A novel spectral framework for second-order homogenization theories

Composite systems are emerging as materials of choice in many engineering applications. Their increasing popularity is predicated on the capacity of these materials to tailor their proprieties by varying their internal structure. Composites exhibit unique combinations of anisotropic properties that are not achievable by traditional materials. Recently, we have developed a novel mathematical framework, called the Microstructure Sensitive Design (MSD), to represent efficiently the microstructure-property relationships of composites using second-order homogenization theories. The latter are a formidable advancement over the elementary first-order theories that account only for the volume fraction information of the constituents. Second-order theories account for the morphological details of the microstructure in capturing the anisotropic properties of materials. The spectral framework of MSD results in simplified structure-property linkages and identifies the microstructure hull, i.e. the space of all theoretically possible microstructural realizations of a given material system. With this new framework, we have successfully established macroscopic elastic properties of composites and delineated second-order property closures for the first time. We also predicted local properties of composites by casting relevant scale-bridging relations in the spectral framework of MSD. These linkages describe the fourth-rank localization tensors connecting the microscopic stresses or strains to the macroscopic loading conditions. The spectral formulations for the localization tensors result in algebraic expressions whose coefficients do not depend on the microstructural details. These coefficients are called the influence coefficients. We have performed numerical integrations and finite element calibrations to evaluate these influence coefficients. These have resulted in very accurate descriptions of the local elastic stress (or strain) fields in a broad range of composite microstructures.