Finite Element and Vibration Frequency Analysis of Cracked Solid Beams: A Comparison of Iron, Steel, and Titanium Materials

Different solid materials are widely used in various constructions due to their availability and low cost. However, cracks or conversions in the structures can affect their stiffness and vibration signatures. This research aims to evaluate the load distribution, deformation, and the effects of the cracks on the natural frequencies and deformations of iron, steel, and titanium beams. A ﬁ nite element-based model and COMSOL Multiphysics software were employed to measure and compare the frequencies and strengths of the beams. The results showed that the frequencies increased with the load, and titanium beams had the highest frequencies and de ﬂ ections, while steel beams had the highest stress resistance. This frequency analysis can help to detect very small cracks (less than 0.05 mm) in the beams. The study concluded that steel is the most suitable material for construction due to its elasticity, availability, and low cost.


Introduction
Structural health monitoring and failure detection are important research topics in engineering [1].Many civil structures and constructions around the world may suffer damage during their service life, which can endanger human lives [2].Cracking is a common damage indicator that can reduce the stress capacity and stability of the structures.Therefore, detecting and analyzing cracks are essential for structural safety [3].One way to detect cracks is to use vibration-based methods, which measure how cracks affect the local stiffness, natural frequency, and mode shape of the structures [4].Many researchers have used different models and methods to study cracks in the various structures.Some of their findings are summarized as follows: Patil and Maiti [5] used frequency measurement to detect multiple cracks in a beam.They showed how frequency depends on crack size and location.Darpe et al. [6] studied a cracked rotor under surface loading.They found that the crack did not change the vibration direction, but it changed the rotation speed.Chasalevris and Papadopoulos [7] studied multiple cracks in beams under bending.They used a matrix method to model how each crack affects the beam's motion.They could determine the size, depth, and location of each crack.Darpe [8] studied a side crack in a rod under bending and friction.He found that the crack changed the rod's vibration pattern.Prabhakar [9] studied a beam with two side cracks using vibration analysis.He used matrix methods to model the cracks based on their size and location.He found that the cracks changed the beam's stiffness and vibration mode.Dackermann [10] used dynamic fingerprints and artificial intelligence to identify defects in structures.He trained his system to recognize different types of damage using vibration data.Georgantzinos and Anifantis [11] studied a crack in a cylindrical beam with different shapes.They found that the shape of the crack affected its opening and closing behavior.Shahbazpanahi and Kamgar [12] modeled crack growth in steel using an interface element with springs that can soften or harden.He used VCCT to estimate the energy release rates and applied fracture criteria to analyze crack growth.Caliò et al. [13] studied the vibration frequencies of spatial arches with and without damage.He compared different parameters and showed how some results can be misleading for inverse problem solutions.Jirásek [14] proposed an isotropic damage model that uses two scalar variables for the damaged stiffness tensor, based on the initial elastic stiffness tensor and the standard isotropic elasticity constants.Cervera and chiumenti [15] reviewed the discrete and smeared crack approaches for tensile cracking problems in the last 40 years.He focused on the smeared approach and pointed out its main limitations, such as the mesh-size and mesh-bias dependence.
This paper aims to simulate and compare the vibration and deformation behavior of uncracked and cracked cylindrical iron, steel, and titanium beams with half-open micro cracks.It also analyses the relationship between the modal natural frequencies and different crack positions for different materials (iron, steel and titanium) based on vibration analysis by using simulation.

Mathematical Modeling
Vibration-based methods are popular for crack detection in the structures, because they are effective and reliable [16].This method depends on how the physical responses, such as natural frequency and crack placement criterion, change [17].A solid cylindrical beam of iron, steel, and titanium with a crack in its body is considered.Bending deformations involve changes in shape and curvature of the structure, which depend on the transversal displacements and bending stiffness.Transversal displacements are the movements perpendicular to the direction of the applied force, and bending stiffness is a measure of how much resistance a structure offers to bending.The governing equation system for the deformation can be written as follows [18]: Since the domain is free on both ends, so the boundary condition will be: r x 0 ð Þ¼0, r x L ð Þ¼0, and This equation relates the bending moment and the curvature of the beam as follows: Strain energy is the energy stored by a deformed material or structure due to external loads.A crack creates a high stress field near its tip, where the strain energy concentrates.The stress intensity factor, K, measures this stress field and depends on the crack geometry, the applied load, and the loading mode [19].The strain energy release rate at the cracked section is, where the stress intensity factors are K I1 , K I2 of Mode I (opening of the crack) under load P 1 and P 2 , respectively.There are three loading modes: Mode I (opening mode), Mode II (sliding mode), and Mode III (tearing mode).For a Mode I crack, the stress intensity factor can be written as follows: where σ is the applied stress, and a is the crack length.The strain energy release rate, G, is a measure of the energy available for crack growth per unit increase in crack area.
For a mode I crack, G can be related to K I by: where É is an effective modulus that depends on the material's Young's modulus, E, and Poisson's ratio, ν.For plane stress conditions, É ¼ E 1−ν 2 and for plane strain conditions, É ¼ E.
Now, suppose we have a cracked body under two different loads, P 1 and P 2 , that produce two different stress intensity factors, K I1 and K I2 , respectively.The total strain energy stored in the body can be written as follows: where U 0 is the strain energy without load or crack, U 1 is the strain energy due to load P 1 , and U 2 is the strain energy due to load P 2 .U 0 is negligible compared to the other terms.The loads are applied independently and do not interact.Hence, the equation is: where δ 1 and δ 2 are the displacements at the crack tip due to loads P 1 and P 2 , respectively.Using Castigliano's theorem, these displacements can be express as follows: The expression for stress intensity factors from earlier studies of Irwin et al. [3] are, 2 Mathematical Problems in Engineering Here, W = width of the domain containing the crack H = height of the domain containing the crack h = distance from the crack tip to the free surface of the domain F 1 and F 2 are applied forces acting on the crack edges Defining the flexibility influence coefficient C ij per unit depth [20], where U c represents the strain energy of the considered domain and J C ¼ δU C δh is strain energy release rate.So that, Using the stress intensity rate (J C ), it is found that, Here, Ė is the crack growth rate, which is the change of crack length per unit time.The local stiffness matrix can be obtained by taking inverse of compliance matrix [18], The stiffness matrix to detect first crack is, The stiffness matrix to detect second crack is, In this study a solid cylindrical beam of iron, steel, or titanium with a length of 0.60 m and a radius of 0.015 m has been considered as the domain as shown in Figure 1. Figure 2 represents the mesh design and Figure 3 represents the internal computational geometry of the domain for the uncracked and cracked beams, which were created using COMSOL Multiphysics software [21].Tables 1 and 2 enlisted the properties of the mesh for the domain.

Results and Discussion
Stress analysis can be used to explain how bodies can deform and fracture as well as find cracks [22].The frequency and load distribution of iron, steel, and titanium beams with half circular double cracks have been studied using the finite element method.A solid cylindrical iron, steel, or titanium beams were modeled using a number of the parameters listed in Tables 3 and 4.
Figure 4 shows how a load of 500 N is applied on the edge of the beam's crown.Figure 5 illustrates the deformation of the body after the load is applied, which is simulated using COMSOL Multiphysics.
Figure 6 shows the pattern of stress distribution in the entire domain for all materials.The stress is maximum at the crack location for all metals, as shown in Figure 6(a)-6(i).The iron, steel, and titanium beams that are not cracked transfer the weight evenly to the end of the body, as shown in Figure 6(j)-6(l).No matter where the crack was, the titanium body flexed more than any other body.
Figure 7 shows how the load is distributed in a crosssection of iron, steel, and titanium beams.The load is highest at the crack location for all metals, as seen in Figure 7(a)-7(i).Iron and titanium beams have more stress at the crack location than steel beams, which have a uniform stress distribution in the body.The uncracked beams have a constant stress on the top edge of the domain, as seen in Figure 7(j)-7(l).
Figure 8 illustrates the stress absorption at various crack positions (0.01 and 0.10 m from the first end of the beam).The steel beam, which bends the most, pass the load to the    Mathematical Problems in Engineering lower part of the crack and disperses it evenly to the edge of the domain.The iron and titanium beams act in a similar way, but they do not distribute the load as evenly as steel does throughout the whole body.Figure 9 shows the load absorption nearer to crack positions (at 0.0085 and 0.085 m) from the beam's starting end.The bottom of the domain receives the most stress.The load was applied at the top edge of the domain, but as it moved to the bottom edge, it created significant, periodic vibration, particularly where the crack existed.
Figure 10 shows the relationship between the frequency of iron, steel, and titanium beams and the load, for a constant crack position.Iron has a higher frequency than steel and titanium for any load, because iron takes more load at the crack location (at 0.01 and 0.10 m from starting point), resulting in more vibration.As the load increases, each beam's frequency curve rises because the stress distribution does as well.This means that when a load is applied to a cracked body, iron is more likely to endure fracture than steel or titanium.
Figure 11 illustrates the deflection of different materials (iron, steel, and titanium) at the crack location and also for intact beam under stress.Titanium bends the most among all materials for both cracked and uncracked cases.Iron and steel have almost the same deflection and load absorption for uncracked cases.However, titanium takes too much stress at the crack location.
Figure 12 displays the deflection of three different materials (iron, steel, and titanium) without any cracks.The graph shows how the deflection of the materials changes at different locations along the body's length, from 0 to 0.3 m.The graph indicates that titanium bends the most, followed by steel and then iron.The graph also indicates that the deflection is not consistent and differs depending on the location.
The deflection of three different materials (iron, steel, and titanium) with cracks at 0.10 m positions is displayed in Figure 13.The graph demonstrates how the deflection of the materials varies at different locations along the length of the body, from 0 to 0.3 m.The graph reveals that steel has the most bending, followed by iron and then titanium.The graph also reveals that the bending is greatest at the positions where the cracks are, which are 0.10 and 0.20 m from the first end of the domain.

Conclusions
This study used a finite element-based model and COMSOL Multiphysics software to simulate the effects of cracks on the natural frequencies, deflections, and stresses of the solid cylindrical beams made of iron, steel, and titanium.The   Mathematical Problems in Engineering study focuses on method for crack detection using vibration analysis, based on the assumption that cracks affect the frequency and strength of the beams.The results showed that the material, size, and position of the cracks influenced the dynamic behavior of the beams.Steel beams had lower deflection and higher stress resistance than iron and titanium beams, while titanium beams had higher frequencies and deflections than iron and steel beams.The analysis was able to detect very small cracks (less than 0.05 mm) in the beams.These results suggest that steel is the best material for construction because of its elasticity, availability, and low cost.This study contributes to the field of structural health  Mathematical Problems in Engineering monitoring by presenting the method for crack detection using vibration analysis.

FIGURE 4 :
FIGURE 4: Applying force on the apex of computational domain.

FIGURE 5 :
FIGURE 5: Deformation of the domain after applying load.

FIGURE 6 :FIGURE 7 :FIGURE 8 :FIGURE 9 :FIGURE 10 :
FIGURE 6: Phase of the stress absorbance at different crack position.(a) Position of first crack at 0.01 m and second crack at 0.10 m of iron.(b) Position of first crack at 0.01 m and second crack at 0.10 m of steel.(c) Position of first crack at 0.01 m and second crack at 0.10 m of titanium.(d) Position of first crack at 0.10 m and second crack at 0.20 m of iron.(e) Position of first crack at 0.10 m and second crack at 0.20 m of steel.(f ) Position of first crack at 0.10 m and second crack at 0.20 m of titanium.(g) Position of first crack at 0.20 m and second crack at 0.30 m of iron.(h) Position of first crack at 0.20 m and second crack at 0.30 m of steel.(i) Position of first crack at 0.20 m and second crack at 0.30 m of titanium.(j) Uncracked iron beam.(k) Uncracked steel beam.(l) Uncracked titanium beam.

TABLE 1 :
Mesh properties of the domain.

TABLE 2 :
Geometrical properties of computational domain.

TABLE 3 :
Properties of computational domain.

TABLE 4 :
Properties of materials.