Lamb's problem for a half-space coupled to a generic distribution of oscillators at the surface
LLamb’s problem for a half-space coupled to a generic distribution of oscillatorsat the surface
Xingbo Pu a,1 , Antonio Palermo a,1 , Alessandro Marzani a, ∗ a Department of Civil, Chemical, Environmental and Materials Engineering, University of Bologna, 40136 Bologna, Italy
Abstract
We propose an analytical framework to model the effect of single and multiple mechanical surface oscillatorson the dynamics of vertically polarized elastic waves propagating in a semi-infinite medium. The formulationextends the canonical Lamb’s problem, originally developed to obtain the wavefield induced by a harmonic linesource in an elastic half-space, to the scenario where a finite cluster of vertical oscillators is attached to themedium surface. In short, our approach utilizes the solution of the classical Lamb’s problem as Green’s functionto formulate the multiple scattered fields generated by the resonators. For an arbitrary number of resonators,arranged atop the elastic half-space in an arbitrary configuration, the displacement fields are obtained in closed-form and validated with numerics developed in a two-dimensional finite element environment.
Keywords:
Elastic waves, Lamb’s problem, seismic metamaterials, metasurfaces
1. Introduction
Modeling the propagation of mechanical surface waves in an elastic half-space is a long-lasting topic inphysics and engineering. A cornerstone of this research topic is the seminal work by Lamb [1] which describesthe fundamental solution for a harmonic load applied on the surface of an elastic medium, a scenario currentlyknown as the Lamb’s problem. Since then, numerous researchers have enriched the complexity of this problem accounting for the presence of inclusions, obstacles, profile and material discontinuities along and within theelastic medium [2, 3, 4, 5, 6, 7, 8].A canonical problem of particular interest concerns the propagation of elastic waves in a semi-infinite sub-strate supporting a cluster of resonant elements. This configuration can indeed illustrate problems of technolog-ical relevance across different length-scales, as seismic waves interacting with the built environment [9, 10, 11] or surface waves propagating in micro-mechanical resonant systems [12, 13]. Additionally, periodic clustersof surface resonators have been recently explored to realize novel devices for surface wave manipulation, theso-called elastic metasurfaces. Among these periodic configurations, arrays of beams or pillars [14, 15, 16], andmass-spring resonators [17, 13, 18] have shown the capabilities to shape both the direction of propagation andthe frequency content of elastic waves. Pivotal in all these coupled substrate-resonators engineering problems is the knowledge of both dispersion relation and wavefield.Several analytical formulations are currently available to derive the dispersive properties [14] and transmis-sion coefficients of metasurfaces [10, 19]. In most cases, these approaches describe the collective behavior of ∗ Corresponding author
Email address: [email protected] (Alessandro Marzani) Equal contributors.
Preprint submitted to Elsevier January 26, 2021 a r X i v : . [ phy s i c s . a pp - ph ] J a n n infinite array of oscillators with the aid of an effective medium approach [13, 20], or via asymptotic andhomogenization techniques [10, 19, 21]. The calculation of the elastic wavefield of a finite-size, arbitrarily distributed cluster of resonators is insteadobtained via numerical techniques (like standard [22] or spectral FEM [23]), since no closed-form formulationis currently available to this purpose. Nonetheless, only the knowledge of the wavefield can shed light on thedestructive or constructive wave interference generated by the resonators array which is in turn responsible forpeculiar wave phenomena like surface-to-bulk wave conversion [14, 18], rainbow trapping and wave localization [23, 24]. Despite the possibility to obtain actual results for specific configurations by means of numericalschemes, analytical treatment of this elastodynamic problem can allow (i) to better comprise the nature of thesephenomena, (ii) to guide the optimal design of waves control devices, and (iii) to derive general conclusions onthe interaction problem between closed resonators mechanically coupled by an elastic substrate.Hence, in this work we develop an exact formulation which extends the classical Lamb’s problem to the case of an elastic half-space coupled to an arbitrary cluster of vertical surface resonators. To this purpose, wecalculate the incident wavefield generated by a harmonic source following the approach by Lamb. The Lamb’ssolution is also used as Green’s function to describe the scattered field generated by each resonator when excitedby a harmonic motion at its base. The overall substrate wavefield is then obtained as solution of the coupledproblem due to the interference of the incident field and the multiple scattered fields of the oscillators. Our formulation can tackle a generic number of different resonators located at arbitrary distances from the source,as illustrated in the various examples discussed in the work and validated against numerical results (FEM).The article is organized as follows. In Section 2, we present our analytical formulation. We begin bydescribing the solution of the Lamb’s problem for a harmonic source applied on the free surface of a semi-infinite elastic substrate (Section 2.1). Then, we recall the response for an oscillator subjected to a harmonic motion at its base (Section 2.2) and formulate the interaction problem between resonators and the half-space(Section 2.3). In Section 3, we calculate the response of an elastic substrate with a single, a pair and a cluster ofsurface resonators and validate our predictions against numerics. Finally, in Section 4 we summarize the mainfindings of our work.
2. Analytical framework We develop an analytical framework to calculate the response of an isotropic, linear elastic half-space coupledwith N oscillators and excited by a harmonic line source (see Fig. 1).Our investigation begins recalling (i) the solution of the Lamb’s problem for a harmonic line load applied atthe free surface of an elastic, isotropic half-space [1] and (ii) the response of a vertical oscillator to an imposedharmonic base motion. The response of the coupled system is obtained by formulating the interaction problem between the source-generated wavefield, the solution of the Lamb’s problem, and the summation of the scatteredwavefields generated by the motion of the surface resonators. Let us consider a time-harmonic force per unit length Q e i ωt applied normal to the free surface of an isotropicelastic medium. We resort to a two-dimensional (2D) plane-strain formulation in the x − z plane, where the x -axis is directed along the wave propagation and the z -axis is perpendicular to the free surface (see Fig. 1). In2 .. N K K K c N c N x x m m N m , x u , z w c a x i e t Q Fig. 1.
Schematic of Rayleigh wave interacting with resonators on an elastic half-space. absence of resonators, free stress boundary conditions ( σ zz = τ zx = 0) characterize the elastic half-space alongits whole surface except for the source location where: σ zz ( x, z, t ) = Qδ ( x )e i ωt , τ zx ( x, z, t ) = 0 , for x = 0 , z = 0 (1)Given the plane-strain conditions, the displacement vector lies in the plane x − z with non-null componentsdenoted as u ( x, z, t ) = [ u, w ]. It is here convenient to use the Helmholtz decomposition u = ∇ Φ + ∇ × Ψ and to express the displacement components as: u = ∂ Φ ∂x − ∂ Ψ y ∂z , w = ∂ Φ ∂z + ∂ Ψ y ∂x (2)where Φ( x, z, t ) is a scalar potential and Ψ y ( x, z, t ) is a component of the vector potential Ψ .Restricting our interest to the steady-state condition, the potentials assume the form:Φ( x, z, t ) = Φ( x, z )e i ωt , Ψ y ( x, z, t ) = Ψ y ( x, z )e i ωt (3)and satisfy the wave equations [25]: ∇ Φ + k p Φ = 0 , ∇ Ψ y + k s Ψ y = 0 (4)in which k p and k s denote, respectively, the wave number of pressure and shear waves in the substrate, namely: k p = ωc p , k s = ωc s (5)where: c p = (cid:115) λ + 2 µρ , c s = (cid:114) µρ (6)are the pressure and shear wave velocities, respectively, λ and µ the Lam´e constants and ρ the mass density ofthe substrate. According to the Hooke’s law and employing the Helmholtz decomposition in Eq. (2), the in-plane compo-3ents of the stress tensor σ can be expressed as function of the potentials: σ zz = − µ (cid:20) k s Φ + 2 (cid:18) ∂ Φ ∂x − ∂ Ψ y ∂x∂z (cid:19)(cid:21) , τ zx = − µ (cid:20) k s Ψ y − (cid:18) ∂ Φ ∂x∂z − ∂ Ψ y ∂z (cid:19)(cid:21) (7)At this stage, we seek for the solutions of the wave Eqs. (4) by means of the Fourier transform along the x -direction: F{∇ Φ + k p Φ } = ∂ ¯Φ ∂z − ( k − k p ) ¯Φ = 0 , F{∇ Ψ y + k s Ψ y } = ∂ ¯Ψ y ∂z − ( k − k s ) ¯Ψ y = 0 (8)which admit general solutions of the form:¯Φ( k, z ) = A e − pz + B e pz , ¯Ψ y ( k, z ) = A e − qz + B e qz (9)with: p = (cid:113) k − k p , q = (cid:112) k − k s (10)and where the coefficients A and A in Eq. (9) must be equal to zero to avoid unbounded responses atincreasing depth z . The remaining coefficients B and B are determined by imposing the boundary conditions.Fourier transforming the stress components in Eq. (7), and making use of the boundary conditions in Eq. (1), lead to the following expressions: (2 k − k s ) B + 2i kqB = Q/µ (11a) − kpB + (2 k − k s ) B = 0 (11b)Solutions of Eqs. (11a) and (11b) provide the coefficients: B = Qµ (2 k − k s ) R ( k ) , B = Qµ kpR ( k ) (12)where R ( k ) denotes the so-called Rayleigh function: R ( k ) ≡ (2 k − k s ) − k pq (13)The inverse Fourier transform of Eq. (9) provides the expression of the potentials in the plane x − z :Φ( x, z ) = F − { ¯Φ } = Q πµ (cid:90) ∞−∞ k − k s R ( k ) e pz +i kx d k (14a)Ψ y ( x, z ) = F − { ¯Ψ y } = Q πµ (cid:90) ∞−∞ kpR ( k ) e qz +i kx d k (14b)At last, by substituting Eqs. (14a) and (14b) into Eq. (2), the displacement components of the wavefieldinduced by the time-harmonic line load are obtained as: u ( f ) ( x, z ) = i Q πµ (cid:90) ∞−∞ k (2 k − k s )e pz − kpq e qz R ( k ) e i kx d k (15a)4 ( f ) ( x, z ) = Q πµ (cid:90) ∞−∞ p (2 k − k s )e pz − k p e qz R ( k ) e i kx d k (15b)where the superscript ( f ) is used to label these displacement components of the free field (no resonators). FromEqs. (15a) and (15b) the free field displacement components at z = 0 can be expressed as [1]: u ( f ) ( x,
0) = i Q πµ (cid:90) ∞−∞ k (2 k − k s − pq ) R ( k ) e i kx d k (16a) w ( f ) ( x,
0) = − Q πµ (cid:90) ∞−∞ k s pR ( k ) e i kx d k (16b) We now consider the steady-state dynamics of mass-spring-dashpot resonators located atop an elastic half-space under harmonic motion. The set O = { x , x , · · · , x N | N ∈ Z + } ⊂ R is introduced to collect the x -coordinate of the resonators. For each resonator we identify a footprint area S with a length 2 a along the x -direction.The dynamic equilibrium equation of each resonator reads: m n ¨ W n + c n ( ˙ W n − ˙˜ w ( x n , K n ( W n − ˜ w ( x n , x n ∈ O (17)where m n , c n and K n are the n -th resonator mass, viscous damping coefficient and spring stiffness, W n isthe absolute vertical displacement of the n -th mass, and ˜ w ( x n ,
0) is the average vertical displacement of the resonator footprint: ˜ w ( x n ,
0) = 12 a (cid:90) x n + ax n − a w ( x,
0) d x for x n ∈ O (18)The use of an average base displacement is motivated by both physical and mathematical arguments. Froma physical point, the average displacement represents the mean motion at the finite-size base of the oscillator.Mathematically, it allows to eliminate the divergence of the Green’s function at the origin.According to Eq. (18), the absolute vertical displacement of the generic n -th resonator excited by a harmonic base motion ˜ w ( x n ,
0) of circular frequency ω reads: W n = m n ω rn + i ωc n m n ( ω rn − ω ) + i ωc n ˜ w ( x n , ≡ T Rn ˜ w ( x n ,
0) for x n ∈ O (19)where T Rn denotes the so-called transmissibility [26] of a damped resonator, and where ω rn = (cid:112) K n /m n is theangular resonant frequency of the n -th resonator. Accordingly, the normal force applied by the resonator to thesubstrate can be written as: F n = m n ω W n = m n ω ( m n ω rn + i ωc n ) m n ( ω rn − ω ) + i ωc n ˜ w ( x n , ≡ Ω n ˜ w ( x n ,
0) for x n ∈ O (20)and the uniform stress exerted by each resonator over the contact area reads: σ zz ( x,
0) = F n S = Ω n ˜ w ( x n , a , τ zx ( x,
0) = 0 , for x ∈ ( x n − a, x n + a ) , x n ∈ O (21)5hese harmonic normal stresses behave as sources of additional wavefields in the half-space and interactwith the free field generated by the source. The nature and implication of this interaction is described in thenext section. As anticipated in the previous section, the resonators excited by a harmonic base motion generate additionalwavefields in the half-space. We label the j -th resonator-induced wavefield u ( s ) j ( x, z ), where the superscript ( s )is used to denote scattered field, so that the total displacement field of the coupled problem can be written as: u ( x, z ) = u ( f ) ( x, z ) + N (cid:88) j =1 u ( s ) j ( x, z ) ≡ u ( f ) ( x, z ) + u ( s ) ( x, z ) (22)where u ( s ) ( x, z ) is the total wavefield. The free wavefield u ( f ) ( x, z ), except for the point of application of theharmonic force ( x = 0), is characterized by null stress components at the surface: σ ( f ) zz ( x,
0) = τ ( f ) zx ( x,
0) = 0 for x ∈ R \ { } (23)hence, the stress at each resonator footprint depends only on the scattered wavefield, namely: σ zz ( x,
0) = σ ( f ) zz ( x,
0) + σ ( s ) zz ( x,
0) = σ ( s ) zz ( x,
0) for x ∈ ( x n − a, x n + a ) , x n ∈ O (24a) τ zx ( x,
0) = τ ( f ) zx ( x,
0) + τ ( s ) zx ( x,
0) = 0 for x ∈ ( x n − a, x n + a ) , x n ∈ O (24b)In force of Eq. (21) and Eq. (22), the scattered normal stress at the resonator footprint in Eq. (24a) can be written as: σ ( s ) zz ( x,
0) = Ω n ˜ w ( x n , a = Ω n [ ˜ w ( f ) ( x n ,
0) + ˜ w ( s ) ( x n , a for x ∈ ( x n − a, x n + a ) , x n ∈ O (25)where the average free-field vertical displacement ˜ w ( f ) ( x n ,
0) at the footprint of the n -th resonator can beobtained exploiting Eq. (15b) as:˜ w ( f ) ( x n ,
0) = 12 a − Q πµ (cid:90) x n + ax n − a (cid:90) ∞−∞ k s pR ( k ) e i kx d x d k = − Q πµ (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i kx n d k for x n ∈ O (26)The average scattered field ˜ w ( s ) ( x n ,
0) can be instead obtained as described in the following. First Eqs.(15a) and (15b) are used as Green’s functions to express the displacement components of the scattered field u ( s ) j ( x, z ) generated by the normal stress exerted by the j -th resonator: u ( s ) j ( x, z ) = i (cid:90) x j + ax j − a σ ( s ) zz ( η, πµ (cid:90) ∞−∞ k (2 k − k s )e pz − kpq e qz R ( k ) e i k ( x − η ) d η d k = iΩ j [ ˜ w ( f ) ( x j ,
0) + ˜ w ( s ) ( x j , πµ (cid:90) ∞−∞ k (2 k − k s )e pz − kpq e qz R ( k ) sin( ka ) ka e i k ( x − x j ) d k for x j ∈ O (27a)6 ( s ) j ( x, z ) = (cid:90) x j + ax j − a σ ( s ) zz ( η, πµ (cid:90) ∞−∞ p (2 k − k s )e pz − k p e qz R ( k ) e i k ( x − η ) d η d k = Ω j [ ˜ w ( f ) ( x j ,
0) + ˜ w ( s ) ( x j , πµ (cid:90) ∞−∞ p (2 k − k s )e pz − k p e qz R ( k ) sin( ka ) ka e i k ( x − x j ) d k for x j ∈ O (27b)Note that the calculation of each scattered wavefield u ( s ) j ( x, z ), and w ( s ) j ( x, z ), as per Eqs. (27a) and (27b),requires only the knowledge of the average vertical displacement, free ˜ w ( f ) ( x j ,
0) and scattered ˜ w ( s ) ( x j , x j ∈ O .Since the free field component ˜ w ( f ) ( x j ,
0) is known from Eq. (26), the only left unknowns in Eqs. 27a and27b are the vertical scattered displacements ˜ w ( s ) ( x j , x n according to Eq. (22):˜ w ( s ) ( x n ,
0) = N (cid:88) j =1 ˜ w ( s ) j ( x n ,
0) for n = 1 , , · · · , N (28)which can be equivalently rewritten as: ˜ w ( s ) ( x ,
0) = ˜ w ( s )1 ( x ,
0) + ˜ w ( s )2 ( x ,
0) + · · · + ˜ w ( s ) N ( x , w ( s ) ( x ,
0) = ˜ w ( s )1 ( x ,
0) + ˜ w ( s )2 ( x ,
0) + · · · + ˜ w ( s ) N ( x , w ( s ) ( x N ,
0) = ˜ w ( s )1 ( x N ,
0) + ˜ w ( s )2 ( x N ,
0) + · · · + ˜ w ( s ) N ( x N ,
0) (29)Then, we substitute Eq. (27b) into Eq. (29) and obtain the following equations: ˜ w ( s ) ( x ,
0) = β [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , β [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , · · · + β N [ ˜ w ( f ) ( x N ,
0) + ˜ w ( s ) ( x N , w ( s ) ( x ,
0) = β [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , β [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , · · · + β N [ ˜ w ( f ) ( x N ,
0) + ˜ w ( s ) ( x N , w ( s ) ( x N ,
0) = β N [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , β N [ ˜ w ( f ) ( x ,
0) + ˜ w ( s ) ( x , · · · + β NN [ ˜ w ( f ) ( x N ,
0) + ˜ w ( s ) ( x N , where: β nj = − Ω j πµ a (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka d k (cid:90) x n + ax n − a e i k ( x − x j ) d x = − Ω j πµ (cid:90) ∞−∞ k s pR ( k ) sin ( ka )( ka ) e i k ( x n − x j ) d k for n, j = 1 , , · · · , N (31)For simplicity, we express Eq. (30) in matrix form as: Ax = b (32)7here the corresponding coefficients are: A = (1 − β ) − β · · · − β N − β (1 − β ) · · · − β N ... ... . . . ... − β N − β N · · · (1 − β NN ) ∈ C N × N (33a) x = ˜ w ( s ) ( x , w ( s ) ( x , w ( s ) ( x N , ∈ C N × , b = β ˜ w ( f ) ( x ,
0) + β ˜ w ( f ) ( x ,
0) + · · · + β N ˜ w ( f ) ( x N , β ˜ w ( f ) ( x ,
0) + β ˜ w ( f ) ( x ,
0) + · · · + β N ˜ w ( f ) ( x N , β N ˜ w ( f ) ( x ,
0) + β N ˜ w ( f ) ( x ,
0) + · · · + β NN ˜ w ( f ) ( x N , ∈ C N × (33b)The solution of Eq. (32) provides the sought average vertical displacement components of the scattered field atthe resonator footprint locations x n . When the matrix A has a nonzero determinant, the system in Eq. (32)has a unique solution x with components: ˜ w ( s ) ( x n ,
0) = ( A − b ) n for n = 1 , , · · · , N (34)At this stage, by substituting Eq. (34) into Eqs. (27a), (27b), we can obtain the j -th scattered field components u ( s ) j ( x, z ) and w ( s ) j ( x, z ) at any point of the x − z plane. The total wavefield is then obtained as the summationof the free and scattered fields: u ( x, z ) = u ( f ) ( x, z ) + u ( s ) ( x, z ) = u ( f ) ( x, z ) + N (cid:88) j =1 u ( s ) j ( x, z ) (35a) w ( x, z ) = w ( f ) ( x, z ) + w ( s ) ( x, z ) = w ( f ) ( x, z ) + N (cid:88) j =1 w ( s ) j ( x, z ) (35b)Equations (35a) and (35b) can be rewritten in integral form as follows: u ( x, z ) = i Q πµ (cid:90) ∞−∞ k (2 k − k s )e pz − kpq e qz R ( k ) e i kx d k + i2 πµ N (cid:88) j =1 Ω j (cid:20) − Q πµ (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i kx j d k + ( A − b ) j (cid:21) (cid:90) ∞−∞ k (2 k − k s )e pz − kpq e qz R ( k ) sin( ka ) ka e i k ( x − x j ) d k (36) w ( x, z ) = Q πµ (cid:90) ∞−∞ p (2 k − k s )e pz − k p e qz R ( k ) e i kx d k + 12 πµ N (cid:88) j =1 Ω j (cid:20) − Q πµ (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i kx j d k + ( A − b ) j (cid:21) (cid:90) ∞−∞ p (2 k − k s )e pz − k p e qz R ( k ) sin( ka ) ka e i k ( x − x j ) d k (37) z = 0 to obtain the surface displacementcomponents: u ( x,
0) = i Q πµ (cid:90) ∞−∞ k (2 k − k s − pq ) R ( k ) e i kx d k + i2 πµ N (cid:88) j =1 Ω j (cid:20) − Q πµ (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i kx j d k + ( A − b ) j (cid:21) (cid:90) ∞−∞ k (2 k − k s − pq ) R ( k ) sin( ka ) ka e i k ( x − x j ) d k (38) w ( x,
0) = − Q πµ (cid:90) ∞−∞ k s pR ( k ) e i kx d k − πµ N (cid:88) j =1 Ω j (cid:20) − Q πµ (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i kx j d k + ( A − b ) j (cid:21) (cid:90) ∞−∞ k s pR ( k ) sin( ka ) ka e i k ( x − x j ) d k (39)The closed-form Eqs. (36), (37), (38), (39) are evaluated numerically via Gauss-Kronrod quadrature. Inthe next section we use the developed approach to predict the wavefield of half-spaces with different sets ofresonators including uniform, graded and disordered arrays.
3. Case studies
In this section, we discuss the behavior of a single resonator, a couple of resonators, and a graded array of resonators placed atop an elastic half-space and excited by a far field source. Our aim is twofold: (i) to testand validate the accuracy of our formulation against a numerical (Finite Element) solution; (ii) to discuss thecoupling between free and scattered wavefields for different mechanical parameters and layout of the resonators.Before tackling these tasks, let us provide some useful parameters to facilitate our calculations and ease thefurther discussions: • to generalize our conclusions, we introduce the dimensionless mass parameter ˆ m j = m j / ( ρSλ rj ), with ρ being the density of the half-space and λ rj = c R /f rj the Rayleigh wavelength at resonant frequency f rj ; this parameter compares the mass of the resonator to the mass of a portion of substrate under theresonator footprint area, with volume V = S × λ rj . Additionally, we introduce the parameter ˆ d mn = d mn /λ rj = ( x n − x m ) /λ rj to denote the normalized distance between the m -th and n -th resonator. • to account for wave energy dissipation, we assume a non-null damping ratio ζ for the resonator response anda non-null hysteretic damping ξ for the elastic substrate. The damping coefficient of the j -th resonatoris calculated as c j = 2 m j ω rj ζ [26], while the complex moduli of the substrate as λ (cid:48) = λ (1 + 2i ξ ) and µ (cid:48) = µ (1 + 2i ξ ) [27]; the introduction of damping in the system allows also to avoid numerical instabilitiesby removing the poles of the integrands in Eqs. (38, 39); • to quantify the contribution of the resonators scattered field at the half-space surface ( z = 0), we introducethe following amplitude ratio ( A R ) [28]: A R = w ( x, w ( f ) ( x,
0) = 1 + w ( s ) ( x, w ( f ) ( x,
0) (40)9nd the normalized distance between the receiver and the first resonator ˆ d = d/λ r = ( x − x ) /λ r where A R will be computed. We begin our investigation considering the case of a single surface oscillator. The resonator has a normalizedmass ˆ m = 1, is located at a distance x = 6 λ r from the harmonic source, and lies over a half-space characterized by the mechanical properties collected in Table 1.The total wavefield generated by a harmonic source at f = f r , computed by using Eqs. (36) and (37), isshown in Fig. 2a in the domain x = [4 , λ r , z = [ − , λ r . The analytically predicted wavefield is in excellentagreement with the one displayed in Fig. 2b, calculated with harmonic FE simulations (see Appendix A fordetails on the FE model). To quantify the effect of the resonator on the surface wavefield, we calculate and show in Fig. 2c the amplitude ratio | A R | for harmonic sources with frequencies f = [0 , f r at two positions,namely ˆ d = 0 . d = 6 (red line), in the scattered far field. Table 1: Mechanical parameters of the resonators and the elastic half-space. The elastic parameters are taken from Ref. [29].
Parameter ValueLine load amplitude, Q N/mFirst resonator frequency, f r a ζ ρ Young modulus of substrate, E
46 MPaPoisson ratio of substrate, ν ξ | A R | >
1, followed by a higher frequency region with a significant deamplification ofthe signal. The two regimes are separated by the frequency f c which corresponds to the resonance of the oscillator coupled to the elastic substrate. In the far field the signal amplification disappears and we observeonly a deamplification of the signal with a maximum drop occurring exactly at the coupled frequency f c .To interpret the different responses observed near and far from the resonator, we expand the expression of | A R | given in Eq. (40) as: | A R | = (cid:115) (cid:12)(cid:12)(cid:12)(cid:12) w ( s ) w ( f ) (cid:12)(cid:12)(cid:12)(cid:12) + 2Re (cid:18) w ( s ) w ( f ) (cid:19) = (cid:115) (cid:12)(cid:12)(cid:12)(cid:12) w ( s ) w ( f ) (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12) w ( s ) w ( f ) (cid:12)(cid:12)(cid:12)(cid:12) cos(∆ θ ) (41)where ∆ θ is the relative phase between the scattered and free vertical displacements, namely, w ( s ) /w ( f ) . According to the Eq. (41), both the amplitude and the relative phase of the scattered field play a role inthe amplitude ratio | A R | : the scattered field amplitude shows a similar trend both in the near and in the farfield and reaches its peak exactly at the coupled frequency f c (see Fig. 2d); conversely, the relative phase ∆ θ significantly changes depending on the receiver location. In the near field, around the coupling frequency, therelative phase (see inset in Fig. 2d) shows that the scattered and free responses are approximately orthogonal. Hence, the contribution of the scattered waves to the value of | A R | is negligible. Conversely, in the far field,scattered and free responses are out-of-phase, leading to destructive interference between the two wavefields andto a minimum value of the amplification ratio | A R | . 10 a) (b)(c)(e) (f)(d) Fig. 2.
Rayleigh wave interaction with a single resonator on an elastic half-space ( ˆ m = 1, x = 6 λ r ). Total wavefield for aharmonic source at f = f r computed using the (a) proposed analytical solution, (b) FE simulation. (c) Amplitude ratio | A R | inthe near ˆ d = 0 . d = 6 (red line). (d) Spectrum and phase of the scattered field vs. the free field. (e)Dynamic amplification factor of the resonator on a rigid (dashed line) and elastic substrate (red line). (f) Coupled frequency f c vs. the mass of the resonator. For comparison, in (c), (d) and (e) we also provide the FE solutions denoted by dots. Let us now investigate in more detail the variation of the coupled frequency f c with respect to the mechanicalparameters of the system, namely substrate and oscillator. To this purpose, we calculate the absolute verticaldisplacement W of a resonator located on the half-space surface at distance d = 0 from the source and compareits response to the one of an identical resonator placed on a rigid substrate. The two resonators verticalresponses, normalized with respect to the relative base displacements, are plotted in Fig. 2e. As expected, theresponse of the resonator on a rigid substrate provides the transmissibility factor | T R | (dashed line), whereasthe maximum amplitude response of the resonator lying on the half-space (red continuous line) does not occurat its natural frequency f r but is shifted towards the coupled resonant frequency f c . This frequency shiftcan be predicted utilizing a lumped-mass model, in which the contribution of the half-space is lumped in the11dditional stiffness K h (see schematic in Fig. 2f). Hence, the coupled frequency can be calculated as: f c = f r (cid:115) K h m ω r + K h (42)where the equivalent substrate stiffness K h can be estimated from Eq. (27b) by considering the averaged stressover the contact area ( z = 0) and a null source-response distance ( x = x j ):1 K h = − πµ (cid:90) ∞−∞ k s pR ( k ) sin ( ka )( ka ) d k (43)Note that the integral in Eq. (43) is frequency dependent, so strictly speaking the coupling frequency f c inEq. (42) is frequency dependent too. As an approximation, here we calculate the value of K h at f = f r and use it to predict the value of f c for different resonator mass.The value of the coupled frequency vs. the resonator mass as predicted from Eq. (42) is plotted in Fig. 2fas a continuous black line. The prediction agrees well with the frequency values at which the response of thedifferent resonators reach its maximum amplitude as calculated from Eq. (19) (marked in Fig. 2f by coloredcircles). In particular, the color of the circles shows the value of the amplification factor ( W/w ( f ) ) at resonance, and highlights that a change of mass produces a change in the effective quality factor of the resonator too.To conclude this section, we remark that all these results have been validated with FE models (see black dotssuperimposed onto all the curves in Figs. 2c,d,e). Studies on the dynamics of coupled oscillators on elastic supports are receiving renovate attention both in the geophysical context, to assess the response of close buildings and their influence on ground vibrations, and inthe design of SAW devices, where micro/nano resonators are proposed for application in classical and quantuminformation processing [30]. In what follows, we show how our analytical formulation allows to properly analyzethe mutual interaction between close resonators and to quantify its effects on the half-space and resonatorresponses.
We begin our investigation considering two identical resonators, with parameters given in Table 1 and witha relative spacing d = 0 . λ r , excited by a far field ( x = 6 λ r ) harmonic source. We calculate the amplituderatio | A R | for a receiver located at ˆ d = 6 from the first resonator. The presence of two oscillators atop thehalf-space leads to a significant reduction in the amplitude ratio which shows its minimum value at a frequencydifferent from the resonator coupled frequency ( f c = f c ). We remark that the system response cannot bepredicted from the simple superposition of the single resonator scenarios, which would neglect the cross-couplinginteraction between the two resonators (see the | A R | of each single resonator scenario (blue lines) in Fig. 3a).To investigate further the coupling between the responses of the resonators, we expand the expression of theamplitude ratio as: | A R | = (cid:118)(cid:117)(cid:117)(cid:116) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )1 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )2 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )1 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos(∆ θ ) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )2 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos(∆ θ ) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )1 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w ( s )2 w ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cos(∆ θ − ∆ θ )(44)where ∆ θ and ∆ θ denote the phase of w ( s )1 /w ( f ) and w ( s )2 /w ( f ) , respectively. The above equation clearlyhighlights the presence of the cross-coupling contribution, namely, the last term in Eq. (44), in addition to the12ndependent contributions of each resonator.This mutual interaction modifies also the resonators responses, as highlighted by Fig. 3b where the uncoupledand coupled oscillators amplification factors are shown. In particular, by looking at the maximum of the | A R | factor, one can observe a shift in the resonance of both oscillators. Additionally, the first resonatorshows a characteristic “frequency splitting” behavior, recently observed experimentally in couples of micropillarsattached to an elastic substrate [30].An effective mutual interaction requires resonators with similar (possibly identical) resonance frequencieslocated at a relatively short distance. That is because the scattered fields are maximized at the coupled resonance frequencies and in the near fields of the resonators, as shown in Fig. 2d for a single resonator. To evidencethis effect, let us consider a configuration of two resonators with different natural frequencies, e.g., f r = 2 f r ,keeping the other parameters unchanged. Fig. 3c shows the amplitude ratio at ˆ d = 6. The reader can appreciatethat the system response is now simply the envelope, namely the superposition of the two single case scenarios.Similarly, the amplification factors of the two resonators, reported in Fig. 3d, confirm that no significant shift occurs in the coupled frequencies of each resonator.We remark that the extraction of these features is eased by our analytical framework which allows toinvestigate and distinguish the contribution of each scattered field to the total response. (a) (b)(c) (d) Fig. 3.
Rayleigh wave interaction with two resonators on an elastic half-space ( ˆ m = ˆ m = 1 , x = 6 λ r , d = 0 . λ r ). (a)Amplitude ratio | A R | computed at ˆ d = 6 for two identical resonators ( f r = f r ). (b) Uncoupled and coupled amplification factorsfor two identical resonators ( f r = f r ). (c) Amplitude ratio | A R | computed at ˆ d = 6 for two different resonators ( f r = 2 f r ). (d)Uncoupled and coupled amplification factors for two different resonators ( f r = 2 f r ). .3. Cluster of resonators So far we have gained some physical insights into the behavior of single and coupled oscillators lying over an elastic substrate. When the number of resonators increases to form a cluster, or a so-called metasurface,the propagation of surface waves in the elastic substrate can be characterized by intriguing phenomena, suchas surface-to-bulk wave conversions (from classical [13] and Umklapp scattering [16]) and wave localization (viaclassical [23] and topological rainbow trapping [24]). Although the extraction of the dispersive properties ofsuch graded clusters can be typically inferred from analytical models developed for infinite regular arrays [14], the evaluation of the related wavefields require the use of numerical schemes. In what follows, we will show thatour analytical framework can properly capture these phenomena.We begin by considering an array of 40 identical resonators ( ˆ m j = 0 . , f rj = f r ), arranged periodically witha lattice spacing 0 . λ r (see Fig. 4a). Similar arrays have been analyzed to assess the surface wave filteringcapabilities of a periodic metasurface [23, 18]. For such a configuration, we expect the existence of a band gap in a narrow frequency region above the oscillator resonance and the occurrence of surface-to-shear waveconversion.To visualize this phenomenon, we consider an incident Rayleigh wave with frequency f = 1 . f r , namelyin the metasurface band gap, which is excited by the harmonic source Q e i ωt sufficiently far away from themetasurface ( x = 6 λ r ). The displacement wavefield is displayed in Fig. 4a and shows how the incident Rayleigh wave is converted into a downward propagating shear wave.The same wave-conversion phenomenon can be extended over a broader frequency range by utilizing a seriesof resonators with a graded variation of frequency along the array [23]. To achieve this purpose, we modelan array of 40 resonators ( ˆ m j = 0 .
5) with resonant frequencies linearly increasing along the array with astep ∆ f r = 0 . f r , for an overall working bandwidth between f r and 3 f r . Alike the periodic metasurface previously discussed, we set a constant spacing between the resonators. Indeed, this graded configuration, alsoknown in the literature as metawedge [23], supports both a mode conversion and a wave localization, dependingon the direction of the incoming excitation. These two distinct wavefields, as calculated using our analyticalframework, are shown respectively in Figs. 4b and 4c.As last example, we calculate the wavefield of a disordered metasurface, constructed by randomly changing the order of resonators considered in the discussed metawedge. The investigation of such a non-periodic resonantsystem is attracting increasing interest in the research community as a design strategy to enlarge the filteringbandwidth of metamaterials [31, 32]. Note that a random configuration can be easily modeled using the proposedanalytical approach which allows to set frequency and location of the resonating scatters at will. In Fig. 4d thereader can appreciate the related wavefield for an incident Rayleigh wave at f = 1 . f r . We also provide a comparison of the amplitude ratio | A R | calculated at ˆ d = 6 in the frequency range f = [0 , f r . The result is shown in Fig. 4e and highlights that the disordered metasurface provides similarfiltering performance to the two analyzed graded systems.14 a) (b)(c) (d)(e) Fig. 4.
Rayleigh wave interaction with finite-size metasurfaces ( N = 40) on an elastic half-space ( ˆ m = 0 .
5, resonator spacing0 . λ r , incident frequency f = 1 . f r ). Calculated wave field for: (a) a periodic metasurface. (b) an inverse metawedge. (c) aclassical metawedge. (d) a disordered metasurface. In the schematics the length of the resonators represent the value of the relatednatural frequencies. (e) Amplitude ratio | A R | computed at ˆ d = 6 in the frequency range f = [0 , f r of the four considered cases.
4. Conclusions
This work proposes an analytical formulation to model and study the interaction of vertically polarized elastic waves with surface resonators. In particular, we have exploited the Green’s functions of the canonicalLamb’s problem to setup a coupled problem between the incident field generated by the source and the scatteredwavefields generated by a set of resonators placed atop the half-space. The formulation can handle an arbitrarynumber of resonators arranged on the surface of the half-space in a generic configuration. The capabilities ofthe developed methodology have been discussed by modeling the dynamics of a single, a couple and a cluster of resonators arranged over an isotropic homogeneous half-space, and have been validated against finite elementsimulations.The method allows to capture the frequency shift of a resonator coupled to the elastic substrate, the mutualinteraction between a couple of close resonators in terms of frequency splitting and amplitude variation and the15ollective response of arrays of resonators, e.g., metasurfaces interacting with Rayleigh waves. Future research efforts will be devoted to extend the methodology to a 3D scenario and exploit its capability to design SAWdevices, waveguides and interpret the dynamics of a cluster of buildings interacting with seismic waves andurban vibrations.
CRediT authorship contribution statementXingbo Pu:
Conceptualization, Methodology, Investigation, Software, Data curation, Writing - original draft.
Antonio Palermo:
Conceptualization, Investigation, Validation, Writing - review & editing, Co-supervision.
Alessandro Marzani:
Conceptualization, Investigation, Writing - review & editing, Supervision, Fundingacquisition.
Declaration of competing interest
The authors declare that they have no conflict of interest.
Acknowledgments
This project has received funding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska Curie grant agreement No 813424. A.P. acknowledges the support of the Universityof Bologna - DICAM through the research fellowship “Metamaterials for seismic waves attenuation”.
Appendix A. Details on the FE model
In this Appendix, we provide details of the FE model (see Fig. A.1) used to validate our analytical solutionsin Section 3.1. First, to simulate the uniform vertical force imposed on the footprint, the mass-dashpot-springoscillator is discretized as an ensemble of 11 truss elements, namely the truss spacing 0 . a (cid:28) λ r . This procedureresults in each point mass m p = m/
11, and the Young modulus of each truss E t = ( m p ω r + i ωc ) /A t , where A t isthe cross-sectional area of a truss. The incident Rayleigh wave is excited by the harmonic load with amplitude Q at a sufficient distance (6 λ r ) from the oscillator. To model the elastic half-space and to avoid unnecessaryreflections, we add Perfectly Matched Layers (PMLs) to the vertical and bottom edges.16 x u , z w a PML
Point mass T r u ss m K c a i e t Q Fig. A.1.
Schematic of FE model.
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