Scale effects in the optimal design of a microstructured medium against buckling

Abstract

Certain classes of problems in optimal structural design lead naturally to the introduction of periodic microstructured media as the basis material for the construction of a mechanical element. The unit cell size of these microstructures cannot be arbitrarily small, as suggested by the pertaining optimization analyses up to date, and has to be related to the structure's overall dimensions. One physically important mechanism that provides such microstructure size limitations is elastic buckling.An analytically tractable model of an infinite periodic rectangular planar frame with axially compressed beams is used to study the optimal buckling loads. For any given design, one can find a critical stress above which buckling instability occurs. In addition one can also find the region in the design space for which the optimal critical mode is a global one, i.e. its characteristic length is much larger than the unit cell size. In this region of the design space one can safely use the homogenized material properties to describe the medium, for they provide all the information needed to predict a global buckling instability. In addition to the detailed parametric study for the model problem investigated here, implications for the optimal design against buckling of more general structures are also briefly discussed.