  3D Computer simulations and experiments are employed to study random packings
of compressible spherical grains under external confining stress. Of particular
interest is the rigid ball limit, which we describe as a continuous transition
in which the applied stress vanishes as (\phi-\phi_c)^\beta, where \phi is the
(solid phase) volume density. This transition coincides with the onset of shear
rigidity. The value of \phi_c depends, for example, on whether the grains
interact via only normal forces (giving rise to random close packings) or by a
combination of normal and friction generated transverse forces (producing
random loose packings). In both cases, near the transition, the system's
response is controlled by localized force chains. As the stress increases, we
characterize the system's evolution in terms of (1) the participation number,
(2) the average force distribution, and (3) visualization techniques.
