Odd elasticity

Abstract

Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials.

15 pages including Supplementary Information, 5 figures. See https://www.youtube.com/playlist?list=PLnzJw7B7T8cIc1EjLVq3AU_hOe1PEAdt9 for Supplementary Movies

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Cited by in corpus (138)

• Exceptional Topology of Non-Hermitian Systems
• Non-Hermitian Physics
• Non-reciprocal phase transitions
• Topological active matter
• Odd dynamics of living chiral crystals
• Odd Viscosity and Odd Elasticity
• Scalar Active Mixtures: The Non-Reciprocal Cahn-Hilliard Model
• Non-Hermitian band topology and skin modes in active elastic media
• Realization of active metamaterials with odd micropolar elasticity
• Selective and Collective Actuation in Active Solids
• Statistical mechanics of a chiral active fluid
• Motile dislocations knead odd crystals into whorls
• Non-equilibrium stationary states of quantum non-Hermitian lattice models
• Variational Onsager Neural Networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs
• Clustering and flocking of repulsive chiral active particles with non-reciprocal couplings
• Response theory: a trajectory-based approach
• Nonreciprocal pattern formation of conserved fields
• Limit cycles turn active matter into robots
• Self-Aligning Polar Active Matter
• Topological defects in solids with odd elasticity
• Active viscoelasticity of odd materials
• Collisions enhance self-diffusion in odd-diffusive systems
• Stokes flows in three-dimensional fluids with odd and parity-violating viscosities
• Non-Hermitian Topological Metamaterials with Odd Elasticity
• Autonomous waves and global motion modes in living active solids
• Shear-driven solidification and nonlinear elasticity in epithelial tissues
• Dynamical Degeneracy Splitting and Directional Invisibility in Non-Hermitian Systems
• Thermodynamics of active field theories: Energetic cost of coupling to reservoirs
• Non-reciprocal response of a two-dimensional fluid with odd viscosity
• Active cholesterics: odder than odd elasticity
• The Extreme Mechanics of Viscoelastic Metamaterials
• Non-reciprocal frustration: time crystalline order-by-disorder phenomenon and a spin-glass-like state
• Soft Metamaterials: Adaptation and Intelligence
• Passive odd viscoelasticity
• Active Rheology in Odd Viscosity Systems
• Active Spaghetti: Collective Organization in Cyanobacteria
• Chiral active matter in external potentials
• Self-organized swimming with odd elasticity
• Oscillatory force autocorrelations in equilibrium odd-diffusive systems
• Lift force in odd compressible fluids
• Chiral active hexatics: Giant number fluctuations, waves and destruction of order
• Mapping of Elastic Properties of Twisting Metamaterials onto Micropolar Continuum using Static Calculations
• Simple views on symmetries and dualities in the theory of elasticity
• Adiabatic edge-to-edge transformations in time-modulated elastic lattices and non-Hermitian shortcuts
• Odd Cosserat elasticity in active materials
• Non-reciprocal alignment induces asymmetric clustering in active mixtures
• Time-correlation functions for odd Langevin systems
• Dual gauge theory formulation of planar quasicrystal elasticity and fractons
• Edge theory of non-Hermitian skin modes in higher dimensions
• Extreme Spontaneous Deformations of Active Crystals
• Non-Bloch band theory for non-Hermitian continuum systems
• Phase Coexistence Implications of Violating Newton's Third Law
• Odd elastohydrodynamics: non-reciprocal living material in a viscous fluid
• A mechanism for anomalous transport in chiral active liquids
• Impact of chirality on active Brownian particle: Exact moments in two and three dimensions
• Flutter instability in solids and structures, with a view on biomechanics and metamaterials
• Anomalous elasticity of cellular tissue vertex model
• The Onsager-Machlup Integral for Non-reciprocal Systems with Odd Elasticity
• Odd elasticity and topological waves in active surfaces
• Odd Microswimmer
• Onsager's variational principle for nonreciprocal systems with odd elasticity
• Algebraic non-Hermitian skin effect and generalized Fermi surface formula in arbitrary dimensions
• Probe particles in odd active viscoelastic fluids: how activity and dissipation determine linear stability
• Dissipative Pairing Interactions: Quantum Instabilities, Topological Light, and Volume-Law Entanglement
• Enhanced stability and chaotic condensates in multi-species non-reciprocal mixtures
• Equilibrium forces on non-reciprocal materials
• Generalized Three-Sphere Microswimmers
• Generalizing odd elasticity theory to odd thermoelasticity for planar materials
• Dynamical structures in phase-separating non-reciprocal polar active mixtures
• Charge Conservation Beyond Uniformity: Spatially Inhomogeneous Electromagnetic Response in Periodic Solids
• Odd elasticity of a catalytic micromachine
• Coexistence of stable and unstable population dynamics in a nonlinear non-Hermitian mechanical dimer
• Non-reciprocal interactions spatially propagate fluctuations in a 2D Ising model
• Rayleigh Edge Waves in Two-Dimensional Crystals with Lorentz Forces -- from Skyrmion Crystals to Gyroscopic Media
• Single active particle engine utilizing a non-reciprocal coupling between particle's position and self-propulsion
• Non-reciprocal Phase Separations with Non-conserved Order Parameters
• Sociohydrodynamics: data-driven modelling of social behavior
• Universal spectral moment theorem and its applications in non-Hermitian systems
• Supersonic kinks and solitons in active solids
• Anomalous grain dynamics and grain locomotion of odd crystals
• Exotic Topology in Ordinary Soft Matter
• Complete absorption of topologically protected waves
• Hydrodynamics of thermal active matter
• Photoinduced non-reciprocal magnetism
• Molecular modelling of odd viscoelastic fluids
• The XY model with vision cone: non-reciprocal vs. reciprocal interactions
• Peristalsis by pulses of activity
• Alignment Destabilizes Crystal Orders in Active Systems
• Chirality and odd mechanics in active columnar phases
• Simulating structured fluids with tensorial viscoelasticity
• Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids
• Network Model of Active Fluctuations of Thin Elastic Shells Swollen by Hydrostatic Pressure
• Symmetry-driven Phononic Metamaterials
• Density Mediated Spin Correlations Drive Edge to Bulk Flow Transition in Active Chiral Matter
• Simulations of Odd Microswimmers
• Hydrodynamic flow field and frictional resistance coefficient of a disk rotating steadily in a compressible fluid layer with odd viscosity on a rigid substrate
• Inertial Spinner Swarm Experiments: Spin Pumping, Entropy Oscillations and Spin Frustration
• Nonreciprocal amplification toward chaos in a chain of Duffing oscillators
• Optical non-Hermitian skin effect in two-dimensional uniform media
• Rigid flocks, undulatory gaits, and chiral foldamers in a chemically active polymer
• Chiral flows can induce neck formation in viscoelastic surfaces
• Mechanochemical feedback drives complex inertial dynamics in active solids
• Hermitian and non-Hermitian topology in active matter
• Flocking with random non-reciprocal interactions
• Classifying topological floppy modes in the continuum
• A geometric formulation of Schaefer's theory of Cosserat solids
• Collective excitations in active solids featuring alignment interactions
• Entropy production in non-reciprocal polar active mixtures
• Condensate Formation in a Chiral Lattice Gas
• Active assembly and non-reciprocal dynamics of elastic membranes
• Solving the Kinetic Ising Model with Non-Reciprocity
• Rheologically tuned modes of collective transport in active viscoelastic films
• Unified description of viscous, viscoelastic, or elastic thin active films on substrates
• Reentrant transition to collective actuation in active solids with a polarizing field
• More is less in unpercolated active solids
• Stochastic Thermodynamics of Non-reciprocally Interacting Particles and Fields
• Wave propagation in an elastic lattice with non-reciprocal stiffness and engineered damping
• Chirality, confinement and dimensionality govern re-entrant transitions in active matter
• Anisotropic odd viscosity via time-modulated drive
• Collective actuation in active solids in the presence of a polarizing field: a systematic analysis of the dynamical regimes
• Irreversibility of stochastic state transitions in Langevin systems with odd elasticity
• Statistical mechanics of a cold tracer in a hot bath
• Inertia-chirality interplay in active Brownian motion: exact dynamics and phase maps
• 3D Topologically Polarized Elastic Metamaterials Enable Asymmetric Energy Isolation at Low Frequencies
• A minimal model with stochastically broken reciprocity
• Emergent non-Hermitian conservation laws at exceptional points
• Odd elasticity in Hamiltonian formalism
• Odd elasticity in disordered chiral active materials
• Classification of rotational zero modes in 2D micropolar solids
• From Active to Odd to Smart Matter
• When soft crystals defy Newton's third law: Non-reciprocal mechanics and dislocation motility
• Nonlinear Odd Viscoelastic Effect
• Active elastocapillarity in soft solids with negative surface tension
• Optimal power and efficiency of odd engines
• Hydrodynamic correlation functions of chiral active fluids
• Topological Flux on a Context Manifold Generates Nonreciprocal Collective Dynamics
• State Diagram of the Non-Reciprocal Cahn-Hilliard Model and the Effects of Symmetry
• Kinetic Theory of Chiral Active Disks: Odd Transport and Torque Density