Bloch–Floquet waves in flexural systems with continuous and discrete elements Giorgio Carta, Michele Brun ⇑ Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy article info Article history: Received 30 January 2015 Available online 4 April 2015 Keywords: Flexural waves Bi-coupled structures Transfer matrix Bloch–Floquet conditions Dispersion properties Propagation zones abstract In this paper we describe the dynamic behavior of elongated multi-structured media excited by flexural harmonic waves. We examine periodic structures consisting of continu- ous beams and discrete resonators disposed in various arrangements. The transfer matrix approach and Bloch–Floquet conditions are implemented for the determination of different propagation and non-propagation regimes. The effects of the disposition of the elements in the unit cell and of the contrast in the physical properties of the different phases have been analyzed in detail, using representations in different spaces and selecting a proper set of non-dimensional parameters that fully characterize the structure. Coupling in series and in parallel continuous beam elements and discrete resonators, we have proposed a class of micro-structured mechanical systems capable to control wave propagation within elas- tic structures. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Waves propagating in a homogeneous continuum are non-dispersive, since their phase and group velocities coin- cide. Conversely, in a structured medium (e.g. a rod, a beam or a plate) dispersion occurs due to the presence of physical boundaries. A heterogeneous medium is also characterized by stop-bands, which are intervals of frequencies at which waves decay exponentially. Heterogeneities may be represented either by an intrinsic microstructure or by structural interfaces. Some real structures are made of modular units, equal to each other, that are joined together. Despite being finite in reality, these structures can be analyzed as infinite sequences of identical elements connected to each other (‘‘periodic structures’’), as demonstrated by Wei and Petyt (1997a,b) for beams, by Brun et al. (2011) for bridges and by Carta et al. (2014a,b) for damaged strips. Wave propagation in periodic structures has been extensively studied. In his classical treatise Brillouin (1953) gave a unified formulation for different classes of problems. In the literature, periodic structures are labeled according to the order of the equation describing the motion of the structure. It is usually convenient to express a differential equation of order p as a system of p first- order differential equations as, for example, in Stroh (1962), Lekhnitskii (1963) and Ting (1996). The number of kinematic independent variables (p=2) defines the cou- pling at the interface between two modular units (or ‘‘unit cells’’): ‘‘mono-coupled’’ if p=2 ¼ 1, ‘‘bi-coupled’’ if p=2 ¼ 2, and so on (see Mead, 1975a,b). Mono-coupled periodic structures (such as rods, one-dimensional lattices, etc.) were investigated by Mead (1975a), Faulkner and Hong (1985) and, more recently, by Martinsson and Movchan (2002), Brun et al. (2010) and Carta and Brun (2012). Bi- coupled periodic structures (like beams) were examined by Mead (1975b, 1996), Heckl (2002), Romeo and Luongo (2002) and Carta et al. (2014a,b). In this paper, attention is focused on bi-coupled periodic structures, with particu- lar interest to civil engineering structures (bridges, http://dx.doi.org/10.1016/j.mechmat.2015.03.004 0167-6636/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: giorgio_carta@unica.it (G. Carta), mbrun@unica.it (M. Brun). Mechanics of Materials 87 (2015) 11–26 Contents lists available at ScienceDirect Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat