Asymptotic Analysis for the Eigenvalues of Peridynamic Operators

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of peridynamic operators using hypergeometric functions, revealing boundedness or divergence depending on kernel singularity, with explicit bounds based on dimension and kernel properties.

Contribution

It provides the first asymptotic analysis of eigenvalues for peridynamic operators, connecting hypergeometric function behavior to kernel properties.

Findings

Eigenvalues are bounded for integrable kernels.

Eigenvalues diverge for singular kernels.

Explicit bounds depend on dimension and kernel characteristics.

Abstract

Explicit representations of the eigenvalues of the peridynamic operator have been recently derived in [5]. These representations are given in terms of generalized hypergeometric functions. Asymptotic analysis of the hypergeometric functions is utilized to identify the asymptotic behavior of the eigenvalues. We show that the eigenvalues are bounded when the kernel is integrable and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the spatial dimension, the integral kernel and the peridynamic horizon.