A generalized pattern-based self-consistent scheme

A new general theory based on the concept of 'morphologically representative pattern' has recently been developed for the homogenization of heterogeneous elastic materials. It is able to yield in the same framework the classical Voigt/Reuss and Hashin-Shtrikman bounds, the classical and so-called 'three-phase' self-consistent estimates as well as the classical unit cell models. It allows also to derive new generalized Voigt/Reuss and Hashin-Shtrikman bounds and generalized self-consistent estimates. This theory is first described and then applied to some particular matrix-inclusion type materials. An important step in the derivation of such bounds or estimates is the computation of the stress and strain fields in an arbitrary heterogeneous inclusion embedded in an infinite medium subjected to homogeneous loading at infinity. Since this problem usually has no known analytical solution, it is solved by a numerical procedure based on the finite elements method. This procedure is first tested on Hashin's 'Composite Sphere Assemblage', for which analytical solutions are known; 2D and 3D meshes are used. It is then extended to materials with ellipsoidal inclusions, for which new bounds and estimates are derived.