Understanding non-affine displacement fields of amorphous solids

Numerous previous studies have shown that athermal quasi-static shear of amorphous solids induces non-affine displacement fields, both during the continuous segments of stress versus strain and during stress drops, which exhibit quadrupolar behavior. However, these non-affine displacement fields cannot be represented exactly by the displacement fields induced by a linear superposition of Eshelby inclusions. We take a stepwise approach to understanding the differences in the nonaffine displacement fields of amorphous solids and those obtained from Eshelby inclusions. We first apply a pure shear perturbation to a single triangle within a hexagonal lattice in two dimensions and show that the resulting displacement field matches that from an Eshelby inclusion. We then perform similar perturbations to positionally disordered, but unstressed spring networks with uniform spring constants and find that the displacement fields also match those from a single Eshelby inclusion with similar values of the root-mean-square error. We show that the displacement fields of spring networks can differ from those induced by Eshelby inclusions when we consider spring networks with non-uniform spring constants and residual stress. An improved understanding of when quadrupolar structures are induced will allow us to better characterize shear transformation zones, and non-affine displacement fields more generally, that occur during deformation of amorphous solids.