Elastic guided waves can be used for the inspection of elongated structures. Since these waves are multi-modal and dispersive, numerical modeling tools are often needed for the design and the optimization of inspection systems.

Here we are interested in waves propagating in bi-helical periodic media. Such structures can be encountered in cables used for electricity transportation. One considers a multi-wire structure composed by two layers of helical wires. The layers are twisted in opposite directions yielding a periodic structure along two helical directions. In our work, we want to take advantage of the periodicity properties of this structure by reducing the problem to a single unit cell through the use of the Floquet-Bloch theorem in order to deduce the wave propagation characteristics. However the difficulty lies in that the periodicity occurs along two curved directions, so that the definition of the unit cell becomes complex. Besides, the periodic boundary conditions have to be carefully written.

In this work, we propose a bi-helical coordinate system. First, we prove the existence of Bloch waves in this non-trivial coordinate system. Second, the curved boundaries of the unit cell are defined. The unit cell is then discretized by the finite element method using three-dimensional tetrahedra. Third, the wave finite element method is applied to solve the wave modes of the structure. It is shown that the components of vector fields, initially Cartesian, have to be projected in appropriate bases, namely the covariant and contravariant bases associated with the bi-helical coordinate system. The results are post-processed to find the dispersion relation as well as the group velocity of modes propagating along the straight axial direction of the structure. Numerical validations are carried out on a one-dimensional periodic single wire and on a two-dimensional periodic cylindrical shell before investigating the bi-layered multi-wire structures.