Issue 29

G. Carta et alii, Frattura ed Integrità Strutturale, 29 (2014) 28-36; DOI: 10.3221/IGF-ESIS.29.04 29 with higher-order terms in the asymptotic approximation of the system is proposed in [8] for the analysis of Floquet- Bloch waves. An elongated elastic solid with a deep transverse crack is examined in [9,10] for the cases of static longitudinal and transverse loads, respectively. In [9,10] a reduced model is formulated, in which the cracked region is approximated asymptotically by an elastic connection, which accounts for the decay of the boundary layer arising near the crack. This model is extended in [11] to dynamic problems. The presence of a crack influences the vibration response of a beam, since it locally modifies the flexibility of the structural element [12]. In [13] the effects of both a double-sided and a single-sided crack on the natural frequencies of a cantilever beam are investigated. In [14] the changes in the lowest eigenfrequency of a simply-supported beam produced by a breathing edge crack are discussed in comparison with experimental results. In this paper, we analyze the dynamic response of an elongated elastic solid with a distributed damage, represented by equally-spaced transverse cracks. First, we determine numerically the eigenfrequencies and eigenmodes of cracked solids with different boundary conditions and different lengths, and we compare the results with the dispersion curves computed for solids with infinite length. Successively, we assess the validity of the reduced asymptotic model examined in [11] for different values of the slenderness ratio of the solid, and we evaluate the positions of the stop-bands in relation with the depth of the cracked sections. Finally, we summarize the results and we discuss briefly the practical applications that can be related to this work. D YNAMIC PROPERTIES OF ELONGATED DAMAGED SOLIDS WITH DIFFERENT BOUNDARY CONDITIONS e examine the dynamic behavior of an elongated elastic solid with equally-spaced cracks. An example of such a solid with a rectangular cross-section and simply-supported conditions is drawn in Fig. 1a, where l is the distance between cracks, while L , h and b are the length, the height and the thickness of the solid, respectively. Since the distance between cracks is constant, the solid can be modeled as a sequence of repetitive cells, one of which is shown in Fig. 1b, where s denotes the height of the cracked section. Figure 1 : (a) Simply-supported elongated solid with cracks located at regular intervals of length l ; (b) repetitive cell of the solid. We consider time-harmonic waves propagating along the axis of the solid that consist of oscillations occurring in the direction of the solid height. This assumption allows to study the solid as a two-dimensional strip subjected to transverse oscillations. By using a finite element model developed in the software Comsol Multiphysics , we obtain the eigenfrequencies and eigenmodes of damaged solids with different lengths. In Figs. 2a-2e we report the results relative to five different boundary conditions: a hinge and a roller, both ends fixed, a fixed end and a roller, a slider and a roller, a slider and a fixed end. The parameter n in the horizontal axes stands for the number of repetitive cells, while the quantity ϕ in the vertical axes is a non-dimensional frequency given by ϕ = ( ρAω 2 l 4 / EJ ) 1/4 . W b h L l h (a) (b) l s l l l l /2 l /2