Issue 29

G. Carta et alii, Frattura ed Integrità Strutturale, 29 (2014) 28-36; DOI: 10.3221/IGF-ESIS.29.04 28 Focussed on: Computational Mechanics and Mechanics of Materials in Italy Elastic wave propagation and stop-band generation in strongly damaged solids G. Carta, M. Brun Department of Mechanical, Chemical and Materials Engineering, University of Cagliari, Italy giorgio_carta@unica.it , mbrun@unica.it A.B. Movchan Department of Mathematical Sciences, University of Liverpool, UK abm@liverpool.ac.uk A BSTRACT . In this work, we study the propagation of elastic waves in elongated solids with an array of equally- spaced deep transverse cracks, focusing in particular on the determination of stop-bands. We consider solids with different types of boundary conditions and different lengths, and we show that the eigenfrequencies associated with non-localized modes lie within the pass-bands of the corresponding infinite periodic system, provided that the solids are long enough. In the stop-bands, instead, eigenfrequencies relative to localized modes may be found. Furthermore, we use an asymptotic reduced model, whereby the cracked solid is approximated by a beam with elastic connections. This model allows to derive the dynamic properties of damaged solids through analytical methods. By comparing the theoretical dispersion curves yielded by the asymptotic reduced model with the numerical outcomes obtained from finite element computations, we observe that the asymptotic reduced model provides a better fit to the numerical data as the slenderness ratio increases. Finally, we illustrate how the limits of the stop-bands vary with the depth of the cracks. K EYWORDS . Elastic waves; Dispersion; Stop-bands; Elastic solids; Cracks; Asymptotics. I NTRODUCTION iscontinuities in elastic solids give rise to stop-bands, which are intervals of frequencies for which waves travelling through the solid are attenuated in amplitude, also if damping is absent. Examples of discontinuities are cracks, imperfections and defects. In addition, discontinuities may be represented by non-smooth reductions of cross-sections due to design purposes. For instance, long bridges consisting of several spans that are simply-supported on piers usually present narrower cross-sections in correspondence of the piers, where the spans are connected only by the upper deck. The generation of stop-bands is generally accompanied by localization phenomena, such as trapped modes occurring near discontinuities. Localization has been observed in different elastic systems, like beams [1], plates [2] and micro-structured media [3-5]. Localization around defects in bi-material delaminating systems is investigated in [6,7], where a lower- dimensional asymptotic model is introduced to study the dispersion properties of the medium. An improved formulation D

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