A Lagrangian formulation of metamaterial homogenization

Abstract

Many metamaterials may be well described as multi-scale systems where the large scale is represented by the wavelength and the small scales are represented by inhomogeneity lengths. The effective material properties of a metamaterial of this sort may be derived by considering the averaged behavior of material elements that are considered “infinitesimal” relative to the wavelength but is on the order of the inhomogeneity lengths. The averaging of the small-scale features enables a straightforward process to describe the metamaterial mechanics in terms of a Lagrangian formalism, and the macroscopic constitutive equations may be obtained by deriving the Euler-Lagrange equations for the macroscopic field quantities. This energetic approach to homogenization may be used to estimate both linear and nonlinear material properties. This talk describes this homogenization method for one-dimensional systems and compares the resulting effective material properties with other homogenization methods.