Issue 29

P. Casini et alii, Frattura ed Integrità Strutturale, 29 (2014) 313-324; DOI: 10.3221/IGF-ESIS.29.27 314 reported in the case of fatigue cracks, also when the damage affects only a small portion of the cross section of the structural element; it requires a nonlinear model to take into account its effect on the system dynamics. In fact the breathing crack model considers that, during the vibration cycle of a structure, the edges of the crack come into and out of contact, leading to sudden changes in the dynamic response of the structure. Depending on the crack model, vibration based methods are also classified into two categories: the linear and the nonlinear approaches. The first group of methods can identify only the open cracks once at a developed stage, once changes in modal parameters become significant [6, 7]. For this reason, refined studies focus on the nonlinear response characteristics that can be investigated to identify the presence of the crack in an early stage. In fact, the structures with breathing cracks behave similarly to bilinear systems and hence exhibit nonlinear phenomena in the dynamic response even for low damage. Therefore in the second group of methods the identification can be obtained by assuming as damage indicators some peculiar characteristics of the nonlinear dynamic response such as the presence of sub and super harmonics [8-11], the changes in the phase diagrams [11, 12], the rise of superabundant nonlinear normal modes [13-15] and non-smooth bifurcations [16]. While these nonlinear effects make the response of beams more difficult to model with respect to notched beams, their appearance clearly marks the boundary between undamaged and damaged behaviour. In fact, as known [17], when a system with a breathing crack is excited by a single harmonic force, distinctive nonlinear features appear in the response. The excitation, in fact, forces the crack to open and close and the resulting clapping of the crack’s edges produces harmonics that are integer multiple or fractional multiple of the forcing frequency. These harmonics are commonly referred to as super- harmonics and sub-harmonics, respectively. These features are easily detectable when the excitation frequency is in an integer ratio or is a multiple of a resonance frequency of the system; moreover, these would be much more sensitive to cracks characteristics than the modal properties of a linear system. For this reason, increasingly over recent years [11, 12, 18, 19], attentions have been focused on the investigations of the nonlinear effects caused by the presence of breathing cracks and on the associated applications to the problem of damage detection. According to the previous observations, the dynamic behavior of beam-like structures with fatigue cracks forced by harmonic excitation is characterized by the appearance of sub and super-harmonics in the response even in presence of cracks with small depth. Since the amplitude of these harmonics depends on the position and the depth of the crack, an identification technique based on such a dependency has been developed by the authors [19]. The main advantage of this method relies on the combined use of different modes of the structure, each sensitive to the damage position depending on the curvature of the mode at its location. In this paper the identification method proposed in [19] is extended and detailed through numerical examples tested on structures with a single breathing crack and of increasing complexity to evaluate the applicability of the method to engineering applications. Dynamic vibrations of small amplitude will be considered so that the crack can be assumed to be stable. The amount of data to obtain a unique solution and the optimal choice of the observed quantities are discussed. Finally, a robustness analysis is carried out for each test case to assess the influence of measuring noise on the damage identification; the robustness of the identification, evaluated through a Monte Carlo simulation, is shown to be quite strong to both measuring and modeling errors envisioning the possibility for in-field applications of this method even in the case of very small cracks. As future developments of this study, another field of investigation will cover the robustness of the procedure with respect to other realistic disturbances such as those caused by constraints imperfections or random distributed mass. Two test cases are studied in the present paper: a simply supported beam and a spanned beam; on the first case, the difficulties in detecting the damage in symmetric systems are addressed, while on the second case the difficulties concerning more complex systems with localized modes are considered. S YSTEMS UNDER INVESTIGATION Generalities ig. 1a,b present the systems under consideration: a simply supported beam, Fig. 1a, and a spanned beam, Fig. 1b. Both cases have been modeled using standard one dimensional finite elements, while the crack is modeled as explained in the following subsection. It is assumed that only one element has a single sided edge breathing crack and that the switching between open and closed behavior of the crack occurs in the equilibrium configuration. Furthermore it is supposed that the opening and closing of the crack is primarily driven by the flexural dynamics of the beam and that the crack is the only source of nonlinearity in the structure. The considered beams are made of steel with a total length l = 0.6 m, and a constant rectangular cross-section A , of height h =0.04 m and width w =0.02 m; the properties of the material are: E =206 GPa,  =7800 kg/m 3 ,  =0.3. The beams have a breathing crack of depth a at distance d from the left-end constraint; p = d / l is the non dimensional position F