Manipulation of elastic waves by zero index metamaterials

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Nov 24, 2014 - Citation: Journal of Applied Physics 116, 204501 (2014); doi: 10.1063/1.4902065 ... [This article is copyrighted as indicated in the article. Reuse ...
Manipulation of elastic waves by zero index metamaterials Ziyu Wang, Wei Wei, Ni Hu, Rui Min, Ling Pei, Yiwan Chen, Fengming Liu, and Zhengyou Liu Citation: Journal of Applied Physics 116, 204501 (2014); doi: 10.1063/1.4902065 View online: http://dx.doi.org/10.1063/1.4902065 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Flat lens for pulse focusing of elastic waves in thin plates Appl. Phys. Lett. 103, 071915 (2013); 10.1063/1.4818716 Light amplification in zero-index metamaterial with gain inserts Appl. Phys. Lett. 101, 031907 (2012); 10.1063/1.4737643 Negative refraction experiments with guided shear-horizontal waves in thin phononic crystal plates Appl. Phys. Lett. 98, 011909 (2011); 10.1063/1.3533641 Numerical analysis of negative refraction of transverse waves in an elastic material J. Appl. Phys. 104, 064906 (2008); 10.1063/1.2978379 Long wavelength propagation of elastic waves in three-dimensional periodic solid-solid media J. Appl. Phys. 101, 073515 (2007); 10.1063/1.2715582

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JOURNAL OF APPLIED PHYSICS 116, 204501 (2014)

Manipulation of elastic waves by zero index metamaterials Ziyu Wang,1,2,3 Wei Wei,4 Ni Hu,1,5 Rui Min,1,5 Ling Pei,1,5 Yiwan Chen,1,5 Fengming Liu,1,5,a) and Zhengyou Liu2 1

School of Science, Hubei University of Technology, Wuhan 430068, China School of Physics and Technology, Wuhan University, Wuhan 430072, China 3 Materials and Technology Institute, Dongfeng Motor, Wuhan 430056, China 4 Hubei Cancer Hospital, Wuhan 430079, China 5 Hubei Collaborative Innovation Center for High-efficiency Utilization of Solar Energy, Hubei University of Technology, Wuhan, 430068, China 2

(Received 8 September 2014; accepted 5 November 2014; published online 24 November 2014) In this work, we investigated anti-plane transverse elastic waves transmission through a zero index metamaterials (ZIM) waveguide embedded with defect. Theoretical analysis and numerical simulations show that total transmission and total reflection of the impinging transverse elastic waves can be achieved by simply adjusting the parameters of the defect, and a ZIM waveguide embedded with a free-wall hole can be utilized as an elastic waves cloaking. Moreover, we present a twodimensional (2D) phononic crystal (PC) to exhibit Dirac-like cones dispersion at the zone center. Effective medium theory finds that such PC can have effectively zero reciprocal of shear modulus and zero mass density, thus zero refractive index. Numerical simulations show that the PC system would be a good experimental candidate to achieve the intriguing transmission properties of the C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902065] ZIM waveguide structure. V

I. INTRODUCTION

In the past decade, zero index metamaterials (ZIM), permittivity and permeability were simultaneously or individually near zero, have been studied in both theoretically and experimentally1–10 and showed peculiar ability to control electromagnetic (EM) waves in ways that fundamentally differ from natural material. In ZIM, the phase velocity of waves can approach infinity, thus the phase of waves throughout a piece of ZIM is almost essentially constant. This unique property leads to many intriguing phenomena and applications, such as tailoring the phase pattern of radiation field,2,3 making light channels and bends of arbitrarily shapes,4–6 and manipulating EM waves propagation through ZIM waveguide by tailoring the parameters of the dielectric defects.7–10 Meanwhile, the concept of metamaterials had been extended to acoustic and elastic media.11–43 Much effort focused on realizing of effectively negative elastic parameters11–18 and acoustic cloaking.19–28 Recently, acoustic ZIM has drawn intense attention by several researches, such as acoustic waveguides loaded with membranes and/or Helmholtz resonator,29–32 coiling up space with curled channels,33,34 and 2D acoustic crystals with Dirac-like cones.35–40 However, as most of these works only considered acoustic ZIM, elastic ZIM41–43 still need to be further researched. In this work, we investigated anti-plane transverse elastic waves transmission through the elastic ZIM waveguide embedded with defect. Comprehensive analysis of how the geometric and elastic parameters of the defects affect the transmission was provided, and numerical simulations are then carried out to justify our theory. We also found that incidence waves can totally go round a free boundary defect a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

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in the elastic ZIM waveguide structures. Thus, by simply boring a hole in the elastic ZIM, arbitrary object can be hidden inside the hole. In this sense, the elastic ZIM embedded with a hole can be regarded as an elastic cloaking. In addition, we provided a 2D phononic crystal (PC) with Diraclike cones dispersion at the zone center, which can be mapped to an elastic material with effectively zero reciprocal of shear modulus 1=leff and zero mass density qeff . Numerical simulations show that the PC system can be a good candidate to achieve the ZIM waveguide structure experimentally for its simple manufacturing requirements and no demand of materials with extreme material parameters. II. THEORETICAL ANALYSIS

A 2D elastic waveguide structure, consists of four distinct regions, is illustrated in Fig. 1. The regions 0 and 3 filled with host medium (with mass density q0 , bulk modulus j0 , and shear modulus l0 ) are separated by the elastic ZIM

FIG. 1. Schematic of the 2D waveguide structure with host medium (regions 0 and 3), elastic ZIM (region 1), and an embedded defect (region 2). Perfectly matched layers are used to prevent reflections from the domain boundaries.

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(region 1) with effective mass density q1 , bulk modulus j1 , and shear modulus l1 . And a cylindrical defect (region 2) with radius rd , mass density q2 , bulk modulus j2 , and shear modulus l2 is embedded in the elastic ZIM region. The x-direction walls of the waveguide are set to be traction-free walls and perfectly matched layers (PML) are used to prevent reflections from either side of the domain. It is known that an elastic waveguide with free traction-wall (Fig. 1) supports two independent waves: shear horizontal (SH) waves (with displacement uz along the z-direction) and Lamb waves (with displacements ux and uy in the xy-plane). However, only the fundamental mode of SH waves is non-dispersive and has plane wave front.44 Thus, for simplicity, we consider the fundamental mode of SH waves in this work. The governing equations for SH waves in a homogeneous isotropic solid may be as @vz ¼ r  ^s q @t @^s ¼ lrvz ; @t

(1)

where vz ¼ @uz =@t represents the velocity field, ^s ¼ sxz i^ þ syz j^ represents the stress tensor, and the gradient is @ ^ @ ^ i þ @y j for the 2D waveguide system has defined as r ¼ @x translational symmetry in the z-direction. In the elastic ZIM region (region 1), as 1=l1 tends to zero, we can see that in the governing equations the velocity field v1z must be constant to keep ^s 1 as finite value. Suppose a plane harmonic iðk0 xxtÞ is incident from left transverse elastic wave vinc z ¼ vz e to right inside the waveguide, where vz is the amplitude of the incident field, k0 is the wave vector in the regions 0, and x is the angular frequency. We omit the time variation item in the rest of this paper for convenience. Thus, velocity fields in the region 0 can be written as v0z ¼ vz ½eik0 ðxþd=2Þ þ Reik0 ðxþd=2Þ ;

(2)

while in region 3 the velocity fields must have the form v3z ¼ vz Teik0 ðxd=2Þ ;

(3)

where R and T are the reflection and transmission coefficients. In region 1, for the velocity field in the elastic ZIM maintains a quasi-static situation (v1z ¼ constant). Then by using the continuous boundary condition at x ¼ d=2 and x ¼ d=2, we have vz ð1 þ RÞ ¼ v1z and vz T ¼ v1z Ð , which 45 leads to 1 þ R ¼ T. Applying Newton’s law s  n^dS dS^ Ð 2 ¼ dV q @@t2u dV for the elastic ZIM region, we can find out the transmission coefficient of the waveguide as T¼

1 Þ ; 1  ð1=g0 hv1z Þ dA2 s2rz dl

(4)

pffiffiffiffiffiffiffiffiffiffi where @A2 is the boundary of the defect, g0 ¼ q0 l0 is the impedance of the host medium, and s2rz is the tangential stress tensor in the defect. For simplicity, only one defect is considered here. To evaluate T, we need to find the tangential stress tensor s2rz , which can be obtained by considering

the elastic boundary conditions between the cylinder defect and the elastic ZIM. In region 2, the displacement along the z-axis u2z is described by the elastic wave equation 2 q2 @@tu22z ¼ r  ðl2 ru2z Þ. In cylindrical coordinates, the general solution in the cylinder defect can be expressed as u2z ¼

X n

an3

1 ^ n ðk2t rd Þeinu ; r  r  kJ k2t

(5)

pffiffiffiffiffiffiffiffiffiffiffiffi where k2t ¼ x q2 =l2 , Jn ðxÞ is the Bessel function, and n is the angular quantum number. The elastic boundary conditions require that u1z ¼ u2z s1rz ¼ s2rz :

(6)

We note here, to satisfy the elastic boundary conditions, the angular quantum number n in Eq. (6) can only take 0 due to the fact that displacement u1z is constant in the elastic ZIM region. The elastic boundary conditions lead to the following equations: u1z ¼ C1 a03 s1rz ¼ C2 a03 ; where

C1 ¼ 2J1 ðk2t rd Þkrd2t rd J2 ðk2t rd Þ,

(7) and

C2 ¼ l2

2

ðk2t rd Þ J3 ðk2t rd Þ4k2t rd J2 ðk2t rd Þ rd we obtain s2rz ¼ CC21 u1z .

while n takes 0. By solving Eq. (7),

Finally, the transmission coefficient (Eq. (4)) can be expressed as T¼

1 : 2iprC2 1þ hg0 xC1

(8)

An inspection of Eq. (8) shows the transmission characteristics of the waveguide system. We can see that total transmission arises if C2 ¼ 0, while total reflection arises if beffiffiffiffiffiffiffiffiffiffiffiffi satisfied by choosing suitaC1 ¼ 0. These conditions canp ble combinations of k2t ¼ x q2 =l2 and radius rd of the defect. We also would like to remark that if a defect possessing traction-free wall, i.e., s2rz ¼ 0, Eq. (4) becomes T ¼ 1, which is the same as the case without defect. Therefore, by simply boring a hole in the elastic ZIM, we can hide arbitrary object inside the hole. In this sense, the ZIM embedded with a hole can be utilized as an elastic cloaking. III. NUMERICAL SIMULATION

To verify the analysis, numerical simulations are carried out by using the finite element method (FEM). The reciprocal of shear modulus and mass density of the elastic ZIM region is described by ! ! x2p x2p 1 1 ; q1 ¼ q0 1– ; ¼ 1– xðx þ iCÞ xðx þ iCÞ l1 l0 where x is the excitation angular frequency and C denotes the loss parameter. When the frequency x  xp , Reðq1 Þ and Reð1=l1 Þ are almost zero (assume very small loss C  0).

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FIG. 2. Transmission coefficients as a function of the radius rd (a) and shear modulus l2 (b) of the defect for the elastic ZIM waveguide structure. The solid line represents the numerically calculated result, while the triangles represent the theoretical result.

Figs. 2(a) and 2(b) show the transmission coefficients as a function of the radius rd and shear modulus l2 of the defect for the elastic ZIM waveguide structure, respectively. The solid line represents the numerically calculated result, while

FIG. 3. The numerically simulated displacement field distributions for the elastic ZIM waveguide structure realizing total transmission (a) and total reflection (b). The radii of the defects are rd ¼ 0:33h and rd ¼ 0:41h, respectively. (c) The numerically simulated displacement field distributions for the elastic ZIM waveguide embedded with a free-wall hole. (d) Comparative simulation for the waveguides without elastic ZIM.

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the triangles represent the theoretical result obtained from Eq. (8). The simulated result agrees with the theoretical result very well, which confirms our analysis above. In Fig. 2(a), we have set q2 ¼ q0 , j2 ¼ j0 , and l2 ¼ l0 , while in Fig. 2(b) we have fixed the bulk modulus and the radius of the defect and assumed the shear modulus of the defect can be adjusted freely. Figs. 3(a) and 3(b) show the displacement field distributions (uz ) of the elastic ZIM waveguide embedded with defect to achieve total transmission and total reflection, respectively. In Fig. 3(a), to satisfy C2 ¼ 0, we choose the radius of the defect rd ¼ 0:33h. The plot clearly demonstrates that total transmission of the incident waves occurs and the displacement field u1z is of course uniform in the elastic ZIM region. In Fig. 3(b), we consider a defect with radius rd ¼ 0:41h, which is chosen to satisfy C1 ¼ 0. According to Eq. (7), C1 ¼ 0 means that the displacement u2z is zero at the interface of the defect. Imposed by the elastic boundary condition, u1z should be zero anywhere as u1z is constant in the region (1). Thus, the incident wave will be totally blocked, which indicates total reflection. As expected, one can see from Fig. 3(b) that the displacement field u1z disappears inside region (1) indeed. In Fig. 3(c), we show the displacement field distribution (uz ) of the elastic ZIM waveguide embedded with a hole defect. The incident waves completely transmit through the elastic ZIM region without phase variation. In contrast, a control simulation (Fig. 3(d)) shows that if elastic ZIM is removed, the hole defect strongly scatters the incident waves. In Fig. 4, we show the transmission coefficients as a function of the normalized frequency for the elastic ZIM structure embedded with a hole defect (red solid lines) and for the structure whose elastic ZIM region is removed while the hole defect is kept (black dotted lines). The results of Fig. 4 reveal that around xp , T ¼ 1 for the elastic ZIM structure which verifies our analysis above. Consequently, we prove that a hole in the elastic ZIM can be used to achieve elastic cloaking. In our previous work,46 we have used a 2D acoustic crystals system, which can be mapped to an acoustic material with effectively zero mass density and zero reciprocal bulk modulus, to demonstrate the intriguing transmission properties of the acoustic ZIM waveguide structure. Here, we would

FIG. 4. The transmission coefficients as a function of the normalized frequency for the elastic ZIM structure embedded with a hole defect (red solid lines) and for the structure whose elastic ZIM region is removed while the hole defect is kept (black dotted lines).

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FIG. 5. (a) Band structure of a 2D square lattice PC. Two linear bands intersect at a Dirac-like point x ¼ 0:213ð2pct;Si3 N4 =aÞ, with an additional band with small group velocity. (b)–(d) The displacement field distributions of the eigenmode near the Dirac-like point. (e) The effective mass density qeff (black solid line) and reciprocal of shear modulus 1=leff (red squares) of the PC as a function of frequency. All the effective elastic parameters have been normalized to the elastic parameters of Si3N4. The quantity ct;Si3 N4 is the velocity of transverse waves of Si3N4.

like to design a 2D PC to demonstrate the intriguing transmission phenomena for elastic waves. Fig. 5(a) shows the band structure for the anti-plane transverse waves of a 2D PC calculated using the multiple scattering methods. The PC is composed of lead cylinders embedded in Si3N4 matrix and arranged in square lattice. Here, the lattice constant is denoted by a, the radius of the cylinders is set at R ¼ 0:49a. Both Pb and Si3N4 are taken to be isotropic, the density q of Pb and Si3N4 taken to be 1:16  104 kg=m3 and 3:27  103 kg=m3 , respectively. The velocity of longitudinal waves is 2:49  103 m=s (1:1  104 m=s) and the velocity of transverse waves is 1:132  103 m=s (6:25  103 m=s) in Pb (Si3N4). Two linear branches (forming a Dirac-like cone) and a third flat branch intersect at a triply degenerate point at C point (x ¼ 0:213ð2pct;Si3 N4 =aÞ). In order to understand the physical nature of the eigenstates near the Dirac-like point, we show the magnitude of displacement field distributions (uz ) of these eigenstates. In Figs. 5(b)–5(d), we show that the eigenmode is a combination of a monopole and doubly degenerate dipoles excitations. The displacement field distributions show that the three branches involve only a mixture of monopolar, dipolar modes. We note that the square lattice system has C4v symmetry. The non-degenerate monopolar mode and the doubly degenerate dipolar modes at the Diraclike point can be labeled as A1 and E, respectively, at the zone center.47 The monopolar and dipolar modes are not required by symmetry for having the same frequency but the system parameters (such as the diameter of the cylinder) can be chosen so that they are equal, and the accidental degeneracy of A1 and E states makes possible the existence of the Dirac-like cones. We also note that the energy associated with displacement fields of the eigenstates near the Dirac-like point mainly localize in the Pb cylinders (Figs. 5(b)–5(d)) as the wave velocity of Pb is lower than that of Si3N4, and the displacement fields are fairly homogeneous in the matrix composite. There is a possibility that we can employ an effective medium theory to describe the physics of the PC system.

As we only study the anti-plane transverse waves here, the PC system can be described by two independent effective elastic parameters: the effective mass density qeff and reciprocal of shear modulus 1=leff , which can be obtained using standard effective medium theory48 qef f 4f ¼1þ  2 T03;03 host q ip khost r t

lef f  lhost 4if ¼  2 T13;13 ; host lef f þ l p kthost r

(9)

here f is the filling ratio, r is the radius of the inclusions, kthost is the transverse wave number in the host medium, T03;03 and T13;13 is the n ¼ 0 and n ¼ 1 angular channel Mie scattering coefficient for the anti-plane transverse mode, respectively. It is noted that the n ¼ 0 and n ¼ 1 Mie scattering corresponds to monopolar and dipolar resonances, which is consistent with the numerical analysis of eigenstates near the Dirac-like point (Figs. 5(b)–5(d)). In Fig. 5(e), we show the effective mass density qeff (black solid line) and reciprocal of shear modulus 1=leff (red squares) as a function of frequency, and qeff and 1=leff intersect at zero at the Dirac-like point frequency (x ¼ 0:213ð2pct;Si3 N4 =aÞ). As qeff and 1=leff go through zero simultaneously and linearly, the effective refractive index also goes through zero but the group velocity remains finite. As we have realized an effectively zero refractive index in the PC system, we show below that can achieve unusual wave manipulation for elastic waves. Fig. 6 shows the results of numerical simulations which demonstrate the unusual wave propagation properties of the PC system. We have used the theoretical analysis above to guide the design of the defects and PMMA (qPMMA ¼ 1:2 103 kg=m3 , cl;PMMA ¼ 2:83  103 m=s, and ct;PMMA ¼ 1:16 103 m=s) defects have be used to achieve the total transmission (Fig. 6(a)) and total reflection (Fig. 6(b)), respectively. In Fig. 6(c), the displacement field distributions show that

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FIG. 6. The numerically simulated displacement field distributions (uz ) for the 2D PC waveguide structures embedded with the PMMA defects to achieve total transmission (a), and total reflection (b). (c) The numerically simulated displacement field distributions for the PC waveguide embedded with a hole. (d) Comparative simulations for the waveguides without PC. The host medium is Si3N4 in all the simulations.

the incidence wave is able to pass through preserving its plane-wave characteristic even though there is a hole inside the PC system. Arbitrary object can be concealed inside the hole, thus the PC system can be used as elastic cloaking. Compared to other elastic waves cloaking,25–28 our PC system is easier to realize so that it will be a good candidate to achieve experimentally. IV. SUMMARY

In conclusion, we have demonstrated how to manipulate SH waves propagating in the elastic ZIM waveguide by embedding a defect. Theoretical analysis shows that adjusting the geometric and elastic parameters of the defects can be applied to achieve total reflection and total transmission. Numerical simulations confirm our theory. We also design a 2D PC with Dirac-like cones dispersion at the zone center, which can be mapped to an elastic material with effectively zero refractive index. Numerical simulations show that the PC system can be used to achieve the elastic ZIM waveguide. Our investigation may have potential application in elastic cloaking and on-chip phononic devices. ACKNOWLEDGMENTS

This work was supported by NSFC (Grant No.11304090), the Natural Science Foundation of Hubei (Grant No. 2013CFB060). 1 2

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