Elastic collapse in disordered isostatic networks

Isostatic networks are minimally rigid and therefore have, generically, nonzero elastic moduli. Regular isostatic networks have finite moduli in the limit of large sizes. However, numerical simulations show that all elastic moduli of geometrically disordered isostatic networks go to zero with system size. This holds true for positional as well as for topological disorder. In most cases, elastic moduli decrease as inverse power-laws of system size. On directed isostatic networks, however, of which the square and cubic lattices are particular cases, the decrease of the moduli is exponential with size. For these, the observed elastic weakening can be quantitatively described in terms of the multiplicative growth of stresses with system size, giving rise to bulk and shear moduli of order exp{-bL}. The case of sphere packings, which only accept compressive contact forces, is considered separately. It is argued that these have a finite bulk modulus because of specific correlations in contact disorder, introduced by the constraint of compressivity. We discuss why their shear modulus, nevertheless, is again zero for large sizes. A quantitative model is proposed that describes the numerically measured shear modulus, both as a function of the loading angle and system size. In all cases, if a density p>0 of overconstraints is present, as when a packing is deformed by compression, or when a glass is outside its isostatic composition window, all asymptotic moduli become finite. For square networks with periodic boundary conditions, these are of order sqrt{p}. For directed networks, elastic moduli are of order exp{-c/p}, indicating the existence of an "isostatic length scale" of order 1/p.Comment: 6 pages, 6 figues, to appear in Europhysics Letter

Why Effective Medium Theory Fails in Granular Materials

Experimentally it is known that the bulk modulus, K, and shear modulus, \mu, of a granular assembly of elastic spheres increase with pressure, p, faster than the p^1/3 law predicted by effective medium theory (EMT) based on Hertz-Mindlin contact forces. To understand the origin of these discrepancies, we perform numerical simulations of granular aggregates under compression. We show that EMT can describe the moduli pressure dependence if one includes the increasing number of grain-grain contacts with p. Most important, the affine assumption (which underlies EMT), is found to be valid for K(p) but breakdown seriously for \mu(p). This explains why the experimental and numerical values of \mu(p) are much smaller than the EMT predictions.Comment: 4 pages, 5 figures, http://polymer.bu.edu/~hmaks

The effective temperature

This review presents the effective temperature notion as defined from the deviations from the equilibrium fluctuation-dissipation theorem in out of equilibrium systems with slow dynamics. The thermodynamic meaning of this quantity is discussed in detail. Analytic, numeric and experimental measurements are surveyed. Open issues are mentioned.Comment: 58 page

Directed force chain networks and stress response in static granular materials

A theory of stress fields in two-dimensional granular materials based on directed force chain networks is presented. A general equation for the densities of force chains in different directions is proposed and a complete solution is obtained for a special case in which chains lie along a discrete set of directions. The analysis and results demonstrate the necessity of including nonlinear terms in the equation. A line of nontrivial fixed point solutions is shown to govern the properties of large systems. In the vicinity of a generic fixed point, the response to a localized load shows a crossover from a single, centered peak at intermediate depths to two propagating peaks at large depths that broaden diffusively.Comment: 18 pages, 12 figures. Minor corrections to one figur

Elastic wave propagation in confined granular systems

We present numerical simulations of acoustic wave propagation in confined granular systems consisting of particles interacting with the three-dimensional Hertz-Mindlin force law. The response to a short mechanical excitation on one side of the system is found to be a propagating coherent wavefront followed by random oscillations made of multiply scattered waves. We find that the coherent wavefront is insensitive to details of the packing: force chains do not play an important role in determining this wavefront. The coherent wave propagates linearly in time, and its amplitude and width depend as a power law on distance, while its velocity is roughly compatible with the predictions of macroscopic elasticity. As there is at present no theory for the broadening and decay of the coherent wave, we numerically and analytically study pulse-propagation in a one-dimensional chain of identical elastic balls. The results for the broadening and decay exponents of this system differ significantly from those of the random packings. In all our simulations, the speed of the coherent wavefront scales with pressure as $p^{1/6}$; we compare this result with experimental data on various granular systems where deviations from the $p^{1/6}$ behavior are seen. We briefly discuss the eigenmodes of the system and effects of damping are investigated as well.Comment: 20 pages, 12 figures; changes throughout text, especially Section V.

Macroscopic model with anisotropy based on micro-macro informations

Physical experiments can characterize the elastic response of granular materials in terms of macroscopic state-variables, namely volume (packing) fraction and stress, while the microstructure is not accessible and thus neglected. Here, by means of numerical simulations, we analyze dense, frictionless, granular assemblies with the final goal to relate the elastic moduli to the fabric state, i.e., to micro-structural averaged contact network features as contact number density and anisotropy. The particle samples are first isotropically compressed and later quasi-statically sheared under constant volume (undrained conditions). From various static, relaxed configurations at different shear strains, now infinitesimal strain steps are applied to "measure" the effective elastic response; we quantify the strain needed so that plasticity in the sample develops as soon as contact and structure rearrangements happen. Because of the anisotropy induced by shear, volumetric and deviatoric stresses and strains are cross-coupled via a single anisotropy modulus, which is proportional to the product of deviatoric fabric and bulk modulus (i.e. the isotropic fabric). Interestingly, the shear modulus of the material depends also on the actual stress state, along with the contact configuration anisotropy. Finally, a constitutive model based on incremental evolution equations for stress and fabric is introduced. By using the previously measured dependence of the stiffness tensor (elastic moduli) on the microstructure, the theory is able to predict with good agreement the evolution of pressure, shear stress and deviatoric fabric (anisotropy) for an independent undrained cyclic shear test, including the response to reversal of strain

Granular media: some ideas from statistical physics

These lecture notes cover the statics and glassy dynamics of granular media. Most of the lectures were in fact devoted to `force propagation' models. We discuss the experimental and theoretical motivations for these approaches, and their conceptual connections with Edwards' thermodynamical analogy. One of the distinctive feature of granular media (common to many other `jammed' systems) is indeed the large number of metastable states that are macroscopically equivalent. We present in detail the (scalar) $q$-model and its tensorial generalization, that aim at modelling the existence of force chains and arching effects without introducing any displacement field. The contrast between the hyperbolic equations obtained within this line of thought and elliptic (elastic) equations is emphasized. The role of disorder on these hyperbolic equations is studied in details using perturbative and diagrammatic methods. Recent (strong disorder) force chain network models are reviewed, and compared with the experimental determination of the force `response function' in granular materials. We briefly discuss several issues (such as isostaticity and generic marginality) and open problems. At the end of these notes, we also discuss the basic dynamical properties of_weakly tapped_ granular assemblies, and stress the phenomenological analogies with other glassy materials. Simple models that account for slow compaction and dynamical heterogeneities are presented, that are inspired by `free-volume' ideas and Edwards' assumption. A connection with the theory of fluctuating random surfaces, also noted recently by Castillo et al., is suggested. Finally, we discuss how the `trap model' can be adapted to granular materials, such that more subtle `memory' effects can be accounted for.Comment: Slightly revised version, one figure and some references adde