Sand piles do not have a perfectly conical shape as suggested by the concept of the angle of repose but have as we found at the bottom a logarithmic tail which is due to some simple similarity argument.

The figure shows an experimental heap (polenta). In addition to the analytical shape we also see the straight line that defines the angle of repose. For more details see the paper published with Juanjo Alonso in Phys. Rev. Lett. 76 (1996) 4911-4914. Similarly one can obtain the shape in a two dimensional silo as published in Comptes Rendus de l'Academie des Sciences Paris, Vol.326, Serie IIb, p.61-67 (1998) with Yan Grasselli:

Its relation to the angle of repose and the dependence on the filling procedure for silos and heaps was systematically studied in 2-d Hele Shaw cells in the paper published in Europ.J.Phys. B, Vol.10, p.673-679 (1999). With Oron Gadi and Stefano Zapperi we made a model to explain the shapes of Yan Grasselli's two-dimensional heaps as published in Granular Matter Vol.2, p.97-100 (2000).

Three dimensional heaps under the impact of particles falling from a certain height form craters which were measured in the paper published in Granular Matter Vol.3, p.201-204 (2001)

Comparison with experiments made at the University of Habana are given in a preprint having the following figures:

For other papers with Stefano Zapperi see also paper , paper and paper