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

Despite extensive study, the microscopic mechanisms governing the deformation of amorphous solids under applied stress remain elusive. In this work, we investigate two-dimensional jammed disk packings subjected to athermal, quasistatic 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 identify local structural defects, which are defined as missing contacts relative to the fully connected network generated from the Delaunay triangulation of disk centers, that govern the non-affine displacement fields. We show that the emergence of an isolated effective quadrupole requires: (1) relatively extended low-frequency vibrational modes for the jammed packing before the applied deformation and (2) the breaking of a contact that is aligned with the low-frequency vibrational modes. The quadrupole that forms is centered around the newly broken contact, which is not necessarily the softest region (with the smallest local shear modulus).