  We derive exact results for stress correlations in near-crystalline systems
in two and three dimensions. We study energy minimized configurations of
particles interacting through Harmonic as well as Lennard-Jones potentials, for
varying degrees of microscopic disorder and quenched forces on grains. Our
findings demonstrate that the macroscopic elastic properties of such
near-crystalline packings remain unchanged within a certain disorder threshold,
yet they can be influenced by various factors, including packing density,
pressure, and the strength of inter-particle interactions. We show that the
stress correlations in such systems display anisotropic behavior at large
lengthscales and are significantly influenced by the pre-stress of the system.
The anisotropic nature of these correlations remains unaffected as we increase
the strength of the disorder. Additionally, we derive the large lengthscale
behavior for the change in the local stress components that shows a $1/r^d$
radial decay for the case of particle size disorder and a $1/r^{d-1}$ behavior
for quenched forces introduced into a crystalline network. Finally, we verify
our theoretical results numerically using energy-minimised static particle
configurations.
