A new spectral framework for establishing localization relationships for elastic behavior of composites and their calibration to finite-element models

Localization relationships aim to connect the microscale response in a composite material to the macroscale loading conditions, while taking into account the local details of the microstructure at the location of interest. Such linkages are at the core of multi-scale modeling of materials since they provide efficient scale-bridging relationships. These structure–property linkages are expressed through fourth-rank localization tensors derived from higher-order homogenization theories. This paper builds upon a recently developed spectral framework called microstructure-sensitive design that was established to formulate localization relationships for the elastic response in composite materials. The method casts existing higher-order homogenization theory into a Fourier space to achieve substantial computational advantages over other multi-scale modeling approaches. More specifically, it is demonstrated that the spectral approach transforms the localization relationship into a simple algebraic series comprising polynomials of the microstructure coefficients. A remarkable feature of this new method is that the coefficients of the polynomial expression, termed influence coefficients, are completely independent of the morphological details in a specific microstructure. Consequently, they need to be established only once. It is demonstrated in this paper that an appropriately truncated localization relationship can be obtained by calibration of the influence coefficients to the results of finite-element models. These, and other, salient features of the proposed spectral framework are first theoretically established, and then demonstrated with a simple case study.