Static Buckling and Free Vibration Analysis of Bi-Dimensional FG Metal Ceramic Porous Beam

1. Shabani, P. Mehdianfar, K. Khorshidi ^*

Department of Mechanical Engineering, Faculty of Engineering, Arak University, Arak, 38156-88349, Iran

1.     Introduction

In recent years, a new class of composite materials formed that contained two or more phases which are known as functionally graded materials (FGMs). FGMs are made of two or more substances whose characteristics vary continuously from one direction to another. The performances and advantages of FGMs have led to their increasing use in many sectors including hi-tech industries, aerospace, naval, automotive, and civil structures[1]. An important application of FGM in structural components is in beams and beam structures. To support the growing use of FGM beams, the study of their vibration behavior becomes necessary. In recent years, the analysis of the free vibration of FGM beams has been performed using different analytical and numerical methods[2]. buckling and free vibration of a curved Timoshenko FG microbeam is studied by Rahmani et al. [3] based on strain gradient theory (SGT) theory. Compared to conventional materials, porous materials possess both remarkable design flexibility and significant applicability across diverse fields. Consequently, numerous researchers have dedicated their efforts to exploring various porous structures. For instance, Ghasemi and Meskini [4] conducted an investigation into the free vibration analysis of porous laminated rotating circular cylindrical shells. Several studies have been carried out on the behaviors of functionally graded material in recent years [5-10]. Atashipour et al. [11] developed a closed-form 2D elasticity solution for stresses and displacements of a curved FG beam subjected to a shear force. Among the earliest investigators in this field, Masjedi et al. [12], conducted a large deflection of functionally graded porous beams based on a geometrically exact theory. The buckling behavior of engineering structures made of advanced materials has been investigated by many researchers[13-18]. Buckling of Sandwich Structures with Metamaterials Core Integrated by Graphene Nanoplatelets Reinforced Polymer Composite was analyzed by Shabani and Khorshidi[19]. The analytical solution presented by Dym and Williams [20] can be used to estimate the buckling behavior of curved beams. The nondeterministic vibration frequencies and mode shapes of FG porous beams were investigated by Gao et al. [21] using a hybrid Chebyshev surrogate model with a discrete singular convolution method. Large amplitude nonlinear vibration analysis of functionally graded porous Timoshenko beams is examined by Ebrahimi and Zia [22]. Chen et al. [23] investigated free vibration and transient analyses of functionally graded piezoelectric materials curved Timoshenko beams in their work. Mahmoodabadi et al. [24] used a multi-objective optimization process based on the genetic algorithm to study damping the vibrations of a piezo-actuating composite beam. The Peridynamics analytical solutions for the buckling load of beams with various boundary conditions were introduced by Yang et al. [25]. Ly et al. [26] used different hybrid ML models to predict the critical buckling load of I-shaped cellular steel beams with circular openings.

In this study, the effects of variable L/h, porosity coefficient, and gradient indexes in two directions on natural frequency and critical buckling load of 2D-FG porous beams have been investigated. Young’s modulus, mass density, and Poisson's ratio of the beam are assumed to vary along the thickness and length of the beam according to power-laws form and also, they are influenced by the porosity. In this study to achieve better accuracy, modified transverse shear stress has been presented and compared with other theories. In the following, the higher-order governing equations are derived by using Hamilton's principle. In the end, the Galerkin method is employed to solve them, and the non-dimensional frequencies and critical buckling loads are obtained.

2.     Problem and Formulation

2.1. Two Dimensionally Functionally Graded Porous Beams

Consider a beam as shown in Fig. 1. with length L, width b, and thickness h, with Cartesian coordinate system O(x y z) which the origin of coordinate system O is chosen at the left of the beam. The Mechanical properties of the beam, such as Young's modulus E(x, z), shear modulus G(x, z), Poisson's modulus υ(x, z) and mass density ρ(x, z) with the material properties can be varying along length and thickness like Fig. 1.

The effective material properties(P) can be expressed by using the rule of mixture:

where P_c and P_m are epitomes of mechanical properties and R_c and R_m are volume fractions of ceramic and metal. Substitution of Eq. (2) into Eq. (1), the material properties of the 2D-FG beam are obtained as [27]:

As shown in Fig. 2, it is assumed that the porosities in the x and z directions are distributed uniformly. The effective material properties of the even porosity are defined as:

where  is a porous parameter or porosity volume fraction.

The assumed beam is graded from metal at the lower left corner edge to ceramic at the top right corner edge (Fig. 1). The volume fraction of ceramic material is given by:

The k_x and k_z are the power-law of the beam which ascertain properties in length and thickness direction. The mechanical properties of the 2D-FG porous beam can be written as [28]:

2.2. Mathematical Modeling

The displacement filed the present shear deformation beam theories, are given as [29]:

where u(x) and w(x) respectively represent the axial and transverse displacements for the mid-access, and φ is the rotation of the cross-sections. Assuming infinitesimal deformations, strain-displacement relations are[30]:

The stress-strain relations by using Hook's law can be defined as follows:

E and G represent the modulus of elasticity and shear moduli, respectively, while ks denotes the shear correction factor. Where the shear modulus (G) is [31]:

Hamilton's principle is employed to extract equations of motion [32, 33].

where U and T are the strain and kinetic energy, respectively of the beam.  is the variation operator. The strain energy of the beam(U) is calculated as follows [34, 35]:

Finally, variation of strain energy with respect to u(x), w(x) and  is shown as.

where A_xx, A_xz, B_xx, D_xx, S_xz, and T_xz are defined by:

The kinetic energy is obtained as [36]:

The inertia coefficients appearing in Eq. 19 can be defined as:

Finally, the total variation of kinetic energy associated with the sandwich beam in the integral form as:

By substituting strain energy Eq. (13) and kinetic energy Eq. (17) into Hamilton's principal Eq. (11), equations of motion may be expressed as Eqs. (18a) – (18c).

2.3. Analytical Solution

To obtain the theoretical solution, the Galerkin method is considered. According to this method, the displacements functions u(x, t), w(x, t) and  are assumed as follows[37]:

where ,  and are unknown coefficients that will be determined. i = , K denotes the order of series and  is the natural frequency. ,  and are the admissible functions that satisfy the fully clamped boundary conditions.

Free vibration analysis of the bi-dimensional functionally graded sandwich beam can be computed from Eq. (21a)[38]. Also, the amount of critical buckling loads will be calculated by Eq. (21b):

where M is the global mass matrix, K is the stiffness matrix and unknown coefficients of Eq. (21) is .

3.     Numerical Results and Discussion

In this section, the free vibration of the 2D-FG porous beam with respect to porosity coefficients ( ) is studied. Functionally graded material composed of a mixture of alumina and steel as ceramic and metal respectively with the material. Their properties are given in Table 1. The influence of different slenderness ratios, L/h = 5, 10, 15, and 20 on the non-dimensional natural frequency are investigated. The shear correction factor for TBT theory is considered as ks = 5/6 [39].

Table 1. Properties of materials[37]

The dimensionless fundamental frequency and critical buckling load are defined as Eq. 22.

where I represent the moment of inertia and A represents the area of the cross-section of the beam.

In this research, the solution is calculated for different the power-law index between 0 and 10 moreover, porosity coefficients are taken as
 0, 0.1, and 0.2. The total thickness of the beam (h) is constant and it is 0.1 m. The function indexes (p0 and q0), are taken as 2 for satisfying the clamped-clamped boundary condition. Besides, g(z) and f(z) are related parameters to the beam's theory, as they take as 0 and z for TBT, and z and  for NHOBT [40], respectively. Validation of our formulation and the results are obtained and compared with the results of Ref. [41], [42], and [43]. In Table 2, the first frequency of FG porous less beam with
L/h = 5 and k_x = 0 are calculated and they will be compared with references. To verify our new theory, the natural frequency of the FG beam with a greater slender ratio in Table 3 is calculated.

In addition, another comparison with ref [41] and [43] is presented to show the accuracy of the new higher shear deformation beam theory to calculate the critical buckling load. The mechanical properties of the pure beam have changed along the z direction and the ratio of beam's length to height is 10. The results are written in Table 4.

The amounts of Ncr of the FG beam with NHOBT and L/h = 5 are reported in Table 5. By checking both tables, it is clear that there is accuracy and agreement between the references and the present study. Thus, the results of natural frequencies and critical buckling loads will be published under effective parameters such as L/h, porosity (η), and functionally graded power indexes (k_x and k_z).

The results of non-dimensional frequencies of a two-dimensional functionally graded porous beam with several η, k_x, and k_z by using two theories are written in Table 6. The amount of ϖ has been reduced by approximately 37% by changing k_z from 0 to 8. On the other hand, the variation of k_x has affected ϖ by around 34%, indicating that k_z has a greater impact on the results. Additionally, by changing η from 0 to 0.2, the amount of ϖ has increased by approximately 3%.

The amount of Ncr under changing k_x, k_z, and η with TBT and NHOBT is calculated and the consequences are written in Table 7. By changing k_z and k_x between 0 to 8, the amount of Ncr has reduced by about 76 and 69 % respectively. The factor with the greatest impact on the results is k_z, with approximately 7% more effect. Furthermore, the Ncr decreased by about 11% with increasing η from 0 to 0.2.

Table 2. Comparison of the dimensionless natural frequencies (l/h = 5 and η = 0)

Table 3. Comparison of the dimensionless natural frequencies (l/h = 20 and η = 0)

Table 4. Comparison of the results of critical buckling load (l/h = 10 and η = 0)

Table 5. Comparison of the results of critical buckling load (l/h = 5 and η = 0)

Table 6. First non-dimensional frequency (ϖ) of 2D-FG porous beam for various kx, kz, and η (L/h = 10).

Table 7. Critical buckling load (Ncr) of 2D-FG porous beam for various kx, kz, and η (L/h = 10).

The beam is made of two-dimensional functionally graded material; thus, it has been influenced by FG power indexes in x (k_x) and z (k_z) directions. The critical buckling loads (Ncr) and the natural frequencies (ϖ) of the beam are related to the k_x and k_z. In Figure 3a, the relation between ϖ and FG power indexes in a porous less beam is shown, the ratio of length to the height of the beam is considered as 10. In Figure 3b, the effect of FG power indexes on Ncr of the porous beam is under new theory (NHOBT) with η = 0.2 and L/h = 10 illustrated. Generally, the amount of Ncr and ϖ have reduced with increasing the FG power indexes. However, Ncr has been affected more than ϖ by changing k_x and k_z. In the high amount of FG power indexes the variations of Ncr have been moved to near 0. On the other hand, the ϖ has been diminished continuously by reducing the k_x and k_z. Power indexes affect how material properties vary along the length and thickness directions, which influence their stiffness and mass distribution. A higher FG power index reduces the effective stiffness and increases the effective mass of a beam.

The impact of various ratios of the beam's spin to its height on ϖ and Ncr is shown in Figure 4. A beam with n = 0.1 and k_x = 1 is used to analyze the three amounts of L/h = 5, 10, and 20 on the ϖ in the first part of the Figure. A direct connection is seen between the amount of L/h and the ϖ. Additionally, the same study on Ncr has been conducted under unique conditions using a new theory. The results of this study are plotted in Figure 4b and the amount of Ncr has been rose by growing the L/h from 5 to 20. Also, the effect of L/h = 5 on the Ncr and ϖ is higher than 10 and 20. Reducing the strain energy will decrease the bending moment of the beam, so when the L/h is going to be diminished, the amount of Ncr and ϖ will be reduced. A higher slenderness ratio means a longer and thinner beam, which is more flexible and less stiff than a shorter and thicker beam.

Another important and effective parameter is porosity. Figure 5 shows the effect of η = 0, 0.1, and 0.2 on the Ncr and ϖ. When the amount of η is 0, the beam does not have any porosity. In the constant k_z and k_z, the ϖ has shown an inverse direction with η so as to by increasing the η, ϖ has fallen. The same relation is seen between η and Ncr, however, the effect on Ncr is bolder than on ϖ. By reducing the stiffness of the beam, the amount of ϖ and Ncr will be influenced and decreased. Porosity affects the material density and strength along the thickness direction of beams, which in turn affects their stiffness and mass. Higher porosity means more voids or holes in the material, which reduces its weight and resistance to deformation. A higher porosity reduces the effective stiffness and increases the effective mass of a beam under both axial and transverse loading, which lowers its critical buckling load and natural frequency.

4.     Conclusions

In this research paper, we have investigated the impact of various parameters, including porosity (η), slenderness ratios (L/h), and power-law indexes (kx and kz) in both axial and thickness directions, on the natural frequencies and critical buckling load of 2D-FG porous beams. We have utilized a new higher-order beam theory to analyze these effects.

Our findings demonstrate that the power-law indexes (kx and kz) play a significant role in determining the natural frequencies. Increasing the FG power indexes leads to a decrease in the values of the natural frequencies (Ncr) and the critical buckling load (ϖ). This can be explained by the change in material properties along the beam's axial and thickness directions. The power-law indexes alter the distribution of the constituent materials, affecting their stiffness and density. Higher power-law indexes result in a more significant variation of material properties, leading to decreased natural frequencies and critical buckling loads. In the case of high power-law indexes, the variations in Ncr do not exhibit significant changes. Additionally, when considering a constant value for kx and kz, we observe an inverse relationship between the porosity (η) and ϖ. As η increases, ϖ decreases, and the same trend is observed between η and Ncr. This behavior can be attributed to the presence of voids or pores within the material. The presence of voids weakens the overall structural integrity and reduces the effective stiffness of the beam. As a result, the natural frequencies decrease, and the critical buckling load is diminished. Furthermore, frequencies and critical buckling loads are more sensitive to porosity in high power-law indexes. A direct connection is seen between the amount of L/h and the ϖ. The amount of Ncr has been rose by growing the L/h from 5 to 20. Also, the effect of L/h = 5 on the Ncr and ϖ is higher than 10 and 20.

These findings provide valuable insights into the behavior of bi-dimensional FG metal ceramic porous beams and highlight the importance of considering porosity, slenderness ratios, and power-law indexes in their design and analysis. Further research in this area can contribute to the development of more efficient and reliable structures with improved performance characteristics.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.

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