Origin of rigidity in marginal solids

Abstract

Granular materials are seemingly simple — they are a collection of particles large enough to be seen by eye, yet they exhibit a large range of totally different behaviors: liquid, gas and solid. We study various interesting aspects of granular materials: (1)First, analysis of experimental data yields a completely new route in which a granular fluid becomes a solid. These new states, know as shear jammed(SJ) granular matter, are an addition to the jamming framework proposed over 10 years ago. (2)We then focus on the emergence of shear-rigidity in granular materials which have no energetically preferred density modulations. In contrast to traditional solids, the emergence of mechanical rigidity in these marginal granular solids is a collective process, which is controlled solely by boundary forces, the constraints of force and torque balance, and the positivity of the contact forces. We develop a theoretical framework and show that these solids have internal patterns that are most naturally represented in the space of gauge fields that impose the constraints. Broken translational invariance in gauge space is a necessary condition for rigidity in granular solids. We use this theoretical framework to understand the stress fluctuations and the ability to resist deformations in SJ states. (3)We generalize the stress ensemble framework to a full tensor representation. In this framework, the angoricity is an intensive quantity, which is conjugate to the stress rather than the energy. We compare the predictions of this framework in SJ states and explain stress fluctuations in SJ states.(4)We also construct a dynamical framework to model deformation and stress avalanches in a slowly driven granular matter. The model assumes small patches of the driven granular material undergoing a stochastic evolution in a stress landscape, elastic deformations are equivalent to tilting the landscape while rearrangements correspond to hopping between different metastable states in this landscape. All processes in this landscape are aided by the angoricity from the stress ensemble, which is an analog of temperature in a thermal activated process.