Scale-size effects analysis on predicting the mechanical properties of periodic composite materials

The homogenization theory is often used to compute the elastic properties of periodic composite materials based on the shape and periodicity of a unit-cell. The unit-cell is thus representative of the smallest periodic heterogeneity of the composite medium. The resulting properties are to be interpreted as describing an homogeneous medium equivalent to the periodic porous medium of the actual composite. However, this theory hypothesizes that the feature size of the unit-cell, d, is much smaller than the resulting composite global dimension, D, i.e. d / D ® 0 (dimensionless unit-cell), as well as the application of periodic boundary conditions to the unit-cell domain. Due to these hypotheses is critical to investigate the problem of how good homogenization predictions are when compared to the actual properties of a composite generated by the finite spatial repetition of a unit-cell characterized by dimensional quantities. Related research work involving two-dimensional bi-material unit-cells with material symmetry has been already reported. As it follows, one addresses again this research topic although considering here the case of three-dimensional porous unit-cells with no a priori material symmetry (anisotropic case). The unit-cell designs here are obtained maximizing the stiffness objective function subjected to constraints on permeability. The outcome of the present work indicate that, for practical purposes, it is sufficient in these examples to have a low scale factor to replace the non-homogeneous composite by the equivalent homogeneous material with the moduli given by homogenization theory. These observations are also consistent with the previous works although they had been focused on bidimensional microstructures with material symmetry.