On the critical exponents of the yielding transition of amorphous solids

TL;DR

This paper numerically investigates the yielding transition in amorphous solids using a scalar model, revealing that critical exponents depend on microscopic potential details and connecting the model to other elastoplastic frameworks.

Contribution

It introduces a scalar model equivalent to coupled Prandtl-Tomlinson particles, demonstrating the dependence of critical exponents on microscopic potential details and relating it to existing elastoplastic models.

Findings

The strain rate vs stress curve follows a power law near critical stress.

Critical exponents depend on the microscopic plastic potential used.

Static exponents are independent of disorder details.

Abstract

We investigate numerically the yielding transition of a two dimensional model amorphous solid under external shear. We use a scalar model in terms of values of the total local strain, that we derive from the full (tensorial) description of the elastic interactions in the system, in which plastic deformations are accounted for by introducing a stochastic "plastic disorder" potential. This scalar model is seen to be equivalent to a collection of Prandtl-Tomlinson particles, which are coupled through an Eshelby quadrupolar kernel. Numerical simulations of this scalar model reveal that the strain rate vs stress curve, close to the critical stress, is of the form $\dot\gamma\sim (\sigma-\sigma_c)^\beta$. Remarkably, we find that the value of $\beta$ depends on details of the microscopic plastic potential used, confirming and giving additional support to results previously obtained with the full tensorial model. %\cite{Jagla_Yiel}. To rationalize this result, we argue that the Eshelby interaction in the scalar model can be treated to a good approximation in a sort of "dynamical" mean field, which corresponds to a Prandtl-Tomlinson particle that is driven by the applied strain rate in the presence of a stochastic noise generated by all other particles. The dynamics of this Prandtl-Tomlinson particle displays different values of the $\beta$ exponent depending on the analytical properties of the microscopic potential, thus giving support to the results of the numerical simulations. Moreover, we find that other critical exponents that depend on details of the dynamics show also a dependence with the form of the disorder, while static exponents are independent of the details of the disorder. Finally, we show how our scalar model relates to other elastoplastic models and to the widely used mean field version known as the H\'ebraud-Lequeux model.