SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES OF CONVENTIONAL AND AUXETIC CELLULAR STRUCTURES

Transcription

1 INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 2, Number 4, Pages c 2011 Institute for Scientific Computing and Information SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES OF CONVENTIONAL AND AUXETIC CELLULAR STRUCTURES XIUHUI HOU, ZICHEN DENG, AND JIAXI ZHOU Abstract. Based on the structure-preserving characteristics of the symplectic algorithm, the wave propagation problem of auxetic cellular structures is analyzed and compared with conventional cellular materials. The dispersion relations along the first Brillouin zone boundary and the contour plots of phase constant surfaces are obtained using the finite element method. Numerical results reveal the superiority of auxetic re-entrant honeycombs in sound reduction applications compared with conventional hexagon lattices. Band structures of the chiral honeycomb are developed to illustrate the unique feature of the symplectic algorithm at higher frequencies calculation. The highly-directional wave propagation properties of auxetic cellular structures are also analyzed, which will provide invaluable guidelines for the future application of auxetic cellular structures in sound insulation. Key words. Symplectic algorithm, Auxetic cellular structures, Wave propagation, and Chiral honeycombs. 1. Introduction Cellular structures, as one kind of periodic structures, are capable of attenuating elastic waves over certain frequency bands, thus featuring a filtering behavior. The existence of elastic band gaps in periodic structures may lead to many potential applications such as sound shields, acoustic filters, transducers, refractive devices, wave guides, etc [1]. Continuing interest in such structures has seen the birth of re-entrant and chiral structures. These are unique honeycombs and foams which exhibit negative Poisson s ratio [2]. Negative Poisson s ratio foam was first developed by Lakes. Later writers have called such materials anti-rubber, auxetic materials or dilatational materials. Here we study and compare the phononic properties of both the conventional and the negative Poisson s ratio cellular structures under Hamiltonian system according to the structure-preserving properties of the symplectic algorithm. According to the classical elasticity theory, the variation scopes of ν are from -1 to 0.5for three dimensional(3d) isotropicmaterialsand from -1to 1 for twodimensional (2D) isotropic systems [3] based on thermodynamic consideration of strain energy [4]. Thus the negative Poisson s ratio effect is theoretically permissible and exists in many natural materials, such as iron pyrites, pyrolytic graphite, cancellous bone and et al. Because of negative Poisson s ratio effect, auxetic materials exhibit a series of fascinating properties compared with the conventional materials, such as increased shear modulus [5], increased indentation resistance [6], enhanced fracture toughness [7], better energy absorption [8, 9, 10], synclastic curvature [11, 12] and et al. The auxetic effect thus deserves a further research based on these fascinating advantages. The Poisson s ratio of a material influences the transmission and reflection of stress waves [13]. It is expected that the negative Poisson s ratio foams exhibit Received by the editors December 1, 2010 and, in revised form, July 4, Mathematics Subject Classification. 70G45,74J

2 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 299 better sound absorption trait due to the convoluted cell ribs [14]. Thus the dynamic wave dispersion and loss properties of conventional and negative Poisson s ratio cellular structures are worthwhile a further research. Experimental investigations were carried out by Chen and Lakes [15] for the dynamical behavior of conventional and negative Poisson s ratio foamed polymeric materials in torsional vibration, where dispersion of standing waves and cut-off frequencies were observed. Howell et al [16]demonstrated that auxetic foams exhibit better sound absorption capability than unconverted foams at all frequencies and smaller pore-size auxetic foams absorb sound more efficiently at frequencies above 630 Hz than those with larger pores. Similar conclusions were achieved by Scarpa & Tomlinson [17], who revealed that due to the improved stiffness ratios, auxetic honeycombs could offer some advantages in sound reduction applications. Foams with curved or convoluted ribs were found to possess the traits of acoustic waves dispersion and cut-off frequencies, which might lead to applications involving the absorption of sound [18].The structural-acoustic performance of the chiral core was proved to be better through comparisons with the square and hexagonal core topologies[19]. Numerical and experimental results show that auxetic structures absorb noise and vibrations more efficiently than conventional equivalents [20]. The phononic properties of a chiral cellular structure were investigated through the application of Bloch theorem where the in-plane wave propagation problem was analyzed [21]. In a word, auxetic materials show overall better energy absorption capability, such as ultrasonic, acoustic and damping, compared with conventional materials. Based on finite element analysis and symplectic mathematics, a general method is developed here to deduce the dispersion relations for three typical cellular structures, which are hexagon, re-entrant and chiral honeycomb structures. The symplectic eigenvalue problem is introduced to analyze the phase constant surface and the dispersion relations. The characteristic of symplectic algorithm at higher frequencies calculation is emphasized in the band gap analysis. The wave beaming effect in auxetic materials guides the design of cellular structures where waves at certain frequencies do not propagate in specified directions. Numerical analysis reveals better sound insulation property of the auxetic lattice than the conventional hexagon lattice, which is the main conclusion achieved in the paper. Major difficulties in the problem considered here are for the chiral lattice structure analysis and the effective combination of the symplectic algorithm with the Bloch theorem. The findings may provide invaluable guidelines for the future application of cellular structures. 2. Geometry of the cellular structures and Poisson s ratio calculation The behavior of a material under deformation is governed by one of the fundamental mechanical properties: the Poisson s ratio [22]. Poisson s ratio, also called the Poisson coefficient, is the negative ratio of transverse contraction strain to longitudinal extension strain in a stretched bar. Since most common materials become thinner in cross section when stretched, Poisson s ratio for them is positive. Materials with negative Poisson s ratio expand laterally when stretched and contract laterally when compressed, which is different from conventional cellular materials [7]. This unusual characteristic is achieved by forming the cells into a re-entrant shape which bulges inwards [23]. The three kinds of structures discussed in this paper are displayed in Fig.1. For the conventional hexagon and re-entrant honeycombs displayed in Fig.1(a) and (b), the cell rib is treated as a Timoshenko beam of square cross-section with

3 300 X. HOU, Z. DENG, AND J. ZHOU Young s modulus E s, Poisson sratio ν s, thickness t, and second moment of the area I. For a load P in the x direction (Fig.2(d)), the deflection can be calculated as [17]: (1) where δ x δ y = δ b sinθ+δ s sinθ +δ a cosθ = δ b cosθ +δ s cosθ δ a sinθ (2) δ b = PL3 sinθ 12E si δ a = PLcosθ E st 2 δ s = ( ν s )( t L )2PL3 sinθ 12E si In the above expressions, δ b represents the bending deflection, δ s the shear deflection and δ a the axial deflection. Thus the strains (3) ξ x = δ x Lcosθ δ y ξ y = H +Lsinθ The definition of the Poisson s ratio ν xy = ξ y /ξ x leads to [ ] cos 2 θ 1+( ν s )(t/l) 2 (4) ν xy = (H/L+sinθ)sinθ 1+ ( ν s +cot 2 θ ) (t/l) 2 Figure 1. The conventional and auxetic cellular structures. The basic cells are listed on right with displacements indices appended.(a) Conventional hexagon honeycomb; (b) Re-entrant honeycomb; (c) Chiral honeycomb.

4 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 301 The thickness t is small enough compared with the cell length L and thus the expression (t/l) 2 is assumed to be negligible. For the basic cell shown in Fig.2(a),θ = π/6 and H/L = 1; whereas for Fig.2 (b),θ = π/6 and H/L = 2.The cellular structures analyzed in the paper are all isotropic in-plane. Thus ν yx = ν xy = 1 is obtained for the hexagon honeycomb structure and ν yx = ν xy = 1 for the re-entrant honeycomb structure. Although chirality is common in nature and organic chemistry, it is an unusual characteristic in structural materials and components [24].The honeycomb structure shown in Fig.2(c) is composed of circular elements or nodes of equal radius r and thickness t c, joined by straight ligaments or ribs of square cross-section with equal length L. The ligaments are constrained to be tangential to the nodes.the distance between the centers of adjacent nodes is defined as H, while the angle between adjacent ligaments is denoted as 2θ. The angle between the imaginary line connecting the node centers and the ribs is denoted by α. The chiral honeycomb simultaneously possesses hexagonal symmetry and a two-dimensional chiral symmetry. Structures exhibiting hexagonal symmetry are mechanically isotropic in-plane. The relations between the geometry parameters can be described as follows: (5) sinθ = 0.5H H = 0.5 θ = π/6 tanα = r L/2 = 2r/L sinα = r H/2 = 2r/H As shown in Fig.2 (e), the ligament is treated as a Timoshenko beam with Young s modulus E s, Poisson s ratio ν s, length L, thickness t b, and second moment of the area I. Under the condition of the load P and accordingly the torque T, the deformation is dominated by the torsion deflection and can be calculated as follows [2]:(The red imaginary line represents the axis of the ligament after deformation). Figure 2. The geometry of the basic cells: (a) Conventional hexagon honeycomb; (b) Re-entrant honeycomb; (c) Chiral honeycomb; (d)basicbeammemberforthecellsof(a)and(b); (e)deformation for the basic cell of (c). The basic vectors Ξ = (d 1,d 2 )for the direct lattice are also appended in (a)-(c), which will be discussed later.

5 302 X. HOU, Z. DENG, AND J. ZHOU The deflection of Fig.2 (e) is related to the angular deflection η of the ligament by (6) = rsinη then (7) x = cosθ y = sinθ where the angular deflection η = TL/(6E s I). Thus the strains (8) ξ x = x H cosθ = rsinη H, ξ y = y H sinθ = rsinη H Then for loading in both direction ν yx = ξ x /ξ y = 1,ν xy = ξ y /ξ x = 1 3. Finite element modeling Both positive and negative Poisson s ratio honeycomb cellular structures are applied here for investigations and comparisons. As displayed in Fig.1, they are hexagon, re-entrant and chiral honeycomb structures. The basic cells are shown on rightwiththeinternal(u i )andexternaldisplacements(u e,e = 1,2, 6)appended. The structural mechanics behavior of the basic cells is predicted using the finite element method, upon which the stiffness matrices of the members are obtained. For each member of the basic cell as shown in Fig.2,the standard beam model is adopted with axial displacement µ ε,transverse displacement µ τ and rotation ϕ ζ as the nodal degrees of freedom. In the local reference system (ε,τ,ζ) of Fig.3, the equilibrium equation for a beam member can be expressed in the form of (9) K e lu = f (10) u = {µ εi,µ τi,ϕ ζi,µ εj,µ τj,ϕ ζj } T f = {f εi,f τi,m ζi,f εj,f τj,m ζj } T where u and f are the elemental displacement vector and force vector respectively. Here K e l represents the elemental (superscript e ) stiffness matrix in the local reference system (suffix l ) and can be written as: (11) K e l = E s A/l 0 0 E s A/l E s I/l 3 6E s I/l E s I/l 3 6E s I/l 2 0 6E s I/l 2 4E s I/l 0 6E s I/l 2 2E s I/l E s A/l 0 0 E s A/l E s I/l 3 6E s I/l E s I/l 3 6E s I/l 2 0 6E s I/l 2 2E s I/l 0 6E s I/l 2 4E s I/l where l denotes the length of the member, A the cross-section area, I the second moment of the area and E s the elastic modulus. The density is denoted by ρ s.the cross-section of all the members is designated as square with width t.accordingly, the elemental mass matrix M e l possesses the form of

6 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 303 (12) M e l = ρ sl 1/ / /35 11l/ /70 13l/ l/210 l 2 / l/420 l 2 /140 1/ / /70 13l/ /35 11l/ l/420 l 2 / l/210 l 2 /105 The member s mass and stiffness matrices in the global reference system (x, y, z) (suffix g) can be described as (13) M e g = TM e lt T, K e g = TK e lt T where the superscript T represents the transpose of the matrix; T is the transformation matrix defined as (14) T = with (15) R = [ R 0 0 R ] cosβ sinβ 0 sinβ cosβ where β is the angle between the local ε axis and the global x axis (Fig.3), which is a function of the internal angle θ in Fig.2. The member s mass and stiffness matrices in the global reference system are then assembled to obtain the basic cell s mass and stiffness matrices defined as M and K respectively. As seen from Fig.2, the finite element modeling process is easy for the hexagonal and re-entrant honeycomb basic cells. This is because that there are no curved elements and the standard beam theory can be applied directly. In order to avoid the complexity associated with the finite-element discretization process of curved elements, the node circles are approximated as a sequence of straight beams(fig.4)[25] Figure 3. Illustration of a beam member for the finite element modeling.

7 304 X. HOU, Z. DENG, AND J. ZHOU and the convergence of the results is verified through the dispersion relation analysis which will be discussed in detail in the following section. 4. Bloch theorem and the first Brillouin zone For the wave propagation problem of periodic cellular structures analyzed here, the Bloch theorem can be introduced to relate the external displacements u e, which can be partitioned as u e = {u a,u b } T. As seen from Fig.1 (a) and Fig.1 (b), for the conventional hexagon and the re-entrant honeycomb cellular structures, the relation between the external displacement components u a and u b is: (16) u b = A 1 u a where (17) u a = {u 1,u 1 } T, u b = {u 2,u 3 } T, A 1 = while for the chiral honeycomb structure in Fig.1(c) ( e ik 1 I 0 0 e ik2 I ) (18) u b = A 2 u a where (19) u a = {u 1,u 2,u 3 } T, u b = {u 4,u 5,u 6 } T e ik1 I 0 0 A 2 = 0 e i(k1+k2) I e ik2 I The symbols k 1,k 2 in Equation (17) and Equation (19) are wave numbers which vary within the first Brillouin zone as displayed in Fig.5. As is known, the intersubstructure force vectors p a and p b can also be related as follows: (20) p b = A j p a ( j = 1 for the regular hexagon and re-entrant honeycomb j = 2 for the chiral honeycomb ) Figure 4. The finite element modeling for the node circle discretization of the chiral honeycomb structure. (a) 6-node model; (b) 12-node model; (c) 24-node model.

8 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 305 The concept of a Brillouin zone was first developed by Lon Brillouin ( ), a French physicist. In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The importance of the Brillouin zonestemsfrom the Blochwavedescriptionofwavesin aperiodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. We need to define first the reciprocal lattice in order to obtain the Brillouin zone. The set of all wave vectors k that yield plane waves with the periodicity of a given Bravais lattice is known as its reciprocal lattice. The relation between the reciprocal lattices (the vector basis defined as Λ = (c 1,c 2 ), see Fig.5) and the direct lattices (the vector basis defined as Ξ = (d 1,d 2 ), see Fig.2) can be described as: (21) c i d j = δ ij (i,j = 1,2) where δ ij is the Kronecker delta. For the conventional hexagon and re-entrant honeycomb structures (Fig.1 (a) and Fig.1 (b)) (22) then d Ξ 1 = (Lcosθ,H +Lsinθ) T d Ξ 2 = ( Lcosθ,H +Lsinθ)T (23) c Λ 1 = ( ) T 1 2Lcosθ, 1 2(H+Lsinθ) c Λ 2 = ( 1 2Lcosθ, 1 2(H+Lsinθ) while for the chiral honeycomb structure (Fig.1 (c)) ) T (24) then d Ξ 1 = (Hcosθ,H sinθ) T d Ξ 2 = ( Hcosθ,H sinθ)t c (25) Λ 1 = ( 1 2H cosθ, ) 1 T 2H sinθ c Λ 2 = ( 1 2H cosθ, ) 1 T 2H sinθ In the reciprocal lattice, the wave vector ( k = 2πλ ) can be expressed as (26) k = k 1 c 1 +k 2 c 2 From the definition of reciprocal lattice given in Equation (21), we obtain (27) k i = k d i (i = 1,2) While the direct lattice defines the spatial periodicity of the considered domain, the reciprocal lattice describes the periodicity of the frequency/wave number relation [26], which is illustrated in Fig.5 with dashed lines. Given the reciprocal lattice vectors, the first Brillouin zone is obtained by selecting any point in the reciprocal lattice as the origin and by connecting it to all nearest neighboring points. The perpendicular bisectors constructed on the connecting lines, also known as Braggs lines, defined the first Brillouin zone. The irreducible Brillouin zone(the domain of

9 306 X. HOU, Z. DENG, AND J. ZHOU Γ Q M in Fig.5 ) is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice, which is the smallest frequency-wave number space necessary to determine wave dispersion for a given lattice [25]. The coordinates of the boundary points of the irreducible Brillouin zone in the reciprocal lattice frame are appended in Fig.5, upon which the wave numbers k 1,k 2 can be designated. 5. Symplectic analysis of the dispersion relation Application of the finite element methods described in Section 3 yields the following dynamic equation of the basic cell (28) ( K ω 2 M ) u = 0 where ω is the frequency of wave propagation and u is the generalized nodal displacement which is formed by the internal displacements (u i ) and the boundary components (u e,e = 1,2, ) For a given frequency ω #, the expression (29) R(ω # ) = K ω 2 # M is usually called as dynamic stiffness matrix, which is ω dependent and often used in the description of wave propagation and vibration problems [27]. The dynamic Equation (28) can be block partitioned according to the internal (suffix i ) and external (suffix e ) displacement components: (30) [ ][ ] Rii R ie ui = 0 R ee u e R ei Figure 5. The reciprocal lattice Λ = (c 1,c 2 ) and the first Brillouin zone (the hexagonal area surrounded by solid lines) for (a) conventional hexagon and re-entrant honeycombs; (b) chiral honeycomb.

10 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 307 The internal displacement vector can first be eliminated. The symmetric matrix R ii considered here is positive definite, which implies that R ii is invertible. Thus the dynamic equation can be reduced to (31) (R ee R ei [R ii ] 1 R ie ) u e = 0 where the external dynamic stiffness matrix (32) R e = R ee R ei [R ii ] 1 R ie = [ ] Kaa K ab K ba K bb The dynamic potential energyof the system can be expressed as U = n U basic, wherethesuffixndenotesthesizeoftheperiodiccellularstructureandu basic (u a,u b,ω# 2 is the dynamic potential energy of the basic cell: ( ) ( (33) U basic ua,u b,ω# 2 ) H [ 1 ua Kaa = 2 u b K ba ]( ) K ab ua K bb u b The variation principle requires that δu = 0. Thus the dynamical equation in frequency domain can be achieved as (34) K ba u j 1 b +(K aa +K bb )u j b +K abu j+1 b = 0 where the superscript j represents the sequence of the basic cell. The relations u j 1 b = u j a,u j b = uj+1 a exist and can be used in the following analysis. The principle of virtual work can be applied here to obtain the dual variable, the inter-substructure force vector ) (35) p b = U basic / u b = K ba u a +K bb u b Substitution of Equation (35)into expression (34) achieves (36) p a = K aa u a K ab u b With the definition of the state vector (37) v a = {u a,p a } T And from the expressions (35) and (36) we have (38) v b = Sv a where ( (39) S = It is found that S T JS = J where K 1 ab K aa K 1 ab K ba K bb K 1 ab K aa K bb K 1 ab J = [ 0 I I 0 Thus the transfer matrix S possesses the symplectic property and the symplectic eigenvalue problem can be introduced here for the wave propagation analysis. ] )

11 308 X. HOU, Z. DENG, AND J. ZHOU Firstly through the application of the method of separation of variables (40) v = ϕψ where ϕ is a scalar function and Ψ is a vector the same size as v. Insertion of the expression (40) into Equation (38) yields (41) (ϕ b /ϕ a )Ψ = SΨ As the right hand side expression of Equation (41) is invariant form cell to cell, a constant µ = ϕ b /ϕ a is assumed and the Equation (41) can be reduced to (42) SΨ = µψ Then the eigenvalue problem of a symplectic matrix is obtained where µ is the eigenvalue and Ψ is the eigenvector. For a symplectic matrix, it is known that if µ is an eigenvalue then so is 1/µ [28].Therefore the 2m eigenvalues of a symplectic matrix can be classified as two groups [27]: (43) (I) µ i, abs(µ i ) < 1 or abs(µ i ) = 1 Im(µ i ) > 0, (i = 1,2, m) (II) µ m+i, µ m+i = µ i (i = 1,2, m) where the symbol abs represents the absolute value of the eigenvalue and the symbol means both the two expressions should be satisfied simultaneously. The eigenvectorsψ i,ψ m+i correspondingto the eigenvaluesµ i,µ m+i aretermed as symplectic orthogonality-work reciprocity which is not detailed here for the sake of brevity. For waves propagating in the periodic cellular structures of infinite length, it is required that µ = 1, otherwise the waves will scatter at infinity. According to the discussions in Section 4, we know that the eigenvalue Θ(ω) = (µ 1,µ 2, µ i µ 2m ) T of the symplectic matrix S is related to the wave numbers k 1,k 2 by (44) Θ(ω) = (diag(a j (k 1,k 2 )),diag( A j (k 1,k 2 ))) T j = 1,2 Then the dispersion relation ω = f (k 1,k 2 ) can be deduced which implies that different frequencies can be obtained via an assigned pair of propagation constants [29]. 6. Results and discussions The relation ω = f (k 1,k 2 ) is usually called as phase constant surface. The dispersion relations in this section are plotted along the irreducible first Brillouin zone boundaries as shown in Fig.5, where the frequency axis ω = ζω/ω 0 is normalized with respect to the flexural resonance ω 0 of a simply supported lattice beam member:ω 0 = ( π 2 /L 2) EI/(ρt 2 ). The parameter ζ = 100 is applied here for the sake of convenient representation. The material aluminum (density ρ s = 2700kg/m 3, Young s modulus E s = Pa ) is adopted here for the band structures analysis. The geometry parameters for the three lattice configurations are set as follows:

12 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 309 Conventional hexagon honeycomb θ = 30,L = H = m,t = m Re-entrant honeycomb θ = 30,H = m,l = H/2,t = m { θ = α = 30 Chiral honeycomb,l = m t b = t c = m,h = L/2,r = Ltanα/2 The same basic geometry dimensions are chosen for all the structures, thus the influence of different lattice configurations on the wave propagation properties is discussed, as displayed in Fig.6 (a)-fig.6 (b) for the conventional hexagon honeycomb, Fig.6 (c)-fig.6 (d) for the re-entrant lattice and Figs.7-8 for the chiral honeycomb, where the iso-frequency contours are exploited as the representation of dispersion relations. The periodicity of the dispersion relations (frequency/wave number) is obviously viewed from the contour plots of Fig.6-Fig.8. It is also observed that for various lattice configurations, the wave propagation properties are remarkably different. However, for the three typical lattices considered here, the same in-plane isotropic properties are depicted through the iso-frequency contours. The contour plots of the phase constant surfaces corresponding to the first and the second wave modes are presented for the hexagon lattice in Fig.6 (a)-fig.6 (b); While Fig.6 (c)-fig.6 (d) display the first and the second wave modes for the re-entrant lattice. Take a comparison of the plots in Fig.6, it is noted that the curves for the conventional hexagon lattice are denser than the re-entrant lattice, which implies more and wider gaps in the re-entrant lattice. This enhanced sound absorption capability is partly due to the property of negative Poisson s ratio. The relationship between the bulk modulus K, shear modulus G and Poisson s ratio ν for isotropic materials is G = 1.5K(1 2ν)/(1+ν). For the re-entrant lattice considered here, ν xy = ν yx = 1 leads to the relation of G K, which implies that the re-entrant lattice can easily undergo volumetric (bulk) deformation but resist shear deformation. This shape restoring feature is proved to be effective in energy absorbing applications, thus demonstrating better sound reduction characteristic for the re-entrant lattice than the conventional hexagon lattice. The boundaries of the first Brillouin zones superimposed in the Figs.6-7 clearly demonstrate the periodicity of the dispersion relations and the corresponding irreducible Brillouin zones reduced by all of the symmetries can certainly determine the wave dispersion characteristic for the lattices considered here. The different shapes of the first Brillouin zones indicate different wave propagation properties, which can be illustrated further in the subsequent band structures analysis. For the chiral honeycomb, three finite element models (see Fig.4) are chosen to verify the convergence of the node circle discretization process. From Fig.7 we see that the contours of the dispersion surfaces corresponding to the first two wave modes are smooth enough for the 24-node model and no much difference is observed between the results of the 12-node model and the 24-node model. The 24-node model is accordingly accepted for the finite element modeling process of the chiral lattice and the corresponding band gaps and wave beaming effect are analyzed subsequently. The contours of the dispersion surfaces corresponding to the first wave mode along the irreducible Brillouin zone boundary Γ Q M are plotted in Fig.8 for the chiral lattice. It can be shown that the wavegroup velocity at a given frequency lies in the direction of the steepest ascent on the phase constant surfaces, which

13 310 X. HOU, Z. DENG, AND J. ZHOU Figure 6. Contour plots for the hexagon and re-entrant honeycombs: (a) hexagon, 1 st mode; (b) hexagon, 2 nd mode; (c) reentrant, 1 st mode; (d) re-entrant, 2 nd mode. Figure 7. Contour plots for the chiral honeycomb: (a) 12-node model, 1 st mode; (b) 12-nodemodel, 2 nd mode; (c) 24-nodemodel, 1 st mode; (d) 24-node model, 2 nd mode. can be found as the normal to the corresponding iso-frequency contour line in the k 1 k 2 plane. The arrows appended on the magnified pictures of Fig.8 (b), Fig.8 (d) and Fig.8 (f) thus illustrate the directional properties of the wave propagation problem. There are almost only two propagating directions along the irreducible Brillouin zone boundaries, which indicate highly-directional band gaps at certain frequencies for the chiral materials of negative Poisson s ratio. The highly directional property of the cellular structures not only allows the determination of wave propagation directions in the structure, but also identifies

14 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 311 regions within the structure where waves do not propagate.this unique characteristic depends on the geometry of the unit cell and the frequency of wave propagation. Figure 8. Contour plots for the chiral lattice (24-node model, 1 st mode) along the irreducible Brillouin zone boundary: (a) the direction of Γ Q ; (b) magnified picture of (a); (c) the direction of Q M; (d) magnified picture of (c); (e) the direction of M Γ;(f) magnified picture of (e) Q M (a) Hexagon Q M (b) Re-entrant 0 Figure 9. Dispersion relations at the boundary of the irreducible Brillouin zones: (a) Conventional hexagon honeycomb; (b) Reentrant honeycomb.

15 312 X. HOU, Z. DENG, AND J. ZHOU Q M Chiral honeycomb Figure 10. Dispersion relations at the boundary of the irreducible Brillouin zone for the chiral honeycomb (24-node model). The band structures displayed in Fig.9 and Fig.10 effectively attest the results achieved in the phase constant surfaces analysis. There are more and wider gaps for the re-entrant honeycomb compared with the conventional hexagon lattice due to the auxetic effect, which depends on the microstructure of the lattice. Thus the crucial role that a negative Poisson s ratio can play in tailoring the mechanical properties of a structure can be utilized properly to give enhanced sound absorption performance. It is also revealed that the band structures are significantly affected by the cellular structures configuration.as illustrated in Fig.10, for the chiral honeycomb, higher frequencies are computed and more gaps are observed according to the structure-preserving characteristic of the symplectic method. The characteristic of symplectic algorithm at higher frequencies calculation is of vital importance in the sound and vibration absorption applications, which may lead to further development in the existing sound absorbers analysis. 7. Conclusions Based on the finite element analysis, the phononic properties of the conventional and auxetic honeycombs are analyzed with the symplectic method. Three typical lattices including hexagon, re-entrant and chiral honeycombs are chosen for the phase constant surfaces and band structures analysis. Numerical examples are carried out to demonstrate the effectiveness of the theory and algorithm developed. The characteristic of symplectic algorithm at higher frequencies calculation is emphasized in the dispersion relations analysis for the chiral lattice. The contours of dispersion surfaces along the irreducible Brillouin zone boundary are obtained for the chiral lattice and the wave beaming effects are deduced, which indicates highlydirectional properties for the wave propagation problem of auxetic materials. It is also revealed that the auxetic honeycombs are more efficient in sound insulation compared with the conventional materials. Given the appropriate geometries, a

16 SYMPLECTIC ANALYSIS FOR THE WAVE PROPAGATION PROPERTIES 313 re-entrant cell honeycomb offers enhanced sound and vibration absorption capacity compared with the conventional hexagon honeycomb, due to the negative Poisson s ratio effect, which is one of the four foundational mechanical properties of materials. Acknowledgments The authors wish to thank the National Natural Science Foundation of China ( ), the National Basic Research Program of China (2011CB ), the National 111 Project of China (B07050), the Doctoral Program Foundation of Education Ministry of China ( )the Open Foundation of State Key Laboratory of Structural analysis of Industrial Equipment (GZ0802). References [1] Yan, Z.Z. and Wang, Y.S., Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method. Physical Review B (Condensed Matter and Materials Physics), (9): p [2] Prall, D. and Lakes, R.S., Properties of a chiral honeycomb with a poisson s ratio of 1. International Journal of Mechanical Sciences, (3): p [3] Wojciechowski, K.W., Remarks on Poisson Ratio beyond the Limits of the Elasticity Theory. Journal of the Physical Society of Japan, : p [4] Liu, Y.P. and Hu, H., A review on auxetic structures and polymeric materials. Scientific Research and Essays, (10): p [5] Huang, X. and Blackburn, S., Developing a new processing route to manufacture honeycomb ceramics with negative Poisson s ratio. Key Engineering Materials, : p [6] Lakes, R.S. and Elms, K., Indentability of Conventional and Negative Poisson s Ratio Foams. Journal of Composite Materials, : p [7] Lakes, R.S., Foam Structures with a Negative Poisson s Ratio. Science, : p [8] Scarpa, F. and Smith, F.C., Passive and MR Fluid-Coated Auxetic PU Foam-Mechanical, Acoustic and Electromagnetic Properties. Journal of Intelligent Material Systems and Structures, : p [9] Scarpa, F., Bullough, W.A., and Lumley, P., Trends in acoustic properties of iron particle seeded auxetic polyurethane foam. in Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science [10] Scarpa, F., Ciffo, L.G. and Yates, J.R., Dynamic properties of high structural integrity auxetic open cell foam. Smart Materials and Structures, : p [11] Evans, K.E., Tailoring the negative Poisson s ratio. Chemistry and Industry, : p [12] Evans, K.E., Design of doubly-curved sandwich panels with honeycomb cores. Composite Structures, : p [13] Lakes, R.S., Advances in negative poisson s ratio materials. Advanced Materials, : p [14] Lakes, R.S., No contractile obligations. Nature, : p [15] Chen, C.P. and Lakes, R.S., Dynamic wave dispersion and loss properties of conventional and negative Poisson s ratio polymeric cellular materials. Cellular Polymers, : p [16] Howell, B., Prendergast, P., and Hansen, L., Examination of acoustic behaviour of negative Poisson s ratio materials. Applied Acoustics, : p [17] Scarpa, F. and Tomlinson, G., THEORETICAL CHARACTERISTICS OF THE VIBRA- TION OF SANDWICH PLATES WITH IN-PLANE NEGATIVE POISSON S RATIO VAL- UES. Journal of Sound and Vibration, (1): p [18] Yang, W., et al., Review on auxetic materials. Journal of Materials Science, (10): p [19] Spadoni, A. and Ruzzene, M., Structural and Acoustic Behavior of Chiral Truss-Core Beams. Journal of Vibration and Acoustics,Transactions of the ASME, : p [20] Sparavigna, A., Phonons in conventional and auxetic honeycomb lattices. Physical Review B, : p [21] Spadoni, A., et al., Phononic properties of hexagonal chiral lattices. Wave Motion, : p

17 314 X. HOU, Z. DENG, AND J. ZHOU [22] Evans, K.E. and Alderson, A., Auxetic Materials: Functional Materials and Structures from Lateral Thinking! Advanced. Materials, (9): p [23] Lakes, R.S., Deformation mechanisms in negative Poisson s ratio materials:structural aspects. Journal of Materials Science, : p [24] Hassan, M.R., et al., Smart shape memory alloy chiral honeycomb. Materials Science and Engineering: A, : p [25] Spadoni, A., et al., Phononic properties of hexagonal chiral lattices. Wave Motion, (7): p [26] Gonella, S. and Ruzzene, M., Analysis of in-plane wave propagation in hexagonal and reentrant lattices. Journal of Sound and Vibration, (1-2): p [27] Zhang, H.W., et al., Phonon dispersion analysis of carbon nanotubes based on inter-belt model and symplectic solution method. International Journal of Solids and Structures, (20): p [28] Zhong, W.X., Symplectic Solution Methodology in Applied Mechanics. 2006, Beijing (in Chinese): Higher Education Press. [29] Hou, X.-H., et al., Symplectic analysis for elastic wave propagation in two-dimensional cellular structures. Acta Mechanica Sinica. 26(5): p Department of Engineering Mechanics, Northwestern Polytechnical University, Xi an , PR China; Department of Engineering Mechanics, Northwestern Polytechnical University, Xi an , PR China; State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian , PR China; dweifan@nwpu.edu.cn (Zichen Deng). College of Mechanical & Vehicle Engineering, Hunan University Changsha , PR China