We consider a bar with extremities subject to a given displacement and made by two elastic bodies with linear stress-strain relation separated by an adhesive layer of thickness h: The material of the adhesive is characterized by a non convex (piecewise quadratic) strain energy density with elastic modulus k: We flrst discuss the stability and the metastability of the equilibrium conflgurations when a given relative displacement is imposed to the ends of the bar. Due to the non convexity of the energy, there are multiple metastable conflgurations made by a mixture of two phases, each phase corresponding to one ascending branch of the stress-strain curve. We then study the limit problem when the pair (h, k) tends to zero and discuss the asymptotic contact laws corresponding to the stable and metastable equilibrium conflgurations of the bar.
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