Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling

Goodrich, Carl P.; Dagois-Bohy, Simon; Tighe, Brian P.; Hecke, Martin van; Liu, Andrea J.; Nagel, Sidney R.

doi:10.1103/physreve.90.022138

Cited by 112 publications

(141 citation statements)

References 39 publications

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“…By iteratively lowering σ t xy , we create configurations with σ xy spanning many orders of magnitude. This protocol is very similar to that used in isotropic jamming to obtain systems at target pressures [24,32,33]. Although some dependence on the packing fraction is expected, we find that it is weak for the system sizes studied.…”

Section: Mechanical Properties Of the Shear Jammed Statementioning

confidence: 72%

“…In the thermodynamic limit, the bulk modulus jumps to a non-zero value while the shear modulus remains zero, so C αβγδ = Bδ αβ δ γδ . By spatial isotropy, the various shear moduli [33,34] in finite systems near jamming all scale the same way; the SO(3) rotational symmetry of isotropic jamming induces a symmetry in the various elastic moduli, all of which vanish linearly with ∆Z as ∆Z → 0 and N → ∞.…”

Section: Emergent Symmetry At the Shear Jamming Transitionmentioning

confidence: 99%

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Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition

et al. 2017

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Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition the date of receipt and acceptance should be inserted later Abstract We study the shear jamming of athermal frictionless soft spheres, and find that in the thermodynamic limit, a shear-jammed state exists with different elastic properties from the isotropically-jammed state. For example, shear-jammed states can have a non-zero residual shear stress in the thermodynamic limit that arises from long-range stress-stress correlations. As a result, the ratio of the shear and bulk moduli, which in isotropically-jammed systems vanishes as the jamming transition is approached from above, instead approaches a constant. Despite these striking differences, we argue that in a deeper sense, the shear jamming and isotropic jamming transitions actually have the same symmetry, and that the differences can be fully understood by rotating the six-dimensional basis of the elastic modulus tensor.

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Section: Mechanical Properties Of the Shear Jammed Statementioning

confidence: 72%

Section: Emergent Symmetry At the Shear Jamming Transitionmentioning

confidence: 99%

Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition

et al. 2017

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“…Note that B, G, and ΔZ should become independent of N in the large N limit, and it is well-established that ΔZ ∼ p 1=2 , B ∼ ΔZ 0 and G ∼ ΔZ in this regime (3,5,12). Furthermore, recent studies of finite-size effects found that NΔZ = FðNp 1=2 Þ (10, 12).…”

Section: Extracting Exponents and Numerical Verificationmentioning

confidence: 99%

“…The jamming transition marks the onset of rigidity in athermal sphere packings and was originally proposed as a zero-temperature transition (2,3) for soft repulsive spheres in a nonequilibrium "jamming phase diagram" (4) of varying packing density and applied shear. Many studies have documented behaviors characteristic of critical phenomena near the jamming transition, including power law scaling (2,3,5) and scaling collapses (6-13) of numerous properties, with the expression of quantities in terms of scaling functions, diverging length scales (6,(14)(15)(16)(17)(18)(19), and finite-size scaling (10,12,20). Theories have been developed to individually understand and relate some of these power laws (15,16,21,22), but a unified scaling theory has been lacking.…”

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Scaling ansatz for the jamming transition

Goodrich

Liu

Sethna

2016

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

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We propose a Widom-like scaling ansatz for the critical jamming transition. Our ansatz for the elastic energy shows that the scaling of the energy, compressive strain, shear strain, system size, pressure, shear stress, bulk modulus, and shear modulus are all related to each other via scaling relations, with only three independent scaling exponents. We extract the values of these exponents from already known numerical or theoretical results, and we numerically verify the resulting predictions of the scaling theory for the energy and residual shear stress. We also derive a scaling relation between pressure and residual shear stress that yields insight into why the shear and bulk moduli scale differently. Our theory shows that the jamming transition exhibits an emergent scale invariance, setting the stage for the potential development of a renormalization group theory for jamming.jamming transition | scaling ansatz | nonequilibrium critical phenomena T he existence of criticality at the jamming transition suggests that universal physics underlies rigidity in disordered solids ranging from glasses to granular materials (1). The jamming transition marks the onset of rigidity in athermal sphere packings and was originally proposed as a zero-temperature transition (2, 3) for soft repulsive spheres in a nonequilibrium "jamming phase diagram" (4) of varying packing density and applied shear. Many studies have documented behaviors characteristic of critical phenomena near the jamming transition, including power law scaling (2, 3, 5) and scaling collapses (6-13) of numerous properties, with the expression of quantities in terms of scaling functions, diverging length scales (6,(14)(15)(16)(17)(18)(19), and finite-size scaling (10,12,20). Theories have been developed to individually understand and relate some of these power laws (15,16,21,22), but a unified scaling theory has been lacking. Here, we develop such a theory by proposing a scaling ansatz for the jamming critical point in terms of the fields originally identified by the jamming phase diagram, namely density and shear.The critical point scaling ansatz introduced by Widom (23) in the 1960s was a key advance in the theory of critical phenomena that set the stage for the development of the renormalization group. The ansatz writes the state functions near continuous equilibrium phase transitions in terms of power law ratios of the control parameters. By positing a scaling function for the free energy, it exploits the fact that quantities, such as the specific heat, magnetization, and susceptibility, are derivatives of the free energy to derive relations not only among their scaling exponents but among their scaling functions. Thus, the scaling ansatz provides a unified and comprehensive description of systems exhibiting what later was realized to be an emergent scale invariance.Unlike most systems exhibiting a dynamical scale invariance, we find that a jammed system can also be viewed as a material with critical properties that are determined by a state function, analo...

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“…Their findings caused the authors to question, if not the formal validity, then at least the use- fulness of linear elasticity in jammed solids -not just at the jamming point, but anywhere in the jammed phase. Subsequently, Van Deen et al [30] and Goodrich et al [31,32] argued that the situation is not so dire. They demonstrated that coarse grained properties of jammed solids are far less sensitive to contact changes than are the individual trajectories.…”

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Beyond linear elasticity: jammed solids at finite shear strain and rate

et al. 2016

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The shear response of soft solids can be modeled with linear elasticity, provided the forcing is slow and weak. Both of these approximations must break down when the material loses rigidity, such as in foams and emulsions at their (un)jamming point -suggesting that the window of linear elastic response near jamming is exceedingly narrow. Yet precisely when and how this breakdown occurs remains unclear. To answer these questions, we perform computer simulations of stress relaxation and shear startup experiments in athermal soft sphere packings, the canonical model for jamming. By systematically varying the strain amplitude, strain rate, distance to jamming, and system size, we identify characteristic strain and time scales that quantify how and when the window of linear elasticity closes, and relate these scales to changes in the microscopic contact network. Our findings indicate that the mechanical response of jammed solids are generically nonlinear and rate-dependent on experimentally accessible strain and time scales.Linear elasticity predicts that when an isotropic solid is sheared, the resulting stress σ is directly proportional to the strain γ and independent of the strain rateγ,with a constant shear modulus G 0 [1]. The constitutive relation (1) -a special case of Hooke's law -is a simple, powerful, and widely used model of mechanical response in solids. Yet formally it applies only in the limit of vanishingly slow and weak deformations. In practice materials possess characteristic strain and time scales that define a linear elastic "window", i.e. a parameter range wherein Hooke's law is accurate. Determining the size of this window is especially important in soft solids, where viscous damping and nonlinearity play important roles [2]. The goal of the present work is to determine when Hooke's law holds, and what eventually replaces it, in packings of soft frictionless spheres close to the (un)jamming transition. Jammed sphere packings are a widely studied model of emulsions and liquid foams [3][4][5][6] and have close connections to granular media and dense suspensions [7][8][9]. Linear elastic properties of jammed solids, such as moduli and the vibrational density of states, are by now well understood [10,11]. Much less is known about their viscoelastic [7,12] and especially their nonlinear response [13,14]. Yet the jamming transition must determine the size of the linear elastic window, because the shear modulus G 0 vanishes continuously at the jamming point, where the confining pressure p goes to zero. Indeed, recent studies of oscillatory rheology [15] and shocks [16][17][18] have shown that, precisely at the jamming point, any deformation is effectively fast and strong, and neither viscous effects nor nonlinearities can be neglected.

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Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling

Cited by 112 publications

References 39 publications

Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition

Emergent SO(3) Symmetry of the Frictionless Shear Jamming Transition

Scaling ansatz for the jamming transition

Beyond linear elasticity: jammed solids at finite shear strain and rate

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