Effective model for elastic waves propagating in a substrate supporting a dense array of plates/beams with flexural resonances

(1)

HAL Id: hal-02441350

https://hal.archives-ouvertes.fr/hal-02441350

Preprint submitted on 15 Jan 2020

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

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, Sebastien Guenneau

To cite this version:

Jean-Jacques Marigo, Kim Pham, Agnès Maurel, Sebastien Guenneau. Effective model for elastic

waves propagating in a substrate supporting a dense array of plates/beams with flexural resonances.

2020. �hal-02441350�

(2)

E

ffective model for elastic waves propagating in a substrate

supporting 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

ffect of an array of plates or beams over a semi-infinite elastic ground on the

propagation 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 in

the ground. An e

ffective model is obtained using asymptotic analysis and homogenization

tech-niques, which can be expressed in terms of the ground alone with e

ffective dynamic

(frequency-dependent) boundary conditions of the Robin’s type. For an incident plane wave at oblique

incidence, the displacement fields and the reflection coefficients are obtained in closed forms and

their validity is inspected by comparison with direct numerics in a two-dimensional setting.

Keywords:

asymptotic analysis; elastic waves; metamaterials; metasurfaces

2010 MSC:

00-01, 99-00

1. Introduction

1 We are interested in wave propagation in a semi-infinite elastic substrate supporting a

perio-2

dic and dense array of thin or slender bodies. This is the canonic idealized configuration used

3 to illustrate the problem of ”site-city interaction”. Such a problem, recent on the seismology

4 history scale, aims to account for the urban environment as a factor modifying the seismic ground

5 motion. Starting in the 19

th

century, the interest was primarily focused on the motion of the soil

6 elicited by static or dynamic sources being concentrated or distributed on the free surface in the

7 absence of buildings. These studies have led to important results as the Lamb’s problem [1, 2].

8 Then, more realistic configurations have been considered using approximate models to predict

9

(3)

r

p

<latexit sha1_base64="vDSle26A1qO56V2Jw87XGkBaFwg=">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</latexit>

S

<latexit sha1_base64="FeOcx6sCSBGT4eKEM+4hHdq8QtM=">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</latexit>

h

<latexit sha1_base64="1s9K20MusN/DXg5qgRUhUjA0NQ0=">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</latexit>

`

p

= '`

<latexit sha1_base64="Uew7XDlCUBJ0Fu2VD8lq40RK/Mc=">AAAC5HicjVHLSsNAFD2Nr1pfVZcuDBbBVUm1oBuh4MZlBfsAU0oSp+3gNAmTiViKS3fuxK0/4Fa/RfwD/QvvjBF8IDohyZlz7zkz914/FjxRjvOcsyYmp6Zn8rOFufmFxaXi8koziVIZsEYQiUi2fS9hgoesobgSrB1L5g19wVr+2YGOt86ZTHgUHqtRzDpDrx/yHg88RVS3uO4yIbquYhdq7Coejuz4ct8992Q84CZULDllxyz7J6hkoIRs1aPiE1ycIkKAFEMwhFCEBTwk9JygAgcxcR2MiZOEuIkzXKJA2pSyGGV4xJ7Rt0+7k4wNaa89E6MO6BRBrySljU3SRJQnCevTbBNPjbNmf/MeG099txH9/cxrSKzCgNi/dB+Z/9XpWhR62DM1cKopNoyuLshcUtMVfXP7U1WKHGLiND6luCQcGOVHn22jSUztureeib+YTM3qfZDlpnjVt6QBV76P8ydobpcrO+Xto2qpVs1GnccaNrBF89xFDYeoo0HeV7jHAx6tnnVt3Vi376lWLtOs4suy7t4AtgSc1Q==</latexit>

h

`

<latexit sha1_base64="ZBIa8SK7e4XGmjkTnRciRDgOJdk=">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</latexit>

x

3

<latexit sha1_base64="0vhGKRAYP4aQFU4Jyb9pVHHDWK0=">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</latexit>

x

1 x

2

0

<latexit sha1_base64="5mYUOUymVgUg3BFR6u8Av/Go1nA=">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</latexit>

x

3 x

1 x

2

0 (a)

(b)

Figure 1: Geometry of the actual problems: (a) Array of parallel plates infinite along x

3 atop an isotropic substrate; (b)

Doubly periodic array of cylindrical beams atop an isotropic substrate. The scalings are chosen to capture the bending

resonances only; with k

T

the wavenumber in the substrate, k

T

h

= η and k

T

` = O(η

2 ).

the e

ffect of complex soils, including the presence of buried foundations, on the displacements

10 in structures on the ground, see e.g. [3, 4, 5] and [6] for a review. In the classical two-step

11 model, the displacements in the soil without structures above, so-called free fields, were firstly

12 calculated and they were subsequently used as input data to determine the motion within the

13 structure [4, 6]. This means that the interaction, refereed to as the soil-structure interaction, was

14 restricted to the e

ffect of the soil on the structure. In the mid-1970s, Luco and Contesse [7] and

15 Wong and Trifunac [8] studied the interaction between nearby buildings and they evidenced the

16 resulting modification on the ground motion. They termed this mutual interaction the

structure-17

soil-structure interaction, which has been later renamed soil-structure-soil interaction. On the

18 basis of these pioneering works the idea took root that several structures may interact with each

19 other and modify the ground motion, supplied by numerical simulations and direct observations

20 during earthquakes [9, 10, 11, 12, 13, 14, 15, 16]. At the scale of a city with the specificity of

21 the presence of a sedimentary basin, the soil-structure-soil interaction has been called site-city

22 interaction, a term firstly coined bu Gu´eguen [10]. From a theoretical point of view, most of the

23 models encapsulate the response of a building with a single or multi-degree of freedom system

24 [17, 10, 18, 19, 20, 16]. On the basis of this model, Boutin, Roussillon and co-workers have

25 developed homogenized models where the multiple interactions between periodically located

26 oscillators are accounted for from a macroscopic city-scale point of view [19, 21, 22, 23, 24]. In

27 the low frequency limit, that is when the incident wavelength is large compared to the resonator

28 spacing, the e

ffect of the resonators can be encapsulated in effective boundary conditions of

29 the Robin type for the soil, a result that we shall recovered in the present study. Such a

mass-30

spring model has been used in physics for randomly distributed oscillators [25] and periodically

31 distributed oscillators [26, 27, 28] for their influence on surface Love and Rayleigh waves. The

32 ability of arrays of resonators to block Love and Rayleigh waves has been exploited to envision

33 new devices of seismic metasurfaces [29, 30, 31, 32, 33, 34, 35, 36, 37, 38] in analogy with

34 metasurfaces in acoustics [39, 40] and in electromagnetism [41, 42].

35 In this study, we use asymptotic analysis and homogenization techniques to revisit the

prob-lem of the interaction of a periodic array of plates or beams on the propagation of seismic waves

(4)

in three dimensions. We consider slender bodies in the low frequency limit which means two

things. Firstly, the typical wavelength 1/k is much larger than the array spacing `, which is a

classical hypothesis. Secondly, we focus on the lowest resonances of the bodies being flexural

resonances. The first flexural resonances correspond to kh ∼ `p/h, with `p

the body thickness, h

the body height and `p/h the slenderness (in comparison the first longitudinal resonance appears

at kh ∼ 1). Now, we consider dense arrays, which means that `p

∼

`, and ϕ = `p/` ∈ (0, 1)

(Figure 1). Hence the asymptotic analysis is conducted considering that

the wavelength 1/k is large compared to h which is itself large compared to `

p

∼

`.

It is worth noting that assuming ϕ

= O(η

n

) with n ≥ 1 would allow a reduction of model in a first

step, resulting in concentrated force problems, as implicitly considered in [19, 21, 35]. Here on

the contrary, the implementation of the asymptotic method will require that we reconstruct the

asymptotic theory of plates and beams in a low frequency regime, as previously done for a single

body in solid mechanics, see e.g. [43, 44] for plates and [45, 46, 47, 48] for beams. However,

this classical theory has to be complemented with matched asymptotic expansions to link the

behavior in the periodic set of bodies with that in the substrate. This “soil-structure” coupling

requires a specific treatment as used in interface homogenization [49, 50, 51, 52], see also [53]

for a resonant case. In the present case, we shall derive e

ffective boundary and transmission

conditions applying in a homogenized region which replaces the actual array; and in this e

ffective

region the wave equation for flexural waves applies. This problem can be further simplified in

e

ffective boundary conditions of the Robin type applying on the surface of the soil, namely

σ · n = K(ω)u,

(1)

where the frequency-dependent rigidity matrix K depends explicitly on the flexural frequencies of

36 the plates

/beams. The rigidity matrix is diagonal as soon as the bodies have sufficient symmetry,

37 resulting in e

ffective impedance conditions which ressemble those obtained in [22] in the same

38 settings.

39 The paper is organized as follows. In §2, we summarize the result of the asymptotic analysis

40 in the case of an array of plates, whose detailed derivation is given in the §3. The resulting

41 ”complete” formulation (3)-(5) is equivalent to that in (6)-(7) thanks to a partial resolution of

42 the problem. In §4, the accuracy of the e

ffective model is inspected by comparison with direct

43 numerics based on multimodal method [54] for an in-plane incident wave. The strong coupling

44 of the array with the ground at the flexural resonances are exemplified and the agreement between

45 the actual and e

ffective problems is discussed. We finish the study in §5 with concluding remarks

46 and perspectives. We provide in the appendix Appendix B the e

ffective problem for the an array

47 of beams which is merely identical to the case of the plates with some specificities which are

48 addressed.

49 2. The actual problem and the e

ffective problem

50 We consider in this section the asymptotic analysis of an array of parallel plates atop an

51 isotropic elastic substrate. We note that the problem splits in the in-plane and out-of-plane

po-52

larizations. The latter case has been already addressed in [36]. We focus in this section on the

53 former, in-plane, case. We further note that the asymptotic analysis of a doubly periodic array

54 of cylinders atop an isotropic substrate is a fully coupled elastodynamic wave problem, which is

55 thus slightly more involved and addressed in the Appendix.

56

(5)

2.1. The physical problem

57 We consider the equation of elastodynamics for the displacement vector u the stress tensor σ

and the strain tensor ε











in the substrate, x1

∈ (−∞, 0) :

divσ

+ ρsω

2 _u

_{= 0, σ = 2µsε + λstr(ε)I,}

_{ε =}

1

2 (∇u

+

t

_∇u),

in the plates, x1

∈ (0, `) :

divσ

+ ρpω

2 u

= 0, σ = 2µpε + λptr(ε)I,

(2)

with the Lam´e coe

fficients (λp, µp) of the plates and (λs, µs) of the substrate, ω the angular

fre-58

quency and I stands for the identity matrix. In three dimensions with x

= (x1

, x

2 , x

3), stress free

59 conditions σ · n

= 0 apply at each boundary between an elastic medium (the plates or the

sub-60

strate) and air, with n the normal to the interface. Eventually, the continuity of the displacement

61 and the normal stress apply at each boundary between the parallel plates and the substrate. This

62 problem can be solved once the source u

inc

_{has been defined and accounting for the radiation}

63 condition for x

1 → −∞ which applies to the scattered field (u − u

inc

).

64 2.2. The e

ffective problem

65 Below we summarize the main results of the analysis developed in the §3 and which provides

66 the so-called ”complete formulation” where the array of parallel plates is replaced by an

equiva-67

lent layer associated to e

ffective boundary and transmission conditions (Figure 2(a)). Owing to

68 a partial resolution, this formulation can be simplified to an equivalent ”impedance formulation”

69 set on the substrate only (Figure 2(b)). We note that all three components of the displacement

70 field appear in this section, and the reader should be aware that we make use of variables x

=

71 (x1, x2, x3) and x

0 _{= (x2, x3).}

72 2.2.1. Complete formulation

73 The e

ffective problem reads as follow











In the substrate, x1

∈ (−∞, 0) :

divσ

+ ρsω

2 _u

_{= 0, σ = 2µsε + λstr(ε)I,}

In the region of the plates, x

1 ∈ (0, h) :

∂

4 _u

2 ∂x

4

1 −

κ

4 u2

= 0, κ =

ρpω

2 _`p

Dp

!

1/4

,

u

1 (x

1 , x

0 )

= u1

(0, x

0 ),

u

3 (x

1 , x

0 )

= u3

(0, x

0 ),

(3)

with x

0 _{= (x2}

_{, x}

3 ),

Dp

=

Ep

(1 − ν

2 p

)

`

3 p

12 ,

(4)

the flexural rigidity of the plates (ρ

p

the mass density, E

p

the Young’s modulus and ν

p

the Poisson’s

ratio), complemented with boundary conditions at x1

= 0 and x1

= h of the form











σ

11 (0

−

, x

0 )

= ρp

ω

2 ϕh u

1 (0, x

0 ),

σ

12 (0

−

, x

0 )

= −

Dp

`

∂

3 _u

2 ∂x

3

1 (0

+

, x

0 ),

σ

13(0

−

, x

0 )

= ϕh







Ep

∂

2 _u

3 ∂x

2

3 (0

−

, x

0 )

+ ρp

ω

2 u3(0

−

, x

0 )







,

u

2 (0

+

, x

0 )

= u2

(0

−

, x

0 ),

∂u2

∂x

1 (0

+

, x

0 )

= 0,

∂

2 _u2

∂x

2

1 (h, x

0 )

=

∂

3 _u2

∂x

3

1 (h, x

0 )

= 0.

(5)

4

(6)

These e

ffective conditions express (i) at x1

= 0 a balance of the stress, prescribed displacements

74 and vanishing rotation and (ii) at x

1 = h, free end conditions with vanishing bending moment

75 and shearing force. One notes that all three components of the displacement field u appear in (5)

76 which involves partial derivatives on x1

and x3

only.

77 x

3 x

1 x

2

0 h

x

3 x

1 x

2

0 (a)

(b)

Figure 2: Geometry in the e

ffective problems for an array of plates. (a) In the complete formulation, the region of the

array is replaced by an equivalent layer where (3) applies, complemented by the transmission conditions in (5). (b) In

the impedance formulation, the problem is reduced to effective boundary conditions (6) which hold at x

1 = 0. The same

holds for the array of beams, with (a) (B.1)-(B.3) and (b) (B.5).

2.2.2. Impedance formulation

78 From (3), the problem in x

1 ∈ (0, h) can be solved owing to the linearity of the problem with

respect to u

2 (0

−

, x

2 ), see Appendix A. Doing so, the problem can be thought in the substrate only

along with the boundary conditions of the Robin’s type, namely











divσ

+ ρsω

2 u

= 0, σ = 2µsε + λstr(ε)I,

x1

∈ (−∞, 0),

σ11(0, x

0 ₎

_{= Z u1(0, x}

0 _),

_σ12

_{(0, x}

0 ₎

_{= Z f (κh) u2(0, x}

0 _),

σ13(0, x

0 ₎

_{= ϕh}







Ep

∂

2 _u3

∂x

2

3 (0, x

0 )

+ ρpω

2 u3(0, x

0 )







,

(6)

with the following impedance parameters

Z

= ρp

ω

2 ϕh,

f

(κh)

=

shκh cos κh

+ chκh sin κh

κh(1 + chκh cos κh)

,

(7)

(we have used that Dpκ

4 _{= ρpω}

2 _{`p). The conditions on (σ11, σ12}

_{) encapsulate the e}

_{ffects of the}

79 in-plane bending of the plates while the condition on σ13

can be understood as the equilibrium

80 of an axially loaded bar (in the absence of substrate, we recover the wave equation for

quasi-81

longitudinal waves). It is worth noting that for out-of-plane displacements, u3(x1, x2) and u1

=

82 u2

= 0, the boundary conditions simplify to σ13(0, x2)

= ρpϕhω

2 u3(0, x2). This corresponds to

83 the impedance condition which can be deduced from the analysis conducted in [36] and resulting

84 in σ13(0, x2)

= µp

ϕk

T

tan(kTh)u3(0, x2) and obtained here in the limit kTh

1.

85

(7)

3. Derivation of the e

ffective problem

86 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 array

spacing ` ∼ `p. Hence, with kT

= ω pρs/µs

and kL

= ω pρs/(λs

+ 2µs) of the same order of

magnitude, we define the small non-dimensional parameter η as

η =

p

kT`,

and

kTh

= 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 asymptotic

analysis is conducted using the rescaled height ˆh of the plates and array’s spacing ˆ` defined by

(ˆh, ˆ`)

=

h

η

,

`

η

2 !

,

which models an array of densely packed thin plates. We also define the associated rescaled

spatial coordinates

y1

=

x1

η

,

z

=

x

η

2 .

(8)

3.1. E

ffective wave equation in the region of the plates

87 3.1.1. Notations

88 In the region of the array of parallel plates, the displacements and the stresses vary in the

horizontal direction over small distances dictated by `, and over large distances dictated by the

incoming waves; these two scales are accounted for by the two-scale coordinates (x

0 _{, z2), with}

x

0 _{= (x2, x3). In the vertical direction, the variations are dictated by h only and this is accounted}

for by the rescaled coordinate y1. It follows that the fields (u, σ) are written of the form

u

=

X

n≥0

η

n

_w

n

(y1

, z

2 , x

0 ),

σ =

X

n≥0

η

n

_π

n

(y1

, z

2 , x

0 ),

(9)

with the three-scale di

fferential operator reading

∇ →

e

1 η

∂

∂y1

+

e

2 η

2 ∂

∂z2

+ ∇x

0 ,

(10)

where e1

= (1, 0, 0) and e2

= (0, 1, 0). Now, we introduce the strain tensor with respect to x

0 ε

x

0 (u)

=

1

2 





0 ∂

x2

u

1 ∂

x3

u

1 ∂

x2

u

1 2∂

x2

u

2 ∂

x3

u

2 + ∂x2

u

3 ∂x

₃

u1

∂x

₃

u2

+ ∂x

₂

u3

_2∂x

3 u3







,

(11)

and the strain tensors with respect to the rescaled coordinates y1

and z2,

ε

y1

_(u)

₌

1

2 





2∂y1

w1

∂

y1

u2

∂

y1

u3

∂

y1

u2

0

0 ∂

y1

u

3

0

0 





,

ε

z2

_(u)

₌

1

2 





0 ∂

z2

u1

0 ∂

z2

u1

2∂

z2

u2

∂

z2

u3

0 ∂

z2

u

3

0 





.

(12)

6

(8)

The system in the region of the plates reads, from (2),











(E1)

1 η

∂y

1 σ11

+ ∂x

α

σ1α

+

1 η

2 ∂z

2 σ12

+ ρpω

2 _u1

_{= 0,}

(Eα)

1 η

∂

y1

σ

α1

+ ∂xβ

σ

αβ

+

1 η

2 ∂

z2

σ

α2

+ ρp

ω

2 uα

= 0,

(C)

σ =

1 η

2µpε

y1

+ λptr(ε

y1

)

+

2µpε

x

0 + λptr(ε

x

0 )

+

1 η

2 2µpε

z

0 + λptr(ε

z

0 ) ,

(13)

with the convention on the Greek letters α

= 2, 3, the same for β, and where ε stands for ε(u).

We shall use the stress-strain relation written in the form

(C

0 )

1 η

ε

y

1 + ε

x

0 +

_η

1 ₂

ε

z

2 =

(1

+ νp)

Ep

σ −

ν

p

Ep

tr(σ) I.

(14)

Eventually, the boundary conditions read

σ2i

= 0, i = 1, 2, 3, at z2

= ±ϕˆ`/2,

(15)

and are complemented by boundary conditions at y1

= 0, ˆh assumed to be known (they will be

justified later). We seek to establish the e

ffective behaviour in the region of the array in terms of

macroscopic averaged fields which do not depend anymore on the rapid coordinate z2

associated

with the small scale ˆ` as the following averages taken along rescaled variable z2. These fields are

defined at any order n as

w

n

_(y

1 , x

0 )

=

1 ϕˆ`

Z

Y

w

n

(y

1 , z

2 , x

0 ) dz

2 ,

π

n

(y

1 , x

0 )

=

1 ˆ

`

Z

Y

π

n

_(y

1 , z

2 , x

0 ) dz

2 ,

with Y

= {z2

∈ (−ϕ ˆ`/2, ϕ ˆ`/2)} the segment shown in figure 3.

89 y

1

<latexit sha1_base64="rscvIIyw1lNkyGDvcuGTjnw6YgY=">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</latexit>

z

2

<latexit sha1_base64="uge7jGvMFNOVIwWvyCfVpooJ40Y=">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</latexit>

0 Y

<latexit sha1_base64="azaGNRY+cDRFSARVYNpE8SeH83k=">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</latexit>

'ˆ

<latexit sha1_base64="q0ypYcmmnOvGpdL+KGmEr57gdNA=">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</latexit>

`

ˆ

`

<latexit sha1_base64="j4UxaUH8AdkQOiiTjXMVS7Glju0=">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</latexit>

ˆ

h

<latexit sha1_base64="0Oqdtpn+DVTHf646QmAzOpdfcxo=">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</latexit>

Figure 3: Analysis of a single plate in rescale coordinates, (9) with Y

= {z

2 ∈ (−ϕ ˆ`/2, ϕ ˆ`/2)}; the analysis holds within

the plate far from its boundaries at y

1 = 0, ˆh.

(9)

3.1.2. Sequence of resolution and main results of the analysis

90 We shall derive the equation satisfied in the region of the array, and additional results on the

91 stresses (π

0 1i

, π

1 1i

), i

= 1, 2, 3, required to establish the effective boundary conditions at y1

= 0, ˆh.

92 The main results will be obtained following the procedure :

93 1. We establish the following properties on π

0 π

0

11 = 0, π

0 2i

= 0, i = 1, 2, 3, π

0

13 = 0.

(16)

2. Then we derive the dependence of (w

0 _{, w}

1 _{) on z2}

_{which have the form}











w

0

1 = W

0

1 (x

0 _),

_w

0

2 = W

0

2 (y

1 , x

0 ),

w

0

3 = W

0

3 (x

0 _),

w

1 ₁

= W

₁

1 (x

0 ) −

λ

p

2(λ

p

+ µp

)

∂W

0

3 ∂x

3 (x

0 ) y1

−

∂W

0

2 ∂y

1 (y1, x

0 _{) z2,}

_w

1 α

= W

α

1 (y1, x

0 ),

(17)

and

π

0

11 = −

E

p

1 − ν

2 p

∂

2 _W

0

2 ∂y

2

1 (y1, x

0 _{) z2,}

_π

0

33 = ϕEp

∂W

0

3 ∂x3

(x

0 ).

(18)

3. Eventually, we identify the form of π

1 1i

, i

= 1, 2, 3, and the Euler -Bernoulli equation

governing the bending W

0

2 . Specifically











π

1

11 (y1, x

0 ₎

_{= ρpω}

2 _{ϕ W}

0

1 (x

0 _{) (ˆh − y1),}

π

1

12 (y1, x

0 )

= −

Ep

(1 − ν

2 p

)

ϕ

3 _`

_ˆ

2

12 ∂

3 _W

0

2 ∂y

3

1 (y1, x

0 ),

π

1

13 (y

1 , x

0 )

= ϕ







E

p

∂

2 _W

0

3 ∂x

2

3 (x

0 )

+ ρp

ω

2 W

₃

0 (x

0 )







ˆh − y1 ,

(19)

and

Ep

(1 − ν

2 p

)

ϕ

2 _`

_ˆ

2

12 ∂

4 _W

0

2 ∂y

4

1 −

ρ

_p

ω

2 W

₂

0 = 0.

(20)

94 In the remainder of this section, we shall establish the above results. We shall denote (E

1 )

n

, (E

α

)

n

95 and (C)

n

_{the equations which correspond to terms in (13) factor of η}

n

_.

96 3.1.3. First step: properties of

π

0 _in

₍₁₆₎

97 From (E)

−2

in (13), we have that ∂z2

π

0 _2i

= 0, which along with the boundary conditions at z2

=

98 ±ϕˆh/2 leave us with π

0 2i

= 0. Next from (E1

)

−1

_{and (E}

3 )

−1

, we also have that ∂

y1

π

0 ₁₁

+ ∂z2

π

1

12 = 0

99 and ∂

y1

π

0 ₁₃

+ ∂z2

π

1

23 = 0; by averaging these relations over Y and accounting for π

1 2i|∂Y

= 0, we

100 get that π

0 ₁₁

and π

0 ₁₃

do not depend on y1. We now anticipate the boundary condition π

0 ₁₁

(ˆh, x

0 )

=

101 π

0

13 (ˆh, x

0 ₎

_{= 0 that we shall prove later on (see forthcoming (35)), we get π}

0

11 = π

0

13 = 0 in Y. We

102 have the properties announced in (16).

103

(10)

3.1.4. Second step:

(w

0 _{, w}

1 _{) in (17) and (π}

0

11 , π

0

33 ) in (18)

104 Some of the announced results are trivially obtained. From (C

0 )

−2

_{in (14), we get that ∂}

z2

w

0 i

=

0, and from (C

0 ₁₁

)

−1

_{and (C}

0

13 )

−1

_{that ∂}

y1

w

0

1 = ∂y1

w

0

3 = 0, which leaves us with the form of w

0 _in

(17). Next (C

0 _αα

)

−1

_{tells us that ∂}

z2

w

1 α

= 0, in agreement with the form of w

1 α

in (17). We have yet

to derive the form of w

1

1 , which is more demanding. From (C

0

12 )

−1

_{, ∂}

z2

w

1

1 = −∂y1

w

0

2 and thus

w

1 ₁

= W1(y1, x

0 ) −

∂W

0

2 ∂y1

(y1, x

0 _{) z2,}

(21)

but we can say more on W

1 . Let us consider the system provided by (C

11 )

0 and (C

22 )

0 , specifically











π

0

11 = (λp

+ 2µp)∂y1

w

1

1 + λp

∂xα

w

0 α

+ ∂z2

w

2

2 ,

0 =

(λ

p

+ 2µp

)

∂z2

w

2

2 + ∂x2

w

0 2 + λp

∂y1

w

1 ₁

+ ∂x3

w

0 ₃

,

(22)

where we have used that π

0

22 = 0. After elimination of ∂z2

w

2

2 and owing to the form of w

0 α

in (16)

and w

1

1 in (21) (at this stage), we get π

0

11 = a(y1

, x

0 _)z

2 + b(y1

, x

0 ) with

a

= −

E

p

1 − ν

2 p

∂

2 _W

0

2 ∂y

2

1 (y1, x

0 _),

_b

₌

2µ

p

(λp

+ 2µp)







2(λp

+ µp)

∂W1

∂y1

(y1, x

0 )

+ λp

∂W

0

3 ∂x3

(x

0 )







,

(23)

(we have used that Ep/(1 − ν

2 p

)

= 4µp(µp

+ λp)/(λp

+ 2µp)). It is now su

fficient to remark that

π

0

11 = 0 imposes b = 0. This immediately provides the form of π

0

11 in (18) and

W1(y1, x

0 ₎

_{= W}

1

1 (x

0 _{) −}

λp

2(λp

+ µp)

∂W

0

3 ∂x3

(x

0 ) y1,

(24)

which along with (21) leaves us with the form of w

1

1 in (17). The same procedure is used to get

π

0

33 , which from (C

33 )

0 , reads

π

0

33 = (λp

+ 2µp

)∂

x3

w

0

3 + λp

∂y1

w

1

1 + ∂x2

w

0

2 + ∂z2

w

2

2 .

(25)

Using that π

0 ₂₂

= 0 to eliminate ∂z

₂

w

2 ₂

, we get

π

0

33 = Ep

∂W

0

3 ∂x3

(x

0 _{) −}

2µpλp

λp

+ 2µp

∂

2 _W

0

2 ∂y

2

1 (y

1 , x

0 ) z

2 ,

(26)

which after integration over Y leaves us with π

0 ₃₃

in (18). Incidentally, w

2 ₂

can be determined from

(22) and we find

w

2 ₂

= −







∂W

0

2 ∂x2

(y1

, x

0 )

+

λp

2(λp

+ µp)

∂W

0

3 ∂x3

(x

0 )







z2

+

λp

+ 2µp

∂

2 _W

0

2 ∂y

2

1 z

2 ₂

2 + W

2

2 (y1

, x

0 ).

(27)

3.1.5. Third step: the

π

1 1i

in

(19) and the Euler-Bernoulli equation in (20).

105 We start with (E)

0 _{in (13) integrated over Y, specifically,}

∂π

1

11 ∂y1

+ ρp

ω

2 _{ϕ W}

0

1 = 0,

∂π

1

12 ∂y1

+ ρp

ω

2 _{ϕ W}

0

2 = 0,

∂π

1

13 ∂y1

+

∂π

0

33 ∂x3

+ ρp

ω

2 W

₃

0 = 0,

(28)

9

(11)

where we have used (16) and π

2 _{· n}

|

∂Y

= 0. Since W

1

0 and W

0

3 depend on x

0 _{only, and accounting}

106 for π

0 ₃₃

(x

0 ) in (18), we get by integration the forms of π

1 ₁₁

and of π

1 ₁₃

announced in (19). Note that

107 we have anticipated the boundary conditions π

1 1i

= 0 at y1

= ˆh, see forthcoming (35).

108 The equation on π

1 ₁₂

in (28) will provide the Euler-Bernoulli equation once π

1 ₁₂

has been

determined (the integration is not possible since W

₂

0 depends on y1). To do so, we use, that

∂

y1

π

0 ₁₁

+∂z2

π

1

12 = 0, from (E1)

−1

_{, along with π}

0

11 in (18). After integration and using the boundary

condition of vanishing π

1

12 at z

2 = ±ϕˆ`/2, we get that

π

1

12 =

E

p

2(1 − ν

2 p

)

∂

3 _W

0

1 ∂y

3

1 (y1, x

0 ₎

_z

2

2 −

ϕ

2 _`

_ˆ

2

4 !

,

(29)

hence the form of π

1

12 in (19). It is now su

fficient to use π

1

12 in (28) to get the Euler-Bernoulli

109 announced in (20).

110 3.2. Effective boundary conditions at the top of the array

111 To derive the transmission conditions at the top of the array, we perform a zoom by

substi-tuting y

1 used in (9) by z

1 = y1

/η, see Figure 4a. Accordingly, the expansions of the fields are

sought of the form

u

=

X

n≥0

η

n

_v

n

_(z

0 _{, x}

0 _),

_{σ =}

X

n≥0

η

n

_τ

n

_(z

0 _{, x}

0 _),

₍₃₀₎

where we denote z

0 = (z1

, z

2). The coordinate z1

∈ (−∞, 0) accounts for small scale variations

of the evanescent fields at the top of the plates. Next, the boundary conditions will be obtained