A One-Dimensional Peridynamic Model of Defect Propagation and its Relation to Certain Other Continuum Models

TL;DR

This paper develops a one-dimensional peridynamic model to study defect propagation, deriving conditions for equilibrium and propagation speed, and relates it to differential-equation models for improved numerical analysis of complex defect interactions.

Contribution

It introduces a peridynamic framework for defect propagation in 1D, deriving a formula for the micromodulus function and establishing its equivalence with certain differential models.

Findings

Critical force for defect equilibrium is calculated.

Propagation speed as a function of applied force is derived.

Peridynamic and differential models are shown to be equivalent with proper micromodulus.

Abstract

The peridynamic model of a solid does not involve spatial gradients of the displacement field and is therefore well suited for studying defect propagation. Here, bond-based peridynamic theory is used to study the equilibrium and steady propagation of a lattice defect -- a kink -- in one dimension. The material transforms locally, from one state to another, as the kink passes through. The kink is in equilibrium if the applied force is less than a certain critical value that is calculated, and propagates if it exceeds that value. The kinetic relation giving the propagation speed as a function of the applied force is also derived. In addition, it is shown that the dynamical solutions of certain differential-equation-based models of a continuum are the same as those of the peridynamic model provided the micromodulus function is chosen suitably. A formula for calculating the micromodulus function of the equivalent peridynamic model is derived and illustrated. This ability to replace a differential-equation-based model with a peridynamic one may prove useful when numerically studying more complicated problems such as those involving multiple and interacting defects.