On the mathematical foundations of the self-consistent clustering analysis for non-linear materials at small strains

We investigate, both mathematically and numerically, the self-consistent clustering analysis recently introduced by Liu-Bessa-Liu and, independently, by Wulfinghoff-Cavaliere-Reese. We establish, in the small strain setting and non-softening material behavior, existence and uniqueness of the solution to the discretized equations for fixed (possibly anisotropic) reference material, by a constructive method. Furthermore, we establish convergence of the solution to the discretized equation to the continuous solution upon refinement of the clusters. Thus we generalize the work of Tang-Zhang-Liu to spatial dimensions larger than 1. We explore the specific structure of the Lippmann-Schwinger equation governing clustering analysis, proving strict equivalence to a problem of Eyre-Milton type. For the latter formulation, existence and uniqueness are easily established based on recent progress in the understanding of polarization schemes in FFT-based computational homogenization methods. Last but not least, for elasto-viscoplastic constituents we clarify the relationship of the self-consistent clustering analysis to the transformation field analysis of Dvorak-Benveniste. Our theoretical considerations are confirmed by pertinent numerical experiments. (C) 2019 Elsevier B.V. All rights reserved.