In this example, a 2D AMMs, which consisted of straight and infinite cylinders arranged in the square matrix, as shown in Figure 1, with the lattice constant a = 15.5   mm and radius r = 5   mm , is computed for its band structure. The materials of the cylinders and square matrix are lead and silicone rubber, respectively. The silicone rubber is a kind of viscoelastic material, and its viscous/damping effect is usually considered(Guo et al., 2019; Li et al., 2019). However, the effect of viscous/damping is neglected during the displacement tendency of the modes’ shape. The physical parameters of the lead and silicone rubber are listed in Table 1. The corresponding 3D specimen models with different heights ( h ) are computed for their band structure. These specimen models can be considered as extending forms of the 2D AMM model in z -axis, as shown in Figure 2. The height parameter h is set as a × 10 − 1 , a × 10 − 0.5 , a × 10 0 , a × 10 0.5 , and a × 10 1 , respectively. The boundary conditions are set as described in the aforementioned mathematical models of AMMs.

In addition to the band structure, the transmittance along the Γ   - X direction is calculated by FEM to verify the accuracy of results of band structure. Mathematically, transmittance can be defined by T L = 10   log 10 W o u t W i n (10)

Here, W i n and W o u t are the power of the incident wave and transmitted wave though the structure, respectively, and can be obtained by W i n = ∭ Ω in 1 2 ρ i n v i n 2 d V (11) W o u t = ∭ Ω out 1 2 ρ o u t v o u t 2 d V (12) where, Ω in and Ω out are the integration domains of the incident wave and transmitted wave, ρ i n and ρ o u t are the density of these two domains, v i n and v o u t are wave velocity of these two domains, respectively. The finite structure with 5 unit cells of 2D AMMs is constructed for the transmittance computation in Γ  - X direction, as shown in Figure 6. The incident wave is a plane wave propagating in the x -direction (from left to right in Figure 6). In order to avoid the effect of wave reflection interference, the periodic boundary condition and the perfect matched layers are set in the 2D AMMs finite structure. The right domain is set to calculate the power of transmitted waves passing through 5 unit cells. For the 3D specimen model, the exactitude of their band structures is also substantiated by calculating the transmittance from the similar finite structure of 5 unit cells.

Based on the aforementioned structure, the band structure and the transmittance are computed. The dynamic responses of the system are investigated via FEM using the commercial software COMSOL Multiphysics. For wave propagation problems, using 12 degrees of freedom can generally obtain a solution with an error rate of less than 1%. With the quadratic elements used in FEM models, 6 second-order elements are sufficient for each wave period. Therefore, in order to determine the mesh size, the wavelength can be obtained by λ l = C l f (13) λ t = C t f (14) where, λ l and λ t are wavelength of longitudinal waves and transverse waves, respectively, and f is wave frequency. To ensure the accuracy of the results in the entire frequency range, f should choose the maximum frequency 1200 Hz. The C l and C t are longitudinal speed of elastic wave and transverse speed of elastic wave, respectively, and can be expressed as C l = λ + 2 μ ρ (15) C t = μ ρ (16)

In order to get clarification for the difference in values, the quasi-2D AMMs model is considered. The ideal 2D AMMs models have no displacement along the z-axis direction. For specimen models, they are completely free and can have displacement in any direction. In order to make the connection between the two cases more clearly, a quasi-2D AMMs model as a midpoint is created. The quasi-2D AMMs model is based on the 3D specimen model and is restricted in the displacement along the z-axis direction. Thus, the additional boundary condition is u z = 0 (17) where, u z is the medium displacement along the z -axis direction.

By sweeping the Bloch wave vector along the contour of an irreducible Brillouin zone (Figure 5), the corresponding band structure is obtained. By using different frequencies of the incident wave in the finite structure, the transmittance can be obtained. Figure 7 shows the band structure of the 2D AMMs model, quasi-2D model, and 3D specimen models with the given height of a × 10 − 1 , a × 10 − 0.5 , a × 10 0 , a × 10 0.5 , and a × 10 1 , respectively. The band structure of the quasi-2D model and specimen model are shown on the left and right sides of each subfigure, respectively. The orange bars are used to show clearly the bandgap, as shown in Figure 7.

The band structure of the 2D AMMs (Figure 7A) and quasi-2D model with smaller heights [Figure 7 (b1, c1, d1, e1, f1)] are similar. To evaluate band structure difference between 2D AMMs model and quasi-2D model quantitatively, the error rate of the energy band can be obtained by ϵ i = | f quasi − 2 D − f 2 D | f quasi − 2 D (18) where, ϵ is the error rates, the subscript i represents i ^th energy band, and f represents the Eigen frequency. The Eigen frequency of the 2D AMMs and quasi-2D model are denoted by subscript 2D and quasi-2D, respectively. When the height of quasi-2D model is a × 10 − 1 , the average error rate of the first 20 band structure is ϵ a v e = ∑ i = 1 20 ϵ i / 20 = 0.0062 . According to the average error rate, there is nearly no difference between the 2D AMMs model and quasi-2D model with smaller heights. Figure 7 shows that with the height increases, band structure have greater differences. At the higher frequency, the variation in height is more sensitive. Though the change in band structure is significantly large, but the bandgap showed almost no change.

The band structure of the quasi-2D model [Figure 7 (b1, c1, d1, e1, f1)] and 3D specimen model [Figure 7 (b2, c2, d2, e2, f2)] with different heights showed a significant difference. Even at smaller heights [Figure 7 (b1, b2)], the band structure is significantly different. At a height of a × 10 − 1 , the quasi-2D model has one bandgap while the specimen model has three bandgaps under 1200 Hz. The band structure of the specimen model is more complicated. With increasing height, the quasi-2D model exhibited only one bandgap, however, the bandgap of the 3D specimen model is reduced from three to one. For the 3D specimen model, the high frequency of the band structure is more sensitive to create a variation in the height. This is similar to the quasi-2D model. In the frequency range below 300   Hz , the band structures are indeed different, however, subtle variation is observed. In the frequency range from 300   Hz to 1200   Hz , all the 3D specimen models with different heights showed more energy bands as compared to the 2D AMMs model. In this frequency range, the higher height of the 3D specimen model resulted in a large number of energy bands. With increasing height in the 3D specimen model, the range of the complete bandgap becomes smaller. The overall bandgap edge frequency of the 3D specimen model is lower than the 2D AMMs’.

In order to understand the physical mechanism due to the variation in the band structure, the mode shapes at the edges of the first bandgap are presented in Figure 8. By comparing the 2D AMMs and quasi-2D models, the obtained mode shape at the edges of the first bandgap are similar [Figure 8 (a1–f1), (a2–f2)]. The mode shapes of both upper edges and lower edges are observed as the translational motion mode. The movement of scatterer is dominant at the lower edges. At the upper edges, the movement of the matrix becomes dominant. This is a typical local resonance phenomenon. Further, by comparing the quasi-2D model and 3D specimen model, the observed mode shapes of both upper and lower edges showed a significant variation. The displacement fields of the quasi-2D model at the high-symmetry point X are a translational motion mode [Figure 8 (b1–f1), (b2–f2)]. However, the corresponding bandgap at the lower edge mode shape is a torsional mode, as can be seen in [Figure 8 (b3, c3, d3)]. At the large height, the corresponding mode shape even becomes out-of-plane mode, as shown in [Figure 8 (e3, f3)]. These different mode shapes are caused by different heights of the 3D specimen model with respect to the different Eigen frequencies. The similarity between the quasi-2D model and 2D AMMs model creates an almost unchanged bandgap. Specifically, a noticeable large change in mode shapes of specimen models leads significant difference in the bandgap.

According to the band structure and mode shapes of the different models above, a method to reduce the discrepancies between 2D AMMs and corresponding 3D specimen model is proposed. It is clear that at the lower height ( a × 10 − 1 and a × 10 − 0.5 ), the band structure of quasi-2D model [Figure 7 (b1, c1)] and 2D AMMs model (Figure 7A) are almost the same. Compared with the 3D specimen model, the quasi-2D model displacement along z-axis is constrained. Thus, it is possible to use a lower height specimen and add roller constraint at upper and lower surfaces, which is perpendicular to the z-axis, to reduce the discrepancies. Even in a higher height, although the band structure of quasi-2D model [Figure 7(d1, e1, f1)] are different from 2D AMMs’, their bandgaps are almost the same. In the experiments, the transmittance usually be measured to verify the bandgaps. In this way, the experiments result from quasi-2D model will match the 2D AMMs’ well. Therefore, with the roller constraint at upper and lower surfaces, even at a higher height the discrepancies between 2D AMMs and 3D specimen model can be reduced.

The transmittance along Γ − X direction is compared to the band structure in Figure 9. For 2D AMMs model (Figure 9A), quasi-2D model at lower height [Figure 9 (b1, b2)] and 3D specimen model [Figure 9 (c1–c5)], it is seen that two different kinds of results match well. This outcome makes two computation results mutually verify their correctness. While for quasi-2D model with higher height [Figure 9 (b3–b5)], the transmittance cannot match the Γ − X bandgap. However, the transmittance of quasi-2D model consistent with the 2D AMMs’ at any height. This is reasonable because the mode shapes of 2D AMMs and quasi-2D model are similar. So their characteristics should be consistent. This phenomenon further validates feasibility of the method mentioned proposed above to reduce the discrepancies between 2D AMMs and 3D specimen model. This can also be verified from the analysis of the cause of the difference below.

The Γ   - X bandgap frequency range of the 3D specimen models with respect to the different heights is illustrated in Figure 10. It is clearly seen that with variable height, the bandgap of the specimen models demonstrated an obvious different frequency range. It confirms a significant divergence in physical properties by means of the 3D specimen models with different heights.

In order to verify the prevalence of differences in the 3D specimen models with different heights, the investigations by considering two AMMs’ specimen models are carried out. Figure 11 shows the structures of two AMMs. The physical parameters of the matrix and scatterer are the same as the previous model i.e., silicone rubber and the lead, respectively, as listed in Table 1. The Γ   - X bandgap frequency range of these two specimen models with different heights is illustrated in Figure 12. It is similar to the previously mentioned results (Figure 10) that the bandgap frequency range is notably disparate with different heights.

To investigate the exact cause for the difference, the displacement along the z -axis of the 2D AMMs model and 3D specimen model are shown in Figure 13, and the stress distribution of the different sections along z -direction is illustrated in Figure 14. Firstly, as per the 2D AMMs model, it can be assumed as infinite in the z -axis direction. Therefore, there will be no displacement along the z -axis direction, as shown in Figure 13A. However, as per the 3D specimen model, the elastic waves not only propagate in the x   - y plane but also in z-direction. The propagation of elastic waves caused medium displacement in the different sections along the z -axis. The variation in medium caused different motion modes that yield displacement along the z -axis direction, as shown in Figure 13B.

On the other hand, in the 2D AMMs model, stress distribution of each section along the z -axis is uniform due to the expended z-direction to the infinity, as shown in Figure 14A. However, in the 3D specimen model case, when the elastic waves propagate through the 3D specimen, which has finite height along z -direction, the stress distribution of the different sections along the z -axis is observed with variations, as shown in Figure 14B. This phenomenon can be verified by the model shapes of the mentioned two different cases. For an ideal 2D AMMs model, the model shape is devoid of bending along the z -direction owing to an infinite direction (Figure 14C). However, in counterparts of the case of 3D specimen models, the bending along z-direction exists in the model shape (Figure 14D). These different shapes cause the varied stress distribution and eventually resulted in the divergence in band structures.

As revealed above, the 2D AMMs and quasi-2D models showed quite similarities in results up to a certain extent. While the 2D AMMs and corresponding 3D specimen models showed a significant difference in the physical properties, and 3D specimen models with different heights have significant divergence in physical properties of materials. Therefore, the computation results of the 2D AMMs can be useful to get insight information from experimental results, meanwhile, data computed with 3D specimen models with different heights can provide reliability in terms of theoretical and experiments study. After noticeable differences between the 2D and 3D cases, 3D specimen models are indispensable especially for the larger heights and softer materials.