  The behavior of shear-oscillated amorphous materials is studied using a
coarse-grained model. Samples are prepared at different degrees of annealing
and then subject to athermal and quasistatic oscillatory deformations at
various fixed amplitudes. The steady-state reached after several oscillations
is fully determined by the initial preparation and the oscillation amplitude,
as seen from stroboscopic stress and energy measurements. Under small
oscillations, poorly annealed materials display shear-annealing, while
ultra-stabilized materials are insensitive to them. Yet, beyond a critical
oscillation amplitude, both kind of materials display a discontinuous
transition to the same mixed state composed by a fluid shear-band embedded in a
marginal solid. Quantitative relations between uniform shear and the
steady-state reached with this protocol are established. The transient regime
characterizing the growth and the motion of the shear band is also studied.
