Abstract
In contrast to the highly successful theory of crystalline elasticity, there is no well-established, universal, theoretical framework that describes stress transmission in disordered, jammed solids. The puzzle is rooted in the question of how non-periodic networks, which are apparently indistinguishable from a snapshot of a fluid, sustain shear. There have been many approaches to addressing this puzzle, including the tools of rigidity percolation, the concept of non-affine strain and plasticity, and a quest for a stress-only framework that completely eliminates the notion of strain. Here, we present a stress-only theory of elasticity for a non-thermal amorphous assembly of grains in a jammed solid, where each grain is subjected to mechanical constraints of force and torque balance. These grain-level constraints lead to the Gauss's law of an emergent $U(1)$ tensor electromagnetism, which then accounts for the mechanical response of such solids. This formulation of amorphous elasticity has several immediate consequences: the mechanical response maps exactly to the static, dielectric response of this tensorial electromagnetism with the polarizability of the medium mapping to an emergent elastic modulus tensor; external forces act as vector electric charges whereas the tensorial magnetic fields are sourced by momentum density. In addition to presenting this general framework of amorphous elasticity, we present detailed simulation results of frictionless granular packings to test the predictions in the static limit of the theory. In addition, we have also demonstrated that some aspects of the theory work remarkably well in predicting the stress-state of experimentally prepared frictional granular packings.