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Zbl 1103.35017
Sivaji Ganesh, Sista; Vanninathan, Muthusamy
Bloch wave homogenization of linear elasticity system. (English)
[J] ESAIM, Control Optim. Calc. Var. 11, 542-573 (2005). ISSN 1292-8119; ISSN 1262-3377/e

Summary: 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.

MSC 2000:: *35B27 Homogenization, etc.
49J45 Optimal control problems inv. semicontinuity and convergence
74J05 Linear waves
74Q15 Effective constitutive equations
35Q72 Other PDE from mechanics

Keywords: directionally regular Bloch spectrum; spectral gap

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