Numerical Investigation on Free Vibration Response of Bi-Directional Porous Functionally Graded Circular/Annular Plates

1. Vasara^a, S. Khare^a*, A. Malguri^b, R. Kumar^a

^a  Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Surat-395007, India

 ^b Oil and gas, HEPIPL, Adani Group, Ahmedabad – 380009, India

1.     Introduction

Lightweight structures are significant in aerospace, marine, nuclear, and civil engineering because new composite materials have a high strength-to-weight ratio. Lightweight isotropic and microscopically inhomogeneous functionally graded materials are made up of two or more phases with smooth material characteristics changing in specific directions. As a result, functionally graded materials can minimize inter-laminar stresses, delamination failure, and thermal deformation while also providing design ability to satisfy specific mechanical requirements.

Many authors have emphasized examining the free vibration of circular plates made up of functionally graded materials in the last decades, particularly in structural vibration. Ebrahimi and Rastgo [1] discussed the natural frequency behavior of thin functionally graded (FG) circular plates using the classical plate theory (CPT). Allahverdizadeh et al. [2] analyzed the axisymmetric vibration behavior of FG circular plate using a semi-analytical approach. Wirowski [3] studied the free vibration behavior of the FG annular plate based on the tolerance averaging technique using the Finite Difference Method (FDM).

Alipour et al. [4] studied the free vibration characteristics of the FG circular and annular plate resting on an elastic foundation with different boundary conditions. Further, The natural frequencies of thin circular FG plates resting on an elastic foundation were investigated using the differential transform method (DTM) by Shariyat and Alipour [5]. Hamzehkolaei et al. [6] used the DQM and concluded that it is an efficient and healthy numerical approach for the axisymmetric bending analysis of functionally graded circular/annular plates. Liu et al. [7] investigated the forced vibration of the FG cylindrical shell using the Pseudo-arclength continuation method. Also, Liu et al. [8] investigated the nonlinear dynamic response of porous FG sandwich shells resting on the Winkler-Pasternak foundation.

Chu et al. [9] introduced the Hermite radial basis collocation method (HRBCM) to determine the natural frequencies of inhomogeneous FGM plates. Lal and Ahlawat [10] proposed a semi-analytical solution for axisymmetric vibrations of circular FGM plates using the CPT.

Li et al. [11] derived a scaling factor for modal analysis of the isotropic thin FG plate and the homogeneous FG plates using classical plate theory. Yin et al. [12] determined vibration characteristics of FG Circular plates using an isogeometric approach based on Kirchoff’s plate theory. Lal and Ahlawat [13] presented the modal analysis of FG circular plate using CPT. Swaminathan et al. [14] reviewed various methods to investigate the different behavior of functionally graded plates. Merazi et al. [15] examined the static behavior of FG plates using the new hyperbolic shear deformation plate theory on neutral surface position. Arefi [16] studied the free vibration of FG circular and annular plates embedded with piezoelectric layers. Wu and Liu [17] utilized the state space-based differential reproducing kernel method (DRK) to analyze the free vibration behavior of FGMs circular/annular plates considering different boundary conditions. Żur [18] utilized the Quasi-Green function approach for investigating the free vibration behavior of the FG circular plates using CPT. Baltac [19] studied the natural frequency behavior of laminated annular/annular sector plates based on Love’s shell theory and the first-order shear deformation theory. The Weak form of finite annular prism methods utilized by Wu and Yu [20] to study the static/dynamic characteristics of FG circular plates. Arshid et al. [21] investigated the natural frequency behavior of the shear deformable elastic FG plate. Javani et al. [22] presented nonlinear frequencies of elastically founded FG circular plate using Differential Quadrature Method (DQM). Singh and Azam [23] determined the natural frequencies of FGM plates, which were resting on elastic foundations under a hygro-thermal environment.

Porosity plays a significantly important role in structural specifications, and taking it into account will result in each analysis being quite similar to the experimental tests. Because of their widespread use, researchers studied the effect of the porosity of porous materials on the mechanical behavior of various structural elements such as beams, plates, and shells. Kumar et al. [24] analyzed nonlinear central deflection and stress analysis of porous FGM plates using the multiquadric radial basic function meshfree method. Mouaici et al. [25] investigated the influence of porosity distribution on the free vibration response of FGMs rectangular plates using the shear deformation plate theory. Akbaş [26] determined the natural frequency of simply-supported porous FG plate using the first-order shear deformation (FSDT) theory. Kiran and Kattimani [27] studied the porosity effect on FGM plates’ free vibration characteristics using the finite element method (FEM). Barati and Zenkour [28] presented the thermo-electrical vibration behavior of piezoelectric FG plates using the refined shear deformation plate theory. Zhao et al. [29] determined the natural frequencies of thick porous FG plates with arbitrary boundary conditions on the periphery using the Fourier series method. Yousfi et al. [30] presented natural frequencies of FG porous plates using higher-order hyperbolic shear deformation plate theory. The higher-order shear deformation theory (HSDT) was utilized by Slimane [31] to investigate the natural frequencies of porous FG plates with simply supported boundary conditions. Demirhan and Taskin [32] presented a state-space-based numerical approach to analyze the effect of porosity on free vibration characteristics of FG plate using the four variable plate theory. The free vibration behavior of levy-type porous FG plates using the dynamic stiffness method was investigated by Ali and Azam [33]. Xue et al. [34] employed the isogeometric approach to determine the natural frequencies of porous FG square, rectangular and circular plates with simply supported boundary conditions using FSDT. Hadji et al. [35] predicted the influence of porosity on natural frequencies of the functionally graded rectangular plate using hyperbolic-shear deformation plate theory. Kumar et al. [36] evaluated the natural frequencies of porous FGMs rectangular plates using the inverse hyperbolic shear deformation theory. Solanki et al. [37] utilized the multi quadratic radial basic function meshfree method to investigate laminated plates’ linear and nonlinear flexure analysis. The effect of porosity on the natural frequencies of Sigmoid-FGM plates was investigated by Singh and Harsha [38] using Galerkin’s Vlasov Method. Balak et al. [39] determined the natural frequencies of elliptical porous sandwich plates based on CPT. Bansal et al. [40] presented Navier’s solution to determine the free vibration response of FG rectangular plate having geometric discontinuities. Van Vinh and Huy  [41] proposed a new hyperbolic shear deformation theory to investigate the Static and dynamic behavior of porous FG sandwich plates using FEM.

Arshid and Khorshidvand [42] explored the free vibration behavior of porous FG circular plates using CPT. Żur and Jankowski [43] studied the influence of uniform and non-uniform porosity on the free vibration behavior of FGM Circular plates applying the CPT. Arshid et al. [44] compared the various displacement field theories for the vibration analysis of saturated porous FGM plates. Żur and Jankowski [45] presented FG porous circular plates’ natural frequencies using CPT. Emdadi et al. [46] analyzed the natural frequency behavior of the circular sandwich plate using modified coupled stress theory and FSDT. Van Vinh and Tounsi [47] utilized nonlocal FSDT to investigate the natural frequencies of FG doubly curved nanoshells. Heshmati and Jalali [48] studied the free vibration response of FG porous circular/annular sandwich plate using the pseudospectral method. Amir et al. [49] investigated the free vibration behavior of FG saturated porous sandwich circular plate using modified coupled stress theory. Bennai et al. [50] presented the free vibration behavior of FG plates using higher-order hyperbolic shear deformation theory. Gao et al. [51] demonstrated a wave propagation of rectangular porous FG graphene platelets plates. Kumar and Singh [52] investigated buckling and free vibration response of the porous FGM plate using inverse hyperbolic higher-order shear deformation theory and governing equation solved by multi quadratic radial basic function meshfree method.  The 3D plate theory was employed by Kaddari et al. [53] to investigate the vibrational behavior of the composite structures. Safarpour et al. [54] studied natural frequencies analysis of Functionally graded reinforced composite porous circular and annular plates using DQM. Shojaeefard et al. [55] studied the free vibration characteristics of FG porous circular plate subjected to thermal load using FSDT using the DQM method.

Many studies have been published to demonstrate and evaluate the mechanical characteristics of FGM structures. However, when material properties are distributed in various directions, axial or nonaxial FGMs are inappropriate for aeronautical applications, medical components, and higher temperature sustainable materials. The bi-directional FGMs (BDFGMs) were designed to full fill the requirements of those above-mentioned applications.

Thom et al. [56] investigated the bending and buckling response of BDFGM plates using the finite element method based on HSDT. Non-uniform rational B-spline-based material modeling was introduced by Lieu et al. [57], [58] to analyze the free vibration and buckling behavior of BDFGM rectangular plates. Hong [59] investigated natural frequencies of BDFGMs rectangular plates using FEM. Vinh [60] analyzed the bi-directional functionally graded sandwich plates using FEM based on the HSDT. Van Vinh [61] investigated the free vibration behavior of the BDFG sandwich plate on Pasternak’s elastic foundation using hybrid quasi-3D theory. Many researchers have presented the free vibration response of composite materials with general boundary conditions [62] [63]. Qin et al. [64] compared a unified solution for the free vibration behavior of cylindrical shells considering arbitrary boundary conditions. using the Mindlin plate theory, Qin et al. [65] investigated the natural frequencies response of rotating cylindrical shells coupled with moderately thick annular plates. A general approach was provided by [66] to investigate the free vibration behavior of rotary FGCNT cylindrical shells considering arbitrary boundary conditions.  Qin et al. [67] presented a unified solution for vibration analysis of laminated FG shallow shells with general boundary conditions using FSDT. The present study considered CC and SS boundary conditions for the particular results.

Shariyat and Alipour [68] presented the semi-analytical solution for free vibration analysis of BDFGMs circular plates resting on an elastic foundation. Tahouneh [69] studied the free vibration characteristics of BDFGMs annular plates using DQM. Lal and Ahlawat [70] utilized the CPT to determine the free vibration response of BDFGMs circular plates. Mahinzare et al. [71] analyzed the natural frequencies for BDFG piezoelectric circular plates using DQM. Shojaeefard et al. [72] investigated the free vibration characteristics of BDFG circular microplates using DQM based on FSDT. Ahlawat [73] presented a free vibration response of BDFGMs circular plate using CLPT. Molla-Alipour et al. [74] proposed the analytical formulation for natural frequencies of BDFGM annular plates by applying the differential transform method.

Numerical solutions already play a crucial role in the advancement of engineering research due to improvements in computer technology. Various numerical approaches are utilized to evaluate the solution using numerical methods, and those are regarded as effective techniques to solve the ordinary and partial differential equations (PDEs). The finite difference method (FDM), the finite volume method (FVM),  the finite element method (FEM), and Galerkin and collocation methods which are types of methods of weighted residuals (MWR), are all examples of numerical simulation approaches. Among the methodologies mentioned above, FEM and FDM are the tools that are mostly used to solve the equations. Although the most commonly used method is FEM, stress concentrations, sharp changes, singularities, and large deformation can be challenging to assess using FEM. To minimize the constraints of conventional FEM, novel approaches like wavelet-based numerical methods, mesh-free methods, the discrete singular convolution (DSC) algorithm, and other comparable techniques were developed, even if this technique overcomes the constraints of the FEM methodology. Still, these methods take very long to approximate the solution.

The differential quadrature method (DQM) is developed to eliminate some of the drawbacks of the methods mentioned above and deal with complex geometry and boundary constraints. The extra features of DQM are easy to use and capable of producing highly accurate numerical results with minimal computational effort [75]. For vibration study of a transversely isotropic hollow toroid, semi-analytical DQM was used by Jiang and Redekop [76]. Ferreira et al. [77] used the DQM approach to examine the free vibration analysis of laminated composite rectangular plates. Alibeigloo and Alizadeh [78] also utilized DQM to investigate the free vibration response of sandwich FG rectangular plates. Arani et al. [79] studied the natural frequency behavior of porous FG plates using the DQM method based on HSDT. Nejati et al. [80] utilized DQM to investigate the vibration behavior of an FG laminated rectangular plate. To prepare the in-plane natural frequencies of elastically constrained FG rectangular plates, Ji et al. [81] utilized the DQM approach.

Using the harmonic differential quadrature method, Civalek and Ulker [82] examined the bending analysis of a circular plate. Mirtalaie and Hajabasi [83] determined the natural frequency behavior of FG annular-sector plates using the DQM. Mirtalaie et al. [84] determined the natural frequencies of annular/sector FG plate using DQM. The state-space-based DQM and artificial neural network (ANN) were compared by Jodaei et al. [85] for the free vibration response of the FG sector plate. Ahlawat and Lal [86] investigated the axisymmetric vibrations of circular functionally graded plates using the generalized DQM technique. The modal analysis of rotating annular discs was accomplished by Shahriari et al. [87] using generalized DQM. Mohammadimehr et al. [88] used DQM to investigate the natural frequencies of annular/ sector FGMs plates. Lal and Saini [89] employed DQM to determine the fundamental frequencies of FG circular plates considering the thermal environment. Behravan Rad [90] investigated static analysis for porous circular BDFGMs plates using DQM. Li et al. [91] investigated the structural behavior of porous BDFGMs rectangular plates using an isogeometric approach. However, to the best of the authors’ knowledge, no findings on the Bidirectional functionally graded porous circular/annular plate have been published. The porosity has a significant impact on the material’s properties and strength, and it is desirable to investigate the influence of porosity on the BDFGM circular/annular plate.

The present work mainly focused on the free vibration response of bi-directional porous FGM circular and annular plates. The DQM technique is used to solve the governing equations based on the FSDT. The natural frequencies of functionally graded (FG) porous circular plates and annular plates are determined, while uniform and non-uniform porosity distributions in the thickness direction are considered. The effects of porosity distributions, porosity volume fraction, material gradation index, radius to thickness ratio, radius ratio, and boundary conditions on the free vibration response of porous bi-directional functionally graded material plates are extensively explored.

2.     Mathematical Formulation

In this paper, consider BDFG porous circular/annular plates instead of unidirectional FG plates in which the material property varies in thickness and radial directions. The coordinate axis is taken at the center of the plate. The inner and outer radiuses of the plate are denoted by "b" and "a", respectively, and thickness is denoted by "h," as illustrated in Fig. 1.

FGMs are one type of composite material in which the mechanical properties vary continuously in the thickness direction. However, this study presents a structural analysis of FG porous circular/annular plate, whose mechanical properties vary continuously in radial and thickness directions, i.e., r and z coordinates. The Young's modulus, Poisson's ratio, and mass density of BDFGMs are expressed as:

Fig.1 Annular plate

Fig. 2 Porosity distribution in the plate (uniform and non-uniform porosity distribution))          

where the subscripts m and c correspond to metal and ceramic phases, V_i is the volume fraction of the internal phase, and  stand for porosity volume fraction. The volume fraction of metal and ceramic phases phase varies in a power-law which is given as [92],

where,

 Here, n₁ and n₂ are the grading indexes in thickness and radial direction. By using Eqs. (1)-(3) one can derive the effective materials properties (P) of the BDFG circular plate by considering two types of porosity distributions, as shown in Fig. 2 [27-29]. The material property of FGMs can be rewritten as below:

PD1: Uniform Porosity Distribution

PD2:Non-uniform porosity distribution

where, P stands for the property of the material in which young modulus (E), Poisson’s ratio (v), and density (ρ) are considered, P_c and P_m refer to ceramic and metal materials, respectively.

The effect of uniform and non-uniform porosity distribution on young modulus (E) in the thickness direction is represented in Figs. 3-4 respectively.

2.1. Governing equation of the BDFGM circular/annular plate

The displacement field for the given plate defines based on the FSDT theory. According to FSDT, displacement in r, θ, and z-direction is provided below by Akbari and Asanjarani [93]:

(a)

(b)

(c)

Fig. 3 (a-c) Variation of Young modulus through the thickness with different porosity volume fractions for uniform porosity distribution (n₁=1).

The above equation   denotes midplane displacement in the radial, circumferential and transverse directions, respectively, and   are considered as rotation in rz and θz planes, and variable “t” represents the time.

The following are the component of strain in the displacement field in terms of mid-surface deflection and curvature effect:

    (a)

(b)

(c)

Fig. 4 (a-c) Variation of Young modulus through the thickness with different porosity volume fractions for non-uniform porosity distribution (n₁=1).

The above Eq. (7)  are the normal strains in the respective directions and   are transverse strains in the respective directions.

Further terms of Eq. (7) are explored as below:

where,

, , . ,  , , , , and .

In the displacement field of the FGM plate, in-plane resultant forces ( , , ), in-plane resultant moment ( ), and shear forces ( ) are given by the equation below[94].

and

In Eqs. (9) and (10),  matrices stand for extensional stiffness matrix, the bending-extensional coupling stiffness matrix, and the bending stiffness matrix respectively. and k_s=5/6 refers to Shear Correction Factor [95].  matrices can be referred to Molla-Alipour et al. [74]:

       In above Eq. (11), Q_ij is the stiffness constant for the FGM plate. Element of Q_ij matrix is mentioned below:

In Eq. 12, the directions r, θ, and z  are represented as 1, 2,  and 3, respectively.

The governing equation of the FG circular/annular plate can be derived from Hamilton’s principle. An analytical form of Hamilton’s principle is stated below:

where,  denotes variation in strain energy and kinetic energy, respectively. Using Hamilton’s principle, five equations of motion in terms of the stress resultants can be written as below:

where I₁ and I₂ stand for mass moment of inertia which defines below:

2.2.             Boundary Condition and Method of Solution

Bellman and his colleagues [96] introduced the DQM in 1972. The DQM method is mostly used to solve ordinary and partial differential equations. The basic idea about DQM is based on the guess quadrature integration method. It may be effectively stated extending the guess quadrature to evaluate the higher-order derivatives of differentiable function gives rise to DQM. In simple words, derivatives of various order can be found by weighted linear sums of the function values at all grid points inside the domain. As discussed in the literature, the DQM is theoretically simple, and it produces very accurate solutions with less computational effort.

The solution domain is divided into several points r_i (i = 1, 2,..., N). The p^th-order derivative of transverse displacement with respect to r at point r_i (with M divisions in that r-direction) is approximated by [97]:

• is the p^th order derivative weighting coefficient of the function ϕ  at the i^th point in the space domain.
• is the function value at the i^th point in the space domain.

The weighting coefficients  can be calculated using the recurrence relations [97],

where  i = 1, 2, . . ., M,  k = 1, 2, . . ., M with  and , p = 1, 2, . . ..M – 1.

where i = 1, 2,. . ., M with p = 1, 2,. . .. M – 1

where  i= 1, 2,. . ., M ,   k = 1, 2,. . ., M    with .

The cosine rule is used to create grid point coordinates.

for m = 1, 2,. . ., M.

       The gridpoint distribution on the peripheral (theta) direction is obtained using the following relation for m= 1, 2,…, N.

The elastic boundary conditions are considered along the edge of the FGM plate are as follows:

The stiffness constants ( and j = 1, 2, 3, 4) for the respective displacements at the four boundaries (j = 1 for r = b, j = 2 for = 0, j = 3 for r = a and j = 4 for are shown in Fig. 5.

Using the formulation described above, one algebraic equation is obtained for each displacement component at each domain. These equations are derived from either an equilibrium equation or a boundary condition. The following eigenvalue problems were derived from the resulting set of algebraic equations.

Here, for example, the vector q_b contains the amplitudes for the displacement component. MATLAB software is used to calculate natural frequencies from the eigenvalue equation specified by Eq. (27).

3.     Numerical Results and Discussion

This work presents the free vibration responses of BDFG circular and annular plates with thickness “h”, outer radius “a”, and inner radius “b”. The circular/annular plates examined in this article are made of two distinct material sets. The material properties of these two sets of material for the BDFG circular/annular plates are reported in Table 1.

Fig. 5. Stiffness constants along the radial and circumferential edges

Table 1. Functionally Graded Material Property

The non-dimensional elastic foundation coefficients are defined as:

Thus, considering an annular plate with both edges clamped (CC),

and for an annular plate with both edges simply supported (SS),

3.1.             Convergence and validation study

In order to demonstrate the validation of the present DQ method, firstly, consider a homogeneous circular plate with clamped and simply supported boundary conditions. The circular plate made up of Material 1 with outer radius (a) = 1.0 m, radius to thickness ratio (a/h) = 100.  In Fig. 6., the numerical results are compared to analytical solutions [9], the Hermite radial basis collocation method (HRBCM) [9], and the isogeometric approach (IGA) [12] with respect to the number of nodes in the radial (M) and circumferential (N) directions. The normalized natural frequency is considered as where . It is observed that the results obtained by DQM are in good agreement at (MxN=14x12) with the analytical solutions, HRBCM and IGA.

Fig. 6. Convergence study on normalized natural frequencies of the isotropic circular plate for clamp and simply supported boundary condition

*a,*b: Chu et al. [9]; *c: Yin et al. [12]

Fig. 7. depicts the convergence and comparison studies for grading index in the thickness direction (n₁=0.2, Material 2) with Żur and Jankowski [45] considering clamp (CC) and simply supported (SS) boundary condition. The plate is made up of Material 2. The fundamental frequencies agree in the range of 2%, at (M x N =14x12).

 Tables 2 and 3 demonstrate the convergence results compared to the results of Yin et al. [12] for the first four modes normalized frequencies in the account of different gradient indexes in the thickness direction (n₁), considering both simply supported and clamped boundary condition. The result is obtained by taking the material property of the plate as material 1 with thickness to radius ratio (h/a)=0.01 and grading index in the radial direction (n₂)=0. It is observed that the results obtained by DQM are in good agreement at (MxN=14x12) with the Yin et al. [12] for clamped and simply-supported FGM circular plates.

3.2.             Parametric Study

After validating the present formulation, the results of certain parametric studies are conducted to discuss the effect of various porosity distributions (uniform and non-uniform distribution), porosity volume fraction, radius to thickness ratio (a/h), material gradation index (thickness direction (n₁), and radial direction(n₂)), radius ratio (b/a), and boundary conditions on normalized natural frequencies of porous bi-directional FG plates.

Fig. 7.  Convergence study on the normalized natural frequency of FGM circular plate in z-direction for clamp and simply-supported boundary condition

*p: Żur and Jankowski [45]

3.2.1.         Influence of porosity distribution

The influence of different porosity distributions, such as uniform (PD1) and non-uniform (PD2) porosity distribution, on normalized natural frequency parameters, is shown in Figs. 8-9 for both simply supported and clamped boundary conditions with Material 1. The material grading index in thickness direction (n₁) and in radial direction (n₂) are considered as (n₁,n₂)=(2,0), (0,2), (2,2), (5,0), (0,5), (5,5)and radius to thickness ratio is taken as (a/h)=10.The normalized natural frequency parameter is considered as,  where .

Figs. 8 (a-b) show the effect of porosity volume fraction (δ) on normalized natural frequencies for clamped porous circular plates with uniform (PD1) and non-uniform (PD2) porosity distribution, respectively. While Figs. 9 (a)-(b) show the effect of porosity volume fraction (δ) on normalized natural frequencies for simply supported porous circular plates with uniform (PD1) and non-uniform (PD2) porosity distribution, respectively.

It is observed that for uniform porosity distribution, the natural frequency decreases with an increase in porosity volume fraction, whereas for non-uniform porosity distribution, the normalized natural frequency increase with the increase in porosity volume fraction for both the clamped and simply supported boundary condition. It is also observed that natural frequency parameters in case of BDFG plates ((n₁,n₂)= (2,2), (5,5)) are less than unidirectional FG plates(n₁,n₂)=(2,0), (0,2), (5,0), (0,5) for both the clamped and simply supported boundary condition.

Table 2. Convergence and comparison study on the normalized natural frequency of simply supported FGM circular plate with various grading indexes (n₁) in the thickness direction

Table 3. Convergence and comparison study on the normalized natural frequency of clamped FGM circular plate with various grading indexes in the thickness direction

 Table 4. The normalized natural frequencies of the clamped porous FG circular and annular plate for different material grading indexes in the thickness direction

Table 5. The normalized natural frequencies of clamped porous FG circular and annular plate for different material grading indexes in the radial direction

Table 6. The normalized natural frequencies of the clamped BDFG circular and annular plate with grading  index in thickness and radial direction

Fig. 8a. Effect of porosity volume fraction on the normalized natural frequency of clamped (CC) BDFG plates with uniform porosity distribution

Fig. 8b. Effect of porosity volume fraction on the normalized natural frequency of clamped (CC) BDFG plates with non-uniform porosity distribution

3.2.2.         Influence of radius to thickness ratio

The effect of radius to thickness ratio (a/h) on normalized natural frequencies of BDFG porous circular plate with clamped and simply-supported boundary conditions are shown in Fig. 10(a-b). In the parametric study, values of the material grading index in thickness (n₁) and radial (n₂) direction are taken as (n₁,n₂) = (1,0), (0,1), (1,1) and the value of porosity volume fraction is taken as (δ) =0.2. The circular BDFG porous plate is made up of material 1.

           It is observed that fundamental frequencies increase with radius to thickness ratio (a/h), but no noticeable variation exceeds (a/h) =20. Again, fundamental frequencies for porous BDFG plates considering both boundary conditions,  are lower than unidirectional FGMs.

Fig. 9a. Effect of porosity volume fraction on the normalized natural frequency of simply supported (SS) BDFG plates with uniform porosity distribution

Fig. 9b. Effect of porosity volume fraction on the normalized natural frequency of simply-supported (SS) BDFG plates with non-uniform porosity distribution

Fig. 10a. Effect of radius to thickness ratio (a/h) on normalized natural frequencies of porous BDFG plates for clamped boundary condition

Fig. 10b. Effect of radius to thickness ratio on the normalized natural frequency of porous BDFG plates for simply supported boundary condition

3.2.3.         Influence of different material grading indexes

Table 4 shows the effect of different material grading indexes in the thickness direction(n₁) on normalized natural frequencies of clamped circular and annular plates considering uniform (PD1) and non-uniform (PD2) porosity distribution. Table 5 shows the effect of different material grading indexes in the radial direction(n₂) on normalized natural frequencies of clamped circular and annular plates considering uniform and non-uniform porosity. Table 6 shows the effect of different material grading indexes on normalized natural frequencies of the clamped BDFG circular and annular plate with grading index in thickness (n₁) and radial direction (n₂). The radius ratio are taken as (b/a) = 0, 0.1, 0.2, 0.3 and 0.5 and the value of radius to thickness ratio is taken as (a/h) = 5. The value of porosity volume fraction is taken as (δ) =0.1 with material 1 for Tables 4-6.  

It is seen from Tables 4-6 that normalized natural frequencies increases with the increase in radius ratio(b/a) and decrease with the increase in grading index for both the porosity distribution PD1 and PD2. Furthermore, the natural frequencies of BDFG circular/annular plates are lower than those of unidirectional FG circular/annular plates, and the effect of the grading index on normalised natural frequencies in the radial direction is stronger than the effect of the grading index on normalised natural frequencies in the thickness direction.

3.2.4.         Influence of radius ratio

Now the effect of radius ratio (b/a) on normalized natural frequencies of simply supported BDFG porous plate with uniform (PD1) and non-uniform (PD2) porosity distribution are examined and presented in Figs. 11. (a-b). Fig. 11(a) presents the effect of radius ratio on natural frequencies of simply supported BDFG porous plate with uniform (PD1) porosity distribution, whereas Fig. 11(b) presents the non-uniform (PD2) porosity distribution. the radius ratio are taken as (b/a) = 0, 0.1, 0.2, 0.3 and 0.5 and the value of radius to thickness ratio is taken as (a/h) = 5. The value of porosity volume fraction is taken as (δ) =0.1.  

It is observed that for both porosity distribution natural frequencies increase with increase in radius ratio and BDFG plates {(n₁, n₂) = (2, 2)} fundamental frequencies is lower than that of unidirectional plates {(n₁, n₂) =(2, 0) or (n₁, n₂) = (0, 2)}. It is observed that in unidirectional FG simply supported porous circular and annular plates, the normalized fundamental frequencies in the thickness direction are more dominant than that of normalized fundamental frequencies in a radial direction.

4.     Conclusions

This study used a differential quadrature method for the free vibration characteristics of bi-directional FGM porous circular/Annular plates. To investigate the effect of porosity on the frequency parameter, two porosity distribution models, uniform and non-uniform, were considered. The power-law method changes the grading index of material properties of a bi-directional FGM plate in the radial and thickness directions utilized to determine material property. The influence of material grading index, types of porosity distribution, porosity volume fraction,  aspect ratios, and boundary conditions over normalized frequency of bi-directional FGM porous pate have also been examined. The conclusions of this numerical analysis can be written as follows:

1. The present solution technique is simple, with a fast computational speed with good accuracy.

Fig. 11a. Effect of radius ratio on fundamental frequencies of simply supported BDFGs circular and annular plates with uniform porosity distribution

Fig. 11b. Effect of radius ratio on fundamental frequencies of simply supported BDFGs circular and annular plates with non-uniform porosity distribution

2. The natural frequency decrease with an increase in porosity volume fraction for uniform porosity distribution, whereas for non-uniform porosity distribution, the normalized natural frequency increase with the increase in porosity volume fraction for non-uniform porosity distribution.
3. The natural frequency increases with increasing (n₁ and n₂) of BDFG plates for both boundary conditions.
4. The normalized natural frequencies increase with the increase in radius ratio(b/a) and decrease with the increase in grading index for uniform and non-uniform porosity distribution PD1 and PD2.
5. The natural frequencies increase with the radius to thickness ratio (a/h), but no noticeable variation exceeds (a/h) =20.
6. The normalized natural frequencies increased with the increase in radius ratio(b/a) and decreased with an increase in grading index for both the porosity distribution PD1 and PD2.

The aforementioned free vibration response may be served as a benchmark for further investigating the BDFGM plates.

5.     References

[1]          Ebrahimi, F. and Rastgo, A., 2008. An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory. Thin-Walled Structures, 46(12), pp.1402–1408.

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