Study on Harmonic Analysis of Functionally Graded Plates Using Fem

Abstract

This paper presents the harmonic and vibration analysis of functionally graded plates using the finite element method. Initially, the plates are assumed isotropic and the material properties of it are assumed to vary continuously through their thickness direction according to a power-law distribution of the volume fractions of the plate constituents. The four noded shell element is used to analyse the functionally graded plates. Four functionally graded plates-Al/Al 2 O 3 , Al/ZrO 2 , Ti–6Al–4V/Aluminium oxide, and SUS304/Si 3 N 4 are considered in the study, and their results are obtained so that the right choice can be made in applications in high temperature environment and in reducing the vibration amplitudes in applications such as aircrafts, rockets, missiles, etc. Numerical results for the natural frequency and harmonic response amplitude are presented. Results are compared and validated with available results in the literature. Effects of boundary conditions, material and damping on natural frequency and harmonic response of the functionally graded plates are also investigated.

Full text

Int. J. of Applied Mechanics and Engineering, 2018, vol.23, No.4, pp.941-961
DOI: 10.2478/ijame-2018-0053
STUDY ON HARMONIC ANALYSIS OF FUNCTIONALLY GRADED
PLATES USING FEM
A.K. SHARMA*
Rajkiya Engineering College
Mainpuri, INDIA - 205119
E-mail: avadesh_sharma@rediffmail.com
P. SHARMA
Madhav Institute of Technology and Science
Gwalior, INDIA
P.S. CHAUHAN
IPS- College of Technology and Management
Gwalior, INDIA
S.S. BHADORIA
Dr. B.R. Ambedkar National Institute of Technology
Jalandhar, INDIA
This paper presents the harmonic and vibration analysis of functionally graded plates using the finite element
method. Initially, the plates are assumed isotropic and the material properties of it are assumed to vary
continuously through their thickness direction according to a power-law distribution of the volume fractions of
the plate constituents. The four noded shell element is used to analyse the functionally graded plates. Four
functionally graded plates-Al/Al2O3, Al/ZrO2, Ti–6Al–4V/Aluminium oxide, and SUS304/Si3N4 are considered in
the study, and their results are obtained so that the right choice can be made in applications in high temperature
environment and in reducing the vibration amplitudes in applications such as aircrafts, rockets, missiles, etc.
Numerical results for the natural frequency and harmonic response amplitude are presented. Results are compared
and validated with available results in the literature. Effects of boundary conditions, material and damping on
natural frequency and harmonic response of the functionally graded plates are also investigated.
Key words: finite element method, functionally graded plate, free vibration.

1. Introduction
Functionally graded materials (FGMs) are special composites with material properties that vary
continuously through their thickness resulting in corresponding changes in the properties of the material.
FGMs are usually made of a mixture of ceramic and metal, and can thus resist high-temperature conditions
while maintaining toughness. The concept of functionally graded materials (FGMs) was first introduced in
1984 by a group of material scientists in Japan, as ultrahigh temperature resistant materials for aircraft, space
vehicles and other engineering applications [1-2]. Functionally graded material (FGM) may be characterized
by the variation in composition and structure gradually over volume, resulting in corresponding changes in
the properties of the material. The materials can be designed for specific function and applications. Various
approaches based on the bulk (particulate processing), layer processing and melt processing are used to


* To whom correspondence should be addressed

942 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
fabricate the functionally graded materials [3]. There are many areas of application for FGMs. The concept is
to make a composite material by varying the microstructure from one material to another material with a
specific gradient. This enables the material to have the best of both materials. If it is for thermal, or corrosive
resistance or malleability and toughness both strengths of the material may be used to avoid corrosion,
fatigue, fracture and stress corrosion cracking [4]. The transition between the two materials can usually be
approximated by means of a power series. The aircraft and aerospace industry and the computer circuit
industry are very interested in the possibility of having materials that can withstand very high thermal
gradients. This is normally achieved by using a ceramic layer connected with a metallic layer [5-6]. The
dynamic behaviour of each element is derived and the solution of the structure as a whole is reconstituted by
attaching each element together at its node points. Predicting the dynamic response of a vibrating system
generally involves determining the equations of motion of the structure and solving them for given boundary
conditions [7]. Finite element methods are one of the most widely used deterministic techniques. For this
technique, a structure is divided into a number of elements. These methods are extensively used to predict
the linear dynamic response of structures in the low frequency region. For more complex systems, the
equations of motion can be approximated using various deterministic modelling techniques such as the finite
element analysis [8]. In engineering design, it is important to calculate the response quantities such as the
displacement, stress, vibration frequencies, and mode shapes of a given set of design parameters [9]. The
natural frequencies and mode shapes can then be used to predict the response due to an applied excitation.
The vibration and harmonic responses of composite plates and FGMs have been extensively studied by a
number of researchers [12-16]. However, the aim of the work is to study the effect of functionally graded
materials on the vibration behavior by making a modal and harmonic analysis of the models.
2. Functionally graded material properties
A functionally graded plate (shown in Fig.1) is considered to be a single-layered plate of uniform
thickness that is made of ceramic and metal. The material property is assumed to be graded through the
thickness in accordance with a power-law distribution that is expressed as
P(z) = (Pc - Pm)Vc+ Pm,
Vc = (1/2 + z/h)n, (n ≥ 0)
where, P represents the effective material property, Pc and Pm denote the properties of the ceramic and metal,
respectively, Vc is the volume fraction of the ceramic, h is the thickness of the plate, and n is the volume fraction
exponent. Figure 2 shows the variation of the volume fraction through the thickness for different exponent n.
Fig.1. Functionally graded plate.

Study on harmonic analysis of functionally graded plates using FEM 943
Fig.2. Variation of volume fraction Vc through the thickness.
3. Methodology
3.1. Background
ANSYS 15.0 with mechanical APDL is used to analyse the functionally graded plate. The analysis
was done on moderately thick square FG plates with various combinations of simply supported and clamped
boundary conditions. It consisted of building geometry of the model and distributing the FGM material
properties along the thickness of the model, meshing the model with a proper smart sized mesh types,
applying loads on the model, setting boundary conditions on the model, and finally running and solving the
model. Modal analysis is done to find the system vibration parameters (i.e., natural frequencies, and mode
shapes). And then for all these combinations harmonic analysis is done to find out the frequency response of
plate.
3.2. Modelling
According to the material properties shown in Tab.1 and using ANSYS 15.0 with mechanical APDL
capabilities, a harmonic analysis problem has been solved for FGM models. A square FG plate of 304.8 mm
x 304.8 mm x 2 mm is considered in the study which is subjected to a force of 1 N at the node location (75
mm, 152.4 mm), as shown in Fig.4. The element SHELL181 was chosen to mesh the model. It is a four-node
element with six degrees of freedom at each node: three translations in the x, y, and z directions, and three
rotations about the x, y, and z-axis. It is suitable for analyzing thin to moderately-thick shell structures and
can be used for layered applications for modelling composites. The degenerate triangular option should only
be used as filler elements in mesh generation. This makes the comparison between the different FG models
easy.

944 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Fig.3. SHELL181 Geometry.
In this section, the frequency characteristics of four types of square plates- Al/Al2O3, Al/ZrO2, Ti–
6Al–4V/ Aluminium oxide, and SUS304/Si3N4- are investigated. The properties of the constituents are
provided in Tab.1. The volume fraction exponent n is taken as 1. Two types of boundary conditions—all
edges clamped (CCCC) and all edges simply supported (SSSS) are considered.
The simply supported boundary condition is expressed as
At x = 0, a: v0= w0= θy=0.
At y = -b/2, b/2: u0= w0= θx= 0.
The boundary conditions at the clamped edges are given by
u0 = v0 = w0 = θx = θy = 0.
Figures 4a and 4b shows the geometry and loads for the FG plate at the SSSS and CCCC boundary
condition respectively.
Fig.4a. Fig.4b.

Study on harmonic analysis of functionally graded plates using FEM 945
Modal Analysis is done to compute the first 10 natural frequencies for each model. The solver used
for modal analysis is Block Lanczos. For the plates with the CCCC and SSSS boundary condition, the
frequency decreases, as the volume fraction exponent n increases. This is expected because a larger volume
fraction exponent means that the plate has a smaller ceramic component, and that its stiffness is thus reduced.
The last one is Harmonic Analysis, and the results were a comparison between the frequency response
amplitude values in each model, and this will be discussed in the following section.
Table 1. Material properties of FGM materials.
Material Elasticity modulus (GPa) Poisson’s ratio Density
Aluminium (Al) 70 0.3 2707
Alumina (Al2O3) 380 0.3 3800
Ti-6Al-4V 105.7 0.298 4429
Aluminium oxide 320.2 0.26 3750
Zirconia (ZnO2) 151 0.3 3000
Stainless steel SUS304 207.78 0.3177 8166
Silicon nitride 322.27 0.24 2370
4. Results
4.1.1. Modal analysis
The variation of the frequency parameter with the boundary condition and with the volume fraction
exponent for plates made of Al/Al2O3, Al/ZrO2, Ti–6Al–4V/Aluminium oxide andSUS304/Si3N4, is
described in Tabs 2-5, respectively. They show a comparison of the fundamental natural frequency
parameters of the four plates. It can be seen that all of the frequencies that represent the various combinations
show a similar behaviour, with the frequencies dropping as the volume fraction exponent increases.
Table 2. Variation of the frequency parameter with the volume fraction exponent n for the square Al/Al2O3
FG plate.
B.C. MODE
n 1 2 3 4 5 6 7 8 9 10
CCCC n=0 374.3 768.64 768.64 1132.1 1395.6 1402.5 1739.8 1739.8 2271.8 2271.8
n=1 311.3 639.2 639.2 941.4 1160.5 1166.1 1446.8 1446.8 1889.3 1889.3
n=10 254.2 522.0 522.0 768.9 947.9 952.4 1181.7 1181.7 1543.1 1543.1
SSSS n=0 204.8 514.5 514.5 822.9 1039.5 1039.5 1345.7 1345.7 1792.9 1792.9
n=1 170.3 427.9 427.9 684.34 864.4 864.4 1119.1 1119.1 1491.0 1491.0
n=10 139.1 349.5 349.5 558.9 706.0 706.0 914.0 914.0 1217.8 1217.8

946 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Table 3. Variation of the frequency parameter with the volume fraction exponent n for the square Ti–6Al–
4V/Aluminium oxide FG plate.
B.C. MODE
n 1 2 3 4 5 6 7 8 9 10
CCCC n=0 341.7 701.7 701.7 1033.6 1274.1 1280.2 1588.5 1588.5 2074.2 2074.2
n=1 268.3 551.0 551.0 811.6 1000.4 1005.3 1247.3 1247.3 1628.6 1628.6
n=10 221.5 454.8 454.8 670.0 825.9 829.8 1029.6 1029.6 1344.5 1344.5
SSSS n=0 187.0 469.7 469.7 751.2 948.9 948.9 1228.5 1228.5 1636.8 1636.8
n=1 146.8 368.8 368.8 589.9 745.1 745.1 964.6 964.6 1285.3 1285.3
n=10 121.23 304.52 304.52 487.0 615.1 615.1 796.3 796.3 1061.1 1061.1
Table 4. Variation of the frequency parameter with the volume fraction exponent n for the square
Al/ZrO2FG plate.
B.C. MODE
n 1 2 3 4 5 6 7 8 9 10
CCCC n=0 265.5 545.3 545.3 803.1 990.0 994.8 1234.3 1234.3 1611.8 1611.8
n=1 232.9 478.3 478.3 704.5 868.4 872.6 1082.7 1082.7 1413.7 1413.7
n=10 209.9 431.1 431.1 635.0 782.8 786.6 975.9 975.9 1274.4 1274.4
SSSS n=0 145.3 365.0 365.0 583.8 737.4 737.4 954.6 954.6 1272.0 1272.0
n=1 127.4 320.2 320.2 512.0 646.8 646.8 837.3 837.3 1115.7 1115.7
n=10 114.9 288.6 288.6 461.6 583.0 583.0 754.8 754.8 1005.7 1005.7
Table 5. Variation of the frequency parameter with the volume fraction exponent n for the square
SUS304/Si3N4 FG plate.
B.C. MODE
n 1 2 3 4 5 6 7 8 9 10
CCCC n=0 429.0 880.8 880.8 1297.5 1599.4 1607.1 1994.1 1994.1 2603.7 2603.7
n=1 263.7 541.5 541.5 797.6 983.3 988.0 1225.9 1225.9 1600.7 1600.7
n=10 216.5 444.5 444.5 654.8 807.1 811.0 1006.3 1006.3 1314.0 1314.0
SSSS n=0 234.7 589.6 589.6 943.0 1191.2 1191.2 1542.1 1542.1 2054.7 2054.7
n=1 144.33 362.5 362.5 579.8 732.4 732.4 948.1 948.1 1263.3 1263.3
n=10 118.4 297.6 297.6 475.9 601.2 601.2 778.3 778.3 1037.0 1037.0
Verification of modal analysis
Table 6 shows the variation of the non dimensional frequency parameter with the volume fraction
exponent for the Al/Al2O3 plates (a/h = 10). Only the results for the first four modes are computed. The

Study on harmonic analysis of functionally graded plates using FEM 947
results as listed in the table, show that the natural frequencies are in good agreement with the results of other
authors.
Table 6. Variation of the frequency parameter with the volume fraction exponent n for the square Al/Al2O3
FG plate.
B.C. MODE
n 1 2 3 4
CCCC n=0, Present 9.85 18.81 18.81 26.36
n=0, Zhao [10] 9.63 18.31 18.31 25.49
Error 2.28% 2.73% 2.73% 4.05%
SSSS n=0, Present 5.54 13.46 13.46 20.44
n=0, Zhao [10] 5.67 13.53 13.53 20.63
Error 2.29% 0.51% 0.51% 0.92%
Results of modal analysis show that the natural frequencies for the plates Ti–6Al–4V/Aluminium
oxide, and SUS304/Si3N4 nearly overlap, whereas the plate Al/Al2O3 has the highest values and the plate
Al/ZrO2 has lowest. A comparison of the 10 natural frequencies of the FG models with CCCC boundary
condition is shown in Fig.5.
Fig.5. Natural frequencies of the FG plate with CCCC boundary condition.
A comparison of the 10 natural frequencies of the FG models with SSSS boundary condition is
shown in Fig.6.

948 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Fig.6. Natural frequencies of the FG platewith SSSS boundary condition.
4.1.2. Harmonic analysis
To obtain the FRF plot, harmonic analysis is to be done by providing the range of natural frequency
of 0 Hz to 2000Hz and 100 substeps. It will generate the FRF plot (linear) on graph of amplitude to
frequency, frequency (Hz) is taken on the x-axis and amplitude (m) on the y-axis. From this graph we come
to know its resonance point. It can be said that the overall response of the system can be known. These
frequency response amplitudes can also be reduced by providing a damping constant. In the study, the
constant damping ratio of 1% to 4% or 0.01 to 0.04 is taken. The effect of the damping constant on the
frequency response amplitude for different sets of FG plates with different boundary conditions and different
volume fraction exponent is shown in Tabs 7-10, respectively.
Table 7. Effect of the damping constant on the frequency response amplitude for the square Al/Al2O3 FG
plate.
B.C. MODE
n D=0 D=0.01 D=0.02 D=0.03 D=0.04
CCCC n=0 0.0291 0.0262 0.0203 0.0147 0.0106
n=1 0.0264 0.0261 0.0249 0.0232 0.0218
n=10 0.0544 0.0519 0.0456 0.0380 0.0307
SSSS n=0 0.0518 0.0496 0.0438 0.0367 0.0299
n=1 0.0353 0.0350 0.0343 0.0335 0.0331
n=10 0.550 0.333 0.152 0.0802 0.0482
The FRF plot of amplitude (m) to frequency (Hz) for Al/Al2O3 FG plate at a different damping ratio
is shown on linear scale. It shows that on increasing the value of damping constant, the maximum amplitude
at resonance point is going down.

Study on harmonic analysis of functionally graded plates using FEM 949
Figs 7a-7e show the FRF plots without damping (D=0) and with damping(D=0.01 to 0.04) for Al/Al2O3 FG
plate at CCCC boundary condition.
Fig.7a Fig.7b
Fig.7c Fig.7d
Fig.7e

950 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Figs 8a-8e shows the FRF plots without damping (D=0) and with damping(D=0.01 to 0.04) for Al/Al2O3 FG
plate at SSSS boundary condition.
Fig.8a Fig.8b
Fig.8c Fig.8d
Fig.8e

Study on harmonic analysis of functionally graded plates using FEM 951
Table 8. Effect of the damping constant on the frequency response amplitude for the square Ti–6Al–
4V/Aluminum oxide FG plate.
B.C. MODE
n D=0 D=0.01 D=0.02 D=0.03 D=0.04
CCCC n=0 0.1141 0.0534 0.0217 0.0108 0.0077
n=1 0.0258 0.0253 0.0245 0.0231 0.0217
n=10 0.1602 0.1056 0.0521 0.0282 0.0171
SSSS n=0 0.0401 0.0394 0.0374 0.0344 0.0310
n=1 0.0482 0.0478 0.0473 0.0459 0.0440
n=10 0.304 0.244 0.153 0.0949 0.0618
Figs 9a-9e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for Ti–6Al–
4V/Aluminium oxide FG plate at CCCC boundary condition.
Fig.9a Fig.9b
Fig.9c Fig.9d

952 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Fig.9e
Figs 10a-10e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for Ti–6Al–
4V/Aluminium oxide FG plate at SSSS boundary condition.
Fig.10a Fig.10b

Study on harmonic analysis of functionally graded plates using FEM 953
Fig.10c Fig.10d
Fig.10e
Table 9. Effect of damping constant on the frequency response amplitude for square Al/ZrO2 FG plate.
B.C. MODE
n D=0 D=0.01 D=0.02 D=0.03 D=0.04
CCCC n=0 0.0530 0.0501 0.0430 0.0348 0.0275
n=1 0.0496 0.0487 0.0473 0.0445 0.0420
n=10 0.0411 0.0406 0.0392 0.0371 0.0346
SSSS n=0 0.0847 0.0831 0.0787 0.0722 0.0648
n=1 0.0732 0.0728 0.0722 0.0703 0.0682
n=10 0.1172 0.1158 0.1118 0.1057 0.0981
Figs 11a - 11e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for
Al/ZrO2FG plate at CCCC boundary condition.

954 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Fig.11a Fig.11b
Fig.11c Fig.11d
Fig.11e

Study on harmonic analysis of functionally graded plates using FEM 955
Figs 12a-12e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for
Al/ZrO2FG plate at SSSS boundary condition.
Fig.12a Fig.12b
Fig.12c Fig.12d
Fig.12e

956 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Table 10. Effect of damping constant on the frequency response amplitude for the square SUS304/Si3N4FG
plate.
B.C. MODE
n D=0 D=0.01 D=0.02 D=0.03 D=0.04
CCCC n=0 0.0460 0.0244 0.0209 0.0169 0.0133
n=1 0.0450 0.0426 0.0368 0.0311 0.0265
n=10 0.0446 0.0408 0.0323 0.0240 0.0177
SSSS n=0 0.0654 0.0624 0.0548 0.0455 0.0368
n=1 0.0597 0.0580 0.0563 0.0535 0.0494
n=10 0.1536 0.1343 0.0964 0.0656 0.0453
Figs 13a-13e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for
SUS304/Si3N4 FG plate at CCCC boundary condition.
Fig.13a Fig.13b
Fig.13c Fig.13d

Study on harmonic analysis of functionally graded plates using FEM 957
Fig.13e
Figs 14a-14e shows the FRF plots without damping (D=0) and with damping (D=0.01 to 0.04) for
SUS304/Si3N4 FG plate at SSSS boundary condition.
Fig.14a Fig.14b
Fig.14c Fig.14d

958 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
Fig.14e
Verification of harmonic analysis
A square plate with all edge fixed boundary condition, of dimension 304.8×304.8×2 mm of steel
material with properties of ρ=7.86×10-9 tonne/mm3, ν = 0.3, Y=2e5MPa was created. A force of 1 N is applied
at the node location (75 mm, 152.4 mm) of all the plates. The frequency range is given from 0 to 3000 Hz.
Harmonic analysis of the plate is solved in ANSYS 15 software using the full method. The full method uses
the full system matrices to calculate the harmonic response (no matrix reduction). During the analysis, it is
found that the fundamental mode for the plate was excited at the frequency 188.94 Hz, where the highest
value of amplitude 1.8e-2mm is obtained (as shown in Tab.11).
Table 11. Verification of the frequency response amplitude for the square steel plate.
B.C. Mode Frequency (Hz) Amplitude (mm)
CCCC Mode 1, Present 188.94 1.8e-2
Mode 1, Khan
[11]
188.06 1.8e-2
Error 0.06% 0.00%
Harmonic analysis of the plates gives the frequency response amplitudes at their fundamental mode.
Results of harmonic analysis show that the plate Al/ZrO2 has the highest amplitude value and the plate
Al/Al2O3 has the lowest amplitude value. Also, on providing the damping constant, the maximum amplitude
value at the resonance point on decreases, and it further on decreases, as the value of damping constant is
increased. The effect of damping on frequency response amplitude of the FG models with CCCC boundary
condition is shown in Fig.15.

Study on harmonic analysis of functionally graded plates using FEM 959
Fig.15. Effect of damping on FG plates with CCCC boundary condition.
Effect of damping on frequency response amplitude of the FG models with SSSS boundary condition
is shown in Fig.16.
Fig.16. Effect of damping on FG plates with SSSS boundary condition.
5. Conclusion
In this study, vibration and harmonic analyses of FGM plates are mode. The effective material
properties of functionally graded materials for the plate structures are assumed to vary continuously through

960 A.K.Sharma, P.Sharma, P.S. Chauhan and S.S.Bhadoria
the plate thickness and are graded in the plate thickness direction according to a volume fraction power law
distribution. Four functionally graded plates of different sets of materials are considered and their natural
frequencies and frequency response amplitude at the fundamental mode are determined, also effects of
damping on dynamic characteristics of the plate are shown in the study. Convergence tests and comparison
studies have been carried out with the commercially available software (ANSYS). A four noded layered shell
element (SHELL181) is used throughout the problem. The obtained results are in good agreement with those
available in the literature for different volume fraction indices, thickness ratios, aspect ratios and different
support conditions. The following concluding remarks can be made for thin to thick FGM plates.
 During the analysis, it was found that all the FGP combinations show a similar behaviour, with the
frequencies dropping as the volume fraction exponent increases.
 Results of modal analysis show that the natural frequencies for the plates Ti–6Al–4V/Aluminium oxide,
and SUS304/Si3N4 nearly overlap, whereas the plate Al/Al2O3 has the highest values and the plate
Al/ZrO2 has the lowest.
 For all boundary conditions, vibration amplitude increases as the volume fraction index increases.
 Results of harmonic analysis show the plate Al/ZrO2 has the highest amplitude values and the plate
Al/Al2O3 has the lowest values.
 Also, on providing the damping constant, the maximum amplitude value at the resonance point decreases,
and it further decreases, as the value of the damping constant is increased. So it becomes very important
to consider the damping constant in the plate design. It helps in vibration control, noise reduction,
stability of the system, fatigue and impact resistance.
6. Future scope
 Different geometric structures can be modelled such as cylindrical, spherical, conical, hyperboloid etc.
 Temperature dependent material property can be considered.
 Different types of analysis such as non-linear, buckling, post buckling, etc., can also be performed using
the presented model.
Nomenclature
a, b  length and width of plate
C  symbol for denoting clamped edge condition
D  damping
F  symbol for denoting free edge condition
h  thickness of a plate
n  volume fraction exponent
S  symbol for denoting simply supported edge condition
t  time
u
0, v0, w0  linear displacement components of a point on the plane z = 0 in x, y, z directions
Vc
 volume fraction of the ceramic
Y  Young Modulus of FG Plate
z  coordinate in transverse direction
ν  Poisson’s ratio
ρ  density
σij  stress components
References
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10.

Study on harmonic analysis of functionally graded plates using FEM 961
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Received: June 30, 2017
Revised: June 22, 2018