Trees as large-scale natural metamaterials for low-frequency vibration reduction

Construction and Building Materials 199 (2019) 737–745

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Trees as large-scale natural metamaterials for low-frequency vibration reduction Yi-fan Liu a, Jian-kun Huang a,⇑, Ya-guang Li a, Zhi-fei Shi b a b

Key Laboratory of State Forestry Administration on Soil and Water Conservation, Department of Civil Engineering, Beijing Forestry University, Beijing 100083, China Institute of Smart Materials and Structures, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

h i g h l i g h t s
Urban trees are regarded as natural metamaterials to reduce ground vibrations.
The influences of material and geometrical parameters on the surface-wave attenuation zone are investigated.
The feasible application of periodically arranging urban trees is illustrated via 3D FEM.
The band gap generation mechanism is analyzed in detail.

a r t i c l e

i n f o

Article history: Received 17 August 2018 Received in revised form 3 December 2018 Accepted 12 December 2018

Keywords: Urban forest Metamaterials Dispersion curve Band gap Rayleigh wave

a b s t r a c t Urban trees are often planted in a periodic arrangement in space. This study takes urban trees as natural metamaterials. Based on the metamaterial theorem, a three-dimensional unit cell is used to represent the entire forest, and finite element analysis via COMSOL software is implemented to identify dispersion relationships. The surface-wave band gaps of urban forests are identified using the sound cone method and strain energy method. The influences of soil elastic modulus, tree spacing, trunk radius, and tree height on band gaps are discussed. Finally, a three-dimensional simulation model is established to verify the effect of urban trees on vibration reduction. The results show that numerical frequency-reduction zones are consistent with theoretical surface-wave band gaps. An increasing soil modulus results in a wider and higher-frequency band gap. Urban forests with larger trunk diameters and smaller tree distance can generate lower-frequency and wider band gaps. It is beneficial to obtain low-frequency band gaps with increasing tree height. A manmade engineered array of trees can be designed with Rayleigh wave band gaps at a low frequency of 80 Hz. This study provides a new concept for the quantitative design of urban forests to reduce ground vibration in specific frequency ranges. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Noise and vibration are common phenomena in engineering and cause high-precision system malfunctions and extensive sound environment pollution. Some engineering measures for reducing vibration and noise, such as open or filled trenches [1–3] and pile barriers [4–6], have some effect; however, they are costly and run counter to the concept of ecological construction. Urban forests have attracted increasing attention as an essential component of cities, serving to reduce traffic vibrations and noise [7–11]. Urban forests can reduce vibration and noise and improve the ecological environment along transportation lines. ⇑ Corresponding author. E-mail addresses: [email protected] (Y.-f. Liu), [email protected]. cn (J.-k. Huang), [email protected] (Y.-g. Li), [email protected] (Z.-f. Shi). https://doi.org/10.1016/j.conbuildmat.2018.12.062 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

They thus offer an efficient, economic, and ecological means of reducing vibration and controlling noise. Much research has focused on identifying logical vegetation collocation to solve vibration and noise problems since discovering sound attenuation in the jungle [12]. Several interesting and fundamental design problems, including the function of urban forests in vibration and noise reduction, have been implemented in recent studies. The attenuation of low-frequency noise is caused by indirect interactions among plants and ground reflections [7]. Noise reduction in plant communities is strongly associated with plant morphology such as leaf area, leaf fresh weight, and leaf shape [8]. The influence of structural parameters including different lengths, widths, and arrangements of green belts on noise reduction has also been investigated [9]. Within a certain frequency range, periodically arranged trees demonstrate better attenuation effects on vibration and noise than those are arranged in a

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disorderly manner [13]. However, research on urban forests has been mainly empirical and the reduction mechanism remains unclear. It lacks a physical model to illustrate the dispersion of vibration and noise and the quantitative characteristics have not been systematically considered. Metamaterials are artificially engineered composites which exhibit superior properties not observed in the constituent materials or nature. Essentially, the term of ‘‘metamaterial” is also a novel idea for the material design. This idea breaks through the limitation of apparent nature rule by the order structure in certain key physical scale in order to acquire extraordinary material functions. Trees as natural materials, are artificially arranged in a periodic distribution. This artificial periodic distribution has a unique bandgap characteristic compared to the disordered distribution in the natural state [13]. Therefore, we regard this arrangement of trees as natural metamaterials. Research on metamaterials originated from locally resonant phononic crystals. When the design size of elastic structural elements is reduced to sub-wavelength dimensions, then artificial periodic structures can generate low-frequency band gaps [14], extraordinary material parameters [15], and negative refraction [16] and wave-mode conversion effects [11]. The unique low-frequency sound absorption characteristics of acoustic metamaterials have led to extensive research on applications of these metamaterials in vibration and noise reduction. The finite element method is commonly used to analyze the energy dissipation mechanism in locally resonant periodic structures, and the influence of various parameters on locally resonant sound absorption materials has also been investigated [17,18]. The cylindrical local resonant structure can further improve low-frequency sound absorption performance compared to spherical local resonant structures [19]. Network-based sound-absorbing structures can achieve broadband absorbing effects in frequency bands above 5 kHz [20]. The above-mentioned research demonstrates the reliability of applying metamaterials theory to nano- or meso-scale fields, and observed frequencies in the kHz to GHz range. Considering the common tree height in the city (8–15 m) and surface wave wavelengths (k = five to a few tens of meters), trees can be considered sub-wavelength local resonant units. Specifically designed forests could lead to traffic vibration reduction and protect the surrounding buildings in lower frequency ranges. Although forests have been considered phononic crystals [13] and used as seismic cloaks [21], the common calculation model is a simple two-dimensional (2D) system [11,21]. The thickness is infinite in the out-of-plane direction, and the trees were simplified as a multi-row of ‘‘tree walls” in the 2D model. In addition, 2D model is based on the assumption of plane strain model and only two degrees exist in this type of model. Vibrations can only occur in the in-plane, thus losing the information on the out-of-plane in the 2D model, which results in losing some important model information [22]. A dedicated design is hence needed to achieve useful vibration reduction with limited tree rows. The Rayleigh-wave is a three-dimensional (3D) elastic wave problem. Therefore, the 2D model is extended to a 3D solution for the band calculation. The 3D model is used to calculate dispersion curves and simulate the wave field, which can objectively illustrate Rayleigh wave propagation. This article further regards the forest as a natural metamaterial and controls the band gap within a low frequency range (80 Hz), which is involved in traffic vibration control [23–26]. The influences and mechanism of the band gap were investigated comprehensively and indicate the feasibility of using periodic trees as large-scale natural metamaterials. The remainder of this article is organized as follows. First, a reasonable ‘‘tree-soil” unit cell was selected to investigate the characteristics of Rayleigh-wave dispersion curves via finite element analysis. In addition, the sound cone method and strain energy

method were used to distinguish the surface-wave bands. Second, band gap features were analyzed by altering the geometric parameters of trees. Finally, a finite periodic trees model was used to numerically validate the effectiveness of the band gap in terms of vibration control, and the generation mechanism was investigated by illustrating vibration modes. 2. Model and method 2.1. Research object This paper provides a new perspective for studying traffic vibration and noise reduction from urban forests. As urban forests are often planted in a periodic arrangement around a building, as shown in Fig. 1(a), this arrangement can be considered a natural metamaterial. The resonance generated by the trees interacts with elastic waves in soil (i.e., Rayleigh waves), thereby suppressing Rayleigh wave propagation at specific frequencies. This attenuation phenomenon is also referred to as a band gap. Band gap generation depends on the self-resonance characteristics of scatterers and interactions with elastic waves in the substrate. The frequency range of a band gap is closely related to the inherent vibration characteristics of a given scatterer. Each tree has its unique characteristics, including material and geometrical properties. However, trees in urban greening forests often share similar varieties and shapes in the actual situation. Therefore, we assume that the trees have the same material and geometrical properties in the modelling. A minimal repeated unit can be selected as the unit cell to replace the entire urban forest according to Bloch’s theory, as shown in Fig. 1(b). Periodic boundary conditions were imposed on the boundaries of the unit cell, which accurately reflects the complexity of the actual site. The periodic arrangement of urban forests can theoretically inhibit the propagation of elastic waves in the band gap. 2.2. Model parameters The ‘‘tree-soil” unit cell constitutes a locally resonant metamaterial that can be simplified as a resonant system consisting of bending beams. Considering the composition of tree biomass, the trunk occupies 80% of the total mass of the upper tree. Urban trees can be approximated as vertical resonators under the observed wavelength (k up to 10 m); therefore, the influence of branches and crowns were ignored when establishing the calculation model in this paper. The tree acts as a cantilever beam, whose roots mainly play a role in connecting tree and soil. The influence of tree roots was also negligible, as the scattering effect had a lower order compared to the trunk [21]. In addition, the shape of root is complex. Therefore, the influence of tree roots is ignored in the present research. Considering the actual growth of common tree species, the shape of the tree was approximated as a circular-truncated cone. The tree spacing (i.e., periodic constant) was a = 2 m; the tree height was l = 12 m; the radius of the bottom truck was r = 0.3 m; and the radius of the top trunk was 0.5r as shown in Fig. 1(b). The motion of surface waves decays exponentially with the depth of the media, whereas the surface-wave energy is principally localized within a Rayleigh wavelength [27]. Based on this feature, adequate soil depth [28] is needed to decouple surface waves from bulk waves [29]. The soil depth h should be sufficiently large compared to the tree spacing [30,31]. Hence, a unit cell in this paper with a height of 30 m (i.e., 10–15a) was considered reasonable for band gap calculation. The 3D unit cell model is shown in Fig. 1(b). As in previous studies [21,32], the elastic modulus of the tree was E1 = 1.67 GPa, Poisson’s ratio was t1 = 0.3, and tree density was q1 = 700 kg/m3. The soil elastic modulus was

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Fig. 1. (a) Rayleigh waves caused by traffic vibrations and forests with periodic arrangement; (b) unit cell; (c) wave vector in first Brillouin zone (dashed line) and first irreducible Brillouin zone (solid line).

E2 = 0.1E1, Poisson’s ratio was t2 = 0.35, and soil density was q2 = 1700 kg/m3. 2.3. Calculation method

2.4. Post-processing methods

Assuming the material is continuous, elastic, and undamped, the harmonic motion of a 3D unit cell can be drawn as follows [33]:

r  ðCðrÞ : ruðrÞÞ þ qðrÞx2 uðrÞ ¼ 0

ð1Þ

where r is a differential operator; r = (x, y, z) is the position vector; u(r) is the displacement vector; x is the circular frequency; C(r) and q (r) are the position-dependent elastic stiffness tensor and mass density, respectively. According to Bloch theory, the displacement field in the periodic system can be expressed as

uðr; tÞ ¼ eiðKrxtÞ uK ðrÞ

ð2Þ

where K is the reduced wave vector, and uK(r) is a periodic function with the same periodicity. Thus, uK(r) can be written as

uK ðrÞ ¼ uK ðr þ aÞ

ð3Þ

Substituting Eq. (3) into Eq. (2), the periodic boundary condition is obtained:

uðr þ aÞ ¼ eiKa uðrÞ

The theoretical calculations can be implemented by choosing the appropriate operations.

ð4Þ

The dispersion curves calculated by the 3D model included surface-wave modes and bulk-wave modes. Given the unique characteristics of surface wave propagation, the sound cone method [35] was used to obtain pure surface-wave modes. The sound line limiting the sound cone corresponds to the smallest phase velocity of the soil. Sound lines are calculated with different directions in the first Brillouin zone and form a sound cone by the formula pffiffiffiffiffiffiffiffiffi x ¼ K  cs , where cs ¼ l=q. cs is the shear wave velocity of the soil, l is the soil shear modulus, and q is the soil density. The region below the shear wave velocity of the soil in the dispersion curves contains only surface wave modes; therefore, surface wave modes can be identified from all dispersion curves of the sound cone as shown in Fig. 2. The black thick line is the sound line, and the gray range below the sound cone is the surface-wave band gap. The area beyond the sound line is a continuous region indicating all possible modes of bulk waves and higher-frequency surface-wave bands. The first 100 bands were preserved in this paper. The sound cone method is suitable for a locally resonant system. Locally resonant and Bragg scattering systems have no sharp

According to the above formula, the infinite periodic structure can be reduced to a unit cell for calculation. The cell should satisfy the periodic boundary conditions shown in Eq. (4). Then, the eigenvalue equation for the unit cell can be written as

ðXðKÞ  x2 MÞ  u ¼ 0

ð5Þ

where the stiffness matrix X is a function of the wave vector, and M is the mass matrix. Due to the symmetry of the Brillouin zone, the wave vector must only be defined in the first irreducible Brillouin zone, as shown in Fig. 1(c). COMSOL, a software which provides a set of predefined physics equations for simulating various physical phenomena, is commonly applied to calculate the dispersion curves for periodic systems [1,33]. The eigenvalue mode in COMSOL is used to implement the formula derived as Eq.(5) [34]. The calculation of dispersion curves were based on COMSOL software, which can solve the eigenvalue problem with complex boundaries as Eq.(4). The top surface of the model was free, whereas the periodic boundary conditions (i.e., Eq. [4]) and the fixed boundary condition (i.e., u = 0) can be directly imposed on the boundaries of the typical cell.

Fig. 2. Dispersion curves of ‘‘tree-soil” unit cell.

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division between each other. Although the ‘‘tree-soil” system mainly reflects the resonant characteristic, the Bragg scattering feature still exists in the system. The sound cone method omits the surface-wave modes beyond the shear wave. Therefore, the strain energy method is also presented here to comprehensively investigate higher-frequency surface modes. Surface waves propagate in the soil surface, and bulk waves propagate into the deep layer. The energy of surface waves is mainly concentrated on the ground surface within one or two Rayleigh wavelengths [27]. Therefore, we can observe the energy distribution of the ‘‘treesoil” system and verify the dispersion curves via the strain energy method as in the following equation:

RRR

1 T S dxdydz h 2 ij ij DOE ¼ RRR 1 1 T S dxdydz h 2 ij ij

ð6Þ

where T ij is the stress tensor; Sij is the conjugate for strain tensor; DOE denotes the distribution of energy. The integral domain of the numerator corresponds to h1 (h1 = 2a range from the top surface of soil), and the integral domain of the denominator corresponds to h (i.e., the entire soil), as shown in Fig. 1(b). The strain energy density is integrated to obtain the elastic strain energy [33]. The value of DOE varies from 0 to 1. When the DOE is up to a critical value, such as 0.9, the modes are saved as the surface-wave mode due to the concentration of surface waves near the ground [27]. The stress and strain were also obtained by COMSOL. In fact, we can conveniently integrate the strain energy density of the specified area directly by COMSOL and obtain the corresponding strain energy. Higher-frequency surface-wave modes were distinguished by the strain energy method marked by the red dots in Fig. 2. The internal dispersion curves of the sound cone coincide exactly with the surface wave modes identified by the strain energy method; therefore, the sound cone method can be used to identify surface-wave dispersion curves from all curves for locally resonant bands. The strain energy method was used to complement the sound cone method to obtain surface-wave dispersion dots for higher-frequency bands. Surface-wave modes can be more accurately determined by combining the sound cone method and strain energy method. 2.5. Case verification Research on metamaterials has mainly focused on microscopic fields such as electromagnetics and materials. This paper extends such research from the microscopic field to the macroscale field. To verify the accuracy of the present method, the previous research, which was on a microscopic scale but similar in that it evaluated large-scale natural trees, was studied once more. Fig. 3

displays the dispersion curves of a phononic crystal with pillars on the surface [35]. The substrate and scatterer are composed of Sih1 0 0i. The sound cone method and strain energy method were used to obtain the dispersion curves. A ‘‘CX” directional band gap appeared, and the present solutions of the band structure (i.e., black solid thin lines and hollow circles) concurred with previous results (i.e., solid circles), thereby validating the feasibility of the present method.

3. Results and discussions The dispersion curves of metamaterials depend on the material and geometrical parameters. To investigate the variation of band gaps in ‘‘tree-soil” metamaterials, four major parameters (soil elastic modulus, tree spacing, trunk radius, and tree height) were selected to illustrate band gap adjustability. In Figs. 4–7, the Xth means the Xth band in the dispersion curves from the low frequency range to the high frequency range. The interval between two parts of dispersion curves is the band gap, which is marked by gray shadow. Parts of low-order bands cannot produce band gaps; therefore, these bands don’t appear in Figs. 4-7.

3.1. Effect of soil elastic modulus After substantial trial and error, the material properties of the substrate were found to significantly influence the dispersion curves. The soil included three primary parameters, namely elastic modulus, Poisson’s ratio, and density. Poisson’s ratio and soil density often vary slightly, and the soil elastic modulus can vary in a wide range under different site conditions, which determine the shape of the sound cone. Therefore, the effect of the soil modulus on the band gap is shown in Fig. 4 (a = 2, r = 0.3, l = 12). As the soil elastic modulus increased, the width of the second band gap increased from 5.39 Hz at 100 MPa to 14.84 Hz at 200 MPa, and the first band gap gradually narrowed and tended to disappear. The lower bound frequency (LBF) of the second band gap and the upper bound frequency (UBF) of the first band gap exhibited little change. The main reason for the increasing band-gap width was that the UBF of the second band gap increased continuously. According to the calculation formula of the sound cone, when the soil elastic modulus increased, so did the shear wave velocity of the substrate, causing an uplift in the sound line that resulted in a larger sound cone area. Since the geometrical parameters of the model had not changed, the position of the dispersion curves in the sound cone did not vary drastically. The larger soil elastic

Frequency/(Hz)

50

11th-12th 9th-10th

45 40

2nd BG

35 1st BG

30 100

125

150

175

200

Soil elastic modulus/(MPa) Fig. 3. Dispersion curves of phononic crystal with pillars on the surface.

Fig. 4. Influence of soil elastic modulus on band gaps.

Y.-f. Liu et al. / Construction and Building Materials 199 (2019) 737–745

90

Frequency/(Hz)

75

15th-16th 13th-14th 11th-12th 9th-10th

4th BG

60

3rd BG

45

2nd BG

30

1st BG

1.0

1.5 2.0 Tree spacing/(m)

2.5

Fig. 5. Influence of tree spacing on band gaps.

50

Frequency/(Hz)

45

4th BG 3rd BG

40

2nd BG

15th-16th 13th-14th 11th-12th 9th-10th

35

0.10

0.15

1st BG

0.20 0.25 Tree radius/(m)

0.30

741

fraction has often been used as a parameter to study the band gap. The filling fraction can be increased by either reducing the tree spacing or increasing the trunk radius. The wave vector K has a direct relationship with tree spacing a. When the tree spacing changes, so do the wave vectors K, thus affecting the position of the sound line. If the trunk radius changes, the position of the dispersion curves changes as well; however, the coverage area of the sound cone remains the same. As an example, if the trunk radius r is reduced from 0.3 m to 0.2 m and tree spacing a varies from 3 m to 2 m, then the model has the same filling fraction (F = 3.14%). However, the dispersion curves, band gap position, and band gap width of these two models are completely different. Therefore, the tree spacing and trunk radius are investigated individually in this section. Considering the actual site conditions, the soil elastic modulus was set as 167 MPa. The tree height (b) and trunk radius (r) were kept as 12 m and 0.3 m, respectively. The tree spacing was increased from 1 m to 2.5 m as shown in Fig. 5. When the tree spacing declined, the LBF of band gaps increased slightly. When the width increased to a certain point, a new higher-frequency band gap began to appear. As the tree spacing continued to decrease, the bandwidth of the previous band gap remained almost unchanged, and the new higher-frequency band gap reappeared. The multiple band gap feature appeared as the tree spacing reduced, extending the range of vibration reduction considerably. The mutative tree spacing altered the range of the first Brillouin zone; however, the geometric parameters of the scatterers exhibited no change, and the position of the dispersion curves thus varied slightly. When the size of the tree model was constant, higher-frequency and wider band gaps were generated due to smaller tree spacing. This pattern suggests that urban forests should be planted closely together. By considering the actual size of trees, lighting, and nutrients, a tree spacing of a = 2 m was chosen as a reasonable planting distance in the following discussions. 3.3. Effect of trunk radius

Fig. 6. Influence of trunk radius on band gaps.

modulus displayed more broadband vibration reduction, and the starting frequency of vibration reduction was higher.

The influence of trunk radius on the band gap is shown in Fig. 6. The tree bottom radius varied from 0.1 m to 0.3 m, and the corresponding top radius was equal to half of the bottom radius. In Fig. 6, the dispersion curves were sensitive to the change in trunk radius. The end frequency of the band gaps floated around 44 Hz, and the starting frequency of the band gaps declined continuously as the trunk radius increased. The total width of the band gaps showed a substantial overall increase with increasing trunk radius. The 4th band gap began to appear once the trunk radius increased to a certain value. For the 3rd band gap, when the trunk radius reached up to 0.15 m, the band gap began to form and gradually replaced the 4th band gap. When the trunk radius was up to 0.20 m, the 2nd band gap appeared; the bandwidth of the band gaps tended to reach a peak at r = 0.3 m. It is easier to open the band gap by low-order vibrations when the trunk radius is larger. Conversely, the smaller the radius, the higher order vibrations can open the band gap. In general, each band gap underwent a progressive process from generation to growth and disappearance, but this process was complemented by lower-frequency band gaps as the trunk radius increased. In fact, an increase in the trunk radius can be regarded as a process of normal tree growth. During this process, the damping effect of trees became increasingly obvious, and the attenuation range covered the original area and expanded to the lower-frequency range.

3.2. Effect of tree spacing

3.4. Effect of tree height

The filling fraction (F = pr2/a2) is an important parameter for opening the band gap and controlling the bandwidth. The filling

The effect of tree height on the band gap is shown in Fig. 7. The tree spacing a was equal to 2 m, and the trunk radius r was equal to

50

Frequency/(Hz)

45 40

1st BG

2nd BG

3rd BG

35 13th-14th 11th-12th 9th-10th

30 25

9

10

11 12 13 Tree height/(m)

14

15

Fig. 7. Influence of tree height on band gaps.

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0.3 m. The end frequency of the band gaps varied slightly along 47 Hz with an increase in tree height from 9 m to 15 m. The LBF of the band gap declined rapidly, and the entire band gap width became larger. If the tree was too short, it is disadvantage to generate band gaps. The band gap tended to appear when the tree height reached up to 9 m. As the tree height increased, the LBF of the 1st band gap decreased rapidly. The bandwidth increased continuously to the peak and then receded. The band gap width was compressed continuously; at the same time, the 2nd band gap began to appear, which compensated the shrinkage of the 1st band gap. When the tree height increased up to 12 m, the first band gap tended to disappear. The width of the 2nd band gap peaked, and the 3rd band gap appeared to remedy the defect. In this case, the ‘‘tree-soil” unit cell can be simplified as a resonant system consisting of bending beams, and the tree acts as a cantilever beam. When the tree height (i.e., the length of beam) increased, its self-resonant frequencies decreased. The total reduction region extended further as the tree height increased. Therefore, urban trees appeared to have a better reduction effect in line with trees’ natural growth.

4. Numerical simulation The theoretically calculated band gap exists in a perfect, infinitely repeated structure; however, actual urban trees have a limited number of rows. To verify the effectiveness of band gaps in urban forests, a 3D simulation model with multiple rows of trees was established. Dynamic responses were calculated using the finite element software ANSYS17.1. A schematic diagram of the 3D model is shown in Fig. 8(a), taking eight rows of trees as an example. The geometric parameters of the simulation model were as follows: tree spacing a = 2 m; trunk radius r = 0.3 m; tree height l = 10 m; and soil height h = 30 m. The vibration source and observation line were at a distance of 2a away from the neighboring unit cell.

In the simulation, it is inevitable to intercept the finite-size calculation model. In order to eliminate the wave reflections caused by truncated boundaries, viscoelastic boundary conditions were imposed on the simulation models via ANSYS Parametric Design Language. Viscoelastic boundary elements have been found to possess high precision, adequate stability and robustness [36]. The boundary elements were added on the surfaces perpendicular to the x-direction and the bottom surface as shown in Fig. 8(b). Moreover, periodic boundary conditions were added on the surface perpendicular to the y-direction to minimize the calculation model. Fig. 9 shows the frequency response function (FRF) and dispersion curves of urban trees with l = 10 m. The theoretical band gap of urban trees within the sound cone was from 38.34 Hz to 47.94 Hz, and the vibration attenuation zone of the FRF aligned well with the theoretical band gap. In Fig. 9(a), the vibration began to decay in the band gap range after two rows of trees. The vibration attenuation was more obvious as the number of tree rows increased. The vibration was attenuated substantially after eight rows of trees. In practical engineering applications, additional rows of trees require more space and cost. Three or four rows of trees with a periodic arrangement should be considered in urban forests, which can still obtain a reasonable attenuation effect. The frequency range of the locally resonant band gap was lower than that of the Bragg scattering band gap [37]. In Fig. 9(a), two main frequency reduction zones appeared in the FRF curves. The lower-frequency zone was the locally resonant band gap; the higher-frequency zone was the Bragg scattering band gap, although this band gap was a ‘‘CX” directional gap. Therefore, locally resonant and Bragg scattering phenomena each existed in the tree-soil system. Beyond the sound line, the Bragg scattering band gap was consistent with the higher-frequency reduction zone. The tree-soil system also indicated that the locally resonant and Bragg scattering systems were not sharply divided. To further illustrate the attenuation features, the dynamic responses of periodic trees with another tree height l = 12 m are shown in Fig. 10. The Fano phenomenon can be observed in

l

Z

k3 c2 Y

h

c3

k1 c1

k2

z x

L

y a

(a)

(b)

Fig. 8. Schematic diagram of periodic trees. (a) 3D model; (b) viscous-spring boundary.

X

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Max

(a)

(b)

Fig. 9. (a) Frequency response function and (b) dispersion curves of periodic trees with l = 10 m.

Min

(c)

(d)

Fig. 11. Displacement distribution diagram (a) without and (b) with tree in the band gap (f = 42 Hz); (c) without and (d) with tree outside the band gap (f = 35 Hz).

5. Band gap generation mechanism analysis To illustrate the generation mechanism of the band gap, the vibration modes (the red points in Fig. 9[b]) corresponding to the specific frequencies of the bands were calculated via COMSOL 5.3a. The modes for the Bragg scattering band gap are shown in Fig. 12(a) and (b). Dark and light grey indicate the maximum and minimum of the mode vibrations, respectively. The motion of the unit cell shows that the vibration was associated with soil and tree

Fig. 10. (a) Frequency response function and (b) dispersion curves of periodic trees with l = 12 m.

Fig. 9(a) and Fig. 10(a). The attenuation was asymmetrical in the lower-frequency band gap and concentrated in a small region near the starting frequency of the band gap [38–40]. The reduction effect shrank near the end frequency of the band gap. This phenomenon also illustrates the lower frequency reduction zone, referred to as the locally resonant band gap. In particular, the Fano phenomenon became more obvious as the tree reached a higher height, even though the band gap width increased in this case as shown in Fig. 10. To demonstrate the wave-inhibited feature of the band gap in tree metamaterials, the displacement distributions in specific frequencies are shown in Fig. 11. Displacement distributions were observed at 42 Hz (in the band gap) and 35 Hz (outside the band gap), respectively. Wave energy was concentrated on the ground surface, revealing surface waves as the main wave modes. In Fig. 11(a) and (c), if no trees were on the ground surface, then the wave could propagate along the ground without attenuation. In Fig. 11(b), if the wave frequency was in the band gap, then the surface waves propagated within only three or four rows of trees, and the waves were prohibited in the case of tree metamaterials. In Fig. 11(d), the surface waves propagated through all trees, and the trees had no effect on forbidding wave propagation. The results also support the attenuation feature in FRF curves and the band gap feature in dispersion curves. Propagation of the surface waves was inhibited in the band gap, while the surface waves passed through the trees outside the band gap.

Max

Min

(a)

(b)

(c)

(d)

Fig. 12. Respective vibration modes of periodic trees corresponding to points B1, B2, L5, and L6 in Fig. 9.

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Table 1 Resonance modes of flat-straight bands. Order of vibration

1st

2nd

3rd

4th

Theoretical formula calculation (Hz) Numerical result (Hz) Vibration modes (corresponding to points L1, L2, L3, and L4 in Fig. 9, respectively)

0.97 1.59

6.10 6.69

17.07 16.62

33.45 31.04

vibrations. For the starting frequency in Fig. 12(a), the tree and soil each exhibited a degree of deformation. For the end frequency in Fig. 12(b), the tree demonstrated substantially sinusoidal deformation, and the soil showed extremely strong surface deformation approximated as strong surface-wave ground motion under the scattering wave by trees. The tree thus served as a scatterer. The surface wave at this frequency was therefore reflected, refracted, and diffracted in the unit cell. Half-wave loss was found in this case and resulted in inhibition of the elastic wave at this specific frequency. The wavelength corresponding to the center frequency (fm) of the first Bragg scattering band gap was twice that of the periodic constant [41]. This wavelength can also be expressed as k ¼ cR =f m . The Rayleigh wave speed was nearly equal to 90% of the shear wave speed; however, the existence of trees enhanced stiffness of the soil, and the effective Rayleigh wave speed of the unit cell was hence approximate to the soil shear wave speed in this case. In Fig. 9(b), the wavelength was k ¼ cs =f m ¼ 190:74=48:92 ¼ 3:90 (m), nearly approximate to twice the periodic constant (2a = 4 m). This finding also validates this band gap as being due to the Bragg scattering mechanism. Limited by the model configuration, the bands were not completely opened, and only a ‘‘CX” directional gap appeared. Given that the incident waves were mainly along the x direction, an obvious attenuation also appeared in this frequency range as shown in Figs. 9 and 10. The starting frequency of the locally resonant band gap appeared at the ‘‘L5” point, and the corresponding vibration mode is shown in Fig. 12(c). The end frequency appeared at the ‘‘L6” point with the corresponding vibration mode shown in Fig. 12(d). In Fig. 12(c), the mode presented a vibration of the tree in terms of longitudinal low-order resonance whereas the motion of the soil remained nearly still. In Fig. 12(d), the tree vibration was due to a clear flexural mode, whereas the energy of the soil was localized on the ground surface. As expected, the LBF of the band gap was due to the local resonance of the tree. The longitudinal selfqffiffiffi E resonant frequencies of the tree can be expressed as f l ¼ ð2i1Þ 4l q, where i is taken as 1; therefore, the first-order longitudinal resonance frequency of the tree was 38.61 Hz, concurring with the numerical result (f L5 ¼ 38:34Hz). The LBF of the band gap was due to the first-order longitudinal self-resonance of the tree. The UBF of the band gap resulted from the tree flexural resonance and surface wave, representing a hybridization vibration mode. Tree resonance and hybridization resonance determined the LBF and UBF of the band gap. This band gap was thus due to the locally resonant mechanism. However, not all resonance modes of the tree could generate locally resonant band gaps; below the locally resonant band gap,

several flat-straight bands were observed in Fig. 9(b). These bands corresponded to the first-, second-, third-, and fourth-order resonance modes of the tree, respectively, from low frequency to high frequency. To further demonstrate the mechanism of these bands, the modes of the flat-straight bands are listed in Table 1. The flexural self-resonant frequencies can be approximately expressed as qffiffiffiffi 2 EI nÞ f f ¼ ðb qA, where bn is the mode coefficient (b1 = 1.875, 2pl2 b2 = 4.694, b3 = 7.855, b4 = 10.996) [42], and A is approximately set as the average section area of the trunk. The theoretical formula calculation and numerical results are compared in Table 1. The above comparison demonstrates that the low-order selfresonance of the tree resulted in flat-straight bands. The tree simply served as a self-resonator in flat-straight bands; even as the tree exhibited intense deformation, the soil remained still. These bands indicate energy propagation in these frequencies. The bands were not opened at these frequency ranges; vibration amplification may have corresponded to the resonant peaks in the FRF curves in Figs. 9 and 10. Therefore, the tree metamaterials demonstrated local resonance and the Bragg scattering mechanism, which are well implicated in controlling elastic wave propagation, especially surface waves. 6. Conclusions Based on the metamaterial theorem, this paper regards periodically arranged trees as natural metamaterials at the geophysics scale, which induce large band gaps for Rayleigh waves at extremely low frequencies of tens of Hz. The effects of four key parameters on the band gap, including soil elastic modulus, tree spacing, trunk radius, and tree height, were investigated. The waveinhibited characteristics of multiple rows of urban trees were verified and confirmed via simulation. The mechanism of vibration reduction in urban forests was illustrated by calculating the modes. This paper extends the understanding of trees in a vibration environment from the perspective of electromagnetic metamaterials translating to elastic waves. Conclusions are as follows: (1) Hard soil sites result in high-frequency and wide band gaps. Compact tree spacing is beneficial to obtaining wide band gaps, with the starting frequency varying only slightly. (2) Over time, trees grow up naturally. The trunk constantly becomes wider and taller; this phenomenon results in lower-frequency and wider band gaps, which expand the range of vibration attenuation. (3) The damping effect of urban trees increases with the number of rows. In practice, three or four rows of green trees are effective and economic, and vibration attenuation is reasonable.

Y.-f. Liu et al. / Construction and Building Materials 199 (2019) 737–745

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