A Green’s function-based approach to the concentration tensor fields in arbitrary elastic microstructures

Abstract

Computational homogenization based on FEM models is the gold standard when it comes to homogenization over a representative volume element (RVE), of so-called complex material microstructures, i.e., such which cannot be satisfactorily represented by an assemblage of homogeneous subdomains called phases. As a complement to the aforementioned models, which depend on the boundary conditions applied to the representative volume element and which, as a rule, do not give direct access to the macro-micro-relations in terms of concentration tensors, we here introduce a Green’s function-based homogenization method for arbitrary inhomogeneous microstructures: Inspired by the ideas underlying traditional phase-based homogenization schemes, such as the Mori-Tanaka or the self-consistent model, the new method rests on mapping, through the strain average rule, the microscopic strain fields associated with an auxiliary problem to the macroscopic strains subjected to the RVE. Thereby, the auxiliary problem is defined on a homogeneous infinite matrix subjected to homogeneous auxiliary strains and to inhomogeneous (fluctuating) polarization stresses representing the fluctuations of the microstiffness field, i.e., the complex microstructure within the RVE. The corresponding microscopic strains appear as the solution of a Fredholm integral equation, delivering a multilinear operator linking the homogeneous auxiliary strains to the microscopic strains. This operator, together with the aforementioned mapping, eventually allows for completing the model in terms of concentration tensor and homogenized stiffness quantification. This is illustrated by example of a sinusoidally fluctuating microstructure, whereby the corresponding singular convolution integrals are analytically evaluated from the solution of the Poisson’s equation, and this evaluation strategy is then analytically verified through a Cauchy principal value analysis, and numerically validated by a state-of-the-art FFT homogenization procedure. For the given example, the novel analytical method is several thousand times faster than an FTT-based computational homogenization procedure.