Supplemental Material for : “ Driving rate dependence of avalanche statistics and shapes at the yielding transition ”

We study the scalar elasto-plastic model in two (2d) and three dimensions (3d) under the presence of an imposed shear-rate, following the modifications proposed by Nicolas et al. [1] to the model of Picard and coworkers [2, 3]. An amorphous system is represented by a coarsegrained scalar field σ(r, t), denoting the instantaneous deviatoric shear stress of the system at spatial position r and time t upon the application of a simple shear. An over-damped dynamics is imposed for this scalar quantity following some basic rules: (i) The stress loads locally in an elastic manner while it is below a certain yield stress σY(r). (ii) When the local stress overcomes the local yield stress, a plastic event occurs. Dissipation occurs locally, expressed as a progressive drop of the local stress, together with a redistribution of the stresses in the rest of the system, provided by a long-range elastic perturbation. This process stops when a criterion for the accumulated local strain is met, the region recovers its elastic properties and a new local yield threshold is chosen from a given distribution. The shear stress perturbation caused on the system is computed within the framework of tensorial linear elasticity assuming an isotropic incompressible material [2]. A Green’s function G(r, r′) relates the stress variation δσ at each point in space with the corresponding component of the plastic strain γpl(r′; t) associated with a plastic event occurring at r′. The perturbation given by the elastic propagator G(r, r′) can be approximated by the far field expression [2, 3] of the continuum mechanics solution [4]