  Attractive colloids display two distinct amorphous solid phases: the
attractive glass, due to particle bonding, and the repulsive glass, due to the
hard core repulsion. By means of a microscopic mean field approach, we analyze
their response to a quasi-static shear strain. We find that the presence of two
distinct interaction length scales may result in a sharp two-step yielding
process, which can be associated with a hysteretic stress response, or with a
reversible but non-monotonic stress-strain curve. We derive a generic phase
diagram characterized by two distinct yielding lines, an inverse yielding, and
a critical point separating the hysteretic and reversible regimes. Our results
should be applicable to a large class of glassy materials characterized by two
distinct interaction length scales.
