A comparison of different approaches in the multi-scale computation of configurational forces

In the present work the FE2 scheme is extended towards the homogenization of material quantities like the Eshelby stress and material node point forces. Therefore, in contrast to standard computational homogenization schemes volume forces on the micro-and on the macro-level have to be taken into account which emerge in the material motion problem due to inhomogeneities in the material. Different approaches in the determination of the material stresses are compared and it is shown that the direct calculation of the material stresses in terms of averaged material quantities requires an internal extra term to fulfill the energy consistency represented by a Hill-Mandel type condition. Furthermore, two approaches based on the average of the material two-point stress and the Eshelby stress are compared which require a further scale-transition which is performed within a postprocessing step. The influence of different micro-structures onto the macroscopic material quantities is studied within numerical examples.