Elastic theory of unconstrained non-Euclidean plates

Abstract

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

References in corpus (2)

• Buckling of swelling gels
• Lower bounds for the energy in a crumpled elastic sheet - A minimal ridge

Cited by in corpus (112)

• A Geometric Theory of Growth Mechanics
• Shape-morphing architected sheets with non-periodic cut patterns
• The von Karman equations for plates with residual strain
• Morphing of Geometric Composites via Residual Swelling
• Director Deformations, Geometric Frustration, and Modulated Phases in Liquid Crystals
• Programmed buckling by controlled lateral swelling in a thin elastic sheet
• Curvature generation in nematic surfaces
• Curvature-Induced Instabilities of Shells
• Geometric frustration and solid-solid transitions in model 2D tissue
• Printing non-Euclidean solids
• A Riemannian Approach to Reduced Plate, Shell, and Rod Theories
• Curvature-Driven Morphing of Non-Euclidean Shells
• Frustrated Self-Assembly of Non-Euclidean Crystals of Nanoparticles
• Ground States of Crystalline Caps: Generalized Jellium on Curved Space
• Elastic interactions between 2D geometric defects
• Geometry of wave propagation on active deformable surfaces
• Geometric charges and nonlinear elasticity of soft metamaterials
• Multiparameter actuation of a neutrally-stable shell: a flexible gear-less motor
• Buckling Thin Disks and Ribbons with Non-Euclidean Metrics
• Subdivision Shell Elements with Anisotropic Growth
• Oscillating membranes: modeling and controlling autonomous shape-transforming sheets
• Overcurvature induced multistability of linked conical frusta: How a `bendy straw' holds its shape
• Pathways to faceting of vesicles
• Cumulative geometric frustration in physical assemblies
• Wrinkle patterns in active viscoelastic thin sheets
• Elasticity and Fluctuations of Frustrated Nano-Ribbons
• Optimal control of plate shape with incompatible strain fields
• The Mechanics of Spacetime - A Solid Mechanics Perspective on the Theory of General Relativity
• Nonlinear mechanics of thin frames
• Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets
• Theory of Liquid Crystal Elastomers: From Polymer Physics to Differential Geometry
• Snapping of Bistable, Prestressed Cylindrical Shells
• Spontaneous deformation of flexible ferromagnetic ribbons induced by Dzyaloshinskii-Moriya interaction
• Model-free thermodynamics of fluid vesicles
• Smart helical structures inspired by the pellicle of euglenids
• Morphoelasticity of Large Bending Deformations of Cell Sheets during Development
• Pattern selection and multiscale behavior in metrically discontinuous non-Euclidean plates
• Contrasting bending energies from bulk elastic theories
• Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
• Wrinkling instability in unsupported epithelial sheets
• Variational models for prestrained plates with Monge-Ampère constraint
• Nonlinear elastic model for faceting of vesicles with soft grain boundaries
• On the role of curvature in the elastic energy of non-Euclidean thin bodies
• The nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature
• Elasticity in Curved Topographies: Exact Theories and Linear Approximations
• Curvature-driven wrinkling of thin elastic shells
• Strain tensor selection and the elastic theory of incompatible thin sheets
• Manipulating the shape of flexible magnetoelastic nanodiscs with meron-like magnetic states
• Elastic capsules at liquid-liquid interfaces
• Multi-valued inverse design: multiple surface geometries from one flat sheet
• Fractional solitons in non-Euclidian elastic plates
• Derivation of an effective plate theory for parallelogram origami from bar and hinge elasticity
• Nonuniform growth and topological defects in the shaping of elastic sheets
• A Riemannian approach to the membrane limit in non-Euclidean elasticity
• Membrane flutter in three-dimensional inviscid flow
• Geometric Theory of Mechanical Screening in two-dimensional solids
• Dilation-invariant bending of elastic plates, and broken symmetry in shells
• Curvature induced magnetization of altermagnetic films
• Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
• Fracton-elasticity duality on curved manifolds
• The Föppl-von Kármán equations of elastic plates with initial stress
• Nature's forms are frilly, flexible, and functional
• Hierarchy of Geometrical Frustration in Elastic Ribbons: shape-transitions and energy scaling obtained from a general asymptotic theory
• Quadratic-stretch elasticity
• A tug-of-war between stretching and bending in living cell sheets
• Shape and Fluctuations of Positively Curved Ribbons
• Universal Moiré Buckling of Freestanding 2D Bilayers
• Minimal resonances in annular non-Euclidean strips
• Some observations on variational elasticity and its application to plates and membranes
• Energies for elastic plates and shells from quadratic-stretch elasticity
• Exact Solution for Elastic Networks on Curved Surfaces
• Mechanics of metric frustration in contorted filament bundles: From local symmetry to columnar elasticity
• Semi-implicit methods for the dynamics of elastic sheets
• Midveins regulate the shape formation of drying leaves
• Toward the mechanics of fractal materials: mechanics of continuum with fractal metric
• Limits of elastic models of converging Riemannian manifolds
• A continuum geometric approach for inverse design of origami structures
• A ribbon model for nematic polymer networks
• The plane stress state of residually stressed bodies: a stress function approach
• Anisotropic swelling of anisotropic elastic panels
• On Saint-Venant compatibility and stress potentials in manifolds with boundary and constant sectional curvature
• Distributed branch points and the shape of elastic surfaces with constant negative curvature
• Deformation of an asymmetric thin film
• A remark on constrained von Karman theories
• Numerical analysis of the LDG method for large deformations of prestrained plates
• Asymptotic rigidity for shells in non-Euclidean elasticity
• Variational Convergence of Discrete Geometrically-Incompatible Elastic Models
• Plates with incompatible prestrain of higher order
• On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapes
• LDG approximation of large deformations of prestrained plates
• Continuum mechanics of moving defects in growing bodies
• Nonlinear mechanics of rigidifying curves
• Models for elastic shells with incompatible strains
• Nonlinear extension of Kolosov-Muskhelishvili stress function formalism
• Stability of isometric immersions of hypersurfaces
• A Weyl geometric model for thermo-mechanics of solids with metrical defects
• On the Bending Energy of Buckled Edge-Dislocations
• General Solution for Elastic Networks on Arbitrary Curved Surfaces in the Absence of Rotational Symmetry
• Odd elasticity in Hamiltonian formalism
• From Volterra dislocations to strain-gradient plasticity
• The mechanics of anisotropic active plates with applications to cell alignment on curved substrates
• Geometric approach to mechanical design principles in continuous elastic sheets
• The linearized Kirchhoff theory for plates with incompatible prestrain
• Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case
• Buckling transition and boundary layer in non-Euclidean plates
• The Intermittent Nature of Leaf Growth Fields
• A Critical Study of Efrati et al.'s Elastic Theory of Unconstrained non-Euclidean Plates
• Large-strain poroelastic plate theory for polymer gels with applications to swelling-induced morphing of composite plates
• Isometric Incompatibility in Growing Elastic Sheets
• Nonlinear bending theories for non Euclidean plates
• Geometrical dynamics of edge-driven surface growth
• A three-dimensional morphoelastic model for self-oscillations in polyelectrolyte hydrogel filaments