A Timoshenko-like equivalent beam model for the static analysis of a chiral metamaterial

Abstract In this paper, a one-dimensional Timoshenko-like equivalent beam model, embedded in a 2D space, is developed to analyze the static response of a chiral metamaterial, first presented in Misra et al. (Contin Mech Thermodyn 32:1497–1513, 2020). This is a beam-like structure, here referred to as the ‘grain beam’, which is made of a periodic assembly of a grain-pair interconnected with solid bars, exhibiting chirality through shear and axial strains coupling. The derivation of the beam model is conducted within the framework of the direct one-dimensional approach, wherein the constitutive law is established through a suitable homogenization procedure, that relies on an energy equivalence between a cell of the periodic model and a segment of the solid beam. Based on the geometry of the interconnecting elements of a grain-pair of the microstructure, analytical as well as numerical approaches are developed to identify the constitutive parameters, the latter grounded on a FE analysis of the cell. The analytical identification procedure, which shed the light on the dependence of the elastic coefficients upon proper defined nondimensional parameters, as the cell aspect ratio and the slenderness of horizontal and vertical fibers, is shown to be effective in designing grain beams with targeted stiffnesses, chirality and macroscopic behavior. The linear static response of some grain beams, taken as case studies, is analyzed and the results obtained by the Timoshenko-like beam model and the FE analyses, these latter carried out on refined models of grain beams, are compared. The effectiveness of the beam model, its limits of applicability, and the associated error magnitude in describing the static behavior of grain beams are investigated and discussed.