D. V. Grinev

Granular materials: Towards the statistical mechanics of jammed configurations

The properties of granular materials can be well defined, that is be a branch of physics, but conventional statistical mechanics is inadequate to handle what amounts to the physics of disordered packings of hard-core particles, either static or driven by external forces. A new approach that employs statistical-mechanical concepts is offered for the description of such systems. An analysis of the stress field in static granular packings is given within the framework of this approach. There are more conventional systems such as polymer glasses which have a rather similar statistical physics to granular media, and some speculative ideas are offered which are a real departure from conventional glass theories.

Transmission of stress in granular materials as a problem of statistical mechanics

We consider the problem of stress transmission in granular materials. We formulate the simplest statically determinate problem of stress transmission through a static granular material. This is the case when grains are rigid and have an average coordination number of z=d+1. Under this condition the system of Newtons equations of interparticle force and torque balance is complete. This means that there exists a complete set of equations for the macroscopic stress tensor σij(r) i.e., the d (where d is the dimension of the problem) equations of force balance ∇jσij(r)=gi(r) have to be supported by d(d−1)/2 equations. These equations have their origin in Newtons laws of interparticle force and torque balance and incorporate tensorial geometrical characteristics of the packing. We conjecture that in order to have a coherent and self-consistent continuum theory of stress transmission in static granular media it is necessary to link the averaging procedure to the concept of compactivity. We emphasize that although real granular materials have many features ignored within the proposed framework it is essential for making progress to derive equations of stress transmission for the simplest model, as opposed to guessing and postulating.

The statistical-mechanical theory of stress transmission in granular materials

We propose a statistical-mechanical theory of stress transmission in disordered arrays of rigid grains with perfect friction. Starting from the equations of intergranular force and torque balance we derive the fundamental equations that govern the propagation of stress through an aggregate. We illustrate the validity of our approach on two simple geometries.

The tensorial formulation of volume function for packings of particles

Abstract A statistical–geometrical theory of packings of particles is given. Probability distribution of density is written in terms of the probability distribution of contact points. Formulation of volume function is presented in terms of configuration tensors. Model volume function is discussed for a simple packing. Links between the invariants of configuration tensors, pair correlation functions and orientational order parameter are highlighted.

The stress transmission universality classes of periodic granular arrays

The transmission of stress is analysed for static periodic arrays of rigid grains, with perfect and zero friction. For minimal coordination number (which is sensitive to friction, sphericity and dimensionality), the stress distribution is soluble without reference to the corresponding displacement fields. In non-degenerate cases, the constitutive equations are found to be simple linear in the stress components. The corresponding coefficients depend crucially upon geometrical disorder of the grain contacts.

Jammed systems in slow flow need a new statistical mechanics

The slow dynamics of granular flow is studied as an extension of static granular problems, which, as a consequence of shaking or related regimes, can be studied by the methods of statistical mechanics. For packed (i.e. ‘jammed’), hard and rough objects, kinetic energy is a minor and ignorable quantity, as is strain. Hence, in the static case, the stress equations need supplementing by ‘missing equations’ depending solely on configurations. These are in the literature; this paper extends the equilibrium studies to slow dynamics, claiming that the strain rate (which is a consequence of flow, not of elastic strain) takes the place of stress, and as before, the analogue of Stokess equation has to be supplemented by new ‘missing equations’ which are derived and which depend only on configurations.

The missing stress-geometry equation in granular media

The simplest solvable problem of stress transmission through a static granular material is when the grains are perfectly rigid and have an average coordination number of z=d+1. Under these conditions there exists an analysis of stress which is independent of the analysis of strain and the d equations of force balance ∇jσij(r)=gi(r) have to be supported by d(d−1)/2 equations. These equations are of purely geometric origin. A method of deriving them has been proposed in an earlier paper (Edwards and Grinev, Phys. Rev. Lett. 82 (1999) 5397). In this paper alternative derivations are discussed and the problem of the “missing equations” is posed as a geometrical puzzle which has yet to find a systematic solution as against sensible but fundamentally arbitrary approaches.

Theory of Vibration-Induced Density Relaxation of Granular Material

We propose a simple theory which describes the settling of loosely packed, cohesionless granular material under mechanical vibrations. Using thermal analogies and basing the theory on an entropic concept, formula are derived for how the density of a powder depends on its history. Comprehensive data from the Chicago group show how an initially deposited powder changes its density under carefully controlled packing to reach a terminal density at a terminal tapping rate. A reduction in the tapping rate moves the system to a higher density, but whereas the first measurements follow an irreversible curve, the second measurements have established a reversible curve. These two regimes of behaviour are analyzed theoretically and a qualitative understanding of them is offered.