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

¹ 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.
² Université Pierre et Marie Curie (Paris VI), Laboratoire Jacques-Louis Lions, 4 Place Jussieu, 75252 Paris Cedex 05, France; cioran@ann.jussieu.fr.
³ 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.

Received: 26 June 2001
Revised: 28 May 2002
Revised: 4 December 2002

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.