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.