On the density of shear transformations in amorphous solids

We study the stability of amorphous solids, focussing on the distribution P(x) of the local stress increase x that would lead to an instability. We argue that this distribution behaves as P(x) similar to x(theta), where the exponent theta is larger than zero if the elastic interaction between rearranging regions is non-monotonic, and increases with the interaction range. For a class of finite-dimensional models we show that stability implies a lower bound on theta, which is found to lie near saturation. For quadrupolar interactions these models yield. theta approximate to 0.6 for d = 2 and. theta approximate to 0.4 in d = 3 where d is the spatial dimension, accurately capturing previously unresolved observations in atomistic models, both in quasi-static flow and after a fast quench. In addition, we compute the Herschel-Buckley exponent in these models and show that it depends on a subtle choice of dynamical rules, whereas the exponent theta does not. Copyright (c) EPLA, 2014