Scaling Description of Non-Local Rheology

Non-locality is crucial to understand the plastic flow of an amorphous material, and has been successfully described by the fluidity, along with a cooperativity length scale {\xi}. We demonstrate, by applying the scaling hypothesis to the yielding transition, that non-local effects in non-uniform stress configurations can be explained within the framework of critical phenomena. From the scaling description, scaling relations between different exponents are derived, and collapses of strain rate profiles are made both in shear driven and pressure driven flow. We find that the cooperative length in non-local flow is governed by the same correlation length in finite dimensional homogeneous flow, excluding the mean field exponents. We also show that non-locality also affects the finite size scaling of the yield stress, especially the large finite size effects observed in pressure driven flow. Our theoretical results are nicely verified by the elasto-plastic model, and experimental data.