Bloch wave homogenization of linear elasticity system

Sista Sivaji Ganesh; Muthusamy Vanninathan

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 4, page 542-573
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Ganesh, Sista Sivaji, and Vanninathan, Muthusamy. "Bloch wave homogenization of linear elasticity system." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 542-573. <http://eudml.org/doc/90777>.

@article{Ganesh2010,
abstract = { 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. },
author = {Ganesh, Sista Sivaji, Vanninathan, Muthusamy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Bloch waves; homogenization; linear elasticity.; directionally regular Bloch spectrum; spectral gap},
language = {eng},
month = {3},
number = {4},
pages = {542-573},
publisher = {EDP Sciences},
title = {Bloch wave homogenization of linear elasticity system},
url = {http://eudml.org/doc/90777},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 542
EP - 573
AB - 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.
LA - eng
KW - Bloch waves; homogenization; linear elasticity.; directionally regular Bloch spectrum; spectral gap
UR - http://eudml.org/doc/90777
ER -

References

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