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

Luciano Carbone; Doina Cioranescu; Riccardo De Arcangelis; Antonio Gaudiello

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 10, Issue: 1, page 53-83
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topCarbone, Luciano, et al. "Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 53-83. <http://eudml.org/doc/90721>.

@article{Carbone2010,

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.
},

author = {Carbone, Luciano, Cioranescu, Doina, De Arcangelis, Riccardo, Gaudiello, Antonio},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Homogenization; gradient constrained variational problems; nonlinear elastomers; homogenization},

language = {eng},

month = {3},

number = {1},

pages = {53-83},

publisher = {EDP Sciences},

title = {Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set},

url = {http://eudml.org/doc/90721},

volume = {10},

year = {2010},

}

TY - JOUR

AU - Carbone, Luciano

AU - Cioranescu, Doina

AU - De Arcangelis, Riccardo

AU - Gaudiello, Antonio

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 10

IS - 1

SP - 53

EP - 83

AB -
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.

LA - eng

KW - Homogenization; gradient constrained variational problems; nonlinear elastomers; homogenization

UR - http://eudml.org/doc/90721

ER -

## References

top- 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).
- L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, An Approach to the Homogenization of Nonlinear Elastomers via the Theory of Unbounded Functionals. C. R. Acad. Sci. Paris Sér. I Math.332 (2001) 283-288.
- 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.
- L. Carbone, D. Cioranescu, R. De Arcangelis and A. Gaudiello, An Approach 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.
- L. Carbone and R. De Arcangelis, On Integral Representation, Relaxation and Homogenization for Unbounded Functionals. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl.8 (1997) 129-135.
- L. Carbone and R. De Arcangelis, On the Relaxation of Some Classes of Unbounded Integral Functionals. Matematiche51 (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.
- L. Carbone and R. De Arcangelis, 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.
- L. Carbone and R. De Arcangelis, 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).
- A. Corbo Esposito and R. De Arcangelis, The Lavrentieff Phenomenon and Different Processes of Homogenization. Comm. Partial Differential Equations17 (1992) 1503-1538.
- A. Corbo Esposito and R. De Arcangelis, Homogenization of Dirichlet Problems with Nonnegative Bounded Constraints on the Gradient. J. Analyse Math.64 (1994) 53-96.
- A. Corbo Esposito and F. Serra Cassano, A Lavrentieff Phenomenon 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).
- C. D'Apice, T. Durante and A. Gaudiello, Some New Results on a Lavrentieff Phenomenon for Problems of Homogenization with Constraints on the Gradient. Matematiche54 (1999) 3-47.
- E. De Giorgi and T. Franzoni, 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).