Which part of a chain breaks

TL;DR

This paper models a harmonic chain under increasing force, deriving a Green's function to predict where a spring breaks, revealing how the breaking point shifts along the chain depending on the force application rate.

Contribution

It introduces an analytical framework for predicting break points in a harmonic chain under dynamic loading, linking force rate to breaking location.

Findings

Large force rate causes break near the pulling end.

Small force rate results in break near the wall.

Breaking point oscillates with changing force application rate.

Abstract

This work investigates the dynamics of a one-dimensional homogeneous harmonic chain on a horizontal table. One end is anchored to a wall, the other (free) end is pulled by external force. A Green's function is derived to calculate the response to a generic pulling force. As an example, I assume that the magnitude of the pulling force increases with time at a uniform rate $\beta$. If the number of beads and springs used to model the chain is large, the extension of each spring takes a simple closed form, which is a piecewise-linear function of time. Under an additional assumption that a spring breaks when its extension exceeds a certain threshold, results show that for large $\beta$ the spring breaks near the pulling end, whereas the breaking point can be located close to the wall by choosing small $\beta$. More precisely, the breaking point moves back and forth along the chain as $\beta$ decreases, which has been called "anomalous" breaking in the context of the pull-or-jerk experiment. Although the experiment has been explained in terms of inertia, its meaning can be fully captured by discussing the competition between intrinsic and extrinsic time scales of forced oscillation.