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1. Cap 5 a a fF In the cvasistatic approximation the forces of gravity ad inertia within the solid medium are neglected the fundamental principle of dynamics applied to a solid particle is written pita og SH 6 aep UT Be Bice where tf k 1 3 and the position vector will be note T x Yz sz p is the mass density Due to the spatial periodicity of the phononic structure Figure 2 the material constants alr Cita F can be expanded in the Fourier series with respect to the two dimensional reciprocal lattice vectors RLV where amp 6 G is the 2D reciprocal vector air e apliit 6 x wt T Cogirl y explilt Gx wije 8 K EE Ep E3 9 is the Bloch wave vector and the frequency a Inf ri v 10 The components of the displacement vector can be written as uir ty Aeppli lEn xp wt 11 used the Bloch theorem it rewrites ufri Dhar de enp lil E Gya wt lenp iiza 12 The wave propagation is chosen independent of the direction z The components of displacements on the directions x u and u2 are decoupled from u3 So there is a decoupling of transverse vertical TV which are polarized along the direction z from other waves called longitudinal waves L and horizontal shear waves TH where the speeds of this waves are Y rite rr Cristina PACHIU Victor MOAGAR 196 2 2 BOUNDARY CONDITIONS AND PROPAGATING MODES For Lamb modes propagating in a2. and the filling fraction is f 2 18 RESULTS The dispersion relations in 2D PhnCs for plate wave s propagation Eq 19 are solved by the program MATLAB The program is designed for modeling engineering problems described by partial differential equations and it is possible to combine different physical models and solve physics problems To validate the numerical model developed we chose for the first time a structure one dimensional with silicon isotropic material where the physical characteristics are summarized in Table 1 and the dispersion curves are presented in Figure 3 The band gap in the structure chosen as model is located near the frequency 1 4 MHz and the results are valid with the literature 20 197 Wave propagation in two dimensional phononic crystal plate 2 Symmetry rae Elastic constants x10 N m SL___sotrope fest 6 te L a a SA CCN lt gt OL A E a ee Ge a A O e e Table 1 The elastic proprieties of Si and LiNbO3 3 0 5 0 1 0 2 0 3 0 4 0 5 Reduced wave vector Figure 3 Dispersion curves of Lamb wave modes for the 1D two component composite plate with unit cell area 0 2 length 0 1 mm height 0 02 mm and lattice parameter 2 mm The shaded region represents the first band gap of Lamb waves On abcisa is frequency in MHz In general we follow the realization a SAW devices with phononic crystals in the active zone so we are interested to study the frequency band gaps in piezoe
3. composites Surface waves propagating at the surface of a two dimensional Al Hg phononic crystal have been observed for first time by Torres et al 19 This work is to investigate theoretically and numerically the characteristics of the surface elastic acoustic waves and plane elastic waves in 2D periodic anisotropic structures based on the PWE method In the theoretical study we consider 2D phononic crystal with general anisotropy and square lattice in Brillouin zone where the host material the circular cylinders of radius r are embedded periodically in background in a solid matrix with lattice spacing a Figure 2 We choose the x y plane as the plane of the waves guide and z is parallel to the cylinder All the elastic parameters are independent of z because the 2D phononic medium is invariant in that direction After this the symmetry of materials is lowered to trigonal and isotropic for the first numerical results obtained with Matlab program Figure 1 Schematic description of a periodic phononic crystal one two and three dimensions of the typical structure 2 PLANE WAVE EXPANSION IN PLANE STRUCTURE In the first subsection are essentially a summary of the PWE method originally exposed in Ref 5 in the context of general anisotropy in materials Subsection II 2 are extensions of the theory to the representation of air holes in a phononic crystal and to the problem of identifying plane modes respectively Consider a phononic cr
4. SISOM 2011 and Session of the Commission of Acoustics Bucharest 25 26 May WAVE PROPAGATION IN TWO DIMENSIONAL PHONONIC CRYSTAL PLATE Cristina PACHIU Victor MOAGAR National Institute for R amp D in Microtechnologies Erou Iancu Nicolae 126A Bucharest 077190 Romania Corresponding author Cristina Pachiu 021 2690772 cristina pachiu imt ro The studies of propagation elastic waves into composite materials have become considerably more frequent in the last years especially for the composite materials called phononic crystals PhnCs which show recurrent variations of the elastic constants and density The field of PhnCs is only about 10 years old and the search for the best phononic structure is still ongoing Phononic crystals will provide new components in acoustics and ultrasonic fields offering functionalities and level of control comparable to the light field The sizes of the crystals are directly proportional with the wave length therefore making it possible to create crystals which vary from macro meters to nanometers and frequencies which vary from Hz to THz Considering the dimensionality rule these materials can be used in manufacturing from phonic isolating systems to filters multiplexers or sensors This study is focused on opening the banned frequency bands into a phononic 2D material through theoretical and numerical studies on surface elastic waves emission We expose the first results of the numerical simulation conducte
5. chez Dehesa Two dimensional phononic crystals studied using a variational method Application to lattices of locally resonant materials J Phys Rev B 67 pp 144301 2003 3 Martinez Sala R Sancho J Sanchez J V Gomez V Llinares J Meseguer F Sound attenuation by sculpture Nature 378 241 1995 4 Klironomos A D Economou E N Elastic wave band gaps and single scattering Solid State Commun 105 327 1998 5 Sigalas M M Soukoulis C M Elastic wave propagation through disordered and or absorptive layered systems Phys Rev B 51 2780 1995 6 Wu T T Huang Z G Lin S Surface and bulk acoustic waves in two dimensional phononic crystal consisting of materials with general anisotropy Physical review B 69 094301 2004 7 Liu Z Chan Sheng C T Goertzen A L Page J H Elastic wave scattering by periodic structures of spherical objects Theory and experiment Phys Rev B 62 pp 2446 2000 199 Wave propagation in two dimensional phononic crystal plate 8 Yang S Page Liu J H Cowan M L Chan C T Sheng P Ultrasound Tunneling through 3D Phononic Crystals Phys Rev Lett 88 104301 2002 9 Kafesaki M Economou E N Classical vibrational modes in phononic lattices theory and experiment Phys Rev B 60 pp 10 11 12 13 14 15 16 I7 18 19 20 11993 1999 Psarobas I E Sigalas M M Phon
6. curves for square lattice air holes LiNbO3 PhnCs with filling factor 64 CONCLUSIONS In this work a plane wave expansion method suited to the analysis of acoustic wave propagation in phononic crystals has been described for two dimensional phononic crystal consisting of different materials with general anisotropy This model brings crowns synthesis of various results so far in the specialty literature by various authors Our contribution comes in theoretical model for obtained the surface modes of a square lattice PhnCs made of trigonal structure of lithium niobate material with air circular inclusions A band gap for surface waves with a filling factor 64 has been found and will continue to optimize the model calculations This analytical study is a first stage of our further investigations to obtain SAW devices with PhnCs materials In the future work we do make theoretical model for a piezoelectric material PhnCs structure to optimize the geometry of the structure for high frequency band gap up to several GHz ACKNOWLEGMENTS The research presented in this paper is supported by the Sectoral Operational Programme Human Resources Development SOP HRD financed from the European Social Fund and by the Romanian Government under the contract number POSDRU 89 1 5 S 63700 REFERENCES 1 Liu Z Zhang X Mao Y Zhu Y Y Yang Z Chan C Sheng T Locally Resonant Sonic Materials Science 289 pp 1734 2000 2 Goffaux C San
7. d on dispersion curves for a PhnCs structure obtained using the plane wave expansion method PWE Key words Phononic crystal Plane wave expansion method Phononic band gaps 1 INTRODUCTION The study of elastic acoustic wave s propagation in periodic structures has originated many novel discoveries in physics in the past decade 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 PhnCs 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 17 by analogy with the electronic or photonic band gap in natural or artificial crystals Phononic crystals are similar to photonic crystals but for the peculiarities of elastic as compared to optical waves These can be created by a one dimensional two dimensional or three dimensional periodic arrangement of inclusions in a host matrix Figure 1 Phononic crystals will provide new components in acoustics and ultrasonic fields offering functionalities and level of control comparable to the light field The sizes of the crystals are directly proportional with the wave length therefore making it possible to create crystals which vary from macro meters to nanometers and frequencies which vary from Hz to THz Considering the dimensionality rule th
8. efects Phys Rev Lett 82 pp 3054 1999 Vasseur J O Hladky Hennion A C Djafari Rouhani B Duval F Dubus B Pennec Y Locally resonant phononic crystals with multilayers cylindrical inclusions Journal of applied physics 101 pp 114904 2007
9. ese materials can be used in manufacturing from phonic isolating systems to filters multiplexers or Sensors Several theoretical methods have been used to study the elastic acoustic structures such as the transfer matrix TM method 5 the multiple scattering theory MST 7 11 the plane wave expansion method PWE 12 14 the finite difference time domain method FDTD 15 17 and finite element method FEM 18 In refs 7 11 the MST theory was applied to study the band gaps of three dimensional periodic acoustic composites and the band structure of a phononic crystal consisting of complex and frequency dependent Lame coefficients Garcia Pablos used the FDTD method to interpret the experimental Cristina PACHIU Victor MOAGAR 194 data of two dimensional systems consisting of cylinders of fluids Hg air and oil inserted periodically in a finite slab of Al host FEM is one such method which solves the partial differential equations numerically by subdividing the whole domain into finer meshes called elements Finite element method is a powerful technique to solve partial derivation equations and has long been used by engineers to solve mechanical structural and electrical problems There exists a lot of literature on usage of FEM to model acoustic wave devices but many of them are limited to bulk wave type 18 Kushwaha utilized the PWE method to calculate the first full band structure of the transverse polarization mode for periodic elastic
10. lectric materials like as PZT LiNbO or quartz In this work we are not discussed the piezoelectric proprieties and we are focusing to develop the computer algorithm only for PhnCs with high anisotropy Still we exemplify the theoretical theory PWE in the case of trigonal symmetry for lithium niobate with the square phononic structure where the wave s propagation in the plane of the surface are along the plan x y In Figures 4 and 5 the dispersion curves of elastic waves for two structures PhnCs with different filling ratios are plotted Holes in the structure have a diameter of 412 nm a respectively 654 nm b We observed an increase in the frequency band gap with increasing filling ratio of the structure For a report of 64 was obtained the frequency band gap until 180 MHz and 270 MHz Figure 3 b In future simulations we will watch for the different geometry of PhnCs filling factor to obtain an optimal structure with frequency band gap as high close to GHz specify the type of LiNbO and useful in designing the GHz SAW devices 9 e a ee a NSwennesel Sgeceen a5e5 Normalized frequency Ww ia Or a gt P 3 4 5 6 7 Reduced wavevector Figure 4 Dispersion curves for square lattice air holes LiINbO PhnCs with filling factor 49 Cristina PACHIU Victor MOAGAR 198 r ki E E E E E Ep a nnet TIT Lett a efeeeeeee Eiis m ltt LET Normalized frequency Reduced wavevector Figure 5 Dispersion
11. onic crystals with planar defects Phys Rev B 66 pp 052302 2000 Hoskinson Z Ye Band gaps and localization in acoustic propagation in water with air cylinders Phys Rev Lett 77 pp 4428 2000 Kushwaha M S Djafari Rouhani B Acoustic Band Structure of Periodic Elastic Composites J Appl Phys 84 pp 4677 1998 Li X Wu F Hu H Zhong S Liu Y Large acoustic band gaps created by rotating square rods in two dimensional periodic composites J Phys D Appl Phys 36 010015 2003 Wu F Hou Z Liu Z Liu Y Stop Gaps and Single Defect States of Acoustic Waves in Two Dimensional Lattice of Liquid Cylinders Phys Lett A 292 198 2001 Garc a Pablos D Sigalas M Montero de Espinosa F R Torres Kafesaki M M Garc a N Phys Rev Lett 84 4349 2000 Sigalas M M Garcia N Appl Phys Lett 76 pp 2307 2000 Tanaka Y Tomoyasu Y Tamura S Band structure of acoustic waves in phononic lattices Two dimensional composites with large acoustic mismatch Phys Rev B 62 pp 7387 2000 Lerch R Simulation of Piezoelectric Devices by Two and Three Dimensional Finite Elements IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 37 pp 233 247 1990 Torres M Montero de Espinosa F R Garcia Pablos D Garcia N Phnonic Band Gaps in Finite Elastic Media Surface States and Localization Phenomena in Linear and Point D
12. plate structure h 2 lt z lt h 2 h is the plate thickness all solutions are kept 1 e a set of 6n partial waves The displacement vectors can be expressed as u r ft 2 exp i k Gy et iw 3 gt A exp i Ha Thexw lilk Ehe iwe DPA of exp tact 13 where is the associated eigenvector of the eigenvalue i The prime of the summation denotes that the sum over is truncated up to n X is the undetermined saan coefficient which can be determined from the traction free boundary conditions on the surface z 0 KA z0 roma ty e he O7 13 14 i rz Substituting Eq 13 into Eq 14 we have _ mi MG MiS A MS ow MEMS eas 15 ly ag tt ita Mie Mg Ma Min M isa3mX3m matrix and its components are MEL CF 5 C8 t 28 ee Ge 16 M e EH i Jet 4 Be ad 17 MA eh eh e 2 Cis 6 et C2 402 402 Ore Ge em ten ea pae es ed eA For the existence of a nontrivial solution of X the following condition must be satisfied 1 e det 0 19 Equation 19 is the dispersion relation for surface waves propagating in two dimensional phononic crystals with both the filling material and the background material belong to the triclinic system In this work we are considered the phononic structures with square lattice In figure XX are the Brillouin region of the square lattice where the reciprocal lattice vector G is E p fo ae i 20 N N 0 21 2
13. ystal composed of a two dimensional periodic array x y plane of material A embedded in a background material B Figure 2 Both materials are supposed to be homogenous and continuous with the highest symmetry 1 e belonging to the triclinic symmetry a b Figure 2 a Square lattice for 2D PhnCs with of cylindrical holes in solid matrix b First irreducible Brillouin zone corresponding of this periodical structure 195 Wave propagation in two dimensional phononic crystal plate 2 2 PWE BASICS METHOD In an elastic solid medium the mechanical properties of the material used to connect the stress to deformation by a law called the law of behavior l hypoth se in small strain this relationship is linear Sap Ce rer kt 1 where i f k b 1 3 fran is the elastic stiffness tensor and 5 is the strain displacement The strain tensor is defined in relation to movement u r A fie Le Substituting Eq 2 into Eq 1 we obtain the law of Hooke L fun fur SIE 4 L z ax t Eny a Voigt notation is the standard mapping for tensor indices and coupling indices ij and kl as follows Gyz Cami yZ Zy XZ ZX XY YX j l 2 3 4 5 6 It is then possible to use contracted notation and Hooke s law is written aag Caps 4 where 7 6 The elasticity matrix with its 21 elasticity coefficients represents the most general case of crystal system triclinic symmetry This matrix takes the form t Eut
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