rspa.royalsocietypublishing.org Research Cite this article: Carta G, Brun M, Movchan AB. 2014 Dynamic response and localization in strongly damaged waveguides. Proc. R. Soc. A 470: 20140136. http://dx.doi.org/10.1098/rspa.2014.0136 Received: 18 February 2014 Accepted: 31 March 2014 Subject Areas: wave motion, mechanics, structural engineering Keywords: elastic periodic structures, dispersion, band-gaps, localization, asymptotics, damage Author for correspondence: M. Brun e-mail: mbrun@unica.it Dynamic response and localization in strongly damaged waveguides G. Carta 1,2 , M. Brun 1,2 and A. B. Movchan 2 1 Dipartimento di Ingegneria Meccanica, Chimica e dei Materiali, Università di Cagliari, Cagliari, Italy 2 Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK In this paper, we investigate the formation of band-gaps and localization phenomena in an elastic strip nearly disintegrated by an array of transverse cracks. We analyse the eigenfrequencies of finite, strongly damaged, elongated solids with reference to the propagation bands of an infinite strip with a periodic damage. Subsequently, we determine analytically the band-gaps of the infinite strip by using a lower-dimensional model, represented by a periodically damaged beam in which the small ligaments between cracks are modelled as ‘elastic junctions’. The effective rotational and translational stiffnesses of the elastic junctions are obtained from an ad hoc asymptotic analysis. We show that, for a finite frequency range, the dispersion curves for the reduced beam model agree with the dispersion data determined numerically for the two-dimensional elastic strip. Exponential localization, boundary layers and standing waves in strongly damaged systems are discussed in detail. 1. Introduction Localization around defects in solids is of high importance in mathematical models of elastic Bloch waves as well as in practical applications of engineering designs. Localization phenomena, in particular trapped modes appearing near defects, can occur in elastic structures with defects, cracks or discontinuities such as beams [1], plates [1–3] and microstructured media [4–7]. In addition, the dynamic response of elongated solids with a distribution of crack-like defects is used in the practical evaluation of properties of composite body armour as well as protection sheets and windscreens of armoured vehicles. 2014 The Author(s) Published by the Royal Society. All rights reserved.