Quasilocalized states of self stress in packing-derived networks

TL;DR

This paper investigates quasilocalized states of self stress in packing-derived networks, revealing their size scaling with network connectivity and confirming their analogy to local mechanical responses, thus advancing understanding of network mechanics.

Contribution

It demonstrates that the lengthscale of quasilocalized SSS scales as 1/√(z_c - z), clarifies their relation to local dipolar responses, and shows this lengthscale depends solely on network connectivity.

Findings

The lengthscale scales as 1/√(z_c - z) with z near 4.

Quasilocalized SSS are analogous to local dipolar force responses.

The lengthscale depends only on network connectivity, not on mechanical coupling.

Abstract

States of self stress (SSS) are assignments of forces on the edges of a network that satisfy mechanical equilibrium in the absence of external forces. In this work we show that a particular class of quasilocalized SSS in packing-derived networks, first introduced in [D. M. Sussman, C. P. Goodrich, and A. J. Liu, Soft Matter 12, 3982 (2016)], are characterized by a lengthscale $\ell_c$ that scales as $1/\sqrt{z_c-z}$ where $z$ is the mean connectivity of the network, and $z_c\!\equiv\!4$ is the Maxwell threshold in two dimensions, at odds with previous claims. Our results verify the previously proposed analogy between quasilocalized SSS and the mechanical response to a local dipolar force in random networks of relaxed Hookean springs. We show that the normalization factor that distinguishes between quasilocalized SSS and the response to a local dipole constitutes a measure of the mechanical coupling of the forced spring to the elastic network in which it is embedded. We further demonstrate that the lengthscale that characterizes quasilocalized SSS does not depend on its associated degree of mechanical coupling, but instead only on the network connectivity.