Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T14:11:02.613Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of thin piezoelectric perforated shells

Published online by Cambridge University Press:  23 October 2007

Marius Ghergu ,
Georges Griso ,
Houari Mechkour  and
Bernadette Miara
Marius Ghergu
Affiliation:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy, PO Box 1-764, RO-014700, Bucharest, Romania. marius.ghergu@imar.ro
Georges Griso
Affiliation:
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris VI), 4 Place Jussieu, 75252 Paris, France. georges.griso@wanadoo.fr
Houari Mechkour
Affiliation:
École Polytechnique, Centre de Mathématiques Appliquées, CMAP (CNRS UMR 7641), 91128 Palaiseau, France. mechkour@cmap.polytechnique.fr
Bernadette Miara
Affiliation:
Laboratoire de Modélisation et Simulation Numérique, ESIEE, 2 Boulevard Blaise Pascal, 91360 Noisy-Le-Grand, France. miarab@esiee.fr
Get access

Abstract

We rigorously establish the existence of the limithomogeneous constitutive law of a piezoelectric composite made of periodicallyperforated microstructures and whose reference configuration is athin shell with fixed thickness. We deal with an extension of theKoiter shell model in which the three curvilinear coordinates ofthe elastic displacement field and the electric potential arecoupled. By letting the size of themicrostructure going to zero and by using the periodicunfolding method combined with the Korn's inequality in perforateddomains, we obtain the limit model.

Keywords

Computational solid mechanicshomogenizationperforationspiezoelectricityshells.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 5 , September 2007 , pp. 875 - 895
Copyright
© EDP Sciences, SMAI, 2007

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

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. CrossRef
Arbogast, T., Douglas, J. and Hornung, U., Derivation of the double porosity model of single phase flow in homogenization theory. SIAM J. Math. Anal. 21 (1990) 823–836. CrossRef
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic methods in periodic media. North Holland (1978).
Bourgeat, A., Castillero, J.B., Otero, J.A. and Ramos, R.R., Asymptotic homogenization of laminated piezocomposite materials. Int. J. Solids Structures 35 (1998) 527–541.
Caillerie, D. and Sanchez-Palencia, E., A new kind of singular stiff problems and application to thin elastic shells. Math. Models Methods Appl. Sci. 5 (1995) 47–66. CrossRef
Caillerie, D. and Sanchez-Palencia, E., Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Math. Models Methods Appl. Sci. 5 (1995) 473–496. CrossRef
A. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C. R. Acad. Sci. Paris, Sér. I 335 (2002) 99–104.
D. Cioranescu and P. Donato, An introduction to homogenization. Oxford University Press (1999).
D. Cioranescu and P. Donato, The periodic unfolding method in perforated domains,Portugaliae Mathematica, Vol. 63, Fasc. 4 (2006) 467–496.
D. Cioranescu and J. Saint-Jean Paulin, Homogenization of reticulated structures. Springer-Verlag, New-York (1999).
D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains. Porth. Math. N.S. 63 (2006) 467–496.
E. Dieulesaint and D. Royer, Ondes élastiques dans les solides, application au traitement du signal. Masson, Paris (1974).
C. Haenel, Analyse et simulation numérique de coques piézoélectriques. Ph.D. thesis, Université Pierre et Marie Curie, France (2000).
T. Ikeda, Fundamentals of piezoelectricity. Oxford University Press (1990).
Koiter, W.T., On the foundations of the linear theory of thin elastic shell. Proc. Kon. Ned. Akad. Wetensch. B73 (1970) 169–195.
T. Lewiński and J.J. Telega, Plates, laminates and shells. Asymptotic analysis and homogenization, Advances in Mathematics for Applied Sciences. World Scientific (2000).
Luckhaus, S., Bourgeat, A. and Mikelic, A., Convergence of the homogenization process for a double porosity model of immiscible two phase flow. SIAM J. Math. Anal. 27 (1996) 1520–1543.
H. Mechkour, Homogénéisation et simulation numérique de structures piézoeléctriques perforées et laminées. Ph.D. thesis, ESIEE-Paris (2004).
Miara, B., Rohan, E., Zidi, M. and Labat, B., Piezomaterials for bone regeneration design. Homogenization approach. J. Mech. Phys. Solids 53 (2005) 2529–2556. CrossRef
Nguetseng, G., A general convergence result for a functional related to the theory of homogenisation. SIAM J. Math. Anal. 20 (1989) 608–623. CrossRef
A. Preumont, A. François and P. de Man, Spatial filtering with piezoelectric films via porous electrod design, in Proc. of 13th Int. Conf. on Adaptive Structures and Technologies, Berlin (2002).
J. Sanchez-Hubert and E. Sanchez-Palencia, Introduction aux méthodes asymptotiques et à l'homogénéisation. Application à la Mécanique des milieux continus. Masson, Paris (1992).
J. Sanchez-Hubert and E. Sanchez-Palencia, Coques élastiques minces. Propriétés asymptotiques. Masson, Paris (1997).