Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05 42             3 3 3 3 3 1,12 2,12 12,12 1 2 3 , = , =              T γ X x Λ x x x x (21)   3 i  x , =1,.., 3 i , being periodic functions evaluated in the UC when 0   . Procedure C: analysis of the perturbation field in the RVE The characterization of the perturbation field  u is performed in [7] by evaluating its actual distribution. To this end, a RVE obtained as assemblage of a large number of UCs for a selected two-phase periodic composite material is considered and remote fully displacement BCs are prescribed. In particular, the micropolar deformation modes are imposed on the boundary, according to the kinematic map in Eq. (8), and the RVE response is evaluated by finite element method. Hence, the distribution of the perturbation field arising in the central UC of the RVE is taken as the benchmark. Thus, the suitable BCs to impose on the single UC are derived, in order to reproduce, with a satisfactory level of accuracy, the actual distribution of the perturbation field. Some selected two-phase composite materials, characterized by material symmetries, ranging from cubic to orthotropic, are analyzed. In all the cases considered, similar distributions of the perturbation displacement fields on the UC boundary emerge. Differently from the case of the first order homogenization procedure, where periodic BCs are suitably adopted, in the analyzed cases more complex BCs have to be considered, which are different for the two components of  u . In Fig. 1 the derived BCs are summarized. In the first row, the applied Cosserat macroscopic deformation components are reported; in the second row the BCs for the component  1 u along the horizontal and vertical edges of the UC are schematically reported, while in the third row those for the displacement component 2 u  are shown. The symbol “p” indicates periodic BCs , see Eq. 12; “s” refers to skew-periodic BCs, that is:             1 2 1 2 2 2 2 1 2 1 2 1 1 1 , , , , , , , , a x a x x a a x a x a x a a             u u u u     (22) Finally, “0” indicates zero perturbation displacement BCs. Figure 1 : Boundary conditions to impose on the UC to evaluate the perturbation fields, derived in [7]. Comparison between Procedures A, B and C The approaches presented were compared by carrying out some numerical tests. This subsection is devoted to a qualitative discussion of the results obtained that are reported in detail in [11]. Reference is made to a specific two-phase composite material, characterized by cubic symmetry, whose texture is made from a soft matrix with stiff square inclusions, both isotropic and regularly spaced. The volume fraction , f defined as the ratio between the area of the inclusions and the total area of the UC, is set equal to 36%. As reference solution the distribution of the perturbation field arising in the central UC of a sufficiently large RVE, undergoing remote fully displacement boundary conditions, is considered. The three procedures lead to the same correct results, if the classical macroscopic strain components 1 , E 2 E and 12  are activated. For the components 1 K , 2 K and ,  different considerations apply. For both components 1 K and  Procedure B provides the best estimate of the perturbation fields. In the case of 1 K only the vertical component of the perturbation field, evaluated along the horizontal edges of the UC, slightly differs from the referential solution, while in the case of  a very good approximation is guaranteed anywhere on the edges.