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Original Article | Open Access | Br. J. Arts Humanit., 5(5), 243-252 | doi: 10.34104/bjah.02302430252

Vibration Analysis of Two-dimensional Functionally Graded Plate with Piezoelectric Layers using the Classical Theory of Plates

Mohammad Sharafi* Mail Img Orcid Img

Abstract

In this article, two-dimensional functional materials have been used using the power law in them, which is a good measure to obtain the properties of a composite material of metal and ceramic. At first, the equations of motion were obtained using Hamiltons method and solved by the GDQ method, and finally, the accuracy of the obtained answers was compared with the existing articles. In the following, the dynamic model of the sheet with two piezoelectric actuator layers at the top and bottom was investigated and the obtained equations were solved using the Ritz method. 

INTRODUCTION

Rectangular plates are one of the mechanical elements that are widely used in various industries such as oil and gas and petrochemical industries. For this reason, investigating the phenomenon of vibrations in this type of plates is a necessary subject, therefore, in this thesis; the vibrations of such plates under the effect of piezoelectric layers that actuate them have been investigated. Examining the vibrations of sheets leads to a series of differential equations. Numerical solution of differential equations is widely used in various fields of engineering. Common methods for solving these equations are FEM, GDQ and RITZ, which are among the numerical methods for solving equations with partial derivatives. In this article, the stability of the sheet with the boundary conditions cccc, ccss, ccsc, cscs, sss and csss has been investigated. The variable of the problem is the piezoelectric thickness and the aim is to find the natural frequency of the sheet along with the piezoelectric. Investigating the piezoelectric effect on plate vibrations and applying the GDQ method to rectangular plates made of two-dimensional FGM materials are new aspects of the work. Research objectives investigating the natural frequencies of two-dimensional FGM sheet under the effect of piezo-electric layers is one of the research objectives. 
Hypotheses or research questions 1- The sheet is assumed to be made of two-dimensional graded material. 2- The power law is used to obtain the mechanical properties of the FGM sheet structure. 3- To obtain the stress and strain tensors in the main structure and the piezoelectric layer, the classic sheet model is used. 4- When using the piezoelectric layer, they are used in pairs at the top and bottom of the sheet (the layers are assumed as operators). 5- Poissons ratio is considered the same. Classical plate theory: In this theory, only strain is ignored and the displacement field inside the plate changes as a linear function of thickness. Hamiltons principle: Hamiltons principle is used as a reference to obtain the equations of motion of the sheet will be GDQ: It is a numerical method in the definition of the derivative and it is used to solve the problem numerically.
Review of Literature 
In 2009, Liu (Zhao and Liew, 2009) investigated the free vibrations of a rectangular sheet made of graded functional materials using the Ritz method. The properties of the functional material are considered according to the variable tracking power model in the direction of the sheet thickness. The differential equations of motion of the problem are obtained based on the first-order deformation theory. Finally, the solution of the obtained equations has led to the solution of an eigenvalue problem that has been solved by the Ritz method. In 2012, Akbari and his colleagues (Alashti and Khorsand, 2012)  analyzed the vibrations of a cylindrical shell made of graded functional materials based on three-dimensional elasticity with piezoelectric layers and taking into account dynamic and thermal loads, and the equations obtained by the differential quadratic method have been solved and compared with the finite difference method. Akhwan and his colleagues (Akhavan et al., 2009) in 2008 have investigated the exact solution of the buckling analysis of a rectangular sheet under load in an elastic bed. The analysis method is based on the theory of rectangular sheet taking into account the first order shear deformation effect. In reference number (Yas and Aragh, 2010) Yas and his colleagues in 2010 have analyzed the free vibration of a plate reinforced with continuous fiber in an elastic bed. The equations of motion based on three-dimensional elasticity have been solved using the differential quadratic method, and finally the frequencies of the system have been investigated by changing the volume of fiber and composite. In 2009, Nemat-Alla and his colleagues (Nemat-Alla et al., 2009) analyzed the plastic elastic analysis of functionally graded materials under thermal loading. The stress-strain relationships based on the elastic-plastic rule of functionally graded materials under thermal loading have been introduced in this model, and the effects of thermal load on functionally graded material and stress-strain changes in the material have been investigated. In 2009, Malekzadeh (Malekzadeh, 2009) analyzed the vibration of a thick plate with a graded function on an elastic bed. The formulation is based on three-dimensional elasticity and FGM relationships are assumed to be exponential. Shaaban and his colleagues (Sharma and Singh, 2011) in 2011 presented an analytical solution for the free vibration of a thick sheet of functionally graded material in an elastic bed with elastic edges. Governing motion equations based on the first order theory of shear deformation and have been solved numerically and the effects of different parameters have been evaluated. Hashemi (Hashemi et al., 2010) in 2010 presented the free vibrations of a rectangular sheet of functionally graded material using the first-order theory of shear deformation, and a new formula for shear correction of the sheet theory is presented in this paper. Hong et al. (Huang et al., 2011) in 2011 have investigated crack vibrations in rectangular thin sheet of functionally graded material. Hu and his colleagues (Thai and Choi, 2011) in 2011 have presented the vibration of a rectangular sheet of functionally graded material in an elastic bed based on the first-order theory of shear deformation. Nezahat and his colleagues (Shirazi et al., 2011) in 2011 investigated the control of active vibrations of a functional rectangular sheet using the fuzzy control method and comparing it with PID controllers and using classical theory and considering a series of local functions of equations obtained the motion and obtained the natural frequencies using the Fourier series. 
In 2013, Sadek and his colleagues (Sadek et al., 2003)  analyzed the vibrations of a sheet with simple supports based on the feedback control method, which was controlled by sensors and piezoelectric movements, and obtained the equations from the Greens function. The controls are based on displacement and speed feedbacks and solved the equations using a numerical method. In 2013, Rehmat Talebi and his colleagues (Talabi and Saidi, 2013) analyzed the vibrations of complete circular and mounted circular plates made of functional materials due to the effect of two layers of piezoelectric actuators on the top and bottom of the plate using the third order theory of shear and with a new method 5 The main equation of motion is obtained by considering Maxwells equation. Hashemi and his colleagues (Hosseini Hashemi, 2019) in this paper, the analysis and free vibrations of the targeted relatively thick rectangular plate (FG) with smart layers based on Mendelin theory and providing an exact closed-response solution are investigated. This structure consists of a FG sheet and two piezoelectric layers. 
Shariayat and his colleagues (Shariayat, 2018) inves-tigated the effects of using piezoelectric sensor and actuator layers on the vibrations of the quad-rangular FGM sheet, studied free vibrations, forced vibrations and active control of transient vibrations in this regard. Graded functional material Sheets are structures whose initial shape is flat and their thickness is very small. The behavior of the sheets can be distinguished depending on the type of material and the structure of their formation. Graded functional materials are materials whose mechanical properties change gently and continuously. In recent years, with the develop-ment of electric motors, turbines and reactors, it seems necessary to use materials that have high thermal and mechanical resistance. And in this chapter, the introduction of the graded functional materials is discussed. Then, how to make and uses of this material will be discussed. This class of materials is introduced here. Among the main applications of these graded functional materials are the use of nuclear reactors (components of the inner wall of the reactor), use in chemical industries (membranes and catalysts) use in medical engineering (implantation of artificial teeth, bones or artificial organs) and mentioned other new technologies such as ceramic motors and coating against corrosion and heat. 
Also, these materials can be widely used in making plates and shells of tanks, reactors, turbines and other components of machines exposed to high heat. Because these parts are highly prepared for failure due to thermal buckling. Another advantage of FG materials compared to layered composite materials is the lack of discontinuity at the junction of layers, because as mentioned in FGM materials, the combina-tion of elements is continuous and gradual. Among the other advantages of graded functional materials, we can mention their use in the construction of thermal insulation coatings, which are widely used. Graded functional materials are composite materials that are microscopically inhomogeneous and their structural characteristics such as the type of distribution, the size of phases, gradually change from one surface to another, and this gradual change leads to a gradual change in their properties. These materials are generally made of a mixture of ceramic with metal or a combination of different metals (with different thermal coefficients). 
The ceramic material has high temperature resistance due to its low thermal conductivity, and the metal material prevents breakage or cracking due to thermal stress due to its malleability. In the simplest FGMs, two different material components are continuously changed from one to the other. The transverse distri-bution of electric potential satisfies the Maxwell equation as well as the electrical boundary conditions for both closed and open circuit piezoelectric layers. (Farsangi and Saidi, 2012) In the micro-electro-mechanical systems (MEMS), Coulombs force plays a major role as a mechanism in sensing and actuation. MEMS devices due to their high sensitivity are widely used in different fields of science (Kazemi et al., 2019). The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configure-tion considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by the means of the 3D exact solution (Brischetto et al., 2016; Akter et al., 2023).

METHODOLOGY

Validation of two-dimensional functional sheet vibration with existing articles to validate the results of this research, we first run the written program in the special case of one-dimensional graded material and without piezoelectric. Then we compare the obtained results with references 7, 8 and 11. In these references, the function sheet one dimension is made of aluminum metal and aluminum oxide ceramic with the following specifications. 

Table 1: specifications of one-dimensional functional grade material (AL/AL2O3).

material

70 e9 2702

380 e9 3800

The results of this review are as follows 

Table 2: Comparison of natural four-frequency with dimension of present research with existing articles for one-dimensional functional material SSSS و (  ) و(  ) و ( (

Present 4492 11092 11128 16880

X. Zhao[11] 4347.4 10416 10416 15936

Error (%) 3.326126 6.490015 6.835637 5.923695

H. Matsunaga[7] 4427 10630 10630 16200

Error (%) 1.468263 4.34619 4.684854 4.197531

Table 2 Comparison of the main natural frequency with the dimension of the current research with existing articles for the one-dimensional functional material AL/AL2O3 for boundary conditions SSSS و (  ) و( )و( (

0 0.5 1 4 10

Present 5950 5225 4492 4070 3855

H. Matsunaga [7] 5777 4917 4427 3811 3642

error 2.994634 6.263982 1.468263 6.796117 5.848435

Sh. H.Hashemi [8] 5769 4920 4454 3825 3627

error 3.137459 6.199187 8.53166 6.405229 6.286187

X. Zhao [11] 5676.3 4820.9 4347 - 3592.3

error 4.821803 8.382252 3.335634 - 7.312864

Investigation of two-dimensional functional sheet vibration (without piezoelectric)   

Table 3: Specifications of two-dimensional graded material.

material  

70 2702

200 7800

380 3800

205 8900

Table 4: Checking the vibration of two-dimensional graded material with support conditions SSSS.

ssss

0 0.5 1 4 10 0 0.5 1 4 10 0 0.5 1 4 10

10 2973 3229.3 3344.1 3536.3 3605.6 3021.9 3415.2 3590 3880.06 3980.7 3251.4 3915.3 4190 4640.2 4790.5

20 1495.

623 1624.56 1682.312 1779.002 1813.865 1520.223 1718.08 1806.017 1951.937 2002.566 1635.678 1969.665 2107.858 2334.339 2409.951

30 998.

064 1084.

185 1122.

905 1186.995 1209.694 1014.754 1146.939 1205.688 1303.158 1336.538 1092.196 1314.507 1406.636 1558.181 1608.252

40 748.548 813.1388 842.1788 890.2463 907.2705 761.0655 860.2043 904.266 977.3685 1002.404 819.147 985.8803 1054.977 1168.636 1206.189

50 598.

8384 650.

511 673.

743 712.

197 725.

8164 608.8524 688.1634 723.4128 781.8948 801.

9228 655.3176 788.7042 843.9816 934.9086 964.9512

80 374.

274 406.5694 421.

0894 445.1231 453.6353 380.5328 430.1021 452.133 488.6843 501.2018 409.5735 492.9401 527.4885 584.3179 603.0945

100 299.

4192 325.

2555 336.8715 356.0985 362.9082 304.4262 344.0817 361.7064 390.9474 400.9614 327.6588 394.3521 421.9908 467.4543 482.4756

The resulting graphs from this table are as follows

Fig. 1: The original natural frequency with 2D functional sheet dimension with SSSS boundary conditions.

Fig. 2: The original natural frequency with the dimension of the two-dimensional functional sheet with boundary SSSSو ( ) conditions.

According to the graph (1), it can be seen that with the increase of the ratio or the decrease of the thickness, the natural frequency of the two-dimensional functional sheet decreases. 2- According to the diagram (2), it can be seen that it increases with increasing (constant) frequency. 3- According to the graph (2), it can be seen that it increases with the increase (constant) of the frequency. This increase is faster for values less than one, and for values greater than one, this increase is slower. 4- According to the diagram (1) by comparing the two cases, it can be seen that the increase has a greater effect on the frequency increase than the increase. 5- According to the diagram (1), it can be seen that the frequency reduction is faster before and after this ratio, the frequency reduction speed becomes lower.

Table 5: Checking the vibration of two-dimensional graded material with SCSC support conditions of the main natural frequency with the dimension of the two-dimensional functional sheet with SCSC boundary conditions.

scsc

0 0.5 1 4 10 0 0.5 1 4 10 0 0.5 1 4 10

10 4428.9 4796.3 4975 5295.2 5385.4 4500.3 5070.1 5345 5820 5955.7 4835.1 5805 6240 6975.3 7175

20 2228 2412.824 2502.721 2663.8 2709.176 2263.918 2550.562 2688.853 2927.806 2996.071 2432.343 2920.26 3139.091 3508.84 3609.452

30 1486.324 1609.622 1669.594 1777.052 1807.322 1510.286 1701.509 1793.764 1953.173 1998.713 1622.644 1948.139 2094.123 2340.787 2407.906

40 1117.621 1210.333 1255.428 1336.23 1358.991 1135.639 1279.426 1348.796 1468.661 1502.905 1220.125 1464.876 1574.647 1760.123 1810.592

50 894.1684 968.3442 1004.423 1069.069 1087.28 908.5836 1023.623 1079.123 1175.023 1202.42 976.1777 1171.995 1259.819 1408.211 1448.589

80 558.8902 605.253 627.8034 668.21 679.5925 567.9002 639.8042 674.4943 734.4353 751.5596 610.1492 732.5425 787.4358 880.1867 905.425

100 447.1479 484.2411 502.2829 534.6107 543.7175 454.3565 511.8843 539.6386 587.5953 601.2957 488.1584 586.0809 629.9991 704.2057 724.398

The resulting graphs from this table are as follows

Fig. 3: The original natural frequency with the dimension of the two-dimensional functional sheet with boundary SCSC conditions.

Fig. 4: Checking the vibration of two-dimensional graded material with CCCS support conditions along with piezoelectric.

Table 6: main natural frequencies of two-dimensional functional sheet with thickness (  ) under the effect of piezoelectric layer with thickness (  ) with CCCS boundary conditions.

CCCS(h=0.1)

0 1 4 1 4

0 4945 5840 6365 6800 7610

0.01 4830 5530 5990 6210 6910

0.02 4840 5425 5815 5955 6550

0.03 4940 5455 5780 5890 6395

0.04 5110 5570 5845 5950 6370

0.05 5325 5745 5970 6085 6435

0.06 5575 5960 6150 6270 6565

0.07 5845 6205 6335 6495 6735

0.08 6130 6470 6600 6735 6940

0.09 6425 6745 6855 6995 7170

0.1 6725 7025 7115 7265 7410

The graph resulting from this table is as follows

Fig. 5: The main natural frequencies of the two-dimensional functional sheet with thickness (  ) under the effect of piezoelectric layer with thickness (  ) with CCCS boundary conditions.

In the case of increasing the piezoelectric thickness ( ), the frequency decreases until the thickness and the frequency increases from this thickness. 2- In the case of ( ) increasing the piezoelectric thickness ( ), the frequency decreases ( ) to the thickness ( ) and the frequency increases from this thickness. 3- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness ( ) and the frequency increases ( ) from this thickness. 4- In the case of increasing the piezoelectric thickness, the frequency decreases to the thickness and the frequency increases from this thickness. 5- In the case of increasing the piezoelectric thickness ( ), the frequency decreases to the thickness ( ) and the frequency increases from this thickness. 6- By increasing the thickness of the piezo ( ), all the graphs are closer to each other and finally one be. 7- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the case of frequency reduction, it is equal to 5.3% and for the case of equal to 8.6% is. 8- As the effect of piezoelectric increases in the initial reduction of frequencies, it increases. For example, in the mode of reducing the frequency in the mode equal to 5.3% and for the mode equal to 5.8% is. 9- The effect of piezoelectric in the initial reduction of frequencies is greater than For example, in the mode of reducing the frequency in the mode equal to 5.8% and for the mode equal to 8.6% is 6-3-5- Investigating the vibration of two-dimensional graded material with CCCC support conditions along with piezoelectric.

Table 7: main natural frequencies of two-dimensional functional sheet with thickness (  ) under the effect of piezoelectric layer with thickness ( ) with CCCC boundary conditions.


CCCC(h=0.1)

0 1 4 1 4

0 5610 6655 7155 7750 8520

0.01 5475 6305 6745 7080 7760

0.02 5488 6180 6560 6790 7370

0.03 5600 6205 6525 6715 7205

0.04 5790 6330 6600 6775 7185

0.05 6035 6525 6750 6920 7265

0.06 6315 6765 6950 7125 7415

0.07 6620 7040 7195 7370 7610

0.08 6945 7335 7460 7640 7845

0.09 7275 7640 7745 7930 8095

0.1 7610 7955 8040 8225 8365

The graph resulting from this table is as follows

Fig. 6: The main natural frequencies of the two-dimensional functional sheet with thickness ( ) under the effect of piezoelectric layer with thickness (  ) with CCCC boundary conditions.

Comparing the vibration of two-dimensional graded material with two piezoelectric layers with SSSS, SCSC, SSSC, SSCC, SCCC and CCCC boundary conditions.

Fig. 7: The original natural frequency with the dimension of a two-dimensional functional sheet with two piezoelectric layers with different boundary conditions in the state.

RESULTS

According to the diagram above, in a specific state (here for), the percent increase and decrease in frequency caused by changes in the piezoelectric thickness is almost constant and uniform. So that all six lines in the above diagram move parallel to each other. Investigating the causes of errors 1- The classical theory of sheets is only valid for thin sheets. In this theory, shear force and rotational inertia are zero. 2- To consider zero and to simplify the equations. 3- Using the numerical method (GDQ) to solve equations: the causes of errors in this part include two parts becomes: a: The smaller the number of points in the grid of the sheet, the greater the error. b: In this research, the method is used to apply the boundary conditions. In this method, we apply the boundary conditions regardless of the delta distance from the edges.

Fig. 8: Boundary condition in the state.

CONCLUSION

According to the tables and charts, it was observed that: For all boundary conditions and different power coefficients, natural frequency decreases with increa-sing ratio (  ). They increase with increasing and frequencies. It should be noted that the frequency changes depend on the arrangement of metals and ceramics in the two-dimensional graduated functional material.  As expected, in a certain case, the thickness of the functional sheet and the power coefficients are ordered from higher to lower according to the boun-dary conditions as follows: from CCCC, CCCS, SCSC, CCSS, SSSC and SSSS. The effect of the presence of two piezoelectric layers on the top and bottom of the sheet in the (close-circuit) mode is such that with the increase in thickness, the piezoelectric frequencies decrease up to a certain thickness and then increase. The piezo effect in a certain state of plate thickness and power coefficients is completely similar for different boundary conditions. This means that the frequencies decrease and then increase at a certain rate. 

ACKNOWLEDGEMENT

We are grateful to all the dear professors for providing their information regarding this research. 

CONFLICTS OF INTEREST

Conflicts of interest are declared obviously in the manuscript. Authors also state separately that they have all read the manuscript and have no conflict of interest. 

Article References:

  1. Akter M, Sarker SPK, and Alam MM. (2023). Magnetohydrodynamics (MHD) effects on heat generation and joule heating with non-uniform surface temperature and natural convection flow over a vertical flat plate, Int. J. Mat. Math. Sci., 5(2), 09-18. https://doi.org/10.34104/ijmms.023.09018  
  2. Brischetto, S., Fantuzzi, N., & Viola, E. (2016). 3D exact and 2D generalized differential quadra-ture models for free vibration analysis of func-tionally graded plates and cylinders. Meccanica, 51, 2059-2098. 
  3. C. S. Huang, O. G. McGee, M. J. Chang. (2011). Vibrations of cracked rectangular FGM thick plates. Composite Structures, P. 1747-1764. 
  4. Farsangi, M. A., & Saidi, A. R. (2012). Levy type solution for free vibration analysis of functionally graded rectangular plates with piezoelectric layers. Smart materials and structures, 21(9), 094017. https://doi.org/10.1088/0964-1726/21/9/094017 
  5. H. Akhavan, SH. Hosseini Hashemi, H, A. Alibeigloo, SH. Vahabi. (2009). Exact solutions for rectangular Mindlin plates under in plane loads resting. Computational Materials Science, 44, 968-978. 
  6. Hossain Nezhad Shirazi, H. R. Owji, M. Rafeeyan, (2011). Active Vibration Control of an FGM Rectangular Plate using Fuzzy Logic Con-trollers. Procedia Engineering, 14, 3019-3026. https://doi.org/10.1016/j.proeng.2011.07.380 
  7. Huu- Tai Thai, Dong- Ho Choi. (2011). A refined shear deformation theory for free vibration of functionally graded plates on elastic foundation. Composites Part B: Engineering. https://doi.org/10.1016/j.compositesb.2011.11.062 
  8. I.S. Sadek. (2003). Feedback control of vibrating plates using piezoelectric patch sensors and actuators. Composite Structures, 62, 397-402. 
  9. Kazemi, A., Vatankhah, R., & Farid, M. (2019). Vibration analysis of size-dependent functionally graded micro-plates subjected to electrostatic and piezoelectric excitations. European J. of Mecha-nics- A/Solids, 76, 46-56. 
  10. M. Rahmat Talabi, and A.R. Saidi. (2013). An explicit exact analytical approach for free vibra-tion of circular/annular functionally graded plates bondedto piezoelectric actuator/sensor layers based on Reddys plate theory. Applied Mathema-tical Modelling, 37, 7664-7684. https://doi.org/10.1016/j.apm.2013.03.021 
  11. M. Shaban, M. M. Alipour. (2011). Semi-analy-tical solution for free vibration of think fun-ctionally graded plates rested on elastic function with elastically restrained edge. Acta Mechanica Solid a Sinica, P. 340-354. 
  12. M.H. Yas, B. Sobhani Aragh, (2010). Free vib-ration analysis of continuous grading fiber rein-forced plates on elastic function. Inter J. of Engineering Science, 48, 1881-1895. https://doi.org/10.1016/j.ijengsci.2010.06.015 
  13. Mahmoud Nemat-Alla, Khaled I. E. Ahmed, Ibraheen Hassab-Allah, (2009). Elastic plastic analysis of two-dimensional functionally grade materials under thermal loading. Intern J. of Solids and Structures, 46, 2774-2786. 
  14. Mohammad Shariayat, (2018). Investigating the effect of using piezoelectric layers on forced and free vibrations of functional quadrilateral sheets, Sharif Scientific Research J., 51. 
  15. P. Malekzadeh, (2009) Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations. Composite Structures, 89, 367-373. https://doi.org/10.1016/j.compstruct.2008.08.007  
  16. R. Akbari Alashti, M. Khorsand, (2012). Three-dimensional dynamo-thermo-elastic analysis of a functionally graded cylindrical shell with piezo-electric layers by DQ-FD coupled. Inter. J. of Pressure Vessels and Piping, 96-97 49e67. 
  17. Sh. Hossein Hashem, H. Rokni Damavandi Taher, H. Akhavan, M. Omidi, (2010). Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Applied Mathematical Modelling, P.1276-1291. https://doi.org/10.1016/j.apm.2009.08.008 
  18. Shahrokh Hosseini Hashemi, (2019). Vibration analysis of precise response of relatively thick rectangular sheets made of targeted materials with piezoelectric layer. Modares Mechanical Engine-ering Research J., 11(3). 
  19. X. Zhao, Y. Y. Lee, K. M. Liew, (2009). Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J. of sound and vibration, 319, 918-939. https://doi.org/10.1016/j.jsv.2008.06.025 

Article Info:

Academic Editor

Dr. Sonjoy Bishwas, Executive, Universe Publishing Group (UniversePG), California, USA.

Received

September 18, 2023

Accepted

October 9, 2023

Published

October 20, 2023

Article DOI: 10.34104/bjah.02302430252

Corresponding author

Mohammad Sharafi*
English Language Teaching, Faculty of Humanities, Sepidan Azad University, Tehran, Iran.

Cite this article

Sharafi M. (2023). Vibration analysis of two-dimensional functionally graded plate with piezoelectric layers using the classical theory of plates, Br. J. Arts Humanit., 5(5), 243-252. https://doi.org/10.34104/bjah.02302430252 

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