Macroscopically Isotropic and Cubic-Isotropic Two-Material Periodic Structures Constructed by the Inverse-Homogenization Method

The present paper deals with the inverse homogenization problem: to reconstruct the layout of two well-ordered elastic and isotropic materials characterized by bulk and shear moduli (κ, μ) and given volume fraction ρ, within a 2D periodicity cell, corresponding to the predefined values of the effective isotropic (κ*, μ*) or cubic symmetry (κ*, μ*, α*) periodic composites. The algorithm used follows from imposing the finite element approximations on the solutions to the basic cell problems of the homogenization theory. The isotropy or cubic symmetry conditions, usually explicitly introduced into the inverse homogenization formulation, do not appear explicitly, as being fulfilled by the special microstructure construction. The square or hexagonal basic cell with the proper internal symmetry is uniformly divided into finite elements each of different element-wise constant material properties. In order to recover the periodic structure of the assumed properties a variety of composites are constructed with different underlying microstructures i.e. the 1-parameter SIMP-like isotropic mixture and 2nd rank orthogonal laminates.