Natural seismic metamaterials: the role of tree branches in the birth of Rayleigh wave bandgap for ground born vibration attenuation

Abstract

Key message

Clusters of forest trees can be assumed as naturally available seismic metamaterials capable of mitigating low-frequency earthquake waves. Strategic afforestation could be an effective way to safeguard a region from earthquake hazards.

Abstract

At the subwavelength scale, forest trees interact with the Rayleigh wave inducing extremely wide locally resonant bandgap in the earthquake frequency range of interest. To better understand this mechanism, we have considered the effect of bulky side branches connected with the main stem. We considered two types of trees configurations and the study is conducted by a computational approach based on the finite element method. The Rayleigh wave band structure is obtained by sound cone and elastic strain energy density methods. The later technique is found effective in detecting the higher surface Rayleigh modes that are undetectable by the sound cone technique. At the resonator attachment point, the wave mode coupling occurs between longitudinal modes of the resonator trees and vertical component of the Rayleigh wave. This coupling results in a $\pi$ phase shift for incident Rayleigh wave and the periodic arrangement of trees act constructively to generate a wide bandgap. The tree bulky side branches seemed to further enhance the coupling strength. By constructing a finite length model and performing frequency domain and time history analyses based on the actual ground earthquake records, the performance and efficiency of the bandgap are validated. Both displacement field plots, time history signals and their Fourier spectra indicate significant amount of vibration mitigation within the bandgap frequency.

Appendix

Wave equation and periodic boundary condition

The governing Navier equation of motion and the constitutive relation for a heterogenous medium with isotropic material properties are given by Maurel et al. (2018) as

$$\rho \frac{{\partial^{2} {\mathbf{u}}\left( {\mathbf{r}} \right)}}{{\partial t^{2} }} = {\text{div}} {{\varvec{\upsigma}}}\quad ;\quad {{\varvec{\upsigma}}} = {\mathbf{C}}:\nabla {\mathbf{u}}\left( {\mathbf{r}} \right),$$

(1)

where $\rho$, ${\mathbf{u}}\left( {\mathbf{r}} \right)$, $t$ are mass density, position dependent displacement vector and time, respectively. Here ${\mathbf{r}} = \left( {x,y,z} \right)$ is the position vector, ${{\varvec{\upsigma}}}$ is the second order stress tensor, $\nabla$ is vector differential operator and assuming zero boundary force ${\mathbf{C}} = \left( {C_{ijkl} } \right)$ is fourth order elasticity tensor that satisfies the generalized Hooke’s law $C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu \left( {\delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} } \right)$ where $i,j,k = 1,2,3$ and $\left\{ {\begin{array}{*{20}c} {\delta_{ij} = 1,\,\,\,\,i = j} \\ {\delta_{ij} = 0,\,\,\,\,i \ne j} \\ \end{array} } \right\}$, in which $\mu$ and $\lambda$ are the Lame coefficients. For wave propagation in an elastic medium, the Navier equation for in-plane waves has displacement component ${\mathbf{u}} = \left( {u_{x} ,u_{y} } \right)$ and the out-of-plane displacement ${\mathbf{u}} = u_{z}$ is not considered. We mainly focused on in-plane waves that propagate on the surface of a semi-infinite substrate (vertical component of the Rayleigh waves). Assuming time harmonic dependence in the form of $e^{ - i\omega t}$, where $\omega$ is angular frequency, Eq (1) can be simplified as

$${\text{div}} {{\varvec{\upsigma}}} + \rho \omega^{2} u = 0\quad ;\quad {{\varvec{\upsigma}}} = \mu \nabla u.$$

(2)

Throughout this study, we consider spacing between the trees as $a$ and it varies for different configurations considered. To study the characteristics of wave propagation in an infinite domain, the Floquet–Bloch periodicity condition as explained in Hofstadter (1976) is implemented. The theory received enormous attenuation with increasing research growth in the field of photonic crystal, phononic crystal, and acoustic metamaterials. Initially, the theory was developed to solve differential equations in particle physics, today it is widely adopted for solving differential equation in other branches of physics and mathematics. According to this theory, the solution of Eq. (1) can be expressed as

$${\mathbf{u}}\left( {{\mathbf{r}},t} \right) = {\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right)e^{{i\left( {{\mathbf{k}}.{\mathbf{r}} - \omega t} \right)}} ,$$

(3)

where ${\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right)$ is the displacement modulation function. Because we consider periodicity in the x-direction, only wavenumber $k$ in this direction is used. We consider only the $\Gamma - X$ direction of the irreducible Brillouin zone for band structure study. The Brillouin zone is shown in Fig. 1c. Further details about the Brillouin reciprocal lattices and Brillouin zone can be found in Muhammad et al. (2019). In Eq. 3, the periodicity of the modulation constant ${\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right)$ is the same as that of the lattice constant $a$ of the primitive cell. Thus, one can consider

$${\mathbf{u}}_{{\mathbf{k}}} \left( {{\mathbf{r}} + a} \right) = {\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right).$$

(4)

Substitution of Eq. 4 into Eq. 3 yields

$${\mathbf{u}}\left( {{\mathbf{r}} + a,t} \right) = {\mathbf{u}}_{{\mathbf{k}}} \left( {{\mathbf{r}} + a} \right)e^{{i\left( {{\mathbf{k}}\left( {{\mathbf{r}} + a} \right) - \omega t} \right)}} = e^{{i{\mathbf{k}}a}} {\mathbf{u}}_{{\mathbf{k}}} \left( {\mathbf{r}} \right)e^{{i\left( {{\mathbf{k}}.{\mathbf{r}} - \omega t} \right)}} = e^{{i{\mathbf{k}}.a}} {\mathbf{u}}\left( {{\mathbf{r}},t} \right),$$

(5)

which is a periodic boundary condition that makes the primitive unit cell infinite along the x-direction, see Fig. 1c. The equation further implies that the difference between the input and output wave is a scalar parameter ${\mathbf{k}}.{\mathbf{a}}$ where ${\mathbf{a}} = \left( {a_{x} } \right)$ due to single direction periodicity. Because of this property, the wave mode in a single unit cell can be replicated in an infinite domain. This duplication of a single unit cell eigenmodes in an infinite domain is represented by a dispersion plot.

Substituting Eq. 5 into Eq. 1 results in an eigenvalue problem of the form

$$\left( {{{\varvec{\Phi}}}\left( {\mathbf{k}} \right) - \omega^{2} {\mathbf{M}}} \right).{\mathbf{U}} = 0,$$

(6)

where ${{\varvec{\Phi}}}$ is the stiffness matrix that is function of the wavevector ${\mathbf{k}}$ and ${\mathbf{M}}$ is the mass matrix. The dispersion plot/spectra is a graphical representation of the wavenumber swept at the boundary of irreducible Brillouin zone versus eigenmodes of the structure Muhammad et al. (2019). Physically it shows the response of eigenmodes or natural frequency of the structure for varying wave speed. There are two major approaches that are adopted to present the dispersion curve. The first technique is the $k\left( \omega \right)$ approach where for a known wavenumber, the unknown eigenmodes are determined. In this case, wavenumber $k$ and frequency $\omega$ are both real. The frequency could have an imaginary part depending upon the modal boundary condition and/or complexity. The second approach refers to $\omega \left( k \right)$ where a range of frequency is swept and the corresponding wavenumber $k = k_{R} + ik_{I}$ is calculated. The real part of the wavenumber $k_{R}$ shows the characteristics of propagating waves while the imaginary part $k_{I}$ indicates the evanescent modes. Hence, all wavenumbers that have $k_{R} > 0$ and $k_{I} = 0$ are called the passband that allows wave propagation in the medium while for wavenumber $k_{R} = 0$ and $k_{I} > 0$, it is the bandgap with evanescent wave modes and decaying energy field that do not allow wave propagation in the infinite domain. In this work, we adopt the $k\left( \omega \right)$ approach and solve the eigenvalue problem of Eq. 6 using FEM.

Sound cone technique

The prior section explained the propagation of waves in an elastic medium along with the Floquet–Bloch periodicity condition and dispersion relation. This section is dedicated to the explanation of sound cone technique that is effective in separating the pure surface modes which have the highest amount of energy concentration at the free surface from the bulk and other leaky mixed modes. The bulk modes include mixed wave modes where energy is distributed toward the depth of unit cell. Upon solving the eigenvalue equation, the result is post-processed for the surface modes study. The boundary of the sound line is defined by developing an algorithm in MATLAB. For details, one can refer to Muhammad et al. (2019). Since the periodicity condition is applied in one direction only, we calculate the eigenmodes in the $\Gamma - X$ direction of Brillouin zone only. The bounding velocity of the sound line can be determined by $\omega = {\mathbf{k}}.c_{s}$ where $c_{s} = \sqrt {\mu_{s} /\rho_{s} }$ and $c_{s}$, $\mu_{s}$, $\rho_{s}$ are shear wave velocity, soil shear modulus and mass density, respectively. Here we assume the Rayleigh wave velocity $c_{R}$ is equivalent to $c_{s}$ i.e. $c_{R} \sim c_{s}$ due to medium homogeneity although in literature $c_{R} \cong 0.9c_{s}$. The sound line divides the dispersion plot into the radiative and non-radiative regions Muhammad et al. (2019). All eigenmodes that lie below the sound line have velocity smaller and/or equivalent to $c_{R}$, thus they can be considered surface modes while the eigenmodes with a velocity greater than $c_{s}$ or $c_{R}$ are the non-surface bulk modes. In this way, the surface modes are separated from the bulk modes. Throughout the study, the sound line is presented in blue, see “Results and discussion” section.

Elastic strain energy density method

Although the sound cone technique is a famous method for studying surface waves band structures as reported in Khelif et al. (2010); Muhammad et al. (2019), for some cases like a pillared structure on the surface of a semi-infinite substrate proposed by Graczykowski et al. (2016) and periodic pile barriers studied by Pu and Shi (2017), the sound cone technique fails to capture the higher surface modes. The elastic strain energy density (SE) method initially proposed by Graczykowski et al. (2016); Pu and Shi (2017) is another approach recently reported that is found effective in capturing those higher surface modes which lie above the sound line. The SE method is solely based on the distribution of elastic strain energy density across the domain of the unit cell. An energy distribution parameter $\xi$ is defined and for all eigenmodes, $\xi$ is calculated. Mathematically, it can be expressed as

$$\xi = \frac{{\int_{{S\left( {2\lambda } \right)}} {U_{\varepsilon } ds} }}{{\int_{{S\left( {H_{s} + h_{mb} } \right)}} {U_{\varepsilon } ds} }},$$

(7)

where $U_{\varepsilon }$ is the strain energy density which is calculated for a height of $2\lambda$ and another complete unit cell height $h_{mb} + H_{s}$ including the soil substrate. Here, $\lambda$ is the wavelength considered while $H_{s} + h_{mb}$ is the total height of the unit cell. The calculation shows for $0 < \xi < 1$ and all eigenmodes that have $\xi > 0.8$, they are considered as surface modes where the highest amount of energy is concentrated inside the tree branches. One can replace the strain energy density with displacement field distribution and a similar conclusion will be obtained. This is because the strain energy is actually derived from the displacement field. In this study, we determine the surface wave band structures using both methods and for some configurations, the SE method is found useful in capturing the higher surface modes. Below the sound line region, both methods give excellent agreement. Our study for the band structure is limited to these two numerical approaches. The analytical solution of the surface waves in periodic structure is still a big challenge. Mostly the numerical and experimental techniques are adopted to study the surface waves in an elastic homogenous periodic system.