HILL’s LEMMA FOR THE AVERAGE-FIELD THEORY OF COSSERAT CONTINUUM

The Hill-Mandel condition, i.e. the micro-macro energy equivalence condition, should be satisfied in the multi-scale homogenization process based on the average-field theory. It needs firstly the derivation of Hill’s lemma. The macroscopic average couple stress tensor can be defined in two different ways in the average-field theory of Cosserat continuum. That leads to difficulties in deriving the Hill’s lemma. This paper constructs a transition form of Hill’s lemma without considering the specific expressions of macro average couple stress tensor. Substitution of the two existing definitions of average couple stresses into the transition form leads to two different versions of Hill’s lemma, respectively. The comparison and analysis are also performed on the presented versions. This paper provides the theoretical basis for further research on the multi-scale modeling of heterogeneous Cosserat continuum.