Variational Principles for Granular Matter

Abstract

In the last few years research on so-called soft materials, as opposed to classical states like solids, fluids, and gases, has attracted much attention. Among various types of foams, polymers, etc., granular materials appear as most interesting in view of a variety of surprising experiments, novel connections to mathematical fields like geometry, partial differential equations, particle systems, and also industrial applications [9] [16] [15]. It is surprising that granular matter became an attractive research field only recently since basic experiments on equilibrium configurations have been performed a century ago [1]. In some situations granular matter can be seen as a granular gas (Boltzmann approach for discrete particles with free path length [20]) or as a granular fluid (Savage-Hutter model for height and speed of avalanches [22] [23], an adaptation of the shallow water or St. Venant equations). If large amounts of sand are slowly accumulated then continuum models are appropriate which use the (geometry of sand) and the distinction of a standing layer and a rolling layer [8] [5] [6] [10] [11] [12] [13]. These models are based on the angle of repose of the material considered and two functional laws which describe how the speed of rolling grains is related to the slope of the standing material and how grains are deposited or start moving on a slope. Also in stationary situations with a non-vanishing rolling layer these two parameter functions play a role [12]. Models with instantaneous deposition have been studied in [2] [3] [19]. Another approach to stationary problems has been designed in [21].