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

The Yield-Strain and Shear-Band Direction in Amorphous Solids Under General Loading