Fast Fourier Transform Method in Peridynamic Micromechanics of Composites

Abstract

Abstract We consider a static linear bond-based peridynamic (proposed by Silling, see J. Mech. Phys. Solids 2000; 48:175–209) composite materials (CMs) of a periodic structure. In the framework of the second background of micromechanics (called also computational analytical micromechanics), one proved that local micromechanics (LM) and peridynamic micromechanics (PM) are formally similar to each other for CM of both random and periodic structures. It gives an opportunity for straightforward generalization of LM methods (including fast Fourier transform, FFT) to their PM counterparts. So, in the PM counterpart of the implicit periodic Lippmann-Schwinger (L-S) equation in LM, we have three convolution kernels corresponding to the properties of the matrix, inclusions, and interaction interface. Eshelby tensor in LM depending on the inclusion shape is replaced by PM counterparts depending on the shapes of inclusions and interaction interface (although the Eshelby concept of homogeneous eigenfields does no work in PM). The mentioned tensors are estimated one time (as in LM) in a frequency domain (also by the FFT method). The polarization schemes of the solution of L-S equation in the Fourier space have one primary unknown variable (polarization) whereas the PM counterpart contains three primary ones estimated at each step which are formally similar to LM case. Computational complexities O(Nlog2N) of the FFT methods are the same in both LM and PM.