Effective model for elastic waves propagating in a substrate supporting a dense array of plates/beams with flexural resonances
Jean-Jacques Marigo, Kim Pham, Agnès Maurel, Sébastien Guenneau
EE ff ective model for elastic waves propagating in a substratesupporting a dense array of plates / beams with flexural resonances Jean-Jacques Marigo
Laboratoire de M´ecanique des solides, Ecole Polytechnique,Route de Saclay, 91120 Palaiseau, France
Kim Pham
IMSIA, ENSTA ParisTech - CNRS - EDF - CEA, Universit´e Paris-Saclay,828 Bd des Mar´echaux, 91732 Palaiseau, France
Agn`es Maurel
Institut Langevin, ESPCI ParisTech, CNRS,1 rue Jussieu, Paris 75005, France
S´ebastien Guenneau
Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France
Abstract
We consider the e ff ect of an array of plates or beams over a semi-infinite elastic ground on thepropagation of elastic waves hitting the interface. The plates / beams are slender bodies with fle-xural resonances at low frequencies able to perturb significantly the propagation of waves inthe ground. An e ff ective model is obtained using asymptotic analysis and homogenization tech-niques, which can be expressed in terms of the ground alone with e ff ective dynamic (frequency-dependent) boundary conditions of the Robin’s type. For an incident plane wave at obliqueincidence, the displacement fields and the reflection coe ffi cients are obtained in closed forms andtheir validity is inspected by comparison with direct numerics in a two-dimensional setting. Keywords: asymptotic analysis; elastic waves; metamaterials; metasurfaces
1. Introduction
We are interested in wave propagation in a semi-infinite elastic substrate supporting a perio-dic and dense array of thin or slender bodies. This is the canonic idealized configuration usedto illustrate the problem of ”site-city interaction”. Such a problem, recent on the seismologyhistory scale, aims to account for the urban environment as a factor modifying the seismic groundmotion. Starting in the 19 th century, the interest was primarily focused on the motion of the soilelicited by static or dynamic sources being concentrated or distributed on the free surface in theabsence of buildings. These studies have led to important results as the Lamb’s problem [1, 2].Then, more realistic configurations have been considered using approximate models to predict Preprint submitted to Journal of the Mechanics and Physics of Solids January 20, 2020 a r X i v : . [ phy s i c s . c l a ss - ph ] J a n p
2, we summarize the result of the asymptotic analysisin the case of an array of plates, whose detailed derivation is given in the §
3. The resulting”complete” formulation (3)-(5) is equivalent to that in (6)-(7) thanks to a partial resolution ofthe problem. In §
4, the accuracy of the e ff ective model is inspected by comparison with directnumerics based on multimodal method [54] for an in-plane incident wave. The strong couplingof the array with the ground at the flexural resonances are exemplified and the agreement betweenthe actual and e ff ective problems is discussed. We finish the study in § ff ective problem for the an arrayof beams which is merely identical to the case of the plates with some specificities which areaddressed.
2. The actual problem and the e ff ective problem We consider in this section the asymptotic analysis of an array of parallel plates atop anisotropic elastic substrate. We note that the problem splits in the in-plane and out-of-plane po-larizations. The latter case has been already addressed in [36]. We focus in this section on theformer, in-plane, case. We further note that the asymptotic analysis of a doubly periodic arrayof cylinders atop an isotropic substrate is a fully coupled elastodynamic wave problem, which isthus slightly more involved and addressed in the Appendix.3 .1. The physical problem
We consider the equation of elastodynamics for the displacement vector u the stress tensor σ and the strain tensor ε in the substrate , x ∈ ( −∞ ,
0) : div σ + ρ s ω u = , σ = µ s ε + λ s tr( ε ) I , ε = ( ∇ u + t ∇ u ) , in the plates , x ∈ (0 , (cid:96) ) : div σ + ρ p ω u = , σ = µ p ε + λ p tr( ε ) I , (2)with the Lam´e coe ffi cients ( λ p , µ p ) of the plates and ( λ s , µ s ) of the substrate, ω the angular fre-quency and I stands for the identity matrix. In three dimensions with x = ( x , x , x ), stress freeconditions σ · n = apply at each boundary between an elastic medium (the plates or the sub-strate) and air, with n the normal to the interface. Eventually, the continuity of the displacementand the normal stress apply at each boundary between the parallel plates and the substrate. Thisproblem can be solved once the source u inc has been defined and accounting for the radiationcondition for x → −∞ which applies to the scattered field ( u − u inc ). ff ective problem Below we summarize the main results of the analysis developed in the § ff ective boundary and transmission conditions (Figure 2(a)). Owing toa partial resolution, this formulation can be simplified to an equivalent ”impedance formulation”set on the substrate only (Figure 2(b)). We note that all three components of the displacementfield appear in this section, and the reader should be aware that we make use of variables x = ( x , x , x ) and x (cid:48) = ( x , x ). The e ff ective problem reads as follow In the substrate , x ∈ ( −∞ ,
0) : div σ + ρ s ω u = , σ = µ s ε + λ s tr( ε ) I , In the region of the plates , x ∈ (0 , h ) : ∂ u ∂ x − κ u = , κ = (cid:32) ρ p ω (cid:96) p D p (cid:33) / , u ( x , x (cid:48) ) = u (0 , x (cid:48) ) , u ( x , x (cid:48) ) = u (0 , x (cid:48) ) , (3)with x (cid:48) = ( x , x ), D p = E p (1 − ν p ) (cid:96) p , (4)the flexural rigidity of the plates ( ρ p the mass density, E p the Young’s modulus and ν p the Poisson’sratio), complemented with boundary conditions at x = x = h of the form σ (0 − , x (cid:48) ) = ρ p ω ϕ h u (0 , x (cid:48) ) , σ (0 − , x (cid:48) ) = − D p (cid:96) ∂ u ∂ x (0 + , x (cid:48) ) ,σ (0 − , x (cid:48) ) = ϕ h E p ∂ u ∂ x (0 − , x (cid:48) ) + ρ p ω u (0 − , x (cid:48) ) , u (0 + , x (cid:48) ) = u (0 − , x (cid:48) ) , ∂ u ∂ x (0 + , x (cid:48) ) = ,∂ u ∂ x ( h , x (cid:48) ) = ∂ u ∂ x ( h , x (cid:48) ) = . (5)4hese e ff ective conditions express (i) at x = x = h , free end conditions with vanishing bending momentand shearing force. One notes that all three components of the displacement field u appear in (5)which involves partial derivatives on x and x only. x
0. The sameholds for the array of beams, with (a) (B.1)-(B.3) and (b) (B.5).
From (3), the problem in x ∈ (0 , h ) can be solved owing to the linearity of the problem withrespect to u (0 − , x ), see Appendix A. Doing so, the problem can be thought in the substrateonly along with the boundary conditions of the Robin’s type, namely div σ + ρ s ω u = , σ = µ s ε + λ s tr( ε ) I , x ∈ ( −∞ , ,σ (0 , x (cid:48) ) = Z u (0 , x (cid:48) ) , σ (0 , x (cid:48) ) = Z f ( κ h ) u (0 , x (cid:48) ) ,σ (0 , x (cid:48) ) = ϕ h E p ∂ u ∂ x (0 , x (cid:48) ) + ρ p ω u (0 , x (cid:48) ) , (6)with the following impedance parameters Z = ρ p ω ϕ h , f ( κ h ) = sh κ h cos κ h + ch κ h sin κ h κ h (1 + ch κ h cos κ h ) , (7)(we have used that D p κ = ρ p ω (cid:96) p ). The conditions on ( σ , σ ) encapsulate the e ff ects of thein-plane bending of the plates while the condition on σ can be understood as the equilibriumof an axially loaded bar (in the absence of substrate, we recover the wave equation for quasi-longitudinal waves). It is worth noting that for out-of-plane displacements, u ( x , x ) and u = u =
0, the boundary conditions simplify to σ (0 , x ) = ρ p ϕ h ω u (0 , x ). This corresponds tothe impedance condition which can be deduced from the analysis conducted in [36] and resultingin σ (0 , x ) = µ p ϕ k T tan( k T h ) u (0 , x ) and obtained here in the limit k T h (cid:28) . Derivation of the e ff ective problem As previously said, the asymptotic analysis is conducted considering that the typical wave-length 1 / k is large compared to the plate height h which is itself large compared to the arrayspacing (cid:96) ∼ (cid:96) p . Hence, with k T = ω (cid:112) ρ s /µ s and k L = ω (cid:112) ρ s / ( λ s + µ s ) of the same order ofmagnitude, we define the small non-dimensional parameter η as η = (cid:112) k T (cid:96), and k T h = O ( η ) , (note that to excite both the bending and the longitudinal modes another scaling is required with kh = O (1), and this is a higher frequency regime studied in [35]). Accordingly, the asymptoticanalysis is conducted using the rescaled height ˆ h of the plates and array’s spacing ˆ (cid:96) defined by(ˆ h , ˆ (cid:96) ) = (cid:32) h η , (cid:96)η (cid:33) , which models an array of densely packed thin plates. We also define the associated rescaledspatial coordinates y = x η , z = x η . (8) ff ective wave equation in the region of the plates3.1.1. Notations In the region of the array of parallel plates, the displacements and the stresses vary in thehorizontal direction over small distances dictated by (cid:96) , and over large distances dictated by theincoming waves; these two scales are accounted for by the two-scale coordinates ( x (cid:48) , z ), with x (cid:48) = ( x , x ). In the vertical direction, the variations are dictated by h only and this is accountedfor by the rescaled coordinate y . It follows that the fields ( u , σ ) are written of the form u = (cid:88) n ≥ η n w n ( y , z , x (cid:48) ) , σ = (cid:88) n ≥ η n π n ( y , z , x (cid:48) ) , (9)with the three-scale di ff erential operator reading ∇ → e η ∂∂ y + e η ∂∂ z + ∇ x (cid:48) , (10)where e = (1 , ,
0) and e = (0 , , x (cid:48) ε x (cid:48) ( u ) = ∂ x u ∂ x u ∂ x u ∂ x u (cid:0) ∂ x u + ∂ x u (cid:1) ∂ x u (cid:0) ∂ x u + ∂ x u (cid:1) ∂ x u , (11)and the strain tensors with respect to the rescaled coordinates y and z , ε y ( u ) = ∂ y w ∂ y u ∂ y u ∂ y u ∂ y u , ε z ( u ) = ∂ z u ∂ z u ∂ z u ∂ z u ∂ z u . (12)6he system in the region of the plates reads, from (2), (E ) 1 η ∂ y σ + ∂ x α σ α + η ∂ z σ + ρ p ω u = , (E α ) 1 η ∂ y σ α + ∂ x β σ αβ + η ∂ z σ α + ρ p ω u α = , (C) σ = η (cid:0) µ p ε y + λ p tr( ε y ) (cid:1) + (cid:16) µ p ε x (cid:48) + λ p tr( ε x (cid:48) ) (cid:17) + η (cid:16) µ p ε z (cid:48) + λ p tr( ε z (cid:48) ) (cid:17) , (13)with the convention on the Greek letters α = ,
3, the same for β , and where ε stands for ε ( u ).We shall use the stress-strain relation written in the form(C (cid:48) ) 1 η ε y + ε x (cid:48) + η ε z = (1 + ν p ) E p σ − ν p E p tr( σ ) I . (14)Eventually, the boundary conditions read σ i = , i = , , , at z = ± ϕ ˆ (cid:96)/ , (15)and are complemented by boundary conditions at y = , ˆ h assumed to be known (they will bejustified later). We seek to establish the e ff ective behaviour in the region of the array in terms ofmacroscopic averaged fields which do not depend anymore on the rapid coordinate z associatedwith the small scale ˆ (cid:96) as the following averages taken along rescaled variable z . These fields aredefined at any order n as w n ( y , x (cid:48) ) = ϕ ˆ (cid:96) (cid:90) Y w n ( y , z , x (cid:48) ) d z , π n ( y , x (cid:48) ) = (cid:96) (cid:90) Y π n ( y , z , x (cid:48) ) d z , with Y = { z ∈ ( − ϕ ˆ (cid:96)/ , ϕ ˆ (cid:96)/ } the segment shown in figure 3. y
3, required to establish the e ff ective boundary conditions at y = , ˆ h .The main results will be obtained following the procedure :1. We establish the following properties on π π = , π i = , i = , , , π = . (16)2. Then we derive the dependence of ( w , w ) on z which have the form w = W ( x (cid:48) ) , w = W ( y , x (cid:48) ) , w = W ( x (cid:48) ) , w = W ( x (cid:48) ) − λ p λ p + µ p ) ∂ W ∂ x ( x (cid:48) ) y − ∂ W ∂ y ( y , x (cid:48) ) z , w α = W α ( y , x (cid:48) ) , (17)and π = − E p − ν p ∂ W ∂ y ( y , x (cid:48) ) z , π = ϕ E p ∂ W ∂ x ( x (cid:48) ) . (18)3. Eventually, we identify the form of π i , i = , ,
3, and the Euler -Bernoulli equationgoverning the bending W . Specifically π ( y , x (cid:48) ) = ρ p ω ϕ W ( x (cid:48) ) (ˆ h − y ) ,π ( y , x (cid:48) ) = − E p (1 − ν p ) ϕ ˆ (cid:96) ∂ W ∂ y ( y , x (cid:48) ) ,π ( y , x (cid:48) ) = ϕ E p ∂ W ∂ x ( x (cid:48) ) + ρ p ω W ( x (cid:48) ) (cid:16) ˆ h − y (cid:17) , (19)and E p (1 − ν p ) ϕ ˆ (cid:96) ∂ W ∂ y − ρ p ω W = . (20)In the remainder of this section, we shall establish the above results. We shall denote (E ) n , (E α ) n and (C) n the equations which correspond to terms in (13) factor of η n . π in (16)From (E) − in (13), we have that ∂ z π i =
0, which along with the boundary conditions at z = ± ϕ ˆ h / π i =
0. Next from (E ) − and (E ) − , we also have that ∂ y π + ∂ z π = ∂ y π + ∂ z π =
0; by averaging these relations over Y and accounting for π i | ∂ Y =
0, weget that π and π do not depend on y . We now anticipate the boundary condition π (ˆ h , x (cid:48) ) = π (ˆ h , x (cid:48) ) = π = π = Y . Wehave the properties announced in (16). 8 .1.4. Second step: ( w , w ) in (17) and ( π , π ) in (18)Some of the announced results are trivially obtained. From (C (cid:48) ) − in (14), we get that ∂ z w i =
0, and from (C (cid:48) ) − and (C (cid:48) ) − that ∂ y w = ∂ y w =
0, which leaves us with the form of w in(17). Next (C (cid:48) αα ) − tells us that ∂ z w α =
0, in agreement with the form of w α in (17). We have yetto derive the form of w , which is more demanding. From (C (cid:48) ) − , ∂ z w = − ∂ y w and thus w = W ( y , x (cid:48) ) − ∂ W ∂ y ( y , x (cid:48) ) z , (21)but we can say more on W . Let us consider the system provided by (C ) and (C ) , specifically π = ( λ p + µ p ) ∂ y w + λ p (cid:16) ∂ x α w α + ∂ z w (cid:17) , = ( λ p + µ p ) (cid:16) ∂ z w + ∂ x w (cid:17) + λ p (cid:16) ∂ y w + ∂ x w (cid:17) , (22)where we have used that π =
0. After elimination of ∂ z w and owing to the form of w α in (16)and w in (21) (at this stage), we get π = a ( y , x (cid:48) ) z + b ( y , x (cid:48) ) with a = − E p − ν p ∂ W ∂ y ( y , x (cid:48) ) , b = µ p ( λ p + µ p ) λ p + µ p ) ∂ W ∂ y ( y , x (cid:48) ) + λ p ∂ W ∂ x ( x (cid:48) ) , (23)(we have used that E p / (1 − ν p ) = µ p ( µ p + λ p ) / ( λ p + µ p )). It is now su ffi cient to remark that π = b =
0. This immediately provides the form of π in (18) and W ( y , x (cid:48) ) = W ( x (cid:48) ) − λ p λ p + µ p ) ∂ W ∂ x ( x (cid:48) ) y , (24)which along with (21) leaves us with the form of w in (17). The same procedure is used to get π , which from (C ) , reads π = ( λ p + µ p ) ∂ x w + λ p (cid:16) ∂ y w + ∂ x w + ∂ z w (cid:17) . (25)Using that π = ∂ z w , we get π = E p ∂ W ∂ x ( x (cid:48) ) − µ p λ p λ p + µ p ∂ W ∂ y ( y , x (cid:48) ) z , (26)which after integration over Y leaves us with π in (18). Incidentally, w can be determined from(22) and we find w = − ∂ W ∂ x ( y , x (cid:48) ) + λ p λ p + µ p ) ∂ W ∂ x ( x (cid:48) ) z + λ p λ p + µ p ∂ W ∂ y z + W ( y , x (cid:48) ) . (27) π i in (19) and the Euler-Bernoulli equation in (20) . We start with (E) in (13) integrated over Y , specifically, ∂π ∂ y + ρ p ω ϕ W = , ∂π ∂ y + ρ p ω ϕ W = , ∂π ∂ y + ∂π ∂ x + ρ p ω W = , (28)9here we have used (16) and π · n | ∂ Y = . Since W and W depend on x (cid:48) only, and accountingfor π ( x (cid:48) ) in (18), we get by integration the forms of π and of π announced in (19). Note thatwe have anticipated the boundary conditions π i = y = ˆ h , see forthcoming (35).The equation on π in (28) will provide the Euler-Bernoulli equation once π has beendetermined (the integration is not possible since W depends on y ). To do so, we use, that ∂ y π + ∂ z π =
0, from (E ) − , along with π in (18). After integration and using the boundarycondition of vanishing π at z = ± ϕ ˆ (cid:96)/
2, we get that π = E p − ν p ) ∂ W ∂ y ( y , x (cid:48) ) (cid:32) z − ϕ ˆ (cid:96) (cid:33) , (29)hence the form of π in (19). It is now su ffi cient to use π in (28) to get the Euler-Bernoulliannounced in (20). ff ective boundary conditions at the top of the array To derive the transmission conditions at the top of the array, we perform a zoom by substi-tuting y used in (9) by z = y /η , see Figure 4a. Accordingly, the expansions of the fields aresought of the form u = (cid:88) n ≥ η n v n ( z (cid:48) , x (cid:48) ) , σ = (cid:88) n ≥ η n τ n ( z (cid:48) , x (cid:48) ) , (30)where we denote z (cid:48) = ( z , z ). The coordinate z ∈ ( −∞ ,
0) accounts for small scale variationsof the evanescent fields at the top of the plates. Next, the boundary conditions will be obtainedby matching the solution in (30) for z → −∞ with that in (9) valid far from the boundary for y → ˆ h . This means that we ask the two expansions to satisfy v ( z , z , x (cid:48) ) + η v ( z , z , x (cid:48) ) + · · · ∼ z →−∞ w (ˆ h + η z , z , x (cid:48) ) + η w (ˆ h + η z , z , x (cid:48) ) + · · · , (and the same for the stress tensors); note that we have used that y = η z . It results that lim z →−∞ v ( z (cid:48) , x (cid:48) ) = w (ˆ h , x (cid:48) ) , lim z →−∞ (cid:32) v ( z (cid:48) , x (cid:48) ) − z ∂ w ∂ y (ˆ h , z , x (cid:48) ) (cid:33) = w (ˆ h , z , x (cid:48) ) , lim z →−∞ τ ( z (cid:48) , x (cid:48) ) = π (ˆ h , z , x (cid:48) ) , lim z →−∞ (cid:32) τ ( z (cid:48) , x (cid:48) ) − z ∂ π ∂ y (ˆ h , z , x (cid:48) ) (cid:33) = π (ˆ h , z , x (cid:48) ) . (31)According to the dependence of the fields in (30) on ( z (cid:48) , x (cid:48) ), the di ff erential operator reads asfollows ∇ → η ∇ z (cid:48) + ∇ x (cid:48) , (32)and we shall need only the first equation of (2), which reads(e) 1 η div z (cid:48) σ + div x (cid:48) σ + ρ p ω u = , (33)where div z (cid:48) and div x (cid:48) means the divergence with respect to the coordinate z (cid:48) and x (cid:48) respectively.In (33), (e) − and (e) − tell us that div z (cid:48) τ = div z (cid:48) τ = , that we integrate over Z = { z ∈
0) the terms in the expansion (30) are assumed to be periodicwith respect to z ∈ ( − ˆ (cid:96)/ , ˆ (cid:96)/
2) while for z ∈ (0 , ∞ ) we have z ∈ ( − ϕ ˆ (cid:96)/ , ϕ ˆ (cid:96)/ ff erent notations for the expansions and for z since their meaning is di ff erent;for simplicity, we keep the same. The transmission conditions will be obtained by matching thesolution in (30) for z → + ∞ with that in (9) for x → + , and for z → −∞ with that in (39) for x → − . Matching the solutions hence means, with z = ( z , z ), v ( z (cid:48) , x (cid:48) ) + η v ( z (cid:48) , x (cid:48) ) + · · · ∼ z →−∞ u ( η z , x (cid:48) ) + η u ( η z , x (cid:48) ) + · · · , v ( z (cid:48) , x (cid:48) ) + η v ( z (cid:48) , x (cid:48) ) + · · · ∼ z → + ∞ w ( η z , z , x (cid:48) ) + η w ( η z , z , x (cid:48) ) + · · · , where we have used that x = η z and y = η z . It results that lim z →−∞ v ( z (cid:48) , x (cid:48) ) = u (0 − , x (cid:48) ) , lim z →−∞ v ( z (cid:48) , x ) = u (0 − , x (cid:48) ) , lim z →−∞ τ ( z (cid:48) , x (cid:48) ) = σ (0 − , x (cid:48) ) , lim z →−∞ τ ( z (cid:48) , x (cid:48) ) = σ (0 − , x (cid:48) ) , (40)and that lim z → + ∞ v ( z (cid:48) , x (cid:48) ) = w (0 + , x (cid:48) ) , lim z → + ∞ (cid:32) v ( z (cid:48) , x (cid:48) ) − z ∂ w ∂ y (0 + , z , x (cid:48) ) (cid:33) = w (0 + , z , x (cid:48) ) , lim z → + ∞ τ ( z (cid:48) , x (cid:48) ) = π (0 + , z , x (cid:48) ) , lim z → + ∞ (cid:32) τ ( z (cid:48) , x (cid:48) ) − z ∂ π ∂ y (0 + , z , x (cid:48) ) (cid:33) = π (0 + , z , x (cid:48) ) . (41)Eventually, with the di ff erential operator in (32), (33) applies; we shall also need from (2) that(c) η σ = µ a ε z (cid:48) + λ a tr( ε z (cid:48) ) I , (c (cid:48) ) 1 η ε y + ε x (cid:48) + η ε z (cid:48) = (1 + ν a ) E a σ − ν a E a tr( σ ) I . (42)( ε stands for ε ( u )) applying in the substrate, a = s, and in the plate, a = p, where we have defined ε z (cid:48) ( u ) = ∂ z u (cid:0) ∂ z u + ∂ z u (cid:1) ∂ z u (cid:0) ∂ z u + ∂ z u (cid:1) ∂ z u ∂ z u ∂ z u ∂ z u . (43)The continuity of the displacement is easily deduced. From (c (cid:48) ) − in (42), v does not dependon z (cid:48) , and ( v , v ) correspond to a rigid body motion, i.e. v = Ω a z + V a and v = Ω a z + V a , with( Ω a , V a ) independent of z (cid:48) . The periodic boundary conditions in the substrate impose Ω s = z = Ω p = v = V p = V s is independent of z (cid:48) . From (40)-(41), u (0 − , x (cid:48) ) = v = w (0 + , z , x (cid:48) ), and making use of (17) u (0 − , x (cid:48) ) = W ( x (cid:48) ) , u (0 − , x (cid:48) ) = W (0 + , x (cid:48) ) . (44)For the same reasons, v is a constant displacement, hence u (0 − , x (cid:48) ) = w (0 + , z , x (cid:48) ), but thishas now a consequence. Indeed, from (41) for the displacement at order 1, we have necessar-ily ∂ y w (0 + , z , x (cid:48) ) = to ensure that w (0 + , z , x (cid:48) ) is finite; from (17), we already know that ∂ y w = ∂ y w = w = W ( y , x (cid:48) ), hence ∂ W ∂ y (0 + , x (cid:48) ) = . (45)We now move on the e ff ective conditions on the force. From (c) − in (33), div z (cid:48) τ = thatwe integrate over Z = { z ∈ (0 , z m ) , z ∈ Y } ∪ { z ∈ ( − z m , , z ∈ ( − ˆ (cid:96)/ , ˆ (cid:96)/ } . Accounting fori) τ · n continuous at z =
0, ii) τ · n = τ periodic at z = ± ˆ (cid:96)/ (cid:82) ˆ (cid:96)/ − ˆ (cid:96)/ τ i ( − z m , z , x (cid:48) ) d z = (cid:82) Y τ i ( z m , z , x (cid:48) ) d z . In the limit z m → ∞ in (40) - (41) along with π i = σ i (0 − , x (cid:48) ) = , i = , , , (46)which tells us that the plates do not couple to the substrate at the dominant order. The couplingappears at the next order, starting with div z (cid:48) τ = from (c) − . As for τ and using again that π i =
0, we get that σ i (0 − , x (cid:48) ) = π i (0 + , x (cid:48) ); eventually, using π i in (19), we get σ (0 − , x (cid:48) ) = ρ p ω ϕ ˆ h W ( x (cid:48) ) , σ (0 − , x (cid:48) ) = − E p (1 − ν p ) ϕ ˆ (cid:96) ∂ W ∂ y (0 + , x (cid:48) ) ,σ (0 − , x (cid:48) ) = ϕ ˆ h E p ∂ W ∂ x ( x (cid:48) ) + ρ p ω W ( x (cid:48) ) (47) The e ff ective problem (3) is obtained for ( u = u , σ = σ + η σ ) in the substrate for x < u = W , σ = π + η π ) in the region of the array for x >
0. Remembering that y = x /η and ˆ h = h /η , ˆ (cid:96) = (cid:96)/η , it is easy to see that (i) the Euler-Bernoulli equation in (3) is obtainedfrom (20), (ii) the e ff ective boundary conditions announced in (5) from (36), (38), (44), (45) and(46)-(47).
4. Numerical validation of the e ff ective problem for a two-dimensional problem In this section, we inspect the validity of the e ff ective problem in a two-dimensional settingfor in-plane waves ( u =
0, hence ∂ x = x → −∞ at oblique incidence on the free surface supportingthe array of plates, and Lamb waves are excited in the plates. This is done using a multimodalmethod with pseudo-periodic solutions in the soil and Lamb modes in the plates; the method isdetailed in [54]. In the e ff ective problem, the solution is explicit, from (3) - (5) or equivalently(6)-(7) when the solution in the plates is disregarded.13e set the material properties for the elastic substrate: ν s = . E s = ρ s = − , and for the plates : ν p = . E p = ρ p =
500 Kg.m − , and ϕ = .
5. We choose (cid:96) = η = √ k T (cid:96) = .
37 ( ω =
124 rad.s − ), hence κ = .
64 m − . We shall consider h ∈ (0 ,
30) m resulting in κ h ∈ (0 ,
20) which includes the first 6 bending modes for h = h n , n = , · · · ,
6, and h (cid:39) h (cid:39) . h (cid:39) . h = . h = . h = . x appears for h = π/ (2 ω ) (cid:112) E p /ρ p (cid:39) . ff ective problem We define the potentials ( φ, ψ ) using the Helmholtz decomposition, with u = ∇ φ + ∇ × ( ψ e ).The incident wave in the substrate is defined in terms of the incident potentials φ inc ( x , x ) = A incL e i α L x e i β x , ψ inc ( x , x ) = A incT e i α T x e i β x , with ( α L , β ) = k L (cos θ L , sin θ L ) , ( α T , β ) = k T (cos θ T , sin θ T ) , (48)with k L = (cid:113) ρ s λ s + µ s ω and k T = (cid:113) ρ s µ s ω . The solution in the substrate reads φ ( x , x ) = φ inc ( x , x ) + (cid:0) R LL A incL + R LT A incT (cid:1) e − i α L x e i β x ,ψ ( x , x ) = ψ inc ( x , x ) + (cid:0) R TL A incL + R TT A incT (cid:1) e − i α T x e i β x . (49) x x
0) m and x ∈ (0 , Obviously, the same reflection coe ffi cients are obtained by solving the complete problem(3)-(5); we get the displacement fields in the region of the plates x ∈ (0 , h ), with u ( x , x ) = u (0 , x ) , u ( x , x ) = u (0 , x ) V ( x ) , u (0 , x ) = i ξ k L D (cid:2) cos θ L (cos 2 θ T − i cos θ T a f ) A incL + (sin 2 θ T cos θ L − i sin θ L cos θ T a f ) A incT (cid:3) e i β x , u (0 , x ) = ik T D (cid:2) sin 2 θ L (2 cos θ T − ia ) A incL − ξ cos θ T ( ξ cos 2 θ T − ia cos θ L ) A incT (cid:3) e i β x , V ( x ) = v ( κ h ) [ch κ ( x − h ) + cos κ ( x − h )] + v ( κ h ) [sh κ ( x − h ) + sin κ ( x − h )]2 (1 + ch κ h cos κ h ) , with v ( κ h ) = (ch κ h + cos κ h ) , v ( κ h ) = (sh κ h − sin κ h ) . (51)15s a reference case, typical displacement fields ( u , u ) for a surface on its own ( h =
0) arereported in figure 6. The incident wave is of the form (48) with A incL = / (2 β ) , A incT = − / (2 α T )producing an incident horizontal displacement equal to unity at x =
0; three incident angles θ L are reported. It is worth noting that with a = ffi cients fora flat interface, see e.g. [55]. The e ff ect of the array is encapsulated in the impedance parameters ( a , f ), or equivalently( Z = k T ρ s a , f ), whose variations versus k T h are reported in figure 7. The parameter Z = ρ p ω ϕ h tells us that heavier plates and higher frequency produce more pronounced coupling with thesubstrate, which is not surprising. The parameter f encapsulates the e ff ects of the bending reso-nances and it diverges when approaching them. This occurs at the frequencies corresponding toa clamped- free single plate, in other words ch κ h cos κ h = −
0. The resulting patterns, not reported, are indeed similar to those obtainedfor a flat interface in figure 6. Since there is not much to be said on the field in the substrate, wefocus on the capability of the complete e ff ective solution to reproduce the actual displacementin the plates. Figure 8 show a small region of the displacement fields near the interface ( h =
5m resulting in k T h = . θ L = ◦ ). From what we have said (the interaction is weak),the displacements in the substrate are neatly reproduced. More interestingly, the displacementsin the plates are also accurately reproduced in an ”averaged” sense which clearly appears forthe displacement u : in the actual problem, u varies linearly with x within a single plate, inagreement with (21); this variation at the small scale is superimposed to a variation at large scale,16rom one plate to the other. The small scale variations do not appear in the homogenized solutionsince they vanish on average while the large scale variation is captured. The same occurs for u but in this case, the small scale variations are less visible because they appears at the order 2 (see(17) and (27)). direct numerics
5) mand x ∈ (0 ,
20) m ( h = θ L = ◦ ). Strong coupling in the vicinity of the bending resonances can be measured by the amplitudesof the displacements in the plates. We report in figure 9 the amplitudes of the horizontal displace-ments against h , at the bottom and at the top of a single plate. In the actual problem these am-plitudes are calculated by averaging over x ∈ ( − (cid:96) p / , (cid:96) p /
2) the profiles | u (0 , x ) | and | u ( h , x ) | obtained numerically. In the homogenized problem | u (0 , x ) | and | u ( h , x ) | = | V ( h ) u (0 , x ) | aregiven in closed-forms from (51).
22 m arecompensated (dashed line). The inset shows the actual variation of u ( h , x ) within a single plate with variations as smallas about 10 − with respect to the mean value. h ∈ (0 ,
15) m, the first three bending resonances are visible by means of high displace-ments at the top of the plates (up to 40 times the amplitude of the incident wave in the reportedcase). It is also visible by means of vanishing amplitude at the bottom of the plate, in agreementwith (51) for f → ∞ . Hence, near the bending resonances, the plates are clamped and theyimpose a vanishing horizontal displacement at the interface with the substrate, a fact alreadymentioned in [22]. In the substrate, the resulting displacements are significantly impacted. Largevalues of f ( κ h ) produce R LL (cid:39) − R TT (cid:39) R TL (cid:39) R LT (cid:39) u ( x , x ) (cid:39) ik L cos θ L (cid:0) A incL cos α L x + A incT tan θ L cos α T x (cid:1) e i β x , u ( x , x ) (cid:39) k T cos θ T (cid:0) − A incL tan θ L sin α L x + A incT sin α T x (cid:1) e i β x , (52)corresponding to a superposition of standing waves. Examples of resulting patterns are shownin figure 10 for the first three bending resonances to be compared with those obtained for a flatinterface in figure 6. It is worth noting that in figure 10 we have accounted for the shift in h n , d i r ec t nu m e r i c s
22 m in the present case.
To go further in the analysis, we report in figure 11 the reflection coe ffi cients against h ∈ (0 ,
25) m and θ ∈ (0 , ◦ ). We represent the real and imaginary parts of the 4 reflection coe ffi -cients. As previously said, our analysis does not hold at and above the first longitudinal reso-nance, which appears for h (cid:39) . κ h = . ff ective model indeedbreaks down at this high value but it remains surprisingly accurate up to h ∼
15 m (hence k T h ∼
5. Conclusion
We have studied the interaction of an array of plates or beams with an elastic half-space u-sing asymptotic analysis and homogenization techniques. The resulting models (3)-(5) for platesand (B.1)-(B.3) for beams provide one-dimensional propagation problems which in their simplerform consist in e ff ective boundary conditions at the surface of the ground, (6) for plates and (B.5)for beams. The exception for plates in the boundary condition σ in (5) is incidental for in-planeincidence but it is interesting since it provides non trivial coupling for arbitrary incidence. For in-plane incidence, the model has been validated by comparison with direct numerical simulationswhich show an overall good agreement. In particular, the displacement fields obtained in aclosed-form accurately reproduce the actual ones; this is of practical importance for applicationsto site-city interaction where the displacements at the bottom and at the top of buildings arerelevant quantities to measure the risk of building damage.Our models have been obtained owing to a deductive approach which applies to a wide va-riety of problems. An important point is that the analysis does not assume a preliminary modelreduction for the resonator on its own and as such, it can be conducted at any order. Higherorder models would involve enriched transmission and boundary conditions able to capture moresubtle e ff ects as the shift in the resonance frequencies visible in the figure 9 or the presenceof heterogeneity at the roots and at the top of the bodies as it has been done in [36]. Next,we have considered bodies with su ffi cient symmetry resulting in a diagonal rigidity matricesand which allow for easier calculations. When symmetries are lost, and the simplest case isthat of beams with rectangular cross-sections, the calculations are similar; they will producecouplings for incidences as soon as the horizontal component does not coincide with one ofthe two principal directions. Additional complexities can be accounted for straightforwardly, asorthotropic anisotropy along x or slow variations in the cross-section. Eventually, the modelsare restricted to the low frequency regime where only the flexural resonances take place. At thethreshold of the first longitudinal resonance, they fail as illustrated in figure 12. Extension of the20resent work consists in adapting the homogenization procedure in order to capture both flexuraland longitudinal resonances at higher frequencies. Acknowledgements
A. M. and S.G. acknowledge insightful discussions with Philippe Roux at the Institute IS-Terre of the University of Grenoble-Alpes. S.G. is also thankful for a visiting position in thegroup of Richard Craster within the Department of Mathematics at imperial College London in2018-2019.
Appendix A. Remark on the solution in the region of the plates
From the boundary conditions (5), ∂ u ∂ x ( h , x ) = ∂ u ∂ x ( h , x ) = u (0 + , x ) = u (0 − , x ), ∂ u ∂ x (0 + , x ) =
0, the general solution for x ∈ (0 , h ) reads as follows u ( x , x ) = A ( x ) { a ( κ h ) [ch κ ( x − h ) + cos κ ( x − h )] + b ( κ h ) [sh κ ( x − h ) + sin κ ( x − h )] } , with a ( κ h ) = (ch κ h + cos κ h ) , b ( κ h ) = (sh κ h − sin κ h ) . (A.1)The displacement u is continuous at x = A ( x )(1 + ch κ h cos κ h ) = u (0 , x ),hence u ( x , x ) = u (0 , x ) V ( x ) , V ( x ) = a ( κ h ) [ch κ ( x − h ) + cos κ ( x − h )] + b ( κ h ) [sh κ ( x − h ) + sin κ ( x − h )]2 (1 + ch κ h cos κ h ) . (A.2)Obviously, this holds except at the resonance frequencies of the plates for ch κ r h cos κ r h = − u (0 , x ) = σ (0 − , x (cid:48) ) in (5) becomes σ (0 − , x (cid:48) ) = − ( D p /(cid:96) ) V (cid:48)(cid:48)(cid:48) (0) u (0 , x ), with V (cid:48)(cid:48)(cid:48) (0) = − κ h f ( κ h ), with f ( κ h ) in (6). With σ (0 − , x (cid:48) ) = ( κ D p h /(cid:96) ) f ( κ h ) u (0 , x ) and κ D p = ρ p ω ϕ(cid:96) ,we recover the form announced in (6). (1 + ch h cos h ) / ch h
0) : div σ + ρ s ω u = , σ = µ s ε + λ s tr( ε ) I , In the region of the beams , x ∈ (0 , h ) : ∂ u α ∂ x − κ u α = , α = , , κ = ρ p ω π r p D p / , (B.1)where D p = E p π r p , (B.2)is the flexural rigidity of the circular beams, complemented by the boundary conditions σ (0 − , x (cid:48) ) = ρ p ω ϕ h u (0 , x ) ,σ α (0 − , x (cid:48) ) = − D p S ∂ u α ∂ x (0 + , x ) , u α (0 + , x (cid:48) ) = u α (0 − , x (cid:48) ) , ∂ u α ∂ x (0 + , x (cid:48) ) = ,∂ u α ∂ x ( h , x (cid:48) ) = ∂ u α ∂ x ( h , x (cid:48) ) = , α = , . (B.3)where ϕ = π r p / S . It follows that the problem can be thought in the substrate only, withdiv σ + ρ s ω u = , σ = µ s ε + λ s tr( ε ) I , x ∈ ( −∞ , , (B.4)along with the boundary conditions of the Robin’s type σ (0 , x (cid:48) ) = Zu (0 , x (cid:48) ) ,σ α (0 , x (cid:48) ) = Z f ( κ h ) u α (0 , x (cid:48) ) , (B.5)where Z and f ( κ h ) are defined in (7) (we used that D p κ = ρ p ω π r p ). Appendix B.1. E ff ective wave equation in the region of the beamsAppendix B.1.1. Notations We shall use the same expansions as in (9) but now, the terms w n and π n depend on z (cid:48) = ( z , z ) (and not only on z ) and we seek to establish the e ff ective behaviour in the region of thearray in terms of macroscopic averaged fields w n ( y , x (cid:48) ) = ϕ ˆ S (cid:90) Y w n ( y , z (cid:48) , x (cid:48) ) d z (cid:48) , π n ( y , x (cid:48) ) = S (cid:90) Y π n ( y , z (cid:48) , x (cid:48) ) d z (cid:48) , (B.6)22here x (cid:48) = ( x , x ) and Y represents the circular section of the beam Y = (cid:26) (cid:113) z + z ≤ ˆ r (cid:27) , withˆ r = r p /η . It is worth noting that it is su ffi cient to replace z by z (cid:48) in (10) to (15); in particular,we have ∇ → e η ∂∂ y + η ∇ z (cid:48) + ∇ x (cid:48) . (B.7)and ε z (cid:48) ( w ) = ∂ z w ∂ z w ∂ z w ∂ z w (cid:0) ∂ z w + ∂ z w (cid:1) ∂ z w (cid:0) ∂ z w + ∂ z w (cid:1) ∂ z w . (B.8) Appendix B.1.2. Sequence of resolution and main results of the analysis
The analysis is made more involved since the problem is two-dimensional in the rescaledcoordinate z (cid:48) . The procedure is thus more complex. It is as follow:1. We establish that π = , (B.9)and the dependence of w on ( z , z ), specifically w = W ( x (cid:48) ) , w α = W α ( y , x (cid:48) ) , (B.10)2. We deduce the form of π and of w π = − E p ∂ W α ∂ y ( y , x (cid:48) ) z α , π α = π αβ = , w = W ( x (cid:48) ) − ∂ W α ∂ y ( y , x (cid:48) ) z α , w α = W α ( y , x (cid:48) ) . (B.11)3. eventually the form of π i and the Euler-Bernoulli equation for the bending W α , α = , π ( y , x (cid:48) ) = ρ p ω ϕ W ( x (cid:48) ) (ˆ h − y ) ,π α ( y , x (cid:48) ) = − E p ϕ ˆ r ∂ W α ∂ y ( y , x (cid:48) ) , (B.12)and E p ˆ r ∂ W α ∂ y − ρ p ω W α = . (B.13) Appendix B.1.3. First step: π in (B.9) and w in (B.10)This step is not very demanding. From (E ) − in (13), ∂ y π + ∂ z α π α =
0, which afterintegration over Y leaves us with ∂ y π =
0; anticipating π = π = (cid:48) α ) − in (14), ∂ z α w = (cid:48) ) − ∂ y w =
0. It follows that w dependsonly on x (cid:48) , in agreement with (B.10). From (C (cid:48) αβ ) − , w α is a rigid body motion i.e.w α = W α ( y , x (cid:48) ) + Ω ( y , x (cid:48) )( e , z , e α ) , (B.14)with ( e , z , e α ) = e · ( z × e α ) being the triple product, and we shall establish that Ω =
0. To doso we infer, from (C (cid:48) α ) − , that ∂ z α w + ∂ y w α = → ∂ y (cid:16) ∂ z w − ∂ z w (cid:17) = . (B.15)Inserting (B.14) in (B.15) tells us that Ω does not depend on y and anticipating the matchingcondition with the displacement in the substrate which imposes that Ω = y =
0, we deducethat Ω = w α announced in (B.10). Appendix B.1.4. Second step: ( π , w ) in (B.11)We start by determining w incompletely (compared to what is announced in (B.10)). For w , we come back to ∂ z α w + ∂ y w α = w α in (B.10) provides us with w = W ( y , x (cid:48) ) − ∂ W α ∂ y ( y , x (cid:48) ) z α , (B.16)and it remains for us to show that W does not depend on y ; this will be done after π has beendetermined. Next, from (C (cid:48) αβ ) − , w α is a rigid body motion, hence w α = W α ( y , x (cid:48) ) + Ω ( y , x (cid:48) )( e , z , e α ) . (B.17)Now, we shall prove that Ω =
0; this will be done once π α have been determined. For thetime being, we pursue the calculations by setting the boundary value problem set in Y on theunknowns ( π αβ , w α ). From (E α ) − and (C αβ ) in (13), it reads ∂ z β π αβ = , π αβ = µ p (cid:16) ε x (cid:48) αβ ( w ) + ε z (cid:48) αβ ( w ) (cid:17) + λ p (cid:16) ∂ y w + ε x (cid:48) γγ ( w ) + ε z (cid:48) γγ ( w ) (cid:17) δ αβ , in Y ,π αβ n β = ∂ Y , (B.18)with w known from (B.10) and w from (B.16) at this stage. It is easy to check that the solutionof this boundary value problem is π αβ = , w α = − e x (cid:48) αβ ( w ) z β − λ p µ p + λ p ) z α ∂ W ∂ y ( y , x (cid:48) ) + λ p µ p + λ p ) g α + W α ( y , x (cid:48) ) + Ω ( y , x (cid:48) )( e , z , e α ) , (B.19)where g = z − z ∂ W ∂ y + z z ∂ W ∂ y , g = z − z ∂ W ∂ y + z z ∂ W ∂ y . (B.20)The above form of w α along with w α in (B.10) can now be used to find π = (cid:0) µ p + λ p (cid:1) ∂ y w + λ p (cid:16) ∂ x α w α + ∂ z α w α (cid:17) , and we get π = E p ∂ W ∂ y ( y , x (cid:48) ) − ∂ W α ∂ y ( y , x (cid:48) ) z α , (B.21)24here we used that E p = µ p (2 µ p + λ p ) / ( µ p + λ p ). It is now su ffi cient to remember that π = ∂ y W =
0, hence the above expression of π simplifies to the form announced in (B.11)and w in (B.16) simplifies to to that in (B.11).We now use the boundary value problem set in Y on the unknowns ( π α , w ). From (E ) − and (C α ) in (13), it reads as follows (cid:40) ∂ z α π α = , π α = µ p (cid:16) ∂ y w α + ∂ x α w + ∂ z α w (cid:17) , in Y ,π α n α = , on ∂ Y , (B.22)with w known from (B.10) and w α from (B.17) at this stage. The solution is again found to beof the form π α = µ p ∂ Ω ∂ y ( y , x (cid:48) )( e , z , e α ) , w = W ( y , x (cid:48) ) − z α ∂ W α ∂ y ( y , x (cid:48) ) + ∂ W ∂ x α ( x (cid:48) ) , (B.23)and we see that Ω = π α =
0. To show that Ω =
0, we use ∂ y π α + ∂ z β π αβ = α ) − . Multiplying by v = v α e α with the triple product v α = ( e , z , e α ) andintegrating over Y , we find that (cid:90) Y v α ∂ y π α d z (cid:48) + (cid:90) ∂ Y v α π αβ n β d l − (cid:90) Y ∂ z β v α π αβ d z (cid:48) = . (B.24)Since π is symmetric and ∇ v is anti-symmetric, we have ∂ z β v α π αβ =
0, and π αβ n β = ∂ Y .Hence, (B.24) reduces to (cid:90) Y v α ∂π α ∂ y d z (cid:48) = → ∂ Ω ∂ y ( y , x (cid:48) ) (cid:90) Y ( e , z , e α ) d z (cid:48) = . (B.25)Next, with ( e , z , e α ) = z whose integral does not vanish, we obtain that ∂ y Ω does not dependon y ; anticipating that ∂ y Ω (ˆ h , x (cid:48) ) = Ω (ˆ h , x (cid:48) ) =
0, we deduce that Ω = π α =
0, from (B.23), and that w α = W α ( y , x (cid:48) ), from (B.17), in agreement with(B.11). Appendix B.1.5. Third step: π i in (B.12) and the Euler-Bernoulli equations in (B.13)This starts with (E) in (13) integrated over Y , specifically ∂π ∂ y + ρ p ω ϕ W = , ∂π α ∂ y + ρ p ω ϕ W α = , (B.26)where we have used that π = π n | ∂ Y = . Since W depends onlyon x (cid:48) , and anticipating that π (ˆ h , x (cid:48) ) =
0, we obtain by integration the form of π in (B.12).To get π α , we multiply (E ) − (which reads ∂ y π + ∂ z β π β =
0, with π in (B.11)) by z α and integrate over Y to find that E p ∂ W β ∂ y ( y , x (cid:48) ) (cid:90) Y z α z β d s + (cid:90) Y π β δ αβ d s = , (B.27)25here we have used that π β n β | ∂ Y =
0. For the circular cross-section of the beams, (cid:82) Y z z d z (cid:48) = (cid:82) Y z d z (cid:48) = (cid:82) Y z d z (cid:48) = π ˆ r /
4. It follows thatˆ S π α ( y , x (cid:48) ) = − E p π ˆ r ∂ W α ∂ y ( y , x (cid:48) ) , (B.28)in agreement with (B.12) (with ϕ ˆ S = π ˆ r ). Coming back to (B.26) with the above form of π α ,we deduce that E p π ˆ r ∂ W α ∂ y − ρ p ω ϕ ˆ S W α = , (B.29)in agreement with (B.13). Appendix B.2. E ff ective boundary conditions at the top of the array of beams As we have done in (30), we consider the following expansions for the displacement andstress u = (cid:88) n ≥ η n v n ( z , x (cid:48) ) , σ = (cid:88) n ≥ η n τ n ( z , x (cid:48) ) . (B.30)We use (e) in (33) (with z (cid:48) → z ) which provide us with div z τ = div z τ = , and this makesthe calculations identical to those conducted in § Z = { z ∈ ( −∞ , , z (cid:48) ∈ Y } . We thus obtain π i (ˆ h , z (cid:48) , x (cid:48) ) = π i (ˆ h , z (cid:48) , x (cid:48) ) = , i = , , , (B.31)(see (35)). The conditions on π i are consistent with (B.9) and (B.11). The condition on π isthat anticipated to find (B.12). Eventually, the condition on π α combined with (B.12) providesthe conditions of zero shear force ∂ W α ∂ y (ˆ h , x (cid:48) ) = . (B.32)To derive the condition of zero bending moment, we proceed the same as we have done in (B.24);with v = v α e α and v α = ( e , z , e α ), we consider the vanishing integral (cid:82) Z v α ∂ z i τ i α d v =
0, hence (cid:90) ∂ Z v α τ i α n i = , (B.33)where we have used that ∂ z i v α τ i α = τ · n vanishes on ∂ Z except atthe bottom face z = − z m and passing to the limit z m → + ∞ , this integral reduces to (cid:90) Y v α π α (ˆ h , z (cid:48) , x (cid:48) ) d s = . (B.34)Making use of (B.23) leads to the anticipated boundary condition ∂ Ω ∂ y (ˆ h , x (cid:48) ) = , (B.35)26hat we have used to get π α =
0. It remains to derive the condition of zero bending moment. Byconsidering a = z α e − z e α and integrating over Z the scalar a · div z τ (since div z τ = ), wefound that 0 = (cid:90) ∂ Z a i τ i j n j d s = − (cid:90) Y z α τ | z = − z m d s − z m (cid:90) Y τ α | z = − z m d s . (B.36)Since we have in addition 0 = (cid:82) ∂ Z τ i j n j d s = (cid:82) Y τ α | z = − z m d s , we can pass to the limit z m → ∞ ,and get 0 = (cid:82) Y z α τ → z α π (ˆ h , x (cid:48) ). Now accounting for π in (B.21), we obtain the expectedboundary condition ∂ W α ∂ y (ˆ h , x (cid:48) ) = . (B.37) Appendix B.3. E ff ective transmission conditions between the substrate and the array In the vicinity of the interface between the substrate and the array, we consider the sameexpansions as in (B.30), and at the dominant order, we still have div z τ = div z τ = . Thecalculations are identical to that conducted in § Z = { z ∈ (0 , + ∞ ) , z (cid:48) ∈ Y } ∪ { z ∈ ( −∞ , , z (cid:48) ∈ ( − ˆ (cid:96)/ , ˆ (cid:96)/ } , and we find σ i (0 − , x (cid:48) ) = , i = , , , (B.38)which are consistent with (B.9) and (B.11). Next, using π i in (B.12), we find σ (0 − , x (cid:48) ) = ρ p ω ϕ ˆ hW ( x (cid:48) ) , σ α (0 − , x (cid:48) ) = − E p π ˆ r S ∂ W α ∂ y (0 , x (cid:48) ) . (B.39)We have yet to establish the continuity of the displacement. From the counterpart of (c (cid:48) ) in (42)(with z (cid:48) → z ), it is easily seen that we have at the dominant orders ε z ( v ) = ε z ( v ) = . (B.40)Therefore v and v are piecewise rigid body motions, namely v = Ω ( x (cid:48) ) × z + V ( x (cid:48) ), thesame for v . Invoking the periodicity of v i , i = , z and z for z < v i at z =
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