#### Abstract

Cyclic loadings cause fatigue to the elements of machines leading to crack initiation and propagation. This phenomenon decreases the age of the elements. In particular, cracks decrease the stiffness of the parts and lower the parts natural frequency, leading to failure under normal working conditions. This paper introduces a new application to carbon nanotube (CNT) composites in the repairing process of a cracked specimen to restore the natural frequency of the specimen. Commonly, patches are made of high strength and high stiffness materials. This paper shows that even low stiffness materials, such as epoxy reinforced with CNT, can contribute to the repair of a cracked specimen. A 2D finite element (FE) simulation is used to study the effects of bonding CNT composite patches over the crack location to repair cracked metal specimens. The effects of the patch thickness, length, and CNTs weight concentration ratio are investigated. Results showed an increase in the natural frequency of 31% compared to the cracked specimen at a crack depth of 70% of the beam depth and at a distance of 20% of the total beam length from the support.

#### 1. Introduction

Vibrational faults in machines are of major concern in the engineering filed. In machines, cracked parts due to fatigue loading can cause a drop in the part natural frequency. This drop can lead to resonant vibrations under normal loading conditions, which may cause failure to the part. A lot of work had been conducted on the change in the natural frequency of a cracked beam compared to a healthy beam. Khaji et al. [1] constructed a closed form analytical solution to the change in the natural frequency of a cracked uniform beam. The crack was molded using a massless, linear, elastic, and rotational spring under six different boundary conditions and different parameters (such as crack location and crack depth). Kim and Stubbs [2] and Lee [3] conducted an experimental work on the detection of crack locations and its size by relating the fractional changes in the natural frequency of the cracked beam to the healthy beam due to the presence of cracks. Zheng and Kessissoglou [4] studied the free vibration of a cracked beam using the finite elements analysis (FEA) to obtain the natural frequencies and mode shapes of the beam. Loya et al. [5] studied the natural frequency of simply supported cracked Timoshenko beam by modeling the cracked region with a massless torsional and normal spring (for the vertical displacement and rotational displacement). Zhong and Oyadiji [6] constructed an analytical model to help in the prediction of the natural frequencies of a cracked simply supported beam with a stationary roving mass. The roving mass helps in detecting the presence of cracks by roving the mass from one end of the beam to another. Since cracks cause a reduction in the stiffness of the beam, this causes a reduction in the beam natural frequency when the roving mass is located near the crack. Matbuly et al. [7] studied the natural frequency of a functionally graded cracked beam resting on a Winkler-Pasternak foundation using the line spring model for formulating and applying the method of differential quadrature to solve it. For the strength of bonded patches, much research had been conducted, with different materials, to evaluate its effectiveness under various conditions. Osnes et al. [8] had studied the strength of bonded joints using the traditional strength of materials approach, which did not agree with experimental results, and inelastic fracture mechanics, which showed good agreement with the experimental results. Hosseini-Toudeshky et al. [9] studied how the curing temperature affects the residual stresses in fiber metal laminate (FML) patches, which showed no effect on the repairing process for the cracked panels. Nahas [10] performed an experimental investigation on the repairing effects of fiber composite patches on the fatigue life of a cracked aluminum specimen. Ariaei et al. [11] used piezoelectric patches to repair cracked beams by applying an external voltage to actuate the piezoelectric patches bonded on the beam. Then the patches reduce the singularity produced by the crack tip. This helped in decreasing the maximum deflection of the beam, due to the reduction of the beam stiffness, to approach the maximum deflection of a healthy beam. Brown et al. [12, 13] used self-healing epoxy matrix composite that incorporates healing agent to hinder fatigue cracks. Self-healing was also used by Blaiszik et al. [14] and Williams et al. [15].

In this paper, a FEA is conducted to examine the repair of the natural frequency of a cracked steel beam (at different locations and depth) by adhesively bonding an epoxy patch (with and without multiwalled carbon nanotubes (MWCNTs) reinforcement). The properties of the steel specimen, epoxy, and multiwalled carbon nanotubes (MWCNTs) are discussed in Section 2. The modeling of the parts and the boundary conditions are discussed in Section 3. The FEA results of the analysis are summarized in Section 4. Discussions of the results are given in Section 4 and conclusions of the results are given in Section 5.

#### 2. Mechanical Properties

CNTs are known for their high mechanical properties [16]. They have shown great potential in the reinforcement of composite materials. Many research works were carried out to predict the mechanical properties of CNT reinforced composite [17–22]. Omidi et al. [23] experimented on the effects of adding MWCNTs into an epoxy patch on Young’s modulus and strength of the CNT/Epoxy composite. These results were used with the work done by Rokni et al. [24, 25] to construct a finite elements model to determine the optimum distribution of CNTs to obtain the highest natural frequency in a microbeam. Different weight percentage (wt.%) of CNTs was used (from 0.5% to 10%). The mechanical properties used in this paper (Young’s modulus , density , and Poisson’s ratio) of the epoxy patch (with different wt.% of CNTs) and the steel specimen are shown in Table 1.

#### 3. Modeling and FE Idealization

A 2D plane stress FE model was constructed in ABAQUS/Standard with dimensions similar to those in [1] and was verified with the results in the literature. Figure 1 shows the dimensions of the beam used in the FE analysis where is the beam length which is 100 mm, is the beam depth which is 25 mm, and is the beam width which is 12.5 mm. The ratio (the ratio of the crack depth to the beam depth) was taken as 0.7 and the crack position was taken at a location of 20% of the beam total length away from the support location. These values, according to [1], show large decrease in the beam original natural frequency compared to the other values used in the work. The epoxy patch has a length of and thickness of .

Figure 2 shows the FE models and the boundary conditions used in the analysis. The nodes to the left end were constrained from movement in the and directions, while the nodes to the right end were left unconstrained. An 8-node quadrilateral element was used in the model. The repair of the natural frequency was investigated under different patch thickness and length ( and ) values with different CNT wt.% (2 wt.%, 4 wt.% and 6 wt.%). values were taken from 0.5 to 3 mm, while was taken from 10 to 30 mm. Then the effects of the crack depth and location were investigated when was 1.5 mm and was 20 mm.

**(a)**

**(b)**

**(c)**

#### 4. Results and Discussions

The first natural frequency was obtained for the model of the original beam (the uncracked beam), and then a crack with a value of and was introduced. The natural frequency was then obtained from ABAQUS for the cracked beam, firstly without the epoxy patch and then with the epoxy patch. Figure 3 shows the FE contour plot of the first mode shape of the beam. The epoxy patch was added to the model at different values (0.5, 1, 1.5, 2, 2.5, and 3 mm). Figure 4 shows the graph of the increase of natural frequency of the cracked beam with different and CNT wt.%. The graph shows that, with the increase of the patch thickness, more repairing is observed in the natural frequency of the patched cracked beam in comparison to the unrepaired beam. A patch thickness of 3 mm increased the natural frequency of the cracked beam by 41% at 6 wt.% CNT, but the increase in can lead to stress concentrations, so was taken as 1.5 mm for the rest of the analysis.

**(a)**

**(b)**

**(c)**

Different values (10, 15, 20, 25, and 30 mm) were used in order to investigate the effects of the patch length on the increase in the natural frequency of the beam. Figure 5 shows that has no effect on the increase in the natural frequency of the cracked beam. Only CNT’s concentration increased the natural frequency, but short length patches are weaker than long ones, due to the small interfacial area between the patch and the beam, so was taken as 20 mm in further analysis.

To investigate the changes in the natural frequency of the cracked beam with the repairing patch at different crack depths () and crack locations (), a patch with = 1.5 mm and = 20 mm was used. was changed from 20% to 70% (0.2, 0.35, 0.5, 0.6, and 0.7) and from 20% to 80% (0.2, 0.4, 0.5, 0.6, and 0.8). Figure 6 shows the relative natural frequency of the cracked beam with and without CNT composite patches at different crack depths. It is clear from the graph that at small crack depths, the effects of the epoxy patches are negligible since the decrease in the natural frequency is also low. Figure 7 shows the increase in the natural frequency of the repaired beam in comparison to the unrepaired one. An increase of 31% in the natural frequency is observed at 6 wt.% CNTs and . Also, a noticeable increase in the repairing takes place when goes above 0.5.

Figure 8 shows the relative natural frequency but with different crack locations. The same is also observed here; the repair has no significant change in the crack locations that produce low decrease in the natural frequency of the beam, while at , the relative natural frequency of the repaired beam increased with increasing CNTs wt.%. Figure 9 shows the increase in natural frequency with different values; the higher the CNTs wt%, the higher the increase in the natural frequency of the beam.

It is noticed from the graphs that increasing the contents of CNT in the epoxy matrix does not always cause a noticeable increase in the repair of the cracked beam natural frequency. Figure 10 shows the increase in the natural frequency of the cracked beam with different concentration ratios at , , mm, and mm. The graph shows that, after 4.0 wt.% CNTs, the increase in the natural frequency converges to 31%; further increase will cause no significant change in the natural frequency of the repaired beam.

The present work is a start for more work on the repair of materials using CNT composite patches. In particular, the authors intend to conduct the following two studies. (a)Experimental investigation of cracked specimens repaired by CNT composite patches to restore the natural frequency of the specimens. The CNT patch will be bonded over the crack location to repair it. The specimens will then be subjected to cyclic loadings to study the influence of the repair on the natural frequency. Different patch thickness and patch length values will be considered. In addition, different CNTs weight concentration ratios will be used. The facilities and equipment available at the Nanocenter of King Abdulaziz University will be used in this study.(b)On the theoretical side, the authors will conduct further analysis on the stress intensity factor at the crack tip as was done by Bachir et al. [26] and will use the *J*-integral concept of Eshelby-Rice to add results for fractal cracks. The *J*-integral for a fractal is path dependent [27]. This is why a *J*-integral fractal should be the rate of release of potential energy per unit of measurement of the fractal crack growth [28]. The authors also intend to compare how well will the CNT reinforced epoxy patches perform when compared with highly stiff composites such as carbon epoxy [29], boron epoxy [30, 31], and graphite epoxy [32].

#### 5. Conclusion

A numerical FEA using ABAQUS/Standard was carried out to observe the repair of the natural frequency of a cracked steel cantilever beam with an adhesively bonded epoxy patch (with and without CNT reinforcement at different wt.%). The first natural frequency was obtained for the original beam and then the natural frequency of the cracked beam ( and ) with different and was obtained to determine the suitable values of and . It is shown that has no effect on the repair of the natural frequency, while as increases, the natural frequency increases. The effect of and was investigated at and values of 1.5 and 20 mm, respectively. The results show that the repair had insignificant effect on locations and depths that produced low decrease in the beam natural frequency no matter which CNT wt.% was used. But at locations where there was a significant drop in the natural frequency, the effects of CNT’s wt.% became insignificant at concentrations more than 4.0 wt.%. Further investigation can be carried out to see the effect of layered epoxy patches with different to reduce stress concentration when the patch thickness increases as in [8]. Experimental validation of the findings will be carried out in more details as the facilities and equipment are available for the authors. Further investigation will be done by inspecting the effects of the patches on the strain energy release rate at the crack tip which can be used as a supporting result on the claim of the effects of the epoxy patches.

#### Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (21-135-D1432). The authors, therefore, acknowledge with thanks DSR technical and financial support.