  It is well known experimentally that well-quenched amorphous solids exhibit a
plastic instability in the form of a catastrophic shear localization at a well
defined value of the external strain. The instability may develop to a
shear-band that in some cases is followed by a fracture. It is also known that
the values of the yield-strain (and yield-stress), as well as the direction of
the shear band with respect to the principal stress axis, vary considerably
with variations in the external loading conditions. In this paper we present a
microscopic theory of these phenomena for 2-dimensional athermal amorphous
solids that are strained quasi-statically. We present analytic formulae for the
yield-strains for different loading conditions, and well as for the angles of
the shear bands. We explain that the external loading conditions determine the
eigenvalues of the quadrupolar Eshelby inclusions which model the non-affine
displacement field. These inclusions model elementary plastic events and
determine both the yield-strain and the direction of the shear-band. We show
that the angles of the shear bands with respect to the principal stress axis
are limited theoretically between $30^o$ and $60^o$. Available experimental
data conform to this prediction.
