Limit behavior of Koiter model for long cylindrical shells and Vlassov model

We propose in this article to consider the limit behavior of the Koiter shell model when one of the characteristic length of the middle surface becomes very large with respect to the other. To do this, we perform a dimensional analysis of Koiter formulation which involves dimensionless numbers characterizing the geometry and the loading. Once reduced to a one-scale problem corresponding to thin-walled beams (long cylindrical shell), using asymptotic expansion technique, we address the limit behavior of Koiter model when the aspect ratio of the shell tends to zero. We prove that at the leading order, Koiter shell model degenerates to a one dimensional thin-walled beam model corresponding to the Vlassov one. Moreover, we obtain a general analytical expression of the geometric constants involved, that improves the empirical expression given by Vlassov.