Numerical homogenization of the Eshelby tensor at small strains

Numerical homogenization methods, such as the FE2 approach, are widely used to compute the effective physical properties of microstructured materials. Thereby, the macroscopic material law is replaced by the solution of a microscopic boundary value problem on a representative volume element in conjunction with appropriate averaging techniques. This concept can be extended to configurational or material quantities, like the Eshelby stress tensor, which are associated with configurational changes of continuum bodies. In this work, the focus is on the computation of the macroscopic Eshelby stress tensor within a small-strain setting. The macroscopic Eshelby stress tensor is defined as the volume average of its microscopic counterpart. On the microscale, the Eshelby stress tensor can be computed from quantities known from the solution of the physical microscopic boundary value problem. However, in contrast to the physical quantities of interest, i.e. stress and strain, the Eshelby stress tensor is sensitive to rigid body rotations of the representative volume element. In this work, it is demonstrated how this must be taken into account in the computation of the macroscopic Eshelby stress tensor. The theoretical findings are illustrated by a benchmark simulation and further simulation results indicate the microstructural influence on the macroscopic configurational forces.