Determination of metamaterial parameters by means of a homogenization approach based on asymptotic analysis

By using modern additive manufacturing techniques, a structure at the millimeter length scale (macroscale) can be produced showing a lattice substructure of micrometer dimensions (microscale). Such a system is called a metamaterial at the macroscale, because its mechanical characteristics deviate from the characteristics at the microscale. Consequently, a metamaterial is modeled by using additional parameters. These we intend to determine. A homogenization approach based on asymptotic analysis establishes a connection between these different characteristics at micro- and macroscales. A linear elastic first-order theory at the microscale is related to a linear elastic second-order theory at the macroscale. Small strains (and, correspondingly, small gradients) are assumed at both scales. A relation for the parameters at the macroscale is derived by using the equivalence of energy at macro- and microscales within a so-called representative volume element (RVE). The determination of the parameters becomes possible by solving a boundary value problem within the framework of the finite element method. The proposed approach guarantees that the additional parameters vanish if the material is purely homogeneous; in other words, it is fully compatible with conventional homogenization schemes based on spatial averaging techniques. Moreover, the proposed approach is reliable, because it ensures that the obtained additional parameters are insensitive to choices of the RVE consisting of a repetition of smaller RVEs depending upon the intrinsic size of the structure.