Computational homogenization of materials with small deformation to determine configurational forces

In this work the mechanical boundary condition for the micro problem in a two-scaled homogenization using a FE² approach is discussed. The strain tensor is often used in the literature for small deformation problem to determine the boundary conditions for the boundary value problem on the micro level. This strain tensor based boundary condition gives consistent homogenized mechanical quantities, e.g. stress tensor and elasticity tensor, but the present work points out that it leads to unphysical homogenized configurational forces. Instead, we propose a displacement gradient based boundary condition for the micro problem. Results show that the displacement gradient based boundary condition can give not only the consistent homogenized mechanical quantities but also the appropriate homogenized configurational forces. The interpretation of the displacement gradient based boundary condition is discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)