Indeterminacy and the onset of motion in a simple granular packing.

We examine the relatively simple problem of a disk placed in a symmetric V-shaped channel, and subjected to gravity and a torque. We obtain analytic predictions of the contact forces and disk motion using two different models. In the first model, the disk is assumed to be perfectly rigid, leading to force indeterminacy. In the second model, the disk is assumed to interact with the walls of the channel via linear springs, leading to a unique solution for the contact forces. The results of these two models are compared. It is shown that there are two possible ways motion can occur--through the appearance of a null eigenvector of the stiffness matrix, or through an instability. When motion occurs through an instability, the first model cannot predict when the disk will rotate; it is necessary to know the undetermined forces in order to predict the motion of the disk. It is also shown how indeterminacy in the first model is linked to memory in the second. The analytical results are also compared with numerical simulations using two different methods, each related to one of the models.