Kirigami-based Elastic Metamaterials with Anisotropic Mass Density for Subwavelength Flexural Wave Control

A novel design of an elastic metamaterial with anisotropic mass density is proposed to manipulate flexural waves at a subwavelength scale. The three-dimensional metamaterial is inspired by kirigami, which can be easily manufactured by cutting and folding a thin metallic plate. By attaching the resonant kirigami structures periodically on the top of a host plate, a metamaterial plate can be constructed without any perforation that degrades the strength of the pristine plate. An analytical model is developed to understand the working mechanism of the proposed elastic metamaterial and the dispersion curves are calculated by using an extended plane wave expansion method. As a result, we verify an anisotropic effective mass density stemming from the coupling between the local resonance of the kirigami cells and the global flexural wave propagations in the host plate. Finally, numerical simulations on the directional flexural wave propagation in a two-dimensional array of kirigami metamaterial as well as super-resolution imaging through an elastic hyperlens are conducted to demonstrate the subwavelength-scale flexural wave control abilities. The proposed kirigami-based metamaterial has the advantages of no-perforation design and subwavelength flexural wave manipulation capability, which can be highly useful for engineering applications including non-destructive evaluations and structural health monitoring.


Results
Design of kirigami-based elastic metamaterial. Figure 1 shows the schematic of the proposed kirigami cell design. We first cut a 0.4 mm-thick aluminum (Al) thin plate (Fig. 1a) and then fold it along the dashed lines ( Fig. 1b) to form a 3D structure (Fig. 1c). The 3D Al structure is attached onto a 1 mm-thick Al substrate with a heavy mass block (18 g) glued on its top (Fig. 1d). The lattice constant of the kirigami-plate unit cell is a. The geometrical parameters and the material properties of the kirigami cell is listed in Table 1. It should be noted the shapes and widths of the kirigami legs along the x-and y-directions are made different intentionall. As a result, different resonant frequencies can be expected for the flexural wave propagations along the two principle directions of the EMM. The legs with a larger width and a U-shaped cross section possess larger bending stiffness and therefore, introduce high resonant frequency along the y-direction. On the contrary, the other two legs with a smaller width and a flat-rectangle cross section have smaller bending stiffness and therefore, reduce the x-directional resonant frequency. It should be mentioned that the size of the mass attached on top of the kirigami structure can be designed to lower both x-and y-directional resonant frequencies into a desired frequency regime in order to ensure that the EMM unit cell vibrates in a subwavelength scale.
The effective mass density of the proposed kirigami EMM can be calculated by using a computational-based effective medium method 34 . By applying out-of-plane displacement excitations W x and W y along the x and y directions, the resultant forces of the metamaterial unit cell along the two directions can be numerically obtained Scientific REPORtS | (2018) 8:483 | DOI: 10.1038/s41598-017-18864-z as F Z,x and F Z,y , respectively. Thus, the AMD tensor composed of the x-and y-directional effective mass densities, ρ eff , x and ρ eff , y , can be readily determined from the following relation 11,34 where ω represents the excitation angular frequency, and V denotes the volume of the unit cell. Figure 2 shows the dynamic ρ eff , x and ρ eff , y of the attached kirigami structure as a function of ω, given the parameters listed in Table 1. Here, ρ eff , x and ρ eff , y values are normalized with the static effective mass density of the EMM unit cell (i.e., ρ eff , x and ρ eff , y at ω = 0). In Fig. 2, it is evident that this kirigami structure exhibits the anisotropy in its effective mass densities, though the general trends of ρ eff , x and ρ eff , y are similar. Again, the discrepancy of these two curves stems from the distinctive leg stiffness of the kirigami structure. Also, it can be found that ρ eff , x and ρ eff , y have different signs in the frequency range from f = 5630 Hz to f = 7150 Hz, which is highlighted in gray. This is an interesting feature which will eventually enable us to achieve the directional propagation of flexural waves. Lastly, in this shaded frequency range, the wavelength of the A0-mode flexural wave propagating in the thin substrate structure is λ = approximatley 41mm at 6 kHz. This wavelength corresponds to almost four times of the characteristic size of the kirigami cell represented by the lattice constant (a = 10 mm in this study). This manifests the subwavelength nature of the designed kirigami cell. We will verify these unique features of the kirigami-based EMM in the following part of the manuscript.
A mass-spring-plate model was also developed to analyze the band structures of the kirigami metamaterial, as shown in Fig. 3a. Consider an infinitely thin plate lying in the x-y plane with periodically attached resonators consisting of two pairs of springs k x and k y and a top mass m R , as shown in Fig. 3b. Here, a is the lattice constant of the unit cell, and a x and a y are the distances between the two points where the springs with coefficients k x and k y are attached along the x-and y-directions, respectively. Inside the unit cell, the four attachment points can be denoted in vector forms with the origin being at the center of the unit cell, as shown in Fig. 3c:  The detailed derivations of the governing equations and the extended plane wave expansion method for band structure calculation are in the Methods section. The band structures along both x-and y-directions are calculated with the two spring constants being determined by conducting finite element-based static analysis on the kirigami legs. As a result, k x = 9.5 MPa and k y = 21 MPa are retrieved from the force-displacement curves, as shown in Fig. 4a and b. The band structures obtained from the mass-spring-plate model are plotted in Fig. 4c alongside with the effective mass densities. It can be found that the frequency difference between the x-directional and the y-directional band gap (light gray shaded area) overlaps with the frequency range where ρ eff , x and ρ eff , y , have different signs (light yellow shaded area).
Note that it is not only the difference between the two spring constants k x and k y , but also the difference between the a x and a y that plays an important role in creating the AMD in the metamaterial unit cell. In our  proposed kirigami structure, the difference between k x and k y indicates different geometric shapes and thicknesses of the kirigami legs, while the difference between a x and a y represents variations in bending angles and leg-attached positions on the bottom plate. Figure 5 shows the effect of different parameter ratios (k x /k y and a x /a y ) on the band structure of the anisotropic kirigami metamaterial. Here, Δω * is the width of the strong-AMD frequency band (i.e., highlighted frequency band in Fig. 4c) where ρ eff , x and ρ eff , y have opposite signs which is the critical condition to realize hyperlens. It can be found that Δω * increases dramatically when either k x /k y or a x /a y starts to increase from a unity. However, when the difference between the spring coefficients or their attachment  locations is large enough, Δω * increases very slowly. It is also notice that increasing a x /a y is more efficient in creating large strong-AMD region with Δω * being up to 0.22 while increasing k x /k y brings a slower change in Δω * and a relatively smaller strong-AMD region with Δω * being up to 0.14.
Directional flexural wave propagation. Now we demonstrate numerically that the proposed kirigami-based EMM can propagate flexural waves directionally by leveraging the anisotropic effective mass density explained in the previous section. Figure 6a and b show full-scale simulation results of elastic waves propagating in a regular thin plate and a kirigami-decorated plate, respectively (software: COMSOL). In both cases, a pair of piezoelectric disks (PZT-5H) are surface-bonded at the center of the plate, with one on the top and the other on the bottom of the plate. Same amplitude but opposite signs of the voltage excitations are applied to the PZT pair for the asymmetric deformation, which excites purely flexural wave (A0-mode Lamb wave) in the plate 35 . Perfectly matched layers (PMLs) are built around the plate region to eliminate any unwanted boundary reflections. Without the kirigami array attached, the flexural wave (f = 6.15 kHz) propagates isotopically in the thin Al plate, as shown in Fig. 6a. With the kirigami array attached around the PZT disk on the top surface of the plate, the flexural wave propagates only along the y-direction in the bottom plate, as shown in Fig. 6b. This is because in the formed metamaterial plate as shown in Fig. 2. That is, the positive effective mass density supports the propagation of waves, while the negative one forms evanescent waves that decay exponentially in space. We again note in passing that the characteristic length of the kirigami cell (10 mm  Demonstration of flexural wave hyperlens. Elastic hyperlens, designed by the EMM with the mass density being positive in one direction while negative in the orthogonal direction, can not only permit the transmission of the evanescent waves carrying the subwavelength information to the near field, but also magnify it, thereby converting it to propagating waves in the far field 36,37 . Although previous acoustic and elastic/acoustic hyperlenses have been demonstrated for the subwavelength pressure waves 13 and in-plane longitudinal elastic waves 9-11 , respectively, the hyperlens for flexural wave has not been fully explored due to the lack of proper subwavelength architectures that can lead to the required AMD. Our proposed kirigami structure can not only generate the AMD for flexural waves but also provide a surface-attached hyperlens solution that can be the first step towards the development of a scanning super-resolution damage imaging system. Figure 7 shows the design of the attachable elastic hyperlens with the proposed kirigami structures. The 180° sector-shaped hyperlens consists of 115 kirigami units, which are arranged in seven layers (the numbers of the units in each layer are marked in Fig. 7a). The inner and outer radii of the hyperlens are 18 mm and 106 mm, respectively, which indicates a magnification ratio of six for the enlarged subwavelength feature. The distance between neighboring unit cells along the circumferential direction is the same as that along the radial direction. To demonstrate its super-resolution imaging ability for flexural waves, two pairs of 2 mm-radius piezo disks are bonded to the top and bottom surfaces of the plate with their voltage excitations tuned for dominated flexural wave generation. The distance between the two disks on the top surface of the plate is 13 mm, which is less than 1/3 λ at 6.15 kHz. Thus, this subwavelength separation of the excitation sources cannot be detected in the regular thin Al plate without attached kirigami hyperlens (Fig. 7b). Figure 7c shows the simulation result of the kirigami-decorated plate, given the same excitation condition described above. Here, the distance between the two sources and the first layer of the hyperlens is 1.8 mm, which is kept small enough for the lens to pick up the evanescent wave that carries the subwavelength feature information. Thanks to AMD, the evanescent wave reaching the inner array of the kirigami cells is converted to the propagating wave that is transmitted the outer region along the radial direction, as shown in Fig. 7c. Therefore, the subwavelength information of the two wave sources can be channeled out to the outer boundary of the hyperlens without significant attenuation. Meanwhile, the distance between the two wave sources is enlarged six times as defined by the magnification ratio, which enables the far field super-resolution imaging since the enlarged feature is now larger than the diffraction limit. Note that this super-resolution capability is achieved only in the frequency range of the AMD. For example, Fig. 7d shows the simulation results at f = 8 kHz where both ρ eff , x and ρ eff , y , have negative values. Therefore, both propagating and evanescent flexural waves generated by the two wave sources are totally blocked by the attached kirigami hyperlens.

Discussion
In this paper, an attachable elastic mechanical metamaterial (EMM) design has been proposed based on kirigami to control subwavelength flexural waves propagating in a thin substrate structure. By strategic design of the kirigami cells, this EMM structure can exhibit the anisotropic mass density (AMD). We calculated the AMD of the kirigami EMM by the numerical-based effective medium method. We also obtained the frequency band structure of the anisotropic flexural wave propagation by using a lumped mass-spring-plate model. Good agreement in the strong-AMD frequency region can be found between the results of the numerical method and the analytical model. Based on this, the directional flexural wave propagation and the concept of the attachable super-resolution elastic hyperlens were demonstrated numerically. The proposed kirigami-based EMM with the AMD has the advantages of no-perforation design and potentially attachable solution for the subwavelength flexural wave manipulations, which can be highly useful features for engineering applications in the fields of non-destructive evaluations and SHM. Moreover, nonlinearity including large deformation or even bucking of the kirigami legs can be easily introduced, which could lead to tunable elastic hyperlens.

Methods
Extended plane wave expansion method for mass-spring-plate model. By using the Kirchhoff plate theory and the Dirac delta function to couple the flexural wave motion of the bottom plate and the vertical vibrations of the attached resonators, the governing equation of the metamaterial plate can be written as To form an eigenvalue problem from governing equation Eq. (3) and obtain the band structure of the metamaterial plate, the plane wave expansion method 38 is extended to deal with the continuous plate system with discretely positioned spring-mass resonant elements. The out-of-plane displacement W of the bottom plate can be assumed as where W 0 is the corresponding Fourier coefficients of the displacement W, → k is the flexural wave vector and → G is the reciprocal-lattice vector as: x y a (0, 2 / ) y y being the basis vectors of the reciprocal lattice, ∼ m and  n being an integer. Here, a x and a y are the distances between the spring attachment points along the x-and y-directions, respectively (Fig. 3c). By applying the periodic condition:    9) can eventually be represented in a matrix form as: