This study deals with a linear elastic body consisting of two solids connected by a thin adhesive interphase with a small thickness e. The three parts have similar elastic moduli. It is proposed to model the limit behavior of the interphase when e -> 0. It has been established [1], using matched asymptotic expansions, that at order zero, the interphase reduces to a perfect interface, while at order one, the interphase behaves like an imperfect interface, with a transmission condition involving the displacement and the traction vectors at order zero. The perfect interface model is exactly recovered using a C-convergence argument. At a higher order, a new model of imperfect interface is obtained by studying the properties of a suitable (weakly converging) sequence of equilibrium solutions. Some analytical examples are given to illustrate the results obtained.