Linear peridynamics Fourier multipliers and eigenvalues

TL;DR

This paper characterizes the Fourier multipliers and eigenvalues of linear peridynamic operators, providing explicit formulas and revealing their structure, with implications for understanding their convergence to classical elasticity models.

Contribution

It introduces explicit formulas and hypergeometric representations for the eigenvalues of linear peridynamic operators, connecting nonlocal models to classical elasticity.

Findings

Eigenvalues derived from Fourier multipliers are explicitly characterized.

The spectrum converges to that of the Navier operator as nonlocality vanishes.

Hypergeometric functions describe the structure of Fourier multipliers.

Abstract

A characterization for the Fourier multipliers and eigenvalues of linear peridynamic operators is provided. The analysis is presented for state-based peridynamic operators for isotropic homogeneous media in any spatial dimension. We provide explicit formulas for the eigenvalues in terms of the space dimension, the nonlocal parameters, and the material properties. The approach we follow is based on the Fourier multiplier analysis developed for the nonlocal Laplacian. The Fourier multipliers of linear peridynamic operators are second-order tensor fields, which are given through integral representations. It is shown that the eigenvalues of the peridynamic operators can be derived directly from the eigenvalues of the Fourier multiplier tensors. We reveal a simple structure for the Fourier multipliers in terms of hypergeometric functions, which allows for providing integral representations as well as hypergeometric representations of the eigenvalues. These representations are utilized to show the convergence of the eigenvalues of linear peridynamics to the eigenvalues of the Navier operator of linear elasticity in the limit of vanishing nonlocality. Moreover, the hypergeometric representation of the eigenvalues is utilized to compute the spectrum of linear peridynamic operators.