Open access peer-reviewed chapter

Second Order Shear Deformation Theory (SSDT) for Free Vibration Analysis on a Functionally Graded Quadrangle Plate

By A. Shahrjerdi and F. Mustapha

Submitted: November 18th 2010Reviewed: March 3rd 2011Published: September 9th 2011

DOI: 10.5772/22245

Downloaded: 3159

1. Introduction

Studies of vibration of plates have matured and are a well-established branch of research in structural dynamics. They have a vast range of applications in engineering and technology. But not much work can be found on vibration analysis of Functionally Graded Materials (FGMs) as compared to isotropic and composite plates and shells. FGMs are those in which the volume fraction of the two or more constituent materials is varied, as a power-law distribution, continuously as a function of position along certain dimension(s) of the structure [1], [2].

From the perspective of finite element method (FEM) studies of FGM, Praveen and Reddy [3], studied the static and dynamic responses of functionally graded (FG) ceramic-metal plate accounting for the transverse shear deformation, rotary inertia and moderately large rotations in the Von-Karman sense, in which the effect of an imposed temperature field on the response of the FG plate was discussed in detail. Ng et al. [4] dealt with the parametric resonance of FG rectangular plates under harmonic in-plane loading. Ferreira and Batra [5] provided a global collocation method for natural frequencies of FG plates by a meshless method with first order shear deformation theory (FSDT). Woo et al. [6] presented an analytical solution for the nonlinear free vibration behavior of FGM plates, where the fundamental equations were obtained using the Von-Karman theory for large transverse deflection, and the solution was based in terms of mixed Fourier series. Zhao et al. [7] studied the free vibration analysis of metal and ceramic FG plates using the element-free kp-Ritz method. The FSDT was employed to account for the transverse shear strain and rotary inertia, mesh-free kernel particle functions were used to approximate the two-dimensional displacement fields and the eigen-equation was obtained by applying the Ritz procedure to the energy functional of the system. Batra and Jin [8] used the FSDT coupled with the FEM to study the free vibrations of an FG anisotropic rectangular plate with various edge conditions. Also, Batra and Aimmanee [9] studied a higher order shear and normal deformable plate theory by FEM. Many studies conducted on FGMs are related to the analysis of free vibration by applying FSDT (see [10]-[12] and the references there in).

Other forms of shear deformation theory, such as the third order-shear deformation theory (TSDT) that accounts for the transverse effects, have been considered. Cheng and Batra [13] applied Reddy's third order plate theory to study buckling and steady state vibrations of a simply supported FG isotropic polygonal plate [14]. Vel and Batra [14] dealt with the three-dimensional exact solution for free and forced vibrations of simply supported FGM rectangular plates using FDST and TSDT by employing the power series method. Nonlinear vibration and dynamic response of FGM plates in thermal environments were studied by Huang et al. [15] based on the higher-order shear deformation plate theory and general Von-Karman type equation. Static analysis of FG plates using TSDT and a meshless method were also presented by Ferreira et al. [16].

As for the first-order shear deformation plate theory (FSDT), the theory extends the kinematics of the classical plate theory (CPT) by relaxing the normality restriction and allowing for arbitrary but constant rotation of transverse normals. On the other hand, the second and third order shear deformation plate theory further relaxes the kinematic hypothesis by removing the straightness assumption; i.e., the straight normal to the middle plane before deformation may become cubic curves after deformation. The most significant difference between the classical and shear deformation theories is the effect of including transverse shear deformation on the predicted deflections, frequencies, and buckling loads [19].

A unified derivation of various shear-deformation models consists of Kirchhoff-Love type, Mindlin-Reissner type theory, third order theory, Layer-Wise theory and Exact-Solution. Librescu et al. [22] studied the correlation between two apparently different higher-order theories and First order transverse shear deformation theory (FSDT) of anisotropic plates. The Kirchhoff-Love assumptions were developed by Librescu and Schmidt [23]. The theory incorporates normal and shear deformation (transverse) as well as the higher-order effects, and accounts for small strains and moderate rotations of the normal.

For experimental work, shear deformation validation and compared structural theories, Stoffle [20] measured and simulated vibrations of viscoplastic plates under impulsive loading and determined how accurately the measured deformations can be calculated by the chosen constitutive and structural theories. He assumed a first-order shear deformation shell theory and applied small strains and moderate rotations and viscoplastic laws. He applied short time measurement techniques to shock tubes in order to record fast loading processes and plate deformations.

As mentioned above, shear deformation theories have been applied to consider transverse shear strains and rotation. Axisymmetric bending and stretching of functionally graded solid circular and annular plates were studied using the second-order shear deformation plate theory by Saidi and Sahraee [21]. Khdeir and Reddy [17] studied the free vibration of laminated composite plates using SSDT. Bahtuei and Eslami [18] also investigated the coupled thermoelastic response of a FG circular cylindrical shell by considering SSDT.

To the authors’ knowledge, not much work has been done in the area of the dynamic stability of FG plate by using SSDT. In this study, the free vibration of FG plates (rectangular and square) by using SSDT is presented. The material properties of the plates are graded along the thickness direction according to a volume fraction power law distribution. Classical elasticity is considered and the complete governing equations are presented. Navier's method is applied to solve the equations. This work aims to investigate the effect of some basic factors such as material properties, side-to-side and side-to-thickness ratio for FG quadrangular plates on simply supported boundary conditions.

2. Gradation relations

The most commonly used models for most of the literature that express the variation of material properties in FGMs is the power law distribution of the volume fraction. According to this model, the material property gradation through the thickness of the plate is assumed to be in the following form [10]:

E=E(x3)=(EcEm)(x3/h+1/2)p+EmE1
ρ=ρ(x3)=(ρcρm)(x3/h+1/2)p+ρmE2

Here Eand ρdenote the modulus of elasticity and density of FG structure, while the parameters with subscript mor crepresent the material properties of a pure metal and pure ceramic plate, respectively. The thickness coordinate variable is presented by x3whileh2x3h2, where his the total thickness of the plate as shown in Figure 1. p0is the volume fraction exponent (also called grading index in this paper); (x3/h+1/2)pdenotes the volume fraction of the ceramic.

Figure 1.

Functionally graded plate.

A FG rectangular is considered as shown in Figure 1. The material in the top surface and in the bottom surface is Full-Ceramic and Full-Metal respectively, and between these two pure materials, the power law distribution of material is applied. The most well-known FGM is compositionally graded from a ceramic to a metal to incorporate such diverse properties as heat, wear and oxidation resistance of ceramics with the toughness, strength, machinability and bending capability of metals [7].

3. Elastic equations

Under consideration is a thin FG plate with constant thicknessh, width,a, and length,b, as shown in Figure 1. Cartesian coordinate system (x1,x2,x3) is used.

3.1. Displacement field and strains

The SSDT is based on the following representation of the displacement field:

u1=u+x3ϕ1+x32ϕ2E3
u2=v+x3ψ1+x32ψ2E4
u3=wE5

Where (u1,u2,u3) denote the displacement components in the (x1,x2,x3) directions respectively; (u,v,w) are the displacements of a point on the mid plane (x1,x2,0). All displacement components (u,v,w,ϕ1,ϕ2,ψ1,ψ2) are functions of position (x1,x2) and timet.

The strain-displacement equations of the linear strain are given by [19].

{ε11ε22γ12}={ε110ε220ε120}+x3{κ11κ22κ12}+x32{κ11'κ22'κ12'}E6
{γ23γ13}={γ230γ130}+x3{γ231γ131}E7

where

ε110=ux1,κ11=ϕ1x1,κ11'=ϕ2x1ε220=vx2,κ22=ψ1x2,κ22'=ψ2x2ε120=ux2+vx1,κ12=ϕ1x2+ψ1x1,κ12'=ϕ2x2+ψ2x1γ230=ψ1+wx2,γ130=ϕ1+wx1,γ231=2ψ2,γ131=2ϕ2E8

3.2. Stress-strain relations

The stress-strain relations are given by [17], [19].

{σ11σ22σ23σ13σ12}={q11q12000q12q2200000q4400000q5500000q66}{ε11ε22γ23γ13γ12}E9

where qijare the material constants given by

q11=q22=E1ν2q12=νq11q44=q55=q66=E2(1+ν)E10

Hence, it follows that

{σ11σ22σ23σ13σ12}=[q11q12000q12q2200000q6600000q5500000q44]({ε110ε220γ230γ130ε120}+x3{κ11κ22γ231γ131κ12}+x32{κ11'κ22'00κ12'})E11

3.3. Equations of motion

For the case of a rectangular plate, K, Uand Vare the kinetic, strain and potential energies of the body, respectively. The summation of the potential energy of external forces and strain energy,U+V, is the total potential energy, Π, of the body. Hamilton's principle for an elastic body is given by,

t1t2(δKδΠ)dt=0E12

The inertias are defined by

Ii=h2h2ρ0(x3)idx3(i=0,1,2,...,6)E13

Hamilton’s principle, equation (Eq. 8), along with the SSDT, given by equation (Eq. 2), yields the complete form of the equilibrium equations:

Q13x1+Q23x2=I0w¨M11x1+M12x2Q13=I2ϕ¨1+I1u¨+I3ϕ¨2L11x1+L12x22R13=I2u¨+I4ϕ¨2+I3ϕ¨1M22x2+M12x1Q23=I2ψ¨1+I1v¨+I3ψ¨2L22x2+L12x12R23=I2v¨+I4ψ¨2+I3ψ¨1E14

whereN,M,L,QandRare the stress resultants. These parameters can be represented by

{N11N22N12}=h2h2{σ11σ22σ12}dx3{N11N22N12}=[A11ε110+B11κ11+D11κ11'+A12ε220+B12κ22+D12κ22'A12ε110+B12κ11+D12κ11'+A22ε220+B22κ22+D22κ22'A66ε120+B66κ12+D66κ12']E15
{M11M22M12}=h2h2{σ11σ22σ12}x3dx3{M11M22M12}=[B11ε110+D11κ11+E11κ11'+B12ε220+D12κ22+E12κ22'B12ε110+D12κ11+E12κ11'+B22ε220+D22κ22+E22κ22'B66ε120+D66κ12+E66κ12']E16
{L11L22L12}=h2h2{σ11σ22σ12}x23dx3{L11L22L12}=[D11ε110+E11κ11+F11κ11'+D12ε220+E12κ22+F12κ22'D12ε110+E12κ11+F12κ11'+D22ε220+E22κ22+F22κ22'D66ε120+E66κ12+F66κ12']E17
{Q13Q23}=h2h2{σ13σ23}dx3{Q13Q23}=[A55γ130+B55γ131A44γ230+B44γ231]E18
{R13R23}=h2h2{σ13σ23}x3dx3{R13R23}=[B55γ130+D55γ131B44γ230+D44γ231]E19

where

Aij,Bij,Dij,Eij,Fij=h2h2qij(1,x3,x32,x33,x34)dx3E20

Here Aij,Bij,Dij,EijandFijare the plate stiffnesses.

For{Aij,Dij,Fij(i,j=1,2,4,5,6)Eij,Bij(i,j=1,2,6)E21

By substituting equation (Eq. 4) into equation (Eq. 11) and then into equation (Eq. 10) and also by applying definition (Eq. 12), Navier’s equations for FG plates are obtained as follows:

A112ux12+A662ux22+(A12+A66)2vx1x2+B112ϕ1x12+B662ϕ1x22+D112ϕ2x12+D662ϕ2x22+(B12+B66)2ψ1x1x2+(D12+D66)2ψ2x1x2=I0u¨+I2ϕ¨2+I1ϕ¨1E22
A122ux1x2+A662ux1x2+A222vx22+A662vx12+(B12+B66)2ϕ1x1x2+(D12+D66)2ϕ2x1x2+B662ψ1x12+B222ψ1x22+D662ψ2x12+D222ψ2x22=I0v¨+I2ψ¨2+I1ψ¨1E23
A55ϕ1x1+A552wx12+2B55ϕ2x1+A44ψ1x2+A442wx22+2B44ψ2x2=I0w¨E24
B112ux12+B662ux22+(B12+B66)2vx1x2+D112ϕ1x12+D662ϕ1x22+E112ϕ2x12+E662ϕ2x22+(D12+D66)2ψ1x1x2+(E12+E66)2ψ2x1x2A55(wx1)A55ϕ12B55ϕ2=I2ϕ¨1+I1u¨+I3ϕ¨2E25
D112ux12+D662ux22+(D12+D66)2vx1x2+E112ϕ1x12+E662ϕ1x22+F112ϕ2x12+F662ϕ2x22+(E12+E66)2ψ1x1x2+(F12+F66)2ψ2x1x22(B55(wx1)+2D55ϕ2+B55ϕ1)=I2u¨+I4ϕ¨2+I3ϕ¨1E26
(B12+B66)2ux1x2+B222vx22+B662vx12A44wx2+(D12+D66)2ϕ1x1x2+(E12+E66)2ϕ2x1x2+D662ψ1x12+D222ψ1x22+E662ψ2x12+E222ψ2x22A44ψ12B44ψ2=I2ψ¨1+I1v¨+I3ψ¨2E27
(D12+D66)2ux1x2+D662vx12+D222vx222B44wx2+(E12+E66)2ϕ1x1x2+(F12+F66)2ϕ2x1x2+E662ψ1x12+E222ψ1x22+F662ψ2x12+F222ψ2x222B44ψ14D44ψ2=I2v¨+I4ψ¨2+I3ψ¨1E28

It can be noted by considering zero values forϕ2&ψ2in equations (Eq. 10) and (Eq. 13), the FSDT equations can be obtained [19].

4. Boundary conditions

For the case of simply supported boundary conditions of FG, as shown in Figure 2, the following relations can be written:

Figure 2.

Simply supported boundary condition in FG plates.

x1=0,a{v(0,x2,t)=0v(a,x2,t)=0ψ1(0,x2,t)=0ψ1(a,x2,t)=0ψ2(0,x2,t)=0ψ2(a,x2,t)=0{M11(0,x2,t)=0N11(0,x2,t)=0M11(a,x2,t)=0N11(a,x2,t)=0{w(0,x2,t)=0w(a,x2,t)=0E29
x2=0,b{u(x1,0,t)=0u(x1,b,t)=0ϕ1(x1,0,t)=0ϕ1(x1,b,t)=0ϕ2(x1,0,t)=0ϕ2(x1,b,t)=0{M22(x1,0,t)=0N22(x1,0,t)=0M22(x1,b,t)=0N22(x1,b,t)=0{w(x1,0,t)=0w(x1,b,t)=0E30

5. Method of solution

The Navier method is used for frequency analysis of a simply supported FG plate. The displacement field can be assumed to be given by:

u=n=1m=1umn(t)cosαx1sinβx2,umn(t)=UeiωtE31
v=n=1m=1vmn(t)sinαx1cosβx2,vmn(t)=VeiωtE32
w=n=1m=1wmn(t)sinαx1sinβx2,wmn(t)=WeiωtE33
ϕ1=n=1m=1ϕ1mn(t)cosαx1sinβx2,ϕ1mn(t)=Φ1eiωtE34
ϕ2=n=1m=1ϕ2mn(t)cosαx1sinβx2,ϕ2mn(t)=Φ2eiωtE35
ψ1=n=1m=1ψ1mn(t)sinαx1cosβx2,ψ1mn(t)=Ψ1eiωtE36
ψ2=n=1m=1ψ2mn(t)sinαx1cosβx2,ψ2mn(t)=Ψ2eiωtE37

where

α=mπa,β=nπbE38

For natural vibrations, substituting equation (Eq. 15) into the equations of motion (Eq. 13), these equations reduce to the following forms:

[C]{UVWΦ1Φ2ψ1ψ2}ω2[M]{UVWΦ1Φ2ψ1ψ2}={0000000}E39

where ωis the natural frequency and

C11=A11α2+A66β2C12=(A12+A66)αβC13=0C14=B11α2+B66β2C15=D11α2+D66β2C16=(B12+B66)αβC17=(D12+D66)αβC21=(A12+A66)αβC22=A66α2+A22β2C23=0C24=(B12+B66)αβC25=(D12+D66)αβC26=B66α2+B22β2C27=D66α2+D22β2C31=0C32=0C33=A55α2+A44β2C34=A55αC35=2B55αC36=A44βC37=2B44βE40
C41=B11α2+B66β2C42=(B12+B66)αβC43=A55αC44=D11α2+D66β2+A55C45=E11α2+E66β2+2B55C46=(D12+D66)αβC47=(E12+E66)αβC51=D11α2+D66β2C52=(D12+D66)αβC53=2B55αC54=E11α2+E66β2+2B55C55=F11α2+F66β2+4D55C56=(E12+E66)αβC57=(F12+F66)αβE41
C61=(B12+B66)αβC62=B66α2+B22β2C63=A44βC64=(D12+D66)αβC65=(E12+E66)αβC66=D66α2+D22β2+A44C67=E66α2+E22β2+2B44C71=(D12+D66)αβC72=D66α2+D22β2C73=2B44βC74=(E12+E66)αβC75=(F12+F66)αβC76=E66α2+E22β2+2B44C77=F66α2+F22β2+4D44E42

By considering relations (Eq. 18), equation (Eq. 17) can be written as:

|CijMijω2|=0E43

By solving equation (Eq. 19) and considering appropriate values for n and m in equation (Eq. 16) the fundamental frequency of a quadrangle FG plate can be obtained.

6. Validation and numerical results

6.1. Validation

The results obtained for a FG plate by applying SSDT are compared with the results obtained by using TSDT as in Ref [5] and the exact solution of [14]. The following non-dimensional fundamental frequencies in Table 1 and Table 2 are obtained by considering material properties the same as [5].

Results in Table 1 and Table 2 show that the values obtained by SSDT are greater than those from TSDT and the exact solution. This is due to the fact that the transverse shear and rotary inertia will have more of an effect on a thicker plate. For the thick plates considered in this

h/a=0.05h/a=0.1h/a=0.2
Present
study
Ref.
[5]
Exact
[14]
Present
study
Ref.
[5]
Exact
[14]
Present
study
Ref.
[5]
Exact
[14]
0.01580.01470.01530.06210.05920.05960.23060.21880.2192

Table 1.

Dimensionless fundamental frequency (ω¯=ωhρmEm) of a simply supported square (Al/Zro2) FG plate (p=1).

case, there is insignificant difference between the result predicted by SSDT and TSDT; SSDT slightly over predicts frequencies. It can be seen that there are good agreements between our results and other results.

p=2p=3p=5
Present
study
Ref.
[5]
Exact
[14]
Present
Study
Ref.
[5]
Exact
[14]
Present
Study
Ref.
[5]
Exact
[14]
0.22920.21880.21970.23060.22020.22110.23240.22150.2225

Table 2.

Dimensionless fundamental frequency (ω¯=ωhρmEm) of a simply supported square (Al/Zro2) FG Plate, thickness-to-side is:h/a=0.2.

Material propertyE(Gpa)ρ(Kg/m3)ν
SUS 304, Metal201.0481660.33
Aluminum, Metal68.927000.33
Zirconia, Ceramic211.045000.33
Si3N4, Ceramic348.4323700.24

Table 3.

Properties of materials used in the numerical example.

6.2. Numerical example

For numerical illustration of the free vibration of a quadrangle FG plate with Zirconia and silicon nitride as the upper-surface ceramic and aluminum and SUS 304 as the lower-surface metal are considered the same as [10]:

6.2.1. Results and discussion for the first ten modes in quadrangular FG plates

In the following Tables, free vibrations are presented in dimensionless form for square and rectangular FG plates.

Tables 4 and 5 show the dimensionless frequency in square (a=b) SUS 304/Si3N4, FG plates. It can be noted that for the same values of grading indexP, the natural frequency increases with increasing mode. The effect of grading index can be shown by comparing the frequency value for the fixed value of mode and changing the values of grading indexp. It can be seen that, the frequency decreases with the increase of the grading index due to the stiffness decreases from pure ceramic to pure metal.

Tables 6 and 7 show the dimensionless frequency in rectangular (b=2a) SUS 304/Si3N4, FG plates. The effect of grading index can be shown by comparing the frequency for the same value of mode and considering different values of grading index pas shown in Table 5. It is clearly visible that the frequency decreases with the increasing grading index, caused by the stiffness decreasing with increasing grading index. For the same value ofp, it can be said that the natural frequency increases with increasing mode. By comparing Tables 6, 7 and 4, 5 it can be observed that for the same values of grading index and mode, the fundamental frequency in square FG plates are greater than those in rectangular FG plates and by

m×nmodep=0p=0.5p=1p=2p=4p=6p=8p=10
1x115.763.9043.3933.0272.7952.6972.6382.597
1x2213.8469.3668.1397.2596.7006.4646.3236.227
2x1313.8469.3668.1397.2596.7006.4646.3236.227
2x2421.35314.44112.54711.18710.3219.9579.7419.593
2x3532.85922.22019.30517.20315.86315.30014.96714.741
3x2632.85922.22019.30517.20315.86315.30014.96714.741
3x3743.36929.32325.47222.68920.91120.16719.72919.431
3x4856.79838.40533.36229.70327.35626.37725.80125.412
4x3956.79838.40533.36229.70327.35626.37725.80125.412
4x41069.05446.69040.55536.09133.22132.02631.32730.856

Table 4.

Variation of the frequency parameter (ϖ=ωa2/hρc/Ec) with the grading index (p) for square. SUS304/Si3N4FG square plates (a/h=10,a=b).

m×nmodep=0p=0.5p=1p=2p=4p=6p=8.p=10.
1x115.3383.6103.1372.7962.5802.4892.4352.398
1x2211.8368.0036.9536.1935.7065.5025.3825.301
2x1311.8368.0036.9536.1935.7065.5025.3825.301
2x2417.26311.67210.1389.0228.3058.0067.8317.714
2x3524.88116.82814.62113.00211.95011.51311.25811.089
3x2624.88116.82814.62113.00211.95011.51311.25811.089
3x3731.35421.20918.42616.37515.034314.47714.15613.943
3x4839.18026.50823.04120.47118.77018.06217.65617.388
4x3939.18026.50823.04120.47118.77018.06217.65617.388
4x41046.02031.14127.06724.03622.02021.18120.70220.387

Table 5.

Variation of the frequency parameter (ϖ=ωa2/hρc/Ec) with the grading index (p) for SUS 304/Si3N4FG square plates (a/h=5,a=b).

m×nmodep=0p=0.5p=1.p=2.p=4p=6p=8p=10
1x113.4612.3412.0341.8141.6741.6161.5801.556
1x225.3383.6103.1372.7962.5802.4892.4352.39
2x1310.3346.9846.0655.4024.9804.8044.700
2x2411.8368.006.9486.1885.7025.4995.3805.300
2x3514.1999.5998.3377.4226.8366.5926.4496.552
3x2620.48413.84512.02010.6899.8359.4829.2769.139
3x3722.37315.12513.13311.67810.74010.35210.1269.976
3x4824.88116.82414.61112.98911.94011.50511.25411.085
4x3931.65621.40918.58516.50615.15714.60214.28214.071
4x41033.71522.80519.80217.58716.14215.54715.20514.979

Table 6.

Variation of the frequency parameter (ϖ=ωa2/hρc/Ec) with the grading index (p) for SUS304/Si3N4FG rectangular plate (a/h=5,a=0.5×b).

m×nmodep=0p=0.5p=1p=2p=4p=6p=8p=10
1x113.6452.4672.1441.9131.7661.7041.6671.642
1x225.7693.9043.3933.0272.7952.6972.6382.597
2x1311.8858.0396.9866.2315.7525.5495.4295.346
2x2413.8469.3658.1387.2586.6996.4636.3236.227
2x3517.03711.52310.0128.9288.2397.9497.7767.658
3x2626.09217.64015.32513.65912.60012.15611.89311.713
3x3728.95819.57817.00815.15813.98113.48713.19512.995
3x4832.85922.21519.29917.19715.85815.29714.96514.739
4x3943.87329.65325.75422.93721.14220.39319.95119.652
4x41047.34432.00227.79424.71522.80921.99921.52221.199

Table 7.

Variation of the frequency parameter (ϖ=ωa2/hρc/Ec) with the grading index (p) for SUS304/Si3N4FG rectangular plate (a/h=10,a=0.5×b).

increasing the side-to-thickness ratio, the frequency also increases. It is evident that the grading index and side-to-thickness ratio effects in frequency are more significant than the other conditions.

6.2.2. Results and discussion for the natural frequency in quadrangular FG (SUS 304/Si3N4) plates

Fig. 3 and fig. 4 illustrate the dimensionless frequency versus grading index (p), for different values of side-to-thickness ratio (a/h) and side-to-side ratio (b/a), respectively.

In Figure 3, the effect of grading index (p) and side-to-thickness ratio (a/h) on dimensionless fundamental frequency of FG (SUS 304/Si3N4) plate is shown. It can be seen that the frequency decreases with increasing grading index, due to degradation of stiffness by the metallic inclusion. It can be observed that the natural frequency is maximum for full-ceramic (p=0.0) and this value increases with the increase of the side-to-thickness ratio, since the stiffness of thin plates is more effectively than the thick plates. It is seen that for the values (p), for 0<p<2the slope is greater than other parts (p>2). It can be said that for side-to-thickness ratios greater than twenty (a/h>20), the frequencies will be similar for different values of grading index. It can be noted that the difference between frequencies in a/h=5and a/h=10are greater than differences of frequency between a/h=10and other curves for the same values of grading indexp. And also it can be concluded that fora/h>20, the difference between the frequencies is small for the same value of grading index.

The effect of grading index (p) and side-to-side ratio (b/a) on dimensionless fundamental frequency of FG (SUS 304/Si3N4) plate can be seen in figure 4. It can be noted that the frequency increases with the increase of the b/asince rectangular plates can be treated as a one-dimensional problem for example, beams or plate strips. It can be observed that the frequency is almost constant for different values of grading index.

Figure 3.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus grading index (p) for different values of side-to-thickness ratio (a/h) in square (b=a) FG (SUS 304/Si3N4) plates.

Figure 4.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus grading index (p) for different values of side-to-side ratio (b/a) FG (SUS 304/Si3N4) plates when a/h=10.0

Fig. (5) and fig. (6) show variation of dimensionless fundamental frequency of FG (SUS 304/Si3N4) plate with side-to-thickness ratio (a/h), for different values of grading index (p) and side-to side ratio (b/a), respectively.

It is seen from figure 5, the fundamental frequency increases with the increase of the value of side-to-thickness ratio (a/h). It is shown that the frequency decreases with the increase of the values of side-to-side (b/a). It can be noted that the slope of frequency versus side-to-thickness ratio (a/h) for part 5<a/h<10is greater than those in another part (a/h>10).

Figure 5.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus side-to-thickness ratio (a/h) for different values of side-to-side ratio (b/a) FG (SUS 304/Si3N4) plates whenp=5.

Figure 6.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus side-to-thickness ratio (a/h) for different values of grading index (p) in square (b=a) FG (SUS 304/Si3N4) plates.

The variation of frequency with side-to-thickness ratio (a/h) for different values of grading index (p) is presented in Figure 6. As expected, by increasing the value of grading index (p) the values of frequency decrease due to the decrease in stiffness. Similarly, in figure (5) while the5<a/h<10, the slope is greater than another ratios. It can be noted that for the values of grading indexp>30, the results for frequency are similar.

Figures 7 and 8 present the variation of dimensionless frequency of FG (SUS 304/Si3N4) plate versus side-to-side ratio(b/a)for different values of grading index(p)and side-to-thickness ratio(a/h), respectively.

Figure 7.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus side-to-side ratio (b=a) for different values of grading index (p) FG (SUS 304/Si3N4) plates whena/h=100.

Figure 8.

Dimensionless frequency (ϖ=ωa2/hρc/Ec) versus side-to-side ratio (b/a) for different values of side-to-thickness ratio (a/h) FG (SUS 304/Si3N4) plates whenp=5.

In figure 7, it is shown that the frequency decreases with the increase of the value of side-to-side ratio (b/a)for all values of grading index(p). It is seen that the frequencies for FG quadrangular plates are between that of a full-ceramic plate and full-metal plate. As expected the frequencies in a full-ceramic plate are greater than those in a full-metal plate.

The results for dimensionless frequency versus side-to-side ratio (b/a)for different values of side-to-thickness ratio (a/h)in FG plate while grading index p=5are shown in figure 8. It is seen that by increasing the value ofb/a, the frequency decreases for all values ofa/h. It can be noted for a/h>10the results are similar.

7. Conclusions

In this chapter, free vibration of FG quadrangular plates were investigated thoroughly by adopting Second order Shear Deformation Theory (SSDT). It was assumed that the elastic properties of a FG quadrangular plate varied along its thickness according to a power law distribution. Zirconia and Si3N4 were considered as a ceramic in the upper surface while aluminum and SUS304 were considered as metals for the lower surface. The complete equations of motion were presented using Hamilton’s principle. The equations were solved by using Navier’s Method for simply supported FG plates.

Some general observations of this study can be deduced here:

  • The decreasing slope of the fundamental frequency for0<p<2, is greater than another part (p>2) for all values of side-to-thickness ratio(a/h)in square FG plate.

  • It was found that the fundamental frequency of the FG plate increases with the increase of the value of side-to-side ratio (b/a).

  • For FG plates, the slope of increasing frequency versus side-to-thickness (a/h)when 5<a/h<10is greater than another part (a/h>10)for any value of grading index and side-to-side ratio.

  • The fundamental frequency versus side-to-side ratio (b/a) for FG quadrangular plates are between those of a full-ceramic plate and full-metal plate whena/h=10.

From the numerical results presented here, it can be proposed that the gradations of the constitutive components are the significant parameter in the frequency of quadrangular FG plates.

Acknowledgments

The authors would like to thank Universiti Putra Malaysia for providing the research grant (FRGS 07-10-07-398SFR 5523398) for this research work.

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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A. Shahrjerdi and F. Mustapha (September 9th 2011). Second Order Shear Deformation Theory (SSDT) for Free Vibration Analysis on a Functionally Graded Quadrangle Plate, Recent Advances in Vibrations Analysis, Natalie Baddour, IntechOpen, DOI: 10.5772/22245. Available from:

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