Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions

Composites Part B xxx (2015) 1e15

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Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions Guoyong Jin*, Zhu Su*, Tiangui Ye, Siyang Gao College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 December 2014 Received in revised form 8 April 2015 Accepted 7 August 2015 Available online xxx

The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of threedimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied. © 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Plates B. Vibration C. Analytical modelling Functionally graded materials

1. Introduction Functionally graded materials (FGMs) belong to a new class of composite materials and their material properties vary continuously along desired directions. Such materials offer a number of advantages over laminated composite materials including improved fatigue resistance, reduction in the thermal, residual and interlaminar stresses, and more desirable joining capabilities [1]. In recent years, FGMs have been utilized to build various plate structures [2e5]. As one of common plate structures, FGM annular sector plates are extensively used in various engineering fields due to their design flexibility and high load-carrying capacity, especially in aerospace, mechanic and marine industries. These applications are often exposed to severe vibration conditions. Therefore, the accurate prediction of the dynamic behavior of such plates becomes significant for designers and engineers. Typically, the analysis of annular sector plates is performed using two-dimensional (2-D) plate theories, such as the classical plate

* Corresponding authors. College of Power and Energy Engineering, Harbin Engineering University, Harbin, 150001, PR China. Tel.: þ86 451 82589199. E-mail addresses: [email protected] (G. Jin), [email protected] (Z. Su).

theory (CPT) [6e9], first-order shear deformation theory (FSDT) [10e14] and higher-order shear deformation theory (HSDT) [15]. It is should noted that those 2-D plate theories are developed on the basis of certain assumptions which simplify the formulation and solution in analytical and computational procedure, but also inherently bring out errors at the same time. As a result, in order to more accurate representation for free vibration of annular sector plates, some investigations were carried out by three-dimensional (3-D) elasticity theory [16e19]. Apart from the plate theories, a number of analytical and numerical methods have been proposed and developed to deal with the vibration problems of annular sector plates, such as finite element method [7,17], variational method [8,9], Rayleigh-Ritz method [10,16,18], differential quadrature method [12,14,19]. According to a comprehensive survey of literature, it is found that a huge amount of research effort has been devoted to analyze FGM annular sector plates [20e39]. Hosseini-Hashemi et al. [20] investigated bucking and free vibration behaviors of radially functionally graded circular and annular sector plate subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation by means of differential quadrature method and CPT, and various combinations of simply supported and clamped boundary conditions are considered. The differential quadrature method was also used by Mirtalaie and Hajabasi [22] to study the

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free vibration of functionally graded thin annular sector plates under various boundary conditions whose material properties are assumed to vary continuously through the thickness. Baferani et al. [23] presented an exact analytical method for free vibration analysis of functionally graded thin annular sector plates resting on Winkler and Pasternak foundations. The simply supported radial edges and arbitrary conditions along the circular edges are considered. Free vibration analysis of moderately thick functionally graded conical, cylindrical shell and annular plate structures was carried out by Tornabene et al. [25] using the differential quadrature (DQ) method based on the first-order shear deformation theory in which two different power-law distributions were took into account. Saidi et al. [29] analyzed free vibration of moderately thick functionally graded annular sector plates on the basis of the firstorder shear deformation plate theory, and the accurate natural frequencies of the FGM annular sector plates with simply supported radial edges were presented. Nie and Zhong [35] studied the free and forced vibration of functionally graded annular sectorial plates with simply supported radial edges and arbitrary circular edges using the combination of the state space method and differential quadrature method based on the three-dimensional theory of elasticity. Tahouneh and Yas [36,37] investigated the free vibration of thick functionally graded annular sector plates with simply supported radial edges in the context of three-dimensional theory of elasticity and a semi-analytical approach composed of the different quadrature method and series was adopted. The above review of the literature reveals that most investigations concerning FGM annular sector plate vibrations are focused on 2-D theories and the 3-D elasticity solution for free vibration of such plates seems to be limited, and the literature on radially functionally graded is also limited. Moreover, it also should noted that most of previous research efforts were restricted to limited sets of classical boundary conditions such as free, simply supported and clamped boundary conditions. However, there exist some deviations from those ideal boundary conditions in practical engineering applications, such as elastic restraints. Consequently, the main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general boundary conditions in the context of three-dimensional theory of elasticity. The material properties of functionally graded sector plates are assumed to vary continuously in thickness or radial direction and two kinds of material distribution are considered. The formulation is derived by the variational principle in conjunction with modified Fourier series which consists of threedimensional Fourier series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given, which can be serve as benchmark solutions. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.

2. Theoretical formulations 2.1. Functionally graded annular sector plates Consider a functionally graded annular sector plate with inner radius R1, outer radius R2, thickness h and sector angle a, as shown in Fig. 1. The cylindrical coordinate system (r, q, z) is taken to describe the displacement components u, v and w in the radial, circumferential and thickness directions which is local on the bottom surface. The annular sector plate domain is bounded by 0  r  R2  R1 ; 0  q  a; 0  z  h.

Fig. 1. The coordinate system and geometry of a thick annular sector plate.

Typically, the fabrication of FGMs can be considered by mixing two discrete phases of materials, for example, a distinct mixture of a ceramic and a metal. The effective material properties of FGMs are usually assumed to vary continuously in the thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction, when FGMs are exposed to high temperature, chemical reactions, high-level radioactivity, and so on [20,21]. In this work, the material properties of functionally graded sector plates are assumed to vary continuously in thickness or radial direction, as shown in Fig. 2. For type I the material properties vary smoothly and continuously in thickness direction; For Type II the material properties vary smoothly and continuously in radial direction. Several models have been used over the years to estimate the effective material properties of FGMs, such as Voigt model, MoriTanaka scheme and the self-consistent method. The detailed descriptions on those models can be founded in monograph by Shen [40]. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties of FGMs. The Voigt model is simple and convenient to use for predicting the overall material properties. The effective material properties of FGMs can be expressed according to:

Ef ¼ ðE1  E2 ÞV1 þ E2 ;

rf ¼ ðr1  r2 ÞV1 þ r2 ;

¼ ðm1  m2 ÞV1 þ m2

mf (1.a-c)

where the subscripts 1 and 2 denote the ceramic and metallic constituents, respectively. Ei, ri and mi (i ¼ 1 and 2) are the Young's modulus, mass density and Poisson ratio. V1 is the volume fraction of ceramic, and the sum of the volume fractions of both the constituent materials makes one. The Mori-Tanaka scheme is more accurate micromechanics model to predict the material properties of graded microstructure which have a well-defined continuous matrix and discontinuous particulate phase. It is assumed that the subscripts 1 and 2 denote the particulate and matrix phases. The effective local buck modulus Kf and the shear modulus Gf obtained by Mori-Tanaka scheme are given by

Kf  K2 V1 ¼ K1  K2 1 þ ð1  V1 ÞðK1  K2 Þ=ðK2 þ 4G2 =3Þ

(2.a)

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3

Fig. 2. Functionally graded annular sector plate (a) Type I; (b) Type II.

Table 1 Material properties of the FGM components. Properties

Aluminum (Al)

Zirconia (ZrO2)

Alumina (Al2O3)

Titanium (Ti)

Silicon carbide (SiC)

E (GPa)

70 0.3 2707

168 0.3 5700

380 0.3 3800

105.8 0.3 4420

410 0.17 3100

m r (kg/m3)


constant lf is related to the bulk and the shear scheme. The Lame modui by lf ¼ Kf  2Gf =3. In this work, the volume fraction of M1 is given by the powerlaw distribution

Gf G2 V1 ¼ G1 G2 1þð1V1 ÞðG1 G2 Þ=fG2 þG2 ð9K2 þ8G2 Þ=½6ðK2 þ2G2 Þg (2.b) where Ki, Gi (i ¼ 1 and 2) are the bulk modulus and shear modulus, defined by Ki ¼ Ei =½3ð1  2mi Þ Gi ¼ Ei =½2ð1 þ mi Þ. The effective mass density defined by Eq. (1.b) is also used in Mori-Tanaka

Type I :

V1 ¼


z p h

ð0  z  hÞ

(3.a)

Table 2 Convergence of frequencies (Hz) for the completely free FGM annular sector plates with different thickness-to-radius ratios (R1 ¼ 1 m, R2/R1 ¼ 2, a ¼ 120 , p ¼ 1; M-T solution). h/R1

Type I 0.1

Q

8

10

0.5

8

10

Type II 0.1

8

10

0.5

8

10

MN

Mode 1

2

3

4

5

6

7

8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14

               

8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14

89.21 89.04 88.97 88.93 89.13 88.99 88.92 88.89 391.33 391.27 391.25 391.24 391.31 391.26 391.24 391.23

139.89 139.58 139.44 139.36 139.81 139.52 139.39 139.33 536.68 536.60 536.57 536.55 536.66 536.59 536.56 536.54

210.76 210.36 210.16 210.05 210.66 210.29 210.12 210.01 573.73 573.71 573.71 573.71 573.73 573.71 573.71 573.71

306.72 306.34 306.20 306.13 306.67 306.30 306.17 306.11 820.66 820.48 820.40 820.36 820.63 820.46 820.39 820.35

386.34 385.48 385.03 384.76 386.18 385.39 384.97 384.72 1128.00 1127.81 1127.75 1127.73 1127.99 1127.81 1127.75 1127.73

504.66 503.74 503.41 503.27 504.56 503.68 503.37 503.23 1243.31 1243.21 1243.18 1243.16 1243.31 1243.21 1243.17 1243.16

574.78 574.76 574.75 574.75 574.78 574.76 574.75 574.75 1306.62 1306.53 1306.49 1306.47 1306.61 1306.52 1306.49 1306.47

8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14

               

8 10 12 14 8 10 12 14 8 10 12 14 8 10 12 14

92.16 92.07 92.02 91.99 92.04 91.97 91.94 91.92 412.03 412.01 412.00 411.99 411.99 411.97 411.97 411.96

141.51 141.30 141.20 141.15 141.36 141.17 141.08 141.04 525.61 525.61 525.61 525.61 525.60 525.59 525.59 525.59

212.44 212.29 212.21 212.16 212.19 212.07 212.01 211.98 529.75 529.71 529.69 529.68 529.68 529.65 529.64 529.63

323.73 323.47 323.35 323.28 323.53 323.28 323.16 323.10 857.61 857.59 857.57 857.56 857.49 857.47 857.46 857.46

394.43 394.21 394.09 394.01 393.81 393.64 393.55 393.50 1135.43 1135.42 1135.42 1135.42 1135.36 1135.36 1135.35 1135.35

525.63 525.63 525.63 525.63 525.62 525.61 525.61 525.61 1143.72 1143.69 1143.67 1143.66 1143.59 1143.56 1143.54 1143.54

535.77 535.38 535.20 535.09 535.18 534.82 534.65 534.56 1414.45 1414.45 1414.45 1414.45 1414.45 1414.45 1414.45 1414.45

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Type II :

V1 ¼


r p R

ð0  r  RÞ

(3.b)

where R ¼ R2- R1. The gradient index p (0  p  ∞) dictates the material variation profile.

2.2. Energy functional

vu u 1 vv vw ; εz ¼ ; εr ¼ ; εq ¼ þ vr r þ R1 r þ R1 vq vz vv 1 vw vu vw gqz ¼ þ ; grz ¼ þ ; vz r þ R1 vq vz vr 1 vu vv v þ  grq ¼ r þ R1 vq vr r þ R1

(4)

C13 C23 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

9 38 0 > εr > > > > > εq > > 0 7 > > > 7> < = 7 0 7 εz 0 7 g > > 7> > qz > > 0 5> grz > > > > > : ; grq C66

Ef mf 
; 1  2mf

(6.a-c)

For an annular sector plates, they can have any combinations of boundary conditions. Considering classical boundary conditions, each boundary must specify three conditions [41]:

srz ¼ 0 u¼0 sqz ¼ 0

(5)

1 þ mf

Ef  C44 ¼ C55 ¼ C66 ¼ Gf ¼
2 1 þ mf

u¼0

where εr, εq, εz, gqz, grz and grq are the normal and shear strain components. The constitutive equations for a FGM annular sector plates are defined as:

C12 C22 C23 0 0 0

C11 ¼ C22 ¼ C33


 Ef 1  mf 
; ¼ lf þ 2Gf ¼
1 þ mf 1  2mf

C12 ¼ C13 ¼ C23 ¼ lf ¼


Within the context of 3-D theory of elasticity, the linear straindisplacement relations of an annular sector plate are given as [36]:

8 9 2 C11 sr > > > > > > > 6 C12 sq > > > > > 6 < = 6 sz C13 ¼6 6 0 t > qz > > > 6 > > 4 > t > 0 > > > : rz > ; trq 0

where sr, sq, sz, tqz, trz and trq are the normal and shear stress components, and Cij are material elastic stiffness coefficients, given as:

or

sr ¼ 0; v ¼ 0

or

srq ¼ 0; w ¼ 0

or

sr ¼ 0; w ¼ 0

or

at r ¼ constant or

srq ¼ 0; v ¼ 0

or

at q ¼ constant

(7.a)

(7.b)

Boundaries also may be elastically constrained, with the constraints being represented as three sets of independent linear springs at edges. In such cases, the boundary conditions are generalized to include elastic constraints, given as:

Fig. 3. Effect of elastic restraints on frequency of the Type I FGM annular sector plates (p ¼ 1, R1 ¼ 1 m, R2/R1 ¼ 2, h/R1 ¼ 0.5): (a) a ¼ 160 ; (b) a ¼ 270 .

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G. Jin et al. / Composites Part B xxx (2015) 1e15

kur0 u ¼ sr ;

kvr0 v ¼ srq ;

kwr0 w ¼ srz

at r ¼ 0

(8.a)

1 T¼ 2

Zh Za ZR 0

kur1 u ¼ sr ;

kvr1 v ¼ srq ;

kwr1 w ¼ srz

at r ¼ R

0

kvq0 v ¼ sq ;

kwq0 w ¼ sqz

at q ¼ 0

(8.c)

kuq1 u ¼ srq ;

kvq1 v ¼ sq ;

kwq1 w ¼ sqz

at q ¼ a

(8.d)

where kur0, kvr0, kwr0, kur1, kvr1, kwr1, kuq0, kvq0, kwq0, kuq1, kvq1 and kwq1 are boundary springs. It is worth noting that the classical boundary conditions can be obtained by assigning the boundary springs at zero or infinity (5  1014). The linear elastic strain energy U of the FGM annular sector plate is expressed as:

(10) where t donate the time. In this work, the boundary conditions are simulated by linear springs, and the potential energy P stored in the boundary springs can be given as:

1 P¼ 2

Zh Za h
  R1 kur0 u2 þ kvr0 v2 þ kwr0 w2  0

0

þ R2 þ

1 2




ðsr εr þ sq εq þ sz εz þ tqz gqz þ trz grz

r¼R

r¼0

dqdz

0

i q¼a

q¼0

ðR1 þ rÞdrdz

(11)

The energy functional P of the FGM annular sector plate can be written as

0

þ trq grq ÞðR1 þ rÞdrdqdz

i

  kuq0 u2 þ kvq0 v2 þ kwq0 w2 


 þ kuq1 u2 þ kvq1 v2 þ kwq1 w2 

Zh Za ZR 0

  kur1 u2 þ kvr1 v2 þ kwr1 w2 

Zh ZR h
0

0

"   2  2 # vu 2 vv vw rf þ þ ðR1 þ rÞdrdqdz vt vt vt

(8.b)

kuq0 u ¼ srq ;

1 U¼ 2

0

5

(9)

The kinetic energy T of the FGM annular sector plate is depicted

P¼T UP

(12)

as:

Fig. 4. Effect of elastic restraints on frequency of the Type II FGM annular sector plates (p ¼ 1, R1 ¼ 1 m, R2/R1 ¼ 2, h/R1 ¼ 0.5): (a) a ¼ 160 ; (b) a ¼ 270 .

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Table 3 pffiffiffiffiffiffiffiffiffiffiffi Comparison of frequency parameters J ¼ uR22 =p2 rh=D for isotropic annular sector plates with different boundary conditions (R1/R2 ¼ 0.4, m ¼ 0.3, a ¼ 90 ). h/R2 BC

0.1

0.2

Mode

SSSS

Ref [16] Present SSFF Ref [16] Present CCFF Ref [16] Present CCSS Ref [16] Present CCCC Ref [16] Present SSSS Ref [16] Present SSFF Ref [16] Present CCFF Ref [16] Present CCSS Ref [16] Present CCCC Ref [16] Present

1

2

3

3.4476 3.4481 2.6606 2.6623 5.2651 5.2684 5.6650 5.6662 5.9306 5.9329 2.9973 2.9969 1.5608 1.5659 3.8839 3.8901 4.1722 4.1771 4.3755 4.3814

5.5699 6.0808 5.5725 6.0746 3.1100 3.2111 3.1209 3.2154 5.5023 6.5224 5.5086 6.5285 7.1019 9.8012 7.1043 9.8032 7.8931 10.918 7.8997 10.932 3.0424 4.5870 3.0392 4.5870 2.3558 2.7829 2.3564 2.7838 4.0336 4.7411 4.0399 4.7570 5.2634 5.5973 5.2675 5.5941 5.6945 7.2911 5.7030 7.2906

4

5

6

8.6811 8.6852 4.8673 4.8727 8.6305 8.6410 11.195 11.188 13.141 13.147 4.9818 4.9797 2.9136 2.9158 5.2979 5.3058 7.0733 7.0698 7.5643 7.5773

9.9621 9.9578 5.8231 5.8278 10.590 10.605 13.007 13.013 14.490 14.516 5.5973 5.5940 4.0289 4.0295 6.2429 6.2497 7.1145 7.1180 8.5621 8.5728

10.248 10.248 7.4718 7.4828 11.735 11.743 13.307 13.315 14.565 14.565 6.7040 6.7044 4.6820 4.6835 6.7281 6.7381 8.4995 8.5093 9.4678 9.4635

Table 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparison of fundamental frequency parameters 6 ¼ uR22 r0 h=D0 for Type II FGM annular sector plates with different boundary conditions (R1/R2 ¼ 0.5, h/ R2 ¼ 0.01, m ¼ 0.3). BC

(g1,g2)

a 30

SSSS

SCSC

CCCC

(-1,-0.5) (0,0) (1,0.5) (-1,-0.5) (0,0) (1,0.5) (-1,-0.5) (0,0) (1,0.5)

Ref [21] 85.2474 103.437 125.648 135.945 165.600 201.729 157.849 192.161 233.896

60 Present 85.5238 103.743 125.971 136.163 165.572 201.137 158.295 192.236 233.179

Ref [21] 46.3607 55.9566 67.5675 55.9329 67.4451 81.3775 87.6075 106.136 128.786

90 Present 46.6747 56.3111 67.9678 56.2912 67.8163 81.7351 88.3290 106.799 129.255

Ref [21] 39.0753 47.0887 56.6953 41.9997 50.5886 60.9022 78.7021 95.2199 115.434

Present 39.3210 47.3683 57.0169 42.2524 50.8709 61.2151 79.3869 95.8634 115.920

2.3. Admissible functions and solution procedure As in previous studies [42e44], the assumed solutions in the variational method are expressed in the form of modified Fourier series which consists of standard 3-D Fourier cosine series plus several auxiliary functions introduced to remove the potential discontinuities with the displacement and their derivatives at edges. In this work, each of displacements of the FGM annular sector plate is given as follows:

9 8 2 X N X M Q P X N M P P > > l > > > > A cos l r cos l q cos l z þ a cos l r cos l qx ðzÞþ m n q m n lz > > mn = < q¼0 n¼0 m¼0 mnq l¼1 n¼0 m¼0 uðr; q; z; tÞ ¼ ejut Q X 2 X N Q X 2 M > > P P P > > l l > > ~nq xlr ðrÞcos ln qcos lq z > > amq cos lm rxlq ðqÞcos lq z þ a ; : l¼1 q¼0 m¼0

l¼1 q¼0 n¼0

9 8 2 X N X M Q P N M X P P > > l > > > > B cos l r cos l q cos l z þ b cos l r cos l qx ðzÞþ m n q m n lz > > mn = < q¼0 n¼0 m¼0 mnq l¼1 n¼0 m¼0 vðr; q; z; tÞ ¼ ejut Q X 2 X N Q X 2 P M > > P P l l > > ~ > > > > bnq xlr ðrÞcos ln qcos lq z bmq cos lm rxlq ðqÞcos lq z þ ; : l¼1 q¼0 m¼0

(13.a-c)

l¼1 q¼0 n¼0

9 8 2 X N X M Q P X N M P P > > l > > > > C cos l r cos l q cos l z þ c cos l r cos l qx ðzÞþ mnq m n q m n lz > > mn = < q¼0 n¼0 m¼0 l¼1 n¼0 m¼0 wðr; q; z; tÞ ¼ ejut Q 2 X X N Q X 2 P M > > P P > > l l > > ~cnq xlr ðrÞcos ln qcos lq z > > cmq cos lm rxlq ðqÞcos lq z þ ; : l¼1 q¼0 m¼0

l¼1 q¼0 n¼0

Table 4 Comparison of the first three frequency parameters U for Type I FGM sector plates with different boundary conditions (R1/R2 ¼ 0.5, h/R ¼ 0.1, a ¼ 120 ). BC

CSCS

SSSS

CSSS

SSCS

n ¼ 0.5

n¼1

n¼2

Ref [29]

Voigt

M-T

Ref [29]

Voigt

M-T

Ref [29]

Voigt

M-T

142.48 153.42 174.72 71.898 91.015 122.19 99.833 113.25 137.94 107.84 124.49 153.45

143.39 154.42 175.86 72.024 91.298 122.95 100.24 113.79 138.63 108.24 125.04 154.19

125.18 134.80 153.49 62.921 79.750 107.37 87.511 99.338 121.02 94.561 109.21 134.63

128.76 138.65 157.89 64.818 82.062 110.46 90.102 102.21 124.49 97.320 112.36 138.52

129.55 139.52 158.86 64.926 82.317 110.87 90.456 102.68 125.09 97.678 112.85 139.16

114.84 123.63 140.73 57.849 73.306 98.653 80.379 91.225 111.11 86.846 100.28 123.57

116.93 125.89 143.34 58.895 74.551 100.33 81.845 92.833 113.05 88.403 102.05 125.78

117.25 126.23 143.68 58.937 74.701 100.55 81.994 93.053 113.33 88.549 102.27 126.04

107.96 116.19 132.22 54.740 69.327 93.208 75.833 86.039 104.75 81.934 94.561 116.42

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7

Table 6 Fundamental frequencies (Hz) of Type I FGM annular sector plates with boundary condition (R1 ¼ 1 m, R2/R1 ¼ 2, M-T solution).

a

h/R1

p

SSSS

CCCC

SCSC

CSCS

E1E1E1E1

E2E2E2E2

E3E3E3E3

120

0.2

0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5

801.11 710.58 585.84 1154.27 1010.27 765.95 1143.09 994.65 759.54 763.29 677.06 558.48 1039.76 912.10 694.23 1032.72 901.84 690.59 746.42 662.09 546.25 986.79 866.79 661.43 981.45 858.75 659.02

1464.03 1289.44 1027.33 2253.09 1957.51 1478.09 2523.66 2172.83 1636.58 1438.65 1267.13 1009.46 2211.03 1920.43 1448.20 2466.96 2123.70 1597.88 1429.15 1258.80 1002.82 2196.16 1907.34 1437.31 2446.01 2105.03 1582.65

822.69 729.56 600.70 1227.87 1075.21 816.88 1214.66 1056.73 809.40 769.54 682.59 562.85 1066.43 935.62 712.57 1058.83 924.57 708.64 748.20 663.68 547.51 995.24 874.24 667.21 989.74 865.97 664.70

1452.16 1278.89 1018.73 2228.10 1935.02 1461.48 2480.21 2133.78 1616.87 1435.41 1264.25 1007.11 2203.89 1913.91 1443.30 2449.48 2106.10 1590.60 1428.25 1258.00 1002.16 2194.17 1905.52 1435.92 2439.75 2097.83 1580.24

801.51 711.02 586.27 1225.61 1086.11 853.38 1602.55 1464.54 1259.09 763.70 677.51 558.92 1117.94 994.99 789.15 1525.06 1403.38 1218.30 746.85 662.56 546.71 1069.06 953.88 760.79 1493.10 1378.97 1203.71

1315.28 1162.82 899.33 1390.57 1240.63 990.69 1761.37 1593.69 1342.87 885.17 783.63 607.74 990.67 893.53 730.37 1469.42 1352.64 1163.86 597.52 530.15 413.61 742.08 679.68 575.05 1313.48 1224.01 1072.08

127.12 123.16 116.10 485.79 467.13 432.14 1204.50 1130.94 992.15 119.90 116.13 109.39 456.10 438.39 405.53 1138.61 1071.97 946.37 114.86 111.22 104.72 436.49 419.81 389.25 1103.75 1041.89 924.37

0.5

1.0

180

0.2

0.5

1.0

270

0.2

0.5

1.0

2  q x1q ðqÞ ¼ q 1 ; a

~lnq , Bmnq , where lm ¼ mp=R, ln ¼ np=a, lq ¼ qp=h. Amnq , almn , almq , a l ~l , Cmnq , cl , cl , ~cl are generalized coordinate variblmn , bmq , b nq nq mq mn ables. M, N, Q are the truncated numbers for Fourier series. xlr ðrÞ, xlq ðqÞ and xlz ðzÞ represent a set of closed-form supplementary functions in r, q and z directions, respectively. The special expressions of the supplementary functions are given below:

x1r ðrÞ ¼ r


r R

1

2

;

x2r ðrÞ ¼

 r2
r 1 R R

x1z ðzÞ ¼ z


z h

2 1 ;

x2q ðqÞ ¼

x2z ðzÞ ¼

  q2 q 1 a a

 z2
z 1 h h

(14.c-d)

(14.e-f)

By choosing the supplementary functions in such a way, the continuity of the displacements and their derivatives can be guaranteed in the three-dimensional elasticity theory because it is easy to verify that

(14.a-b)

Table 7 Fundamental frequencies (Hz) of Type II FGM annular sector plates with boundary condition (R1 ¼ 1 m, R2/R1 ¼ 2, M-T solution).

a

h/R1

p

SSSS

CCCC

SCSC

CSCS

E1E1E1E1

E2E2E2E2

E3E3E3E3

120

0.2

0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5 0.5 1 5

808.99 692.65 520.90 984.50 856.75 710.14 985.77 858.15 711.84 768.14 655.32 492.38 931.94 813.41 655.80 933.60 815.17 657.90 749.89 638.53 479.50 917.61 800.76 632.07 919.47 802.69 634.38

1453.77 1273.96 1006.51 2187.35 1906.33 1492.09 2490.30 2166.70 1674.49 1425.43 1250.03 990.39 2137.95 1865.58 1465.21 2426.30 2115.16 1639.42 1414.70 1240.89 984.26 2119.07 1849.80 1455.11 2400.43 2094.09 1625.48

831.19 711.43 534.48 1114.74 963.23 764.13 1116.70 965.30 766.34 774.58 660.83 496.38 974.74 847.07 673.67 976.55 848.96 675.89 751.72 640.10 480.62 928.84 809.49 637.42 930.71 811.43 639.75

1441.03 1263.56 999.34 2164.89 1888.85 1479.10 2465.91 2148.73 1659.60 1421.93 1247.15 988.41 2131.43 1860.46 1461.44 2418.31 2109.23 1634.43 1413.73 1240.08 983.71 2117.22 1848.33 1454.04 2398.01 2092.29 1623.96

809.36 693.04 521.34 1068.37 946.43 804.43 1507.38 1394.24 1243.20 768.51 655.72 492.84 1019.65 906.93 756.44 1471.50 1366.30 1212.08 750.28 638.95 479.97 1007.10 896.17 736.38 1465.98 1362.50 1202.73

1398.86 1251.41 953.81 1472.18 1326.66 1037.93 1852.52 1696.92 1388.97 937.23 840.01 641.58 1040.28 946.72 759.20 1534.00 1429.26 1204.79 631.11 566.87 435.39 774.27 714.71 594.46 1368.60 1291.79 1113.02

128.21 124.45 117.03 490.42 473.01 437.50 1216.88 1148.35 1010.72 120.91 117.35 110.34 460.22 443.91 410.83 1150.06 1089.26 966.11 115.80 112.39 105.68 440.23 424.94 394.15 1115.00 1059.19 944.72

0.5

1.0

180

0.2

0.5

1.0

270

0.2

0.5

1.0

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G. Jin et al. / Composites Part B xxx (2015) 1e15

0

x1r ð0Þ ¼ 1

0

x2r ðRÞ ¼ 1

x1r ð0Þ ¼ x1r ðRÞ ¼ x1r ðRÞ ¼ 0; x2r ð0Þ ¼ x2r ðRÞ ¼ x2r ð0Þ ¼ 0;

0

0

(15.a) (15.b)

Similar conditions exist for xlq ðqÞ and xlz ðzÞ. Substituting Eqs (4)e(6), (9)e(11) into Eq (12) together with the displacement functions defined in Eq (13), and setting the variation of energy functional P to zero respect to the generalized coordinate variables, the equations of motion can be obtained as

h i K  u2 M X ¼ 0

(16)

where X is the generalized coordinate variable vector, given as l ~l ; Cmnq ; cl ; cl ; ~cl T . K is ~lnq ; Bmnq ; blmn ; bmq ; b X ¼ ½Amnq ; almn ; almq ; a nq mn mq nq the generalized stiffness matrix which is obtained from strain energy U and potential energy P. M is the generalized mass matrix which is obtained from kinetic energy T. The detailed expressions for the elements in above matrices M and K are given in Appendix A. 3. Results and discussions Several numerical examples will be presented in this section to verify the accuracy and reliability of the current method. In identifying the boundary conditions, the symbols F, C and S denote free,

clamped and simply supported boundary conditions, respectively. It is noted that different interpretations have been given to the simply supported boundary conditions, and in this paper the simply supported boundary are defined as: sr ¼ 0; v ¼ w ¼ 0 for the edges r ¼ constant; sq ¼ 0; u ¼ w ¼ 0 for the edges q ¼ constant. The letters E1, E2 and E3 denote the elastic restraints. Taking edges r ¼ constant for example, the elastic restraints are defined as: us0; v ¼ w ¼ 0 for E1, vs0; u ¼ w ¼ 0 for E2, ws0; u ¼ v ¼ 0 for E3. In the numerical example presented, a letter string will be used to describe the combination of boundary conditions, for example, the symbol SFCE1 identifies the annular sector plate having simple supported boundary condition at r ¼ 0, free boundary condition at q ¼ 0, clamped boundary condition at r ¼ R, elastic restraint I at q ¼ a. The material properties for FGM annular sector plates are given in Table 1. 3.1. Convergence study In order to demonstrate the convergence of the present method, free vibration of completely free SiC/Ti annular sector plates with different thickness-to-radius ratios is investigated. Table 2 shows the first seven frequencies of Type I and II FGM annular sector plates with p ¼ 1. The geometrical parameters of FGM annular sector plates are given as: R1 ¼ 1 m, R2/R1 ¼ 2, a ¼ 120 , h/R1 ¼ 0.1, 0.5. The Mori-Tanaka model is employed to evaluate the effective material properties. From Table 2, it is evident that frequencies

Fig. 5. Mode shapes of Type I FGM annular sector plates with different boundary conditions (p ¼ 1, R1 ¼ 1 m, R2/R1 ¼ 2, h/R1 ¼ 0.5): (a) a ¼ 160 and FCFC; (b) a ¼ 270 and CFCF; (c) a ¼ 340 and CCCC.

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G. Jin et al. / Composites Part B xxx (2015) 1e15

9

Fig. 6. Mode shapes of Type II FGM annular sector plates with different boundary conditions (p ¼ 1, R1 ¼ 1 m, R2/R1 ¼ 2, h/R1 ¼ 0.5): (a) a ¼ 160 and CFFF; (b) a ¼ 270 and SCSC; (c) a ¼ 340 and FSFS.

converge monotonically as the truncated numbers increase. In the following cases, the truncated numbers will be uniformly selected as M  N  Q ¼ 12  12  10 (see Table 2). Secondly, the effects of elastic restraints on the frequency parameters of the SiC/Ti annular sector plates are studied. Three nondimensional elastic restraint parameters Gu, Gv and Gw are introduced here which are defined as the ratios of the corresponding spring stiffness to the bend stiffness D2 ¼ E2 h3 =12ð1  m22 Þ, i.e., Gu ¼ ku/D2, Gv ¼ kv/D2, Gw ¼ kw/D2. The variations of first three pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi frequency parameters U ¼ uR22 r2 h=D2 of Sic/Ti annular plates with various elastic restraints are illustrated in Figs. 3 and 4. The boundary condition is considered to be that only one kind of elastic restraint parameters at edges r ¼ constant varies from extremely small (102) to extremely large (106) while the rest groups of elastic restraint parameters are assigned at zero. The geometrical dimensions used for analysis are: R1 ¼ 1 m, R2/R1 ¼ 2, h/R1 ¼ 0.5, a ¼ 160 and 270 . The gradient index is taken to be as one. It is observed that the increase of elastic restraint parameter leads to the increase of the frequency parameters in a certain range, while beyond the range there little variation on frequency parameters. In the following analysis, the elastic restraint parameters are uniformly selected as Gu ¼ Gv ¼ Gw ¼ 10. 3.2. FGM annular sector plates with various boundary conditions In this subsection, the comparisons of the present results with those available in literature are presented. Table 3 shows the first

pffiffiffiffiffiffiffiffiffiffiffi six frequency parameters J ¼ uR22 =p2 rh=D for isotropic aluminum annular sector plates with different boundary conditions (i.e. SSSS, SSFF, CCFF, CCSS and CCCC). The geometrical dimensions are given as: R1/R2 ¼ 0.4, h/R2 ¼ 0.1 and 0.2, a ¼ 90 . The results are compared with other 3-D elasticity solutions determined by Liew et al. [16] using the Ritz method. Table 4 shows the first three frequency parameters U for Type I Al2O3/Al sector plates with different boundary conditions. Both Voigt model and Mori-Tanaka scheme are considered. The geometrical dimensions are given as: R1/R2 ¼ 0.5, h/R2 ¼ 0.1, a ¼ 120 . Free vibration analysis for the same problem has been reported by Saidi et al. [29] using the FSDT. Table 5 shows comparison of fundamental frequency paramepthe ffiffiffiffiffiffiffiffiffiffiffi ters 6 ¼ uR22 rh=D for Type II FGM annular sector plates derived by the present method with CPT solutions by Hosseini-Hashemi et al. [21]. The material properties are considered to vary exponentially along the radial direction. The geometrical parameters are given as: R1/R2 ¼ 0.5, h/R2 ¼ 0.01, m ¼ 0.3. a ¼ 30 , 60 and 90 . Three kinds of boundary conditions (i.e. SSSS, SCSC, and CCCC) are considered. . From Tables 3e5, it can be seen that a good agreements can be achieved. The slight discrepancies may due to the different solutions strategies and plate theories employed in those literature. Several new numerical results for FGM annular sector plates with various boundary conditions are presented. In the analysis, the Mori-Tanaka scheme is employed. The annular sector plates are made of Silicon carbide (SiC) and Titanium (Ti). Tables 6 and 7 show the fundamental frequencies of Type I and II FGM annular sector

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G. Jin et al. / Composites Part B xxx (2015) 1e15

Fig. 7. Effect of the gradient index p on the fundamental frequency of Type I FGM sector plates with various boundary conditions: (a) CCCC; (b) SSSS; (c) CSCS; (d) E1E1E1E1.

plates with different boundary conditions (i.e. SSSS, CCCC, SCSC, CSCS, E1E1E1E1, E2E2E2E2 and E3E3E3E3), respectively. The gradient indexes are taken to be 0.5, 1 and 5. The geometrical dimensions are given as: R1 ¼ 1 m, R2/R1 ¼ 2, a ¼ 120 , 180 and 270 , h/R1 ¼ 0.2, 0.5 and 1.0. The results show that the fundamental frequencies decrease as the sector angle a increases irrespective of the value of other geometry parameters and the type of boundary constraint. It is observed that the fundamental frequencies increase as the thickness-to-radius ratios H/R1 increases except for the case of Type I plates with SSSS boundary condition. It can be seen from tables that for a FGM annular sector plate of fix sector angle a and thickness-to-radius ratios H/R1, both the gradient index p and boundary condition can alter the value of the fundamental frequency. Some mode shapes of FGM annular sector plates are illustrated in Figs. 5e6.. 3.3. Parameter studies In this subsection, the influence of the gradient index on the fundamental frequency is studied for free vibration the FGM annular sector plates with various boundary conditions and sector angles. The Mori-Tanaka scheme is adopted to evaluate the effective material properties of FGMs. Fig. 7 shows the variation of

fundamental frequency parameters of Type I FGM annular sector plates against the gradient index. Four types of boundary conditions (i.e. CCCC, SSSS, CSCS and E1E1E1E1) are considered. For all considered cases, it is observed that the fundamental frequency parameters decrease as the gradient index p increases. The fundamental frequency parameter of isotropic annular sector plates is obtained when p ¼ 0. When the gradient index is minor, the fundamental frequency parameters decrease quickly, while when the gradient index raises, the change of fundamental frequency parameters decrease slowly. The fundamental frequency parameter of FGM annular sector plate with a ¼ 160 is larger than those of plates with a ¼ 270 and 340 . The fundamental frequency parameters of plate with a ¼ 270 and 340 are similar but the former is slightly larger than the latter. Fig. 8 shows the variation of fundamental frequency parameters of Type I FGM annular sector plates against the gradient index. The vibration characteristics of the Type II FGM annular sector plate are same as those of Type I one. 4. Conclusions The main purpose of this paper is to investigate free vibration behaviors of functionally graded sector plates with general

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11

Fig. 8. Effect of the gradient index p on the fundamental frequency of Type II FGM sector plates with various boundary conditions: (a) CCCC; (b) SSSS; (c) CSCS; (d) E1E1E1E1.

boundary conditions in the context of three-dimensional theory of elasticity. Generally, the material properties of functionally graded sector plates are assumed to vary continuously and smoothly in thickness direction. However, the changes in the material properties may occur in the other directions, such as radial direction. Therefore, two types of functionally graded annular sector plates are considered in the paper. In this work, both the Voigt model and Mori-Tanaka scheme are adopted to evaluate the effective material properties. Each of displacements of annular sector plate, regardless of boundary conditions, is expressed as modified Fourier series which consists of three-dimensional Fourier cosine series plus several auxiliary functions introduced to overcome the discontinuity problems of the displacement and its derivatives at edges. To ensure the validity and accuracy of the method, numerous examples for isotropic and functionally graded sector plates with various boundary conditions are presented. Furthermore, new results for functionally graded sector plates with elastic restraints are given, which can be serve as benchmark solutions. The effects of the material profiles and boundary conditions on the free vibration of the functionally sector plates are also studied.

Acknowledgment The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 51175098 and 51279035) and the Fundamental Research Funds for the Central Universities of China (No. HEUCFQ1401).

Appendix A The generalized mass and stiffness matrixes of an FGM annular sector plate in Eq. (16) are respectively given by

2

Muu M¼4 0 0

0 Mvv 0

3 0 0 5 Mww

2

Kuu T K ¼ 4 Kuv T Kuw

Kuv Kvv KTvw

3 Kuw Kvw 5 Kww (A.1 e2)

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12

G. Jin et al. / Composites Part B xxx (2015) 1e15

The elements of sub-matrix Muu are:

n o 11 Muu

a;b

Zh Za ZR ¼

rf cos lm r cos lm1 r cos ln q cos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz 0

n o 12 Muu

a;b1

0

Zh Za ZR ¼ 0

n o 13 Muu

a;b2

a;b3

a1 ;b

a1 ;b1

a1 ;b2

a1 ;b3

a2 ;b

a2 ;b1

a2 ;b2

a2 ;b3

a1 ;b

a3 ;b1

rf cos lm r cos lm1 r cos ln q cos ln1 qxlz ðzÞcos lq1 zðR1 þ rÞdrdqdz

(A.7)

0

0

rf cos lm r cos lm1 r cos ln q cos ln1 qxlz ðzÞxl1 z ðzÞðR1 þ rÞdrdqdz

(A.8)

rf cos lm r cos lm1 r cos ln qxl1 q ðqÞxlz ðzÞcos lq1 zðR1 þ rÞdrdqdz

(A.9)

rf cos lm rxl1 r ðrÞcos ln q cos ln1 qxlz ðzÞcos lq1 zðR1 þ rÞdrdqdz

(A.10)

0

0

0

Zh Za ZR ¼ 0

0

Zh Za ZR rf cos lm r cos lm1 rxlq ðqÞcos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz

¼ 0

(A.11)

0

Zh Za ZR ¼ 0

rf cos lm r cos lm1 rxlq ðqÞcos ln1 q cos lq zxl1 z ðzÞðR1 þ rÞdrdqdz

(A.12)

rf cos lm r cos lm1 rxlq ðqÞxl1 q ðqÞcos lq z cos lq1 zðR1 þ rÞdrdqdz

(A.13)

rf cos lm rxl1 r ðrÞxlq ðqÞcos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz

(A.14)

rf xlr ðrÞcos lm1 r cos ln q cos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz

(A.15)

0

Zh Za ZR ¼ 0

0

Zh Za ZR ¼ 0

0

Zh Za ZR ¼ 0

n o 42 Muu

0

¼

0

n o 41 Muu

(A.6)

Zh Za ZR

0

n o 34 Muu

rf cos lm rxl1 r ðrÞcos ln q cos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz 0

Zh Za ZR

0

n o 33 Muu

0

¼

0

n o 32 Muu

(A.5)

Zh Za ZR

0

n o 31 Muu

rf cos lm r cos lm1 r cos ln qxl1 q ðqÞcos lq z cos lq1 zðR1 þ rÞdrdqdz 0

¼

0

n o 24 Muu

(A.4)

Zh Za ZR

0

n o 23 Muu

0

¼

0

n o 22 Muu

rf cos lm r cos lm1 r cos ln q cos ln1 q cos lq zxl1 z ðzÞðR1 þ rÞdrdqdz 0

Zh Za ZR

0

o n 21 Muu

0

¼ 0

n o 14 Muu

(A.3)

0

0

0

Zh Za ZR ¼

rf xlr ðrÞcos lm1 r cos ln q cos ln1 q cos lq zxl1 z ðzÞðR1 þ rÞdrdqdz 0

0

(A.16)

0

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G. Jin et al. / Composites Part B xxx (2015) 1e15

n

43 Muu

Zh Za ZR

o a3 ;b2

¼ 0

n

44 Muu

13

0

rf xlr ðrÞcos lm1 r cos ln qxl1 q ðqÞcos lq z cos lq1 zðR1 þ rÞdrdqdz

(A.17)

rf xlr ðrÞxl1 r ðrÞcos ln q cos ln1 q cos lq z cos lq1 zðR1 þ rÞdrdqdz

(A.18)

0

Zh Za ZR

o a3 ;b3

¼ 0

0

0

Muu ¼ Mvv ¼ Mww The elements of the first row of the generalized stiffness matrix K are:

n

11 Kuu

o a;b

9 8 C11 ðR1 þ rÞlm lm1 sin lm r sin lm1 r cos ln q cos ln1 q cos lq z cos lq1 z > > > > > > > > > > > > C l sin l r cos l r cos l q cos l q cos l z cos l z m m1 n n1 q q1 > > 12 m > > > > > > > > > > C l cos l r sin l r cos l q cos l q cos l z cos l zþ > > m m n n q q 12 m1 1 1 1 > > h a R > > Z Z Z < Zh Za h = C22 R1 kur0 ¼ drdqdz þ > > R þ r cos lm r cos lm1 r cos ln q cos ln1 q cos lq z cos lq1 zþ > > 1 > > > > 0 0 0 > 0 0 > > > > > > > > C55 ðR1 þ rÞlq lq1 cos lm r cos lm1 r cos ln q cos ln1 q sin lq z sin lq1 zþ > > > > > > > > > > > C66 > > ; : ln ln1 cos lm r cos lm1 r sin ln q sin ln1 q cos lq z cos lq1 z R1 þ r Zh ZR h i kuq0 þ R2 kur1 ð  1Þmþm1 cos ln q cos ln1 q cos lq z cos lq1 zdqdz þ 0

nþn1

þ kuq1 ð  1Þ

n

12 Kuu

o a;b1

i cos lm r cos lm1 r cos lq z cos lq1 zðR1 þ rÞdrdz

mþm1

Zh ZR h i cos ln q cos ln1 q cos lq zxl1 z ðzÞdqdz þ kuq0 0

þ kuq1 ð  1Þ

13 Kuu

o a;b2

(A.19)

9 8 C ðR þ rÞlm lm1 sin lm r sin lm1 r cos ln q cos ln1 q cos lq zxl1 z ðzÞ > > > > > > 11 1 > > > > > > l sin l r cos l r cos l q cos l q cos l zx ðzÞ C > > m m m n n q l1 z 1 1 > > 12 > > > > > > > > > > C l cos l r sin l r cos l q cos l q cos l zx ðzÞþ m m n n q 12 m1 l z 1 1 > > 1 > > > Zh Za ZR > Zh Za h = < C22 R1 kur0 ¼ drdqdz þ cos l r cos l r cos l q cos l q cos l zx ðzÞ m m n n q l1 z 1 1 > > > > R1 þ r > > > 0 0 0 > 0 0 > > 0 > > > > > > C ðR þ rÞlq cos lm r cos lm1 r cos ln q cos ln1 q sin lq zxl1 z ðzÞþ > > > > 55 1 > > > > > > > > > > C66 > > ln ln1 cos lm r cos lm1 r sin ln q sin ln1 q cos lq zxl1 z ðzÞ ; : R1 þ r þ R2 kur1 ð  1Þ

n

0

nþn1

i cos lm r cos lm1 r cos lq zxl1 z ðzÞðR1 þ rÞdrdz

0

9 8 C11 ðR1 þ rÞlm lm1 sin lm r sin lm1 r cos ln qxl1 q ðqÞcos lq z cos lq1 z > > > > > > > > > > > > C l sin l r cos l r cos l qx ðqÞcos l z cos l z m m m n q q > > 12 l q 1 1 1 > > > > > > > > > > C l cos l r sin l r cos l qx ðqÞcos l z cos l zþ m m1 n l1 q q q1 > > 12 m1 > > > Zh Za ZR > Zh Za h = < C 22 R1 kur0 ¼ drdqdz þ cos lm r cos lm1 r cos ln qxl1 q ðqÞcos lq z cos lq1 zþ > > > > R1 þ r > > > 0 0 0 > 0 0 > > > > > C55 ðR1 þ rÞlq lq1 cos lm r cos lm1 r cos ln qxl1 q ðqÞsin lq z sin lq1 z > > > > > > > > > > > > > > > 0 C 66 > > ; : ln cos lm r cos lm1 r sin ln qxl1 q ðqÞcos lq z cos lq1 z R1 þ r i þ R2 kur1 ð  1Þmþm1 cos ln qxl1 q ðqÞcos lq z cos lq1 zdqdz

(A.20)

(A.21)

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14

n o 14 Kuu

o n 11 Kuv

o n 12 Kuv

o n 13 Kuv

o n 14 Kuv

G. Jin et al. / Composites Part B xxx (2015) 1e15

9 8 C11 ðR1 þ rÞlm sin lm rx’l1 r ðrÞcos ln q cos ln1 q cos lq z cos lq1 z > > > > > > > > > > > > > > l sin l rx ðrÞcos l q cos l q cos l z cos l zþ C m l1 r n n1 q q1 > > 12 m > > > > > > > > > > C cos l rx’ ðrÞcos l q cos l q cos l z cos l zþ > > m n n q q 12 > > 1 1 r l h a R 1 > Z Z Z > Zh ZR h = < C22 kuq0 ¼ drdqdz þ > > > > R þ r cos lm rxl1 r ðrÞcos ln q cos ln1 q cos lq z cos lq1 zþ > 1 > > 0 0 0 > 0 0 > > > > > > > > > C55 ðR1 þ rÞlq lq1 cos lm rxl1 r ðrÞcos ln q cos ln1 q sin lq z sin lq1 zþ > > > > > > > > > > > > > C66 > > ; : ln ln1 cos lm rxl1 r ðrÞsin ln q sin ln1 q cos lq z cos lq1 z R1 þ r i þ kuq1 ð  1Þnþn1 cos lm rxl1 r ðrÞcos lq z cos lq1 zðR1 þ rÞdrdz

a;b3

a;b

(A.22)

9 8 C12 lm ln1 sin lm r cos lm1 r cos ln q sin ln1 q cos lq z cos lq1 z > > > > > > > > > > > > C > > 22 > > h a R l cos l r cos l r cos l q sin l q cos l z cos l zþ > > Z Z Z < n1 m m1 n n1 q q1 = R1 þ r ¼ drdqdz > > > C66 ln lm1 cos lm r sin lm1 r sin ln q cos ln1 q cos lq z cos lq1 zþ > > > > 0 0 0 > > > > > > > C > > > ; : 66 ln cos lm r cos lm1 r sin ln q cos ln1 q cos lq z cos lq1 z > R1 þ r

(A.23)

8 9 C12 lm ln1 sin lm r cos lm1 r cos ln q sin ln1 q cos lq zxl1 z ðzÞ > > > > > > > > > > > > C22 > > > > l cos l r cos l r cos l q sin l q cos l zx ðzÞþ > Zh Za ZR > n m m n n q l z 1 1 1 C66 ln lm1 cos lm r sin lm1 r sin ln q cos ln1 q cos lq zxl z ðzÞþ > > > 1 > > > 0 0 0 > > > > > > > C > > > : 66 ln cos lm r cos lm1 r sin ln q cos ln1 q cos lq zxl1 z ðzÞ > ; R1 þ r

(A.24)

9 8 0 C12 lm sin lm r cos lm1 r cos ln qxl1 q ðqÞcos lq z cos lq1 zþ > > > > > > > > > > > > 0 C > > 22 > > > cos l r cos l r cos l qx ðqÞcos l z cos l zþ Zh Za ZR > m m1 n l1 q q q1 = > > C66 ln lm1 cos lm r sin lm1 r sin ln qxl1 q ðqÞcos lq z cos lq1 zþ > > > > 0 0 0 > > > > > > > > > > ; : C66 ln cos lm r cos lm1 r sin ln qxl q ðqÞcos lq z cos lq1 z > 1 R1 þ r

(A.25)

9 8 C12 lm ln1 sin lm rxl1 z ðzÞcos ln q sin ln1 q cos lq z cos lq1 z > > > > > > > > > > C22 > > > > > > l cos l rx ðzÞcos l q sin l q cos l z cos l z n m n n q q > l z 1 1 1 Zh Za ZR > 1 = < R1 þ r ¼ drdqdz 0 > > C66 ln cos lm rxl1 z ðzÞsin ln q cos ln1 q cos lq z cos lq1 zþ > > > > > > 0 0 0 > > > > > > > > > > C66 : ln cos lm rxl1 z ðzÞsin ln q cos ln1 q cos lq z cos lq1 z ; R1 þ r

(A.26)

8 9 Zh Za ZR < C13 ðR1 þ rÞlm lq1 sin lm r cos lm1 r cos ln q cos ln1 q cos lq z sin lq1 z = C lq cos lm r cos lm1 r cos ln q cos ln1 q cos lq z sin lq1 zþ ¼ drdqdz : 23 1 ; ðR þ rÞl l cos l r sin l r cos l q cos l q sin l z cos l z C q m m m n n q q 55 1 1 1 1 1 0 0 0

(A.27)

a;b1

a;b2

a;b3

o n 11 Kuw

a;b

o n 12 Kuw

8 9 0 Zh Za ZR > < C13 ðR1 þ rÞlm sin lm r cos lm1 r cos ln q cos ln1 q cos lq zxl1 z ðzÞþ > = 0 ¼ drdqdz C23 cos lm r cos lm1 r cos ln q cos ln1 q cos lq zxl1 z ðzÞþ > > : ; C55 ðR1 þ rÞlq lm1 cos lm r sin lm1 r cos ln q cos ln1 q sin lq zxl1 z ðzÞ 0 0 0

(A.28)

o n 13 Kuw

8 9 Zh Za ZR < C13 ðR1 þ rÞlm lq1 sin lm r cos lm1 r cos ln qxl1 q ðqÞcos lq z sin lq1 z = C lq cos lm r cos lm1 r cos ln qxl1 q ðqÞcos lq z sin lq1 zþ ¼ drdqdz : 23 1 ; C55 ðR1 þ rÞlq lm1 cos lm r sin lm1 r cos ln qxl1 q ðqÞsin lq z cos lq1 z 0 0 0

(A.29)

a;b1

a;b2

Please cite this article in press as: Jin G, et al., Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions, Composites Part B (2015), http://dx.doi.org/10.1016/j.compositesb.2015.08.032

G. Jin et al. / Composites Part B xxx (2015) 1e15

n

14 Kuw

o a;b3

9 8 Zh Za ZR < C13 ðR1 þ rÞlm lq1 sin lm rxl1 r ðrÞcos ln q cos ln1 q cos lq z sin lq1 z = C23 lq1 cos lm rxl1 r ðrÞcos ln q cos ln1 q cos lq z sin lq1 z ¼ drdqdz 0 ; : C 55 ðR1 þ rÞlq cos lm rxl1 r ðrÞcos ln q cos ln1 q sin lq z cos lq1 z 0 0 0

where a ¼ qðM þ 1ÞðN þ 1Þ þ nðM þ 1Þ þ m þ 1; b ¼ qðM þ 1ÞðN þ 1Þ þ n1 ðM þ 1Þ þ m1 þ 1 a1 ¼ ðl  1ÞðM þ 1ÞðN þ 1Þ þ nðM þ 1Þ þ m þ 1; b1 ¼ ðl1  1ÞðM þ 1ÞðN þ 1Þ þ n1 ðM þ 1Þ þ m1 þ 1 a2 ¼ ðl  1ÞðM þ 1ÞðQ þ 1Þ þ qðM þ 1Þ þ m þ 1; b2 ¼ ðl1  1ÞðM þ 1ÞðQ þ 1Þ þ q1 ðM þ 1Þ þ m1 þ 1 a3 ¼ ðl  1ÞðN þ 1ÞðQ þ 1Þ þ qðN þ 1Þ þ n þ 1; b3 ¼ ðl1  1ÞðN þ 1ÞðQ þ 1Þ þ q1 ðN þ 1Þ þ n1 þ 1

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Please cite this article in press as: Jin G, et al., Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions, Composites Part B (2015), http://dx.doi.org/10.1016/j.compositesb.2015.08.032