As one of major failure modes of mechanical structures subjected to periodic loads, embedded cracks due to fatigue can cause catastrophic failure of machineries.Understanding the dynamic characteristics of a structure with an embedded crack is helpful for early crack detection and diagnosis. In this work, a new three-segment beam model with local flexibilities at crack tips is developed to investigate the vibration of a cantilever beam with a closed, fully embedded horizontal crack, which is assumed to be not located at its clamped or free end or distributed near its top or bottom side. The three-segment beam model is assumed to be a linear elastic system, and it does not account for the nonlinear crack closure effect; the top and bottom © 2016. This manuscript version is made available under the Elsevier user license http://www.elsevier.com/open-access/userlicense/1.0/ 2 segments always stay in contact at their interface during the beam vibration. It can model effects of local deformations in the vicinity of the crack tips, which cannot be captured by previous methods in the literature. The middle segment of the beam containing the crack is modeled by a mechanically consistent, reduced bending moment. Each beam segment is assumed to be an Euler-Bernoulli beam, and the compliances at the crack tips are analytically determined using a J-integral approach and verified using commercial finite element software. Using compatibility conditions at the crack tips and the transfer matrix method, the nature frequencies and mode shapes of the cracked cantilever beam are obtained. The three-segment beam model is used to investigate the effects of local flexibilities at crack tips on the first three natural frequencies and mode shapes of the cracked cantilever beam. A stationary wavelet transform (SWT) method is used to process the mode shapes of the cracked cantilever beam; jumps in single-level SWT decomposition detail coefficients can be used to identify the length and location of an embedded horizontal crack. Closed crack; J-integral; Three-segment beam model J J J J J J      (2) where the superscript r denotes the right part. For segments BC and DE, dY=0 and T i =0; hence 0 r BC J  , 0 r DE J  (3) For segment AB, one has c c c c J J J

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This study addresses the crack surfaces closure effects of a cracked cantilever beam subjected to transverse force applied at its free end under static loading. A crack closure method is developed as needed to account for crack surface contact in regions wherein the linear solutions predict physically inadmissible crack surface inter penetration. An efficient finite element algorithm is developed accordingly to solve the static problem of the beam containing a fully embedded sharp crack in a linear, elastic, homogeneous and isotropic system. Emphasis is placed on the comparison between the near-tip fracture characteristics estimates, e.g. the normalized energy release rate, Mode I and Mode II stress intensity factors (SIF), and mode mixity, with closure effects and those without closure effects reported elsewhere. The crack length, orientation, and crack center location appear as the studied model parameters of a beam of fixed beam aspect ratio. In addition, based on the study observations of pure Mode II crack, a case study on the beam with an embedded vertical crack in the compressive regime is reported to show the deformed profiles due to full crack surfaces closure during loading. The curvature profile of the cracked beam with and without closure effects are compared to those of the healthy beam for the application of closure effects in improving the results accuracy.