Free Vibration Analysis of 2D Functionally Graded Porous Beams Using Novel Higher-Order Theory

1. C. M. Reddy ^a*, N. V. Kumar ^b

^a Research Scholar, Department of Mechanical Engineering, School of Technology, GITAM, Hyderabad, 502329, India.

^b Professor, Department of Mechanical Engineering, School of Technology, GITAM, Hyderabad, 502329, India.

1.     Introduction

Materials have played a significant role in the development of society throughout human history. But in the present scenario, conventional engineering materials are deficient in fulfilling the desired properties, demanded in various applications. To overcome the aforesaid deficiency, advancements in the field of materials are needed.

FGMs are such materials that have a good number of advantages over conventional materials. They largely preserve the characteristics of the components that make them up. According to Li et al. [1], the continuous transition of materials minimizes residual strain, and stress concentration, and offers a good strength-to-weight ratio. These features make FGMs attractive to structural engineers and researchers, and they are leading to be used in broad engineering fields, such as Aerospace, Mechanical, Civil, and Nuclear domains.

1.1. FGMs

Asemi et al. [2] “for the transverse vibration of double-piezoelectric-nano plate systems with initial stress under an external electric voltage, a nonlocal continuum plate model was created”. To account for the impact of shearing between two piezoelectric nanoplates in addition to the typical behavior of coupling elastic medium, the Pasternak foundation model was used. Aydogdu et al. [3] based on exponential and power law, the elastic modulus of the beam was varied in a thickness direction. Using shear deformation shell theory, the free vibration behavior of a simply supported FGB was determined. Al-Zahrani et al. [4] Finite element method was used for investigating free vibration in axial and thickness directions of bi-directional FGBs. To describe the changes of volume fraction in metal and ceramic based on the Vogit model.

Barati et al. [5] their paper examined “bi-directional Functionally Graded (FG) nanobeams” exposed to a longitudinal magnetic field in terms of transverse vibrations. The small-scale effect was taken into account using the nonlocal elasticity hypothesis. Bellifa et al. [6] For bending and dynamic behavior of FG plates, a first-order theory incorporating shear deformation was developed. The governing equations of axial and transverse deformations of functionally graded plates (FGPs) were developed using 1^st-order plate theory with shear deformation. Daneshmehr et al. [7], their study examined the small-scale effects on free vibration behavior. The small-scale effects on natural frequencies were investigated using Eringen's nonlocal hypothesis. While stocky and short nanoplates were taken into account, higher-order shear deformation plate theory was adapted to provide more precise results when analyzing the nanoplate.  Ebrahimi et al. [8] based on 3^rd-order beam theory with shear deformation, free vibration characteristics of FG nanobeams were investigated by presenting a Navier-type solution. Along the thickness direction, the material properties of FG nanobeam were changed continuously as per power law. Hadji et al. [9] “in their study, attention was given to examining nano beam's dynamic stability in a generic condition of non-uniform bending stress. Higher-order deformation theory was used to develop governing equations of motion for FGPs. Zaoui et al. [10] using “an improved hyperbolic shear deformation theory, developed analytical solutions to free vibration analysis of FGBs that included stretching effect”.

Haghshenas Gorgani et al. [11], various higher-order shear deformation theories have been established for bending and free vibration of FGPs. The pull-in behavior of FGM cantilever micro/nano-beams under the influence of electrostatic force was studied. By adopting a skew-symmetric portion of rotation gradients, the coupling tensor becomes skew-symmetric fulfilling consistent couple stress theory.  Hebbar et al. [12] 2D and quasi-3D theories of shear deformation were used to analyze the behavior of free vibration, static bending, and elastic buckling of FGBs with simply supported. Huang et al. [13] studied composite nanoplate's nonlinear vibration analysis, lipid face sheets and an FG core were used to create a composite nanoplate. The FG core material characteristics vary in three different directions. The viscoelastic impact of lipid layers was investigated using the Kelvin-Voigt model. The “nonlinear differential equation of vibration analysis of composite” nanoplate was obtained” utilizing Von-Karman hypotheses. Jha et al. [14] using “higher-order theory of shear and normal deformation, free vibration analysis of FG elastic, rectangular, and simply supported plates as described”. Although heterogeneous, mechanical characteristics of FGM were modified smoothly concerning spatial coordinates. Armagan Karamanli [15], two-directional FGBs behavior of free vibration was presented with various boundary conditions. The properties of beam material were changed by accommodating various gradation exponents in x and z directions. Ketabdari et al. [16], “based on Winkler and Pasternak elastic foundation, the free vibration of homogeneous and FGPs” were tested. The elastic foundation was a combination of Pasternak and Winkler electric support with parabolically and linearly variable stiffness coefficients along the directions. Larbi et al. [17] presented a bending analysis of the free vibration of FGB on the natural surface position of shear deformation theory. Boundary settings were satisfied with no shear correction factor. Mohammadi et al. [18], the “free vibration behavior of a rectangular graphene sheet subjected to a shear in-plane force investigated. “The vibration analysis of “orthotropic single-layered graphene sheets exposed to shear in-plane” force has been studied using nonlocal elasticity theory.” Ohab-Yazdi et al. [19], based on the generalized differential quadrature method and minimum total potential energy, derived boundary conditions, and governing equations. In their investigation vibration analysis of 2D, FG nanobeams were done by employing the Euler-Bernoulli theory. Nguyen et al. [20], 1^st order beam theory with shear deformation was developed to determine the static and vibration of FGBs. Transverse shear stiffness was improved by using plane stress and equilibrium equation.

Rahmani et al. [21] discussed the effect of size dependency in FGM based on the beam theory of Timoshenko. Along with the thickness, the material properties of FG nanobeams varied based on the power law. Safarabadi et al. [22] considered the surface effect in an investigation of vibration frequencies of nanobeams and for the satisfaction of surface balance equations of continuum surface elasticity, to propose the Gurtin-Murdoch model. Şimşek [23], the fundamental frequency of FGB was investigated using classical, first-order, and third-order theories with various boundary settings. Babaei et al. [24], this study developed a computational 3D finite element approach to examine the static responses and natural frequencies of a saturated FG porous annular elliptical sector plate. Utilizing the Rayleigh-Ritz energy model, the finite element method was used to construct the governing equations. Talha et al. [25], FGM plates were analyzed to determine the behavior of vibration and static, based on a theory of HSD with modification of transverse displacement by finite element model. Thai et al. [26], FGBs were analyzed to determine static bending and vibration analysis with various theories of HSD. With transverse shear strain, boundary settings satisfy the top and bottom surfaces of the beam. Vo, et al. [27], “an improved theory of shear deformation was developed for the analysis of static bending and vibration of FGBs. The shear correction factor was not necessary by shear deformation theory”. Şimşek [28], “Free and forced vibration of a Timoshenko beam with bi-directional FGM was explored under the influence of a moving load”. In both axial and thickness dimensions, the beam characteristics varied exponentially. Sayyad et al. [29], the theory of unified shear deformation was developed based on displacement for analysis of advanced composite plates and beams. This theory considered shape function in terms of the transverse shear deformation effect. Razouki et al. [30], under UDL, developed exact analytical results of a simply supported beam. Specific analytical formulas were expressed by using 3^rd-order shear deformation and compared to existing numerical results and classical analytical ones.

1.2. Porous FGMs

Slimane et al. [31], using “higher order shear deformation theory, free vibration analysis of a simply supported FGP with porosity was investigated. The material characteristics of FG porous plate changed over a thickness of plate”. Mehdianfar et al. [32], investigated on analysis of free vibration of 2D porous FG sandwich beams by using the Galerkin method. From Hamilton’s principle, the motion equations of the beam were derived. This theory didn't require a shear correction factor. In this, the analytical values were compared with Timoshenko's first-order parabolic shear deformation theory. Asemi et al. [33], this research investigated for the first time static, dynamic, and natural frequency assessments of FG porous annular sector plate reinforced by graphene nanoplatelets. The plate was made up of a layered model with a metallic matrix that has open-cell interior pores and graphene platelets that are dispersed uniformly or unevenly. Babaei et al. [34], this research investigated natural frequencies and dynamic responses of thick beams formed of saturated porous materials resting on a viscoelastic foundation. Higher-order beam theory was utilized to model the beam. The viscoelastic basis was modeled using the Kelvin-Voight equation. Mohammadi et al. [35], Lipid face sheets and an FG core were used to create a composite nanoplate to study the composite nanoplate's nonlinear vibration analysis. The FG core's material characteristics vary in three different directions. The viscoelastic impact of lipid layers was investigated using the Kelvin-Voigt model. “Nonlinear differential equation of vibration analysis for composite nanoplate was obtained utilizing Von-Karman hypotheses.

Kim et al. [36] by “using classical and first-order shear deformation plate theories, the bending, free vibration, and buckling responses of FG porous micro-plates were investigated”.  Chen et al. [37] presented free vibration analysis of linear and nonlinear rotating FG porous beams by employing Timoshenko beam theory, modified couple of stresses, and assumptions of Von Karman Geometric nonlinearity. Li et al. [38] investigated buckling, free vibration, and bending analysis of 2D FGPs by using isogeometric analysis and first-order shear deformation theory. The governing equations of porous 2D FG plates were derived by employing Hamilton's principle. Bathini et al. [39] investigated the free vibration behavior of bi-directional FGPs from a refined theory of shear deformation with first order. From Lagrange equations, equations of motion were obtained. Kumar et al. [40], based on classical plate theory to investigated transverse vibration in thin isotropic simply supported FG rectangular plates with the effect of porosity. The plate is positioned elastically against rotation. Vasara et al. [41], investigated non-axisymmetric free vibration in 2D FGPs with porosity by using first-order theory. By employing gradient index in both directions and porosity distribution in different ways.

The fabrication of FGMs can be done in a variety of ways, including vapor deposition, self-propagating high-temperature synthesis, powder metallurgy, non-pressure sintering, and multistep sequential infiltration. Porosities and micro-voids, however, may appear in the material created during the sintering process when FGMs are produced. This is a result of the ceramic phase being at a comparatively low temperature while the metal phase coagulated at a very high temperature. The strength of the material will be exotically weakened by the pores. When constructing bidirectional FGM components, it is essential to evaluate the porosity effect.

From the literature, it is observed that most studies dealt with analysis on non-porous FGM such as beams, and plates using first and second order theories in one direction”. This intrigued us to investigate the effect of porosity in two directions in FGM adapting n^th order theory.

“The objective of this paper is to focus on the free vibration behavior of 2D-FGBs with porosity based on a power-law variation of material properties with various end conditions, aspect ratios, gradient indexes, and porosity index".  A unique shear shape function was derived and n^th order theory was adapted to take into account the effect of transverse shear deformation to get zero shear stress conditions at the top and bottom surfaces of FGB.

2.     Materials and Methods

In this research work, a rectangular beam of FGM with length L in the x-direction, width b in the y-direction, and thickness h in the z-direction is considered as shown in Fig. 1. Variation of beam material along the length direction, in this situation the material gradient is changed from left (-L/2) to right (L/2) and in the thickness direction, FGM rectangular beam is created by grading ceramic and metal phases. Here, the upper surface (z= +h/2) with metal and the lower surface (z=-h/2) with ceramic. The middle surface of the beam is the reference surface i.e. (z=0).

Fig. 1. Geometry of Functionally Graded Beam

Material properties of FGBs are the function of the volume fraction of constituent materials. The functional relationship between thickness coordinates and material properties is assumed. The volume fraction of material according to power-law distribution in two directions (x and z) is expressed in Eq. 1[42].

where, h and z represent the thickness of the beam and thickness coordinate, and L and x represent the length of the beam and length coordinate respectively. Origin (O) is the rectangular beam's mid surface (x, y) thus
z 𝜖 [-h/2, h/2]. ‘p’ indicates the volume fraction behavior along the beam's thickness and length. Figure 2 shows a variation of volume fractions of metal in thickness and length direction.

Fig. 2. Volume fractions of metal in thickness (z/h) and length (x/L) direction.

2.1. Formulation for Functionally Graded Porous Beams

Porosities appear as a defect in FGBs because of technical and penetration problems in the production process. Porosities in the beam are two types namely even and uneven as shown in Fig. 3. “The properties of efficient material of FG beams such as Modulus of elasticity E, Poisson's ratio μ and mass density ρ, are to be found by using a modified rule of the mixture in which the porosity is represented by α, which affects averagely on the material volume fraction of each constituent”. As a result, the material property P(x, z) can be written for each type of porosity in coordinates of the x and z directions.

Fig. 3. Representation of bi-directional FGB with even and uneven porosity distributions

Material properties of the FGM porous beam (even distribution) are presented in Eq.2 [43].

here α represents the coefficient of porosity which can be defined as the ratio between void volume and complete volume (0 ≤ α< 1). “The subscripts m denotes the metal and c denote ceramic phases. ‘P_x’ and 'P_z' is non-negative variables that define AFG (along an axis) and FG (along thickness) power indexes, respectively. These are related to volume fraction change along axis and thickness". The material properties of the beam, i.e. Young’s Modulus ‘E’ and mass density ‘ρ’ of FGM porous beam (even) are given in Eq. 3, Eq. 4.

The material properties of the beam, i.e. Young’s Modulus ‘E’ and mass density ‘ρ’ of FGM porous beam (uneven) are given below in Eq. 5, Eq. 6.

2.2. Displacement Field and Constitutive Equations

Consider the FG rectangular beam as shown in Fig.1. for an analytical result of vibration analysis.  The accurate vibration of beams depends upon transverse shear and normal deformation. Therefore any refinement of classical beam theory is generally meaningless. In this regard, the effect of transverse shear and normal strain is considered. The present theory has important features as follows:

The displacement equations are based on Reddy’s advanced refined higher-order beam theory.

From the above equations, u is axial displacement, w is transverse displacement, and u₀, w₀ is the axial displacement at any point on the neutral axis,  is the bending slope and ϕ is a shear slope. For determining the distribution of transverse shear deformation shape function .

The relationship between stress and strain of two directional FGM beam coordinate axes is given by,

2.3. Governing Equations of Motion

From the principle of Hamilton, the equations of motion are derived and can be stated in time intervals [0,t] as,

where δU is the variation of strain energy, and δk is the variation of kinetic energy respectively.

2.4. Formulation of Free Vibration

The strain energy of bi-directional FGB can be written as:

Substituting Eq. (9), Eq. (11), Eq. (14), and Eq.(15) into Eq. (17) the strain energy can be written as,

The kinetic energy of bi-directional FGB can be written similarly:

The kinematic boundary conditions expressed in Lagrange equations from Hamilton’s principle are expressed in infinity dimensions in terms of generalized coordinators and are presented in displacement functions as,

Proposed , , and  are shape functions in the natural frequency of beam, and  complex numbers are used in determining unknown coefficients , .

By substituting equations 20, 21, and 22 into 18 & 19 and got governing equations of motion.

The values of A_j, B_j, and C_j represented with q_j, lead to:

The stiffness and mass matrices are [Ski] and [Mki], respectively. The stiffness and mass matrices should be symmetric and in max size. The stiffness and mass matrix components are given by,

3.     Results and Discussion

Free vibration analysis of 2D FGBs, “which are affected by thickness ratio, aspect ratio, gradation indexes, type of porosity, and volume fraction porosity, is presented. The numerical investigations on Simply Supported (SS), Clamped-Clamped (CC), and Clamped-Free (CF) beam at different boundary conditions are carried out” as shown in Table 1.

Table 1. Various kinematic boundary conditions for numerical computations.

Free vibration behavior is presented to discuss and validate the accuracy of the current theory. FGM porous beam is considered for numerical results, made of Alumina and Aluminum with material properties as follows:

According to power-law distribution, the FGB material properties are varying in thickness (h) and axial (L) directions. For a representation of results, the following dimensionless natural frequency (λ) parameter is used:

Consider the homogeneous beam and a different number of terms with displacement functions. The result of dimensionless free vibration is presented with various gradient indexes in x and z directions, aspect ratios, and boundary conditions. For comparison purpose, previous result [15, 28] was used in terms of free vibration as shown in Table 2, Table 3, Table 4, Table 5, Table 6, and Table 7. It can be observed that in dimensionless free vibration of SS, CC, and CF beams, the response was very quick and polynomial expansion is at 6 in the SS beam and 4 in CC and CF beams. But for better accuracy purposes, polynomial expansion at 12 terms was considered. In terms of aspect ratios (L/h=5 and L/h=20) and gradient indexes in both directions, dimensionless free vibration decreases in SS, CC, and CF beams as the gradient indexes increases in both directions. On the other hand, the dimensionless natural frequency of the CC beam increases in the x-direction and decreases in the z-direction as the gradient index value increase, this is due to the end conditions and reactions.

From the result, the elasticity modulus and stiffness of the beam increase subsequently with an increase in gradient indexes. The frequencies have raised as since the gradient indexes and elasticity modulus increase along with the stiffness of the beam. Of course, this is accurate provided that the beam's mass remains constant. Anyhow in the study mass is not consistent and it is increased by gradient indexes of material. According to vibration theory, the vibration frequency is inversely proportional to mass and directly proportional to the rigidity (also known as elasticity modulus). Then it can be inferred that the effect of modulus of elasticity is more on dimensionless fundamental frequency and the effect of mass is less on the dimensionless fundamental frequency. But the valves of elasticity modulus and mass can compensate for the dimensionless fundamental frequency. However, it is observed that the effect of mass is slightly more than the effect of elasticity modules on frequency. Another important observation is that the effect of gradient indexes (p_x, p_z) is different on frequencies. This effect is dependent upon the boundary conditions. From observation, the effect of gradient indexes on the frequency of SS and CC beam is the same, but the gradient index in the x-direction is more effective than the gradient index in the z-direction as shown in Table 4 and Table 5. In the CF beam, the effect of the gradient index in the z-direction is more than in the x-direction of the gradient index on frequency. Finally, the frequency is more in the CC beam and followed by SS and CF beams as shown in Table 2, Table 3, Table 4, Table 5, Table 6, and Table 7. It is found that the aspect ratio effect becomes very important because, the free vibration increases as the aspect ratio increases, refer to Fig. 4. Fig.5, and Fig. 6.

Fig. 4. Changes in dimensionless natural frequencies of SS beam at various aspect ratios along the x-direction gradient index and z-direction gradient index.

Fig. 5. Changes in dimensionless natural frequencies of CC beam at various aspect ratios along the x-direction gradient index and z-direction gradient index.

Table 2. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of SS bi-directional FGB, L/h=5

Table 3. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of SS bi-directional FGB, L/h=20

Table 4. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of CC bi-directional FGB, L/h=5.

Table 5. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of CC bi-directional FGB, L/h=20

Table 6. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of CF bi-directional FGB, L/h=5.

Table 7. Influence of gradient indexes and aspect ratio on dimensionless natural frequencies of CF bi-directional FGB, L/h=20.

Table 8. Influence of gradient indexes and porosity distribution on dimensionless natural frequencies
of SS bi-directional FGB at aspect ratio L/h=5

Table 9. Influence of gradient indexes and porosity distribution on dimensionless natural frequencies
 of CC bi-directional FGB at aspect ratio L/h=5

Table 10. Influence of gradient indexes and porosity distribution on dimensionless natural frequencies
 of CF bi-directional FGB at aspect ratio L/h=5

Fig. 12. Dimensionless natural frequencies of CF beam with uneven porosity at aspect ratio L/h=5

Figure 7 and Figure 8 show the dimensionless free vibration versus the gradient indexes in length (p_x) and thickness (_Pz) direction for different porosity distribution patterns when α = 0.1. It can be observed that the difference between different porosity distribution patterns can be increased with an increase in the porosity index (α). “Which indicates that the porosity distribution in thickness (z) direction has a greater influence on vibration than axial porosity (x direction) distribution.” On the other hand, free vibration is more in even porosity distribution than in uneven porosity distribution.

Figure 9, Figure 10, Figure 11, and Figure 12 illustrate the dimensionless free versus gradient indexes in length (P_x) and thickness (P_z) direction of the beam and the total volume fraction of porosity in different modes, respectively. It can be seen that the sensitivity of free vibration to the gradient index of the beam and total volume fraction of porosity has improved with an increase in the vibration mode number.

Table 8 shows the free vibration value of the SS beam which increases with an increase in porosity index, and decreases with an increase in gradation exponents in the x and z directions. Vibration value is more in even porosity distribution as compared with uneven porosity distribution.

Table 9 shows the free vibration value of the CC beam which increases with an increase in porosity index, and decreases with an increase in gradation exponents in the x and z directions. Vibration value is more in even porosity distribution as compared with uneven porosity distribution.

Table 10 shows the free vibration value of the CF beam which increases with an increase in porosity index, and decreases with an increase in gradation exponents in the x and z directions. Vibration value is more in even porosity distribution as compared with uneven porosity distribution.

It can be noted that free vibration increases with the porosity coefficient and this impact is more significant at a high porosity value, which is caused by a reduction in flexible rigidity of the FG beam at a high porosity value. In addition, it can be observed that the vibration decreases with an increase in gradient index, this is for even and uneven types of porosity. When the gradient index is zero (p=0) the beam is pure metal with less stiffness. As the gradient index (p) tends to infinity, a beam is a pure ceramic with high stiffness which means less vibration.

4.     Conclusions

Two directional FG porous beams were analyzed for the behavior of free vibration, subjected to various boundary conditions (SS, CC, and CF).

Considering these boundary conditions with different aspect ratios and gradation exponents in x and z directions. n^th-order shear deformation theory was adapted to determine free vibration with even and uneven porosity distribution. Based on power-law distribution, the effective properties of FG porous beams in two directions were determined.

The effect of boundary conditions, distribution of porosity, aspect ratios, and gradation exponents on free vibration analysis through several numerical illustrations was highlighted.

The computed results are compared to those with earlier investigations in terms of dimensionless free vibration. The calculated outcomes are found to have a very good correlation with earlier ones. Aspect ratios, gradient indexes, and boundary conditions' impact on 2D-FGBs' free vibration were explored.

The most significant findings w.r.t nonporous FGBs are listed below:

• The gradient indexes have a significant impact on the dimensionless free vibration of 2D-FGBs. However, the gradient index's impact in the x-direction is more profound than its impact in the z-direction.
• The shear deformation effect on the vibration of 2D-FGBs increases as the aspect ratio increases. CC 2D-FGB is found to be significantly more susceptible to the shear deformation effect than the other 2D-FGB models.
• CC beams experience the highest first free vibration, followed by SS and CF beams.
• By choosing appropriate gradient indexes, vibration behaviors of 2D-FGBs may be regulated to match the design requirements.
• The shear deformation impact is quite significant, especially for thick beams, and the proposed theory yields accurate findings and is effective in resolving vibration behaviors of 2D-FGBs.
• The suggested two-directionally porous beam model is used to examine the free vibration behavior of two-directional porous beams.
• The most significant findings w.r.t porous FGBs are listed below:
• As the volume percentage of porosity increases close to the middle surface, free vibration will increase for the same total volume fraction of porosity.
• The effect of porosity distribution in a thickness direction is more dominant than the effect of axial direction on free vibration.
• The “porosity parameter is a crucial parameter that must be considered in the design of modern structures and the percentage of porosity in structure can be affected considerably in its performance and response”. The proposed method shall also be useful to analyze the shear deformation of FGBs, where these FGB surfaces are subjected to high temperatures at one end and low temperatures at the other end.

Nomenclature

References

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