L. <i>On the calculation of the equilibrium and stiffness of frames</i>

Maxwell, James Clerk

doi:10.1080/14786446408643668

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“…We, for the first time, implement a generalized pebble game for 2d frictional systems and use it to identify rigid clusters in very slowly sheared packings. As we show below, this allows us to identify a second-order rigidity transition and to link stresses and nonaffine motions to the microscopic structure of frictionally jammed packings.Generalized isostaticity: To establish context, we first review the application of Maxwell constraint counting to jamming [18]. For N particles in d dimensions and a mean number of contacts per particle z, interparticle forces yield dzN/2 constraints.…”

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Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

Henkes¹,

Quint²,

Fily³

et al. 2016

Phys. Rev. Lett.

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We study the nature of the frictional jamming transition within the framework of rigidity percolation theory. Slowly sheared frictional packings are decomposed into rigid clusters and floppy regions with a generalization of the pebble game including frictional contacts. We discover a second-order transition controlled by the emergence of a system-spanning rigid cluster accompanied by a critical cluster size distribution. Rigid clusters also correlate with common measures of rigidity. We contrast this result with frictionless jamming, where the rigid cluster size distribution is noncritical.The interplay of constraints, forces, and driving gives rise to the jamming transition in granular media. It is now wellestablished that the frictionless jamming transition has characteristics of both second-and first-order transitions. Both the average coordination number and the largest rigid cluster size jump at the transition, yet there exists a diverging lengthscale [1][2][3][4]. Frictional jamming is more puzzling: The hysteresis observed in the stress-strain rate curves of stresscontrolled flow simulations [5][6][7][8] and experiments [9] has lead to an interpretation as a first-order transition. Yet, signs of second-order criticality appear when treating the fraction of contacts at the Coulomb threshold as an additional parameter [10][11][12].To elucidate the frictional jamming transition from a microscopic viewpoint, we extend concepts and tools from rigidity percolation, i.e., the onset of mechanical rigidity in disordered spring networks [13][14][15][16], to frictional packings. The former is driven by the emergence of a system-spanning rigid cluster that can be mapped out (in 2d) using the pebble game [17], an improved constraint counting method that goes beyond meanfield by identifying redundant constraints. We, for the first time, implement a generalized pebble game for 2d frictional systems and use it to identify rigid clusters in very slowly sheared packings. As we show below, this allows us to identify a second-order rigidity transition and to link stresses and nonaffine motions to the microscopic structure of frictionally jammed packings.Generalized isostaticity: To establish context, we first review the application of Maxwell constraint counting to jamming [18]. For N particles in d dimensions and a mean number of contacts per particle z, interparticle forces yield dzN/2 constraints. Since each particle has 1 2 d(d + 1) translational and rotational degrees of freedom, there are 1 2 (N − 1)d(d + 1) total degrees of freedom (subtracting out global degrees of freedom). When these match the force constraints, we arrive at the isostatic criterion, or dzN/2 = 1 2 (N − 1)d(d + 1). In the limit N → ∞, z iso = d + 1 for frictional granular materials. For frictionless packings, we ignore rotations and obtain the familiar z iso = 2d.Despite being mean field, i.e. neglecting spatial correla- tions, isostaticity works seemingly well to locate the jamming transition in static frictionless systems [1]. For frictional syste...

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confidence: 99%

“…Generalized isostaticity: To establish context, we first review the application of Maxwell constraint counting to jamming [18]. For N particles in d dimensions and a mean number of contacts per particle z, interparticle forces yield dzN/2 constraints.…”

mentioning

confidence: 99%

Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

Henkes¹,

Quint²,

Fily³

et al. 2016

Phys. Rev. Lett.

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“…Near the unstable fixed point, P n+1 (p) = λ 2 P n (p), where λ 2 = 1/4[5p 4 c + 13p 6 c − 14p 7 c ] = 0.9554, showing that the probability of a bond being in the percolating cluster renormalizes to zero at the phase transition as expected for a second-order phase transition. From figure 4, we can see how the singular behaviour at the phase transition develops as n increases: n = 12 appears very close to giving the full singularity.…”

Section: Hierarchical Modelsmentioning

confidence: 99%

“…For the diluted case, a bond is present with probability p (concentration) and absent with probability 1 − p, so the probability of the two solid dots being rigidly connected in the second panel of figure 2 using the weights from figure 3 is p = p 8 + 8p 7 ( 8 . This leads to the relationship between the probabilities p n+1 , p n of rigidity Here, (a) has all eight bonds present and is rigid with one redundant edge and has probability p 8 , (b) has any single edge missing and has probability 8p 7 (1 − p), (c) has any pair of edges missing from the three lower (shown) or the three upper ones and has probability 6p 6 (1 − p) 2 and (d) has a triple of edges missing either from the lower or upper part of the graph and has probability 2p 5 (1 − p) 3 .…”

Section: Hierarchical Modelsmentioning

confidence: 99%

“…This leads to the relationship between the probabilities p n+1 , p n of rigidity Here, (a) has all eight bonds present and is rigid with one redundant edge and has probability p 8 , (b) has any single edge missing and has probability 8p 7 (1 − p), (c) has any pair of edges missing from the three lower (shown) or the three upper ones and has probability 6p 6 (1 − p) 2 and (d) has a triple of edges missing either from the lower or upper part of the graph and has probability 2p 5 (1 − p) 3 . The number of edges in the rigid cluster is indicated by the number under each graph.…”

Section: Hierarchical Modelsmentioning

confidence: 99%

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Two exactly soluble models of rigidity percolation

Thorpe

Stinchcombe

2014

Phil. Trans. R. Soc. A.

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We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination 〈 r 〉=4.41, which is above the ‘mean field’ value 〈 r 〉=4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β =0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν =3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination 〈 r 〉=3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

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Fragile Glass Formers: Evidence for a New Paradigm, and a New Relation to Strong Liquids

Angell¹

2012

Structural Glasses and Supercooled Liquids

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L. On the calculation of the equilibrium and stiffness of frames

Cited by 913 publications

References 0 publications

Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

Rigid Cluster Decomposition Reveals Criticality in Frictional Jamming

Two exactly soluble models of rigidity percolation

Fragile Glass Formers: Evidence for a New Paradigm, and a New Relation to Strong Liquids

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