Our aim is to validate the effectiveness of a matched asymptotic expansion method introduced in \cite{geymonat:inria-00540992}, \cite{geymonat:hal-00757005}. A simplified model for the influence of small identical heterogeneities periodically distributed on an internal surface is obtained and applied to the overall response of a linearly elastic body. A careful numerical study compares the solution obtained by a standard method on a fine mesh to the one obtained by asymptotic expansion. We compute both the zero th and first order terms in the expansion. To efficiently compute the first order term we introduce a domain decomposition method.