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Broadband surface wave attenuation in periodic trench barriers
Journal Pre-proof Broadband surface wave attenuation in periodic trench barriers Xingbo Pu, Zhifei Shi
PII:
S0022-460X(19)30693-5
DOI:
https://doi.org/10.1016/j.jsv.2019.115130
Reference:
YJSVI 115130
To appear in:
Journal of Sound and Vibration
Received Date: 25 January 2019 Revised Date:
28 November 2019
Accepted Date: 30 November 2019
Please cite this article as: X. Pu, Z. Shi, Broadband surface wave attenuation in periodic trench barriers, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/j.jsv.2019.115130. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Broadband surface wave attenuation in periodic trench barriers Xingbo Pu and Zhifei Shi* Institute of Smart Materials and Structures, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
Abstract Surface wave manipulation in periodic trench barriers based on real band structures has its inherent limitations. This work revisits this problem from the perspective of complex band structures. To this end, a numerical model using the finite element method is first derived based on a physical model and formulations, which is validated through good correlations between its results and those from existing studies. Next, the model is used to conduct a parametric study to evaluate the effect of different geometrical parameters and material damping. Results indicate that the complex band structure can accurately capture the decay of surface waves and characterize periodic systems with damping. Wide evanescent surface-wave regions, including the band gap and leaky surface modes, exist in periodic trench barriers. These regions are more robust when proper geometrical and material parameters are selected, thus more effectively enabling the realization of broadband surface wave attenuation using periodic trench barriers, which is confirmed in both frequency and time domains.
Keywords: Rayleigh wave; complex band structure, ground vibration isolation; metamaterial; phononic crystal; EPS geofoam.
*
Correspondence to: Professor Zhifei Shi, E-mail address: [email protected]. 1
1. Introduction Artificial periodic structures/materials, known as phononic crystals or elastic metamaterials, can exhibit frequency band gaps under certain conditions, within which the wave propagation can be prohibited [1, 2]. Therefore, they can control sound and vibration in ways that cannot be achieved using natural materials [3]. Such exceptional feature enables a wide range of their applications, including waveguides [4, 5], acoustic cloaks [6, 7], elastic wave obstacles [8, 9], and seismic wave isolation systems [10, 11]. The research effort on this topic has dramatically increased in recent years. In general, the band gap of periodic structures can be described using the dispersion equation, which can be derived from a unit cell utilizing the Bloch-Floquet theorem due to the spatial periodicity. The dispersion equation can be transformed into an eigenvalue problem, which can be solved using two methods. The first method is to use real wavenumber k in the first Brillouin zone to solve unknown real frequencies ω , which is typically referred to as the traditional ω ( k ) method and applies to the real band structure. The other way, in contrast, is to solve unknown wavenumbers based on a fixed frequency, which is termed as k ( ω ) method and applies to the complex band structure since the wavenumbers are usually complex
values. The real band structure, as the name implies, reveals propagating modes due to real-valued wavenumbers and frequencies only. Although it can define a band gap as a frequency range in which no eigenmodes exist, it cannot explain the decay form of waves in the band gap [12]. Also, it is challenging to consider material damping using this method, especially with frequency-dependent viscous losses [13]. Fortunately, these two obstacles can be tackled using a complex band structure, which can capture both propagating and evanescent modes. The two modes constitute a complete solution for periodic structure problems [1]. It has been theoretically and experimentally proven that evanescent waves play an essential role in wave propagation problems [14, 15].
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Extensive studies have been conducted on complex band structures in recent years, mainly on bulk waves in two-dimensional (2D) or three-dimensional (3D) periodic structures [16-20] and Lamb waves in periodic plates [21-23]. As a typical guided surface wave, the Rayleigh wave has attracted extensive attention as it carries most of the vibratory energy and decays much slower than bulk waves. The corresponding dispersion property has been thoroughly investigated by researchers using the Plane Wave Expansion (PWE) method [24-26] and Finite Element Method (FEM) [27, 28], mainly for real band structures. Less work, however, has been devoted to complex band structure considering surface waves [29]. Regarding applications, extensive work has been conducted to control the propagation of surface waves. On a smaller scale (elastic waves at MHz frequencies), applications are being explored to design high-resolution superlens for high precision measurements [30, 31] and to filter or guide surface waves in the field of mechanical engineering [32-34]. On a larger scale, controlling low-frequency (below 100 Hz) surface waves to protect infrastructure is of typical interest. Brule et al. [35] confirmed with a large-scale experiment that periodic boreholes at about one-meter scale could be used to block incoming surface waves. Colombi et al. [36, 37] carried out a series of interesting studies and demonstrated theoretically and experimentally that forests could act as natural seismic metamaterials. According to the local resonance mechanism, Palermo et al. [38, 39] pointed out that resonant barriers buried under soil surface are promising for seismic surface wave isolation because band gaps can be tuned to below 10 Hz. In addition, periodic trench barriers [40] and periodic pile barriers [41-43] have been developed to reduce traffic-induced ground vibrations. Thanks to the Bragg scattering mechanism, these wave barriers can be designed to open a wider band gap in the middle-frequency range. The aforementioned studies were mainly focused on surface wave manipulation based on real band structures. As stated earlier, this method has inherent limitations. For example, for both 1D and 2D periodic wave barriers, an interesting phenomenon reported by [44, 45] is that the incident surface wave
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can also be attenuated in a frequency range outside the band gap in real band structures. Unfortunately, an accurate explanation for this feature cannot be provided within the context of real band structures. Therefore, this study will revisit the surface wave attenuation in periodic trench barriers using complex band structures. Until now, there is limited research in this area. It is expected that this work can provide a better understanding of broadband surface wave isolation using periodic wave barriers. The paper is organized as follows. After this introduction, Section 2 describes the model and method employed to obtain complex band structures for surface waves. Next, in Section 3, results obtained with the present methodology are compared with those in related studies to validate the present method and physical model. Section 4 presents the complex band structure for surface waves and reveals the character of evanescent surface waves. It is found that there exist two wide frequency ranges of evanescent surface waves, indicating that periodic trench barriers can be designed to achieve broadband surface wave isolation. Section 5 discusses the influence of geometrical parameters and material damping on complex band structures and on screening effectiveness in the frequency domain. In Section 6, broadband surface wave attenuation in periodic trench barriers is further confirmed in the time domain by investigating the reduction of train-induced ground vibrations. Finally, the main conclusions are summarized in Section 7.
2. Model and method The physical model considered in this work is shown in Fig. 1. Since analytical solutions are extremely difficult to obtain, the analyses are performed using numerical techniques, including eigenvalue and response studies, which provide solutions for infinite and finite periodic structures, respectively. 2.1 Eigenvalue study The problem can be stated as follows. Considering trench barriers embedded in a semi-infinite soil substrate periodically, under the assumption of an infinite periodic extent, the dispersion analysis of 4
elastic waves can be represented by a unit cell shown in Fig. 1(a), in which Bloch-Floquet conditions are applied to the left and right boundaries. The trench depth, width, and lattice constant are assumed to be h , w and a , respectively. In the absence of body force, the governing equation of motion can be
written as ∇ ⋅ σ (r ) = ρ (r )
Where ∇
is the gradient operator; σ (r )
∂ 2 u (r ) ∂t 2
(1)
is the stress tensor that can be expressed as
σ ij = λδ ij ε kk + 2µε ij ; λ , µ are the Lamé constants; ε ij = ( ui , j + u j ,i ) 2 is the strain tensor; ρ (r) is the
mass density; r = ( x, y) is the position vector; u = (u1 , u2 ) is the displacement vector. According to Bloch-Floquet theorem, the displacement field can be expressed as the product of a periodic function and an exponential item as u(r) = U(r)ei(k ⋅r −ωt )
(2)
In which ω is the angular frequency; k is the Bloch wave vector; U(r) is a periodic function about the lattice. Substituting the Bloch-Floquet condition in Eq. (2) into Eq. (1), the governing equation can be transformed into an algebraic eigenvalue problem following FEM discretization procedure as Φ ( Λ, ω ) = 0
(3)
In which Λ = −ik is the eigenvalue to be solved for a specific frequency ω . The eigenequation of Eq. (3) is a typical k (ω ) formulation. In general, the wavenumber k = k Re + ik Im is a complex value. The real part kRe represents the propagation of a wave mode; and the imaginary part kIm denotes the attenuation property, which is termed as the attenuation coefficient. Usually, a wave mode can be treated as an evanescent wave mode if k Im ≠ 0 [12]. Eq. (3) can be solved with the aid of Partial Differential Equations (PDEs) in Comsol Multiphysics [46], in which an eigenvalue problem in a domain Ω has the general expression Λ 2 ea U − Λd a U + ∇ ⋅ ( −c∇U − α U + γ ) + β ⋅ ∇U + bU = f 5
(4)
Comparing Eqs. (1) and (2) with Eq. (4), we get the nonzero coefficients in Eq. (4) as follows, − ( λ + 2 µ ) 0 0 −λ −µ 0 0 −µ c= 0 −µ 0 −µ 0 − λ + 2µ −λ 0 ( ) ρω 2 b= 0
0 ρω 2
( λ + 2µ ) 0 ea = 0 µ ( λ + 2µ ) Λ 0 α = 0 λΛ − ( λ + 2 µ ) Λ 0 β = 0 −µΛ
(5)
(6)
(7)
0 µΛ µΛ 0
(8)
0 −λΛ −µΛ 0
(9)
For a rectangular unit cell Ω = [ 0, a ] × [0, H ] shown in Fig. 1(a), the periodic boundary conditions are U x =0 = U x = a −n ⋅ (−c∇U − α U ) x =0 = n ⋅ (−c∇U − α U) x = a
(10)
Where n is the outward unit normal vector on the domain boundary. Besides, the stress-free boundary condition is −n ⋅ (−c∇U − α U) y = H = 0
(11)
U y=0 = 0
(12)
and the fixed boundary condition is
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It should be noted that the system considered herein yields both surface waves and other modes. To obtain pure surface modes from the solution, we implement a post-processing technique based on the center of strain energy along the y-axis, 1 ξ= H
∫
H
y ⋅ σ ij ε ij 2 dS
0
∫
H
0
σ ij ε ij 2 dS
(13)
Where H is the total height of the unit cell. Because the energy of Rayleigh waves is localized near the free surface, we can assume that the corresponding modes are Rayleigh wave modes if ξ ≥ 0.8 [28]. 2.2 Numerical modeling scheme Theoretically, the band structure of a unit cell is suitable for an infinite periodic system. The number of unit cells, however, is always finite in practice. Therefore, transmission analysis is carried out to assess the isolation performance of multi-row trenches. Although it is a 3D problem in practice, it can be simplified as a 2D plane strain problem [47, 48] when the barrier length is sufficiently long. The numerical model (40 m ×15 m) is depicted in Fig. 1(b). The incident surface wave is modeled by applying a vertical harmonic load F0 = 1000 N on the free surface. Although the load generates both surface waves propagating along the free surface and bulk waves in the soil substrate, displacements at the surface are dominated by surface waves in the far-field of the source. Thus, the effect of bulk waves is negligible [45, 49]. Perfectly Matched Layers (PMLs) are applied to the left, right, and bottom sides of the model to simulate half-space boundary conditions. The transmission coefficient is evaluated in terms of the amplitude reduction factor ( AR ) defined by Woods [50]: AR =
u y1 uy0
(14)
Where u y 0 and u y1 denote reference value without barriers and the real part of the vertical displacement with barriers at the detection point, respectively. Obviously, a smaller value of AR results
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in a better vibration isolation effect. For example, AR = 0.3 indicates that wave barriers can reduce 70% of vertical displacement.
3. Comparison and verification To validate the numerical model derived in Section 2, we recalculate two numerical examples published in related studies. The first one is the harmonic response of an open trench subjected to a vertical harmonic load at f = 50 Hz . The trench’s width w = 0.1λR and depth h = 1λR , where λR is the corresponding Rayleigh wavelength. The results obtained from the present numerical model are compared with those from Ahmad et al. [51] and Bordon et al. [52] using conventional and dual Boundary Element Method (BEM), respectively, where good correlations can be observed among the three results, as shown in Fig. 2. The second comparative study is surface-wave band structures in a 1D periodic system consisting of rubber and polyfoam materials with dimensions a = 0.4 m and w = 0.2 m, which was studied by Huang et al. [44] using the traditional ω(k ) approach. As can be seen in Fig. 3, the results from the present model based on complex band structure agree with those based on the conventional real band structure. Another notable feature of the present model is that, other than real part of the wavenumber, it contains an imaginary part, which can characterize the attenuation of surface wave and will be discussed in more detail in the next section.
4. Complex band structures and evanescent surface waves In this section, we consider another complex band structure for surface waves, where the half-space and backfill materials are assumed to be isotropic and linearly elastic, with material parameters provided in Table 1. The geometrical parameters are a = 1m , h = 2 m , w = 0.3 m and H = 20a (see Fig. 1a). The unit cell is discretized into quadrilateral elements with cubic Lagrange shape function. Fig. 4 displays the complex band structure for surface waves. Unlike the traditional ω(k ) approach, the 8
complex band structure contains two adjacent panels, where the left and right panels show the real and imaginary part of the wavenumber, respectively. As pointed out earlier, the non-zero imaginary value represents the evanescent mode that decays exponentially in the direction of propagation. Two features can be immediately identified in Fig. 4. First, the lower and upper bounds of the band gap are connected by evanescent surface waves (see the lower shaded area). The real parts of these evanescent modes are located at the boundary point k = π a of the Brillouin zone. The other interesting feature is that there also exist evanescent surface waves above the band gap (see the upper shaded area), but their real parts are located inside the first Brillouin zone. In order to explain the physical mechanism, we display the dispersion curves of bulk waves in the soil substrate. It can be observed that the second branch of the surface-wave dispersion curves intersects the dispersion curve for longitudinal waves. This means that those modes inside the upper shaded area are leaky or pseudo surface waves because they contain small bulk-wave components that leak energy into the substrate. Mathematically, the leaky surface wave arises from complex wavenumber solutions. Therefore, it is an evanescent wave in a broad sense [53]. Further information on pseudo surface waves in phononic crystals can be found in [25, 28, 29, 54, 55]. To further confirm these evanescent modes, we investigate the transmission coefficient of six rows of trench barriers, with results plotted in Fig. 5. It can be seen that the ranges of transmission reduction agree with the areas where the evanescent waves exist, which implies that broadband surface wave isolation can be achieved using the evanescent wave properties. To further understand the evanescent surface wave, we display its decay process in Fig. 6(a), which shows the change of vertical displacement amplitude along the direction of wave propagation in the band gap at f = 54 Hz . Intuitively, the amplitude decays exponentially inside the trench barriers denoted by the shaded area. To validate this point, we select the peak points to fit an exponential function pe−qX , with the fitting results plotted in solid line, where p = 1.78 ×10−5 m and q = 0.503 m−1 . Apparently, the decay rate q = 0.503 m−1 agrees with the imaginary wavenumber Im(k) = 0.505 m−1 in the complex band 9
structure in Fig. 4. In addition, the displacement amplitude at the local coordinate X = 0 is u0 = 1.86 ×10−5 m , which is close to the fitting parameter p . As another example, we consider an
evanescent mode beyond the band gap at f = 74 Hz . The fitting result is shown in Fig. 6(b), and the corresponding parameters are p = 1.52 ×10−5 m and q = 0.257 m−1 . Similarly, the two parameters are also in agreement with the displacement amplitude u0 = 1.54 ×10−5 m and the imaginary wavenumber
Im(k) = 0.253m−1 in the complex band structure in Fig. 4. Based on the above findings, we can conclude that, unlike traditional real band structures, the complex band structure can accurately describe the exponential decay of evanescent waves. More importantly, there exist wide frequency ranges of evanescent surface waves, indicating that broadband surface wave isolation using periodic trench barriers is feasible.
5. Parametric analysis As described above, the attenuation of surface waves can be explained in terms of the complex dispersion relations because the imaginary part of the wavenumber characterizes the property of evanescent surface waves. Since dispersion relations are determined by geometrical and material parameters, periodic trench barriers can be designed to exhibit low-frequency and broadband evanescent wave properties based on the analysis of complex band structures. Using this concept, we will show how the design, including the geometrical and material parameters of a periodic trench barrier, can affect the frequency range of evanescent surface waves. 5.1. Influence of trench depth Earlier work [40, 44, 56] indicated that the trench depth plays a significant role in surface wave isolation. Specifically, these studies found that the surface wave screening efficiency increases with the increase of the trench depth and then remains almost constant. Although an explanation was provided qualitatively, a quantitative and accurate physical interpretation is still lacking, especially from the
10
perspective of real band structures. Thus, we examine the effect of trench depth on evanescent modes of complex dispersion relations. The lattice constant a and trench width w are assumed to be 1 m and 0.3 m, respectively. Fig. 7 shows the result, which indicates that the increase of trench depth does not affect the lower and upper boundary frequencies of the band gap. The other message from Fig. 7 is that the increase in trench depth indeed results in a significant increase in the imaginary part of the wavenumber. However, when the trench depth is increased to h = 2a , the imaginary wavenumber remains unchanged. This feature means that the vibration isolation efficiency can be improved by increasing the trench depth until it reaches h = 2a , beyond which further increase in the trench depth no longer changes the efficiency of the barrier. This point is further confirmed by the harmonic analysis with six rows of trench barriers in Fig. 7. As expected, it can be observed that the incident surface waves attenuate greatly in the evanescent wave regions denoted by the shaded areas. More importantly, the magnitude of AR reduces with the increase in trench depth and then remains the same when it exceeds the depth h = 2a . The underlying mechanism can be explained as follows. Because the Rayleigh-wave energy is mainly concentrated in the depth range of one wavelength, the isolation efficiency does not increase significantly when the barrier depth exceeds this value. For the wavenumber at the boundary of the first Brillouin zone, i.e., k = π a , the corresponding wavelength is exactly equal to 2a . Besides, one can find that in the frequency range between 60-70 Hz, a trench with a shorter depth, i.e., h = a, provides a more considerable attenuation. It is difficult to explain this counter-intuitive result accurately. Although an in-depth discussion of this phenomenon is beyond the scope of this study, a brief qualitative explanation is presented here. The Rayleigh wavelength is 1.33-1.55 m in that frequency range, which is larger than h = a and less than h = 2a. When a Rayleigh wave is incident on a trench, it will give rise to reflected and transmitted Rayleigh waves, reflected and transmitted body waves, and body waves that radiate downward from the trench. When the trench depth is about one
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Rayleigh wavelength, it radiates downward body waves better [51]. Therefore, the transmitted Rayleigh wave is less than that in h = 2a. 5.2. Influence of trench width The trench width is an important geometrical parameter that characterizes the filling fraction. The lattice constant a and trench depth h are assumed to be 1 m and 2 m, respectively. As shown in Fig. 8, increasing the trench width does not affect the lower bound frequency of the band gap for surface waves, but can significantly increase the gap width and the imaginary part of the wavenumber. Similarly, we carry out the harmonic analysis of six rows of trench barriers to confirm this result. As expected, the corresponding transmission coefficients in Fig. 8 show the same phenomena, which is consistent with what was found for a single in-filled trench [51, 57]. This is because of the destructive interference of elastic waves, which is enhanced when the filling fraction is increased. Based on the above analyses, it can be stated that the increase of the trench width can facilitate broadband surface wave screening. 5.3. Influence of lattice constant The lattice constant is another important parameter that characterizes the geometrical property. Fig. 9 shows the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different lattice constants. The trench width w and depth h are assumed to be 0.3a and 2a , respectively. It can be seen that the lower and upper bounds of the band gap decrease as the lattice constant increases, which can be explained in terms of the Bragg condition f ∝ 1 a [1], i.e., the increase of the lattice constant naturally leads to the decrease in frequency. On the other hand, it can be seen again that the increase in the lattice constant does not decrease the imaginary part of the wavenumber, but expands the frequency range of evanescent surface waves. This feature implies that the increase of lattice constant is advantageous for achieving broadband surface wave isolation. Similar to what is observed above, the transmission coefficients of six rows of trench
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barriers strengthen this viewpoint, as shown in Fig. 9, where the attenuation zones in transmission coefficients agree with the evanescent surface-wave regions predicted by the Bloch-Floquet theorem. 5.4. Influence of shear wave velocity The effect of shear wave velocity of soil on complex band structure and amplitude reduction is evaluated. The shear wave velocity is varied from 110 m/s to 150 m/s by changing the soil modulus while keeping other parameters as constant. It can be observed from Fig. 10 that the lower boundary frequency of the band gap increases as shear wave velocity increases. The reason is that the increase in shear-wave velocity means that the stiffness of the system increases, thereby resulting in an increase in frequency. It can also be observed that the bandwidth and attenuation coefficient also increase with the increase of the shear-wave velocity, which is consistent with the harmonic response of six rows of trench barriers in terms of amplitude reduction (AR) in Fig. 10. This phenomenon can be understood as follows. It is known from elastic wave theory that the impedance ratio of materials on both sides governs the transmission and reflection at the interface [58]. For materials considered in this paper, the impedance ratio is less than one, meaning that the infill is a soft material. The impedance ratio decreases as the shear-wave velocity of soil increases. For trenches filled with soft material, the decrease in the impedance ratio results in better vibration screening, because a very soft barrier is close to the condition of an open trench that has a very low transmission. 5.5. Influence of soil damping The foregoing analyses are carried out in the absence of material damping. Materials in practice always exhibit dissipative behavior. In dissipative materials, the frequency or the wavenumber are complex values, i.e., all waves are evanescent modes. Since real band structure cannot display this feature, the complex band structure is used to reveal the spatial attenuation characteristic of evanescent waves. In engineering practice, the dissipation is described using the hysteretic material damping ratio
η and the complex Lamé constants, which are expressed as λ* = λ (1 + 2ηi) and µ * = µ (1 + 2ηi) [59].
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Neglecting the backfill material damping, we show the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different damping ratios in Fig. 11. As expected, it can be observed that there exists a non-zero imaginary part of the wavenumber for every frequency, indicating that all surface waves are evanescent waves in the context of soil damping. Strictly speaking, it is difficult to define band gaps since all wavenumbers are complex values. For convenience, the band gap in the absence of material damping is displayed as a reference (see the shaded area). Another notable feature found in Fig. 11 is that the imaginary part of the wavenumber remains almost constant in the band gap, but increases with the increase of damping ratio outside the band gap. The same characteristic can be found in the transmissions in Fig. 11. This phenomenon could be explained as follows. Damping in soil always acts as dissipative effects on mechanical energy. Thus, the inhibition of wave motion is enhanced as the damping ratio increases. In addition, waves decay exponentially in the band gap caused by Bragg scattering. In contrast, the effect of low damping is no longer obvious. 5.6. Influence of backfill material damping In order to evaluate the influence of backfill material damping, we neglect the soil damping. Fig. 12 shows the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different damping ratios of the backfill material. Similarly, all surface waves are evanescent waves with consideration of backfill material damping. Similar to the results when considering soil damping in Fig. 11, the imaginary part of the wavenumber remains almost unchanged in the band gap, but increases with the increase of damping ratio outside the band gap. The most striking difference is that the degree of damping effects in backfill materials is not as high as that in soils. This result can be explained as follows. From the perspective of the whole model, the area occupied by the trench barriers is small. Therefore, the influence of the backfill material damping is naturally not as big as that caused by the soil damping. Overall, damping is always advantageous for wave attenuation for Bragg scattering wave barriers.
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6. Application to train-induced ground-borne vibration isolation The aforementioned analysis is extended to an application in Civil Engineering in this section. Doing so, we consider the isolation of ground-borne vibration induced by moving trains, since it is an important environmental issue, especially in densely populated areas [60, 61]. To this end, we measured the vertical acceleration of train-induced ground vibrations in Beijing, China, with measurement sites shown in Fig. 13. Fig. 14 displays the time history and corresponding Fourier spectra of ground vibrations induced by subway lines 1 and 13. As can be seen from Figs. 14(b) and (d), the train-induced vibrations contain many frequencies, with the main frequencies ranging from 30 Hz to 80 Hz. For the two acceleration inputs, numerical calculations are performed in the time domain. The boundary condition of the source is u&&y = Acc(t ) , where the function Acc(t ) is the time history at the original site. Fig. 15 shows vertical acceleration responses and corresponding Fourier spectra at the detection point from the numerical model in Fig. 1. The solid and dashed lines denote the results obtained with and without six rows of trenches (a = 1.6 m, h = 2a, w = 0.3a), respectively. Obviously, the ground-borne vibration is significantly mitigated with the presence of six rows of trench barriers. The reduction of the vertical acceleration amplitude is about 70% and 90% for subway lines 1 and 13 respectively, when compared with those without trench barriers. More importantly, it can be seen from the Fourier spectra that the frequency ranges of vibration reduction are in agreement with the evanescent surface-wave regions calculated using the Bloch-Floquet theorem (see shaded areas in Figs. 15b and 15d). Again, the results in the time domain confirm the existence of wide evanescent surface-wave regions in periodic trench barriers and the feasibility of applicability of broadband surface wave isolation in practice.
7. Conclusions This paper numerically studies the attenuation behavior of surface waves in periodic trench barriers, where both elastic and dissipative cases are considered. The spatial decay of surface waves is
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investigated using complex band structures through solving the dispersion equation for assigned real-valued frequencies. It is found that the periodic trench barriers possess broadband evanescent surface-wave regions, including the band gap and leaky surface modes. The transmission analysis shows that the incident surface wave decays exponentially in wave barriers when its frequency is within evanescent surface-wave regions. In particular, all surface waves become evanescent modes when considering material damping. It is difficult to identify the band gap without the complex band structure of an undamped system for reference. Based on the results from the complex dispersion and the transmission analysis, it can be concluded that the periodicity-induced wave scattering dominates the attenuation of surface waves, and the influence of damping is small in the band gap. While outside the band gap, the attenuation caused by damping is dominant. The trench depth is an important design parameter, which is recommended to be twice the lattice constant based on the complex dispersion and transmission analysis. Moreover, the results from the parametric study indicate that periodic trench barriers are capable of blocking broadband surface waves by tuning geometrical parameters, simultaneous increase of the lattice constant, and the trench width. The feasibility is further validated in the time domain by evaluating the performance of railway induced ground-borne vibration isolation.
Acknowledgments This work was supported by the National Natural Science Foundation of China (51878030). We wish to thank the editor and the anonymous reviewers for their constructive comments.
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[53] Y. Zhou, M.-H. Lu, L. Feng, X. Ni, Y.-F. Chen, Y.-Y. Zhu, S.-N. Zhu, N.-B. Ming, Acoustic surface evanescent wave and its dominant contribution to extraordinary acoustic transmission and collimation of sound, Phys. Rev. Lett., 104 (2010) 164301. [54] N.E. Glass, A.A. Maradudin, Leaky surface‐elastic waves on both flat and strongly corrugated surfaces for isotropic, nondissipative media, J. Appl. Phys., 54 (1983) 796-805. [55] A.A. Maznev, A.G. Every, Surface acoustic waves in a periodically patterned layered structure, J. Appl. Phys., 106 (2009) 113531. [56] A. Saikia, Numerical study on screening of surface waves using a pair of softer backfilled trenches, Soil Dyn. Earthq. Eng., 65 (2014) 206-213. [57] M. Adam, O. von Estorff, Reduction of train-induced building vibrations by using open and filled trenches, Comput. Struct., 83 (2005) 11-24. [58] X. He, Traffic induced environmental vibrations and controls, Science Press Ltd, Beijing, 2010. [59] S. Kramer, Geotechnical earthquake engineering, Prentice-Hall, Upper Saddle River, New Jersey, 1996. [60] D.J. Thompson, J. Jiang, M.G.R. Toward, M.F.M. Hussein, A. Dijckmans, P. Coulier, G. Degrande, G. Lombaert, Mitigation of railway-induced vibration by using subgrade stiffening, Soil Dyn. Earthq. Eng., 79 (2015) 89-103. [61] D.J. Thompson, J. Jiang, M.G.R. Toward, M.F.M. Hussein, E. Ntotsios, A. Dijckmans, P. Coulier, G. Lombaert, G. Degrande, Reducing railway-induced ground-borne vibration by using open trenches and soft-filled barriers, Soil Dyn. Earthq. Eng., 88 (2016) 45-59.
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Tables and captions Table 1. Parameters of the elastic half-space and the trench barrier [40]. Materials
Density (kg/m³)
Young modulus (MPa)
Poisson ratio
Soil EPS geofoam
1800 60
46 37
0.25 0.32
21
Figures and captions
Fig. 1. 2D plane strain models. (a) The unit cell used for band structure calculations. (b) Multi-rows of trench barriers embedded in a semi-infinite soil substrate. ( l = 46 m , l1 = 21 m , l2 = 3 m and l3 = 18 m ).
Fig. 2. Comparative study for ground vibration screening using open trench.
22
Fig. 3. Comparative study for band structure of surface waves in a periodic rubber-polyfoam system.
Fig. 4. Complex band structure for surface waves in a periodic system composed of soil and geofoam-filled trenches (a = 1 m, h = 2a, w = 0.3 m). The normalized displacement field of three eigenmodes labeled by points A, B, and C characterizes the feature of Bloch surface waves.
23
Fig. 5. Transmission coefficient as a function of frequency for six rows of trench barriers (a = 1 m, h = 2a, w = 0.3 m). The shaded areas correspond to the evanescent surface-wave regions in Fig. 4.
Fig. 6. Harmonic responses with six rows of trenches (a = 1 m, h = 2a, w = 0.3 m): (a) f = 54 Hz, (b) f = 74 Hz. Trenches are located in the shaded area. The solid line represents fitting results. 24
Fig. 7. Influence of trench depth (a = 1 m, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
Fig. 8. Influence of trench width (a = 1 m, h = 2a): (a) w = 0.2 m, (b) w = 0.25 m, (c) w = 0.35 m, (d) w = 0.4 m. Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
25
Fig. 9. Influence of lattice constant (w = 0.3a, h = 2a): (a) a = 1.2 m, (b), a = 1.4 m, (c) a = 1.6 m, (d) a = 2 m. Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
26
Fig. 10. Influence of shear wave velocity. Six rows of trench barriers are considered for harmonic analysis.
Fig. 11. Influence of soil damping (a = 1 m, h = 2a, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded area represents the surface-wave band gap in the absence of material damping.
Fig. 12. Influence of backfill material damping (a = 1 m, h = 2a, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded area represents the surface-wave band gap in the absence of material damping.
27
Fig. 13. Vibration measurement sites in Beijing, China: (a) subway line 1, (b) subway line 13.
Fig. 14. The vertical acceleration record of train-induced ground vibrations and corresponding Fourier spectra: (a-b) subway line 1, (c-d) subway line 13.
28
Fig. 15. Vertical acceleration responses at the detection point and corresponding Fourier spectra with and without six rows of trenches (a = 1.6 m, h = 2a, w = 0.3a): (a-b) subway line 1, (c-d) subway line 13. The shaded areas represent the evanescent surface-wave regions predicted by the Bloch-Floquet theorem.
29
Author Contribution Statement Xingbo: Calculating, Analyzing and Writing. As the corresponding author, Zhifei financially support the research and is responsible for the quality of the paper.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
No
PII:
S0022-460X(19)30693-5
DOI:
https://doi.org/10.1016/j.jsv.2019.115130
Reference:
YJSVI 115130
To appear in:
Journal of Sound and Vibration
Received Date: 25 January 2019 Revised Date:
28 November 2019
Accepted Date: 30 November 2019
Please cite this article as: X. Pu, Z. Shi, Broadband surface wave attenuation in periodic trench barriers, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/j.jsv.2019.115130. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Broadband surface wave attenuation in periodic trench barriers Xingbo Pu and Zhifei Shi* Institute of Smart Materials and Structures, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China
Abstract Surface wave manipulation in periodic trench barriers based on real band structures has its inherent limitations. This work revisits this problem from the perspective of complex band structures. To this end, a numerical model using the finite element method is first derived based on a physical model and formulations, which is validated through good correlations between its results and those from existing studies. Next, the model is used to conduct a parametric study to evaluate the effect of different geometrical parameters and material damping. Results indicate that the complex band structure can accurately capture the decay of surface waves and characterize periodic systems with damping. Wide evanescent surface-wave regions, including the band gap and leaky surface modes, exist in periodic trench barriers. These regions are more robust when proper geometrical and material parameters are selected, thus more effectively enabling the realization of broadband surface wave attenuation using periodic trench barriers, which is confirmed in both frequency and time domains.
Keywords: Rayleigh wave; complex band structure, ground vibration isolation; metamaterial; phononic crystal; EPS geofoam.
*
Correspondence to: Professor Zhifei Shi, E-mail address: [email protected]. 1
1. Introduction Artificial periodic structures/materials, known as phononic crystals or elastic metamaterials, can exhibit frequency band gaps under certain conditions, within which the wave propagation can be prohibited [1, 2]. Therefore, they can control sound and vibration in ways that cannot be achieved using natural materials [3]. Such exceptional feature enables a wide range of their applications, including waveguides [4, 5], acoustic cloaks [6, 7], elastic wave obstacles [8, 9], and seismic wave isolation systems [10, 11]. The research effort on this topic has dramatically increased in recent years. In general, the band gap of periodic structures can be described using the dispersion equation, which can be derived from a unit cell utilizing the Bloch-Floquet theorem due to the spatial periodicity. The dispersion equation can be transformed into an eigenvalue problem, which can be solved using two methods. The first method is to use real wavenumber k in the first Brillouin zone to solve unknown real frequencies ω , which is typically referred to as the traditional ω ( k ) method and applies to the real band structure. The other way, in contrast, is to solve unknown wavenumbers based on a fixed frequency, which is termed as k ( ω ) method and applies to the complex band structure since the wavenumbers are usually complex
values. The real band structure, as the name implies, reveals propagating modes due to real-valued wavenumbers and frequencies only. Although it can define a band gap as a frequency range in which no eigenmodes exist, it cannot explain the decay form of waves in the band gap [12]. Also, it is challenging to consider material damping using this method, especially with frequency-dependent viscous losses [13]. Fortunately, these two obstacles can be tackled using a complex band structure, which can capture both propagating and evanescent modes. The two modes constitute a complete solution for periodic structure problems [1]. It has been theoretically and experimentally proven that evanescent waves play an essential role in wave propagation problems [14, 15].
2
Extensive studies have been conducted on complex band structures in recent years, mainly on bulk waves in two-dimensional (2D) or three-dimensional (3D) periodic structures [16-20] and Lamb waves in periodic plates [21-23]. As a typical guided surface wave, the Rayleigh wave has attracted extensive attention as it carries most of the vibratory energy and decays much slower than bulk waves. The corresponding dispersion property has been thoroughly investigated by researchers using the Plane Wave Expansion (PWE) method [24-26] and Finite Element Method (FEM) [27, 28], mainly for real band structures. Less work, however, has been devoted to complex band structure considering surface waves [29]. Regarding applications, extensive work has been conducted to control the propagation of surface waves. On a smaller scale (elastic waves at MHz frequencies), applications are being explored to design high-resolution superlens for high precision measurements [30, 31] and to filter or guide surface waves in the field of mechanical engineering [32-34]. On a larger scale, controlling low-frequency (below 100 Hz) surface waves to protect infrastructure is of typical interest. Brule et al. [35] confirmed with a large-scale experiment that periodic boreholes at about one-meter scale could be used to block incoming surface waves. Colombi et al. [36, 37] carried out a series of interesting studies and demonstrated theoretically and experimentally that forests could act as natural seismic metamaterials. According to the local resonance mechanism, Palermo et al. [38, 39] pointed out that resonant barriers buried under soil surface are promising for seismic surface wave isolation because band gaps can be tuned to below 10 Hz. In addition, periodic trench barriers [40] and periodic pile barriers [41-43] have been developed to reduce traffic-induced ground vibrations. Thanks to the Bragg scattering mechanism, these wave barriers can be designed to open a wider band gap in the middle-frequency range. The aforementioned studies were mainly focused on surface wave manipulation based on real band structures. As stated earlier, this method has inherent limitations. For example, for both 1D and 2D periodic wave barriers, an interesting phenomenon reported by [44, 45] is that the incident surface wave
3
can also be attenuated in a frequency range outside the band gap in real band structures. Unfortunately, an accurate explanation for this feature cannot be provided within the context of real band structures. Therefore, this study will revisit the surface wave attenuation in periodic trench barriers using complex band structures. Until now, there is limited research in this area. It is expected that this work can provide a better understanding of broadband surface wave isolation using periodic wave barriers. The paper is organized as follows. After this introduction, Section 2 describes the model and method employed to obtain complex band structures for surface waves. Next, in Section 3, results obtained with the present methodology are compared with those in related studies to validate the present method and physical model. Section 4 presents the complex band structure for surface waves and reveals the character of evanescent surface waves. It is found that there exist two wide frequency ranges of evanescent surface waves, indicating that periodic trench barriers can be designed to achieve broadband surface wave isolation. Section 5 discusses the influence of geometrical parameters and material damping on complex band structures and on screening effectiveness in the frequency domain. In Section 6, broadband surface wave attenuation in periodic trench barriers is further confirmed in the time domain by investigating the reduction of train-induced ground vibrations. Finally, the main conclusions are summarized in Section 7.
2. Model and method The physical model considered in this work is shown in Fig. 1. Since analytical solutions are extremely difficult to obtain, the analyses are performed using numerical techniques, including eigenvalue and response studies, which provide solutions for infinite and finite periodic structures, respectively. 2.1 Eigenvalue study The problem can be stated as follows. Considering trench barriers embedded in a semi-infinite soil substrate periodically, under the assumption of an infinite periodic extent, the dispersion analysis of 4
elastic waves can be represented by a unit cell shown in Fig. 1(a), in which Bloch-Floquet conditions are applied to the left and right boundaries. The trench depth, width, and lattice constant are assumed to be h , w and a , respectively. In the absence of body force, the governing equation of motion can be
written as ∇ ⋅ σ (r ) = ρ (r )
Where ∇
is the gradient operator; σ (r )
∂ 2 u (r ) ∂t 2
(1)
is the stress tensor that can be expressed as
σ ij = λδ ij ε kk + 2µε ij ; λ , µ are the Lamé constants; ε ij = ( ui , j + u j ,i ) 2 is the strain tensor; ρ (r) is the
mass density; r = ( x, y) is the position vector; u = (u1 , u2 ) is the displacement vector. According to Bloch-Floquet theorem, the displacement field can be expressed as the product of a periodic function and an exponential item as u(r) = U(r)ei(k ⋅r −ωt )
(2)
In which ω is the angular frequency; k is the Bloch wave vector; U(r) is a periodic function about the lattice. Substituting the Bloch-Floquet condition in Eq. (2) into Eq. (1), the governing equation can be transformed into an algebraic eigenvalue problem following FEM discretization procedure as Φ ( Λ, ω ) = 0
(3)
In which Λ = −ik is the eigenvalue to be solved for a specific frequency ω . The eigenequation of Eq. (3) is a typical k (ω ) formulation. In general, the wavenumber k = k Re + ik Im is a complex value. The real part kRe represents the propagation of a wave mode; and the imaginary part kIm denotes the attenuation property, which is termed as the attenuation coefficient. Usually, a wave mode can be treated as an evanescent wave mode if k Im ≠ 0 [12]. Eq. (3) can be solved with the aid of Partial Differential Equations (PDEs) in Comsol Multiphysics [46], in which an eigenvalue problem in a domain Ω has the general expression Λ 2 ea U − Λd a U + ∇ ⋅ ( −c∇U − α U + γ ) + β ⋅ ∇U + bU = f 5
(4)
Comparing Eqs. (1) and (2) with Eq. (4), we get the nonzero coefficients in Eq. (4) as follows, − ( λ + 2 µ ) 0 0 −λ −µ 0 0 −µ c= 0 −µ 0 −µ 0 − λ + 2µ −λ 0 ( ) ρω 2 b= 0
0 ρω 2
( λ + 2µ ) 0 ea = 0 µ ( λ + 2µ ) Λ 0 α = 0 λΛ − ( λ + 2 µ ) Λ 0 β = 0 −µΛ
(5)
(6)
(7)
0 µΛ µΛ 0
(8)
0 −λΛ −µΛ 0
(9)
For a rectangular unit cell Ω = [ 0, a ] × [0, H ] shown in Fig. 1(a), the periodic boundary conditions are U x =0 = U x = a −n ⋅ (−c∇U − α U ) x =0 = n ⋅ (−c∇U − α U) x = a
(10)
Where n is the outward unit normal vector on the domain boundary. Besides, the stress-free boundary condition is −n ⋅ (−c∇U − α U) y = H = 0
(11)
U y=0 = 0
(12)
and the fixed boundary condition is
6
It should be noted that the system considered herein yields both surface waves and other modes. To obtain pure surface modes from the solution, we implement a post-processing technique based on the center of strain energy along the y-axis, 1 ξ= H
∫
H
y ⋅ σ ij ε ij 2 dS
0
∫
H
0
σ ij ε ij 2 dS
(13)
Where H is the total height of the unit cell. Because the energy of Rayleigh waves is localized near the free surface, we can assume that the corresponding modes are Rayleigh wave modes if ξ ≥ 0.8 [28]. 2.2 Numerical modeling scheme Theoretically, the band structure of a unit cell is suitable for an infinite periodic system. The number of unit cells, however, is always finite in practice. Therefore, transmission analysis is carried out to assess the isolation performance of multi-row trenches. Although it is a 3D problem in practice, it can be simplified as a 2D plane strain problem [47, 48] when the barrier length is sufficiently long. The numerical model (40 m ×15 m) is depicted in Fig. 1(b). The incident surface wave is modeled by applying a vertical harmonic load F0 = 1000 N on the free surface. Although the load generates both surface waves propagating along the free surface and bulk waves in the soil substrate, displacements at the surface are dominated by surface waves in the far-field of the source. Thus, the effect of bulk waves is negligible [45, 49]. Perfectly Matched Layers (PMLs) are applied to the left, right, and bottom sides of the model to simulate half-space boundary conditions. The transmission coefficient is evaluated in terms of the amplitude reduction factor ( AR ) defined by Woods [50]: AR =
u y1 uy0
(14)
Where u y 0 and u y1 denote reference value without barriers and the real part of the vertical displacement with barriers at the detection point, respectively. Obviously, a smaller value of AR results
7
in a better vibration isolation effect. For example, AR = 0.3 indicates that wave barriers can reduce 70% of vertical displacement.
3. Comparison and verification To validate the numerical model derived in Section 2, we recalculate two numerical examples published in related studies. The first one is the harmonic response of an open trench subjected to a vertical harmonic load at f = 50 Hz . The trench’s width w = 0.1λR and depth h = 1λR , where λR is the corresponding Rayleigh wavelength. The results obtained from the present numerical model are compared with those from Ahmad et al. [51] and Bordon et al. [52] using conventional and dual Boundary Element Method (BEM), respectively, where good correlations can be observed among the three results, as shown in Fig. 2. The second comparative study is surface-wave band structures in a 1D periodic system consisting of rubber and polyfoam materials with dimensions a = 0.4 m and w = 0.2 m, which was studied by Huang et al. [44] using the traditional ω(k ) approach. As can be seen in Fig. 3, the results from the present model based on complex band structure agree with those based on the conventional real band structure. Another notable feature of the present model is that, other than real part of the wavenumber, it contains an imaginary part, which can characterize the attenuation of surface wave and will be discussed in more detail in the next section.
4. Complex band structures and evanescent surface waves In this section, we consider another complex band structure for surface waves, where the half-space and backfill materials are assumed to be isotropic and linearly elastic, with material parameters provided in Table 1. The geometrical parameters are a = 1m , h = 2 m , w = 0.3 m and H = 20a (see Fig. 1a). The unit cell is discretized into quadrilateral elements with cubic Lagrange shape function. Fig. 4 displays the complex band structure for surface waves. Unlike the traditional ω(k ) approach, the 8
complex band structure contains two adjacent panels, where the left and right panels show the real and imaginary part of the wavenumber, respectively. As pointed out earlier, the non-zero imaginary value represents the evanescent mode that decays exponentially in the direction of propagation. Two features can be immediately identified in Fig. 4. First, the lower and upper bounds of the band gap are connected by evanescent surface waves (see the lower shaded area). The real parts of these evanescent modes are located at the boundary point k = π a of the Brillouin zone. The other interesting feature is that there also exist evanescent surface waves above the band gap (see the upper shaded area), but their real parts are located inside the first Brillouin zone. In order to explain the physical mechanism, we display the dispersion curves of bulk waves in the soil substrate. It can be observed that the second branch of the surface-wave dispersion curves intersects the dispersion curve for longitudinal waves. This means that those modes inside the upper shaded area are leaky or pseudo surface waves because they contain small bulk-wave components that leak energy into the substrate. Mathematically, the leaky surface wave arises from complex wavenumber solutions. Therefore, it is an evanescent wave in a broad sense [53]. Further information on pseudo surface waves in phononic crystals can be found in [25, 28, 29, 54, 55]. To further confirm these evanescent modes, we investigate the transmission coefficient of six rows of trench barriers, with results plotted in Fig. 5. It can be seen that the ranges of transmission reduction agree with the areas where the evanescent waves exist, which implies that broadband surface wave isolation can be achieved using the evanescent wave properties. To further understand the evanescent surface wave, we display its decay process in Fig. 6(a), which shows the change of vertical displacement amplitude along the direction of wave propagation in the band gap at f = 54 Hz . Intuitively, the amplitude decays exponentially inside the trench barriers denoted by the shaded area. To validate this point, we select the peak points to fit an exponential function pe−qX , with the fitting results plotted in solid line, where p = 1.78 ×10−5 m and q = 0.503 m−1 . Apparently, the decay rate q = 0.503 m−1 agrees with the imaginary wavenumber Im(k) = 0.505 m−1 in the complex band 9
structure in Fig. 4. In addition, the displacement amplitude at the local coordinate X = 0 is u0 = 1.86 ×10−5 m , which is close to the fitting parameter p . As another example, we consider an
evanescent mode beyond the band gap at f = 74 Hz . The fitting result is shown in Fig. 6(b), and the corresponding parameters are p = 1.52 ×10−5 m and q = 0.257 m−1 . Similarly, the two parameters are also in agreement with the displacement amplitude u0 = 1.54 ×10−5 m and the imaginary wavenumber
Im(k) = 0.253m−1 in the complex band structure in Fig. 4. Based on the above findings, we can conclude that, unlike traditional real band structures, the complex band structure can accurately describe the exponential decay of evanescent waves. More importantly, there exist wide frequency ranges of evanescent surface waves, indicating that broadband surface wave isolation using periodic trench barriers is feasible.
5. Parametric analysis As described above, the attenuation of surface waves can be explained in terms of the complex dispersion relations because the imaginary part of the wavenumber characterizes the property of evanescent surface waves. Since dispersion relations are determined by geometrical and material parameters, periodic trench barriers can be designed to exhibit low-frequency and broadband evanescent wave properties based on the analysis of complex band structures. Using this concept, we will show how the design, including the geometrical and material parameters of a periodic trench barrier, can affect the frequency range of evanescent surface waves. 5.1. Influence of trench depth Earlier work [40, 44, 56] indicated that the trench depth plays a significant role in surface wave isolation. Specifically, these studies found that the surface wave screening efficiency increases with the increase of the trench depth and then remains almost constant. Although an explanation was provided qualitatively, a quantitative and accurate physical interpretation is still lacking, especially from the
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perspective of real band structures. Thus, we examine the effect of trench depth on evanescent modes of complex dispersion relations. The lattice constant a and trench width w are assumed to be 1 m and 0.3 m, respectively. Fig. 7 shows the result, which indicates that the increase of trench depth does not affect the lower and upper boundary frequencies of the band gap. The other message from Fig. 7 is that the increase in trench depth indeed results in a significant increase in the imaginary part of the wavenumber. However, when the trench depth is increased to h = 2a , the imaginary wavenumber remains unchanged. This feature means that the vibration isolation efficiency can be improved by increasing the trench depth until it reaches h = 2a , beyond which further increase in the trench depth no longer changes the efficiency of the barrier. This point is further confirmed by the harmonic analysis with six rows of trench barriers in Fig. 7. As expected, it can be observed that the incident surface waves attenuate greatly in the evanescent wave regions denoted by the shaded areas. More importantly, the magnitude of AR reduces with the increase in trench depth and then remains the same when it exceeds the depth h = 2a . The underlying mechanism can be explained as follows. Because the Rayleigh-wave energy is mainly concentrated in the depth range of one wavelength, the isolation efficiency does not increase significantly when the barrier depth exceeds this value. For the wavenumber at the boundary of the first Brillouin zone, i.e., k = π a , the corresponding wavelength is exactly equal to 2a . Besides, one can find that in the frequency range between 60-70 Hz, a trench with a shorter depth, i.e., h = a, provides a more considerable attenuation. It is difficult to explain this counter-intuitive result accurately. Although an in-depth discussion of this phenomenon is beyond the scope of this study, a brief qualitative explanation is presented here. The Rayleigh wavelength is 1.33-1.55 m in that frequency range, which is larger than h = a and less than h = 2a. When a Rayleigh wave is incident on a trench, it will give rise to reflected and transmitted Rayleigh waves, reflected and transmitted body waves, and body waves that radiate downward from the trench. When the trench depth is about one
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Rayleigh wavelength, it radiates downward body waves better [51]. Therefore, the transmitted Rayleigh wave is less than that in h = 2a. 5.2. Influence of trench width The trench width is an important geometrical parameter that characterizes the filling fraction. The lattice constant a and trench depth h are assumed to be 1 m and 2 m, respectively. As shown in Fig. 8, increasing the trench width does not affect the lower bound frequency of the band gap for surface waves, but can significantly increase the gap width and the imaginary part of the wavenumber. Similarly, we carry out the harmonic analysis of six rows of trench barriers to confirm this result. As expected, the corresponding transmission coefficients in Fig. 8 show the same phenomena, which is consistent with what was found for a single in-filled trench [51, 57]. This is because of the destructive interference of elastic waves, which is enhanced when the filling fraction is increased. Based on the above analyses, it can be stated that the increase of the trench width can facilitate broadband surface wave screening. 5.3. Influence of lattice constant The lattice constant is another important parameter that characterizes the geometrical property. Fig. 9 shows the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different lattice constants. The trench width w and depth h are assumed to be 0.3a and 2a , respectively. It can be seen that the lower and upper bounds of the band gap decrease as the lattice constant increases, which can be explained in terms of the Bragg condition f ∝ 1 a [1], i.e., the increase of the lattice constant naturally leads to the decrease in frequency. On the other hand, it can be seen again that the increase in the lattice constant does not decrease the imaginary part of the wavenumber, but expands the frequency range of evanescent surface waves. This feature implies that the increase of lattice constant is advantageous for achieving broadband surface wave isolation. Similar to what is observed above, the transmission coefficients of six rows of trench
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barriers strengthen this viewpoint, as shown in Fig. 9, where the attenuation zones in transmission coefficients agree with the evanescent surface-wave regions predicted by the Bloch-Floquet theorem. 5.4. Influence of shear wave velocity The effect of shear wave velocity of soil on complex band structure and amplitude reduction is evaluated. The shear wave velocity is varied from 110 m/s to 150 m/s by changing the soil modulus while keeping other parameters as constant. It can be observed from Fig. 10 that the lower boundary frequency of the band gap increases as shear wave velocity increases. The reason is that the increase in shear-wave velocity means that the stiffness of the system increases, thereby resulting in an increase in frequency. It can also be observed that the bandwidth and attenuation coefficient also increase with the increase of the shear-wave velocity, which is consistent with the harmonic response of six rows of trench barriers in terms of amplitude reduction (AR) in Fig. 10. This phenomenon can be understood as follows. It is known from elastic wave theory that the impedance ratio of materials on both sides governs the transmission and reflection at the interface [58]. For materials considered in this paper, the impedance ratio is less than one, meaning that the infill is a soft material. The impedance ratio decreases as the shear-wave velocity of soil increases. For trenches filled with soft material, the decrease in the impedance ratio results in better vibration screening, because a very soft barrier is close to the condition of an open trench that has a very low transmission. 5.5. Influence of soil damping The foregoing analyses are carried out in the absence of material damping. Materials in practice always exhibit dissipative behavior. In dissipative materials, the frequency or the wavenumber are complex values, i.e., all waves are evanescent modes. Since real band structure cannot display this feature, the complex band structure is used to reveal the spatial attenuation characteristic of evanescent waves. In engineering practice, the dissipation is described using the hysteretic material damping ratio
η and the complex Lamé constants, which are expressed as λ* = λ (1 + 2ηi) and µ * = µ (1 + 2ηi) [59].
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Neglecting the backfill material damping, we show the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different damping ratios in Fig. 11. As expected, it can be observed that there exists a non-zero imaginary part of the wavenumber for every frequency, indicating that all surface waves are evanescent waves in the context of soil damping. Strictly speaking, it is difficult to define band gaps since all wavenumbers are complex values. For convenience, the band gap in the absence of material damping is displayed as a reference (see the shaded area). Another notable feature found in Fig. 11 is that the imaginary part of the wavenumber remains almost constant in the band gap, but increases with the increase of damping ratio outside the band gap. The same characteristic can be found in the transmissions in Fig. 11. This phenomenon could be explained as follows. Damping in soil always acts as dissipative effects on mechanical energy. Thus, the inhibition of wave motion is enhanced as the damping ratio increases. In addition, waves decay exponentially in the band gap caused by Bragg scattering. In contrast, the effect of low damping is no longer obvious. 5.6. Influence of backfill material damping In order to evaluate the influence of backfill material damping, we neglect the soil damping. Fig. 12 shows the imaginary part of the wavenumber and transmission coefficients as a function of frequency for different damping ratios of the backfill material. Similarly, all surface waves are evanescent waves with consideration of backfill material damping. Similar to the results when considering soil damping in Fig. 11, the imaginary part of the wavenumber remains almost unchanged in the band gap, but increases with the increase of damping ratio outside the band gap. The most striking difference is that the degree of damping effects in backfill materials is not as high as that in soils. This result can be explained as follows. From the perspective of the whole model, the area occupied by the trench barriers is small. Therefore, the influence of the backfill material damping is naturally not as big as that caused by the soil damping. Overall, damping is always advantageous for wave attenuation for Bragg scattering wave barriers.
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6. Application to train-induced ground-borne vibration isolation The aforementioned analysis is extended to an application in Civil Engineering in this section. Doing so, we consider the isolation of ground-borne vibration induced by moving trains, since it is an important environmental issue, especially in densely populated areas [60, 61]. To this end, we measured the vertical acceleration of train-induced ground vibrations in Beijing, China, with measurement sites shown in Fig. 13. Fig. 14 displays the time history and corresponding Fourier spectra of ground vibrations induced by subway lines 1 and 13. As can be seen from Figs. 14(b) and (d), the train-induced vibrations contain many frequencies, with the main frequencies ranging from 30 Hz to 80 Hz. For the two acceleration inputs, numerical calculations are performed in the time domain. The boundary condition of the source is u&&y = Acc(t ) , where the function Acc(t ) is the time history at the original site. Fig. 15 shows vertical acceleration responses and corresponding Fourier spectra at the detection point from the numerical model in Fig. 1. The solid and dashed lines denote the results obtained with and without six rows of trenches (a = 1.6 m, h = 2a, w = 0.3a), respectively. Obviously, the ground-borne vibration is significantly mitigated with the presence of six rows of trench barriers. The reduction of the vertical acceleration amplitude is about 70% and 90% for subway lines 1 and 13 respectively, when compared with those without trench barriers. More importantly, it can be seen from the Fourier spectra that the frequency ranges of vibration reduction are in agreement with the evanescent surface-wave regions calculated using the Bloch-Floquet theorem (see shaded areas in Figs. 15b and 15d). Again, the results in the time domain confirm the existence of wide evanescent surface-wave regions in periodic trench barriers and the feasibility of applicability of broadband surface wave isolation in practice.
7. Conclusions This paper numerically studies the attenuation behavior of surface waves in periodic trench barriers, where both elastic and dissipative cases are considered. The spatial decay of surface waves is
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investigated using complex band structures through solving the dispersion equation for assigned real-valued frequencies. It is found that the periodic trench barriers possess broadband evanescent surface-wave regions, including the band gap and leaky surface modes. The transmission analysis shows that the incident surface wave decays exponentially in wave barriers when its frequency is within evanescent surface-wave regions. In particular, all surface waves become evanescent modes when considering material damping. It is difficult to identify the band gap without the complex band structure of an undamped system for reference. Based on the results from the complex dispersion and the transmission analysis, it can be concluded that the periodicity-induced wave scattering dominates the attenuation of surface waves, and the influence of damping is small in the band gap. While outside the band gap, the attenuation caused by damping is dominant. The trench depth is an important design parameter, which is recommended to be twice the lattice constant based on the complex dispersion and transmission analysis. Moreover, the results from the parametric study indicate that periodic trench barriers are capable of blocking broadband surface waves by tuning geometrical parameters, simultaneous increase of the lattice constant, and the trench width. The feasibility is further validated in the time domain by evaluating the performance of railway induced ground-borne vibration isolation.
Acknowledgments This work was supported by the National Natural Science Foundation of China (51878030). We wish to thank the editor and the anonymous reviewers for their constructive comments.
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Tables and captions Table 1. Parameters of the elastic half-space and the trench barrier [40]. Materials
Density (kg/m³)
Young modulus (MPa)
Poisson ratio
Soil EPS geofoam
1800 60
46 37
0.25 0.32
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Figures and captions
Fig. 1. 2D plane strain models. (a) The unit cell used for band structure calculations. (b) Multi-rows of trench barriers embedded in a semi-infinite soil substrate. ( l = 46 m , l1 = 21 m , l2 = 3 m and l3 = 18 m ).
Fig. 2. Comparative study for ground vibration screening using open trench.
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Fig. 3. Comparative study for band structure of surface waves in a periodic rubber-polyfoam system.
Fig. 4. Complex band structure for surface waves in a periodic system composed of soil and geofoam-filled trenches (a = 1 m, h = 2a, w = 0.3 m). The normalized displacement field of three eigenmodes labeled by points A, B, and C characterizes the feature of Bloch surface waves.
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Fig. 5. Transmission coefficient as a function of frequency for six rows of trench barriers (a = 1 m, h = 2a, w = 0.3 m). The shaded areas correspond to the evanescent surface-wave regions in Fig. 4.
Fig. 6. Harmonic responses with six rows of trenches (a = 1 m, h = 2a, w = 0.3 m): (a) f = 54 Hz, (b) f = 74 Hz. Trenches are located in the shaded area. The solid line represents fitting results. 24
Fig. 7. Influence of trench depth (a = 1 m, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
Fig. 8. Influence of trench width (a = 1 m, h = 2a): (a) w = 0.2 m, (b) w = 0.25 m, (c) w = 0.35 m, (d) w = 0.4 m. Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
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Fig. 9. Influence of lattice constant (w = 0.3a, h = 2a): (a) a = 1.2 m, (b), a = 1.4 m, (c) a = 1.6 m, (d) a = 2 m. Six rows of trench barriers are considered for harmonic analysis. The shaded areas represent the evanescent surface-wave regions.
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Fig. 10. Influence of shear wave velocity. Six rows of trench barriers are considered for harmonic analysis.
Fig. 11. Influence of soil damping (a = 1 m, h = 2a, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded area represents the surface-wave band gap in the absence of material damping.
Fig. 12. Influence of backfill material damping (a = 1 m, h = 2a, w = 0.3 m). Six rows of trench barriers are considered for harmonic analysis. The shaded area represents the surface-wave band gap in the absence of material damping.
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Fig. 13. Vibration measurement sites in Beijing, China: (a) subway line 1, (b) subway line 13.
Fig. 14. The vertical acceleration record of train-induced ground vibrations and corresponding Fourier spectra: (a-b) subway line 1, (c-d) subway line 13.
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Fig. 15. Vertical acceleration responses at the detection point and corresponding Fourier spectra with and without six rows of trenches (a = 1.6 m, h = 2a, w = 0.3a): (a-b) subway line 1, (c-d) subway line 13. The shaded areas represent the evanescent surface-wave regions predicted by the Bloch-Floquet theorem.
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Author Contribution Statement Xingbo: Calculating, Analyzing and Writing. As the corresponding author, Zhifei financially support the research and is responsible for the quality of the paper.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
No