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.
Mots clés : computational solid mechanics, homogenization, perforations, piezoelectricity, shells
@article{M2AN_2007__41_5_875_0,
author = {Ghergu, Marius and Griso, Georges and Mechkour, Houari and Miara, Bernadette},
title = {Homogenization of thin piezoelectric perforated shells},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {875--895},
publisher = {EDP-Sciences},
volume = {41},
number = {5},
year = {2007},
doi = {10.1051/m2an:2007046},
mrnumber = {2363887},
zbl = {1138.74039},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007046/}
}
TY - JOUR AU - Ghergu, Marius AU - Griso, Georges AU - Mechkour, Houari AU - Miara, Bernadette TI - Homogenization of thin piezoelectric perforated shells JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 875 EP - 895 VL - 41 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007046/ DO - 10.1051/m2an:2007046 LA - en ID - M2AN_2007__41_5_875_0 ER -
%0 Journal Article %A Ghergu, Marius %A Griso, Georges %A Mechkour, Houari %A Miara, Bernadette %T Homogenization of thin piezoelectric perforated shells %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 875-895 %V 41 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007046/ %R 10.1051/m2an:2007046 %G en %F M2AN_2007__41_5_875_0
Ghergu, Marius; Griso, Georges; Mechkour, Houari; Miara, Bernadette. Homogenization of thin piezoelectric perforated shells. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 5, pp. 875-895. doi : 10.1051/m2an:2007046. http://www.numdam.org/articles/10.1051/m2an:2007046/
[1] , Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | Zbl
[2] , and, Derivation of the double porosity model of single phase flow in homogenization theory. SIAM J. Math. Anal. 21 (1990) 823-836. | Zbl
[3] , and, Asymptotic methods in periodic media. North Holland (1978). | MR
[4] ,, and, Asymptotic homogenization of laminated piezocomposite materials. Int. J. Solids Structures 35 (1998) 527-541. | Zbl
[5] and, A new kind of singular stiff problems and application to thin elastic shells. Math. Models Methods Appl. Sci. 5 (1995) 47-66. | Zbl
[6] and, Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Math. Models Methods Appl. Sci. 5 (1995) 473-496. | Zbl
[7] , and, Periodic unfolding and homogenization. C. R. Acad. Sci. Paris, Sér. I 335 (2002) 99-104. | Zbl
[8] and, An introduction to homogenization. Oxford University Press (1999). | MR | Zbl
[9] and, The periodic unfolding method in perforated domains,Portugaliae Mathematica, Vol. 63, Fasc. 4 (2006) 467-496. | Zbl
[10] and, Homogenization of reticulated structures. Springer-Verlag, New-York (1999). | MR | Zbl
[11] , and, The periodic unfolding method in perforated domains. Porth. Math. N.S. 63 (2006) 467-496. | Zbl
[12] and, Ondes élastiques dans les solides, application au traitement du signal. Masson, Paris (1974).
[13] , Analyse et simulation numérique de coques piézoélectriques. Ph.D. thesis, Université Pierre et Marie Curie, France (2000).
[14] , Fundamentals of piezoelectricity. Oxford University Press (1990).
[15] , On the foundations of the linear theory of thin elastic shell. Proc. Kon. Ned. Akad. Wetensch. B73 (1970) 169-195. | Zbl
[16] and, Plates, laminates and shells. Asymptotic analysis and homogenization, Advances in Mathematics for Applied Sciences. World Scientific (2000). | MR | Zbl
[17] , and, Convergence of the homogenization process for a double porosity model of immiscible two phase flow. SIAM J. Math. Anal. 27 (1996) 1520-1543. | Zbl
[18] , Homogénéisation et simulation numérique de structures piézoeléctriques perforées et laminées. Ph.D. thesis, ESIEE-Paris (2004).
[19] ,, and, Piezomaterials for bone regeneration design. Homogenization approach. J. Mech. Phys. Solids 53 (2005) 2529-2556.
[20] , A general convergence result for a functional related to the theory of homogenisation. SIAM J. Math. Anal. 20 (1989) 608-623. | Zbl
[21] , and, Spatial filtering with piezoelectric films via porous electrod design, in Proc. of 13th Int. Conf. on Adaptive Structures and Technologies, Berlin (2002).
[22] and, Introduction aux méthodes asymptotiques et à l'homogénéisation. Application à la Mécanique des milieux continus. Masson, Paris (1992).
[23] and, Coques élastiques minces. Propriétés asymptotiques. Masson, Paris (1997). | Zbl
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