Accurate evaluation of lowest band gaps in ternary locally resonant phononic crystals

Transcription

1 Vol 15 No 8, August 6 c 6 Chin. Phys. Soc /6/15(8)/ Chinese Physics and IOP Publishing Ltd Accuate evaluation of lowest band gaps in tenay locally esonant phononic cystals Wang Gang( ) a)b), Shao Li-Hui( ) c), Liu Yao-Zong( ) a)b), and Wen Ji-Hong( ) a)b) a) Institute of Mechatonical Engineeing, National Univesity of Defense Technology, Changsha 4173,China b) Photonic/Phononic Band Gap Reseach Cente, National Univesity of Defense Technology, Changsha 4173,China c) Astonaut Cente of China, Beijing 194, China (Received 16 Septembe 5; evised manuscipt eceived 9 Apil 6) Based on a bette undestanding of the lattice vibation modes, two simple sping mass models ae constucted in ode to evaluate the fequencies on both the lowe and uppe edges of the lowest locally esonant band gaps of the tenay locally esonant phononic cystals. The paametes of the models ae given in a easonable way based on the physical insight into the band gap mechanism. Both the lumped-mass methods and ou models ae used in the study of the influences of stuctual and the mateial paametes on fequencies on both edges of the lowest gaps in the tenay locally esonant phononic cystals. The analytical evaluations with ou models and the theoetical pedictions with the lumped-mass method ae in good ageement with each othe. The newly poposed heuistic models ae helpful fo a bette undestanding of the locally esonant band gap mechanism, as well as moe accuate evaluation of the band edge fequencies. Keywods: phononic cystals, locally esonant, analog model PACC: 43, 4335, 6, 816H 1. Intoduction In ecent yeas, a geat deal of wok has been devoted to the study of the popagation of elastic o acoustic wave in peiodic stuctues composed of diffeent mateials, [1 16] which ae efeed to as the phononic cystals (PCs) [1] by analogy with the photonic cystals [17,18] fo electomagnetic waves. Because of the peiodicity in such stuctues, thee exist fequency anges in which elastic waves ae fobidden. This phenomenon can be of eal inteest because of the ich physics of elastic system, whee the wave can mix longitudinal modes with tansvese modes and a lage contast between the elastic paametes is allowed as well. A well-known example is the locally esonant (LR) PCs consisting of vey soft ubbe [,5] (with an elastic constant of five odes lowe than that of common solids) and othe components. These new mateials ae most likely to obtain the low-fequency gaps with stuctues of small dimensions, leading to pomising applications such as low-fequency vibation/noise insulations. The lumped-mass (LM) method is employed in the study of two-dimensional (D) and theedimensional (3D) tenay LR PCs in this pape, fistly poposed in the 1D fom [9] and ecently extended to two- [11,13] and thee-dimensional foms. The method is based on the idea of the discetization of the continuous system. It has been concluded [11,13] that the LM method conveges faste than the taditional techniques and its convegence is insensitive to the shap vaiation in elastic constant on the intefaces inside PCs. Moeove, it can be easily used in the band gap calculation of PCs with complex lattice stuctues. These advantages ae unique in compaison with the othe wok [5,14] on the impovement of taditional methods. Seveal analogous models have been used in the eseach of the mechanism of low-fequency band gaps in the LR PCs. [5 7,1] A simple mechanical model of mass and sping was fist used by Goffaux, et al [5,6] in ode to fit the tansmission specta nea the fist esonant peak and the edges of the fist band gaps in a D tenay LR PCs. We also establish a simila model in ode to claify the oigin of the LR band gaps in D binay LR PCs. [1] Howeve, the paametes of these models [5,6,1] ae given, not calculated with a easonable method based on physical insights. Poject suppoted by the National Natual Science Foundation of China (Gant No 5575) and the State Key Development Pogam fo Basic Reseach of China (Gant No 5137). wang-g@vip.sina.com

2 1844 Wang Gang et al Vol.15 Hisekon used a diffeent simple model [7] to evaluate the fequencies on the lowe edges of the band gaps in D tenay LR PCs, and a easonable deivation of the paametes in the model is given. Howeve, the thin ubbe layes in ac shape ae teated as simple equalthickness flat layes in his model, [7] and the model can be used only to simulate the lowe edges of the band gaps with slight mismatches. But Hisekon s wok [7] does motivate the pesent study. Hee, in ode to study the oigin of the LR band gaps moe sufficiently, we estudy the lattice vibation modes that ae elated to the fomation of the lowest LR band gaps of typical [] tenay PCs with the LM method. An additional analogous model is constucted to epesent the vibation modes on the uppe edges of the LR band gaps. The paametes of the two models ae calculated based on a clea undestanding of the physical insight. The two models and the aithmetic of its paametes ae veified with the LM method by changing the stuctual and mateial paametes of the tenay LR PCs.. Lattice vibation modes of the tenay LR PCs Figue 1 illustates the band stuctue of typical 3D tenay LR PCs [] that is composed of simple cubic aays of coated Pb sphees immesed in epoxy. Figue illustates the lattice vibation modes coesponding to points L 1,,3, L 4,5,6 and L 4,5,6 in Fig.1. Fig.1. The band stuctue of typical 3D [] tenay LR PCs calculated with the 3D LM method. The shadowed egion epesents the lowest band gap. Fig.. Lattice displacement vectos in thee pependicula cutaway intefaces inside the 3D tenay LR PCs coesponding to (a) L 1,,3, (b) L 4,5,6 and (c) L 4,5,6 in Fig.1 espectively. The diection and the length of each aow epesent the diection and amplitude of the displacement vecto at the stating point of the aow. The shadowed egion, the inside and the outside epesent the coating ubbe, Pb coe and epoxy host espectively.

3 No. 8 Accuate evaluation of lowest band gaps in Simila to the case of D tenay LR PCs, [1] Fig.3(a) illustates a clockwok-like tosion LR mode that cannot geneate a gap. [1] Howeve, The lowest LR band gaps in 3D tenay LR PCs ae geneated by the 4 6th LR modes L 4,5,6 that ae combined togethe and illustated in Fig.(b), whee the coe vibates as a igid sphee and the coating acts as spings. Dynamic balance of the whole system is insued by the fixed delay in phase of adjacent lattices. As to the othe thee vibation modes at points L 4,5,6 in Fig.1 that ae illustated in Fig.(c) (on the uppe edge of the lowest band gap), the host media vibates with an opposite phase to the coe. The synchonous vibation of the whole system insues the dynamic balance. model illustated in Fig.3(b). In the model, the new paticles m epesents the equivalent mass of host media in a lattice. As the phenomena in the D and 3D tenay LR PCs aound the lowest band gaps can be easily descibed with the sample models in Fig.3, they may be used to evaluate the fequencies on the edges of the lowest band gap. This evaluation is vey helpful fo the simplification of the design of the LR PCs. Howeve, accuate evaluations need the coect calculations of the paametes in the analogous models. Hee, thee calculation methods ae used and compaed in the evaluation of fequencies of the lowest band gaps in D and 3D tenay LR PCs. Based on the detailed vibation modes illustated in Fig., we can see that only ubbes in egion A (illustated in Fig.4) ae mainly compessed and stetched when the coe vibates up and down. So stiffness k in Fig.3 can be calculated with paametes in egion A. Fig.3. Analogous models coesponding to vibation modes on the (a) lowe and (b) uppe edges of the lowest LR band gap. The coesponding lattice vibation modes of the D LR PCs have been discussed patly befoe. [1] As the lattice vibation modes ae simila to those in the coesponding 3D case, they ae elided in this pape. 3. Analogous models and the calculation of elevant paametes Based on the above discussion, we can conclude as follows. 1) The vibation modes on the lowe edges of the lowest band gaps in both the D and the 3D LR PCs coespond to the mass sping model illustated in Fig.3(a). In the model, paticles m 1 epesents the equivalent mass of one oscillato and sping k epesents the equivalent stiffness of the oscillato. ) The vibation modes on the uppe edges of the lowest band gaps in both the D and the 3D LR PCs can be descibed with a new mass sping mass Fig.4. (a) The coss sections of the D and 3D tenay LR PCs. (b) and (c) Sketch maps explaining the calculation methods of paametes in the analogous models coesponding to D and 3D tenay LR PCs espectively. Fistly, Hisekon s method [7] is used as method I in this pape. In method I, the m 1 and m in Fig.3

4 1846 Wang Gang et al Vol.15 ae calculated with the mass of the coe and epoxy host in a lattice espectively, and the ubbe in egion A is teated as two layes with equal-thickness fo simplicity. Thus, fo the typical D tenay LR PC, we have m 1 = m coe = ρ coe π coe, (1) m = m host = ρ host (a πext ), () k = 4C 11 coe ext coe, (3) and fo the typical 3D one, m 1 = m coe = 4 3 ρ coeπ 3 coe, (4) m = m host = ρ host (a π3 ext ), (5) Fig.6. Fequencies on the edges of the lowest band gaps in the typical (a)d and (b)3d tenay LR PCs at seveal densities of the host mateial. k = C 11π coe ext coe, (6) whee C 11 = λ coating + µ coating ; λ coating and µ coating ae the Lame constants of the ubbe in the coating laye. With the two models illustated in Fig.3, the fequencies on the edges of the lowest LR band gap can be evaluated with f 1 = 1 k ; f = 1 k(m 1 + m ). (7) π m 1 π m 1 m Using method I and the model in Fig.3, we evaluate the fequencies of the lowest band gap of the typical 3D tenay LR PCs with diffeent stuctual and mateial paametes (dotted lines in Figs.5 9) and compaed them with that calculated with the LM method (solid lines with dots in Figs.5 9). Fig.7. Fequencies on the edges of the lowest band gaps in the typical (a)d and (b)3d tenay LR PCs at seveal densities of the coating mateial. Fig.8. Fequencies on the edges of the lowest band gaps in the typical (a)d and (b)3d tenay LR PCs at seveal adiuses of the coe. Fig.5. Fequencies on the edges of the lowest band gaps in the typical 3D tenay LR PCs at seveal densities of the coe.

5 No. 8 Accuate evaluation of lowest band gaps in Fig.9. Fequencies on the edges of the lowest band gaps in the typical (a)d and (b)3d tenay LR PCs at seveal oute adiuses of the coating laye. Fom these figues, we can see that the fequencies evaluated with method I ae highe than the theoetical ones, especially on the uppe edge of the band gap. The analogous models with paametes calculated with method I match with the D and 3D LR PCs basically only on the lowe edges of thei band gaps. These mismatches ae due to the undeestimate of stiffness k, i.e. the equivalent stiffness of coating laye in egion A in Fig.4. Hisekon s simplification, whee the ubbe is teated as a laye with an equalthickness, is not accuate because the thickness (along the diection of wave popagation) in the coating laye is only at the cente of egion A and longe than at othe pats in egion A. In method II, we impove it by egading each slende ba along the diection of wave popagation in egion A as a tiny sping, and stiffness k as the sum of the stiffnesses of all these tiny spings in paallel connection. Thus, using the integal technique, we have the new effective stiffness k fo D case as k = 4C 11 coe and fo the 3D case as k = C 11 = C 11 dx ext x coe x = C 11 π (x +y ) coe coe [ ] coe + π coe/ + extacsin( coe / ext ) ext coe ext, (8) coe 1 ext x y coe x y dxdy ddθ ext coe = 4πC 11 3( ext coe ) [ 3 ext + 3 coe ( ext coe )3/]. (9) Using method II, we evaluate again the fequencies of the lowest band gaps in D and 3D tenay LR PCs with diffeent stuctual and mateial paametes (dashed lines in Figs.5 9) and compae them with the calculated with the LM method (solid lines with dots in Figs.5 9). We can see that the fequencies evaluated with method II ae much moe accuate than those evaluated with method I. Howeve, thee still exists a lage mismatch accoding to the uppe edges of the band gap. Especially in Fig.7, when the density of the coating laye changes, the evaluated fequencies should emain unchanged accoding to methods I and II while the actual values decease obviously in the invese popotion to the density. This is due to the fact that the density of coating laye is not taken into consideation in the calculation of paametes in models illustated in Fig.3. We can find that when the LR PCs vibate as they ae modelled in Fig.3(b) (on the uppe edge of the band gap), thee exists a standing point that is immovable in the coesponding vibation modes. Thus the mass of the coating laye in egion A can be divided into two masses and they ae added to m 1 and m in Fig.3(b) espectively. Based on the detailed vibation modes illustated in Fig., we can see that the coating laye in egion B vibates as the attachment of the host laye does, which means that the mass of egion B can be added to m diectly. Thus in method III, the expessions fo the calculation of m 1 and m ae changed into α m 1 = m coe + m A 1 + α, 1 (1) m = m host + m B + m A 1 + α, whee α = m /m 1 = m A + m B + m host m A + m coe. (11) Fo the D case, the mass of the coating laye in egion A and that in egion B, espectively, ae

6 1848 Wang Gang et al Vol.15 m A = ρ coating π(ext coe ) m B, ext ) m B = ρ coating ext x dx = ρ coating (accos( coe / ext ) ext coe ext coe. (1) coe As fo the 3D case, we have m A = 4 3 ρ coatingπ( 3 ext 3 coe) m B, m B = ρ coating π ext coe ext ddθ = 4 3 ρ coatingπ( ext coe )3/. (13) The fequencies evaluated with method III and illustated as solid lines in Figs.5 9 match well with the theoetical esults on both the uppe and the lowe edges of the LR band gaps. As fo the D tenay LR PCs, we can obtain simila conclusions. By using the method III, we can also evaluate the fequencies on both the lowe and uppe edges of the fist band gaps of the D tenay LR PCs successfully. 4. Conclusions The vibation modes on the lowe and the uppe edges of lowest band gaps of the tenay locally esonant phononic cystals can be descibed with the mass sping fixtue and the mass sping mass models espectively. The paametes of the models ae given in a easonable way based on the physical insight into the band gap mechanism. The analytical estimations with the models and the theoetical pedictions with the lumped-mass method ae in good ageement with each othe, especially fo the uppe edge of the locally esonant band gap. The poposals of these models ae heuistic and helpful fo a bette undestanding of the locally esonant band gap mechanism as well as accuate evaluation of the coesponding band edge fequencies. The latte makes it possible to make a simple design of locally esonant gaps in phononic cystals, which is impotant fo thei applications in low fequency vibation/noise sheltes. 5. Acknowledgment One of the authos, G. Wang, wishes to thank M M. Hisekon fo his heuistic discussion. The authos would like to thank the anonymous efeee fo valuable comments and suggestions. Refeences [1] Kushwaha M S, Halevi P, Dobzynski L and Djafai- Rouhani B 1993 Phys. Rev. Lett. 71 [] Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T and Sheng P Science [3] Matínez-Sala R, Sancho J, Sánchez J V, Gómez V, Llinaes J and Mesegue F 1995 Natue (London) [4] Goffaux C, Sanchez-Dehesa J and Lambin P 4 Phys. Rev. B [5] Goffaux C and Sánchez-Dehesa J 3 Phys. Rev. B [6] Goffaux C, Sánchez-Dehesa J, Yeyati A L, Lambin P, Khelif A, Vasseu J O and Djafai-Rouhani B Phys. Rev. Lett [7] Hisekon M 4 Appl. Phys. Lett [8] Hisekon M, Delsanto P P, Bata N K and Matic P 4 Ultasonics 4 31 [9] Wen J H, Wang G, Liu Y Z and Yu D L 4 Acta Phys. Sin (in Chinese) [1] Wang G, Liu Y Z, Wen J H and Yu DL 6 Chin. Phys [11] Wang G, Wen J and Wen X 5 Phys. Rev. B [1] Wang G, Wen X, Wen J, Shao L and Liu Y 4 Phys. Rev. Lett [13] Wang G, Wen J, Liu Y and Wen X 4 Phys. Rev. B [14] Cao Y, Hou Z and Liu Y 4 Phys. Lett. A [15] Hua J, Zhang S and Cheng J C 5 Acta Phys. Sin (in Chinese) [16] Qin B, Chen J J and Cheng J C 5 Chin. Phys [17] Yablonovitch E 1987 Phys. Rev. Lett [18] John S 1987 Phys. Rev. Lett