Finding the minimal unit cell – Bloch theorem for structures with translation and reflection or rotation symmetries

Finding the minimal unit cell – Bloch theorem for structures with translation and reflection or rotation symmetries Bloch theorem provides a useful tool to analyze wave propagation in periodic systems. While this method has been developed for structures periodic by translation, the goal of this session is to generalize the transfer matrix and the inverse Bloch method to structures with glide (translation and reflection) or screw (translation and rotation) symmetries by deriving appropriate boundary conditions. In this work, quasi-one-dimensional wave propagation (one dimensional waves propagating in two or three dimensional structures, e.g. undulated beam, twisted wire) and quasi-two-dimensional wave propagation (e.g. undulated plate, core panel, weave fabric, pipe, nanotube) is investigated. By considering a smaller periodicity, the computational cost is decreased and the interpretability of the dispersion curves is improved since the number of folding curves and non-interacting intersecting curves is reduced. While the size of a unit cell can be divided by a factor two when glide symmetries are considered, this ratio is significantly increased for screw or rotational symmetries. Moreover, the proposed revisited Bloch method is applicable to screw symmetric structures that do not possess purely translational symmetries, and for which the classical method cannot be used (e.g. chiral nanotube, wrinkled helicoid, tetrahelix). F. Maurin, Bloch theorem with revised boundary conditions applied to glide and screw symmetric, quasi-one-dimensional structures, Wave Motion 61 (2016) 20–39. F. Maurin, C. Claeys, L. Van Belle, W. Desmet, Bloch theorem with revised boundary conditions applied to glide, screw and rotational symmetric structures, Computer Methods in Applied Mechanics and Engineering 318 (2017), 497-513.status: accepte