Selection of two-dimensional nonlinear strain waves in micro-structured media

Alexey V. Porubov; Franco Pastrone; Gérard A. Maugin

doi:10.1016/j.crme.2004.02.020

Selection of two-dimensional nonlinear strain waves in micro-structured media
[Selection d'ondes de déformation non linéaires à deux dimensions dans des milieux à microstructure]

Présenté par : Évariste Sanchez-Palencia

Alexey V. Porubov ¹ ; Franco Pastrone ² ; Gérard A. Maugin ³

¹ Ioffe Physico-Technical Institute of the Russian Academy of Sciences, St. Petersburg 194021, Russia
² Dipartimento di Matematica, Università di Torino, Via C. Alberto 10, 10123 Torino, Italy
³ Laboratoire de modélisation en mécanique associé au CNRS, université Pierre et Marie Curie, Tours 65-55, case 162, place Jussieu 4, 75252 Paris cedex 05, France

Comptes Rendus. Mécanique, Volume 332 (2004) no. 7, pp. 513-518.

Résumé
Abstract

On montre que dans un milieu élastique à microstructure le gradient de micro-déplacement permet la propagation de longues ondes localisées non linéaires de déformation en deux dimensions. Ces ondes penvent exister même en présence de dissipation et d'apport d'énergie mais pour des valeurs précises de l'amplitude de l'onde et sa vitesse de propagation. Une augmentation on une diminution de ces deux quantités se produit plus rapidement dans la phase initiale de propagation que pour une onde plane localisée. Cependont, les valeurs stationnaires sélectionnées par l'apport et perte d'énergie sont plus élevées que pour les ondes planes.

It is shown that the micro-displacement gradient allows the propagation of two-dimensional localized long nonlinear strain waves in a medium with microstructure. These waves may exist even in the presence of dissipation and energy input in the microstructured medium but with selected values of the wave amplitude and velocity. An increase or a decrease in the wave amplitude and velocity happens faster at the initial stage than that of the plane localized wave. However, their steady values selected by the energy input/output, are higher for the plane waves.

Reçu le : 2003-12-16
Accepté le : 2004-02-25
Publié le : 2004-04-30

DOI : 10.1016/j.crme.2004.02.020

Keywords: Computational solid mechanics, Nonlinear wave, Microstructure
Mots-clés : Mécanique des solides numérique, Ondes non linéaires, Microstructure

Affiliations des auteurs :

Alexey V. Porubov ¹ ; Franco Pastrone ² ; Gérard A. Maugin ³

¹ Ioffe Physico-Technical Institute of the Russian Academy of Sciences, St. Petersburg 194021, Russia
² Dipartimento di Matematica, Università di Torino, Via C. Alberto 10, 10123 Torino, Italy
³ Laboratoire de modélisation en mécanique associé au CNRS, université Pierre et Marie Curie, Tours 65-55, case 162, place Jussieu 4, 75252 Paris cedex 05, France

@article{CRMECA_2004__332_7_513_0,
     author = {Alexey V. Porubov and Franco Pastrone and G\'erard A. Maugin},
     title = {Selection of two-dimensional nonlinear strain waves in~micro-structured media},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {513--518},
     publisher = {Elsevier},
     volume = {332},
     number = {7},
     year = {2004},
     doi = {10.1016/j.crme.2004.02.020},
     language = {en},
}

TY  - JOUR
AU  - Alexey V. Porubov
AU  - Franco Pastrone
AU  - Gérard A. Maugin
TI  - Selection of two-dimensional nonlinear strain waves in micro-structured media
JO  - Comptes Rendus. Mécanique
PY  - 2004
SP  - 513
EP  - 518
VL  - 332
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crme.2004.02.020
LA  - en
ID  - CRMECA_2004__332_7_513_0
ER  -

%0 Journal Article
%A Alexey V. Porubov
%A Franco Pastrone
%A Gérard A. Maugin
%T Selection of two-dimensional nonlinear strain waves in micro-structured media
%J Comptes Rendus. Mécanique
%D 2004
%P 513-518
%V 332
%N 7
%I Elsevier
%R 10.1016/j.crme.2004.02.020
%G en
%F CRMECA_2004__332_7_513_0

Alexey V. Porubov; Franco Pastrone; Gérard A. Maugin. Selection of two-dimensional nonlinear strain waves in micro-structured media. Comptes Rendus. Mécanique, Volume 332 (2004) no. 7, pp. 513-518. doi : 10.1016/j.crme.2004.02.020. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.02.020/

Bibliographie
Cité par

[1] G.A. Maugin Nonlinear Waves in Elastic Crystals, Oxford University Press, UK, 1999

[2] V.I. Erofeev Wave Processes in Solids with Microstructure, World Scientific, Singapore, 2003

[3] A.V. Porubov; F. Pastrone Nonlinear bell-shaped and kink-shaped strain waves in microstructured solids, Int. J. Nonlinear Mech., Volume 40 (2004) (available online)

[4] B.B. Kadomtsev; V.I. Petviashvili On the stability of solitary waves in a weakly dispersive media, Sov. Phys. Dokl., Volume 15 (1970), pp. 539-541

[5] M.J. Ablowitz; H. Segur Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981

[6] A.C. Eringen Theory of micropolar elasticity (H. Liebowitz, ed.), Fracture. An Advanced Treatise, vol. 2, Mathematical Fundamentals, Academic Press, New York, 1968

[7] R.D. Mindlin Microstructure in linear elasticity, Arch. Rational Mech. Anal., Volume 1 (1964), pp. 51-78

[8] F.D. Murnaghan Finite Deformations of an Elastic Solid, Wiley, New York, 1951

[9] D.R. Bland The Theory of Linear Viscoelasticity, Oxford University Press, Oxford, 1960

Haiyong Qin; Mostafa M. A. Khater; Raghda A. M. Attia Copious closed forms of solutions for the fractional nonlinear longitudinal strain wave equation in microstructured solids, Mathematical Problems in Engineering, Volume 2020 (2020), p. 8 (Id/No 3498796) | DOI:10.1155/2020/3498796 | Zbl:1544.74045
Igor A. Gula; Alexander M. Samsonov Bulk Nonlinear Elastic StrainWaves in a Bar with Nanosize Inclusions, Generalized Models and Non-classical Approaches in Complex Materials 1, Volume 89 (2018), p. 395 | DOI:10.1007/978-3-319-72440-9_21
C. I. Christov On the pseudolocalized solutions in multi-dimension of Boussinesq equation, Mathematics and Computers in Simulation, Volume 127 (2016), pp. 19-27 | DOI:10.1016/j.matcom.2012.06.015 | Zbl:1520.35105
Ivan Sertakov; Jüri Engelbrecht; Jaan Janno Modelling 2D wave motion in microstructured solids, Mechanics Research Communications, Volume 56 (2014), p. 42 | DOI:10.1016/j.mechrescom.2013.11.007
Christo I. Christov Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations, Mathematics and Computers in Simulation, Volume 82 (2012) no. 6, pp. 1079-1092 | DOI:10.1016/j.matcom.2010.07.030 | Zbl:1320.76031
C. I. Christov; J. Choudhury Perturbation solution for the 2D Boussinesq equation, Mechanics Research Communications, Volume 38 (2011) no. 3, pp. 274-281 | DOI:10.1016/j.mechrescom.2011.01.014 | Zbl:1272.76182
Franco Pastrone Hierarchical structures in complex solids with microscales, Proceedings of the Estonian Academy of Sciences, Volume 59 (2010) no. 2, pp. 79-86 | DOI:10.3176/proc.2010.2.04 | Zbl:1375.74014
A. Casasso; F. Pastrone Wave propagation in solids with vectorial microstructures, Wave Motion, Volume 47 (2010) no. 6, pp. 358-369 | DOI:10.1016/j.wavemoti.2009.12.006 | Zbl:1231.74243
Yibin Fu Linear and nonlinear wave propagation in coated or uncoated elastic half-spaces, Waves in nonlinear pre-stressed materials. Papers based on the presentation at CISM course, Udine, Italy, September 2006., Wien: Springer, 2007, pp. 103-127 | DOI:10.1007/978-3-211-73572-5_4 | Zbl:1167.74022
O. Ilison; A. Salupere Propagation of $s e c h^{2}$ -type solitary waves in higher-order KdV-type systems, Chaos, Solitons and Fractals, Volume 26 (2005) no. 2, pp. 453-465 | DOI:10.1016/j.chaos.2004.12.045 | Zbl:1153.74340

Cité par 10 documents. Sources : Crossref, zbMATH

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: