Issue 29

M.L. De Bellis et alii, Frattura ed Integrità Strutturale, 29 (2014) 37-48; DOI: 10.3221/IGF-ESIS.29.05 38 scales. By adopting generalized continua this limit is overcome. Many authors have focused on coupling different continuum models at the two scales. In most cases, at the microscopic level, the classical Cauchy continuum is adopted, especially because nonlinear constitutive relationships are well-established in this framework. Among various generalized continua (second-gradient, couple stress, micropolar or multifield), a micropolar Cosserat continuum at the macro-level and a Cauchy continuum at the micro-level are used here to study the homogenized response of periodic composite materials. The computational homogenization technique adopted effectively predicts the macroscopic behavior of composite materials [5, 6]. Since composite materials, characterized by regular textures are analyzed, a Unit Cell (UC) is selected at the micro-level. Consistently with the strain-driven approach, the two levels are linked through a kinematic map based on a third order polynomial expansion, previously derived by the authors in [7], from an original idea developed in [1]. The displacement field at the micro-level is represented as the superposition of the kinematic map and an unknown perturbation field, due to the heterogeneous nature of the material. For classical first order computational homogenization the perturbation fields are periodic [8], but this is not true when higher order polynomial terms are considered, as underlined in [9,10]. In this study, the problem of determining the displacement perturbation fields, with particular reference to its influence on the structural response evaluation, is investigated. To this end, the three following techniques are adopted. The first is based on the solution of the Boundary Value Problem (BVP) by applying periodic boundary conditions on the UC. The second is the 3 steps homogenization, presented by the authors in [11], and based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components. Finally the BVP is solved by applying special boundary conditions on the UC, as derived in [7]. Furthermore, the identification of the homogenized linear elastic 2D Cosserat constitutive parameters is performed, by using the Hill-Mandel technique, based on the generalized macro-homogeneity condition presented in [11]. As known, this technique has inherent limitations, leading to physically inconsistent results [9, 12, 13]. For example, higher order constitutive components are identified, also when a homogeneous elastic material at the micro-level is considered. Despite the drawbacks, this technique has been widely used, at least when asymptotic techniques [14] cannot be applied, as in the case of coupling micropolar and classical continua. The influence of the selection of the UC is analyzed and some key issues are outlined. By considering two different UCs, selected for representing the composite texture, it emerges that the constitutive response of the homogenized medium depends on the choice of the cell. Indeed, while the elastic Cauchy coefficients are irrespective of the UC selected, this does not occur for the bending and skew-symmetric shear Cosserat coefficients, at least with regard to computational homogenization. This fact is also confirmed by the results obtained from the structural applications. Two numerical tests are presented to highlight the main aspects of the presented micropolar computational homogenization technique and to emphasize the differences obtained using the three procedures to describe the perturbation fields. M ICROPOLAR HOMOGENIZATION he computational homogenization technique used here adopts a 2D micropolar continuum at the macro-level and a classical 2D Cauchy continuum at the micro-level. At typical macro-level material point, the displacement vector 1 2 ={ , , } T U U  U is defined, where 1 U and 2 U are the translational degrees of freedom and  is the rotational degree. The micropolar strain vector is characterized by six components as follows: 1 2 12 1 2 = E E K K                     E (1) where 1 E , 2 E and 12  are the axial and symmetric shear strains, 1 K and 2 K are the curvature components, while  is the skew-symmetric shear component. The compatibility equations can be written in compact form as: T