The regular packing of fibres in three dimensions

Abstract

The possible tightest packing of bars of variously shaped cross–sections is investigated in detail. The bars are taken to lie along well–defined directions, e.g. cube edges, face diagonals, etc. The results are used to explore the feasibility of designing a fibrous composite which is both elastically isotropic and contains an appreciable volume fraction of reinforcement. Such is shown not to be possible for a regular array of bars all of the same cross–section. A volume fraction of about 0.25 can be obtained by arranging bars in a quasi–periodic fashion, akin to a three–dimensional Penrose tiling. Large volume fractions, greater than one half, together with isotropy may be obtained with two sets of bars each with two different cross–sections, e.g. 60% parallel to cube body diagonals and 40% parallel to cube edges.