Ultrasonics 52 (2012) 920–924
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Lamb waves in phononic crystal slabs: Truncated plane parallels to the axis of periodicity Jiujiu Chen ⇑, Yunjia Xia, Xu Han, Hongbo Zhang State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanics and Vehicle Engineering, Hunan University, Changsha City 410082, PR China
a r t i c l e
i n f o
Article history: Received 8 January 2012 Received in revised form 29 February 2012 Accepted 29 February 2012 Available online 19 March 2012 Keywords: Lamb waves Phononic crystal Band structures
a b s t r a c t A theoretical study is presented on the propagation properties of Lamb wave modes in phononic crystal slabs consisting of a row or more of parallel square cylinders placed periodically in the host material. The surfaces of the slabs are parallel to the axis of periodicity. The dispersion curves of Lamb wave modes are calculated based on the supercell method. The ﬁnite element method is employed to calculate the band structures and the transmission power spectra, which are in good agreement with the results by the supercell method. We also have found that the dispersion curves of Lamb waves are strongly dependent on the crystal termination, which is the position of the cut plane through the square cylinders. There exist complete or incomplete (truncated) layers of square cylinders with the change of the crystal termination. The inﬂuence of the crystal termination on the band gaps of Lamb wave modes is analyzed by numerical simulations. The variation of the crystal termination leads to obvious changes in the dispersion curves of the Lamb waves and the widths of the band gaps. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction It may be well-known that elastic vibrations in the periodic arrays called phononic crystals satisfy a frequency band structure. Phononic crystals (PCs) consist of composite materials with elastic coefﬁcients which vary periodically in space. During the past decades, the propagation of elastic waves in PCs has attracted much attention [1,2]. The occurrence of band gaps, where the propagation of acoustic or elastic waves with frequencies within the gaps is forbidden regardless of the direction, suggests numerous technological applications such as acoustic ﬁlters, ultrasonic silent blocks, focus lens and so on [3–9]. More recently, the properties of the plate-mode waves in two-dimensional (2D) phononic crystal (PC) slabs have been intensively studied because of their potential practice . Unlike the inﬁnite PCs for bulk waves, which are inﬁnite along three dimensions in real space, the PCs studied for the platemode waves are taken to be a ﬁnite size system in one direction. Hou et al. have investigated the propagation of elastic wave modes in a two-layer free standing plate composed of a one-dimensional PC thin layer coated on a uniform substrate . Chen et al. have investigated the propagation of Lamb waves in one-dimensional PC plate coated symmetrically with solid loading layers on both sides . It has been shown that Lamb waves can be supported in 2D PC slabs with the slab surfaces perpendicular to the axis of
periodicity and the ratio of the thickness of the slabs to the lattice period inﬂuences the band gaps of Lamb waves [13–15]. However, another conﬁguration that the surfaces of slabs are parallel to the axis of a row of square cylinders also supports Lamb waves. The PC slabs of this conﬁguration have the potential applications in high-frequency wireless communication and sensing systems and we studied the structure for precast-slab-type construction inspection and other reliable nondestructive evaluation. In this paper, we investigate the characteristics of Lamb waves in PC slabs consisting of a row or more of square cylinders placed periodically in the host material as shown in Fig. 1. This paper is organized as follows. In Section 2, the theory of the supercell method is explained. In Section 3, the geometry model of calculation will be exhibited. The dispersion curves of the conventional structure are calculated by the supercell method and the ﬁnite element (FE) method (Comsol Multiphysics 3.5a), which are in good agreement with the transmission power spectra (TPS). The inﬂuence of the crystal termination on the band gaps of Lamb wave modes is analyzed by numerical simulations. The variation of the crystal termination leads to obvious changes in the locations and the widths of the band structures. Conclusions drawn in this paper are given in Section 4.
2. Theory ⇑ Corresponding author. E-mail address: [email protected]
(J. Chen). 0041-624X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2012.02.015
The equation governing the motion of displacement vector u(r, t) can be written as
J. Chen et al. / Ultrasonics 52 (2012) 920–924
periodic system and the stress-free boundary condition for Lamb waves is satisﬁed.
3. Numerical results
3.1. Geometry and method of calculation
Fig. 1. (a) Auxiliary superstructure for calculations of Lamb wave. (b) Unit cell of (a), two vacuum layers with thickness h1 are added on their outer surfaces. The h2 equals to the lattice space a and the side length of the elastic square cylinders (denoted by B) is d.
qðrÞu€ p ¼ @ q ½cpqmn ðrÞ@ n um ðp ¼ 1; 2; 3Þ;
where r=(x, z)=(x, y, z) is the position vector, q(r) and cpqmn(r) are the position-dependent mass density and elastic stiffness tensor, respectively, and the summation convention over repeated indices is assumed in Eq. (1). Due to the spatial periodicity, the material constants, q(r) and cpqmn(r), can be expanded in the Fourier series with respect to the 2D reciprocal lattice vectors (RLV), G = (Gx, Gy), as the follows:
P jGx e qG
cpqmn ðrÞ ¼
P jGx pqmn e cG
Utilizing the Bloch theorem and expanding the displacement vector u(r, t) in Fourier series, one obtains
uðr; tÞ ¼
where k = (kx, ky) is a Bloch wave vector and x is the circular frequency. Note that q and cpqmn are independent on the z-direction because of the homogeneity of the system along the cylinder axis. By substituting Eqs. (2)–(4) into Eq. (1), it can be shown that the wave motion polarized in the z-direction, namely the SH wave, decouples from the wave motions polarized in the x- and ydirections, namely, the P and the SV waves. We focus our attentions to the P and the SV waves because it is relatively simple to discuss the SH wave. The Lamb waves can be obtained by the coupling between the P and the SV waves since two vacuum layers (denoted by C in Fig. 1b) with thickness h1 are used for designing the imaginary
We ﬁrst brieﬂy introduce the PC slabs, which are composed of unit cells consisting of embedded elastic square cylinders (denoted by B) of side length d and matrix material (denoted by A) with lattice spacing a. The material A and B are the Tungsten and the Vacuum, respectively. The elastic parameters used in the calculations 12 44 are C 11 A ¼ 502 GPa, C A ¼ 199 GPa and C A ¼ 152 GPa and mass 3 density qA ¼ 19200Kg=m for the Tungsten. Firstly the band structures of the Lamb wave modes were computed using the supercell method, which has been proven in previous works to be an efﬁcient method for obtaining the curves of plate-mode waves in PC slabs. The key point of the supercell method is to design an appropriate auxiliary inﬁnite periodic superstructure in order to apply the Bloch theorem. The ﬁlling rate f is 2 deﬁned by f ¼ ðd Þ=ða2 Þ and the thickness of slabs h2 equals to the lattice spacing, as shown in Fig. 1. The reciprocal lattice vector is G ¼ ð2pN 1 =a; 2pN 2 =ðh2 þ 2h1 Þ; with N1 and N2 integers, respectively. In the calculation, f = 0.09 and 529 RLV are used to reach good convergence. Secondly, we have made the calculation of the dispersion curves for PC slabs by using the FE method with the Comsol Multiphysics 3.5a. The calculation is performed by solving the eigenvalue problem of the unit cell. The main step of the calculation will be presented. First, the periodic boundary conditions are applied to the sides of unit cell along the x-direction based on the Bloch’s theorem. The periodic boundary conditions in terms of displacement up could be deﬁned as up ðx þ nd; yÞ ¼ up ðx; yÞ exp½jðkx ndÞ; where kx is the wave vector in the x-direction and n is an arbitrary integer. Moreover, the top and button surfaces are deﬁned free boundary conditions. Second, the 2D cross section of the supercell in the x–y plane is meshed and divided into ﬁnite elements. The ﬁnite elements are triangles with three nodes that has two degrees of freedoms ux and uy. Finally, the FE method transforms the analysis into a generalized elastic eigenvalue problem expressed by the linear equations ½Kðkx Þ w2 Mu ¼ 0; where K(kx) and M are the stiffness and mass matrices of the system. The relationship between the angular frequency w and the wave number kx is inherent in these equations. Then the band structure w = w(kx) can be built by sweeping kx through the entire ﬁrst Brillouin zone. An alternative searching arithmetic is employed to determine the eigenfrequency w. Fig. 2a shows the low-frequency part of the dispersion curves of the Lamb wave modes with f = 0.09, d = 0.3 mm, and a = 1.0 mm. The horizontal axis is the reduced wave number k ¼ ka=p. As shown in Fig. 2a, the data which is computed by the supercell method is marked with the solid lines, and the points represent the data calculated using the FE method. We can see that the locations and widths of band gaps of the dispersion curves from FE analysis are in good agreement with the results by the supercell method, and we ﬁnd a band gap (shaded region) which is from 1.372 to 1.745 MHz between the third and the fourth transmission band. The value of the gap width is 0.3724 MHz and the corresponding gap/midgap ratio is approximately 0.2389. In the band gap, all acoustic modes are suppressed. Finally, in order to further demonstrate the existence of the band gaps for the lower-order modes in the PC slabs, the FE method was employed to calculate the TPS for the ﬁnite periodic structure. The TPS of this structure corresponding to Fig. 2a has been performed as shown in Fig. 2b. There is a broad region ranged from 1.375 to 1.752 MHz for which the transmission is less than –
J. Chen et al. / Ultrasonics 52 (2012) 920–924
Reduced Wave Vector
0 -50 -100 -150
Firstly, we investigate the band structures of PC slabs whose thickness is an integral multiple of the lattice spacing a. In this case, the PC slabs have complete square cylinders. As shown in Fig. 3, the thickness of slabs has three values (h2 = a, h2 = 2a and h2 = 3a), where f = 0.09. There is a square cylinder in the y-direction when h2 = a, while there are two (three) complete square cylinders at h2 = 2a (h2 = 3a). From the Fig. 2a, we can see a complete band gap between the third and the fourth transmission band, where h2 = a, d = 0.3 mm, and a = 1.0 mm. The width of the band gap is 0.3724, and the middle of band gap is 1.5585 MHz. However, when h2 = 2a or h2 = 3a, the band structures change dramatically, and there is not band gap in the low-frequency part of the dispersion curves of the Lamb wave modes. So we can conclude that the width of the band gaps markedly decrease with the increase of the thickness of the slabs. Moreover, in order to further investigate the variation of the band gaps as a function of the crystal termination of the slabs, we used the parameter s(0 6 s 6 1). With the variation of the parameter s, there exist complete or incomplete (truncated) layers of square cylinders. The meaning of the parameter s can be understood from Fig. 4. When s = 0 (as shown in Fig. 4a) the crystal terminates with h2 = a. The termination changes with the increase of the parameter s. For example s = 0.5, the surface cuts in the half another cells, leaving cylinders of half the nominal cross section. If s = 1 (as shown in Fig. 4d), h2 = 3a and there are three complete square cylinders in the y-direction. To investigate the effects of the parameter s on the band gaps, the band structures are presented at s = 0.1, s = 0.5 and s = 0.8. When s = 0.1, the band structures are shown in Fig. 5a. From the
Fig. 2. (a) The dispersion curves calculated by the supercell method (solid lines) and the FE method (points) of the unit cell for Lamb wave modes. (b) The TPS computed by the FE method with f = 0.09, a = 1.0 mm, and h2 = 1.0 mm. (c) The displacement ﬁeld of the conventional structure at 0.8 MHz.
200 dB. We also notice a slight dip centered at about 0.8 MHz in Fig. 2b. This dip is attributed to the band gap of antisymmetric lamb wave modes. As shown in Fig. 2c, we can observe that the lamb wave mode at the frequency of 0.8 MHz is antisymmetric at the front of the periodical structure and the propagation of antisymmetric Lamb wave is forbidden after through the system. The result is consistent with the band structures computed by the supercell method or the FE method.
(a) h2 = a
(b) h2 = 2a
(c) h2 = 3a
Fig. 3. The structures of the phononic unit cell with three values ((a) h2 = a, (b) h2 = 2a and (c) h2 = 3a) of the thickness (h2) of the slabs.
3.2. The variation of the band structures In order to further investigate the band structures of PC slabs whose surfaces are parallel to the axis of periodicity, we changed the thickness of the plates so as to the existence of more complete or incomplete (truncated) layers of square cylinders placed periodically in PCs. With the variation of the crystal termination, the widths and the locations of the band structures which are completed using the FE method remarkably change.
(b) =(a-d)/ 2a
(c) =(a+d)/ 2a
Fig. 4. Variation of the position of the unit cell’s termination, marked with two arrows. (a) The crystal terminates with complete cells at the surface. (b) The host material has continuous maximum layers without another cylinder at the surface. (c) The surfaces are tangent to the left (right) face of the complete cylinder at the surface. (d) The crystal terminates again with complete cells at the surface; however, tow additional cells have been added.
J. Chen et al. / Ultrasonics 52 (2012) 920–924
2.5 2 1.5 1 0.5 0
Reduced Wave Rector
2.5 2 1.5 1 0.5 0
Reduced Wave Rector
2.5 2 1.5
and the ﬁfth (the sixth and the seventh) transmission band extends from 1.476 to 1.494 MHz (2.004–2.013 MHz). The band gap width is only 0.018 MHz (0.009 MHz) and the corresponding gap/midgap ratio is only 0.012 (0.004). However, Fig. 5c shows there is only one shallow gap that locates between the sixth and the seventh transmission band extends from 1.520 to 1.627 MHz When s = 0.8. The gap width is 0.107 MHz and the corresponding gap/midgap ratio is about 0.068. From above we can conclude that the locations and the widths of the band structures vary obviously with the parameter s. According above stated, the parameter s has crucial effect on the band structures. So we further investigate the band structure with the parameter s ranging from 0 to 1. Fig. 6 depicts the width of the lowest band gap as a function of the parameter s. We can speak of six regions involved in the problem. In the ﬁrst region where 0 6 s 6 0.18, the gaps exist for all s. The width of the lowest band gap decrease linearly with the increase of the thickness, and the band gap is closed at s = 0.18. The phenomenon is attributed to the upward movement of the third transmission band and downward movement of the fourth transmission band. When 0.18 6 s 6 0.36, in the second region, there is not gap due to h2 increasing continuously. In the third region where 0.36 6 s 6 0.6, there exist small gaps between the fourth and the ﬁfth transmission band owing to local resonance of the stubbed waveguides with the increase of h2. In the fourth region deﬁned by 0.6 6 s 6 0.68, there is also not gap. But in the ﬁfth region where 0.68 6 s 6 0.98, there exist band gaps between the sixth and the seventh transmission band and the maximum width of the gap is 0.1146 MHz when s = 0.82. It can be explained that the ﬁrst band gap is closed and the second band gap is pushed to low frequency region with the increasing thickness of the slabs. Finally, in the sixth region deﬁned by 0.98 6 s 6 1, the gap disappears. A physical explanation for this phenomenon can be found from that the cutoff frequencies of higher order modes of Lamb wave decreases and eventually the second band gap disappears with the increasing s. At last, in order to further demonstrate the existence of the band gaps for the lower-order modes with the varying s, we take a further step to investigate the displacement ﬁeld of wave propagation in the periodic structures. The displacement ﬁelds are plotted in Fig. 7a, c, e at the load frequencies of 1.54 MHz (1.49 MHz and 1.56 MHz), where s = 0.1 (s = 0.5 and s = 0.8). By contrast, the displacement ﬁelds at the load frequencies of 1.45 MHz (1.5 MHz) are plotted in Fig. 7b and d, where s = 0.3 (s = 0.65). As shown in Fig. 7a, we observe obviously the propagation of Lamb waves are forbidden at the frequency of 1.54 MHz. Fig. 7c and e
Reduced Wave Rector Fig. 5. Dispersion relations of Lamb waves modes for (a) s = 0.1, (b) s = 0.5 and (c) s = 0.8.
picture, we can ﬁnd two band gaps (shaded region). The ﬁrst (second) band gap located between the third and the fourth (the sixth and the seventh) transmission band extends from 1.441 to 1.596 MHz (2.340–2.410 MHz). The band gap width is 0.155 MHz (0.07 MHz) and the corresponding gap/midgap ratio is 0.1017 (0.029). As shown in Fig. 5b, we observe two shallow band gaps at s = 0.5. The ﬁrst (second) band gap located between the fourth
Fig. 6. The width of the lowest band gap versus the value of s.
J. Chen et al. / Ultrasonics 52 (2012) 920–924
τ = 0.1 1.54MHz
the Lamb wave modes and the width of the ﬁrst complete band gap vary observably with the parameter s. Thus, we can achieve tunable band gap for Lamb wave modes by varying the parameter s. Acknowledgments
(b) τ = 0.3 1.45MHz
(c) τ = 0.5 1.49MHz
(d) τ = 0.65 1.5MHz
(e) τ = 0.8 1.56MHz Fig. 7. Displacement ﬁelds for (a) s = 0.1, (b) s = 0.3, (c) s = 0.5, (d) s = 0.65 and (e) s = 0.8.
shows the same phenomenon at the frequency of 1.49 MHz (1.56 MHz) when s = 0.5 (s = 0.8). However, the displacement can be observed in the receiving side of the PC slabs in Fig. 7b and d at the frequency of 1.45 MHz (1.5 MHz) when s = 0.3 (s = 0.65). The results are in good agreement with those in Fig. 6. 4. Conclusion We have examined the band structures of lower-order Lamb wave modes propagating in PC slabs consisting of a row or more of parallel square cylinders placed periodically in the host material based on the supercell method or the FE method for inﬁnitely long periodic systems and have calculated the TPS for ﬁnite systems by using the FE method. The result shows that the dispersion curves of
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