Effect of Elastically Restrained Edges on Free Transverse Vibration of Functionally Graded Porous Rectangular Plate

1. Kumar ^*

Department of Mathematics, Government Girls Degree College, Behat- 247121, India

1.     Introduction

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 first-order shear deformation theory and the element-free kp-Ritz 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 straight-sided plates with circular and non-circular cut-outs. 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 element-free Galerkin method. Baferani et. al. [9] investigated free vibration of FG rectangular plates based on first-order shear deformation theory. Chakraverty and Pradhan [10-11] investigated free vibration of thin FG rectangular plates incorporating the effects of Winkler foundation and thermal environment using the Rayleigh-Ritz method. Pradhan and Chakraverty [12] dealt with static analysis of thin FG rectangular plates under mechanical load using the Rayleigh-Ritz 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 first-order 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, simply-supported, and fluid-saturated 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 third-order shear deformation theory. Mojahedin et. al. [24] studied the buckling of FG circular porous plates using the energy method based on higher-order 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 higher-order shear deformation theory. Mechab et. al. [27-28] studied free vibration/probabilistic analysis of FG nanoplate with porosities resting on Winkler-Pasternak 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 four-variable theory to study the buckling of FG piezoelectric porous plates resting on an elastic foundation. Barati and Zenkour [31] explored the electro-thermo-mechanical vibrational behavior of FG piezoelectric plates with porosity using a refined four-variable plate theory. Mouaici et. al. [32] used hyperbolic shear deformation theory to examine the effect of porosity on the vibration of non-homogeneous plates. Ebrahimi and Jafari [33] presented an analytical solution to study the buckling characteristics of porous magneto-electro-elastic FG plates. In a series of papers, Rezaei and co-workers [34-40] 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 in-plane 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 first-order shear deformation theory. Barati and co-workers [44-47] have presented vibration analysis of smart/nano FG porous plates using a refined four-variable theory. Ali et. al. [48] studied free vibration of the embedded porous plate using higher order shear deformation theory. Wang and Zu [49-51] 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. Electro-mechanical 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 magneto-electro-elastic 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 high-order 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 non-polynomial higher-order 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 higher-order shear deformation theory.  Du et. al. [60] performed a free vibration analysis of rectangular plates with three types of porosity distributions using the Rayleigh-Ritz method based on first-order shear deformation theory. Rjoub and Alshatnawi [61] predicted the natural frequencies of a simply-supported functionally graded porous cracked plate using the Artificial Neural Network technique. Bansal et. al. [62] provided Navier solution and FEM-based 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 edge-based smoothed finite element method. Chai and Wang [64] investigated traveling wave vibration of spinning graphene platelets reinforced porous joined conical-cylindrical shells using the power series method.

An up-to-date review of works pertaining to the application of the Rayleigh-Ritz method in vibration analysis of structural elements is given by Kumar [65] and Pablo et. al. [66]. In their two papers, Wang and co-workers [67-68] used the Rayleigh-Ritz 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 simply-supported 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 Rayleigh-Ritz method incorporating boundary characteristic orthogonal polynomials as eigenfunctions is used to obtain the first three natural frequencies and mode shapes.

2.     Formulation of the problem

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 mid-plane of the plate. The distance between the geometrical mid-plane 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.

According to Hook’s law

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

The expression for the strain energy is

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

Fig. 1. (i) FG porous elastically restrained rectangular plate with physical neutral surface and geometrical middle surface (ii) plate with even porosity (iii) cross-section of the plate

The kinetic energy of the plate is given as

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:

where  are young’s moduli of ceramic and metal; are densities of ceramic and metal;  is the non-negative 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

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

and

where

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

where  are the rotational spring constants.

Introducing the non-dimensional variables  together with

We obtain the standard eigenvalue problem as follows:

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

and  is the frequency parameter.

The orthonormal polynomials  are generated using the Gram-Schmidt process (Singh and Chakraverty [75]).

3.     Results

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 simply-supported 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 (3-5) and Figs. (2-7).

Table 1. Convergence of first three values of frequency parameter Ω of FG porous plate with increasing value of

Table 2. Comparison of frequency parameter Ω of simply-supported FG plate

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

Table 4. First three values of frequency parameter Ω of FG porous plate for

Table 5. Proportionality factor for simply-supported FG porous plate

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 non-dimensional 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 frequency-response 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 mid-plane 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.

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.

4.     Conclusion

Free transverse vibration of a thin isotropic FG rectangular porous plate is studied here. A simply-supported 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 Rayleigh-Ritz 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 present technique is simple, straightforward forward, and provides good accuracy. The use of orthonormal polynomials results in a standard eigenvalue problem which can be easily solved for frequency parameters.
• The frequency of porous plate is lower than that of FG plate.
• Frequency generally decreases with increasing value of porosity volume fraction . But, the variation of frequency with porosity volume fraction is not monotonic for all the three modes as can be seen for . Here, frequencies in first and second modes decrease continuously but frequency in third mode first increases up to and then decreases. The porosity has a considerable effect on frequency in the case of thin plates.
• The frequencies of isotropic simply-supported FG porous plate are proportional to those of simply-supported homogeneous isotopic plate and the proportionality factor is independent of the aspect ratio

The results presented here may serve as a benchmark for further studies dealing with FG porous rectangular plates with elastically restrained edges.

Nomenclature

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.497-508.

[2] Abrate, S., 2006. Free vibration, buckling, and static deflections of functionally graded plates. Composite Science and Technology, 66, pp.2383-2394.

[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.593-600.

[4] Zhao, X., Lee, Y.Y., Liew, K.M., 2009. Mechanical and thermal buckling analysis of functionally graded plates. Composite Structures, 90, pp.161-171.

[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.3991-4011.

[6] Janghorban, M., Zareb, A., 2011. Thermal effect on free vibration analysis of functionally graded arbitrary straight-sided plates with different cutouts. Latin American Journal of Solids and Structures, 8, pp.245-257.

[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, 108-109, pp.93-99.

[8] Jaberzadeh, E., Azhari, M., Boroomand, B., 2013. Thermal buckling of functionally graded skew and trapezoidal plates with different boundary conditions using the element-free Galerkin method. European Journal of Mechanics A/Solids, 42, pp.18-26.

[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 Rayleigh-Ritz 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.132-156.

[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.721-734.

[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.3401-3423.

[14] Pham, T.T., 2016. Analytical solution for thermal buckling analysis of rectangular plates with functionally graded coatings. Aerospace Science and Technology, 55, pp.465-473.

[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.177-187.

[16] Young-Hoon Li, Seok-In, B., Ji-Hwan, K., 2016. Thermal buckling behavior of functionally graded plates based on neutral surface. Composite Structures, 137, pp.208-214.

[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.39-53.

[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.191-203.

[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.1-18.

[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.19-31.

[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.111-120.

[23] Rezaei, A., Saidi, A., 2015. Exact solution for free vibration of thick rectangular plates made of porous materials. Composite Structures, 134, pp.1051-1060.

[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. Thin-Walled Structures, 99, pp.83-90.

[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.14-22.

[26] Ebrahimi, F., Habibi, S., 2016. Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate. Steel and Composite Structures, 20(1), pp.205-225.

[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 two-variable refined plate theories. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(8), pp.2193-2211.

[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 Winkler-Pasternak elastic foundations. Applied Mathematical Modelling, 40(2), pp.738-749.

[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.2190-2205.

[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.307-320.

[31] Barati, M. R., Zenkour, A. M., 2016. Electro-thermoelastic vibration of plates made of porous functionally graded piezoelectric materials under various boundary conditions. Journal of Vibration and Control, 24(10), pp.1910-1926.

[32] Mouaici, F., Benyoucef, S., Atmane, H.A., Tounsi, A., 2016. Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory. Wind Structures, 22, pp.429-454.

[33] Ebrahimi, F., Jafari, A., 2016. Buckling behavior of smart EEE-FG porous plate with various boundary conditions based on refined theory. Advances in Material Research, 5(4), pp.279-298.

[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.1051-1060.

[35] Rezaei, A.S., Saidi, A.R., 2016. Application of Carrera unified formulation to study the effect of porosity on natural frequencies of thick porous-cellular plates. Composites Part B: Engineering, 91, pp.361-370.

[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. Thin-Walled Structures, 120, pp.366-377.

[37] Rezaei, A.S., Saidi, A.R., 2017. On the effect of coupled solid-fluid deformation on natural frequencies of fluid saturated porous plates. European Journal of Mechanics – A/Solids, 63, pp.99-109.

[38] Rezaei, A.S., Saidi, A.R., 2017. Buckling response of moderately thick fluid-infiltrated porous annular sector plates. Acta Mechanica, 228(11), pp.3929-3945.

[39] Rezaei, A.S., Saidi, A.R., 2017. An analytical study on the free vibration of moderately thick fluid-infiltrated porous annular sector plates. Journal of Vibration and Control, 24(18), pp.4130-4144.

[40] Askari, M., Saidi, A.R., Rezaei, A.S., 2017. On natural frequencies of Levy-type thick porous cellular plates surrounded by piezoelectric layers. Composite Structures, 179, pp.340-354.

[41] Kamranfard, M.R., Saidi, A.R., Naderi, A., 2017. Analytical solution for vibration and buckling of annular sectorial porous plates under in-plane uniform compressive loading. Proceedings of the Institution of Mechanical Engineering Part C: Journal of Mechanical Engineering Sciences, 232(12), pp.2211-2228.

[42] Şimşek, M., Aydın, M., 2017. Size-dependent 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.408-421.

[43] Akbaş, Ş.D., 2017. Vibration and static analysis of functionally graded porous plates. Journal of Applied and Computational Mechanics, 3(3), pp.199-207.

[44] Ebrahimi, F., Jafari, A., Barati, M.R., 2017. Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin-Walled Structures, 119, pp.33-46.

[45] Shahverdi, H., Barati, M.R., 2017. Vibration analysis of porous functionally graded nanoplates. International Journal of Engineering Sciences, 120, pp.82-99.

[46] Barati, M.R., Shahverdi, H., Zenkour, A.M., 2017. Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory. Mechanics of Advanced Materials and Structures, 24(12), pp.987-998.

[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.1865-1881. 

[48] Ali, G.A., Zahra, K.M., Mehdi, K., Iman, A., 2017. Free vibration of embedded porous plate using third-order shear deformation and poroelasticity theories. Journal of Engineering, pp.1-13, 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.550-562.

[50] Wang, Y.Q., Zu, J.W., 2017. Large-amplitude vibration of sigmoid functionally graded thin plates with porosities. Thin-Walled Structures, 119, pp.911-924.

[51] Wang, Y.Q., Zu, J.W., 2017. Porosity-dependent 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.1461-1480.

[53] Wang, Y.Q., 2018. Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica, 143, pp.263-271.

[54] Kiran, M.C., Kattimani, S.C., Vinyas, M., 2018. Porosity influence on structural behavior of skew functionally graded magneto-electro-elastic plate. Composite Structures, 191, pp.36-77.

[55] Arshid, E., Khorshidvand, A. R., 2018. Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method. Thin-Walled Structures, 125, pp.220-233.

[56] Arani, A.G., Khani, M., Maraghi, Z.K., 2018. Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory. Journal of Vibration and Control, 24, pp.3698-3713.

[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.106-120.

[59] Daikh, A.A., Zenkour, A. M., 2019. Free vibration and buckling of porous power-law and sigmoid functionally graded sandwich plates using a simple higher-order shear deformation theory. Material Research Express, 6, DOI: 10.1088/2053-1591/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 functionally-graded porous cracked plates. Structures, 28, pp.2392-2403.

[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.4149-4170.

[63] Tran, T.T., Quoc-Hoa Pham, 2021. Static and free vibration analyses of functionally graded porous variable-thickness plates using an edge-based smoothed finite element method. Defence Technology, 17(3), pp.971-986.

[64] Chai, Q., & Wang, Y.Q., 2022. Traveling wave vibration of grapheme platelet reinforced porous joined conical-cylindrical shells in a spinning motion. Engineering Structures, 252(1), DOI: 10.1016/ j.engstruct.2021.113718.

[65] Kumar, Y., 2018. The Rayleigh-Ritz method for linear dynamic, static and buckling behavior of beams, shells and plates: A literature review. Journal of Vibration and Control, 24(7), pp.1205-1227.

[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.785-815.

[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.625-646.

[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.1587-1604.

[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.257-267.

[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.127-134.

[71] Kumar, Y., 2012. Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using Rayleigh-Ritz method. Earthquake Engineering and Engineering Vibration, 11(2), DOI: 10.1007/s11803-012-0117-1.

[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.537-552.

[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.157-178.

[76] Leissa, A.W., 1973. The free vibration of rectangular plates. Journal of Sound and Vibration, 31(3), pp.257-293.