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Random Elasticity
Take a random packing of particles in a state of mechanical equilibrium.
These particles might represent grains in a granular packing, bubbles in a dense emulsion, or atoms in a metallic glass.
Now apply some homogeneous shear to the boundaries of the system and all the particles.
If the particles happen to form a Bravais crystal, then the sheared system (still a Bravais crystal) will still be in a state of mechanical equilibrium.
On the other hand if the particles start out in a configuration which is not a Bravais crystal, such as the red discs shown in the figure, they will need to make some corrections to the homogeneous shear (represented in the figure by the black arrows) in order to remain in equilibrium.
These corrections can reduce the shear modulus of the system away from the naive expectation, and they completely dominate the behavior in purely repulsive systems near the onset of rigidity.
In, [2], Anaël Lemâitre and I showed how these inhomogeneous corrections are responsible for nucleating plasticity in slowly driven systems.
In, [3], we showed how the corrections to the shear modulus can come from a rather broad range of the vibrational spectrum.
In, [6], I showed how simple assumptions about the nature of the eigenmodes of the system lead to a particularly simple form for the real-space correlations of the elastic displacement field.