Tunable guided waves in a soft phononic crystal with a line defect

Phononic crystals (PnCs) involve perfect translational symmetry or periodicity and exhibit many anomalous properties.^1–3 They have aroused intense research interest in the past few decades. When a defect, such as a point or a line defect generated by removing or changing one or an array of scatterers, is introduced in a perfect PnC, the translational symmetry will be broken and localized states may occur.^1–3 Defects in the realm of PnCs⁴ have been proven to be fruitful in the aspects of guiding waves,⁵ harvesting energy,⁶ and trapping waves.⁷ To design the PnC devices with defect states to meet different working conditions, it seems very important to make them tunable.^8–10 Recent years have witnessed a lot of PnCs with tunable properties.¹¹ As one kind of tunable PnC, soft PnCs, which can undergo finite and reversible elastic deformation due to the hyperelasticity of soft materials, have received considerable attention.^12–14 Under tensile^15,16 or compressive¹⁷ deformation, the geometrical configuration and the instantaneous stiffness of soft PnCs can be altered effectively, which further affects bandgap properties.

If the defected structure comprises a part made of soft materials with hyperelasticity, it may undergo finite and reversible elastic deformation as well. In this regard, new possibilities for designing tunable waveguides emerge. Recently, soft porous PnCs with different arrangements of inclusions have been investigated in terms of post-buckling deformation, tunable Poisson’s ratio, and wave propagation.^18–20 Since the inclusions can be inserted into the porous matrix flexibly, it is easy to introduce the defects into the soft porous PnCs by removing/adding inclusions in an appropriate pattern. Then, the defects may be utilized to trigger the local or global instability by subjecting the structure to a certain load (usually compressive in nature). Furthermore, defects and instability could be used, in a positive manner, to alter the characteristics of wave propagation instead of causing the failure of the structure.

The purpose of this study is then to harness the buckling instability and the defect states of a two-dimensional (2D) soft porous periodic structure with hard inclusions to design a tunable elastic waveguide. To this end, we first investigate the wave propagation in such a soft PnC structure without defects under uniaxial compression by numerical and experimental methods. Then, we remove some of the hard inclusions to generate a line defect and explore the buckling instability and tunability of guided waves in the deformed structure.

The designed 2D perfect PnC is shown in Fig. 1(a), which consists of the soft porous matrix with embedded hard inclusions. The porous matrix is made of rubber [the black part in Fig. 1(a)] with cylindrical holes [the white parts in Fig. 1(a)] in a triangular lattice; the hard inclusions are made of steel cylinders and are inserted in every other hole [the gray and orange parts in Fig. 1(a)]. It has been demonstrated numerically and experimentally that the post-buckling deformation of the designed PnC in Fig. 1(a) is robust, and it is appropriate to treat the part enclosed by the red dashed line [as shown in Fig. 1(c)] as the representative volume element (RVE) of the periodic structure when the post-buckling deformation is considered.¹⁹ The width and height of the RVE are denoted as a and b, respectively, and the corresponding first Brillouin zone and irreducible part are shown in Fig. 1(d). Based on the perfect PnC, we design a structure shown in Fig. 1(b) with a line defect by removing an array of hard inclusions [the orange parts in Fig. 1(a)]. In this article, experimental and numerical investigations will be conducted for both the perfect PnC [Figs. 1(a)] and the defected PnC [Fig. 1(b)].

The experiments comprise two parts: compressional buckling and wave propagation tests. The experimental setup is shown in Fig. 2(a), and the defected PnC sample is shown in Fig. 2(b). We manufactured the porous rubber samples with 10 × 12 cylindrical holes through a mold-casting process and inserted the steel cylinders into every other hole very tightly to ensure perfect contact between the cylinders and the rubber. The porosity of the rubber samples is ϕ = 70% with the diameter of the hole d = 8 mm. The thickness of the sample is about 50 mm, which is thick enough such that the assumption of a plane-strain state is valid.

For the compressional buckling and post-buckling deformation tests, the sample was uniaxially compressed along the vertical direction. Lubricant was applied to the surfaces between the loading blocks and rubber samples to reduce the friction effects as less as possible. In the wave propagation test, to facilitate the implementation of the excitation to the sample, a plastic block, which was connected to a steel excitation bar, was glued to the sample [see the part enclosed by the red box on the right side of Fig. 2(b)]. An accelerometer (4517-002/4518-003 Brüel & Kjær) was inserted between the block and the excitation bar to detect the input signal. To record the output signal, another accelerometer (4517-002/4518-003 Brüel & Kjær) was attached to the left end of the sample. The two accelerometers are at the same vertical height. The signal generator (DG4162, RIGOL) generated a white noise in a certain range of frequency (0–1200 Hz) as the input signal, which was amplified by using the power amplifier (2718, Brüel & Kjær) to control the shaker (4809, Brüel & Kjær). Both input and output signals were recorded via a data acquisition module (3050-A-060, Brüel & Kjær). Finally, the transmittance was calculated by $T=20lgAout/Ain$⁠, where A_in and A_out are the acceleration amplitudes of the input and output signals, respectively.

To further explore the characteristics of deformation and wave propagation of the designed PnCs, we utilized the finite element method (FEM) with commercial software ABAQUS and Python scripting to implement the buckling and post-buckling analyses, band structure calculation, and steady-state response (i.e., transmittance) computation. It is important to indicate that we combine the general theory of nonlinear elastic deformation^21,22 with the small-on-large theory²³ to solve the problems of linear wave propagation in a deformed structure and calculate the band structure.²⁴ In FEM simulations, to be specific, we apply periodic boundary conditions to the RVE for the deformation analysis. Then, the stress and strain fields of the deformed structure are introduced into the wave model. According to the method of applying the Bloch boundary conditions to the RVE,^24,25 we establish two instances with identical mesh and material properties to represent the real and imaginary parts of the corresponding fields, respectively. Specifically, for the deformed RVE, we apply the Bloch conditions to the nodes periodically located on the RVE boundary in the FEM simulations. Then, the complex eigenvalue problem can be solved. For details, the reader is referred to Refs. 24 and 25.

As noticed earlier, the plane-strain state is assumed, and the element types CPE6H and CPE6 in ABAQUS are adopted for the rubber matrix and steel inclusions, respectively. Perfect contact with the continuous displacement and traction is considered between the rubber matrix and the inserted steel cylinders.

To describe the hyperelasticity of the rubber, we used the neo-Hookean model with the corresponding strain energy density function given by¹⁹ 

where $Ī1$ is the first deviatoric strain invariant, J is the local volume change, and μ_m0 and K_m0 are the initial shear and bulk moduli, respectively. For the rubber of the samples, μ_m0 = 0.5 MPa, K_m0 = 50 MPa, and the density ρ_m0 = 1300 kg/m³. In addition, damping in soft materials plays a significant role in wave propagation. Hence, we consider the damping in the rubber whose loss factor is 0.061 at room temperature (25 °C) when conducting the frequency response analysis.¹⁹ However, because of the difficulty in implementation of complex band structure calculation for finitely deformed elastic bodies in ABAQUS, we do not consider the viscoelasticity of rubber materials when calculating the band structure. Young’s modulus of the steel is E_s = 194.02 GPa, the density is ρ_s = 7930 kg/m³, and Poisson’s ratio is ν_s = 0.3. Here, we focus on the uniaxial compression in the vertical direction. The displacement load is considered in the FEM simulations as in the experiments and is defined as λ = −ɛ_yy, where ɛ_yy is the nominal strain in the vertical direction.

We first pay our attention to the wave propagation in the structure without defects, i.e., the structure depicted in Fig. 1(a). To calculate the band structures, we focus on the RVE shown in Fig. 1(c) and scan the wave vector along the path M-G-X-M-Y-G in the irreducible Brillouin zone (BZ) [see Fig. 1(d)]. Furthermore, the transmission spectra along the G-X direction are calculated with consideration of the damping effect by applying the same excitation to the finite structure as what we did in the experiments. It has been numerically demonstrated in our previous study that the applied loads can effectively tune the bandgaps.¹⁹ Here, we experimentally verify this phenomenon. Figure 3(a) exhibits the band structure of the undeformed sample without defects and the corresponding transmittance obtained from experiments and simulations. It is clearly shown that there is a broad and low-frequency complete bandgap from 314 to 1143 Hz. The numerical and experimental results show a good agreement, which further validates our calculations. When the structure undergoes finite deformation, e.g., in Fig. 3(b) where the displacement load is λ = 0.1, the bandgap found in the deformed structure is narrowed to 380–1067 Hz. In addition, a narrow complete bandgap appears in the relatively high frequency region from 1123 to 1185 Hz, and there is a directional bandgap in the G-X direction at lower frequencies, which can be further confirmed by the transmission spectra obtained from the experiment and the numerical simulation except for some inevitable errors caused by some reasons (e.g., the experimental results and the FEM simulations for transmittance spectra are based on the finite structure, but the band structures are based on the infinite structure; we neglect the viscoelasticity of the rubber when calculating the band structures; we adopt point excitation in the frequency domain analyses, which may not excite the pure plane waves as assumed in the model). For the structure without embedded hard inclusions, as can be seen in Figs. 3(c) and 3(d), there is no bandgap even at higher frequencies whether it is subjected to a displacement load [Figs. 3(d)] or not [Figs. 3(c)]. In addition, in our previous work, we have summarized the variation of the bandgap with the applied load for the structure with hard inclusions, which indicates that there is a broad bandgap covering the frequency range of 400–1000 Hz as long as the applied load λ is less than 0.1.¹⁹ Therefore, the comparison demonstrates the advantages of the structure containing hard inclusions, that is, a very wide bandgap can be generated, which provides the possibility to design elastic waveguides. Only in the frequency range of this wide bandgap can the vibration be localized at the line defect, which is of critical importance for the subsequent design of tunable waveguides.

Next, we turn our attention to the deformation and wave characteristics of the defected PnC [i.e., the structure with a line defect shown in Fig. 1(b)]. Figure 4(a) shows the experimental result for the defected structure under uniaxial compression with λ = 0.1, which has an excellent agreement with the numerical simulation exhibited in Fig. 4(b). It is notable that the deformation fashion of the parts enclosed by the upper and lower red boxes in Fig. 4(b) is almost the same as that of the structure shown in Fig. 1(a) under uniaxial compression (cf. Fig. 10 in Ref. 19). In addition, the deformation in the vertical direction is not homogenous due to the existence of the line defect. The large deformation is prone to localizing at the part enclosed by the blue dashed box in Fig. 4(b) (which contains the line defect), which implies that the nominal strain of the parts enclosed by the red boxes in Fig. 1(c) is smaller than 0.1 or, equivalently, the nominal strain of the defect part is larger than 0.1. In fact, the deformation pattern of the defect part is similar to that of the structure without inclusions (i.e., the porous matrix) under uniaxial compression; see Fig. 9 in Ref. 19.

To analyze the wave behavior in the structure with a line defect, we calculate the dispersion relations of the supercells [see the right insets shown in Figs. 5(a) and 5(b)]. Here, only the wave propagation in the horizontal direction is considered. Therefore, the Bloch condition is applied to the left and right boundaries of the supercell, and the periodic boundary condition is applied to the top and bottom boundaries. Since for the structure without the line defect there is a broad bandgap covering 400–1000 Hz as long as the applied load λ does not exceed 0.1,¹⁹ here, we mainly show the band structure in this frequency region. It is demonstrated from Figs. 5(a) and 5(b) that many defect bands appear in the bandgap of the perfect PnCs. These defect bands correspond to the guided wave modes, which are localized at and propagate along the line defect. Examples of these localized mode maps are shown in Figs. 5(c) and 5(d) for points K₁ and K₂ marked in Figs. 5(a) and 5(b), respectively. It is also noticed that some bandgaps for the guided wave modes appear and present remarkable changes with the load applied. For instance, we can find a bandgap (directional) in the frequency range of 655–786 Hz in Fig. 5(a), which can be verified by the transmission spectra. When the uniaxial compression with λ = 0.1 is applied, this bandgap disappears, and two other bandgaps emerge in the lower frequency regions; see Fig. 5(b).

To further demonstrate the tunability of the guided wave properties by applying external loads, we plot the displacement distributions (at the same frequency of 440 Hz) of the finite structure in Figs. 5(e) and 5(f) under two states: undeformed (λ = 0) and deformed with λ = 0.1. For the undeformed structure, on the one hand, 440 Hz is on a passband of the guided wave mode, and thus, the elastic waves can propagate along the line defect; see Fig. 5(e). On the other hand, 440 Hz is in the range of the guided mode bandgap of the deformed structure, and hence, the wave propagation is suppressed at this frequency, and the waves cannot be guided by the line defect.

Furthermore, we investigate the dispersion relations of the structures under different applied loads to analyze the variation trend of the guided mode bandgaps in Fig. 6. As the applied load increases, we find that some of the bands no longer intersect with each other and the bands tend to be flat, which implies that the group velocity is slowed down gradually. For example, with the increase in the applied load, the red band gradually moves up to generate a bandgap. In addition, the blue band goes down and becomes flat and the green band becomes flat as well, which forms the bandgaps gradually.

The above results indicate that whether the elastic waves in a certain frequency range can propagate along the line defect depends on the magnitude of the applied load (or, equivalently, the deformation). In addition, the applied load can modulate the speed and vibration mode of the guided wave. This provides us an effective way to design tunable waveguides.

In conclusion, a tunable waveguide has been designed numerically and experimentally by introducing a line defect into a soft PnC structure with hard inclusions and exploiting the finite deformation of the soft porous matrix. First, we have experimentally demonstrated the effectiveness of tuning bandgaps by the post-buckling deformation of the perfectly periodic structure comprising the soft porous matrix and hard inclusions. Then, the static and dynamic localized states have been found by introducing a line defect in the structure, that is, the deformation tends to localize at the defect (static localized state) and the vibration is prone to localizing at the defect as well (dynamic localized state). From the experimental observations and numerical simulations, we conclude that utilizing a line defect and instability of a soft PnC could be an effective approach to the tuning of the static and dynamic responses of the defected PnC. Since the proposed method is readily applicable to design the elastic waveguides to meet different working conditions, we envision it to be a guide to make tunable waveguides in the practical engineering applications.

Y.W. and J.L. contributed equally to this work.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11532001, 12021002, and 12072315). W.C. would also like to acknowledge the support from the Natural Science Foundation of Zhejiang Province (Grant No. LD21A020001).

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