A fast algorithm for the elastic fields due to a single fiber break in a periodic fiber-reinforced composite

Abstract

The stress state in a shear-lag model of a unidirectional fiber composite with an arbitrary configuration of fiber breaks is obtained by the weighted superposition of the stress state due to a single broken fiber. In a periodic patch comprised of N fibers located at the points of a regular lattice, a method to determine the stress state due to a single break was proposed by Landis et al. (J Mech Phys Solids 48(3):621–648, 2000). This method entails the determination of the eigenspace of an \(N\times N\) matrix, at a computational cost of \(O(N^3)\). In the present work, an alternative algorithm is proposed. This algorithm exploits the circulant structure of the matrix describing the inter-fiber interactions. The asymptotic computational complexity of the present algorithm equals that of the discrete Fourier transform: \({O}(N \log N)\). Run times of the present method with the eigensolution based method are compared, and shown to be very favorable for the present method, even for small N. Power-law scaling of the overloads due to a single break to much larger distances than previously possible has been verified using the present method.