Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-05T04:35:31.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set

Published online by Cambridge University Press:  15 February 2004

Luciano Carbone ,
Doina Cioranescu ,
Riccardo De Arcangelis  and
Antonio Gaudiello
Luciano Carbone
Affiliation:
Università di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy; carbone@biol.dgbm.unina.it., dearcang@unina.it.
Doina Cioranescu
Affiliation:
Université Pierre et Marie Curie (Paris VI), Laboratoire Jacques-Louis Lions, 4 Place Jussieu, 75252 Paris Cedex 05, France; cioran@ann.jussieu.fr.
Riccardo De Arcangelis
Affiliation:
Università di Napoli “Federico II”, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, via Cintia, Complesso Monte S. Angelo, 80126 Napoli, Italy; carbone@biol.dgbm.unina.it., dearcang@unina.it.
Antonio Gaudiello
Affiliation:
Università di Cassino, Dipartimento di Automazione, Elettromagnetismo, Ingegneria dell'Informazione e Matematica Industriale, via G. Di Biasio 43, 03043 Cassino (FR), Italy; gaudiell@unina.it.
Get access

Abstract

The paper is a continuation of a previous work of the same authors dealing with homogenization processes for some energies of integral type arising in the modeling of rubber-like elastomers. The previous paper took into account the general case of the homogenization of energies in presence of pointwise oscillating constraints on the admissible deformations. In the present paper homogenization processes are treated in the particular case of fixed constraints set, in which minimal coerciveness hypotheses can be assumed, and in which the results can be obtained in the general framework of BV spaces. The classical homogenization result is established for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained, and with an explicitly computed homogeneous density.

Keywords

Homogenizationgradient constrained variational problemsnonlinear elastomers
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 53 - 83
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Claredon Press, Oxford Math. Monogr. (2000).
A.H.T. Banks, N.J. Lybeck, B. Munoz and L. Yanyo, Nonlinear Elastomers: Modeling and Estimation, in Proc. of the “Third IEEE Mediterranean Symposium on New Directions in Control and Automation”, Vol. 1. Limassol, Cyprus (1995) 1-7.
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. North Holland, Stud. Math. Appl. 5 (1978).
A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford Lecture Ser. Math. Appl. 12 (1998).
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Longman Scientific & Technical, Pitman Res. Notes Math. Ser. 207 (1989).
Carbone, L., Cioranescu, D., De Arcangelis, R. and Gaudiello, A., Approach, An to the Homogenization of Nonlinear Elastomers via the Theory of Unbounded Functionals. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 283-288. CrossRef
L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, Homogenization of Unbounded Functionals and Nonlinear Elastomers. The General Case. Asymptot. Anal. 29 (2002) 221-272.
Carbone, L., Cioranescu, D., De Arcangelis, R. and Gaudiello, A., Approach, An to the Homogenization of Nonlinear Elastomers in the Case of the Fixed Constraints Set. Rend. Accad. Sci. Fis. Mat. Napoli (4) 67 (2000) 235-244.
Carbone, L. and De Arcangelis, R., Integral Representation, On, Relaxation and Homogenization for Unbounded Functionals. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8 (1997) 129-135.
Carbone, L. and De Arcangelis, R., On the Relaxation of Some Classes of Unbounded Integral Functionals. Matematiche 51 (1996) 221-256; Special Issue in honor of Francesco Guglielmino.
L. Carbone and R. De Arcangelis, Unbounded Functionals: Applications to the Homogenization of Gradient Constrained Problems. Ricerche Mat. 48-Suppl. (1999) 139-182.
Carbone, L. and De Arcangelis, R., On the Relaxation of Dirichlet Minimum Problems for Some Classes of Unbounded Integral Functionals. Ricerche Mat. 48 (1999) 347-372; Special Issue in memory of Ennio De Giorgi.
Carbone, L. and De Arcangelis, R., On the Unique Extension Problem for Functionals of the Calculus of Variations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 12 (2001) 85-106.
L. Carbone and S. Salerno, Further Results on a Problem of Homogenization with Constraints on the Gradient. J. Analyse Math. 44 (1984/85) 1-20.
D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford University Press, Oxford Lecture Ser. Math. Appl. 17 (1999).
Corbo Esposito, A. and De Arcangelis, R., The Lavrentieff Phenomenon and Different Processes of Homogenization. Comm. Partial Differential Equations 17 (1992) 1503-1538. CrossRef
Corbo Esposito, A. and De Arcangelis, R., Homogenization of Dirichlet Problems with Nonnegative Bounded Constraints on the Gradient. J. Analyse Math. 64 (1994) 53-96. CrossRef
Corbo Esposito, A. and Serra Cassano, F., Lavrentieff Phenomenon, A for Problems of Homogenization with Constraints on the Gradient. Ricerche Mat. 46 (1997) 127-159.
G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser-Verlag, Progr. Nonlinear Differential Equations Appl. 8 (1993).
D'Apice, C., Durante, T. and Gaudiello, A., Some New Results on a Lavrentieff Phenomenon for Problems of Homogenization with Constraints on the Gradient. Matematiche 54 (1999) 3-47.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975) 842-850.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Grundlehren Math. Wiss. 219 (1976).
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton Math. Ser. 28 (1972).
L.R.G. Treloar, The Physics of Rubber Elasticity. Clarendon Press, Oxford, First Ed. (1949), Third Ed. (1975).
W.P. Ziemer, Weakly Differentiable Functions. Springer-Verlag, Grad. Texts in Math. 120 (1989).