Construction of a bipotential representing a linear non-associated constitutive law

Arjdal El Hanafi; Jamal Chaoufi; Claude Vallée; Arnaud Germaneau; Kossi Atchonouglo; Hassan Fatmaoui; Abdelaaziz Ghafiri

doi:10.1016/j.crme.2013.09.002

Construction of a bipotential representing a linear non-associated constitutive law
[Construction dʼun bipotentiel représentant une loi linéaire non associée]

Arjdal El Hanafi ¹ ; Jamal Chaoufi ¹ ; Claude Vallée ² ; Arnaud Germaneau ² ; Kossi Atchonouglo ³ ; Hassan Fatmaoui ¹ ; Abdelaaziz Ghafiri ¹

¹ Laboratory of Electronics, Signal Processing and Modelling Physics, Department of Physics, University Ibn Zohr, 80000 Agadir, Morocco
² University of Poitiers, Institut P′, UPR CNRS 3346, SP2MI, téléport 2, 11, bd Marie-et-Pierre-Curie, BP 30179, 86962 Futuroscope-Chasseneuil, France
³ University of Lomé, Department of Physics, BP 1515, Lomé, Togo

Comptes Rendus. Mécanique, Volume 341 (2013) no. 9-10, pp. 667-671.

Résumé
Abstract

Nous considérons un matériau dont la relation contraintes–déformations est linéaire, mais pas symétrique. La recherche dʼun potentiel étant vaine, nous tentons de représenter la loi de comportement par un bipotentiel. La méthode de Fitzpatrick conduit à construire une suite croissante appropriée de bipotentiels. La technique est illustrée sur lʼexemple des lois coaxiales.

We consider a material whose stress–strain relation is linear but not symmetrical. As the research of a potential is futile, we attempt to represent the constitutive law by a bipotential. Fitzpatrick method leads us to construct a suitable increasing sequence of bipotentials. The technique is exemplified on coaxial constitutive laws.

Reçu le : 2013-06-12
Accepté le : 2013-09-12
Publié le : 2013-09-01

DOI : 10.1016/j.crme.2013.09.002

Keywords: Non-associated constitutive law, Non-standard materials, Implicit standard materials, n-Monotonicity, Bipotentials
Mot clés : Lois de comportement non associées, Matériaux non standard, Matériaux standard implicites, Matériaux n-monotones, Bipotentiels

Affiliations des auteurs :

Arjdal El Hanafi ¹ ; Jamal Chaoufi ¹ ; Claude Vallée ² ; Arnaud Germaneau ² ; Kossi Atchonouglo ³ ; Hassan Fatmaoui ¹ ; Abdelaaziz Ghafiri ¹

¹ Laboratory of Electronics, Signal Processing and Modelling Physics, Department of Physics, University Ibn Zohr, 80000 Agadir, Morocco
² University of Poitiers, Institut P′, UPR CNRS 3346, SP2MI, téléport 2, 11, bd Marie-et-Pierre-Curie, BP 30179, 86962 Futuroscope-Chasseneuil, France
³ University of Lomé, Department of Physics, BP 1515, Lomé, Togo

@article{CRMECA_2013__341_9-10_667_0,
     author = {Arjdal El Hanafi and Jamal Chaoufi and Claude Vall\'ee and Arnaud Germaneau and Kossi Atchonouglo and Hassan Fatmaoui and Abdelaaziz Ghafiri},
     title = {Construction of a bipotential representing a linear non-associated constitutive law},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {667--671},
     publisher = {Elsevier},
     volume = {341},
     number = {9-10},
     year = {2013},
     doi = {10.1016/j.crme.2013.09.002},
     language = {en},
}

TY  - JOUR
AU  - Arjdal El Hanafi
AU  - Jamal Chaoufi
AU  - Claude Vallée
AU  - Arnaud Germaneau
AU  - Kossi Atchonouglo
AU  - Hassan Fatmaoui
AU  - Abdelaaziz Ghafiri
TI  - Construction of a bipotential representing a linear non-associated constitutive law
JO  - Comptes Rendus. Mécanique
PY  - 2013
SP  - 667
EP  - 671
VL  - 341
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crme.2013.09.002
LA  - en
ID  - CRMECA_2013__341_9-10_667_0
ER  -

%0 Journal Article
%A Arjdal El Hanafi
%A Jamal Chaoufi
%A Claude Vallée
%A Arnaud Germaneau
%A Kossi Atchonouglo
%A Hassan Fatmaoui
%A Abdelaaziz Ghafiri
%T Construction of a bipotential representing a linear non-associated constitutive law
%J Comptes Rendus. Mécanique
%D 2013
%P 667-671
%V 341
%N 9-10
%I Elsevier
%R 10.1016/j.crme.2013.09.002
%G en
%F CRMECA_2013__341_9-10_667_0

Arjdal El Hanafi; Jamal Chaoufi; Claude Vallée; Arnaud Germaneau; Kossi Atchonouglo; Hassan Fatmaoui; Abdelaaziz Ghafiri. Construction of a bipotential representing a linear non-associated constitutive law. Comptes Rendus. Mécanique, Volume 341 (2013) no. 9-10, pp. 667-671. doi : 10.1016/j.crme.2013.09.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2013.09.002/

Bibliographie
Cité par

[1] K. Hackl Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids, Volume 45 (1997) no. 5, pp. 667-688

[2] B. Halphen; N.Q. Son On the generalized standard materials, J. Méc., Volume 14 (1975), pp. 39-63 (in French)

[3] J.J. Moreau Application of convex analysis to the treatment of elasto-plastic systems (P. Germain; B. Nayroles, eds.), Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Mathematics, vol. 503, Springer, Berlin, 1976

[4] P.G. Ciarlet; G. Geymonat; F. Krasucki Legendre–Fenchel duality in elasticity, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 9–10, pp. 597-602

[5] Convex functionals, Séminaire Jean-Leray. Sur les équations aux dérivées partielles, vol. 2, Collège de France, Paris, 1966, pp. 1-108 http://archive.numdam.org (in French) 2003. Reprint: Istituto poligrafico e zecca dello stato S.p.A., Roma

[6] M. Buliga; G. de Saxcé; C. Vallée Non maximal cyclically monotone graphs and construction of a bipotential for the Coulombʼs dry friction law, J. Convex Anal., Volume 17 (2010) no. 1, pp. 81-94

[7] G. de Saxcé; L. Bousshine Implicit standard materials (D. Weichert; G. Maier, eds.), Inelastic Behaviour of Structures under Variable Repeated Loads — Direct Analysis Methods, CISM Courses and Lectures IV, vol. 432, International Centre for Mechanical Sciences, Springer, Vienna, New York, 2002

[8] G. de Saxcé; J.B. Tritsch Shakedown with non-associated flow rule (D. Weichert; G. Maier, eds.), Inelastic Behaviour of Structures under Variable Repeated Loads — Direct Analysis Methods, CISM Courses and Lectures IV, vol. 432, International Centre for Mechanical Sciences, Springer, Vienna, New York, 2002

[9] S. Bartz; H.H. Bauschke; J.M. Borwein; S. Reich; X. Wang Fitzpatrick function, cyclic monotonicity and Rockafellar antiderivative, Nonlinear Anal., Volume 66 (2007), pp. 1198-1223

[10] C. Vallée; C. Lerintiu; D. Fortuné; K. Atchonouglo; M. Ban Representing a non–associated constitutive law by a bipotential issued from a Fitzpatrick sequence, Arch. Mech., Volume 61 (2009) no. 3–4, pp. 325-340

[11] C. Vallée; C. Lerintiu; J. Chaoufi; D. Fortuné; M. Ban; K. Atchonouglo A class of non-associated materials: n-monotone materials — Hookeʼs Law of elasticity revisited, J. Elast., Volume 112 (2013) no. 2, pp. 111-138

[12] S. Fitzpatrick Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization, Canberra (S.P. Fitzpatrick; J.R. Giles, eds.) (Proceedings of the Centre for Mathematical Analysis of the Australian National University), Volume vol. 20 ( August 8–24, 1988 ), pp. 59-65

[13] N. Point; S. Erlicher Pseudo-potentials and bipotentials: A constructive procedure for non-associated plasticity and unilateral contact, Discrete Contin. Dyn. Syst., Ser. S, Volume 6 (2013) no. 2, pp. 567-590

[14] A. Visintin Structural stability of rate-independent nonpotential flows, Discrete Contin. Dyn. Syst., Ser. S, Volume 6 (2013) no. 1, pp. 257-275

[15] A. Visintin Variational formulation and structural stability of monotone equations, Calc. Var. Partial Differ. Equ., Volume 47 (2013) no. 1–2, pp. 273-317

Cité par Sources :

Commentaires - Politique