More than one century ago Janssen [Z. Ver. Dtsch. Ing. 39, 1045 (1895)] proposed an elegant model to describe the pressure variations in a vertical container filled with a granular material at rest. In the present paper we build up a dynamical version of this model. We analyze the propagation of a pressure front in a dry granular medium inside a cylinder, taking into account the solid friction that exists between the grains and the cylinder walls. Assuming that the granular material under pressure has a linear elastic behavior, we derive a linear partial differential equation for the pressure field. Using the Green function method, we determine analytically the behavior of the granular medium undergoing a pressure step. We find in particular that a pressure front propagates at speed c, the speed of sound in the granular material (within the linear elasticity framework, c is a constant). Due to friction at the cylinder walls, the front amplitude decays exponentially. We also show that a stopping front starts after a certain time lag and propagates behind the pressure front, at a speed larger than c. When reached by this second front, the grains stop and do not move any more. The final pressure profile that we predict when all grains have eventually stopped is similar, but not identical, to the pressure profile determined by the Janssen model.

DOI:https://doi.org/10.1103/PhysRevE.55.5759