Analysis of Bloch’s Method and the Propagation Technique in Periodic Structures

Abstract

Bloch analysis was originally developed by Bloch to study the electron behavior in crystalline solids. His method has been adapted to study the elastic wave propagation in periodic structures. The absence of a rigorous mathematical analysis of the approach, as applied to periodic structures, has resulted in mistreatment of internal forces and misapplication to nonlinear media. In a previous article (Farzbod and Leamy, 2009, “The Treatment of Forces in Bloch Analysis,” J. Sound Vib., 325(3), pp. 545–551), we clarified the treatment of internal forces. In this article, we borrow the insight from the previous work to detail a mathematical basis for Bloch analysis and thereby shed important light on the proper application of the technique. For example, we conclusively show that translational invariance is not a proper justification for invoking the existence of a “propagation constant,” and that in nonlinear media, this results in a flawed analysis. We also provide a simple, two-dimensional example, illustrating what the role stiffness symmetry has on the search for a band gap behavior along the edges of the irreducible Brillouin zone. This complements other treatments that have recently appeared addressing the same issue.