In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process.
Mots clés : Bloch waves, homogenization, linear elasticity
@article{COCV_2005__11_4_542_0,
author = {Ganesh, Sista Sivaji and Vanninathan, Muthusamy},
title = {Bloch wave homogenization of linear elasticity system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {542--573},
publisher = {EDP-Sciences},
volume = {11},
number = {4},
year = {2005},
doi = {10.1051/cocv:2005018},
mrnumber = {2167874},
zbl = {1103.35017},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2005018/}
}
TY - JOUR AU - Ganesh, Sista Sivaji AU - Vanninathan, Muthusamy TI - Bloch wave homogenization of linear elasticity system JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2005 SP - 542 EP - 573 VL - 11 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2005018/ DO - 10.1051/cocv:2005018 LA - en ID - COCV_2005__11_4_542_0 ER -
%0 Journal Article %A Ganesh, Sista Sivaji %A Vanninathan, Muthusamy %T Bloch wave homogenization of linear elasticity system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2005 %P 542-573 %V 11 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2005018/ %R 10.1051/cocv:2005018 %G en %F COCV_2005__11_4_542_0
Ganesh, Sista Sivaji; Vanninathan, Muthusamy. Bloch wave homogenization of linear elasticity system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 4, pp. 542-573. doi : 10.1051/cocv:2005018. http://www.numdam.org/articles/10.1051/cocv:2005018/
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