R Rizzoni

S Dumont
email: serge.dumont@unimes.fr

F Lebon
email: lebon@lma.cnrs-mrs.fr

On Saint Venant -Kirchhoff imperfect interfaces

Using matched asymptotic expansions with fractional exponents, we obtain original transmission conditions describing the limit behavior for soft, hard and rigid thin interphases obeying the Saint Venant-Kirchhoff material model. The novel transmission conditions, generalizing the classical linear imperfect interface model, are discussed and compared with existing models proposed in the literature for thin films undergoing finite strain. As an example of implementation of the proposed interface laws, the uniaxial tension and compression responses of butt joints with soft and hard interphases are given in closed form.

Introduction

Adhesive bonding technology is widely employed in engineering structural assembly and especially in aeronautics industry, where the use of composite materials is necessary to lighten structures. Due to the presence of the adhesive layer, adhesive bonding joints are subjected to a complex state of stress with high stress concentrations and, consequently, accurate analysis and modeling of adhesive materials and bonded joints are required.

Because the adhesive layer is usually soft and very thin when compared with the characteristic dimensions of the structure, a relatively large number of elements in the thickness direction is necessary to achieve sufficiently accurate calculations in standard existing finite element codes. This gives rise to a large number of degrees of freedom and high simulation costs. To successfully deal with this difficulty, interphase modeling has to precede the computation of the numerical solution. A classical modeling approach consists in describing the adhesive as a distinct lower-dimensional continuum, i.e. a material surface.

A simplified two-dimensional modeling can be achieved by introducing suitable assumptions concerning the displacement and the stress fields inside the adhesive [START_REF] Goland | The stresses in cemented joints[END_REF][START_REF] Edlund | Analysis of elastic and elastic-plastic adhesive joints using a mathematical programming approach[END_REF][START_REF] Edlund | A geometrically nonlinear model of the adhesive joint problem and its numerical treatment[END_REF] or by applying the asymptotic expansion method [START_REF] Klarbring | Derivation of the adhesively bonded joints by the asymptotic expansion method[END_REF][START_REF] Klarbring | Asymptotic modelling of adhesive joints[END_REF][START_REF] Geymonat | Mathematical analysis of a bonded joint with a soft thin adhesive[END_REF][START_REF] Lebon | Asymptotic behavior of a hard thin linear interphase: an energy approach[END_REF]. The asymptotic expansion method provides a systematic and rigorous approach to obtain interfacial laws describing the mechanical behavior of the limit material surface accounting for the elastic properties of an elastic thin adhesive. In the small deformation theory, interfacial laws appropriate for linear elastic adhesives have been obtained by many authors, for a not exhaustive list see the reference works of Klarbring [START_REF] Klarbring | Derivation of the adhesively bonded joints by the asymptotic expansion method[END_REF][START_REF] Klarbring | Asymptotic modelling of adhesive joints[END_REF], Caillerie [START_REF] Caillerie | The effect of a thin inclusion of high rigidity in an elastic body[END_REF], Geymonat [START_REF] Geymonat | Asymptotic analysis of the behaviour of two bonded plates [Analyse asymptotique du comportement en flexion de deux plaques collées[END_REF][START_REF] Geymonat | Mathematical analysis of a bonded joint with a soft thin adhesive[END_REF], Licht [START_REF] Licht | Some New Mathematical Modelings of Junctions, East-West[END_REF], and also [START_REF] Abdelmoula | Comportement asymptotique d'une interface mince[END_REF][START_REF] Benveniste | An O h ( ) N interface model of a three-dimensional curved interphase in conduction phenomena[END_REF][START_REF] Krasucki | Yield design of bonded joints[END_REF][START_REF] Lebon | Numerical study of soft adhesively bonded joints in finite elasticity[END_REF][START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF][START_REF] Lebon | Asymptotic behavior of a hard thin linear interphase: an energy approach[END_REF][START_REF] Lebon | First-order numerical analysis of linear thin layer[END_REF][START_REF] Lebon | Asymptotic modelling of interface taking into account contact conditions: asymptotic expansions and numerical implementation[END_REF][START_REF] Rizzoni | Asymptotic analysis of an elastic thin interphase with mismatch strain[END_REF][START_REF] Schmidt | Modelling of adhesively bonded joints by an asymptotic method[END_REF][START_REF] Serpilli | Limit models in the analysis of three different layered elastic strips[END_REF][START_REF] Serpilli | An overview of different asymptotic models for anisotropic three-layer plates with soft adhesive[END_REF][START_REF] Zaittouni | Etude théorique et numérique du comportement d'un assemblage de plaques[END_REF].

The modeling of thin adhesives in finite elasticity has received much less attention than adhesives undergoing small displacements [START_REF] Åslund | Asymptotic analysis of adhesively bonded nonlinearly elastic plates[END_REF][START_REF] Edlund | A geometrically nonlinear model of the adhesive joint problem and its numerical treatment[END_REF][START_REF] Ganghoffer | Geometrically non-linear modelling of contact problems involving thin elastic layers[END_REF][START_REF] Ganghoffer | Modelling of the mechanical behaviour of joints bonded by a nonlinear incompressible elastic adhesive[END_REF][START_REF] Krasucki | Mathematical analysis of nonlinear bonded joint models[END_REF][START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]. In [START_REF] Edlund | A geometrically nonlinear model of the adhesive joint problem and its numerical treatment[END_REF], an elastic adhesive joint is considered, with an adhesive made of isotropic Saint Venant-Kirchhoff material and flexible as compared to the adherents. The large displacement-small deformation problem is addressed by introducing a displacement linearly varying through the thickness of the adhesive. A weak formulation is then obtained for a geometrically non linear two-dimensional description of the adhesive.

In [START_REF] Ganghoffer | Geometrically non-linear modelling of contact problems involving thin elastic layers[END_REF], the asymptotic expansion method is applied to study the mechanical behavior of a thin nonlinear elastic adhesive made of a material much softer than those of the two adherents. The Saint Venant-Kirchhoff material model is assumed for both the adherents and the adhesive and a two dimensional simplified model for the adhesive is obtained. The convergence of a three-dimensional solution towards the limit solution is also given, together with error estimates.

In [START_REF] Ganghoffer | Modelling of the mechanical behaviour of joints bonded by a nonlinear incompressible elastic adhesive[END_REF], a thin adhesive layer made of a nonlinear incompressible elastic material is considered. The three-dimensional equilibrium problem, posed in a mixed variational form, is analyzed by using the asymptotic expansion method. Several limit two-dimensional models are obtained for the adhesive, according to the values of a parameter representing its elastic properties. The existence and the uniqueness of the solution of the limit problems are established and Γ-convergence techniques are applied in order to prove the convergence of the asymptotic expansion.

In [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF], an adhesive bonded joint made of nonlinear elastic materials with nonconvex energy density is studied by using Γconvergence techniques. In the limit problem, the adhesive layer is replaced by a constraint condition in the form of a contact law depending on the relative behavior of the two small parameters, the thickness and the stiffness of the adhesive.

Within the framework of nonlinear elasticity, two stored energy functions for the adhesive, the material model of Saint Venant-Kirchhoff and the model of Ciarlet-Geymonat, are studied in [START_REF] Krasucki | Mathematical analysis of nonlinear bonded joint models[END_REF]. Using the asymptotic expansions method, the limit energies associated to the two stored energy functions are computed and a rigorous mathematical analysis of the two limit models is presented.

A composite structure consisting of two nonlinearly elastic plates bonded by a thin and soft adhesive layer is studied in [START_REF] Åslund | Asymptotic analysis of adhesively bonded nonlinearly elastic plates[END_REF]. The materials of the plates are Saint Venant-Kirchhoff materials, while a more general nonlinear relation is used for the adhesive. A two-dimensional plate model for the compound structure is obtained, in which the adhesive is taken into account only through its material response to a pure shear load.

In the present paper, we consider a joint made of two adherents and a thin adhesive modeled as Saint Venant-Kirchhoff materials, the simplest hyperelastic material model extending the linear elastic material to the nonlinear regime [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. The stored energy density of the Saint Venant-Kirchhoff model is

∑ ∑ W μ E λ E E u u u ( ( )) = ( ( )) + 2 ( ( )) i j ij i ii , =1 3 2 =1 3 2
(1.1)

where λ μ , are positive elastic constant called the Lamé's constants, and

E u u u u u ( ) = 1/2(∇ + (∇ ) + (∇ ) ∇ ) T T (1.2)
are the components of the Green-Lagrange strain tensor for a displacement field u.

Three different adhesive types are studied: in the first model the adhesive is "soft", i.e. the elastic coefficients of the adhesive, λ and μ, rescale as its thickness ε; in the second model the adhesive is "hard", i.e. λ and μ are independent of ε; in the third model the adhesive is "rigid", i.e. λ and μ rescale as the inverse of ε.

To obtain transmission conditions mechanically equivalent to the behavior of the three types of adhesives, an asymptotic method is proposed, using classical expansions in the hard and rigid case and fractional power series in the soft case. This proposal is mainly motivated by the analysis of Licht and Michaille [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF], which identifies ε , p-1 with p the exponent entering the growth conditions on the adhesive stored energy, as a critical size of the adhesive stiffness. In particular, above this critical size the stiffness is large enough to provide a limit model of perfect interface, below this critical size the stiffness is too small to maintain perfect adherence and at the critical size an imperfect (soft) interface model that allow displacement discontinuities in the adhesive is obtained. The choice of expansions with fractional powers in the soft case is also motivated by a simple one-dimensional example presented in Section 2. The example shows that for a soft adhesive the jump of the displacement rescales like ε ,

2/3
for the Saint Venant-Kirchhoff energy (1.1) being p -1 = 3. The example also gives insights into the types of transmission conditions arising from different rescaling of the adhesive elastic stiffness and it highlights the role of the load rescaling and the possibly occurrence of multiple solutions due to the failure of quasiconvexity of the energy (1.1). The three-dimensional equilibrium problem of the adhesive joint is considered in Section 3, where the strong and weak formulations of the mixed boundary value problem are introduced.

Section 4 is devoted to the asymptotic analysis, which is based on fractional matched asymptotic expansions. In Section 5, the transmission conditions obtained via the asymptotic expansion method are summarized, rewritten as interface laws and discussed in light of the existing results for elastic adhesives undergoing small and finite strains [START_REF] Abdelmoula | Comportement asymptotique d'une interface mince[END_REF][START_REF] Bhattacharya | A theory of thin films of martensitic materials with applications to microactuators[END_REF][START_REF] Geymonat | Mathematical analysis of a bonded joint with a soft thin adhesive[END_REF][START_REF] Goland | The stresses in cemented joints[END_REF][START_REF] Klarbring | Derivation of the adhesively bonded joints by the asymptotic expansion method[END_REF][START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF][START_REF] Lebon | Asymptotic behavior of a hard thin linear interphase: an energy approach[END_REF][START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces Int[END_REF].

In Section 6, the interface laws calculated for the soft interface are compared with the results obtained via Γ-convergence techniques by Licht and Michaille in [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]. We show that the interface laws calculated in the present paper at the order zero for the cases of soft and hard interphase are in agreement with the results of Licht and Michaille, provided the interphase elastic stiffness appropriately rescales with its thickness. In particular, if the stiffness rescales like ε , 3 then the minimization of the Γ-limit gives the variational forms of the zero and higher order interface laws that we calculate for the soft interphase. The higher order laws of imperfect interface that we calculate for the two cases of hard and rigid interphases do not find counterparts in [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]. Section 7 proposes the analysis of uniaxial tension and compression of a butt joint as an example of implementation of our contact laws. The example generalizes some analogous results given in [START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces Int[END_REF] for small strains. The macroscopic response of the joint is calculated and plotted for the two cases of soft and hard interface taking into account different ratios of the adhesive/adherent thickness and stiffness. The example could serve also as elasticity solution benchmark, the uniaxial tension and compression responses being given in closed form.

In the paper, the usual summation convention is used. Latin indices take the values 1, 2, 3 and Greek indexes the values 1, 2.

A one-dimensional example

Consider a thin adhesive layer occupying the reference configuration given by interval ε (0, ) in contact with an adherent occupying the reference configuration ε l ( , ), as depicted in Fig. 1. The materials of the adhesive and the adherent are both nonlinear and modeled as Saint Venant-Kirchhoff materials. The bar is fixed at one end, say x = 0, and subjected to a force Q at the other end, x l = . The thickness εis small, e.g. ε l / ⪡1. The equilibrium problem of the composite bar, stated on the reference configuration, takes the form

s su ε ε l ( + ′)′ = 0 in (0, ) ∪ ( , ), (2.1
)

s E u u ε = ( ′ + ( ′) ) in (0, ), ε 1 2 2 (2.2) s E u u ε l = ( ′ + ( ′) ) in ( , ), 1 2 2 (2.3) u x = 0 = 0, (2.4
)

s su x ε [ + ′] = 0 = , (2.5
)

u x ε [ ] = 0 = , (2.6
)

s su Q x l + ′ = = , (2.7)
where a prime denotes the first derivative, the symbol

f f ε f ε [ ]≔ ( ) -( ) + -
denotes the jump of f l : (0, ) →  at the point x ε

= and E E , ε denote the elastic moduli of the adhesive and of the adherent, respectively. Integrating (2.1) under the conditions (2.5) and (2.7) gives

s su Q l + ′ = in [0, ]. (2.8)
Substituting (2.2) and (2.3) into (2.8) yields the differential equations

u u u Q E ε ( ′) + 3( ′) + 2 ′ = 2 in (0, ), ε 3 2
(2.9)

u u u Q E ε l ( ′) + 3( ′) + 2 ′ = 2
in ( , ). These equations imply that u′ is piecewise constant in the bar and its values are solutions of the above cubic equations which admit multiple solutions, depending on the values of Q E / ε , Q/E. The multiplicity of solutions is due to the non monotonicity of the (cubic) stress-strain response function in the Saint Venant-Kirchhoff material model. When Q > 0, it is immediately seen that Eqs. (2.9) and (2.10) have a unique solution u

z ′ = ( ) > 0 F with z Q E = / ε and z Q E = / , respectively. Integrating u z ′ = ( ) F under the boundary conditions (2.3)-(2.7) gives the solution u x Q E x x ε Q E x ε Q E ε x ε l ( ) = ( / ) , ∈ [0, ), ( / )( -) + ( / ) , ∈ [ , ]. ε ε ⎧ ⎨ ⎩ F F F (2.11)
The solution (2.11) allows to evaluate the difference u ε u u ( ) -(0)≕  giving the jump of the displacement due to the presence of the deformable adhesive between the point x ε = of the adherent and the constraint at x=0.

For a soft adhesive, e.g.

E εE = , ε  the solution (2.11) yields u Q εE ε = . soft ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ F    (2.12)
For a hard adhesive, e.g.

E E = , ε  the solution (2.11) yields u Q E ε = . hard ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ F    (2.13)
Finally, for a rigid adhesive, e.g.

E E ε = , ε -1  one has u Qε E ε = . rigid ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ F    (2.14)
Eq. (2.12) deserves some discussion. In view of the smallness of ε, one could use the asymptotic behavior y y

( ) ∼ (2 ) 1/3 F as y → + ∞ to conclude that u Q εE ε Q εE ε Q E ε Q εE = ∼ 2 = 2 a s →+∞ . soft 1/3 1/3 2/3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ F       (2.15)
Thus, this transmission condition can be interpreted as arising from a force scaling Q ε ∼ q with q < 1, which incorporates the special case q=0 of a force Q independent of ε. Notably, Eq. (2.15) cannot be used to recover the classical soft interface law of linear elasticity. The latter can however be reobtained from (2.12) by linearizing F about the origin, y y ( ) ∼ F as y → 0, thus

u Q εE ε Q εE ε Q E Q εE = ∼ = a s →0 . soft ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ F       (2.16)
Clearly, this last result relies on a force scaling Q ε ∼ q with q > 1. In the remaining part of the paper, we will assume that the loads are independent of ε. This choice is simple and it brings to a non linear behavior of the adherents in the limit as ε → 0.

The three-dimensional problem

Let Ω be a composite body comprising two adherents, Ω , 3 is introduced, and let x x x ( , , )

1 2
3 be taken to denote the three coordinates of a particle in Ω. The origin lies at the center of the interphase midplane and the x 3 - axis runs perpendicular to the interphase midplane, thus the domains Ω ε ± and B ε are defined by

Ω x x x Ω x ε = ( , , ) ∈ : ± > 2 , ε ± 1 2 3 3 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ (3.1) B x x x Ω x ε = ( , , ) ∈ : | | < 2 . ε 1 2 3 3 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ (3.2)
Let S ε ± denote the interfaces between the adherents and the interphase:

S x x x Ω x ε = ( , , ) ∈ : = ± 2 . ε ± 1 2 3 3 ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ (3.3)
On a part Γ 1 of Ω ∂ , an external load g is applied, and on a part

Γ 0 of Ω ∂ such that Γ Γ ∩ = ∅ 0 1 the displacement is imposed to vanish. It is also assumed that Γ B ∩ = ∅ ε 0 and Γ B ∩ = ∅ ε 1 . A body force f is applied in Ω ε ± .
The equations governing the equilibrium problem of the composite structure are written as follows:

s s u f Ω s s u n g Γ s s u B s s u S u S u Γ s A E u Ω s A E u B ( + ) + = 0 in , ( + ) = on , ( + ) = 0 in , + =0 o n , = 0 on , = 0 on , = ( ) i n , = ( ) i n , ij ε kj ε i k ε j i ε ij ε kj ε i k ε j i ij ε kj ε i k ε j ε i ε k ε i k ε ε i ε ε i ε ij ε ijhk hk ε ε ij ε ijhk ε hk ε ε , , ± , 1 , , 3 3 , ±
± 0 ± ± ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪     (3.4)
where s ε is the second Piola-Kirchhoff stress tensor, u E( ) ε is the Green-Lagrange strain tensor defined in (1.2), A ± , A ε are the elasticity tensors of the deformable adherents and the interphase, respectively. In the sequel, they will be considered isotropic, with Lamé's coefficients equal to λ μ , ± ± in the adherents and λ μ , ε ε in the interphase.

The existence and uniqueness of the problem (3.4) is not generally guaranteed. Indeed, the Saint-Venant Kirchhoff energy is not even rankone convex [START_REF] Dret | The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function[END_REF], and thus the direct method of the calculus of variations does not apply. By using the implicit function theorem, an existence result can be established for the case when the pure displacement boundary value condition is considered and the body force is sufficiently small [START_REF] Dacorogna | Existence of minimizers for non-quasiconvex integrals[END_REF][START_REF] Mardare | Existence of minimizers for the pure displacement problem in nonlinear elasticity[END_REF][START_REF] Li | Existence of minimizers and microstructure in nonliner elasticity[END_REF].

Asymptotic analysis

Because the thickness of the interphase is very small, it is natural to seek the solution of problem (3.4) by using asymptotic expansions with respect to the small parameter ε. The domain is rescaled using a classical procedure [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. First, the following sets are introduced:

• Ω x x x Ω x = {( , , ) ∈ : ± > } ± 1 2 3 3 1
2 (the rescaled adherents);

• B x x x Ω x = {( , , ) ∈ : | | < } 1 2 3 3 1
2 (the rescaled interphase);

• S x x x Ω x = {( , , ) ∈ : = ± } ± 1 2 3 3 1 2 .
Next, the interphase is rescaled into a domain of unit thickness (cf. Fig. 2). In particular,

• in the interphase, the following change of variable is introduced

x x x B z z z B z z z x x x ε ( , , ) ∈ → ( , , ) ∈ , with ( , , ) = , , ε 1 2 3 1 2 3 1 2 3 1 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
and it is set

z z z x x x u u ( , , ) = ( , , ), ε ε 1 2 3 1 2 3  (4.1) z z z x x x s s ^( , , ) = ( , , ). ε ε 1 2 3 1 2 3 (4.2)
• In the adherents, the following change of variable is introduced

x x x Ω z z z Ω ( , , ) ∈ → ( , , ) ∈ , ε 1 2 3 ± 1 2 3 ± (4.3) with z z z x x x ε ( , , ) = ( , , ± 1/2 ∓ /2), 1 2 3 1 2 3
and it is set

z z z x x x u u ( , , ) = ( , , ), ε ε 1 2 3 1 2 3 (4.4) z z z x x x s s ( , , ) = ( , , ). ε ε 1 2 3 1 2 3 (4.5)
External forces are assumed to be independent of ε. As a consequence, it is set f z z z f x x x ( , , ) = ( , , )

1 2 3 1 2
3 and g z z z g x x x ( , , ) = ( , , )

ij kj i k j i ij kj i k j i iα i β βα α ε i α α i iβ β ε i i k i k i α i α ε i i i i ij ijhk hk ij ijhk ε hk , , ± , 1 , , 1 ,3 3 , 3 , 3 ,3 1 ,3 33 ,3 3 3 , 3 3 , 1 33 ,3 ±
± 0 ± ± 2 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪                     (4.6)
where E E ,  are the Green-Lagrange strain tensors in the rescaled adherents and interphase domains, respectively, having components:

E u u u u i j u ( ) = 1 2 ( + + ), , = 1, 2, 3, ij i j j i k j k j , , , , (4.7
)

E u u u u α β u ( ) = 1 2 ( + + ), , = 1, 2, αβ α β β α k α k β , , , ,      (4.8) E ε u u ε u u α u ( ) = 1 2 1 + + 1 , = 1, 2, α α α k α k 3 , 3 3 , , , 3 ⎛
⎝ ⎜ ⎞ ⎠ ⎟       (4.9) E ε u u u u ( ) = 1 2 ( + ). k k 33 2 3,3 ,3 ,3      (4.10)
If the interphase is isotropic, the constitutive equation of the rescaled interphase takes the form

μ λ s E u E u I E u I ^( ( )) = 2 ( ) + ( • ( )) , ε ε       (4.11)
with I the identity tensor. In the following, to simplify the notation, we drop the dependence of s ^on E u ( )   and later on of s on E u ( )). Substituting (4.8)-(4.10) into (4.11), one obtains

s μ u u λ u u ε u ε δ α β u u u = ( + + • ) + + 1 2 + 1 + 1 2 , , =1,2, αβ ε α β β α α β ε γ γ γ αβ , , , , , , 2 3,3 2 ,3 2 ⎛
⎝ ⎜ ⎞ ⎠ ⎟          (4.12) s μ u ε u ε α u u = + 1 + 1 • , = 1, 2, α ε α α α 3 3 , , 3 , , 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟      (4.13) s μ λ ε u ε λ u u u = (2 + ) 1 + 1 2 + + 1 2 . ε ε ε γ γ γ 33 3, 3 2 ,3 2 , , 2 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟      (4.14)
Motivated by (2.12), the following asymptotic series with fractional powers for the second Piola-Kirchhoff stress and the displacement in the original unrescaled domain are assumed:

s s ε s ε s εs ε s ε s ε εs o ε = + + + + + + + ( ), ij ε ij ij ij ij ij ij ij 2 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 (4.15) u u ε u ε u εu ε u ε u ε u o ε = + + + + + + + ( ), i ε i i i i i i i 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 2 (4.16)
for i j , = 1, 2, 3. According to (4.15) and (4.16), the following asymp- totic series in the rescaled interphase and adherents domains are obtained:

s s ε s ε s εs ε s ε s ε s o ε = + + + + + + + ( ), ij ε ij ij ij ij ij ij ij 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 2         (4.17) u u ε u ε u εu ε u ε u ε u o ε = + + + + + + + ( ), i ε i i i i i i i 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 2         (4.18) s s ε s ε s εs ε s ε s ε s o ε = + + + + + + + ( ), ij ε ij ij ij ij ij ij ij 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 2 (4.19) u u ε u ε u εu ε u ε u ε u o ε = + + + + + + + ( ), i ε i i i i i i i 0 1/3 1/3 2/3 2/3 1 4/3 4/3 5/3 5/3 2 2 2 (4.20)
for i j , = 1, 2, 3.

Expansions of the equilibrium equations in the interphase

Substituting the expansions (4.17) and (4.18) into the equilibrium equations of the rescaled interphase (third equation in (4.6)), we obtain that the following conditions hold in B:

• Order -2: u s ( ) =0, i,3 0 33 0 ,3   (4.21)
• Order -5/3:

u s u s ( + ) =0 , i i ,3 0 33 1/3 ,3 1/3 33 0 ,3     (4.22)
• Order -4/3:

u s u s u s ( + + ) =0 , i i i ,3 0 33 2/3 ,3 1/3 33 1/3 ,3 2/3 33 0 ,3       (4.23)
• Order -1:

u s s u s u s u s u s u s ( ) + ( + ) + ( + + + ) = 0, i αα i i β β i i i i ,3 0 3 0 , 3 0 , 0 3 0 ,3 ,3 0 33 1 , 3
1/3 33 2/3 ,3 2/3 33 1/3 ,3 1 33 0 ,3              (4.24)
• Order -2/3:

u s u s s u s u s u s u s u s u s u s ( + ) +( + + ) + ( + + + + ) = 0, i α i α α i iβ β iβ β i i i i i ,3 0 3 1/3 ,3 1/3 3 0 , 3 1/3 , 0 3 1/3 , 1/3 3 0 ,3 ,3 0 33 4/3 ,3 1/3 33 1 ,3 2/3 33 2/3 ,3 1 33 1/3 ,3 4/3 33 0 ,3                    (4.25)
• Order -1/3:

u s u s u s s u s u s u s u s u s u s u s u s u s ( + + ) +( + + + ) + ( + + + + + ) = 0, i α i α i α α i i β β i β β i β β i i i i i i ,3 0 3 2/3 ,3 1/3 3 1/3 ,3 2/3 3 0 , 3 2/3 , 0 3 2/3 , 1/3 3 1/3 , 2/3 3 0 ,3 ,3 0 33 5/3 ,3 1/3 33 4/3 ,3 2/3 33 1 ,3 1 33 2/3 ,3 4/3 33 1/3 ,3 5/3 33 0 ,3 
                        (4.26)
• Order 0:

s u s u s u s u s u s s u s u s u s u s u s u s u s u s u s u s u s ( + ) +( + + + ) +( + + + + ) + ( + + + + + + ) = 0. i α iββ αα i α i α i α i αα i i β β i β β i β β i β β i i i i i i i 0 , 0 0 , , 3 0 3 1 ,3 1/3 3 2/3 ,3 2/3 3 1/3 ,3 1 3 0 , 3 1 , 0 3 1 , 1/3 3 2/3 , 2/3 3 1/3 , 1 3 0 ,3 ,3 0 33 2
                                  (4.27)

Expansions of the equilibrium equations in the adherents

Substituting the expansions (4. [START_REF] Geymonat | Asymptotic analysis of the behaviour of two bonded plates [Analyse asymptotique du comportement en flexion de deux plaques collées[END_REF]) and (4.20) into the equilibrium equations of the rescaled interphase (first equation in (4.6)), we obtain the following conditions:

P f Ω + = 0 in , ij j i , 0 ± (4.28) P Ω = 0 in , ij j , 1/3 ± (4.29) P Ω = 0 in , ij j , 2/3 ± (4.30) P Ω = 0 in , ij j , 1 ± (4.31)
with P , ij l l = 0, 1/3, 2/3, 1, the components of the first Piola-Kirchhoff stress tensor in the adherents:

P s u s = + , ij ij i k kj 0 0 , 0 0 (4.32) P s u s u s = + + , ij ij i k kj i k kj 1/3 1/3 , 0 1/3 , 1/3 0 (4.33) P s u s u s u s = + + + , ij ij i k kj i k kj i k kj 2/3 2/3 , 0 2/3 , 1/3 1/3 , 2/3 0 (4.34) P s u s u s u s u s = + + + + . ij ij i k kj i k kj i k kj i k kj 1 1 , 0 1 , 1/3 2/3 , 2/3 1/
u s 0 = , i,3 0 33 0   (4.40)
• Order -2/3:

u s u s 0 = + , i i ,3 0 33 1/3 ,3 1/3 33 0     (4.41)
• Order -1/3:

u s u s u s 0 = + + , i i i ,3 0 33 2/3 ,3 1/3 33 1/3 ,3 2/3 33 0       (4.42)
• Order 0:

P s u s u s u s u s u s = + + + + + , i i i β β i i i i 3 0 3 0 , 0 3 0 ,3 0 33 1 , 3
1/3 33 2/3 ,3 2/3 33 1/3 ,3 1 33 0            (4.43)
• Order 1/3:

P s u s u s u s u s u s u s u s = + + + + + + + , i i i ββ i β β i i i i i 3 1/3 3 1/3 , 0 3 1/3 , 1/3 3 0 ,3 0 33 4/3 ,3 1/3 33 1 ,3 2/3 33 2/3 ,3 1 33 1/3 ,3 4/3 33 0
               (4.44)
• Order 2/3:

P s u s u s u s u s u s u s u s u s u s = + + + + + + + + + , i i i β β i β β i β β i i i i i i 3 2/3 3 2/3 , 0 3 2/3 , 1/3 3 1/3 ,
                   (4.45)
• Order 1:

P s u s u s u s u s u s u s u s u s u s u s u s = + + + + + + + + + + + . i i i β β i β β i β β i β β i i i i i i i 3 1 3 1 , 0 3 1
                       (4.46)

Expansions of the constitutive equations of the interphase

The equations written so far are general in the sense that they are independent of the constitutive behavior of the material. Three specific cases of elastic material are now considered for the interphase: a "soft" material, characterized by elastic moduli linearly rescaling with the thickness ε, a "hard" material, characterized by elastic moduli inde- pendent of the thickness ε, and a "rigid" material, characterized by elastic moduli linearly rescaling with ε .

-1

For the soft case, the full expansions (4.17)-(4.20) including the terms with fractional powers have been considered.

For the hard and the rigid cases, classical the terms with fractional powers have been eliminated when using (4.17)-(4.20), because they give rise to many useless equations. In other words, classical expansions have been used for the hard and the rigid cases.

Interphase made of a "soft" material

The Lamé's coefficients are assumed as follows:

λ λ ε μ μ ε = , = . ε ε   (4.47)
Substituting the expansion (4.17) into the constitutive Eqs. (4.12)-(4.14) and using (4.47), one obtains the following conditions:

• Order -1:

u u 0 = , k k ,3 0 ,3 0   (4.48)
• Order -2/3:

u u 0 = , k k ,3 0 ,3 1/3   (4.49)
• Order -1/3:

u u u u 0 = 2 + , k k k k ,3 0 ,3 2/3 ,3 1/3 ,3 1/3     (4.50)
• Order 0:

s λ u u u u u δ = ( + + ) , αβ k k k k αβ 0 3,3 0 ,3 0 ,3 1 ,3 1/3 ,3 2/3        (4.51) s μ u u u = ( + ), α α k αk 3 0 ,3 0 , 0 ,3 0      (4.52) s μ λ u u u u u = (2 + )( + + ), k k k k 33 0 3,3 0 ,3 0 ,3 1 ,3 1/3 ,3 2/3         (4.53)
• Order 1/3:

s λ u u u u u u u δ α β = ( + + + ) , , = 1, 2, αβ k k k k k k αβ 1 3,3 1/3 ,3 0 ,3 4/3 ,3 1/3 ,3 1 1 2 ,3 2/3 ,3 2/3          (4.54) s μ u u u u u α = ( + + ), = 1, 2, α α k αk k αk 3 1/3 ,3 1/3 , 0 ,3 1/3 , 1/3 ,3 0        (4.55) s μ λ u u u u u u u = (2 + )( + + + ). k k k k k k 33 1/3 3,3 1/3 ,3 0 ,3 4/3 ,3 1/3 ,3 1 1 2 ,3 2/3 ,3 2/3           (4.56)
The conditions at the next orders are not considered here because they are expected to yield transmission conditions of higher orders.

From (4.48)-(4.55) it follows that

B u u = 0 in ⇒ [ ] = 0, ,3 0 0   (4.57) B u u = 0 in ⇒ [ ] = 0, ,3 1/3 1/3   (4.58) s i j B = 0, , = 1, 2, 3, in , ij 0  (4.59) s α B = 0, = 1, 2, in , α3 1/3  (4.60)
where, given any

f B R : ↦ , 3 it has been set f z z f z z f z z [ ]( , )≔ ( , , (1/2) ) -( , , ( -1/2) )
λ λ μ μ = , = . ε ε   (4.64)
Substituting the expansion (4.17) deprived of the terms with fractional exponents into the constitutive Eqs. (4.12)-(4.14) and using (4.64), we obtain that the following conditions hold in B:

• Order -2:

u u 0 = , k k ,3 0 ,3 0   (4.65)
• Order -1:

u u u 0 = + , k k ,3 0 ,3 1 3,3 0    (4.66) u u u 0 = + , α kα k ,3 0 , 0 ,3 0    (4.67)
• Order 0:

s μ u u u u λ u u u δ λ u u u δ α β = ( + + ) + + 1 2 ( ) + + 1 2 , , =1,2, αβ α β β α k α k β γ γ k γ k γ αβ k k αβ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 3,3 1 ,3 1 ,3 1 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟               (4.68) s μ u u u u u u α = ( + + + ), = 1, 2, α α α k αk k αk 3 0 3, 0 , 3 1 , 0 ,3 1 , 1 ,3 0
        (4.69) s μ λ u u u λ u u u = (2 + ) + 1 2 + + 1 2 ( ) . k k γ γ k γ k γ 33 0 3,3 1 , 3 1 ,3 1 , 0 , 0 , 0 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟           (4.70)
As before, conditions at the next orders are not considered because they are expected to yield transmission conditions of higher orders. Eq. (4.65) implies that u = 0

μ λ s H H H H IH H I ^= ( + + ) + • + 1 2 T T 0 0 0 0 0 0 0 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟         (4.72)
with I the identity matrix and

H u e u e u e ≔ ⊗ + ⊗ + ⊗ . 0 , 1 0 1 , 2 0 2 , 3 1 3 
   (4.73)
For later use, we note in passing that Eq. (4.72) takes the component form In the following, we assume that the hypotheses of the Lemma are satisfied. The condition that ϕ ∇ be invertible is considered an accep- table hypothesis in view of the large-displacement small-strain situation depicted by the Saint Venant-Kirchhoff material model. The question of the invertibility of ϕ ϕ s K I (∇ (∇ ) + )

s μ u u u u λ u u u δ λ u u u δ α β = ( + + ) + + 1 2 ( ) + + 1 2 ( ) , , =1 ,2 , αβ α β β α k α k β α α k α k α αβ k k αβ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 3,3 1 ,3 1 ,3 1 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟               (4.74) s μ u u u u α = + + 1 2 ( ) , =1 ,2 , α α α k αk 3 0 ,3 1 3, 0 , 0 ,3 1 ⎛
⎝ ⎜ ⎞ ⎠ ⎟       (4.75) s μ λ u u u λ u u u = (2 + ) + 1 2 ( ) + + 1 2 ( ) . k k γ γ k γ k γ 33 0 3,3 1 , 3 1 ,3 1 , 0 , 0 , 0 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟           ( 4
T 33 0  appears to be more complicated and it is not addressed here.

In view of the Lemma, the displacement vector field u 1  can be represented in the form

z S u u u = [ ] + ( ), 1 1 3 1    (4.80)
where it has been set

S f z z f z z f z z ( )( , )≔1/2( ( , , 1/2) + ( , , -1/2)) 1 2 1 2 1 2
for a given f B R : ↦ .  is deduced:

μ λ P e I H H H H H I H H I e = ( + ) ( + + ) + • + 1 2 T T 0 3 0 0 0 0 0 0 0 2 3 ⎪ ⎪ ⎪ ⎪ ⎧ ⎨ ⎩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎫ ⎬ ⎭          (4.83)
Finally, integrating (4.27) with respect to z 3 and using (4.46) and the Lemma, we obtain

P s u s u s [ ] = -( + + [ ] ) , i i α i ββ α i αα 3 1 0 , 0 0 1 3 0 ,      (4.84)
which, using (4.72), gives the condition

μ λ P e I H H H H H I H H I [ ] = -div ( + ) ( + + ) + • + 1 2 . p T T 1 3 0 0 0 0 0 0 0 2 ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟          (4.85)
The notation div p indicates the divergence in the plane of the interphase, e.g.

P Pe

Pe div = ( ) + ( ) .

λ ε λ μ ε μ = 1 , = 1 . ε ε   (4.87)
Substituting the expansion (4.17) deprived of the terms with fractional exponents into the constitutive Eqs. (4.12)-(4.14) and using (4.87), the following conditions are found to hold in B:

• Order -3:

u u 0 = , k k ,3 0 ,3 0   (4.88) • Order -2 u u u 0 = + , k k ,3 0 ,3 1 3,3 0    (4.89) u u u 0 = + , α kα k ,3 0 , 0 ,3 0    (4.90) • Order -1 μ u u u u λ u u u δ λ u u u u u δ α β 0 = ( + + ) + ( + ( )) + ( + + ) , , = 1, 2, α β β α k α k β γ γ k γ k γ αβ k k k k αβ , 0 , 0 , 0 , 0 , 0 1 2 , 0 , 0 3,3 1 ,3 0 ,3 2 1 2 ,3 1 ,3 1
               (4.91) μ u u u u u u α 0 = ( + + + ), = 1, 2, α α kα k kα k 3, 0 , 3 1 , 0 ,3 1 , 1 ,3 0
       (4.92) μ λ u u u u u λ u u u 0 = (2 + ) + + 1 2 + + 1 2 ( ) . k k k k γγ kγ kγ 3,3 1 , 3 0 ,3 2 ,3 1 ,3 1 , 0 , 0 , 0 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟            (4.93)
Eq. ( 4.88) imply that u = 0

μ λ E IEI 0 + ( • ) = ,     (4.94)
where the components of E  are defined as follows: which are equivalent to the following conditions

E u u u u α β u ( ) = ( + ) + , , = 1, 2, αβ α β β α k α k β 0 1 2 , 0 , 0 1 2 , 0 , 0       (4.95) E u u u u α u u ( , ) = 1 2 ( + ) + = 1,2, αβ α α k α k 0 1 3, 0 , 3 1 1 2 , 0 ,3 1 
      (4.96) E u u u u ( ) = + . k k 33 1 3,3 1 1 2 ,3 1 ,3 1 
u u + =0 , 1,1 0 1 2 ,1 0 2   (4.101) u u + =0 , 2,2 0 1 2 ,2 0 2   (4.102) u u u u + + • =0, 1,2 0 2,1 0 ,1 0 ,2 0 

Matching external and internal expansions

The transmission conditions obtained in Section 4.4 are appropriate for the rescaled equilibrium problem, prescribing the jump defined as

z z z z z z f f f [ ]( , )≔ ( , , 1/2) -( , , -1/2), 1 2 1 2 1 2
with B R f: ↦ . 3 In this Section, the transmission conditions are related to interface laws appropriate for the limit equilibrium problem, in which the interphase is replaced by the limit interface

S x x x Ω x = {( , , ) ∈ : = 0} 0 1 2 3 3 (4.107)
and the adherents by the domains

Ω x x x Ω x = {( , , ) ∈ : ± > 0}. ± 0 1 2 3 3 (4.108)
Taking into account the asymptotic expansion (4.16) and assuming that the displacement in the adherent u ε can be expanded in a Taylor series representation along the x -3 direction (external expansion), it results:

ε ε ε ε ε u x u x u x u x u x u x u x u x , ± 2 = ( , 0 ) ± 2 ( , 0 ) + ⋯ = ( , 0 ) + ( , 0 ) + ( ,0)+ ( ,0)± 1 2 ( , 0 ) + ⋯ ε ε ε ± ,3 ± 0 ± 1 / 3 1 / 3 ± 2/3 2/3 ± 1 ± ,3 0 ± ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (`)
In view of the continuity of the displacements at the interfaces S ε ± and S ± we also have

ε ε ε u x u x u x u x u x ( , 0 ) + ( , 0 ) + ( , 0 ) + ( ( , 0 ) ± ( , 0 )) + ⋯ 0 ± 1/3 1/3 ± 2/3 2/3 ± 1 ± 1 2 ,3 0 ± (4.110) ε ε ε u z u z u z u z = ,± + ,± 1 2 + ,± 1 2 + ,± +⋯ 0 1 2 1/3 1/3 2/3 2/3 1 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟     (4.111) ε ε ε u z u z u z u z = ,± 1 2 + ,± 1 2 + ,± + ,± +⋯ 0 1/3 1/3 2/3 2/3 1 2 1 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (4.112)
Identifying the terms in the same powers of ε in the above external expansion and in the asymptotic expansions for u ε  (internal expansion) and for u ε , it is deduced that:

u x u z u z u x u z u z u x u z u z u x u x u z u z ( , 0 ) = , ± = , ± 1 2 ,
( , 0 ) = , ± = , ± ,

( , 0 ) = , ± = , ± 1 2 ,
( , 0 ) ± ( , 0 ) = , ± = , ± . Then, the contact conditions appropriate for the limit equilibrium problem, i.e. expressed in terms of the fields defined on Ω Ω ∪ , + 0 -0 can be obtained by substituting the following relations into the interphase laws: (4.6)) and taking into account that they have to be satisfied for any value of ε give the simple relations

0 ± 0 1 2 0 1/3 ± 1/3 1 2 1/3 1 2 2/3 ± 2/3 1 2 2/3 1 ± 1 2 ,3 0 ± 1 1 2 1 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟     ( 4
( , 0 ) = , ± 1 2 = ,± 1 2 ,
( , 0 ) = , ± = , ± , ( , 0 ) = , ± = , ± , ( , 0 ) ± ( , 0 ) = , ± = , ± . 0 ± 3 0 3 0 3 1/3 ± 3 1/3 1 2 3 1/3 1 2 3 2/3 ± 3 2/3 1 2 3 2/3 1 2 3 1 ± 3 1 2 ,3 0 ± 3 1 1 2 3 1 1 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟     (
l u u [ ] = = 0, 1/3, 2/3, l l   (4.117) u u u [ ] = + ( ),
s A E u l Ω = ( ) = 0, 1/3, 2/3, 1, in . ij l ijhk hk l ± ± (4.121)

Summary and discussion of the contact laws

This Section summarizes the transmission conditions obtained for the three material models of interphase considered in this paper: "soft", "hard" and "rigid". Next, using the matching conditions studied in the previous Section, interface conditions are obtained from the proposed transmission conditions. Lastly, these interface conditions are discussed in view of the laws calculated for soft and hard interphases in linear elasticity and modeling an imperfect contact because they allow for displacement discontinuities.

Interface laws for soft interphases

For the soft interphase, our asymptotic analysis yields the following results up to the second order: (5.7)

ε Oε P e P e = + () , ε 3 2/3 2 3     (5.8) ε Oε u u = + () , ε 2/3 2     (5.9)
which, substituted into (5.4)-(5.6), give

o ε P e = 0 + ( ), ε 3   (5.10) ε μ λ o ε P e u u = 1 2 (2 + ) + ( ). ε ε ε ε ε 3 3 2 1 / 3     (5.11)
These equations can be viewed as extending to the three-dimensional case the interface law (2.15) obtained in the one-dimensional case. Moreover, even though the Saint Venant-Kirchhoff material reduces to the linear elastic material under the approximation of small strains [START_REF] Ciarlet | Mathematical Elasticity[END_REF], a simple inspection of the transmission conditions (5.10), (5.11) shows that they do not reduce to the classical imperfect contact laws for soft interfaces in linear elasticity

Te Te K u = 0, = , 3 3 33     (5.12)
with T the Cauchy stress. This occurrence is completely analogous to the behavior of the one-dimensional interface law (2.15).

Interface laws for hard interphases

For the case of a hard interphase, our asymptotic analysis yields the following transmission conditions up to the third order:

u Pe [ ] = 0, [ ] = 0, , 0 0 3
(5.13)

μ λ u I H H H H H IH H I e [ ] = ( + ) ( + + ) + • + 1 2 , T T 1 0 0 0 0 0 0 0 2 3 ⎪ ⎪ ⎪ ⎪ ⎧ ⎨ ⎩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎫ ⎬ ⎭   (5.14) μ λ P e I H H H H H I H H I [ ] = -div ( + ) ( + + ) + • + 1 2 , p T T 1 3 0 0 0 0 0 0 0 2 ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟   (5.15) with H u e u e u e = ⊗ + ⊗ + [ ] ⊗ . 0 , 1 0 1 , 2 0 2 1 3
(5.16)

In (5.13), the continuity conditions (4.113) and (4.114) have been taken into account. Using the matching conditions (4.117)-(4.120), the interface conditions in the final configuration can be rewritten in the following form:

u Pe = 0, = 0, 0 0 3     (
T T 0 3 0 0 0 0 0 0 0 2 3 ⎪ ⎪ ⎪ ⎪ ⎧ ⎨ ⎩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎫ ⎬ ⎭   (5.
( + + ) + • + 1 2 -( ), p T T 1 3 0 0 0 0 0 0 0 2 ,3 0 3  ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟     (5.19)
ε Oε u u = + ( ) , ε 3     (5.22)
which, substituted into (5.17) and (5.15), allow to rewrite them in the form

μ λ O ε P e I H H H H H I H H I e = ( + ) ( + + ) + • + 1 2 + ( ), ε ε ε ε T ε T ε ε ε ε 0 3 2 3 ⎪ ⎪ ⎪ ⎪ ⎧ ⎨ ⎩ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎫ ⎬ ⎭ (5.23) ε μ λ O ε

P e I H H H H H I H H I P e

= -div ( + ) ( +

+ ) + • + 1 2 -( ) + ( ), ε p ε ε ε ε T ε T ε ε ε ε 3 2 ,3 0 3  ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟ ⎞ ⎠ ⎟ ⎟   (5.24) with ε H u e u e u u e = ⊗ + ⊗ + 1 -( ) ⊗ . ε ε ,1 0 1 , 2 0 2 , 3 0 3  ⎛ ⎝ ⎜ ⎞ ⎠ ⎟   (5.25)
Eqs. (5.23), (5.24) have two peculiarities. First, they prescribe the jump of the displacement and of the traction vector implicitly, the jump of the displacement field entering both right-hand sides of the equations in a nonlinear way. Next, for "small" enough strains H ⪡1 ε and negligeable higher order terms in H , ε Eqs. (5.23) and (5.24) formally reduce to the higher order interface laws obtained for a hard, linear elastic interphase in [START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF][START_REF] Lebon | Asymptotic behavior of a hard thin linear interphase: an energy approach[END_REF][START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces Int[END_REF]:

u ε μ σ u u Oε [ ] = 1 - -( ) + ( ) ε ε 1 1 3 0 3,1 0 1,3 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.26) u ε μ σ u u Oε [ ] = 1 - -( ) + ( ) ε ε 2 2 3 0 3,2 0 2,3 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.27) u ε μ λ σ λ μ λ u u u Oε [ ] = 1 (2 + ) - (2 + ) ( + ) -( ) + ( ) ε ε ε ε ε ε 3 3 3 0 1,1 0 2,2 0 3,3 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
(5.28)

σ ε μ μ λ μ λ u μu μ μ λ μ λ u λ μ λ σ σ Oε [ ] = - 4 ( + ) (2 + ) - - (2 + 3 ) (2 + ) - ( + ) -( ) + ( ), ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 13 1,11 0 1,22 0 2,21 0 33,1 0 13,3 0 ⎛
⎝ ⎜ ⎞ ⎠ ⎟ (5.29) σ ε μ μ λ μ λ u μu μ μ λ μ λ u λ μ λ σ σ Oε [ ] = - 4 ( + ) ( 2 + ) - - (2 + 3 ) (2 + ) - (2 +
) -( ) + ( ), ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 23 2,22 0 2,11 0 1,12 0 33,2 0 23,3 0 ⎛ ⎝
⎜ ⎞ ⎠ ⎟ (5.30)
σ εσ σ σ Oε [ ] = -( - - (
)) + ( ),

ε 33 13,1 0 23,2 0 33,3 0 (5.31)
with σ ij ε the components of the Cauchy stress in the adherents.

Interface laws for rigid interphases

For a rigid interphase, the asymptotic analysis yields the following transmission conditions up to the second order:

u e + =1, , 1 0 1 2  (5.33
)

u e + =1, , 2 0 2 2  (5.34
)

u e u e ( + )•( + ) = 0, ,1 0 1 , 2 0 2   (5.35)
implying that the deformation associated to u (•, ± 1/2) 0  is an isometric mapping of S into .

( + ) ∧ ( + ) , 1 3 ,1 0 1 , 2 0 2 ,1 0 1 , 2 0 2 
    (5.36)
implying that the relative position vector (at the first order ) between points on the top and the bottom of the interphase remains perpendicular to the deformed middle surface (at the zeroth order ) without stretching.

The lowest order interface model corresponding to the rigid adhesive is thus the perfect interface model described by the classical continuity conditions u Pe = 0, = 0,

0 0 3    
augmented by the further restriction that u 0 corresponds to an isometric mapping.

Comparison of the soft interface laws with the limit model of Licht and Michaille

In [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF], Licht and Michaille consider an elastic body constituted of adherent and interphase hyperelastic materials with nonconvex bulk energy density. In our notations and for the case of homogeneous materials, the total energy that they consider is

∫ ∫ h d V ρ b d V u ux ux u ( )≔ (∇ ( )) + (∇ ( )) -( ), ε ρ Ω ε B x x , ε ε ε ± (6.1)
where ρ ε is a small parameter taking into account the low stiffness of the interphase and

∫ ∫ dV dA u f x u x g x u x ( )≔ ( )• ( ) + ( )• ( ) Ω Γ x x 1 (6.2)
is the loading potential. Licht and Michaille identify several limit problems depending on the relative order of magnitude of ρ ε with respect to ε . 3 In [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF], three regimes are identified; in particular, the debonding phenomenon is characterized. For a Saint Venant-Kirchhoff material, [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF] shows that 1. for ρ ε = ε r , r 0 < < 3, the glue stiffness is sufficiently high to maintain adhesion. Licht and Michaille prove that in the limit problem the jump of the displacement at the interface vanishes (see space V 0 in [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]) and that the limit energy consists of the joint energy of the adhesives. In other words, the limit model of the thin adhesive layer is a perfect interface. 2. for ρ ε = ε 3 , the three-body limit problem obtained in [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF] for a general energy contains the energy term

∫ L Qb dA u x e ( ( ) ⊗ ) , S p x ∞, 3   (6.3)
where

L ρ ε 2 = lim /( ) ∈ [0, + ∞) ε ε p →0 -1 + , p is the growth exponent of b, b p
∞, is the density of the surface energy defined as follows:

b t b t F F ( )≔ lim 1 ( ), p t p ∞, →+∞ (6.4)
and Qb p ∞, is its quasiconvex envelope. For the Saint Venant-Kirchhoff energy density, one has p=4 (cf. (1.1)). Indeed, by evaluating the energy (1.1) at t E E F = ( ), with F the deformation gradient associated to u, one can easily show that leading term of the energy

W t E F ( ) when t → ∞ is t μ λ F F F 4 + 2 . T 4 2 4 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ (6.5)
Now, using (6.6) and assuming to identify ρ ε with the elastic constant μ ,

ε one finds b λ μ F F F F ( ) = 1 4 + 8 , p T ∞, 2 4   (6.6)
where λ μ ,   are the rescaled Lam'e constants and the same rescaling with εhas been assumed for the two constants. The energy density (6. (2 + ) ( ) .

S x 4      (6.8)
3. for ρ ε = ε r , r > 3, [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF] shows that adhesion is lost. In the limit problem L=0 and that there is no energy of the interphase left in the limit problem but the bodies can separate.

The case of a soft adhesive studied in the present paper is concerned with ρ ε = ε which is a subcase of the first case above studied by Licht and Michaille. This is a case where Licht and Michaille obtain perfect adhesion, so does the present paper (cf. (5.4)). The case of a hard adhesive ρ = 1 ε and the case of a rigid adhesive ρ ε = ε are not strictly sensu studied in [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF] since Licht and Michaille choose the glue stiffness to vanish (Fig. 3).

Uniaxial tension and compression of a butt joint

In this Section, two nonlinear elastic isotropic parallelepipeds Ω - On the remaining part of the boundary, Ω ∂ l 0 , the surface forces are taken to vanish. The load intensity Q is assumed to be independent of ε and body forces are null. The parallelepipeds are taken to be made of the same Saint Venant-Kirchhoff material, with Lamé constants λ μ , , but in the analysis, the related elastic constants In the next Subsections, the equilibrium problem of the composite body made of two blocks joined by a non linear elastic interface is studied in the following two cases: (i) the interface behavior is soft and it is described by the interface laws (5.10) and (5.11); (ii) the interface behavior is hard and it is described by the interface laws (5.23) and (5.24).

Butt joint with soft interface behavior

Neglecting the higher order terms in ε ε , 1/3 in (5.10) and (5.11), the equilibrium problem of the joined structure is written as

Ω Ω Ω Ω Ω Ω S μ
ε ε ε E ν ε νE ν ν ε ε ε ε T ε T ε ε ε ε ε ε ε ε ε α l ε + 0 - 0 1 + (1 + )(1 -2 ) + 0 - 0 + 0 - 0 3 0 3 1 2 2 0 0 3 3 1 ± 3 ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟       (7.3) For the displacement field Ω Ω u : ∪ ↦ ε + 0 - 0 3
 we seek solutions of the form constants to be determined, representing the stretches parallel to the x 1 and x 3 axes, respectively, and the u ∈ , with The above condition restricts the stretches λ 1 and λ 3 to take values in the intervals ν (0, 1 + ) and ν ν (0, (1 + )/ ), respectively. The occurrence of limit stretches can be interpreted as the failure of existence of solutions as in (7.4) for large strains, possibly related to the development of microstructure in the adherents [START_REF] Li | Existence of minimizers and microstructure in nonliner elasticity[END_REF]. Solving (7.8) with respect to λ 1 gives λ ν νλ = 1 + -.

P Eλ ν ν λ νEλ ν ν λ = 2(1 + )(1 -2 ) ( -1) + 2(1 + )(1 -2 ) ( -1), ε 11 1 1 2 1 3 2 (7.6) P Eλ ν ν λ νEλ ν ν λ = 4(1 + )(1 -2 ) ( -1) + (1 + )(1 -2 ) ( -1).
E Q ν ν ν E - 3 9 ≤ ≤ (1 + )(1 + 2 ) 2 . 3 (7.
12)

The solution of (7.11) is

λ Q E Q E Q E Q E ν ν ν = 2 3 cos 1 3 arccos 3 3 if - 3 9 ≤ < 3 9 , 2 3 cosh arccosh 3 3 if 3 9 ≤ < (1 + )(1 + 2 ) 2 . 3 1 3 3 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ (7.13)
The jump u ε   is determined through the interface condition in (7.3), which gives

u α u ε μ λ Q Q = 0, = 1, 2, = [ 3]2 [ 3]2 + . α ε ε ε ε 3 2/3    (7.14)
In view of (7.4), the macroscopic stretch along the x 3 axes, Λ, is In Fig. 4, the Young modulus of the adherents is 30 times greater than the Young modulus of the adhesive, E E / = 30, ε and the thin dashed curves correspond to increasing values of the adhesive thickness, namely ε l / = {0.005; 0.025; 0.05}, the dashing space increasing with increasing ε. Fig. 4 shows that the (finely dashed) curve for ε l / = 0.005 and the (solid) curve for ε = 0 run very close to each other and that the adhesive thickness has a remarkable effect on the macroscopic response for ε l / = 0.025, 0.05. In Fig. 5, the adhesive thickness is set ε l l = 0.025/( + ) ε Both Figures also evidentiate a particular feature of the response curves taking into the presence of the adhesive near the origin: they fail to reproduce the behavior of the dashed curve at small Λ. This is related to the inability of (5.10), (5.11) to reduce to the classical imperfect contact laws for soft interfaces at small strains, as already remarked in Section 5.

Butt joint with hard interface behavior

Neglecting the higher order terms in εin the interface laws (5.23) and (5.24), the equilibrium problem of the structure made of two identical blocks joined by a hard interface is written as

+ ( • + ) } on , = , = 1, 2, on ∂ , ± =± on , ε ε ε E ν ε νE ν ν ε ε ε ε T ε T ε ε p ε ε ε ε T ε T ε ε ε ε ε ε ε ε T ε T ε ε ε ε ε α l ε + 0 - 0 1 + (1 + )(1 -2 ) + 0 - 0 + 0 - 0 3 1 2 2 ,3 0 3 0 0 3 1 2 2 3 0 0 3 3 1 ±  ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟   (7.16)
with H ε as in (5.25) and u P , 0 0 the displacement and the corresponding first Piola-Kirchhoff stress tensor solution of the equilibrium problem without the interface. In view of the results obtained in the previous subsection, the displacement u 0 is given by (7.4), with u = 0 ε   and λ λ , 1 3 satisfying (7.9), (7.11). The tensor P 0 is given by (7.10).

For the displacement field

Ω Ω u : ∪ ↦ ε + 0 - 0 3
 we seek again a solution of the form (7.4). The constants λ λ , 1 3 are still chosen to satisfy (7.9) and (7.11), in order to match the constitutive equations of the adherents and the boundary conditions. Thus,

Ω u u u = ± 1 2 in , ε ε 0 ± 0   (7.
⊗ ) + 1 -( -1) ⊗ . ε ε 1 1 1 2 2 3 3 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟   (7.18)
The first Piola-Kirchhoff stress corresponding to (7.17

μ λ u ε u λ 0 = 2 + 1 - , ε ε ε 1 1 3 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟     (7.19) μ λ u ε u λ 0 = 2 + 1 - ε ε ε 1 2 3 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟     (7.20) Q ν E ν ν ε u λ ε u λ ν ν λ = (1 -) (1 + )(1 -2 ) 2 + 1 - × 2 + 1 - -1- 4 (1 -) (1 -) . ε ε ε ε ε ε ε ε 3 3 3 3 2 1 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎞ ⎠ ⎟ ⎟     (7.21)
The first two of these conditions are satisfied by taking u

u = 0 = ε ε 1 2     .
In view of (7.11), the solution of the third (cubic) equation determines u ε 3   as a function of λ 3 . In general, the solution to (7.21) is not unique and, as a selection criterion, the root of smallest modulus can be considered. This provides the continuity of the response curve Q Λ / through the origin, with Λ given again by (7.15). Figs. 6 and7 shown the macroscopic responses Q/ϵ and q/ϵ, with q Q λ = / 1 2 the load per unit area in the deformed configuration and Λ ϵ = ln the logaritmic strain. The thick solid line corresponds to the macroscopic response calculated without taking into account the presence of the hard interface. To plot the Figures, the following values of the elastic constants have been assumed: ν = 0.33, ν = 0.4. and an increasing dashing corresponds to increasing values of E E / ε in the set {0.5, 10, 50}. From the Figures, it can be noted that the curves are almost overlapped, meaning that the presence of the hard interphase scarcely affects the macroscopic response for the given set of geometric and material parameters.

Conclusion

Using matched asymptotic expansions with fractional exponents, we have obtained original transmission conditions, appropriated for soft, hard and rigid adhesive materials obeying the Saint Venant-Kirchhoff model. The particular type of expansion chosen in the present paper, cf. (4.15) and (4.16), is strictly related to the exponent (p=4) appearing in the growth conditions of the Saint Venant-Kirchhoff energy density. The same exponent enters the transmission conditions calculated for the soft adhesive, cf. (5.10), (5.11).

In the present paper, we restrict to a Saint Venant-Kirchhoff constitutive model, for which the exponent 1/3 of the small parameter εappears, following the soft case of the one-dimensional example. In a more general situation, the fractional exponent of the asymptotic expansion is expected to depend on the exponent p appearing in the growth conditions of the energy density [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF].

The transmission conditions proposed in the present paper find agreement with the results obtained via Γconvergence techniques by Licht and Michaille for the case of a soft adhesive [START_REF] Licht | A modelling of elastic adhesive bonded joints[END_REF]. Conditions (5.23) and (5.24) obtained for a hard interphase do not find analogous counterparts.

The nonlinear contact law calculated by Ganghoffer and Schultz in [START_REF] Ganghoffer | Geometrically non-linear modelling of contact problems involving thin elastic layers[END_REF], which is similar to the one obtained by Edlund and Klarbring in [START_REF] Edlund | A geometrically nonlinear model of the adhesive joint problem and its numerical treatment[END_REF], can not be compared with the interface laws obtained in the present paper for a soft interface. Indeed, an appropriate rescaling of the out-of-plane deformation component inside the interphase is assumed in [START_REF] Ganghoffer | Geometrically non-linear modelling of contact problems involving thin elastic layers[END_REF], which is not used in our analysis.

Transmission conditions for a Saint Venant-Kirchhoff soft interface have been obtained also in [START_REF] Krasucki | Mathematical analysis of nonlinear bonded joint models[END_REF]. Under the assumption of a linear scaling of the charges with the adhesive thickness, the limit behavior in the adherents is that of linear elasticity whereas it remains nonlinear in the adhesive. After linearization, the transmission conditions calculated in [START_REF] Krasucki | Mathematical analysis of nonlinear bonded joint models[END_REF] provide the classical linear contact laws of spring-like type. The limit problem and the transmission conditions for a nonlinear Saint Venant-Kirchhoff soft interface obtained in this paper differ from the ones obtained in [START_REF] Krasucki | Mathematical analysis of nonlinear bonded joint models[END_REF], having been obtained without applying any load scaling. In our approach, the limit behavior of the adherents remains nonlinear. As already remarked in Section 5.1, by linearizing the transmission conditions of the soft interface (Eqs. (5.11)) for small strains, one cannot recover the classical contact laws of a linear elastic soft interface.

The situation is different for the case of a hard interface. Indeed, by linearizing the transmission conditions (5.23), (5.24) for small strains, one recovers the transmissions conditions calculated in [START_REF] Abdelmoula | Comportement asymptotique d'une interface mince[END_REF][START_REF] Lebon | Asymptotic analysis of a thin interface: the case involving similar rigidity[END_REF][START_REF] Lebon | Asymptotic behavior of a hard thin linear interphase: an energy approach[END_REF][START_REF] Rizzoni | Higher order model for soft and hard elastic interfaces Int[END_REF] in the linearly elastic setting and generalizing the perfect interface case by taking into account higher order terms. Thus, the transmission conditions calculated in the present paper for a hard adhesive can be viewed a generalization of the transmission conditions for a hard interface in linear elasticity.

In [START_REF] Bessoud | Plate-like and shell-like inclusions with high rigidity[END_REF][START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF][START_REF] Bessoud | Asymptotic analysis of shell-like inclusions with high rigidity[END_REF][START_REF] Chapelle | Modeling of the inclusion of a reinforcing sheet within a 3D medium[END_REF], different cases of plate-like and shell-like linear elastic interphases are considered by scaling the intermediate layer stiffness with ε 1/ (membrane interface) and ε 1/ 3 (inextensible flexural interface). The external loads, applied to the adherents, remain unscaled with respect to ε. Moreover, the asymptotic models are mathematically justified by virtue of a strong convergence argument. For an intermediate layer stiffness scaling with ε 1/ , in [START_REF] Bessoud | Multi-materials with strong interface: variational modelings[END_REF] it is found that the interphase behaves as an elastic membrane in the limit. This is different from the result obtained in the present paper, where it has been found that the rigid interface behaves as a perfect interface model at the zeroth order with the restriction that the deformation associated with the limit displacement is an isometric mapping. We believe that the difference may be due to our choice of the leading order in the inner expansion of the second Piola-Kirchhoff stress tensor (4.17), where the leading order has been simply chosen ε .

0 A different choice of the leading order in (4.17) is expected to give rise to a completely different limit interface model for the rigid case.

The interface laws calculated in the present paper are expected to find significant applications in different contexts; definitely, they should be of importance in the analysis of adhesive joints, especially for all those applications requiring an accurate modeling of the nonlinear pre-peak behavior of the adhesive [START_REF] Albarella | A 3D multiscale cohesive zone model for quasi-brittle materials accounting for friction, damage and interlocking[END_REF][START_REF] Ascione | On the flexural behaviour of GFRP beams obtained by bonding simple panels: an experimental investigationl[END_REF][START_REF] Cognard | Créac'hcadec, A study of the non-linear behaviour of adhesively-bonded composite assemblies[END_REF][START_REF] Ivanov | Computational models of laminated glass plate under transverse static loading[END_REF][START_REF] Lenci | Nonlinear free dynamics of a two-layer composite beam with different boundary conditions[END_REF][START_REF] Valoroso | Identification of mode-I cohesive parameters for bonded interfaces based on DCB test[END_REF].

The proposed interface laws could also serve as generalization of the classical linear spring-type interface model in simulations of imperfect nonlinear bonding between the constitutive components of composites, in particular to study the influence of interfacial imperfections on the effective macroscopic behavior [START_REF] Li | Interfacial imperfection coupling model with application to the in-plane fracture problem of a multiferroic composite[END_REF][START_REF] Li | A new interfacial imperfection coupling model (IICM) and its effect on the facture behavior of a layered multiferroic composite: anti-plane case[END_REF].

Fig. 1 .

 1 Fig. 1. Reference configuration of the one-dimensional composite bar made of a Saint Venant-Kirchhoff material.

ε 2 

 2 ± joined by an interphase, B , ε as represented in Fig.2. The interphase occupies a cylindrical region of height εand cross-section S ⊂ , with S ∂ a smooth boundary. An orthonormal Cartesian basis O e

  change of variables, the domains Γ 0 and Γ 1 are transformed into the domains denoted by Γ 0 and Γ , 1 respectively.

Fig. 2 .

 2 Fig. 2. Reference configuration of the joint viewed as a composite body made of two adherents in contact via a thin adhesive layer (left) and rescaled configuration of the joint (right).

  result, the remaining conditions (4.68)-(4.70) simplify as

3 ( 4 1  are independent of z . 3 and u , 3 1

 34133 Integrating the latter equation with respect z 3 and using (4.32) and (4.43), one obtains the first contact conditions .78) implying the continuity of the traction vector at the order zeroTo complete the analysis and obtain the remaining contact condi-tions, a first step is to prove that, the vectors s e ^0 3 and u ,3 The following Lemma, whose proof is postponed in Appendix, shows that this is true under suitable assumptions. Lemma 1. Let K be taken to denote the matrix are independent of z .

3

 3

3 and s 33 0

 33 Next, assuming that the hypotheses of the Lemma hold, we integrate (4.77) with respect to z 3 under the condition of continuity of the traction at S ± and we use (4.80) and the definition (4.73) to get: in (4.81) by using (4.75), (4.76) and (4.80), the following relation between the traction P e

3 .

 3 Interphase made of a "rigid" material The Lamé's coefficients are now assumed as follows:

1 

 1 24)-(4.26) and using (4.43)-(4.45) the transmission conditions (4.78) are reobtained. Note that (4.104)-(4.106) imply that u [ ] is independent of z , 3 thus integrating (4.24) with respect to z 3 under the condition of continuity of the traction at S ± and using (4.43), (4.71), (4.73) yields again (4.81). The latter allows to evaluate s e ^0 3 and shows that it is independent of z 3 . Using these results together with (4.27) and (4.46), we reobtain Eq. (4.84), in which s

  ( ( , 0 ) + ( , 0 )).

3 )

 3 Using the matching relations (4.117) and (4.119), the transmission conditions for the soft interphase can be rewritten in the final configuration Ω

  account the expansions (4.15), (4.16)) and the relations (4.32)-(4.34), one finds

32 )

 32 In addition to this equation, the two sets of conditions (4.101)-(4.103) and (4.104)-(4.106) have been obtained. Conditions (4.101)-(4.103) can be restated as

3 

 3 Physically, the deformation associated to u 0  belongs to the class of "paper-folding" deformations, the deformations that a flat sheet of paper having the shape of S can undergo.Conditions (4.104)-(4.106) give the additional restriction

  Therefore, if L ≠ + ∞ the limit surface energy (6.3) for b p ∞, as in (6.6) takes the form ∫

3 -

 3 respectively, and joined by an interface along a common face S 0 . In the reference configuration, the composite structure is subjected to a tensile (compressive) load Q > 0 (Q < 0) aligned parallel to e 3 and acting on the upper and the lower bases Γ x x x Ω x l = {( , , ) ∈ : = }

Fig. 3 .

 3 Fig. 3. Reference configuration of the butt joint studied in the example.

ε 3 

 3   a constant vector to be determined, representing the jump of the displacement at the interface S 0 .The Piola-Kirchhoff stress tensor corresponding to (7.4) is

  divergence of P ε vanishes together with the jump of P ε at the interface S .0 To meet the natural boundary conditions on Ω ∂ ,

11 ) 3 ,

 113 This equation determines λ 3 as a function of the load Q and, in view of the restriction λ ν ν ∈ (0, (1 + )/ ) imposes the following restriction on the load

3 + 3 -

 33 The macroscopic response to uniaxial tension/compression is shown in Figs. 4, 5 in terms of applied surface force Q (divided by E) per unit area in the reference configuration and applied surface force q Q λ = / 1 2 (divided by E) per unit area in the deformed configuration versus the logaritmic strain Λ ϵ = ln . In the Figures, the Poisson's

Fig. 4 .

 4 Fig. 4. Uniaxial tension and compression response of a butt joint with a soft thin interphase. Normalized applied surface force per unit area in the reference configuration, Q E / , and normalized applied surface force per unit area in the deformed configuration, q E / , versus the macroscopic logaritmic strain ϵ for the values E E / = 30 ε

17

 17

  ) is still given by (7.10), i. e. P P = .ε 0 The (constant) jump u ε   is determined in order to satisfy the interface laws in(7.16). Since H ε and P 0 are constant tensors, the first interface law reduces to P e = 0, satisfied. The second interface law gives the following conditions:

ε 1 ε 3 + 3 - 3 + 3 -

 13333 In Fig.6, it has been set E E / = and an increasing dashing corresponds to increasing values of ε l l /( + ) in the set {0.005, 0.025, 0.05}. In Fig.7, it has been set ε l l /( + ) = 0.025

Fig. 5 .

 5 Fig. 5. Uniaxial tension and compression response of a butt joint with a soft thin interphase. Normalized applied surface force per unit area in the reference configuration, Q E/ , and normalized applied surface force per unit area in the deformed configuration, q E / , versus the macroscopic logaritmic strain ϵ for the adhesive/adherents thickness ratio ε l / = 0.025 and Poisson's ratios ν = 0.33, ν = 0.4.εThe thick solid line corresponds to the response without the thin interphase, the thin dashed curves to increasing ratios E E / = {10, 30, 150} ε

Fig. 6 . 1 ε

 61 Fig. 6. Uniaxial tension and compression response of a butt joint with a hard thin interphase. Normalized applied surface force per unit area in the reference configuration, Q E / , and normalized applied surface force per unit area in the deformed configuration, q E / , versus the macroscopic logaritmic strain ϵ for the elastic constants E E / = 1 ε , ν = 0.33, ν = 0.4. ε The thick solid line corresponds to the response without the thin interphase, the thin dashed curves to ε l / = {0.005; 0.025; 0.05}, the dashing increasing with increasing thickness ratio ε l / .

Fig. 7 .

 7 Fig. 7. Uniaxial tension and compression response of a butt joint with a hard thin interphase. Normalized applied surface force per unit area in the reference configuration, Q E / , and applied surface force per unit area in the deformed configuration, q E / , versus the macroscopic logaritmic strain ϵ for the thickness ratio ε l / = 0.025, and Poisson's ratios ν = 0.33, ν = 0.4. ε The thick solid line corresponds to the response without the thin interphase, the thin dashed curves to increasing ratios E E / = {0.5, 10, 50}, ε the dashing space increasing with increasing E E / ε .

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and constructive comments that greatly contributed to improving the final version of the paper.

 is invertible, then the system (A.6) admits only the trivial solution u , , [START_REF] Lenci | Nonlinear free dynamics of a two-layer composite beam with different boundary conditions[END_REF] 1

 is independent of z .

3 Therefore, in view of (A.5) and of the invertibility of ϕ ∇ , s e ^0 3 is also independent of z 3 . □


Giuseppe Valenzise

Andrei Purica

Vedad Hulusic

Marco Cagnazzo

Quality Assessment of Deep-Learning-Based Image Compression

Image compression standards rely on predictive coding, transform coding, quantization and entropy coding, in order to achieve high compression performance. Very recently, deep generative models have been used to optimize or replace some of these operations, with very promising results. However, so far no systematic and independent study of the coding performance of these algorithms has been carried out. In this paper, for the first time, we conduct a subjective evaluation of two recent deeplearning-based image compression algorithms, comparing them to JPEG 2000 and to the recent BPG image codec based on HEVC Intra. We found that compression approaches based on deep auto-encoders can achieve coding performance higher than JPEG 2000, and sometimes as good as BPG. We also show experimentally that the PSNR metric is to be avoided when evaluating the visual quality of deep-learning-based methods, as their artifacts have different characteristics from those of DCT or wavelet-based codecs. In particular, images compressed at low bitrate appear more natural than JPEG 2000 coded pictures, according to a no-reference naturalness measure. Our study indicates that deep generative models are likely to bring huge innovation into the video coding arena in the coming years.

I. INTRODUCTION

Current image compression methods rely substantially on transform coding: instead of searching for an optimal vector quantizer in the pixel domain, which is a hard problem [START_REF] Gersho | Vector Quantization and Signal Compression[END_REF], they transform the input data into an alternative representation, where dependencies among pixels are greatly reduced and an ensemble of independent scalar quantizers can be used. A proper choice of the transformed domain is fundamental to compact the energy of the signal into a few coefficients, and to enable perceptual scaling. Traditionally, image compression standards have been mainly employing linear representations, such as the discrete cosine transform (DCT) [START_REF] Ahmed | Discrete cosine transform[END_REF], used in block-based coding algorithms such as JPEG and in video codecs such as H.264/AVC and HEVC, as well as wavelet transforms [START_REF] Mallat | A wavelet tour of signal processing[END_REF], used in JPEG 2000. These representations are optimal only under strong assumptions about the underlying distribution of image data, e.g., local stationarity, smoothness or piece-wise smoothness. Failure to meet these assumptions is the cause of typical visual artifacts, such as ringing and blur. More recent techniques, such as dictionary learning and sparse coding [START_REF] Tosic | Dictionary learning[END_REF], have shown that it is possible to build more flexible representations by replacing a fixed transform by a non-linear optimization process with a sparsity constraint. However, these methods still represent the input data with linear combinations of the dictionary atoms.

Recently proposed image compression algorithms based on deep neural networks [START_REF] Theis | Lossy image compression with compressive autoencoders[END_REF], [START_REF] Agustsson | Soft-to-hard vector quantization for endto-end learning compressible representations[END_REF], [START_REF] Minnen | Spatially adaptive image compression using a tiled deep network[END_REF], [START_REF] Rippel | Real-time adaptive image compression[END_REF], [START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF], [START_REF] Agustsson | Generative adversarial networks for extreme learned image compression[END_REF] leverage much more complex, and highly non-linear, generative models. The goal of deep generative models is to learn the latent data-generating distribution, based on a very large sample of images. Specifically, a typical architecture consists in using auto-encoders [START_REF] Kingma | Auto-encoding variational bayes[END_REF], [START_REF] Goodfellow | Deep learning[END_REF], which are networks trained to reproduce their input. The structure of an auto-encoder includes an information bottleneck, i.e., one or more layers with fewer elements than the input/output signals. This forces the autoencoder to keep only the most relevant features of the input, producing a low-dimensional representation of the image. Alternative approaches use generative adversarial networks (GAN) to achieve extremely low bitrates [START_REF] Santurkar | Generative compression[END_REF]; however, their training process tends to be unstable and their applicability to practical image coding is still under study at the time of this writing.

While several deep-learning-based image compression methods have been proposed recently, little has been done to assess the performance of these techniques compared to traditional coding methods. Most of these works report only objective results in terms of PSNR and MS-SSIM [START_REF] Ballé | End-to-end optimized image compression[END_REF], [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], [START_REF] Agustsson | Soft-to-hard vector quantization for endto-end learning compressible representations[END_REF]. Minnen et al. [START_REF] Minnen | Spatially adaptive image compression using a tiled deep network[END_REF] ran a pairwise comparisons test with 10 observers, to assess the preference of their method over JPEG, getting better results at a rate of 0.25 and 0.5 bpp. Theis et al. [START_REF] Theis | Lossy image compression with compressive autoencoders[END_REF] conducted a single-stimulus rating test with 24 viewers, comparing to [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], JPEG and JPEG 2000. Their proposal outperforms the other methods, including JPEG 2000 at bitrates of 0.375 and 0.5 bpp.

To the authors' knowledge, this is the first independent study to assess the performance of deep-learning-based image compression. Specifically, we propose the three following contributions: i) we evaluate the rate-distortion performance of two recent image compression methods based on deep auto-encoders [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], [START_REF] Ballé | End-to-end optimized image compression[END_REF], compared to JPEG 2000 and BPG (Better Portable Graphics, a variant of HEVC Intra, which currently yields state-of-the-art image coding performance); ii) we evaluate the accuracy of 9 fidelity metrics in predicting mean opinion scores for deep-learning compressed images; iii) we assess the naturalness of deep-learning compressed images, using an opinion-and distortion-unaware metric. Our results show that, at least in some cases, deep-learning-based compression achieve performance as good as BPG, yielding more natural compressed images than JPEG 2000. In addi-tion, our analysis suggests that the different kind of artifacts produced by some deep-learning-based methods are difficult to gauge using metrics such as PSNR. The subjectively annotated dataset and the metrics will be made available online for the research community.

II. DEEP-LEARNING-BASED IMAGE COMPRESSION

In this paper, we selected two popular deep-learning-based compression algorithms: the auto-encoder-based method of Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF], and the approach based on residual autoencoders with recurrent neural networks, proposed by Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF]. This choice is motivated, on one hand, by the fact that these two methods were amongst the first to produce (at least in some cases) results with higher visual quality than JPEG or JPEG2000 compression. On the other hand, more recent methods are somehow inspired by these approaches, but differently from [START_REF] Ballé | End-to-end optimized image compression[END_REF] and [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], the code to reproduce their results is not publicly available at the time of this writing, which makes it difficult to carry out a fair evaluation.

A. Ballé et al. (2017)

Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF] propose an image compression algorithm consisting of a nonlinear analysis/synthesis transform and a uniform quantizer. The nonlinear transform is implemented through a convolutional neural network (CNN) with three stages. Each stage is composed by a convolution layer, downsampling and a non-linearity. Differently from conventional CNN's, which employ standard activation functions such as ReLU or tanh, in [START_REF] Ballé | End-to-end optimized image compression[END_REF] the non-linearity is biologically inspired and implements a sort of local gain control by means of a generalized divisive normalization (GDN) transform. The parameters of the GDN are tuned locally, at each scale, mimicking somehow local adaptation.

The authors of [START_REF] Ballé | End-to-end optimized image compression[END_REF] directly optimize the rate-distortion function D + λR, assuming uniform quantization. However, the gradient of the quantization function would be zero almost everywhere, thus hindering learning. Therefore, the authors approximate quantization noise with i.i.d. uniform noise; this corresponds to smoothing the discrete probability mass function of the transformed coefficients with a box filter. The bitrate is then approximated with the differential entropy of the smoothed, continuous distribution. The distortion is computed as the mean squared error (MSE) between the original and reconstructed samples. The whole coding scheme is optimized end-to-end, for a given value of λ, resulting in different operational points over the D(R) curve. Using MSE distortion, this formulation can be shown to be equivalent to a variational auto-encoder with a uniform approximate posterior [START_REF] Kingma | Auto-encoding variational bayes[END_REF], with the important difference that a generative model tends to minimize distortion (λ → 0), while in [START_REF] Ballé | End-to-end optimized image compression[END_REF] the R/D tradeoff is optimized.

A Matlab code of this codec, together with trained models for gray-scale and color images for six values of λ, can be found at http://www.cns.nyu.edu/ ∼ lcv/iclr2017/.

B. Toderici et al. (2017)

One of the limitations of [START_REF] Ballé | End-to-end optimized image compression[END_REF] is that it requires a separate model, and thus a new training, for each value of λ. Instead, Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF] follow a different approach based on a single model. This is obtained by making the encoding and decoding processes progressive: after the first coding/decoding iteration, the residue with respect to the original is computed; afterwards, this residue is further encoded/decoded, and the difference with respect to previous residue is found. This scheme is applied on 32 × 32 pixel patches. The compression model is also coupled with a binarizer.

The model used by authors is based on Recurrent Neural Networks (RNN). However, convolutions are used to replace multiplications in the RNN traditional models. Several architectures derived from the well known LSTM networks were tested and the Gated Recurrent Units were found to provide the best results. The encoder uses 4 convolutional layers with RNN elements, and a resolution reduction of 2 is achieved after each layer by using a stride of 2 × 2. As such, for a 32 × 32 × 3 input image, the output after a single iteration will be a 2 × 2 × 32 binary representation.

A Python implementation of the codec, in tensorflow framework, can be found at https://github.com/tensorflow/models/ tree/master/research/compression/image encoder/. The network architecture and weights are given in a binary format.

C. Coding rate computation

For each of the tested methods we need to compute the number of bits used to represent the encoded images at various quality level. This is straightforward for the standard methods JPEG, JPEG2000 and BPG, which actually produce the compressed file. On the other hand, the available implementations of both methods by Ballé et al. and by Toderici et al. do not directly provide an encoded file. In the first case, the authors implemented a CABAC-like entropy coder to encode the quantized transform coefficients, and the rate-distortion results they provide are based on this encoder. However, the latter is not available in their code. Therefore we performed an entropy estimation and implemented a simple entropy encoder (EE) using run-length encoding (RLE) to effectively represent the long runs of zeros coming from null channels in the tensor produced by this method. We found that the RLE+EE gave coding rates close to the estimated entropy, therefore in the following we use the former as coding rate. As for the Toderici et al. method, it produces a tensor of values within {±1}, which can be considered as binary symbols of an encoded stream. Thus, the number of symbols can directly be used as size (in bits) of each encoded layer.

III. SUBJECTIVE EVALUATION

In this section, we describe the subjective experiment we conducted in order to assess the quality of images compressed with the methods in [START_REF] Ballé | End-to-end optimized image compression[END_REF] and [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF]. [START_REF] Froehlich | Creating cinematic wide gamut HDR-video for the evaluation of tone mapping operators and HDR-displays[END_REF]. MasonLake and SouthBranchKingsRiver are tone-mapped images from the Fairchild HDR photographic survey [START_REF] Fairchild | The HDR photographic survey[END_REF].

A. Experiment setup 1) Material: For the study, we selected 6 uncompressed images of size 736×960 pixels, shown in Figure 1. The Bistro and Showgirl are cropped frames from high dynamic range (HDR) video sequences in [START_REF] Froehlich | Creating cinematic wide gamut HDR-video for the evaluation of tone mapping operators and HDR-displays[END_REF], which have been tone mapped using the display adaptive tone mapping in [START_REF] Mantiuk | Display adaptive tone mapping[END_REF]. MasonLake and SouthBranchKingsRiver are also HDR images, cropped and tone mapped with [START_REF] Mantiuk | Display adaptive tone mapping[END_REF]. Computer was acquired by the authors, using a Canon EOS 700D camera in raw mode, and applying the camera native response curve and white balancing. Finally, Screenshot is a screen capture, cropped to match the resolution of test stimuli. These images were selected out of 19 candidate pictures, on the basis of their spatial information, key and colorfulness [START_REF] Hulusic | A model of perceived dynamic range for HDR images[END_REF], as well as on their semantics (outdoor, people/faces, man-made objects). Screenshot was selected to include an example of synthetic image, which might be representative of a screen-content compression scenario.

2) Stimuli: Starting from these 6 pristine contents, we generated 113 compressed stimuli, in a such a way to span uniformly the impairment scale, as described in Section III-A3. Specifically, we used 4 compression methods: JPEG 2000, BPG, Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF] and Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF]. For JPEG 2000 and BPG we used the openJPEG library available for download at http://www.openjpeg.org/ and the BPG implementation found at https://bellard.org/bpg/, while for the last two algorithms we used the implementation publicly available from the authors. We selected 5 bitrates, corresponding to 5 different quality levels, for JPEG 2000 and BPG. For Ballé and Toderici, it is not possible to fix an arbitrary bitrate for coding, as only an ensemble of predefined bitrates was available. In particular, in some cases we could not find images corresponding to the highest quality level ("Imperceptible").

3) Design: In the study we employ the double stimulus impairment scale (DSIS) methodology [START_REF] Itu-R | Methodology for the subjective assessment of the quality of television pictures[END_REF], Variant I with a side-by-side presentation. In each trial, a pair of images with same content, one being original -reference and one compressed -test was displayed, and the participants were asked to evaluate the level of degradation of the test image relative to the reference. A continuous impairment scale ([0,100], 100 corresponding to "Imperceptible" and 0 to "Very annoying") was utilized. Each participant evaluated 113 images, where the pairs were selected randomly with a single constraintthe same content could not appear twice consecutively.

4) Participants and apparatus: There were 23 participants (15 male, 8 female) with an average age of 32. The experiment was conducted in a dark and quiet room. The stimuli were displayed at full HD on a Dell Ultrasharp U2410 24" display. The ambient illumination in the room, measured between the screen and participants, was 2.154 lux. The distance from the screen was fixed to 70 cm (approximately three times the height of the pictures on the display) with the eyes in the middle of the display, both horizontally and vertically.

5) Procedure: Prior to the experiment, the participants were verbally explained the experimental procedure. This was followed by a training session with a stimulus that was not used in the main study, showing all the levels of distortion across different compression methods. Upon completion of the training, they were left in the room to do the main test. There were no time constraints for the image observation before evaluating it. The images were shown side-by-side while the slider was on the right edge all the time, allowing them to vote when they made a decision. Once rated, the next image pair was displayed. The average duration of the test was approximately 22 minutes.

B. Results

Before looking at the data from the subjective experiment, screening of the observers for detection of potential outliers was performed, as proposed in the R-REC-BT.500-13 [START_REF] Itu-R | Methodology for the subjective assessment of the quality of television pictures[END_REF]. The procedure detected no outliers.

Following this, mean opinion score (MOS) values and confidence intervals (CI) were computed for all 113 test conditions. Rate distortion curves with computed MOS values for all images used in the experiment are provided in Figure 2. The results for the lowest bitrates are cluttered and have lowest CIs, which is expected and confirms that this distortion level corresponds unanimously to the "Very annoying" level on the rating scale. However, there are several notable results visible in the plots for the other bitrates.

Toderici method seems to result in highest perceived visual quality in case of MasonLake, and for SouthBranchKingsRiver, but in this case only at medium-high bitrate. At the same time, the same method performs the worst for Bistro and Showgirl scenes. Furthermore, images compressed using Ballé method got better subjective scores than JPEG 2000 in all but SouthBranchKingsRiver case. This suggests a strong content dependency in the performance of these methods, compared to the more stable traditional codecs. It seems that for natural images with a lot of textures the gains are more evident for deep-learning-based methods. We further analyze the artifacts of learning-based methods in Section IV-B.

IV. OBJECTIVE EVALUATION

In this section we report objective metrics results for the stimuli of the study. Specifically, we evaluate the performance of popular full-reference metrics employed for traditional compression to predict visual quality of the coding methods in [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF] and [START_REF] Ballé | End-to-end optimized image compression[END_REF]. Afterwards, we present a qualitative analysis of the artifacts produced by deep-learning-based coding.

A. Fidelity metrics

We include in our evaluation nine commonly used fullreference image quality metrics, see Tables I andII. We used the publicly available implementation of these metrics in the MeTriX MuX library for Matlab, available at http: //foulard.ece.cornell.edu/gaubatz/metrix mux/. We evaluate fidelity metrics in terms of prediction accuracy, prediction monotonicity, and prediction consistency, as recommended in [START_REF] Itu-T | Methods, metrics and procedures for statistical evaluation, qualification and comparison of objective quality prediction models[END_REF]. For prediction accuracy, Pearson correlation coefficient (PCC), and root mean square error (RMSE) are computed. Spearman rank-order correlation coefficient (SROCC) is used for prediction monotonicity, and outlier ratio (OR) is calculated to determine the prediction consistency. These performance metrics have been computed after performing a non-linear regression on objective quality metric scores using a logistic function.

The results are reported in Table I, per compression method and over all contents, and in Table II, where: i) conventional (JPEG 2000 and BPG) and deep-learning-based (Toderici et al., Ballé et al.) are grouped; and ii) all methods and contents are considered together. While for standard codecs the trends are similar as in previous studies [START_REF] Sheikh | A statistical evaluation of recent full reference image quality assessment algorithms[END_REF], an analysis of the results in Table II reveals that the PCC of all metrics but SSIM, VIF, UQI and IFC significantly drops when evaluated on Ballé and Toderici, compared to BPG and JPEG 2000 (p < 0.05). In particular, the reduction of prediction accuracy is highest for PSNR. A probable explanation for this drop is the different kinds of artifacts produced by these two compression methods. To confirm this behavior, we report in Figure 3(a) the MOS with respect to PSNR values. We see from the scatter plot that for Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF] and Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], stimuli with similar PSNR values might be significantly different in terms of perceived visual quality. This phenomenon is much less present when using different metrics, such as MS-SSIM and VIF (Figures 3(a) and (b), respectively). We thus recommend to avoid using PSNR when evaluating compression performance of future deep-learning-based image or video compression techniques. Table II suggests instead to employ VIF and MS-SSIM. and straight lines are generally well preserved. In part (b) of Figure 4, these characteristic artifacts are shown in comparison with a BPG-compressed image at similar bitrate, which has higher MOS. Notice again the sort of "brush" effect for Ballé, and the little presence of artifacts on high-contrast edges. Finally, in Figure 4(c) we show an example for Toderici against JPEG 2000. The two images have similar bitrates; however, JPEG 2000 has a better PSNR score while Toderici has a higher MOS score. This is interesting to note as the image compressed with Toderici et al. has more details but loses in PSNR most likely due to blocking artifacts. It should be noted that these type of artifacts have been partially suppressed in their follow-up work [START_REF] Johnston | Improved lossy image compression with priming and spatially adaptive bit rates for recurrent networks[END_REF]. However, we did not have access to the updated implementation.

B. Qualitative results

In order to show examples of the artifacts of deep-learningbased methods, we report in

In order to investigate further the distortion of Ballé and Toderici's methods, we computed the natural image quality evaluator (NIQE) metric [START_REF] Mittal | Making a "completely blind" image quality analyzer[END_REF] for all stimuli but Screenshot (which is a synthetic image). NIQE measures the distance of the distribution of mean-contrast normalized coefficients of the image under study with respect to a reference distribution of the same coefficients found on a large dataset of pristine images, and thus is both opinion-and distortion-unaware. Lower values of the metric indicate higher naturalness. Notice that stimuli compressed with both Ballé and Toderici tend to be more natural, i.e., they follow better the statistics of natural images, than images coded with JPEG 2000.

V. DISCUSSION AND CONCLUSIONS

This is the first, independent study to evaluate subjectively and objectively the quality of two deep-learning-compressed images. On our dataset, both methods achieve results similar to or better than JPEG 2000 in many cases, and in other cases their rate-distortion performance is equivalent to that of BPG. These results are somewhat justified by the more natural appearance of images compressed by the two deep-learningbased methods, compared, e.g., to JPEG 2000. The reason for this is most probably related to the much more powerful generative model expressed by deep auto-encoders, compared to simple image transforms such as DCT and wavelets (in the case of BPG, the use of in-loop filters considerably reduces the visibility of blocking, at the advantage of naturalness). A direct consequence is that simple pixel-based metrics such as the PSNR, which are still widely used in image and video coding, are much less accurate to judge visual quality with the new methods.

A more detailed analysis would require coding images at the same (or as close as possible) bitrate and conduct pairwise comparison tests to evaluate precisely the preferences of observers. However, this is difficult at this stage as the available implementations of Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF] and Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF] do not enable a fine-granularity rate control. Yet, our results are still surprising, when thinking that BPG and JPEG 2000 are the product of decades of coding optimization and engineering, while the methods of Ballé et al. [START_REF] Ballé | End-to-end optimized image compression[END_REF] and Toderici et al. [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF] are proofs of concept developed in the last year. This suggests a great potential in using deep generative models for next-generation image and video codecs.

Fig. 1 .

 1 Fig.1. Test images used in the study. Bistro and Showgirl are tone-mapped images from the Stuttgard HDR video database[START_REF] Froehlich | Creating cinematic wide gamut HDR-video for the evaluation of tone mapping operators and HDR-displays[END_REF]. MasonLake and SouthBranchKingsRiver are tone-mapped images from the Fairchild HDR photographic survey[START_REF] Fairchild | The HDR photographic survey[END_REF].

Fig. 2 .

 2 Fig. 2. MOS vs. bitrate. Error bars indicate 95% confidence intervals.

Figure 4

 4 some details of coded images. Part (a) of the figure demonstrates a case where Ballé et al. achieve clearly better visual quality than JPEG 2000 at a higher bitrate. Notice that there are no visible, unnatural ringing or mosquito noise artifacts, nor blocking, and although high-frequency details are somehow blurred, edges

Fig. 3 .

 3 Fig. 3. Scatter plots for MOS against different objective metrics.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Examples of images (details) coded with different methods. (a) A case where an image coded with the method of Ballé et al. has better quality than JPEG 2000 at a similar bitrate. (b) An example where Ballé et al. has worse quality than BPG. (c) An example of Toderici et al. has better quality in MOS (12.4 vs 8.1) but lower in PSNR (20.85dBs vs 21.35dBs).

TABLE I STATISTICAL

 I ANALYSIS OF OBJECTIVE QUALITY METRICS ON THE PROPOSED DATASET (I). BEST METRICS VALUES PER COLUMN ARE HIGHLIGHTED IN

									BOLD.
			JPEG2K			BPG				Balle				Toderici
	Metric	PCC	SROCC	RMSE	OR	PCC	SROCC	RMSE	OR	PCC	SROCC	RMSE	OR	PCC	SROCC	RMSE	OR
	PSNR	0.860	0.873	15.1	0.14	0.851	0.854	16.2	0.20	0.621	0.574	19.7	0.20	0.699	0.707	19.9	0.18
	SSIM	0.894	0.892	13.5	0.07	0.915	0.901	12.5	0.07	0.860	0.863	12.8	0.08	0.836	0.824	15.3	0.11
	MS-SSIM	0.961	0.958	8.3	0.00	0.971	0.956	7.4	0.00	0.936	0.939	8.9	0.00	0.923	0.898	10.7	0.00
	VSNR	0.903	0.907	12.9	0.07	0.889	0.882	14.1	0.10	0.750	0.728	16.6	0.16	0.768	0.738	17.8	0.18
	VIF	0.958	0.956	8.7	0.03	0.968	0.949	7.8	0.00	0.929	0.924	9.3	0.04	0.940	0.911	9.5	0.07
	UQI	0.833	0.833	16.7	0.23	0.792	0.770	18.9	0.33	0.760	0.752	16.3	0.20	0.863	0.863	14.1	0.04
	IFC	0.921	0.915	11.7	0.07	0.931	0.915	11.3	0.07	0.880	0.876	11.9	0.04	0.964	0.932	7.4	0.04
	NQM	0.881	0.884	14.3	0.10	0.916	0.905	12.4	0.10	0.793	0.804	15.3	0.12	0.848	0.817	14.7	0.11
	WSNR	0.946	0.949	9.8	0.00	0.962	0.953	8.5	0.00	0.896	0.889	11.2	0.04	0.890	0.866	12.7	0.11

TABLE II STATISTICAL

 II ANALYSIS OF OBJECTIVE QUALITY METRICS ON THE PROPOSED DATASET (II). RESULTS ARE GROUPED CONSIDERING: STANDARD IMAGE CODECS (JPEG 2000 AND BPG); DEEP-LEARNING-BASED COMPRESSION ALGORITHMS; AND ALL THE 113 STIMULI OF THE DATASET. BEST METRICS VALUES PER COLUMN ARE HIGHLIGHTED IN BOLD.

		JPEG2K & BPG	Balle & Toderici		All methods
	Metric	PCC	SROCC	RMSE	PCC	SROCC	RMSE	PCC	SROCC	RMSE
	PSNR	0.858	0.870	15.676	0.652	0.638	20.156	0.763	0.766	18.619
	SSIM	0.902	0.908	13.154	0.829	0.830	14.864	0.866	0.871	14.392
	MS-SSIM	0.964	0.957	8.170	0.917	0.907	10.6	0.941	0.936	9.77604
	VSNR	0.888	0.896	14.011	0.740	0.731	17.881	0.815	0.815	16.677
	VIF	0.962	0.953	8.348	0.931	0.919	9.740	0.944	0.936	9.516
	UQI	0.812	0.802	17.813	0.815	0.821	15.423	0.813	0.807	16.751
	IFC	0.925	0.917	11.615	0.922	0.907	10.282	0.922	0.910	11.163
	NQM	0.897	0.899	13.499	0.803	0.794	15.842	0.852	0.848	15.078
	WSNR	0.953	0.955	9.261	0.866	0.851	13.318	0.910	0.908	11.914