Document Type : Research Paper
Author
Department of Mathematics, Government Girls Degree College, Behat 247121, India
Abstract
Keywords
Main Subjects
Effect of Elastically Restrained Edges on Free Transverse Vibration of Functionally Graded Porous Rectangular Plate
Department of Mathematics, Government Girls Degree College, Behat 247121, India
KEYWORDS 

ABSTRACT 
Functionally graded; Porous rectangular; Restrained; Physical neutral surface; RayleighRitz. 
In this paper, the author studied free transverse vibration of a thin isotropic simplysupported functionally graded (FG) rectangular plate with porosity effect based on classical plate theory. The plate is considered to be elastically restrained against rotation. It is assumed that the material properties of the graded plate are porositydependent. An even porosity distribution is considered for analysis purposes. Due to the asymmetry of material in the thickness direction, the neutral surface is not the same as the geometrical midplane of the plate. The concept of the physical neutral surface of the FG plate along with classical plate theory is used to formulate the problem. Hence, the physical neutral surface is taken as the reference plane. The first three dimensionless frequencies of the plate are obtained using the RayleighRitz method. Boundary characteristic orthogonal polynomials (eigenfunctions), generated using the GramSchmidt process, are used in the RayleighRitz method. A parametric study shows that porosity and material distribution parameters have remarkable effects on the free vibration response of the plate. Results are compared with those of simplysupported FG plates. 
Functionally graded materials, generally made of ceramic and metal, are nonhomogeneous materials in which material properties vary continuously in appropriate directions. Free transverse vibration analysis of FG rectangular plates has gained attention of many researchers. A few papers about free vibration of FG plates have appeared in the literature and are summarized as follows: Dynamic response of initially stressed FG rectangular thin plates resting on elastic foundation has been studied by Yang and Shen [1]. Abrate [2] studied free vibrations, buckling, and static deflections of FG rectangular plates. He concluded that the natural frequencies of FG plates are proportional to those of homogeneous isotropic plates. Ferreira et. al. [3] computed natural frequencies of square FG plates employing the asymmetric collocation method. Zhao et. al. [4] presented the mechanical and thermal buckling analysis of FG rectangular plates using firstorder shear deformation theory and the elementfree kpRitz method. Talha and Singh [5] studied static and free vibration of FGM plates using higher order shear deformation theory in conjunction with FEM. Janghorban and Zareb [6]^{ }investigated the thermal effect on free vibration of FG arbitrary straightsided plates with circular and noncircular cutouts. Ghannadpour et. al. [7] used the finite strip method to analyse the buckling behaviour of FG rectangular plates under thermal loading. The plates were subjected to distributed impulsive loads. Thermal buckling of FG skew and trapezoidal plates has been investigated by Jaberzadeh et. al. [8] using the elementfree Galerkin method. Baferani et. al. [9] investigated free vibration of FG rectangular plates based on firstorder shear deformation theory. Chakraverty and Pradhan [1011] investigated free vibration of thin FG rectangular plates incorporating the effects of Winkler foundation and thermal environment using the RayleighRitz method. Pradhan and Chakraverty [12] dealt with static analysis of thin FG rectangular plates under mechanical load using the RayleighRitz method. Khorshidi and Bakhsheshy [13]^{ }investigated the vibration analysis of FG rectangular plates partially in contact with a fluid. Pham [14] developed an analytical solution to investigate the thermal buckling of imperfect rectangular plates with FG coating under uniform temperature rise. Atmane et. al. [15] studied thermal buckling of a simply supported sigmoid FG rectangular plate employing firstorder shear deformation theory. Lee et. al. [16] presented a thermal buckling analysis of FG rectangular plates based on the neutral surface of a structure. Kumar et. al. [17] investigated free vibration of thin FG rectangular plates using the dynamic stiffness method.
The materials having pores are termed as porous materials. The application of these materials in the aeronautical industry, energy absorbing systems, sound absorbers, insulating materials, heat exchangers, construction materials, and electromagnetic shielding has necessitated the study of different behaviors of structures made of porous materials in recent years [18]. A significant number of works dealing with the static, bending, vibration, and buckling problems of porous beams and plates are reviewed as follows: Theodorakopoulos and Beskos [19] studied flexural vibration of thin, rectangular, simplysupported, and fluidsaturated porous plates. Leclaire et. al. [20]^{ }presented a simple model of the transverse vibration of a thin rectangular porous plate saturated by a fluid. The vibration of a clamped rectangular porous plate using Galerkin’s variational method has been presented by Leclaire et. al. [21]. Vibration analysis of porous FG beams was presented by Wattanasakulpong and Ungbhakorn [22]. Razaei and Saidi [23] presented an exact solution for the vibration of rectangular porous plates using Reddy’s thirdorder shear deformation theory. Mojahedin et. al. [24] studied the buckling of FG circular porous plates using the energy method based on higherorder shear deformation theory. Chen et. al. [25] presented free and forced vibration of FG porous beams with different kinds of porosity distributions. Ebrahimi and Habibi [26] presented a finite element formulation for deflection and vibration of FG porous plates based on higherorder shear deformation theory. Mechab et. al. [2728]^{ }studied free vibration/probabilistic analysis of FG nanoplate with porosities resting on WinklerPasternak elastic foundation. Jahwari and Naguib [29]^{ }presented an analysis of FG viscoelastic porous structure with a higher order plate theory. Barati et. al. [30] used a refined fourvariable theory to study the buckling of FG piezoelectric porous plates resting on an elastic foundation. Barati and Zenkour [31] explored the electrothermomechanical vibrational behavior of FG piezoelectric plates with porosity using a refined fourvariable plate theory. Mouaici et. al. [32]^{ }used hyperbolic shear deformation theory to examine the eﬀect of porosity on the vibration of nonhomogeneous plates. Ebrahimi and Jafari [33] presented an analytical solution to study the buckling characteristics of porous magnetoelectroelastic FG plates. In a series of papers, Rezaei and coworkers [3440] studied the free vibration and buckling behavior of porous plates employing various plate models. Kamranfard et. al. [41] presented an analytical solution for vibration and buckling of porous annular sector plates under inplane uniform compressive loads. Şimşek and Aydın [42] used a modified couple stress theory to study the forced vibration of FG microplates with porosity effects. Akbas [43] dealt with free vibration and static bending of simply supported FG plates with porosity effect incorporating firstorder shear deformation theory. Barati and coworkers [4447]^{ }have presented vibration analysis of smart/nano FG porous plates using a refined fourvariable theory. Ali et. al. [48] studied free vibration of the embedded porous plate using higher order shear deformation theory. Wang and Zu [4951] studied free/forced vibration of FG rectangular porous plates with different complicating effects. Wang and Yang [52] investigated the nonlinear vibration of moving FG plates in contact with liquid and containing porosities. Electromechanical vibration analysis of FG piezoelectric porous plates in the translation state has been presented by Wang [53]. Kiran et. al. [54] studied the effect of porosity on the structural behavior of skew functionally graded magnetoelectroelastic plates. Free vibration analysis of saturated porous circular plates made of FG material integrated with a piezoelectric actuator is presented by Arshid and Khorshidvand [55] using the differential quadrature method. Arani et. al. [56]^{ }dealt with the dynamic analysis of rectangular porous plates resting on the Pasternak foundation using highorder shear deformation theory. Gupta and Talha [57] presented the influence of porosity on flexural and free vibration of FG plates in a thermal environment based on nonpolynomial higherorder shear and normal deformation theory. Zhao et. al. [58] studied free vibration of functionally graded porous rectangular plates by means of an improved Fourier series method considering three types of porosity distributions. Daikh and Zenkour [59] obtained a Navier solution of free vibration and mechanical buckling of porous functionally graded sandwich plates using higherorder shear deformation theory. Du et. al. [60] performed a free vibration analysis of rectangular plates with three types of porosity distributions using the RayleighRitz method based on firstorder shear deformation theory. Rjoub and Alshatnawi [61] predicted the natural frequencies of a simplysupported functionally graded porous cracked plate using the Artificial Neural Network technique. Bansal et. al. [62] provided Navier solution and FEMbased solution of vibration of porous functionally graded plates with geometric discontinuities and partial supports based on the refined exponential shear deformation theory. Tran et. al. [63] presented static and free vibration of functionally graded porous plates using an edgebased smoothed finite element method. Chai and Wang [64] investigated traveling wave vibration of spinning graphene platelets reinforced porous joined conicalcylindrical shells using the power series method.
An uptodate review of works pertaining to the application of the RayleighRitz method in vibration analysis of structural elements is given by Kumar [65] and Pablo et. al. [66]. In their two papers, Wang and coworkers [6768] used the RayleighRitz method to analyze the vibration of longitudinally moving plate submerged in an infinite liquid domain and that of FG cylindrical shells with porosities.
In some engineering problems, the boundary conditions along the edges of the plate are assumed to be either clamped or simply supported. But the actual boundary conditions tend to be in between these two limiting cases. To achieve these boundary conditions, analysis is done by modeling the edge conditions as a collection of elastic springs whose combined effect could vary from zero to infinity. Very few researchers (Laura and Grossi [69], Okan [70], Kumar [71], Zhang et. al. [72], He et. al. [73]) have considered the effect of elastically restrained edges against rotation on the vibration of plates.
The objective of this work is to study the free transverse vibration of a thin isotropic FG rectangular plate with even porosity distribution. The plate is simplysupported and elastically restrained against rotation along the edges. Material properties of the plate, continuously varying in the thickness direction, are assumed to be dependent on porosity. The RayleighRitz method incorporating boundary characteristic orthogonal polynomials as eigenfunctions is used to obtain the first three natural frequencies and mode shapes.
Let us consider a thin isotropic FG elastically restrained against a rotation rectangular plate made of porous material with length ‘a’ taken in the x direction, breadth ‘b’ in the y direction and thickness ‘h’ in z direction as shown in Fig. 1. The top surface (h/2) is ceramic rich while bottom surface (h/2) is metal rich.
The physical neutral surface does not coincide with the geometrical midplane of the plate. The distance between the geometrical midplane and the physical neutral surface is considered to be . There exists uniformly distributed (even) porosity in the plate. In this model, porosity spreads uniformly through the thickness direction.
The strains are defined as: 


(1) 
According to Hook’s law

(2) 
Using relation (1) in (2), we obtain

(3) 
The expression for the strain energy is

(4) 
Using relations (1) and (3) in (4), strain energy becomes

(5) 
Fig. 1. (i) FG porous elastically restrained rectangular plate with physical neutral surface and geometrical middle surface (ii) plate with even porosity (iii) crosssection of the plate
The kinetic energy of the plate is given as

(6) 
where is the displacement, is Young’s modulus, is the density, is the Poisson’s ratio, the subscript following a variable denotes differentiation of the variable w.r.to the subscript following it, and is the time.
The effective material properties viz. Young’s modulus and density are assumed to be graded in the thickness direction according to the power law (Wattanasakulpong and Ungbhakorn [22]) as follows:

(7) 

(8) 
where are young’s moduli of ceramic and metal; are densities of ceramic and metal; is the nonnegative volume fraction index which describes the material distribution across the thickness of the plate and is porosity volume fraction. The value of equal to 0 corresponds to the perfect FG plate. The plate becomes isotropic homogeneous if either (fully ceramic) or (fully metal).
For harmonic solution, the displacement is assumed to be

(9) 
where is the circular frequency, represents the maximum transverse displacement at the point and .
Using relations (7), (8), and (9), the expressions for maximum strain energy and kinetic energy of the plate become

(10) 
and

(11) 
where

(12) 

(13) 

(14) 

(15) 

The maximum strain energy (Warburton and S.L. Edney [74]) associated with the rotational restraints in the edges is given by

(16) 
where are the rotational spring constants.
Introducing the nondimensional variables together with

(17) 
We obtain the standard eigenvalue problem as follows:

(18) 
where is the order of approximation to get the desired accuracy, are orthonormal polynomials, are unknowns,

(19) 




, 



and is the frequency parameter.
The orthonormal polynomials are generated using the GramSchmidt process (Singh and Chakraverty [75]).
In this modal analysis, a plate made of functionally graded material (Aluminium/ Alumina, i.e., ) is considered. Here, the symbol is used for and represents The following values of material coefficients (Talha and Singh [5])^{ }for the FG plate and other parameters are taken:





We also assume that the Poisson’s ratio remains constant along the thickness direction as the plate is considered to be thin. The first three values of the frequency parameter have been calculated from the standard eigenvalue problem given by (18). For this purpose, a computer program has been developed by the author in C++. Table 1 shows the convergence of frequency parameter with increasing . To achieve an accuracy of four decimal places, the value of has been fixed as 26. A comparison of frequencies of simplysupported isotropic FG rectangular/square plates is shown in Table 2. The results are in good agreement with those available in the literature. The results have been reported in Tables (35) and Figs. (27).
Table 1. Convergence of first three values of frequency parameter Ω of FG porous plate with increasing value of





Mode 
10 
15 
20 
24 
25 
26 


0.3 
I 
89.7801 
10.2914 
10.2914 
10.2914 
10.2914 
10.2914 
II 
179.1420 
179.1420 
21.2307 
21.2264 
21.2264 
21.2264 

III 
913.5480 
179.1420 
21.2390 
21.2264 
21.2264 
21.2264 



0.3 
I 
8923.51 
10.2931 
10.2931 
10.2931 
10.2931 
10.2931 
II 
17796.3 
17796.3 
21.2429 
21.2429 
21.2429 
21.2429 

III 
91336.8 
17796.3 
21.2430 
21.2429 
21.2429 
21.2429 



0.1 
I 
16078 
64.7397 
64.7397 
64.7397 
64.6713 
64.6713 
II 
30095.9 
29615.5 
83.7024 
83.7024 
83.7024 
83.7024 

III 
107816 
30095.9 
171.346 
171.3450 
119.9290 
119.9290 



0.2 
I 
29.5262 
29.5218 
29.5160 
29.5160 
29.5160 
29.5160 
II 
47.3251 
47.3251 
47.2301 
47.2262 
47.2262 
47.2262 

III 
99.1221 
77.5665 
77.5665 
76.8723 
76.7577 
76.7577 



0.2 
I 
15.2531 
15.0241 
15.0241 
15.0238 
15.0238 
15.0238 
II 
32.9505 
32.9504 
28.4808 
28.4808 
28.4808 
28.4808 

III 
36.5462 
36.5462 
35.9856 
35.9856 
35.9856 
35.9856 
Table 2. Comparison of frequency parameter Ω of simplysupported FG plate


Reference 
Mode I 
Mode II 
Mode III 
0 
0.4 
Leissa [76] 
11.4487 
16.1862 
24.0818 


Present 
11.4487 
16.1863 
24.2984 

1.0 
Leissa [76] 
19.7392 
49.3480 
49.3480 


Present 
19.7392 
49.3490 
49.3490 

1.5 
Leissa [76] 
32.0762 
61.6850 
98.6960 


Present 
32.0762 
61.6860 
98.6982 
0.2 
1.0 
Kumar et. al. [17] 
18.3177 
45.7942 
45.7942 


Present 
18.3177 
45.7951 
45.7951 
1.0 
0.5 
Kumar et. al. [17] 
9.4131 
15.0610 
24.4741 


Present 
9.4131 
15.0612 
24.7371 

1.0 
Kumar et. al. [17] 
15.0610 
37.6524 
 


Present 
15.0610 
37.6531 
 

2.0 
Kumar et. al. [17] 
37.652 
60.244 
97.896 


Present 
37.6524 
60.2447 
97.9168 
5.0 
1.0 
Kumar et. al. [17] 
12.9831 
32.4578 
32.4578 


Present 
12.9831 
32.4584 
32.4584 
Table 3. First three values of frequency parameter Ω of FG porous plate for





0 
1 
2 
3 
4 
5 

Mode 


0.0 
I 
20.3576 
16.7595 
15.5579 
15.1744 
15.0228 
14.9330 
II 
28.1742 
22.5141 
20.7440 
20.1829 
19.9590 
19.8262 

III 
45.3146 
37.7248 
35.6836 
35.1296 
34.9489 
34.8525 

0.1 
I 
20.8079 
16.6007 
14.9632 
14.3858 
14.1615 
14.0502 
II 
28.7066 
22.1958 
19.8414 
19.0234 
18.7051 
18.5464 

III 
46.2599 
37.7577 
35.1488 
34.3706 
34.1258 
34.0306 

0.2 
I 
21.3262 
16.2883 
13.8835 
12.8726 
12.4421 
12.2443 
II 
29.3263 
21.6684 
18.2965 
16.9092 
16.3217 
16.0518 

III 
47.3702 
37.6793 
34.1196 
32.8494 
32.3928 
31.9499 

0.3 
I 
21.9326 
15.6980 
11.6935 
9.3033 
7.8542 
6.9551 
II 
30.0596 
20.7689 
15.3005 
12.1211 
10.2133 
9.0352 

III 
48.6954 
37.3800 
30.5239 
24.3207 
20.5589 
18.2235 




0.0 
I 
34.6795 
26.9424 
24.5964 
23.8533 
23.5532 
23.3746 
II 
70.9287 
55.1290 
50.3468 
48.8330 
48.2222 
47.8587 

III 
70.9287 
55.1290 
50.3468 
48.8330 
48.2222 
47.8587 

0.1 
I 
35.2494 
26.4107 
23.3518 
22.3010 
21.8902 
21.6839 
II 
72.0935 
54.0525 
47.8166 
45.6753 
44.8387 
44.4189 

III 
72.0935 
54.0525 
47.8166 
45.6753 
44.8387 
44.4189 

0.2 
I 
35.9161 
25.6153 
21.3390 
19.6197 
18.8964 
18.5636 
II 
73.4572 
52.4395 
43.7197 
40.2136 
38.7386 
38.0605 

III 
73.4572 
52.4395 
43.7197 
40.2136 
38.7386 
38.0605 

0.3 
I 
36.7098 
24.3647 
17.6371 
13.8647 
11.6389 
10.2761 
II 
75.0818 
49.8999 
36.1736 
28.4688 
23.9184 
21.1299 

III 
75.0818 
49.8999 
36.1736 
28.4688 
23.9184 
21.1299 




0.0 
I 
81.4036 
67.0071 
62.2002 
60.6664 
60.0599 
59.7007 
II 
112.6970 
90.0566 
82.9760 
80.7316 
79.8361 
79.3049 

III 
167.1160 
130.9970 
120.0990 
116.6610 
115.2840 
114.4660 

0.1 
I 
83.2031 
66.3703 
59.8209 
57.5116 
56.6146 
56.1699 
II 
114.8260 
88.7832 
79.3657 
76.0936 
74.8202 
74.1856 

III 
169.9390 
128.7410 
114.4610 
109.5510 
107.6400 
106.6850 

0.2 
I 
85.2746 
65.1195 
55.5028 
51.4611 
49.7399 
48.9492 
II 
117.3050 
86.6735 
73.1861 
67.6369 
65.2870 
64.2072 

III 
173.2520 
125.2630 
105.1450 
96.9921 
93.5559 
91.9784 

0.3 
I 
87.6980 
62.7578 
46.7478 
37.1944 
31.4035 
27.8106 
II 
120.2380 
83.0756 
61.2021 
48.4844 
40.8531 
36.1407 

III 
177.2070 
119.6390 
87.5955 
69.2985 
58.3839 
51.6567 
Table 4. First three values of frequency parameter Ω of FG porous plate for





0 
1 
2 
3 
4 
5 

Mode 


0.0 
I 
24.6096 
18.7771 
17.0717 
16.5339 
16.3162 
16.1866 
II 
31.8669 
24.3144 
22.1060 
21.4097 
21.1278 
20.9599 

III 
65.2334 
49.7732 
45.2527 
43.8272 
43.2501 
42.9065 

0.1 
I 
24.9625 
18.3586 
16.1601 
15.4106 
15.1175 
14.9701 
II 
32.3237 
23.7724 
20.9256 
19.9551 
19.5756 
19.3847 

III 
66.1687 
48.6639 
42.8363 
40.8496 
40.0728 
39.6820 

0.2 
I 
25.3817 
17.7570 
14.7195 
13.5109 
13.0037 
12.7705 
II 
32.8666 
22.9934 
19.0602 
17.4952 
16.8385 
16.5364 

III 
67.2800 
47.0693 
39.0177 
35.8139 
34.4697 
33.8514 

0.3 
I 
25.8882 
16.8410 
12.1213 
9.5079 
7.9736 
7.0364 
II 
33.5225 
21.8074 
15.6958 
12.3118 
10.3250 
9.1114 

III 
68.6226 
44.6413 
32.1304 
25.2031 
21.1361 
18.6517 




0.0 
I 
36.0000 
27.4679 
24.9732 
24.1865 
23.8680 
23.6784 
II 
74.2966 
56.6881 
51.5395 
49.9160 
49.2587 
48.8673 

III 
74.2966 
56.6881 
51.5395 
49.9160 
49.2587 
48.8673 

0.1 
I 
36.5161 
26.8557 
23.6396 
22.5432 
22.1145 
21.8989 
II 
75.3618 
55.4247 
48.7874 
46.5246 
45.6399 
45.1948 

III 
75.3618 
55.4247 
48.7874 
46.5246 
45.6399 
45.1948 

0.2 
I 
37.1294 
25.9757 
21.5323 
19.7642 
19.0224 
18.6812 
II 
76.6274 
53.6085 
44.4383 
40.7894 
39.2584 
38.5541 

III 
76.6274 
53.6085 
44.4383 
40.7894 
39.2584 
38.5541 

0.3 
I 
37.8703 
24.6357 
17.7314 
13.9085 
11.6641 
10.2931 
II 
78.1565 
50.8431 
36.5941 
28.7044 
24.0724 
21.2429 

III 
78.1565 
50.8431 
36.5941 
28.7044 
24.0724 
21.2429 




0.0 
I 
98.4856 
75.1444 
68.3194 
66.1674 
65.2961 
64.7773 
II 
127.4670 
97.2574 
88.4241 
85.6387 
84.5110 
83.8396 

III 
182.6360 
139.3510 
126.6940 
122.7030 
121.0870 
120.1250 

0.1 
I 
99.8976 
73.4696 
64.6713 
61.6719 
60.4991 
59.9091 
II 
129.2950 
95.0897 
83.7024 
79.8203 
78.3024 
77.5388 

III 
185.2550 
136.2450 
119.9290 
114.3660 
112.1910 
111.0970 

0.2 
I 
101.5750 
71.0620 
58.9063 
54.0694 
52.0399 
51.1064 
II 
131.4660 
91.9737 
76.2409 
69.9806 
67.3539 
66.1457 

III 
188.3660 
131.7800 
109.2380 
100.2680 
96.5038 
94.7726 

0.3 
I 
103.6020 
67.3964 
48.5082 
38.0498 
31.9098 
28.1590 
II 
134.0900 
87.2294 
62.7829 
49.2469 
41.3000 
36.4455 

III 
192.1250 
124.9820 
89.9542 
70.5587 
59.1716 
52.2155 
Table 5. Proportionality factor for simplysupported FG porous plate
Mode 







Proportionality factor 
% change 
I 
0.5 
0 
0 
12.3370 
0 
0.1 
12.5139 
1.0143 
1.4 
II 



19.7395 


20.0225 
1.0143 
1.4 
III 



32.4210 


32.8858 
1.0143 
1.4 
I 




1 

9.2033 
0.7460 
25.4 
II 






14.7255 
0.7460 
25.4 
III 






24.1858 
0.7460 
25.4 
I 




2 

8.1012 
0.6567 
34.3 
II 






12.9620 
0.6567 
34.3 
III 






21.2894 
0.6567 
34.3 
I 




5 

7.5046 
0.6083 
39.2 
II 






12.0076 
0.6083 
39.2 
III 






19.7217 
0.6083 
39.2 
I 
1.0 


19.7392 
0 
0.1 
20.0222 
1.0143 
1.4 
II 



49.3480 


50.0565 
1.0143 
1.4 
III 



49.3480 


50.0565 
1.0143 
1.4 
I 




1 

14.7253 
0.7460 
25.4 
II 






36.8139 
0.7460 
25.4 
III 






36.8139 
0.7460 
25.4 
I 




2 

12.9619 
0.6567 
34.3 
II 






32.4053 
0.6567 
34.3 
III 






32.4053 
0.6567 
34.3 
I 




5 

12.0074 
0.6083 
39.2 
II 






30.0191 
0.6083 
39.2 
III 






30.0191 
0.6083 
39.2 
I 
2.0 


49.3480 
0 
0.1 
50.0556 
1.0143 
1.4 
II 



78.9579 


80.0899 
1.0143 
1.4 
III 



128.3320 


130.1720 
1.0143 
1.4 
I 




1 

36.8132 
0.7460 
25.4 
II 






58.9019 
0.7460 
25.4 
III 






95.7344 
0.7460 
25.4 
I 




2 

32.4047 
0.6567 
34.3 
II 






51.8482 
0.6567 
34.3 
III 






84.2698 
0.6567 
34.3 
I 




5 

30.0185 
0.6083 
39.2 
II 






48.0302 
0.6083 
39.2 
III 






78.0645 
0.6083 
39.2 
Figure 2 shows the effect of the volume fraction index on the first three values of the frequency parameter for .
It is observed that frequency decreases with increasing value of . This is due to the fact that a higher value of introduces more metal components and reduces the stiffness of the plate, i.e., elasticity modulus and bending rigidity. The variation of nondimensional frequency with porosity volume fraction for is shown in Fig. 3.
The frequency decreases with increasing value of . With the increase in , the strength of the material decreases. It is concluded that even porosity distribution lowers the natural frequency.
The effect of aspect ratio on frequency for is shown in Fig. 4 and it is observed that frequency increases with increasing value of .
Fig. 5 demonstrates the frequencyresponse variation with for The frequency first increases and then becomes constant.
The variation of , i.e., the distance between the physical neutral surface and the geometrical midplane with porosity volume fraction k for different values of volume fraction index is shown in Fig. 6.
The value of increases with the increase in k. It is also observed that the value of for remains lower than that for up to k=0.09 and remains higher for k >0.09.

Fig. 2. Frequency parameter of isotropic FG porous plate for : First mode , 

Fig. 3. Frequency parameter of isotropic FG porous plate for : First mode , 

Fig. 4. Frequency parameter of isotropic FG porous plate for : First mode , 

Fig. 5 Frequency parameter of isotropic FG porous plate : First mode , Second mode , Third mode 

Fig. 6 Variation of with volume fraction index g for different values of porosity volume fraction k; 
The value of for remains lower than that for up to and remains higher for k>0.16. Variation of proportionality factor with is shown in Fig. 7.
The first three mode shapes for FG porous square plate are shown in Fig. 8.

Fig. 7 Variation of proportionality factor with for simplysupported FG porous plate without restraint edges for k=0.1 (□), k=0.2 (∆), k=0.3 (O) 

Fig. 8 First three mode shapes for FG porous square plate for 
Free transverse vibration of a thin isotropic FG rectangular porous plate is studied here. A simplysupported plate having all the edges elastically restrained against rotation is considered. Material properties are assumed to be graded in the thickness direction and are dependent on even porosity distribution. The first three frequencies are obtained using the RayleighRitz method and boundary characteristic orthogonal polynomials. The effects of volume fraction index, porosity volume index, aspect ratio, and restraint parameters are studied on the frequencies. It is concluded that
The results presented here may serve as a benchmark for further studies dealing with FG porous rectangular plates with elastically restrained edges.
Nomenclature

Length of the plate 

Breadth of the plate 

Volume fraction index 

Thickness of the plate 

Porosity volume fraction 

Time 

Displacement 

Order of approximation 

Strain and kinetic energies of the plate 

Young’s modulus 

Young’s modulus of metal 

Young’s modulus of ceramic 

Rotational spring constants 

Maximum strain and kinetic energies of the plate 

Maximum strain energy associated to the rotational restraints 

Maximum transverse displacement 

Density of the plate material 

Poisson’s ratio 

Frequency parameter 

Aspect ratio 

Circular frequency 

Density of metal 

Density of ceramic 

Distance between the geometrical midplane and the physical neutral surface 

Normal and shear strains 

Normal and shear stresses 

j^{th} Orthogonal polynomial 

j^{th} Orthonormal polynomial 

The Kronecker delta 
Acknowledgments
The author is thankful to the learned reviewers for their constructive comments.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript.
References
[1] Yang, J., Shen, H.S., 2001. Dynamic response of initially stressed functionally graded rectangular thin plates. Composite Structures, 54, pp.497508.
[2] Abrate, S., 2006. Free vibration, buckling, and static deflections of functionally graded plates. Composite Science and Technology, 66, pp.23832394.
[3] Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian L.F., Jorge, R.M.N., 2006, Natural frequencies of functionally graded plates by a meshless method. Composite Structures, 75, pp.593600.
[4] Zhao, X., Lee, Y.Y., Liew, K.M., 2009. Mechanical and thermal buckling analysis of functionally graded plates. Composite Structures, 90, pp.161171.
[5] Talha, M., Singh, B.N., 2010. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied Mathematical Modelling, 34, pp.39914011.
[6] Janghorban, M., Zareb, A., 2011. Thermal eﬀect on free vibration analysis of functionally graded arbitrary straightsided plates with diﬀerent cutouts. Latin American Journal of Solids and Structures, 8, pp.245257.
[7] Ghannadpour, S.A.M., Ovesy, H.R., Nassirnia, M., 2012. Buckling analysis of functionally graded plates under thermal loadings using finite strip method. Computers and Structures, 108109, pp.9399.
[8] Jaberzadeh, E., Azhari, M., Boroomand, B., 2013. Thermal buckling of functionally graded skew and trapezoidal plates with different boundary conditions using the elementfree Galerkin method. European Journal of Mechanics A/Solids, 42, pp.1826.
[9] Hasani Baferani, A., Saidi A.R., Ehtesham, H., 2013. On free vibration of functionally graded Mindlin plate and effect of in plane displacements. Journal of Mechanics, 29(2), pp.373 384.
[10] Chakraverty, S., Pradhan, K.K., 2014. Free vibration of functionally graded thin rectangular plates resting on Winkler foundation with general boundary conditions using RayleighRitz method. International Journal of Applied Mechanics, 6(4), DOI: 10.1142/S1758825114500434.
[11] Chakraverty, S., Pradhan, K.K., 2014. Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions. Aerospace Science and Technology, 36, pp.132156.
[12] Pradhan, K.K., Chakraverty, S., 2015. Static analysis of functionally graded thin rectangular plates with various boundary conditions. Archive of Civil and Mechanical Engineering, 15(3), pp.721734.
[13] Khorshidi, K., Bakhsheshy, A., 2015. Free vibration analysis of a functionally graded rectangular plate in contact with a bounded ﬂuid. Acta Mechanica, 226(10), pp.34013423.
[14] Pham, T.T., 2016. Analytical solution for thermal buckling analysis of rectangular plates with functionally graded coatings. Aerospace Science and Technology, 55, pp.465473.
[15] Atmane, H.A., Bedia, E.A.A., Bouazza, M., Tounsi, A., Fekrar, A., 2016. On the thermal buckling of simply supported rectangular plates made of a sigmoid functionally graded Al/Al2O3 based material. Mechanics of Solids, 51(2), pp.177187.
[16] YoungHoon Li, SeokIn, B., JiHwan, K., 2016. Thermal buckling behavior of functionally graded plates based on neutral surface. Composite Structures, 137, pp.208214.
[17] Kumar, S., Ranjan, V., Jan, P., 2018. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Composite Structures, 197, pp.3953.
[18] Porous Materials,
http://www.uio.no/studier/emner/matnat /kjemi/KJM5100/h06/undervisningsmateriale/16KJM5100 2006 porous e.pdf.
[19] Theodorakopoulos, D.D., Beskos, D.E., 1994. Flexural vibration of poroelastic plates. Acta Mechanica, 103, pp.191203.
[20] Leclaire, P., Cummings, A., Horoshenkov, K.V., 2001. Transverse vibration of a thin rectangular porous plate saturated by a fluid. Journal of Sound and Vibration, 247, pp.118.
[21] Leclaire, P., Horoshenkov, K.V., Swift, M.J., Hothersall, D.C., 2001. The vibrational response of a clamped rectangular porous plate. Journal of Sound and Vibration, 247, pp.1931.
[22] Wattanasakulpong, N., Ungbhakorn, V., 2014. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities. Aerospace Science and Technology, 32(1), pp.111120.
[23] Rezaei, A., Saidi, A., 2015. Exact solution for free vibration of thick rectangular plates made of porous materials. Composite Structures, 134, pp.10511060.
[24] Mojahedin, A., Jabbari, M., Khorshidvand, A., Eslami, M., 2016. Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory. ThinWalled Structures, 99, pp.8390.
[25] Chen, D., Yang, J., Kitipornchai, S., 2016. Free and forced vibrations of shear deformable functionally graded porous beams. International Journal of Mechanical Sciences, 108, pp.1422.
[26] Ebrahimi, F., Habibi, S., 2016. Deﬂection and vibration analysis of higherorder shear deformable compositionally graded porous plate. Steel and Composite Structures, 20(1), pp.205225.
[27] Mechab, I., Mechab, B., Benaissa, S., Serier, B., Bouiadjra, B.B., 2016. Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on twovariable refined plate theories. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(8), pp.21932211.
[28] Mechab, B., Mechab, I., Benaissa, S., Ameri, M., Serier, B., 2016. Probabilistic analysis of effect of the porosities in functionally graded material nanoplate resting on WinklerPasternak elastic foundations. Applied Mathematical Modelling, 40(2), pp.738749.
[29] Jahwari, A.F., Naguib, H.E., 2016. Analysis and homogenization of functionally graded viscoelastic porous structures with a higher order plate theory and statistical based model of cellular distribution. Applied Mathematical Modelling, 40(3), pp.21902205.
[30] Barati, M.R., Sadr, M.H., Zenkour, A.M., 2016. Buckling analysis of higher order graded smart piezoelectric plates with porosities resting on elastic foundation. International Journal of Mechanical Sciences, 117, pp.307320.
[31] Barati, M. R., Zenkour, A. M., 2016. Electrothermoelastic vibration of plates made of porous functionally graded piezoelectric materials under various boundary conditions. Journal of Vibration and Control, 24(10), pp.19101926.
[32] Mouaici, F., Benyoucef, S., Atmane, H.A., Tounsi, A., 2016. Effect of porosity on vibrational characteristics of nonhomogeneous plates using hyperbolic shear deformation theory. Wind Structures, 22, pp.429454.
[33] Ebrahimi, F., Jafari, A., 2016. Buckling behavior of smart EEEFG porous plate with various boundary conditions based on refined theory. Advances in Material Research, 5(4), pp.279298.
[34] Rezaei, A.S., Saidi, A.R., 2015. Exact solution for free vibration of thick rectangular plates made of porous materials. Composite Structures, 134, pp.10511060.
[35] Rezaei, A.S., Saidi, A.R., 2016. Application of Carrera uniﬁed formulation to study the eﬀect of porosity on natural frequencies of thick porouscellular plates. Composites Part B: Engineering, 91, pp.361370.
[36] Rezaei, A.S., Saidi, A.R., Abrishamdari, M., Mohammadi, M.H.P., 2017. Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: An analytical approach. ThinWalled Structures, 120, pp.366377.
[37] Rezaei, A.S., Saidi, A.R., 2017. On the eﬀect of coupled solidﬂuid deformation on natural frequencies of ﬂuid saturated porous plates. European Journal of Mechanics – A/Solids, 63, pp.99109.
[38] Rezaei, A.S., Saidi, A.R., 2017. Buckling response of moderately thick ﬂuidinﬁltrated porous annular sector plates. Acta Mechanica, 228(11), pp.39293945.
[39] Rezaei, A.S., Saidi, A.R., 2017. An analytical study on the free vibration of moderately thick ﬂuidinﬁltrated porous annular sector plates. Journal of Vibration and Control, 24(18), pp.41304144.
[40] Askari, M., Saidi, A.R., Rezaei, A.S., 2017. On natural frequencies of Levytype thick porous cellular plates surrounded by piezoelectric layers. Composite Structures, 179, pp.340354.
[41] Kamranfard, M.R., Saidi, A.R., Naderi, A., 2017. Analytical solution for vibration and buckling of annular sectorial porous plates under inplane uniform compressive loading. Proceedings of the Institution of Mechanical Engineering Part C: Journal of Mechanical Engineering Sciences, 232(12), pp.22112228.
[42] Şimşek, M., Aydın, M., 2017. Sizedependent forced vibration of an imperfect functionally graded (FG) microplate with porosities subjected to a moving load using the modified couple stress theory. Composite Structures, 160, pp.408421.
[43] Akbaş, Ş.D., 2017. Vibration and static analysis of functionally graded porous plates. Journal of Applied and Computational Mechanics, 3(3), pp.199207.
[44] Ebrahimi, F., Jafari, A., Barati, M.R., 2017. Vibration analysis of magnetoelectroelastic heterogeneous porous material plates resting on elastic foundations. ThinWalled Structures, 119, pp.3346.
[45] Shahverdi, H., Barati, M.R., 2017. Vibration analysis of porous functionally graded nanoplates. International Journal of Engineering Sciences, 120, pp.8299.
[46] Barati, M.R., Shahverdi, H., Zenkour, A.M., 2017. Electromechanical vibration of smart piezoelectric FG plates with porosities according to a refined fourvariable theory. Mechanics of Advanced Materials and Structures, 24(12), pp.987998.
[47] Ebrahimi, F., Jafari, A., Barati, M.R., 2017, Free vibration analysis of smart porous plates subjected to various physical fields considering neutral surface position. Arabian Journal for Science and Engineering, 42(5), pp.18651881.
[48] Ali, G.A., Zahra, K.M., Mehdi, K., Iman, A., 2017. Free vibration of embedded porous plate using thirdorder shear deformation and poroelasticity theories. Journal of Engineering, pp.113, DOI: 10.1155/2017/1474916.
[49] Wang, Y.Q., Zu, J.W., 2017. Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerospace Science and Technology, 69, pp.550562.
[50] Wang, Y.Q., Zu, J.W., 2017. Largeamplitude vibration of sigmoid functionally graded thin plates with porosities. ThinWalled Structures, 119, pp.911924.
[51] Wang, Y.Q., Zu, J.W., 2017. Porositydependent nonlinear forced vibration analysis of functionally graded piezoelectric smart material plates. Smart Materials and Structures, 26(10), pp.105014.
[52] Wang, Y.Q., Yang, Z., 2017. Nonlinear vibrations of moving functionally graded plates containing porosities and contacting with liquid: internal resonance. Nonlinear Dynamics, 90(2), pp.14611480.
[53] Wang, Y.Q., 2018. Electromechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica, 143, pp.263271.
[54] Kiran, M.C., Kattimani, S.C., Vinyas, M., 2018. Porosity influence on structural behavior of skew functionally graded magnetoelectroelastic plate. Composite Structures, 191, pp.3677.
[55] Arshid, E., Khorshidvand, A. R., 2018. Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via diﬀerential quadrature method. ThinWalled Structures, 125, pp.220233.
[56] Arani, A.G., Khani, M., Maraghi, Z.K., 2018. Dynamic analysis of a rectangular porous plate resting on an elastic foundation using highorder shear deformation theory. Journal of Vibration and Control, 24, pp.36983713.
[57] Gupta, A., Talha, M., 2018. Influence of porosity on the flexural and free vibration responses of functionally graded plates in thermal environment. International Journal of Structural Stability and Dynamics, 18 (1), DOI:10.1142/S021945541850013X.
[58] Zhao, J., Wang, Q., Deng, X., Choe, K., Zhong, R., Shuai, C., 2019. Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions. Composites Part B: Engineering, 168, pp.106120.
[59] Daikh, A.A., Zenkour, A. M., 2019. Free vibration and buckling of porous powerlaw and sigmoid functionally graded sandwich plates using a simple higherorder shear deformation theory. Material Research Express, 6, DOI: 10.1088/20531591/ab48a9.
[60] Du, Y., Wang, S., Sun, L., Shan, Y., 2019. Free vibration of rectangular plates with porosity distributions under complex boundary constraints. Shock and Vibration, DOI: 10.1155/2019/6407174.
[61] Al Rjoub, Y. S., Alshatnawi, J.A., 2020. Free vibration of functionallygraded porous cracked plates. Structures, 28, pp.23922403.
[62] Bansal, G., Gupta, A., Katiyar, V., 2020. Vibration of porous functionally graded plates with geometric discontinuities and partial supports. Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 234(21), pp.41494170.
[63] Tran, T.T., QuocHoa Pham, 2021. Static and free vibration analyses of functionally graded porous variablethickness plates using an edgebased smoothed finite element method. Defence Technology, 17(3), pp.971986.
[64] Chai, Q., & Wang, Y.Q., 2022. Traveling wave vibration of grapheme platelet reinforced porous joined conicalcylindrical shells in a spinning motion. Engineering Structures, 252(1), DOI: 10.1016/ j.engstruct.2021.113718.
[65] Kumar, Y., 2018. The RayleighRitz method for linear dynamic, static and buckling behavior of beams, shells and plates: A literature review. Journal of Vibration and Control, 24(7), pp.12051227.
[66] Pablo, M.G., Jose, V.A.S., Harmani, L., 2017. A review and study on Ritz method admissible functions with emphasis on buckling and free vibration of isotropic and anisotropic beams and plates. Archive of Computational Methods in Engineering, 25(3), pp.785815.
[67] Wang, Y.Q., Zu, J.W., 2017. Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain. Applied Mathematics and Mechanics, 38(5), pp.625646.
[68] Wang, Y.Q., Ye, C., Zu, J.W., 2018. Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities. Applied Mathematics and Mechanics, 39(11), pp.15871604.
[69] Laura, P.A.A., & Grossi, R., 1979. Transverse vibrations of rectangular anisotropic plates with edges elastically restrained against rotation. Journal of Sound and Vibration, 64(2), pp.257267.
[70] Okan, M.B., 1982. Free and damped vibrations of an orthotropic rectangular plate with one edge free while the rest are elastically restrained against rotation. Ocean Engineering, 9(2), pp.127134.
[71] Kumar, Y., 2012. Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using RayleighRitz method. Earthquake Engineering and Engineering Vibration, 11(2), DOI: 10.1007/s1180301201171.
[72] Zhang, Y., & Zhang, S., 2019. Free transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained and remaining others free. Applied Sciences, 9, DOI: 10.3390/app9010022.
[73] He, Y., Duan, M., & Su, J., 2021. Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions. Engineering Structures, 247, DOI: 10.1016/j.engstruct.2021.113129.
[74] Warburton, G.B., Edney, S.L., 1984. Vibrations of rectangular plates with elastically restrained edges. Journal of Sound and Vibration, 95(4), pp.537552.
[75] Singh, B., Chakraverty, S., 1994. Flexural vibration of skew plates using boundary characteristic orthogonal polynomials in two variables. Journal of Sound and Vibration, 173(2), pp.157178.
[76] Leissa, A.W., 1973. The free vibration of rectangular plates. Journal of Sound and Vibration, 31(3), pp.257293.