Odd dipole screening in disordered matter

Disordered solids, straddling the solid-fluid boundary, lack a comprehensive continuum mechanical description. They exhibit a complex microstructure wherein multiple meta-stable states exist. Deforming disordered solids induces particles rearrangements enabling the system to transition between meta-stable states. A dramatic consequence of these transitions is that quasistatic deformation cycles modify the reference state, facilitating the storage and release of mechanical energy. Here we develop a continuum mechanical theory of disordered solids, which accounts for the absence of a reference state and the lack of conserved potential energy. Our theory, which introduces a new modulus describing non-conservative mechanical screening, reduces to classical elasticity in the absence of screening. We analytically derive predictions for the deformation field for various perturbations and geometries. While our theory applies to general disordered solids, we focus on a two-dimensional disordered granular system and predict accurately the non-affine displacement fields observed in experiments for both small and large deformations, along with the observable vanishing shear modulus. The new proposed moduli satisfy universal relations that are independent of the specific experimental realization. Our work thus forms the basis of an entirely new family of continuum descriptions of the mechanics of disordered solids.