Stress distribution in sandpiles – A variational approach (original) (raw)

Analytic solutions for the stress field in static sandpiles

Mechanics of Materials, 2016

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.

Granular elasticity: General considerations and the stress dip in sand piles

Physical Review E, 2006

Granular materials are predominantly plastic, incrementally nonlinear, preparation-dependent, and anisotropic under shear. Nevertheless, their static stress distribution is well accounted for, in the whole range up to the point of failure, by a judiciously tailored isotropic nonanalytic elasticity theory termed granular elasticity. The first purpose of this paper is to carefully expound this view. Then granular elasticity is employed to consider the stress distribution in two-dimensional sand piles ͑or sand wedges͒. Starting from a uniform density, the pressure at the bottom of the pile is found to show a single central peak. It turns into a pressure dip, if some density inhomogeneity, with the center being less compact, is assumed. These two pressure distributions are remarkably similar to recent measurements, made in piles obtained, respectively, by rainlike pouring and funneling. In an accompanying paper, the stress distributions in silos and under point loads, calculated using the same method, are also found to agree with experiments.

On the stress depression under a sandpile

Powder Technology, 1994

The observed minimum in the normal stress beneath the highest portion of a 'sandpile' (a pile of granular material) is a counter-intuitive result that has long remained unexplained.

The role of particle shape on the stress distribution in a sandpile

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008

The results of an experimental investigation into the counter-intuitive phenomenon that a local minimum in the normal stress profile is sometimes found under the apex of a sandpile are presented. Specifically, the effects of particle shape on the stress distribution are studied and it is shown that anisotropy of the particles significantly enhances the dip. This amplification is attributed to the mechanical stability induced by boundary alignment of the anisotropic particles. Circular, ellipsoidal and pear-shaped cylinders are used and the stress propagates principally towards the sides of the pile through primary stress chains. Secondary chains are also present and we propose that the relationship between the magnitudes of the ratio of primary to secondary chains is correlated with the size of the dip.

Simulation study on micro and macro mechanical behaviour of sand piles

Powder Technology, 2010

We investigate numerically the micro and macro mechanical behaviour of non-cohesive granular materials, especially in the static limit. To achieve this goal we performed numerical simulations generating twodimensional "sand piles" from several thousands of convex polygonal particles with varying shapes, sizes and corner numbers, using a discrete element approach based on soft particles. We emphasize that the displacement (strain) fields inside sand piles have not been measured in experiments on sand piles. Averaging is made reproducible by introducing a representative volume element (RVE), the size of which we determine by careful measurements. Stress tensors are studied for both symmetric and asymmetric sand piles in two-dimensional systems, where the particles are dropped from a point source. Furthermore, we determine the fabric tensor inside the sand piles. A surprising finding is the behaviour of the contact density in this kind of heap, which increases where the pressure is at a minimum. The fabric is linearly proportional to the product of the volume fraction and the mean coordination number for a pile consisting of monodisperse mixture of particles. We observe that the macroscopic stress, strain and fabric tensors are not collinear in the sand piles.

Variational formulation of sandpile dynamics

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996

Principle of stationary action was applied to derive a system of governing equations and boundary conditions describing dynamics of sandpile avalanches. The derived general system of equations for the sandpile dynamics is demonstrated to include equations of flow of granular material down a rigid wedge as a particular case. It is shown that the variational principle can be readily implemented in a numerical algorithm.

Numerical solutions of the variational equations for sandpile dynamics

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

A variational form of the energy conservation equation is applied to derive a system of variational inequalities describing the dynamics of sandpile growth on an arbitrary rigid support surface due to an external source of granular material. Water transport through the river networks and formation of lakes is also investigated as a limiting case of the sandpile evolution problem. A numerical algorithm is suggested for the solution of the obtained system of variational inequalities. The developed numerical procedure is applied for the investigation of the pile growth on a number of rigid support surfaces, water transport, and formation of lakes.

Forces in piles of granular material: an analytic and 3D DEM study

Granular Matter, 2001

We investigate the stress distribution at the base of a conical sandpile using both analytic calculations and a three dimensional discrete element code. In particular, we study how a minimum in the normal stress can occur under the highest part of the sandpile. It is found that piles composed of particles with the same size do not show a minimum in the normal stress. A stress minimum is only observed when the piles are composed of particles with different sizes, where the particles are size segregated in an ordered, symmetric, circular fashion, around the central axis of the sandpile. If a pile is composed of particles with different sizes, where the particles are randomly distributed throughout the pile, then no stress dip is observed. These results suggest that the stress dip is due to ordered, force contacts between equiheight particles which direct stress to the outer parts of the pile.

Micro and Macro Aspects of the Elastoplastic Behaviour of Sand Piles

Micro-Macro-interaction, 2008

We use a discrete element method to simulate the dynamics of granulates made up from arbitrarily shaped particles. Static and dynamic friction are accounted for in our force laws, which enables us to simulate the relaxation of (two-dimensional) sand piles to their final static state. Depending on the growth history, a dip in the pressure under a heap may or may not appear. Properties of the relaxed state are measured and averaged numerically to obtain the values of field quantitities pertinent for a continuum description. In particular, we show that it is possible to obtain not only stresses but also displacements in the heap, by judicious use of an adiabatic relaxation experiment, in which gravity is slowly changed. Hence the full set of variables of the theory of elastiticity is available, allowing comparison with elastoplastic models for granular aggregates. A surprising finding is the behaviour of the material density in a heap with dip, which increases where the pressure is minimum.