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A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites

by H. Kang and S. Yu

(Report number 2017-31)

Abstract
We prove in a mathematically rigorous way the asymptotic formula of Flaherty and Keller on the effective property of densely packed periodic elastic composites with hard inclusions. The proof is based on the primal-dual variational principle, where the upper bound is derived by using the Keller-type test functions and the lower bound by singular functions made of nuclei of strain. Singular functions are solutions of the Lam\'{e} system and capture precisely singular behavior of the stress in the narrow region between two adjacent hard inclusions.

Keywords: Flaherty-Keller formula, densely packed composite, effective elastic modulus, primal-dual principle, singular functions

@Techreport{KY17_727,
  author = {H. Kang and S. Yu},
  title = {A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-31},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-31.pdf },
  year = {2017}
}

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