Higher-order effective modeling of periodic heterogeneous beams. I. Asymptotic expansion method

This paper is concerned with the elastostatic behavior of heterogeneous beams with a cross-section and elastic moduli varying periodically along the beam axis. By using the two-scale asymptotic expansion method, the interior solution is formally derived up to an arbitrary desired order. In particular, this method is shown to constitute a systematic way of improving Bernoulliʹs theory by including higher-order terms, without any assumption, in contrast to Timoshenkoʹs theory or other refined beam models. Moreover, the incompatibility between the interior asymptotic expansions and the real boundary conditions is emphasized, and the necessity of a specific treatment of edge effects is thus underlined.