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Large complete band gap in two-dimensional phononic crystal slabs with elliptic inclusions Yongsen Li a, Jiujiu Chen a,n, Xu Han a, Kan Huang b, Jianguo Peng b a b

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, PR China Hunan Provincial Communication Planning Survey and Design Institute, Changsha 410008, PR China

a r t i c l e i n f o

abstract

Article history: Received 13 September 2011 Received in revised form 1 December 2011 Accepted 9 January 2012 Available online 14 January 2012

Phononic band structure with periodic elliptic inclusions for the square lattice is investigated based on the plane wave expansion method. The numerical results show the systems composed of tungsten (W) elliptic rods embedded in a silicon (Si) matrix can exhibit a larger complete band gap than the conventional circular phononic crystal (PC) slabs. The phononic band structure of the plate-mode waves and the width of the ﬁrst complete band gap can be tuned by varying the ratio of the minor axis and the major axis, the orientation angle of the elliptic rods and the thickness of the PC slabs. We also study the band structure of plate-mode waves propagating in two-dimensional (2D) slabs with periodic elliptic inclusions coated on uniform substrate. & 2012 Elsevier B.V. All rights reserved.

Keywords: Elliptic inclusions Band gaps Phononic crystal

1. Introduction During the past two decades, there have been considerable interests in the elastic wave propagation in the periodic composite media (called a PC) due to their rich physics and potential applications. Phononic crystals (PCs) are composite materials with elastic coefﬁcients, which vary periodically in space. A most interesting aspect of these materials arises from the possibility of frequency regions, known as absolute phononic band gaps, over which there can be no propagation of elastic waves in the crystal, whatever the direction of propagation [1–4]. More recently, the properties of the plate-mode waves in 2D PC slabs have been a well studied topic because of their potential practice. It has been shown that plate-mode waves can be supported in 2D PC slabs and the ratio of the thickness of the plates to the lattice period inﬂuences the band gaps of plate-mode waves [5–10]. Unlike the inﬁnite PC for the bulk wave, in the ﬁnite structure plate-mode waves can be scattered not only by periodically arranged scatterers but also by the surface of the plate. The elastic waves in a composite plate can be very complicated due to the coupling of longitudinal and transversal strain components with complex wave vectors reﬂected by plate boundaries. It is difﬁcult to get a larger complete band gap in 2D PC slabs. To obtain a larger complete band gap for circular inclusions,

n

Corresponding author. Tel./fax: þ 86 731 88823993. E-mail address: [email protected] (J. Chen).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.01.089

anisotropic inclusions [11] have been used. Despite the progress that has been given to the band gaps in plate-mode waves 2D PCs. The search for a new system that can produce larger complete band gaps of plate-mode waves at some desired frequency range remains to be an important issue. In this paper, we theoretically study the dispersion curves of plate-mode waves propagating in 2D slabs with periodic elliptic inclusions for the square lattice. The dispersion curve of the platemode waves and the width of the ﬁrst complete band gap can be tuned by changing the ratio of the minor axis and the major axis, the orientation angle of the elliptic rods , the thickness of PC plate and the thickness of the uniform substrate layer.

2. Formulations Our calculations are based on supercell PWE method [6]. In this method, an imaginary three-dimensional (3D) periodic system is constructed by stacking the studied thin plates and vacuum layers alternately. The difference between the imaginary periodic system and true 3D one is that the Bloch feature of the wave along the thickness direction is broken in the virtual system. The geometry of the considered structure, which is stacked by a PC layer with thickness tp and an uniform substrate layer with thickness ts, is shown in Fig. 1. In an inhomogeneous linear elastic medium with no body force, the equation of motion for displacement vector u(r,t) can be

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Y. Li et al. / Physica B 407 (2012) 1191–1195

Q ¼ ðf v þ f s Þ

tp

ts

Substrate 2Rx

3. Numerical results 3.1. Compared the band structures of elliptic inclusions with that of the cylinder

tv 2Ry

tp ts tv x

y Fig. 1. (a) Square-lattice two-dimensional PC slabs with elliptic inclusions coated on uniform substrate, tp and ts are the thickness of PC and substrate layer, respectively. (b) Unit cell of (a), two vacuum layers with thickness tv are added on their outer surfaces. Rx is the major axis and Ry is the minor axis. The lattice constant in xy plane is a.

written as

rðrÞu€ i ¼ @j ½cijmn ðrÞ@n um , i,j ¼ x,y,z

ð1Þ

The conﬁned wave in each plate can be express as: X XX uj ðr,zÞ ¼ ujGz ðrÞ eiGz z ¼ uGr ,Gz eiðkr þ Gr Þr eiGz z

ð2Þ

Gz

aðr,zÞ ¼

Gr

X

aGz ðrÞ eiGz z ¼

Gz

Gz

XX

aGr ,Gz eiGr r eiGz z

Gr

ð3Þ

Gz

where r¼(x,y),kr ¼(kx,ky) is the Bloch wave vector in xy plane. Gr ¼(Gx,Gy), Gx,Gy ¼2np/a, Gz ¼2np/az (n ¼0,71,72,y,7N) with az ¼2tv þtp þts and a can be r or cijkl.aGr ,Gz are the Fourier coefﬁcients of elastic constant. In our considered system, if the PC layer is consisted of elliptic cylinders arranged squarely in matrix and the substrate layer is uniform, the Fourier coefﬁcients in Eq. (3) can be calculated by for Gr ¼0 ( Gz ¼ 0,f p ½am þ f ðae am Þ þ f s as aGr ,Gz ¼ ð4Þ Gz a 0,P½am þ f ðae am Þ þ as Q and for Gr a 0 8 < Gz ¼ 0,2f f p ðae am Þ J1 ðGÞ G aGr ,Gz ¼ : Gz a0,2f Pðae am Þ J1 ðGÞ G With qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G ¼ R2x ðGx cos y þGy sin yÞ2 þ R2y ðGy cos yGx sin yÞ2 P ¼ ðf v þ f p þ f s Þ

ð8Þ

where Rx is the major axis and Ry is the minor axis, fv ¼tv/az,fp ¼ tp/az, fs ¼ts/az,f¼ pRxRy/a2, y is the rotation angle and a is the lattice constant. ae and am are the constants of the embedded materials and matrix materials (r or cijkl.), respectively. as is the constants of the substrate materials (r or cijkl.) To obtain Eq. (6) we set the origin of the coordinate system at the center of the unit cell in xy plane and at the bottom in z direction.

PC

z

eiGz ðtv þ ts Þ eiGz tv f v iGz ðt v þ t s Þ iGz t v

eiGz ðtv þ tp þ ts Þ eiGz ðtv þ ts Þ ðf v þf s Þ , iGz ðt v þt p þ t s Þ iGz ðt v þ t s Þ

ð5Þ

ð6Þ

ð7Þ

In order to make the solution convergent, an enough larger tv must be used in the calculation. For the studied system with small thickness 2tv ¼1.0a is enough [6]. The elastic parameters used in 12 44 the calculations are C 11 A ¼ 50:2,C A ¼ 19:9,C A ¼ 15:2 (in units of 11 2 10 dyn/cm ) and mass density rA ¼19.2 g/cm3 for W. 12 44 11 C 11 dyn/cm2) B ¼ 16:57,C B ¼ 6:39, C B ¼ 7:956 (in units of 10 3 and mass density rB ¼2.332 g/cm for Si. The numbers of plane wave are chosen as Gx,Gy,Gz ¼0,71, 72 can ensure the convergence of the calculation due to the small discrepancy between both solid materials used. We ﬁrst compare the cylinder inclusions with the elliptic structure when ts ¼0. Fig. 2(a) depicts the low-frequency part of the dispersion curves of the plate-mode waves along the boundaries of the irreducible part of the Brillouin zone with ﬁlling ratio f¼0.1675 for elliptic inclusions. The thickness of PC plate is a with the y ¼0 and T¼0.47. T is the ratio of the minor axis and the major axis (T¼ Ry/Rx). The vertical axis is the normalized frequency on ¼ oa/ct, where ct is equal to ct ¼ ðððcA44 cB44 Þf þ cB44 Þ=ððrA rB Þf þ rB ÞÞ0:5 . A complete band gap exists between the ﬁfth and the sixth bands and the value of the normalized gap is 0.7069. Fig. 2(b) represents the case for cylinders with the same ﬁlling ratio. From Fig. 2(b), it can be seen that there also exists a complete band gap between the ﬁfth and the sixth bands, but the value of the normalized gap is only 0.3666. In order to give a better insight into the inﬂuence of the ﬁlling fractions on the width of the ﬁrst complete band gap, Fig. 2(c) presents the width of the ﬁrst complete band gap as a function of the ﬁlling fraction for the elliptic inclusions and the cylinders. From the Fig. 2(c), we can see that for the case of elliptic inclusions the ﬁrst complete band gap is monotonically increasing for the ﬁlling fraction within the range from 0.03 to 0.35. The maximum value of gap width occurs at the optimal ﬁlling fraction f¼0.35, leading to a maximum width 1.02. But for the cylinders, the ﬁrstly complete band gap exists for the ﬁlling fraction within a range between 0.05 and 0.27. The optimal ﬁlling fraction that gives a maximum value of the width 0.3666 occurs at f ¼0.1675. The range of the ﬁlling fraction that renders the existence of the ﬁrst band gap for cylinders is narrower than for elliptic inclusions. Moreover, the width of the band gap for elliptic inclusions is larger than that of the system for cylinders with the same ﬁlling fraction. The effect of ellipticity on the band gap is obvious and we can easily obtain wider band gap by elliptic inclusions instead of cylinders. 3.2. Effect of the ratio of the minor axis and the major axis (T¼ Ry/Rx) on the band structures of elliptic inclusions with no substrate layer Now we analyze the change of band structure for the platemode waves in 2D PC slabs with elliptic inclusions by varying the

Y. Li et al. / Physica B 407 (2012) 1191–1195

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Fig. 2. (a) Low-frequency part of the dispersion curves of the plate-mode waves along the boundaries of the irreducible part of the Brillouin zone with ﬁlling ratio f¼ 0.1675 for elliptic inclusions with ts ¼ 0. (b) Same as in Fig. 2(b) but for cylinders. (c) The ﬁrst band gap width as a function of the ﬁlling fraction for the elliptic inclusions and the cylinders.

3.3. Effect of the rotation Angle y on the band structures of elliptic inclusions with no substrate layer In the next step, we analyze the inﬂuence of the rotation angle on the width of the ﬁrst complete band gap. The results are illustrated in Fig. 4 with different values of y(p/12,p/6,p/4). From fmax ¼ pT/4(T2 sin2 y þcos2 y), we can know that different angles corresponding to different maximum ﬁlling ratio. In order to facilitate comparative analysis, we still take the ﬁlling ratio from 0 to 0.35 and the thickness of plate is a. From the Fig. 4, it can

easily be seen that for every discussed angle y the width of the ﬁrst band gap in such system are all increasing as the ﬁlling fraction is increasing. When the ﬁlling fraction is from 0 to 0.1, the width of the ﬁrst complete band gap is almost the same for three y. However, the difference between their gap widths is apparent with the increase in ﬁlling fraction. The gap width for y ¼ p/12 is larger than the other cases with the same ﬁlling fraction. Another feature is that the width of the ﬁrst band gap is reducing as the rotation angle y increasing, but the change is not signiﬁcant. This implies that a larger rotation angle y does not

1

0.8 T=0.5 Gap Width

orientation angle of the elliptic rods. The normalized gap width of the ﬁrst band gap as a function of the ﬁlling fraction for different value of T is shown in Fig. 3. From fmax ¼ pT/4(T2 sin2 y þcos2 y), when y ¼0 and T ¼0.5,0.6,0.7, the corresponding maximum ﬁlling fraction are 0.3925, 0.471, 0.5495, respectively. Here we take the ﬁlling fraction from 0 to 0.35. It can easily be seen that for every dissuced T there is certain ﬁlling fraction for opening the gap. For T¼0.5,0.6,0.7, the corresponding band gaps open at the ﬁlling fraction f¼0.04,0.045,0.05, respectively. For larger ﬁlling fractions, the values of the gap width in three cases are much different from each other. Especially for T¼0.5, the width of ﬁrst complete band gap enlarges with the ﬁlling fraction increasing and reaches its maximum value at f¼0.35. But for T¼0.6 and T¼0.7, the result is some changed and there is somewhat similarity between T¼ 0.6 and T¼0.7. Obviously, their band gap widths are all increase at ﬁrst and then decrease as the ﬁlling fraction increasing; the maximum value is obtained at f¼0.3 for T¼0.6 and f¼0.25 for T¼0.7.With the increase of the value of T, their band gap width become smaller with the same ﬁlling fraction. It is for T¼1, especially, which is the cylinder case. This means that the value of T has a great impact on the band gap. We can change the value of T to design a wider band gap.

0.6 T=0.6 T=0.7 0.4

0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f Fig. 3. First band gap width as a function of the ﬁlling fraction for different value of T.

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Y. Li et al. / Physica B 407 (2012) 1191–1195

result in a wider band gap. Effect of the rotation angle y on the band structures of elliptic inclusions is not obvious.

1 tp=0.75a

3.4. Effect of the thickness of plate tp on the band structures of elliptic inclusions with no substrate layer

3.5. Effect of the thickness of substrate layer ts on the band structures of elliptic inclusions Finally, we considered the system, which is sticked by a PC layer coated on an uniform substrate layer. Fig. 6 presents the normalized gap width of the ﬁrst complete band gap as a function of the ﬁlling fraction for different ts. Compared with Fig. 5, it can be seen that the normalized width of the ﬁrst complete band gap increases approximately linearly with the increasing ﬁlling fraction for all discussed ts. These properties can be applied to design acoustic sensors for multilayer materials detection and aircraft icing forecast. The ﬁrst complete band gap opens at the ﬁlling fraction f¼0.05,0.10 and 0.25, for ts ¼0.25a,0.50a and 0.75a,

Gap Width

tp=1.0a 0.6

0.4 tp=1.25a 0.2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f Fig. 5. First band gap width is plotted as a function of the ﬁlling fraction for different thickness of PC slabs with ts ¼0.

1

0.8

Gap Width

Fig. 5 shows that the gap width depends on the thickness of plate.In fact,a plate can support a number of plate-mode waves depending on the value of the ratio tp/l, where l is the acoustic wavelength. When the wavelengths of these plate-mode waves match the lattice spacing, stop bands appear in the plate-mode waves dispersion curves [12–14]. We study the situations for tp ¼0.75a,1.0a,1.25a. For tp ¼0.75a, the ﬁrst complete band gap open at the ﬁlling fraction f ¼0.05. It is noted that the value of the normalized gap width of the ﬁrst complete band gap in this system increases with the increase of the value of the ﬁlling fraction until a critical value and then decreases. The maximum width of the ﬁrst complete band gap appears at around f ¼0.2. For tp ¼1.0a, the ﬁrst complete band gaps open at the ﬁlling fraction f¼0.04 and the width of the ﬁrst complete band gap increases with the ﬁlling fraction increasing. Moreover, for tp ¼1.25a, the relation of the width of the ﬁrst band gap and the ﬁlling fraction is more complex than the other two cases. With the ﬁlling fraction increasing, the width of the ﬁrst complete band gap increases at ﬁrst with the maximum located at f¼0.05 and then decreases until f¼0.1. However, the width of the ﬁrst band gap is increasing when the ﬁlling fraction larger than 0.1. As a result, we can conclude that width of the ﬁrst complete band gap can be tuned by changing the thickness of slabs.

0.8

ts=0.25a

0.6

ts=0.50a 0.4

0.2

ts=0.75a

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f Fig. 6. First band gap width is plotted as a function of the ﬁlling fraction for different thickness of uniform substrate.

respectively. The width of the ﬁrst complete band gap decreases with the increase in ts.

1 = /12

4. Conclusion

0.8

Gap Width

= /6

= /4

By using of the plane-wave expansion method, we present the larger complete plate-mode wave band gap in the 2D PC plate with W elliptic inclusions embedded in Si matrix compared to the W cylinders embedded in a Si matrix. The inﬂuence of the value of T, rotation angley, the thickness of slabs t p and the thickness of uniform substrate layer t s are analyzed. The phononic band gap analysis with the variation in ellipticity can prove helpful in the design of the phononic band gap based devices.

0.6

0.4

0.2

Acknowledgments

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

f Fig. 4. First band gap width is plotted as a function of the ﬁlling fraction for different rotation angle.

Financial Supported form the National Science Foundation of China under Grant no 10902035, the Research Fund for the Doctoral Program of Higher Education under Grant no 2009016112009 and

Y. Li et al. / Physica B 407 (2012) 1191–1195

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