Modeling macroscopic extended continua with the aid of numerical homogenization schemes

Even in the range of small elastic deformations the behavior of foams is not well described by only two elastic constants. Usually the manufacturers give values of the material parameters depending on the loading conditions. This problem is investigated on a microscopic scale by a simple beam model and on the macroscopic scale by an extended continuum model. It has been found that this approach shows the size effect [J. Mater. Sci. 18 (1983) 2572] that cannot be described within the framework of the standard continuum mechanical setting. The existence of the size effect within this model can be explained by independent rotations which do not scale with the displacement field. macroscopic material parameters are generally unknown for foams the macroscopic properties are derived from the microscale where the parameters are assumed to be known. After evaluation of the microscopic constitutive equations, which are also considered to be known, the quantities are mappped back to the macroscale by a homogenization procedure. This approach is known from literature as FE2 model, see e.g. [V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials, PhD-thesis, Technical University of Eindhoven, 2002], [Int. J. Numer. Meth. Eng., 54 (2002) 1235] or [Arch. Appl. Mech., 72 (2002) 300]. It is shown that the Cosserat continuum and the FE2 model are able to describe the same effects qualitatively.