Micropolar Asymptotic Homogenization for Periodic Cauchy Materials

We present a micropolar-based asymptotic homogenization approach [1,2] for the analysis of composite materials with periodic microstructure. The up-scaling relations are inspired by those originally proposed by [3] in the framework of the computational homogenization, expressing the local displacement field as a function of a cubic polynomial kinematic map depending on first, second and third order homogeneous tensors directly related to the classical and micropolar 2D deformation modes [4,5,6]. The local displacement field is described as superposition of the macroscopic driven kinematic map and local periodic perturbation fields. These perturbation functions are inherently related to the heterogeneous nature of the composite medium and are derived from the solution of recursive cell problems. The down-scaling relations are derived from a newly proposed third order asymptotic expansion of the local displacement field in terms of the macroscopic displacement and its first, second and third order gradients. The overall micropolar elastic tensors derive from a properly conceived energy equivalence between the macroscopic point and a representative portion of the heterogeneous material at the microscopic scale. Different applications to bi-phase orthotropic layered material to are proposed in order to exploit the capabilities of the proposed approach.