The transmission of stress through a marginally stable granular pile in two dimensions is exactly formulated in terms of a vector field of loop forces, and thence in terms of a single scalar potential.The loop force formulation leads to a local constitutive equation coupling the stress tensor to fluctuations in the local geometry. For a disordered pile of rough grains (even with simple orientational order) this means the stress tensor components are coupled in a frustrated manner, analogous to a spin glass.If the local geometry of the pile of rough grains has long range staggered order, frustration is avoided and a simple linear theory follows. Known exact lattice solutions for rough grains fall into this class.We show that a pile of smooth grains (lacking friction) can always be mapped into a pile of unfrustrated rough grains. Thus it appears that the problems of rough and smooth grains may be fundamentally distinct.64.60. Ak, 05.10.c 61.90.+d It has long been recognized in engineering practice that granular materials which lack cohesion cannot be regarded entirely as solids, nor (until 'fluidised' or yielding) can they be regarded entirely as fluids [1]. In this Letter we explore the scenario that there exists a marginal state of granular matter in between solid and liquid. Most particularly, we are concerned with how that intermediate state transmits stress.The marginal state is readily characterised in terms of the mean coordination numberz for mechanical equilibrium. If external loads, such as gravity, are exerted on a pile of N grains [2] then the Nz/2 intergranular forces must be able to adjust so as to achieve N d(d + 1)/2 constraints balancing the force and torque on each grain in d dimensions. For perfectly rough grains, where every contact supports friction, this gives a critical coordination number z c = d + 1, whilst for ideally smooth asymmetric grains with no frictionFor mean coordination numbersz < z c the pile cannot be stable and under a general loading it must rearrange or consolidate. By contrast forz > z c the intergranular forces are underdetermined by the conditions of force balance alone and the deformation of individual grains together with their local constitutive equations becomes relevant. In the first case we have granular motion and the pile is fluid or yielding, whereas in the latter case intragranular deformation is crucial and we have a solid of constitution influenced by that of the individual grains. It is interesting to note that for ideally smooth grains, z c matches the maximum mean coordination attainable by sequential packing (and in two dimensions it is the absolute maximum possible [6]), so a (random) pile of smooth grains is probably never solid.The marginal casez = z c is special in that the grains do not need to move and the intergranular forces, and hence distribution of stress, are determined by conditions of force balance alone. This has been identified as a paradigm problem of theoretical granular mechanics [1], [7], [8]. The macroscopic analogues of balancing ...

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.

Understanding the response of granular matter to intrusion of solid objects is key to modelling many aspects of behaviour of granular matter, including plastic flow. Here we report a general model for such a quasistatic process. Using a range of experiments, we first show that the relation between the penetration depth and the force resisting it, transiently nonlinear and then linear, is scalable to a universal form. We show that the gradient of the steady-state part, Kϕ, depends only on the medium’s internal friction angle, ϕ, and that it is nonlinear in μ = tan ϕ, in contrast to an existing conjecture. We further show that the intrusion of any convex solid shape satisfies a modified Archimedes’ law and use this to: relate the zero-depth intercept of the linear part to Kϕ and the intruder’s cross-section; explain the curve’s nonlinear part in terms of the stagnant zone’s development.

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.

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.

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.

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 oftenobserved force chains be predicted from a first-principles continuous theory? (iii) How to 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 non-isostatic and more realistic granular assemblies. 46.05.+b, 62.25.+g, 81.40.Jj Granular systems have become a subject of intensive research in recent years both due to their enormous technological importance and the fundamental theoretical challenges that they pose [1]. In particular, stress transmission has focused much attention following experimental [2] [3] and numerical [4] observations that arching effects give rise to nonuniform stress fields [5] and in particular to chain-like regions of large forces which cannot be straightforwardly described by conventional approaches [6]. It has been recognized that to fully understand this phenomenon in general granular packings it is essential to first understand stress transmission in isostatic systems [5]. Isostatic states are configurations of grains in which the intergranular contact forces can be determined directly from statics, namely, force and torque balance, without reference to stress-strain relations. These states are characterized by low mean coordination number, which depends on the dimensionality and the roughness of the grains. These states have been shown to be easy to approach experimentally [7]. Several empirical [8] and statistical [5] [9] models have been proposed for the macroscopic equations that govern the stress field in such systems. Very recently, however, the two-dimensional case has been solved from first principles on the scale of a few grains [10]. The main result of the new theory is an equation that relates directly between the stress tensorσ and a rank-two symmetric fabric tensorP which characterizes the microstructure:

Ball and Blumenfeld Reply: We are grateful for Marder's Comment [l] on our Letter [2], and explain below why the numerical data presented by him do not contradict our conclusions. The essential point is that the logarithmic stress oscillations we have found are an exact result and solve the field equations with the proper boundary conditions. However, we do agree that they are difficult to observe numerically. Whether these oscillations bias the pattern formation is a question not addressed by Marder's simulations, but on which we will remark. We also explain the unaccounted for phenomena that he observes, namely, nearly linear oscillations in the simulations and the power-law decay of the curvature of the simulated tip with the order of the approximation.Using the elegant method of Muskhelishvili [1], Marder searches for our predicted oscillations in lnr by approaching the tip of the wedge along a ray [3]. The general form of the oscillations we found is