[Une équation des ondes dispersive utilisant l'élasticité non-locale]
La mécanique des milieux continus non-locaux permet de prendre en compte des effets d'échelle qui peuvent être significatifs lorsque l'on s'intéresse aux structures à faible échelle (micro ou nano-structures). Cette Note s'intéresse à un modèle de propagation d'ondes dans un milieu élastique non-local. Nous montrons qu'une équation d'ondes dispersive est obtenue à partir d'une loi constitutive non-locale, basée sur une combinaison des déformations locales et non-locales. Le modèle comprend à la fois le modèle au gradient classique et le modèle intégral d'Eringen. Les propriétés dynamiques du modèle sont discutées et corroborent des résultats récents permettant d'unifier les approches au gradient en régime statique et dynamique. De plus, ce modèle permet de décrire de manière très précise les courbes de dispersion du modèle de Born–Kármán.
Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This Note investigates a model of wave propagation in a nonlocal elastic material. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. This model comprises both the classical gradient model and the Eringen's integral model. The dynamic properties of the model are discussed, and corroborate well some recent theoretical studies published to unify both static and dynamics gradient elasticity theories. Moreover, an excellent matching of the dispersive curve of the Born–Kármán model of lattice dynamics is obtained with such nonlocal model.
Accepté le :
Publié le :
Mots-clés : Milieux continus, Modèle non-local, Modèle au gradient, Équation des ondes, Élasticité, Matériau hétérogène, Dispersion, Modèle de Born–Kármán
Noël Challamel 1 ; Lalaonirina Rakotomanana 2 ; Loïc Le Marrec 2
@article{CRMECA_2009__337_8_591_0, author = {No\"el Challamel and Lalaonirina Rakotomanana and Lo{\"\i}c Le Marrec}, title = {A dispersive wave equation using nonlocal elasticity}, journal = {Comptes Rendus. M\'ecanique}, pages = {591--595}, publisher = {Elsevier}, volume = {337}, number = {8}, year = {2009}, doi = {10.1016/j.crme.2009.06.028}, language = {en}, }
TY - JOUR AU - Noël Challamel AU - Lalaonirina Rakotomanana AU - Loïc Le Marrec TI - A dispersive wave equation using nonlocal elasticity JO - Comptes Rendus. Mécanique PY - 2009 SP - 591 EP - 595 VL - 337 IS - 8 PB - Elsevier DO - 10.1016/j.crme.2009.06.028 LA - en ID - CRMECA_2009__337_8_591_0 ER -
Noël Challamel; Lalaonirina Rakotomanana; Loïc Le Marrec. A dispersive wave equation using nonlocal elasticity. Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 591-595. doi : 10.1016/j.crme.2009.06.028. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2009.06.028/
[1] Nonlinear Waves in Elastic Crystals, Oxford University Press, 1999
[2] Nonlocal Continuum Field Theories, Springer, New York, 2002
[3] A Geometric Approach to Thermomechanics of Dissipating Continua, Progress in Mathematical Physics, Birkhäuser, Boston, 2004
[4] Gradient elasticity theories in statics and dynamics – A unification of approaches, Int. J. Fracture, Volume 139 (2006), pp. 297-304
[5] On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Int. J. Solids Structures, Volume 43 (2006), pp. 1404-1421
[6] On causality of the gradient elasticity models, J. Sound Vibration, Volume 297 (2006), pp. 727-742
[7] On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., Volume 54 (1983), pp. 4703-4710
[8] Gradient-enhanced damage for quasi-brittle materials, Int. J. Num. Meth. Engng., Volume 39 (1996), pp. 3391-3403
[9] Plastic failure of nonlocal beams, Phys. Rev. E, Volume 78 (2008), p. 026604
[10] Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., Volume 135 (2009) no. 3, pp. 117-131
[11] Theory of nonlocal elasticity and some applications, Res. Mech., Volume 21 (1987), pp. 313-342
[12] The small length scale effect for a nonlocal cantilever beam: A paradox solved, Nanotechnology, Volume 19 (2008), p. 345703
[13] Update on a class of gradient theories, Mech. Mat., Volume 35 (2003), pp. 259-280
[14] One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure – Part 1: Generic formulation, Eur. J. Mech. A/Solids, Volume 21 (2002), pp. 555-572
[15] Théorie de l'intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Comptes Rendus Hebdomadaires de l'Académie des Sciences de Paris, Volume 72 (1871), pp. 755-759
[16] Torsional surface waves in a gradient-elastic half-space, Wave Motion, Volume 31 (2000), pp. 333-348
- Buckling of micromorphic Timoshenko columns, European Journal of Mechanics - A/Solids, Volume 111 (2025), p. 105537 | DOI:10.1016/j.euromechsol.2024.105537
- Green’s functions of size-dependent Timoshenko beams: Gradient elasticity versus stress-driven nonlocal theories, International Journal of Solids and Structures, Volume 314 (2025), p. 113308 | DOI:10.1016/j.ijsolstr.2025.113308
- Two-Phase Peridynamic Elasticity with Exponential Kernels. I: Statics and Vibrations of Axial Rods, Journal of Engineering Mechanics, Volume 151 (2025) no. 5 | DOI:10.1061/jenmdt.emeng-8252
- Mathematical Modeling of Thermal and Deformation Fields in Non-Metallic Heterogeneous Materials During Control by the Thermal Imaging Method, Journal of Engineering Sciences, Volume 12 (2025) no. 1, p. C12 | DOI:10.21272/jes.2025.12(1).c2
- Equivalence between micromorphic, nonlocal gradient, and two-phase nonlocal beam theories, Acta Mechanica (2024) | DOI:10.1007/s00707-024-04180-x
- Free vibration response of micromorphic Timoshenko beams, Journal of Sound and Vibration, Volume 591 (2024), p. 118602 | DOI:10.1016/j.jsv.2024.118602
- A re-look into the modeling aspects of Eringen’s strain-driven nonlocal Euler-Bernoulli nanobeam bending problems, Mechanics of Advanced Materials and Structures, Volume 31 (2024) no. 28, p. 10484 | DOI:10.1080/15376494.2023.2290695
- Discrete and continuous models of linear elasticity: history and connections, Continuum Mechanics and Thermodynamics, Volume 35 (2023) no. 2, p. 347 | DOI:10.1007/s00161-022-01180-x
- Modelling non-local elasticity in 1D vibrating rods using Corrected Smoothed Particle Hydrodynamics method, European Journal of Mechanics - A/Solids, Volume 91 (2022), p. 104403 | DOI:10.1016/j.euromechsol.2021.104403
- A fractional nonlocal elastic model for lattice wave analysis, Mechanics Research Communications, Volume 126 (2022), p. 103999 | DOI:10.1016/j.mechrescom.2022.103999
- Large Amplitude Free Vibration Analysis of Isotropic Curved Nano/Microbeams Using a Nonlocal Sinusoidal Shear Deformation Theory-Based Finite Element Method, International Journal of Structural Stability and Dynamics, Volume 21 (2021) no. 05, p. 2150074 | DOI:10.1142/s0219455421500747
- Modelling flexural wave propagation by the nonlocal strain gradient elasticity with fractional derivatives, Mathematics and Mechanics of Solids, Volume 26 (2021) no. 10, p. 1538 | DOI:10.1177/1081286521991206
- A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type: part I—analytical formulation and thermodynamic framework, Meccanica, Volume 56 (2021) no. 3, p. 629 | DOI:10.1007/s11012-020-01297-w
- A generalized integro-differential theory of nonlocal elasticity of n-Helmholtz type—part II: boundary-value problems in the one-dimensional case, Meccanica, Volume 56 (2021) no. 3, p. 651 | DOI:10.1007/s11012-020-01298-9
- Lattice-Based Nonlocal Elastic Structural Models, Size-Dependent Continuum Mechanics Approaches (2021), p. 1 | DOI:10.1007/978-3-030-63050-8_1
- Transition from Discrete to Continuous Media: The Impact of Symmetry Changes on Asymptotic Behavior of Waves, Symmetry, Volume 13 (2021) no. 6, p. 1008 | DOI:10.3390/sym13061008
- An analytical study on wave propagation in functionally graded nano-beams/tubes based on the integral formulation of nonlocal elasticity, Waves in Random and Complex Media, Volume 30 (2020) no. 3, p. 562 | DOI:10.1080/17455030.2018.1543979
- Propagation of weakly nonlinear waves in nanorods using nonlocal elasticity theory, Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Volume 21 (2019) no. 1, p. 190 | DOI:10.25092/baunfbed.543422
- Long-Range Retarded Elastic Metamaterials: Wave-Stopping, Negative, and Hypersonic or Superluminal Group Velocity, Physical Review Applied, Volume 11 (2019) no. 1 | DOI:10.1103/physrevapplied.11.014041
- Vibration of a Timoshenko beam supporting arbitrary large pre-deformation, Acta Mechanica, Volume 229 (2018) no. 1, p. 109 | DOI:10.1007/s00707-017-1953-x
- A nonlocal higher-order model including thickness stretching effect for bending and buckling of curved nanobeams, Applied Mathematical Modelling, Volume 57 (2018), p. 121 | DOI:10.1016/j.apm.2017.12.025
- Stress gradient, strain gradient and inertia gradient beam theories for the simulation of flexural wave dispersion in carbon nanotubes, Composites Part B: Engineering, Volume 153 (2018), p. 285 | DOI:10.1016/j.compositesb.2018.08.083
- Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models, Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, p. 320 | DOI:10.1016/j.crme.2017.12.014
- Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories, European Journal of Mechanics - A/Solids, Volume 67 (2018), p. 200 | DOI:10.1016/j.euromechsol.2017.09.004
- Statics and dynamics of nanorods embedded in an elastic medium: Nonlocal elasticity and lattice formulations, European Journal of Mechanics - A/Solids, Volume 67 (2018), p. 254 | DOI:10.1016/j.euromechsol.2017.09.009
- A numerical study on the buckling and vibration of nanobeams based on the strain and stress-driven nonlocal integral models, International Journal of Computational Materials Science and Engineering, Volume 07 (2018) no. 03, p. 1850016 | DOI:10.1142/s2047684118500161
- Nonlinear Wave Modulation in Nanorods Based on Nonlocal Elasticity Theory by Using Multiple-Scale Formalism, International Journal of Engineering and Applied Sciences, Volume 10 (2018) no. 3, p. 140 | DOI:10.24107/ijeas.422906
- Application of Green's function method to bending of stress gradient nanobeams, International Journal of Solids and Structures, Volume 143 (2018), p. 209 | DOI:10.1016/j.ijsolstr.2018.03.009
- Exact and Nonlocal Solutions for Vibration of Axial Lattice with Direct and Indirect Neighboring Interactions, Journal of Engineering Mechanics, Volume 144 (2018) no. 5 | DOI:10.1061/(asce)em.1943-7889.0001441
- Solvability and microlocal analysis of the fractional Eringen wave equation, Mathematics and Mechanics of Solids, Volume 23 (2018) no. 10, p. 1420 | DOI:10.1177/1081286517726371
- Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects, Meccanica, Volume 53 (2018) no. 13, p. 3415 | DOI:10.1007/s11012-018-0887-2
- Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity, Multidiscipline Modeling in Materials and Structures, Volume 14 (2018) no. 5, p. 810 | DOI:10.1108/mmms-09-2017-0109
- Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach, Applied Physics A, Volume 123 (2017) no. 5 | DOI:10.1007/s00339-017-0887-4
- Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, Volume 165 (2017), p. 250 | DOI:10.1016/j.compstruct.2017.01.032
- Instead of Introduction, Internal Variables in Thermoelasticity, Volume 243 (2017), p. 1 | DOI:10.1007/978-3-319-56934-5_1
- Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects, International Journal of Mechanical Sciences, Volume 120 (2017), p. 159 | DOI:10.1016/j.ijmecsci.2016.11.025
- Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures, Volume 88 (2017), p. 194 | DOI:10.1016/j.physe.2017.01.006
- “Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal, Zeitschrift für angewandte Mathematik und Physik, Volume 68 (2017) no. 2 | DOI:10.1007/s00033-017-0795-7
- , Volume 1790 (2016), p. 150031 | DOI:10.1063/1.4968770
- A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section, Acta Mechanica, Volume 227 (2016) no. 9, p. 2543 | DOI:10.1007/s00707-016-1618-1
- Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science, Volume 112 (2016), p. 282 | DOI:10.1016/j.commatsci.2015.10.044
- Size dependency in vibration analysis of nano plates; one problem, different answers, European Journal of Mechanics - A/Solids, Volume 59 (2016), p. 124 | DOI:10.1016/j.euromechsol.2016.03.011
- Free vibration analysis of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Volume 102 (2016), p. 77 | DOI:10.1016/j.ijengsci.2016.02.010
- Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material, International Journal of Engineering Science, Volume 107 (2016), p. 77 | DOI:10.1016/j.ijengsci.2016.07.011
- Bending of Euler–Bernoulli beams using Eringen’s integral formulation: A paradox resolved, International Journal of Engineering Science, Volume 99 (2016), p. 107 | DOI:10.1016/j.ijengsci.2015.10.013
- Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory, International Journal of Mechanical Sciences, Volume 115-116 (2016), p. 135 | DOI:10.1016/j.ijmecsci.2016.06.011
- Nonlocal or gradient elasticity macroscopic models: A question of concentrated or distributed microstructure, Mechanics Research Communications, Volume 71 (2016), p. 25 | DOI:10.1016/j.mechrescom.2015.11.006
- Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory, Microfluidics and Nanofluidics, Volume 20 (2016) no. 5 | DOI:10.1007/s10404-016-1739-9
- General investigation for longitudinal wave propagation under magnetic field effect via nonlocal elasticity, Applied Mathematics and Mechanics, Volume 36 (2015) no. 10, p. 1305 | DOI:10.1007/s10483-015-1985-9
- Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory, Composite Structures, Volume 133 (2015), p. 1079 | DOI:10.1016/j.compstruct.2015.08.014
- Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua, European Journal of Mechanics - A/Solids, Volume 53 (2015), p. 107 | DOI:10.1016/j.euromechsol.2015.03.003
- Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory, International Journal of Engineering Science, Volume 97 (2015), p. 84 | DOI:10.1016/j.ijengsci.2015.08.013
- Electromechanical behavior, end enhancements and radial elasticity of single-walled CNTs: A physically-consistent model based on nonlocal charges, International Journal of Solids and Structures, Volume 72 (2015), p. 190 | DOI:10.1016/j.ijsolstr.2015.07.016
- Nonlocal Equivalent Continua for Buckling and Vibration Analyses of Microstructured Beams, Journal of Nanomechanics and Micromechanics, Volume 5 (2015) no. 1 | DOI:10.1061/(asce)nm.2153-5477.0000062
- Reflections on mathematical models of deformation waves in elastic microstructured solids, Mathematics and Mechanics of Complex Systems, Volume 3 (2015) no. 1, p. 43 | DOI:10.2140/memocs.2015.3.43
- Discrete systems behave as nonlocal structural elements: Bending, buckling and vibration analysis, European Journal of Mechanics - A/Solids, Volume 44 (2014), p. 125 | DOI:10.1016/j.euromechsol.2013.10.007
- A generalized nonlocal elasticity solution for the propagation of longitudinal stress waves in bars, European Journal of Mechanics - A/Solids, Volume 45 (2014), p. 75 | DOI:10.1016/j.euromechsol.2013.11.014
- Bibliography, Fractional Calculus With Applications in Mechanics (2014), p. 379 | DOI:10.1002/9781118909065.biblio
- Bibliography, Fractional Calculus with Applications in Mechanics (2014), p. 289 | DOI:10.1002/9781118577530.biblio
- Nonlocal elasticity defined by Eringen’s integral model: Introduction of a boundary layer method, International Journal of Solids and Structures, Volume 51 (2014) no. 9, p. 1758 | DOI:10.1016/j.ijsolstr.2014.01.016
- Analytical length scale calibration of nonlocal continuum from a microstructured buckling model, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Volume 94 (2014) no. 5, p. 402 | DOI:10.1002/zamm.201200130
- Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Composite Structures, Volume 105 (2013), p. 351 | DOI:10.1016/j.compstruct.2013.05.026
- Transverse waves propagating in carbon nanotubes via a higher-order nonlocal beam model, Composite Structures, Volume 95 (2013), p. 328 | DOI:10.1016/j.compstruct.2012.07.038
- Flexural waves in multi-walled carbon nanotubes using gradient elasticity beam theory, Computational Materials Science, Volume 67 (2013), p. 188 | DOI:10.1016/j.commatsci.2012.08.035
- Out-of-Plane Buckling of Microstructured Beams: Gradient Elasticity Approach, Journal of Engineering Mechanics, Volume 139 (2013) no. 8, p. 1036 | DOI:10.1061/(asce)em.1943-7889.0000543
- Virus sensor based on single-walled carbon nanotube: improved theory incorporating surface effects, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 371 (2013) no. 1993, p. 20120424 | DOI:10.1098/rsta.2012.0424
- Bibliography, Reinforced Concrete Beams, Columns and Frames (2013), p. 279 | DOI:10.1002/9781118639511.biblio
- Bibliography, Reinforced Concrete Beams, Columns and Frames (2013), p. 291 | DOI:10.1002/9781118635360.biblio
- A more general investigation for the longitudinal stress waves in microrods with initial stress, Acta Mechanica, Volume 223 (2012) no. 9, p. 2065 | DOI:10.1007/s00707-012-0682-4
- Bibliography, Carbon Nanotubes and Nanosensors (2012), p. 325 | DOI:10.1002/9781118562000.biblio
- On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale, European Journal of Mechanics - A/Solids, Volume 36 (2012), p. 25 | DOI:10.1016/j.euromechsol.2012.02.004
- Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, International Journal of Engineering Science, Volume 56 (2012), p. 17 | DOI:10.1016/j.ijengsci.2012.02.004
- The investigation of the nonlocal longitudinal stress waves with modified couple stress theory, Acta Mechanica, Volume 221 (2011) no. 3-4, p. 321 | DOI:10.1007/s00707-011-0500-4
- BUCKLING OF COMPOSITE NONLOCAL OR GRADIENT CONNECTED BEAMS, International Journal of Structural Stability and Dynamics, Volume 11 (2011) no. 06, p. 1015 | DOI:10.1142/s0219455411004452
- Boundary-Layer Effect in Composite Beams with Interlayer Slip, Journal of Aerospace Engineering, Volume 24 (2011) no. 2, p. 199 | DOI:10.1061/(asce)as.1943-5525.0000027
- On the propagation of localization in the plasticity collapse of hardening–softening beams, International Journal of Engineering Science, Volume 48 (2010) no. 5, p. 487 | DOI:10.1016/j.ijengsci.2009.12.002
- A variationally based nonlocal damage model to predict diffuse microcracking evolution, International Journal of Mechanical Sciences, Volume 52 (2010) no. 12, p. 1783 | DOI:10.1016/j.ijmecsci.2010.09.012
- Bending, Buckling, and Vibration of Micro/Nanobeams by Hybrid Nonlocal Beam Model, Journal of Engineering Mechanics, Volume 136 (2010) no. 5, p. 562 | DOI:10.1061/(asce)em.1943-7889.0000107
- Buckling of elastic beams on non-local foundation: A revisiting of Reissner model, Mechanics Research Communications, Volume 37 (2010) no. 5, p. 472 | DOI:10.1016/j.mechrescom.2010.05.007
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