  We study theoretically the yielding of sheared amorphous materials as a
function of increasing levels of initial sample annealing prior to shear, in
three widely used constitutive models and three widely studied annealing
protocols. In thermal systems we find a gradual progression, with increasing
annealing, from smoothly "ductile" yielding, in which the sample remains
homogeneous, to abruptly "brittle" yielding, in which it becomes strongly shear
banded. This progression arises from an increase with annealing in the size of
an overshoot in the underlying stress-strain curve for homogeneous shear, which
causes a shear banding instability that becomes more severe with increasing
annealing. "Ductile" and "brittle" yielding thereby emerge as two limiting
cases of a continuum of yielding transitions, from gradual to catastrophic. In
contrast, athermal systems with a stress overshoot always show brittle yielding
at low shear rates, however small the overshoot.
