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Mathematical Physics Seminar

Emergence of Elasticity in Amorphous Solids

Bulbul Chakraborty, Brandeis University

Date & time: Wednesday, 27 October 2021 at 10:45AM - 11:45AM

The theory of elasticity of crystalline solids is one of the best-known field theories in physics.  It  emerges from the spontaneous breaking of translation symmetry and provides a complete description of the stress response based on  the principle of momentum conservation, which relates the divergence of the stress tensor to external forces, and a constitutive relation between stress and the strain field, which is defined in terms of the displacements from the unique reference structure defined by the broken-symmetry phase.  Applying this classical  paradigm to non-thermal (non-Brownian)  disordered solids is fraught with  difficulties.  To begin with, there is no obvious broken symmetry and hence there is no unique reference structure about which displacements can be defined.   Alternatively stated, each mechanically balanced jammed configuration is equally eligible as a  reference configuration allowing for a redundancy in definition of displacements that the course-grained observables are expected to be insensitive to.  In addition, there is no free-energy functional since these solids are not in thermal equilibrium, and thus a rigorous basis for a stress-strain constitutive relation is lacking.    

In this talk, I will discuss some of our recent work on the development of a theory of elasticity for such solids. Central to this framework is a gauge theoretic structure that arises from-- (1) the lack of a well-defined zero-stress reference configuration, and, (2) the local mechanical equilibrium of each grain in a non-thermal solid with the latter serving as a Gauss's law for a tensorial electric field. In this mapping, the forces act as charges of the so-called vector charge theory of a tensor electromagnetism first developed in the context of spin liquids, and force balance and torque balance are inherently incorporated in the structure of the theory. Computing the stress distribution in such amorphous solids then maps to solving the problem of electrostatics of this vector-charge theory in the presence of a dielectric. This stress-only framework is completely devoid of any reliance on a reference structure or displacement fields, which should not have any measurable consequences in amorphous solids. I will compare the theoretical predictions to experimental and numerical measurements of stress-stress correlations in jammed frictional and frictionless solids.