Micro-structure in linear elasticity

1964/01/01 by R. D. Mindlin · 3,940 citations
Engineering · Materials Science · Mathematics · #Artificial intelligence #Complex system #Composite Structure Analysis and Optimization #Computer science #Elasticity (physics) #Finite element method #Linear elasticity #Linear system #Materials science #Mathematical analysis #Mathematics #Nonlinear system #Nonlocal and gradient elasticity in micro/nano structures #Physics #Quantum mechanics #Statistical physics #Thermodynamics #Thermoelastic and Magnetoelastic Phenomena

paper · doi:10.1007/bf00248490

published in Archive for Rational Mechanics and Analysis 16(1), 51-78 (Springer Science+Business Media)

openalex publication_date 1964/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Cited by

• Theories of elasticity with couple-stress
• A nonlinear theory of elastic materials with voids
• On the thermodynamics of systems with nonlocality
• Size-Dependent Effects and Surface Roughness in Contact Mechanics
• Bernoulli–Euler Dielectric Beam Model Based on Strain-Gradient Effect
• Mixed-Mode Loading of a Straight Crack With Surface Strain-Gradient Elasticity
• A gradient-enriched continuum model for magneto-elastic coupling: Formulation, finite element implementation and in-plane problems
• Emergent quantization from a dynamic vacuum
• A Lowest-Order Robust Mixed Finite Element Method with a Third-Order Tensor Variable for Strain Gradient Elasticity
• Recovering nonsmooth coefficients for higher-order perturbations of a polyharmonic operator
• Torsional and SH surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin-Mindlin gradient theory
• Exact and numerical solutions to a Mindlin microcontinuum model
• Commentary about the paper "Continuum Mech. Thermodyn. (2010) 22:163-176" by Podio Guidugli and Vianello and some related papers on higher gradient theories
• Polyharmonic Nonlinear Scalar Field Equations
• Static size-effects meet the dynamic scattering properties of finite-sized mechanical metamaterials: a relaxed micromorphic study with parameter identification via two-stage static-dynamic optimization
• Material data identification in generalized continua
• Symmetry classes for even-order tensors
• Cosserat micropolar and couple-stress elasticity models of flexomagnetism at finite deformations
• A screw dislocation in a functionally graded material using the translation gauge theory of dislocations
• Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model
• Size Effects on Mechanical Properties of Micro/Nano Structures
• Representation of Material Elements and Geometry of Substructural Interactions
• Novel weak form quadrature elements for non-classical higher order beam and plate theories
• Nonlinear vibration of functionally graded nonlocal nanobeam with thermal effect: analytical model versus finite element approach
• A modified micromorphic model based on micromechanics for granular materials
• Nonlocal metamaterials and metasurfaces
• Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids
• Staggered time discretization in finitely-strained heterogeneous visco-elastodynamics with damage or diffusion in the Eulerian frame
• Generic stability in dissipative generalized mechanics
• Is a `hadronic' shear current one of the sources in metric-affine gravity?
• Recent Developments in CAD/analysis Integration
• Breaking Scale Separation in Metamaterials' homogenization: Interface-Inertia-Enhanced Relaxed Micromorphic Model
• At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola
• Discrete and continuous models of linear elasticity: history and connections
• Generalized Hooke's law for isotropic second gradient materials
• Shell energies derived from three-dimensional isotropic strain-gradient elasticity
• The problem of sharp notch in microstructured solids governed by dipolar gradient elasticity
• Effective elastic properties of planar SOFCs: A non-local dynamic homogenization approach
• Unified nonlocal rational continuum models developed from discrete atomistic equations
• Scattering of Antiplane Shear Waves by Fractals in Strain Gradient Elasticity
• A New Framework for Unidimensional Structures Based on Generalised Continua
• Relativistic second gradient theory of continuous media
• Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena
• A review of inverse problems for generalized elastic media: formulations, experiments, synthesis
• Non-classical solutions of a continuum model for rock descriptions
• A variant of the linear isotropic indeterminate couple stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and complete traction boundary conditions
• Relaxed micromorphic broadband scattering for finite-size meta-structures -- a detailed development
• Mixed Finite element method for stress gradient elasticity
• Local micromorphic non-affine anisotropy for materials incorporating elastically bonded fibers
• A non-singular theory of dislocations in anisotropic crystals
• Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits
• Granular micromechanics based micromorphic model predicts frequency band gaps
• A Family of Solitary Waves in Frenkel-Kontorova Lattice
• Geometric theory of generalised continua using moving frames
• Novel differential quadrature element method for higher order strain gradient elasticity theory
• Propagation of Love waves in linear elastic isotropic Cosserat materials
• Core energy and regularization parameters of non-singular continuum theories of dislocations using atomistic simulations
• An Optimal and Robust Nonconforming Finite Element Method for the Strain Gradient Elasticity
• Second-gradient models for incompressible viscous fluids and associated cylindrical flows
• Propagation of elastic waves in a flexomagnetic solid
• Size effect in micro-scale cantilever beam bending
• Investigation of static buckling and bending of nanoplates made of new functionally graded materials considering surface effects on an elastic foundation
• Displacement general solutions in strain gradient elasticity: review and analysis
• Hyperfluid — a model of classical matter with hypermomentum
• WITHDRAWN: On gradient enriched elasticity theories: A reply to “Comment on ‘On non-singular crack fields in Helmholtz type enriched elasticity theories’ ” and important theoretical aspects
• Anti-plane analysis of an infinite plane with multiple cracks based on strain gradient theory
• The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure
• Interaction of cracks with dislocations in couple-stress elasticity. Part I: Opening mode
• Defects in unidimensional structures
• Analytical Solutions for Long Cylindrical Shells Under Radial Deformations Based on the Isotropic Relaxed Micromorphic Continuum
• An immersed boundary hierarchical B-spline method for flexoelectricity
• On the polar nature and invariance properties of a thermomechanical theory for continuum-on-continuum homogenization
• Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality
• On doubly critical polyharmonic double phase problems: Existence and non-existence of solutions
• Time discretization in convected linearized thermo-visco-elastodynamics at large displacements
• Computational homogenization of higher-order continua
• Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams
• On solutions of a Boussinesq-type equation with displacement-dependent nonlinearities: the case of biomembranes
• Identification of higher-order continua equivalent to a Cauchy elastic composite
• The gauge theory of dislocations: Static solutions of screw and edge dislocations
• A model for the interaction of dislocations with planar defects based on Allen–Cahn type microstructure evolution coupled to strain gradient elasticity
• Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations
• On admissible external loadings within the first and second strain gradient elasticity
• Modelling the Size Effects on the Mechanical Properties of Micro/Nano Structures. [europepmc]
• The postulations á la D'Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. [europepmc]
• Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage. [europepmc]
• Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. [europepmc]
• Mixed finite-element formulations in piezoelectricity and flexoelectricity. [europepmc]
• A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects. [europepmc]
• Real wave propagation in the isotropic-relaxed micromorphic model. [europepmc]
• On the role of micro-inertia in enriched continuum mechanics. [europepmc]
• Asymmetry of the atomic-level stress tensor in homogeneous and inhomogeneous materials. [europepmc]
• Vibration and Buckling of Shear Deformable Functionally Graded Nanoporous Metal Foam Nanoshells. [europepmc]
• Scale Effects in Orthotropic Composite Assemblies as Micropolar Continua: A Comparison between Weak- and Strong-Form Finite Element Solutions. [europepmc]
• Reduced micromorphic model in orthogonal curvilinear coordinates and its application to a metamaterial hemisphere. [europepmc]
• A Micromorphic Beam Theory for Beams with Elongated Microstructures. [europepmc]
• Exponential stability in Mindlin's Form II gradient thermoelasticity with microtemperatures of type III. [europepmc]
• Non-Conventional Thermodynamics and Models of Gradient Elasticity. [europepmc]
• Phenomenological Thermodynamics of Irreversible Processes. [europepmc]
• Torsional Characteristics of Carbon Nanotubes: Micropolar Elasticity Models and Molecular Dynamics Simulation. [europepmc]
• Bio-Inspired Design of a Porous Resorbable Scaffold for Bone Reconstruction: A Preliminary Study. [europepmc]
• Hygro-Thermal Vibrations of Porous FG Nano-Beams Based on Local/Nonlocal Stress Gradient Theory of Elasticity. [europepmc]
• Analysis of planes within reduced micromorphic model. [europepmc]
• A discrete-to-continuum model of protein complexes. [europepmc]
• Size-Dependent Buckling and Post-Buckling Analysis of the Functionally Graded Thin Plate Al-Cu Material Based on a Modified Couple Stress Theory. [europepmc]
• Analysis of Size-Dependent Linear Static Bending, Buckling, and Free Vibration Based on a Modified Couple Stress Theory. [europepmc]
• Interpretation of Second Law of Thermodynamics in Extended Procedures for the Exploitation of the Entropy Inequality: Korteweg Fluids and Strain-Gradient Elasticity as Examples. [europepmc]
• Band gaps of elastic waves in 1-D dielectric phononic crystal with the flexoelectric and strain gradient effects consideration. [europepmc]