## 1. Introduction

Most of the members of engineering structures operate under loading conditions, which may cause damages or cracks in overstressed zones. The presence of cracks in a structural member, such as a beam, causes local variations in stiffness, the magnitude of which mainly depends on the location and depth of the cracks. These variations, in turn, have a significant effect on the vibrational behavior of the entire structure. To ensure the safe operation of structures, it is extremely important to know whether their members are free of cracks, and should any be present, to assess their extent. The procedures often used for detection are direct procedures such as ultrasound, X-rays, etc. However, these methods have proven to be inoperative and unsuitable in certain cases, since they require expensive and minutely detailed inspections [1]. To avoid these disadvantages, in recent decades, researchers have focused on more efficient procedures in crack detection using vibration-based methods [2]. Modelling of a crack is an important aspect of these methods.

The majority of published studies assume that the crack in a structural member always remains open during vibration [3-7]. However, this assumption may not be valid when dynamic loadings are dominant. In this case, the crack breathes (opens and closes) regularly during vibration, inducing variations in the structural stiffness. These variations cause the structure to exhibit non-linear dynamic behavior [8]. The main distinctive feature of this behavior is the presence of higher harmonic components. In particular, a beam with a breathing crack shows natural frequencies between those of a non-cracked beam and those of a faulty beam with an open crack. Therefore, in these cases, vibration-based methods should employ breathing crack models to provide accurate conclusions regarding the state of damage. Several researchers [9-11] have developed breathing crack models considering only the fully open and fully closed crack states. However, experiments have indicated that the transition between these two crack states does not occur instantaneously [12]. In reference [13] represented the interaction forces between two segments of a beam, separated by a crack, using time-varying connection matrices. These matrices were expanded in Fourier series to simulate the alternation of a crack opening and closing. However, the implementation of this study requires excessive computer time. In references [14, 15] considered a simple periodic function to model the time-varying stiffness of a beam. However, this model is limited to the fundamental mode, and thus, the equation of motion for the beam must be solved.

A realistic model of a breathing crack is difficult to create due to the lack of fundamental understanding about certain aspects of the breathing mechanism. This involves not only the identification of variables affecting the breathing crack behavior, but also issues for evaluating the structural dynamic response of the fractured material. It is also not yet entirely clear how partial closure interacts with key variables of the problem. The actual physical situation requires a model that accounts for the breathing mechanism and for the interaction between external loading and dynamic crack behavior. When crack contact occurs, the unknowns are the field singular behavior, the contact region and the distribution of contact tractions on the closed region of the crack. The latter class of unknowns does not exist in the case without crack closure. This type of complicated deformation of crack surfaces constitutes a non-linear problem that is too difficult to be treated with classical analytical procedures. Thus, a suitable numerical implementation is required when partial crack closure occurs.

In reference [16] constructed a lumped cracked beam model from the three-dimensional formulation of the general problem of elasticity with unilateral contact conditions on the crack lips. The problem of a beam with an edge crack subjected to a harmonic load was considered in [17]. The breathing crack behavior was simulated as a frictionless contact problem between the crack surfaces. Displacement constraints were applied to prevent penetration of the nodes of one crack surface into the other crack surface. In reference [18] studied the problem of a cantilever beam with an edge crack subjected to a harmonic load. The breathing crack behavior was represented via a frictionless contact model of the interacting surfaces. In [19] studied the effect of a helicoidal crack on the dynamic behavior of a rotating shaft. This study used a very accurate and simplified model that assumes linear stress and strain distributions to calculate the breathing mechanism. The determination of open and closed parts of the crack was performed through a non-linear iterative procedure.

This chapter presents the vibrational behavior of a beam with a non-propagating edge crack. To treat this problem, a two-dimensional beam finite element model is employed. The breathing crack is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into a number of conventional finite elements. This non-linear dynamic problem is solved using an incremental iterative procedure. This study is applied for the case of an impulsive loaded cantilever beam. Based on the derived time response, conclusions are extracted for the crack state (i.e. open or closed) over the time. Furthermore, the time response is analyzed by Fourier and continuous wavelet transforms to show the sensitivity of the vibrational behavior for both a transverse and slant crack of various depths and positions. Comparisons are performed with the corresponding vibrational behavior of the beam when the crack is considered as always open. To assess further the validity of this technique, the quasi-static problem of a three-dimensional rotating beam with a breathing crack is also presented. The formulation of this latter problem is similar to the former one. The main differences are: the inertia and damping terms are ignored, any possible sliding occurs in two dimensions and the iterative procedure is applied to load instead of time increment. The flexibility of the rotating beam and the crack state over time are presented for both a transverse and slant crack of various depths. The validation of the present study is demonstrated through comparisons with results available from the literature.

## 2. Finite element formulation

In the following, both a two and three-dimensional beam models with a non-propagating surface crack are presented. For both models the crack surfaces are assumed to be planar and smooth and the crack thickness negligible. The beam material properties are considered linear elastic and the displacements and strains are assumed to be small. The region around the crack is discretized into conventional finite elements. The breathing crack behavior is simulated as a full frictional contact problem between the crack surfaces, which is an inherently non-linear problem. Any possible sliding is assumed to obey Coulomb’s law of friction, and penetration between contacting areas is not allowed. The non-linear dynamic problem is discussed for the two-dimensional model and the corresponding quasi-static for the three-dimensional model. Both problems are solved utilizing incremental iterative procedures. For completeness reasons, the contact analysis in three dimensions and the formulation and solution of the non-linear dynamic problem are presented below.

Figure 1 illustrates a two-dimensional straight cantilever beam with a rectangular cross-section

In the finite element method (FEM) framework, the equilibrium equation governing the dynamic behavior of the model is:

where

Figure 2 depicts a three-dimensional cantilever beam with length

The equilibrium equation governing the quasi-static behavior of the model is:

where

The equilibrium equation governing the quasi-static behavior of the model is:

where

Considering the three-dimensional model, the crack is composed of two surfaces, which intersect on the crack front. Parts of these two surfaces may come into contact on an interface. The size of the interface can vary during the interaction between the load and the structure, but the interface is usually comprised of two parts, i.e., an adhesive part and a slipping part, depending on the friction conditions maintained between the contacting surfaces. In the open crack state, the corresponding part of the crack surface is subjected to traction-free conditions. The so-called slave–master concept that is widely used for the implementation of contact analysis is adopted in this work for prediction of the crack-surface interference. One of the two crack surfaces is considered as the master surface, with the other as the slave. Both master and slave crack surfaces are defined by the local coordinate systems

Recalling the equilibrium condition, the force between the components is always expressed by the following equations:

In the open crack state, the following traction-free conditions are held between the components:

From the definition of adhesion, the displacement components on the corresponding crack surfaces are interconnected by the equations:

When an initial gap

The slip state does not prohibit the existence of a gap between the crack surfaces, so equation (5) is still valid in this case. However, the tangential force component is defined in terms of friction as:

where

The simulation of the breathing crack behavior as a full frictional contact problem constitutes the present study as a non-linear dynamic problem. In the FEM framework, a non-linear dynamic problem described by an equation such equation (1) is solved using an implicit direct integration scheme [20]. According to this method, the solution time interval of interest

where the left-hand subscripts denote the time.

The solution to this non-linear problem requires an iterative procedure. Employing the modified Newton-Raphson iteration method [20], the displacement vector

while equations (8) are written as:

where the right-hand subscripts in brackets represent the iteration number, with

Employing for example an implicit time integration scheme, formulas are implemented that relate the nodal accelerations, velocities, and displacement vectors at time

where the matrix

where

Considering that the problem has been solved for time

For reasons of simplicity, the iteration number has been omitted from equations (13)-(17). However, the formulation given below is repeated for all iterations. These equations are transformed to the global Cartesian coordinate system

## 3. Local flexibilities in cracked beams

A crack introduces local flexibilities in the stiffness of the structure due to strain energy concentration. Although local flexibilities representing the fracture in stationary structures are constant for open cracks, the breathing mechanism causes their time dependence. In a fixed direction, local flexibilities of rotating beams change also with time due to the breathing mechanism. Evidently, the vibrational response of a rotating cracked beam depends on the crack opening and closing pattern in one cycle.

Since the torsional and bending vibrations are dominant in rotating beams, in this chapter it is assumed that the corresponding local flexibilities are also dominant in the local flexibility matrix, neglecting the cross-coupling terms. Numerical results showed that the off-diagonal coefficients of this matrix are at least two orders of magnitude lower than the diagonal ones, and thus are considered negligible. Therefore, the presence of the crack can equivalently represented by a diagonal local flexibility matrix, independent of the crack contact conditions.

The exact relationship between the fracture characteristics and the induced local flexibilities is difficult to be determined by the strain energy approach, because, stress intensity factor expressions for this complex geometry are not available. For the computation of the local crack compliance, a finite element method was used. According to the point load displacement method, if at some node preferable lying on the tip of the beam is applied the external load vector

In equation (18),

where

where

where

where

Equation (23) is used to compute the coefficients of the local flexibility matrix

## 4. Response analysis

The response derived from equations (9) and (11) cannot be examined directly to distinguish the breathing crack effects. For this reason, fast Fourier and continuous wavelet transforms are employed. These two popular transforms in signal analysis are briefly discussed below for reasons of completeness, and more information can be found in references [21, 22].

The fast Fourier transform (FFT) is a perfect tool for finding the frequency components in a signal of stationary nature. Unfortunately, FFT cannot show the time point at which a particular frequency component occurs. Therefore, FFT is not a suitable tool for a non-stationary signal, such as the impulsive response of the cracked cantilever beam considered in this study, which requires time-frequency representation. To overcome this FFT deficiency, the short time Fourier transform (STFT) could be adopted, which maps a signal into a two-dimensional function of time and frequency. This windowing technique analyzes only a small section of the signal at a time. However, the information about time and frequency that is obtained has a limited precision that is determined by the size of the window, which is the same for all frequencies.

Wavelet transforms are a novel and precise way to analyze signals and can overcome the problems that other signal transforms exhibit. The most important advantage of wavelet transformations is that they have changeable window dimensions. For low frequencies, the window is wide, while for high frequencies, it is narrow. Thus, maximum time frequency resolution is provided for all frequency intervals.

The continuous wavelet transform (CWT), as employed in this study, is defined mathematically as:

where

The CWT has an inverse that permits recovery of the signal from its coefficient

## 5. Results and discussions

To demonstrate the accuracy of the presented study, a two-dimensional beam is considered with length

To demonstrate further the accuracy of the presented study, a three-dimensional beam with length

Figure 4 presents the variation of the flexibility coefficient

Figure 5 illustrates predictions of the crack closure portion for a transverse crack with depth

Figure 6 depicts this situation for cracks having slope

Further knowledge regarding the crack contact state can be also derived studying a strip under reversing loadings. Figure 8a illustrates the distinction between the open and partially closed crack configurations as defined by the crack opening displacements (CODs). In this figure the normalized displacements of the crack surface over the strip height *h*, close to the crack tip is shown. It is apparent that direct bending opens the crack, although load reversing causes partial crack closure. Solid lines represent crack opening mode and dashed lines partial crack closure, respectively. Negative values in this figure represent sign convention between the axes and not penetrating crack surfaces. Numerical results show that open crack displacements are two orders of magnitude higher than those corresponding to partially closed cracks. Figure 8b shows the transverse to crack line CODs, for two cracks with *θ*=135° and depths

Figure 9 depicts the impulsive displacement response in the transverse direction versus time

It is also visible from Figure 10a that for the deep crack the beam portion located on the left of the crack is nearly undeformed due to the dominance of the support conditions in the vibrational behavior of the beam at the considered time. Based on the Figures 10b-10c, it is observed that the distance between crack surfaces increases as the crack depth increase. This is reasonable since the small crack reduces slightly the local stiffness at crack position. This slight change of local stiffness matrix is time dependent and varies from zero to a maximum value depending on the breathing mechanism. On the contrary, for the deep crack the range of local stiffness change is wider than that of the small crack cases. For this reason, the effect of the crack is more important. The deformed shape of beam and the corresponding detail of the crack opening deformation for the two considered crack cases are also presented in Figure 11, which is magnified as Figure 10, for time

Figure 12 depicts the FFT of the transverse acceleration response for the two-dimensional beam with a breathing crack of

Figure 13 shows the sensitivity of beam vibrational behavior in terms of crack position. It seems that the crack position affects all but the first natural frequency. The absolute percentage differences are

Figure 14a shows a contour map of the CWT for the transverse displacement response of the two-dimensional beam with a breathing crack of

Higher numbers of ridges appear in the first region while there are fewer in the third. A comparison of these three regions shows differences in the magnitude of the wavelet coefficients. Figure 14b shows the contour map of the CWT for the transverse displacement response for a fully open crack. This contour map consists of three visible horizontal regions. The first two regions are similar to those of the breathing crack model (Figure 14a) in respect to their extent and the number and extent of ridges. On the contrary, the magnitude of the wavelet coefficients appears differences. The third region consists of fewer and wider ridges than that of the breathing model (Figure 14a).

The higher number of ridges in the breathing model most likely represents changes to stiffness matrix due to the breathing of the crack. Similar conclusions regarding the magnitude of the wavelet coefficients, the extent of the regions, and the number and shape of ridges per region can be made from Figure 15, which shows the sensitivity analysis with respect to crack depth. Although, the effect of the crack depth in the morphology of the Figure 15 is visible, a qualitative analysis is required to correlate the natural frequencies obtained from the FFT to the extent of the horizontal regions and the number and the extent of ridges per region. Furthermore, the magnitude of the wavelet coefficients in the domain

## 6. Conclusions

In this chapter, finite element procedures able to approach the vibrations of a beam with a breathing crack were presented. Although the developed models were discretized into a number of conventional finite elements, the breathing was treated as a full frictional contact problem between the crack surfaces. Quasi-static and non-linear dynamic analyses were performed aiming the prediction of vibration characteristics of cracked beams. The solutions were obtained using incremental iterative procedures. The results show good agreement with experimental or theoretical ones found in the open literature. The assesment of crack contact state in the different phases of the breathing mechanism gives comprehesive answers for the local flexibility variations and concequently for the vibrational response of a cracked beam. The derived time response of the two-dimensional beam was analyzed by various integral transforms including FFT and CWT. This model was assessed for the case of a cantilever beam subjected to an impulse loading. The results show the sensitivity of vibrational behavior with respect to crack characteristics. Although a qualitative analysis was required to interpret the results exactly, this study proved that FFT and CWT can be used as supplementary tools for crack detection techniques.