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Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates
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Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates Yilan Huang , Yang Huang , Weiqiu Chen , Ronghao Bao PII: DOI: Reference:
S0020-7403(19)32424-5 https://doi.org/10.1016/j.ijmecsci.2019.105348 MS 105348
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
5 July 2019 20 November 2019 27 November 2019
Please cite this article as: Yilan Huang , Yang Huang , Weiqiu Chen , Ronghao Bao , Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105348
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Highlights
We design a soft tunable 1D phononic crystal by connecting two different periodic plates made of hyperelastic materials. The interface state wave modes observed inside the system can be either activated or deactivated by an applied pre-stretch on the structures with different initial geometrical parameters. Due to topological protection, the system provides a robust way of manipulating waves propagation in real time.
1
Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates Yilan Huang1, Yang Huang1, Weiqiu Chen1,2,3,*, Ronghao Bao1,3,* 1. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province and Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China; 2. State Key Lab of CAD & CG, Zhejiang University, Hangzhou 310058, China; 3. Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China
Abstract: In this paper, we investigate the possibility of flexible manipulation of elastic waves propagating in soft periodic plates through topological state shifting. We design a soft 1D phononic crystal by connecting two different periodic plates made of hyperelastic materials. By choosing different geometrical parameters of the two plates, topologically protected state of elastic waves is observed at the interface between them. Since the hyperelastic material can undergo large elastic deformation, flexible and efficient shifting of topological state can be realized by applying appropriate pre-stretch on the structure. The interface state wave modes can be either activated or deactivated by an applied pre-stretch on the structures with different initial geometrical parameters. Due to the advantageous property that topologically protected wave modes are stable against any defects and disorders, it provides a robust way of manipulating waves propagation in real time. The system and the strategy proposed here have great potential in constructing various novel acoustic devices.
Keywords: periodic structure; topologically protected; interface state; wave manipulation; large deformation.
*
Corresponding author. E-mail: [email protected]; Tel./Fax: 86-571-87951866.
*
Corresponding author. E-mail: [email protected]; Tel./Fax: 86-571-87951866. 2
1.
Introduction
Phononic crystals (PnCs), since initially proposed and studied by Kushwaha et al. in 1993 [1], have been vastly researched in the recent several decades, due to their wonderful acoustic properties caused by the unique ability of manipulating wave propagation through band gaps
[2,3]
. Recent
studies further revealed that, by introducing topological states into the traditional PnC structures, advantageous local acoustic phenomena (e.g. edge modes and interface states) that are hardly seen in normal structures can be observed. Topological phenomena are ubiquitous in the physics world. In condensed matter physics, topological phenomena in quantum Hall and topological insulators have already been extensively studied
[4-6]
. By making analogies, topological concepts have been
further extended to bosonic systems such as photonic
[7-9]
and phononic structures
[10-11]
. The most
promising properties of those structures are topologically protected states existing at the boundaries or interfaces, which are robust against defects and disorders. Thus, topological wave mechanics has brought in another revolution of manipulation and control of acoustic waves. In one-dimensional (1D) phononic crystal systems, interface states with large energy enhancement may be induced when connecting different PnC structures with opposite topological invariancies around particular bandgaps. The topological interface states of acoustic waves were first realized in fluid systems
[12,13]
, and recently were also observed in solid systems
[14,15]
. For 1D phononic
systems with inversion symmetry, the topological properties of frequency modes and band gaps are characterized by the well-known Zak phase, which was initially proposed to study the energy bands of electrons in solid structures
[16,17]
. Recently, the relationship of Zak phase and Bloch
bands in acoustic phononic systems is also well studied [12,18]. Another inspiring research area in PnCs is tunable phononic systems made of soft hyperelastic materials. Solid phononic systems made of traditional materials are hardly able to change their material properties and geometric structures. As a result, their mechanical and acoustic properties are fixed after manufacture, inevitably limiting their applications in frequency-changing environments. While tunable hyperelastic materials give rise to the potential of designing PnC systems with alterable acoustic properties along the deformation trajectory, enormously enhancing the practical applications of PnCs as adaptive acoustic devices, including sound filters, acoustic 3
mirrors, acoustic wave guides, and vibration isolators
[19-23]
. Abundant researches already showed
both theoretically and experimentally that soft materials can undergo large deformation repeatedly under different external stimuli, including electric fields mechanical loadings
[24-27]
, magnetic fields, temperature and
[28-38]
, providing efficient strategies for modifying band gaps and
manipulating wave propagations in soft PnCs. This kind of flexible tunability of band gaps is mainly caused by the change of structural geometry and material effective stiffness [39]. Existing researches have shown that the topological characteristics of PnCs are sensitively affected by the geometry of the structures and altering the geometric parameters is the main strategy to shift the interface states in 1D systems. In this paper, we want to explore the feasibility of flexible modification of the topological states of PnCs under pre-stretch due to external mechanical loadings. First, we study the topological states of in-plane waves in a 1D soft thick plate. A simple 1D acoustic system is proposed by connecting two phononic crystals with similar geometries but quite distinct topological properties. We show that topological interface mode inside a particular band gap can be observed at the interface of the two PnCs. Besides, an edge mode in a specific structure is also observed. Then we study the effect of large deformation on the topological properties and the interface mode by applying a pre-stretch on the 1D plate. Results show that, by properly designing the initial structure, the topological properties can be efficiently modified and the interface states can be activated or deactivated under pre-stretch. In our work, all the simulations of pre-deformation, band gaps and transmission spectra are based on the commercial software ABAQUS. Python-based scripts are used to perform the related modeling, meshing and calculations. The proposed structures can be used to construct advantageous acoustic devices, such as energy harvesters, wave sensors and acoustic filters, especially in dynamic situations, where convenient and efficient modification of wave propagation is required.
2. Summary of models and calculation methods
The phononic system considered in this paper is composed of two thick periodic plates connected to each other. We assume the thickness of the plate is much larger than the length of the unit cell, and consider in-plane waves propagating from one end of the system to the other end. The 3D 4
sketch of the system is shown in Figure 1(a), and the cross section of one single unit cell is schematically shown in Figure 1(b). The unit cell includes two different parts with different material properties and different geometric sizes. The length and width of part A are L and D respectively, while those of part B are l and d. The total length of the cell therefore is a = l + L. The unit cell satisfies inversion symmetry with respect to its central plane. The assembled system is shown in Figure 2. The left and right sides of the dashed line represent two periodic plates, and each of them contains a finite number (here denoted by N) of unit cells (denoted by Cell 1 and Cell 2 respectively). The whole plate is made of soft hyperelastic materials, which can sustain substantial elastic deformation. The material behavior is described by the neo-Hookean model whose strain energy density function takes the following form:
C10 I1 3
1 2 J 1 , D1
(1)
where I1 is the first deviatoric strain invariant, J indicates the local volume change, and C10 and D1 are two material constants. The initial shear modulus and bulk modulus are calculated through 0 2C10 , K0 2 / D1 . To perform the computation with finite element method, the commercial software ABAQUS is used. Python-based scripts are adopted to facilitate the related steps of modeling, meshing, calculation, and results analyzing. Based on the thick plate assumption, we use 2D plane-strain models to conduct all simulations. First, to study the band structure of a periodic plate, a model of a single unit cell is created. The Bloch boundary conditions are imposed in the periodic direction using specific python scripts, and a frequency step is used to calculate the dispersion relations. By analyzing the band structure and the frequency modes, the topological characteristics of particular band gap can be determined. Then to study the interface states, a supercell containing the whole structure is created (including N Cell 1 units and N Cell 2 units), and its band structure is also calculated using the same method. In order to verify the results, the transmission behaviors of the finite structure (see Fig. A1 in Appendix A) are further examined, for which the harmonic analysis is used with the frequency sweeping through the range of interest. The transmission spectra are calculated according to the ratio between the average displacement amplitudes at the input and 5
output regions of the finite structure. It is noted that, throughout this study, there is no consideration of application of acoustic load and radiation of the structure, which however may be of significance in practical applications. Next, the effect of large deformation induced by external mechanical load on band structure and on interface state is explored. In the case of a unit cell (or a supercell), periodic boundary conditions are applied on the boundaries in the periodic direction using python scripts; and in the case of a finite structure, a total displacement is applied to the boundaries in the stretching direction (i.e. in the x direction as shown in Figure 1(a)). To calculate the band gaps and transmission spectra of a pre-stretched model, the computational processes are identical, except that there is a need to introduce the results of deformation and effective stiffness into the model as the initial state. A more detailed description of the computational processes is provided in Appendix A. Further information may be found in Refs. [33, 34].
Fig. 1. Schematics of the phononic system: (a) The system contains two periodic thick plates connected to each other; (b) The cross section of a single unit cell.
Fig. 2. A cross section view of the whole system, containing N Cell 1 units and N Cell 2 units. At 6
the center is the interface between the two periodic plates.
3. Numerical results
To demonstrate the tunability of topological characteristics in our systems, first we consider a unit cell. The geometric parameters and material properties are as follows: L=10.66mm, l=7.38mm, D=16.4mm, and d=8.594mm-d. C10A=5.5 × 105Pa, D1A=3.636 × 10-8Pa-1, C10B=9.35 × 105Pa, D1B=2.14×10-8Pa-1, and 0 A 0 B 1050 kg/ m3 , where C10A, D1A, and C10B, D1B are the material parameters of part A and part B, respectively. 0 A and 0B represent the initial densities of part A and part B, respectively. Here the geometry of the unit cell is changed by the variable d. In Figure 3(a), the evolution of the first band gap (more specifically, the band gap between the third and fourth frequency modes) with respect to d is presented. Figures 3(b)-(d) show the dispersion relations for the cases of d=-0.3mm, 0, and 0.3mm respectively. As d increases from -0.3mm to 0.3mm, the band gap closes at first, but then reopens. At the degenerate point of d=0, this band gap completely disappears. One interesting phenomenon is that, during the process of closing and reopening, the topological feature of this band gap is changed. For 1D PnC structures, the Zak phase is usually introduced to describe the topological state of an isolated frequency mode. It can be calculated by
nZak i
/a
/ a
where
U n, k | k | U n, k dk ,
(2)
U n, k is the normalized displacement amplitude of the n-th frequency state with wave
number k. Due to the inversion symmetry of the unit cell with respect to its central plane, the displacement field of a frequency mode is either symmetric or antisymmetric. According to the symmetry property of the frequency mode, the Zak phase must be either 0 or
[14-17]
. The
topological state of a band gap is calculated by summing over all the Zak phases of the frequency modes below that gap, denoted as
n
iZak mod 2 , which can be either 0 or . Although
i 1
the Zak phase is gauge-dependent, varying with the choice of the origin of the unit cell, the 7
difference between two topological states of band gaps in different unit cells depends solely on the structure. A detailed description of the topological state of band gaps is provided in Appendix B. The contour plot of the edge frequency modes at X (the right edge of the irreducible Brillouin zone) around the first band gap is also presented as the insets of Figure 3(a). The topological feature of the band gap is related to its edge frequency modes. Comparing the band gaps of d <0 and d >0, the lower edge mode and the upper edge mode swap with each other and the degenerate point at d =0 is a topological transition point. On the left of the transition point, the topological state of the band gap is
, and on the right it is 0. This phenomenon is a critical
signature in a 1D topological phononic system that promises the existence of interface mode. Interface modes can be activated by connecting two different PnCs with different topological states of band gaps overlapping with each other. However, the band gaps with different topological states (d <0 and d >0) in Figure 3 do not overlap well. To satisfy the well-overlapping requirements, two different kinds of PnC plates are proposed by elaborately designing their geometries. The unit cell of the first kind (C1) is the same as the one described above with d =0.3mm. The geometry of the unit cell of the second plate (C2) is: L=10.88mm, l=6.4mm, D=16mm, and d=8mm. The material properties are the same as before. The dispersion relations and band gaps of C1 and C2 are shown in Figures 4(a) and (b), respectively. The band gap of C1 covers the frequency range from 1394 Hz to 1430 Hz, and that of C2 covers from 1390 Hz to 1438 Hz. The band edge frequency modes at X are shown by the insets in Figures 4(a) and (b). Besides, the corresponding transmission spectra are also presented in the right panels of Figures 4(a) and (b), which are calculated based on the finite structures consisting of 200 unit cells. In Figure 4(c), we calculate the band structure of a supercell which is assembled by connecting 100 C1 and 100 C2 sequentially. The dark gray region in Figure 4(c) indicates the overlapped frequency range covered by the band gaps both in C1 and C2. The light gray region is the frequency range covered by either the band gap in C1 or that in C2. A single wave mode is found at the frequency of 1417 Hz, which locates inside the overlapped band gap range. The corresponding displacement amplitude is shown in Figure 4(d). As is seen, the displacement amplitude near the interface is much larger than that in the rest region, suggesting it be an interface state mode. 8
In Figure 4(c), the transmission spectra of a finite structure including 100 C1 and 100 C2 are also presented. To well observe the transmission properties as well as the interface state, two transmission spectra are considered: the red line represents the transmission spectrum for the wave propagating from the left side to the interface, while the blue one represents that from the left side to the right side. The specific details can be found in Appendix A. For the transmission to the interface (the red line), one peak is observed inside the band gap, at the frequency of 1416 Hz. The peak frequency agrees well with the frequency of the interface state as shown in the band structure. While in the case of transmission to the right side (the blue line), instead of only one peak, two peaks are observed inside the band gap. One is at the frequency of 1416 Hz, corresponding to the interface state mode; another one is at the frequency of 1421 Hz. If we take a further look at the transmission spectrum of C2, one peak at exactly the same frequency (i.e. 1421 Hz) is also observed, as shown in Figure 4(b). We believe this frequency corresponds to an edge state mode of C2. Edge state wave modes are usually excited at the edge of a finite phononic structure and are sensitive to the number of unit cells in the structure. In Figures 5(b)-(d), the transmission spectra of finite PnCs with different numbers of C2 are presented. For the case of 200 unit cells (Figure 5(b)), although a peak is observed, its maximum transmission ratio is still below -5, which means that any substantial signals can be hardly detected at the right side. As the number of unit cells reduces, the transmission ratio of the peak increases. In the case of 50 unit cells (Figure 5(d)), the edge state is more than a peak, expanding to a narrow frequency range, in which waves can fully propagate through the structure. In Figure 5(e), the displacement amplitude in the structure of 50 unit cells is shown, at the frequency of 1422 Hz. As the number of unit cells further reduces, this finite phononic structure becomes fully transmittable. Based on the above analysis of the edge mode, the difference between the two transmission spectra shown in Figure 4(c) can be well explained. Considering the case of transmission from the left side to the interface, since the PnC plate of C 1 exhibits no edge mode, the only peak is caused by the interface mode, as shown by the red line. While as wave transmits to the right side, both the interface mode and the edge mode of C2 can be excited as expected, corresponding to the two peaks shown by the blue line. Interestingly, this mechanism can be utilized to design different finite PnC structures with different acoustic properties. Two more primitive examples are shown in 9
Figure 6. In the previous example, a finite system composed of 100 C1 connected to 100 C2 is considered. Now we consider a system that connects 50 C1, 100 C2, and again 50 C1 sequentially. For dispersion relation analysis of the supercell, the result is the same as the previous case (Figure 6(a)), since the corresponding infinite periodic structure does not change. But the transmission spectrum is totally different, as shown in Figure 6(b). Instead of one interface and one edge of C2, this finite structure contains two interfaces but no edge of C2, and both its left and right ends are edges of C1. As a result, no edge mode at frequency 1421 Hz is observed, but the interface mode is much stronger. Another case is connecting 50 C2, 100 C1, and again 50 C2 sequentially. Now there are two interfaces and two edges of C2 in the whole structure, so the interface mode as well as the edge mode should be much stronger than the original ones. The results are shown in Figure 6(c). In both Figures 6(b) and (c), the red lines show the transmission ratio from the left side to the center, after propagating through 100 unit cells. These results reveal that not only the number of unit cells, but also the arrangement of these unit cells have great effects on the transmission properties of the designed system.
Fig. 3. (a) Edge frequency modes around the first band gap as a function of the variable d. Red and blue lines represent Zak phases of
and 0, respectively. The insets show the contour plots
of the edge frequency modes. (b)-(d) Dispersion relations around the first band gap, for d being -0.3mm, 0, and 0.3mm, respectively.
10
Fig. 4. (a) Dispersion relation and transmission spectrum of C1 around the first band gap. The insets show the contour plots of the edge frequency modes at X. (b) Dispersion relation and transmission spectrum of C2 around the first band gap. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, in the same frequency range. Red line in the transmission spectra shows waves transmitting from the left side to the interface, while blue line shows that from the left side to the right side. (d) Displacement amplitude of the interface state indicated in the dispersion relation of the supercell.
11
Fig. 5. (a) Dispersion relation of C2. (b)-(d) Transmission spectra of finite structures containing various numbers of unit cells C2: 200 in (b), 100 in (c) and 50 in (d). (e) Displacement amplitude of the edge mode for the case of 50 C2.
12
Fig. 6. (a) Dispersion relation of a supercell containing 100 C1 and 100 C2. (b)-(c) Transmission spectra of finite structures with different arrangements of unit cells: 50C 1+100C2+50C1 in (b), and 50C2+100C1+50C2 in (c).
Until now, we have discussed the effects of geometric designing on the topological features and elastic wave properties of the structures, both in the unit-cell level and in the global-structural level. In the next part of this section, we are going to show that large finite deformation can be also utilized to shift the topological characteristics of the structure, providing an efficient way to modify wave propagation in real time. Structures made of soft hyperelastic material can undergo very large elastic deformation repeatedly without any damages. In Figure 7(a), we show the evolution of the band gap in C1 as the unit cell is stretched. To simulate periodic stretch of the unit cell, a periodic boundary condition with nominal strain
is applied on the left and right sides,
as discussed in more details in Appendix A1. As the pre-stretch increases, the band gap fully closes at
0.07 , but reopens immediately. During this process, the edge frequency wave 13
modes of the lower and upper edges switch with each other, as shown in the insets of Figure 7(a). The topological state of this band gap is changed, from 0 to
, due to the pre-stretch applied to
the structure. The change of topological feature caused by stretching is similar to that caused by geometric change, as we showed in the previous examples. By comparison, Figure 7(b) shows the evolution of the band gap in unit cell C2 under stretching, whose geometric parameters are described in the previous examples. In this case, the band gap widens gradually and the topological features of its lower and upper edge modes do not change. Besides, after stretching, both the band gaps of C1 and C2 still locate in the similar frequency range, but have the same topological state. As a result, if we globally stretch the whole structure containing C1 and C2, the interface state mode would disappear. As before, we consider a structure containing 100 C1 and 100 C2, but a global pre-stretch of nominal strain
0.15 is applied to it, as a way to induce large deformation. Notice that, since
the geometries of unit cells C1 and C2 are different while the forces applied on them are the same, the nominal strains caused by the global pre-stretch inside them are different. By calculation, the effective nominal strain of C1 is 0.142, and that of C2 is 0.159. The post-stretching band structures and transmission properties are shown in Figure 8. Since now the band gaps of C1 and C2 have the same topological states, there is no interface state mode inside the overlapped frequency range of the band gaps, as exactly shown in Figure 8(c). Comparing the transmission spectra in Figure 8(c), the red one, which describes waves propagating halfway to the interface, decays in the frequency range inside the band gap of C1, while the blue one, which describes waves propagating all the way to the right side, decays in the whole frequency range inside the band gaps of both C1 and C2. Furthermore, the blue one decays more intensively than the red one. Another interesting thing to notice is that, the edge mode inside the original band gap of C 2 disappears in Figure 8(b) after stretching. That’s because the edge mode is not caused by the topological properties of the structure, and its frequency is hardly affected by the pre-stretch applied to the structure. Although during stretching the topological state of the band gap does not change, the edge mode moves upwards relating to the band gap and finally moves out of the band gap as the frequency range of the band gap goes downwards. In Figure 7(b), the black line inside the band gap depicts the 14
evolution of the edge mode, whose frequency is calculated from the transmission analysis. It can be seen that the edge mode frequency shifts downwards gradually, much slower than the upper edge frequency mode. The mechanism that causes the edge mode requires further research. By combining geometric designing with pre-stretch, it can be very flexible to design PnC structures with different acoustic properties. In the previous example, we showed that the pre-stretch can deactivate the interface state mode without any change of the structure itself, which provides a means of convenient modification of the topological properties of the structure. The reversion can also be achieved. Here we design two new unit cells (namely C1 and C2 ), with different geometric properties as: L=9.84mm, l=6.56mm, D=18.37mm, and d=10.50mm for
C1 , and L=9.84mm, l=6.56mm, D=17.38mm, and d=9.43mm for C2 . The evolution of the corresponding topological band gaps under pre-stretch is presented in Figure 9. Initially at the stretch-free state, the band gaps in C1 and C2 have the same topological state (0). As the pre-stretch
is applied, the topological state of C1 band gap does not change, as suggested by
the edge frequency modes shown by the insets in Figure 9. However, the topological state of C2 band gap shifts approximately at the transition point
0.13 , suggesting that an interface
mode will emerge after this point. The band structures and transmission spectra are shown in Figures 10 and 11, for the stretch-free structure and post-stretching structure, respectively. Initially, when
0 , in the overlapped frequency range of the two band gaps, there is also a complete
band gap in the supercell. While after stretching ( equals 0.18 for the supercell/finite structure, and equals 0.17 and 0.19 for unit cells C1 and C2 , respectively.), an interface state mode is observed in the overlapped frequency range of these band gaps. In the transmission spectra, a peak is also observed at the same frequency of the interface state mode. Comparing to the blue line, the peak of the red line almost has no decay, suggesting the displacement amplitude at the interface is very large.
15
Fig. 7. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets. The
black line inside the band gap in (b) shows the evolution of the edge mode in C2 as
varies.
Fig. 8. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.142 for C1, and =0.159 for C2. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, with
16
=0.15.
Fig. 9. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets.
Fig. 10. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 .
17
Fig. 11. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.17 for C1 , and =0.19 for C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 , with
=0.18.
4. Conclusions
In summary, we have designed a 1D phononic system constructed by connecting two periodic plates made of soft materials with different geometrical parameters. The acoustic properties of this system are thoroughly studied numerically using ABAQUS combined with the secondary development scripts written in Python. From the results, the topological protected state of elastic waves is observed at the interface between the two plates when proper parameters are chosen. Besides, the edge mode is also observed for specific finite periodic plates. Hyperelastic materials can withstand large elastic deformation, and the band structure in the unit cells is sensitive to deformation caused by the pre-stretch applied to the system. As a result, the topological states of band gaps can be switched, and the interface state modes can be either activated or deactivated. Interface state with topological protection is very stable against defects and disorders in the system, providing a robust way of convenient and efficient modification of wave propagation inside 1D PnC systems. The strategy proposed in this paper utilizes large deformation to control the topological properties and to manipulate wave propagation through PnC systems. The study has great potential in constructing various advantageous acoustic devices, such as energy harvesters, 18
wave sensors and acoustic filters where a flexible and efficient control of wave propagation is required. These large-deformation-based devices that can easily switch between two states of wave propagation also fulfill the goal of one structure with two and even more functions, therefore highly enhancing the efficiency in practical application.
Conflict of interest statement
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Nos. 11532001 and 11621062). Partial support from the Fundamental Research Funds for the Central Universities (No. 2016XZZX001-05) and the Shenzhen Scientific and Technological Fund for R & D (No. JCYJ20170816172316775) is also acknowledged. The work was also supported by the open project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) under Grant No. KFJJ16-04M.
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Author statement Manuscript title: Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates I have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; AND I have drafted the work and revised it critically for important intellectual content; AND I have approved the final version to be published; AND I agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons who have made substantial contributions to the work reported in the manuscript, including those who provided editing and writing assistance but who are not authors, are named in the Acknowledgments section of the manuscript and have given their written permission to be named. If the manuscript does not include Acknowledgments, it is because the authors have not received substantial contributions from nonauthors.
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Appendix A
In this appendix, we describe the computational processes of the simulation in detail. All computations are carried out under the 2D plane-strain assumption, using commercial software ABAQUS.
A.1 Simulations of pre-stretch and finite deformation
In calculating dispersion relation and band structure, we assume the structure is infinite and periodic. A unit cell as shown in Figure 1(b) is adopted. When the infinite structure is under global pre-stretch, the left and right boundaries of a unit cell should satisfy the following boundary conditions: U (a, y ) U (0, y ) 1 , a . V (a, y ) V (0, y ),
(A.1)
where U and V are the displacement components in the x and y directions, respectively.
is the global nominal strain that is applied to the structure. For the unit cell, its deformed state is also geometrically periodic, which means for a pair of points on the left and right boundaries, their displacements in the y direction must be the same, while the difference between their displacements in the x direction equals a . To apply the above boundary conditions to the unit cell, pointwise multi-point constraints are created to constrain the left boundary nodes with their corresponding right boundary nodes, with the help of scripts written in Python [40,41]. When studying the interface properties, a supercell or finite structure that contains a finite number of unit cells is considered. To simulate the pre-stretch, periodic boundary conditions (for supercell) or global boundary conditions (for finite structure) are applied using the exact same method.
A.2 Dispersion relations and band gaps
The transmissibility of waves in periodic structures can be studied with dispersion relation analysis. In theory, for a unit cell (or supercell) of the periodic structure, a base vector a1 is used
21
to describe its periodicity (for 2D or 3D structures, it’s a set of base vectors, ai ), whose length equals the length of a unit cell. A reciprocal vector b1 is further introduced to describe the wave number domain, satisfying
a1 b1 2 . The Bloch condition states that,
waves that propagate
inside a 1D periodic structure satisfy equation
u( x0 + r ) u( x0 )ei ( k r ) ,
(A.2)
where u is displacement amplitude, and k, in the reciprocal domain, is the wave number. Let x0 represent a point on the left side boundary, and let r=a, Eq. (A.2) becomes a k-related Bloch boundary condition. Let the wave number k go through the irreducible Brillouin zone (IBZ,
k [0,b1 / 2] ), and by substituting the boundary conditions into the wave equation, the dispersion relation in IBZ can be calculated as an eigen frequency problem. In FEM simulation using ABAQUS, two instances are created, representing the real part and imaginary part of the wave equation respectively, and the Bloch boundary condition can be applied to the boundaries of these two instances accordingly
[42]
. The whole processes are coded into Python-based scripts
using the secondary development function provided by ABAQUS. This method can be used to calculate dispersion relations for the unit cell (or supercell) with or without pre-stretch. When pre-stretch is considered, the states of deformation and stress should be imported as initial states from the previously calculated results.
A.3 Calculations of transmission spectra
Analyzing the transmission spectra of finite cells can be used to verify the results of dispersion relations, such as band gaps and interface state modes. It also can be used to detect edge modes which can not be seen in the dispersion relations, since they are only properties of a finite structure. A basic schematic of a finite structure is shown in Figure A1. It can be constructed by one class of unit cell, or by arrangement of several classes of unit cells. The black dashed line in the center represents an interface between the two different unit cells. A force stimulus is applied at the left boundary of the structure, and waves propagate from the left to the right. Steady-state dynamics analysis step in ABAQUS is used with the frequency sweeping through the range of interest. Then 22
the displacement amplitudes at the three detection lines (marked in red lines in Figure A1) are extracted to compute the transmission spectra. The transmission spectrum of waves propagating from the left to the center is calculated by comparing the average displacement amplitudes at the central and the left detection lines as:
mean(uc ) Tc =log , mean(ul )
(A.3)
and that of waves propagating from the left to the right is calculated by comparing the average displacement amplitudes at the right and left detection lines:
mean(ur ) Tr =log , mean(ul )
(A.4)
where ul , uc , and ur are points in the left, central, and right detection lines, respectively.
mean( ) is the average function. Due to the existence of interface state or edge state phenomena, the transmission properties of Tc and Tr can be substantially different.
Fig. A1. Schematic of a finite structure for transmission analysis.
Appendix B
For a specific band gap, the topological state is determined by the summation of Zak phases of all the frequency modes below this band gap (mod by 2 ). In Figure 3, we show the dispersion relations of the unit cell around its first band gap. The dispersion relations in the whole frequency range below 1800 Hz are shown in Figures B1(a) and (b), when d equals -0.3mm and 0.3mm. The frequency modes in blue (red) have Zak phases with value 0 ( ). The topological state of the first-order band gap is determined by the first three 23
frequency modes below it. It equals 0 if there are two frequency modes with the Zak phases being
, and equals if there is only one. As d increases from negative to positive, the edge frequency modes around the band gap swap with each other, causing the Zak phase of the lower frequency mode to shift from 0 to , further causing the topological state of the band gap to shift from
to 0. Figure B1(c) further demonstrates the analyzing of the Zak phase of a frequency
mode. When the unit cell of the structure has inversion symmetry, the Zak phase of a frequency mode can either be 0 or , depending on the symmetry of the corresponding displacement fields [16, 17]
. For example, the four displacement vector plots at the four points labeled in Figure B1(a) are
shown in Figure B1(c), where the length and direction of an arrow represent the amplitude and direction of the displacement at the corresponding point. Plot 1 and Plot 3 share the same symmetry of the displacement field, so the Zak phase of the corresponding frequency mode is 0; while Plot 2 and Plot 4 have different symmetries of the displacement field, and the Zak phase becomes . More detailed analysis of the Zak phase can be found in Refs. [14-17].
Fig. B1. (a)-(b) Dispersion relations of unit cells with d being -0.3mm and 0.3mm. The Zak phase of the frequency mode in blue (red) is 0 ( ). (c) Displacement vector plots at the four points labeled in (a), indicating a bending characteristic of these two frequency modes.
For reference, in Figures B2 and B3, we show the Zak phase of each frequency mode and the topological states of band gaps in those four unit cells ( C1 , C 2 , C1 , and C2 ) appearing in this paper, before and after pre-stretch. When the two PnC plates with different unit cells connected to 24
each other, the existence of interface state is dependent on the topological states of the overlapped band gaps: if these two band gaps have the same topological state, there is no interface state, while if they have different topological states, one interface state is guaranteed. For the case of C1 + C 2 , initially, without pre-stretch, the band gaps in C1 and C 2 have topological states of 0 and
respectively. One single interface state therefore exists. After pre-stretch, the topological state of the band gap in C1 transfers from 0 to
, and as a result, the interface state disappears. For the
case of C1 + C2 , initially both band gaps have the topological state of 0, while after pre-stretch, the topological state of the band gap in C2 transfers from 0 to
. So the system has no
interface state at first, but one will emerge due to pre-stretch.
Fig. B2. Dispersion relations and band gaps of unit cells C1 and C 2 , before and after pre-stretch.
25
Fig. B3. Dispersion relations and band gaps of unit cells C1 and C2 , before and after pre-stretch.
26
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Figure caption list Fig. 1. Schematics of the phononic system: (a) The system contains two periodic thick plates connected to each other; (b) The cross section of a single unit cell. Fig. 2. A cross section view of the whole system, containing N Cell 1 units and N Cell 2 units. At the center is the interface between the two periodic plates. Fig. 3. (a) Edge frequency modes around the first band gap as a function of the variable d. Red and blue lines represent Zak phases of and 0, respectively. The insets show the contour plots of the edge frequency modes. (b)-(d) Dispersion relations around the first band gap, for d being -0.3mm, 0, and 0.3mm, respectively. Fig. 4. (a) Dispersion relation and transmission spectrum of C1 around the first band gap. The insets show the contour plots of the edge frequency modes at X. (b) Dispersion relation and transmission spectrum of C2 around the first band gap. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, in the same frequency range. Red line in the transmission spectra shows waves transmitting from the left side to the interface, while blue line shows that from the left side to the right side. (d) Displacement amplitude of the interface state indicated in the dispersion relation of the supercell. Fig. 5. (a) Dispersion relation of C2. (b)-(d) Transmission spectra of finite structures containing various numbers of unit cells C 2: 200 in (b), 100 in (c) and 50 in (d). (e) Displacement amplitude of the edge mode for the case of 50 C2. Fig. 6. (a) Dispersion relation of a supercell containing 100 C1 and 100 C2. (b)-(c) Transmission spectra of finite structures with different arrangements of unit cells: 50C 1+100C2+50C1 in (b), and 50C2+100C1+50C2 in (c). Fig. 7. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets. The
black line inside the band gap in (b) shows the evolution of the edge mode in C2 as
varies.
Fig. 8. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.142 for C1, and =0.159 for C2. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, with
=0.15. 31
Fig. 9. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets.
Fig. 10. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 . Fig. 11. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.17 for C1 , and =0.19 for C2 . (c) Dispersion relation and transmission spectra of
a supercell containing 100 C1 and 100 C2 , with
=0.18.
Fig. A1. Schematic of a finite structure for transmission analysis. Fig. B1. (a)-(b) Dispersion relations of unit cells with d being -0.3mm and 0.3mm. The Zak phase of the frequency mode in blue (red) is 0 ( ). (c) Displacement vector plots at the four points labeled in (a), indicating a bending characteristic of these two frequency modes. Fig. B2. Dispersion relations and band gaps of unit cells C1 and C 2 , before and after pre-stretch. Fig. B3. Dispersion relations and band gaps of unit cells C1 and C2 , before and after pre-stretch.
32
Graphical abstract
33
Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates Yilan Huang , Yang Huang , Weiqiu Chen , Ronghao Bao PII: DOI: Reference:
S0020-7403(19)32424-5 https://doi.org/10.1016/j.ijmecsci.2019.105348 MS 105348
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
5 July 2019 20 November 2019 27 November 2019
Please cite this article as: Yilan Huang , Yang Huang , Weiqiu Chen , Ronghao Bao , Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.105348
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Highlights
We design a soft tunable 1D phononic crystal by connecting two different periodic plates made of hyperelastic materials. The interface state wave modes observed inside the system can be either activated or deactivated by an applied pre-stretch on the structures with different initial geometrical parameters. Due to topological protection, the system provides a robust way of manipulating waves propagation in real time.
1
Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates Yilan Huang1, Yang Huang1, Weiqiu Chen1,2,3,*, Ronghao Bao1,3,* 1. Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province and Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China; 2. State Key Lab of CAD & CG, Zhejiang University, Hangzhou 310058, China; 3. Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China
Abstract: In this paper, we investigate the possibility of flexible manipulation of elastic waves propagating in soft periodic plates through topological state shifting. We design a soft 1D phononic crystal by connecting two different periodic plates made of hyperelastic materials. By choosing different geometrical parameters of the two plates, topologically protected state of elastic waves is observed at the interface between them. Since the hyperelastic material can undergo large elastic deformation, flexible and efficient shifting of topological state can be realized by applying appropriate pre-stretch on the structure. The interface state wave modes can be either activated or deactivated by an applied pre-stretch on the structures with different initial geometrical parameters. Due to the advantageous property that topologically protected wave modes are stable against any defects and disorders, it provides a robust way of manipulating waves propagation in real time. The system and the strategy proposed here have great potential in constructing various novel acoustic devices.
Keywords: periodic structure; topologically protected; interface state; wave manipulation; large deformation.
*
Corresponding author. E-mail: [email protected]; Tel./Fax: 86-571-87951866.
*
Corresponding author. E-mail: [email protected]; Tel./Fax: 86-571-87951866. 2
1.
Introduction
Phononic crystals (PnCs), since initially proposed and studied by Kushwaha et al. in 1993 [1], have been vastly researched in the recent several decades, due to their wonderful acoustic properties caused by the unique ability of manipulating wave propagation through band gaps
[2,3]
. Recent
studies further revealed that, by introducing topological states into the traditional PnC structures, advantageous local acoustic phenomena (e.g. edge modes and interface states) that are hardly seen in normal structures can be observed. Topological phenomena are ubiquitous in the physics world. In condensed matter physics, topological phenomena in quantum Hall and topological insulators have already been extensively studied
[4-6]
. By making analogies, topological concepts have been
further extended to bosonic systems such as photonic
[7-9]
and phononic structures
[10-11]
. The most
promising properties of those structures are topologically protected states existing at the boundaries or interfaces, which are robust against defects and disorders. Thus, topological wave mechanics has brought in another revolution of manipulation and control of acoustic waves. In one-dimensional (1D) phononic crystal systems, interface states with large energy enhancement may be induced when connecting different PnC structures with opposite topological invariancies around particular bandgaps. The topological interface states of acoustic waves were first realized in fluid systems
[12,13]
, and recently were also observed in solid systems
[14,15]
. For 1D phononic
systems with inversion symmetry, the topological properties of frequency modes and band gaps are characterized by the well-known Zak phase, which was initially proposed to study the energy bands of electrons in solid structures
[16,17]
. Recently, the relationship of Zak phase and Bloch
bands in acoustic phononic systems is also well studied [12,18]. Another inspiring research area in PnCs is tunable phononic systems made of soft hyperelastic materials. Solid phononic systems made of traditional materials are hardly able to change their material properties and geometric structures. As a result, their mechanical and acoustic properties are fixed after manufacture, inevitably limiting their applications in frequency-changing environments. While tunable hyperelastic materials give rise to the potential of designing PnC systems with alterable acoustic properties along the deformation trajectory, enormously enhancing the practical applications of PnCs as adaptive acoustic devices, including sound filters, acoustic 3
mirrors, acoustic wave guides, and vibration isolators
[19-23]
. Abundant researches already showed
both theoretically and experimentally that soft materials can undergo large deformation repeatedly under different external stimuli, including electric fields mechanical loadings
[24-27]
, magnetic fields, temperature and
[28-38]
, providing efficient strategies for modifying band gaps and
manipulating wave propagations in soft PnCs. This kind of flexible tunability of band gaps is mainly caused by the change of structural geometry and material effective stiffness [39]. Existing researches have shown that the topological characteristics of PnCs are sensitively affected by the geometry of the structures and altering the geometric parameters is the main strategy to shift the interface states in 1D systems. In this paper, we want to explore the feasibility of flexible modification of the topological states of PnCs under pre-stretch due to external mechanical loadings. First, we study the topological states of in-plane waves in a 1D soft thick plate. A simple 1D acoustic system is proposed by connecting two phononic crystals with similar geometries but quite distinct topological properties. We show that topological interface mode inside a particular band gap can be observed at the interface of the two PnCs. Besides, an edge mode in a specific structure is also observed. Then we study the effect of large deformation on the topological properties and the interface mode by applying a pre-stretch on the 1D plate. Results show that, by properly designing the initial structure, the topological properties can be efficiently modified and the interface states can be activated or deactivated under pre-stretch. In our work, all the simulations of pre-deformation, band gaps and transmission spectra are based on the commercial software ABAQUS. Python-based scripts are used to perform the related modeling, meshing and calculations. The proposed structures can be used to construct advantageous acoustic devices, such as energy harvesters, wave sensors and acoustic filters, especially in dynamic situations, where convenient and efficient modification of wave propagation is required.
2. Summary of models and calculation methods
The phononic system considered in this paper is composed of two thick periodic plates connected to each other. We assume the thickness of the plate is much larger than the length of the unit cell, and consider in-plane waves propagating from one end of the system to the other end. The 3D 4
sketch of the system is shown in Figure 1(a), and the cross section of one single unit cell is schematically shown in Figure 1(b). The unit cell includes two different parts with different material properties and different geometric sizes. The length and width of part A are L and D respectively, while those of part B are l and d. The total length of the cell therefore is a = l + L. The unit cell satisfies inversion symmetry with respect to its central plane. The assembled system is shown in Figure 2. The left and right sides of the dashed line represent two periodic plates, and each of them contains a finite number (here denoted by N) of unit cells (denoted by Cell 1 and Cell 2 respectively). The whole plate is made of soft hyperelastic materials, which can sustain substantial elastic deformation. The material behavior is described by the neo-Hookean model whose strain energy density function takes the following form:
C10 I1 3
1 2 J 1 , D1
(1)
where I1 is the first deviatoric strain invariant, J indicates the local volume change, and C10 and D1 are two material constants. The initial shear modulus and bulk modulus are calculated through 0 2C10 , K0 2 / D1 . To perform the computation with finite element method, the commercial software ABAQUS is used. Python-based scripts are adopted to facilitate the related steps of modeling, meshing, calculation, and results analyzing. Based on the thick plate assumption, we use 2D plane-strain models to conduct all simulations. First, to study the band structure of a periodic plate, a model of a single unit cell is created. The Bloch boundary conditions are imposed in the periodic direction using specific python scripts, and a frequency step is used to calculate the dispersion relations. By analyzing the band structure and the frequency modes, the topological characteristics of particular band gap can be determined. Then to study the interface states, a supercell containing the whole structure is created (including N Cell 1 units and N Cell 2 units), and its band structure is also calculated using the same method. In order to verify the results, the transmission behaviors of the finite structure (see Fig. A1 in Appendix A) are further examined, for which the harmonic analysis is used with the frequency sweeping through the range of interest. The transmission spectra are calculated according to the ratio between the average displacement amplitudes at the input and 5
output regions of the finite structure. It is noted that, throughout this study, there is no consideration of application of acoustic load and radiation of the structure, which however may be of significance in practical applications. Next, the effect of large deformation induced by external mechanical load on band structure and on interface state is explored. In the case of a unit cell (or a supercell), periodic boundary conditions are applied on the boundaries in the periodic direction using python scripts; and in the case of a finite structure, a total displacement is applied to the boundaries in the stretching direction (i.e. in the x direction as shown in Figure 1(a)). To calculate the band gaps and transmission spectra of a pre-stretched model, the computational processes are identical, except that there is a need to introduce the results of deformation and effective stiffness into the model as the initial state. A more detailed description of the computational processes is provided in Appendix A. Further information may be found in Refs. [33, 34].
Fig. 1. Schematics of the phononic system: (a) The system contains two periodic thick plates connected to each other; (b) The cross section of a single unit cell.
Fig. 2. A cross section view of the whole system, containing N Cell 1 units and N Cell 2 units. At 6
the center is the interface between the two periodic plates.
3. Numerical results
To demonstrate the tunability of topological characteristics in our systems, first we consider a unit cell. The geometric parameters and material properties are as follows: L=10.66mm, l=7.38mm, D=16.4mm, and d=8.594mm-d. C10A=5.5 × 105Pa, D1A=3.636 × 10-8Pa-1, C10B=9.35 × 105Pa, D1B=2.14×10-8Pa-1, and 0 A 0 B 1050 kg/ m3 , where C10A, D1A, and C10B, D1B are the material parameters of part A and part B, respectively. 0 A and 0B represent the initial densities of part A and part B, respectively. Here the geometry of the unit cell is changed by the variable d. In Figure 3(a), the evolution of the first band gap (more specifically, the band gap between the third and fourth frequency modes) with respect to d is presented. Figures 3(b)-(d) show the dispersion relations for the cases of d=-0.3mm, 0, and 0.3mm respectively. As d increases from -0.3mm to 0.3mm, the band gap closes at first, but then reopens. At the degenerate point of d=0, this band gap completely disappears. One interesting phenomenon is that, during the process of closing and reopening, the topological feature of this band gap is changed. For 1D PnC structures, the Zak phase is usually introduced to describe the topological state of an isolated frequency mode. It can be calculated by
nZak i
/a
/ a
where
U n, k | k | U n, k dk ,
(2)
U n, k is the normalized displacement amplitude of the n-th frequency state with wave
number k. Due to the inversion symmetry of the unit cell with respect to its central plane, the displacement field of a frequency mode is either symmetric or antisymmetric. According to the symmetry property of the frequency mode, the Zak phase must be either 0 or
[14-17]
. The
topological state of a band gap is calculated by summing over all the Zak phases of the frequency modes below that gap, denoted as
n
iZak mod 2 , which can be either 0 or . Although
i 1
the Zak phase is gauge-dependent, varying with the choice of the origin of the unit cell, the 7
difference between two topological states of band gaps in different unit cells depends solely on the structure. A detailed description of the topological state of band gaps is provided in Appendix B. The contour plot of the edge frequency modes at X (the right edge of the irreducible Brillouin zone) around the first band gap is also presented as the insets of Figure 3(a). The topological feature of the band gap is related to its edge frequency modes. Comparing the band gaps of d <0 and d >0, the lower edge mode and the upper edge mode swap with each other and the degenerate point at d =0 is a topological transition point. On the left of the transition point, the topological state of the band gap is
, and on the right it is 0. This phenomenon is a critical
signature in a 1D topological phononic system that promises the existence of interface mode. Interface modes can be activated by connecting two different PnCs with different topological states of band gaps overlapping with each other. However, the band gaps with different topological states (d <0 and d >0) in Figure 3 do not overlap well. To satisfy the well-overlapping requirements, two different kinds of PnC plates are proposed by elaborately designing their geometries. The unit cell of the first kind (C1) is the same as the one described above with d =0.3mm. The geometry of the unit cell of the second plate (C2) is: L=10.88mm, l=6.4mm, D=16mm, and d=8mm. The material properties are the same as before. The dispersion relations and band gaps of C1 and C2 are shown in Figures 4(a) and (b), respectively. The band gap of C1 covers the frequency range from 1394 Hz to 1430 Hz, and that of C2 covers from 1390 Hz to 1438 Hz. The band edge frequency modes at X are shown by the insets in Figures 4(a) and (b). Besides, the corresponding transmission spectra are also presented in the right panels of Figures 4(a) and (b), which are calculated based on the finite structures consisting of 200 unit cells. In Figure 4(c), we calculate the band structure of a supercell which is assembled by connecting 100 C1 and 100 C2 sequentially. The dark gray region in Figure 4(c) indicates the overlapped frequency range covered by the band gaps both in C1 and C2. The light gray region is the frequency range covered by either the band gap in C1 or that in C2. A single wave mode is found at the frequency of 1417 Hz, which locates inside the overlapped band gap range. The corresponding displacement amplitude is shown in Figure 4(d). As is seen, the displacement amplitude near the interface is much larger than that in the rest region, suggesting it be an interface state mode. 8
In Figure 4(c), the transmission spectra of a finite structure including 100 C1 and 100 C2 are also presented. To well observe the transmission properties as well as the interface state, two transmission spectra are considered: the red line represents the transmission spectrum for the wave propagating from the left side to the interface, while the blue one represents that from the left side to the right side. The specific details can be found in Appendix A. For the transmission to the interface (the red line), one peak is observed inside the band gap, at the frequency of 1416 Hz. The peak frequency agrees well with the frequency of the interface state as shown in the band structure. While in the case of transmission to the right side (the blue line), instead of only one peak, two peaks are observed inside the band gap. One is at the frequency of 1416 Hz, corresponding to the interface state mode; another one is at the frequency of 1421 Hz. If we take a further look at the transmission spectrum of C2, one peak at exactly the same frequency (i.e. 1421 Hz) is also observed, as shown in Figure 4(b). We believe this frequency corresponds to an edge state mode of C2. Edge state wave modes are usually excited at the edge of a finite phononic structure and are sensitive to the number of unit cells in the structure. In Figures 5(b)-(d), the transmission spectra of finite PnCs with different numbers of C2 are presented. For the case of 200 unit cells (Figure 5(b)), although a peak is observed, its maximum transmission ratio is still below -5, which means that any substantial signals can be hardly detected at the right side. As the number of unit cells reduces, the transmission ratio of the peak increases. In the case of 50 unit cells (Figure 5(d)), the edge state is more than a peak, expanding to a narrow frequency range, in which waves can fully propagate through the structure. In Figure 5(e), the displacement amplitude in the structure of 50 unit cells is shown, at the frequency of 1422 Hz. As the number of unit cells further reduces, this finite phononic structure becomes fully transmittable. Based on the above analysis of the edge mode, the difference between the two transmission spectra shown in Figure 4(c) can be well explained. Considering the case of transmission from the left side to the interface, since the PnC plate of C 1 exhibits no edge mode, the only peak is caused by the interface mode, as shown by the red line. While as wave transmits to the right side, both the interface mode and the edge mode of C2 can be excited as expected, corresponding to the two peaks shown by the blue line. Interestingly, this mechanism can be utilized to design different finite PnC structures with different acoustic properties. Two more primitive examples are shown in 9
Figure 6. In the previous example, a finite system composed of 100 C1 connected to 100 C2 is considered. Now we consider a system that connects 50 C1, 100 C2, and again 50 C1 sequentially. For dispersion relation analysis of the supercell, the result is the same as the previous case (Figure 6(a)), since the corresponding infinite periodic structure does not change. But the transmission spectrum is totally different, as shown in Figure 6(b). Instead of one interface and one edge of C2, this finite structure contains two interfaces but no edge of C2, and both its left and right ends are edges of C1. As a result, no edge mode at frequency 1421 Hz is observed, but the interface mode is much stronger. Another case is connecting 50 C2, 100 C1, and again 50 C2 sequentially. Now there are two interfaces and two edges of C2 in the whole structure, so the interface mode as well as the edge mode should be much stronger than the original ones. The results are shown in Figure 6(c). In both Figures 6(b) and (c), the red lines show the transmission ratio from the left side to the center, after propagating through 100 unit cells. These results reveal that not only the number of unit cells, but also the arrangement of these unit cells have great effects on the transmission properties of the designed system.
Fig. 3. (a) Edge frequency modes around the first band gap as a function of the variable d. Red and blue lines represent Zak phases of
and 0, respectively. The insets show the contour plots
of the edge frequency modes. (b)-(d) Dispersion relations around the first band gap, for d being -0.3mm, 0, and 0.3mm, respectively.
10
Fig. 4. (a) Dispersion relation and transmission spectrum of C1 around the first band gap. The insets show the contour plots of the edge frequency modes at X. (b) Dispersion relation and transmission spectrum of C2 around the first band gap. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, in the same frequency range. Red line in the transmission spectra shows waves transmitting from the left side to the interface, while blue line shows that from the left side to the right side. (d) Displacement amplitude of the interface state indicated in the dispersion relation of the supercell.
11
Fig. 5. (a) Dispersion relation of C2. (b)-(d) Transmission spectra of finite structures containing various numbers of unit cells C2: 200 in (b), 100 in (c) and 50 in (d). (e) Displacement amplitude of the edge mode for the case of 50 C2.
12
Fig. 6. (a) Dispersion relation of a supercell containing 100 C1 and 100 C2. (b)-(c) Transmission spectra of finite structures with different arrangements of unit cells: 50C 1+100C2+50C1 in (b), and 50C2+100C1+50C2 in (c).
Until now, we have discussed the effects of geometric designing on the topological features and elastic wave properties of the structures, both in the unit-cell level and in the global-structural level. In the next part of this section, we are going to show that large finite deformation can be also utilized to shift the topological characteristics of the structure, providing an efficient way to modify wave propagation in real time. Structures made of soft hyperelastic material can undergo very large elastic deformation repeatedly without any damages. In Figure 7(a), we show the evolution of the band gap in C1 as the unit cell is stretched. To simulate periodic stretch of the unit cell, a periodic boundary condition with nominal strain
is applied on the left and right sides,
as discussed in more details in Appendix A1. As the pre-stretch increases, the band gap fully closes at
0.07 , but reopens immediately. During this process, the edge frequency wave 13
modes of the lower and upper edges switch with each other, as shown in the insets of Figure 7(a). The topological state of this band gap is changed, from 0 to
, due to the pre-stretch applied to
the structure. The change of topological feature caused by stretching is similar to that caused by geometric change, as we showed in the previous examples. By comparison, Figure 7(b) shows the evolution of the band gap in unit cell C2 under stretching, whose geometric parameters are described in the previous examples. In this case, the band gap widens gradually and the topological features of its lower and upper edge modes do not change. Besides, after stretching, both the band gaps of C1 and C2 still locate in the similar frequency range, but have the same topological state. As a result, if we globally stretch the whole structure containing C1 and C2, the interface state mode would disappear. As before, we consider a structure containing 100 C1 and 100 C2, but a global pre-stretch of nominal strain
0.15 is applied to it, as a way to induce large deformation. Notice that, since
the geometries of unit cells C1 and C2 are different while the forces applied on them are the same, the nominal strains caused by the global pre-stretch inside them are different. By calculation, the effective nominal strain of C1 is 0.142, and that of C2 is 0.159. The post-stretching band structures and transmission properties are shown in Figure 8. Since now the band gaps of C1 and C2 have the same topological states, there is no interface state mode inside the overlapped frequency range of the band gaps, as exactly shown in Figure 8(c). Comparing the transmission spectra in Figure 8(c), the red one, which describes waves propagating halfway to the interface, decays in the frequency range inside the band gap of C1, while the blue one, which describes waves propagating all the way to the right side, decays in the whole frequency range inside the band gaps of both C1 and C2. Furthermore, the blue one decays more intensively than the red one. Another interesting thing to notice is that, the edge mode inside the original band gap of C 2 disappears in Figure 8(b) after stretching. That’s because the edge mode is not caused by the topological properties of the structure, and its frequency is hardly affected by the pre-stretch applied to the structure. Although during stretching the topological state of the band gap does not change, the edge mode moves upwards relating to the band gap and finally moves out of the band gap as the frequency range of the band gap goes downwards. In Figure 7(b), the black line inside the band gap depicts the 14
evolution of the edge mode, whose frequency is calculated from the transmission analysis. It can be seen that the edge mode frequency shifts downwards gradually, much slower than the upper edge frequency mode. The mechanism that causes the edge mode requires further research. By combining geometric designing with pre-stretch, it can be very flexible to design PnC structures with different acoustic properties. In the previous example, we showed that the pre-stretch can deactivate the interface state mode without any change of the structure itself, which provides a means of convenient modification of the topological properties of the structure. The reversion can also be achieved. Here we design two new unit cells (namely C1 and C2 ), with different geometric properties as: L=9.84mm, l=6.56mm, D=18.37mm, and d=10.50mm for
C1 , and L=9.84mm, l=6.56mm, D=17.38mm, and d=9.43mm for C2 . The evolution of the corresponding topological band gaps under pre-stretch is presented in Figure 9. Initially at the stretch-free state, the band gaps in C1 and C2 have the same topological state (0). As the pre-stretch
is applied, the topological state of C1 band gap does not change, as suggested by
the edge frequency modes shown by the insets in Figure 9. However, the topological state of C2 band gap shifts approximately at the transition point
0.13 , suggesting that an interface
mode will emerge after this point. The band structures and transmission spectra are shown in Figures 10 and 11, for the stretch-free structure and post-stretching structure, respectively. Initially, when
0 , in the overlapped frequency range of the two band gaps, there is also a complete
band gap in the supercell. While after stretching ( equals 0.18 for the supercell/finite structure, and equals 0.17 and 0.19 for unit cells C1 and C2 , respectively.), an interface state mode is observed in the overlapped frequency range of these band gaps. In the transmission spectra, a peak is also observed at the same frequency of the interface state mode. Comparing to the blue line, the peak of the red line almost has no decay, suggesting the displacement amplitude at the interface is very large.
15
Fig. 7. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets. The
black line inside the band gap in (b) shows the evolution of the edge mode in C2 as
varies.
Fig. 8. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.142 for C1, and =0.159 for C2. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, with
16
=0.15.
Fig. 9. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets.
Fig. 10. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 .
17
Fig. 11. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.17 for C1 , and =0.19 for C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 , with
=0.18.
4. Conclusions
In summary, we have designed a 1D phononic system constructed by connecting two periodic plates made of soft materials with different geometrical parameters. The acoustic properties of this system are thoroughly studied numerically using ABAQUS combined with the secondary development scripts written in Python. From the results, the topological protected state of elastic waves is observed at the interface between the two plates when proper parameters are chosen. Besides, the edge mode is also observed for specific finite periodic plates. Hyperelastic materials can withstand large elastic deformation, and the band structure in the unit cells is sensitive to deformation caused by the pre-stretch applied to the system. As a result, the topological states of band gaps can be switched, and the interface state modes can be either activated or deactivated. Interface state with topological protection is very stable against defects and disorders in the system, providing a robust way of convenient and efficient modification of wave propagation inside 1D PnC systems. The strategy proposed in this paper utilizes large deformation to control the topological properties and to manipulate wave propagation through PnC systems. The study has great potential in constructing various advantageous acoustic devices, such as energy harvesters, 18
wave sensors and acoustic filters where a flexible and efficient control of wave propagation is required. These large-deformation-based devices that can easily switch between two states of wave propagation also fulfill the goal of one structure with two and even more functions, therefore highly enhancing the efficiency in practical application.
Conflict of interest statement
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Nos. 11532001 and 11621062). Partial support from the Fundamental Research Funds for the Central Universities (No. 2016XZZX001-05) and the Shenzhen Scientific and Technological Fund for R & D (No. JCYJ20170816172316775) is also acknowledged. The work was also supported by the open project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) under Grant No. KFJJ16-04M.
19
Author statement Manuscript title: Flexible manipulation of topologically protected waves in one-dimensional soft periodic plates I have made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data for the work; AND I have drafted the work and revised it critically for important intellectual content; AND I have approved the final version to be published; AND I agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons who have made substantial contributions to the work reported in the manuscript, including those who provided editing and writing assistance but who are not authors, are named in the Acknowledgments section of the manuscript and have given their written permission to be named. If the manuscript does not include Acknowledgments, it is because the authors have not received substantial contributions from nonauthors.
20
Appendix A
In this appendix, we describe the computational processes of the simulation in detail. All computations are carried out under the 2D plane-strain assumption, using commercial software ABAQUS.
A.1 Simulations of pre-stretch and finite deformation
In calculating dispersion relation and band structure, we assume the structure is infinite and periodic. A unit cell as shown in Figure 1(b) is adopted. When the infinite structure is under global pre-stretch, the left and right boundaries of a unit cell should satisfy the following boundary conditions: U (a, y ) U (0, y ) 1 , a . V (a, y ) V (0, y ),
(A.1)
where U and V are the displacement components in the x and y directions, respectively.
is the global nominal strain that is applied to the structure. For the unit cell, its deformed state is also geometrically periodic, which means for a pair of points on the left and right boundaries, their displacements in the y direction must be the same, while the difference between their displacements in the x direction equals a . To apply the above boundary conditions to the unit cell, pointwise multi-point constraints are created to constrain the left boundary nodes with their corresponding right boundary nodes, with the help of scripts written in Python [40,41]. When studying the interface properties, a supercell or finite structure that contains a finite number of unit cells is considered. To simulate the pre-stretch, periodic boundary conditions (for supercell) or global boundary conditions (for finite structure) are applied using the exact same method.
A.2 Dispersion relations and band gaps
The transmissibility of waves in periodic structures can be studied with dispersion relation analysis. In theory, for a unit cell (or supercell) of the periodic structure, a base vector a1 is used
21
to describe its periodicity (for 2D or 3D structures, it’s a set of base vectors, ai ), whose length equals the length of a unit cell. A reciprocal vector b1 is further introduced to describe the wave number domain, satisfying
a1 b1 2 . The Bloch condition states that,
waves that propagate
inside a 1D periodic structure satisfy equation
u( x0 + r ) u( x0 )ei ( k r ) ,
(A.2)
where u is displacement amplitude, and k, in the reciprocal domain, is the wave number. Let x0 represent a point on the left side boundary, and let r=a, Eq. (A.2) becomes a k-related Bloch boundary condition. Let the wave number k go through the irreducible Brillouin zone (IBZ,
k [0,b1 / 2] ), and by substituting the boundary conditions into the wave equation, the dispersion relation in IBZ can be calculated as an eigen frequency problem. In FEM simulation using ABAQUS, two instances are created, representing the real part and imaginary part of the wave equation respectively, and the Bloch boundary condition can be applied to the boundaries of these two instances accordingly
[42]
. The whole processes are coded into Python-based scripts
using the secondary development function provided by ABAQUS. This method can be used to calculate dispersion relations for the unit cell (or supercell) with or without pre-stretch. When pre-stretch is considered, the states of deformation and stress should be imported as initial states from the previously calculated results.
A.3 Calculations of transmission spectra
Analyzing the transmission spectra of finite cells can be used to verify the results of dispersion relations, such as band gaps and interface state modes. It also can be used to detect edge modes which can not be seen in the dispersion relations, since they are only properties of a finite structure. A basic schematic of a finite structure is shown in Figure A1. It can be constructed by one class of unit cell, or by arrangement of several classes of unit cells. The black dashed line in the center represents an interface between the two different unit cells. A force stimulus is applied at the left boundary of the structure, and waves propagate from the left to the right. Steady-state dynamics analysis step in ABAQUS is used with the frequency sweeping through the range of interest. Then 22
the displacement amplitudes at the three detection lines (marked in red lines in Figure A1) are extracted to compute the transmission spectra. The transmission spectrum of waves propagating from the left to the center is calculated by comparing the average displacement amplitudes at the central and the left detection lines as:
mean(uc ) Tc =log , mean(ul )
(A.3)
and that of waves propagating from the left to the right is calculated by comparing the average displacement amplitudes at the right and left detection lines:
mean(ur ) Tr =log , mean(ul )
(A.4)
where ul , uc , and ur are points in the left, central, and right detection lines, respectively.
mean( ) is the average function. Due to the existence of interface state or edge state phenomena, the transmission properties of Tc and Tr can be substantially different.
Fig. A1. Schematic of a finite structure for transmission analysis.
Appendix B
For a specific band gap, the topological state is determined by the summation of Zak phases of all the frequency modes below this band gap (mod by 2 ). In Figure 3, we show the dispersion relations of the unit cell around its first band gap. The dispersion relations in the whole frequency range below 1800 Hz are shown in Figures B1(a) and (b), when d equals -0.3mm and 0.3mm. The frequency modes in blue (red) have Zak phases with value 0 ( ). The topological state of the first-order band gap is determined by the first three 23
frequency modes below it. It equals 0 if there are two frequency modes with the Zak phases being
, and equals if there is only one. As d increases from negative to positive, the edge frequency modes around the band gap swap with each other, causing the Zak phase of the lower frequency mode to shift from 0 to , further causing the topological state of the band gap to shift from
to 0. Figure B1(c) further demonstrates the analyzing of the Zak phase of a frequency
mode. When the unit cell of the structure has inversion symmetry, the Zak phase of a frequency mode can either be 0 or , depending on the symmetry of the corresponding displacement fields [16, 17]
. For example, the four displacement vector plots at the four points labeled in Figure B1(a) are
shown in Figure B1(c), where the length and direction of an arrow represent the amplitude and direction of the displacement at the corresponding point. Plot 1 and Plot 3 share the same symmetry of the displacement field, so the Zak phase of the corresponding frequency mode is 0; while Plot 2 and Plot 4 have different symmetries of the displacement field, and the Zak phase becomes . More detailed analysis of the Zak phase can be found in Refs. [14-17].
Fig. B1. (a)-(b) Dispersion relations of unit cells with d being -0.3mm and 0.3mm. The Zak phase of the frequency mode in blue (red) is 0 ( ). (c) Displacement vector plots at the four points labeled in (a), indicating a bending characteristic of these two frequency modes.
For reference, in Figures B2 and B3, we show the Zak phase of each frequency mode and the topological states of band gaps in those four unit cells ( C1 , C 2 , C1 , and C2 ) appearing in this paper, before and after pre-stretch. When the two PnC plates with different unit cells connected to 24
each other, the existence of interface state is dependent on the topological states of the overlapped band gaps: if these two band gaps have the same topological state, there is no interface state, while if they have different topological states, one interface state is guaranteed. For the case of C1 + C 2 , initially, without pre-stretch, the band gaps in C1 and C 2 have topological states of 0 and
respectively. One single interface state therefore exists. After pre-stretch, the topological state of the band gap in C1 transfers from 0 to
, and as a result, the interface state disappears. For the
case of C1 + C2 , initially both band gaps have the topological state of 0, while after pre-stretch, the topological state of the band gap in C2 transfers from 0 to
. So the system has no
interface state at first, but one will emerge due to pre-stretch.
Fig. B2. Dispersion relations and band gaps of unit cells C1 and C 2 , before and after pre-stretch.
25
Fig. B3. Dispersion relations and band gaps of unit cells C1 and C2 , before and after pre-stretch.
26
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Figure caption list Fig. 1. Schematics of the phononic system: (a) The system contains two periodic thick plates connected to each other; (b) The cross section of a single unit cell. Fig. 2. A cross section view of the whole system, containing N Cell 1 units and N Cell 2 units. At the center is the interface between the two periodic plates. Fig. 3. (a) Edge frequency modes around the first band gap as a function of the variable d. Red and blue lines represent Zak phases of and 0, respectively. The insets show the contour plots of the edge frequency modes. (b)-(d) Dispersion relations around the first band gap, for d being -0.3mm, 0, and 0.3mm, respectively. Fig. 4. (a) Dispersion relation and transmission spectrum of C1 around the first band gap. The insets show the contour plots of the edge frequency modes at X. (b) Dispersion relation and transmission spectrum of C2 around the first band gap. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, in the same frequency range. Red line in the transmission spectra shows waves transmitting from the left side to the interface, while blue line shows that from the left side to the right side. (d) Displacement amplitude of the interface state indicated in the dispersion relation of the supercell. Fig. 5. (a) Dispersion relation of C2. (b)-(d) Transmission spectra of finite structures containing various numbers of unit cells C 2: 200 in (b), 100 in (c) and 50 in (d). (e) Displacement amplitude of the edge mode for the case of 50 C2. Fig. 6. (a) Dispersion relation of a supercell containing 100 C1 and 100 C2. (b)-(c) Transmission spectra of finite structures with different arrangements of unit cells: 50C 1+100C2+50C1 in (b), and 50C2+100C1+50C2 in (c). Fig. 7. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets. The
black line inside the band gap in (b) shows the evolution of the edge mode in C2 as
varies.
Fig. 8. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.142 for C1, and =0.159 for C2. (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2, with
=0.15. 31
Fig. 9. Edge frequency modes around the first band gaps of C1 (a) and C2 (b), respectively, as a function of the effective nominal strain
. Edge frequency modes are shown by the insets.
Fig. 10. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 . (c) Dispersion relation and transmission spectra of a supercell containing 100 C1 and 100 C2 . Fig. 11. (a)-(b) Dispersion relations and transmission spectra of C1 and C2 under pre-stretch, with
=0.17 for C1 , and =0.19 for C2 . (c) Dispersion relation and transmission spectra of
a supercell containing 100 C1 and 100 C2 , with
=0.18.
Fig. A1. Schematic of a finite structure for transmission analysis. Fig. B1. (a)-(b) Dispersion relations of unit cells with d being -0.3mm and 0.3mm. The Zak phase of the frequency mode in blue (red) is 0 ( ). (c) Displacement vector plots at the four points labeled in (a), indicating a bending characteristic of these two frequency modes. Fig. B2. Dispersion relations and band gaps of unit cells C1 and C 2 , before and after pre-stretch. Fig. B3. Dispersion relations and band gaps of unit cells C1 and C2 , before and after pre-stretch.
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Graphical abstract
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