In this paper a composite body is considered. This body is made of three solids: two linear elastic adherents and a piecewise linear thin adhesive. The composite occupies a bounded domain depending naturally on a small parameter ε, which is the thickness, assumed constant, of the adhesive. Classically, it is possible to derive an interface imperfect law using asymptotic expansions as the thickness ε tends to zero. In this work, the material in the interphase is assumed to be graded, i.e. its elasticity properties vary along the thickness. Moreover, an unilateral condition is considered to avoid penetrations. A first result of the paper is that it is possible to apply the above methodology based on asymptotic expansions to this kind of material. Then, a finite element method is introduced to solve the initial problem (with three layers) and the limit problem (with two layers in imperfect contact). Various types of graded materials are numerically analyzed. In particular, different types of stiffness distributions are studied in detail.