Compressive stiffness of staggered woodpile lattices: Mechanics, measurement, and scaling laws

Abstract

Lattices in woodpile arrangement are of interest in many applications such as tissue engineering scaffolds, elastic metamaterials and lightweight structures: the choice in lattice arrangement and stacking parameters facilitate innovative material design. Additive manufacturing has enabled fabrication of such structured materials with tunable properties. Here, the elastic response of woodpile lattices is studied analytically, numerically, and experimentally when they are compressed in the stacking direction, with struts staggered in alternating layers. Expressions for the apparent Young’s modulus, and its dependence on porosity, are derived from the analysis of a periodically-supported, periodically-loaded, elastic filament. A fifth power law relating the apparent Young’s modulus with the volume fraction is obtained in the asymptotic limit of high porosity, which is consistent with scaling arguments presented here. When the stacking is asymmetric, the apparent stiffness is presented in terms of an analytically known function of the skewness parameter α. For dense lattices, departure from the proposed power law is observed in computational simulations, as well as laboratory experiments on polylactic acid (PLA) 3D-printed woodpiles. Variations from power law can be attributed to unaccounted for effects in the micromechanics of the filaments, e.g. filament shear and diametrical compression. The experimentally obtained relationship, between the apparent modulus and porosity, is in excellent agreement with our analysis and numerical results.