The present paper proposes an approximation of the influence of a small inclusion, having different stiffness, in an elastic medium restricting its cinematics to a finite dimensional space. The problem, for the unknown displacement, is written in accordance of transmission conditions between elastic medium and inclusion. The multi-scale asymptotic expansions techniques, by using some corrective terms, are introduced to model the inclusion influence in both initial and approximated problems. Finally, some error estimates for asymptotic expansions are presented and different strategies are compared.