Asymptotic analysis of plates in static and dynamic strain gradient elasticity

Christian Licht; Thibaut Weller

doi:10.5802/crmeca.118

Note

Asymptotic analysis of plates in static and dynamic strain gradient elasticity

Christian Licht ^1,² ; Thibaut Weller ²

¹ Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
² LMGC, Université de Montpellier, CNRS, Montpellier, France

Comptes Rendus. Mécanique, Volume 350 (2022), pp. 325-342.

Résumé

We study the steady-state and transient responses of a second-order elastic plate by implementing an asymptotic analysis of the three-dimensional equations with respect to two geometric characteristics seen as parameters: the thickness of the plate and an inner material length. Depending on their ratio, four different models arise. Conditions under which Reissner–Mindlin kinematics may appear are discussed while the influence of crystalline symmetries is studied. The transient situation is solved through Trotter’s theory of approximation of semi-groups of operators acting on variable spaces.

Reçu le : 2022-06-02
Accepté le : 2022-06-03
Publié le : 2022-07-11

DOI : 10.5802/crmeca.118

Mots clés : Asymptotic analysis, Strain gradient elasticity, Plate models, Transient problems, $m$-dissipative operators, Approximation of semi-groups in the sense of Trotter

Affiliations des auteurs :

Christian Licht ^{1, 2} ; Thibaut Weller ²

¹ Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand
² LMGC, Université de Montpellier, CNRS, Montpellier, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMECA_2022__350_G2_325_0,
     author = {Christian Licht and Thibaut Weller},
     title = {Asymptotic analysis of plates in static and dynamic strain gradient elasticity},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {325--342},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {350},
     year = {2022},
     doi = {10.5802/crmeca.118},
     language = {en},
}

TY  - JOUR
AU  - Christian Licht
AU  - Thibaut Weller
TI  - Asymptotic analysis of plates in static and dynamic strain gradient elasticity
JO  - Comptes Rendus. Mécanique
PY  - 2022
SP  - 325
EP  - 342
VL  - 350
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.118
LA  - en
ID  - CRMECA_2022__350_G2_325_0
ER  -

%0 Journal Article
%A Christian Licht
%A Thibaut Weller
%T Asymptotic analysis of plates in static and dynamic strain gradient elasticity
%J Comptes Rendus. Mécanique
%D 2022
%P 325-342
%V 350
%I Académie des sciences, Paris
%R 10.5802/crmeca.118
%G en
%F CRMECA_2022__350_G2_325_0

Christian Licht; Thibaut Weller. Asymptotic analysis of plates in static and dynamic strain gradient elasticity. Comptes Rendus. Mécanique, Volume 350 (2022), pp. 325-342. doi : 10.5802/crmeca.118. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.118/

Bibliographie
Cité par

[1] P. Germain La méthode des puissances virtuelles en mécanique des milieux continus premiere partie : théorie du second gradient, J. de Méc., Volume 12 (1973), pp. 235-274 | Zbl

[2] P. G. Ciarlet Mathematical Elasticity. Volume II: Theory of Plates, North-Holland, Amsterdam, The Netherlands, 1997

[3] M. Serpilli; F. Krasucki; G. Geymonat An asymptotic strain gradient Reissner–Mindlin plate model, Meccanica, Volume 48 (2013), pp. 2007-2018 | DOI | MR | Zbl

[4] N. Auffray; H. Le Quang; Q. C. He Matrix representations for 3D strain-gradient elasticity, J. Mech. Phys. Solids, Volume 61 (2013), pp. 1202-1223 | DOI | MR | Zbl

[5] R. Paroni; P. Podio-Guidugli; G. Tomassetti A justification of the Reissner–Mindlin plate theory through variational convergence, Anal. Appl., Volume 5 (2007), pp. 165-182 | DOI | Zbl

[6] C. Licht; T. Weller Asymptotic modeling of piezoelectric plates with electric field gradient, C. R. Méc., Volume 340 (2012), pp. 405-410

[7] C. Licht; T. Weller Modeling of linearly electromagneto-elastic thin plates, C. R. Méc., Volume 335 (2007), pp. 201-206 | Zbl

[8] G. Rossi; L. Placidi; N. Auffray On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure, Eur. J. Mech. A: Solids, Volume 69 (2017), pp. 179-191 | DOI | MR

[9] O. Iosifescu; C. Licht Transient response of a thin linearly elastic plate with Norton or Tresca friction, Asymptot. Anal., Volume 128 (2022) no. 4, pp. 555-570 | Zbl

[10] H. F. Trotter Approximation of semi-groups of operators, Pac. J. Math., Volume 28 (1958), pp. 897-919 | MR | Zbl

[11] C. Licht; T. Weller Approximation of semi-groups in the sense of Trotter and asymptotic mathematical modeling in physics of continuous media, Discrete Contin. Dyn. Syst., Volume 12 (2019), pp. 1709-1741 | MR | Zbl

[12] T. Weller; C. Licht Asymptotic modeling of thin piezoelectric plates, Ann. Solid Struct. Mech., Volume 1 (2010), pp. 173-188 | DOI

Cité par Sources :

Commentaires - Politique