  We perform computational studies of repulsive, frictionless disks to
investigate the development of stress anisotropy in mechanically stable (MS)
packings. We focus on two protocols for generating MS packings: 1) isotropic
compression and 2) applied simple or pure shear strain $\gamma$ at fixed
packing fraction $\phi$. MS packings of frictionless disks occur as geometric
families (i.e. parabolic segments with positive curvature) in the
$\phi$-$\gamma$ plane. MS packings from protocol 1 populate parabolic segments
with both signs of the slope, $d\phi/d\gamma >0$ and $d\phi/d\gamma <0$. In
contrast, MS packings from protocol 2 populate segments with $d\phi/d\gamma <0$
only. For both simple and pure shear, we derive a relationship between the
stress anisotropy and dilatancy $d\phi/d\gamma$ obeyed by MS packings along
geometrical families. We show that for MS packings prepared using isotropic
compression, the stress anisotropy distribution is Gaussian centered at zero
with a standard deviation that decreases with increasing system size. For shear
jammed MS packings, the stress anisotropy distribution is a convolution of
Weibull distributions that depend on strain, which has a nonzero average and
standard deviation in the large-system limit. We also develop a framework to
calculate the stress anisotropy distribution for packings generated via
protocol 2 in terms of the stress anisotropy distribution for packings
generated via protocol 1. These results emphasize that for repulsive
frictionless disks, different packing-generation protocols give rise to
different MS packing probabilities, which lead to differences in macroscopic
properties of MS packings.
