  Solids are distinguished from fluids by their ability to resist shear. In
traditional solids, the resistance to shear is associated with the emergence of
broken translational symmetry as exhibited by a non-uniform density pattern. In
this work, we focus on the emergence of shear-rigidity in a class of solids
where this paradigm is challenged. Dry granular materials have no energetically
or entropically preferred density modulations. We show that, in contrast to
traditional solids, the emergence of shear rigidity in these granular solids is
a collective process, which is controlled solely by boundary forces, the
constraints of force and torque balance, and the positivity of the contact
forces. We develop a theoretical framework based on these constraints, which
connects rigidity to broken translational symmetry in the space of forces, not
positions of grains. We apply our theory to experimentally generated
shear-jammed (SJ) states and show that these states are indeed characterized by
a persistent, non-uniform density modulation in force space, which emerges at
the shear-jamming transition.
