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Dense granular gases: Can they jam?

Wednesday 7th January 2009 - 10:30 to 10:55
INI Seminar Room 1
Dry granular matter is described naturally by a graph. Two grains in repulsive contact in a granular fluid, or just about to collide in a gas, are represented by two vertices connected by an edge. There are no attractive, cohesive forces between grains and contacts between grains are imposed by external forces (gravity or boundaries) or by collisions. In hard dry granular materials with infinite tangential friction, the forces are scalar, geometrical contacts between grains. The elementary excitations are non-slip rotations of a grain another if it is permitted. Then, the granular material flows like a three-dimensional bearing. If not, it is jammed. In a granular gas, collisions (removal of a contact and its replacement by another) are additional elementary excitations. In the graph, the vertices are the grains, and the edges, the contacts represented by the adjacency matrix. Circuits of grains in contact can be even or odd, and the material is essentially discrete. Its physical properties are given by the eigenvalues and eigenvectors of the adjacency matrix. that is by linear algebra of the graph. Thus, whereas the statics is nonlinear, the topological dynamics, and the generic physical behavior of the granular material is a problem of linear algebra, because we have replaced material elements by geometrical objects, in a structure that is a graph. A granular material is therefore a metamaterial, with generic physical behavior given by its structure rather than by the chemistry or hardness of its constituents. In fact, it behaves like soft matter: large, extended deformations with localized critical stresses. The graph structure has the essential elements of discreteness (granularity), odd circuits (''arches'') and disorder. There can be no defect-free, continuous model of granular material, no constitutive equation. The two possible states of (disordered) granular matter, dry fluid or collision-dominated gas, and jammed, rigid but fragile solid are direct results of the topological dynamics of the graph, where the elements responsible for the jammed state are odd circuits, circuits with an odd number of grains in contact. The granular material (n grains) is jammed by the c odd circuits that frustrate the non-slip rotation. The lowest eigenvalue of the dynamical matrix is 4c/n. Odd circuits are the « arches » holding the granular packing together. The odd vorticity, core of odd circuits, forms closed (R-) loops, that are large in disordered granular materials of size L. (In ordered materials, their size is limited by the periodicity of the close-packed structure). The disordered granular material is a fragile, jammed solid stabilized by c odd circuits. The ''order'' parameter 4c/n ~ 1/L is extended over the entire material, and the unjamming transition between fragile solid and dry fluid is a second-order, scaling transition with intermittence (The large R-loops have a small line tension ~ 1/L2 and cannot be driven to shrink).
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons