Asymptotic analysis of an elastic thin interphase with mismatch strain

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Asymptotic analysis of an elastic thin interphase with mismatch strain

Raffaella Rizzoni, Frédéric Lebon

To cite this version:

Raffaella Rizzoni, Frédéric Lebon. Asymptotic analysis of an elastic thin interphase with mismatch strain. European Journal of Mechanics - A/Solids, Elsevier, 2012, 36, pp.1-8.

�10.1016/j.euromechsol.2012.02.005�. �hal-00694059�

Asymptotic analysis of an adhesive joint with mismatch strain

Raffaella Rizzoni

^a,*

, Frédéric Lebon

^b

aDipartimentodiIngegneria,UniversitàdiFerrara,ViaSaragat1,44122Ferrara,Italy

bLaboratoiredeMécaniqueetd’Acoustique,UniversitéAix-Marseille,31CheminJoseph-Aiguier,13402MarseilleCedex20,France

This paper proposes the study of the equilibrium problem, where two elastic bodies are bonded to a thin elasticfilm under mismatch strain conditions resulting in a state of residual stress. The asymptotic behavior of thefilm/adherent system is modeled as thefilm thickness tends to zero, using a method based on asymptotic expansions and energy minimization procedures. This method yields a family of non-local imperfect interface laws, which define a jump in the displacement and the traction vector fields. The amplitudes of the jumps turn out to be correlated with the state of residual stress and the elastic properties of the materials. As an example, the interface law is calculated at order zero in the case of a pure homogeneous mismatch strain and a thin isotropicfilm consisting of Blatz-Ko material.

1. Introduction

A thin layer is afinefilm of material deposited onto or bonded to another material, called the adherent. The purpose of the thin layer is to endow the surface of the adherent with specific properties. At the same time it takes advantage of the massive properties of the adherent, such as its mechanical strength and/or thermal proper- ties (Park and Park, 1999; Fang and Lo, 2000; Resel, 2003). Thin films are used in microelectronic integrated circuits, magnetic information storage systems, optical coatings, wear resistant coat- ings, corrosion resistant coatings, etc. (see (Hu and Wang, 2006; Nie et al., 2006; Gan, 2008) and references therein). There is a large number of (physical, chemical, etc.) deposition processes. From a mechanical point of view, a thinfilm is a layer of material ranging in thickness from just a few fractions of a nanometer (in the case of a monolayer) to several micrometers. It is therefore characterized by at least one small dimension, which is much smaller than the other two.

Over the last few decades, many experiments have been per- formed and many mathematical models and numerical approaches have been developed to study the mechanical behavior of thinfilms (Huang and Rosakis, 2005; Mishnaevsky and Gross, 2005; Ngo et al., 2007; Cheng and Lee, 2008; Janssen et al., 2009; Pureza et al., 2009; Steigmann, 2009; Pureza et al., 2010; Xie and Fan, 2010). In particular, it has now been established that there are

residual stresses multilayered structures and that their presence plays an important role in the ability of these structures to with- stand external loads (Freund and Suresh, 2003). The aim of this study is to present a new method of modeling of thinfilms in the context of multilayered structures.

Many factors are known to influence the distribution of residual stresses, i.e., the mechanical properties of thefilms and adherents, mismatches between the thermal expansion offilms and those of the adherents, processing parameters such as the deposition rate, film thickness, adherent temperature and chamber pressure during deposition, andfilm/adherents adhesion characteristics. Here, we focus on the mechanical properties of both the film and the adherents and on the strain mismatch in thefilm, which can be of various origins (thermal, chemical, etc.). The aim is to apply asymptotic techniques to predict the response of thefilm/adherent system under conditions where a mismatch strain results in a state of residual stress. In particular, asymptotic techniques are used to calculate an interface law to substitute the thinfilm and to consider the mismatch strain and the elastic properties of thefilm/adherent system. Asymptotic techniques have been previously used by the authors to model the behavior of thinfilms with a similar rigidity to that of the adherents (Lebon and Rizzoni, 2010, 2011a), to study soft thinfilms (Lebon et al., 2004, 1997) and thin adhesives governed by a non convex energy (Lebon and Rizzoni, 2008), and to analyze imperfect adhesion between adhesive and adherents (Lebon and Zaittouni, 2010).

The approach presented in (Paroni and Rizzoni, 2005) and recalled in Section2is used to model the effects of the mismatch strain. This strain is viewed as afinite deformation f₀³ from the natural configuration of a thin interphase to a stressed

*Corresponding author.

E-mail addresses:[email protected],[email protected](R. Rizzoni), [email protected](F. Lebon).

configuration, which is compatible with that of the adherents and maintained by external constraints. The stressed interphase is then bonded to the adherents, the constraints are released and thefilm/

adherent system deforms. The interphase and the adherents are assumed to be hyperelastic and the deformation of the film/

adherent system, f, is assumed to be infinitesimal. The latter assumption appears to be justified whenfilm strain is not changed appreciably as a result of the deformation of the adherents. Within these and further regularizing assumptions, the equilibrium problem is expressed as the minimization of the total energy given by the sum of two terms: the deformation energy of the interphase after the infinitesimal deformationfhas been superimposed on the state of strain f₀³; and the deformation energy of the adherents undergoing the small strainf. Bothf₀³ and the deformation energy in thefilm can depend on the thickness 3 of the interphase.

In Section3, the results obtained in (Paroni, 2006) are recal- led, which provide suitable conditions for ensuring the existence of at least one minimizer of the total energy. Section 4 deals with the asymptotic analysis. In particular, the asymptotic expansion method is introduced and a minimization strategy is used to obtain interface conditions. The strategy presented in (Lebon and Rizzoni, 2011a) is based on two assumptions: first, we assume the existence of expansions in series of the displacement and residual stress vector fields in terms of the small parameter describing the thickness; next, we assume that we can obtain the fields which are stationary points of the three dimensional energy byfinding the stationary points of the energies obtained

at the various levels of the expansion. At order zero, this method yields a continuous displacement vectorfield at the interface and a jump in the traction vectorfield; the amplitude of the jump is directly related to the residual stress and thefilm elastic prop- erties (cfr. (50)). At higher orders, non-local imperfect laws defining the jumps in both the displacement and traction vector fields are obtained.

For the sake of completeness, an example is presented in the last section of the paper. The interface law calculated at order zero is restricted to the simple case of a pure homogeneous mismatch strain and an isotropicfilm consisting of Blatz-Ko material.

2. Three-dimensional energy of a mismatch strained interphase

Let us consider a thin interphase with a reference configuration B₀³ placed between two bodiesU³ 3R³, as depicted inFig. 1a. The configurationB₀³ is a natural state for the interphase, and there is a mismatch strain between the two bodies. We assume there is a deformation f₀³ from B₀³ to a configuration B³3R³ that is compatible with U³ , is assumed to exist. Some constraints are assumed to act onB³to maintain the mismatch strainf₀³ prescribed.

The strained interphase is then assumed to be brought into contact with the two bodies and bonded to them (Fig. 1b). After the bonding, the constraints are released and the joint deforms. Let f³:B³ 1R³describe its deformation (Fig. 1c).

B

.

^X

.

^x

+

B

.

^z 1

f0 f , u

p

x3

x 0

z3

z 0

+

B

p^-1

u ,u±

a b c

d

Fig. 1.(a) natural state of a thin interphase having a mismatch strain between two bodies; (b) the strained configuration of the thin interphase just before it is bonded to the two bodies; (c) thefinal deformed configuration of the joint; (d) the rescaled configuration of the interphase.

LetXand xindicate points in B₀³ and inB³, respectively. The deformationf³ is assumed to be infinitesimal, i.e.

f³ðxÞ ¼ xþu³ðxÞ ¼ f₀³ðXÞ þu³ f₀³ðXÞ

(1) whereu³:B³ 1R³ is a small displacement superimposed on the state of strainf₀³. The interphase is assumed to be elastic, and the strain energy density per unit reference volumedXis written as w³ðVf ³ðXÞÞ ¼ w³

Vf₀³ðXÞ þVu³ðxÞVf₀³ðXÞ

: (2)

To simplify the notation we writeF³,F₀³ instead ofVf³,Vf₀³;and we omit the variablesX,x. Take the expansion

w³ðF³Þ ¼w³ F₀³

þw_F³ F₀³

$Vu³F₀³ þ1 2w_{F F}³

F₀³ Vu³F₀³

$Vu³F₀³ þo

jVu³j²

; (3)

where the indexFdenotes differentiation and the dot indicates the scalar product between tensors. Without any loss of generality, we setw³ðF₀³Þ ¼0. The displacement gradientVu³ is assumed to be infinitesimal, so that all the terms of an order higher than one in Vu³ can be neglected. All the subsequent formulae are valid only up to an error which tends to zero,jVu³j². Using the tensor product ðA@BÞH ¼A H B^T(Del Piero, 1979), and introducing the tensors T₀³ :¼

detF₀³1

w_F³ F₀³

F₀^3T (4)

b₀³ :¼

detF₀³1 I@F₀³

w_{F F}³ F₀³

I@F₀^3T

(5)

defining the Cauchy stress tensor and a fourth order elastic tensor, we obtain

w³ðF³Þ ¼ detF₀³

T₀³$Vu³ þ1

2b₀³ ðVu³Þ$Vu³

: (6)

After integrating over the domain B³, we obtain the total deformation energy of the interphase subjected to the displace- mentu³:

Z

B³

T₀³$Vu³ þ1

2b₀³ ðVu³Þ$Vu³

dx: (7)

In (Del Piero, 1979; Del Piero and Rizzoni, 2008) it is proved that ifw³ is frame-indifferent, thenb₀³ can be decomposed into

b₀³ ¼ c₀³ þI@T₀³; (8)

whereIis the second-order identity tensor andc₀³ is a tensor with two minor symmetries. Using these symmetries, the deformation energy of the interphase takes the form

Z

B³

T₀³$eðu³Þ þ1

2Vu³T₀³$Vu³ þ1

2c₀³ ðeðu³ÞÞ$eðu³Þ

dx; (9)

whereeðu³Þ ¼ 1=2ðVu³ þ ðVu³Þ^TÞis the symmetric part ofVu³.

3. Statement of the three dimensional problem

LetS3R²be an open bounded set with a Lipschitz boundary. In the rest of the paper, we takeB³ to be restricted to the cylindrical domain with a constant cross-section S and a constant small thickness 3 1. Let S³ denote the flat interfaces between the interphase and the two bodiesU³ and letU^{3 ¼}U³ WS³ WB³ denote the composite comprising the interphase and the two bodies. Let (O,x1,x2, x3) denote an orthogonal frame with its origin at the

center of the interphase midplane and with the x₃-axis perpen- dicular to the interfacesS³ (Fig. 1b).

Letu³ :U³1R³be the displacement vectorfield fromU³, i.e., the displacement undergone by the adherents after being bonded to the strained interphase. For the continuity of the displacementfield across the surfacesS³ , it is necessary that

½u³ ¼ 0 on S³ (10)

where

½u³:¼ u³ x₁;x₂;

³ 2

þ u³

x₁;x₂; ³

2

(11)

stands for the jumps in the displacement acrossS³ . In(11),u³(x1,x2, (3/2)^þ) (resp.u³(x1,x2, (3/2)) indicates the limit ofu³(x1,x2,x3) asx3

tends to (3/2) andx₃(3/2) (resp.x₃(3/2)).

The adherents are assumed to be homogeneous and linear elastic with elasticity tensorsa. A body force density,f, is applied to U³ and a surface force density, g, to Gg3vU³. Homogeneous boundary conditions are prescribed onGu ¼vU³=ðGg=ðvU³XvB³ÞÞ:

u³ ¼ 0 on

G

u: (12)

We also make the following assumptions:

H1 8>

>>

>>

<

>>

>>

>:

T₀³;c₀³ ˛L^NðB³Þ;a˛L^N

U

³

;

ðaÞ_ijkl ¼ ðaÞ_klij ¼ ðaÞ_jilk ¼ ðaÞ_ijlk; c₀³

ijkl ¼ c₀³

klij ¼ c₀³

jilk ¼ c₀³

ijlk; d

h

_;

h

₀³ _>0:aðeÞ$e

h

_e$e;

c₀³ ðxÞðeÞ$e

h

₀³ e$e ce˛R⁹;e ¼ e^T;x˛B³; H2

d³0:B³X

G

_gWsuppð

f

Þ

¼ B; c³ < 30; H3

f

_˛L²ð

U

^Þ;

g

_˛L²

G

_g:

Assumption (H1) concerns the usual symmetry properties and positive definiteness hypothesis of the elasticity tensors. Assump- tion (H2) means thatGgis located far from the interphase. In (H3), thefields of the external forces are taken to have sufficient regu- larity to ensure the existence of equilibrium configurations.

The equilibrium configurations of the composite body are the minimizers of the total energy

E³ðu³Þ ¼ Z U³

1

2aðeðu³ÞÞ$eðu³Þ

f

$u³

dx Z Gg

g

$u³dsx

þ Z

B³

T₀³$eðu³Þ þ1

2Vu³T₀³$Vu³ þ1

2c₀³ ðeðu³ÞÞ$eðu³Þ

dx

(13) in the space of kinematically admissible displacements

V³ ¼ u˛H

U

³:u¼ 0 on

G

u : (14) HereH(U³) is the space of vector-valued functions in the setU³ which are continuous and differentiable as many times as necessary.

Based on a result proved inParoni (2006, Thm. 3.2), the coer- civity of the energyE³ (and hence the existence result, via the Lax- Milgram theorem) holds if

h

₀³ >c_k

s

₀³

32; (15)

where c_k is a constant which is independent of 3 ands₀^{3 is the} essential sup of the minimum eigenvalue ofT₀³

s

₀³ :¼ essinf_x˛B3

min

a˛R³

T₀³ðxÞa$a : (16)

As pointed out in Paroni (2006), condition (16) requires“the compression due to the residual stress”not be too large. Note that if the stress tensorT₀³ is non-negative definite, thens0³ will vanish and, in view of (15), coercivity simply follows from the positive defi- niteness ofh₀³.

4. Asymptotic analysis

In this Section, the asymptotic expansion method is used to obtain the interface conditions giving the effects of the interphase on the mechanical behavior of thefilm/adherent systemU^{3 (Lebon} and Rizzoni, 2010, 2011a,b). In order to reformulate the equilibrium problem in a domain which is independent of 3, the following change of variables is made:

ð^z;z₃Þ ¼ pð^x;x₃Þ ¼ 8<

:

^x;x3³1

ð^x;x3Þ˛B³;

^x;x₃H2³ 1 2

ð^x;x3Þ˛

U

³ ; (17)

where^x¼ ðx1;x₂Þ;^z¼ ðz1;z₂Þ. With this change of variables,B³ is rescaled by a factor 31 along the interphase thickness and the bodies U³ are translated 1/2(13) in the same direction, as depicted inFig. 1d. In the new coordinate system, the interphase occupies the domain

B ¼

ðz₁;z₂;z₃Þ˛R³:ðz₁;z2Þ˛S;jz₃j<1 2

; (18)

and the adherents occupy the domainsU ¼ U³ 1=2ð1 3Þe₃, where e3 denotes the unit vector along the z3-axis. Let S ¼ fð^z;z₃Þ˛R³: ^z˛S;z₃ ¼ 1=2gdenote the interfaces between the interphase and the adherents after rescaling, and let U ¼UþWUWBWSþWS denote the configuration of the composite body after the change of variables. Lastly, letGu andGg denote the translations ofGuandGg, respectively. We take

~

u³ ð^z;z₃Þ:¼ u³+p¹

ð^z;z₃Þ; ð^z;z₃Þ˛

U

; (19) to denote the displacement in the adherents adjacent to the rescaled interphase, and

u³ð^z;z3Þ:¼ u³+p¹

ð^z;z₃Þ; ð^z;z₃Þ˛B; (20) to denote the displacement in the rescaled interphase. Continuity of the displacements at the interfacesS³ entails that

u~³

^z;1 2

¼ u³

^z;1 2

; ^z˛S: (21)

Note also that in view of the change of variables, we can write u³

^x; ³ 2

¼ u~³

^z;1 2

; ^x;^z˛S: (22)

u³

^x; ³ 2

¼ u³

z₁;z₂;1 2

; ^x;^z˛S: (23)

Letf~_:¼f₊p¹ and g~_:¼g₊p¹ denote the rescaled external forces and let~T₀³;~c₀³ denote the rescaled Cauchy stress tensor and the rescaled elastic tensor, respectively.

We rewrite the assumption (H2) as follows:

H2⁰Þ BX

G

_gWsupp

f

~ ¼ B: (24)

To rewrite the energy in an appropriate form, we introduce the notation

t₀^3;i

j:¼

~T₀³

ij; i;j ¼ 1;2;3;

K₀^3;ij

kl:¼

~ c₀³

kiljþ T~₀³

ij

d

_kl; (25)

wheredklis the Kronecker delta. In view of the symmetry of~c₀³ and ofT~₀³;we have the property

K₀^3;ij_T

¼ K₀^3;ji: (26)

After changing the variables and using the definitions(25)and (26), we express the energy(13)over thefixed domainU^: E³ðu³Þ ¼

Z U

1 2a

e

~ u³

$e

~ u³

f

~$~u³

dz

Z Gg

~

g

$~u³ dszþ Z

B

3t₀^3;a$u_;a³ þt₀^3;3$u_;3³

dz

þ Z

B

3

2K₀^3;ab

u_;a³

$u_;b³ þK₀^3;a3

u_;a³ $u_;3³

þ1 23K₀^3;33

u_;3³

$u_;3³

dz ¼:E³ u³;~u³

(27)

The standard summation convention over repeated indices is used throughout this paper and the indexesa,brange from 1 to 2.

With the assumptions (H1)O(H3) and (H2⁰), we minimize the rescaled energy(27)in the following set of displacements:

V ¼n u³;~u³

˛HðBÞ Hð

U

þÞ Hð

U

Þ: u~³ ¼ u³ onS;~u³ ¼ 0 on

G

u

o: (28)

Since we are looking for the behavior of the minimizers of(13) when the interphase thickness 3 is small, we assume that the minimizing displacements can be expressed as the sum of the series

~

u³ ¼ u~⁰þ 3~u¹þ 32~u²þo

32

; (29)

u³ ¼ u⁰þ 3u¹þ 32u²þo

32

; (30)

where the displacement vectors~u⁰;u~¹;.are independent of 3. We also assume that the Cauchy stress~T₀³ and the elastic tensor~c₀³ can be expressed as the sum of the series:

T~₀³ ¼ 1

3 T₀¹þT₀⁰þ 3T₀¹þ 32T₀²þo

32

;

~ c₀³ ¼ 1

3 c¹₀ þc⁰₀þ 3c¹₀þ 32c²₀þo

32

;

(31)

and set

K₀^I;ij

kl:¼

c^I₀

kiljþ T₀^I

ij

d

_kl; t₀^I;i

j:¼ T₀^I

ij; (32)

whereI¼ 1, 0, 1, . andi,j¼1, 2, 3. We then substitute the expansions(29)and(30)into(27)to obtain

E³ u³;u~³

¼ 1

32E²ðu⁰Þ þ1

3E¹ðu⁰;u¹Þ þ E⁰

u⁰;u¹;~u⁰

þ 3E¹

u⁰;.;u³;~u⁰;~u¹ þ 32E²

u⁰;.;u⁴;u~⁰;.;u~² þo

32

;

(33)

where the energiesE^I,I¼ 2,1, 0, 1, 2 are independent of 3 and are defined as follows:

Let us now consider the minimization of each of these ener- gies. The function class in which we look for the solution of each energy minimization problem is the natural class of displace- ments with finite energy; kinematic constraints are added whenever they follow from the minimization of the energies at lower orders. In the following analysis, we also make the simplifying assumption

t₀^1;3 ¼ 0 (39)

i.e., at the lowest order, plane stress state is assumed to exist.

4.1. Minimization ofE²

Since~c₀³ is a positive definite tensor and t¹₀ ^;3 is assumed to vanish, the tensorK^1;33₀ is also positive definite. The energyE²is therefore non negative and the minimizers are independent ofz3: u⁰_i;3 ¼ 0; i ¼ 1;2;3; a:e: in B: (40)

In view of the continuity conditions(21), (22)and(23), we also have

~ u⁰_þ

^z;þ1 2

¼ ~u⁰

^z;1 2

; ^z˛S; (41)

u⁰

^x;0^þ

¼ u⁰

^x;0

; ^x˛S: (42)

4.2. Minimization ofE¹

Based on(40), we obtain that the energyE¹vanishes.

4.3. Minimization ofE⁰

In view of (39) and (40), the energy E⁰ simplifies as follows:

E²ðu⁰Þ:¼ Z

B

1 2

K₀^1;33 u⁰_;3

$u⁰_;3

dz; (34)

E¹ðu⁰;u¹Þ:¼ Z

B

0

@t₀^1;3$u⁰_;3þK₀^1;a3 u⁰_;a

$u⁰_;3þ1 2

X⁰

I¼ 1

X^I

J¼0

K₀^I;33 u^ðIJÞ_;3

$u^J_;3 1

Adz; (35)

E⁰

u⁰;u¹;u²;u~⁰

:¼ Z U

1 2a

e

~u⁰

$e

~u⁰

f

~$~u⁰

dz Z G~g

~

g

$u~⁰dszþ Z

B

t^1;₀ ^a$u⁰_;aþ1 2K₀^1;ab

u⁰_;a

$u⁰_;bþ X⁰

I¼ 1

t₀^I;3$u^ðIÞ_;3

! dz

þ Z

B

0

@ X⁰

I¼ 1

X^I

J¼0

K₀^I;a3 u^ðIJÞ_;a

$u^J_;3þ1 2

X¹

I¼ 1

X^1I

J¼0

K₀^I;33

u^ð1IJÞ_;3

$u^J_;3 1

Adz; (36)

E¹

u⁰;u¹;u²;u³;~u⁰;u~¹

:¼ Z U

a

e

u~⁰

$e

u~¹

f

~$u~¹

dz Z G~g

~

g

$~u¹dszþ

Z

B

0

@X⁰

I¼1

t₀^I;a$u^ðIÞ_;a þX¹

I¼1

t^I;3₀ $u^ð1IÞ_;3

þ1 2

X⁰

I¼1

X^I

J¼0

K₀^I;ab u^ðIJÞ_;a

$u^J_;b 1 Adzþ

Z

B

0

@X¹

I¼1

X^1I

J¼0

K₀^I;a3

u^ð1IJÞ_;a

$u^J_;3þ1 2

X²

I¼1

X^2I

J¼0

K₀^I;33

u^ð2IJÞ_;3

$u^J_;3 1 Adz;

(37) E²

u⁰;u¹;u²;u³;u⁴;~u⁰;u~¹;u~²

:¼ Z U

a

e

~u⁰

$e

~u²

f

~_$~u²

dz Z G~g

~

g

_$~u²dszþ Z U

1 2a

e

u~¹

$e

u~¹

dz

þ Z

B

X¹

I¼ 1

t₀^I;a$u^ð1IÞ;a þ X²

I¼ 1

t₀^I;3$u^ð2IÞ_;3

! dzþ

Z

B

0

@1 2

X¹

I¼ 1

X^1I

J¼0

K₀^I;ab

u^ð1IJÞ_;a

$u^J_;b

þX²

I¼ 1

X^2I

J¼0

K₀^I;a3

u^ð2IJÞ_;a

$u_;3^J 1 Adzþ

Z

B

0

@1 2

X³

I¼ 1

X^3I

J¼0

K₀^I;33

u^ð3IJÞ_;3

$u^J_;3 1

Adz: (38)

and becomes independent of u². We seek the minimizers of this energy in the class of displacements

V₀ ¼

u⁰;u¹;u~⁰

˛HðBÞ HðBÞ Hð

U

Þ: ~u⁰

^z;1 2

¼ u⁰

^z;1 2

; ^z˛S; u_;3⁰ðzÞ ¼ 0; z˛B; ~u⁰ðzÞ ¼ 0; z˛

G

~u

: (44) By minimizing(43)with respect tou¹;we obtain

u¹_;3 ¼

K₀^1;331

K₀^1;a3 u⁰_;a

; (45)

which, after integration with respect toz₃, gives

½u¹ ¼ <

K₀^1;331

K₀^1;a3>

u⁰_;a

: (46)

Here<$>denotes the average across the thickness ofB. After substituting(46)back into thefirst variation of(43), it turns out that the minimizers of(43)have to satisfy the equilibrium equations div

a

e

~ u⁰

þ

f

~ ¼ 0 in

U

(47) a

e

u~⁰

n ¼ ~

g

on

G

~g; (48)

a

e

u~⁰

n ¼ 0 on v

U

=

G

~g; (49)

and the following jump condition h

s

~⁰ⁱn ¼

<t₀^1;a>þK^^{1;b a}

0

u⁰_;b

;a (50)

where½~s⁰:¼ ~s⁰_þs~⁰_; ~s⁰_:¼aðeð~u⁰ÞÞand K^^1;₀ ^{b a}:¼<K₀^{1;b a}><

K₀^1;a3_T

K₀^1;33₁

K₀^1;b3>: (51)

Condition(50)states that the jump in the traction vector across the rescaled interphaseBdoes not vanish. Note that conditions(41) and (50) are mixed transmission conditions involving both the displacement vector at order zero and its derivatives and the lower order terms in the expansions ofT~oand~c₀³:

4.4. Minimization ofE¹

In view of(39), (40), (45), (47)e(50), the energyE¹turns out to depend only onu⁰and can be thereforeE¹regarded as a constant term.

4.5. Minimization ofE²

In view of(39), (40), (45), (47)e(50), the energyE²simplifies into the sum of a term depending only onu₀;plus the energy E²⁰

u¹;u²;~u¹

:¼ Z U

1 2a

e

~ u¹

$e

~ u¹

dz

þ Z

B

t₀^0;aþK~^0;ba

u⁰_;b þ1

2K₀^1;ba

u¹_;b

$u¹_;adz

þ Z

B

t^0;3₀ þK~^0;a3 u⁰_;a

þK₀^1;a3 u¹_;a

þ1 2K₀^1;33

u²_;3

$u²_;3dz; (52) where

K~^0;ba:¼ K^0;₀^baK₀^0;3a

K₀^1;331

K₀^1;b3; (53) K~^0;a3:¼ K₀^0;a3K₀^0;33

K^1;33₀ ¹

K₀^1;a3: (54)

The energy term depending only on u₀ can be taken to be a constant term because u₀ is completely determined by the minimization of the energyE⁰(cfr.(41), (47)e(50)).

The energyE²⁰can be simplified by noting that in view of(46), the vectorfieldu¹can be written in the form:

u¹ð^z;z3Þ ¼ ½u¹ð^zÞz₃þ1 2S

u~¹

ð^zÞ (55)

whereSð~u¹Þð^zÞ:¼~u¹ð^z;ð1=2Þ^þÞ þ~u¹ð^z;ð1=2ÞÞ. Substituting(55) and(46)into the expression forE²⁰and integrating with respect to z3, we eliminate the dependence onu¹and reduce the minimiza- tion problem ofE²⁰to the minimization of the following energy E⁰

u⁰;u¹;u²;~u⁰

¼ Z U

1 2a

e

~ u⁰

$e

~ u⁰

f

~$~u⁰

dz

Z G~g

~

g

$~u⁰ds_zþ Z

B

t^1;₀ ^a$u⁰_;aþ1 2K₀^1;ab

u⁰_;a

$u⁰_;b

þK₀^1;a3 u⁰_;a

$u¹_;3þ1 2K₀^1;33

u¹_;3

$u¹_;3

dz (43)

E²⁰

u²;~u¹

:¼ Z U

1 2a

e

u~¹

$e

u~¹

dzþ Z

S

<

t^0;₀^aþK~^0;ba u⁰_;b

>$

1 2S

~u¹

;a

dsz

Z

S

<

z₃K₀^1;ba

>

K₀^1;33₁ K₀^1;g3

u⁰_;g

;b$ 1

2S

~ u¹

;a

ds_zþ Z

S

<K₀^1;ba>

1 2S

~ u¹

;b

$ 1

2S

~ u¹

;a

þ<

K₀^1;3a

>

u²_;3

$1 2S

~ u¹

;a

ds_zþ Z

B

t^0;3₀ þK~^0;a3 u⁰_;a

þ1

2K₀^1;a3S

~ u¹

;a

$u²_;3dz

Z

B

K₀^1;a3

<

K₀^1;331

K₀^1;g3>

u⁰_;g

;az₃

$u²_;3dz: (56)

The energy E²⁰⁰ is minimized in the following class of displacements:

V₁ ¼

u²;u~¹

˛HðBÞ Hð

U

Þ: ~u¹ðzÞ ¼ 0; z˛

G

~u

: (57)

After minimizing with respect tou²;we obtain u²_;3 ¼

K₀^1;331

t₀^0;3þK~^0;a3 u⁰_;a

þ1

2K₀^1;a3S

~u¹

;a K₀^1;a3

<

K₀^1;331

K₀^1;g3>

u⁰_;g

;az₃

: (58)

The remaining Euler-Lagrange equations give the equilibrium equations

div

a

e

~ u¹

¼ 0 in

U

; (59)

a

e

~u¹

n¼ 0 on v

U

; (60)

and the following jump condition h~

s

¹ⁱe₃ ¼

<

t₀^0;aþK~^0;ba u⁰_;b

>2<

K₀^1;ba

>

S

~u¹

;b

;a

<

z₃K₀^1;ba

>

K₀^1;33₁ K₀^1;g3

u⁰_;g

;b

;a þ

<

K₀^1;3a

>

K₀^1;331

t₀^0;3þK~^0;a3 u⁰_;a þK₀^1;a3

K₀^1;331 K₀^1;g3

u⁰_;g

;a

;a (61)

where½~s¹:¼~s¹_þ~s¹ands~¹_:¼aðeð~u¹ÞÞ.

5. An example of the interface law in the case of a thin isotropic interphase

In this section, the general result obtained(50)will be applied to the simple special case of a mismatch strain which is homogeneous and independent of 3 and an isotropic interphase with deformation energy scaling like 31. In particular, the strained configuration is taken to beB³ related toB₀³ by a pure homogeneous strain with the deformation gradient

F₀ ¼ X³

i¼1

l

_iðe_i5eiÞ; (62)

where e1, e2, e3 are the directions parallel to axes 1, 2 and 3, respectively, andli,i ¼1, 2, 3, are constant stretches which are independent of 3. As an example, we take afilm of Blatz-Ko material (Holzapfel, 2000)

w³ðFÞ ¼

a

³ 1

2ðF$F3Þ þ

g

¹ðdetFÞ¹1

; (63)

where

a

³ ¼ A

3 (64)

andA,gare positive constants. InDel Piero and Rizzoni (2008)it was established that the Cauchy stress and the elastic tensorc₀³ (see (8)) can be written as

T₀³ðF₀Þ ¼ A

3 ðdetF₀Þ¹

F₀F₀^T ðdetF₀Þ^gI

; (65)

c₀³ ðF₀Þ ¼ A

3 ðdetF₀Þ^1gð2Sþ

g

I5IÞ; (66)

whereSis the symmetry mapping on the set of all second-order tensors (Del Piero, 1979). ForF0as in(62), we obtain

T₀³ ¼ A

3

D

X³

i¼1

l

²_i

D

^gðei5e_iÞ; (67)

c₀³ ¼ A

3

D

^1þgð2Sþ

g

I5I;Þ (68)

whereD^:¼l1l2l3. We take the stretchesl1,l2,l3such that

l

₃ ¼

D

^g; (69)

l

₁

l

₂

l

₃>0; (70)

l

₂>max (

l

2ð1þgÞ g

1 ;

l

2ð1þggÞ 1

)

¼ 8>

><

>>

:

l

2ð1þgÞ g

1 ;

g

2;

l

2ð1þggÞ

1 ;

g

_>2:

(71)

With this choice, the condition (39) is satisfied and the Cauchy stress T₀³ turns out to be non-negative definite. As dis- cussed in Section3, the existence of energy minimizers of(13) is therefore guaranteed, provided that the smallest eigenvalue ofc₀³

h

₀³ ¼ 2A

3

D

^{1þg (72)}

is positive. The positivity of h₀³ follows from (70) and from the positivity ofA.

Let us now look at the form taken by(50). From(25), K₀^1;aa ¼ A

D

^1þg

g

ea5eaþ

1þ

l

²_a

D

^gI;

a

¼ 1;2; (73)

K₀^1;33 ¼ A

D

^1þgð

g

e₃5e3þ2IÞ; (74) K₀^1;ij ¼ A

D

^1þg

2e_i5e_jþ

g

e_j5e_i

; isj;i;j ¼ 1;2;3; (75) and from(51),

K^^1;11₀ ¼ A

D

^1þg

2þ3

g

2þ

g

e₁5e1þe₂5e2e₃5e3þ

l

²₁

D

^gI; (76) K^^1;12₀ ¼ A

D

^1þg

2e₁5e2þ 2

g

2þ

g

e₂5e1

; (77)

K^^1;21₀ ¼ A

D

^1þg

2

g

2þ

g

e₁5e2þ2e₂5e1

; (78)

K^^1;22₀ ¼ A

D

^1þg

2þ3

g

2þ

g

e₂5e₂þe₁5e₁e₃5e₃þ

l

²₂

D

^gI: (79) The interface law(50)therefore takes the form