On the bounds of applicability of two-step homogenization technique for porous materials

We discuss applicability of the two-step homogenization procedure to microstructures formed by spherical pores of two distinct sizes with total porosity of 40%. Results of one- and two-step homogenizations utilizing Non-Interaction Approximation (NIA), Mori-Tanaka-Benveniste Scheme (MTB) and Differential Scheme (DS) are compared with numerical data obtained by finite element simulations. A modified collective rearrangement method powered by computationally efficient hierarchical k-means tree algorithm is developed for generating microstructures containing spherical pores of different sizes with prescribed partial porosities. Two-step procedure turns to be almost commutative with respect to the sequence of homogenization step showing 0.2% as a maximum relative error. Sensitivity of approximated overall elastic properties to the size difference of spherical inhomogeneities is observed with the shear modulus showing stronger dependence than the bulk modulus. The two-step MTB can be used to approximate effective properties of solids with spherical pores of distinct size as long as this size difference between pore families is larger than 10 times.