Origin of Rigidity in Dry Granular Solids Bulbul Chakraborty, Brandeis University, DMR 0905880

1. Origin of Rigidity in Dry Granular SolidsBulbul Chakraborty, Brandeis University, DMR 0905880 Dry grains interact via purely repulsive, contact interactions. Yet, externally imposed stresses lead to the onset of solidity (the jamming transition). A basic question in granular physics is what imparts shear rigidity to a collection of dry grains. Conventional solids, both crystalline and amorphous, possess permanent spatial density modulations that reflect energetic preferences. Their rigidity comes from the energetic cost of straining or deforming these permanent density modulations. Dry granular solids, however, do not have a well-defined zero stress states characterized by energetically preferred density modulations in space. What is paramount in determining structures in granular solids are the constraints of force and torque balance. These constraints lead to the existence of gauge fields much like the situation in topologically ordered states such as the quantum hall states. In our recent work, we argue that shear rigidity in dry-granular solids is a reflection of order in gauge space. The gauge fields can be represented by tilings as illustrated in the figure. The positivity constraint on contact forces makes the gauge-field tiling behave like an elastic network. Increasing stresses on a granular solid translates to an elastic strain on the gauge-field tilings. Rigidity is a consequence of permanent density modulations in gauge-space. • Due to Newton's second law, the forces acting on a grain must sum up to zero in a packing, leading to the definition of a gauge (height) field residing on the voids between grains (red points). • Each grain is defined by a force tile. The vertices of this force-tile correspond to the values of the heights. • A typical tiling constructed from force-tiles corresponding to granular solid where each grain has 6 contacts. The non-cohesive nature of contact forces guarantee the convexity of the force tiles. The boundary is defined by externally imposed stresses. • Illustration of permanent density modulation in height space. Violating the convexity constraint leads to the disappearance of this pattern

2. Broader ImpactsBulbul Chakraborty, Brandeis University, DMR 0905880 Chakraborty Group (Summer 2012) Back Row (L-R): Victoria Wu (REU student from Smith College), Sumantra Sarkar, Danny Goldstein (graduate students) Front Row (L-R): Dapeng Bi, Michael Giver (Graduate students), Michal Dichter (Undergraduate working on hopper project with Shubha Tewari and the PI) • Vice-Chair (Chair-elect) of • 2012 Gordon Conference on Granular & Granular-Fluid Flow • Organizer and PI of first NSF-funded Advanced Studies Institute in India (to be held in June, 2013) • Co-PI of Brandeis IGERT on Geometry and Dynamics