Novel Frame-Type Seismic Surface Wave Barrier with Ultra-Low-Frequency Bandgaps for Rayleigh Waves

Abstract

Seismic surface waves carry significant energy that poses a major threat to structures and may trigger damage to buildings. To address this issue, the implementation of periodic barriers around structures has proven effective in attenuating seismic waves and minimizing structural dynamic response. This paper introduces a framework for seismic surface wave barriers designed to generate multiple ultra-low-frequency band gaps. The framework employs the finite-element method to compute the frequency band gap of the barrier, enabling a deeper understanding of the generation mechanism of the frequency band gap based on vibrational modes. Subsequently, the transmission rates of elastic waves through a ten-period barrier were evaluated through frequency–domain analysis. The attentional effects of the barriers were investigated by the time history analysis using site seismic waves. Moreover, the influence of the soil damping and material damping are separately discussed, further enhancing the assessment. The results demonstrate the present barrier can generate low-frequency band gaps and effectively attenuate seismic surface waves. These band gaps cover the primary frequencies of seismic surface waves, showing notable attenuation capabilities. In addition, the soil damping significantly contributes to the attenuation of seismic surface waves, resulting in an attenuation rate of 50%. There is promising potential for the application of this novel isolation technology in seismic engineering practice.

1. Introduction

Earthquakes release vast amounts of energy that can inflict significant damage to buildings [1,2,3,4], with the primary source of impact being the seismic waves they generate. Seismic waves can be categorized into surface waves and body waves, while seismic surface waves have the majority of the released energy [5,6,7]. However, the surface waves pose a significant threat to the safety of structures due to the propagation and attenuation characteristics of the seismic surface waves. Seismic barriers have proven to be an effective solution to mitigate the effect of seismic waves on existing buildings. However, conventional barriers with dimensions of tens of meters can only attenuate low-frequency seismic surface waves, which has a limitation in practical engineering [8,9]. Therefore, it is necessary to develop a small-sized barrier for reducing seismic waves.

Phononic crystals [10,11,12,13,14,15,16,17] represent periodic structures composed of two or more elastic solids. These crystals exhibit the ability to influence the propagation of elastic waves by generating distinctive dispersion relationships and frequency bandgaps [18,19,20,21,22,23,24,25]. The bandgap mechanism can be classified into two principal types: Bragg scattering [11,26,27,28,29,30,31,32,33] and local resonance [8,30,34,35,36,37]. Bragg scattering involves the scattering of elastic waves when they encounter the barrier structure. On the other hand, local resonance is the vibration within the barrier structure when an elastic wave occurs, attenuating the energy carried by the elastic wave. The phononic crystals can reduce the size of the barrier against the attenuating seismic surface waves and meet practical engineering applications. Consequently, numerous researchers [38,39,40,41,42] have incorporated phononic crystals into the design of seismic surface wave barriers.

Numerous researchers have focused on the development of seismic surface wave barriers. A 1D periodic composite wall structure composed of concrete and rubber was used to attenuate seismic waves due to the low-frequency band gaps. Enhanced attenuation capabilities were observed when these structures were arranged in positive and negative gradients [34,43,44]. However, the practical implementation of such barriers with a large size poses significant engineering challenges. Consequently, scholars [45,46,47,48,49] have turned their attention towards the exploration of 2D and 3D barrier configurations, seeking innovative solutions to overcome size constraints and maximize effectiveness in mitigating seismic wave impacts.

Huang et al. [28] and Cheng et al. [26] have presented results showing that square columns exhibit superior efficacy in attenuating elastic waves compared to cylindrical columns, whereas hexagonal columns yielded wider bandgaps. Du et al. [27] and Zeng et al. [33] have conducted comparisons on the bandgap properties of hollow, filled, and solid columns, revealing that hollow and filled columns possess the capacity to generate wider bandgaps. Hollow columns filled with various materials can generate different sizes of frequency band gaps [27,29,30,33,50]. The results revealed that a greater difference in Young’s modulus between the filler material and the outer frame of the composited structure leads to some wider frequency bandgaps at low frequencies.

To expand the low-frequency bandgap of elastic waves, a cross-shaped trenchless structure was developed to generate a low-frequency bandgap [30], which offered insights for designing shielding structures against seismic surface waves. The soil body at the real site consists of multiple soil layers rather than a single layer of soil. Some researchers [31,51] have constructed a multi-layer soil model to obtain a more realistic numerical simulation result. It was indicated that the elastic wave is concentrated in the first two layers of soil in the multi-layer soil model, while the lower layer of soil is almost unaffected by the propagation of the elastic waves. Refraction and reflection phenomena of elastic waves occur predominantly within these upper layers, forming the accumulation of energy at the soil surface. Analysis of the transmission spectrum revealed a notable mitigation in the efficacy of barriers for attenuating elastic waves, which explains the complexity of seismic wave propagation through stratified soil structures.

Furthermore, researchers [21,31,52,53,54,55] have designed two-dimensional columnar periodic structures such as “I” and “T” types as well as columnar periodic structures with bases. However, the findings show that these structures can produce some small ranges of bandgap within the ultra-low frequency. Li et al. [56] proposed a resonator structure featuring two internal vibrators based on local resonance theory. This configuration demonstrated the ability of the structure to generate an ultra-low frequency bandgap. Nonetheless, it was noteworthy that the bandwidth of the bandgap remained relatively narrow. Bai et al. [43] proposed a seismic metamaterial wall based on soil expended with concrete and rubber, which could generate several ultra-low frequency bandgaps in the range of 0 Hz–10 Hz. It was indicated that this metamaterial wall could effectively attenuate the frequency from 0 Hz to 10 Hz, but the size of it was too large to be used in engineering. A novel meter-scale seismic metamaterial [29] could generate one wide ultra-low bandgap with frequency from 1.132 Hz to 12.695 Hz. Although this metamaterial was designed for Lamb waves, and its ability to attenuate real seismic waves was not tested, this barrier provides a new reference for the study of barriers at ultra-low frequencies. Chen et al. [46] proposed a 3D metaconcrete that could generate a wide ultra-low frequency bandgap with a width of 4.6 Hz when the barriers were arranged according to decreasing or increasing dimensions. However, its size is also too large for engineering. Brûlé et al. [57] conducted a full-scale test on periodic cylindrical barriers, and the results showed that the barriers could attenuate elastic waves within the bandgap. It is proved that the cylindrical barriers can attenuate surface seismic waves and provided a practical basis for subsequent research. Many researchers [8,51,54] have conducted bench-scale testing to verify the attenuation effect of the numerically simulated barrier structure on elastic waves. The results demonstrated that the barrier effectively attenuated elastic waves within the bandgap, providing a foundation for practical applications.

Although previous studies have shown that 1D [34,43,58,59], 2D [29,60,61,62,63], or 3D [45,46,47] barriers can create narrow bandgaps, there is still a need for further exploration, particularly regarding ultra-low frequency bandgaps. Therefore, we studied a seismic surface wave barrier with ultra-low frequency bandgaps. Most of these studies have focused on cylindrical barriers, leaving other shapes underexplored. References [26,27] compared cylindrical barriers with square barriers and concluded that the square barrier had wider bandgaps than the cylindrical barrier. Therefore, we propose investigating the potential of a hollow frame-type square column combined with a solid square column barrier. Furthermore, it is worth noting that previous research has typically assumed soil to be linearly elastic, ignoring its damping properties in reality. To address this, we conducted time history analyses of the barrier structure on damped soil and barrier materials, aiming to provide more realistic and applicable insights.

This study is organized as follows. Section 2 describes the theoretical method for bandgaps. Section 3 shows the finite-element calculation method used in this study, calculates the bandgap, and analyzes the mechanism of the bandgap. Then, the transmission spectrum and time history analysis of the 10-cycle barrier are calculated. The effect of the soil and material damping are discussed. Finally, the main conclusions are given in Section 4.

2. Theory and Models of the Novel Barrier

This study proposes a novel frame-type barrier for Rayleigh waves. The unit cell of the novel frame-type barrier is shown schematically in Figure 1a–d. a is defined as the lattice constant, b and c are defined as the outer width and the thickness of the hollow column, d and e represent the width and length of the four columns, and h is the height of the barrier, respectively. Oudich et al. [64] pointed out that if the height of the unit cell is greater than 10 times the lattice constant, the height will not affect the bandgap results; therefore, the height used in this study is 16 times the lattice constant, H = 16a. The light yellow, blue, and orange areas in Figure 1b–d represent soil, rubber, and steel, respectively, and all of materials are linearly elastic and undamped. As shown in Figure 1d, each frame-type barrier is periodically arranged in the x–y directions to form a complete seismic surface wave barrier around the building. The geometric parameters and material properties [60] corresponding to this study are shown in

Table 1. Geometric parameters of the unit cell.

a (m)	b (m)	c (m)	d (m)	e (m)	h (m)	f (m)	H (m)
2.5	2	0.1	0.2	1.2	3	0.6	40

and

Table 2. Material parameters of the unit cell.

Material	Density (kg/m3)	Young’s Modulus (Pa)	Poisson’s Ratio
Clayey silt	1939	4.4 × 107	0.4
Rubber	1300	7.7 × 105	0.49
Steel	7784	2.07 × 1011	0.3

, respectively.

The novel periodically arranged frame-type barrier can be simplified to a square unit cell in Figure 1e. The shaded area is the first Brillouin zone, and the eigenfrequencies are obtained by scanning the eigenvectors along Γ-Χ-Μ-Γ. The soil and material properties are assumed to be linearly elastic, homogeneous, and isotropic, so the governing equations for elastic wave propagation can be written as follows [61]:

ρ (r) u (r) = ∇ [(λ (r) + 2μ (r)) (∇∙u (r))] − ∇ × [μ (r) ∇ × u (r)]

(1)

where r (r = (x, y, z)) is the position vector, u (u = (u, v, w)) is the displacement vector, ρ is the mass density, and t is the time variable. According to the Floquet–Bloch theorem, the displacement field of an elastic wave in a periodic structure can be written as below:

u (r, t) = exp (i (k ∙ r − ω t)) u_k (r)

(2)

where i² = −1, and k is the elastic wave vector. The periodic boundary condition can be set on the unit cell and combined with Equation (2) to solve Equation (1). The displacement amplitude condition can be written as follows:

(r + a) = exp (i (k ∙ a)) u (r)

(3)

where a is the lattice constant. The eigenvalue equation for the unit cell can be written as given below:

(K − ω² M) U = 0

(4)

where K and M are the stiffness and mass matrices of the unit cell, respectively. The eigenfrequencies of the unit cell can be obtained by solving Equations (1), (3), and (4).

To eliminate the effect of the dimensions, the relative bandgap width f_w can be used to describe the bandgap characteristics of the unit cell and can be written as follows:

f_w = 2 (f_u − f_l)/(f_u + f_l)

(5)

where f_u and f_l are the upper- and lower-boundary frequencies of the bandgap, respectively.

In this study, COMSOL Multiphysics 5.6 finite-element software was used to calculate the bandgaps, transmission spectrum, and time history analysis.

3. Results and Discussion

3.1. Frequency Bandgaps

Seismic waves are divided into body waves and surface waves, and most energy released by earthquakes is concentrated in Rayleigh waves [5,6,7]. Rayleigh waves are interference waves generated by the body waves at the free surface and propagated along the free surface, and their orbits are inverse elliptical. Rayleigh waves have almost no attenuation in the horizontal direction and decay exponentially with depth. Moreover, the bandgaps for Rayleigh waves can be obtained by the eigenfrequencies of the unit cell. The eigenfrequencies can be calculated by the eigenfrequency in the Solid Mechanics module using COMSOL.

To verify the accuracy of the finite-element software used in this study, the model and the bandgaps are plotted in Figure 2 [65]. The periodic constant of the unit, the radius of pile, and the parameter about depth are taken as a = 0.8 m, r = 0.3 m, and h = 20a, respectively. The model comprises foam columns that are completely buried in the soil. The Young’s modulus E, mass density ρ, and the Poisson ratio υ of foam and soil are taken as 37 Mpa and 20 Mpa, 60 kg/m³ and 1800 kg/m³, and 0.32 and 0.35, respectively. The two materials are also assumed to be linearly elastic and undamped. The COMSOL was used for the eigenfrequencies calculation, and the energy distribution parameter identification method was used for surface wave identification. It can be seen from Figure 2b that the bandgaps of the barrier are in agreement with the previous paper, which proves that the numerical method used in this study is correct and can be used to calculate the bandgap of the surface wave barrier.

To simulate the periodically aligned barrier structures, it is necessary to set the periodic boundary condition (Periodic BC) around the unit cell based on Bloch’s period theory, and a fixed boundary condition (Fixed BC) and a free boundary condition (Free BC) are set at the bottom and top of the unit cell to simulate an infinitely deep soil and surface soil, respectively, in Figure 3a. The eigenfrequencies calculated by COMSOL include both surface and body waves and pseudo-surface waves. The surface waves can be identified using the energy distribution parameter identification method [65].

The size of the grid cell has a significant impact on the accuracy of the results. It is essential that the cell size be considerably smaller than the minimum geometric size and the shortest wavelength of the model. In order to verify the convergence of the mesh size, the maximum cell size was chosen as 1/2, 1/4, and 1/8 of the minimum wavelength. The cell sizes at the minimum geometry, i.e., the thickness of the frame, were c, c/2, and c/4, respectively. These cell sizes were defined as mesh 1, mesh 2, and mesh 3, as shown in Figure 3b. Mesh 1 contains 12,058 domain cells, 4644 boundary cells, and 568 edge cells. Mesh 2 contains 14,568 domain cells, 5586 boundary cells, and 700 edge cells. Mesh 3 contains 40,060 domain cells, 8492 boundary cells, and 884 edge cells. The dispersion curves were calculated using the aforementioned three gridding methods, and the results are presented in Figure 3c. Figure 3c illustrates that the discrepancy between the eigenvalues of mesh 1 and mesh 2 and 3 at high frequencies are considerable, whereas the eigenvalues of mesh 2 and mesh 3 are identical. This suggests that the eigenvalues of mesh 2 are highly satisfactory with regard to convergence, and mesh 2 was selected for this paper.

Figure 3d shows the frequency bandgap of the barrier using the Rayleigh wave theory of the unit cell. As shown in Figure 3d, the surface waves are expressed by the dot–dash line. It can be seen that the unit cell has a complete bandgap with a relative width of 0.47 Hz according to Equations (2) and (5), and the upper- and lower-boundary frequencies are 2.7922 Hz and 4.4875 Hz, respectively. Simultaneously, the surface wave barriers produce seven directional bandgaps in the Γ–Χ direction. The second bandgap is a directional bandgap with a relative width of 0.24 Hz, and its upper- and lower-boundary frequencies are 6.7351 Hz and 8.5685 Hz, respectively.

3.2. Vibration Modes

In this section, the bandgap generation mechanism of the unit cell is investigated by analyzing the vibration modes of the first two bandgaps’ boundaries in the Γ–Χ direction. Figure 3 and Figure 4 show the points selected for the vibration mode diagram, and the vibration mode diagrams are shown in Figure 4.

It can be seen from Figure 4a that the displacements of points A1, A2, A3, and A4 are mainly concentrated in the central square column. The internal structure rotates clockwise in the x–z plane, and the magnitude of the displacement decreases from top to bottom. In Figure 4b, the displacement is mainly concentrated in the rubber, and the internal structure rotates counter-clockwise in the x–y plane. In Figure 4c, the displacement is mainly concentrated in the rubber, while the internal structure moves in a positive direction along the z-axis, and the magnitude of the displacement gradually decreases from top to bottom. In Figure 4d, the displacement is mainly concentrated in the two rubbers along the y-axis direction, while these two rubbers move in a positive direction along the y-axis. The reason is that the energy is absorbed by the internal structure through horizontal displacement at points A1 and A4 and rotational and vertical displacement at points A2 and A3, respectively. Additionally, the rotational displacement at A2 is uniformly distributed in the i-axis direction, indicating its independence from the height of the barrier structure. The displacements at points A1, A3, and A4 are related to the height of the structure, which are mainly concentrated in the upper part of the structure and gradually decrease with increasing depth. Furthermore, the vibration mode diagrams indicate that the displacements are concentrated in the upper part of the unit cell structure, particularly in the internal structure. These four vibration modes are the main causes of the Rayleigh wave bandgaps.

3.3. Transmission Spectrum Analysis

In this section, the attenuation effects of the surface wave barrier are studied by using frequency domain analysis. In theory, the periodic barrier is infinitely arranged in a half-infinite space of soil, while the finite-unit cell of the barrier is applied in practical engineering. The transmission spectrum of finite-period barriers on elastic waves must be calculated. The array transmission model of the barrier, which consists of ten-unit cells periodically embedded in the soil along the horizontal x direction, is shown in Figure 5. The load excitation location is 16a away from the barrier, a distance that should be sufficient for dissipating bulk waves in the soil, thereby ensuring that the barrier is exclusively affected by Raleigh waves; the output response position is 2a away from the last unit cell. Periodic boundary conditions are adopted along the z directions to meet the requirement of infinite periodicity (Oudich and Badreddine Assouar 2012), and a perfectly matched layer (PML) of 3a thickness around the soil is applied to mitigate the impact of waves on the soil surface. The transmission spectrum is defined as follows:

FRF = 20 × log₁₀ (u₂/u₁)

(6)

where u₁ and u₂ are the displacements at the pickup points with and without the barriers, respectively. The transmission spectrum is shown in Figure 6.

In Figure 6, it can be seen that the barrier structure can effectively attenuate the elastic wave when the main frequency falls in the bandgap of the barrier. In particular, in the second bandgap range (from 6.7351 Hz and 8.5685 Hz), the attenuation value reaches −33.17 dB at 7.55 Hz.

The attenuation mechanism at special points can be better analyzed by the overall and internal vibration modals. The special points are 2 Hz, which is outside of the bandgaps, and 7.5 Hz, which is within the two bandgaps, and these are shown in Figure 7. At 2 Hz, the barrier structure has little effect on the vibration displacement of the soil surface at the pick-up point, which indicates that the barrier cannot attenuate the elastic wave at 2 Hz. Figure 7b shows that the soil around the barrier experiences a significant vibrational displacement compared to that not around the barrier. Additionally, Figure 7e indicates that only the first six unit-cell barriers are disturbed by the elastic wave. The combined effect of the barrier structure and the surrounding soil does not change the overall propagation of the elastic wave. This indicates that the barrier neither attenuates the elastic wave nor creates an elastic wave bandgap, which is consistent with the bandgap results obtained in Section 2. As depicted in Figure 7c, the elastic wave gradually transforms into body and surface waves when the frequency is 7.5 Hz, with the majority of the energy concentrated in the surface wave. Figure 7d demonstrates that the vibration displacement on the soil surface behind the barrier structure is considerably reduced. The barrier structure not only disrupts the propagation path of the elastic wave but also transforms a portion of the surface wave into the body wave. The vibrational displacement generated by the body wave in the soil does not increase significantly, and there is no significant vibrational displacement in the soil at the barrier. Figure 7f shows that the energy of the surface wave is concentrated in the internal structure of the barrier, particularly in the first eight unit-cell structures. The vibration displacement generated by the internal structure absorbs the energy of the surface wave, indicating that the barrier structure effectively blocks the propagation of the elastic wave and attenuates its energy. Therefore, the barrier can be used to mitigate the surface waves and protect the buildings.

Figure 6 shows the effects of the barrier in attenuating Rayleigh waves for all the frequencies, except for the 2 Hz. Specifically, it was found that the FRF is small in 0 Hz–7.4 Hz and large in 7.4 Hz–20 Hz. The reasons for the phenomenon are that the elastic waves propagate in the form of alternating maximum and minimum, and the barrier structure alters the propagation path of the elastic wave, thereby changing the vibration displacement value at the pickup point. Therefore, the FRF of the surface waves may also appear negative when the main frequency falls outside the bandgap. In addition, the lower the frequency, the larger the wavelength and the wider the range of vibration amplitudes. As illustrated in Figure 7e, the first six unit-cell structures in the barrier structure at 2 Hz can be considered one cycle of the Rayleigh wave, and the two unit-cell structures in the barrier structure at 7.5 Hz similarly constitute one cycle of the Rayleigh wave, as shown in Figure 7f. Therefore, the attenuation effect of the barrier on low-frequency elastic waves is better than the effect on ultra-low frequency waves.

3.4. Time History Analysis

To further verify the attenuation of the 10-cycle barrier on the surface waves, the time history analysis for the artificial waves and El Centro seismic wave are calculated in this section. The mode for the time history analysis is the same as in Section 3.3, but PML cannot be used in time history analysis; thus, the low-reflection boundary condition was set as the boundary condition, as shown in Figure 8. The Hanning modulation window function was selected for the artificial wave, allowing for the application of a center frequency excitation at specific frequencies, which can be written as follows:

$$A_{in}=\left\{\begin{array}{l}{\displaystyle \frac{1}{A_{\max }}}\left[1-\cos ({\displaystyle \frac{2\pi f_{c}t}{C}})\right]\times \sin (2\pi f_{c}t),0\le t\le {\displaystyle \frac{C}{f_c}}\\ 0,t\ge {\displaystyle \frac{C}{f_c}}\end{array}\right\}$$

(7)

where A_max is the maximum amplitude of the artificial wave; f_c and C are the central frequency and the periodic number, respectively. In this section, the main frequencies of the artificial waves are 2 Hz and 7.5 Hz, as shown in Figure 9. The El Centro seismic waves are shown in Figure 10.

The artificial and seismic waves are normalized to eliminate the differences in magnitude. The artificial and El Centro seismic waves are applied at the excitation point, and the resulting acceleration in the x-direction is measured at the pickup point. The corresponding results are depicted in Figure 11 and Figure 12, respectively.

As shown in Figure 11a, at the central frequency of 2 Hz, the acceleration response at the output position is larger than that of the structure without barriers, which means the surface wave barrier may amplify the acceleration when the main frequency falls outside the bandgap, with a maximum amplification of 67.64%, as confirmed by the transmission spectrum results. At the same time, the amplification continuously affects the soil surface at a rate of 35.5%, resulting in a vibratory displacement of the soil. As shown in Figure 11a, the surface wave can be attenuated effectively by the barrier when the central frequency is 7.5 Hz. In particular, the maximum attenuation of the acceleration reaches 84.48%, and the total attenuation reaches 80.49% in the same range of time (0.9 s–1.82 s). However, the barrier structure amplifies the elastic wave with a maximum amplification of 23.19% and continually acts on the soil surface with an amplification rate of 8% when the time is more than 1.82 s. Although the barrier may amplify the surface wave, the maximum acceleration decreases by 76.81% with the barrier. In addition, the barrier can effectively attenuate the elastic wave with a central frequency of 7.5 Hz, and the small-amplitude vibration cannot damage the building.

The time history analysis and acceleration response spectrum of El Centro are shown in Figure 12. In Figure 12a, it can be seen that the barrier structure significantly attenuates the maximum acceleration with a reduction ratio of 90%. Although the barrier structure amplifies the El Centro seismic waves when the frequency falls in the frequency range of 0–3.01 Hz, the maximum acceleration is only 0.45 and is evenly distributed. It can be concluded that the barrier structure attenuates 55% of the El Centro seismic waves. As seen from Figure 12b, the El Centro seismic waves can be significantly attenuated with a reduction of 14.91% when the main frequency falls into the bandgap at 8.5–20 Hz. Additionally, the attenuation ratio of the barrier reaches 65.53% in 2.8–3.56 Hz and 75.84% in 6.2–20 Hz. The barrier structure amplifies the acceleration response, which is consistent with the transmission spectrum results only by 27.15%, and the maximum acceleration value is only 0.2484. Overall, the barrier’s mitigation ratio of the acceleration of the El Centro seismic wave is 14.91%. It is concluded that the barrier structure can effectively attenuate El Centro seismic waves when the main frequency falls into the bandgap.

3.5. Effect of Damping

As shown above, the soil parameters are assumed to be linearly elastic and undamped, whereas the actual soil is damping. The factors influencing soil damping include the physical properties of the soil (density, viscosity, and angle), the geometry of the soil body (depth and shape), and the characteristics of the dynamic loads (frequency, magnitude, and duration). In the finite-element software COMSOL, Rayleigh damping is used, and the equation for Rayleigh damping is written as follows:

C = α_dMM + β_dKK

(8)

where α_dM is the mass damping coefficient, and β_dK is the stiffness damping coefficient. Rayleigh damping is affected by the frequency range and damping ratio of the structure. In this study, the frequency range of the structure is 0.1–20 Hz, and they can be written as follows:

$$\{\begin{array}{l}{f}_1=0.1 \mathrm{Hz}\\ f_2=20 \mathrm{Hz}\end{array}$$

(9)

where f₁ and f₂ are the lower and upper boundaries of the structural frequency, respectively. The angular frequencies can be calculated:

$$\{\begin{array}{l}{\omega }_1=2\pi f_1=0.628\\ {\omega }_2=2\pi f_2=125.664\end{array}$$

(10)

where ω₁ and ω₂ are the lower and upper boundaries of the angular frequency of the barrier, respectively. The Rayleigh damping coefficients are written as given below:

$$\{\begin{array}{l}{\displaystyle \frac{{\alpha }_{dM}}{2{\omega }_1}}+{\displaystyle \frac{{\beta }_{dK}{\omega }_1}{2}}=\xi \\ {\displaystyle \frac{{\alpha }_{dM}}{2{\omega }_2}}+{\displaystyle \frac{{\beta }_{dK}{\omega }_2}{2}}=\xi \end{array}$$

(11)

where ζ is the damping ratio.

The soil damping ratio is from 0.01 to 0.08, with a general value of 0.05 taken in this study. The mass and stiffness damping coefficient of Rayleigh damping was calculated to be 0.0625 and 0.0008, respectively. The time history analysis of the El Centro seismic wave was calculated with soil damping, and the results are shown in Figure 13. Figure 13a illustrates that the soil damping significantly attenuates the El Centro seismic wave, with an attenuation value reaching 50%. Figure 13b illustrates that soil damping attenuates frequencies within 3.66–20 Hz, with the greatest attenuation observed within 4.9–2.13 Hz, reaching a value of 62.45%. However, soil damping does not attenuate frequencies within 0–3.66 Hz. It can be concluded that the presence of soil damping can accelerate the attenuation of elastic waves.

At the same time, the rubber and steel are still damping. The damping coefficient of a composite material is closely related to its internal structure and composition, primarily due to the internal friction of the material itself and the friction between the material and the external environment. It is challenging to determine the composite material damping for a barrier with rubber and steel, as proposed in this study. Consequently, the general damping ratio of the material was used in this study, which is consistent with the soil damping. The time history analysis of the barrier to El Centro was conducted with both soil and barrier damping, and the results are shown in Figure 14. The attenuation by the barrier with both soil and material damping is almost the same as it is with only soil damping. It is indicated that material damping has a negligible effect on the attenuation of seismic waves.

4. Conclusions

The present study introduces a novel framework for a seismic surface wave barrier composed of steel and rubber. The band structure of the barrier was computed using finite-element analysis. Additionally, frequency–domain and time–domain analyses were conducted for a 10-cycle potential barrier to verify its attenuation characteristics on seismic waves. The influence of soil damping and material damping on the attenuation effects was also considered. The results indicate the follows:

(1)

The barrier exhibits all directional ultra-low-frequency band gaps ranging from 2.7922 to 4.4875 Hz, along with multiple directional band gaps. These gaps effectively cover the 3–8 Hz dominant frequency of seismic waves such as those from the El Centro earthquake. And the soil damping plays a beneficial role in attenuating seismic waves;

(2)

It can significantly reduce structural damage during earthquakes and holds promising potential as a novel seismic isolation technology for engineering applications. It is of great significance for the seismic protection of existing important buildings, especially ultra-high-rise buildings and places where precision instruments are manufactured;

(3)

The buried frame-type seismic surface wave barrier proposed in this paper has good attenuation only for seismic wave-dominant frequency in the range of 2.7922 Hz–4.4875 Hz, but there is almost no attenuation for lower-frequency seismic waves. Therefore, some further research is needed on barriers with ultra-low-frequency bandgaps;

(4)

The buried frame-type barrier proposed in this paper is of the Bragg scattering type, and there is also a possibility for a local resonance-type barrier according to the mechanism of bandgap generation. Therefore, the local resonance-type barrier will be researched in future work.