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Tuning of band structures in porous phononic crystals by grading design of cells
Ultrasonics xxx (2015) xxx–xxx
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Tuning of band structures in porous phononic crystals by grading design of cells Kai Wang a, Ying Liu a,⇑, Qin-shan Yang b a b
Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 21 January 2015 Received in revised form 26 February 2015 Accepted 28 February 2015 Available online xxxx Keywords: Porous phononic crystal Band structure Grading design FEM
a b s t r a c t As the results of the evolution of species, grading structures widely exist in the nature and display distinguish advantages. In this manuscript, grading concept is introduced to redesign the topological structure of pores with the aim to see the effects of grading on the band structure in porous phononic crystals. Circular pores are considered and the crossing grading is made. The wave dispersion in graded structures is investigated comparatively to the normal ones under the same porosity. The band gaps in grading structures are given, as well as the vibration modes of the unit cell corresponding to the absolute band gap (ABG) edges. The results show that the grading structure greatly decreases the critical porosity for the opening of the ABGs. Wider ABGs could be obtained at lower frequencies along with the increase of the porosity. Through controlling the topological parameters of the grading structure, the band structure could be tuned. These results will provide an important guidance in the band tuning in porous phononic crystals by grading design of cells. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction As one kind of novel materials, porous phononic crystals (PPCs), which have periodically distributed open or closed pores in bulk materials, have been widely studied due to the light weight and potential applications in the multi-functional fields, which include mechanical, thermal, and acoustic, etc. [1]. Especially for one important characteristic, the existence of the absolute band gaps (ABGs) [2–4], which represents frequency regions where propagating elastic waves do not exist, attracts more and more attention. Liu et al. [5,6] investigated the influence of pore shapes, lattice structures and lattice transformation on the band structures in 2D normal porous phononic crystals. Wang et al. [7] discussed the ABG structures in phononic crystals with cross-like holes. Yan and Zhang [8] analyzed the wave localization in two-dimensional porous phononic crystals with one-dimensional aperiodicity. For super phononic crystals, Liu et al. [9] investigated the band gaps in Kagome honeycomb, which is constituted by triangular and hexagonal component cells. Three-dimensional Kagome truss, which is composed by periodically distributed two head-to-head regular tetrahedrons, is also investigated to clarify the band structures in 3D PPCs [10]. These results show that the band structures ⇑ Corresponding author. Tel.: +86 10 51688763; fax: +86 10 51682094. E-mail address: [email protected] (Y. Liu).
are sensitive to the pore topological parameters. Through the pore topology design, the band gaps in PPCs could be tuned. Actually, porous materials widely exist in the nature, from the human bone to the tree trunk. If we observe these natural porous materials carefully, we can see that some are with uniform pores, such as honeycombs, some are heterogeneous, such as bones, depending on the functional or physiological purpose. Researches show that grading structure is one typical topology in natural porous materials, which is the results of natural evolution, and displays noticeably advantageous [11]. In the present discussion, by adopting the concept of grading, we make a try to grade the normal circular porous materials with smaller pores to see the effects of grading on the wave dispersion in porous materials, and hope to provide a new strategy in the pore topological design.
2. Theory Fig. 1 is a typical representation of a porous phononic crystal with circular pores (denoted by A) periodically arranged in the 2D space, and the z-coordinate is set parallel to the axes of the pores, which are treated as vacuum. Then if the elastic waves propagate in the transverse plane (x0y plane) with the displacement vectors independent of the z-coordinate, they can be decoupled into the mixed in-plane mode and the anti-plane shear mode.
http://dx.doi.org/10.1016/j.ultras.2015.02.022 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.
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the grading structure). The position of the center circular pore, named main pore, is fixed at the center of the unit cell. The sizes of the five pores and positions of the surrounding four circular pores, named minor pores, could be adjusted. This is just one kind of grading strategy. More detailed discussion about grading structure is out of the scope of the present manuscript and will be given in the forthcoming papers. The Acoustic Module operating under the 2D plane strain Application Mode (ACPN) in COMSOL is applied to solve the governing equations. The discrete form of the eigenvalue equations in the unit cell can be written as
ðK x2 MÞU ¼ 0;
Fig. 1. The transverse cross section of a representative 2D phononic crystal and the first Brillouin zone of the 2D square lattice. a is the lattice constant.
Accordingly, the in-plane wave equations are expressed in the frequency domain as
qðrÞx2 ui ¼ r ðlðrÞrui Þ þ r þ
@ ðkðrÞr uÞ; @xi
lðrÞ
@ u @xi
ði ¼ x; yÞ
ð1Þ
In Eq. (1), r = (x, y) denotes the position vector, x is the angular frequency, q is the mass density, k and l are the Lamé constant and shear modulus, u = (ux, uy) is the displacement vector in the transverse plane, and r ¼ ð@=@x; @=@yÞ is the 2D vector differential operator. Moreover, the above equation is also available for the solid phononic crystals in which A becomes a solid phase, and the material parameters in Eq. (1) are replaced by the parameters of component A. According to Bloch theorem, the displacement field can be expressed as
uðrÞ ¼ eiðkrÞ uk ðrÞ;
ð2Þ
where k = (kx, ky) is the wave vector limited to the first Brillouin zone of the reciprocal lattice and uk(r) is a periodical vector function with the same periodicity as the crystal lattice. In the present discussion, a simple grading structure is adopted. Fig. 2a displays the squarely distributed 2D normal (solid yellow circles) or graded (translucent yellow circles) circular porous phononic crystals. Under the same porosity, a circular pore (Fig. 2b, unit cell of the normal structure) is sub-graded into five circular pores crossing arranged as shown in Fig. 2c (unit cell of
ð3Þ
where U is the displacement at the nodes, K and M are the stiffness and mass matrices of the unit cell, respectively. The free boundary condition is imposed on the surface of the pore, and the Bloch boundary conditions on the two opposite boundaries of the unit cell, yielding
Uðr þ aÞ ¼ eiðkaÞ UðrÞ;
ð4Þ
where r is located at the boundary nodes and a is the vector that generates the point lattice associated with the phononic crystals. Through the maximum cell size and the changing rate controlling, the representative cell is meshed by using the triangular Lagrange quadratic elements provided by COMSOL. Since the present results are compared with the ones given Ref. [9], the detailed convergence analysis of the calculation is not given here anymore. Eigenfrequency analysis is chosen as the solver mode, and the direct SPOOLES is selected as the linear system solver. Moreover, the Hermitian transpose of the constrain matrix and parameter settings in symmetry direction in the advanced solver is required. The model built in COMSOL is saved as a MATLAB-compatible ‘.m’ file. The file is programmed to let the wave vector k sweep the edges of the irreducible Brillouin zone, so that we can obtain the whole dispersion relations. 3. Numerical examples and discussion Seen as Fig. 2c, the geometrical parameters of the grading structures could be changed under certain porosity. In this section, we will discuss the effects of the topology variation (size and position) on the band structures in grading PPCs. The results for PPC shown in Fig. 2b are not given here anymore. The detailed results please refer to Ref. [5]. In the calculation, the matrix material is Aluminum, and the pore is vacuum. The elastic parameters used are: k = 68.3 GPa, l = 28.3 GPa, and q = 2730 kg/m3. The lattice constant a = 0.2 m. 3.1. Influence of the porosity on the band structures in grading PPCs For normal PPC shown in Fig. 2b, the critical porosity for the opening of the ABG is f = 0.43 with the porosity given as
f ¼ Apore =Aunit ¼ npr 2 =a2 ;
Fig. 2. (a) Diagrammatic sketch of 2D phononic crystals with pores squarely distributed. Solid circles represent the initial structure; (b) unit cell of the normal structure; (c) unit cell of the grading structure.
ð5Þ
where Apore is the area of the pores, and Aunit the area of the unit cell. n is the number of the circular pores and r the radius. Keeping this porosity and considering the symmetry of the structure, we are crossing-grading the normal circular pore into five small ones. The centers of the circular pores are then determined (Fig. 2c). Then we change the radii of the pores simultaneously to obtain different porosities from 0.175 to 0.43 by considering the geometrical limitation of the lattice. Of course, other grading strategy could be chosen. The influence of the grading strategy is out of the purpose of this paper and does not discuss here extensively.
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Fig. 3 shows the band structures in grading PPCs with different porosities. In the figures, the gray areas between two lines represent the possible ABGs. The dimensionless frequency xa/2pVt is used in the y axis, with Vt = 3110 m/s, the velocity of the elastic transverse waves in the Al matrix. The gap width Dx and the central frequency xg of the ABGs are determined by
Dx ¼ xBu xBL ;
ð6aÞ
xg ¼ ðxBu þ xBL Þ=2:
ð6bÞ
where xBu and xBL are the maximum and lowest frequencies of the ABG, that is, the upper and lower bounds of the ABG. Seen as Fig. 3, grading greatly decreases the critical porosity for the opening of the ABGs. It is observed that as in the normal structure, the ABG is also opened between the 3rd and the 4th bands, but the opening porosity f = 0.175 (Fig. 3a), which is far smaller than 0.43 in the normal ones. The central frequency is 0.76 with the gap width Dx = 0.775–0.741 = 0.034. Along with the increase of the porosity, the central frequency of the ABG is dropped with wider gap width. It is decreased from 0.69 with the gap width Dx = 0.776–0.605 = 0.171 at f = 0.3–0.58 with the gap width Dx = 0.754–0.415 = 0.339 at f = 0.4. Simultaneously, a new ABG is opened between the 9th and the 10th bands. When f = 0.43, more ABGs are opened at higher frequencies. The central frequency of the lowest ABG is 0.47 with the largest gap width Dx = 0.699– 0.247 = 0.452. The central frequencies for the new ABGs are 0.826, 0.913, 0.958, 1.15, with the gap width 0.062, 0.069, 0.023, 0.05, respectively. The vibration modes of the unit cell at the edges of the ABGs, which are marked out by Bu and BL, are also shown in Fig. 3. For the upper-edge mode, the unit cell vibrates symmetrically to the diagonal dash dot line plotted in Fig. 3a for Bu. For the lower-edge mode (Fig. 3, BL), the unit cell vibrates symmetrically to the vertical dash dot line shown in BL. Comparison between Fig. 3a and b shows that along with the increase of the porosity, the vibration modes at the gap edges are not changed. Fig. 4 shows the comparison of the first ABG (between the 3rd and the 4th bands) in the normal structure (Fig. 2b) and the grading one (Fig. 2c). The shadow parts display the upper and lower edges of the first ABG. It is seen clearly that the critical porosity for the opening of the ABG is greatly reduced in the grading structure. The ABG bounds are descended and the gap width is
Fig. 3. Variation of the band structures with respect to the porosity. (a) f = 0.175; (b) f = 0.3; (c) f = 0.4; (d) f = 0.43.
Fig. 4. Comparison of the first ABG in the normal (Fig. 2a) and grading PPCs (Fig. 2b). In the figure, the red color represents the normal structure, and the blue is for the grading one. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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increased. It is noticed that along with the increase of the porosity, the upper bounds of the ABGs in the grading structures are less affected but the lower bounds are sensitive to this variation and dropped shapely along with the increase of the porosity. The relative ratio of the gap width and the central frequency, Dx/xg, is also plotted in Fig. 4 by red solid and blue dash lines. It is obvious that along with the increase of the porosity, Dx/xg in the grading structure increases quickly compared to that in the normal one, which indicates that we can obtain the ABG at much lower frequencies with wider gap width in the grading structure. 3.2. Influence of the relative radius ratio of the main and minor pores on the band structures in grading PPCs Band structure is sensitive to the topology of the grading structures. For certain porosity, we can change the radii, the central distance, as well as the circumferential positions of the minor pores to obtain different topological parameters of the grading structure. In this section, the influence of the relative size of the main and minor pores on the band structures in the grading PPCs is firstly investigated. Let nR = Rc/Rs, where Rc is the radius of the main pore and Rs the radius of the minor one. Diagrammatically shown in Fig. 5, we take the topology (given in Fig. 2b) at f = 0.3 as the initial structure with Rc = 0.0077 m and Rs = 0.0307 m. By keeping f = 0.3, and changing the radii Rc and Rs, we have nR varying from 0.25 to 2 to see the effects of the relative size of the main and minor pores on the band structures in PPCs. Fig. 6 displays the band structures in PPCs with different nR. It is seen that when nR < 1, that is, the main pore is smaller than the minor one, the ABG opens between the 3rd and the 4th bands. The central frequency of the ABG is 0.672 with the gap width Dx = 0.749 0.594 = 0.155 (Fig. 6a). When nR is increased, the gap width is widened. It is increased from the gap width Dx = 0.776 0.605 = 0.171 with the central frequency 0.76 (Fig. 6b, nR = 1), to the gap width Dx = 0.803 0.575 = 0.228 with the central frequency 0.689 (Fig. 6c, nR = 1.5). When nR = 2, the central frequency of the ABG is 0.667 with the gap width Dx = 0.817 0.517 = 0.3 (Fig. 6d). The vibration modes of the unit cell at the edges of the ABG, which are marked out by Bu and BL, are given in Fig. 6a–c, respectively. It is seen that for the upper-edge mode, the unit cell vibrates
Fig. 5. Diagrammatic sketch of the relative size variation of the main and minor pores.
Fig. 6. Variation of the band structures with respect to the relative radius ratio of the main and minor pores. (a) nR = 0.25; (b) nR = 1; (c) nR = 1.5; (d) nR = 2.
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symmetrically to the diagonal dash dot line shown in Bu, whilst for the lower-edge mode, the unit cell vibrates symmetrically to the vertical dash dot line shown in BL. Comparing the vibration modes given in Fig. 6a–c shows that the upper-edge vibration modes are less affected by the relative size of the main and minor pores. But when nR > 1, that is, the main pore is bigger than the minor one, the lower-edge mode is upside down compared to the ones when nR 6 1. Fig. 7 gives the variation of the ABG with respect to the relative radius ratio nR. It is noticed that although the porosity keeps constant (f = 0.3), the parameters of the ABGs are also changed with respect to the variation of nR. When the main pore is smaller than the minor ones, that is, nR < 1, the ABG is relatively narrow. Along with the increase of nR, the upper edge of the ABG is rising with lower edge dropping down. As a result, the gap width is increased whilst the central frequency is almost not changed. That is, when the main pore is bigger than the minor ones, the ABG is widened with decreased lower gap edge. Relative bigger main pores are more advantageous for wider ABGs (see Fig. 7).
5
We take the configuration at f = 0.3 and nR = 1.5 as the original configuration. At this time, the distance between the main and minor pores is d0 = 6.6 mm (Fig. 8). Let RD = d/d0, where d is the central distance between the main and minor pores. Fig. 9 shows the ABGs in grading PPCs along with the variation of the pore central distance. It is seen that when RD = 0.94 (Fig. 9a), the central frequency of the ABG (between the 3rd and the 4th bands) is 0.661 with the gap width Dx = 0.807 0.515 = 0.292. Along with the increase of RD, the gap width is increased (Fig. 3b–d) from Dx = 0.803 0.575 = 0.228 at RD = 1 to Dx
3.3. Influence of the relative center distance between the main and minor pores on the band structures in grading PPCs In this section, the influence of the center distance between the main and minor pores on the ABG in grading PPCs is investigated.
Fig. 7. Variation of the ABGs with respect to the relative radius ratio nR of the main and minor pores.
Fig. 8. Diagrammatic sketch of the relative distance variation between the main and minor circular pores.
Fig. 9. Variation of the band structures with respect to the relative central distance of the main and minor pores. (a) RD = 0.94; (b) RD = 1.07; (c) RD = 1.12.
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Fig. 10. Variation of the ABGs with respect to the relative distance RD between the main and minor pores.
Fig. 11. Diagrammatic sketch of the relative angle between the main and minor circular pores.
= 0.797 0.560 = 0.237 when RD = 1.12. Fig. 10 shows clearly this variation with respect to RD. It is seen that the upper edge of the ABG is less affected by the variation of the relative distance; whist the lower gap edge is increased along with the increase of RD, and then keeps almost constant when RD P 1. That is, when the pore radius is fixed, the closer of the pores may cause the dropping of the central frequency of the ABG; whilst the looser between the main and minor pores has less effects on the ABGs.
3.4. Influence of the relative angle between the main and minor pores on the band structures in grading PPCs Seen as Fig. 2c, crossing grading is made in the present discussion. The minor pores are just on the vertical and horizontal symmetric axes of the unit cell. Actually, the positions of the minor pores could be adjusted by rotating them around the main one. In order to see clearly the influence of the relative angle between the main and minor pores on the band structures in grading PPCs, in the present discussion, the minor pores are rotated clockwise around the main one from 0° to 45° considering the symmetry of the structure, which is diagrammatically sketched in Fig. 11. During this process, the minor pores are gradually moved from
Fig. 12. Variation of the band structures with respect to the relative angle between the main and minor circles. (a) h = 0°; (b) h = 11.25°; (c) h = 33.75°; (d) h = 45°.
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the vertical and horizontal symmetry axes to the diagonal symmetrical axes of the unit cell. Moreover, we have nR = 1, and f = 0.3. As shown in Fig. 12a, when h = 0°, the ABG is opened between the 3rd and the 4th bands. The central frequency is 0.691 with the gap width Dx = 0.776 0.606 = 0.17. Along with the increase of h, the band gap width is decreased. When h = 11.25°, the central frequency of the ABG (Fig. 12b, between the 3rd and the 4th bands) is 0.689 with the gap width Dx = 0.743 0.635 = 0.108. When h is increased to 33.75°, the ABG is opened between the 4th and the 5th bands. The central frequency (Fig. 12c) is increased to 0.776 with the gap width decreased to Dx = 0.786 0.767 = 0.019. It is noticed that a new ABG is opened between the 5th and the 6th bands, with the central frequency 0.875 and the gap width Dx = 0.896 0.854 = 0.042. When h reaches the maximum value, that is, h = 45°, the gap width (Fig. 12d, between 5th and 6st bands) achieves the minimum value with the central frequency 0.851 and the gap width Dx = 0.854 0.848 = 0.006. The ABG between the 4th and the 5th bands has been closed. The vibration modes of the unit cell at the band edges are given in Fig. 12b–c. The vibration mode for the unit cell at h = 0° please refers to Fig. 3b. In the figures, the black dash dot lines show the symmetry axes of the unit cell, and the red solid lines the symmetry axes of the grading pores. It is seen that along with the rotation of the minor pores around the main one, the symmetric axes of the pores and the unit cell are no longer coincided to each other. When h 6 11.25°, the ABG is opened between the 3rd and the 4th bands, and the vibration modes are almost symmetric to the diagonal dash dot line for the upper-edge mode and vertical dash dot line
7
for the lower-edge mode, just as the situation when h = 0°. Along with the further rotation, the vibration modes of the pores are changed with the aim to keep the symmetric vibration modes of the unit cell, that is, for the upper-edge mode, symmetrically to the diagonal dash dot line, whilst for the lower-edge mode, to the vertical dash dot line. It is seen that at this time, that is, h > 11.25°, the ABG is opened between the 4th and the 5th bands. When h > 33.75°, the symmetric axes of the unit cell and the pores tend to coincide again, and a new ABG is opened between the 5th and the 6th bands and the ABG opened between 4th and 5th bands is closed. Seen as Fig. 12d, when h = 45°, that is, when the minor pores lies on the diagonal symmetric axes of the unit cell, the upper-edge vibration mode for the unit cell is symmetry to the vertical dash dot line, and the lower-edge mode, symmetry to the horizontal dash dot line. Fig. 13 gives the variation of the ABG with respect to the relative angle h between the main and minor circular pores. Along with the rotation of the minor pores around the main one, the upper bound of the ABG is dropped slightly, whilst the lower bound is increased yielding an increased central frequency. When h = 40°, the first ABG is closed. It is seen that when h = 33.75°, a second ABG is opened between the 5th and the 6th bands. Along with the further increase of the relative angle h, the central frequency of the second ABG is dropped from 0.875 to 0.851 when h = 45° (Fig. 13a). In order to see the influence of the relative size ratio, that is nR, on the band structures during the rotation of the minor pores, Fig. 13b gives the variation of the ABG with respect to h when nR = 1.5 under the same porosity. It is seen that the basic rules are the same. Along with the increase of the relative angle, the central frequency of the first ABG is increased from 0.69 at h = 0° to 0.762 at h = 33.75°, and then closed. The second ABG is opened when h = 22.5°. The central frequency is dropped along with the increase of the relative angle, from 0.88 to 0.82, and closed when h = 45°. This further indicates that even the porosity is the same, the topological parameters will affect the ABG structures. The coincidence of the symmetry axes of the unit cell and the grading pores are more advantageous for wider ABGs with lower central frequencies.
4. Summary In this paper, the band gap structures in two-dimensional graded circular porous phononic crystals are investigated by using finite element (FE) simulation. Summarizing the results above we can conclude that
Fig. 13. Variation of the ABGs with respect to the relative angle h between the main and minor pores h. (a) f = 0.3, nR = 1; (b) f = 0.3, nR = 1.5. The blue color represents the first ABG, and the orange one is for the second ABG.
(1) Grading decreases the critical porosity for the opening of the ABG. The ABGs could be opened at lower frequencies. Along with the increase of the porosity, we can obtain ABGs with lower central frequencies and wider band gaps. (2) Under certain porosity, grading topological parameters will affect ABG properties. Bigger main pore (nR > 1) and smaller pore distance (Rd < 1) are advantageous for a wider ABG with a relative lower central frequency. When the grading pores and the lattice structure has the same symmetric axes, we can obtain the ABG at lower bands with a wider band gap. When the symmetric axes do not coincide with each other, the central frequency of the ABG is increased with a narrower band gap. When the symmetric axes of grading pores are turned to diagonal directions, the ABG is almost closed. That is, the coincidence of the symmetric axes of the lattice and grading pores are favor for the opening of ABG. It is noticed that the upper bound of the ABG is less affected by the variation of the topological parameters due to the relative stable vibration modes at the upper band edges.
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The results indicate clearly that ABGs are sensitive to the variation of the grading topological parameters. Wider ABGs with much lower central frequencies could be obtained compared to the normal structures. In the present discussion, only one simple grading strategy is proposed. More detailed discussion on the optimization of the grading strategy should be further concerned. Acknowledgements The financial supports from the Fundamental Research Funds for the Central Universities (2014JBZ014), the National Science Foundation of China (No. 11272046), 111 project, and the Program for New Century Excellent Talents in University (NCET11-0566), are acknowledged.
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Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ultras.2015.02. 022.
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Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Tuning of band structures in porous phononic crystals by grading design of cells Kai Wang a, Ying Liu a,⇑, Qin-shan Yang b a b
Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
a r t i c l e
i n f o
Article history: Received 21 January 2015 Received in revised form 26 February 2015 Accepted 28 February 2015 Available online xxxx Keywords: Porous phononic crystal Band structure Grading design FEM
a b s t r a c t As the results of the evolution of species, grading structures widely exist in the nature and display distinguish advantages. In this manuscript, grading concept is introduced to redesign the topological structure of pores with the aim to see the effects of grading on the band structure in porous phononic crystals. Circular pores are considered and the crossing grading is made. The wave dispersion in graded structures is investigated comparatively to the normal ones under the same porosity. The band gaps in grading structures are given, as well as the vibration modes of the unit cell corresponding to the absolute band gap (ABG) edges. The results show that the grading structure greatly decreases the critical porosity for the opening of the ABGs. Wider ABGs could be obtained at lower frequencies along with the increase of the porosity. Through controlling the topological parameters of the grading structure, the band structure could be tuned. These results will provide an important guidance in the band tuning in porous phononic crystals by grading design of cells. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction As one kind of novel materials, porous phononic crystals (PPCs), which have periodically distributed open or closed pores in bulk materials, have been widely studied due to the light weight and potential applications in the multi-functional fields, which include mechanical, thermal, and acoustic, etc. [1]. Especially for one important characteristic, the existence of the absolute band gaps (ABGs) [2–4], which represents frequency regions where propagating elastic waves do not exist, attracts more and more attention. Liu et al. [5,6] investigated the influence of pore shapes, lattice structures and lattice transformation on the band structures in 2D normal porous phononic crystals. Wang et al. [7] discussed the ABG structures in phononic crystals with cross-like holes. Yan and Zhang [8] analyzed the wave localization in two-dimensional porous phononic crystals with one-dimensional aperiodicity. For super phononic crystals, Liu et al. [9] investigated the band gaps in Kagome honeycomb, which is constituted by triangular and hexagonal component cells. Three-dimensional Kagome truss, which is composed by periodically distributed two head-to-head regular tetrahedrons, is also investigated to clarify the band structures in 3D PPCs [10]. These results show that the band structures ⇑ Corresponding author. Tel.: +86 10 51688763; fax: +86 10 51682094. E-mail address: [email protected] (Y. Liu).
are sensitive to the pore topological parameters. Through the pore topology design, the band gaps in PPCs could be tuned. Actually, porous materials widely exist in the nature, from the human bone to the tree trunk. If we observe these natural porous materials carefully, we can see that some are with uniform pores, such as honeycombs, some are heterogeneous, such as bones, depending on the functional or physiological purpose. Researches show that grading structure is one typical topology in natural porous materials, which is the results of natural evolution, and displays noticeably advantageous [11]. In the present discussion, by adopting the concept of grading, we make a try to grade the normal circular porous materials with smaller pores to see the effects of grading on the wave dispersion in porous materials, and hope to provide a new strategy in the pore topological design.
2. Theory Fig. 1 is a typical representation of a porous phononic crystal with circular pores (denoted by A) periodically arranged in the 2D space, and the z-coordinate is set parallel to the axes of the pores, which are treated as vacuum. Then if the elastic waves propagate in the transverse plane (x0y plane) with the displacement vectors independent of the z-coordinate, they can be decoupled into the mixed in-plane mode and the anti-plane shear mode.
http://dx.doi.org/10.1016/j.ultras.2015.02.022 0041-624X/Ó 2015 Elsevier B.V. All rights reserved.
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the grading structure). The position of the center circular pore, named main pore, is fixed at the center of the unit cell. The sizes of the five pores and positions of the surrounding four circular pores, named minor pores, could be adjusted. This is just one kind of grading strategy. More detailed discussion about grading structure is out of the scope of the present manuscript and will be given in the forthcoming papers. The Acoustic Module operating under the 2D plane strain Application Mode (ACPN) in COMSOL is applied to solve the governing equations. The discrete form of the eigenvalue equations in the unit cell can be written as
ðK x2 MÞU ¼ 0;
Fig. 1. The transverse cross section of a representative 2D phononic crystal and the first Brillouin zone of the 2D square lattice. a is the lattice constant.
Accordingly, the in-plane wave equations are expressed in the frequency domain as
qðrÞx2 ui ¼ r ðlðrÞrui Þ þ r þ
@ ðkðrÞr uÞ; @xi
lðrÞ
@ u @xi
ði ¼ x; yÞ
ð1Þ
In Eq. (1), r = (x, y) denotes the position vector, x is the angular frequency, q is the mass density, k and l are the Lamé constant and shear modulus, u = (ux, uy) is the displacement vector in the transverse plane, and r ¼ ð@=@x; @=@yÞ is the 2D vector differential operator. Moreover, the above equation is also available for the solid phononic crystals in which A becomes a solid phase, and the material parameters in Eq. (1) are replaced by the parameters of component A. According to Bloch theorem, the displacement field can be expressed as
uðrÞ ¼ eiðkrÞ uk ðrÞ;
ð2Þ
where k = (kx, ky) is the wave vector limited to the first Brillouin zone of the reciprocal lattice and uk(r) is a periodical vector function with the same periodicity as the crystal lattice. In the present discussion, a simple grading structure is adopted. Fig. 2a displays the squarely distributed 2D normal (solid yellow circles) or graded (translucent yellow circles) circular porous phononic crystals. Under the same porosity, a circular pore (Fig. 2b, unit cell of the normal structure) is sub-graded into five circular pores crossing arranged as shown in Fig. 2c (unit cell of
ð3Þ
where U is the displacement at the nodes, K and M are the stiffness and mass matrices of the unit cell, respectively. The free boundary condition is imposed on the surface of the pore, and the Bloch boundary conditions on the two opposite boundaries of the unit cell, yielding
Uðr þ aÞ ¼ eiðkaÞ UðrÞ;
ð4Þ
where r is located at the boundary nodes and a is the vector that generates the point lattice associated with the phononic crystals. Through the maximum cell size and the changing rate controlling, the representative cell is meshed by using the triangular Lagrange quadratic elements provided by COMSOL. Since the present results are compared with the ones given Ref. [9], the detailed convergence analysis of the calculation is not given here anymore. Eigenfrequency analysis is chosen as the solver mode, and the direct SPOOLES is selected as the linear system solver. Moreover, the Hermitian transpose of the constrain matrix and parameter settings in symmetry direction in the advanced solver is required. The model built in COMSOL is saved as a MATLAB-compatible ‘.m’ file. The file is programmed to let the wave vector k sweep the edges of the irreducible Brillouin zone, so that we can obtain the whole dispersion relations. 3. Numerical examples and discussion Seen as Fig. 2c, the geometrical parameters of the grading structures could be changed under certain porosity. In this section, we will discuss the effects of the topology variation (size and position) on the band structures in grading PPCs. The results for PPC shown in Fig. 2b are not given here anymore. The detailed results please refer to Ref. [5]. In the calculation, the matrix material is Aluminum, and the pore is vacuum. The elastic parameters used are: k = 68.3 GPa, l = 28.3 GPa, and q = 2730 kg/m3. The lattice constant a = 0.2 m. 3.1. Influence of the porosity on the band structures in grading PPCs For normal PPC shown in Fig. 2b, the critical porosity for the opening of the ABG is f = 0.43 with the porosity given as
f ¼ Apore =Aunit ¼ npr 2 =a2 ;
Fig. 2. (a) Diagrammatic sketch of 2D phononic crystals with pores squarely distributed. Solid circles represent the initial structure; (b) unit cell of the normal structure; (c) unit cell of the grading structure.
ð5Þ
where Apore is the area of the pores, and Aunit the area of the unit cell. n is the number of the circular pores and r the radius. Keeping this porosity and considering the symmetry of the structure, we are crossing-grading the normal circular pore into five small ones. The centers of the circular pores are then determined (Fig. 2c). Then we change the radii of the pores simultaneously to obtain different porosities from 0.175 to 0.43 by considering the geometrical limitation of the lattice. Of course, other grading strategy could be chosen. The influence of the grading strategy is out of the purpose of this paper and does not discuss here extensively.
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Fig. 3 shows the band structures in grading PPCs with different porosities. In the figures, the gray areas between two lines represent the possible ABGs. The dimensionless frequency xa/2pVt is used in the y axis, with Vt = 3110 m/s, the velocity of the elastic transverse waves in the Al matrix. The gap width Dx and the central frequency xg of the ABGs are determined by
Dx ¼ xBu xBL ;
ð6aÞ
xg ¼ ðxBu þ xBL Þ=2:
ð6bÞ
where xBu and xBL are the maximum and lowest frequencies of the ABG, that is, the upper and lower bounds of the ABG. Seen as Fig. 3, grading greatly decreases the critical porosity for the opening of the ABGs. It is observed that as in the normal structure, the ABG is also opened between the 3rd and the 4th bands, but the opening porosity f = 0.175 (Fig. 3a), which is far smaller than 0.43 in the normal ones. The central frequency is 0.76 with the gap width Dx = 0.775–0.741 = 0.034. Along with the increase of the porosity, the central frequency of the ABG is dropped with wider gap width. It is decreased from 0.69 with the gap width Dx = 0.776–0.605 = 0.171 at f = 0.3–0.58 with the gap width Dx = 0.754–0.415 = 0.339 at f = 0.4. Simultaneously, a new ABG is opened between the 9th and the 10th bands. When f = 0.43, more ABGs are opened at higher frequencies. The central frequency of the lowest ABG is 0.47 with the largest gap width Dx = 0.699– 0.247 = 0.452. The central frequencies for the new ABGs are 0.826, 0.913, 0.958, 1.15, with the gap width 0.062, 0.069, 0.023, 0.05, respectively. The vibration modes of the unit cell at the edges of the ABGs, which are marked out by Bu and BL, are also shown in Fig. 3. For the upper-edge mode, the unit cell vibrates symmetrically to the diagonal dash dot line plotted in Fig. 3a for Bu. For the lower-edge mode (Fig. 3, BL), the unit cell vibrates symmetrically to the vertical dash dot line shown in BL. Comparison between Fig. 3a and b shows that along with the increase of the porosity, the vibration modes at the gap edges are not changed. Fig. 4 shows the comparison of the first ABG (between the 3rd and the 4th bands) in the normal structure (Fig. 2b) and the grading one (Fig. 2c). The shadow parts display the upper and lower edges of the first ABG. It is seen clearly that the critical porosity for the opening of the ABG is greatly reduced in the grading structure. The ABG bounds are descended and the gap width is
Fig. 3. Variation of the band structures with respect to the porosity. (a) f = 0.175; (b) f = 0.3; (c) f = 0.4; (d) f = 0.43.
Fig. 4. Comparison of the first ABG in the normal (Fig. 2a) and grading PPCs (Fig. 2b). In the figure, the red color represents the normal structure, and the blue is for the grading one. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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increased. It is noticed that along with the increase of the porosity, the upper bounds of the ABGs in the grading structures are less affected but the lower bounds are sensitive to this variation and dropped shapely along with the increase of the porosity. The relative ratio of the gap width and the central frequency, Dx/xg, is also plotted in Fig. 4 by red solid and blue dash lines. It is obvious that along with the increase of the porosity, Dx/xg in the grading structure increases quickly compared to that in the normal one, which indicates that we can obtain the ABG at much lower frequencies with wider gap width in the grading structure. 3.2. Influence of the relative radius ratio of the main and minor pores on the band structures in grading PPCs Band structure is sensitive to the topology of the grading structures. For certain porosity, we can change the radii, the central distance, as well as the circumferential positions of the minor pores to obtain different topological parameters of the grading structure. In this section, the influence of the relative size of the main and minor pores on the band structures in the grading PPCs is firstly investigated. Let nR = Rc/Rs, where Rc is the radius of the main pore and Rs the radius of the minor one. Diagrammatically shown in Fig. 5, we take the topology (given in Fig. 2b) at f = 0.3 as the initial structure with Rc = 0.0077 m and Rs = 0.0307 m. By keeping f = 0.3, and changing the radii Rc and Rs, we have nR varying from 0.25 to 2 to see the effects of the relative size of the main and minor pores on the band structures in PPCs. Fig. 6 displays the band structures in PPCs with different nR. It is seen that when nR < 1, that is, the main pore is smaller than the minor one, the ABG opens between the 3rd and the 4th bands. The central frequency of the ABG is 0.672 with the gap width Dx = 0.749 0.594 = 0.155 (Fig. 6a). When nR is increased, the gap width is widened. It is increased from the gap width Dx = 0.776 0.605 = 0.171 with the central frequency 0.76 (Fig. 6b, nR = 1), to the gap width Dx = 0.803 0.575 = 0.228 with the central frequency 0.689 (Fig. 6c, nR = 1.5). When nR = 2, the central frequency of the ABG is 0.667 with the gap width Dx = 0.817 0.517 = 0.3 (Fig. 6d). The vibration modes of the unit cell at the edges of the ABG, which are marked out by Bu and BL, are given in Fig. 6a–c, respectively. It is seen that for the upper-edge mode, the unit cell vibrates
Fig. 5. Diagrammatic sketch of the relative size variation of the main and minor pores.
Fig. 6. Variation of the band structures with respect to the relative radius ratio of the main and minor pores. (a) nR = 0.25; (b) nR = 1; (c) nR = 1.5; (d) nR = 2.
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symmetrically to the diagonal dash dot line shown in Bu, whilst for the lower-edge mode, the unit cell vibrates symmetrically to the vertical dash dot line shown in BL. Comparing the vibration modes given in Fig. 6a–c shows that the upper-edge vibration modes are less affected by the relative size of the main and minor pores. But when nR > 1, that is, the main pore is bigger than the minor one, the lower-edge mode is upside down compared to the ones when nR 6 1. Fig. 7 gives the variation of the ABG with respect to the relative radius ratio nR. It is noticed that although the porosity keeps constant (f = 0.3), the parameters of the ABGs are also changed with respect to the variation of nR. When the main pore is smaller than the minor ones, that is, nR < 1, the ABG is relatively narrow. Along with the increase of nR, the upper edge of the ABG is rising with lower edge dropping down. As a result, the gap width is increased whilst the central frequency is almost not changed. That is, when the main pore is bigger than the minor ones, the ABG is widened with decreased lower gap edge. Relative bigger main pores are more advantageous for wider ABGs (see Fig. 7).
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We take the configuration at f = 0.3 and nR = 1.5 as the original configuration. At this time, the distance between the main and minor pores is d0 = 6.6 mm (Fig. 8). Let RD = d/d0, where d is the central distance between the main and minor pores. Fig. 9 shows the ABGs in grading PPCs along with the variation of the pore central distance. It is seen that when RD = 0.94 (Fig. 9a), the central frequency of the ABG (between the 3rd and the 4th bands) is 0.661 with the gap width Dx = 0.807 0.515 = 0.292. Along with the increase of RD, the gap width is increased (Fig. 3b–d) from Dx = 0.803 0.575 = 0.228 at RD = 1 to Dx
3.3. Influence of the relative center distance between the main and minor pores on the band structures in grading PPCs In this section, the influence of the center distance between the main and minor pores on the ABG in grading PPCs is investigated.
Fig. 7. Variation of the ABGs with respect to the relative radius ratio nR of the main and minor pores.
Fig. 8. Diagrammatic sketch of the relative distance variation between the main and minor circular pores.
Fig. 9. Variation of the band structures with respect to the relative central distance of the main and minor pores. (a) RD = 0.94; (b) RD = 1.07; (c) RD = 1.12.
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Fig. 10. Variation of the ABGs with respect to the relative distance RD between the main and minor pores.
Fig. 11. Diagrammatic sketch of the relative angle between the main and minor circular pores.
= 0.797 0.560 = 0.237 when RD = 1.12. Fig. 10 shows clearly this variation with respect to RD. It is seen that the upper edge of the ABG is less affected by the variation of the relative distance; whist the lower gap edge is increased along with the increase of RD, and then keeps almost constant when RD P 1. That is, when the pore radius is fixed, the closer of the pores may cause the dropping of the central frequency of the ABG; whilst the looser between the main and minor pores has less effects on the ABGs.
3.4. Influence of the relative angle between the main and minor pores on the band structures in grading PPCs Seen as Fig. 2c, crossing grading is made in the present discussion. The minor pores are just on the vertical and horizontal symmetric axes of the unit cell. Actually, the positions of the minor pores could be adjusted by rotating them around the main one. In order to see clearly the influence of the relative angle between the main and minor pores on the band structures in grading PPCs, in the present discussion, the minor pores are rotated clockwise around the main one from 0° to 45° considering the symmetry of the structure, which is diagrammatically sketched in Fig. 11. During this process, the minor pores are gradually moved from
Fig. 12. Variation of the band structures with respect to the relative angle between the main and minor circles. (a) h = 0°; (b) h = 11.25°; (c) h = 33.75°; (d) h = 45°.
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the vertical and horizontal symmetry axes to the diagonal symmetrical axes of the unit cell. Moreover, we have nR = 1, and f = 0.3. As shown in Fig. 12a, when h = 0°, the ABG is opened between the 3rd and the 4th bands. The central frequency is 0.691 with the gap width Dx = 0.776 0.606 = 0.17. Along with the increase of h, the band gap width is decreased. When h = 11.25°, the central frequency of the ABG (Fig. 12b, between the 3rd and the 4th bands) is 0.689 with the gap width Dx = 0.743 0.635 = 0.108. When h is increased to 33.75°, the ABG is opened between the 4th and the 5th bands. The central frequency (Fig. 12c) is increased to 0.776 with the gap width decreased to Dx = 0.786 0.767 = 0.019. It is noticed that a new ABG is opened between the 5th and the 6th bands, with the central frequency 0.875 and the gap width Dx = 0.896 0.854 = 0.042. When h reaches the maximum value, that is, h = 45°, the gap width (Fig. 12d, between 5th and 6st bands) achieves the minimum value with the central frequency 0.851 and the gap width Dx = 0.854 0.848 = 0.006. The ABG between the 4th and the 5th bands has been closed. The vibration modes of the unit cell at the band edges are given in Fig. 12b–c. The vibration mode for the unit cell at h = 0° please refers to Fig. 3b. In the figures, the black dash dot lines show the symmetry axes of the unit cell, and the red solid lines the symmetry axes of the grading pores. It is seen that along with the rotation of the minor pores around the main one, the symmetric axes of the pores and the unit cell are no longer coincided to each other. When h 6 11.25°, the ABG is opened between the 3rd and the 4th bands, and the vibration modes are almost symmetric to the diagonal dash dot line for the upper-edge mode and vertical dash dot line
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for the lower-edge mode, just as the situation when h = 0°. Along with the further rotation, the vibration modes of the pores are changed with the aim to keep the symmetric vibration modes of the unit cell, that is, for the upper-edge mode, symmetrically to the diagonal dash dot line, whilst for the lower-edge mode, to the vertical dash dot line. It is seen that at this time, that is, h > 11.25°, the ABG is opened between the 4th and the 5th bands. When h > 33.75°, the symmetric axes of the unit cell and the pores tend to coincide again, and a new ABG is opened between the 5th and the 6th bands and the ABG opened between 4th and 5th bands is closed. Seen as Fig. 12d, when h = 45°, that is, when the minor pores lies on the diagonal symmetric axes of the unit cell, the upper-edge vibration mode for the unit cell is symmetry to the vertical dash dot line, and the lower-edge mode, symmetry to the horizontal dash dot line. Fig. 13 gives the variation of the ABG with respect to the relative angle h between the main and minor circular pores. Along with the rotation of the minor pores around the main one, the upper bound of the ABG is dropped slightly, whilst the lower bound is increased yielding an increased central frequency. When h = 40°, the first ABG is closed. It is seen that when h = 33.75°, a second ABG is opened between the 5th and the 6th bands. Along with the further increase of the relative angle h, the central frequency of the second ABG is dropped from 0.875 to 0.851 when h = 45° (Fig. 13a). In order to see the influence of the relative size ratio, that is nR, on the band structures during the rotation of the minor pores, Fig. 13b gives the variation of the ABG with respect to h when nR = 1.5 under the same porosity. It is seen that the basic rules are the same. Along with the increase of the relative angle, the central frequency of the first ABG is increased from 0.69 at h = 0° to 0.762 at h = 33.75°, and then closed. The second ABG is opened when h = 22.5°. The central frequency is dropped along with the increase of the relative angle, from 0.88 to 0.82, and closed when h = 45°. This further indicates that even the porosity is the same, the topological parameters will affect the ABG structures. The coincidence of the symmetry axes of the unit cell and the grading pores are more advantageous for wider ABGs with lower central frequencies.
4. Summary In this paper, the band gap structures in two-dimensional graded circular porous phononic crystals are investigated by using finite element (FE) simulation. Summarizing the results above we can conclude that
Fig. 13. Variation of the ABGs with respect to the relative angle h between the main and minor pores h. (a) f = 0.3, nR = 1; (b) f = 0.3, nR = 1.5. The blue color represents the first ABG, and the orange one is for the second ABG.
(1) Grading decreases the critical porosity for the opening of the ABG. The ABGs could be opened at lower frequencies. Along with the increase of the porosity, we can obtain ABGs with lower central frequencies and wider band gaps. (2) Under certain porosity, grading topological parameters will affect ABG properties. Bigger main pore (nR > 1) and smaller pore distance (Rd < 1) are advantageous for a wider ABG with a relative lower central frequency. When the grading pores and the lattice structure has the same symmetric axes, we can obtain the ABG at lower bands with a wider band gap. When the symmetric axes do not coincide with each other, the central frequency of the ABG is increased with a narrower band gap. When the symmetric axes of grading pores are turned to diagonal directions, the ABG is almost closed. That is, the coincidence of the symmetric axes of the lattice and grading pores are favor for the opening of ABG. It is noticed that the upper bound of the ABG is less affected by the variation of the topological parameters due to the relative stable vibration modes at the upper band edges.
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The results indicate clearly that ABGs are sensitive to the variation of the grading topological parameters. Wider ABGs with much lower central frequencies could be obtained compared to the normal structures. In the present discussion, only one simple grading strategy is proposed. More detailed discussion on the optimization of the grading strategy should be further concerned. Acknowledgements The financial supports from the Fundamental Research Funds for the Central Universities (2014JBZ014), the National Science Foundation of China (No. 11272046), 111 project, and the Program for New Century Excellent Talents in University (NCET11-0566), are acknowledged.
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Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ultras.2015.02. 022.
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