Isostaticity, auxetic response, surface modes, and conformal invariance in twisted kagome lattices

Abstract

Model lattices consisting of balls connected by central-force springs provide much of our understanding of mechanical response and phonon structure of real materials. Their stability depends critically on their coordination number $z$. $d$-dimensional lattices with $z=2d$ are at the threshold of mechanical stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency "floppy" modes that provide avenues for lattice collapse. The physics of systems as diverse as architectural structures, network glasses, randomly packed spheres, and biopolymer networks is strongly influenced by a nearby isostatic lattice. We explore elasticity and phonons of a special class of two-dimensional isostatic lattices constructed by distorting the kagome lattice. We show that the phonon structure of these lattices, characterized by vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity, depends sensitively on boundary conditions and on the nature of the kagome distortions. We construct lattices that under free boundary conditions exhibit surface floppy modes only or a combination of both surface and bulk floppy modes; and we show that bulk floppy modes present under free boundary conditions are also present under periodic boundary conditions but that surface modes are not. In the the long-wavelength limit, the elastic theory of all these lattices is a conformally invariant field theory with holographic properties, and the surface waves are Rayleigh waves. We discuss our results in relation to recent work on jammed systems. Our results highlight the importance of network architecture in determining floppy-mode structure.

12 pages, 7 figures

References in corpus (3)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• Elasticity of Stiff Polymer Networks
• Elasticity and Response in Nearly Isostatic Periodic Lattices

Cited by in corpus (103)

• Topological Phononic Crystals with One-Way Elastic Edge Waves
• Topological Boundary Modes in Isostatic Lattices
• Modeling semiflexible polymer networks
• Topological modes bound to dislocations in mechanical metamaterials
• Nonlinear conduction via solitons in a topological mechanical insulator
• Selective buckling via states of self-stress in topological metamaterials
• Transformable topological mechanical metamaterials
• Non-Hermitian band topology and skin modes in active elastic media
• Mechanical Weyl Modes in Topological Maxwell Lattices
• Topological phonons and Weyl lines in 3 dimensions
• Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots
• Chiral Assemblies of Pinwheel Superlattices on Substrates
• Lattice Mechanics of Origami Tessellations
• Rigidity Loss in Disordered Systems: Three Scenarios
• Dualities and non-Abelian mechanics
• A discretized Chern-Simons gauge theory on arbitrary graphs
• Geared topological metamaterials with tunable mechanical stability
• Conformal Elasticity of Mechanism-Based Metamaterials
• Mechanics of elastic networks
• Knotted Defects in Nematic Liquid Crystals
• Silicene vs. ordered 2D silicide: the atomic and electronic structure of the Si-$(\sqrt{19}\times \sqrt{19})R23.4^{\circ }$/Pt(111) surface reconstruction
• Edge Modes and Asymmetric Wave Transport in Topological Lattices: Experimental Characterization at Finite Frequencies
• Topological edge floppy modes in disordered fiber networks
• Band theory and boundary modes of high-dimensional representations of infinite hyperbolic lattices
• Fracturing of topological Maxwell lattices
• Enumerating rigid sphere packings
• Elasticity of Filamentous Kagome Lattice
• Geometry and the onset of rigidity in a disordered network
• Mechanical instability at finite temperature
• Phonon analogue of topological nodal semimetals
• Chiral edge mode in the coupled dynamics of magnetic solitons in a honeycomb lattice
• Reconstruction of moiré lattices in twisted transition metal dichalcogenide bilayers
• Microtwist elasticity: A continuum approach to zero modes and topological polarization in Kagome lattices
• Length scales and self-organization in dense suspension flows
• Tunable Band Topology in Gyroscopic Lattices
• Directional mechanical response in the bulk of topological metamaterials
• Universal Continuum Theory for Topological Edge Soft Modes
• Autonomous actuation of zero modes in mechanical networks far from equilibrium
• Self-assembly of three-dimensional open structures using patchy colloidal particles
• Buckling of thermalized elastic sheets
• Topological mechanics in quasicrystals
• Localizing softness and stress along loops in three-dimensional topological metamaterials
• Geometric auxetics
• Topologically protected interface phonons in two-dimensional nanomaterials: hexagonal boron nitride and silicon carbide
• Rigidity percolation by next-nearest-neighbor braces on generic and regular isostatic lattices
• Theory of spin waves in a hexagonal antiferromagnet
• Simple views on symmetries and dualities in the theory of elasticity
• Topological boundary modes in jammed matter
• Theorem on the Compatibility of Spherical Kirigami Tessellations
• Mechanical Graphene
• Entropic effects in the self-assembly of open lattices from patchy particles
• Soft mechanical metamaterials with transformable topology protected by stress caching
• Odd Cosserat elasticity in active materials
• Existence of Corner Modes in Elastic Twisted Kagome Lattices
• Breakdown of Conventional Winding Number Calculation in One-Dimensional Lattices with Interactions Beyond Nearest Neighbors
• Symmetry detection of auxetic behaviour in 2D frameworks
• Deterministic and stochastic control of kirigami topology
• Shear shocks in fragile networks
• Synthetically Non-Hermitian Nonlinear Wave-like Behavior in a Topological Mechanical Metamaterial
• Hybrid topological magnon-phonon modes in honeycomb and kagome lattices
• Two-dimensional Mechanical Metamaterials with Unusual Poisson Ratio Behavior
• Symmetry of the phonon landscape across the duality boundary of twisted kagomes lattices
• Penrose Tilings as Jammed Solids
• Multi-functional Twisted-Kagome lattices: Tuning by Pruning Mechanical Metamaterials
• Finite temperature mechanical instability in disordered lattices
• Self-Assembly of Isostatic Self-Dual Colloidal Crystals
• Scale without conformal invariance in membrane theory
• Topological Flexural Modes in Polarized Bilayer Lattices
• Topology and geometry of spin origami
• Algebraic non-Hermitian skin effect and generalized Fermi surface formula in arbitrary dimensions
• Some results on the Guest-Hutchinson modes and periodic mechanisms of the Kagome lattice metamaterial
• Dynamics of self-dual kagome metamaterials and the emergence of fragile topology
• Finite-Frequency Topological Maxwell Modes in Mechanical Self-Dual Kagome Lattices
• Realizing negative Poisson's ratio in unstressed spring networks
• Disordered auxetic networks with no re-entrant polygons
• Discharging dynamics of topological batteries
• Duality,Hidden Symmetry and Dynamic Isomerism in 2D Hinge Structures
• Empty perovskites as Coulomb floppy networks: entropic elasticity and negative thermal expansion
• Rayleigh surface waves of extremal elastic materials
• Rayleigh Edge Waves in Two-Dimensional Crystals with Lorentz Forces -- from Skyrmion Crystals to Gyroscopic Media
• Patterns on a Roll: A Method for Continuous Feed Nanoprinting
• High Bulk Modulus Pentamodes: the Three-Dimensional Metal Water
• Mechanics of metric frustration in contorted filament bundles: From local symmetry to columnar elasticity
• Capillary-driven binding of thin triangular prisms at fluid interfaces
• Self-assembled multi-layer simple cubic photonic crystals of oppositely charged colloids in confinement
• Architecting mechanisms of damage in topological metamaterials
• Phonons and elasticity in critically coordinated lattices
• Omnidirectional domain wall modes protected by fragile topological states
• Ultrarigid periodic frameworks
• Curvature screening in draped mechanical metamaterial sheet
• The first-order flexibility of a crystal framework
• Sparse phonon modes of a limit-periodic structure
• Cell augmentation framework for topological lattices
• Localized interfacial Phonon Modes at the Electronic Axion Domain Wall
• Liftings and stresses for planar periodic frameworks
• Non-local twist sequences in floppy kagome chains
• Shape Entropy of a Reconfigurable Ising Surface
• Classification of rotational zero modes in 2D micropolar solids
• Transformable Super-Isostatic Crystals Self-Assembled from Segment Colloidal Rods
• Kagome modes, a new route to ultralow thermal conductivity?
• Geometric approach to mechanical design principles in continuous elastic sheets
• Hierarchical Topological States without Dimension Reduction
• Viscoelastic scaling regimes for marginally-rigid fractal spring networks