Penrose tilings as jammed solids

Mechanics of fiber networks under a bulk strain

Biopolymer networks are common in biological systems from the cytoskeleton of individual cells to collagen in the extracellular matrix. The mechanics of these systems under applied strain can be explained in some cases by a phase transition from soft to rigid states. For collagen networks, it has been shown that this transition is critical in nature and it is predicted to exhibit diverging fluctuations near a critical strain that depends on the network's connectivity and structure. Whereas prior work focused mostly on shear deformation that is more accessible experimentally, here we study the mechanics of such networks under an applied bulk or isotropic extension. We confirm that the bulk modulus of subisostatic fiber networks exhibits similar critical behavior as a function of bulk strain. We find different nonmean-field exponents for bulk as opposed to shear. We also confirm a similar hyperscaling relation to what was previously found for shear.

Scaling to zero of compressive modulus in disordered isostatic cubic networks

Networks with as many mechanical constraints as degrees of freedom and no redundant constraints are minimally rigid or isostatic. Isostatic networks are relevant in the study of network glasses, soft matter, and sphere packings. Because of being at the verge of mechanical collapse, they have anomalous elastic and dynamical properties not found in the more commonly occurring hyperstatic networks. In particular, while hyperstatic networks are only slightly affected by geometric disorder, the elastic properties of isostatic networks are dramatically altered by it. In this paper, we show how disorder and system size strongly affect the ability of isostatic networks to sustain a compressive load. We develop an analytic method to calculate the bulk compressive modulus B for various boundary conditions as a function of disorder strength and system size. For simplicity, we consider square and cubic lattices with L^{d} sites, each having d mechanical degrees of freedom, and dL^{d} rotatable springs in the presence of hot-solid disorder of magnitude ε. Additionally, ∼L^{θ} sites may be fixed, thus introducing a nonextensive number of redundancies, either in the bulk or on the boundaries of the system. In all cases, B is analytically and numerically shown to decay as L^{-μ} with μ_{large}=d-θ for large disorder and μ_{small}=max{(d-θ-1),0} for small disorder. Furthermore B(L,ε)L^{μ_{small}}=g(λ) with λ=L^{(μ_{large}-μ_{small})}ε^{2} a scaling variable such that λ<<1 is small disorder and λ>1 is large disorder. The faster decay to zero of B in the large disorder regime results from a broad distribution of spring tensions, including tensions of both signs in equal proportions, which is remarkable since the system is under a purely compressive load. Notably, the bulk modulus is discontinuous at ε=0, a consequence of the fact that the regular network sits at an unstable degenerate configuration.

Signatures of Topological Phonons in Superisostatic Lattices

Soft topological surface phonons in idealized ball-and-spring lattices with coordination number z=2d in d dimensions become finite-frequency surface phonons in physically realizable superisostatic lattices with z>2d. We study these finite-frequency modes in model lattices with added next-nearest-neighbor springs or bending forces at nodes with an eye to signatures of the topological surface modes that are retained in the physical lattices. Our results apply to metamaterial lattices, prepared with modern printing techniques, that closely approach isostaticity.

Topological boundary modes in jammed matter

Granular matter at the jamming transition is poised on the brink of mechanical stability, and hence it is possible that these random systems have topologically protected surface phonons. Studying two model systems for jammed matter, we find states that exhibit distinct mechanical topological classes, protected surface modes, and ubiquitous Weyl points. The detailed statistics of the boundary modes shed surprising light on the properties of the jamming critical point and help inform a common theoretical description of the detailed features of the transition.

Phonons and elasticity in critically coordinated lattices

Much of our understanding of vibrational excitations and elasticity is based upon analysis of frames consisting of sites connected by bonds occupied by central-force springs, the stability of which depends on the average number of neighbors per site z. When z  <  zc  ≈  2d, where d is the spatial dimension, frames are unstable with respect to internal deformations. This pedagogical review focuses on the properties of frames with z at or near zc, which model systems like randomly packed spheres near jamming and network glasses. Using an index theorem, N0  -NS  =  dN  -NB relating the number of sites, N, and number of bonds, NB, to the number, N0, of modes of zero energy and the number, NS, of states of self stress, in which springs can be under positive or negative tension while forces on sites remain zero, it explores the properties of periodic square, kagome, and related lattices for which z  =  zc and the relation between states of self stress and zero modes in periodic lattices to the surface zero modes of finite free lattices (with free boundary conditions). It shows how modifications to the periodic kagome lattice can eliminate all but trivial translational zero modes and create topologically distinct classes, analogous to those of topological insulators, with protected zero modes at free boundaries and at interfaces between different topological classes.

Selective buckling via states of self-stress in topological metamaterials

States of self-stress--tensions and compressions of structural elements that result in zero net forces--play an important role in determining the load-bearing ability of structures ranging from bridges to metamaterials with tunable mechanical properties. We exploit a class of recently introduced states of self-stress analogous to topological quantum states to sculpt localized buckling regions in the interior of periodic cellular metamaterials. Although the topological states of self-stress arise in the linear response of an idealized mechanical frame of harmonic springs connected by freely hinged joints, they leave a distinct signature in the nonlinear buckling behavior of a cellular material built out of elastic beams with rigid joints. The salient feature of these localized buckling regions is that they are indistinguishable from their surroundings as far as material parameters or connectivity of their constituent elements are concerned. Furthermore, they are robust against a wide range of structural perturbations. We demonstrate the effectiveness of this topological design through analytical and numerical calculations as well as buckling experiments performed on two- and three-dimensional metamaterials built out of stacked kagome lattices.

Rigidity loss in disordered systems: three scenarios

We reveal significant qualitative differences in the rigidity transition of three types of disordered network materials: randomly diluted spring networks, jammed sphere packings, and stress-relieved networks that are diluted using a protocol that avoids the appearance of floppy regions. The marginal state of jammed and stress-relieved networks are globally isostatic, while marginal randomly diluted networks show both overconstrained and underconstrained regions. When a single bond is added to or removed from these isostatic systems, jammed networks become globally overconstrained or floppy, whereas the effect on stress-relieved networks is more local and limited. These differences are also reflected in the linear elastic properties and point to the highly effective and unusual role of global self-organization in jammed sphere packings.

Rigidity percolation by next-nearest-neighbor bonds on generic and regular isostatic lattices

We study rigidity percolation transitions in two-dimensional central-force isostatic lattices, including the square and the kagome lattices, as next-nearest-neighbor bonds ("braces") are randomly added to the system. In particular, we focus on the differences between regular lattices, which are perfectly periodic, and generic lattices with the same topology of bonds but whose sites are at random positions in space. We find that the regular square and kagome lattices exhibit a rigidity percolation transition when the number of braces is ∼LlnL, where L is the linear size of the lattice. This transition exhibits features of both first-order and second-order transitions: The whole lattice becomes rigid at the transition, and a diverging length scale also exists. In contrast, we find that the rigidity percolation transition in the generic lattices occur when the number of braces is very close to the number obtained from Maxwell's law for floppy modes, which is ∼L. The transition in generic lattices is a very sharp first-order-like transition, at which the addition of one brace connects all small rigid regions in the bulk of the lattice, leaving only floppy modes on the edge. We characterize these transitions using numerical simulations and develop analytic theories capturing each transition. Our results relate to other interesting problems, including jamming and bootstrap percolation.