Two-dimensional elastic Bloch waves in helical periodic structures

Abstract

The goal of this paper is to investigate the propagation of two-dimensional elastic Bloch waves in helical periodic structures, obeying two discrete screw symmetries about the same axis. First, a three-dimensional coordinate system is built from the two helical directions of periodicity of the problem and the radial coordinate originating from the symmetry axis. The existence of Bloch waves in bi-helical structures can be justified owing to the independence of the metric tensor of the coordinate system on both helical coordinates. Considering the elastodynamic equilibrium equations, Bloch theorem is expressed in appropriate bases to project the vector wavefields, namely the covariant/contravariant bases of the bi-helical coordinate system (or, alternatively, the cylindrical basis). From a geometrical point of view, the three-dimensional unit cell is delimited by non-plane boundaries, which must be carefully parametrized. The so-called wave finite element method is then applied to numerically solve the Bloch wave eigenproblem and the implementation of the numerical method in a bi-helical system is detailed. Owing to the cylindrical nature of the geometry, the two-dimensional propagation constants are not independent to each other. The relationship between both constants is established. The calculation of wave mode properties (wavenumbers, group and energy velocities) is performed along the helical propagation directions, as well as the straight axial and circumferential directions. Numerical validations of the overall approach are carried out for cylindrical uniform tubes and a chiral nanotube. Finally, the method is applied to a complex multi-wire structure, often encountered in energy cables, consisting of two layers of helical wires twisted in opposite directions.