Matthieu Malléjac

Théo Cavalieri

V Romero-García

Aurélien Merkel

Daniel Torrent

Johan Christensen

Jensen Li

Jean-Philippe Groby
email: jean-philippe.groby@univ-lemans.fr

Non-locality of the Willis coupling in fluid laminates

Keywords: Willis coupling, Non-locality, Laminate structures 2010 MSC: 00-01, 99-00

The closed form expressions of the effective properties in periodic fluid laminates are derived thanks to the Padé approximation of the transfer matrix. A secondorder Taylor expansion of the transfer matrix elements exhibits Willis coupling. This coupling is the sum of a local term and a nonlocal term. The nonlocal term arises from the apparent bulk modulus in quasi one-dimensional problems.

The nonlocality directly impacts the governing equations modeling the acoustic wave propagation in these Willis materials, which then involve convolution products in space. As an example, a two-orthotropic porous material laminate

Introduction

Since the seminal work of Willis in the 80's [START_REF] Willis | Variational principles for dynamic problems for inhomogeneous elastic media[END_REF], the eponymous materials have received an increasing attention, because of their analogy with bi-isotropic electromagnetic metamaterials [START_REF] Lindell | Electromagnetic waves in chiral and bi-isotropic media[END_REF]. The Willis coupling parameters couple the potential and kinetic energies in the acoustic conservation relations, therefore enhancing the ability to control waves in asymmetric [START_REF] Koo | Acoustic omni meta-atom for decoupled access to all octants of a wave parameter space[END_REF][START_REF] Muhlestein | Experimental evidence of willis coupling in a one-dimensional effective material element[END_REF][START_REF] Liu | Willis metamaterial on a structured beam[END_REF][START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF][START_REF] Ponge | Dynamic homogenization theory for nonlocal acoustic metamaterials[END_REF][START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF][START_REF] Nemat-Nasser | Homogenization of periodic elastic composites and locally resonant sonic materials[END_REF][START_REF] Melnikov | Acoustic meta-atom with experimentally verified maximum willis coupling[END_REF] or non-reciprocal [START_REF] Nassar | Modulated phononic crystals: Nonreciprocal wave propagation and willis materials[END_REF][START_REF] Quan | Nonreciprocal willis coupling in zero-index moving media[END_REF][START_REF] Zhai | Active willis metamaterials for ultracompact nonreciprocal linear acoustic devices[END_REF][START_REF] Cho | Acoustic willis meta-atom beyond the bounds of passivity and reciprocity[END_REF][START_REF] Olivier | Nonreciprocal and even willis couplings in periodic thermoacoustic amplifiers[END_REF] systems. Willis couplings effectively appear along the diagonal of the propagation matrix, i.e., the Hamiltonian, either with opposite signs to account for the structural asymmetry, i.e., the even coupling, or with identical signs to account for the non-reciprocity and/or nonlocality, i.e., the odd coupling [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF]. Non-locality arises from spatial dispersion and translates into the dependence of the effective properties on the wavenumber [START_REF] Landau | Chapter XII -spatial dispersion[END_REF]. In acoustics, only a few articles tackled the problem of non-locality [START_REF] Nemati | Nonlocal description of sound propagation through an array of helmholt z resonators[END_REF][START_REF] Nemati | Nonlocal dynamics of dissipative phononic fluids[END_REF][START_REF] Quan | Passive acoustic metasurface with unitary reflection based on nonlocality[END_REF][START_REF] Geib | Tunable nonlocal purely active nonreciprocal acoustic media[END_REF][START_REF] Lafarge | Acoustic Waves in Periodic Structures, Metamaterials, and Porous Media[END_REF]. In order to investigate the nonlocal Willis coupling, we consider a simple asymmetric fluid laminate excited at oblique incidence, thus extending results derived at normal incidence notably in [START_REF] Groby | Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as willis materials[END_REF]. The effective properties are derived by considering the Padé approximation of the transfer matrix that relates the state vectors at both sides of the unit cell. A second-order Taylor expansion is required to derive effective density tensor, bulk modulus, and Willis coupling. Similar results were derived in electromagnetism in [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF] using the Baker-Campbell-Hausdorff formula [START_REF] Weiss | The baker-hausdorff formula and a problem in crystal physics[END_REF]. Laminate fluid-like porous structures are often used to achieve impedance matching for absorption or insulation purposes [START_REF] Tanneau | Optimization of multilayered panels with poroelastic components for an acoustical transmission objective[END_REF][START_REF] Boulvert | Optimally graded porous material for broadband perfect absorption of sound[END_REF][START_REF] Cavalieri | Graded and anisotropic porous materials for broadband and angular maximal acoustic absorption[END_REF]. As an example, we thus consider a two-orthotropic fluid-like porous material laminate.

The article is organized as follows. The propagation matrix of a general Willis material layer is derived in Subsection 2.1. The formulation to evaluate the effective properties in reciprocal structures is reminded in Subsection 2.2 and applied to a two-material unit cell fluid laminate in Section 3. In Section 4, the apparent and effective properties as well as the scattering coefficients of a two-orthotropic fluid-like porous material laminate are discussed. Finally, the general forms of the effective properties are derived for a general anisotropic fluid laminate in Section 5.

General statement

Wave propagation in a planar Willis medium

Assuming an implicit time dependence e -iωt and locality of the medium, i.e., the intrinsic medium parameters are independent of the spatial coordinate x, the pressure p and velocity V satisfy the following equations in a general Willis fluid medium iω ξp + iω ρ

• V = ∇p, iω ζ • V + iωCp = ∇ • V, (1)
where ξ and ζ are Willis couplings, ρ is the mass density tensor, and C is the compressibility, which is inverse to the bulk modulus B = 1/C. We consider a layer of such material, invariant in the plane x ⊥ = (x 1 , x 2 ) and of thickness L along the x 3 -direction, as sketched in Fig. 1 (a). The Cartesian coordinate system attached to the layer is assumed to match the principal directions of the material such that ρ = diag (ρ j ), j = 1, 2, 3. The spatial Fourier transform of Eq. (1) can thus be cast in the form

∂ ∂x 3 W = A • W, (2)
where W =< p, Ṽ3 > T is the state vector formed by the spatial Fourier transforms p and Ṽ3 of p and V • x 3 = V 3 respectively and A is the propagation matrix that reads as

A = iω    ξ 3 ρ 3 C - (k 1 -ωζ 1 ) (k 1 -ωξ 1 ) ω 2 ρ 1 - (k 2 -ωζ 2 ) (k 2 -ωξ 2 ) ω 2 ρ 2 ζ 3    . (3)
The details of the derivation are provided in Appendix A. Note that each element of the propagation matrix is that along the x 3 direction, but that related to the compressibility, i.e., the a 21 component. This element accounts for both the propagation direction in the plane x ⊥ via its dependence on the in-plane wavenumber components k ⊥ =< k 1 , k 2 > and the properties of the material along these in-plane directions. This element is usually cast in the form of an apparent compressibility [START_REF] Terroir | General method to retrieve all effective acoustic properties of fullyanisotropic fluid materials in three dimensional space[END_REF][START_REF] Jiménez | Acoustic Waves in Periodic Structures, Metamaterials, and Porous Media[END_REF]]

C a = C - (k 1 -ωζ 1 ) (k 1 -ωξ 1 ) ω 2 ρ 1 - (k 2 -ωζ 2 ) (k 2 -ωξ 2 ) ω 2 ρ 2 = C - k 2 1 ω 2 ρ 1 - k 2 2 ω 2 ρ 2 + k 1 ω (ζ 1 + ξ 1 ) ω 2 ρ 1 + k 2 ω (ζ 2 + ξ 2 ) ω 2 ρ 2 - ζ 1 ξ 1 ρ 1 - ζ 2 ξ 2 ρ 2 , (4)
while the material is obviously local. Therefore, this type of system is often 35 described as quasi one-dimensional, because its propagation matrix is formally written as that of a one-dimensional system [START_REF] Brouard | A general method of modelling sound propagation in layered media[END_REF][START_REF] Terroir | General method to retrieve all effective acoustic properties of fullyanisotropic fluid materials in three dimensional space[END_REF][START_REF] Jiménez | Acoustic Waves in Periodic Structures, Metamaterials, and Porous Media[END_REF]. The solution to Eq. (2)

relates the state vector at x 3 = L to that at x 3 = 0 via the matrix exponential, i.e., the transfer matrix,

W(l) = exp (Al) W(0) = T l W(0). (a) (b) (c)

Calculation of the effective properties from the knowledge of the unit cell transfer matrix

We assume a quasi one-dimensional reciprocal and asymmetric system composed of a d-periodic unit cell of respective propagation matrix A e . The state vectors at both sides of the unit cell, W(d) and W(0), are related to each other through the 2×2 transfer matrix

T d = exp (A e d) of elements t ij , (i, j) ∈ [1, 2] 2 .
Following Ref. [START_REF] Groby | Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as willis materials[END_REF], the propagation matrix is correctly approximated by the inversion of the first-order Padé approximation of the matrix exponential,

A e ≈ 2 d (T d + I) -1 (T d -I) ≈ 2 d 1 2 + t 11 + t 22   t 11 -t 22 2t 12 2t 21 t 22 -t 11   , (5)
where the reciprocity of the system has been accounted for by imposing det (T d ) =

1 and I is the identity matrix. This form directly provides the apparent elements of the propagation matrix of a quasi one-dimensional Willis material, Eq. ( 3), from which the Willis parameters and the other effective properties can be efficiently calculated, as sketched in Fig. 1 (b). The effective parameters derived following this procedure are indicated by a superscript e in the following.

Effective properties of the reciprocal laminated two-material unitcell structure

We consider a laminated unit cell of thickness d composed of a layer of material M (1) of density tensor ρ(1) , compressiblity C (1) , and thickness l (1) , and a layer of material M (2) of density tensor ρ(2) , compressiblity C (2) , and thickness l (2) = d -l (1) , as sketched in Fig. 1 (c). The state vectors at both sides of the unit cell are thus related by the total transfer matrix composed of the multiplication of the transfer matrices modeling the propagation in each layer

W(d) = T d W(0) = T l (2) T l (1) W(0), (6)
where T l (j) = exp A (j) l (j) , j = 1, 2, is the transfer matrix of the layer of material M (j) over the thickness l (j) , with A (j) the propagation matrix of each anisotropic layer of the laminate simply obtained when the Willis couplings in Eq. (3) vanish, i.e., ξ = ζ = 0 [START_REF] Terroir | General method to retrieve all effective acoustic properties of fullyanisotropic fluid materials in three dimensional space[END_REF]. These transfer matrices read as

T l (j) =    cos k (j) a l (j) iZ (j) a sin k (j) a l (j) i Z (j) a sin k (j) a l (j) cos k (j) a l (j)    , (7)
where k

(j) a = ω C (j) a ρ (j)
3 is nothing but the projection of the wavenumber along the x 3 -axis, i.e., k

(j) a = k (j)
3 , and

Z (j) a = ρ (j) 3 /C (j) a , j = 1, 2. Assuming λ > > d in such a way that k (1)
a l (1) and k

(2) a l (2) are much smaller than 1 and making use of Eq. ( 5) leads to the following propagation matrix components a e ij , (i,

j) ∈ [1, 2] 2 a e 11 = -a e 22 = ω 2 l (1) l (2) 2d C (2) a ρ (1) 3
-C (1) a ρ (2) 3 = ω 2 l (1) l (2) 2d C (2) ρ (1) 3 -C (1) ρ (2) 3 + (k 1 ) 2 ω 2 ρ (2) 3 ρ (1) 1 - ρ (1) 3 ρ (2) 1
+ (k 2 ) 2 ω 2 ρ (2) 3 ρ (1) 2 - ρ (1) 3 ρ (2) 2
, a e 12 = iω ρ (1)

3 l (1) + ρ (2) 3 l (2) d , a e 21 = iω C (1)
a l (1) + C

(2)

a l (2) d = iω C (1) l (1) + C (2) l (2) d - i ω l (1)
dρ

(1) 1 + l (2) dρ (2) 1 (k 1 ) 2 - i ω l (1)
dρ

(1) 2 + l (2) dρ (2) 2 (k 2 ) 2 . ( 8
)
The details of the calculation are provided in Appendix B. Note that these expressions are identical to those derived via the Baker-Campbell-Hausdorff formula in Appendix C. By identifying Eq. ( 8) with Eq. ( 3), we obtain on the one hand

ρ e 3 = ρ (1) 3 l (1) + ρ (2) 3 l (2) d , 1 ρ e 1 = l (1)
dρ

(1) 1 + l (2) dρ (2) 1 , 1 ρ e 2 = l (1)
dρ

(1) 2 + l (2) dρ (2) 2
,

C e = C (1) l (1) + C (2) l (2) d , (9)
which are classical results from the homogenization theory, when first-order Taylor expansion is considered, i.e., O (kd) 2 , and on the other hand

ξ e 1 = -ζ e 1 = 0, ξ e 2 = -ζ e 2 = 0, ζ e 3 = -ξ e 3 = -iωl (1) l (2) 2d C (2) ρ (1) 3 -C (1) ρ (2) 3 + (k 1 ) 2 ω 2 ρ (2) 3 ρ (1) 1 - ρ (1) 3 ρ (2) 1 + (k 2 ) 2 ω 2 ρ (2) 3 ρ (1) 2 - ρ (1) 3 ρ (2) 2 , (10)
when second-order Taylor expansion is considered, i.e., O (kd) 3 . This last result applies several comments.

First, the structure being symmetric along the x 1 and x 2 directions, the results ξ e 1 = -ζ e 1 = 0 and ξ e 2 = -ζ e 2 = 0 were expected. Nevertheless, this result must be tempered by analyzing Eq. ( 4) in terms of power of ω. Assuming ζ j and ξ j , j = 1, 2, of O ω 2 as it is usually the case for reciprocal Willis material [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF][START_REF] Groby | Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as willis materials[END_REF], the last two terms of Eq. ( 4) are O ω 3 , which would imply a third-order Taylor expansion of the transfer matrix elements, i.e., O (kd) 4 . Rigorously, we can only conclude that ξ e 1 = -ζ e 1 and ξ e 2 = -ζ e 2 in the absence of analysis of the current system, i.e., by noticing that the structure is symmetric along the x 1 and x 2 directions. In the general case, the propagation of a plane wave along each principal direction, i.e., along the directions x 1 , x 2 , and x 3 must be successively considered. This induces a permutation of the indices in the propagation matrix, thus allowing a direct determination of the remaining Willis coupling elements which are then located along the diagonal of the propagation matrix. Nevertheless, this point should be moderated, because all elements of the propagation matrix are O (kd) 3 . Thus, these two last terms could be avoided because they are O (kd) 4 . The Willis couplings along these directions are O ω 2

as the other, except the last two terms of the propagation matrix should not impact the scattering coefficients, because they become of higher order.

Second, the structure being reciprocal, we have ξ e = -ζ e .

Third, the even Willis coupling ζ e 3 is the sum of a local term, i.e.,

-iωl (1) l (2) 2d C (2) ρ (1) 3 -C (1) ρ (2) 3
, and of a nonlocal term, i.e., -iωl (1) l (2) 2d

(k 1 ) 2 ω 2 ρ (2) 3 ρ (1) 1 - ρ (1) 3 ρ (2) 1 + (k 2 ) 2 ω 2 ρ (2) 3 ρ (1) 2 - ρ (1) 3 ρ (2) 2
.

Note that the local term is that derived in Ref. [START_REF] Groby | Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as willis materials[END_REF]. The initial form of the whole Willis coupling is that of the local term where the compressibilities are the apparent compressibilities, i.e., C

a , j = 1, 2. The non-locality thus arises from the dependence of the apparent compressibilities on k ⊥ = (k 1 , k 2 ). Nevertheless, this non-locality is effective and not apparent.

Fourth, the nonlocal Willis coupling is even in that formulation and not odd as proposed in Ref. [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF]. Nevertheless, both Willis couplings, i.e., the local and the nonlocal couplings, are the difference of properties between the material M (1) and M (2) , while all the other effective properties are weighted sums of both material properties. The x 3 -direction of propagation is thus implicit and is accounted for when solving the bounded system. This was already pointed out in Ref. [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF], where the odd feature is attributed to the phase change over the unit cell, which will change sign with the opposite propagation direction.

Fifth, the Willis coupling is a linear function of the frequency in the absence of dispersion. While the first-order Taylor expansion gives rise to constitutive equations that are characteristic of symmetric structures, i.e., classical homoge-

nization with A H =< [0 iωρ e 3 ],
[iωC e a 0] > T , the second-order Taylor expansion reveals Willis coupling that is characteristic of asymmetric structures. In the quasi-static limit, an asymmetric laminated structure generally falls back to a symmetric structure.

All these remarks suggest that the governing equations in a Willis material composed of a fluid laminate unit cell should rather be of the form

-iω ζ ⊥ p + iω ρ • V = ∇p, iω ζ ⊥ •V + iωCp = ∇ • V, (11)
where ⊥ expresses the convolution product in the in-plane space, as detailed in Appendix D. This last system highlights the dependence of the Willis coupling on the in-plane coordinates and thus the non-locality features. Nevertheless, this dependence does not take the usual form of non-locality, which would rather be on the whole Cartesian coordinates and in particular on the x 3 -coordinate. It is more closely related to a dependence of the Willis coupling on the angle of incidence and thus suggests that the form of the governing equations, although 100 providing the accurate propagation matrix, could be inadequate to describe the three-dimensional acoustic wave propagation in fluid laminates. This is also in accordance with the fact that this non-locality only relies on the dependence of the apparent compressibility on k ⊥ .

Results and discussion

105

We consider a fluid-like laminated structure composed of two air-saturated porous materials as depicted in Fig. 2 (a). Both materials consist in the periodic repetition of the micro-structure unit-cell sketched in the inset of Fig. 2 (a). This micro-structure unit-cell is generated by extruding an ellipsoid from a cuboid, the semi-axes of which are longer than the associated cuboid sides [START_REF] Terroir | General method to retrieve all effective acoustic properties of fullyanisotropic fluid materials in three dimensional space[END_REF]. This assumption is fundamental to form an open porosity material. The respective micro-structure unit cells of the two materials M (1) and M (2) differ in the cuboid and ellipsoid dimensions. The skeletons are assumed motionless. The resulting equivalent fluids are orthotropic because the ellipsoid is centered in the cuboid and the in-plane dimensions are imposed as identical. Each material is thus fully characterized by the porosity φ, the thermal characteristic length Λ , the thermal permeability Θ, the in-plane and out-of-plane tortuosities α ⊥ = α 1 = α 2 and α = α 3 , the in-plane and out-of-plane viscous characteristic lengths Λ ⊥ and Λ , and the in-plane and out-of-plane viscous permeabilities K ⊥ and K . These porous material properties are scaled by the thicknesses l (1) and M (2) material micro-structures and are calculated with a two-scale homogenization procedure [START_REF] Cavalieri | Graded and anisotropic porous materials for broadband and angular maximal acoustic absorption[END_REF][START_REF] Terroir | General method to retrieve all effective acoustic properties of fullyanisotropic fluid materials in three dimensional space[END_REF][START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF]. They are listed in Table 1. The complex and frequency dependent effective densities and bulk modulus take the forms [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF]

ρ h ρ 0 = α h φ + i η/K h ωρ 0 1 - iωρ 0 η 2α h K h φΛ h 2 , h = , ⊥ , B 0 B = φ   γ + (γ -1)   1 + i φη/Θ ωPrρ 0 1 - iωPrρ 0 η 2Θ φΛ 2   -1    , (12)
M

M (2) l c (µm) 300 75

φ 0.6718 0.6718 Λ (m) 0.8637 × l (1) c 0.8906 × l (2) c Θ (mm 2 ) 0.08545 × l (1) c 2 0.1220 × l (2) c 2 α ⊥ 1.3572 1.3486 α 2.8903 3.1570 Λ ⊥ (µm) 0.5713 × l (1) c 0.6030 × l (2) c Λ (µm) 0.7599 × l (1) c 1.00175 × l (2) c K ⊥ (mm 2 ) 0.01612 × l (1) c 2 0.01766 × l (2) c 2 K (mm 2 ) 0.03568 × l (1) c 2 0.06464 × l (2) c 2
Table 1: Acoustic properties of the fluid-like porous materials.

where the air density is ρ 0 = 1.213 kg.m -3 , the adiabatic constant is γ = 1.4, the dynamic viscosity is η = 1.839 × 10 -5 Pa.s, the Prandtl number is Pr = 0.71, and the air bulk modulus is B 0 = γP 0 , with the atmospheric pressure P 0 = 1.013 × 10 5 Pa.

The layer thicknesses are l (1) = 2 cm and l (2) = 1 cm, such that d = l (1) + l (2) = 3 cm. The effective and apparent properties as calculated from

Eqs. [START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF][START_REF] Nemat-Nasser | Homogenization of periodic elastic composites and locally resonant sonic materials[END_REF], and (10) are compared to those as numerically calculated from log (T L ) = A e num L. The structure is L = N d-thick, N ∈ N, i.e., an integer number of repetition of the unit cell, and is excited by a plane incident wave initially propagating in the air medium that surrounds the laminate, with θ and ψ the elevation and azimutal angles such that k 1 = k 0 sin (θ) cos (ψ), k 2 = k 0 sin (θ) sin (ψ), and k (0) 3 = k 0 cos (θ), as shown in Fig. 1 (a). The scattering coefficients are the reflection coefficients R + , when the structure is excited from the x 3 positive axis, and R -, when the structure is excited from the x 3 negative axis, and the transmission coefficient T . These scattering coefficients are calculated from the transfer matrix via

R + =
where Z0 = Z 0 / cos (θ) is the reduced impedance that accounts for the elevation angle, with the air impedance Z 0 = √ B 0 ρ 0 . The transfer matrix elements used in Eq. ( 13) are either those of T L or those of T e L = exp (A e L). Note that R + = R -= R H , when classical first-order Taylor expansion homogenization is employed, because the structure is symmetric with ζ e 3 = 0. We first analyzed the apparent elements as they appear in Eq. ( 8) and as calculated numerically.

The normalized values ζ e 3 c 0 and B e a /γP 0 are depicted in Figs. 2 (e,f) and Figs. 2 (g,h) for (θ, ψ) = (0, 0) and (θ, ψ) = ( π 3 , π 6 ) respectively. Figure 2 (b) depicts the normalized density ρ e 3 /ρ 0 , which does not depend on the angle of incidence. The dependence of both ζ e 3 and B e a on k ⊥ is clearly visible. The effective and apparent properties directly calculated from Eqs. [START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF][START_REF] Nemat-Nasser | Homogenization of periodic elastic composites and locally resonant sonic materials[END_REF] are in excellent agreement with those numerically calculated, thus validating the theoretical expressions.

We also numerically verify that ζ e 3,num = -ξ e 3,num . While the imaginary part of ζ e 3 c 0 vanishes at low frequency, its real part goes to a finite value because of the viscous regime of the porous materials, i.e., ρ

i /ρ 0 ≈ α (j) i φ (j) + i η/Ki ωρ0 , i = , ⊥, and j = 1, 2. This prevents the opening of a band gap at low frequency, which could have resulted from a purely imaginary value of ζ e 3 . Only ζ e 3 starts to deviate from ζ e 3,num for frequencies higher than 400 Hz. This should be mitigated by the fact that the dispersion relation k e = ω (ζ e 3 ) 2 + ρ e /B e a is correctly modeled by the effective properties as shown in Figs. 2 (i,j). Note that Re (k e d) becomes larger than ≈ 0.2 for f > 400 Hz. The low frequency requirement is thus theoretically no longer satisfied in this frequency range. Both the properties and the scattering coefficients, depicted in Figs. 2 (c,d), seem to be correctly modeled even for much larger values of Re (k e d), while it is usually accepted that low frequency approximation is valid for Re (k e d) < 0.1. For the sake of comparison, the reflection and transmission coefficients, R H and T H , calculated with ζ a 3 = 0, i.e., classical homogenization results, are plotted on Figs. 2 (c,d). Although the transmission coefficient is correctly modeled whatever the order of the Taylor expansion considered to derive the effective properties, the reflection coefficients become different for f > 200 Hz, which roughly corresponds to Re (k e d) < 0.1.

The Willis coupling is thus a key parameter to model laminated structures, allowing to correctly account for the geometrical and material asymmetry. The frequency range of validity and accuracy of the effective properties is widened simply because a second-order instead of a first-order Taylor expansion is considered. This was already pointed out in Ref. [START_REF] Sieck | Origins of willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization[END_REF] and is also partly due to the non-resonant feature of the present structure. The dispersion relations calculated in the absence of the Willis coupling, i.e., classical homogenization, are also plotted in Figs. 2 (i,j) and are found in good agreement with those numerically To get a grip on the effective parameters, the normalized Willis coupling ζ e 3 , in-plane ρ e ⊥ /ρ 0 = ρ e 2 /ρ 0 = ρ e 1 /ρ 0 and out-of-plane ρ e /ρ 0 = ρ e 3 /ρ 0 normalized densities, and the normalized value of the bulk modulus B e /γP 0 are plotted Figs. 3 (a,b,c). To make it clearer and because the considered Willis material is orthotropic, the Willis coupling is decomposed into a local part and a nonlocal part as

ζ e 3 = ζ e,l 3 + (k 1 /k 0 ) 2 ζ e,nl 3
+ (k 2 /k 0 ) 2 ζ e,nl

General form of the effective properties of a laminate

To derive the general form of the effective properties of a laminate, we consider a laminated unit cell of thickness d composed of N layers of material M (j) of density tensor ρ(j) , compressiblity C (j) , and thickness l

(j) , j ∈ N = [1, N ],
such that d = j∈N l (j) . Following the procedure described in Section 2.2, we end with

ρ e 3 = j∈N ρ (j) 3 l (j) d , 1 ρ e 1 = j∈N l (j) dρ (j) 1 , 1 ρ e 2 = j∈N l (j) dρ (j) 2 , C e = j∈[1,N ] C (j) l (j) d , ξ e 1 = -ζ e 1 = 0, ξ e 2 = -ζ e 2 = 0, ζ e 3 = -ξ e 3 = j∈N i>j -iωl (j) l (i) 2d ρ (j) 3 C (i) a -ρ (i) 3 C (j) a = j∈N i>j -iωl (j) l (i) 2d ρ (j) 3 C (i) -ρ (i) 3 C (j) + (k 1 ) 2 ω 2 ρ (i) 3 ρ (j) 1 - ρ (j) 3 ρ (i) 1 + (k 2 ) 2 ω 2 ρ (i) 3 ρ (j) 2 - ρ (j) 3 ρ (i) 2 . ( 14
)
Note that the derivation of these general formulae are straigthfoward via the Padé approximation and seems more complex via the Baker-Campbell-Hausdorff formula [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF].

Conclusion

The closed forms of the effective properties of an anisotropic two-material Eqs.(A.1) reads as

iωξ 1 p + iωρ 1 Ṽ1 = ik 1 p , iωξ 2 p + iωρ 2 Ṽ2 = ik 2 p , iωξ 3 p + iωρ 3 Ṽ3 = ∂ p ∂x 3 , iωζ 1 V 1 + iωζ 2 V 2 + iωζ 3 Ṽ3 + iωC p = ik 1 Ṽ1 + ik 2 Ṽ2 + ∂ Ṽ3 ∂x 3 . (A.2)
From the first two equations, it is clear that Ṽ1 = (k 1 -ωξ 1 ) p/ωρ 1 and Ṽ2 = (k 2 -ωξ 2 ) p/ωρ 2 . The introduction of these expressions in the fourth equation leads, together with the third equation, to the following first order differential system The total transfer matrix of the two-fluid laminate unit cell is

∂ p ∂x 3 = iωξ 3 p + iωρ 3 Ṽ3 , ∂ Ṽ3 ∂x 3 = iωC p + i (ωζ 1 -k 1 ) Ṽ1 + i (ωζ 2 -k 2 ) Ṽ2 + iωζ 3 Ṽ3 = iωC - i (k 1 -ωζ 1 ) (k 1 -ωξ 1 ) ωρ 1 - i (k 2 -ωζ 2 ) (k 2 -ωξ 2 ) ωρ 2 p + iωζ 3 Ṽ3 , (A.
T d = T l (2) T l (1) =      c (1) c (2) - s (1) s (2) Z (2) a Z (1) a iZ (1) a s (1) c (2) + iZ (2) a s (2) c (1) ic (1) s (2) Z (2) a + f racic (2) s (1) Z (1) a c (1) c (2) - s (1) s (2) Z (1) a Z (2) a      , (B.1)
where c (j) = cos k (j) a l (j) and s (j) = sin k (j) a l (j) , j = 1, 2. Second order expansion of each element t ij , (i, j) ∈ [1, 2] 2 thus reads as

t 1,1 = 1 - k (1) a l (1) 2 2 - k (2) a l (2) 2 2 - Z (2)
a k

(1)

a l (1) k (2) a l (2) Z (1) a + O (kd) 3 , t 1,2 = iZ (1) a k (1) a l (1) + iZ (2) a k (2) a l (2) + O (kd) 3 , t 2,1 = ik (1) a l (1) Z (1) a + ik (2) a l (2) Z (2) a + O (kd) 3 , t 2,2 = 1 - k (1) a l (1) 2 2 - k (2) a l (2) 2 2 - Z (1)
a k

(1)

a l (1) k (2) a l (2) Z (2) a + O (kd) 3 , (B.2)
which leads to

a 1,1 = 2 (t 1,1 -t 2,2 ) d (2 + t 1,1 + t 2,2 ) = k (1) a l (1) k (2) a l (2) 2d Z (1) a Z (2) a - Z (2) a Z (1) a + O (kd) 3 , a 1,2 = 4t 1,2 d (2 + t 1,1 + t 2,2 ) = i Z (1) a k (1)
a l (1) + Z

(2) Appendix C. Derivation of the effective properties using the Baker-Campbell-Hausdorff formula Inspired by [START_REF] Liu | Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers[END_REF], we derive the effective properties of a two-material fluid laminate by the Baker-Campbell-Hausdorff formula [START_REF] Weiss | The baker-hausdorff formula and a problem in crystal physics[END_REF] in order to valid our analytical results. The Baker-Campbell-Hausdorff formula perfectly fits our problem, because the total transfer matrix reads as a multiplication of matrix exponentials. The story is completely different if the configuration incorporates point resonators whose transfer matrix does not take the form of a matrix exponential [START_REF] Groby | Analytical modeling of one-dimensional resonant asymmetric and reciprocal acoustic structures as willis materials[END_REF]. The Baker-Campbell-Hausdorff formula is concerned with exp (Z) = exp B (1) exp B (2) , (C.1)

a l (2) d + O (kd) 3 , a 2,1 = 4t 2,1 d (2 + t 1,1 + t 2,2 ) = i d k (1) a l (1) Z (1) a + k (2) a l (2) Z (2) a + O (kd) 3 , a 2,2 = 2 (t 2,2 -t 1,1 ) d (2 + t 1,1 + t 2,2 ) = k (1) a l (1) k (2) a l (2) 2d Z (2) a Z (1) a - Z (
where B (1) and B (2) are square matrices. The analytical expression of Z = log exp B (1) exp B (2) is approximated at the first-order by Z ≈ B (1) + B (2) + 1 2 B (1) , B (2) , (C.2)

where B (1) , B (2) = B (1) B (2) -B (2) B (1) is the commutator of B (1) and B (2) .

Applying this formula to our problem, i.e., Eq. ( 6) 2) exp A (1) l (1) , (C.3) directly provides

T d = exp (A e d) = T l (2) T l (1) = exp A (2) l (
A e =      ω 2 l (1) l (2) 2d C (2) a ρ (1)
3 -C (1) a ρ

(2) 3

iω ρ

(1)

3 l (1) + ρ

(2)

3 l (2) d iω C (1)
a l (1) + C

a l (2) d -ω 2 l (1) l (2) 2d C (2) a ρ

(1) where the spatial dependence has been highlighted,

3 -C (1) a ρ (2) 3 
ζ 3 (x ⊥ ) = ∞ -∞ ζ3 (k ⊥ )e ik ⊥ •x ⊥ dx ⊥ /4π 2 ,
and ⊥ is a convolution product along the in-plane component x ⊥ . This system can thus be cast in the form of Eq. ( 11).

Figure 1 :

 1 Figure 1: [Color online] Sketch of a L-thich layer of a general Willis fluid (a), of the d-thick unit cell of the Willis fluid, and of the two-orthotropic fluid laminate unit cell (c).

  µm of the M

(ζ e 3 ) 2 + ρ e /B e a ± ζ e 3 translating

 323 evaluated and calculated with the effective Willis coupling. Thus, this coupling mostly impacts the impedances Z ± a = B e a the structural asymmetry into different reflection coefficients when the laminate is excited from the positive or the negative x 3 -directions.

Figure 2 :

 2 Figure 2: [Color online] Sketch of the laminate unit cell together with the micro-structure unit cell (a). Absolute value of the reflection and transmission coefficients (c-d) of a L = dthick laminate (N = 1), normalized Willis coupling (e-f), normalized apparent bulk modulus (g-h), and dispersion relation -black lines depict the classical homogenization results-(i,j) for (θ, ψ) = (0, 0) and (θ, ψ) = ( π 3 , π 6 ). The normalized density along the x 3 -direction is depicted in (b). Real (blue curves) and imaginary (red curves) parts of each element are depicted. Elements calculated with the effective parameters are plotted in solid line and those numerically evaluated are plotted with circle symbols.

3 . 3 ,Figure 3 :

 333 Figure 3: [Color Online] Normalized effective properties of the laminate porous material: local (solid line) and nonlocal (dashed line) Willis couplings (a), out-of-plane (dashed line) and inplane (solid lines) densities (b), and bulk modulus (c). Real parts are plotted in blue lines and imaginay parts in red lines. The absolute values of the reflection and transmission coefficients of a L = 5d-thick laminate (N = 5) for (θ, ψ) = ( π 3 , π 6 ) as calculated via Eqs. (8, 9, 10) and via the total transfer matrix (d). Coefficients calculated with the effective parameters are plotted in solid lines and the numerically evaluated elements are plotted with circle symbols.

  fluid laminate are derived thanks to the Padé approximation of the total transfer matrix. These closed forms rely on second-order Taylor expansions of the transfer matrix elements, allowing the derivation of the Willis coupling. This Willis coupling is composed of a purely local term and nonlocal terms that depend on the projection of the wavenumber on the in-plane directions. The nonlocality directly impacts the governing equations modeling the acoustic wave propagation in these Willis materials, which then involve convolution products in space. This highlights the dependence of the Willis coupling on the in-plane coordinates and thus on the angle of incidence, suggesting that the form of the governing equations, although providing the accurate propagation matrix, could be inadequate to describe the three-dimensional acoustic wave propagation in fluid laminates. The nonlocal Willis coupling arises from the apparent compressibility. While the effective density and bulk modulus are weighted sum of the material properties that compose the laminate, the Willis coupling involves their difference. Thus, the derived closed form requires orientating the laminate, which is the reason why the nonlocal Willis coupling is even in our formulation.In practice, the Willis coupling widens the frequency range of validity and accuracy of the effective properties and thus of the calculated scattering coefficients when compared to classical homogenization results for which the Willis coupling is absent. This widening is mostly attributed to a better modeling of the material impedance thanks to the Willis coupling. As an example, a two-orthotropic porous material laminate is considered. All effective properties and scattering coefficients are found in good agreements. The general forms of the effective properties of a laminate are provided. Beyond easing the engineering use of Willis materials, this article also aims at providing the basis of the experimental characterization of 2 or 3 dimensional planar Willis materials by inverting the propagation matrix derived in the present article.

3 )

 3 which can then be cast in the form of Eqs. (2) and (3). Appendix B. Derivation of the effective properties from the total transfer matrix of a two-fluid laminate 210

p ∂x 3 ,Ṽ3 ∂x 3 ,-iωζ 3

 333 to the expressions provided in Eq.[START_REF] Shuvalov | Effective willis constitutive equations for periodically stratified anisotropic elastic media[END_REF].215Appendix D. Rewritting the constitutive equations when the Willis parameters depend on x ⊥ From Eqs. (9) and[START_REF] Melnikov | Acoustic meta-atom with experimentally verified maximum willis coupling[END_REF], it is clear that C e and ρe do not depend on k ⊥ , butthat ζ e 3 (k ⊥ ) and thus ζe (k ⊥ ) = -ξe (k ⊥ ) =< 0, 0, ζ e 3 (k ⊥ ) > T do.Therefore, the Willis coupling is an in-plane spatial Fourier transform and should be ζe rather than ζe . Each step of the derivation of the propagation matrix from the governing equations (detailed in Appendix A) is the same and can thus be followed in reverse, but the inverse spatial Fourier transform of the governing equations. Effectively, the system Eq. (A.2) becomesiωρ 1 Ṽ1 = ik 1 p , iωρ 2 Ṽ2 = ik 2 p , -iω ζ3 p + iωρ 3 Ṽ3 = ∂ iω ζ3 Ṽ3 + iωC p = ik 1 Ṽ1 + ik 2 Ṽ2 + ∂ (x ⊥ ) ⊥ p(x) + iωρ 3 Ṽ3 = ∂ p(x) ∂x 3 , iωζ 3 (x ⊥ ) ⊥ Ṽ3 (x) + iωCp(x) = ∂V 1

  t 11 -t 12 / Z0 + Z0 t 21 -t 22 t 11 -t 12 / Z0 -Z0 t 21 + t 22 , R -= -t 11 -t 12 / Z0 + Z0 t 21 + t 22 t 11 -t 12 / Z0 -Z0 t 21 + t 22 ,

	T	=	2 t 11 -t 12 / Z0 -Z0 t 21 + t 22	,

Acknowledgments J.-P.G., M.M., T.C. and V.R.-G. would like to acknowledge the support of the ANR-RGC METARoom project (ANR-18-CE08-0021). J.L. would like to acknowledge the support of the Research Grants Council in Hong Kong (Grant No. 16302218). D.T. acknowledges financial support through the "Ramón y Cajal" fellowship under grant number RYC-2016-21188.

Appendix A. Derivation of the propagation matrix

The system Eqs.( 1) is first expanded as

ãe ik1x1+ik2x2 dx 1 dx 2 , the in-plane spatial Fourrier transform of


Pierre Charbonnier

For an Ecological Realpolitik

On September 22, 2020, Xi Jinping, the chairman of the PeopleÕs Republic of China, announced a plan to reduce greenhouse gas emissions with the aim of achieving carbon neutrality before 2060. Here, then, is China, the worldÕs largest CO2 producer and leading industrial power, sometimes dubbed the Òchimney of the world,Ó seemingly embarking on an unprecedented path of development. ÊÊÊÊÊÊÊÊÊÊSince at least the 1990s, Western environmentalism has been the subject of scathing criticism, notably from India. Ramachandra Guha, for instance, exposed the colonial and racist imaginary of the ÒwildernessÓ that enabled Americans to cleanse their urban and industrial guilty conscience by way of natural parks, which were established by evicting indigenous populations. This colonial disorder, which accompanies the environmental policies of the wealthy, continues to a certain extent with the paradox of the Green New Deal. There has long been a gap between ecologyÕs universalist, moral discourse, including when it is linked to social issues, and the darker reality of the structural, material inequities that it struggles to offset. We know therefore that ecologyÕs moral superiority does not amount to much, that it is something to be forged rather than posited. Peaceful ideas are often intimately bound up with a violent world.

And in this respect, too, the Chinese decision has upended the game. Indeed, the plan Xi announced to phase out fossil fuel dependence is based neither on a moral argument with regard to the environmental ravages caused by extractive industrialism, nor on the desire to curb or abolish the system of capitalist exploitation. It simply seeks to modify its material foundation, in what could be called an eco-modernist perspective, which is not incompatible with power ambitions. It so happens that, because of the Chinese economyÕs weight on a global scale, this plan Ð decided in a vertical, top-down fashion Ð is likely to have beneficial consequences for the global climate, and hence for all of humanity (which is what distinguishes it from a similar plan adopted in France, for example). At the same time, the plan is but a lateral consequence of global power-game decisions made in Beijing Ð a game the chairman of China knows how to play well.

We Europeans tend to think (and I am no exception) that the ecological question has taken over from a liberating movement that has run out of steam. We think, in other words, that environmentalism enshrines the social demands of equality and freedom in a new regime of production and consumption that could loosen the hold of economic exploitation and individualist anomie. In short, the point is to promote the emergence of a new social type, breaking with the one that accompanied the period of rapid growth, and rely on this to reactivate the process of democratization and social inclusion that has come to a standstill. This project can be used to disqualify the Chinese announcement, to assert that it does not rise to the challenge or that it resolves the problem through authoritarian means. That may well be. But by adopting this strategy (and I believe that this is the prevailing attitude in these spheres), we run the risk of not fully grasping the geopolitical and ideological waters in which we are navigating willy-nilly, and hence of not grasping the historical sense of our own project.

ÊÊÊÊÊÊÊÊÊÊIndeed, it is simplistic to imagine that the conflict in which we are caught pits exploiting, alienating, and extractive capitalism against a political ecology of reconciliation between human beings, and between humans and nonhumans. This would be the consequence of conflating the countercultural lexicon of environmentalism with the lexicon of social critique in the red-green universe: ecology or barbarism. But now we find ourselves in a situation where aging fossil capitalism, mired in its material and social contradictions, coexists with a state capitalism engaged in accelerated decarbonization, and with the more demanding and radical path of reinventing the meaning of progress and the social value of production. If we accept this description of the situation, as clearly rudimentary as it is, EuropeÕs red-green left takes on a different significance. It is then no longer locked in a binary confrontation with capitalism (reputed to be unfailingly fossil) in which it embodies the frontline of progress, invested as it is with a universal mission. The Chinese model that is being developed provides a third term, a third model of development, which is both compatible with the global climate aims defined in the 2016 Paris Agreement and possibly in tension with the green ideal of democracy that the social-ecological movement advocates.

Otherwise put, political ecology loses its status as the unique countermodel; it loses its ability to impose itself in debates as an anti-hegemonic political form. Two questions follow from this.

First, what kind of alliance will it establish with the Chinese model to safeguard at least what is essential on a strictly climatic level, at the risk of no longer having Òclean handsÓ? And, symmetrically, how will it make its specificity heard with regard to this new paradigm?

The European social-ecological left must figure out whether the Chinese announcement has Òstolen the spotlight,Ó so to speak, by embodying the central path towards breaking the climate impasse, or whether, by a more complex game of three players, which also involves relations with the United States, it opens a breach that must be entered without delay. This breach is quite simply the definitive weakening of fossil capitalism, that is, of the American way of life (indeed, the US appears to be the weakest player on the global political and economic scene right now), consequently opening the possibility of a more direct debate between China and Europe.

To put the question even more simply: What political forms should undergird the ecological turn? European ecology must take a turn towards realism. This does not mean it has to embark on an aggressive, pugnacious debate with other geopolitical players, but it must abandon its harmful habit of expressing itself in consensual, pacifying, and even moralizing terms, and agree to play on a complex political terrain.

After all, this dimension has always been present in the history of social welfare, even though we donÕt always like to be reminded of these things.

The development of systems of protection began in Prussia; and, in a way, Xi Jinping is a little like the Bismarck of ecology: he does not so much listen to the demands of environmental justice as he anticipates them in order to silence them. The postwar advances in social rights in Europe are incomprehensible outside the geopolitical game that combines the specter of fascism, the war to be stamped out, the Bolshevik possibility, and American influence. As a British political representative put it, ÒThe National Health Service is a by-product of the blitz.Ó 2 The fact is that emancipation is not always, and not even primarily, won through expressions of moral generosity; it is also a matter of power. The figure of Lenin seems to be making a return to favor in critical thought, perhaps precisely because ecology has not yet found its Lenin.

The ecology movement should therefore agree to talk about strategy, conflict, and security; it should present itself as a dynamics of building a political form that assumes the idea of power without scaling back on social and democratic demands. In fact, these demands can only be achieved if they are invested into specifically political reflections and practices. But for this to be possible, we have to leave behind our tendency toward moral depoliticization, because we no longer have a monopoly on the critique of the fossil development paradigm. A new arena is emerging, and we have no choice but to launch ourselves into it.

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