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Bloch wave homogenization of linear elasticity system

Published online by Cambridge University Press:  15 September 2005

Sista Sivaji Ganesh  and
Muthusamy Vanninathan
Sista Sivaji Ganesh
Affiliation:
IISc-TIFR Mathematics Programme, TIFR Centre, P.O. Box 1234, Bangalore 560012, India; siva@math.tifrbng.res.in; vanni@math.tifrbng.res.in
Muthusamy Vanninathan
Affiliation:
IISc-TIFR Mathematics Programme, TIFR Centre, P.O. Box 1234, Bangalore 560012, India; siva@math.tifrbng.res.in; vanni@math.tifrbng.res.in
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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.

Keywords

Bloch waveshomogenizationlinear elasticity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 4 , October 2005 , pp. 542 - 573
Copyright
© EDP Sciences, SMAI, 2005

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