Second order homogenization method based on higher order finite elements

Modeling materials with lattice-like microstructures like open-cell foams requires an extended continuum mechanical setting on the macroscopic scale, e. g. a micropolar or micromorphic theory. In order to avoid the formulation of constitutive equations a higher order numerical homogenization scheme (FE²) is proposed. Therefore, each integration point possesses its own microstructure which, in the present case, consists of beam-like elements representing the cell walls. In this paper, the microstructures are discretized by continuum-based higher order locking free finite elements with high aspect ratios, leading to a numerically efficient treatment of a local displacement-driven boundary value problem according to the macroscopic strain and curvature. The resulting stress distributions in the microstructures are homogenized to macroscopic stresses and couple stresses. The approach is demonstrated by a numerical example. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)