Summability Methods for Elastic Local Fields in Periodic Heterogeneous Materials

In the past, Eshelby's method for a single inclusion in an infinite medium has been extended to periodic heterogeneous materials for the evaluation of global properties. In this work, the extended Eshelby's method is used for the evaluation of local fields such as strain and stress in periodic heterogeneous materials characterized by unit cells containing multiple fibres or voids with different geometric and mechanical properties. The proposed method provides Fourier coefficients that are used to construct partial sums of three-dimensional trigonometric Fourier series of local fields. These partial sums exhibit unwanted effects such as the Gibbs phenomenon. In order to attenuate these effects, the behaviour of iterated Fejér partial sums and means of the Riesz summability method is investigated. Extensive numerical examples on both a multiphase composite and a material with voids are provided: in the examples, partial sums, iterated Fejér partial sums, and Riesz means for the local stress are compared with FEM solutions. The numerical comparison shows Riesz means perform better than partial sums and iterated Fejér partial sums and are effective in approximating elastic local fields in periodic heterogeneous solids.