Properties of the density of shear transformations in driven amorphous solids

The strain load Δγthat triggers consecutive avalanches is a key observable in the slow deformation of amorphous solids. Its temporally averaged value ⟨Δγ⟩ displays a non-trivial system-size dependence that constitutes one of the distinguishing features of the yielding transition. Details of this dependence are not yet fully understood. We address this problem by means of theoretical analysis and simulations of elastoplastic models for amorphous solids. An accurate determination of the size dependence of ⟨Δγ⟩ leads to a precise evaluation of the steady-state distribution of local distances to instabilityx. We find that the usually assumed formP(x) ∼x(withθbeing the so-called pseudo-gap exponent) is not accurate at lowxand that in generalP(x) tends to a system-size-dependentfinitelimit asx→ 0. We work out the consequences of this finite-size dependence standing on exact results for random-walks and disclosing an alternative interpretation of the mechanical noise felt by a reference site. We test our predictions in two- and three-dimensional elastoplastic models, showing the crucial influence of the saturation ofP(x) at smallxon the size dependence of ⟨Δγ⟩ and related scalings.