Continuous Tuning of Line Defect Modes in Silicon Two dimensional Phononic Crystal

To realize the frequency tuning of phononic crystals (PCs) and the functional design of tunable PCs, acoustic components with more flexible working frequencies are manufactured to meet the various requirements of engineering applications. We proposed a combined tuning method that combines the change of the Young's modulus of the shape memory alloy and the rotation of the scatterer. The tunable band structure and transmission spectra of the method were calculated using the finite element method. We analyzed the effect of fill rate and viscosity of matrix on the band structure and studied the regulation law of the dual regulation mode. The numerical results show that the double tuning method makes up for the shortcomings of the single tuning method and has the characteristics of widely tuning range, continuous adjustment, and more tuning modes. In addition, a PC waveguide is constructed by using this combined tuning method, which realizes the flexible construction of waveguide channels and the continuous tuning of wide frequency range. It is an important guideline for the research of tunable waveguides, the design of acoustic components, and the application of practical engineering.

I. INTRODUCTION

Section:

In recent decades, there has been growing interest in the propagation of elastic/acoustic waves in phononic crystals (PCs) and acoustic metamaterials. ^1–5 1. M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys. Rev. Lett. 71, 141 (1993). https://doi.org/10.1103/physrevlett.71.2022 2. M. Kafesaki, M. M. Sigalas, and N. García, Phys. Rev. Lett. 85, 4044 (2000). https://doi.org/10.1103/physrevlett.85.4044 3. T. Miyashita, Meas. Sci. Technol. 16, R47 (2005). https://doi.org/10.1088/0957-0233/16/5/r01 4. Y.-F. Wang, Y.-Z. Wang, B. Wu, W. Chen, and Y.-S. Wang, Appl. Mech. Rev. 72, 040801 (2020). https://doi.org/10.1115/1.4046222 5. Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, Surf. Sci. Rep. 65, 229 (2010). https://doi.org/10.1016/j.surfrep.2010.08.002 When elastic/acoustic waves propagate through PCs, they are affected by the internal periodic structures of the PCs and can form bandgaps that are able to control the propagation of the elastic wave/acoustic waves. The application of this unique property in negative refraction, ^6–8 6. X. N. Liu, G. K. Hu, G. L. Huang, and C. T. Sun, Appl. Phys. Lett. 98, 251907 (2011). https://doi.org/10.1063/1.3597651 7. B. Morvan, A. Tinel, A.-C. Hladky-Hennion, J. Vasseur, and B. Dubus, Appl. Phys. Lett. 96, 101905 (2010). https://doi.org/10.1063/1.3302456 8. R. Zhu, X. N. Liu, G. K. Hu, C. T. Sun, and G. L. Huang, Nat. Commun. 5, 5510 (2014). https://doi.org/10.1038/ncomms6510 filters, ^5,9,10 5. Y. Pennec, J. O. Vasseur, B. Djafari-Rouhani, L. Dobrzyński, and P. A. Deymier, Surf. Sci. Rep. 65, 229 (2010). https://doi.org/10.1016/j.surfrep.2010.08.002 9. S. Benchabane, A. Khelif, J.-Y. Rauch, L. Robert, and V. Laude, Phys. Rev. E 73, 065601(R) (2006). https://doi.org/10.1103/physreve.73.065601 10. T.-T. Wu, L.-C. Wu, and Z.-G. Huang, J. Appl. Phys. 97, 094916 (2005). https://doi.org/10.1063/1.1893209 and waveguides ^11–14 11. F.-C. Hsu, C.-I. Lee, J.-C. Hsu, T.-C. Huang, C.-H. Wang, and P. Chang, Appl. Phys. Lett. 96, 051902 (2010). https://doi.org/10.1063/1.3298643 12. M. Ghasemi Baboly, C. M. Reinke, B. A. Griffin, I. El-Kady, and Z. C. Leseman, Appl. Phys. Lett. 112, 103504 (2018). https://doi.org/10.1063/1.5016380 13. A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, and V. Laude, Appl. Phys. Lett. 84, 4400 (2004). https://doi.org/10.1063/1.1757642 14. J.-E. Wu, R. Hu, B. Tang, X. Wang, H. Jia, K. Deng, Z. He, and H. Zhao, J. Appl. Phys. 125, 215101 (2019). https://doi.org/10.1063/1.5085800 has attracted wide attention. Among them, PC waveguides have been focused on as they can be used to transmit acoustic energy and acoustic information within specific frequency ranges. Waveguide research is generally used to introduce defects in perfect PCs, so that the elastic/acoustic wave can be limited to the defect, or along the defect path of propagation. ^13,15 13. A. Khelif, A. Choujaa, S. Benchabane, B. Djafari-Rouhani, and V. Laude, Appl. Phys. Lett. 84, 4400 (2004). https://doi.org/10.1063/1.1757642 15. X. Zhang, Z. Liu, Y. Liu, and F. Wu, Solid State Commun. 130, 67 (2004). https://doi.org/10.1016/j.ssc.2004.01.007 However, in most cases, the use of PC waveguides is very passive because the corresponding operating frequency is fixed once the structure is designed, which greatly limits the application of PC waveguides.

Researchers have recently begun to focus on the active tuning of PCs and acoustic metamaterials. Casadei et al. ¹⁶ 16. F. Casadei, T. Delpero, A. Bergamini, P. Ermanni, and M. Ruzzene, J. Appl. Phys. 112, 064902 (2012). https://doi.org/10.1063/1.4752468 actively tuned the bandgap using piezoelectric patches with an adjustable stiffness and realized a tunable acoustic waveguide on a two-dimensional plate structure by a piezoelectric vibrator shunted through passive inductive circuits. Shu et al. ¹⁷ 17. H. Shu, W. Liu, S. Li, L. Dong, W. Wang, S. Liu, and D. Zhao, J. Vib. Control 22, 1777 (2016). https://doi.org/10.1177/1077546314544694 used the voltage feedback tuning method to change the equivalent shear stiffness of the piezoelectric ring and then realized the adjustment of the torsional vibration bandgap in the ring-shaped piezoelectric PC. However, it is necessary to accurately tune the negative capacitance of the shunt circuit, because, in practice, there is instability in the active circuit. ¹⁸ 18. R. Zhu, Y. Y. Chen, M. V. Barnhart, G. K. Hu, C. T. Sun, and G. L. Huang, Appl. Phys. Lett. 108, 011905 (2016). https://doi.org/10.1063/1.4939546 Wang et al. ^19,20 19. Y. Wang, T. Wang, Y. Wang, and V. Laude, Phys. Rev. Appl. 8, 014006 (2017). https://doi.org/10.1103/physrevapplied.8.014006 20. Y.-F. Wang, T.-T. Wang, J.-P. Liu, Y.-S. Wang, and V. Laude, Compos. Struct. 206, 588 (2018). https://doi.org/10.1016/j.compstruct.2018.08.088 established a reconfigurable coupled resonant waveguide by adjusting the liquid characteristics to realize the flexible tuning of the elastic wave according to the liquid flow. Furthermore, Jin et al. ²¹ 21. Y. Jin, N. Fernez, Y. Pennec, B. Bonello, R. P. Moiseyenko, S. Hemon, Y. Pan, and B. Djafari-Rouhani, Phys. Rev. B 93, 054109 (2016). https://doi.org/10.1103/physrevb.93.054109 introduced whispering-gallery modes (WGMs). By adjusting the inner radius of the hollow column, the wavelength division in multiplexer devices was designed. de Sousa et al. ²² 22. V. C. de Sousa, D. Tan, C. De Marqui, Jr., and A. Erturk, Appl. Phys. Lett. 113, 143502 (2018). https://doi.org/10.1063/1.5050213 introduced shape-memory alloy (SMA) into acoustic metamaterials, using the change of the Young's modulus in the shape-memory effect to tune the working frequency of the metamaterial beam. However, because the Young's modulus of SMA can remain stable only in the determined phase, the gradual change of the Young's modulus in the intermediate state of the phase transition cannot truly be applied to the experiment. Therefore, the above-mentioned tuning method is also passive and cannot achieve the effect of continuous tuning. In summary, piezoelectric material tuning, mechanical reconfigurable tuning, and intelligent material tuning methods all have their corresponding drawbacks, and the use of a single tuning method is not ideal. Therefore, the combination of multiple tuning methods to achieve functional complementarity needs to be considered.

In this paper, we combined the SMA Young's modulus change with the rotating scatterer to establish a dual control method. First, the effect of the change of the SMA Young's modulus on the tuning range of the PC bandgap under different fill factors was studied. Second, the influence of matrix viscosity on the energy band structure was studied. Finally, a tuning model combining the multi-field coupling reconfiguration of the SMA and the mechanical reconfiguration of the rotating scatterer was established, and its tuning performance was studied. In addition, the defect state characteristics of the double control mode were studied, and the PC waveguide was constructed. Through the supercell method and the finite element method, the transmission characteristics and frequency adjustment of the waveguide were studied.

II. MODEL AND CALCULATION METHOD

Section:

In this study, a two-dimensional two-component solid-fluid PC model was used. The model was composed of a square scatterer "A" material arranged in a square lattice in the matrix "B" material. The scatterer was SMA, and the matrix was water. The unit cell plane is shown in Fig. 1(a) . If the lattice constant is a and the edge length of the scatterer is b, the fill factor f = b ²/a ². This model is for an ideal two-dimensional PC, which is infinitely long in the z-direction and has infinitely many periods in the x- and y-directions.

FIG. 1. (a) Schematic diagram of a PC cell. (b) Supercell model consisting of 1 × 7 cells. The yellow parts are the scatterer and the shape-memory alloy, the red part is the defect scatterer, and the gray part is the water matrix. (c) Bent and (d) flat PC waveguide channels consisting of 19 × 19 cells.

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Many experimental studies on SMAs have shown that the Young's modulus varies with temperature in the phase transition temperature range. The Young's modulus of a complete austenite can reach more than three times that of complete martensite (Table I). Moreover, there is a strong correlation between the Young's modulus and the martensite fraction in the material. SMA has four transformation temperatures that need to especially be considered (ordered from the lowest to the highest temperature): the martensite finish M_f, the martensite start M_s, the austenite start A_s, and the austenite finish A_f. Considering the temperature-dependent phase transformation kinetics of the SMA under low stress, only the low and high temperature phases (the adaptive/twin martensite phase and austenite phase) occur, which are analyzed according to Brinson's model. ²³ 23. L. C. Brinson, J. Intell. Mater. Syst. Struct. 4, 229 (1993). https://doi.org/10.1177/1045389X9300400213

TABLE I. Physical parameters of martensite, austenite, and water.

Material	Temperature (°C)	Density, ρ (kg/m3)	Young's modulus, E (GPa)	Poisson ratio
Martensite	35	6500	30	0.3
Austenite	65	6500	90	0.3
Water	⋯	1000	0.219	⋯

During the transformation from the martensite phase to the austenite phase (the temperature range is M _f ≤ T ≤ M _S ), the proportion of the martensite phase in the material can be expressed as

$$\xi =\frac{1-{\xi }_0}{2}\left[\cos \left(\pi \frac{T-M_f}{M_s-M_f}\right)+1\right],$$

(1)

where ξ is the proportion of the martensite phase in the material, ξ ₀ is the initial value, T is the SMA temperature (°C), and T ₀ is the initial value.

During the transformation from the austenite phase to the martensite phase (the temperature range is A _s ≤ T ≤ A _f ), the proportion of martensite phase in the material can be expressed as

The Young's modulus of the SMA under any state can then be expressed as

where E_A represents the Young's modulus under the austenite phase of the SMA, E_M represents the Young's modulus under the martensite phase, ξ = 1 in the complete martensite phase, and ξ = 0 in the complete austenite phase.

In order to more directly observe the tuning effect of the SMA on the band structure before and after phase transformation, the Young's modulus of the SMA with complete austenite is set to be three times the value of that with complete martensite, ^24–26 24. X.-F. Lv, S.-F. Xu, Z.-L. Huang, and K.-C. Chuang, Phys. Lett. A 384, 126056 (2020). https://doi.org/10.1016/j.physleta.2019.126056 25. M. B. Xu and G. Song, J. Sound Vib. 278, 307 (2004). https://doi.org/10.1016/j.jsv.2003.10.029 26. M. Ruzzene and A. Baz, J. Vib. Acoust. 122, 151 (2000). https://doi.org/10.1115/1.568452 i.e., E _A = 3E _M . Therefore, it is assumed that E _M = 30GPa (35 °C), E _A = 90GPa (65 °C), density ρ = 6500kg/m³, and Poisson's ratio σ = 0.3 (see Table I). The acoustic parameters of the SMA in the two states are calculated by elastic dynamic theory. The dispersion relationship and transmission characteristics of the PCs are studied using the finite element analysis software COMSOL Multiphysics 5.6. In solving the band structure of the PCs, the eigenvalue problem derived from the finite element equation can be obtained,

$$-\frac{{\omega }^2(k)P\left(r\right)}{\rho \left(r\right)c^2(r)}=\nabla \cdot \left[\frac{1}{\rho \left(r\right)}\nabla P\left(r\right)\right],$$

(4)

where ω denotes the eigenfrequency, k denotes the wave vector, $P\left(r\right)$ denotes the sound pressure, $\rho \left(r\right)$ denotes the density, $c\left(r\right)$ denotes the sound velocity, and $\vec{r}$ denotes the position vector. When the martensite change in SMA occurs, the change in Young's modulus causes a change in the speed of sound, thereby moving the energy band of the PCs. Using the k-scan of the irreducible Brillouin zone, the dispersion relationship of the PC can be obtained to describe the band structure.

When solving the transmission characteristics of PCs, a finite periodic structure of 1 × 10 is established along the x-direction. In the y-direction, Bloch periodic boundary conditions are applied on both sides of the model, and the wave vector k is scanned along the $\mathrm{\Gamma }$ -X direction. A Perfectly Matched Layer (PML) is applied to the right end of the x-direction of the finite structure to prevent energy reflection. On the left side of the model, the acoustic line source with a unit amplitude (p ₀ = 1) of lattice constant a is applied and propagates along the x-direction. The energy transfer loss T is defined as T = 20 log₁₀(X _OU /X _IN ), where X _OU and X _IN denote the output and input power, respectively. Similar calculation methods are used to solve the transmission loss of the PC waveguide.

III. RESULTS AND DISCUSSION

Section:

A. Effect of fill factor on the regulation range of the Young's modulus variation

To screen out the model with the optimal regulation range, the regulation range of the band structure with the change of the Young's modulus was studied under different fill factors. Nine sets of models (f = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9) were set up under the two states of complete martensite and complete austenite, and the lattice constant a = 20 mm. It should be emphasized that the geometric size of the PCs can be arbitrarily amplified and reduced without considering the correlation loss. To observe the relationship between the bandgap and the material and structural parameters more directly, the normalized frequency (Ω = ωa/2πc) was used to represent the calculation results.

Through the finite element analysis of the PCs with different fill factors f, it was found that the full bandgap is only opened in the complete martensite state when the fill factor is 0.4–0.7, as shown in Fig. 2(a) . The starting, cutoff, and central frequencies decrease with the increase of the fill factor. To observe the relationship between the fill factor and the tuning range of the Young's modulus change more clearly (and take into account the value of a low frequency bandgap for practical engineering), the directional bandgap (called the first directional bandgap) between the first and second band curves in the $\mathrm{\Gamma }$ -X direction was studied, and the regulation principle of the change of the Young's modulus on the directional bandgap was analyzed.

FIG. 2. Changes in (a) the complete martensite bandgap, (b) the bandgap in the $\mathrm{\Gamma }$ -X direction, and (c) the relative width as a function of fill factor. M, martensite; A, austenite.

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As shown in Fig. 2(b) , in the case of complete martensite, the starting frequency of the bandgap in the first direction decreases rapidly and then slowly increases with the increase of the fill factor. The cutoff frequency first increases and then decreases with the increase of fill factor. Therefore, the bandgap position in the first direction moves to a lower frequency with the increase of fill factor in the complete martensite state. The change rule of the complete austenite is approximately the same. By comparing and analyzing the variation of the first directional bandgap with the fill factor in the two states, it is found that the first directional bandgap starting frequency of complete austenite is slightly higher than that of complete martensite. The cutoff frequency of complete austenite is much higher than that of complete martensite. Furthermore, the central frequency of complete austenite is always higher than that of complete martensite. Therefore, the regulation of the change of the Young's modulus will widen the bandgap for complete austenite, which can be observed more clearly in Fig. 2(c) . The relative width $\mathrm{\Delta }\bar{\omega }/{\bar{\omega }}_c$ is an important comprehensive parameter to evaluate the effectiveness of the bandgap ²⁷ 27. M. Sigalas and E. N. Economou, Solid State Commun. 86, 141 (1993). https://doi.org/10.1016/0038-1098(93)90888-t and refers to the ratio of the bandgap width $\mathrm{\Delta }\bar{\omega }$ to the central frequency ${\bar{\omega }}_c$ . The relative width of bandgap was calculated in two states, and it was found that with the increase of the fill factor, the relative width first increases and then decreases. The full bandgap reaches a peak when the fill factor is 0.5. The relative width of the bandgap in the first direction peaked at a fill factor of 0.3 in the complete martensite state. The relative width of the bandgap in the first direction reaches its peak when the fill factor is 0.5 in the complete austenite. Based on these results, the two-dimensional solid–liquid PC model with a fill factor of 0.4 was used for subsequent experiments.

B. Effect of viscosity on band structure

In the study of fill rates, we only consider the adjustment of the Young's modulus to the band and did not consider the viscosity changes of water in different temperatures. In fact, the temperature change of SMA must cause a change in the nearby water temperature. Therefore, the viscosity of water will also change. When the temperature of the water and the temperature of the SMA are consistent, the viscosity of water changes the most. Then, the impact on the band curve is the largest. It is assumed that the temperature of the complete martensite is 35 °C, the temperature of the complete austenite is 65 °C, and the viscosity of water at the corresponding temperature is 0.725 and 0.44mPas.

In order to study the effect of water viscosity on the band structure, we introduced the thermoviscous loss in the calculation of the band structure of PCs of f = 0.4, as shown in Fig. 3 . The band structure has moved in frequency, but very little. The amount of movement is related to the frequency, the higher the frequency, the more the movement. In other words, the viscosity of water affects the frequency of the band structure, but it is very limited in the SMA martensite transformation temperature range. Therefore, in order to simplify the calculation, the viscosity change of water is no longer considered in the subsequent research.

FIG. 3. Band structure calculation considering viscosity. (a) Martensite. (b) Austenite.

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C. Performance analysis of the rotating scatterer tuning method and the combined tuning method

Previous experimental studies ^22,28 22. V. C. de Sousa, D. Tan, C. De Marqui, Jr., and A. Erturk, Appl. Phys. Lett. 113, 143502 (2018). https://doi.org/10.1063/1.5050213 28. C. Sugino, S. Leadenham, M. Ruzzene, and A. Erturk, J. Appl. Phys. 120, 134501 (2016). https://doi.org/10.1063/1.4963648 have found that the Young's modulus of the SMA can remain stable only in the determined phase. Therefore, in the intermediate state of the phase transformation, the gradual change characteristics of the Young's modulus cannot be truly applied to the experiment. Therefore, the continuous tuning of the bandgap of SMA-PCs cannot be realized, and tuning in only two states is not an ideal result. To realize the continuous tuning of the bandgap, the tuning mode of the rotating square scatterer was combined with the tuning mode of the Young's modulus change. Since the rotation of the square scatterer is symmetrical, it was only analyzed under the counterclockwise rotation of 0–45°, as shown in Fig. 1(a) . In this study, we mainly focus on the interrelationship between the two tuning modes to establish a double tuning method to achieve the optimization for a single mode.

The PC models with different rotation angles at a fill factor of 0.4 were simulated and analyzed. It was found that when the rotation angle was 15°, the first bandgap appeared in the complete austenite (the full bandgap between the first band curve and the second band curve), and until the rotation angle was 20°, the first bandgap appeared in the complete martensite, as shown in Fig. 4(a) . With the increase of the rotation angle, the starting frequency and the central frequency of the first bandgap shift toward lower frequencies, and the cutoff frequency shifts toward higher frequencies. The first bandgap becomes as the rotation angle increases and reaches a maximum at $\mathrm{\theta }$ = 45°. These same results were observed by Wu et al. ²⁹ 29. F. Wu, Z. Liu, and Y. Liu, Phys. Rev. E 66, 046628 (2002). https://doi.org/10.1103/physreve.66.046628 Combined with the tuning effect of the Young's modulus change, the rotation angle and the Young's modulus change are consistent in the tuning of the width and the cutoff frequency of the first bandgap, and they promote each other. Although they opposite in the tuning of the starting frequency, the regulation of the rotation angle plays a major role; hence, overall, the first bandgap broadens and shifts toward lower frequencies.

FIG. 4. Changes in (a) the first bandgap, (b) the first directional bandgap, and (c) $\mathrm{\Delta }\bar{\omega }/{\bar{\omega }}_c$ as a function of the rotation angle θ. A, austenite; M, martensite.

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It can be seen from Figs. 4(a) to 4(c) that the first bandgap and the first directional bandgap are widened with the increase of the rotation angle, and the trends in their changes are the same for the two states of complete martensite and complete austenite. In order to analyze the relationship between the two tuning methods more clearly, we have compared and analyzed the start, center, and cutoff frequencies of the first bandgap and the first direction bandgap regulated by Young's modulus. The bandgap of the austenite phase is always higher than that of the martensite phase, and within the range of rotation 0°–45°, the high range is smaller than the bandgap adjustment range. As shown in Fig. 4(c) , the above conclusions are still applicable in the relative width of the bandgap. This shows that the two tuning methods are independent of each other. The bandgap width in the austenite phase is generally better than that in the martensite phase, and when the scatterer rotates by 15°, the first full bandgap opens in the fully austenite state before it does in the martensite phase.

The tuning mode of the rotating scatterer can compensate for the inability of the tuning mode of the Young's modulus change to be continuously tuned and can increase its tuning range. In turn, the tuning mode of the Young's modulus change also increases the tuning range of the rotating scatterers and add richer tuning modes. Both mutually compensate for and promote the tuning of the bandgap. To observe the tuning law more intuitively, the energy band curve and the transmission loss characteristic curve of complete martensite and complete austenite at θ = 0° and θ = 45° are given, as shown in Fig. 5 . It is found that there are some differences between the energy band curve and the transmission loss characteristic curve. The pressure fields at different positions are given (D1, D2, D3, D4, D5, D6, D7, and D8). They were found to be the "deaf band" because it could not be excited due to its symmetry.

FIG. 5. (a) The energy band curve, transmission characteristic curve, and partial pressure fields of complete martensite and complete austenite at (a) 0° rotation and (b) 45° rotation.

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The combination of the two tuning methods provides a research idea for the optimization and tuning of the bandgap. The effects of the two tuning methods on the bandgap are independent of each other. The combination of the two tuning methods realizes the continuous tuning of the Young's modulus change, increases the width of the bandgap, and improves the practical value of acoustic metamaterials.

D. Design of tunable waveguides

Based on the results of the above tuning methods, two tunable PC waveguide models were designed from a functional aspect. The defect mode was introduced into the PC structure to limit the acoustic wave requirement of a full and large bandgap. Therefore, in the unit cell model, rotating 45° scatterers and water were selected. The supercell model was composed of 1 × 7 cells, as shown in Fig. 1(b) . The bent waveguide and the split waveguide are shown in Figs. 1(c) and 1(d) , respectively. Defects were constructed through the change of the Young's modulus and the rotation of the scatterer, and the energy transmission of the waveguide channel was calculated by the finite element method.

It can be seen from Fig. 6(a) that the bandgap of the perfect PC composed of martensite scatterers appears in the range of 692m/s < fa < 909m/s. The bandgap of the perfect PC composed of austenite scatterers appears in the range of 706m/s < fa < 929m/s [Fig. 6(b) ]. The simulation results show that there is only one band curve in the bandgap due to the introduction of defects, which produces a single mode guided wave that can propagate along the path and avoids the interference caused by multiple bands of traditional defects. Further analysis shows that the value of the Young's modulus of perfect PCs determines the range of the bandgap and the position of the defect mode corresponding to the bandgap. The rotation angle of the defect and the value of the Young's modulus can tune the position of the defect mode corresponding to the bandgap, and the tuning range is very ideal. For the perfect PC in the martensite phase, the tuning range of the waveguide frequency can achieve 837 m/s < fa < 921 m/s, and for the perfect PC in the austenite phase, it can achieve 843 m/s < fa < 929 m/s. Defects have little effect on the bandgap boundary. For convenience, the average value of the bandgap starting/cutoff frequencies of the intact PCs in the martensite/austenite phase were considered as the bandgap starting/cutoff frequencies in the band structure.

FIG. 6. Band structures of PC supercells. (a) Complete PCs consisting of scatterers in the martensite phase. (b) Complete PCs consisting of scatterers in an austenite phase. The purple and green solid lines represent that the defect is martensite of 0° and 10° rotation, respectively. The blue and orange solid lines represent that the defect is austenite of 0° and 10° rotation, respectively. The gray area represents the passband range of the corresponding perfect PCs.

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The bent acoustic waveguide channel is designed by selectively changing the Young's modulus of the scatterers and the angle of the rotating scatterers in the PCs, whereby the acoustic transmission loss curve of the corresponding model is calculated. Since the highest/lowest frequency of the energy band curve of the complete PCs and defects are austenite/martensite, the two models with a rotation of 10° have been omitted, i.e., the complete PCs are austenite and the defects are martensite; the complete PCs are martensite and the defects are austenite. For the remaining six supercell models in Fig. 6 , the bent acoustic waveguide channel [shown in Fig. 1(c) ] and the bisected acoustic waveguide channel [shown in Fig. 1(d) ] were mainly constructed. The frequency range of interest was swept to obtain the transmission loss curve of the corresponding model.

It can be seen from Fig. 7 that a defect mode passband appears in the bandgap of the intact PCs, and the position of the passband frequency can be adjusted by means of Young's modulus changes and rotating scatterers. Combined with the results from Fig. 6 , the frequency position of the passband in the transmission loss curve is in good agreement with the defect passband in the band structure. When the acoustic pressure source with a reduced frequency of fa = 856m/s is applied, both the intact PCs and the point defects are the pressure field distribution of martensite, as shown in Fig. 8(a) . The results clearly show that acoustic waves can only propagate along the waveguide channel.

FIG. 7. Transmission loss curves of the bent waveguide when the perfect PCs are (b) austenite and (b) martensite. (c) Transmission loss curves of the bisected acoustic waveguide. Model 1 corresponds to the perfect PCs being martensite, and the point defect is martensite with 0° rotation. Model 2 indicates that the intact PCs are austenite, and the point defect is martensite with 0° rotation.

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FIG. 8. Pressure field distribution in (a) a bent waveguide channel (the perfect PCs are austenite and the point defects are martensite with 0° rotation) and (b) a planar waveguide channel (the perfect PCs are austenite and the point defects are austenite with 0° rotation).

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Fabricating defects by changing the Young's modulus and the rotating scatterers can set waveguides with different shapes and paths. It can be seen from Fig. 7(c) that the two different channels have almost the same energy transmission, indicating that the acoustic wave is evenly divided at the bifurcation. As shown in Fig. 8(b) , when a sound pressure source with a reduced frequency of fa = 895 m/s is applied, the complete PCs are austenite, and the scatterers are the acoustic energy propagation path of the austenite rotated by 0°. To change the waveguide path, we only need to change the Young's modulus of the SMA by temperature or the rotating scatterer. At the same time, the frequency of the waveguide can also be adjusted synchronously.

IV. CONCLUSIONS

Section:

In summary, we have established a double tuning method and applied it to the tunable waveguide. The double tuning method combines the change of the Young's modulus of the SMA and the rotation of the scatterer. It has the advantage of widely tuning range, continuous adjustment, and more tuning modes. Applying this tuning method to the waveguide can flexibly set the waveguide route, and the two tuning methods can play independent roles. This has significance in directing the research of waveguides, the tuning of waveguide frequencies, the design of acoustic components, and practical engineering applications.

Conflict of Interest

No potential conflict of interest was reported by the authors.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

REFERENCES

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