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. V ibration increases the local height, but is counterbalanced by the nonlinea r coupling, which tends to suppress the growth of the height. The steady st ate heap can be solved in closed form in terms of Jacobian elliptic functio ns. Phenomena such as heap formation and downward and upward heaps can be r eproduced. Our results agree with the experimentally observed change of dow nward to upward steady heaps as the vibration strength is increased. Predic tions from the model compare favorably with experimental results on heap pr ofiles and heaping angles.