Modelling Adhesion by Asymptotic Techniques

In this chapter, a review of theoretical and numerical asymptotic studies on thin adhesive layers is proposed. A general mathematical method is presented for modelling the mechanical behavior of bonding and interfaces. This method is based on a simple idea that the adhesive film is supposed to be very thin; the mechanical problem depends strongly on the thinness of the adhesive. It is quite natural, mathematically and mechanically, to consider the limit problem, that is, the asymptotic problem obtained when the thickness and, possibly, the mechanical characteristics of the adhesive thin layer tend to zero. This asymptotic analysis leads to a limit problem with a mechanical constraint on the surface, to which the layer shrinks. The formulation of the limit problem includes the mechanical and geometrical properties of the layer. This limit problem is usually easier to solve numerically by using finite elements software. Theoretical results (i.e. limit problems) can be usually obtained by using at least four mathematical techniques: gamma-convergence, variational analysis, asymptotic expansions and numerical studies. In the chapter, some examples will be presented: comparable rigidity between the adhesive and the adherents, soft interfaces, adhesive governed by a non convex energy and imperfect adhesion between adhesive and adherents. Some numerical examples will also be given and, finally, an example of a numerical algorithm will be presented.