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

Abstract

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 wedges). 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.

Granular solid hydrodynamics

“…With τ ∼ T −1 g diverging, granular stress persists forever, displaying in essence elastic behavior, see [63][64][65][66][67]. When granular media are being sheared, because T g = 0, the stress relaxes irreversibly.…”

From Elasticity to Hypoplasticity: Dynamics of Granular Solids

“…And since the elastic stress is the only contribution in statics, force balance reads ∇ j σ ij = ∇ j π ij = ρG i . This was solved for three classical cases: silos, sand piles and granular sheets under a point load, resulting in rather satisfactory agreement to experiments, see [4]. Moreover, the energy w (with P ≡ 1 3 π ) is convex only for π s /P ≤ 2/ξ, implying no elastic solution is stable beyond it.…”

“…Next, we compare Eqs (15, 16) to (4,5) in their results with respect to "response envelopes," a standard test in soil mechanics for rating constitutive relations [5]. Axial symmetry of the triaxial geometry is assumed, with σ ij , v ij diagonal, and σ 1 ≡ σ xx = σ yy ,…”

“…The entropy s, on the other hand, is closer to the so-called "configurational entropy," [8] (see section 6 of the first of [4] for a discussion of their relationship). The balance equations are…”

“…Expressed as here, the stress and density dependence are essentially determined by M ijk that (with ξ = 5/3 and B 0 = 8500 MPa) is a known quantity [4]. For the four free constants, we take…”