Research Paper
Vibration attenuation analysis of periodic underground barriers using complex band diagrams

https://doi.org/10.1016/j.compgeo.2020.103821Get rights and content

Abstract

Vibrations propagating underground may have harmful effects on adjacent buildings and human health. Recently, the notion of phononic crystals has been introduced and applied to isolate vibrations below the ground. In this paper, considering the viscoelastic material model, the complex band diagrams are suggested to analyse the transportation of vibrations in the composite foundation consist of soil and different underground structures. In the complex band diagram, the minimum imaginary part of the wave vectors is taken as the index to measure the ability of the pile group to weaken vibrations. Several types of underground barriers, including circular piles, continuous walls, and multi-layer piles, are analysed. The result shows that multi-layer piles can induce local resonance and achieve an attenuation zone at the low-frequency level. The attenuation effect of periodic barriers is then verified by transmission models. The results show that the complex band diagrams can quantitatively evaluate the isolation effect of periodic barriers, which is beyond the capability of classic band diagrams.

Introduction

Environmental vibrations caused by traffic, machinery operation, building construction, or earthquakes have become one of the major environmental hazards (Duggal, 2007, Reitherman, 2012, Yao et al., 2019). Long-term environmental vibrations, especially traffic vibrations at low frequencies, can cause fatigue damage to buildings and pose a risk to human health (Coquel and Fillol, 2017, Hegde and Venkateswarlu, 2019). In practice, many efforts have been made to isolate vibrations. The most common approach is to increase the resistance of buildings by adjusting the stiffness, strength, and structural form of the building: however, these methods cannot be applied to old buildings or historical constructions.

An alternative way is to proactively attenuate vibration waves by installing artificial periodic underground barriers, such as trenches, in the surrounding ground (Al-Hussaini and Ahmad, 1996, Pu and Shi, 2020). In recent years, phononic crystals, as an innovative type of underground vibration barriers, are attracting more and more attention, as shown in Fig. 1. Phononic crystals are functional materials formed by periodically arranged materials in another solid or fluid medium (Lu et al., 2009, Laude et al., 2015). By rationally designing the distribution of the constituent materials, the scattering and resonance of the waves propagating in the phononic crystals will bring many interesting phenomena. One of these characteristics the ability to block the propagation of waves over certain frequency ranges, which are named band gaps.

Recently, some researchers have utilised phononic crystals to isolate vibration by constructing periodic structures underground, including piles, trenches, continuous walls, and geofoams (Al-Hussaini and Ahmad, 1996, Pu and Shi, 2020, Murillo et al., 2009, Turan et al., 2013, Brule et al., 2014, Ekanayake et al., 2014, Yan et al., 2015, Dijckmans et al., 2016, Miniaci et al., 2016, Basack and Nimbalkar, 2018, Meng and Shi, 2019, Zheng et al., 2020). Through field experiments, Brule et al. (2014) experimentally verified the attenuation effect of vibration through the pile group. In addition, the influence of pile shape on the attenuation effect was analysed. Based on phononic crystal theory, Yan et al. (2015) used the finite element method to simulate the behaviour of three-dimensional (3D) periodic foundations. Through experiments, it is proved that the 3D periodic foundations can attenuate seismic waves. By changing the material, size, and shape of the cross-section, Miniaci et al. (2016) performed a numerical simulation to analyse the band diagram of the unit cell structure. Chromatic dispersion analysis and full-scale 3D transient wave propagation simulation were conducted on the finite-size model to explore the attenuation when the seismic wave propagates. Meng and Shi (2019) used the COMSOL partial differential equation (PDE) method to study multiple rows of pile barriers in saturated soil, and analysed the attenuation zones for plane waves.

The aforementioned method provides an innovative way to isolate underground vibrations. However, current researches mainly use the linear elastic material model. The viscosity of both the soil and pile is neglected, which causes differences to arise between model and reality. Since the viscosity of material will attenuate vibrations at all frequencies, the notion of band gaps will not apply when considering the viscosity of materials. On the other hand, previous researches mainly focused on the isolation effect at relatively high frequencies (Huang and Shi, 2013, Huang and Shi, 2013); which is away from the main frequency of seismic waves (<20 Hz) (Gadallah and Fisher, 2005) or traffic vibration (<30 Hz) (Hegde and Venkateswarlu, 2019). The causes of band gaps can be classified as Bragg scattering and local resonance mechanisms. For Bragg scattering; the vibration waves are mainly reflected layer-by-layer at the interfaces between different materials (such as the pile-soil interface) (Martínez-Sala et al., 1995, Kushwaha et al., 1993). The wavelengths of the band gaps are of the same order as the periodicity of the primitive unit cell. For local resonance band gaps, the vibration waves induce the local vibration of massive cores in the phononic crystal, and the wavelengths of band gaps can reach one or two orders of magnitude higher than the lattice constant (the dimension of a unit cell) (Li et al., 2019, Zhengyou et al., 2000). Obviously, for a given lattice constant, the local resonance mechanism is more promising in acquiring band gaps at low-frequency level, but it has not received enough attention in current research.

Therefore, in this research, a viscoelastic material model is used to simulate the periodic structures in soil. The complex band diagrams are employed to analyse the composite foundation. The minimum imaginary part of wave vectors is used as the index to measure the degree of attenuation at different frequencies. Using this approach, several typical periodic structures, including cavities, circular piles, continuous walls, and multi-layer piles, are analysed. Then we propose a multi-layer pile that can induce local resonance that makes it capable of attenuating vibrations at low frequencies. Finally, the effect of the proposed designs is verified by a viscoelastic transmission model, and the feasibility of the attenuation curves we used in this paper is confirmed.

Section snippets

k(ω) method and complex band diagram

For wave propagating in solid media, the vibration of materials is governed by Newton’s second law. The stress-strain relationship of soil is considered to be linear elastic. The governing equation can be expressed as follows (Li et al., 2016):ρ(r)u¨=[(λ(r)+2μ(r))(·u)]-×[μ(r)×u]where u = (ux, uy, uz) is the displacement vector and ρ(r) is the material density. To consider the viscoelastic properties of the materials, we use the complex damping model proposed in Ref. (Yang and Yan, 2009); Ed

Designs of periodic underground barriers

In this section, the complex band diagram is used to analyse circular piles and continuous walls that has been widely discussed in previous research (Turan et al., 2013, Dijckmans et al., 2016, Pu and Shi, 2019). Then, we propose a creative structure with local resonance band gap at low frequency level. Considering the size of artificial structures in practical engineering, we set the lattice constant of unit cell a = 2 m. The pile-soil interfaces are assumed to be perfectly bonded. The

Transmission spectra verification

A 2D plane strain finite element model is established to verify the attenuation effect of the designs above, as shown in Fig. 7(a). The ribbon-like model consists of 8 unit cells. The size and material properties of the unit cell are consistent with the analysis above. Perfect matching layers (PMLs) are set at the two ends of the model to eliminate the reflections. On the other direction, the boundary condition is set to be Bloch-Floquet periodic condition. The pile-soil interfaces are assumed

Conclusion

The notion of phononic crystals has been applied to reform the ground and block underground vibrations, but the viscosity of soil are usually neglected. In this research, the complex band diagrams are introduced to consider the viscoelasticity of the material. The comparison of the complex band diagram and the classic band diagram shows that the minimum imaginary part of complex band diagram, i.e. the attenuation curves, can demonstrate the attenuation of the vibration waves at different

Funding

This research was supported by the Fundamental Research Funds of Central South University [Grant No. 2019zzts609]; the Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences [Grant No. Z018009].

CRediT authorship contribution statement

Youzhi Jiang: Investigation, Writing - original draft. Fei Meng: Validation, Writing - review & editing. Yafeng Chen: Methodology. Yun Zheng: Supervision, Resources, Funding acquisition. Xiaobin Chen: Supervision, Resources, Funding acquisition. Jiasheng Zhang: Supervision, Resources, Funding acquisition. Xiaodong Huang: Conceptualization, Project administration.

Declaration of Competing Interest

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.

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    Youzhi Jiang and Fei Meng contribute equally to this work.

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