Analysis of volume-average relations in continuum mechanics

In this paper, volume-average relations related to the multilevel modelling process in continuum mechanics are analysed and the concept of average consistency is investigated both analytically and numerically. These volume averages are used in the computational homogenization technique, where a transition of the mechanical properties from the local, microscopic, to the global, macroscopic, length scale is obtained. The representative volume element (RVE) is used as a reference placement and the solution, in terms of volume-averaged stress, will depend on which boundary conditions are chosen for the RVE. Three types of boundary conditions - periodic, affine and anti-periodic - are analysed with respect to the average consistence for the kinematical and stress relations used in continuum mechanics. The inconsistence is quantified by introducing the inconsistence ratio. It is shown analytically that some average stress relations are fulfilled, assuming the periodic boundary condition and anti-periodic traction vector, whereas the average relations connected to the deformation are in general not average consistent. The inconsistence is investigated in a plane model using the finite element technique. The numerical investigation has shown that the inconsistence ratios related to the deformation are also average consistent in the examples considered.