Localized modes in imperfect periodic structures

Lack of periodicity in engineering structures can arise because of imperfections in the production process or a particular purpose to produce desirable physical effects. This contribution presents an approach to characterize the influence of the defects on wave propagation by applying the Floquet-Bloch theory in a two-scale framework. Localized defects are introduced periodically on a scale larger than the period of the periodic structure. For different structures with various periodic mechanical or geometric defects, the dispersion curves and shape of the first Bloch modes are analyzed and compared to the reference structure without defects. It is found that such defects can generate non-propagating modes at frequencies located inside the bandgaps of the periodic structures. Parametric studies on the appearance, the eigenfrequency, and the stability of the localized modes are carried out by considering various mechanical or geometric defects. Their appearance is slightly influenced by the amplitude of the defects, but their eigenfrequencies vary monotonically with the amplitude of the defects. With the increase in the scale of the introduction of the defects, the deformations of these non-propagating modes are more and more located around the disturbed cell. Otherwise, the link between the observed localized modes and topological modes is studied and discussed.