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One-dimensional phononic crystals with locally resonant structures
Physics Letters A 327 (2004) 512–521 www.elsevier.com/locate/pla
One-dimensional phononic crystals with locally resonant structures Gang Wang a,b,∗ , Dianlong Yu a,b , Jihong Wen a,b , Yaozong Liu a,b , Xisen Wen a,b a Institute of Mechatronical Engineering, National University of Defense Technology, Changsha 410073, China b PBG Research Center, National University of Defense Technology, Changsha 410073, China
Received 26 March 2004; received in revised form 23 May 2004; accepted 24 May 2004 Available online 5 June 2004 Communicated by R. Wu
Abstract The propagation of longitudinal and transverse elastic waves oblique or perpendicular to the laminations of infinite periodically layered fourfold system is studied. Comprehensive study is performed for the one-dimensional phononic crystals with locally resonant structures. Compared with its three- and two-dimensional counterparts, additional resonance is found in this one-dimensional case. 2004 Elsevier B.V. All rights reserved. PACS: 43.40.+s; 46.40.Cd; 63.20.-e Keywords: Phononic crystal; Band gap; Locally resonant
1. Introduction The study of wave propagation in periodic structures has originated many novel discoveries in physics. Analogies from such subfields in physics have also opened new fruitful avenues in the research on the propagations of classical waves in periodic structures. As an example, the “phononic crystals” (PCs) which is composed of artificial periodic elastic/acoustic structures that exhibit so-called “phononic band gap” (PBG), has received a great deal of attention [1–4,7– 18] by analogy with the electronic or photonic band gap in natural or artificial crystals. The emphasis was
* Corresponding author.
E-mail address: [email protected] (G. Wang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.047
laid on the existence of complete elastic/acoustic band gaps (ABG) within which sound and vibrations are all forbidden. This is of interest for applications such as elastic/acoustic filters, improvements in the design of transducers, noise control and vibration shield; as well as for pure physics concerned with the Anderson localization of sound and vibration [4]. Several theoretical methods have been used to study the elastic/acoustic band structures, such as, the transfer-matrix (TM) method [5], the multiple scattering theory (MST) [7–11], the plane-wave expansion (PWE) method [12–14], and the finite difference time domain (FDTD) method [15–18]. As for the applications in sound or vibration shelter, the size of PCs is of crucial importance. Conventional elastic-wave band-gap material operating under the principle of Bragg reflection mechanism can hardly
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Fig. 1. Infinite extended periodic layered fourfold structure. We consider the longitudinal and transverse waves propagating parallel or oblique to x direction.
fulfill it because the long sound wavelength (about two times of the lattice constant with the Bragg condition) in common solids. This would bring on structures with the size of outdoor sculptures in order to prevent environmental noises [3] or the bigger size for the vibration shielding of common machines. The pioneering work of Liu et al. [1] has motivated the work in this Letter. The authors advanced the locally resonant mechanism of band-gap in researching the three-dimensional (3D) PCs consisting of cubic arrays of coated spheres immersed in an epoxy matrix. They also constructed a simple cubic 3D PCs with lattice constant of 1.55 cm and sharp band-gap at the frequency around 400 Hz. Similar works by Goffaux et al. [2] shows that narrow gap with low frequency also exists in its two-dimensional (2D) counterpart, the array of coated cylinders in epoxy. In this Letter, a comprehensive study is performed for the one-dimensional counterpart of the ternary systems, the so-called locally resonant materials, recently studied by Liu et al. [1] and Goffaux et al. [2]. Compared with its counterparts, the research on 1D PCs which was not the emphasis in the former works is quite different and close to the hypostasis of the locally resonant mechanism in PCs.
longitudinal and transverse modes) and out-of-plane (transverse mode) waves. We initially look for the solution of the in-the-plane one. By assuming that the wave vector k is one component of wave vector k that parallel to the interfaces between different media and kx is the other component of k that vertical to the interfaces, the general solution for j th layer in nth period is clearly [6]: l l − uy (x) = A+ j,n cosh qj x − Aj,n sinh qj x + − + Bj,n cosh qjt x − Bj,n sinh qjt x × ei(k y−ωt ) , (1) l qj + l l ux (x) = A sinh qj x − A− j,n cosh qj x ik j,n ik + − − t Bj,n sinh qjt x − Bj,n cosh qjt x iqj × ei(k y−ωt ) , where
x = x − na −
The wave propagation in a homogeneous solid can be strongly altered by inserting periodical inclusions with different elastic constants. The periodic inclusions in this so-called PCs induced a wave scattering and destructive interferences can appear in some frequency ranges, leading to forbidden band-gaps. Total reflection is then expected in these frequency ranges. Owing to the isotropy of the media there is a decoupling between the in-the-plane (coupling of
j −1
dm ,
m=1
na +
j −1 m=1
2. Theory
(2)
dm < x < na +
ω2 1/2 , qjl = k2 − 2 cl,j
cl,j = (λj + 2µj )/ρj ,
j
dm ,
j = 1, . . . , 4,
m=1
ω2 1/2 qjt = k2 − 2 , ct,j ct,j = µj /ρj .
− There are now four unknown constants A+ j , Aj , − and Bj which determine the displacement field in each layer. To find these constants and to solve the dispersion relations of this 1D PC, we employ the continuity of displacement and stress at the interfaces between the different layers.
Bj+
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p p × sinh q2 d2 sinh q3 d3 p p + (F1 F2 F3 + F4 ) cosh q2 d2 cosh q3 d3 p p × sinh q1 d1 sinh q4 d4 p p + (F2 F3 + F1 F4 ) cosh q1 d1 cosh q3 d3 p p × sinh q2 d2 sinh q4 d4 p p + (F3 + F1 F2 F4 ) cosh q1 d1 cosh q2 d2 p p × sinh q3 d3 sinh q4 d4 p p + (F1 F3 + F2 F4 ) sinh q1 d1 sinh q2 d2 p p × sinh q3 d3 sinh q4 d4 ,
Thus the relationship between layer j and j + 1 (j = 1, . . . , 3) in nth period can be written as ψ j,n = Tj ψ j +1,n ,
(3)
and the relationship between layer 4 in nth period and layer 1 in (n + 1)th period is ψ 4,n = T4 ψ 1,n+1 .
(4)
Thus ψ 1,n = Tψ 1,n+1 ,
(5)
where T = T4 T3 T2 T1 , + − − T ψ j,n = A+ j,n , Bj,n , Aj,n , Bj,n .
where
However, due to the periodicity of the layered structure in the x direction, Bloch theorem [19] guarantees: ψ 1,n = e
−ikx a
ψ 1,n+1 .
(6)
Inserting Eq. (5) into Eq. (6) yields a standard complex eigenvalue problem of 4 × 4 matrix: T − e−ikx a I = 0. (7) For a given k and ω, Eq. (6) gives the four values of kx . When none of the four values is real number, there exists region with no dispersion curves, i.e., the forbidden band. For the out-of-plane modes, one can get the dispersion relations with the similar (more simple) process. When k is zero, the problem becomes two decoupled ones for the longitudinal and the transverse modes, respectively. Thus the dispersion relations for ω (embodied in qj ) as a function of wave vector kx is: cos(kx a)
p p = cosh q1 d1 cosh q2 d2 p p × cosh q3 d3 cosh q4 d4 p 1 + (F1 +F2 F3 F4 ) cosh q3 d3 2 p p p × cosh q4 d4 sinh q1 d1 sinh q2 d2 p p + (F1 F2 + F3 F4 ) cosh q2 d2 cosh q4 d4 p p × sinh q1 d1 sinh q3 d3 p p + (F2 +F1 F3 F4 ) cosh q1 d1 cosh q4 d4
(8)
F1 = F3 =
2 q1 ρ1 cp,1 2 q2 ρ2 cp,2 2 q3 ρ3 cp,3 2 q4 ρ4 cp,4
,
F2 =
,
F4 =
2 q2 ρ2 cp,2 2 q3 ρ3 cp,3
,
1 , F1 F2 F3
(9)
and p denotes l and t for the longitudinal and the transverse modes, respectively. Similar works on periodically layered binary system has been done earlier in Ref. [6], where the dispersion relation of the binary case is given as: cos(kx a)
p p = cosh q1 d1 cosh q2 d2 p p 1 1 + sinh q1 d1 sinh q2 d2 . F1 + 2 F1
(10)
For the finite periodically layered system, the transmitting frequency response function is the primary representation of its elastic wave property, instead of the dispersion relation of infinite system. In this Letter, the method of transfer matrices [5] is used in such case.
3. Result for the binary 1D PCs: 1D periodic structure of Pb and epoxy layers The parameters that include the appearance of band gaps in 1D binary PCs are the filling fraction (the fraction of one material in all), lattice constant, and the contrast of the elastic properties of different materials. The density contrast is a fundamental factor; and high
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Fig. 2. (a) Phononic band structure of out-of-plane (pure transverse) wave oblique to the laminations of 1D binary periodic multi-layer structure of Pb and epoxy layers. (b) Corresponding band structure of longitudinal (continuous lines) and transverse (dashed lines) waves propagating perpendicular to the laminations. (c) Corresponding band structure of in-the-plane (coupling of longitudinal and transverse) waves propagating oblique to the laminations. The filling fraction is 0.5. The shadowed regions define the band gaps. The frequencies ω are given in reduced units. a is the lattice constant, and ct,ave is the average transverse velocity inside the crystal, which is calculated by ct,ave = a/[aPb /ct,Pb + aepoxy /ct,epoxy ], where aPb (aepoxy ) and ct,Pb (ct,epoxy ) are the thickness and transverse velocity of one Pb (epoxy) layer.
contrast of it is more possible to obtain wide band gaps. Here, as a comparison example between Bragg scattering and locally resonant mechanism, we consider a 1D binary periodic system of Pb and epoxy layers. Fig. 2 shows the dispersion relations of elastic modes with a 1D binary PCs having a filling fraction f = aPb /a = 0.5, where aPb is the thickness of one Pb layer, and a = 20 mm is the lattice constant. The elastic parameters employed in the calculations were ρPb = 11600 kg m−3 , ρepoxy = 1180 kg m−3 , λPb = 4.23 × 1010 Pa, λepoxy = 4.43 × 109 Pa, µPb = 1.49 × 1010 Pa, µepoxy = 1.59 × 109 Pa. The modes of longitudinal wave (continuous lines) as well as those of transverse wave (dotted lines) propagating perpendicular to the laminations (k = 0) were illustrated in Fig. 2(b). When k = 0 (oblique incidence), the modes of out-of-plane (pure transverse wave) as well as those of in-the-plane (coupling of longitudinal and transverse waves) were illustrated, respectively, in Fig. 2(a) and (c). Band gaps, resulting from longitudinal and transverse dispersion curves, settle as the shadowed regions in the three figures. The analysis of the band structures in Fig. 2 can be straightforward for the case of pure longitudinal
or transverse modes propagating perpendicular to the laminations. The gaps are the results of destructive interference of the wave reflections in the periodic layers, the so-called Bragg scattering mechanism. So, the magnitude of the pure longitudinal or transverse midgap frequency ωl,mid or ωt,mid can be estimated. Its value should be closer to the frequency where the wave band folding takes place. For example, the estimation of the midgaps for the first pure longitudinal and transverse modes is made as follows: cl,ave 2πct,ave ωl,mid ≈ cl,ave kK = 2ct,ave a 2πct,ave , = 1.0969 (11) a 2πct,ave , ωt,mid ≈ ct,ave kK = 0.5 (12) a where kK = π/a is the wave vector at point K, a −1 ct,ave = aPb aepoxy = 1146.9 ms , ct,Pb + ct,epoxy a −1 cl,ave = aPb aepoxy = 2516.1 ms + cl,Pb cl,epoxy are the average transverse and longitudinal sound velocity inside the crystal.
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(in reduced units), the normalized gap width maximize at fPb = 0.5.
4. Result for the ternary 1D PCs: 1D periodic structure of Pb and epoxy layers separated by soft rubber layers
Fig. 3. (a) The upper (filled circles) and lower (hollow circles) edges, and midgap (cross symbols) frequencies determining the gap in a 1D binary periodic multi-layer structure of Pb and epoxy layers for several Pb filling fractions (fPb ). (b) The corresponding normalized gap width (ω/ωg ), where ωg is the midgap frequency.
In Fig. 2, the two midgap frequencies calculated are ωKL,1 + ωKL,2 2πct,ave (13) = 1.097 , 2 a ωKT,1 + ωKT,2 2πct,ave ωt,mid = (14) = 0.4999 . 2 a The estimated values (in reduced frequencies) accurately agree with the actual ones resulting from the band-structure calculation, 1.0969 and 1.0970 for the pure longitudinal mode with error in 0.01%; 0.5 and 0.4999 for the transversal mode with error in 0.02%. This result illuminates that the gaps created by a Bragg scattering mechanism appear at frequencies relate to the periodicity. So gaps at very low frequencies can only be achieved by using structures of extra large periodicity or materials with very slow elastic wave velocity. For a gap with midgap frequency of 10 Hz, that is a 1D periodic Pb/epoxy multi-layer structure with a lattice parameter of about 100 meters. Another parameter controlling the band gap is the filling fraction f . Fig. 3(a) and (b) show, respectively, the band edges and midgap frequencies of the first gap and its normalized width for the pure transverse modes when k = 0. These results represent typical behaviors associated with Bragg gap in 2D structures, in which a Bragg gap peaks at some intermediate filling fraction. Thus in Fig. 3, with the undulating of midgap frequencies near the estimated Bragg value 0.5 ωl,mid =
Very soft rubber was recently used [1,2] as the coating of heavy (Pb or Au) spherical or cylindrical inclusions arranged in certain lattice in an epoxy host. The very low elastic constants of the coating layer resulted in a strong resonant band structure with a gap at a frequency so-called two orders of magnitude lower than the expected one by Bragg scattering. In analogy with electromagnetic situations in which the dielectric function can be negative, this phenomenon was interpreted as the consequence of effective negative elastic constants in the range of frequencies where the sub-frequency gaps appear. Here, in order to find the key factor of locally resonant mechanism, we analyzed the band-gap with the periodic multi-layer ternary 1D phononic crystal illustrated in the sub-figure of Fig. 4(b), constructed by inserting thin layers of soft rubber with thickness of 1 mm in each interface of 10-mm thickness Pb and 10mm thickness epoxy layers. Thus, we separate Pb and epoxy layers with thin layer of same soft rubber [1], whose elastic parameters are ρrubber = 1300 kg m−3 , λrubber = 6 × 105 Pa, µrubber = 4 × 104 Pa. Fig. 4(a), (b) and (c) also presents the corresponding band structure calculated with Eqs. (7) and (8). Although the thickness of soft rubber is only 1/10 of Pb or epoxy, its sound velocity cannot be ignored in calculating the average velocity and the reduced frequencies of PCs because its longitudinal and transverse velocities are only 22.87 m s−1 and 5.547 m s−1 , i.e., 1/106 and 1/199 of the corresponding velocities in epoxy. For the sake of fair comparison, the reduced frequencies are calculated by ωa/2πcl,ave and ωa/2πct,ave , where cl,ave = ct,ave =
a aPb cl,Pb
+
aepoxy cl,epoxy
aPb ct,Pb
+
aepoxy ct,epoxy
+
2arubber cl,rubber
+
2arubber ct,rubber
,
a
are average longitudinal and transverse velocities.
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Fig. 4. (a) Phononic band structure of out-of-plane (pure transverse) wave oblique to the laminations of 1D ternary PCs. (b) The structure of the 1D ternary PCs (sub-figure), and the corresponding band structure of longitudinal (continuous lines) and transverse (dashed lines) waves propagating perpendicular to the laminations. (c) Corresponding band structure of in-the-plane (coupling of longitudinal and transverse) waves propagating oblique to the laminations. The thickness of Pb, epoxy and rubber are 10 mm, 10 mm and 1 mm. The frequencies ω are given in reduced units ωa/2π ct,ave , where ct,ave is the average transverse elastic wave velocity. The complete flat bands are associated with modes localized in some layers of this 1D phononic crystal (see Fig. 5).
Considered the band structures shown in Fig. 4, it is remarkable for the presence of flat bands crossing the complete BZ. These bands are real and converged, associated with eigenmodes of some layers (including soft rubber or itself only) of this periodic structure as shown in Fig. 5, where the displacements |uy (x)| are associated with the transverse modes of four different flat bands with frequencies (in reduced units ωa/2πct,ave ) 0.050, 0.151, 1.051 and 1.071, respectively, in Fig. 4(b) are represented. The first resonant eigenmode is localized within the rubber–Pb– rubber mass-spring alike structure, in which the Pb layer plays the role of mass and soft rubber layer acts as the spring. The second resonant eigenmode is localized with alike mass-spring structure of rubber– epoxy–rubber, which was not found in 2D or 3D cases [1,2] before. As the mass of epoxy layer is smaller than Pb layer at the same thickness, the frequency of the rubber–epoxy–rubber eigenmode is higher than of rubber–Pb–rubber. The third and the fourth resonant eigenmodes are localized in the soft rubber. The very low elastic constants of soft rubber allow the existence of the first two propagating modes at very low frequencies. Similar flat bands were also found in 2D and 3D locally resonant systems [1,2].
Fig. 5. (a), (b), (c), (d) Modulus of the displacement vector |u(x)| for the transverse modes associated with the first four flat bands (corresponding to KT, 1 , KT, 2 , KT, 3 and KT, 4 in Fig. 4, respectively). The abscissa scale is given in unit of mm. In all cases the displacement occurs only inside certain layers of the periodic structure (see text for details).
One noticeable feature in Fig. 4 is the appearance of complete gap in frequency region about 1/5 of magnitude lower than the expected ones in Bragg conditions. Two orders of magnitude lowness [1,2] in 2D and 3D cases cannot be got here due to the scales of
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Fig. 6. The frequencies of lower four pure longitudinal modes at K point (KL, 1 , KL, 2 , KL, 3 and KL, 4 ) determining the gaps in a 1D ternary phononic crystal of Pb and epoxy layers separated by soft rubber layers for several (a) Pb, (b) epoxy and (c) rubber thickness. The inset of sub-figure (c) shows a zoom of the corresponding total graph in the low frequency region.
different reduced frequencies were used in Refs. [1,2] and the present Letter. In Refs. [1,2], the reduced frequency is calculated by ωa/2πct,epoxy, where the ct,epoxy = 1160.8 m s−1 is the transverse velocity of epoxy. However, for fair comparison and discussion in this Letter, ct,epoxy is replaced with the average transverse velocity ct,ave in the 1D ternary PCs, where ct,ave = 58.2 m s−1 , 20 times lower than the velocity in epoxy. The sub-frequency gap is originated from the resonances arising from the insertion of soft layers, as pointed out by Liu [1,7], Goffaux [2] and their coworkers. In order to support the resonant origin of subfrequency gaps, we plot in Fig. 6(a)–(c) the dependency of band edges frequencies related to thickness of Pb, epoxy and soft rubber layers, respectively. Also we plot in Fig. 7 the dependency of the first sub-frequency band gap related to the assumed density of inserting layer (soft rubber layer).
Fig. 7. The dependency of the first sub-frequency band gap related to the assumed density of inserting layer (soft rubber).
The behavior of the band edges is completely different from that of a Bragg gap. In Fig. 6(a) only one (Pb-edge) of the two lower band-gap edges depends on the thickness of Pb layer, and the other (epoxyedge) changes only with the thickness of epoxy layer in Fig. 6(b). The upper two edges (rubber-edges)
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Fig. 8. (a), (b), (c) Modulus of the displacement vector |u(x)| for the longitudinal modes at A, B and C points in Fig. 6, respectively. The abscissa scales are given in unit of mm. The modes are associated with the first four flat bands.
Fig. 9. (a) The calculated longitudinal band structures of a designed 1D PC of the structure in Fig. 1 with lattice constant of 15.5 mm in which Pb, rubber and epoxy layers are in thickness of 10 mm, 2 mm and 1.5 mm. (b) Corresponding calculated frequency response functions (FRF) of two, three and four periods swatch of the designed 1D PC (solid lines) and a 3-period finite structure except 48 volume % of randomly dispersed thickness of epoxy layers (dashed line). A wide sub-frequency gap from 380 Hz to 2150 Hz can be predicted.
change little with the thickness of Pb and epoxy layers. All the edges have significant changes with the thickness of soft rubber. Farther in Fig. 7, the edges of the first sub-frequency band gap are independent of the density of inserting layers. As the elastic wave velocity depends on its density, this means that the first sub-frequency gap frequency is independent of the Bragg conditions, i.e., not restricted within the so-called two orders of magnitude lowness [1,2] in frequency. These are signatures of their resonant origin. Particularly, the Pb-edge (epoxy-edge/rubberedges) is a consequence of the strong localized character of its associated mode, which consists of vi-
bration of the Pb (epoxy/rubber) layer. Here, the rubber layer acts as a soft spring at the Pb- and epoxy-edges of first sub-frequency gap, and the density of rubber affects as a mass at the rubber-edges. More intuitionistic understanding can be got with Figs. 5(a)–(d) and 7. The transformation course between Bragg scattering and locally resonant mechanism is also analyzed in Figs. 6(c) and 8 by selecting three representative points with different thickness of rubber layer. Point A locates at arubber = 0.0005 mm, where the equivalent stiffness of rubber layer are too big (compared with the epoxy layers) to be treated as a single spring:
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krubber = scell (λrubber + 2µrubber)/arubber N = 1.28 × 1012scell , m kepoxy = scell (λepoxy + 2µepoxy)/aepoxy N = 0.761 × 1012scell , m
(15)
(16)
and no locally resonant evidence can be found in the corresponding modes (plotted in Fig. 8(a)). Points B and C locates at arubber = 0.08 mm and 0.12 mm, respectively, where the equivalent stiffness of rubber layer are small enough for the formation of locally resonant mechanism at the edges of the first subfrequency band gap (shown in Fig. 8(b), (c)). But for the rubber edges, the contrast of mass and stiffness of rubber layers are not big enough for the energy to localize only in the rubber layers at point B. Using the results foregoing, we design a 1D PC of the structure in Fig. 4 with lattice constant of 15.5 mm (composed of Pb, rubber and epoxy layers in thickness of 10 mm, 2 mm and 1.5 mm, respectively). A wide longitudinal subfrequency gap from 380 Hz to 2150 Hz can be predicted in it. Fig. 9 shows the numerical simulation results of the designed 1D PC, i.e., the band structures (Fig. 9(a)), frequency response functions (FRF) of two, three and four periods swatches (solid lines in Fig. 9(b)), and a 3-period finite structure except 48 volume % of randomly dispersed thickness of epoxy layers (dashed lines in Fig. 9(b)).
5. Conclusion In conclusion, we have studied the propagation of longitudinal and transverse elastic waves perpendicular to the laminations of infinite periodically layered fourfold system. One-dimensional phononic crystals composed of periodically layered binary and fourfold system (with low and high contrast of elastic constant) were studied and compared. The band structures and the associated resonant modes of the 1D ternary PCs of locally resonant structures composed of periodic Pb and epoxy layers separated by soft rubber ones (a periodically layered fourfold system). Its resonant mechanism was adequately proved with the evidences of flat bands, lower sub-frequency gaps, the dependency of band edges on rubber’s density and thickness of each layer, and the FRF of finite structure with randomly
periodicity. Different from the 2D or 3D cases, 1D problem shows an additional resonance, i.e. the resonance of epoxy in soft rubber. Finally, with the results in this Letter, we have designed a 1D locally resonant PC with lattice constant of 1.55 cm, and have predicted a wide sub-frequency gap from 380 Hz to 2150 Hz in it. As one-dimensional multi-layered structure is more practicable, these findings will be significant in the application of phononic crystals.
Acknowledgements This work was supported by the State Key Development Program for Basic Research of China (Grant No. 51307). G. Wang gratefully acknowledges Prof. Zhengyou Liu for the valuable discussion with him as well as the helpful suggestion from him about the research works in this work.
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One-dimensional phononic crystals with locally resonant structures Gang Wang a,b,∗ , Dianlong Yu a,b , Jihong Wen a,b , Yaozong Liu a,b , Xisen Wen a,b a Institute of Mechatronical Engineering, National University of Defense Technology, Changsha 410073, China b PBG Research Center, National University of Defense Technology, Changsha 410073, China
Received 26 March 2004; received in revised form 23 May 2004; accepted 24 May 2004 Available online 5 June 2004 Communicated by R. Wu
Abstract The propagation of longitudinal and transverse elastic waves oblique or perpendicular to the laminations of infinite periodically layered fourfold system is studied. Comprehensive study is performed for the one-dimensional phononic crystals with locally resonant structures. Compared with its three- and two-dimensional counterparts, additional resonance is found in this one-dimensional case. 2004 Elsevier B.V. All rights reserved. PACS: 43.40.+s; 46.40.Cd; 63.20.-e Keywords: Phononic crystal; Band gap; Locally resonant
1. Introduction The study of wave propagation in periodic structures has originated many novel discoveries in physics. Analogies from such subfields in physics have also opened new fruitful avenues in the research on the propagations of classical waves in periodic structures. As an example, the “phononic crystals” (PCs) which is composed of artificial periodic elastic/acoustic structures that exhibit so-called “phononic band gap” (PBG), has received a great deal of attention [1–4,7– 18] by analogy with the electronic or photonic band gap in natural or artificial crystals. The emphasis was
* Corresponding author.
E-mail address: [email protected] (G. Wang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.047
laid on the existence of complete elastic/acoustic band gaps (ABG) within which sound and vibrations are all forbidden. This is of interest for applications such as elastic/acoustic filters, improvements in the design of transducers, noise control and vibration shield; as well as for pure physics concerned with the Anderson localization of sound and vibration [4]. Several theoretical methods have been used to study the elastic/acoustic band structures, such as, the transfer-matrix (TM) method [5], the multiple scattering theory (MST) [7–11], the plane-wave expansion (PWE) method [12–14], and the finite difference time domain (FDTD) method [15–18]. As for the applications in sound or vibration shelter, the size of PCs is of crucial importance. Conventional elastic-wave band-gap material operating under the principle of Bragg reflection mechanism can hardly
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Fig. 1. Infinite extended periodic layered fourfold structure. We consider the longitudinal and transverse waves propagating parallel or oblique to x direction.
fulfill it because the long sound wavelength (about two times of the lattice constant with the Bragg condition) in common solids. This would bring on structures with the size of outdoor sculptures in order to prevent environmental noises [3] or the bigger size for the vibration shielding of common machines. The pioneering work of Liu et al. [1] has motivated the work in this Letter. The authors advanced the locally resonant mechanism of band-gap in researching the three-dimensional (3D) PCs consisting of cubic arrays of coated spheres immersed in an epoxy matrix. They also constructed a simple cubic 3D PCs with lattice constant of 1.55 cm and sharp band-gap at the frequency around 400 Hz. Similar works by Goffaux et al. [2] shows that narrow gap with low frequency also exists in its two-dimensional (2D) counterpart, the array of coated cylinders in epoxy. In this Letter, a comprehensive study is performed for the one-dimensional counterpart of the ternary systems, the so-called locally resonant materials, recently studied by Liu et al. [1] and Goffaux et al. [2]. Compared with its counterparts, the research on 1D PCs which was not the emphasis in the former works is quite different and close to the hypostasis of the locally resonant mechanism in PCs.
longitudinal and transverse modes) and out-of-plane (transverse mode) waves. We initially look for the solution of the in-the-plane one. By assuming that the wave vector k is one component of wave vector k that parallel to the interfaces between different media and kx is the other component of k that vertical to the interfaces, the general solution for j th layer in nth period is clearly [6]: l l − uy (x) = A+ j,n cosh qj x − Aj,n sinh qj x + − + Bj,n cosh qjt x − Bj,n sinh qjt x × ei(k y−ωt ) , (1) l qj + l l ux (x) = A sinh qj x − A− j,n cosh qj x ik j,n ik + − − t Bj,n sinh qjt x − Bj,n cosh qjt x iqj × ei(k y−ωt ) , where
x = x − na −
The wave propagation in a homogeneous solid can be strongly altered by inserting periodical inclusions with different elastic constants. The periodic inclusions in this so-called PCs induced a wave scattering and destructive interferences can appear in some frequency ranges, leading to forbidden band-gaps. Total reflection is then expected in these frequency ranges. Owing to the isotropy of the media there is a decoupling between the in-the-plane (coupling of
j −1
dm ,
m=1
na +
j −1 m=1
2. Theory
(2)
dm < x < na +
ω2 1/2 , qjl = k2 − 2 cl,j
cl,j = (λj + 2µj )/ρj ,
j
dm ,
j = 1, . . . , 4,
m=1
ω2 1/2 qjt = k2 − 2 , ct,j ct,j = µj /ρj .
− There are now four unknown constants A+ j , Aj , − and Bj which determine the displacement field in each layer. To find these constants and to solve the dispersion relations of this 1D PC, we employ the continuity of displacement and stress at the interfaces between the different layers.
Bj+
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p p × sinh q2 d2 sinh q3 d3 p p + (F1 F2 F3 + F4 ) cosh q2 d2 cosh q3 d3 p p × sinh q1 d1 sinh q4 d4 p p + (F2 F3 + F1 F4 ) cosh q1 d1 cosh q3 d3 p p × sinh q2 d2 sinh q4 d4 p p + (F3 + F1 F2 F4 ) cosh q1 d1 cosh q2 d2 p p × sinh q3 d3 sinh q4 d4 p p + (F1 F3 + F2 F4 ) sinh q1 d1 sinh q2 d2 p p × sinh q3 d3 sinh q4 d4 ,
Thus the relationship between layer j and j + 1 (j = 1, . . . , 3) in nth period can be written as ψ j,n = Tj ψ j +1,n ,
(3)
and the relationship between layer 4 in nth period and layer 1 in (n + 1)th period is ψ 4,n = T4 ψ 1,n+1 .
(4)
Thus ψ 1,n = Tψ 1,n+1 ,
(5)
where T = T4 T3 T2 T1 , + − − T ψ j,n = A+ j,n , Bj,n , Aj,n , Bj,n .
where
However, due to the periodicity of the layered structure in the x direction, Bloch theorem [19] guarantees: ψ 1,n = e
−ikx a
ψ 1,n+1 .
(6)
Inserting Eq. (5) into Eq. (6) yields a standard complex eigenvalue problem of 4 × 4 matrix: T − e−ikx a I = 0. (7) For a given k and ω, Eq. (6) gives the four values of kx . When none of the four values is real number, there exists region with no dispersion curves, i.e., the forbidden band. For the out-of-plane modes, one can get the dispersion relations with the similar (more simple) process. When k is zero, the problem becomes two decoupled ones for the longitudinal and the transverse modes, respectively. Thus the dispersion relations for ω (embodied in qj ) as a function of wave vector kx is: cos(kx a)
p p = cosh q1 d1 cosh q2 d2 p p × cosh q3 d3 cosh q4 d4 p 1 + (F1 +F2 F3 F4 ) cosh q3 d3 2 p p p × cosh q4 d4 sinh q1 d1 sinh q2 d2 p p + (F1 F2 + F3 F4 ) cosh q2 d2 cosh q4 d4 p p × sinh q1 d1 sinh q3 d3 p p + (F2 +F1 F3 F4 ) cosh q1 d1 cosh q4 d4
(8)
F1 = F3 =
2 q1 ρ1 cp,1 2 q2 ρ2 cp,2 2 q3 ρ3 cp,3 2 q4 ρ4 cp,4
,
F2 =
,
F4 =
2 q2 ρ2 cp,2 2 q3 ρ3 cp,3
,
1 , F1 F2 F3
(9)
and p denotes l and t for the longitudinal and the transverse modes, respectively. Similar works on periodically layered binary system has been done earlier in Ref. [6], where the dispersion relation of the binary case is given as: cos(kx a)
p p = cosh q1 d1 cosh q2 d2 p p 1 1 + sinh q1 d1 sinh q2 d2 . F1 + 2 F1
(10)
For the finite periodically layered system, the transmitting frequency response function is the primary representation of its elastic wave property, instead of the dispersion relation of infinite system. In this Letter, the method of transfer matrices [5] is used in such case.
3. Result for the binary 1D PCs: 1D periodic structure of Pb and epoxy layers The parameters that include the appearance of band gaps in 1D binary PCs are the filling fraction (the fraction of one material in all), lattice constant, and the contrast of the elastic properties of different materials. The density contrast is a fundamental factor; and high
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515
Fig. 2. (a) Phononic band structure of out-of-plane (pure transverse) wave oblique to the laminations of 1D binary periodic multi-layer structure of Pb and epoxy layers. (b) Corresponding band structure of longitudinal (continuous lines) and transverse (dashed lines) waves propagating perpendicular to the laminations. (c) Corresponding band structure of in-the-plane (coupling of longitudinal and transverse) waves propagating oblique to the laminations. The filling fraction is 0.5. The shadowed regions define the band gaps. The frequencies ω are given in reduced units. a is the lattice constant, and ct,ave is the average transverse velocity inside the crystal, which is calculated by ct,ave = a/[aPb /ct,Pb + aepoxy /ct,epoxy ], where aPb (aepoxy ) and ct,Pb (ct,epoxy ) are the thickness and transverse velocity of one Pb (epoxy) layer.
contrast of it is more possible to obtain wide band gaps. Here, as a comparison example between Bragg scattering and locally resonant mechanism, we consider a 1D binary periodic system of Pb and epoxy layers. Fig. 2 shows the dispersion relations of elastic modes with a 1D binary PCs having a filling fraction f = aPb /a = 0.5, where aPb is the thickness of one Pb layer, and a = 20 mm is the lattice constant. The elastic parameters employed in the calculations were ρPb = 11600 kg m−3 , ρepoxy = 1180 kg m−3 , λPb = 4.23 × 1010 Pa, λepoxy = 4.43 × 109 Pa, µPb = 1.49 × 1010 Pa, µepoxy = 1.59 × 109 Pa. The modes of longitudinal wave (continuous lines) as well as those of transverse wave (dotted lines) propagating perpendicular to the laminations (k = 0) were illustrated in Fig. 2(b). When k = 0 (oblique incidence), the modes of out-of-plane (pure transverse wave) as well as those of in-the-plane (coupling of longitudinal and transverse waves) were illustrated, respectively, in Fig. 2(a) and (c). Band gaps, resulting from longitudinal and transverse dispersion curves, settle as the shadowed regions in the three figures. The analysis of the band structures in Fig. 2 can be straightforward for the case of pure longitudinal
or transverse modes propagating perpendicular to the laminations. The gaps are the results of destructive interference of the wave reflections in the periodic layers, the so-called Bragg scattering mechanism. So, the magnitude of the pure longitudinal or transverse midgap frequency ωl,mid or ωt,mid can be estimated. Its value should be closer to the frequency where the wave band folding takes place. For example, the estimation of the midgaps for the first pure longitudinal and transverse modes is made as follows: cl,ave 2πct,ave ωl,mid ≈ cl,ave kK = 2ct,ave a 2πct,ave , = 1.0969 (11) a 2πct,ave , ωt,mid ≈ ct,ave kK = 0.5 (12) a where kK = π/a is the wave vector at point K, a −1 ct,ave = aPb aepoxy = 1146.9 ms , ct,Pb + ct,epoxy a −1 cl,ave = aPb aepoxy = 2516.1 ms + cl,Pb cl,epoxy are the average transverse and longitudinal sound velocity inside the crystal.
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(in reduced units), the normalized gap width maximize at fPb = 0.5.
4. Result for the ternary 1D PCs: 1D periodic structure of Pb and epoxy layers separated by soft rubber layers
Fig. 3. (a) The upper (filled circles) and lower (hollow circles) edges, and midgap (cross symbols) frequencies determining the gap in a 1D binary periodic multi-layer structure of Pb and epoxy layers for several Pb filling fractions (fPb ). (b) The corresponding normalized gap width (ω/ωg ), where ωg is the midgap frequency.
In Fig. 2, the two midgap frequencies calculated are ωKL,1 + ωKL,2 2πct,ave (13) = 1.097 , 2 a ωKT,1 + ωKT,2 2πct,ave ωt,mid = (14) = 0.4999 . 2 a The estimated values (in reduced frequencies) accurately agree with the actual ones resulting from the band-structure calculation, 1.0969 and 1.0970 for the pure longitudinal mode with error in 0.01%; 0.5 and 0.4999 for the transversal mode with error in 0.02%. This result illuminates that the gaps created by a Bragg scattering mechanism appear at frequencies relate to the periodicity. So gaps at very low frequencies can only be achieved by using structures of extra large periodicity or materials with very slow elastic wave velocity. For a gap with midgap frequency of 10 Hz, that is a 1D periodic Pb/epoxy multi-layer structure with a lattice parameter of about 100 meters. Another parameter controlling the band gap is the filling fraction f . Fig. 3(a) and (b) show, respectively, the band edges and midgap frequencies of the first gap and its normalized width for the pure transverse modes when k = 0. These results represent typical behaviors associated with Bragg gap in 2D structures, in which a Bragg gap peaks at some intermediate filling fraction. Thus in Fig. 3, with the undulating of midgap frequencies near the estimated Bragg value 0.5 ωl,mid =
Very soft rubber was recently used [1,2] as the coating of heavy (Pb or Au) spherical or cylindrical inclusions arranged in certain lattice in an epoxy host. The very low elastic constants of the coating layer resulted in a strong resonant band structure with a gap at a frequency so-called two orders of magnitude lower than the expected one by Bragg scattering. In analogy with electromagnetic situations in which the dielectric function can be negative, this phenomenon was interpreted as the consequence of effective negative elastic constants in the range of frequencies where the sub-frequency gaps appear. Here, in order to find the key factor of locally resonant mechanism, we analyzed the band-gap with the periodic multi-layer ternary 1D phononic crystal illustrated in the sub-figure of Fig. 4(b), constructed by inserting thin layers of soft rubber with thickness of 1 mm in each interface of 10-mm thickness Pb and 10mm thickness epoxy layers. Thus, we separate Pb and epoxy layers with thin layer of same soft rubber [1], whose elastic parameters are ρrubber = 1300 kg m−3 , λrubber = 6 × 105 Pa, µrubber = 4 × 104 Pa. Fig. 4(a), (b) and (c) also presents the corresponding band structure calculated with Eqs. (7) and (8). Although the thickness of soft rubber is only 1/10 of Pb or epoxy, its sound velocity cannot be ignored in calculating the average velocity and the reduced frequencies of PCs because its longitudinal and transverse velocities are only 22.87 m s−1 and 5.547 m s−1 , i.e., 1/106 and 1/199 of the corresponding velocities in epoxy. For the sake of fair comparison, the reduced frequencies are calculated by ωa/2πcl,ave and ωa/2πct,ave , where cl,ave = ct,ave =
a aPb cl,Pb
+
aepoxy cl,epoxy
aPb ct,Pb
+
aepoxy ct,epoxy
+
2arubber cl,rubber
+
2arubber ct,rubber
,
a
are average longitudinal and transverse velocities.
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Fig. 4. (a) Phononic band structure of out-of-plane (pure transverse) wave oblique to the laminations of 1D ternary PCs. (b) The structure of the 1D ternary PCs (sub-figure), and the corresponding band structure of longitudinal (continuous lines) and transverse (dashed lines) waves propagating perpendicular to the laminations. (c) Corresponding band structure of in-the-plane (coupling of longitudinal and transverse) waves propagating oblique to the laminations. The thickness of Pb, epoxy and rubber are 10 mm, 10 mm and 1 mm. The frequencies ω are given in reduced units ωa/2π ct,ave , where ct,ave is the average transverse elastic wave velocity. The complete flat bands are associated with modes localized in some layers of this 1D phononic crystal (see Fig. 5).
Considered the band structures shown in Fig. 4, it is remarkable for the presence of flat bands crossing the complete BZ. These bands are real and converged, associated with eigenmodes of some layers (including soft rubber or itself only) of this periodic structure as shown in Fig. 5, where the displacements |uy (x)| are associated with the transverse modes of four different flat bands with frequencies (in reduced units ωa/2πct,ave ) 0.050, 0.151, 1.051 and 1.071, respectively, in Fig. 4(b) are represented. The first resonant eigenmode is localized within the rubber–Pb– rubber mass-spring alike structure, in which the Pb layer plays the role of mass and soft rubber layer acts as the spring. The second resonant eigenmode is localized with alike mass-spring structure of rubber– epoxy–rubber, which was not found in 2D or 3D cases [1,2] before. As the mass of epoxy layer is smaller than Pb layer at the same thickness, the frequency of the rubber–epoxy–rubber eigenmode is higher than of rubber–Pb–rubber. The third and the fourth resonant eigenmodes are localized in the soft rubber. The very low elastic constants of soft rubber allow the existence of the first two propagating modes at very low frequencies. Similar flat bands were also found in 2D and 3D locally resonant systems [1,2].
Fig. 5. (a), (b), (c), (d) Modulus of the displacement vector |u(x)| for the transverse modes associated with the first four flat bands (corresponding to KT, 1 , KT, 2 , KT, 3 and KT, 4 in Fig. 4, respectively). The abscissa scale is given in unit of mm. In all cases the displacement occurs only inside certain layers of the periodic structure (see text for details).
One noticeable feature in Fig. 4 is the appearance of complete gap in frequency region about 1/5 of magnitude lower than the expected ones in Bragg conditions. Two orders of magnitude lowness [1,2] in 2D and 3D cases cannot be got here due to the scales of
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Fig. 6. The frequencies of lower four pure longitudinal modes at K point (KL, 1 , KL, 2 , KL, 3 and KL, 4 ) determining the gaps in a 1D ternary phononic crystal of Pb and epoxy layers separated by soft rubber layers for several (a) Pb, (b) epoxy and (c) rubber thickness. The inset of sub-figure (c) shows a zoom of the corresponding total graph in the low frequency region.
different reduced frequencies were used in Refs. [1,2] and the present Letter. In Refs. [1,2], the reduced frequency is calculated by ωa/2πct,epoxy, where the ct,epoxy = 1160.8 m s−1 is the transverse velocity of epoxy. However, for fair comparison and discussion in this Letter, ct,epoxy is replaced with the average transverse velocity ct,ave in the 1D ternary PCs, where ct,ave = 58.2 m s−1 , 20 times lower than the velocity in epoxy. The sub-frequency gap is originated from the resonances arising from the insertion of soft layers, as pointed out by Liu [1,7], Goffaux [2] and their coworkers. In order to support the resonant origin of subfrequency gaps, we plot in Fig. 6(a)–(c) the dependency of band edges frequencies related to thickness of Pb, epoxy and soft rubber layers, respectively. Also we plot in Fig. 7 the dependency of the first sub-frequency band gap related to the assumed density of inserting layer (soft rubber layer).
Fig. 7. The dependency of the first sub-frequency band gap related to the assumed density of inserting layer (soft rubber).
The behavior of the band edges is completely different from that of a Bragg gap. In Fig. 6(a) only one (Pb-edge) of the two lower band-gap edges depends on the thickness of Pb layer, and the other (epoxyedge) changes only with the thickness of epoxy layer in Fig. 6(b). The upper two edges (rubber-edges)
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Fig. 8. (a), (b), (c) Modulus of the displacement vector |u(x)| for the longitudinal modes at A, B and C points in Fig. 6, respectively. The abscissa scales are given in unit of mm. The modes are associated with the first four flat bands.
Fig. 9. (a) The calculated longitudinal band structures of a designed 1D PC of the structure in Fig. 1 with lattice constant of 15.5 mm in which Pb, rubber and epoxy layers are in thickness of 10 mm, 2 mm and 1.5 mm. (b) Corresponding calculated frequency response functions (FRF) of two, three and four periods swatch of the designed 1D PC (solid lines) and a 3-period finite structure except 48 volume % of randomly dispersed thickness of epoxy layers (dashed line). A wide sub-frequency gap from 380 Hz to 2150 Hz can be predicted.
change little with the thickness of Pb and epoxy layers. All the edges have significant changes with the thickness of soft rubber. Farther in Fig. 7, the edges of the first sub-frequency band gap are independent of the density of inserting layers. As the elastic wave velocity depends on its density, this means that the first sub-frequency gap frequency is independent of the Bragg conditions, i.e., not restricted within the so-called two orders of magnitude lowness [1,2] in frequency. These are signatures of their resonant origin. Particularly, the Pb-edge (epoxy-edge/rubberedges) is a consequence of the strong localized character of its associated mode, which consists of vi-
bration of the Pb (epoxy/rubber) layer. Here, the rubber layer acts as a soft spring at the Pb- and epoxy-edges of first sub-frequency gap, and the density of rubber affects as a mass at the rubber-edges. More intuitionistic understanding can be got with Figs. 5(a)–(d) and 7. The transformation course between Bragg scattering and locally resonant mechanism is also analyzed in Figs. 6(c) and 8 by selecting three representative points with different thickness of rubber layer. Point A locates at arubber = 0.0005 mm, where the equivalent stiffness of rubber layer are too big (compared with the epoxy layers) to be treated as a single spring:
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krubber = scell (λrubber + 2µrubber)/arubber N = 1.28 × 1012scell , m kepoxy = scell (λepoxy + 2µepoxy)/aepoxy N = 0.761 × 1012scell , m
(15)
(16)
and no locally resonant evidence can be found in the corresponding modes (plotted in Fig. 8(a)). Points B and C locates at arubber = 0.08 mm and 0.12 mm, respectively, where the equivalent stiffness of rubber layer are small enough for the formation of locally resonant mechanism at the edges of the first subfrequency band gap (shown in Fig. 8(b), (c)). But for the rubber edges, the contrast of mass and stiffness of rubber layers are not big enough for the energy to localize only in the rubber layers at point B. Using the results foregoing, we design a 1D PC of the structure in Fig. 4 with lattice constant of 15.5 mm (composed of Pb, rubber and epoxy layers in thickness of 10 mm, 2 mm and 1.5 mm, respectively). A wide longitudinal subfrequency gap from 380 Hz to 2150 Hz can be predicted in it. Fig. 9 shows the numerical simulation results of the designed 1D PC, i.e., the band structures (Fig. 9(a)), frequency response functions (FRF) of two, three and four periods swatches (solid lines in Fig. 9(b)), and a 3-period finite structure except 48 volume % of randomly dispersed thickness of epoxy layers (dashed lines in Fig. 9(b)).
5. Conclusion In conclusion, we have studied the propagation of longitudinal and transverse elastic waves perpendicular to the laminations of infinite periodically layered fourfold system. One-dimensional phononic crystals composed of periodically layered binary and fourfold system (with low and high contrast of elastic constant) were studied and compared. The band structures and the associated resonant modes of the 1D ternary PCs of locally resonant structures composed of periodic Pb and epoxy layers separated by soft rubber ones (a periodically layered fourfold system). Its resonant mechanism was adequately proved with the evidences of flat bands, lower sub-frequency gaps, the dependency of band edges on rubber’s density and thickness of each layer, and the FRF of finite structure with randomly
periodicity. Different from the 2D or 3D cases, 1D problem shows an additional resonance, i.e. the resonance of epoxy in soft rubber. Finally, with the results in this Letter, we have designed a 1D locally resonant PC with lattice constant of 1.55 cm, and have predicted a wide sub-frequency gap from 380 Hz to 2150 Hz in it. As one-dimensional multi-layered structure is more practicable, these findings will be significant in the application of phononic crystals.
Acknowledgements This work was supported by the State Key Development Program for Basic Research of China (Grant No. 51307). G. Wang gratefully acknowledges Prof. Zhengyou Liu for the valuable discussion with him as well as the helpful suggestion from him about the research works in this work.
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