A problem related to the stability of force chains

When a load is applied to a granular material, the stress is not uniformly distributed but concentrates into quasi-linear particle assemblies known as ''force chains.'' By the time they can be observed, force chains are apparently quasistatic in the sense that they persist over time scales much longer than elastic timescales (such as a contact time) over which an unstable chain breaks apart. But many force chains attempt to form, but are unstable and break apart after a few contact times. Stability requires that each particle in the chain be pressed against its neighbors by the forces in the chain. Furthermore, each particle must also be in quasistatic equilibrium in the sense that all forces on it must roughly balance. This paper examines a simple linear string of particles in an attempt to estimate how much force is required to hold the chain together and how large a force imbalance can be tolerated. There are two modes of instability, a simple case in which an overloaded contact pushes its particles apart breaking the chain as they separate, and a more complicated mode requiring the interaction of elastic waves traveling along the chain. Overall, dissipation acts as a stabilizing factor, first by reducing the initial energy of the overloaded contact as it unloads, and subsequently, by reducing the energy of the elastic waves.