Particle-scale origin of quadrupolar non-affine displacement fields in granular solids

TL;DR

This paper investigates the microscopic origins of quadrupolar non-affine displacement fields in granular solids, linking local structural defects and contact breaking to vibrational modes, and models these fields using an adapted Eshelby inclusion approach.

Contribution

It introduces a reformulation of Eshelby's inclusion method to model non-affine displacements in jammed disk packings, connecting local defects to vibrational modes and contact breaking.

Findings

Isolated quadrupoles increase with pressure.

Quadrupoles are linked to contact breaking along vibrational modes.

Displacement fields can be reconstructed from Eshelby-like defects.

Abstract

In this work, we identify the local structural defects that control the non-affine displacement fields in jammed disk packings subjected to athermal, quasistatic (AQS) simple shear. While complex non-affine displacement fields typically occur during simple shear, isolated effective quadrupoles are also observed and their probability increases with increasing pressure. We show that the emergence of an isolated effective quadrupole requires the breaking of an interparticle contact that is aligned with low-frequency, spatially extended vibrational modes. Since the Eshelby inhomogeneity problem gives rise to quadrupolar displacement fields in continuum materials, we reformulate and implement Eshelby's equivalent inclusion method (EIM) for jammed disk packings. Using EIM, we show that we can reconstruct the non-affine displacement fields for jammed disk packings in response to applied shear as a sum of discrete Eshelby-like defects that are caused by mismatches in the local stiffnesses of triangles formed from Delaunay triangulation of the disk centers.