Title: Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?,
Abstract: The deformation of disordered solids relies on swift and localised
rearrangements of particles. The inspection of soft vibrational modes can help
predict the locations of these rearrangements, while the strain that they
actually redistribute mediates collective effects. Here, we study soft modes
and strain redistribution in a two-dimensional continuous mesoscopic model
based on a Ginzburg-Landau free energy for perfect solids, supplemented with a
plastic disorder potential that accounts for shear softening and
rearrangements. Regardless of the disorder strength, our numerical simulations
show soft modes that are always sharply peaked at the softest point of the
material (unlike what happens for the depinning of an elastic interface).
Contrary to widespread views, the deformation halo around this peak does not
always have a quadrupolar (Eshelby-like) shape. Instead, for finite and
narrowly-distributed disorder, it looks like a fracture, with a strain field
that concentrates along some easy directions. These findings are rationalised
with analytical calculations in the case where the plastic disorder is confined
to a point-like `impurity'. In this case, we unveil a continuous family of
elastic propagators, which are identical for the soft modes and for the
equilibrium configurations. This family interpolates between the standard
quadrupolar propagator and the fracture-like one as the anisotropy of the
elastic medium is increased. Therefore, we expect to see a fracture-like
propagator when extended regions on the brink of failure have already softened
along the shear direction and thus rendered the material anisotropic, but not
failed yet. We speculate that this might be the case in carefully aged glasses
just before macroscopic failure.