Analysis of in-plane wave propagation in periodic structures with Sierpinski-carpet unit cells

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Analysis of in-plane wave propagation in periodic structures with Sierpinski-carpet unit cells Jiankun Huang a,n, Massimo Ruzzene b, Shengbing Chen c a b c

Department of Civil Engineering, Beijing Forestry University, Beijing 100083, China School of Aerospace Engineering, Georgia Institute of Technology, Atlanta 30318, USA China Aerodynamics Research and Development Center, Mianyang 621000, China

a r t i c l e i n f o

abstract

Article history: Received 10 July 2016 Received in revised form 26 January 2017 Accepted 7 February 2017 Handling Editor: L.G. Tham

In this study, the dispersion relationships and characteristics of directional periodic structure waveguides with Sierpinski carpet unit cells were investigated. Finite element method was used to analyze the Sierpinski-carpet periodic structures with square Pb cylinders inserted in the rubber background. The dispersion curves of unit cells with different stages were investigated first, then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the Sierpinski-carpet periodic structures. The dynamic responses of finite periodic structures were investigated to observe the directional features, and the results were used to support the fact that Sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways; the dispersion curves grew compressed as the stage increased. Multiple band characteristics were observed in the case of the second-stage fractal configuration. The directional propagation characteristics in Sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones. The results of this study have significant potential in terms of the application of Sierpinski-carpet periodic structures for vibration isolation. & 2017 Elsevier Ltd All rights reserved.

Keywords: Periodic structure Sierpinski carpet Band gap Group velocity Directional characteristic

1. Introduction In recent years, the propagation of elastic waves in phononic crystals (or “periodic structure”, as referred to here) has attracted a significant amount of research attention. Periodic structures are periodic arrays of two or more elastic materials with disparate densities and elastic constants, or periodic geometrical configurations. The most unique feature of periodic structures is their band gaps, which are also called attenuation zones (AZs). Elastic waves with frequencies in the band gap ranges cannot propagate in the periodic structures. The existence and location of band gaps are dependent on the material and geometrical parameters of the representative unit cell. Wave propagation and AZs in periodic structures have been researched for at least 50 years [1–4]. Over the past decade, there has been steadily increasing interest in periodic structures due to their interesting physical properties and potential engineering application, including noise control [5–7], vibration reduction [8–11], and seismic isolation [12–15]. Forbidding waves usually require the existence of a full band gap, therefore, most of the earlier work was concerned with a maximization problem of band gaps in periodic structures [12,16,17]. n

Corresponding author. E-mail addresses: [email protected], [email protected] (J. Huang), [email protected] (M. Ruzzene), [email protected] (S. Chen). http://dx.doi.org/10.1016/j.jsv.2017.02.020 0022-460X/& 2017 Elsevier Ltd All rights reserved.

Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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However, a large full band gap can be difficult to guarantee in some practical issues. Fortunately, band gap behavior is complemented by the directional characteristics of the wave propagation in periodic structures; elastic waves only propagate in certain directions of the periodic structures though the frequencies of waves are within the directional AZs or even pass bands [18]. Ruzzene and collaborators [19–22] have conducted extensive research on the application of combined band gaps and directional characteristics for the prediction of periodic structure dynamic behavior, such as two-dimensional (2D) hexagonal and re-entrant lattices, 2D periodically perforated plates and non-linear periodic lattices. More recently, the wave propagation dispersed by fractal or quasi-fractal structures have been explored by many researchers. These fractal or quasifractal structures exist primarily in Sierpinski gasket [23–27] or Sierpinski carpet [27–31] form. In a study on the Sierpinski gasket, Krzysztofik [23] presented a modified Sierpinski fractal monopole antenna and found that fractal technology allows design of smaller, high-performance multiband antennas. Castiñeira-Ibáñez et al. [24,25] developed an optimized Sierpinski fractal technique that enables the creation of a wide band gap in an acoustic case. They determined the influence of filling fraction on the band gaps of sonic crystal in order to obtain a wide and full AZ; moreover, theoretical results were experimentally validated to confirm a significant advantage of fractal geometry in terms of band gap production. PuenteBaliarda et al. [26] experimentally and numerically explored the multi-band behavior of a fractal Sierpinski antenna with self-similarity properties. Both Sierpinski carpet and gasket patch antennas were established by Abdelhak et al. [27], where the antenna structure is replaced by its equivalent rectangular patch. The sound field scattered by a fractal surface in the form of a Sierpinski carpet was calculated by Lyamshev and Urusovskii [28] via perturbation method. Arshad and Xu [29] designed a Sierpinski-carpet fractal antennas with a compact configuration for wideband applications. Using metallized foam technology, Anguera et al. [30] designed a square Sierpinski-inspired carpet monopole which may be useful as an antenna for pico-cell base station applications. Gao et al. [31] found that multi-frequency AZs exist in 2D Sierpinski-fractal phononic crystals and can be pushed toward the low-frequency range by increasing the fractal dimension. Research to date on periodic structures has two points remained to be proven. Firstly, the unit cell or the structure considered to be the Sierpinski carpet has been mainly studied with respect to photonic materials [23,26,27,29,30] while research on Sierpinski-carpet phononic crystals or periodic structures has been fairly rare [28,31]; the properties of elastic wave propagation in periodic structures with a traditional Sierpinski-carpet form were neglected. The dispersion curves of periodic structures due to different stages have not studied comprehensively, either. Whether the characteristics of photonic bands can be compliant to the periodic structures well merits further research. Secondly, with respect to the elastic wave propagation, another important and interesting characteristic of periodic structures is their anisotropic behaviors marked by preferential directions of energy flow. Despite extensive studies on periodic structures, the directional characteristics of Sierpinski-carpet periodic structures at given frequencies have scarcely been considered. The periodic structures suggest spatial tunability via their directional and band gap behaviors, which may have significant application in vibration isolation. In the present study, we not only examined the dispersion relationship of Sierpinski-carpet periodic structures with different stages, but also applied group velocities to investigate the directional waveguides of periodic structures at given frequencies. Sierpinski-carpet unit cells with square Pb cylinders inserted in the rubber background were investigated. In addition, we also used numerical simulation to demonstrate the wave-guiding capability of Sierpinski-carpet periodic structures. The dynamic responses of finite periodic structures predict the potential applications of AZs and the directional characteristics of Sierpinski-carpet periodic structures in terms of vibration reduction. The remainder of this paper is organized as follows. In Section 2, basic concepts including the Bloch theorem are explained and the finite element method (FEM) is used to calculate the dispersion curves of different unit cells. Section 3 presents our analysis of the features of the dispersion relationship and our investigation of the group velocities of Sierpinski-carpets with different stages. In Section 4, finite periodic structures are constructed by FEM and wave propagation in different directions is investigated in order to observe the directional characteristics. Section 5 provides an example of the application of the proposed Sierpinski-carpet periodic structure. Section 6 presents our conclusions.

2. Mathematical formulation Fig. 1 shows schematic plots of the fractal unit cells with different stages; Figs. 1(a)–(c) represent the 1-stage, 2-stage, and 3-stage Sierpinski-carpet unit cells, respectively. The gray and white regions represent different materials. Geometrically, Sierpinski's carpets are constructed by the following procedure [27,28]. A square unit cell with periodic constant a was divided equally into nine small squares, then a small central square was excised and filled with the other material to create the 1-stage Sierpinski carpet as shown in Fig. 1(a). The same procedure was repeated over the eight remaining small squares in turn to construct the unit cell of the 2-stage Sierpinski carpet shown in Fig. 1(b). In reality, the subdivision of squares ends at a certain step L at which point an L-stage Sierpinski-carpet unit cell is obtained. The similarity dimension of the Sierpinski carpet can be defined as [23]:dH = ln 8/ln 3 = 1.8928. In this paper, gray and white regions represent Pb square cylinders and the rubber background, respectively. In order to use a computer program to construct Sierpinski carpet unit cells for our numerical analyses, the position vector derivative method was applied and the descriptions about the particular carpet implementation were listed in the Appendix. Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 1. Models of Pb–rubber Sierpinski-carpet unit cells with different stages. (a) L ¼ 1, (b) L ¼2, (c) L ¼3.

2.1. Governing equation This paper focused on the propagation of elastic plane waves in the fractal periodic structure. Therefore, a plane strain model was utilized to study the propagation of in-plane waves in a 2D periodic structure. The model includes both the longitudinal (P-) wave and transverse (SV-) wave. Many engineering problems would be simplified as a plane strain model, such as the mechanical filters [31–33], periodic pile barriers [9,10] and periodic foundations [12]. This paper can provide the theoretical guidance for the structure design. The materials were assumed to be elastic, without damping, and banded perfectly on the interfaces. Without considering the body force, the harmonic motion of an inhomogeneous elastic medium can be written as follows [12,32]:

⎧ ⎛ ∂u ⎪ ∂uj ⎞ ∂uj ⎞ ⎤ ⎫ ∂ ⎡ 1⎪ ∂ ⎛ ⎢ μ (r) ⎜ i + ⎨ ⎜ λ (r) ⎟+ ⎟ ⎥ ⎬ = − ω2ui2 ⎪ ρ ⎩ ∂xi ⎝ ∂xj ⎠ ∂xj ⎢⎣ ∂xi ⎠ ⎥⎦ ⎪ ⎝ ∂xj ⎭

(i, j = x, y) (1)

where u, ρ, λ, and μ are the displacement, density and two Lames’ parameters of the medium, respectively; r is the position vector and ω is the circular frequency. 2.2. Bloch theory Bloch theorem indicates that the proportionate change in wave amplitude occurring from cell to cell is irrelevant to the unit cell location within the system. Wave propagation in periodic structures is traditionally analyzed according to the Bloch theorem. The solutions to Eq. (1) can be expressed as [12]:

u (r, t ) = uK (r) e i (K·r − ωt )

(2)

where K = [Kx, Ky ] is the Bloch wave vector and the wave amplitude uK (r) is a periodic function with the same periodicity as the unit cell, and t is the time. This makes the following characteristic clear:

uK (r + R) = uK (r)

(3)

where R is the lattice vector and R = n1a1 + n2 a2 + n3 a3. Bloch theorem allows these computations to be undertaken by considering only one of the unit cells of the entire periodic structure, substantially increasing the efficiency of investigation [3]. The boundary conditions of the unit cell can be derived according to Eqs. (2) and (3):

u (r + R, t ) = e i (K·(r + R)− ωt ) ·uK (r + R) = e iK·R ·e i (K·r − ωt ) ·uK (r) = u (r, t )·e iK·R

(4)

Because the periodic structure has a periodic feature in the x–y plane, a unit cell can serve as the computation domain. According to Bloch theory and by factoring out the exp(iωt ), the following relationship must be satisfied at the cell boundary [34].

u (r + a) = e i (K·a) u (r)

(5)

where a is the periodic constant, which is also equal to the side length of the square unit cell. 2.3. Brillouin zone The space defined by the basis vectors (a1, a2) is denoted as the direct lattice. Within a direct lattice space, one may define a reciprocal lattice, which is described by the reciprocal basis vectors (b1, b2). The relationship between the basis vectors and Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 2. First Brillouin zone and irreducible Brillouin zone.

reciprocal basis vectors are given by a i·bj = 2πδ ij , where defined by the reciprocal base vectors b1 and b2 [22]:

δij is the Kronecker delta with i,j¼1,2 [19,22]. The Brillouin zone is

b1 = 2π

a2 × a3 a1·(a2 × a3 )

(6a)

b 2 = 2π

a3 × a1 a1·(a2 × a3 )

(6b)

Taking a3 as the unit vector ez = [0, 0, 1], b1 and b2 were calculated to be b1 = (2π /a, 0) and b2 = (0, 2π /a). The conventional dispersion relationship relating the wave frequencies and wavenumber plays an important part in the wave propagation study [35–37]. Dispersion relationship, which was calculated with the periodic boundary conditions, is also an important aspect of the wave propagation characteristic study in periodic structures [9,10,12,38]. Due to periodicity, it was only necessary to consider the dispersion relationship on the first Brillouin zone. The Bloch wave vector varies within the first Brillouin zone so that complete frequency–wavenumber relation is obtained. The first Brillouin zone is highly symmetrical, so a triangle zone ΓXMΓ can replace the entire Brillouin zone. The irreducible Brillouin zone (ΓXMΓ zone) is the smallest frequency–wavenumber space, which is essential to determine dispersion relationship for a given unit cell. The three edges ΓX, XM and MΓ, which indicate the x-direction, oblique direction less than 45°, and diagonal direction, respectively, control the bounds of band gaps. The first Brillouin zone and irreducible Brillouin zone are shown in Fig. 2. Therefore we simply swept the Bloch wave vectors on the edges of ΓXMΓ to obtain the dispersion curves. 2.4. Calculation of dispersion curves by finite element method Inserting Eqs. (2) and (5) into Eq. (1), the wave equations can then be transformed into an eigenvalue equation. The problem is however generally analyzed through a discretization method, such as the FEM. FEM is a common approach for calculating the dispersion curves of periodic structures. The complexity of the unit cell increases as the stage of the Sierpinski carpet increases. Other approaches including plane wave expansion method [10] and finite-difference time domain method [32] are less appropriate in terms of convergence or modeling for complex structures. FEM is advantaged by compatibility, convergence, accuracy, and suitability for building complex models. In this study, the COMSOL 3.5 FEM software was adopted to calculate the dispersion curves and group velocities of the Sierpinski-carpet unit cell [39]. The unit cell is meshed with square meshes. According to the convergence requirement, the mesh size was set to less than 1/8 the minimum wavelength. The normalized frequency was defined as Ω = ωa/2πct , where ct is the speed of transverse waves of background materials [11]. The observed frequency range was less than Ω = 5. The maximum size of the meshed model was less than 1/54 of the periodic constant, which satisfies the convergence requirement. For the adaptive FEM, 2D 4-node square finite elements and plane strain solution were applied in COMSOL 3.5 [39]. Due to the periodicity of the problem, periodic boundary conditions were required as expressed in Eq. (5). The left and bottom boundary conditions K K uiKx (r) and ui y (r) were connected to the right and top boundary conditions uiKx (r) eiKx a and ui y (r) eiKy a in the unit cell, respectively; the unit cell was then meshed and divided into finite elements via FEM. Different elements were connected by sharing the same nodes, thus, the eigenvalue solutions were obtained as follows [19]:

¯ )u = 0 (K¯ (K) − ω2M

(7)

¯ and M ¯ are stiffness and mass matrices of the unit cell, where u is the displacement at all the nodes of the unit cell; K respectively. Eq. (7) is an eigenvalue problem whose solution represents the dispersion relationship of the periodic structure. Imposing two of the three unknowns (Kx, Ky and ω) results in the solving for the third. Kx and Ky are generally varied within the first irreducible Brillouin zone. Eigenfrequency analysis was chosen as the solver mode in COMSOL 3.5 [12,39]. The process also necessitated Hermitian transpose of the constraint matrix and symmetry detection in the advanced solver parameter settings. A MATLAB-compatible ‘.m’ file which can automatically adjust K values was programmed to sweep the Bloch wave Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 3. Dispersion curves of periodic structure with square Pb cylinders in rubber matrix [34].

vectors K at the edge of the first irreducible Brillouin zone, which allowed us to obtain the dispersion curves ωn (K), where n is the first, second, etc., vibration band. 2.5. Comparison and verification The proposed method was verified to be reasonably accurate by comparing our numerical results against theoretical and experimental results obtained by previous researchers [32,34]. Two sets of results that are representative and correlative with our result are shown in Figs. 3 and 4. Fig. 3 shows the dispersion curves of Pb square cylinders embedded in the rubber matrix, which can be considered a 1-stage quasi-Sierpinski carpet with filling fraction f ¼0.36; a full band gap exists and the dispersion curves we calculated agree well with the results given by Liu et al. [34]. Fig. 4 shows the numerical and experimental dynamic responses of Al matrix with air cylinders, which can also be considered as a 1-stage quasi-Sierpinski carpet. We found several directional AZs existing in the numerical dispersion curves; which are marked by thick lines in the vertical axis, to be generally consistent with the attenuation ranges presented by previous researchers [32]. By comparing this work with previously published results, it is effectually confirmed that the dispersion curves we obtained should indeed be able to determine the dynamic response of the periodic structure with reasonable accuracy. The comparison was also important in regard to establishing the trend that Sierpinski carpets have unique band gap regions for different stages.

3. Properties of dispersion relationship 3.1. Band gap A full band gap corresponds to a frequency range where the elastic wave propagation is reduced in all directions. The placement of an excitation with frequencies in the band gap will lead to a low wave transmission level through the structure. The band gap characteristics of periodic structures are discussed firstly in this section. Fig. 5 shows the dispersion curves of Pb–rubber Sierpinski carpets with various stages. The material parameters are shown in Table 1. The periodic

Fig. 4. Numerical dispersion curves by proposed method (a) and experimental and calculated results for dynamic responses of Al matrix with air cylinders [32] (b) Parameters: a = 2.8025 mm , f¼ 0.4.

Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 5. Dispersion curves of Pb–rubber Sierpinski-carpet unit cell with different stages. (a) L ¼1, (b) L ¼ 2, (c) L ¼ 3. Table 1 Material properties [12,34] (density ρ, elastic modulus E, and Poisson's ratio ν). Material

ρ (kg/m3)

E (GPa)

ν

Pb Rubber

11 600 1300

40.82 1.37  10  4

0.37 0.463

constant a was equal to 1 (dimensionless unit) in the theoretical and numerical discussions of Sections 3 and 4. It can be observed that a small full band gap ranging from Ω = 0.753 to Ω = 0.825 existed on the band structure. For the case of L¼2, the full band gap in the lower-frequency range disappeared due to increase in the bound of the third band and decrease in the bound of the fourth band; however, full band gaps with relatively larger widths appeared in middle and higher frequency ranges. Two large band gaps appeared from Ω = 2.300 to Ω = 2.653 and from Ω = 3.706 to Ω = 3.882. For the case of L ¼3, only two relatively narrow full band gaps (from Ω = 2.288 to Ω = 2.350 and Ω = 2.434 to Ω = 2.508) were found in the middle frequency range. Based on the shapes of these bands, as shown in Fig. 5, the band structures had self-similarity features, especially in the lower frequency range. The first three bands were almost identical in all three cases. That said, in the middle and higher frequency ranges, the band structures did differ and the frequencies of higher order bands for L ¼2 and L¼ 3 were lower than those for L ¼1. Obviously, the filling fraction increased from f ¼0.11 to 0.30 as the stage increased from L ¼1 to 3. A large filling fraction usually results in a wider band gap for non-fractal unit cells. The increasing filling fraction is not the major influence factor on band gaps for higher stage fractal unit cells. Diffusion interference between these scatterers and matrix leads to a more complex pattern which is visible in the dispersion curves shown in Fig. 5. More differences occurred at higher frequencies rather than lower ones, as smaller scatterers can diffuse the smaller wavelengths more intensely than a central square scatterer [33]. The central square scatterer primarily tunes longer wavelengths, therefore, the shapes of the first three bands are similar. In effect, the surrounding small scatterers primarily modulate the bands in the high frequency range while the central scatterer controls the motion of the bands in the low-frequency range. As shown in Figs. 5(a)–(c), the bands were compressed and the low-frequency band gaps disappeared as the stage of the Sierpinski carpet increased. Multi-band properties of fractal structures in photonic crystals appeared in the case of L¼2, but ultimately disappeared as the stage increased (e.g., in the case of L ¼3). As opposed to fractal photonic crystals, increasing the stage for the geometry of a traditional Sierpinski carpet using present materials is not helpful in further obtaining a multiband solution [26,29]. Multi-band characteristics are dependent on the geometry of the unit cell and scatterers as well as the material properties of the medium, so the fractal photonic crystal information is not entirely applicable to Sierpinski-carpet periodic structures. As mentioned above, a lack of full band gaps can be compensated for by the directional characteristics of elastic waves in periodic structures. Therefore, we examined the group velocities of two representative frequencies Ω = 0.580 and Ω = 0.738 (dotted and dashed lines in Fig. 5) as discussed in the following subsection. 3.2. Group velocity The obtained dispersion curves can be used to evaluate the band gaps of the considered periodic structure. Moreover, the group velocity defines the manner in which wave energy flows in the periodic media subjected to an external perturbation [20]. Evaluating group velocities allows the dispersive behavior of the periodic structure and anisotropic characteristics of the domain in the wave propagation plane to be determined. Owing to a high anisotropy, the direction of the wave propagation can be chosen to obtain certain required directional or spatial filters, which can be used to isolate sensitive Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 6. Group velocities of Pb–rubber Sierpinski-carpet unit cells with different stages at Ω = 0.580 . (a) L ¼1, (b) L ¼2, (c) L ¼ 3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

equipment from a localized excitation source [18]. The dispersion relationship directly affects the group velocity and determines the wave motion in different directions. In some cases, the periodic structure only has a narrow full band gap, whose vibration isolation capacity is limited. The application of periodic structures on vibration isolation can be expanded by exploiting the directional propagation of waves in periodic structures. Meanwhile, if the incident wave has a directional feature, the negative group velocity can also further extend the vibration isolation range. All these features can be described by group velocity plots, so we found it crucial to investigate such phenomena through the straightforward evaluation of group velocity. The group velocity in 2D plane structures can be expressed as follows [19,22]:

⎛ ∂ω ∂ω ⎞ ⎟ cg = ⎜ , ⎝ ∂Kx ∂Ky ⎠

(8)

Group velocity is defined as the gradient of dispersion surfaces. The dispersion relations for a 2D periodic structure can be represented as surfaces defining the frequencies associated with the Bloch wave vectors Kx and Ky. At a given frequency, the Bloch wave vector (Kx, Ky ) should be swept in all directions in the first Brillouin zone, then the corresponding in-plane group velocity can be obtained accurately. The amplitude of group velocity is related to the preferential directions of wave propagation, or the existence of “forbidding directions” along which waves cannot propagate. Fig. 6 shows the group velocities of Pb–rubber Sierpinski carpets of different stages at a given frequency Ω = 0.580. (This frequency is marked by a dotted line in Fig. 5.) This frequency falls in a directional band gap for L ¼1 and leads to negative group velocities for L ¼2; meanwhile, major energy directions can be observed in the corresponding group velocity plot for L ¼3. Therefore, this frequency is highly representative and discussed as follows. The plane of this frequency cuts only the third band, thus returning a single group velocity as shown in Fig. 6(a), where the given frequency is in the range of the directional band gap and the shape of the group velocity is clearly aligned to the x - and y -directions; in effect, elastic waves only propagated along the x - and y -directions. The corresponding group velocities of the Bloch wave vectors in the first quadrant are highlighted in green. Group velocities mainly centered around the x - and y -directions, as marked with highlighted dots. A portion of said highlighted dots are in the diagonal direction, however, they are in the third quadrant and thus represent negative group velocity. In other words, waves could only propagate along the x - and y -directions and not through the diagonal direction because the vibration source was located at the center of the periodic structure. (This behavior is further discussed in Section 5.) For the case of L ¼2, we attempted to confirm the existence of “caustics” in x- and y-directions to determine whether strong energy was focused on the corresponding wave directions. Due to the negative group velocity of all the Bloch wave vectors in the first quadrant, the waves could not propagate along the incident wave direction and instead were all reflected. As shown in Fig. 6(c), the group velocity was also mainly aligned to the x - and y -directions and was primarily positive, suggesting that full reflection did not occur. Fig. 7 shows the group velocities of Pb–rubber Sierpinski carpets of different stages at a given frequency Ω = 0.738 (dashed line in Fig. 5). This frequency leads to a clearly negative group velocity feature for L ¼1 and the group velocity exhibits caustics in the diagonal directions for L ¼2 and L ¼3. Therefore, this characteristic frequency is discussed as follows. The plane of this frequency cuts the dispersion surfaces of the bands once, twice, or three times in cases L ¼1, 2, and 3, respectively. Three different types of group velocities are shown in Figs. 7(a)–(c). In Fig. 7(a), the shape of group velocity is a twirled square rather than a circle, indicating that the periodic structure cannot be considered an isotropic material. The plot shows strongly negative group velocity characteristics as well, and interestingly, the existence of corners in x- and y-directions. Such corners are associated with relatively strong energy focused in the propagating wave directions, but shape of Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 7. Group velocities of Pb–rubber Sierpinski-carpet unit cells with different stages at Ω = 0.738. (a) L ¼ 1, (b) L ¼2, (c) L ¼3.

the group velocity plot is still a twirled square. In Fig. 7(b), the group velocity of the third band is characterized by extended “caustics” which resulted from interference between the various wave components propagating in the structure [22]. The caustics occurred in diagonal directions, suggesting that wave energy was aligned predominantly to the diagonal directions. The shape of the group velocity was a square for the fourth band where the wave energy was focused on the corners of the square; group velocity was also positive. In the case of L¼3, as shown in Fig. 7(c), although the shape of group velocity was similar to that of L ¼2, the group velocity of the third band was focused in the x - and y -directions. The shape of the fourth band for L ¼3 was similar to that of the third band for the case of L ¼2, while the shape of the fifth band for L ¼3 was similar to that of the fourth band for the case of L¼2. These observations further indicate that the band structure was compressed as the stage increased. Constructive and destructive interference between these scatterers created a complex response pattern [18]. The physical implications of the geometry of the group velocities were explored by plugging the values into Eq. (8). The directional characteristics are shown in Figs. 6 and 7. The fact that the maximum group velocities increased as stage increased is also quite clear by comparison among Figs. 7(a)–(c). Increase in Pb scatterers resulted in enhanced structural stiffness, therefore increasing the corresponding group velocities.

4. Numerical simulation Group velocity plots represent the energy flow [18,19,22]. The displacement field under the steady-state vibration also presents the vibration energy propagation [18,22]. Therefore, we used group velocity plots to show the theoretical wave propagation features. For the (real) finite periodic structure, we calculated the displacement field to show the real wave propagation. The two methods have a consistent effect in the directional descriptions. The group velocity plots are calculated based on a unit cell using the periodic boundary conditions. The displacement field is calculated based on a (real) finite periodic model. Two methods can be mutually verified and used to describe the directional characteristics of the wave propagation [18]. Therefore, we further investigated the response of the system to harmonic pressure wave excitation from the center of the numerical model. A finite Pb–rubber system was calculated via FEM in ANSYS software. The harmonic pressure wave excitation is expressed as P ¼1000sin(2πf ) [Pa], where f is the frequency of the incident wave. This model contains 50  50 unit cells with a very small hole (radius¼a/27) in the center, plus a harmonic pressure wave on the edge of the small hole as shown in Fig. 8. The wave propagation trend as predicted by the group velocity was verified by the numerical response. This model was also appropriate for validating whether or not “dead zones” or “dead directions” existed at certain frequencies. To consider the effect of the far field during the numerical analysis, viscous-spring boundary elements were added to the boundaries to reduce (or even render negligible) the reflection of waves. Damping constant C and spring constant K are the parameters of the viscous-spring boundary elements, which can be expressed as follows [40]. For the viscous-spring elements perpendicular to the boundary:

CN = ρm cp,

KN = αN

μm RD

(9)

and for the viscous-spring elements parallel to the boundary:

CT = ρm ct ,

KT = α T

μm RD

(10)

where cp and ct are the longitudinal and transverse speed of elastic waves in matrix, respectively; RD is the distance between Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 8. Schematic diagram of finite periodic structure with Sierpinski carpets in the case of L ¼2.

the vibration source and boundary nodes, and ρm and μm are the density and shear modulus of the matrix, respectively. The viscous-spring boundary has favorable robustness to αN and αT , which were considered 1.5 and 0.5, respectively, in the simulation discussed below [40]. The model was reduced to a free-field model including solely matrix material to confirm the accuracy of the proposed simulation model. Fig. 9 shows the responses along the center line when normalized frequency was equal to the central observed frequency Ω = 2.5. The extended solution of the free field with free boundary conditions was used to validate the model's accuracy. The results in t ¼2.4 s are shown, as at that point, waves had already propagated along the whole model with viscous-spring boundaries but without affecting the observed range of the extended model. The analytical results with viscous-spring boundaries agreed well with the extended solution, which validates the feasibility and accuracy of the proposed simulation model. Fig. 9 also shows where the viscous-spring boundary had favorable robustness to RD; therefore, RD was subsequently taken as the half length of the model diagonal. Fig. 10 depicts the numerically determined displacement of the periodic structure with Pb–rubber Sierpinski carpet at Ω ¼0.580. The results clearly illustrate the directional characteristics of the wave propagation in the periodic medium. Snapshots of the numerical wave field at t ¼2 s and t¼ 3 s were shown in Fig. 10(a) and (b), in which the theoretical group velocity profiles (dots) used in Section 3 were also plotted. Figs. 10(a) and (b) show where the energy was focused on the center and propagated only in x - and y -directions. The waves propagation motion created a “ þ” shape, which is consistent with the shape of the group velocity shown in Fig. 6(a). It is clear from the numerical wave field that the expected directional wave propagation profile is observed. The theoretically computed group velocity profiles are consistent to the wave field. By comparing Figs. 10(c) and (d), it is again clear that the wave traveled primarily in x - and y -directions; however, their guide-wave performance was worse than that of L¼ 1 because the normalized frequency fell in the directional band gap for L ¼1 and in the pass band for L ¼2 and L¼ 3. The wave propagation mechanism for L¼2 and L ¼3 was different. The wave field took “þ” shape for L ¼2 due to the full reflection of negative group velocities because the vibration source was located in the center of a strongly symmetrical model. The wave field for L¼3, conversely, was based on the wave propagation of a positive group velocity. In all three cases, the vibration reduction regions (or “dead zones”) occurred in the diagonal

Fig. 9. Displacement responses for a harmonic incident plane wave with Ω = 2.5.

Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 10. Scattering displacement field of a Pb–rubber Sierpinski-carpet periodic structure with different stages at Ω = 0.580 . (Wave field overlaid with the theoretical group velocity profile from Fig. 6(a) (dots) for L ¼ 1.) (a) L ¼ 1 and t ¼2 s, (b) L ¼ 1 and t ¼3 s, (c) L ¼ 2, (d) L ¼3.

directions. The dead zones were more pronounced in the case of L¼ 1 than in L ¼2 and L¼3 due to the existence of a directional band gap in L ¼1, which is in agreement with theoretical analysis depicted in Fig. 6. Fig. 11 shows the numerically determined displacement of the periodic structure with Pb–rubber Sierpinski carpet at Ω ¼ 0.738. These results also clearly indicate the varying directivity of wave energy in the three periodic mediums. Fig. 11 (a) shows where, again, the energy focused on the center and propagated in x - and y -directions. The waves propagated in the shape of a rolled square, which is consistent with the shape of the group velocity shown in Fig. 7(a). As shown in Figs. 11 (b) and (c), waves were aligned predominantly to diagonal directions and the response was characterized by the emergence of dead zones in x - and y -directions. The dead zones in the case of L¼ 2 were more noticeable than those in L ¼3, which is also consistent with the analysis illustrated in Fig. 7. Based on the above numerical results, we confirmed that the real wave field is remarkably consistent with the corresponding shapes of group velocities. In the absence of caustics, the periodic structure response had a smooth spatial distribution although the group velocity was far from circular, reflecting the non-isotropic geometry of the periodic structure. In the presence of caustics, the physical mechanism was dominated by constructive and destructive interference between the scatterers resulting in a complex spatial response distribution [18]. A dead zone in which a relatively low response was obtained formed beyond the caustics. As discussed above, group velocity information is crucial for accurately determining the directional propagation characteristics of Sierpinski-carpet periodic structures. The dead zone information can be used to complement the band gap behavior, which further expands the application of periodic structures for wave tuning. The applications of these Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 11. Scattering displacement field of a Pb–rubber Sierpinski-carpet periodic structure with different stages at Ω = 0.738. (a) L ¼ 1, (b) L ¼2, (c) L ¼3.

characteristics, including the negative group velocity feature, are further discussed in the following section.

5. Application Directional characteristics shown in group velocity plots may also be directly observed from the time domain simulation of wave propagation in the structural system. In the example of the paper, a 60a × 60a model with 15  30 unit cells was examined as shown in Fig. 12. The vibration source was, again, located at the center of the model and the periodic structure was arranged only on the right side of the vibration source. This numerical application example matches conceptually a number of practical vibration isolation or sound attenuation materials in application (e.g., a protective material shielding sensitive equipment from a point loading or pressure wave excitation). This configuration can be used to compare the wave propagation features with and without Sierpinski-carpet periodic structures. The material parameters of the mediums were the same as those discussed above. For practical application, the periodic constant was set to a ¼0.01 m, the size of the model was also normalized by the periodic constant. The elastic displacement distribution for various frequencies is displayed in Fig. 13, which illustrates the interaction of directional wave propagation and characteristic frequencies. Fig. 13(a) corresponds to L ¼1 and Ω = 0.580 in the directional band gap and shows where elastic waves could only propagate through the periodic structure without attenuation in the x -direction. Waves could not propagate along the diagonal directions. In the left side of the model, waves propagate along all the directions due to the absence of periodic structures. Fig. 13(b) shows the elastic displacement pattern for L¼ 3 and Ω = 0.580 in the transmission range. Waves mainly propagated along the x -direction to the side of the periodic structure, Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 12. Calculation model for Sierpinski-carpet periodic structure application in the case of L ¼ 1.

Fig. 13. Elastic displacement distribution of a Sierpinski-carpet periodic structure for different stages at characteristic frequencies. (a) L ¼ 1 and Ω = 0.580 , (b) L ¼3 and Ω = 0.580 , (c) L ¼ 1 and Ω = 0.738, and (d) L ¼3 and Ω = 0.738.

but propagated along all the directions on the other side of the vibration source. As shown in Fig. 13(c), the amplitude was reduced significantly to the side of the periodic structure; waves were almost totally reflected in front of the periodic structures due to the negative group velocity for L¼1 and Ω = 0.738, so waves were unable to propagate through the Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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Fig. 14. Frequency responses for Sierpinski-carpet periodic structure with different stages. (a) L ¼1, (b) L ¼ 2, (c) L ¼3.

periodic structure (almost as if there had been a full band gap). As an important result, the existence of full reflective Bloch wave modes with a negative group velocity was highlighted, especially in Sierpinski carpet periodic structures. Hence the frequency range with negative group velocities is believed to have great interests for designing fractal periodic systems as all directional mechanical filters. As shown in Fig. 13(d), the wave primarily propagated along the diagonal directions while wave propagation in the x -direction was forbidden for L ¼3 and Ω = 0.738. Figs. 13(a)–(d) altogether demonstrate the waveforbidding behavior on the directional band gap and the directional propagation of special frequencies in the pass band. The frequency response function (FRF) was also calculated to further evaluate the screening effectiveness of Sierpinskicarpet periodic structures. FRF is defined as FRF [dB] = 20 log10A /A0 , where A is the amplitude of dynamic response with periodic structures and A0 is the corresponding response without periodic structures. Fig. 14 lists the FRF values for the three types of periodic structures we examined; the analytical band gaps are also shown in Fig. 14 as gray zones. The frequency attenuation regions agreed very well with the AZs, suggesting that the magnitude of vibrations with frequencies in the range of the band gaps decreased significantly. There also were multiple attenuation ranges in the case of L¼2, indicating that the Sierpinski-carpet periodic structures extended the vibration reduction zones without increasing the size of unit cells. In the case of L¼ 1, the full band gap was narrower than the corresponding lower transmission range. Vibrations began to decrease at Ω = 0.708, at which point negative group velocity began to appear. This feature apparently expanded the vibration reduction capacity of Sierpinski-carpet periodic structures. It was particularly interesting to find that vibrations decreased to a certain extent near Ω = 0.738 in the case of L¼ 2 and L¼3 due to the directional propagation characteristics even though these frequencies were in the pass band. The band gap behavior in Sierpinski carpet periodic structures is complemented by unique directional characteristics in the revision. Waves with frequencies in the pass bands may only propagate in certain directions in the periodic system.

6. Conclusions Sierpinski-carpet periodic structures including Pb square cylinders inserted in a rubber matrix were explored at length in this study. The dispersion relationships of Sierpinski carpets with different stages were investigated and the directional characteristics of Sierpinski-carpet periodic structures were analyzed according to group velocities and validated via numerical analysis. A finite periodic structure model subjected to harmonic excitations was used to illustrate the application of Sierpinski-carpet periodic structures for vibration isolation. Three major conclusions based on the results were drawn as follows. (1) The Sierpinski-carpet periodic structure in the case of L ¼2 is beneficial in obtaining multiple band gaps, which can significantly reduce vibrations and expand the range of vibration reduction. (2) Periodic structures with different stages at given frequencies exhibit wave directionality. If the wave frequency is in the directional band gap, wave propagation can be inhibited in the corresponding directions. Wave propagation is isolated to a certain extent in special directions even if the wave frequency is in the pass band. The existence of a negative group velocity in the fractal periodic structure is highlighted, which is almost equal to a full band gap. (3) The corresponding frequencies of each band are compressed as stage increases. The band structure of Sierpinskicarpet periodic structures has self-similar features, especially in the lower frequency range. In the middle and higher frequency ranges, the band structures differ significantly. These observations reflect the unique tunability of Sierpinski-carpet periodic structures, which opens the possibility of designing an elastic wave filter device with a periodic configuration to protect sensitive equipment from a localized vibration source. It is not suggested that Sierpinski-carpet periodic structures are superior to other periodic structures. However, the results discussed above do shed light on fractal periodic structures in general. Although the analysis we Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i

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conducted was restricted to the plane wave propagation in Sierpinski-carpet periodic structures, exploration of fractal periodic beams, plates, and shells merits further research.

Acknowledgments This work was supported by the Fundamental Research Funds for the Central Universities of China (Grant No. BLX201411) and the National Natural Science Foundation of China (Grant Nos. 51678046 and 51278263). The authors would also like to thank Prof. Fabrizio Scarpa of University of Bristol and Dr. Leihong Li of Georgia Institute of Technology for their comments and suggestions in improving our manuscript. Thanks are also due to Prof. Zhifei Shi of Beijing Jiaotong University for helpful remarks.

Appendix A The number of grids on the initial square is N = 9L and the filling fraction of the scatterers can be expressed as:

f=

L

∑k = 1

8L − k 9k − L − 1

(A.1)

If the central position of each sub-square can be exactly described, the Sierpinski carpet can be drawn by computer program. Meanwhile, the position can be used in the dispersion relationship calculation via FEM method. The position vector derivative method presented in this work allows the positions of scatterers to be obtained by the derivative position vectors of smaller sub-squares. If L ¼1, only one sub-square exists in the center of the initial square and the position of the central sub-square is defined as (0,0). If L ≥ 2, the central position (rx, ry) of each central small sub-square can be defined as follows:

ri = rb +

a (s − 1), 3L (3−L

s = 1, …, 3L − 1

(A.2) 3L − 1,

3−1) a/2

where i = x, y ; rb = represents the initial position of the first scale. When s = the end position (re) of the − first scale is (3−1 − 3−L ) a/2. The central position of each surrounding small sub-square can also be defined via Eq. (A.2), however, rb represents the initial position of the k-th scale and must be defined as follows:

rb = rik −

3−L (3l − 2 − 1) a 2

(A.3)

where l = L, L − 1, … , 2; k = 2, 3, … , L while s = 1, … , position of surrounding sub-squares:

(

) (

r k = rxk, r yk = R xk − 1 + R xk , R yk − 1 + R yk

(

)

where the vector Rk = Rxk , R yk belongs to and

(−

3l − 2.

rkx

and

rky

represent the vectors of derivation and control the

)

(A.4)

(R

k x

) ( R , 0), ( R

, R yk ,

k x

k x

)(

)(

)(−

, − R yk , 0, R yk , 0, − R yk ,

)(−

Rxk , R yk ,

)

Rxk , 0 ,

)

Rxk , − R yk , respectively. Rx1 = R1y = 0; Rxk = 3l − L − 1a , R yk = Rxk .

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Please cite this article as: J. Huang, et al., Analysis of in-plane wave propagation in periodic structures with Sierpinskicarpet unit cells, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.020i