The Isostatic Conjecture

Connelly, Robert; Gortler, Steven J.; Solomonides, Evan; Yampolskaya, Maria

doi:10.1007/s00454-018-00051-0

Cited by 2 publications

(9 citation statements)

References 32 publications

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“…In a series of papers, Connelly and co-workers [9,10,15] relate configurations that are locally maximally dense to the rigidity of a related tensegrity (see, e.g., [39]) over the contact graph. Notably, the recent work in [13] proves results about the number of contacts appearing in such locally maximally dense packings under appropriate genericity assumptions.Another motivation comes from geometric constraint solving [34], where combinatorial methods are also applied to structures made of disks and spheres.Laman's Theorem Given the relationship between disk packings and associated frameworks, it is very tempting to go further and apply the methods of combinatorial rigidity (see, e.g., [25]) theory to infer geometric or physical properties from the contact graph alone. Several recent works in the soft matter literature [16,21,30] use such an approach.The combinatorial approach is attractive because we have a very good understanding of framework rigidity in dimension 2, provided that p is not very degenerate.…”

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confidence: 79%

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confidence: 79%

“…Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly et al (Connelly et al 2018 (http://arxiv.org/abs/ 1702.08442)) on the number of contacts in a jammed packing of discs with generic radii.…”

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confidence: 82%

“…The literature considers a number of different boundary shapes, including convex polyhedra (e.g., [9]) and flat tori (e.g., [13,16]). Here, we consider a boundary formed by three large touching exterior disks.…”

Section: Isostatic Jammingmentioning

confidence: 99%

“…The isostatic theorem with a flat torus as the container was proven in [13] using very general results of Guo [20] on circle patterns in piecewise-flat surfaces. Such methods could also be applied to the tri-cusp setting.…”

Section: Relationship To the Flat-torus Isostatic Theoremmentioning

confidence: 99%

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Rigidity for sticky discs

2019

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We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of n disks in the plane with generic radii cannot have more than 2n − 3 pairs of disks in contact.The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n − 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma.Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly, et al.[13] on the number of contacts in a jammed packing of disks with generic radii. Sticky disk and framework rigidity A motion of a packing, called a flex, is one that preserves the radii, the disjointness of the disk interiors, as well as tangency between pairs of disks with a corresponding edge in the contact graph. (The last condition makes the disks "sticky".) A packing is rigid when all its flexes arise from rigid body motions; otherwise, it is flexible.Since any packing has a neighborhood on which the contact graph remains fixed along any flex (one must move at least some distance before a new contact can appear), the constraints on the packing are locally equivalent to preserving the pairwise distances between the circle centers. Forgetting that the disks have radii and must remain disjoint and keeping only the distance contraints between the centers, we get exactly "framework rigidity", where we have a configuration * Figure 1: A rigid sticky disk packing and its underlying bar framework. Disks are the green circles with red center points / joints, the bars are blue segments. p = (p 1 , . . . , p n ) of n points in a d-dimensional Euclidean space and a graph G; the pair (G, p) is called a (bar-and-joint) framework. The allowed flexes of the points are those that preserve the distances between the pairs indexed by the edges of G. As with packings, a framework is rigid when all its flexes arise from rigid body motions and otherwise flexible. Given a packing (p, r) with contact graph G, we call the framework (G, p) its underlying framework.Motivations Rigidity of sticky disk packings, and the relationship to frameworks, has several related, but formally different motivations. The first comes from the study of colloidal matter [31], which is made of micrometer-sized particles that interact with "short range potentials" [2, 33] that, in a limit, behave like sticky disks [22] (spheres in 3d).Secondly, there is a connection to jammed packings. Here one has a "container", which can shrink uniformly and push the disks together. In this setting, we do not require any contact graph to be preserved. A fundamen...

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confidence: 79%

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confidence: 79%

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confidence: 82%

Section: Isostatic Jammingmentioning

confidence: 99%

Section: Relationship To the Flat-torus Isostatic Theoremmentioning

confidence: 99%

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Rigidity for sticky discs

2019

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Shear thickening in dense suspensions driven by particle interlocking

Blair

Ness

2022

J. Fluid Mech.

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The processing of dense suspensions is a crucial step in many industries including mining; the production of ceramics; and the manufacture of pharmaceuticals. It is widely reported that these suspensions exhibit nonlinear behaviours such as shear thinning and thickening, with particle surface contacts recently being accepted as a primary culprit in the latter. In light of this, the modelling community have started to explore the role of particle surface tribology, predominantly by incorporating Coulombic friction models borrowed from the field of dry granular matter. Full details of the interactions between particle surfaces remain unclear, however, and it is suggested that physical interlocking of particle asperities may be key. Here, we use particle-based simulations to explore explicitly the effect of interlocking on the rheology of dense suspensions of micron-sized solids in a Newtonian fluid. Our simplified model recovers shear thinning, thickening and jamming phenomena commonly seen in experiments.

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The Isostatic Conjecture

Cited by 2 publications

References 32 publications

Rigidity for sticky discs

Rigidity for sticky discs

Shear thickening in dense suspensions driven by particle interlocking

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