  Yield stress materials flow if a sufficiently large shear stress is ap-
plied. Although such materials are ubiquitous and relevant for indus- try,
there is no accepted microscopic description of how they yield, even in the
simplest situations where temperature is negligible and where flow
inhomogeneities such as shear bands or fractures are ab- sent. Here we propose
a scaling description of the yielding transition in amorphous solids made of
soft particles at zero temperature. Our description makes a connection between
the Herschel-Bulkley expo- nent characterizing the singularity of the flow
curve near the yield stress {\Sigma}c, the extension and duration of the
avalanches of plasticity observed at threshold, and the density P(x) of soft
spots, or shear transformation zones, as a function of the stress increment x
be- yond which they yield. We argue that the critical exponents of the yielding
transition can be expressed in terms of three independent exponents {\theta},
df and z, characterizing respectively the density of soft spots, the fractal
dimension of the avalanches, and their duration. Our description shares some
similarity with the depinning transition that occurs when an elastic manifold
is driven through a random potential, but also presents some striking
differences. We test our arguments in an elasto-plastic model, an automaton
model similar to those used in depinning, but with a different interaction
kernel, and find satisfying agreement with our predictions both in two and
three dimensions.
