Homogenization of rod-like metamaterials as a special Cosserat rod

Abstract

Rod-like metamaterials are the structures that are obtained by periodically assembling its microstructural unit (network of rods) in just one direction. In this work, we present a scheme for obtaining the nonlinear constitutive response of such structures when homogenized macroscopically as a continuum rod. To capture accurately arbitrary and large deformation, the geometrically exact special Cosserat rod theory is used for modeling the rod at both micro and macro scales. By assuming the metamaterial structure to be strained uniformly (at macroscale) along its arc length, the full structure problem is reduced to just that of its microstructural unit but subjected to helically periodic boundary condition. The microscale problem, consisting of a network of rods and formulated in a variational setting, is solved in the presence of rod joint constraints and helically periodic boundary conditions. The expressions for the macroscale/homogenized rod's stress resultants (internal contact force and moment) and stiffnesses are then obtained. Finally, several numerical examples having different microstructural units/RVEs are presented to demonstrate our method. We start with simpler square and cross RVEs to validate our results with the existing literature. We then take up more complex RVEs such as square RVEs having helical constituent rods which have application as artificial muscle material and eventually we work on the homogenization of auxetic tubular metamaterials. We show how various design parameters of these RVEs can be tuned to obtain the desired macroscopic response.