Analytic solutions for the stress field in static sandpiles

In the present paper we propose a new class of analytical solutions for the equilibrium problem of a prismatic sand pile under gravity, capturing the effects of the history of the sand pile formation on the stress distribution. The material is modeled as a continuum composed by a cohesionless granular material ruled by Coulomb friction, that is a material governed by the Mohr-Coulomb yield condition. The closure of the balance equations is obtained by considering a special restriction on stress, namely a special form of the stress tensor relative to a special curvilinear, locally non-orthogonal, reference system. This assumption generates a class of closed-form equilibrium solutions, depending on three parameters. By tuning the value of the parameters a family of equilibrium solutions is obtained, reproducing closely some published experimental data, and corresponding to different construction histories, namely, for example, the deposition from a line source and by uniform raining. The repertoire of equilibrated stress fields that we obtain in two special cases contains an approximation of the Incipient Failure Everywhere (IFE) solution and a closed-form description of the arching phenomenon.