Issue 29

G. Carta et alii, Frattura ed Integrità Strutturale, 29 (2014) 28-36; DOI: 10.3221/IGF-ESIS.29.04 31 In this paper, E is the Young's modulus, ν is the Poisson's ratio, ρ is the mass density, ω is the angular frequency, while A and J are the area and the second moment of inertia of the cross-section of the solid. The results in Figs. 2a-2e are determined by assigning the following properties to the solid: E = 31 GPa, ν = 0.2, ρ = 2500 kg/m 3 , l = 3 m, h = 0.3 m, s = 0.1 m, b = 0.5 m. The black dots indicate the eigenfrequencies associated with propagating modes, while the grey dots represent the eigenfrequencies corresponding to localized modes. Examples of propagating and localized modes for the different cases are shown in Fig. 3. We point out that the solid with a slider and a roller does not exhibit localized modes (see Fig. 2d). Figure 3 : Examples of propagating eigenmodes ( (a) , (c) , (e) , (g) , (h) , (i) ) and localized eigenmodes ( (b) , (d) , (f) , (j) ) of solids with n = 5 repetitive cells and with different boundary conditions, shown in the insets. The dispersion curves for an infinite cracked solid are shown in Fig. 2f, where k is the wavenumber. The dispersion curves are obtained from a finite element model in Comsol Multiphysics by imposing Floquet-Bloch conditions at the vertical sides of the repetitive cell in Fig. 1b. The frequency ranges for which waves propagate without attenuation are denoted by "pass-bands" and are illustrated in grey color on the right of Figs. 2a-2f. On the other hand, the frequency ranges for which waves decay exponentially are called "stop-bands". From the outcomes in Fig. 2 we infer that the majority of the eigenfrequencies associated with propagating modes fall inside the pass-bands, while most of the eigenfrequencies relative to localized modes lie within the stop-bands. ϕ = 7.043 ϕ = 8.381 ϕ = 7.230 ϕ = 8.368 ϕ = 7.047 ϕ = 8.442 ϕ = 4.329 ϕ = 7.144 ϕ = 6.837 ϕ = 8.434 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)