This work is concerned with the equilibrium problem of a composite body comprising two linear elastic adherents joined by an adhesive of small thickness ɛ and made of a conewise linear elastic material (Curnier et al., 1995). The elasticity coefficients of the adhesive material are chosen to simulate soft behavior in traction and hard behavior in compression, or vice versa. Shear moduli are always assumed to be soft. The method of matched asymptotic expansions is applied to calculate the transmission conditions approximating the behavior of the adhesive in the limit of vanishing ɛ In the plane of the adhesive a classical spring-type law is obtained, linking the tangential components of the stress vector to the jumps of the tangential components of the displacement vector. In the direction perpendicular to the plane of the adhesive, Signorini’s contact conditions of non interpenetration are recovered. However, adhesion can emerge for adhesives showing octantwise linear elastic behavior or particular symmetries of half-spacewise behavior.