Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method

Composite Structures 94 (2012) 1398–1405

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Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method Bahar Uymaz a,⇑, Metin Aydogdu b, Seckin Filiz c,d a

Department of Mechanical Engineering, Namik Kemal University, Corlu, Tekirdag, Turkey Department of Mechanical Engineering, Trakya University, Edirne, Turkey c Natural Science Institute, Trakya University, 22180 Edirne, Turkey d Corlu Vocational School, Namık Kemal University, 59860 Tekirdag, Turkey b

a r t i c l e

i n f o

Article history: Available online 20 November 2011 Keywords: FG plate In-plane material homogeneity Natural frequency Mode shape

a b s t r a c t In this study, functionally graded plates which the properties of material varying through the in-plane direction is considered. The analysis is based on a five-degree-of-freedom shear deformable plate theory with different boundary conditions. The vibration solutions are obtained using the Ritz method and assumed displacement functions are in the form of the Chebyshev polynomials. The material properties are assumed to vary as a power form of the in-plane direction. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. Effects of the different material composition, the Poisson ratio and the plate geometry (side–side, side–thickness) on the free vibration frequencies and mode shapes are investigated. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In Functionally Graded Materials (FGMs) the properties of material varying continuously or step-wisely from one side to the other side. Due to the continuous and gradual changes, one side may have high mechanical strength and the other side may have high thermal resistant property. The gradient may be through the thickness or through the in-plane direction and the properties are graded such as crystal structure and particle distribution. Partial change and continuous grading of properties such as composition and structure of a material provides to be a material with multi-functions. Thus, a FGM by changing structure continuously with making use of nanotechnology. Although adequate attention has been devoted to the free vibration problem of FGM structures (including beams, plates and shells) which the properties of material varying through the thickness direction [1–6], there are few investigations about these structures, which the properties of material varying through the one or two inplane direction [7–10]. Because of this, there is a few knowledge about these structures vibration frequencies and mode shapes. Recently, Lü et al. [11], obtained semi-analytical elasticity solutions for bending and thermal deformations of 2D-FGM beam with different boundary conditions. Liu et al. [12], considered the free vibration of a FG plate with in-plane material variation by using a Levy-type solution. In this paper, the vibration problem of FG plate is considered for the properties of material varying through the one in-plane direc-

⇑ Corresponding author. Tel.: +90 0 282 652 94 76; fax: +90 0 282 652 93 72. E-mail address: [email protected] (B. Uymaz). 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.11.002

tion by using the Ritz method. In Ritz method, the five unknown displacement functions are assumed in the double series form of Chebyshev polynomials of the lateral coordinates of the plates. The free vibration frequencies and mode shapes are investigated for different plate geometry and material compositions. 2. Theoretical formulations Consider a functionally graded rectangular plate having a length a and width b and a constant thickness h. The plate geometry and dimensions are defined with respect to a cartesian coordinate system (x, y, z) and the origin is placed at the center of the plate and the axes are parallel to the edges of the plate. The corresponding displacement components u, v and w along the x, y and z directions, respectively. A linear elastic material behavior is considered. The properties of the plate are assumed to vary through the inplane direction with a desired variation of the volume fractions of the two materials in between two surfaces. The effective material properties of the functionally graded plate is obtained by using a simple power law distribution as follows,

P ¼ V m Pm þ V c Pc

ð1Þ

Vm þ Vc ¼ 1  p x 1 þ Vc ¼ a 2

ð2Þ

 p x 1 PðxÞ ¼ ðPc  Pm Þ þ þ Pm a 2

ð3Þ ð4Þ

where P denotes a material property such as elasticity modulus E, the Poisson ratio m and mass density q and p is the volume fraction

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B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405

uðx; y; z; tÞ ¼ u0 ðx; y; tÞ  zw0;x þ ø1 ðzÞu1 ðx; y; tÞ

Table 1 Boundary functions corresponding different boundary conditions.

v ðx; y; z; tÞ ¼ v 0 ðx; y; tÞ  zw0;y þ ø2 ðzÞv 1 ðx; y; tÞ

Boundary condition

fx1 ðXÞ

fu1 ðXÞ

fu1 ðXÞ

F–F S–F S–S C–F C–S C–C

1 1 1 1+X 1+X 1  X2

1 1+X 1  X2 1+X 1  X2 1  X2

1 1+X 1  X2 1+X 1  X2 1  X2

wðx; y; z; tÞ ¼ w0 ðx; y; tÞ where u0, v0, w0, u1 and v1 are the five unknown displacement functions and U1 and U2 represent the shape functions determining the distribution of the transverse shear stresses along the thickness. These shape functions are defined according to shear deformable plate theories as follows

CPT : U1 ðzÞ ¼ U2 ðzÞ ¼ 0;

index, 0 6 p 6 1. Assume that the plate material is made from ceramic and metal, and it is of full metal at x = a/2 and of full ceramic at x = a/2. The effective material properties of FG plate is the elasticity modulus, Poisons ratio and mass density. From the five-degree-of-freedom shear deformable plate theory the displacement field for the considered plate can be written as

1,8 1,6 1,4 1,2 1,0 0

2

4

p

6

8

10

1,8 1,6 1,4 1,2 1,0 0,8 0,6

12

1,3 FSDPT PSDPT ESDPT2

SSSS

2,0

0

2

4

(a)

6

p

8

10

FSDPT PSDPT ESDPT2

CSFS

1,2

Frequency Parameter

FSDPT PSDPT ESDPT2

Frequency Parameter

Frequency Parameter

SSCS

2,0

0,8

USDPT : U1 ðzÞ ¼ U2 ðzÞ ¼ z; ½13   4z2 ð6Þ PSDPT : U1 ðzÞ ¼ U2 ðzÞ ¼ z 1  2 ; ½14 3h 2ð z Þ2 z h ½15; U1 ðzÞ ¼ U2 ðzÞ ¼ za In/ ; ½16: ESDPT : U1 ðzÞ ¼ U2 ðzÞ ¼ 2ðhz Þ2 e

2,2

2,2

ð5Þ

1,1 1,0 0,9 0,8 0,7 0,6 0,5

12

0

2

4

(b)

p

6

8

10

12

(c)

Fig. 1. Variation of frequency parameter with p index and different theories for different boundary conditions (a/h = 20, a/b = 1).

6

5

a/b=0.5 a/b=1 a/b=2

4 3 2 1 0

0

2

4

p

6

8

10

5 4 3 2 1 0

12

1,8 a/b=0.5 a/b=1 a/b=2

SSSS

Frequency Parameter

SSCS

Frequency Parameter

Frequency Parameter

6

0

2

(a)

4

p

6

8

10

12

CSFS

1,6

a/b=0.5 a/b=1 a/b=2

1,4 1,2 1,0 0,8 0,6 0,4

0

2

(b)

4

p

6

8

10

12

(c)

Fig. 2. Variation of frequency parameter with p index and a/b ratio for different boundary conditions (a/h = 50, ESDPT).

2,2 a/h=10 a/h=20 a/h=50 a/h=100

2,0 1,8 1,6 1,4 1,2 1,0 0,8

0

2

4

p

6

(a)

8

10

12

1,4

2,0

SSSS

Frequency Parameter

SSCS

2,2

Frequency Parameter

Frequency Parameter

2,4

a/h=10 a/h=20 a/h=50 a/h=100

1,8 1,6 1,4 1,2 1,0 0,8 0,6

0

2

4

p

6

(b)

8

10

12

CSFS

1,2

a/h=10 a/h=20 a/h=50 a/h=100

1,0 0,8 0,6 0,4 0,2

0

2

4

p

6

(c)

Fig. 3. Variation of frequency parameter with p index and a/h ratio for different boundary conditions (a/b = 1, ESDPT).

8

10

12

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The stress–strain relationships are given by the generalized Hooke’s law as follows

4,5 a/b=0.5 p=0.5

Frequency Parameter

4,0

2

3,5 3,0

SSCS SSSS CSFS

2,5

Q 11

Q 12

0

Q 11

0

0

0

Q 66

0 Q 66

0 0

sym:

0

0

32

ex

3

76 e 7 76 y 7 76 7 76 eyz 7 76 7 76 7 54 exz 5

ð7Þ

exy

Q 66

where Qij are the stiffnesses defined as follows,

1,5

Q 11 ¼ 0

20

40

60

80

100

120

a/h Fig. 4. Variation of frequency parameter with a/h ratio and different boundary conditions (ESDPT).

1 2

4,5 a/h=50 p=0.5

4,0

Q 12 ¼

mðxÞEðxÞ 1  mðxÞ2

;

Q 66 ¼

EðxÞ 2ð1 þ mðxÞÞ

Z Z Z

ð8Þ

ðrx ex þ ry ey þ rxz exz þ ryz eyz þ rxy exy ÞdV

ð9Þ

V

ex ¼ u0;x  zw0;xx þ U1 ðzÞu1;x ; ey ¼ v 0;y  zw0;yy þ U2 ðzÞv 1;y ; cyz ¼ U02 v 1 ; cxz ¼ U01 u1 ; cxy ¼ u0;y þ v 0;x  2zw0;xx þ U1 u1;y þ U2 v 1;x

3,0 2,5 2,0 1,5

ð10Þ

@ where a prime denotes the derivative with respect to z and, x ¼ @x . The kinetic energy T of the plate can be written as

1,0 0,5

;

eij (i, j = x, y, z) are the strain components as follows

SSCS SSSS CSFS

3,5

EðxÞ 1  mðxÞ2

The linear elastic strain energy Ve for a rectangular plate based on a five-degree-of-freedom shear deformable plate theory can be written as

Ve ¼

Frequency Parameter

2

sxy

2,0

1,0

3

rx

6r 7 6 6 y7 6 7 6 6 6 syz 7 ¼ 6 7 6 6 7 6 6 4 sxz 5 4

0

1

2

a/b

3

4

5

T¼

Fig. 5. Variation of frequency parameter with a/b ratio and different boundary conditions (ESDPT).





qðxÞ ðu;t Þ2 þ ðv ;t Þ2 þ ðw;t Þ2 dV

6,0

5 4 p=0 p=0.5 p=1 p=5 p=10

3 2 1

SSCS 0

5,8 5,6 5,4 5,2 5,0 4,8

20

30

40

4,4

50

p=0 p=0.5 p=1 p=5 p=10

SSCS

4,6

10

0

10

20

30

Ec/Em

Ec/Em

(a)

(b)

40

50

Fig. 6. Variation of frequency parameter with Ec/Em ratio and p index for (a) qc/qm = 1 and (b) qc/qm = Ec/Em (=2, 5, 10, 20, 40), (a/h = 20, a/b = 0.5, ESDPT). 6,0

SSSS

5

Frequency Parameter

Frequency Parameter

6

4 3 2 p=0 p=0.5 p=1 p=5 p=10

1 0

0

10

20

30

40

50

ð11Þ

V

6,2

Frequency Parameter

Frequency Parameter

Z Z Z

where q is the mass density per unit volume. The maximum energy functional P of the rectangular plate is defined as

6

0

1 2

SSSS

5,8 5,6

p=0 p=0.5 p=1 p=5 p=10

5,4 5,2 5,0 4,8 4,6

0

10

20

30

Ec/Em

Ec/Em

(a)

(b)

40

50

Fig. 7. Variation of frequency parameter with Ec/Em ratio and p index for (a) qc/qm = 1 and (b) qc/qm = Ec/Em (=2, 5, 10, 20, 40), (a/h = 20, a/b = 0.5, ESDPT).

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B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405 1,65

CSFS

1,6

Frequency Parameter

Frequency Parameter

1,8

1,4 1,2

p=0 p=0.5 p=1 p=5 p=10

1,0 0,8 0,6 0,4

0

10

20

30

40

1,55 1,50 1,45

p=0 p=0.5 p=1 p=5 p=10

1,40 1,35 1,30 1,25

50

CSFS

1,60

0

10

20

30

Ec/Em

Ec/Em

(a)

(b)

40

50

SSCS

Ec/Em=2 Ec/Em=5 Ec/Em=10 Ec/Em=20 Ec/Em=40

0

10

20

30

40

50

5,6 5,4 5,2 5,0 4,8 4,6 4,4 4,2 4,0 3,8

1,6

SSSS

Frequency Parameter

6,4 6,2 6,0 5,8 5,6 5,4 5,2 5,0 4,8 4,6 4,4

Frequency Parameter

Frequency Parameter

Fig. 8. Variation of frequency parameter with Ec/Em ratio and p index for (a) qc/qm = 1 and (b) qc/qm = Ec/Em (=2, 5, 10, 20, 40), (a/h = 20, a/b = 0.5, ESDPT).

Ec/Em=2 Ec/Em=5 Ec/Em=10 Ec/Em=20 Ec/Em=40

0

10

20

30

40

1,5 CSFS

1,4

Ec/Em=2 Ec/Em=5 Ec/Em=10 Ec/Em=20 Ec/Em=40

1,3 1,2 1,1

50

0

10

20

ρc/ρm

ρc/ρm

ρc/ρm

(a)

(b)

(c)

30

40

50

Fig. 9. Variation of frequency parameter with qc/qm ratio and Ec/Em ratio for different boundary conditions (a/h = 20, a/b = 0.5, p = 0.5, ESDPT).

Table 2 Comparison of frequency parameters of square isotropic plate for SSSS boundary condition with 3D solution. a/h

Liew et al. [17]

PSDPT

ESDPT

2 5 10 100

1.2590 1.7758 1.9342 1.9993

1.4422 1.8466 1.9570 1.9974

1.3038 1.7832 1.9339 1.9974

Table 3 The first five frequency parameters of FG plate for SSCS boundary condition with ESDPT (a/b = 0.5, a/h = 20, ESDPT). Mode

D1 D2 D3 D4 D5

p 0

0.5

1

5

10

5.6356 8.6861 14.3407 20.0231 22.4877

3.9573 6.0745 9.9675 14.0679 14.3512

3.3481 5.1478 8.4172 10.1045 12.0770

2.5000 3.8794 5.6435 6.3639 9.1788

2.4020 3.7282 6.1196 8.6746 9.7494

Table 4 The first five frequency parameters of FG plate for SSSS boundary condition with ESDPT (a/b = 0.5, a/h = 20, ESDPT). Mode

D1 D2 D3 D4 D5

p 0

0.5

1

5

10

4.9739 7.9340 12.9501 16.7049 19.6030

3.4943 5.5407 8.9916 11.7411 13.7770

2.9614 4.7010 7.6091 10.0847 11.7560

2.2293 3.5707 5.8008 7.6745 8.9972

2.1400 3.4351 5.5821 7.2554 8.5486

where x is radial frequency. The displacement amplitude functions U(X, Y), V(X, Y), W(X, Y), U1(X, Y) and V1(X, Y) are written in terms of nondimensional coordinates in the form of Chebyshev polynomials as follows

UðX; YÞ ¼ F u ðX; YÞ

i¼1

VðX; YÞ ¼ F v ðX; YÞ

P ¼ V e max  T max

ð12Þ

According to the vibration problem displacement components are assumed as

1 X 1 X

1 X 1 X

WðX; YÞ ¼ F W ðX; YÞ

U 1 ðX; YÞ ¼ F u1 ðX; YÞ

u1 ðx; y; tÞ ¼ U 1 ðx; yÞ sin xt

m¼1 n¼1 1 X 1 X

C mn Pm ðXÞPn ðYÞ

ð14Þ

Dpq þ Pp ðXÞPq ðYÞ

p¼1 q¼1

v 0 ðx; y; tÞ ¼ Vðx; yÞ sin xt v 1 ðx; y; tÞ ¼ V 1 ðx; yÞ sin xt

Bkl Pk ðXÞPl ðYÞ

k¼1 l¼1 1 X 1 X

u0 ðx; y; tÞ ¼ Uðx; yÞ sin xt w0 ðx; y; tÞ ¼ Wðx; yÞ sin xt

Aij Pi ðXÞPj ðYÞ

j¼1

V 1 ðX; YÞ ¼ F v 1 ðX; YÞ ð13Þ

1 X 1 X

Ers Pr ðXÞPs ðYÞ

r¼1 s¼1

where X = 2x/a, Y = 2y/b and Z = 2z/h are nondimensional parameters, Ps(f) (s = 1, 2, 3, . . .; f = X, Y) is the one-dimensional

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B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405

sth Chebyshev polynomial which can be written in terms of cosine functions as follows

Table 5 The first five frequency parameters of FG plate for CSFS boundary condition with ESDPT (a/b = 0.5, a/h = 20, ESDPT). Mode

Ps ðfÞ ¼ cos½ðs  1Þ arccosðfÞ;

p

D1 D2 D3 D4 D5

0

0.5

1

5

10

1.6792 4.8010 6.6812 9.9006 11.2177

1.3190 2.5514 3.8561 6.4899 7.9186

1.2369 3.3451 5.0498 6.8474 9.2865

0.8485 2.5593 3.4424 5.2868 6.6308

0.7516 2.3216 3.6387 4.8405 6.5405

ðs ¼ 1; 2; 3; . . .Þ

The boundary functions Fg(X, Y); (g = u, v, w, u1, v1), respectively, corresponding to the displacement amplitude functions are written as

F g ðX; YÞ ¼ fg1 ðX; YÞfg2 ðX; YÞ;

SSCS

ðg ¼ u; v ; w; u1 ; v 1 Þ

SSSS

-12

-4

x 10

x 10 1

0

0.5

0

ð16Þ

CSFS

-13

x 10 1

ð15Þ

-2

0

-1

-4

-0.5

-2

-6

-1

-3 1

-1.5 1 0.5 0 -0.5 -1

Y

-1

0

-0.5

-8 1

1

0.5

0.5 0 -0.5 -1

Y

X

Δ1

-1

0

-0.5

1

0.5

0.5 0 -0.5

Δ1

-12

x 10

6

4

-0.5

4

2 0 -2

-2 1

-1 -1.5 -2 -2.5

-4 1 0.5 0 -1

Y

-1

0

-0.5

1

0.5

-3 1 0.5 0 -0.5 -1

Y

X

Δ2

-1

0

-0.5

0.5

1

-12

0 -0.5

0

-5

0.5 0 -0.5

-1 -1

Y

0

0.5

0.5

0

0

-2

-0.5

-4

0 -0.5 -1

Y

X

Δ3

-1

0

-0.5

0.5

1

0.5 0 -0.5

-14

1

2

0

1.5

-1

1

-2

0.5

-3

0

1

-4 1

-0.5 1

Y

-0.5

-1 -1

0

X

Δ4

0.5

1

X

2.5

-2

0

1

x 10

1

0.5

0.5

-4

x 10 2

-1

-1

0

-0.5

Δ3

2

-0.5

-1

Y

X

Δ3

0

X

-6 1 0.5

-11

1

-4

2

1

0.5

x 10

1

-1 1

-10 1

-1

0

-0.5

Δ2

x 10

-0.5

-1

Y

X

-13

x 10

x 10

0.5

Δ2

5

X

x 10 0

0

1

-4

-14

x 10 6

2

-0.5

0.5

Δ1

8

-0.5

-1 -1

Y

X

0

0.5 0 -0.5

Y

-1

-1

Δ4

0

-0.5

X

0.5

1

0.5 0 -0.5

Y

-1

-1

Δ4

Fig. 10. Mode shapes of the first four frequency parameter for U displacement (a/h = 20, a/b = 0.5, p = 0.5, ESDPT).

0

-0.5

X

0.5

1

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B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405

SSCS

SSSS

-13

CSFS

-14

-3

x 10

x 10

x 10

20

4

1

15

2

0.5

10 0

0 5

-2

-0.5

0

-4 1

-5 1 0.5 0 -0.5 -1

Y

-1

0

-0.5

0.5

1

-1 1 0.5 0 -0.5 -1

Y

X

Δ1

-1

0

-0.5

0.5

1

0.5 0 -0.5

Δ1

-13

-14

-4

2

4

0

2 0

-2

-2

-2

-4 1

-4 1

-4 1 0.5 0 -0.5 -1

-1

-0.5

0

0.5

0.5

1

0 -0.5

X

-1

Y

-1

Δ2

0

-0.5

0.5

0.5

1

0 -0.5

-12

2 0

2 0

-2

-2

-1 1

-4 1

-4 1 0.5 0 -1 -1

Y

0

-0.5

0.5

0.5

1

0 -0.5 -1

Y

X

-1

Δ3

0

-0.5

0.5

0.5

1

0 -0.5 -1

Y

X

-1

Δ3

-12

-14

8

2

6

1

0

-0.5

0.5

1

X

Δ3 -4

x 10

x 10

x 10 3

X

-4

4

-0.5

1

4

6

0

-0.5

0.5

x 10

x 10 8

0.5

0

Δ2

-14

x 10

-0.5

-1 -1

Y

X

Δ2

1

X

x 10

6

0

-0.5

1

4

8

2

-1

0.5

Δ1

x 10

x 10 4

Y

-1

Y

X

0

1.5 1 0.5

4

0

0 2

-1

-0.5

-2

0

-1

-3 1

-2 1

-1.5 1

0.5 0 -0.5

Y

-1

-1

0

-0.5

0.5

1

X

0.5 0 -0.5

Y

-1 -1

Δ4

0

-0.5

X

0.5

1

0.5 0 -0.5

Y

-1

-1

Δ4

-0.5

0

0.5

1

X

Δ4

Fig. 11. Mode shapes of the first four frequency parameter for V displacement (a/h = 20, a/b = 0.5, p = 0.5, ESDPT).

Boundary conditions can be written in the following form for x = constant:

Free edgeðFÞ : rx ¼ 0; sxy ¼ 0; sxz ¼ 0 Simply supported edgeðSÞ : w ¼ 0; v ¼ 0; rx ¼ 0 Clamped edgeðCÞ : w ¼ 0; v ¼ 0; u ¼ 0

ð17Þ

The boundary components fgs ðg ¼ u; v ; w; u1 ; v 1 ; s ¼ 1; 2Þ corresponding to different boundary conditions for x-axis are given in Table 1. Substituting the displacement amplitude functions into each energy equation and minimizing the functional P with respect to

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B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405

SSCS

SSSS

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5 1

CSFS 4 3 2 1

-2.5 1 0.5 0 -0.5 -1

Y

-1

0

-0.5

0.5

0 1 0.5

1

0 -0.5 -1

Y

X

Δ1

-1

0

-0.5

0.5

0.5

1

0 -0.5 -1

Y

X

Δ1

2

2

1

1

0

0

-1

-1

-2 1

-2 1

-1

0

-0.5

0.5

1

X

Δ1

4 3 2 1

0.5 0 -0.5 -1

Y

-1

0

-0.5

0.5

1

0 -1 1 0.5 0 -0.5 -1

Y

X

Δ2

-1

0

-0.5

0.5

1

0.5 0 -0.5

Δ2

1

2

3

1.5

2

0

1

1

-0.5

0.5

0

-1

0

-1

-1.5

-0.5

-2

-2 1

-1 1 0.5 0 -1

Y

-1

0

-0.5

0.5

1

0 -0.5 -1

Y

X

Δ3

1

1

0

0

-1

-1

-2 1

-2 1 0.5 0 -0.5 -1

Y

-1

-1

0

-0.5

0.5

1

0

-0.5

X

0.5

1

X

0.5 0 -0.5 -1

Y

X

Δ3

2

-0.5

1

-3 1 0.5

2

-1

0.5

Δ2

0.5

-0.5

-1

Y

X

0

-1

0

-0.5

0.5

1

X

Δ3

3 2 1 0 -1 -2 -3 1 0.5 0 -0.5

Y

-1

Δ4

-1

0

-0.5

0.5

1

0.5 0 -0.5

Y

X

Δ4

-1

-1

0

-0.5

0.5

1

X

Δ4

Fig. 12. Mode shapes of the first four frequency parameter for W displacement (a/h = 20, a/b = 0.5, p = 0.5, ESDPT).

unknown coefficients Aij, Bkl, Cmn, Dpq, Ers the following frequency equation is obtained:

3. Numerical results

  ½K  X2 ½M f½X g ¼ 0

In this section, various numerical results are presented for the following material properties which used in the present study,

where X ¼ xa respectively.

qffiffiffiffi qc Ec

ð18Þ

, [K] and [M] are stiffness and mass sub-matrices,

SUS304 Si3N4

E = 201.04 GPa E = 348.43 Gpa

m = 0.3262 m = 0.2400

q = 8166 kg/m3 q = 2370 kg/m3

B. Uymaz et al. / Composite Structures 94 (2012) 1398–1405

To facilitate comparison, the non-dimensional frequency parameters are expressed as

D2 ¼ X2

" # 2 b D

ð19Þ

p4 2

m Þ where D ¼ 12ð1 . The free vibration frequencies and mode shapes h2 are obtained for different plate geometry, material compositions and different boundary conditions. In Table 2, fundamental frequency parameter of simply supported square isotropic plates are compared with 3D elasticity solutions and good agreement is observed between results. The variation of fundamental frequency with respect to the various parameter like the material composition (p index), the aspect ratio (a/b) and the side-to-thickness ratio (a/h) of considered plate with different boundary conditions and different plate theories are given in Figs. 1 and 5. The frequency parameters obtained with different shear deformation theories are close to each other. It is observed that the frequency parameter decreases for plates with increasing p values and significant differences are obtained for the values of 0 6 p 6 5. This is because the plate rigidity becomes weak with the higher values of p index. The frequency parameter decreases as a/b is increased and the effect of the aspect ratio is predominant for the values of a < b. The results show that in all cases, the frequency parameters are increased when the plates become thinner and increasing constraints at the boundaries increases the frequency parameter (see Figs. 2–4). In Figs. 6–8, effect of the constant qc/qm ratio and variable qc/ qm ratio on frequency parameters are investigated. In the first situation (constant qc/qm ratio), frequency parameter decreases as Ec/Em and p are increased. In the second situation (variable qc/qm ratio), frequency parameter decreases as Ec/Em is increased for p < 5 and increases as Ec/Em is increased for p > 5. With the same Ec/Em ratio, frequency parameter decreases as p index is increased for p < 5 and frequency parameter increases as p index is increased for p > 5. In all cases considered, it is observed that the value of 5 for p index is critical value. Additionally, it is investigated that the effect of the qc/qm ratio on frequency parameter with Ec/Em ratio and it is according to this figure, the frequency parameter increases for plates with increasing qc/qm ratio and Ec/Em ratio and the sharpest increasing is observed for the value of qc/qm ratio is 1 6 qc/qm 6 10 (see Fig. 9 and Table 2). Tables 3–5 show the first five frequency parameters for different values of p index and boundary conditions. It is observed that the high frequencies decrease with increase in the p index same as fundamental frequency parameters. The mode shapes of the first four frequency parameter of considered plate for displacement defined in Eq. (13) are shown in Figs. 10–12.

1405

4. Conclusions The free vibration problem of functionally graded plates with different boundary conditions are investigated using Ritz method. The variation of the effective material properties are assumed that through the one in-plane direction. The formulation is based on the higher-order shear deformation plate theory and the displacement fields are expressed in terms of the Chebyshev polynomials. Numerical results are given for different material composition (p index), the aspect ratio (a/b) and the side-to-thickness ratio (a/h) of considered plate with different boundary conditions and different plate theories. The effect of the variation of the qc/qm ratio on the frequency parameters is investigated and it is observed that the variation of the qc/qm ratio significantly influence the frequency parameters and can not be discounted. Finally, the mode shapes of the first four frequency parameter of considered plate are given for considered boundary conditions. References [1] Abrate S, Vibration Free. Buckling and static deflections of functionally graded plates. Compos Sci Technol 2006;66:2383–94. [2] Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des 2007;28:1651–6. [3] Ferreira AJM, Batra RC, Roque CMC, Qian LF, Jorge RMN. Natural frequencies of functionally graded plates by a meshless method. Compos Struct 2006;75:593–600. [4] Kim YW. Temperature dependent vibration analysis of functionally graded rectangular plates. J Sound Vib 2005;284:531–49. [5] Uymaz B, Aydogdu M. Three-dimensional vibration analysis of functionally graded plates under various boundary conditions. J Reinf Plast Compos 2007;26:1847–63. [6] Uymaz B, Aydogdu M. Vibration analysis of functionally graded plates with simply supported edge condition 4. Ankara international aerospace conference, METU, Ankara; 2007. [7] Aboudi J, Pindera MJ, Arnold S. Thermoelastic theory for the response of materials functionally graded in two directions. Int J Solids Struct 1996;33(7):931–66. [8] Aboudi J, Pindera MJ, Arnold S. Thermoplasticity theory for bi-directionally functionally graded materials. J Therm Stress 1996;19(9):809–61. [9] Nemat-Alla M. Reduction of thermal stresses by developing two-dimensional functionally graded materials. Int J Solids Struct 2003;40(26):7339–56. [10] Qian LF, Ching HK. Static and dynamic analysis of 2-D functionally graded elasticity by using meshless local Petrov–Galerkin method. J Chin Inst Eng 2004;27(4):491–503. [11] Lü CF, Chen WQ, Xu RQ, Lim CW. Semi-analytical elasticity solutions for bidirectional functionally graded beams. Int J Solids Struct Compos Struct 2008;45:258–75. [12] Liu DY, Wang CY, Chen WQ. Free vibration of FGM plates with in-plane material inhomogeneity. Compos Struct 2010;92(5):1047–51. [13] Yang PC, Norris CH, Stavsky Y. Elastic wave propagation in heterogeneous plates. Int J Solids Struct 1966;2(4):665–84. [14] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech-T ASME 1984;51(4):745–52. [15] Karama M, Afaq KS, Mistou S. Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int J Solids Struct 2003;40(6):1525–46. [16] Aydogdu M. A new shear deformation theory for laminated composite plates. Compos Struct 2009;89:94–101. [17] Liew KM, Hung KC, Lim MK. A continuum three-dimensional vibration analysis of thick rectangular plates. Int J Solids Struct 1993;30:3357–79.