Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05 43 To be noted is that the solution corresponding to the curvature 2 K can be obtained by rotating that evaluated for 1 K . Adopting Procedure A , the field  u differs qualitatively from the actual solution and it can be concluded that this procedure leads to grossly erroneous results. Finally, the results obtained by applying Procedure C are close to those evaluated with Procedure B , although the vertical perturbation components along the horizontal lines in presence of 1 K and  are worse approximated. I DENTIFICATION PROCEDURE : H ILL -M ANDEL MACRO - HOMOGENEITY CONDITION he identification procedure adopted in this study is based on the generalized Hill-Mandel macro-homogeneity condition. The virtual work evaluated at the macroscopic Cosserat point is set equal to the average virtual work of the heterogeneous Cauchy medium in the UC. Thus, the following expression holds: = T T  Σ E σ ε (22) where Σ is the micropolar stress vector evaluated at the macroscopic point, while σ is the Cauchy stress vector at the typical point of the UC. After solving the BVP on the UC and determining the microscopic stress and strain fields, σ and ε , the homogenized Cosserat elastic constitutive matrix C can be derived by using Eq. (22). Considering a two-phase composite material with a regular arrangement of the inclusions, characterized by orthotropic texture, the homogenized Cosserat elastic constitutive matrix, expressed in a reference frame aligned with the principal axes of the material, results as: 11 12 12 22 33 44 55 66 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0                     C C C C C C C C C (23) Regarding the two-phase composite medium examined in subsection Comparison between Procedures A, B and C , the macroscopic micropolar elasticity matrix C has been evaluated, referring to two different UCs obtained by translation in the same heterogeneous medium, see Fig. 2. a) b) Figure 2 : a) UC1 texture; b) UC2 texture. The results, presented in [11], are critically discussed to highlight the dependence of the identification equivalent coefficients on the centering of the UC. Indeed, differently from the Cauchy coefficients, which are proved to be irrespective of the choice of UC, for the bending and skew-symmetric shear micropolar coefficients this is not straightforward, at least in the framework of computational homogenization. In the following the focus is on the determination of 44 C , 55 C and 66 C , governing the bending and skew-symmetric shear behavior of the micropolar equivalent medium. T