# Homogenization of thin piezoelectric perforated shells

Marius Ghergu; Georges Griso; Houari Mechkour; Bernadette Miara

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 5, page 875-895
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topGhergu, Marius, et al. "Homogenization of thin piezoelectric perforated shells." ESAIM: Mathematical Modelling and Numerical Analysis 41.5 (2007): 875-895. <http://eudml.org/doc/250048>.

@article{Ghergu2007,

abstract = {
We rigorously establish the existence of the limit
homogeneous constitutive law of a piezoelectric composite made of
periodically
perforated microstructures and whose reference configuration is a
thin shell with fixed thickness. We deal with an extension of the
Koiter shell model in which the three curvilinear coordinates of
the elastic displacement field and the electric potential are
coupled. By letting the size of the
microstructure going to zero and by using the periodic
unfolding method combined with the Korn's inequality in perforated
domains, we obtain the limit model.
},

author = {Ghergu, Marius, Griso, Georges, Mechkour, Houari, Miara, Bernadette},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Computational solid mechanics; homogenization; perforations;
piezoelectricity; shells.; existence; Koiter shell model; Korn's inequality; periodic unfolding method},

language = {eng},

month = {10},

number = {5},

pages = {875-895},

publisher = {EDP Sciences},

title = {Homogenization of thin piezoelectric perforated shells},

url = {http://eudml.org/doc/250048},

volume = {41},

year = {2007},

}

TY - JOUR

AU - Ghergu, Marius

AU - Griso, Georges

AU - Mechkour, Houari

AU - Miara, Bernadette

TI - Homogenization of thin piezoelectric perforated shells

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2007/10//

PB - EDP Sciences

VL - 41

IS - 5

SP - 875

EP - 895

AB -
We rigorously establish the existence of the limit
homogeneous constitutive law of a piezoelectric composite made of
periodically
perforated microstructures and whose reference configuration is a
thin shell with fixed thickness. We deal with an extension of the
Koiter shell model in which the three curvilinear coordinates of
the elastic displacement field and the electric potential are
coupled. By letting the size of the
microstructure going to zero and by using the periodic
unfolding method combined with the Korn's inequality in perforated
domains, we obtain the limit model.

LA - eng

KW - Computational solid mechanics; homogenization; perforations;
piezoelectricity; shells.; existence; Koiter shell model; Korn's inequality; periodic unfolding method

UR - http://eudml.org/doc/250048

ER -

## References

top- G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.
- T. Arbogast, J. Douglas and U. Hornung, Derivation of the double porosity model of single phase flow in homogenization theory. SIAM J. Math. Anal.21 (1990) 823–836.
- A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic methods in periodic media. North Holland (1978).
- A. Bourgeat, J.B. Castillero, J.A. Otero and R.R. Ramos, Asymptotic homogenization of laminated piezocomposite materials. Int. J. Solids Structures35 (1998) 527–541.
- D. Caillerie and E. Sanchez-Palencia, A new kind of singular stiff problems and application to thin elastic shells. Math. Models Methods Appl. Sci.5 (1995) 47–66.
- D. Caillerie and E. Sanchez-Palencia, Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Math. Models Methods Appl. Sci.5 (1995) 473–496.
- A. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization. C. R. Acad. Sci. Paris, Sér. I335 (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).
- W.T. Koiter, 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).
- S. Luckhaus, A. Bourgeat and A. Mikelic, 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).
- B. Miara, E. Rohan, M. Zidi and B. Labat, Piezomaterials for bone regeneration design. Homogenization approach. J. Mech. Phys. Solids53 (2005) 2529–2556.
- G. Nguetseng, A general convergence result for a functional related to the theory of homogenisation. SIAM J. Math. Anal.20 (1989) 608–623.
- 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).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.