales. The basic unit of this bio-inspired OHC metamaterial structure consists of three steel V-shape plates with different length and setting angles (clustered stereocilia) attached to a square elastic rubber membrane (BM), a rubber cylinder (combined with the OHC and OPC) with two conical transitionary segments as the cilium rootlet, connecting the BM with another square elastic rubber membrane (RL), and a plastic frame attached to one side of the BM (Fig. 2a). The RL and the BM share the same rubber material parameters, so did the OHC and the OPC. The lattice constant, the thickness of the BM and the RL are denoted with a = 10 mm, t BM = 0.5 mm and t RL = 0.5 mm, respectively. Meanwhile, the OPC and OHC are connected with a single cylinder with the radius of 3 mm and the length of 20 mm, and there are two conical transitionary segments as the cilium rootlet, where the three different lengths of stereocilia from long to short are l B1 = 6 mm, l B2 = 2.8 mm and l B3 = 1.4 mm, respectively, and the thickness is t B = 1 mm. Additionally, the thickness of the frame and the inner length of the flank are 1 and 9 mm, respectively. In Fig. 2a, the existence is to result in that the structure becomes asymmetrical. For convenience, we have also built a bio-inspired structure without stereocilia shown in Fig. 2b. 3 Simplified theoretical model for negative effective parameters Negative effective parameters are the most typical physical properties of acoustic metamaterials, which have been developed into a mature theory. In order to illustrate the negative effective parameters of the proposed OHC bionic structures, we start by considering a simple but heuristic dissipation-free mass spring mass model, as shown in Fig. 3a. Based on the Newton s second law, the kinetic equation of the model reads M x 1 Kx ð 2 x 1 Þ ¼ F ð1þ m x 2 þ Kx ð 2 x 1 Þ ¼ 0 where x 1 and x 2 are the displacements of the masses M and m, respectively. Assuming x 1, x 2 and F vary time-harmonically with the angular frequency x, we have

3 Cochlear outer hair cell bio-inspired metamaterial with negative effective parameters Page 3 of 8525 Fig. 1 a The structure of the cochlear and BM; b the detailed structure of the Corti organ; c clustered stereocilia of the OHCs; d the structure of the HC and stereocilia x 2 mx 2 þ Kx ð 2 x 1 Þ ¼ 0 ð2þ from which we can obtain x 2 ¼ x2 0 x 2 0 x x2 1 ð3þ p where x 0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffi K=M 2. Eliminating x 2 from Eq. (1) yields K F ¼ M þ x 1 ð4þ x 2 0 x2 Given that observers can only see the exterior, the apparent inertia of the entire system becomes a function of frequency as follows K M eff ¼ M þ x 2 ð5þ 0 x2 where M eff is the effective mass of the system. The function also could be written as: M eff ¼ M þ mx2 0 x 2 ð6þ 0 x2 For the vibration modes in the through-thickness direction (Z modes), the basic unit structure of Fig. 2b could be simplified to a theoretical vibration model with two parallel springs and a mass (Fig. 3b), wherein the RL and the BM provide the stiffness, respectively, i.e., K RL and K BM, OHC and OPC jointly provide the inner equivalent mass m e, and the frames provide the outer equivalent mass M. Since the bio-inspired unit in Fig. 2b is symmetrical, both the inner and the outer equivalent masses could be calculated directly, which make the m e and M approximately equating to and kg, respectively. And the first-order resonant frequency of this equivalent system could be calculated by numerical simulation (the detailed information will be introduced in Sect. 4, i.e., the lower edge of the shaded region in Fig. 4d), where x 0 is close to 57 Hz. The total stiffness of the parallel springs could be written as that K = K RL? K BM. Additionally, based on the theory given in the previous literature [11], the dynamic effective mass M eff and stiffness K eff could be solved by the following two formulas:

4 525 Page 4 of 8 F. Ma et al. reported by the literature [33], where the mechanism mainly depends on a special gated stiffness characteristic of the OHCs, rather than the local resonance. 4 Band structures and formation mechanisms of the bending wave bandgaps Fig. 2 a The bio-inspired OHC structure; b the bio-inspired structure without stereocilia M eff ¼ M þ m ex 2 0 x 2 ð7þ 0 x2 and K K eff ¼ 1 þ m ex 2 0 Mðx 2 0 x2 Þ ð8þ Previous studies suggest that Eq. (7) is right, and by substituting the above parameter values into Eq. (7), the effective mass changing with the frequency can be obtained and shown in Fig. 3c. From the figure it can be clearly seen that there exist negative mass characteristic in the equivalent system. The formation mechanism of negative parameter in Fig. 3c is completely different from that By using the COMSOL Mutiphysics software, the band structure of the OHC unit along the edge of first Brillouin zone is solved. Then, a local resonance bending wave bandgap is obtained, as the shaded regions shown in Fig. 4a, b. From the figures, it could be obtained that there exists a bending wave bandgap within the frequency band lower than 100 Hz, the width of which is 55 Hz from 21 to 76 Hz (i.e., from point A1 (the lower edge) to point F1 (the upper edge), which were the Z modes along membrane thickness direction, as shown in Fig. 5), and a wider bandgap should be achieved through further parameter modifications. The shaded regions in Fig. 4a, b, d represent the frequency ranges of bending wave bandgaps, which means that the bending wave vibration mode (Z mode) is absent in these ranges [23, 25, 26]. Figure 4a shows that compared with the traditional local resonance unit, the width of the bandgaps in the low-frequency range has been effectively expanded. The band structure of the unit without steel stereocilia is also calculated, which suggests that the bending wave bandgap locates in the range of Hz, and the vibration modes in the points of Fig. 4a coincide with the unit with steel stereocilia. In addition, we also design a structure with the same total mass but only a uniform cylinder (that with the radius of 3.5 mm and the length of 20 mm shown in Fig. 2b), whose band structure is shown in Fig. 4d. Comparing the band structures of Fig. 4a, d, it is suggested that the pure rubber structure has the wider bandgap from 57 to 173 Hz than that of the structure with steel stereocilia. However, the lower edge of bandgap increases from 21 to Fig. 3 a A simple mass spring mass model; b the simplified theory model of the OHC bio-inspired unit; c the calculated effective mass

5 Cochlear outer hair cell bio-inspired metamaterial with negative effective parameters Page 5 of 8525 Fig. 4 a The band structure of the bio-inspired structure; b the enlarged band structure near the point C of (a), wherein the horizontal axis denotes the reduced wave vector, the total width between the point C and point X has been normalized to 1, and hereby only the variation in the interval (0, 0.2) is demonstrated; c the first Brillouin zone; d the band structure of the bio-inspired structure without stereocilia (Fig. 2b). For the OHC bioinspired structures, Young s modulus, Poisson s ratio and density of rubber are MPa, and 1300 kg/m 3, respectively; those of plastic are 220 MPa, and 1190 kg/m 3, respectively; and those of steel are 206 GPa, 0.33 and 7850 kg/m 3, respectively 57 Hz, which means that the stereocilia plays the roles of the additional mass to expand the width of the bandgap and the distinct contrast of density between the top steel and rubber cylinder to reduce the lower edge frequency of bandgap. The formation mechanism of the bending wave bandgap has been theoretically and experimentally discussed and verified in our previous research [23, 25, 26], wherein the unit cells of are similar to those of Fig. 2b [25, 26]. Both vibration reduction [23] and sound insulation [25, 26] test results suggest that the coupled components between the XY modes and the Z modes within the bending wave bandgap could not hinder the formation of the bending wave bandgap. In order to reveal the mechanism of the bandgap for the proposed OHC bionic unit, we have diagrammed the mode shapes of some marked points of Fig. 4a in Fig. 5. A1 isthe bending mode of the membrane at point M, where the OHC, OPC and stereocilia translate along Z direction and drive the BM and RL to bend, while the frame remains stationary. B1 and C1 are the swing modes at point C, where the OHC, OPC and stereocilia swing in the XZ and YZ planes, respectively. Similarly, D1 ande1 are the bending modes of the OHC, OPC and stereocilia at point C, where the OHC, OPC and stereocilia bend in the XZ and YZ planes, respectively. F1 is the bending mode of the membrane at point C, where the frame translates along Z direction and drives the BM to bend, while the rest remain stationary. G1andH1are also the bending modes of the OHC and OPC at point M in XZ and YZ planes, respectively. The figures indicate that the bandgap of Fig. 4a is generated by the two Z modes, i.e., modes A1 and F1. Because of the reverse vibration between the mass and the frame, a characteristic of the negative dynamic mass has shown up, which can insulate the mechanical wave propagation in the through-thickness direction of the membrane. In Fig. 4a, two flat bands (B and C) located in the middle of the bandgap correspond to the structure translational vibration modes in the horizontal and vertical directions. Due to the symmetry of cell structure, the horizontal and vertical vibration modes are equivalent to each other, and the two modes generate a band degeneracy in the range of the reciprocal lattice vector space far away from point C. The structure has the same dynamic features under this swing resonant mode, the cylindrical mass block drives the membrane movement, and the surrounding frames almost remain stationary (G1 and H1), which indicates that the frame can be considered as a rigid base to completely localize the inner vibration pattern by unit. Meanwhile, when the frame and the thin-shell parts are connected to each other, the vibration of the frames and plates could not be activated.

6 525 Page 6 of 8 F. Ma et al. Fig. 5 The modes of some key points of Fig. 4a, b. In the top row, the red color represents the largest displacement, and the blue color represents the smallest displacement; in the bottom row, the black outlines represent the simplified structure, and the red outlines represent the deformation Due to the presence of the local resonance and its interaction with the traveling wave in the membrane, the dispersion curves for the original bending wave of the membrane (denoted as A0 in Fig. 4a) and the straight lines representative of the local resonance (denoted as B and C in Fig. 4a, respectively) cut off each other, followed by a bending wave bandgap. For the local resonant mode A and the antisymmetric Lamb wave (A0 mode) in band structure of Fig. 4a, the modal shapes are shown in Fig. 5 (denoted as A1 and F1, respectively). Due to their polarization directions along the through-thickness direction of the membrane (i.e., the Z direction), mode A can be easily excited by the antisymmetric Lamb wave propagation in the membrane, and there is a strong coupling among them. When the corresponding lines intersect with each other, they could be mutually exclusive and truncated in the band structure. As a result, a bandgap (i.e., the bending wave bandgap, which has been marked in Fig. 4a) upon the straight lines is generated along the through-thickness direction of the membrane. Comparison between Fig. 3c and Fig. 4d suggests that a negative mass characteristic is obtained within the bending wave bandgap. And this formation mechanism of negative mass characteristic can also be described by the simple theory model in Fig. 3b. When the mechanical wave transmits to this unit, a force F will be produced and applied to the base of the unit. Meanwhile, the vibration of mass block m e will produce a reaction force f to the base of the unit, which will vibrate under the external excitation F and internal reaction f. When the frequency of the external excitation approaches closely to the natural frequency of the internal unit, the unit will resonate. Since the frequencies are close to each other, f and F always are superposed reversely or offset completely. The composition of forces applied to the base will approach to zero, and thus, the vibration could not spread in the unit, which results in the formation of a bandgap. The frequency of resonance bandgap is dominated by the natural frequencies of the internal unit, and for the vibration system of Fig. 3b, it could be estimated by the following equation: f ¼ 1 2p rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K RL þ K BM m e ð9þ where m e = kg. Not only will the equivalent stiffness K RL and K BM be determined by material modulus and thickness of the RL and BM, but they will also be influenced by the vibration mode and boundary conditions, which is difficult to be solved directly. However, the first resonance frequency f 1 (Z mode) in Eq. (9) equals to 57 Hz solved by numerical simulation, and then, Eq. (9) makes the equivalent stiffness K RL = K BM & 128 N/m. It is noted that the simple theory

7 Cochlear outer hair cell bio-inspired metamaterial with negative effective parameters Page 7 of 8525 model in Fig. 3b is only suitable for Z modes, but not for other type of local resonance modes. 5 Sound insulation property and potential applications In practice, the structures have been always used in the air for sound insulation, so the calculation of STL for the bionic OHC structure is necessary, which could take into account the interactions between the units and the air. Previous literatures have suggested that for the local resonance structures, the vibration property should be mainly determined by the unit structure rather than the periodicity [10, 17]. The bionic OHC structure is relatively complex with the consideration of the calculation of the periodicity array, and we mainly focus on the mechanical characteristic in the through-thickness direction of the membrane, which is of no periodicity unit. Therefore, only a cell unit with fixed boundaries and the surrounding air domain can be considered in such a calculation (Fig. 6a). The calculated STL curve of the unit without steel stereocilia in the range from 20 to 240 Hz has been shown in Fig. 6b. From the figure, it is suggested that at the lower bound of the bandgap in the band structure (at 57 Hz), the STL curve exhibits a sound insulation dip characteristic. And after entering the bandgap frequency ranges, the STL begins to increase. The results show that the bionic OHC structures proposed in this study have excellent sound insulation ability. The average STL is approximately 70 db in the extremely low frequency range below 200 Hz. Employing such lightweight rubber structures could get excellent sound insulation ability, which provides a strong support for the potential application of bionic acoustic metamaterials from simulation perspective. Compared with the previous membrane-type acoustic metamaterials [10, 17, 31], tension is not necessary in these OHC bionic structures, the sound insulation property appears more stable, and the sound insulation amplitudes are significantly improved. Analyzing the band structure of Fig. 4d and the STL curve of Fig. 6b, it is suggested that a sound insulation dip feature has been exhibited at the lower boundary of the bending wave bandgap in the band structure. But after entering the bandgap, the STL began to increase drastically, indicating the effect of the bandgap. In addition, it is also seen that at the upper edge of the bending wave bandgap (approximately 173 Hz), the simulated STL amplitude reaches to the maximal value 80 db, equivalent to that % of the incident sound waves cannot penetrate through this bio-inspired structure. In other words, in the bending wave bandgap, the structure exhibits very excellent sound insulation capacity. In our previous studies, such relationship has been experimentally validated [25, 35]. Considering that in the thickness direction the size is limited by many factors in practical applications, the structural isolation capability for mechanical waves is to be mainly emphasized. Compared with the local resonance unit proposed in previous literatures, such bio-inspired structure has an average density much smaller than that of typographic (Pb) or steel; so the weight can be dramatically reduced. Moreover, this structure has the same millimeter Fig. 6 a The bionic OHC unit without steel stereocilia for sound insulation calculation; in this model, the boundaries of the frames and membranes have been fixed. In addition, the plane wave radiation boundary conditions are applied on both the incidence and transmission surfaces, wherein a unit sound pressure (1 Pa) is applied on the incidence surface. b The STL curves of the bionic structures; the black solid curve denotes the STL of the structure without steel stereocilia, and the red solid curve denotes the STL of the structure with steel stereocilia

8 525 Page 8 of 8 F. Ma et al. size level as the aforementioned local resonance acoustic metamaterials, whereas the frequency of the bandgap is reduced to the lowest level (21 76 Hz) within the hearing threshold, even lower than that of the membrane-type metamaterial. 6 Conclusions This work considers the periodical stereocilia clusters in the cochlea as a type of periodic acoustic metamaterials, and the vibration characteristics are analyzed through the band structure. Then, the design concept of bio-inspired acoustic metamaterials is proposed, and meanwhile, an extremely low-frequency bandgap results from the bio-inspired structure with millimetric length scale and light weight, exhibiting excellent acoustic performance. In addition, it is revealed through an equivalent parallel springs system that such cochlear OCH bio-inspired structure has the characteristics of negative parameter features, which enriches the theoretical system of acoustic metamaterials and provides a new concept to design materials and devices with more excellent acoustic performance. Compared with the local resonance of acoustic metamaterials in previous literatures, such a bio-inspired structure with smaller size and lighter weight achieves a lower bandgap within the hearing threshold, which makes a significant breakthrough in the low-frequency acoustical performance realized. The design concept proposed in this paper could provide an effective method for the acoustic metamaterials to obtain broad bandgaps in low frequency and also have a great number of potential applications in the low-frequency vibration and noise reduction. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No Thanks to the China Digital Science and Technology Museum for providing the open figures for Fig. 1 of this paper. We also thank the reviewer given some instructive suggestions and encourages helping us to improve the paper in depth. References 1. T. Douglas, Science 299, 1192 (2003) 2. T.A. Taton, Nat. Mater. 2, 73 (2003) 3. X.F. Gao, L. Jiang, Nature 432, 36 (2004) 4. P.W.K. Rothemund, Nature 440, (2006) 5. G. Chin, Science 303, 287 (2004) 6. J. Li, L. Bai, Nat. Nanotechnol. 7, (2012) 7. J. Lee, S. Peng, D. Yang, Y.H. Roh, H. Funabashi, N. Park, E.J. Rice, L. Chen, R. Long, M. Wu, D. Luo, Nat. Nanotechnol. 7, (2012) 8. K.L. Young, M.B. Ross, M.G. Blaber, M. Rycenga, M.R. Jones, C. Zhang, A.J. Senesi, B. Lee, G.C. Schatz, C.A. Mirkin, Adv. Mater. 26, (2013) 9. F. Ma, J. Wu, M. Huang, G. Fu, C. Bai, Appl. Phys. Lett. 105, (2014) 10. Z. Yang, J. Mei, M. Yang, N. Chan, P. Sheng, Phys. Rev. Lett. 101, (2008) 11. J. Li, K.H. Fung, Z.Y. Liu, P. Sheng, C.T. Chan, Generalizing the concept of negative medium to acoustic waves (Springer, Berlin, 2007) 12. N. Fang, D. Xi, J. Xu, M. Ambati, W. Srituravanich, C. Sun, X. Zhang, Nat. Mater. 5, (2006) 13. S.H. Lee, C.M. Park, Y.M. Seo, Z.G. Wang, C.K. Kim, Phys. Rev. Lett. 104, (2010) 14. L. Hao, C. Ding, X. Zhao, Appl. Phys. A 106, (2012) 15. X. Zhou, G. Hu, Appl. Phys. Lett. 98, (2011) 16. Z. Liang, J. Li, Phys. Rev. Lett. 108, (2012) 17. Z. Yang, H.M. Dai, N.H. Chan, G.C. Ma, P. Sheng, Appl. Phys. Lett. 96, (2010) 18. M. Oudich, Y. Li, B.M. Assouar, Z. Hou, New J. Phys. 12, (2010) 19. J.C. Hsu, T.T. Wu, Appl. Phys. Lett. 90, (2007) 20. Y. Chen, G. Huang, X. Zhou, G. Hu, C.T. Sun, J. Acoust. Soc. Am. 136, (2014) 21. Y. Zhang, J. Wen, Y. Xiao, X. Wen, J. Wang, Phys. Lett. A 376, (2012) 22. Z. Liu, X. Zhang, Y. Mao, Y. Zhu, Z. Yang, C. Chan, P. Sheng, Science 289, (2000) 23. S. Zhang, J. Wu, Z. Hu, J. Appl. Phys. 113, (2013) 24. H. Larabi, Y. Pennec, B. Djafari-Rouhani, J.O. Vasseur, Phys. Rev. E 75, (2007) 25. F. Ma, J.H. Wu, M. Huang, W. Zhang, S. Zhang, J. Phys. D Appl. Phys. 48, (2015) 26. F. Ma, J.H. Wu, M. Huang, Eur. Phys. J. Appl. Phys. 71, (2015) 27. H. Shen, M.P. Paidoussis, J. Wen, D. Yu, L. Cai, X. Wen, J. Phys. D Appl. Phys. 45, (2012) 28. H. Tian, X. Wang, Y. Zhou, Appl. Phys. A 114, (2014) 29. T.T. Wu, Z.G. Huang, T.C. Tsai, T.C. Wu, Appl. Phys. Lett. 93, (2008) 30. Y. Pennec, B. Djafari-Rouhani, H. Larabi, O.J. Vasseur, A.-C. Hladky-Hennion, Phys. Rev. B 78, (2008) 31. J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Nat. Commun. 3, 1758 (2012) 32. F. Ma, J.H. Wu, M. Huang, J. Phys. D Appl. Phys. 48, (2015) 33. P. Martin, D.A. Mehta, J.A. Hudspeth, Proc. Natl. Acad. Sci. U.S.A. 97, (2000) 34. S.J. Elliott, C.A. Shera, Smart Mater. Struct. 21, (2012) 35. L. Shen, J. Wu, Z. Liu, G. Fu, Int. J. Mod. Phys. B 29, (2015)