SUMMARY

In periodic lattice structures, analysis of wave propagation to uncover dispersion relationships can be greatly simplified by invoking the Floquet-Bloch theorem. The accompanying Bloch formalism, which was first introduced for the study of quantum mechanics and has been ‘borrowed’ in structural analysis, allows a system’s degrees of freedom to be reduced to a small subset contained in a single unit cell. When this is combined with the finite element method, the result is a powerful framework for analyzing wave propagation and dispersion in complex media. Unresolved issues still exist, however. First, the equations of motion governing this subset contain internal force terms (not arising in quantum mechanics), which must be eliminated before establishing an eigenvalue problem for the dispersion relationships. There are subtle issues with regard to the elimination of these forces, which most other researchers have either ignored or bypassed with erroneous conditions. In this thesis, these issues are resolved with the introduction of a convenient transformation technique, which leads to a rigorous elimination of all internal forces. In addition, we employ the transformation technique to prove the mathematical foundation of Bloch as used in structural analysis, and we revisit important properties of the reduced equations of motion. In proposed thesis work, we will use the transformation technique to rigorously include structural damping, which is inherent in real phononic structures but heretofore missing in their analysis. As a culminating exercise, this thesis will address application of Bloch in finding the dispersion curves in reduced-dimension materials described by non-Cartesian coordinates, such as graphene sheets and carbon nanotubes in their natural, ‘wavy’ state.