Comptes Rendus
A dispersive wave equation using nonlocal elasticity
[Une équation des ondes dispersive utilisant l'élasticité non-locale]
Comptes Rendus. Mécanique, Volume 337 (2009) no. 8, pp. 591-595.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2009.06.028
Keywords: Continuum mechanics, Nonlocal model, Gradient model, Wave equation, Elasticity, Heterogeneous material, Dispersive properties, Born–Kármán model
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

1 Université européenne de Bretagne, INSA de Rennes – LGCGM, 20, avenue des buttes de Coësmes, 35043 Rennes cedex, France
2 Université européenne de Bretagne, IRMAR – université de Rennes 1, campus de Beaulieu, 35042 Rennes cedex, France
@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  - 
%0 Journal Article
%A Noël Challamel
%A Lalaonirina Rakotomanana
%A Loïc Le Marrec
%T A dispersive wave equation using nonlocal elasticity
%J Comptes Rendus. Mécanique
%D 2009
%P 591-595
%V 337
%N 8
%I Elsevier
%R 10.1016/j.crme.2009.06.028
%G en
%F CRMECA_2009__337_8_591_0
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] G.A. Maugin Nonlinear Waves in Elastic Crystals, Oxford University Press, 1999

[2] A.C. Eringen Nonlocal Continuum Field Theories, Springer, New York, 2002

[3] L. Rakotomanana A Geometric Approach to Thermomechanics of Dissipating Continua, Progress in Mathematical Physics, Birkhäuser, Boston, 2004

[4] H. Askes; E.C. Aifantis Gradient elasticity theories in statics and dynamics – A unification of approaches, Int. J. Fracture, Volume 139 (2006), pp. 297-304

[5] M. Lazar; G.A. Maugin; E.C. Aifantis On a theory of nonlocal elasticity of bi-Helmholtz type and some applications, Int. J. Solids Structures, Volume 43 (2006), pp. 1404-1421

[6] A.V. Metrikine On causality of the gradient elasticity models, J. Sound Vibration, Volume 297 (2006), pp. 727-742

[7] A.C. Eringen On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., Volume 54 (1983), pp. 4703-4710

[8] R.H.J. Peerlings; R. de Borst; W.A.M. Brekelmans; J.H.P. de Vree Gradient-enhanced damage for quasi-brittle materials, Int. J. Num. Meth. Engng., Volume 39 (1996), pp. 3391-3403

[9] N. Challamel; C. Lanos; C. Casandjian Plastic failure of nonlocal beams, Phys. Rev. E, Volume 78 (2008), p. 026604

[10] S. Forest Micromorphic approach for gradient elasticity, viscoplasticity, and damage, J. Eng. Mech., Volume 135 (2009) no. 3, pp. 117-131

[11] A.C. Eringen Theory of nonlocal elasticity and some applications, Res. Mech., Volume 21 (1987), pp. 313-342

[12] N. Challamel; C.M. Wang The small length scale effect for a nonlocal cantilever beam: A paradox solved, Nanotechnology, Volume 19 (2008), p. 345703

[13] E.C. Aifantis Update on a class of gradient theories, Mech. Mat., Volume 35 (2003), pp. 259-280

[14] A.V. Metrikine; H. Askes 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] J.V. Boussinesq 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] H.G. Georgiadis; I. Vardoulakis; G. Lykotrafitis Torsional surface waves in a gradient-elastic half-space, Wave Motion, Volume 31 (2000), pp. 333-348

  • N. Challamel; S. El-Borgi; M. Trabelssi; J.N. Reddy Buckling of micromorphic Timoshenko columns, European Journal of Mechanics - A/Solids, Volume 111 (2025), p. 105537 | DOI:10.1016/j.euromechsol.2024.105537
  • Noël Challamel; A. Aftabi S. 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
  • Noël Challamel; Massimiliano Zingales 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
  • Vladimir Tonkonogyi; Maryna Holofieieva; Yurii Morozov; Volodymyr Yarovyi; Oksana Bieliavska; Isak Karabegović 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
  • Noël Challamel; C. M. Wang; J. N. Reddy; S. A. Faghidian Equivalence between micromorphic, nonlocal gradient, and two-phase nonlocal beam theories, Acta Mechanica (2024) | DOI:10.1007/s00707-024-04180-x
  • N. Challamel; S. El-Borgi; M. Trabelssi; J.N. Reddy Free vibration response of micromorphic Timoshenko beams, Journal of Sound and Vibration, Volume 591 (2024), p. 118602 | DOI:10.1016/j.jsv.2024.118602
  • Gaurab Kumar Khanra; I. R. Praveen Krishna; P. Raveendranath 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
  • Noël Challamel; Y. P. Zhang; C. M. Wang; Giuseppe Ruta; Francesco dell’Isola 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
  • Rafael C. Deptulski; Magdalena Dymitrowska; Djimédo Kondo 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
  • Noël Challamel; Teodor Atanacković; Y.P. Zhang; C.M. Wang A fractional nonlocal elastic model for lattice wave analysis, Mechanics Research Communications, Volume 126 (2022), p. 103999 | DOI:10.1016/j.mechrescom.2022.103999
  • G. Prateek; De Sarthak; R. Vasudevan; M. Haboussi; M. Ganapathi 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
  • Yishuang Huang; Peijun Wei; Yuqian Xu; Yueqiu Li 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
  • Dario De Domenico; Giuseppe Ricciardi; Harm Askes 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
  • Dario De Domenico; Giuseppe Ricciardi; Harm Askes 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
  • Noël Challamel; Chien Ming Wang; Hong Zhang; Isaac Elishakoff Lattice-Based Nonlocal Elastic Structural Models, Size-Dependent Continuum Mechanics Approaches (2021), p. 1 | DOI:10.1007/978-3-030-63050-8_1
  • Igor Andrianov; Steve Koblik; Galina Starushenko 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
  • A. Norouzzadeh; R. Ansari; H. Rouhi 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
  • Güler GAYGUSUZOĞLU 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
  • A. Carcaterra; F. Coppo; F. Mezzani; S. Pensalfini 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
  • Loïc Le Marrec; Jean Lerbet; Lalaonirina R. Rakotomanana 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
  • M. Ganapathi; O. Polit 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
  • Dario De Domenico; Harm Askes 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
  • Noël Challamel 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
  • Davood Shahsavari; Behrouz Karami; Sima Mansouri 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
  • Noël Challamel; Metin Aydogdu; Isaac Elishakoff 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
  • M. Faraji Oskouie; R. Ansari; H. Rouhi 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
  • Güler GAYGUSUZOĞLU 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
  • Seyed Mojtaba Hozhabrossadati; Noël Challamel; Mohammad Rezaiee-Pajand; Ahmad Aftabi Sani 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
  • Noël Challamel; Chien Ming Wang; Hong Zhang; Sritawat Kitipornchai 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
  • Günther Hörmann; Ljubica Oparnica; Dušan Zorica 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
  • A. Norouzzadeh; R. Ansari; H. Rouhi 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
  • Amir Norouzzadeh; Reza Ansari; Hessam Rouhi 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
  • A. Norouzzadeh; R. Ansari; H. Rouhi 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
  • Xiaobai Li; Li Li; Yujin Hu; Zhe Ding; Weiming Deng 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
  • Arkadi Berezovski; Peter Ván Instead of Introduction, Internal Variables in Thermoelasticity, Volume 243 (2017), p. 1 | DOI:10.1007/978-3-319-56934-5_1
  • Li Li; Yujin Hu 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
  • A. Norouzzadeh; R. Ansari 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
  • Giuseppe Rosi; Luca Placidi; Francesco dell’Isola “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
  • G. J. Tsamasphyros; C. Chr. Koutsoumaris, Volume 1790 (2016), p. 150031 | DOI:10.1063/1.4968770
  • Cédric Forgit; Benoit Lemoine; Loïc Le Marrec; Lalaonirina Rakotomanana 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
  • Li Li; Yujin Hu 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
  • Akbar Jafari; Saeed Shirvani Shah-enayati; Ali Asghar Atai 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
  • Li Li; Xiaobai Li; Yujin Hu 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
  • Li Li; Yujin Hu 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
  • J. Fernández-Sáez; R. Zaera; J.A. Loya; J.N. Reddy 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
  • Li Li; Yujin Hu; Xiaobai Li 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
  • N. Challamel; C.M. Wang; I. Elishakoff 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
  • Li Li; Yujin Hu; Xiaobai Li; Ling Ling 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
  • U. Güven 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
  • Li Li; Yujin Hu; Ling Ling 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
  • Noël Challamel; Vincent Picandet; Bernard Collet; Thomas Michelitsch; Isaac Elishakoff; C.M. Wang 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
  • Li Li; Yujin Hu 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
  • Elena Benvenuti 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
  • Noël Challamel; Zhen Zhang; C. M. Wang 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
  • Jüri Engelbrecht; Arkadi Berezovski 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
  • N. Challamel; C.M. Wang; I. Elishakoff 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
  • U. Güven 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
  • R. Abdollahi; B. Boroomand 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
  • N. Challamel; J. Lerbet; C.M. Wang; Z. Zhang 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
  • Noël Challamel 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
  • Y. Huang; Q.-Z. Luo; X.-F. Li 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
  • J.X. Wu; X.F. Li; W.D. Cao 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
  • Noël Challamel; Mohammed Ameur 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
  • Isaac Elishakoff; Noël Challamel; Clément Soret; Yannis Bekel; Thomas Gomez 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
  • U. Güven 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
  • D.A. Fafalis; S.P. Filopoulos; G.J. Tsamasphyros 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
  • Metin Aydogdu 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
  • U. Güven 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
  • NOËL CHALLAMEL; ISMAIL MECHAB; NOUREDDINE EL MEICHE; BAGHDAD KROUR 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
  • Noël Challamel; Ulf Arne Girhammar 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
  • Noël Challamel; Christophe Lanos; Charles Casandjian 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
  • Noël Challamel 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
  • Y. Y. Zhang; C. M. Wang; N. Challamel 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
  • N. Challamel; S.A. Meftah; F. Bernard 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

Cité par 79 documents. Sources : Crossref

Commentaires - Politique


Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !


Publier un nouveau commentaire:

Publier une nouvelle réponse: