Bloch Wave Homogenization of Linear Elasticity System

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. Mathematics Subject Classification. 35B27, 73B27, 74B05. Received July 12, 2004. Revised December 23, 2004. Introduction In this article, we analyze the homogenization process of periodic structures using Bloch waves in the case of linear elasticity system in three dimensions. As is well known, homogenization process is concerned with macroscopic approximations of heterogeneous media. We refer the reader to the books [4,11,14,23] for a beautiful analysis of this subject. To carry out the homogenization process various methods have been introduced in the literature. They include the methods of multiscale asymptotic expansions [4], oscillating test functions [16], two-scale convergence [1, 17], Γ-convergence [11]. In contrast to the above physical space methods, Conca and Vanninathan, in their paper [8], have followed a purely Fourier approach using Bloch waves in the case of scalar selfadjoint problem. Their analysis has been extended to the non-selfadjoint case in [25]. For applications of the Bloch wave method, we cite a few references [2, 3, 7–9, 24]. This method has also given rise to one fundamental object called Bloch approximation in the context of both theoretical and numerical aspects of homogenization [5,6]. In the literature, one also sees some phase space methods to homogenization: H-measures [26], defect measures [12], Wigner measures [13]. In [8], the authors work with the usual ordered Bloch spectrum and they prove the regularity of the first eigenvalue and eigenmode for small momenta |η| and then use it to prove the required homogenization result.