Raphael Blumenfeld

Granular Entropy: Explicit Calculations for Planar Assemblies

This paper proposes a new volume function for calculation of the entropy of planar granular assemblies. This function is extracted from the antisymmetric part of a new geometric tensor and is rigorously additive when summed over grains. It leads to the identification of a conveniently small phase space. The utility of the volume function is demonstrated on several case studies, for which we calculate explicitly the mean volume and the volume fluctuations.

Stresses in Isostatic Granular Systems and Emergence of Force Chains

Progress is reported on several questions that bedevil understanding of granular systems: (i) Are the stress equations elliptic, parabolic, or hyperbolic? (ii) How can the often-observed force chains be predicted from a first-principles continuous theory? (iii) How do we relate insight from isostatic systems to general packings? Explicit equations are derived for the stress components in two dimensions including the dependence on the local structure. The equations are shown to be hyperbolic and their general solutions, as well as the Green function, are found. It is shown that the solutions give rise to force chains, and the explicit dependence of the force chains trajectories and magnitudes on the local geometry is predicted. Direct experimental tests of the predictions are proposed. Finally, a framework is proposed to relate the analysis to nonisostatic and more realistic granular assemblies.

On Granular Stress Statistics: Compactivity, Angoricity, and Some Open Issues

We discuss the microstates of compressed granular matter in terms of two independent ensembles: one of volumes and another of boundary force moments. The former has been described in the literature and gives rise to the concept of compactivity: a scalar quantity that is the analogue of temperature in thermal systems. The latter ensemble gives rise to another analogue of the temperature: an angoricity tensor. We discuss averages under either of the ensembles and their relevance to experimental measurements. We also chart the transition from the microcanoncial to a canonical description for granular materials and show that one consequence of the traditional treatment is that the well-known exponential distribution of forces in granular systems subject to external forces is an immediate consequence of the canonical distribution, just as in the microcanonical description E = H leads to exp (-H/kT). We also put this conclusion in the context of observations of nonexponential forms of decay. We then present a Boltzmann-equation and Fokker-Planck approaches to the problem of diffusion in dense granular systems. Our approach allows us to derive, under simplifying assumptions, an explicit relation between the diffusion constant and the value of the hitherto elusive compactivity. We follow with a discussion of several unresolved issues. One of these issues is that the lack of ergodicity prevents convenient translation between time and ensemble averages, and the problem is illustrated in the context of diffusion. Another issue is that it is unclear how to make use in the statistical formalism the emerging ability to exactly predict stress fields for given structures of granular systems.

Interdependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems.

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are interdependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle-the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.

Granular matter and the marginal rigidity state

Model experiments are reported on the build-up of granular piles in two dimensions. These show that, as the initial density of falling grains is increased, the resulting pile has decreasing final density and its coordination number approaches the low value predicted for the theoretical marginal rigidity state. This provides the first direct experimental evidence for this state of granular matter. We trace the decrease in the coordination number to the dynamics within an advancing yield front between the consolidated pile and the falling grains. We show that the fronts size diverges as the marginal rigidity state is approached, suggesting a critical phenomenon.

Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions

Abstract.A method is proposed for the characterisation of the entropy of cellular structures, based on the compactivity concept for granular packings. Hamiltonian-like volume functions are constructed both in two and in three dimensions, enabling the identification of a phase space and making it possible to take account of geometrical correlations systematically. Case studies are presented for which explicit calculations of the mean vertex density and porosity fluctuations are given as functions of compactivity. The formalism applies equally well to two- and three-dimensional granular assemblies.

Auxetic strains—insight from iso-auxetic materials

Auxeticity is the result of internal structural degrees of freedom that get in the way of affine deformations. This paper proposes a new understanding of strains in disordered auxetic materials. A class of iso-auxetic structures is identified, which are auxetic structures that are also isostatic, and these are distinguished from conventional elasto-auxetic materials. It is then argued that the mechanisms that give rise to auxeticity are the same in both classes of materials and the implications of this observation on the equations that govern the strain are explored. Next, the compatibility conditions of Saint Venant are demonstrated to be irrelevant for the determination of stresses in iso-auxetic materials, which are governed by balance conditions alone. This leads to the conclusion that elasticity theory is not essential for the general description of auxetic behaviour. One consequence of this is that characterisation in terms of negative Poissons ratio may be of limited utility. A new equation is then proposed for the dependence of the strain on local rotational and expansive fields. Central to the characterization of the geometry of the structure, to the iso-auxetic stress field equations, and to the strain-rotation relation is a specific fabric tensor. This tensor is defined here explicitly for two-dimensional systems, however, disordered. It is argued that, while the proposed dependence of the strain on the local rotational and expansive fields is common to all auxetic materials, iso-auxetic and elasto-auxetic materials may exhibit significantly different macroscopic behaviours.

Stress in planar cellular solids and isostatic granular assemblies: coarse-graining the constitutive equation

A recent theory for stress transmission in isostatic granular and cellular systems predicts a constitutive equation that couples the stress field to the local microstructure (Phys. Rev. Lett. 88 (2002) 115505). The theory could not be applied to macroscopic systems because the constitutive equation becomes trivial upon straightforward coarse-graining. This problem is resolved here for arbitrary planar structures. The solution is based on the observation that the staggered order makes it possible to couple the stress to a reduced geometric tensor that can be coarse-grained. The method proposed here makes it possible to apply this idea to realistic systems whose staggered order is generally ‘frustrated’. This is achieved by a renormalization procedure which removes the frustration and enables the use of the upscalable reduced tensor. As an example we calculate the stress due to a defect in a periodic solid foam.

Universal structural characteristics of planar granular packs.

The dependence of structural self-organization of granular materials on preparation and grain parameters is key to predictive modeling. We study 60 different mechanically equilibrated polydisperse disc packs, generated numerically by two protocols. We show that, for same-variance disc size distributions (DSDs), (1) the mean coordination number of rattler-free packs versus the packing fraction is a function independent of initial conditions, friction, and the DSD, and (2) all quadron volume and cell order distributions collapse to universal forms, also independent of the above. This apparent universality suggests that, contrary to common wisdom, equilibrated granular structures may be determined mainly by the packing protocol and higher moments of the DSD.

da Vinci fluids, catch-up dynamics and dense granular flow

Abstract.We introduce and study a da Vinci fluid, a fluid whose dissipation is dominated by solid friction. We analyse the flow rheology of a discrete model and then coarse-grain it to the continuum. We find that the model gives rise to behaviour that is characteristic of dense granular fluids. In particular, it leads to plug flow. We analyse the nucleation mechanism of plugs and their development. We find that plug boundaries generically expand and we calculate the growth rate of plug regions. In systems whose internal effective dynamic and static friction coefficients are relatively uniform we find that the linear size of plug regions grows as (time)1/3 . The suitability of the model to granular materials is discussed.