Asymptotic analysis of soft thin layers with non convex energy

We consider a bar made by two elastic bodies with linear stress-strain relation separated by an adhesive layer of thickness h. The adhesive is characterized by a non convex (piecewise quadratic) strain energy density with elastic modulus k. Firstly, we considering the equilibrium problem of the bar when its extremities are subject to a relative displacement and and determine the stable and metastable solutions. Then, we let h, k → 0 to obtain the corresponding asymptotic contact laws, linking the stress to the jump of the displacement at the adhesive interface.