About the projection of kinematical quantities within the FE2-approach

Within the FE²-approach strain–like quantities are projected to the boundary of the attached microstructure yielding Dirichlet boundary conditions for the microscopic boundary value problem. By use of the divergence theorem one can show that the projection of gradient information to the boundary of the attached microstructure fulfills the condition that the volumetrical average of the strain–like quantities in the inner of the microstructure is exactly the same as the projected quantity, provided that the microstructure is continous. However, within the Cosserat continuum theory [1] the strain tensor depends directly on the extra rotation and not on the gradient of it. Thus, the projection can not be carried out in the same way as for gradient information. The present approach shows how to circumvent this problem by giving the extra–rotation within the Cosserat continuum theory a strong kinematical interpretation in the form of a displacement field, which shows the same effects as the extra–rotation itself. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)