Non-local and non-linear problems in the mechanics of disordered systems: application to granular media and rigidity problems

Guyon, E.; Roux, Stéphane; Hansen, Alex; Bideau, Daniel; Troadec, Jean-Paul; Crapo, Henry

doi:10.1088/0034-4885/53/4/001

Cited by 119 publications

(82 citation statements)

References 84 publications

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“…Ref. [20] gives a counterexample for a system of discs (like systems A and B, equivalent to a network of articulated bars) on the regular triangular lattice. In specific configurations (like that of a regular triangular lattice), one might exceptionnally have h = k − k 0 > 0 on generically isostatic structures.…”

Section: E Generic Versus Geometric Propertiesmentioning

confidence: 99%

Geometric origin of mechanical properties of granular materials

2000

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Model granular assemblies, in which grains are assumed rigid and frictionless, at equilibrium under some prescribed external load, are shown to possess, under generic conditions, several remarkable mechanical properties, related to isostaticity and potential energy minimization. Isostaticity -the uniqueness of the contact forces, once the list of contacts is known-is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to small load increments, even though grains are perfectly rigid. In general, one gets an upper bound on the contact coordination number. The approximation of small displacements, that is introduced and discussed, allows to draw analogies with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation in rearrangements of the network of contacts. Plasticity and hysteresis are related to the lack of such uniqueness, which can be traced back, apart from intergranular friction, to non-reversible rearrangements of small but finite extent, in which the system jumps between two distinct potential energy minima in configuration space, or to bounded tensile forces, deriving from a non-convex potential, in the contacts. Properties of response functions to load increments are discussed. On the basis of past numerical studies, it is argued that, provided the approximation of small displacements is valid, displacements due to the rearrangements of the rigid grains in response to small load increments, once averaged on the macroscopic scale, are solutions to elliptic boundary value problems (similar to the Stokes problem for viscous incompressible flow).

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Section: E Generic Versus Geometric Propertiesmentioning

confidence: 99%

Geometric origin of mechanical properties of granular materials

2000

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“…Particles within this system are under stress, supporting the weight of the material above them in addition to any applied load. The inter-particle contact forces crucially determine the bulk properties of the assembly, from its load-bearing capability [2,3] to sound transmission [4][5][6] or shock propagation [7,8]. Only in a crystal of identical, perfect spheres is there uniform load-sharing between particles.…”

Section: Introductionmentioning

confidence: 99%

Force distribution in a granular medium

1998

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We report on systematic measurements of the distribution of normal forces exerted by granular material under uniaxial compression onto the interior surfaces of a confining vessel. Our experiments on three-dimensional, random packings of monodisperse glass beads show that this distribution is nearly uniform for forces below the mean force and decays exponentially for forces greater than the mean. The shape of the distribution and the value of the exponential decay constant are unaffected by changes in the system preparation history or in the boundary conditions. An empirical functional form for the distribution is proposed that provides an excellent fit over the whole force range measured and is also consistent with recent computer simulation data.

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“…Interactions between particles are treated as constraints, and the rigidity of a network of these constraints is established when the number of independent constraints equals the number of degrees of freedom (minus those associated with the center of mass motion of the system). These ideas were introduced into the glass field by Phillips (14) and Thorpe (15) and to the problem of granular packing by Guyon et al (16). Once the nature of the constraints associated with each bond is specified, the Maxwell condition establishes the minimum number of bonds required for there to be no internal floppy modes left in a system of particles.…”

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Rigidity percolation and the spatial heterogeneity of soft modes in disordered materials

Souza

Harrowell

2009

Proc. Natl. Acad. Sci. U.S.A.

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We show how the spatial character of unconstrained motion in a network of bonds can be directly inferred from the topological arrangement of constraints. Relaxation time scales of these soft modes are determined, and from this information we generate spatial maps of the heterogeneous distribution of relaxation times in the disordered network. We show that the nature of the dynamic heterogeneity and its sensitivity to changes in bond configuration depends dramatically on the proximity of the system to the rigidity percolation point.dynamic heterogeneity ͉ glass transition ͉ inherent structure ͉ local mode ͉ supercooled liquid T he kinetic behavior of supercooled liquids is dominated by those configurations, known as inherent structures (1), which correspond to the local minima of the potential energy. This perspective, articulated by Goldstein (2) in 1969, is now generally accepted (3-6). The inherent structures can be regarded as zero-temperature disordered solids. The continuous transition from fluidity to rigidity that characterizes the glass transition is accomplished by the supercooled liquid sampling these disordered solids at nonzero temperatures for durations that increase as the temperature drops. Eventually, a low enough temperature is reached such that 1 local minimum persists indefinitely. To understand the kinetic and structural features of the transitions between different inherent structures we must understand the types of motion available to disordered solids. In this article we start from the proposal that constraint networks provide a general description of such solids and then develop a method for identifying the spatial distribution of the motions that remain unconstrained in such materials (i.e., the soft modes) and the spectrum of relaxation times associated with these modes.In the absence of explicit structural correlates for dynamics, the descriptive approach of dynamic heterogeneities has proved a powerful way forward in developing microscopic accounts of glassy relaxation and in articulating questions about determining dynamics from structure (7). Recently, it was shown that the spatial distribution of low-frequency quasi-localized normal modes is strongly correlated with the spatial distribution of irreversible reorganization (8). This study and related work (9-13) underline the importance of understanding the relation between local soft modes and structure in disordered solids. Understanding how features of an inherent structure determine the spatial pattern of motions is of central importance in developing a useful microscopic treatment of glass transitions and is the goal of this article.Faced with the daunting variety of particle arrangements that typically comprise a disordered solid, geometrical characterizations of such solids are difficult to both perform and interpret. An alternative treatment of amorphous rigidity is based on networks of constraints. The constraint approach to rigidity focuses on topology, instead of geometry, and the global balance between constraints and deg...

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Non-local and non-linear problems in the mechanics of disordered systems: application to granular media and rigidity problems

Cited by 119 publications

References 84 publications

Geometric origin of mechanical properties of granular materials

Geometric origin of mechanical properties of granular materials

Force distribution in a granular medium

Rigidity percolation and the spatial heterogeneity of soft modes in disordered materials

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