Towards nonlinear imperfect interface models including micro-cracks and smooth roughness

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Towards nonlinear imperfect interface models including micro-cracks and smooth roughness

Serge Dumont, Frédéric Lebon, Maria Letizia Raffa, Raffaella Rizzoni

To cite this version:

Serge Dumont, Frédéric Lebon, Maria Letizia Raffa, Raffaella Rizzoni. Towards nonlinear imper-

fect interface models including micro-cracks and smooth roughness. Annals of Solid and Structural

Mechanics, Springer Berlin Heidelberg, 2017, 9 (1-2), pp.13-27. �10.1007/s12356-017-0047-8�. �hal-

01697064�

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Towards nonlinear imperfect interface models including micro-cracks and smooth roughness

S. Dumont

¹

· F. Lebon

²

· M. L. Raﬀa

³

· R. Rizzoni

⁴

Keywords Asymptotic analysis · Nonlinear constitutive equations · Roughness · Imperfect interface · Nonlinear contact laws

1 Introduction

Nowadays structural bonding is a very common technique in numerous industrial domains as aeronautics or civil engineering. This technology consists in an adhesive assembly of two bodies. The main function of this assembly is to transmit mechanical stresses in a given environment dur- ing the whole life of the structure. The technology’s ben- efits are obvious: low costs, simplified industrial processes, increased lightness, etc. Conventional adhesives (epoxides, polyamides, acrylic, cyanoacrylates, etc.) come in the form of thin films. Therefore, it is very important to propose simple as well as precise analytical and numerical models to simulate and predict the behavior of thin adhesive layers.

The present work aims at proposing a quite general methodology to describe the behavior of such a structure. Adhe- sive layers being often characterized by their low thickness, it seems natural to introduce an asymptotic model whose small parameter is the layers’ smallest characteristic length 𝜀. The problem of a thin adhesive joint is then classically replaced by a transmission problem with mechanical characteristics obtained from the mechanical and geometrical characteristics of the adhesive thin ﬁlm (cf. [2, 12, 13, 15]).

In former papers by authors ([36, 38, 39, 45] and ref- erences therein), the illustrated methodology has been applied for a planar elastic thin adhesive interphase, pos- sibly anisotropic. Furthermore, in [37] this asymptotic technique within the small strains context was coupled to micromechanical homogenization concepts and Hertzian Abstract The present paper deals with a general asymp-

totic theory aimed at deriving some imperfect interface models starting from thin interphases. The novelty of this work consists in taking into account some non-standard constitutive behaviors for the interphase material. In particular, micro-cracks, surface roughness and geometrical nonlinearity are included into the general framework of the matched-asymptotic-expansion theory. The elastic equilibrium problem of a three-composite body comprising two elastic adherents and an adhesive interphase is investigated.

Higher order interface models are derived within the cases of soft and hard interphase materials. Simple FEM-based numerical applications are also presented.

F. Lebon

lebon@lma.cnrs-mrs.fr S. Dumont

serge.dumont@unimes.fr M. L. Raﬀa

maria-letizia.raﬀa@u-pec.fr R. Rizzoni

rizzoni.raﬀaella@unife.it

1 Université de Nîmes, Institut Montpellierain Alexander Grothendieck, CNRS UMR 5149, CC 051, Place Eugène Bataillon 34090 Montpellier, France

2 Aix-Marseille Univ., CNRS, Centrale Marseille, LMA, Marseille, France

3 CNRS, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

4 Department of Engineering, University of Ferrara, via Saragat 1, 44122 Ferrara, Italy

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theory in order to formulate a model of contact between smoothly-rough surfaces in non-sliding conditions.

In the present paper, several additional new features are introduced. Particularly, an overview of simple techniques accounting for (smooth) roughness, micro-damage (micro- cracks) and geometrical nonlinearities is presented. Two main novelties are presented: (1) roughness is introduced in the asymptotic formulation within the framework of ﬁnite strain [36, 39, 44] as well as in the hard interface formulation; (2) two theories of micromechanical homogenization for micro-cracked materials, i.e., Kachanov’s theory [30, 48] and dilute scheme [4, 35], are studied and compared.

The paper is organized as follows. Firstly, a review of soft and hard imperfect interface models in linear elasticity is presented in Sect. 2. Next, a smooth roughness at the contacting interfaces between the two adherents and the adhesive interphase in introduced in Sect. 3. A simple methodology of micromechanical homogenization able to take into account micro-damage is presented in Sect. 4. In Sect. 5, an interphase consisting of a nonlinear elastic material, the Saint Venant–Kirchhoﬀ material model, is studied.

Lastly, some simple numerical applications are proposed in Sect. 6, to validate the proposed interface models.

2 Soft and hard imperfect interface models in linear elasticity

We begin by specifying what we understand by “soft” and

“hard” interfaces. This notion is related to the stiffness ratio between the adhesive and the adherents. For example, in the case of two pieces of steel (with a stiffness close to 210 GPa) bonded by an araldite glue (a stiffness close to 2 GPa), the interphase is considered as soft. On the contrary, if two samples of maple wood (the stiffness is close to 10

GPa) are bonded by an epoxy resin (the Young’s modulus is close to 4 GPa), the interphase is considered as hard.

2.1 The three-dimensional equilibrium problem A thin interphase, B

^𝜀

, joining two adherents, Ω

^𝜀

±

, is considered occupying a cylindrical region of small thickness and cross-section S, with S an open bounded set in IR

²

with a smooth boundary 𝜕S. Introduced the Cartesian frame ( O , x

1

, x

2

, x

3

) with ( 𝐞

1

, 𝐞

2

, 𝐞

3

) the corresponding orthonor- mal basis, the domains B

^𝜀

and Ω

^𝜀

±

are deﬁned as

with 0 < 𝜀 ∕ h << 1 and h the characteristic length of Ω. The surfaces between the adherents and the interphase of small thickness 𝜀 are

The geometrical conﬁguration described above is shown in Fig. 1.

The adherents are subjected to a body force density 𝐟 :Ω

^𝜀_±

↦ R

³

and to a surface force density 𝐠 : Γ

^𝜀_g

↦ R

³

. Body forces are neglected in the adhesive.

On Γ

^𝜀_u

= 𝜕 Ω

^𝜀

⧵ (Γ

^𝜀_g

∪ 𝜕 Ω

^𝜀

∩ 𝜕 B

^𝜀

), homogeneous boundary conditions are prescribed:

where 𝐮

^𝜀

: Ω

^𝜀

↦ R

³

is the displacement ﬁeld deﬁned on Ω

^𝜀

. Γ

^𝜀_g

is assumed to be located far from the interphase and the ﬁelds of the external forces are endowed with suﬃcient B

^𝜀

= (1)

{

( x

₁

, x

₂

, x

₃

) ∈ Ω: − 𝜀 2 < x

₃

<

𝜀 2

} .

Ω

^𝜀_±

= { (2)

( x

₁

, x

₂

, x

₃

) ∈ Ω: ± x

₃

>

𝜀 2

} ,

S

^𝜀

(3)

±

= {

( x

₁

, x

₂

, x

₃

) ∈ Ω: x

₃

= ± 𝜀 2

} .

𝐮

^𝜀

= 𝟎 on Γ

^𝜀u

, (4)

Fig. 1 Geometrical conﬁgura- tions of the three-composite system within the a the refer- ence model (interphase); b the rescaled model and c the limit model (interface)

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regularity to ensure the existence of equilibrium conﬁgura- tion. In the following, the symbol (, ) means derivative. The equations governing the equilibrium problem of the composite structure are written as follows:

with 𝐮

^𝜀

= u

^𝜀

i

𝐞

_i

a displacement from Ω

^𝜀

±

∪ B

^𝜀

into IR

³

, 𝛔

^𝜀

= 𝜎

^𝜀

ij

𝐞

i

⊗ 𝐞

j

the Cauchy stress, 𝐞 ( 𝐮

^𝜀

) the strain tensor under the small perturbations hypothesis (e

_ij

= 1∕2(u

_i,j

+ u

_j,i

)) and [[f ]]

^±_𝜀

the jump of f at S

^𝜀

±

. The materials of the adherents and of the thin interphase are assumed to be homogeneous and linearly elastic and 𝐚

±

, 𝐛

^𝜀

are taken to denote the fourth-order elasticity tensors of the adherents and of the interphase, respectively. The tensors 𝐚

±

, 𝐛

^𝜀

are assumed to be symmetric, with the minor and major symmetries, and positive deﬁnite. It is well known that the above equilibrium problem admits an unique solution. At least two diﬀerent methods are possible to study asymptotically the problem (5), by using the minimization of the total energy [18, 19, 42, 43, 45] or, directly, by using the equilibrium equations [9, 16, 17, 20–23, 34]. In this paper, the last method was preferred.

2.2 The rescaled equilibrium problem

Because the thickness of the interphase is very small, it is natural to seek an approximated solution of problem (5) as the small parameter 𝜀 vanishes. In this paper the method of matched asymptotic expansions is applied, which is based on a rescaling of the domains and on series expansions of the (rescaled) relevant fields (displacement and stress vector fields) with respect to 𝜀. In the following, the symbol (∧) will indicate the rescaled fields in the adhesive and the symbol (−) the rescaled fields in the adherents.

According to the method of matched asymptotic expansions, a classical change of scale is ﬁrst applied to the adhesive and to the adherents [6]:

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⎧ ⎪

⎪ ⎪

⎨ ⎪

⎪ ⎪

⎩ 𝜎

^𝜀

ij,j

+ f

_i

= 0 in Ω

^𝜀_±

,

𝜎

^𝜀

ij

n

_j

= g

_i

on Γ

_g

,

𝜎

^𝜀

ij,j

= 0 in B

^𝜀

,

[[ 𝜎

^𝜀

i3

]]

^±_𝜀

= 0 on S

^𝜀_±

,

[[u

^𝜀_i

]]

^±_𝜀

= 0 on S

^𝜀_±

, u

^𝜀_i

= 0 on Γ

_u

, 𝜎

^𝜀

ij

= ( a

_±

)

ijkl

e

_kl

( 𝐮

^𝜀

) in Ω

^𝜀_±

, 𝜎

^𝜀

ij

= (b

^𝜀

)

_ijkl

e

_kl

( 𝐮

^𝜀

) in B

^𝜀

,

̂ (6)

𝐩:( x

₁

, x

₂

, x

₃

) ↦ ( z

₁

, z

₂

, z

₃

) = (

x

₁

, x

₂

, x

₃

𝜀 )

,

̄ (7)

𝐩:( x

₁

, x

₂

, x

₃

) ↦ ( z

₁

, z

₂

, z

₃

) = ( x

₁

, x

₂

, x

₃

± 1∕2 ∓ 𝜀 ∕2).

Let 𝐮 ̂

^𝜀

: = 𝐮

^𝜀

◦ 𝐩 ̂

⁻¹

be a displacement from the rescaled interphase

and let 𝐮 ̄

^𝜀

: = 𝐮

^𝜀

◦ 𝐩 ̄

⁻¹

be a displacement from the rescaled adherents

In (7) the plus (minus) sign applied in Ω

₊

(Ω

₋

). Under the change of variables, the domains Γ

u

and Γ

g

are transformed into the domains denoted by Γ ̄

u

and Γ ̄

g

, respectively. The external forces are assumed to be independent of 𝜀 , thus it is set f ̄ (z

₁

, z

₂

, z

₃

) = f (x

₁

, x

₂

, x

₃

), ̄ g(z

₁

, z

2

, z

3

) = g(x

₁

, x

2

, x

3

).

2.3 Asymptotic expansions

Within the rescaled conﬁguration (cf. Fig. 1b), the asymptotic expansions of the displacement ﬁelds in the adherents and in the interphase take the following form:

Starting from Eq. (11), the adhesive’s strain tensor is obtained:

with:

and

where 𝛼 = 1, 2, Sym( ⋅ ) gives the symmetric part of a tensor and k = 0, 1, …

The strain tensor in the adherents can be obtained as:

with:

B = (8) {

( x

₁

, x

₂

, x

₃

) ∈ Ω: − 1

2 < x

₃

< 1 2 }

,

Ω

_±

= { (9)

( x

₁

, x

₂

, x

₃

) ∈ Ω: ± x

₃

> 1 2

} .

(10) 𝐮

^𝜀

(z

₁

, z

₂

, z

₃

) = 𝐮

⁰

+ 𝜀𝐮

¹

+ 𝜀

²

𝐮

²

+ o( 𝜀

²

),

̂ (11)

𝐮

^𝜀

(z

₁

, z

₂

, z

₃

) = 𝐮 ̂

⁰

+ 𝜀 ̂ 𝐮

¹

+ 𝜀

²

𝐮 ̂

²

+ o( 𝜀

²

),

(12)

̄

𝐮

^𝜀

(z

₁

, z

₂

, z

₃

) = 𝐮 ̄

⁰

+ 𝜀 ̄ 𝐮

¹

+ 𝜀

²

𝐮 ̄

²

+ o( 𝜀

²

).

(13) 𝐞 ( 𝐮 ̂

^𝜀

) = 𝜀

⁻¹

𝐞 ̂

⁻¹

+ 𝐞 ̂

⁰

+ 𝜀 ̂ 𝐞

¹

+ O ( 𝜀

²

)

̂ (14) 𝐞

⁻¹

: =

⎡ ⎢

⎢ ⎣

0 1 2 u ̂

⁰

𝛼,3

1 2 u ̂

⁰

𝛼,3

u ̂

⁰

3,3

⎤ ⎥

⎥ ⎦

̂ (15) 𝐞

^k

: =

⎡ ⎢

⎢ ⎣

Sym(̂ u

^k_𝛼,𝛽

)) 1

2 ( u ̂

^k_3,𝛼

+ u ̂

^k+1_𝛼,3

) 1

2 (̂ u

^k_3,𝛼

+ u ̂

^k+1_𝛼,3

) u ̂

^k+1_3,3

⎤ ⎥

⎥ ⎦ ,

(16) 𝐞 ( 𝐮 ̄

^𝜀

) = 𝜀

⁻¹

𝐞 ̄

⁻¹

+ 𝐞 ̄

⁰

+ 𝜀 ̄ 𝐞

¹

+ O ( 𝜀

²

)

̄ (17)

𝐞

⁻¹

= 𝟎,

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and again k = 0, 1, … The stress ﬁelds in the rescaled adhesive and adherents, denoted 𝜎 𝜎 𝜎 ̂

^𝜀

= 𝜎 𝜎 𝜎 ◦ 𝐩 ̂

⁻¹

and 𝜎 𝜎 𝜎 ̄

^𝜀

= 𝜎 𝜎 𝜎 ◦ 𝐩 ̄

⁻¹

respectively, are represented in asymptotic expansions:

As body forces are neglected in the adhesive, the equilibrium equation can be written at the two ﬁrst orders as:

where i = 1, 2, 3 .

Equation (22) shows that the stress vector 𝜎 𝜎 𝜎 ̂

⁰

i3

is constant with respect to z

₃

in the adhesive, and thus it can be written:

where [ . ] denotes the jump between z

₃

=

1

2

and z

₃

= −

1 2

. Analogously, equation (23) can be rewritten in the following integrated form:

All equations written so far are general in the sense that they are independent of the constitutive behavior of the materials.

The equilibrium equation in the adherents can be written as:

As a perfect contact law between the adhesive and the adherents is assumed, the continuity of the displacement and stress vector ﬁelds is enforced. In particular, the continuity of the displacements leads to [45]:

̄ (18) 𝐞

^k

=

⎡ ⎢

⎢ ⎣

Sym(̄ u

^k_𝛼,𝛽

) 1

2 (̄ u

^k_3,𝛼

+ u ̄

^k_𝛼,3

) 1

2 (̄ u

^k_3,𝛼

+ u ̄

^k_𝛼,3

) u ̄

^k_3,3

⎤ ⎥

⎥ ⎦ ,

(19) 𝜎 𝜎 𝜎

^𝜀

= 𝜎 𝜎 𝜎

⁰

+ 𝜀𝜎 𝜎 𝜎

¹

+ O ( 𝜀

²

),

(20)

̂

𝜎 𝜎 𝜎

^𝜀

= 𝜎 𝜎 𝜎 ̂

⁰

+ 𝜀 ̂ 𝜎 𝜎 𝜎

¹

+ O ( 𝜀

²

),

(21)

̄ 𝜎

𝜎 𝜎

^𝜀

= 𝜎 𝜎 𝜎 ̄

⁰

+ 𝜀 ̄ 𝜎 𝜎 𝜎

¹

+ O ( 𝜀

²

) .

̂ (22) 𝜎 𝜎 𝜎

⁰

i3,3

= 0,

̂ (23) 𝜎 𝜎 𝜎

⁰

i1,1

+ 𝜎 ̂ 𝜎 𝜎

⁰

i2,2

+ 𝜎 𝜎 ̂ 𝜎

¹

i3,3

= 0,

[ (24)

̂ 𝜎

⁰

i3

]

= 𝟎 ,

[ 𝜎 𝜎 𝜎 ̂

¹

(25)

i3

] = −[ 𝜎 𝜎 𝜎 ̂

⁰

i1,1

] − [ 𝜎 𝜎 𝜎 ̂

⁰

i2,2

].

̄ (26) 𝜎

⁰

ij,j

+ f

_i

= 0,

̄ (27) 𝜎

¹

ij,j

= 0.

(28) 𝐮

⁰

(̄ 𝐱 , 0

^±

) = 𝐮 ̂

⁰

(

̄ 𝐳 , ± 1

2 )

= 𝐮 ̄

⁰

(

̄ 𝐳 , ± 1

2 )

,

(29) 𝐮

¹

(

𝐱 , 0

^±

)

± 1 2

𝐮

⁰

,3

( 𝐱 , 0

^±

)

= ̂ 𝐮

¹

(

𝐳 , ± 1 2 )

= 𝐮

¹

(

𝐳 , ± 1 2 )

,

and the continuity of the stress vector gives the following conditions:

for i = 1, 2, 3.

Following [45], the matrices 𝐊

^jl𝜀

(with j, l = 1, 2, 3 ) are introduced, whose components are deﬁned by the relation:

In view of the symmetry properties of the elasticity tensor 𝐛

^𝜀

, it results that 𝐊

^jl𝜀

= ( 𝐊

^lj𝜀

)

^T

, with j, l = 1, 2, 3.

2.4 Internal/interphase analysis

In the following, two speciﬁc cases of linearly elastic material are studied for the interphase. One, called “soft” material, is characterized by elastic moduli which are linearly rescaled with respect to the thickness 𝜀 ; the second case, called “hard”

material, is characterized by elastic moduli independent of the thickness 𝜀.

2.4.1 Soft interphase analysis

Assuming that the interphase is “soft”, one deﬁnes:

where the tensor 𝐛 does not depend on 𝜀. Accordingly to position (32), it is set:

Introducing the expansions in the constitutive equations leads to:

and

for j = 1, 2, 3, and

which represents the classical law for a soft interface.

Analogously, it is obtained:

𝜎 (30) 𝜎 𝜎

⁰

i3

( 𝐱 , 0

^±

)

= 𝜎 ̂ 𝜎 𝜎

⁰

i3

( 𝐳 , ± 1

2 )

= 𝜎 𝜎 𝜎

⁰

i3

( 𝐳 , ± 1

2 )

(31) 𝜎 𝜎

𝜎

¹

i3

( 𝐱 , 0

^±

)

± 1 2 𝜎 𝜎 𝜎

⁰

i3,3

( 𝐱 , 0

^±

)

= 𝜎 𝜎 𝜎 ̂

¹

i3

( 𝐳 , ± 1

2 )

= 𝜎 𝜎 𝜎

¹

i3

( 𝐳 , ± 1

2 )

(K

_𝜀^jl

)

_ki

: = b

^𝜀_ijkl

. (32)

𝐛

^𝜀

= 𝜀𝐛 , (33)

K

_ki^jl

: = b

_ijkl

. (34)

(35)

̂ 𝜎 𝜎

𝜎

⁰

+ 𝜀 ̂ 𝜎 𝜎 𝜎

¹

= 𝐛 ( 𝐞 ̂

⁻¹

+ 𝜀 ̂ 𝐞

⁰

) + o ( 𝜀 )

(36) 𝜎 ̂

𝜎 𝜎

⁰

= b ( 𝐞 ̂

⁻¹

)

̂ 𝜎 𝜎

𝜎

¹

= b ( 𝐞 ̂

⁰

)

̂ (37) 𝜎 𝜎

𝜎

⁰

𝐢

_j

= 𝐊

^3j

𝐮 ̂

⁰

,3

̂ (38) 𝜎 𝜎

𝜎

⁰

𝐢

₃

= 𝐊

³³

[ 𝐮 ̂

⁰

] ,

(39) [ 𝜎 𝜎 𝜎 ̂

¹

𝐢

₃

]

= − 𝐊

³¹

[ 𝐮 ̂

⁰

]

,1

− 𝐊

³²

[ 𝐮 ̂

⁰

]

,2

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Denoting < f > ( 𝐳 ̄ ): = 1 2

( f ( 𝐳 ̄ , 1

2 ) + f ( 𝐳 ̄ , − 1 2 ) )

, it is obtained:

where the sum over 𝛼 = 1, 2 is performed.

The complete problem for soft interface can be written at the two ﬁrst orders as:

where << f >> ( 𝐱 ̄ ): = 1 2

( f ( 𝐱 ̄ , 0

⁺

) + f ( 𝐱 ̄ , 0

⁻

) ) . [ (40)

𝐮 ̂

¹

]

= ( 𝐊

³³

)

−1

(

< ̂ 𝛔

¹

𝐢

₃

> − 𝐊

^𝛼3

< ̂ 𝐮

⁰

>

,𝛼

)

(41)

⎧ ⎪

⎪ ⎪

⎨ ⎪

⎪ ⎪

⎩ 𝜎

⁰

ij,j

+ f

_i

= 0 in Ω

_±

,

𝜎

⁰

ij

n

_j

= g

_i

on Γ

_g

,

u

⁰_i

= 0 on Γ

_u

, 𝜎

⁰

ij

= ( a

_±

)

ijkl

e

_kl

( 𝐮

⁰

) in Ω

_±

, [

𝜎 𝜎 𝜎

⁰

i3

] = 𝟎 on S, 𝜎

𝜎 𝜎

⁰

i3

= 𝐊

³³

ik

[ u

⁰_k

]

on S.

(42)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 𝜎¹

ij,j=0 inΩ_±,

𝜎¹

ijn_j=0 onΓ_g,

u¹_i =0 onΓ_u,

𝜎¹

ij=𝜎¹

ij=(

a_±)

ijkle_kl(u¹) inΩ_±,

[𝜎𝜎𝜎¹

i3

]= −𝐊^3𝛼_ik[ u⁰_k]

,𝛼+<< 𝜎𝜎𝜎⁰

i3,3>> onS,

[u¹_i]

=( 𝐊³³)−1

ij

(

<< 𝜎𝜎𝜎¹

j3>>−𝐊^𝛼3_jk <<u⁰_k>>_,𝛼 )

+<<u⁰_i,3>> onS,

Taking into account the ﬁrst equation in (45) and using the positive deﬁniteness of the tensor 𝐛, one obtains:

which corresponds to the kinematics of the perfect interface.

Analogously, it is obtained:

and

The complete problem for hard interface can be written at the two ﬁrst orders as:

̂ (46) 𝐮

⁰

,3

= 0 ⇒ [ 𝐮 ̂

⁰

] = 𝟎,

[ 𝐮 ̂

¹

] = ( 𝐊

³³

)

⁻¹

(47) (

𝜎 ̂

𝜎 𝜎

⁰

𝐢

₃

− 𝐊

^𝛼3

𝐮 ̂

⁰

,𝛼

) ,

[ (48)

̂ 𝜎 𝜎 𝜎

¹

𝐢

₃

]

= (

− 𝐊

^𝛼𝛽

𝐮 ̂

⁰

,𝛽

− 𝐊

^3𝛼

[̂ 𝐮

¹

] )

,𝛼

= ( (49)

− 𝐊

^𝛼𝛽

𝐮 ̂

⁰

,𝛽

− 𝐊

^3𝛼

( 𝐊

³³

)

⁻¹

(

̂ 𝜎

𝜎 𝜎

⁰

𝐢

₃

− 𝐊

^𝛽3

𝐮 ̂

⁰

,𝛽

))

,𝛼

.

(50)

⎧ ⎪

⎪ ⎪

⎨ ⎪

⎪ ⎪

⎩ 𝜎

⁰

ij,j

+ f

_i

= 0 in Ω

_±

,

𝜎

⁰

ij

n

_j

= g

_i

on Γ

_g

,

u

⁰_i

= 0 on Γ

_u

, 𝜎

⁰

ij

= ( a

_±

)

ijkl

e

_kl

(u

⁰

) in Ω

_±

, [

𝜎 𝜎 𝜎

⁰

i3

] = 𝟎 on S, [ u

⁰_i

]

= 0 on S.

(51)

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

𝜎

¹_ij,j

= 0 in Ω

_±

,

𝜎

¹_ij

n

_j

= 0 on Γ

_g

,

u

¹_i

= 0 on Γ

_u

,

𝜎

¹_ij

= ( a

_±

)

ijkl

e

_kl

( 𝐮

¹

) in Ω

_±

,

[ 𝜎 𝜎 𝜎

¹_i3

]

= (

K

_ij^𝛼𝛽

u

⁰_j,_𝛽

+ (

𝐊

^3𝛼

( 𝐊

³³

)

⁻¹

)

ij

(

̂ 𝜎 𝜎 𝜎

⁰_j3

− 𝐊

^𝛽3

jk

u

⁰_k,_𝛽

))

,𝛼

+ << 𝜎 𝜎 𝜎

⁰_i3,3

>> on S, [ u

¹_i

]

= ( 𝐊

³³

)

⁻¹_ij

(

𝜎 𝜎 𝜎

⁰_j3

− 𝐊

^𝛼3

jk

u

⁰_k,𝛼

)

+ << u

⁰_i,3

>> on S.

2.4.2 Hard interphase analysis For a “hard” interphase, it is set:

where the tensor 𝐛 is independent of 𝜀, and 𝐊

^jl

is still taken to denote the matrices such that K

_ki^jl

: = b

_ijkl

.

Taking into account the relations (13) and (20), the stress-strain relationship takes the following form:

which gives 𝐛

^𝜀

= 𝐛 , (43)

(44) 𝜎 ̂

𝜎 𝜎

⁰

+ 𝜀 ̂ 𝜎 𝜎 𝜎

¹

= 𝐛 ( 𝜀

⁻¹

𝐞 ̂

⁻¹

+ 𝐞 ̂

⁰

+ 𝜀 ̂ 𝐞

¹

) + o ( 𝜀 ),

𝟎 = 𝐛 ( 𝐞 ̂

⁻¹

), (45) 𝜎 ̂

𝜎 𝜎

⁰

= 𝐛 ( 𝐞 ̂

⁰

).

3 How introduce roughness?

3.1 The three-dimensional equilibrium problem

In this Section, a general methodology is proposed to take into account the interphase roughness. In particular, two given positive-valued dimensionless roughness functions, 𝜂

^±

∈ C

⁰

( S , IR

²

), describing a smooth (i.e., 𝜂

^±

are independent of 𝜀 ) surface roughness between the adherents and the interphase, are introduced. The domains B

^𝜀

, Ω

^𝜀

±

and S

^𝜀

±

are

then deﬁned as:

(7)

and the interfaces between the adherents and the adhesive interphase are

We introduce two modiﬁed changes of variables:

Note that with these changes of variables, in the adhesive one has dx

₃

= 𝜂

^±

dz

₃

.

As a perfect contact law between the adhesive and the adherents is assumed, the continuity of the displacement and stress vector ﬁelds is enforced. In particular, the continuity of the displacements gives:

Expanding the displacement in the adherent, 𝐮

^𝜀

, in Taylor series along the x

₃

− direction and taking into account the asymptotic expansion, it results:

Thus it is obtained:

Identifying the terms with the same powers of 𝜀, Eq. (61) gives:

(52)

B^𝜀={

(x₁,x₂,x₃) ∈ Ω: − 𝜀

2𝜂⁻(x₁,x₂)<x₃ < 𝜀

2𝜂⁺(x₁,x₂)} ,

Ω

^𝜀_±

= { (53)

( x

₁

, x

₂

, x

₃

) ∈ Ω: ± x

₃

> 𝜀

2 𝜂

^±

( x

₁

, x

₂

) } ,

S

^𝜀

(54)

±

= { ( x

1

, x

2

, x

3

) ∈ Ω: x

3

= ± 𝜀 2 𝜂

^±

( x

1

, x

2

) } .

̂ (55)

𝐩

^𝜂

:(x

₁

, x

₂

, x

₃

) ↦ (z

₁

, z

₂

, z

₃

) = (

x

₁

, x

₂

, x

₃

𝜂

⁺

𝜀

) , x

₃

≥ 0,

̂ (56)

𝐩

^𝜂

:(x

₁

, x

₂

, x

₃

) ↦ (z

₁

, z

₂

, z

₃

) = (

x

₁

, x

₂

, x

₃

𝜂

⁻

𝜀

) , x

₃

≤ 0,

(57)

𝐩̄^𝜂:(x₁,x₂,x₃)↦(z₁,z₂,z₃) = (x₁,x₂,x₃±1∕2−𝜂⁺𝜀∕2),x₃≥0,

(58)

𝐩̄^𝜂:(x₁,x₂,x₃)↦(z₁,z₂,z₃) = (x₁,x₂,x₃±1∕2+𝜂⁻𝜀∕2),x₃≤0.

(59) 𝐮

^𝜀

(

̄ 𝐱 , ± 𝜂

^±

𝜀

2 )

= 𝐮 ̂

^𝜀

(

̄ 𝐳 , ± 1

2 )

= 𝐮 ̄

^𝜀

(

̄ 𝐳 , ± 1

2 )

.

(60)

𝐮^𝜀

( 𝐱,̄ ±𝜂^±𝜀

2 )

=𝐮^𝜀(̄𝐱, 0^±) ±𝜂^±𝜀 2 𝐮^𝜀

,3(𝐱, 0̄ ^±) +⋯

=𝐮⁰(𝐱̄, 0^±) +𝜀𝐮¹(𝐱̄, 0^±) ±𝜂^±𝜀 2 𝐮⁰

,3(𝐱̄, 0^±) +⋯

(61)

𝐮⁰(̄𝐱, 0^±) +𝜀𝐮¹(̄𝐱, 0^±) ±𝜂^±𝜀 2 𝐮⁰

,3

(𝐱, 0̄ ^±) +⋯=̂𝐮⁰

(

̄ 𝐳,±1

2 )

+𝜀 ̂𝐮¹ (

̄ 𝐳,±1

2 )

+⋯

=̄𝐮⁰ (

̄ 𝐳,±1

2 )

+𝜀 ̄𝐮¹ (

̄ 𝐳,±1

2 )

+⋯

(62) 𝐮

⁰

( 𝐱, 0 ̄

^±

) = 𝐮 ̂

⁰

(

̄ 𝐳, ± 1

2 )

= 𝐮 ̄

⁰

(

̄ 𝐳, ± 1

2 )

,

Following a similar analysis for the stress vector, analo- gous results are obtained:

for i = 1, 2, 3. Note that using this technique, the jump the stress vector at order zero is not modiﬁed. However, taking the derivatives along the third axis shows that the matching conditions at order one are modiﬁed as follows:

with <<< f >>> ( ⋅ ) = 1∕2 𝜂

⁺

f ( ⋅ , 0

⁺

) + 1∕2 𝜂

⁻

f ( ⋅ , 0

⁻

).

Now the equilibrium equation at order zero gives:

and thus,

At order one, the equilibrium equation leads to the condition

3.2 Soft interphase analysis

In this section, the interphase is assumed to be “soft”. The notation are the same as in Sect. 2. It is obvious to observe that Eq. (37) becomes

for j = 1, 2, 3, and

where 𝜂 = 𝜂

⁺

+ 𝜂

⁻

2 . Equation (72) describes an imperfect interface mode taking into account roughness.

Combining Eqs. (70) and (71), it is obtained:

(63) 𝐮

¹

( 𝐱 , 0

^±

) ± 1

2 𝜂

^±

𝐮

⁰

,3

( 𝐱 , 0

^±

) = ̂ 𝐮

¹

(

𝐳 , ± 1 2 )

= 𝐮

¹

(

𝐳 , ± 1 2 )

.

𝜎 (64) 𝜎 𝜎

⁰

i3

( 𝐱 ̄ , 0

^±

) = 𝜎 𝜎 𝜎 ̂

⁰

i3

( 𝐳 ̄ , ± 1

2 )

= 𝜎 𝜎 𝜎 ̄

⁰

i3

( 𝐳 ̄ , ± 1

2 )

,

(65) 𝜎 𝜎

𝜎

¹_i3

( 𝐱 ̄ , 0

^±

) ± 1

2 𝜂

^±

𝜎 𝜎 𝜎

⁰_i3,3

(̄ 𝐱 , 0

^±

) = 𝜎 𝜎 𝜎 ̂

¹_i3

(

𝐳 ̄ , ± 1 2 )

= 𝜎 𝜎 𝜎 ̄

¹_i3

(

𝐳 ̄ , ± 1 2 )

,

[ 𝐮

¹

]( 𝐱 ̄ ) = [ 𝐮 ̂

¹

]( 𝐱 ̄ )+ <<< 𝐮

⁰

(66)

,3

>>> ( 𝐱 ̄ ),

[ 𝜎 𝜎 𝜎

¹

𝐞

₃

]( 𝐱 ̄ ) = [ 𝜎 𝜎 𝜎 ̂

¹

𝐞

₃

]( 𝐱 ̄ )+ <<< 𝜎 𝜎 𝜎

⁰

(67)

,3

𝐞

₃

>>> ( 𝐱 ̄ ),

1 (68) 𝜂

^±

𝜎 𝜎 𝜎 ̂

⁰_i3,3

= 𝟎

[ (69)

̂ 𝜎 𝜎 𝜎

⁰

i3

]

= 0.

̂ (70) 𝜎 𝜎

𝜎

¹_i3,3

= − 𝜂

^±

𝜎 𝜎 𝜎 ̂

⁰_i_𝛼,𝛼

.

𝜂

^±

𝜎 𝜎 𝜎 ̂

⁰

𝐢

_j

= 𝐊

^3j

𝐮 ̂

⁰

(71)

,3

̂ (72) 𝜎 𝜎 𝜎

⁰

𝐢

₃

=

1 𝜂 𝐊

³³

[

̂

𝐮

⁰

]

(8)

The constitutive equation gives:

Being 𝜎 𝜎 𝜎 ̂

¹

and 𝐮 ̂

⁰

,𝛼

linear in z

₃

, we have [with j = 3 in Eq.

(74)]:

The complete problem for soft interface can be written at the two ﬁrst orders as:

(73) [ 𝜎 𝜎 𝜎 ̂

¹

𝐢

₃

]

= − 𝐊

³¹

[ 𝐮 ̂

⁰

]

,1

− 𝐊

³²

[ 𝐮 ̂

⁰

]

,2

̂ (74) 𝜎

𝜎 𝜎

¹

𝐢

_j

= 𝐊

^𝛼j

𝐮 ̂

⁰

,𝛼

+ 1 𝜂

^±

𝐊

³^j

𝐮 ̂

⁰

,3

[ (75)

̂ 𝐮

¹

]

= 𝜂 ( 𝐊

³³

)

−1

(

< ̂ 𝜎 𝜎 𝜎

¹

𝐢

₃

> − 𝐊

^𝛼3

< ̂ 𝐮

⁰

>

_,𝛼

)

(76)

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

𝜎

⁰_ij,j

+ f

_i

= 0 in Ω

_±

, 𝜎

⁰_ij

n

_j

= g

_i

on Γ

_g

, u

⁰_i

= 0 on Γ

_u

, 𝜎

⁰_ij

= (

a

_±

)

ijkl

e

_kl

( 𝐮

⁰

) in Ω

_±

, [

𝜎 𝜎 𝜎

⁰_i3

]

= 𝟎 on S, 𝜎

𝜎 𝜎

⁰_i3

= 1 𝜂

𝐊

³³

ik

[ u

⁰_k

]

on S.

The integration of the constitutive equations gives:

and by combining with the equilibrium equation one obtains

The complete problem for hard interface can be written at the two ﬁrst orders as:

4 How introduce microcracks?

A very simple technique is now proposed to account for the presence of microcracks in the interphase. The idea is to introduce these microcracks (or porosity) directly in the elastic constitutive equations. A heterogeneous material is considered made of an elastic material with cracks.

To model the macroscopic behavior of cracked materials, homogenization models can be used and then a new homogeneous equivalent medium can be deﬁned. The stiﬀness tensor depends on the crack length and orientation. In [11, 40, 41], the authors introduce cracks by applying the methodology developed by Kachanov and his collaborators [30, 48]. In this approach, a crack density 𝜌 is introduced. In two dimensions, this density is seen as the ratio between the square of the length of the crack l and the elementary volume describing the material at the microscale. Thus, the [̂ 𝐮

¹

] = 𝜂( 𝐊

³³

)

⁻¹

(79)

(

̂ 𝜎 𝜎

𝜎

⁰

𝐢

₃

− 𝐊

^𝛼3

𝐮 ̂

⁰

,𝛼

)

[ (80)

̂ 𝜎 𝜎 𝜎

¹

𝐢

₃

]

= (

− 𝐊

^𝛼𝛽

𝐮 ̂

⁰

,𝛽

− 𝐊

^3𝛼

[̂ 𝐮

¹

] )

,𝛼

= − 𝜂 (81) (

𝐊

^𝛼𝛽

𝐮 ̂

⁰

,𝛽

+ 𝐊

^𝛼3

( 𝐊

³³

)

⁻¹

(

𝜎 ̂

𝜎 𝜎

⁰

𝐢

₃

− 𝐊

^𝛼3

𝐮 ̂

⁰

,𝛼

))

,𝛼

(82)

⎧ ⎪

⎪ ⎪

⎨ ⎪

⎪ ⎪

⎩ 𝜎

⁰

ij,j

+ f

_i

= 0 in Ω

_±

,

𝜎

⁰

ij

n

_j

= g

_i

on Γ

_g

,

u

⁰_i

= 0 on Γ

_u

, 𝜎

⁰

ij

= ( a

_±

)

ijkl

e

_kl

( 𝐮

⁰

) in Ω

_±

, [

𝜎 𝜎 𝜎

⁰

i3

] = 𝟎 on S, [ u

⁰_i

]

= 0 on S.

(83)

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎩

𝜎

¹_ij,j

= 0 in Ω

_±

,

𝜎

¹_ij

n

_j

= 0 on Γ

_g

,

u

¹_i

= 0 on Γ

_u

,

𝜎

¹_ij

= ( a

_±

)

ijkl

e

_kl

(u

¹

) in Ω

_±

,

[ 𝜎 𝜎 𝜎

¹_i3

]

= 𝜂 (

K

_ij^𝛼𝛽

u

⁰_j,𝛽

+ (

𝐊

^3𝛼

( 𝐊

³³

)

⁻¹

)

ij

(

̂ 𝜎

𝜎 𝜎

⁰_j3

− 𝐊

^𝛽3_jk

u

⁰_k,𝛽

))

,𝛼

+ <<< 𝜎 𝜎 𝜎

⁰_i3,3

>>> on S, [ u

¹_i

]

= 𝜂( 𝐊

³³

)

⁻¹_ij

(

𝜎 𝜎 𝜎

⁰_j3

− 𝐊

^𝛼3_jk

u

⁰_k,𝛼

)

+ <<< u

⁰_i,3

>>> on S.

3.3 Hard interphase analysis

In this section, a “hard” interphase is considered. The jump in the displacements at order zero is not modiﬁed:

giving again the kinematics of the perfect interface.

(77)

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

𝜎_ij,j¹ =0 inΩ_±,

𝜎_ij¹n_j=0 onΓ_g,

u¹_i =0 onΓ_u,

𝜎_ij¹=( 𝐚±

)

ijkle_kl(u¹) inΩ_±,

[𝜎𝜎𝜎¹_i3]

= −𝐊^3𝛼

ik

[u⁰_k]

,𝛼+<<< 𝜎𝜎𝜎⁰_i3,3>>> onS,

[u¹_i]

=𝜂( 𝐊³³)−1

ij

(

<< 𝜎𝜎𝜎¹_j3>>−𝐊^𝛼3

jk <<u⁰_k>>_,𝛼 )

+<<<u⁰_i,3>>> onS.

̂ (78) 𝐮

⁰

,3

= 𝟎 ⇒ [ 𝐮 ̂

⁰

] = 𝟎,