A Re-Examination of Wave Dispersion and on Equivalent Spatial Gradient of the Integral in Bond-Based Peridynamics

Abstract

In this paper, wave dispersion properties in bond-based peridynamics (BBPD) are re-examined. By BBPD formulation, wave dispersion (frequency (ω) - wavenumber (k)) relation is known to be transcendental in nature. In fact, for uniform micromodulus function (C), there exists a frequency bandwidth with multiple k. Further, there is a ω within this bandwidth at which propagating k are infinite in number. Question that needs an answer is can all k propagate in the BBPD? In literature, an agreement between certain continuum gradient models (CM) and BBPD has been reported. This agreement is established in the sense that, for a judicious choice of C, wave dispersion properties are the same between BBPD and CM. This equivalence between a finite-ordered (CM) and an infinite-ordered (BBPD) displacement gradient model motivates to an associated converse question: given a C, does BBPD integral represent an effective displacement gradient of a finite order? In this paper, an attempt has been made to fix an order to the BBPD integral with the help of a Fourier frequency-based spectral study, under a one-dimensional rod setting. Both kinetically local and nonlocal formulations of BBPD are considered. It is argued that only one wave mode may propagate in a BBPD rod. This implies, Neumann force condition at a boundary should read as a spatial gradient of order one. This further implies, the BBPD spatial integral effectively represents a second-order displacement gradient in space. The wave motion study of this paper corroborates with the literature that applies local boundary conditions to nonlocal boundary value problems.