Analyses of elastic waves in Aluminum/Barium sodium niobate and Quartz/Epoxy phononic structures

This study was aimed at using the theory for two-dimensional phononic crystal consisting of materials with general anisotropy to calculate the band gaps phenomena in Aluminum/Barium sodium niobate (Y-cut) with hexagonal lattice and Quartz/Epoxy with square lattice phononic structures. Wave propagation properties of solids in which the periodic modulation occurs along the bounding surface has been discussed in this paper. Especially surface and bulk acoustic wave properties of solids were studied in both square and hexagonal lattices consisting of isotropic, trigonal, and orthorhombic symmetry materials. From the previous laboratory study, we confirmed that the widths of the frequency band gap were strongly affected by the filling ratio, density, and elastic constants matching ratios. In this study, the results have shown that surface wave band gaps could be found along a specific direction. In the materials we investigated in this paper, we found that there is no full band gap for surface waves. Instead, full band gap for bulk acoustic wave can be obtained for transverse polarization mode. Results of this paper can serve as a basis for both numerical and experimental investigations of phononic crystal related structures. Introduction The existence of complete band gaps of electromagnetic waves in photonic structures extending throughout the Brillouin zone has demonstrated a variety of fundamental and practical interests.[1,2] This has led to a rapid growing interests in the analogous acoustic effects in periodic elastic structures called the phononic crystals. Surface wave propagation on layered superlattices with traction free surface parallel to the layers has been explored extensively in the past.[3] However, investigations on surface wave properties of solids in which the periodic modulation occurs on the traction free surface has not started until recently.[4-8] Vinces et al.[4,5] studied experimentally the surface waves generated by a line-focus acoustic lens at the water-loaded surfaces of a number of two-dimensional superlattices that intersect the surface normally. Propagation of Scholte-like acoustic waves at the liquid-loaded surfaces of period structures has also been studied.[6] The superlattices considered in Refs.[4-6,8] are of isotropic materials. As for superlattices consist of anisotropic materials, Tanaka and Tamura [7] reported detail calculations for surface waves on a square superlattice consisting of cubic materials (AlAs/GaAs) and many salient features of surface waves in two-dimensional superlattices have been described. In addition, Tanaka and Tamura[8] also reported detail calculations for surface waves on a hexagonal superlattice consisting of isotropic materials (Al/polymer). For the analysis of bulk acoustic waves [9-15], Kushwaha et al.[9] reported the first full band-structure calculations of the transverse polarization mode for periodic, elastic composite. In the Ref.[10], Kushwaha et al. calculated the band structures for the transverse polarization modes of nickel alloy cylinders in aluminum alloy host, and vice versa. They also investigate the dependence of spectral gap on the filling fraction and on the material parameters. Kafesaki et al.[11] reported the multiple-scattering theory (MST method) for three-dimensional periodic acoustic composites. 1 Corresponding author: [email protected] !"#$%&'(&"")(&'$*+,")(+-.$/0-.1$2345236$724489$::$;;;<5;;2= 0&-(&"$+,$>,,:?@@AAA1.B("&,(C(B1&", D$724489$E)+&.$E"B>$FGH-(B+,(0&.I$JA(,K")-+&L M&-(&"$+N+(-+H-"$.(&B"$2448@OG'@;P !""#$%&'()#$*)*$+*,-#./#01$(#/2#3/4(*4()#/2#('%)#010*$#516#7*#$*0$/,83*,#/$#($14)5%((*,#%4#146#2/$5#/$#76#146#5*14)#9%('/8(#('*#9$%((*4#0*$5%))%/4#/2#('* 087"%)'*$:#;$14)#;*3'#<87"%31(%/4)#=(,>#?9%(@*$"14,>#999-((0-4*(-#ABC:#DEF-DDF-DFG-HIJEKLMELMI>MM:DN:ENO Title of Publication (to be inserted by the publisher) Garcia-Pablos et al.[12] used the FDTD method to interpret experimental data for two-dimensional systems consisting of cylinders of fluids (Hg, air, and oil) inserted periodically in a finite slab of Al host. Psarobas and Stefanou[13] calculated the band structure of a phononic crystal consisting of complex and frequency dependent Lame’ coefficients. Zhengyou Liu et al.[14] extended the multiple-scattering theory for elastic waves by taking into account the full vector character, and presented a comparison between theory and ultrasound experiment for a hexagonal-close-packed array of steel balls immersed in water. Jun Mei et al.[15] reported the same method in Ref.[16] to extend the method in the case of cylinders. However, the bulk waves analysis in phononic structures among the Refs.[9-15] are of isotropic materials only. In this paper, we extended Ref.[7] to study phononic band gaps of elastic/acoustic waves in two-dimensional Aluminum/Barium sodium niobate (Y-cut) (Ba2NaNb5O15) with hexagonal lattice and Quartz/Epoxy with square lattice phononic structures. The explicit formulations of the plane harmonic bulk wave and the surface wave dispersion relations in such a general phononic structure are discussed based on the plane wave expansion method. Equations of Motion of 2-D Phononic Crystals In an inhomogeneous linear elastic anisotropic medium with no body force, the equation of motion for the displacement vector ( ) , t u r can be written as )] , ( ) ( [ ) , ( ) ( t u C t u m n ijmn j i r r r r ∂ ∂ =   ρ (1) where ) , , ( ) , ( z y x z = = x r is the position vector, ( ) ρ r , (r) ijmn C are the position-dependent mass density and elastic stiffness tensor, respectively. In the following, we consider a phononic crystal composed of a two dimensional periodic array (x-y plane) of material A embedded in a background material B. Due to the spatial periodicity, the material constants, ( ) ρ x , ( ) ijmn C x can be expanded in the Fourier series with respect to the two-dimensional reciprocal lattice vectors (RLV), 1 2 ( , ) G G = G , as G G x G x ρ ρ  ⋅ = i e ) ( (2) ijmn i ijmn C e C G G x G x  ⋅ = ) ( (3) where G ρ and ijmn CG are the corresponding Fourier coefficients. On utilizing the Bloch theorem and expanding the displacement vector ( , ) t u r in Fourier series for bulk wave analysis, we have ) ( ) , (  ⋅ − ⋅ = G G x G x k A r u z ik i t i i z e e e t ω (4) where ) , ( 2 1 k k = k is the Bloch wave vector, ω is the circular frequency and z k is the wave number along the z-direction, AG is the amplitude of the displacement vector. We note that as the component of the wave vector z k equals to zero, Eq. (6) degenerates into the displacement vector of a bulk acoustic wave. Substituting Eqs. (2), (3) and (4) into Eq. (1), and after collecting terms systematically, we obtain the generalized eigenvalue problem 0 ) ( 2 = + + U C B A z z k k (5) where A, B and C are 3 3 n n × matrices and are functions of the Bloch wave vector k , components of the two-dimensional RLV, circular frequency ω, the Fourier coefficients of mass density G ρ and components of elastic stiffness tensor ijmn CG . n is the total number of RLV used in the Fourier expansion and T A A A ] [ 3 2 1 ' ' ' G G G U = is the displacement vector. The expressions of the matrices are listed in Ref.[18, 19]. !!"# $%&'()*+,-(,./(%*+012)0-&*,3&'42'0-/(