Symmetric heaping in grains: a phenomenological model

Heap formation of granular materials in a vertical vibrating bed is studied by a simple model using the profile of the heap as the dynamic variable. Vibration increases the local height, but is counterbalanced by the nonlinear coupling, which tends to suppress the growth of the height. The steady state heap can be solved in closed form in terms of Jacobian elliptic functions. Phenomena such as heap formation and downward and upward heaps can be reproduced. Our results agree with the experimentally observed change of downward to upward steady heaps as the vibration strength is increased. Predictions from the model compare favorably with experimental results on heap profiles and heaping angles.