  We study numerically the yielding transition of a two dimensional model glass
subjected to athermal quasi-static cyclic shear deformation, with the aim of
investigating the effect on the yielding behaviour of the degree of annealing,
which in turn depends on the preparation protocol. We find two distinct regimes
of annealing separated by a threshold energy. Poorly annealed glasses
progressively evolve towards the threshold energy as the strain amplitude is
increased towards the yielding value. Well annealed glasses with initial
energies below the threshold energy exhibit stable behaviour, with negligible
change in energy with increasing strain amplitude, till they yield.
Discontinuities in energy and stress at yielding increase with the degree of
annealing, consistently with recent results found in three dimensions. We
observe significant structural change with strain amplitude that closely
mirrors the changes in energy and stresses. We investigate groups of particles
that are involved in plastic rearrangements. We analyse the distributions of
avalanche sizes, of clusters of connected rearranging particles, and related
quantities, employing finite size scaling analysis. We verify previously
investigated relations between exponents characterising these distributions,
and a newly proposed relation between exponents describing avalanche and
cluster size distributions.
