On the gradient elastic wave propagation in cylindrical waveguides with microstructure

The present work describes the development of a complete theoretical framework of wave propagation in cylindrical waveguides possessing microstructure. In parallel, a thorough investigation of the full 3-D model of wave propagation in cylinders is presented. The first step is the spectral decomposition of the boundary value problem emerging via wave propagation analysis. The spectral representation of the specific gradient elasticity problem reflects the ability to construct all the possible propagating modes in cylindrical geometry. Several byproducts arise along the present work, which constitute generalizations of well known important features of classical elasticity and are indispensable for modeling the gradient elasticity problem. We note the construction of the set of dyadic Navier eigenfunctions which constitute the generalization of the Navier eigenvectors. The restriction of the Navier eigendyadics on cylindrical surfaces gives birth to the dyadic cylindrical harmonics, which constitute the generalization of the well known vector harmonics. This set is also a basis in the sense that the trace of every dyadic field on a cylindrical surface can be represented as a countable superposition of dyadic cylindrical harmonics. The method aims at providing the necessary theoretical establishment for the determination of the dispersion curves emerging in cortical bones.