An Asymptotic Approach to Thin Film Adhesion

A common approach to the modeling of very thin adhesive ﬁlms is their replacement with an equivalent contact law, prescribing the jumps in the displacement and traction vector ﬁelds at the limit interface as the ﬁlm thickness goes to zero. From the geometrical point of view, the adhesive ﬁlm is eliminated, although it is accounted for mechanically. Recently, we analyzed several cases to obtain the equivalent contact laws: soft ﬁlms [17, 18]; adhesive ﬁlms governed by a non convex energy [12]; linearly elastic adhesive ﬁlms having stiﬀness comparable with the adherents [19, 13]; imperfect adhesion between ﬂat adhesive ﬁlms and the adherents [14]; joints with mismatch strain between the adhesive and the adherents [16]. In this paper, we extend the results obtained in [13, 14] to the case of a thin curvilinear elastic anisotropic adhesive undergoing plane deformations. The asymptotic method proposed in [14] and based on the energy minimization is used. After obtaining the contact law in a general system of curvilinear coordinates, we analyze the gluing between two circular adherents (ﬁgure 2), a case of signiﬁcant importance for composite materials which often contain ﬁbers or particles.


Introduction
Interphase regions are thin layers of a third material between the composite constituents, fibers or particles and matrix, which are due to mechanical imperfections, fiber treatments, and various modifications of the matrix in the vicinity of the fibers.The presence of interphases influences the load transfers from the matrix to fibers and it can have a strong effect on the overall mechanical response of the composite.The existence of interphase regions of finite but small dimensions poses also several numerical problems: in particular, the number of degrees of freedom adopted in studies using a finite element approach can be very large, which affects the convergence and the accuracy of the solution.
A classical approach is to replace the three-phase problem of two constituents and an interphase perfectly adherent to them with a two-phase problem, in which the thin interphase is substituted by a mechanically equivalent interface of vanishing thickness ε [6,7,8,20,10,11,20,21,22,24,23,25].In particular, several studies were concerned with the derivation of the imperfect interface, modeling the effect of a thin interphase of known elastic properties [1,2,3,9,12,21,25].The method of the derivation can be based on different mathematical techniques: gamma-convergence, variational analysis, asymptotic expansions and numerical studies [15].Using these methods, it has been established that at the first order (ε → 0) flat stiff elastic interphases are equivalent to perfect interfaces, at which the stress and the displacement vectors are continuous.At a higher order (the second term in the expansion), an imperfect interface model is obtained, with a contact condition involving the first order displacement and traction vectors and their derivatives [1,13,14,19].The higher order term, giving rise to the imperfect interface model, can be interpreted as a correction of the leading solution corresponding to the perfect interface model.
In this paper, we extend the previous results to the case of a curved thin adhesive made of linear elastic anistropic material and undergoing plane deformations.Section 2 deals with the equilibrium problem of the interphase/adherents system.The asymptotic analysis is introduced in Section 3, where the existence of expansions in series of the displacements and stress vector fields in terms of the small parameter describing the thickness is assumed.Another assumption is that we can obtain the fields which are stationary points of the energy of the system by finding the stationary points of the energies obtained at each level in the expansion.This method provides two types of relations: either an interface relation or an equilibrium relation.In particular, at the order −1 we obtain the continuity of the displacement vector field at order zero; at the order 1 we obtain the continuity of the traction vector field at order zero and relation (54), which determines the jump at the interface on the displacement vector field at order one; at the order 2 we find relation (64), which prescribes the jump of traction at order one in terms of the displacement and traction vector fields at order zero.Relations (54, 64) are associated with an imperfect interface law, non-local in the sense that it involves tangential derivatives of the fields at order zero.
In Section 4 we specialize the imperfect interface law, given by relations (54, 64) and obtained in a general system of curvilinear coordinates, to the case of a circular inclusion bounded by a thin isotropic adhesive layer.
Figure 1: (a) Initial configuration with a thin interphase B ε of small thickness ε placed between two adherents; (b) limit configuration, where the interphase B ε is replaced by the interface S; (c) rescaled configuration with an interphase B of unit thickness placed between the adherents.

Statement of the problem
Consider a two-dimensional plane strain configuration in which a thin elastic adhesive interphase B ε ∈ R 2 of constant thickness ε separates two elastic adherents Ω ε + , Ω ε − ∈ R 2 , see figure 1.The two interfaces between the three region Ω ε + , Ω ε − and B ε will be denoted S ε + and S ε − .These interfaces are assumed to be perfect with the usual assumption that the displacement vector, u ε , and the stress vector, σ ε n, are continuous on S ε ± .It is the purpose here to substitute the thin interphase B ε by a curve S separating the adherents Ω ε + , Ω ε − , and to determine the interface conditions on the displacement and tractions across S equivalent to the three phase configuration with perfect interface conditions.
The three regions are anisotropic, homogeneous and linear elastic and let a ± denote the elasticity tensors of the regions Ω ε ± , and b the elasticity tensor of the interphase B ε .Denoted e the strain tensor linear elasticity gives the Cauchy stress tensor σ ε as follows: Let us introduce a rectangular system of Cartesian coordinates Ox 1 x 2 x 3 so that the plane x 3 = 0 coincides with the plane of the interphase.The elasticity tensors are assumed to satisfy the following assumptions: The equilibrium configurations of the three-phase composite body are the minimizers of the total energy in the space of kinematically admissible displacements where H(Ω ε ; R 3 ) is the space of the vector-valued functions on the set − , which are continuous and differentiable as many times as necessary.In view of the above regularity assumptions on a ± , b, f, g, the existence of a unique minimizer u ε in V ε is ensured [5,.
We assume that a parametric representation of S has been introduced in R 2 , x = φ(s 1 ), s 1 ∈ (a, b).We also assume that the curves S ε + and S ε − are parallel to one another and we introduce the following representation for B ε : where n is the unit normal vector to S pointing from Ω ε − to Ω ε + .It is convenient to work with the system of planar orthogonal curvilinear coordinates s 1 , s 2 .Let h ε 1 , h ε 2 denote the metric coefficient of the curvilinear coordinated system, and for the present parallel system we have In the curvilinear system, the displacement gradient takes the form: where the notation (a | b | c) indicates the matrix whose columns are the vectors a, b, c, and the vectors U i , i = 1, 2, are defined as follows: In (12,13) a comma is used to denote partial differentiation, and e 3 is the unit vector normal to the plane of the interphase.In curvilinear coordinates, we have where ± are the new domains of integration corresponding to Ω ε ± , after the change of variables from the Cartesian coordinates to the curvilinear ones, and h ε 1 is the Jacobian of the transformation; α ∈ I g := (α 1 , α 2 ) is the parameter of the parametric representation chosen for Γ g , and h ε s is the Jacobian of the surface transformation; U := (U 1 , U 2 ) is a 2 × 1 block matrix; B is a 2 × 2 block matrix whose blocks, B αβ , α, β = 1, 2, are the matrices with components defined by the relations In view of the symmetry properties of the elasticity tensor b, the matrices B αβ have the property that B βα = (B αβ ) T , α, β = 1, 2. Therefore, B is symmetric.

Asymptotic analysis
In the following, we sketch the asymptotic method based on the energy minimization which was introduced in [14].The first step of the method is a change of variables, in order to reformulate the equilibrium problem in an interphase domain independent of ε : After the change of variables, the interphase occupies the domain and the adherents occupy the domains Ω± = Ω ε ± ± 1/2(1 − ε)n .Let S ± denote the interfaces between the interphase and the two bodies after rescaling, and let Ω = Ω+ ∪ Ω− ∪ B ∪ S + ∪ S − denote the configuration of the composite body after the change of variables.Lastly, let Γu and Γg denote the shifts of Γ u and Γ g , respectively.
Let ũε ± = u ε • p−1 be the displacement from the adherents Ω ± adjacent to the rescaled interphase and let ūε = u ε • p−1 be the displacement from the rescaled interphase.In view of the condition of perfect interfaces between the adherents and the rescaled interphase, we have that ũε ± (z) = ūε (z), z ∈ S ± .Let f := f • p−1 and g := g• p−1 denote the rescaled external forces, and hε := the rescaled metric coefficients.With these assumptions and denoted with a tilde domains and fields associated with the adherents and with a bar domains and fields associated with the interphase, the rescaled energy takes the form where D := (a, b) × (−1/2, +1/2), and The rescaled equilibrium problem P ε can be formulated as follows: find the pair (ũ ε ± , ūε ) minimizing the energy (20) in the set of displacements The next step of the asymptotic method followed in the present paper is the expansion of the displacement vectors ũε ± , ūε and of the (rescaled) metric coefficients in power of ε : We also have Substituting the above expansions into the rescaled energy, we obtain 32) where In these relations, the functions Ū l , l = 0, 1, 2, . . ., are the terms of the expansion We note that the terms Ū l , l = 1, 2, . . ., do not enter the energy E −1 because the zero order term of hε 1,2 ( hε 1 ) −1 vanishes.The last step of the asymptotic method is based on the following main assumption: fields which are stationary points of the energy E ε can be obtained by finding the stationary points of the energies E l , l = −1, 0, 1, 2, . . . .The analysis of the stationary points of the energies at the various levels allows to identify the conditions between ū0 , ū1 , ū2 , . . .and the corresponding stress vectors arising from ũ0 ± , ũ1 ± , ũ2 ± , . . .at the interfaces S + , S − .

Stationary points of E −1
In view of (38), the energy E −1 reduces to to be minimized in the class of displacements ū0 ∈ H( B; R 3 ).Since b is a positive definite tensor, the matrix B 22 is also positive definite.Therefore, the energy E −1 is non negative and the minimizers have the property that ū0 ,2 = 0, a.e. in B, i. e. the minimizing displacements are independent of t 2 in the interphase.
Based on this result and on the initial assumption of perfect contact between the interphase and the adherents, we obtain the vanishing of the jump of ũ0 across B :

Stationary points of E 0
From (43) it follows that Ū −1 = 0, and thus the energy E 0 is independent of ū0 , ū1 .With a little abuse of notation, we drop the dependence of the these vector fields from the argument of E 0 , which becomes We seek the energy minimizer in the class of displacements ), t 1 ∈ (a, b), ũ± = 0 on Γu }.
(46) Using standard arguments, we obtain the equilibrium equations div(a ± (e(ũ 0 a ± (e(ũ 0 ± ))n = g on Γg , (48) Denoted σ0 n the traction vector, we also find that its jump across the rescaled curved interphase B vanishes: From the mechanical viewpoint, condition (44) and (50) give a perfect interface condition for the interphase modeling.

Stationary points of E 1
The vanishing of Ū −1 makes E 1 independent of ū2 .Again with a little abuse of notation, we drop the dependence of this vector field in the argument of E 1 , which simplifies as In view of equilibrium equations (47 ÷ 49), the Euler-Lagrange equations of E 1 take the form This relation together with the continuity condition (44) gives the following condition on the jump in the displacement vector field ū1 across the interphase Note that in view of the continuity of the displacement fields at the interfaces S ± , the latter condition can be rewritten in the equivalent form )). (54)

Conclusion
In this paper, we obtained the interface law mechanically equivalent to the effect of a curved thin elastic anisotropic adhesive interphase undergoing plane deformations.The derivation of the interface law was based on an energy approach proposed in [14] and based on two main assumptions: the possible existence of expansions in series in terms of the interphase thickness of the displacement and stress fields, and the assumption that the fields which are stationary points of the energy of the system can be obtained by finding the stationary points of the energies obtained at each level in the expansion.The main result is that at the order zero the interphase behaves as an ideal interface across which the displacements and tractions are continuous (see ( 44) and ( 50)).At the next order the interphase is modeled by an imperfect interface, whose contact law prescribes the jumps of the displacements and of the traction vector fields at order one in terms of the corresponding fields obtained at the order zero (see ( 54) and ( 64)).
As an example, the case of a linear elastic isotropic thin interphase bounding a circular inclusion was considered and the equivalent imperfect interface law was obtained.

Figure 2 :
Figure 2: Rescaled configuration of a circular interphase B of unit thickness placed between two adherents.