  Yielding transitions in athermal amorphous materials resemble critical
phenomena. Historically, they have been described by the Herschel-Bulkley
rheological formula, which implies singular behaviors at yield points. In this
paper, I examine this class of phenomena using an elementary version of the
thermodynamic shear-transformation-zone (STZ) theory, focusing on the role of
the effective disorder temperature, and paying special attention to scaling and
dimensional arguments. I find a wide variety of Herschel-Bulkley-like
rheologies but, for fundamental reasons not specific to the STZ theory,
conclude that the yielding transition is not truly critical. In particular,
there is a correlation length that grows rapidly, but ultimately saturates near
the yield point.
