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In this work we discuss possible physical origins for non-trivial exponentsin the athermal rheology of soft materials at low but finite driving rates. Akey ingredient in our scenario is the presence of a self-consistent mechanicalnoise that stems from the spatial superposition of long-range elastic responsesto localized plastically deforming regions. We study analytically a mean-fieldmodel, in which this mechanical noise is accounted for by a stress diffusionterm coupled to the plastic activity. Within this description we show how adependence of the shear modulus and/or the local relaxation time on the shearrate introduces corrections to the usual mean-field prediction, concerning theHerschel-Bulkley-type rheological response of exponent 1/2. This feature of themean-field picture is then shown to be robust with respect to structuraldisorder and partial relaxation of the local stress. We test this predictionnumerically on a mesoscopic lattice model that implements explicitly thelong-range elastic response to localized shear transformations, and we concludeon how our scenario might be tested in rheological experiments.

Nontrivial rheological exponents in sheared yield stress fluids