Hermite radial basis collocation method for vibration of functionally graded plates with in-plane material inhomogeneity

Computers and Structures 142 (2014) 79–89

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Hermite radial basis collocation method for vibration of functionally graded plates with in-plane material inhomogeneity Fuyun Chu, Lihua Wang ⇑, Zheng Zhong, Jianzhang He School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, PR China

a r t i c l e

i n f o

Article history: Received 6 April 2014 Accepted 9 July 2014 Available online 3 August 2014 Keywords: Hermite radial basis collocation method Free vibration FGM thin plate In-plane inhomogeneity

a b s t r a c t A meshfree Hermite-type collocation method with radial basis functions is presented to investigate the free vibration of thin FGM plates with in-plane material inhomogeneity. For the eigenvalue analysis of thin plate vibration, conventional radial basis collocation method (RBCM) leads to an over-determined system for the eigenproblems, while Hermite radial basis collocation method (HRBCM) introducing more degrees of freedom can result in a determined system. Convergence and comparison studies with analytical solutions demonstrate the effectiveness and accuracy of the proposed method. Effects on the natural frequencies and mode shapes originated from the material inhomogeneity are numerically investigated. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials (FGMs) are composites with compositions varying continuously in spatial directions, which lead to mechanical properties varying smoothly in space for a predetermined functional performance. A number of researches paid attention to the investigation of FGM structures with material inhomogeneity along the thickness direction [1–5]. However, structures of material properties varying in-plane direction are short of exploration. Special mechanical properties can be introduced if the material compositions vary in-plane direction, for example, variable rigidity, internal force distribution, natural frequencies and buckling load, etc. Such mechanical properties provide more design opportunities in engineering applications. The coefficients of the differential equations for Kirchhoff thin plate model of FGMs are variables compared to the constant coefficients for the thin plate model of homogeneous materials. Analytical solutions are difficult to obtain for such partial differential equations with variable coefficients. Therefore, few studies are dedicated to this area. Uymaz et al. [6] investigated the vibration of functionally graded plates with material properties varying through the in-plane direction based on Ritz method, using a five-degree-of-freedom shear deformable plate model. Yu et al. [7] analyzed the bending problem of a thin rectangular plate with in-plane stiffness changing through a power form. A Levy-type

⇑ Corresponding author. E-mail address: [email protected] (L. Wang). http://dx.doi.org/10.1016/j.compstruc.2014.07.005 0045-7949/Ó 2014 Elsevier Ltd. All rights reserved.

form was introduced for the transverse displacement which resulted in a Whittaker equation and then an analytical solution could be attained by solving this equation. Further, a semi-analytical method composed of a Levy-type solution and a particular integration solution was employed by Liu et al. [8] to explore the free vibration of anisotropic rectangular functionally graded plate with in-plane material inhomogeneity. Although analytical solutions can provide benchmark results for assessing approximate theories, only special cases can be analytically solved. Solutions of general cases still lie on the numerical methods. Finite element method (FEM) and boundary element method (BEM) are two common numerical methods. In FEM, construction of C1 conforming finite element approximation causes severe difficulties in variational formulations of thin plate model. The need of higher grid quality and density also increases the workload and complexities for problems of material inhomogeneity. Carrera and Giunta [9] and Carrera [10] proposed powerful unified formulations (CUF) for the analysis of beams, plates and shells for the finite element analysis, in which the governing equations and the boundary conditions were derived in terms of a fundamental nucleo that did not depend upon the approximation order. As a result, the numerical solutions give the complete threedimensional displacement for the cross-section. In BEM, it is quite hard to procure the fundamental solutions for FGM problems. In recent years, meshfree methods have attracted considerable attentions for no need of mesh and higher accuracy because high continuity shape functions can be adopted. Meshfree radial basis collocation method (RBCM) is a truly meshfree method, based on collocation scheme and radial basis approximation which is

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

infinitely differentiable. It is prominent from the meshfree methods for its easy implementation, high accuracy and spectral convergence [11–14], which is an outstanding candidate for solving differential equations with variable coefficients. Radial basis functions (RBFs) were firstly introduced by Kansa [15,16] to solve partial differential equations, where multiquadric radial basis functions (MQ RBFs) were successfully used as the spatial approximation for parabolic, hyperbolic and elliptic Poisson’s equations. After that, RBFs were introduced for elasticity problems of FGMs. Ferreira [17] investigated the free vibration of Timoshenko beams and Mindlin plates based on the radial basis approximation associated with collocation method by solving the corresponding eigenproblems. RBCM was also adopted to analyze the bending, free vibration and buckling of thick homogeneous plates based on higher-order shear deformation theory [18–20]. Associated with some other techniques, such as pseudospectral method [21], differential quadrature collocation [22] and finite differences method [23], RBFs were also extended to solve nonhomogeneous materials problems. Xiang and Wang [24] and Xiang et al. [25] studied the free vibration of laminated composite plates based on trigonometric shear deformation theory and three dimensional isotropic elasticity theory using RBCM. Approximation of RBFs possesses high accuracy and spectral convergence. However, the accuracy of derivatives of interpolating functions is usually very poor on the boundary when collocation method is used. In order to enhance the accuracy for the derivatives of the approximation functions, Wu [26] constructed the Hermite-type RBFs method, in which both the RBFs and their derivatives are employed in the approximation functions. Fasshauer [27] introduced this Hermite type collocation method associated with RBFs to solve partial differential equations. Moreover, Zhang et al. [28] studied a 2D elastic problem and Liu et al. [29] investigated a bending problem of Kirchhoff plates based on Hermitebased collocation. It had been proven in these works that the accuracy of the solutions could be improved significantly by using the Hermite-type collocation method. In this paper, Hermite radial basis collocation method (HRBCM) is introduced for the vibration analysis of FGM plates with in-plane material inhomogeneity. Not only the accuracy can be improved on the boundaries of the domain compared to conventional RBCM, but also Hermite collocation can directly solve the eigenproblem of Kirchhoff plates which is a determined system, compared to an over-determined system introduced by RBCM. The influences of inhomogeneity on the frequencies and vibration modes are also discussed. 2. Collocation with radial basis functions 2.1. Radial basis functions Interpolation of radial basis function for a function u(x) can be expressed as h

uðxÞ  u ðxÞ ¼

Ns X

uI ðxÞaI ; x 2 X

ð1Þ

I¼1

where Ns is the number of source points in X, uI(x) is the radial basis function and aI is the expansion coefficient. Typical RBFs uI(x) include Multiquadric (MQ)

uI ðxÞ ¼ ðr2I þ c2 Þ

n32

;

n ¼ 1; 2; 3; . . .

ð2Þ

Gaussian

 2 r g I ðxÞ ¼ exp  I2 c

ð3Þ

Polyharmonic splines

( g I ðxÞ ¼

r 2n I ln r I

ð4Þ

r I2n1

Thin plate splines

g I ðxÞ ¼ r 2I ln r I

ð5Þ

where r I ¼ kx  xI k denotes the radial distance from the center, and c is the shape parameter. MQ RBF is well known for possessing the following spectral convergence property

ku  uh kL1 ðXÞ  C 1 gc=h kukt

ð6Þ

where h is the characteristic node distance, 0 < g < 1 is a real number, C1 is a generic constant independent of c and h, and k  kt is the induced form defined in [30]. RBFs are radial and positive definite. For a given point set S and a continuous function f, there is a unique interpolant uI(x) from xj  S which agrees with f in the problem domain. Generally, such interpolants can be constructed to interpolate appropriate smooth function on any set of linear functional which are linearly independent in the space of test functions. These flexibilities provide RBFs good potentiality of application in collocation methods. Further, Cheng et al. [31] proposed an error estimate for MQ RBFs with collocation method as pffi c=h

ku  uh kL1 ðXÞ  C 2 g

kukt

ð7Þ

where C2 is also a generic constant independent of c and h. 2.2. Direct collocation Consider a boundary value problem as follows

Lu ¼ f;

in X

Bu ¼ g;

on C

ð8Þ

where L is the differential operator in the open domain X, B is the differential operator on the boundary C, X ¼ X [ C, u is the problem unknown, f is the source term, and g is the term associated with boundary conditions. For the problem unknown u, the approximation represented by RBFs can be expressed as

u  uh ¼

Ns X

uI ðxÞaI ¼ UT a

ð9Þ

I¼1

where uh ¼ ½uh1 ; uh2 ; uh3 , aTI ¼ ½a1I ; a2I ; a3I , aT ¼ ½a1 ; . . . ; aNs , UT ¼ ½U1 ; . . . ; UNs , UI = uII, uI is the radial basis function centered at source Np point xI, and Ns is the number of source points. Define fpI gI¼1 # X, Nq fqI gI¼1 # C as the collocation points in domain X, and on the boundary C, respectively. By introducing approximation (9) into strong form (8) to be evaluated at the collocation points in the domain and on the boundaries, we have

LUT ðpI Þa ¼ fðpI Þ; T

BU ðqI Þa ¼ gðqI Þ;

pI 2 X;

I ¼ 1; . . . ; Np

ð10Þ

q I 2 C;

I ¼ 1; . . . ; Nq

ð11Þ

which can be summarized as

Ka ¼ F

ð12Þ

where

" K¼

LUT ðpI Þ T

BU ðqI Þ

# ;

F¼



fðpI Þ gðqI Þ

 ð13Þ

In order to achieve sufficient accuracy, generally, more collocation points should be used than the source points. Therefore, it should be noted that Eq. (13) is an over-determined system.

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

3. Free vibration analysis of FGM plates with in-plane material inhomogeneity

outward normal n. Mn, Mns and Qn are the moment, torque and shear force on the boundaries, respectively.

3.1. Basic equations

3.2. Free vibration analysis

Consider a FGM thin plate with in-plane material inhomogeneity, as shown in Fig. 1. The material properties of the plate vary in the plane xoy. The flexural rigidity of the plate is D = Eh3/12(1  m2) which is a function of x and y since Young’s modulus E is a variable depending on the position. Poisson’s ratio m is taken as a constant. The moment and shear forces in the thin plate are given as

Assume a harmonic solution of deflection w for the eigenproblem, involving a spatial term W(x, y) and a temporal term eixt where x is the eigenfrequency,

8 2  @ w @2 w > > > M x ¼ Dðx; yÞ @x2 þ m @y2 > < 2  @ w @2 w M y ¼ Dðx; yÞ @y2 þ m @x2 > > > > : M ¼ Dðx; yÞð1  mÞ @ 2 w yx @x@y

ð14Þ

! @ 2 @Dðx;yÞ @ 2 w @2w @Dðx; yÞ @2w r w þ m ð1  mÞ  2 2 @x @x @x @y @y @x@y ! 2 2 @ @Dðx; yÞ @ w @Dðx; yÞ @ w @2w F Sy ¼ Dðx; yÞ r2 w  ð1  mÞ  þ m @y @x @x@y @y @y2 @x2 ð15Þ

where Mx, My, Mxy are the moments, FSx and FSy are the shear forces, and w is the deflection of the plate. Equilibrium equations for the system are

@x

@y

@2D @2W @2D @2W @2D @2W r ðDr WÞ  ð1  mÞ 2 þ 2 2 @y @x @x@y @x@y @x2 @y2

Invoking the boundary conditions in Eq. (24), the free vibration problem can be generally described by

LW  qx2 W ¼ 0 in X

ð25Þ

B1 W ¼ 0 on C

ð26Þ

B2 W ¼ 0 on C

ð27Þ

where L is the differential operator in a bounded domain X, B2 and B3 are the differential operators on the boundary C. These differential operators are defined as

B2 ¼ ðn2x þ mn2y Þ

@2 @2 @2 þ 2nx ny ð1  mÞ þ ðmn2x þ n2y Þ 2 2 @x @x@y @y ð30Þ

(3) On free boundary

B1 ¼ Dðn2x þ mn2y Þ

2 @2 @2 2 2 @ þ 2Dn n ð1  m Þ þ Dð m n þ n Þ x y x y @x2 @x@y @y2

@2 @2 @2 B2 ¼ Dðn2x þ mn2y Þ 2 þ 2Dnx ny ð1  mÞ þ Dðmn2x þ n2y Þ 2 @x @x@y @y ( ! ) @ @D @ 2 @2 @D @2 þ  nx D r2 þ þ m ð1  m Þ @x @x @x2 @y @y2 @x@y ( !) 2 2 @ @D @ @D @ @2 þ þ m  ny D r2 þ ð1  mÞ @y @x @x@y @y @y2 @x2 ( ! ) @ @2 @2 @2 2 2 nx ny Dð1  mÞ  2 þ 2 þ ðny  nx ÞDð1  mÞ  @s @x @y @x@y

ð18Þ

(2) Simply supported boundary

ð19Þ

(3) Free boundary

ð31Þ

ð20Þ

In which equivalent shear force is defined as

3.3. Discretization of the equations and solution of eigenproblem

ð21Þ

and

@ @ @ ¼ ny þ nx @s @x @y

ð29Þ

B1 ¼ 1

(1) Clamed boundary

@ @ @ ¼ nx þ ny ; @n @x @y

ð28Þ

(2) On simply supported boundary

Typical boundary conditions for the thin plate can be expressed

@M ns Qn ¼ þ F Sn @s

!

B1 ¼ 1 @ B2 ¼ @n

as

Mn ¼ 0 Q n ¼ 0

@2D @2 @2D @2 @2D @2 2 þ 2 2 2 @y @x @x@y @x@y @x @y2

L ¼ r2 ðDr2 Þ  ð1  mÞ

(1) On clamped boundary

ð17Þ

Mn ¼ 0 w ¼ 0

¼ qx2 W ð24Þ

Sy

@w ¼0 @n

!

2

ð16Þ

where q is the material density, which is a function of x and y. Introducing (14) and (15) into (16), after some mathematical manipulation we obtain the governing equation of FGM thin plate with in-plane material inhomogeneity as ! @2D @2w @2D @2w @2D @2w @2w r2 ðDr2 wÞ  ð1  mÞ  2 þ þq 2 ¼0 @y2 @x2 @x@y @x@y @x2 @y2 @t

w¼0

ð23Þ

Substituting Eq. (23) into Eq. (17), we obtain the following equation 2

F Sx ¼ Dðx; yÞ

8 2 @F @F Sx > þ @ySy  qðx; yÞ @@tw2 ¼ 0 > > < @x @M @Mx þ @yxy  F Sx ¼ 0 @x > > > : @Mxy þ @My  F ¼ 0

wðx; y; tÞ ¼ Wðx; yÞeixt

ð22Þ

Base on the conventional RBCM, introducing approximation (9) into strong form (25)–(27) and evaluating them at the collocation points in the domain and on the boundary renders

ðLðUT ðpI ÞÞ  qx2 UT ðpI ÞÞa ¼ 0; T

where n defines the outer normal of the boundary, s denotes the tangent, and nx and ny are the components along x and y axes of unit

pI 2 X;

I ¼ 1; . . . ; Np

ð32Þ

B1 U ðqI Þa ¼ 0;

qI 2 C;

I ¼ 1; . . . ; Nq

ð33Þ

B2 UT ðqI Þa ¼ 0;

qI 2 C;

I ¼ 1; . . . ; Nq

ð34Þ

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

3.4. Hermite-type collocation In the Hermite-based interpolation, the problem unknown u can be approximated as

uðxÞ  uh ðxÞ ¼

Nb Ns X X nI ðxÞaI þ AðnI ðxÞÞaNs þI I¼1

ð37Þ

I¼1

where n(x) is the approximation function, aI is the expansion coefficient and A is the differential operator. The approximation represented by RBFs in Hermite form can be expressed as

u h ¼ ð UT

AðUT Þ Þa

Fig. 1. FGM thin plate with in-plane material inhomogeneity.

Eqs. (32)–(34) give the eigenvalue problem

ðK  x2 MÞa ¼ 0

ð35Þ

where

2

3

2

6 7 K ¼ 4 B1 UT ðqI Þ 5;

6 M¼4

LðUT ðpI ÞÞ T

qUT ðpI Þ 0

3 7 5

ð38Þ T

T

where u ¼ aI = [a1I, a2I, a3I] , a ¼ ½a1 ; . . . ; aNs þNb  , UT ¼ ½U1 ; . . . ; UNs , AðU Þ ¼ ½AðU1 Þ; . . . ; AðUNb Þ, UI = uII, uI is the radial basis function centered at source point xI, and Ns is the number of  , Nb is the number of source points source points in the domain X on the boundary C. Collocate the approximation on collocation h

ð36Þ

½uh1 ; uh2 ; uh3 , T

N

N

p q points fpI gI¼1 # X and fqI gI¼1 # C. Introducing Eq. (38) into (8) renders

h i L UT ðpI Þ þ AðUT ðPI ÞÞ a ¼ fðpI Þ;

B UT ðqI Þ þ AðUT ðqI ÞÞ a ¼ gðqI Þ;

pI 2 X;

I ¼ 1; . . . ; Np

ð39Þ

qI 2 C;

I ¼ 1; . . . ; Nq

ð40Þ

0

B2 U ðqI Þ

and K is the stiffness term, M is the inertial term. The dimension of K and M are (Np + 2Nq)  (Nl + Nb), where Nl is the number of source points in the problem domain and Nb is the number of source points on the boundaries and Nl + Nb = Ns. K and M are overdetermined matrices since Np P Nl and Nq P Nb. As a result, eigenvalue analysis cannot be employed based on eigenproblem Eq. (35).

In the numerical analysis of this work

A¼

@ @n

ð41Þ

Eqs. (39) and (40) can be summarized as

 ¼F Ka

ð42Þ

Table 1 1st–6th natural frequencies of FGM square plate with parameter c in different boundary conditions. Boundary condition

c

Methods

SSSS

0.0

Analytical solution HRBCM solution

2.0

SCSS

0.0

2.0

SSSC

0.0

2.0

SCSC

0.0

2.0

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Analytical solution HRBCM solution

Frequency

55 99 13  13 55 99 13  13 55 99 13  13 55 99 13  13 55 99 13  13 55 99 13  13 55 99 13  13 55 99 13  13

1

2

3

4

5

6

19.7392 19.5969 19.6330 19.6632 19.8948 19.6938 19.7611 19.9094

49.3480 50.2762 49.4791 49.3620 49.7215 50.1253 49.8804 49.7151

49.3480 50.4751 49.4833 49.3620 49.7350 50.8915 49.8126 49.7459

78.9568 80.5601 79.1191 78.9490 79.3844 80.6633 79.5267 79.3868

98.6960 103.6460 99.1184 98.8862 99.1042 94.4529 99.4284 99.2417

98.6960 102.6024 99.0993 98.8999 99.2587 97.0848 99.5265 100.6515

23.6463 23.7849 23.5824 23.5985 22.7095 21.8801 22.4528 22.5552

51.6743 52.7855 51.8626 51.7138 51.5913 51.8911 51.5984 51.5602

58.6464 59.9029 58.8607 58.6990 57.4984 58.9278 57.6855 57.5082

86.1345 88.6543 86.5124 86.2154 85.5342 87.5232 85.7929 85.5371

100.2700 101.3714 100.7725 100.5075 100.4570 100.7350 100.8103 100.6539

113.2280 114.7303 113.6242 113.3606 112.0680 106.0261 111.8852 112.1459

23.6463 23.7511 23.5987 23.6005 25.2961 25.7876 25.3534 25.2895

51.6743 52.8721 51.8625 51.7159 52.6318 54.2483 52.9195 52.7194

58.6464 59.9826 58.8616 58.7103 60.9060 62.5891 61.2610 61.0369

86.1345 88.6635 86.5124 86.2378 87.8067 90.8452 88.3790 87.9856

100.2700 103.1876 100.7674 100.4761 100.9490 104.2484 101.5379 101.2682

113.2280 116.8110 113.6981 112.9584 115.7500 118.7173 116.5226 116.1861

28.9509 29.3662 28.9893 28.9445 29.5147 29.9635 29.5555 29.5180

54.7431 56.6238 55.0592 54.8496 55.2498 57.2268 55.5691 55.3641

69.3270 70.8843 69.5511 69.3869 70.0462 70.9042 70.2751 70.1023

94.5853 98.1705 95.1385 94.7423 95.2294 97.6745 95.7691 95.3861

102.2160 106.7399 102.8873 102.4649 102.6770 107.2822 103.3536 103.3145

129.0960 130.5125 129.5384 129.0501 129.8920 127.4881 130.9684 130.2347

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

-0.35

0.00 0.31

-0.50

1.99

-0.65

2.36

h

log10||ω -ω ||

1.60

h

log10||ω -ω ||

-0.40

-0.80 0.50 -0.95 1.18

-0.80 1.32 1.42

-1.20

-1.10 -1.25 -1.15

-1.05

-0.95

-0.85

-0.75

-1.60 -1.15

-0.65

-0.95

-0.85

log10h

(a) SSSS

(b) SCSS

0.00

-0.75

-0.65

-0.75

-0.65

0.0

-0.50

-0.5

log10||ω -ω ||

3.65

-1.00

4.07 -1.0

4.05

h

h

log10||ω -ω ||

-1.05

log10h

1.34 2.99

-1.50

-1.5

4.48

5.87 -2.00

5.87

-2.0

-2.50 -1.15

-1.05

-0.95

-0.85

-0.75

-2.5 -1.15

-0.65

-1.05

-0.95

-0.85

log10h

log10h

(c) SSSC

(d) SCSC

Fig. 2. Convergence of fundamental frequencies in different boundary condition.

25

40 SSSS-Analytical SSSS-HRBCM

24 23

b 2ω

ρ0 D0

SCSS-Analytical SCSS-HRBCM SSSC-Analytical SSSC-HRBCM

37 34

b 2ω

22

ρ0 D0

31

21

28

20

25

19 0

2

4

6

22

8

0

2

47

42 40

45

D0

8

6

8

38

43

ρ0

6

SCSC-Analytical SCSC-HRBCM CSCS-HRBCM

CCCC

b 2ω

4

γ

γ

b 2ω

41 39

ρ 0 36 D0 34 32

37

30

35 0

2

4

6

8

28

0

2

γ Fig. 3. The relationship of fundamental frequencies and the gradient parameter c.

4

γ

84

F. Chu et al. / Computers and Structures 142 (2014) 79–89

b2ω

ρ0 D0

19.90

36.6

19.85

36.5

19.80

36.4

b2ω

19.75

ρ0 D0

19.70

36.2

19.65

36.1

19.60 0.20

0.25

0.30

0.35

36.0 0.20

0.40

D0

0.30

0.35

ν

(a) SSSS

(b) CCCC

0.40

25.5

23.7

ρ0

0.25

ν

24.0

b 2ω

36.3

25.0

23.4

b 2ω 23.1

ρ0 D0

24.5

24.0

22.8

22.5 0.20

0.25

0.30

0.35

23.5 0.20

0.40

0.25

0.30

0.35

ν

ν

(c) SCSS

(d) SSSC

0.40

Fig. 4. The relationship of fundamental frequencies and Poisson’s ratio with different parameter c.

42

10.50 ζ=0 ζ=0.25 ζ=0.5 ζ=0.75 ζ=1.0 ζ=2.0 ζ=5.0 ζ=10.0

8.75

D

7.00 5.25

39 36 33

b 2ω

ρ0 D0

30

SSSS SCSC CSCS CCCC

27

3.50

24 1.75

21

0.00 0.0

0.2

0.4

0.6

0.8

18

1.0

0

2

4

6

8

y

K¼

# LU ðpI Þ L AðU ðpI ÞÞ

; BUT ðqI Þ B AðUT ðpI ÞÞ

"

T

F¼

14

16

18

20

Fig. 6. The relationship of fundamental frequencies and the gradient parameter f.

where T

12

ζ

Fig. 5. Variation of rigidity with different parameter f.

"

10

fðpI Þ

evaluating them at the collocation points in the domain and on the boundary, we obtain the following discrete equation

#

gðqI Þ

ð43Þ

ðLðUT þ AðUT ÞÞ  qx2 ðUT þ AðUT ÞÞÞa ¼ 0; pI 2 X; I ¼ 1; .. .; Np ð44Þ B1 ðUT þ AðUT ÞÞa ¼ 0; qI 2 C; I ¼ 1;. .. ;Nq

3.5. Hermite-type collocation for free vibration analysis Consequently, HRBCM is adopted to analyze the eigenproblem. Introducing approximation (38) into strong form (25)–(27) and

T

T

B2 ðU þ AðU ÞÞa ¼ 0; qI 2 C; I ¼ 1;. .. ;Nq

ð45Þ ð46Þ

Eqs. (44)–(46) give the eigenproblem

 ¼0 ðK  x2 MÞa

ð47Þ

85

F. Chu et al. / Computers and Structures 142 (2014) 79–89

75

160 γ=2.0 ζ=5.0

65

120

55

ρ0

ω

γ=2.0 ζ=5.0

140

45

ω

D0 35

ρ 0 100 D0

80

25

60

15

40

5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

20 0.0

3.5

0.5

1.0

1.5

2.0

a/b

a/b

(a) SSSS

(b) CCCC

2.5

3.0

3.5

Fig. 7. The relationship of fundamental frequencies and Poisson’s ratio with different parameter f.

25 23.5 24

23

23

ω

22

21

21.5

20

21

19 5

20.5

4 3

α

2 1 0 0

1

2

3

ω

22.5

22

25.5 25 24.5 24 23.5 23 22.5 22 5

24 23.5 23 4

5

4

24.5

3

20

α

β

2 1 0 0

(a) SSSS

1

2

3

4

5

β

(b) SCSS 32

43

31

44

42

42

41

40

40

27

38

39

26

36 5

34

30

29

28

28

26

ω

30

32

ω

22.5

24 22 5 4 3

β

2 1 0 0

2

1

3

α

4

5

25 24

38

4

3

α 2

1 0 0

1

2

3

4

5

37

β

(d) CCCC

(c) SSSC

Fig. 8. The relationship of fundamental frequencies and the gradient parameters a and b.

where

4. Numerical examples

3 LðUT þ AðUT ÞÞ 7 6 T T 7 K¼6 4 B1 ðU þ AðU ÞÞ 5; 2

T

T

B2 ðU þ AðU ÞÞ

2 6 M¼6 4

qðUT þ AðUT ÞÞ 0

3 7 7 5

4.1. Free vibration analysis of a square plate with in-plane material inhomogeneity

ð48Þ

0

The dimension of K and M are (Np + 2Nq)  (Nl + 2Nb). When we consider Np = Nl and Nq = Nb, Eq. (47) is a determined system which can be employed for the eigenvalue analysis.

We use C, S and F to represent the clamped, simply supported and free boundary conditions on the boundaries, respectively. Arbitrary combination of C, S and F denotes the boundary conditions on x = 0, y = 0, x = a and y = b successively. For example, CSCS defines clamped boundary condition on boundary x = 0, simply supported boundary condition on boundary y = 0, clamped boundary

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

Fig. 9. The 1st–6th vibration mode shapes of homogeneous and heterogeneous square plates with SSSS.

condition on boundary x = a and simply supported boundary condition on boundary y = b. The material coefficients of the FGM plates are Poisson’s ratio m = 0.3, side length of the square plate a = b = 1.0 m, density q0 = 8000.0 kg/m3 and Young’s modulus E0 = 2.06  1011 Pa. Three cases are studied for evaluation. 4.1.1. Case I Consider the material coefficients as exponential functions of y. Elastic modulus E and density q are given as

E ¼ E0 ecy ;

q ¼ q0 ecy

ð49Þ

where c is a variable parameter. Three different discretization schemes with 5  5, 9  9 and 13  13 refinement are employed for investigation. Numerical solutions obtained from HRBCM are validated by the analytical solutions as shown in Table 1, which demonstrates that the proposed approach for thin plates can procure good accuracy and high convergence. Fig. 2 presents the error estimate which shows that

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

-1.0 -1.2

1.08 1.79

-1.6

h

log10||ω -ω ||

-1.4

-1.8

3.97

-2.0 -2.2 -2.4

Simple Support Clamped

4.79

-2.6 -1.00

-0.95

-0.90

-0.85

-0.80

-0.75

-0.70

-0.65

log10h Fig. 10. A FGM circular plate with in-plane inhomogeneity.

Fig. 12. Convergence of fundamental frequencies in different boundary conditions.

HRBCM can achieve exponential convergence for FGM problems as for homogeneous problems. The relationships of the fundamental frequencies and gradient parameters in problems with different boundary conditions are shown in Fig. 3, which describes that the fundamental frequencies increase with the increment of c except for the case of SCSS. The reason is that the natural frequencies are directly proportional to the rigidity of the plate which is increasing with the increment of parameter c. For SCSS, firstly, the fundamental frequency decreases with the increment of c, then it will increase with the increment of c. A minimum fundamental frequency can be detected with the increment of c. These results illustrate that the boundary conditions can also affect the natural frequencies since the boundary conditions are involved in the solutions of eigenproblem. Although Poisson’s ratio is considered as a constant in this case, different Poisson’s ratios affect the natural frequencies of FGM plates. The relationships of the fundamental frequencies and Poisson’s ratios are given in Fig. 4, which demonstrates that the natural frequencies are nearly linear dependent of Poisson’s ratio. For homogeneous materials plates, that is, gradient parameter is c = 0, the natural frequencies are independent of Poisson’s ratio.

Variation of rigidity with respect to different f is shown in Fig. 5. It is noted that the derivatives of the rigidity D (oD/oy and o2D/oy2) go to infinity at y = 0 when f < 1. However, such derivatives are not involved in the boundary conditions at y = 0. Infinite derivatives of the rigidity on the boundary do not affect the accuracy of solutions when f < 1 compared to that affect the solutions based on standard Runge–Kutta algorithm [8]. The relationships of natural frequencies and parameter f with different boundary conditions are shown in Fig. 6, which shows that there are three phases with the increment of parameter f. Firstly, the natural frequencies decrease with the increment of parameter f; secondly, the natural frequencies increase with f after they reach minimal values; finally, after reaching peak values, the natural frequencies gradually decrease. When studying the relationship of fundamental frequencies and aspect ratio of the plate, we set b ¼ 1:0 m as a constant and a is a variable varying from 0.4 m to 3.0 m. Fig. 7 characterizes the effect of aspect ratio on the natural frequencies, in which the fundamental frequencies are decreasing when a/b is increasing. As shown in Figs. 3 and 6, the gradient parameters can be used to adjust the natural frequencies of the FGM plates.

4.1.2. Case II In case II, we consider the material coefficients of E and q are the power functions of y.

E ¼ E0 V 0 þ Eb V b ; q ¼ q0 V 0 þ qb V b ; yf ð0  y  b; 0  f  1Þ Vb ¼ b

V 0 þ V b ¼ 1; ð50Þ

where f is a variable parameter, V0 and Vb are the volume fractions of the material on the boundaries y = 0 and y = b. Material constants at the edge y = b are qb = 10q0 and Eb = 10E0.

73 Nodes

4.1.3. Case III In case III, we consider the material coefficients of E and q are the exponential functions of x and y as

E ¼ E0 eaxþby ;

q ¼ q0 eaxþby

ð51Þ

where a and b are the variable parameters. Relationships of natural frequencies and variation of parameters are shown in Fig. 8. These results give the selections of parameters for the required natural frequencies in engineering applications. The inhomogeneity for the materials also introduces the change of mode shapes for the FGM plates compared to the homogeneous

121 Nodes Fig. 11. Discretizations of circular plate.

177 Nodes

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F. Chu et al. / Computers and Structures 142 (2014) 79–89

28

17.5

Exponential function

Power function

25

16.0

22 14.5

ω

ρ 0 19

ω

D0

ρ0 D0 13.0

16 11.5

13 10

0

1

2

3

4

5

10.0

6

0

2

γ

4

6

8

10

ζ

Fig. 13. The relationship of fundamental frequency and gradient parameters c and f for circular plate.

Table 2 The fundamental frequency of circular plate in different boundary conditions. Boundary conditions

Analytical solution [32]

Simply supported Clamped

4.977 10.220

HRBCM solutions with different discretization 73

121

177

4.898 10.258

4.922 10.227

4.942 10.223

Fig. 14. The 1st–6th vibration mode shapes of homogeneous and heterogeneous circular plates with simply supported boundary.

materials. The 1st–6th mode shapes with SSSS boundary condition are shown in Fig. 9 for comparisons of homogeneous and heterogeneous plates. Each group of pictures represents the vibration mode shapes of homogeneous plates, unidirectional FGM plates as described in case I and bidirectional FGM plates as described in case III, respectively. These results exhibit that the vibration mode shapes for FGM plates are unsymmetrical compared to the symmetric mode shapes of homogeneous plates. Moreover, inhomogeneity results in the obvious change of mode shapes, especially for the high-order mode shapes. Similar phenomenon can be observed in the FGM plate problems with other boundary conditions.

4.2 Free vibration of circular plate with in-plane material inhomogeneity Consider a circular plate with in-plane material inhomogeneity shown in Fig. 10. Poisson’s ratio is m = 0.3, radius of circular plate is R = 1.0 m, density is q0 = 8000.0 kg/m3, and Young’s modulus is E0 = 2.06  1011 Pa. Refinements of 73, 121 and 177 nodes are shown in Fig. 11. Since there is no analytical solution for the free vibration of circular plate with variable material coefficients, we firstly solve the free vibration equations of circular plate with homogeneous

F. Chu et al. / Computers and Structures 142 (2014) 79–89

material to validate the accuracy of the proposed method. Numerical results compared to the analytical solutions are presented in Table 2 which indicates the good accuracy of HRBCM, and Fig. 12 describes the exponential convergence for HRBCM once again. Two cases of circular plate with in-plane inhomogeneity are studied as follows Case I: E ¼ E0 ecr ; q ¼ q0 ecr .  f Case II: E ¼ E0 V 0 þ ER V R ; q ¼ q0 V 0 þ qR V R ; V 0 þ V R ¼ 1; V R ¼ Rr ð0  r  R;0  f  1Þ. The relationship of fundamental frequencies and gradient parameter c and f are given in Fig. 13. When the material parameters vary in an exponential function as in case I, the fundamental frequency increases with the increment of the gradient parameter c. For case II where the material parameters vary in a power function, firstly, the fundamental frequency increases with the increment of gradient parameter f, then it decreases with the increment of gradient parameter f. Vibration mode shapes of circular plate with homogeneity and in-plane inhomogeneity are compared in Fig. 14. Although the material parameters in FGM circular plate vary as in the square plate, since the inhomogeneous circular plate is centrosymmetric, less variation in the mode shapes can be detected for inhomogeneous circular plate compared to homogeneous circular plate than the variation for inhomogeneous square plate compared to homogeneous square plate. 5. Conclusions The free vibration analysis of FGM plates with in-plane material inhomogeneity is studied based on Hermite-type radial basis collocation method. Derivatives of approximation are employed for the interpolation function, which introduces a determined system for the eigenvalue problem compared to an overdetermined system when using conventional RBCM. Moreover, HRBCM can also improve the computational accuracy. Convergence and comparison studies validate the effectiveness and accuracy of the proposed method. Numerical examples demonstrate that the in-plane material inhomogeneity has a significant effect on the natural frequencies and vibration mode shapes, which provides more formalization for the FGM design when the material inhomogeneity is required in engineering applications. Acknowledgements This work was supported by National Natural Science Foundation of China (Project No. 11202150), Fundamental Research Funds for the Central Universities and Shanghai Leading Academic Discipline Project (Project No. B302). References [1] Zhong Z, Shang E. Closed-form solutions of three-dimensional functionally graded plates. Mech Adv Mater Struct 2008;15:355–63. [2] Zhong Z, Yu T. Vibration of a simply supported functionally graded piezoelectric rectangular plate. Smart Mater Struct 2006;15:1404–12. [3] Aydogdu M, Taskin V. Free vibration analysis of functionally graded beams with simply supported edges. Mater Des 2007;28:1651–6.

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[4] Chen XL, Liew KM. Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Mater Struct 2004;13:1430–7. [5] Gilhooley D, Xiao J, Batra R, McCarthy M, Gillespiejr J. Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions. Comput Mater Sci 2008;41:467–81. [6] Uymaz B, Aydogdu M, Filiz S. Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Compos Struct 2012;94:1398–405. [7] Yu T, Nie G, Zhong Z, Chu F. Analytical solution of rectangular plate with inplane variable stiffness. Appl Math Mech 2013;34:395–404. [8] Liu DY, Wang CY, Chen WQ. Free vibration of FGM plates with in-plane material inhomogeneity. Compos Struct 2010;92:1047–51. [9] Carrera E, Giunta G. Refined beam theories based on a unified formulation. Int J Appl Mech 2010;2:117–43. [10] Carrera E. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Meth Eng 2003;10(3):215–96. [11] Hardy RL. Multiquadric equations of topography and other irregular surfaces. J Geophys Res 1971;76:1905–15. [12] Franke R. Scattered data interpolation: tests of some methods. Math Comput 1982;98:181–200. [13] Wang L, Chu F, Zhong Z. Study of radial basis collocation method for wave propagation. Eng Anal Bound Elem 2013;37:453–63. [14] Jamil M, Ng EYK. Evaluation of meshless radial basis collocation method (RBCM) for heterogeneous conduction and simulation of temperature inside the biological tissues. Int J Therm Sci 2013;68:42–52. [15] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput Math Appl 1990;19:127–45. [16] Kansa EJ. Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 1990;19:147–61. [17] Ferreira AJM. Free vibration analysis of Timoshenko beams and Mindlin plates by radial basis functions. Int J Comput Meth 2005;02:15–31. [18] Ferreira AJM, Roque CMC, Jorge RMN, Fasshauer GE, Batra RC. Analysis of functionally graded plates by a robust meshless method. Mech Adv Mater Struct 2007;14:577–87. [19] Ferreira AJM, Roque CMC. Analysis of thick plates by radial basis functions. Acta Mech 2011;217:177–90. [20] Ferreira AJM, Roque CMC, Neves AMA, Jorge RMN, Soares CMM, Liew KM. Buckling and vibration analysis of isotropic and laminated plates by radial basis functions. Composites Part B 2011;42:592–606. [21] Ferreira AJM, Fasshauer GE. Analysis of natural frequencies of composite plates by an RBF-pseudospectral method. Compos Struct 2007;79:202–10. [22] Rodrigues JD, Roque CMC, Ferreira AJM, Cinefra M, Carrera E. Radial basis functions-differential quadrature collocation and a unified formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami’s Zig-Zag theory. Comput Struct 2012;90–91:107–15. [23] Roque CMC, Rodrigues JD, Ferreira AJM. Analysis of thick plates by local radial basis functions-finite differences method. Meccanica 2012;47:1157–71. [24] Xiang S, Wang K-M. Free vibration analysis of symmetric laminated composite plates by trigonometric shear deformation theory and inverse multiquadric RBF. Thin Wall Struct 2009;47:304–10. [25] Xiang S, Yang M-s, Jiang S-x, Wang K-m. Three-dimensional vibration analysis of isotropic plates by multiquadric and thin-plate spline radial basis functions. Comput Struct 2010;88:837–44. [26] Wu CM. Hermite–Birkhoff interpolation of scattered data by radial basis functions. Approx Theor Appl 1992;8:1–10. [27] Fasshauer GE. Solving partial differential equations by collocation with radial basis functions. In: M’ehaut’e AL, Rabut C, Schumaker LL, editors. Surface fitting and multiresolution methods. Nashville, USA: Vanderbilt University Press; 1997. p. 131–8. [28] Zhang X, Song KZ, Lu MW, Liu X. Meshless methods based on collocation with radial basis functions. Comput Mech 2000;26:333–43. [29] Liu Y, Hon YC, Liew KM. A meshfree Hermite-type radial point interpolation method for Kirchhoff plate problems. Int J Numer Methods Eng 2006;66:1153–78. [30] Madych WR, Nelson SA. Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J Approx Theory 1992;70:94–114. [31] Cheng AHD, Golberg MA, Kansa EJ, Zammito G. Exponential convergence and H-c multiquadric collocation method for partial differential equations. Numer Meth Part D E 2003;19:571–94. [32] Xu Z. Elasticity. 4th ed. Beijing: Higher Education Press; 2006.