Multi-scale modeling of elastic response of three-dimensional voxel-based microstructure datasets using novel DFT-based knowledge systems

In this paper, we present a novel approach for predicting the elastic fields in three-dimensional voxel-based microstructure datasets subjected to uniform periodic boundary conditions. This new formalism has its theoretical roots in the statistical continuum theories developed originally by Kroner. However, in the approach described by Kroner the terms in the series were established by selecting a reference medium and numerically evaluating a complex series of nested convolution integrals. This approach is largely hampered by the principal value problem and exhibits high sensitivity to the properties of the selected reference medium. In the present work, we have recast the same series expressions into much more computationally efficient representations using discrete Fourier transforms (DFTs). The main advantage of the new DFT-based framework is that it allows easy calibration of Kroner’s expansions to results from finite element methods, thereby overcoming all of the main obstacles associated with the principal value problem and the need to select a reference medium. Consequently, the DFT-based approach presented here produces much more accurate predictions. In this paper the new mathematical formalism is first presented in a generalized framework, and its viability is then demonstrated for a selected class of two phase composites.