Keywords: thin film, elasticity, asymptotic analysis.

In [1,2,3,4,5,6] the mechanical behaviour of thin films between elastic adherents has been studied. The analysis is based on the classic idea that a very thin adhesive film can be replaced by a contact law. The contact law describes the asymptotic behaviour of the film in the limit as its thickness goes to zero and it prescribes the jumps in the displacement and traction vector fields at the limit interface. The formulation of the limit problem involves the mechanical and the geometrical properties of the adhesive and the adherents, and in [1,2,3,4,5,6] several cases were considered: soft films [1]; adhesive films governed by a non convex energy [2]; flat linear elastic films having stiffness comparable with that of the adherents and giving rise to imperfect adhesion between the films and the adherents [3,4]; joints with mismatched strain between the adhesive and the adherents [6]. Several mathematical techniques can be used to perform the asymptotic analysis: gamma-convergence, variational analysis, matched asymptotic expansions and numerical studies [5]. In this paper, new results extending those obtained in [4] to curvilinear films in two-dimensional elasticity are presented. The asymptotic method proposed in [4] and based on the energy minimization is used. After obtaining the contact law in a general system of curvilinear coordinates, the gluing between two circular adherents is analysed, a case of significant importance for composite materials which often contain fibres or particles.

1

F. Lebon, R. Rizzoni, S. Ronel-Idrissi, "Analysis of non-linear soft thin interfaces", Computers and Structures, 82, 1929-1938, 2004.

2

F. Lebon, R. Rizzoni, "Asymptotic study of a soft thin layer: the non convex case", Mechanics of Advanced Materials and Structures, 15(1), 12-20, 2008. doi:10.1080/15376490701410521

3

F. Lebon, R. Rizzoni, "Asymptotic analysis of a thin interface: the case involving similar rigidity", International Journal of Engineering Science, 48(5), 473-486, 2010. doi:10.1016/j.ijengsci.2009.12.001

4

F. Lebon, R. Rizzoni, "Asymptotic behavior of a hard thin interphase in linear elasticity: an energy approach", International Journal of Solids and Structures, 48, 441-449, 2011. doi:10.1016/j.ijsolstr.2010.10.006

5

F. Lebon, R. Rizzoni, "Modelling adhesion by asymptotic techniques", in K.A. Wilkinson, D.A. Ordonez, (Editors), "Adhesive Properties in Nanomaterials, Composites and Films", Nova Publisher, 2011.

6

F. Lebon, R. Rizzoni, "Asymptotic analysis of an elastic thin interphase with mismatch strain", European Journal of Mechanics, in press.

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