Natural Frequency Analysis of Functionally Graded Porous Beams Using Hyperbolic Shear Deformation Theory

FG materials (FGMs) have drawn a lot of attention lately because of their special qualities and possible uses in a variety of engineering fields, such as mechanical, civil, and aerospace engineering. FG beams combine the advantageous qualities of several materials to provide better performance, which makes them particularly suitable for use in load-bearing structures exposed to dynamic loading conditions [1], [2], [3].

Porosity may occur due to defects or thermal differences. In other situations, the porosity of FGM is the desired result of its unique abilities [4], [5]. Imitating nature to possess abilities, FGP materials possess distinct advantages, enabling their usage in various engineering applications. The unique mechanical properties of porous FGMs, which are derived from gradation in material composition and porosity, allow them to exhibit superior strength-to-weight ratios and enhanced energy absorption capabilities. These characteristics are precious in fields such as aerospace engineering, where reducing weight while maintaining structural integrity is crucial for performance and fuel efficiency. The ability to tailor material properties, including stiffness and damping characteristics, through the design of the porosity distribution provides engineers with innovative solutions to address specific operational demands [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].

Additionally, in civil engineering, FGP beams can be employed in lightweight structural components, such as bridges and high-rise buildings, where controlling vibrations is vital for the safety and comfort of occupants [17], [18], [19]. The vibration behavior of beams is an essential part of structural analysis because it provides insights into the stability and safety of engineering structures. Traditional beam theories often assume idealized conditions that do not account for shear deformations or variations in material properties. However, as the dimensions of structural elements decrease—particularly in modern engineering applications—shear deformations become increasingly significant. Therefore, it is essential to explore the dynamic behavior of FGP beams with HSDT to design safer and more effective structural elements [20], [21].

This study presents an analysis of the free vibration characteristics of FGP beams via HSDT. A refined beam theory that accounts for both shear deformations and the effects of porosity was employed, aiming to provide more accurate predictions of natural frequencies and mode shapes. This study seeks to enhance the understanding of the vibrational behavior of these advanced materials and evaluate the influence of various parameters, including material grading and porosity distribution, on their dynamic performance.

The results of this study can have practical consequences for the design and usage of FGP beams in engineering practice, in addition to advancing the theoretical understanding of their free vibration.

A straight functionally graded porous (FGP) beam with a rectangular cross-section, characterized by its length $L$, width $b$, and thickness $h$, was considered in the present study. The beam is assumed to be simply supported at both ends. The Cartesian coordinate system O($x,y,z$) is located on the left edge of the central axis of the beam, where $x$-, $y$-, and $z$-axes are taken along the length, width and depth of the beam, as given in Figure 1.

An FG beam is composed of ceramic and metal constituents, where material composition at the upper surface is assumed to be ceramic-rich and varies continuously to the metal-rich one located in the lower surface. Besides, an FGP beam is supposed to include porosities spreading across thickness because of the fault during manufacture and this situation is described with even (IP-I) and uneven (IP-II) porosity models in the study by Wattanasakulpong and Chaikittiratana [22], as shown in Figure 1.

The modified rule of mixture for an FGP beam with a porosity volume fraction, $\alpha(\alpha\ll1)$, can be expressed [22]:

where, $P_m$ and $P_c$ are the material properties of metal and the ceramic; and $V_m$ and $V_c$ are their volume fractions, respectively.

The relation for the total volume fraction of the metal and ceramic constituents is:

The volume fraction of the ceramic constituents obeys the following power-law function for FGP beams:

where, $p$ denotes the power-law index $(0 \leq p \leq \infty)$, and $z$ is the distance from the mid-plane of the FG beam.

The effective material properties of FGP beams with even (IP-I) and uneven (IP-II) porosity models can be expressed as follows, respectively [22]:

It can be noted that, as $\alpha=0$, the FG beam becomes the perfect one. In this case, as $p$=0, the FG beam becomes a fully ceramic one, while it turns into a fully metal one for $p$=$\infty$.

In this study, unless otherwise indicated, FG beams are composed of aluminum (Al) as the metal phase and alumina (Al$_2$O$_3$) as the ceramic phase. The material properties are as follows: $E_m$=70 GPa; $\rho_m$=2702 kg/m$^3$; $E_c$=380 GPa; and $\rho_c$=3960 kg/m$^3$. Figure 2 illustrates the variations of Young’s modulus of FG beams with P, IP-I, and IP-II models across the thickness direction.

Bending ($w_b$) and shear ($w_s$) are two of the components that make up the transverse displacements. Only $x$ and $t$ have any effect on these components.

The axial displacement is made up of extension, bending, and shear components as follows:

The displacements of the bending component, $w_b$, are considered the same as the displacements given by the classical beam theory. Hence, $u_b$ can be described as:

In combination with $w_s$, the shear component ($u_s$) produces a variation in shear strain ($\gamma_{x z}$) and hence in shear stress ($\tau_{x z}$) over the thickness of the beam. Therefore, shear stress ($\tau_{x z}$) is zero at the top and bottom faces of the FG beam. As a result, the expression for $u_s$ may be written as:

where, $f(z)$ is defined as a hyperbolic function [23]:

The field of displacement of the HSDT can be constructed based on the assumptions established in the earlier section as follows:

The displacements are related to the following strains:

and

where,

According to Hooke’s law,

where,

Hamilton’s principle was used to develop the equations of motion [24]:

where, $t, \delta U$ and $\delta K$ are the time and the virtual variations of the strain kinetic energy.

The variation in the strain energy of the beam may be represented as:

where, the stress resultants $N$, $M_b$, $M_s$ and $Q_{xz}$ are specified as:

The variation in kinetic energy may be represented as:

where, the differentiation with respect to the time variable t is indicated by the dot-superscript convention; $\rho$($z$) is the mass density; and ($I_0, I_1, J_1, I_2, J_2, K_2$) are the mass inertia and specified as:

The following equations of motion of the FG beam were found by substituting Eqs. (21) and (23) into Eq. (20), gathering the coefficients of $\delta u_0, \delta w_b$, and $\delta w_s$, and integrating by parts vs. both spatial and time variables:

Eqs. (13)-(16), (17)-(18), and (21) may be used to formulate Eqs. (24)-(26) in terms of displacements ($u_0, w_b, w_s$) as follows:

where, $A_{11}$, $D_{11}$, etc. are the stiffnesses of the beam, and defined as:

and

In the current research, the FG beam is supposed to have simply supported ends. Therefore, the Navier solution technique was used, and the displacement variables are provided by:

where, $U_m$, $W_{bm}$, and $W_{sm}$ are arbitrary parameters to be calculated, $\omega$ is the natural frequency of the FG beam, and $\sqrt{i}$=-1 is the imaginary unit.

The following equations were derived by substituting the expansions of $u_0, w_b$, and $w_s$ from Eq. (32) into the equations of motion Eqs. (27)-(29).

where,

and

This section focuses on the current situation using numerical examples. Comparisons were conducted to demonstrate the existing formulations’ validity, and the following dimensionless parameters were applied:

where, $\Omega$ is the dimensionless natural frequency (DNF).

Table 1 shows the comparison of dimensionless fundamental natural frequencies (DFNFs) of FGP beams versus $p$ for two different $\alpha$ with those proposed in studies by Avcar et al. [25] and Pandey and Pradyumna [26]. The material properties were taken from the related studies, and the span-to-depth ratio ($L/h$=10) was employed.

Table 2 presents the comparison of DFNFs of FG beams versus different $L/h$ with those proposed by several researchers [25], [27], [28]. The material properties were taken from the associated studies as $E_m$=70 GPa; $\rho_m$=2700 kg/m$^3$; $E_c$=380 GPa; $\rho_c$=3800 kg/m$^3$; $v_m$=$v_c$=0.23; $p$=0.3, as well as DNF, which was defined as $\Omega$=$\omega L^2 / h \sqrt{I_0 / A_{11}}$. DFNF of FG beams increased with the increase of $L/h$. Additionally, at the value of $L/h$=100, DFNF values were identical with the results of previous works.

Table 3 compares the DFNFs of FG beams versus $p$ with those proposed by several researchers [25], [29], [30]. The material properties taken from the related studies, such as $L/h$=5, and $\alpha$=0, were considered. DFNFs became lower with the increase of $p$. The biggest decrease of DFNF occurred between $p$ values of 0 and 1, as shown in Table 3.

Figure 3 compares the first three modes of the DNFs of all three types of FG beams for $L/h$=5,$p$=2,and $\alpha$=0.2. It can be concluded that as the Mode Numbers (MNs) increased, the DNFs also increased, and the most significant increase in DNFs occurred between MNs 1 and 2, followed by a smaller increase between MNs 2 and 3. The DNFs were highest for P and lowest for IP-I beams in all MNs. Additionally, the largest overall change in DNF was observed is the $\mathbf{P}$ beam.

Figure 4 shows DFNFs of FGP beams versus $\alpha$ for $L/h$=5, and $p$=2. According to Figure 4, the DFNFs decrease with respect to the increase of $\alpha$, and the difference gets more pronounced for model IP-I than IP-II. Furthermore, at higher values of $\alpha$, the difference of DFNFs is higher.

Figure 5 shows the value of DFNFs of all three types of FG beams versus $p$ for $L/h$=10, and $\alpha$=0.1. With the increase of $p$, DFNF values decreased. The DFNFs of the FG P-type beam were higher than those of the IP-I and IP-II beams for larger values of $p$. Additionally, the impact of the change in $p$ on DFNFs is most pronounced for the IP-I and least for $\mathbf{P}$.

Figure 6 demonstrates DFNFs of all three types of FG beams versus $L/h$ for $p$=2, and $\alpha$=0.1. It can be concluded that with the increase of $L/h$ ratio, DFNFs increased; and the change of DFNFs decreased. DFNFs of IP-I and IP-II became closer for higher values of $L/h$.

In this study, the free vibration behavior of FG beams with different porosity distribution types was studied. HSDT was employed to describe the kinematic relations of shear-deformable FG beams, while Hamilton’s principle was used to derive the equations of motion. Navier’s method was implemented to obtain a solution for the free vibration analysis of the FG beam under simply supported boundary conditions. The accuracy of the formulation was validated through comparison with existing studies.

The results obtained from the current study allow the following conclusions to be made:

a) The fundamental frequencies of FG beams are inversely related to $p$.

b) The fundamental frequencies of FG beams are inversely related to $\alpha$.

c) The increase of $L/h$ leads to an increase in the fundamental frequency.

d) The type of porosity becomes less noticeable at higher values of $L/h$.

e) The effect of change in $p$ on natural frequencies is more pronounced in the porous beams rather than perfect beams.

Based on the conclusions, porosity holds potential for future works for FGMs.