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Semi-analytical solution for three-dimensional vibration of functionally graded circular plates
Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910 www.elsevier.com/locate/cma
Semi-analytical solution for three-dimensional vibration of functionally graded circular plates G.J. Nie *, Z. Zhong School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China Received 9 January 2007; received in revised form 14 May 2007; accepted 19 June 2007 Available online 10 August 2007
Abstract Based on the three-dimensional theory of elasticity and assuming that the mechanical properties of the materials vary continuously in the thickness direction and have the same exponent-law variations, the three-dimensional free and forced vibration analysis of functionally graded circular plate with various boundary conditions is achieved in this paper. A semi-analytical method, which makes use of the state space method and the one-dimensional differential quadrature method, is used to obtain the vibration frequencies and dynamic responses of circular plates. The analytical solution in the thickness direction can be acquired using the state space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the accuracy and superiority of the presented method. The influences of the material property graded index, the circumferential wave number and the forced frequencies on the dynamic behavior for different functionally graded circular plates are also studied. 2007 Elsevier B.V. All rights reserved. Keywords: Functionally graded materials; Circular plates; Free vibration; Forced vibration; Semi-analytical method
1. Introduction Functionally graded materials (FGMs) have gained considerable attention in recent years. FGMs are composite materials that are microscopically inhomogeneous, and the mechanical properties vary continuously in one (or more) direction(s). This is achieved by gradually changing the composition of the constituent materials along one direction, usually in the thickness direction, to obtain smooth variation of material properties and optimum response to externally applied loading. A lot of different applications of FGMs can be found in references [1,2]. Dynamical analysis of FGMs structural components is important in the design of FGMs devices. Several results can be found in the literature. For example, Yang and Shen [3] dealt with the dynamic response of initially stressed functionally graded rectangular thin plates subjected to *
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0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.06.028
partially distributed impulsive lateral loads. Vel and Batra [4] presented a three-dimensional exact solution for free and forced vibrations of simply supported functionally graded rectangular plates. Kitipornchai et al. [5] studied the nonlinear vibration of imperfect shear deformable laminated rectangular plates comprising a homogeneous substrate and two layers of functionally graded materials. Chen [6] investigated the nonlinear vibration of functionally graded plates with arbitrary initial stresses and the effects of the amplitude of vibration, initial condition and volume fraction on nonlinear vibration were studied. Roque et al. [7] performed the free vibration analysis of functionally graded plates by the multiquadric radial basis function method and a higher-order shear deformation theory. Serge [8] considered the problems of free vibrations, buckling, and static deflections of functionally graded plates in which material properties vary through the thickness. Ferreira et al. [9] used the global collocation method, the first and the third-order shear deformation plate theories, the Mori–Tanaka technique to homogenize material
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properties, and approximated the trial solution with multiquadric radial basis functions to analyze free vibrations of functionally graded plates. Woo et al. [10] provided an analytical solution for the nonlinear free vibration behavior of plates made of functionally graded materials. Park and Kim [11] researched the thermal postbuckling and vibration of the functionally graded plate considering the nonlinear temperature-dependent material properties. However, the investigations on three-dimensional vibration of FGMs circular plates are limited in number and are discussed briefly here. Prakash and Ganapathi [12] studied the asymmetric flexural vibration and thermoelastic stability of FGMs circular plates using finite element method. Efraim and Eisenberger [13] presented the vibration analysis of thick annular plates with variable thickness made of isotropic material and FGMs by the exact element method. This paper deals with three-dimensional free and forced vibration for FGMs circular plates. The material properties are assumed to be graded in the thickness direction according to an exponential distribution. The formulations are based on the three-dimensional theory of elasticity. A semi-analytical method, which makes use of the state space method [14,15] and the one-dimensional differential quadrature method [16–20], is employed in the free and forced vibration analysis. Numerical results for the natural frequencies and dynamic responses of circular plates are presented. A parametric study is also given to demonstrate the influences of the material property graded index, the circumferential wave number and the forced frequencies for different functionally graded circular plates. 2. Mathematical formulations 2.1. Governing equations Consider a transversely isotropic circular plate composed of FGMs with radius a and thickness h, as shown in Fig. 1. A cylindrical coordinate system r, h, z with the origin o on the center of the bottom plane is employed to describe the plate displacement. The equations of elastic equilibrium under the cylindrical coordinate system, in the absence of body forces, are
orr 1 osrh osrz rr rh o2 ur þ þ þ ¼q 2 ; r oh or oz r ot 2 osrh 1 orh oshz 2srh o uh þ þ þ ¼q 2 ; r oh or oz r ot osrz 1 oshz orz srz o 2 uz þ þ þ ¼q 2 ; r oh or oz r ot
ð1Þ
where rr, rh, rz, shz, srz, srh are stress components, ur, uh, uz are displacement components, q denotes material density and t is time. The displacement components are related to the strain components through the following relations: our 1 ouz ouh ; chz ¼ þ ; r oh or oz 1 ouh ur our ouz eh ¼ þ ; crz ¼ þ ; r oh r oz or ouz ouh 1 our uh ; crh ¼ þ ; ez ¼ r oh oz or r
er ¼
ð2Þ
where er, eh, ez, chz, crz, crh are strain components. The constitutive equations for a transversely isotropic functionally graded material are rr ¼ c11 er þ c12 eh þ c13 ez ; rh ¼ c12 er þ c11 eh þ c13 ez ; rz ¼ c13 er þ c13 eh þ c33 ez ;
shz ¼ c44 chz ; srz ¼ c44 crz ; c11 c12 crh ; srh ¼ 2
ð3Þ
where c11, c12, c13, c33, c44 are elastic stiffness components. In FGMs, these material property parameters are the functions of the coordinates r, h, z and they vary continuously only along one direction in most real cases. In the present paper, we assume that the material properties have the following exponential distributions in the thickness direction of the plates, i.e., k ¼ k 0 ekðhÞ ; z
ð4Þ 0
where k denotes c11, c12, c13, c33, c44, q, k is the values of material properties at the plane z = 0, k is the material property graded index. The assumption was employed by many researchers (e.g., [15,21], among others). Considering Eqs. (1)–(4) and letting ur ; uh ; uz ; ouozr ; ouozh ; ouozz be the state variables, the governing equations represented by the displacements and their first-order differentials can be written as the state space formulation. 0 D1 P o P ¼ ; ð5Þ oz C C D2 D3 T where P ¼ ½ ur uh uz T , C ¼ ouozr ouozh ouozz , and D1, D2, D3 are given in Appendix A. 2.2. Boundary conditions For a solid circular plate, the boundary conditions are Clamped:
Fig. 1. Geometry of circular plates composed of functionally graded materials.
at r ¼ a;
ur ¼ 0; uh ¼ 0; uz ¼ 0:
ð6Þ
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
Simply supported: at r ¼ a;
rr ¼ 0; uh ¼ 0; uz ¼ 0:
ð7Þ
The regularity conditions of solid circular plate on the central point are ouz ¼ 0: ð8Þ at r ¼ 0; ur ¼ 0; uh ¼ 0; or For an annular plate with the inner radius b and outer radius a, the boundary conditions are Clamped–Clamped: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; ur ¼ 0; uh ¼ 0; uz ¼ 0:
ð9Þ
Clamped–Simply supported: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; rr ¼ 0; uh ¼ 0; uz ¼ 0:
ð10Þ
Clamped–Free: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; rr ¼ 0; srh ¼ 0; srz ¼ 0:
ð11Þ
For free vibration problems, boundary conditions at the top and bottom surfaces are assumed as the following. at bottom surface (z = 0) and top surface (z = h), srz ¼ 0;
shz ¼ 0;
rz ¼ 0:
ð12Þ
For forced vibrations, normal and tangential tractions at the top and bottom surfaces can be assumed as follows: at bottom surface (z = 0) ixm t srz ¼ K ; rm cosðmhÞe ixm t shz ¼ K hm sinðmhÞe ;
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circumferential coordinate h for certain types of vibration using assumed displacements field in the following form: ur ðr; h; z; tÞ ¼ u~r ðr; zÞ cosðmhÞeixt ; uh ðr; h; z; tÞ ¼ ~uh ðr; zÞ sinðmhÞeixt ; uz ðr; h; z; tÞ ¼ ~uz ðr; zÞ cosðmhÞeixt ;
ð14Þ
where m = 0, 1, 2, . . . , 1 is the circumferential wave number. It is obvious that m = 0 means axisymmetric vibration. While interchanging the trigonometric function cos mh and sin mh in Eq. (14), another set of free vibration modes can be obtained. However, in such a case, m = 0 means torsional vibration of no warping. Substituting Eq. (14) into Eq. (5) and using the following non-dimensional parameters and relation, sffiffiffiffiffiffi z r q0 Z ¼ ; R ¼ ; X ¼ xh 0 ; h a c11 ð15Þ c0ij ~ur ~uh ~uz 0 U R ¼ ; U h ¼ ; U Z ¼ ; cij ¼ 0 ; h h h c11 the state space equations can be given as " # " # 0 D1 P o P ¼ ; ð16Þ oZ C D2 D3 C oU h oU Z T R , D2 and where P ¼ ½ U R U h U Z T , C ¼ oU oZ oZ oZ D3 can be derived from D2 and D3. While the partial derivatives with respect to the radial variable R of the unknown displacements UR, Uh, UZ are expressed as the function of the displacements, the solution to Eq. (16) can be obtained. 3.2. Differential quadrature method (DQM)
ixm t rz ¼ K : zm cosðmhÞe
ð13aÞ
at top surface (z = h) ixm t srz ¼ K þ ; rm cosðmhÞe þ shz ¼ K hm sinðmhÞeixm t ;
ixm t rz ¼ K þ : ð13bÞ zm cosðmhÞe
where xm denotes the circular frequency. 3. Semi-analytical method A semi-analytical approach combining state space method (SSM) and differential quadrature method (DQM) is used to solve the governing equations shown in Eq. (5). The semi-analytical approach employs the state space method in the gradient direction (z-direction) of circular plates and uses the one-dimensional differential quadrature rule in the radial direction to establish a linear eigenvalue system from which the fundamental frequencies and the dynamic responses can be obtained. 3.1. State space formulations When the circular plate vibrating with natural frequency x, it is possible to separate the dependence of time and the
The basic idea of differential quadrature method (DQM) is to approximate an unknown function and its partial derivatives with respect to a spatial variable at any discrete point as the linear weighted sums of their values at all the discrete points chosen in the solution domain. According to the differential quadrature rule, the rth-order derivative of an unknown function f(n) at point i can be expressed as [19]. N X or f ðnÞ ðrÞ ¼ Aij f ðnj Þ ði ¼ 1; 2; . . . ; N Þ; ð17Þ onr n¼ni j¼1 where N is the number of discrete points. n is the coordinate of discrete point j and f(nj) is the function value at ðrÞ the discrete point j. Aij is a weighting coefficient matrix of rth-order derivative and the expressions of the weighting coefficients are given in Ref. [19]. Using DQM to the state space equations shown in Eq. (16), the new state space equations with respect to the state variables at discrete points can then be obtained as follows: " # " #" # 0 Di1 Pi o Pi ¼ ; ð18Þ oZ Ci Di2 Di3 Ci
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i ¼ ½ U R1 ; U R2 ; . . . ; U RN where P T U Z1 ; U Z2 ; . . . ; U ZN , Ci ¼
h
oU R R ; . . . ; oU oZ 1 oZ N
oU h oZ
1
;...;
U h1 ; U h2 ; . . . ; U hN
oU h
oU Z
oZ
oZ
N
1
;...;
oU Z iT oZ
N
;
The boundary conditions of the annular plate shown in Eqs. (9)–(11) also can be written as the following discretized forms. Clamped–Clamped:
N is the number of discrete points and the expressions of Di1 ; Di2 ; Di3 are omitted. The solution to Eq. (18) can be written as [22] F i ðZÞ ¼ eAi Z F i ð0Þ;
ð19Þ
Ai Z
is the matrix exponential function, Fi(Z) and where, e Fi(0) are the values of the state variables at arbitrary plane Z and the bottom plane Z = 0, and " # 0 Di1 ; Ai ¼ Di2 Di3 6N 6N h iT i i i F i ðZÞ ¼ U iR ðZÞ U ih ðZÞ U iZ ðZÞ oU R ðZÞ oU h ðZÞ oU Z ðZÞ ; oZ oZ oZ h iT i ð0Þ i ð0Þ i ð0Þ oU oU oU h R Z ; F i ð0Þ ¼ U iR ð0Þ U ih ð0Þ U iZ ð0Þ oZ oZ oZ with T
U iR ðZÞ ¼ ½U R1 ðZÞ; . . . ; U RN ðZÞ ;
T
oU iR ðZÞ oU R ðZÞ oU R ðZÞ ¼ ;...; ; oZ oZ oZ 1 N
T
From Eq. (19), we can get F i ð1Þ ¼ eAi F i ð0Þ;
ð20Þ
where Fi(1) and Fi(0) are the values of the state variables at the top plane and the bottom plane, respectively. The boundary conditions of solid circular plate shown in Eqs. (6)–(8) can be written as the following discretized forms:
U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0:
ð21Þ
Simply supported: rRN ¼ 0; U hN ¼ 0; U ZN ¼ 0:
ð22Þ
Regularity conditions: U R1 ¼ 0; U h1 ¼ 0; U Z1 ¼
ð1Þ N X A1s ð1Þ
s¼2
A11
U Zs ; ð23Þ
ð1Þ Aij
at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; rR1 ¼ 0; U h1 ¼ 0; U Z1 ¼ 0: a
ð25Þ
Clamped–Free: at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; rR1 ¼ 0; sRh1 ¼ 0; sRZ1 ¼ 0: a
ð26Þ
In the following we will consider free vibration problems and forced vibration problems, respectively.
i ¼ 1; 2; . . . ; N : ð27Þ Substituting Eq. (27) and the corresponding boundary conditions shown Eqs. (21)–(26) for different plates into Eq. (20), the algebraic equations for free vibrations can be obtained. B T ¼ 0;
Clamped:
at R ¼ 0;
Clamped–Simply supported:
at Z ¼ 0 and Z ¼ 1 !
ð1Þ N 1 N 1 oU R h X Ai1 X ð1Þ ð1Þ þ A U Zs ð1Þ A1s U Zs ¼ 0; oZ i a s¼2 is A s¼2 11
oU h mh U Zi ¼ 0; oZ i aRi !
N 1 oU Z hc013 X mU hi U Ri ð1Þ ¼ 0; þ A U Rs þ þ oZ i ac033 s¼2 is Ri Ri
U iZ ðZÞ ¼ ½U Z1 ðZÞ; . . . ; U ZN ðZÞ ;
T
oU iZ ðZÞ oU Z ðZÞ oU Z ðZÞ ¼ ;...; : oZ oZ oZ 1 N
at R ¼ 1;
ð24Þ
(1) Free vibration problems The discretized forms of the boundary conditions at the top and bottom surfaces for free vibration problems, shown in Eq. (12), can be expressed as follows:
U ih ðZÞ ¼ ½U h1 ðZÞ; . . . ; U hN ðZÞT ;
T
oU ih ðZÞ oU h ðZÞ oU h ðZÞ ¼ ;...; ; oZ oZ oZ 1 N
at R ¼ 1;
at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; U R1 ¼ 0; U h1 ¼ 0; U Z1 ¼ 0: a
is the weighting coefficient matrix of first-orwhere der derivative.
ð28Þ
where B is a 6(N 2) · 6(N 2) matrix, and T T ¼ U iR ð0Þ U ih ð0Þ U iZ ð0Þ U iR ð1Þ U ih ð1Þ U iZ ð1Þ
with T U iR ð0Þ ¼ U R2 ð0Þ; U R3 ð0Þ; . . . ; U RðN 1Þ ð0Þ ; T U ih ð0Þ ¼ U h2 ð0Þ; U h3 ð0Þ; . . . ; U hðN 1Þ ð0Þ ; T U iZ ð0Þ ¼ U Z2 ð0Þ; U Z3 ð0Þ; . . . ; U ZðN 1Þ ð0Þ ; T U iR ð1Þ ¼ U R2 ð1Þ; U R3 ð1Þ; . . . ; U RðN 1Þ ð1Þ ; T U ih ð1Þ ¼ U h2 ð1Þ; U h3 ð1Þ; . . . ; U hðN 1Þ ð1Þ ; T U iZ ð1Þ ¼ U Z2 ð1Þ; U Z3 ð1Þ; . . . ; U ZðN 1Þ ð1Þ :
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
While setting the determinant of B matrix to zero for non-trivial solutions, the characteristic frequency equation of FGMs circular plate can be gotten. jBj ¼ 0:
ð29Þ
Solving the above frequency equation governing the free vibration of FGMs circular plate, an infinite number of frequencies and corresponding mode shapes can be obtained. (2) Forced vibration problems For forced vibration problems, the discretized forms of the boundary conditions at the top and bottom surfaces, shown in Eqs. (13a) and (13b), can be expressed as follows.At the bottom surfaces (Z = 0): ! ð1Þ N 1 N 1 oU R h X Ai1 X K ð1Þ ð1Þ þ Ais U Zs ð1Þ A1s U Zs 0 rmi0 ¼ 0; oZ i a s¼2 c44 c11 A11 s¼2
oU h mh K U Zi 0 hmi0 ¼ 0; oZ i aRi c44 c11 !
N 1 0 X oU Z hc13 mU hi U Ri K ð1Þ 0 zmi0 ¼ 0; þ 0 Ais U Rs þ þ oZ i ac33 s¼2 Ri Ri c33 c11
i ¼ 1; 2; . . . ; N :
ð30aÞ
At the top surfaces (Z = 1): !
ð1Þ N 1 N 1 oU R h X Ai1 X Kþ ð1Þ ð1Þ þ Ais U Zs ð1Þ A1s U Zs 0 rmi ¼ 0; oZ i a s¼2 c44 c011 ek A11 s¼2
oU h mh Kþ U Zi 0 hmi ¼ 0; oZ i aRi c44 c011 ek !
N 1 oU Z hc013 X ð1Þ mU hi U Ri Kþ þ A U Rs þ þ ¼ 0; 0 zmi oZ i ac033 s¼2 is Ri Ri c33 c011 ek
ð30bÞ
Substituting Eqs. (30a), (30b) and the corresponding boundary conditions shown Eqs. (21)–(26) for different plates into Eq. (20), the algebraic equations for forced vibrations can be obtained B T ¼ Q;
ð31Þ
where B is a 6(N 2) · 6(N 2) matrix, Q is a traction force vector and U ih ð0Þ
Z = 0 are Young’s modulus E = 380 GPa, the density q = 3800 kg/m3, Poisson’s ratio m = 0.3, respectively. The material properties are assumed as the exponential distributions in the thickness of the plates shown in Eq. (4). Equally spaced discrete points are adopted and the number of discrete points in the radial direction is nine. The comparisons of the first five non-dimensional frequencies between the present solutions and the existing solutions are listed in Table 1. And the comparisons of the first nondimensional frequencies for different graded index are shown in Table 2. While using the software ANSYS, threedimensional eight-node solid element (Solid5) is adopted. The plate is divided into ten layers in the thickness and the element size in the radial and circumferential directions is less than 0.1a. And the total numbers of the elements and nodes are 6150 and 7128, respectively. From Tables 1 and 2, it can be found that the present results are in good agreement with the available results either for the isotropic plates or for FGMs plates. On the other hand, the superiority of the present method is also demonstrated through the above examples. Using the semi-analytical method, the analytical solution in the thickness direction is given and an effective approximate solution in the radial direction is obtained by selecting fewer discrete points. However, in order to get the precise results to a certain extent, while using ANSYS, the plate has to be meshed into a lot of elements. Compared with finite element method, the semi-analytical method requires much less computational effort and it has great advantage in the computational efficiency. 4.2. Free vibration
i ¼ 1; 2; . . . ; N :
T ¼ U iR ð0Þ
4905
U iZ ð0Þ
U iR ð1Þ
U ih ð1Þ U iZ ð1Þ
T
:
By solving Eq. (31) for forced vibrations, all state parameters at Z = 0 and Z = 1 are obtained. We can use the Eqs. (14) and (19) to calculate the displacements field in FGMs circular plates.
Example 2. Consider a functionally graded circular plate with radius a and thickness h, which is clamped on the Table 1 Comparison of the first five non-dimensional frequencies of circular plates (h/a = 0.2 and k = 0) m 0
1
Ref. [23] ANSYS Present Ref. [23] ANSYS Present
X3
X4
X5
0.320 0.325 0.326 0.393 0.395 0.405
0.410 0.410 0.410 0.454 0.458 0.459
0.603 0.619 0.615 0.594 0.593 0.614
0.693 0.694 0.695 0.755 0.754 0.745
k
4. Numerical examples 0
Example 1. To demonstrate the accuracy of the method, numerical results are presented for a clamped circular plate with radius a and thickness h. The material constants at
X2
0.097 0.098 0.098 0.192 0.190 0.190
Table 2 Comparison of the first non-dimensional frequencies of circular plates (h/a = 0.2) m
4.1. Comparison results
X1
1 2
ANSYS Present ANSYS Present ANSYS Present
1
2
3
4
5
0.096 0.096 0.186 0.186 0.284 0.277
0.090 0.090 0.176 0.176 0.269 0.262
0.082 0.082 0.161 0.161 0.248 0.241
0.073 0.073 0.144 0.144 0.224 0.218
0.064 0.064 0.128 0.128 0.201 0.195
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circumferential edges. The material properties are the same as Example 1. The lowest non-dimensional frequencies are listed in Table 3. It can be seen from Table 3 that the lowest nondimensional frequency increases with the increase of the thickness-to-width ratio of the plate and the circumferential wave number. And the lowest non-dimensional frequency decreases with the increase of the material property graded index.
Table 3 The lowest non-dimensional frequencies of circular plates with different k, m and h/a
Example 3. Consider a functionally graded annular plate with simply supported at the inner radius (b = 0.1) and clamped outer radius (a = 1.0). The ratio of the thickness and the radius is 0.2. The boundary conditions are shown in Eq. (25). The material properties at Z = 0 are the same as Example 1 and the material graded index k is 1. The displacement modes corresponding to the first three nondimensional frequencies are plotted in Fig. 2. From Fig. 2, it can be seen that the displacement modes of the first non-dimensional frequency are axisymmetric and the displacement modes of the second and the third non-dimensional frequency are asymmetric. The vibrating
0.3
h/a
0.1
0.2
0.4
0.5
m
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
k 0
1
2
3
4
5
0.026 0.053 0.083 0.098 0.190 0.283 0.200 0.368 0.528 0.319 0.560 0.788 0.446 0.758 1.054
0.025 0.052 0.081 0.096 0.186 0.277 0.196 0.361 0.519 0.313 0.551 0.777 0.439 0.747 1.039
0.024 0.049 0.076 0.090 0.176 0.262 0.185 0.343 0.495 0.298 0.527 0.745 0.419 0.718 1.000
0.021 0.045 0.069 0.082 0.161 0.241 0.170 0.318 0.461 0.275 0.493 0.700 0.390 0.676 0.946
0.019 0.039 0.061 0.073 0.144 0.218 0.153 0.289 0.423 0.250 0.454 0.650 0.358 0.630 0.887
0.017 0.034 0.054 0.064 0.128 0.195 0.136 0.261 0.386 0.225 0.416 0.600 0.326 0.583 0.827
modes corresponding to the second and the third frequencies are related to the circumferential wave number m = 1 and m = 2, respectively.
Fig. 2. Displacement modes of FGMs annular plate corresponding to the first three frequencies.
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
4.3. Forced vibration
x = 1.05x0, where x0 is the lowest value of the natural frequency, are shown in Fig. 3.
Example 4. Considering the same material properties as Example 1, numerical results are presented for the forced vibrations of a clamped functionally graded circular plate (h/ a = 0.1 and k = 1) under the following boundary conditions: Case 1. rz ¼ srz ¼ shz ¼ 0 at z ¼ 0; rz ¼ 109 eixt ; srz ¼ 0; shz ¼ 0
at z ¼ h:
Case 2. rz ¼ srz ¼ shz ¼ 0 at z ¼ 0; rz ¼ 109 cosðhÞeixt ; srz ¼ 0;
4907
shz ¼ 0
at z ¼ h:
For Case 1, the through-the-thickness variation of displacements ur and uz, stresses rr, rh, rz and srz, at a chosen position (r/a = 0.5, h = 45), for a static load (x = 0) and for forcing frequencies x = 0.8x0, x = 0.95x0 and
It can be found from Fig. 3 that the displacement UR has a linear variation in the thickness direction while UZ is almost uniform either for a static load (x = 0) or for a forced vibration (x 5 0). The stresses rr, rh, rz, srz are nonlinear functions of the thickness coordinate. For Case 2, the through-the-thickness variation of displacements ur, uh and uz, stresses rr, rh, rz, shz, srz and srh, at a chosen position (r/a = 0.5, h = 45), for different forced frequencies x = 0, x = 0.8x0, x = 0.95x0 and x = 1.05x0, are shown in Fig. 4. From Fig. 4, it can be seen that the displacements and stresses have the same variation as Fig. 3. And UR, Uh vary linearly in the thickness direction while UZ is uniform either for a static load (x = 0) or for a forced vibration (x 5 0). The stresses rr, rh, rz, shz, srz, srh are nonlinear functions of the thickness coordinate. It is easy to observe from Figs. 3 and 4 that the displacements and stresses become larger as the forcing frequency
Fig. 3. Variation of physical quantities with coordinate z at a location (r/a = 0.5, h = 45) for the first frequency of a circular plate with h/a = 0.1 and k = 1 for Case 1: (a) displacement UR (m), (b) displacement UZ (m), (c) stress rr (Pa), (d) stress rh (Pa), (e) stress rz (Pa), (f) stress srz (Pa).
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G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
Fig. 4. Variation of physical quantities with coordinate z at a location (r/a = 0.5, h = 45) for the first frequency of a circular plate with h/a = 0.1 and k = 1 for Case 2: (a) displacement UR (m), (b) displacement Uh (rad), (c) displacement UZ (m), (d) stress rr (Pa), (e) stress rh (Pa), (f) stress rz (Pa), (g) stress shz (Pa), (h) stress srz (Pa), (i) stress srh (Pa).
approaches the natural frequency of the circular plates. In the actual application, it should be avoided that the forcing
frequency is close to the natural frequency of the functionally graded circular plate.
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
5. Conclusions
4909
c011 o2 2 o2 1 o 1 þ þ 2c044 or2 r2 oh2 r or r2
c012 o2 1 o 1 q0 o2 0 2 þ 2 þ 0 2; or r or r 2c44 c44 ot
0 k o kc13 o 1 kc0 o ; d 31 ¼ 0 þ ; d 32 ¼ 130 ; d 23 ¼ hr oh hc33 or r hrc33 oh
c044 o2 1 o2 1 o q0 o2 þ þ : d 33 ¼ 0 þ c33 or2 r2 oh2 r or c033 ot2 The matrix D3 is given as 2 3 d 14 d 15 d 16 6 7 ðA:3Þ D3 ¼ 4 d 24 d 25 d 26 5; d 22 ¼
The three-dimensional free and forced vibration analysis of FGMs circular plates with various boundary conditions are investigated in this paper by a semi-analytical approach (SSM-DQM). The analytical solution in the graded direction can be acquired using the state space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. The numerical results are compared with the existing numerical solutions and good agreement is displayed. Compared with finite element method, the semi-analytical method requires much less computational effort and it has great advantage in the computational efficiency. The lowest non-dimensional frequency of clamped circular plate increases with the increase of the thickness-to-width ratio of the plate and the circumferential wave number. And the lowest non-dimensional frequency decreases with the increase of the material property graded index. For the forced vibration problems, the displacements and stresses become larger as the forcing frequency approaches the natural frequency of the circular plates. In the actual application, it should be avoided that the forcing frequency is close to the natural frequency of the functionally graded circular plate. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 10432030) and the Natural Science Foundation of Shanghai (No. 05zr14122). Appendix A. Expression of the matrixes D1, D2, D3 The matrix 2 1 0 6 D1 ¼ 4 0 1 0 0
D1 is given as 3 0 7 0 5: 1
The matrix D2 2 d 11 d 12 6 D2 ¼ 4 d 21 d 22 d 31 d 32
is given as 3 d 13 7 d 23 5; d 33
ðA:1Þ
ðA:2Þ
where
c011 o2 1 o2 1 o 1 c0 o2 q0 o2 þ 2120 þ þ þ 0 2; 0 2 2 2 2 2 2r oh r or r c44 or 2r c44 oh c44 ot
2 2 0 0 c 1 o 3 o c 1 o 1 o ; d 12 ¼ 11 120 c044 2r oroh 2r2 oh 2c44 r2 oh r oroh k o ; d 13 ¼ h or
c0 1 o2 3 o c0 1 o2 1 o þ 2 2 120 ; d 21 ¼ 110 2c44 r oroh r oh 2c44 r oroh r oh d 11 ¼
d 34
d 35
d 36
where k o c0 o d 14 ¼ ; d 15 ¼ 0; d 16 ¼ 13 ; d 24 ¼ 0; h or c044 or k 1 o c0 o 130 ; d 25 ¼ ; d 26 ¼ h r oh rc44 oh
c0 þ c0 o 1 c0 þ c0 1 o þ ; d 35 ¼ 13 0 44 d 34 ¼ 13 0 44 ; or r r oh c33 c33 k d 36 ¼ : h References [1] C.C.M. Wu, M. Kahn, W. Moy, Piezoelectric ceramics with functional gradients: a new application in material design, J. Am. Ceram. Soc. 79 (1996) 809–812. [2] X.H. Zhu, Z.Y. Meng, Operational principle, fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator, Sensors Actuators A 48 (1996) 169–173. [3] J. Yang, Hui-Shen Shen, Dynamic response of initially stressed functionally graded rectangular thin plates, Compos. Struct. 54 (2001) 497–508. [4] S.S. Vel, R.C. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, J. Sound Vib. 272 (2004) 703–730. [5] S. Kitipornchai, J. Yang, K.M. Liew, Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections, Int. J. Solids Struct. 41 (2004) 2235–2257. [6] C.S. Chen, Nonlinear vibration of a shear deformable functionally graded plate, Compos. Struct. 68 (2005) 295–302. [7] C.M.C. Roque, A.J.M. Ferreira, R.M.N. Jorge, A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory, J. Sound Vib. 1 (2006) 1–23. [8] A. Serge, Free vibration, buckling, and static deflections of functionally graded plates, Compos. Sci. Technol. 66 (2006) 2383–2394. [9] A.J.M. Ferreira, R.C. Batra, C.M.C. Roque, L.F. Qian, R.M.N. Jorge, Natural frequencies of functionally graded plates by a meshless method, Compos. Struct. 75 (2006) 593–600. [10] J. Woo, S.A. Meguid, L.S. Ong, Nonlinear free vibration behavior of functionally graded plates, J. Sound Vib. 289 (2006) 595–611. [11] J.S. Park, J.H. Kim, Thermal postbuckling and vibration analyses of functionally graded plates, J. Sound Vib. 289 (2006) 77–93. [12] T. Prakash, M. Ganapathi, Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method, Composites: Part B 37 (2006) 642–649.
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[13] E. Efraim, M. Eisenberger, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, J. Sound Vib. 299 (2007) 720–738. [14] J.S. Lee, L.Z. Jiang, Exact electroelastic analysis of piezoelectric laminae via state space approach, Int. J. Solids Struct. 33 (1996) 977– 990. [15] Z. Zhong, E.T. Shang, Three-dimensional exact analysis of a simplify supported functionally graded piezoelectric plate, Int. J. Solids Struct. 40 (2003) 5335–5352. [16] G. Karami, P. Malekzadeh, A new differential quadrature methodology for beam analysis and the associated differential quadrature element method, Comput. Methods Appl. Mech. Engrg. 191 (2002) 3509–3526. [17] K.M. Liew, Y.Q. Huang, J.N. Reddy, Vibration analysis of symmetrically laminated plates based on FSDT using the moving least squares differential quadrature method, Comput. Methods Appl. Mech. Engrg. 192 (2003) 2203–2222.
[18] T.C. Fung, Stability and accuracy of differential quadrature method in solving dynamic problems, Comput. Methods Appl. Mech. Engrg. 191 (2002) 1311–1331. [19] C.W. Bert, M. Malik, The differential quadrature method in computational mechanics: a review, Appl. Mech. Rev. 49 (1996) 1– 28. [20] X. Wang, Y. Wang, Free vibration analyses of thin sector plates by the new version of differential quadrature method, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3957–3971. [21] P. Gu, J. Asaro, Crack in functionally graded materials, Int. J. Solids Struct. 34 (1997) 1–17. [22] F.R. Gantmacher, The Theory of Matrix, Chelsea, New York, 1960. [23] C.F. Liu, Y.T. Lee, Finite element analysis of three-dimensional vibrations of thick circular and annular plates, J. Sound Vib. 233 (1) (2000) 63–80.
Semi-analytical solution for three-dimensional vibration of functionally graded circular plates G.J. Nie *, Z. Zhong School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China Received 9 January 2007; received in revised form 14 May 2007; accepted 19 June 2007 Available online 10 August 2007
Abstract Based on the three-dimensional theory of elasticity and assuming that the mechanical properties of the materials vary continuously in the thickness direction and have the same exponent-law variations, the three-dimensional free and forced vibration analysis of functionally graded circular plate with various boundary conditions is achieved in this paper. A semi-analytical method, which makes use of the state space method and the one-dimensional differential quadrature method, is used to obtain the vibration frequencies and dynamic responses of circular plates. The analytical solution in the thickness direction can be acquired using the state space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the accuracy and superiority of the presented method. The influences of the material property graded index, the circumferential wave number and the forced frequencies on the dynamic behavior for different functionally graded circular plates are also studied. 2007 Elsevier B.V. All rights reserved. Keywords: Functionally graded materials; Circular plates; Free vibration; Forced vibration; Semi-analytical method
1. Introduction Functionally graded materials (FGMs) have gained considerable attention in recent years. FGMs are composite materials that are microscopically inhomogeneous, and the mechanical properties vary continuously in one (or more) direction(s). This is achieved by gradually changing the composition of the constituent materials along one direction, usually in the thickness direction, to obtain smooth variation of material properties and optimum response to externally applied loading. A lot of different applications of FGMs can be found in references [1,2]. Dynamical analysis of FGMs structural components is important in the design of FGMs devices. Several results can be found in the literature. For example, Yang and Shen [3] dealt with the dynamic response of initially stressed functionally graded rectangular thin plates subjected to *
Corresponding author. Tel.: +86 21 6598 2383; fax: +86 21 6598 3267. E-mail address: [email protected] (G.J. Nie).
0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.06.028
partially distributed impulsive lateral loads. Vel and Batra [4] presented a three-dimensional exact solution for free and forced vibrations of simply supported functionally graded rectangular plates. Kitipornchai et al. [5] studied the nonlinear vibration of imperfect shear deformable laminated rectangular plates comprising a homogeneous substrate and two layers of functionally graded materials. Chen [6] investigated the nonlinear vibration of functionally graded plates with arbitrary initial stresses and the effects of the amplitude of vibration, initial condition and volume fraction on nonlinear vibration were studied. Roque et al. [7] performed the free vibration analysis of functionally graded plates by the multiquadric radial basis function method and a higher-order shear deformation theory. Serge [8] considered the problems of free vibrations, buckling, and static deflections of functionally graded plates in which material properties vary through the thickness. Ferreira et al. [9] used the global collocation method, the first and the third-order shear deformation plate theories, the Mori–Tanaka technique to homogenize material
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properties, and approximated the trial solution with multiquadric radial basis functions to analyze free vibrations of functionally graded plates. Woo et al. [10] provided an analytical solution for the nonlinear free vibration behavior of plates made of functionally graded materials. Park and Kim [11] researched the thermal postbuckling and vibration of the functionally graded plate considering the nonlinear temperature-dependent material properties. However, the investigations on three-dimensional vibration of FGMs circular plates are limited in number and are discussed briefly here. Prakash and Ganapathi [12] studied the asymmetric flexural vibration and thermoelastic stability of FGMs circular plates using finite element method. Efraim and Eisenberger [13] presented the vibration analysis of thick annular plates with variable thickness made of isotropic material and FGMs by the exact element method. This paper deals with three-dimensional free and forced vibration for FGMs circular plates. The material properties are assumed to be graded in the thickness direction according to an exponential distribution. The formulations are based on the three-dimensional theory of elasticity. A semi-analytical method, which makes use of the state space method [14,15] and the one-dimensional differential quadrature method [16–20], is employed in the free and forced vibration analysis. Numerical results for the natural frequencies and dynamic responses of circular plates are presented. A parametric study is also given to demonstrate the influences of the material property graded index, the circumferential wave number and the forced frequencies for different functionally graded circular plates. 2. Mathematical formulations 2.1. Governing equations Consider a transversely isotropic circular plate composed of FGMs with radius a and thickness h, as shown in Fig. 1. A cylindrical coordinate system r, h, z with the origin o on the center of the bottom plane is employed to describe the plate displacement. The equations of elastic equilibrium under the cylindrical coordinate system, in the absence of body forces, are
orr 1 osrh osrz rr rh o2 ur þ þ þ ¼q 2 ; r oh or oz r ot 2 osrh 1 orh oshz 2srh o uh þ þ þ ¼q 2 ; r oh or oz r ot osrz 1 oshz orz srz o 2 uz þ þ þ ¼q 2 ; r oh or oz r ot
ð1Þ
where rr, rh, rz, shz, srz, srh are stress components, ur, uh, uz are displacement components, q denotes material density and t is time. The displacement components are related to the strain components through the following relations: our 1 ouz ouh ; chz ¼ þ ; r oh or oz 1 ouh ur our ouz eh ¼ þ ; crz ¼ þ ; r oh r oz or ouz ouh 1 our uh ; crh ¼ þ ; ez ¼ r oh oz or r
er ¼
ð2Þ
where er, eh, ez, chz, crz, crh are strain components. The constitutive equations for a transversely isotropic functionally graded material are rr ¼ c11 er þ c12 eh þ c13 ez ; rh ¼ c12 er þ c11 eh þ c13 ez ; rz ¼ c13 er þ c13 eh þ c33 ez ;
shz ¼ c44 chz ; srz ¼ c44 crz ; c11 c12 crh ; srh ¼ 2
ð3Þ
where c11, c12, c13, c33, c44 are elastic stiffness components. In FGMs, these material property parameters are the functions of the coordinates r, h, z and they vary continuously only along one direction in most real cases. In the present paper, we assume that the material properties have the following exponential distributions in the thickness direction of the plates, i.e., k ¼ k 0 ekðhÞ ; z
ð4Þ 0
where k denotes c11, c12, c13, c33, c44, q, k is the values of material properties at the plane z = 0, k is the material property graded index. The assumption was employed by many researchers (e.g., [15,21], among others). Considering Eqs. (1)–(4) and letting ur ; uh ; uz ; ouozr ; ouozh ; ouozz be the state variables, the governing equations represented by the displacements and their first-order differentials can be written as the state space formulation. 0 D1 P o P ¼ ; ð5Þ oz C C D2 D3 T where P ¼ ½ ur uh uz T , C ¼ ouozr ouozh ouozz , and D1, D2, D3 are given in Appendix A. 2.2. Boundary conditions For a solid circular plate, the boundary conditions are Clamped:
Fig. 1. Geometry of circular plates composed of functionally graded materials.
at r ¼ a;
ur ¼ 0; uh ¼ 0; uz ¼ 0:
ð6Þ
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
Simply supported: at r ¼ a;
rr ¼ 0; uh ¼ 0; uz ¼ 0:
ð7Þ
The regularity conditions of solid circular plate on the central point are ouz ¼ 0: ð8Þ at r ¼ 0; ur ¼ 0; uh ¼ 0; or For an annular plate with the inner radius b and outer radius a, the boundary conditions are Clamped–Clamped: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; ur ¼ 0; uh ¼ 0; uz ¼ 0:
ð9Þ
Clamped–Simply supported: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; rr ¼ 0; uh ¼ 0; uz ¼ 0:
ð10Þ
Clamped–Free: at r ¼ a; at r ¼ b;
ur ¼ 0; uh ¼ 0; uz ¼ 0; rr ¼ 0; srh ¼ 0; srz ¼ 0:
ð11Þ
For free vibration problems, boundary conditions at the top and bottom surfaces are assumed as the following. at bottom surface (z = 0) and top surface (z = h), srz ¼ 0;
shz ¼ 0;
rz ¼ 0:
ð12Þ
For forced vibrations, normal and tangential tractions at the top and bottom surfaces can be assumed as follows: at bottom surface (z = 0) ixm t srz ¼ K ; rm cosðmhÞe ixm t shz ¼ K hm sinðmhÞe ;
4903
circumferential coordinate h for certain types of vibration using assumed displacements field in the following form: ur ðr; h; z; tÞ ¼ u~r ðr; zÞ cosðmhÞeixt ; uh ðr; h; z; tÞ ¼ ~uh ðr; zÞ sinðmhÞeixt ; uz ðr; h; z; tÞ ¼ ~uz ðr; zÞ cosðmhÞeixt ;
ð14Þ
where m = 0, 1, 2, . . . , 1 is the circumferential wave number. It is obvious that m = 0 means axisymmetric vibration. While interchanging the trigonometric function cos mh and sin mh in Eq. (14), another set of free vibration modes can be obtained. However, in such a case, m = 0 means torsional vibration of no warping. Substituting Eq. (14) into Eq. (5) and using the following non-dimensional parameters and relation, sffiffiffiffiffiffi z r q0 Z ¼ ; R ¼ ; X ¼ xh 0 ; h a c11 ð15Þ c0ij ~ur ~uh ~uz 0 U R ¼ ; U h ¼ ; U Z ¼ ; cij ¼ 0 ; h h h c11 the state space equations can be given as " # " # 0 D1 P o P ¼ ; ð16Þ oZ C D2 D3 C oU h oU Z T R , D2 and where P ¼ ½ U R U h U Z T , C ¼ oU oZ oZ oZ D3 can be derived from D2 and D3. While the partial derivatives with respect to the radial variable R of the unknown displacements UR, Uh, UZ are expressed as the function of the displacements, the solution to Eq. (16) can be obtained. 3.2. Differential quadrature method (DQM)
ixm t rz ¼ K : zm cosðmhÞe
ð13aÞ
at top surface (z = h) ixm t srz ¼ K þ ; rm cosðmhÞe þ shz ¼ K hm sinðmhÞeixm t ;
ixm t rz ¼ K þ : ð13bÞ zm cosðmhÞe
where xm denotes the circular frequency. 3. Semi-analytical method A semi-analytical approach combining state space method (SSM) and differential quadrature method (DQM) is used to solve the governing equations shown in Eq. (5). The semi-analytical approach employs the state space method in the gradient direction (z-direction) of circular plates and uses the one-dimensional differential quadrature rule in the radial direction to establish a linear eigenvalue system from which the fundamental frequencies and the dynamic responses can be obtained. 3.1. State space formulations When the circular plate vibrating with natural frequency x, it is possible to separate the dependence of time and the
The basic idea of differential quadrature method (DQM) is to approximate an unknown function and its partial derivatives with respect to a spatial variable at any discrete point as the linear weighted sums of their values at all the discrete points chosen in the solution domain. According to the differential quadrature rule, the rth-order derivative of an unknown function f(n) at point i can be expressed as [19]. N X or f ðnÞ ðrÞ ¼ Aij f ðnj Þ ði ¼ 1; 2; . . . ; N Þ; ð17Þ onr n¼ni j¼1 where N is the number of discrete points. n is the coordinate of discrete point j and f(nj) is the function value at ðrÞ the discrete point j. Aij is a weighting coefficient matrix of rth-order derivative and the expressions of the weighting coefficients are given in Ref. [19]. Using DQM to the state space equations shown in Eq. (16), the new state space equations with respect to the state variables at discrete points can then be obtained as follows: " # " #" # 0 Di1 Pi o Pi ¼ ; ð18Þ oZ Ci Di2 Di3 Ci
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i ¼ ½ U R1 ; U R2 ; . . . ; U RN where P T U Z1 ; U Z2 ; . . . ; U ZN , Ci ¼
h
oU R R ; . . . ; oU oZ 1 oZ N
oU h oZ
1
;...;
U h1 ; U h2 ; . . . ; U hN
oU h
oU Z
oZ
oZ
N
1
;...;
oU Z iT oZ
N
;
The boundary conditions of the annular plate shown in Eqs. (9)–(11) also can be written as the following discretized forms. Clamped–Clamped:
N is the number of discrete points and the expressions of Di1 ; Di2 ; Di3 are omitted. The solution to Eq. (18) can be written as [22] F i ðZÞ ¼ eAi Z F i ð0Þ;
ð19Þ
Ai Z
is the matrix exponential function, Fi(Z) and where, e Fi(0) are the values of the state variables at arbitrary plane Z and the bottom plane Z = 0, and " # 0 Di1 ; Ai ¼ Di2 Di3 6N 6N h iT i i i F i ðZÞ ¼ U iR ðZÞ U ih ðZÞ U iZ ðZÞ oU R ðZÞ oU h ðZÞ oU Z ðZÞ ; oZ oZ oZ h iT i ð0Þ i ð0Þ i ð0Þ oU oU oU h R Z ; F i ð0Þ ¼ U iR ð0Þ U ih ð0Þ U iZ ð0Þ oZ oZ oZ with T
U iR ðZÞ ¼ ½U R1 ðZÞ; . . . ; U RN ðZÞ ;
T
oU iR ðZÞ oU R ðZÞ oU R ðZÞ ¼ ;...; ; oZ oZ oZ 1 N
T
From Eq. (19), we can get F i ð1Þ ¼ eAi F i ð0Þ;
ð20Þ
where Fi(1) and Fi(0) are the values of the state variables at the top plane and the bottom plane, respectively. The boundary conditions of solid circular plate shown in Eqs. (6)–(8) can be written as the following discretized forms:
U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0:
ð21Þ
Simply supported: rRN ¼ 0; U hN ¼ 0; U ZN ¼ 0:
ð22Þ
Regularity conditions: U R1 ¼ 0; U h1 ¼ 0; U Z1 ¼
ð1Þ N X A1s ð1Þ
s¼2
A11
U Zs ; ð23Þ
ð1Þ Aij
at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; rR1 ¼ 0; U h1 ¼ 0; U Z1 ¼ 0: a
ð25Þ
Clamped–Free: at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; rR1 ¼ 0; sRh1 ¼ 0; sRZ1 ¼ 0: a
ð26Þ
In the following we will consider free vibration problems and forced vibration problems, respectively.
i ¼ 1; 2; . . . ; N : ð27Þ Substituting Eq. (27) and the corresponding boundary conditions shown Eqs. (21)–(26) for different plates into Eq. (20), the algebraic equations for free vibrations can be obtained. B T ¼ 0;
Clamped:
at R ¼ 0;
Clamped–Simply supported:
at Z ¼ 0 and Z ¼ 1 !
ð1Þ N 1 N 1 oU R h X Ai1 X ð1Þ ð1Þ þ A U Zs ð1Þ A1s U Zs ¼ 0; oZ i a s¼2 is A s¼2 11
oU h mh U Zi ¼ 0; oZ i aRi !
N 1 oU Z hc013 X mU hi U Ri ð1Þ ¼ 0; þ A U Rs þ þ oZ i ac033 s¼2 is Ri Ri
U iZ ðZÞ ¼ ½U Z1 ðZÞ; . . . ; U ZN ðZÞ ;
T
oU iZ ðZÞ oU Z ðZÞ oU Z ðZÞ ¼ ;...; : oZ oZ oZ 1 N
at R ¼ 1;
ð24Þ
(1) Free vibration problems The discretized forms of the boundary conditions at the top and bottom surfaces for free vibration problems, shown in Eq. (12), can be expressed as follows:
U ih ðZÞ ¼ ½U h1 ðZÞ; . . . ; U hN ðZÞT ;
T
oU ih ðZÞ oU h ðZÞ oU h ðZÞ ¼ ;...; ; oZ oZ oZ 1 N
at R ¼ 1;
at R ¼ 1; U RN ¼ 0; U hN ¼ 0; U ZN ¼ 0; b at R ¼ ; U R1 ¼ 0; U h1 ¼ 0; U Z1 ¼ 0: a
is the weighting coefficient matrix of first-orwhere der derivative.
ð28Þ
where B is a 6(N 2) · 6(N 2) matrix, and T T ¼ U iR ð0Þ U ih ð0Þ U iZ ð0Þ U iR ð1Þ U ih ð1Þ U iZ ð1Þ
with T U iR ð0Þ ¼ U R2 ð0Þ; U R3 ð0Þ; . . . ; U RðN 1Þ ð0Þ ; T U ih ð0Þ ¼ U h2 ð0Þ; U h3 ð0Þ; . . . ; U hðN 1Þ ð0Þ ; T U iZ ð0Þ ¼ U Z2 ð0Þ; U Z3 ð0Þ; . . . ; U ZðN 1Þ ð0Þ ; T U iR ð1Þ ¼ U R2 ð1Þ; U R3 ð1Þ; . . . ; U RðN 1Þ ð1Þ ; T U ih ð1Þ ¼ U h2 ð1Þ; U h3 ð1Þ; . . . ; U hðN 1Þ ð1Þ ; T U iZ ð1Þ ¼ U Z2 ð1Þ; U Z3 ð1Þ; . . . ; U ZðN 1Þ ð1Þ :
G.J. Nie, Z. Zhong / Comput. Methods Appl. Mech. Engrg. 196 (2007) 4901–4910
While setting the determinant of B matrix to zero for non-trivial solutions, the characteristic frequency equation of FGMs circular plate can be gotten. jBj ¼ 0:
ð29Þ
Solving the above frequency equation governing the free vibration of FGMs circular plate, an infinite number of frequencies and corresponding mode shapes can be obtained. (2) Forced vibration problems For forced vibration problems, the discretized forms of the boundary conditions at the top and bottom surfaces, shown in Eqs. (13a) and (13b), can be expressed as follows.At the bottom surfaces (Z = 0): ! ð1Þ N 1 N 1 oU R h X Ai1 X K ð1Þ ð1Þ þ Ais U Zs ð1Þ A1s U Zs 0 rmi0 ¼ 0; oZ i a s¼2 c44 c11 A11 s¼2
oU h mh K U Zi 0 hmi0 ¼ 0; oZ i aRi c44 c11 !
N 1 0 X oU Z hc13 mU hi U Ri K ð1Þ 0 zmi0 ¼ 0; þ 0 Ais U Rs þ þ oZ i ac33 s¼2 Ri Ri c33 c11
i ¼ 1; 2; . . . ; N :
ð30aÞ
At the top surfaces (Z = 1): !
ð1Þ N 1 N 1 oU R h X Ai1 X Kþ ð1Þ ð1Þ þ Ais U Zs ð1Þ A1s U Zs 0 rmi ¼ 0; oZ i a s¼2 c44 c011 ek A11 s¼2
oU h mh Kþ U Zi 0 hmi ¼ 0; oZ i aRi c44 c011 ek !
N 1 oU Z hc013 X ð1Þ mU hi U Ri Kþ þ A U Rs þ þ ¼ 0; 0 zmi oZ i ac033 s¼2 is Ri Ri c33 c011 ek
ð30bÞ
Substituting Eqs. (30a), (30b) and the corresponding boundary conditions shown Eqs. (21)–(26) for different plates into Eq. (20), the algebraic equations for forced vibrations can be obtained B T ¼ Q;
ð31Þ
where B is a 6(N 2) · 6(N 2) matrix, Q is a traction force vector and U ih ð0Þ
Z = 0 are Young’s modulus E = 380 GPa, the density q = 3800 kg/m3, Poisson’s ratio m = 0.3, respectively. The material properties are assumed as the exponential distributions in the thickness of the plates shown in Eq. (4). Equally spaced discrete points are adopted and the number of discrete points in the radial direction is nine. The comparisons of the first five non-dimensional frequencies between the present solutions and the existing solutions are listed in Table 1. And the comparisons of the first nondimensional frequencies for different graded index are shown in Table 2. While using the software ANSYS, threedimensional eight-node solid element (Solid5) is adopted. The plate is divided into ten layers in the thickness and the element size in the radial and circumferential directions is less than 0.1a. And the total numbers of the elements and nodes are 6150 and 7128, respectively. From Tables 1 and 2, it can be found that the present results are in good agreement with the available results either for the isotropic plates or for FGMs plates. On the other hand, the superiority of the present method is also demonstrated through the above examples. Using the semi-analytical method, the analytical solution in the thickness direction is given and an effective approximate solution in the radial direction is obtained by selecting fewer discrete points. However, in order to get the precise results to a certain extent, while using ANSYS, the plate has to be meshed into a lot of elements. Compared with finite element method, the semi-analytical method requires much less computational effort and it has great advantage in the computational efficiency. 4.2. Free vibration
i ¼ 1; 2; . . . ; N :
T ¼ U iR ð0Þ
4905
U iZ ð0Þ
U iR ð1Þ
U ih ð1Þ U iZ ð1Þ
T
:
By solving Eq. (31) for forced vibrations, all state parameters at Z = 0 and Z = 1 are obtained. We can use the Eqs. (14) and (19) to calculate the displacements field in FGMs circular plates.
Example 2. Consider a functionally graded circular plate with radius a and thickness h, which is clamped on the Table 1 Comparison of the first five non-dimensional frequencies of circular plates (h/a = 0.2 and k = 0) m 0
1
Ref. [23] ANSYS Present Ref. [23] ANSYS Present
X3
X4
X5
0.320 0.325 0.326 0.393 0.395 0.405
0.410 0.410 0.410 0.454 0.458 0.459
0.603 0.619 0.615 0.594 0.593 0.614
0.693 0.694 0.695 0.755 0.754 0.745
k
4. Numerical examples 0
Example 1. To demonstrate the accuracy of the method, numerical results are presented for a clamped circular plate with radius a and thickness h. The material constants at
X2
0.097 0.098 0.098 0.192 0.190 0.190
Table 2 Comparison of the first non-dimensional frequencies of circular plates (h/a = 0.2) m
4.1. Comparison results
X1
1 2
ANSYS Present ANSYS Present ANSYS Present
1
2
3
4
5
0.096 0.096 0.186 0.186 0.284 0.277
0.090 0.090 0.176 0.176 0.269 0.262
0.082 0.082 0.161 0.161 0.248 0.241
0.073 0.073 0.144 0.144 0.224 0.218
0.064 0.064 0.128 0.128 0.201 0.195
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circumferential edges. The material properties are the same as Example 1. The lowest non-dimensional frequencies are listed in Table 3. It can be seen from Table 3 that the lowest nondimensional frequency increases with the increase of the thickness-to-width ratio of the plate and the circumferential wave number. And the lowest non-dimensional frequency decreases with the increase of the material property graded index.
Table 3 The lowest non-dimensional frequencies of circular plates with different k, m and h/a
Example 3. Consider a functionally graded annular plate with simply supported at the inner radius (b = 0.1) and clamped outer radius (a = 1.0). The ratio of the thickness and the radius is 0.2. The boundary conditions are shown in Eq. (25). The material properties at Z = 0 are the same as Example 1 and the material graded index k is 1. The displacement modes corresponding to the first three nondimensional frequencies are plotted in Fig. 2. From Fig. 2, it can be seen that the displacement modes of the first non-dimensional frequency are axisymmetric and the displacement modes of the second and the third non-dimensional frequency are asymmetric. The vibrating
0.3
h/a
0.1
0.2
0.4
0.5
m
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
k 0
1
2
3
4
5
0.026 0.053 0.083 0.098 0.190 0.283 0.200 0.368 0.528 0.319 0.560 0.788 0.446 0.758 1.054
0.025 0.052 0.081 0.096 0.186 0.277 0.196 0.361 0.519 0.313 0.551 0.777 0.439 0.747 1.039
0.024 0.049 0.076 0.090 0.176 0.262 0.185 0.343 0.495 0.298 0.527 0.745 0.419 0.718 1.000
0.021 0.045 0.069 0.082 0.161 0.241 0.170 0.318 0.461 0.275 0.493 0.700 0.390 0.676 0.946
0.019 0.039 0.061 0.073 0.144 0.218 0.153 0.289 0.423 0.250 0.454 0.650 0.358 0.630 0.887
0.017 0.034 0.054 0.064 0.128 0.195 0.136 0.261 0.386 0.225 0.416 0.600 0.326 0.583 0.827
modes corresponding to the second and the third frequencies are related to the circumferential wave number m = 1 and m = 2, respectively.
Fig. 2. Displacement modes of FGMs annular plate corresponding to the first three frequencies.
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4.3. Forced vibration
x = 1.05x0, where x0 is the lowest value of the natural frequency, are shown in Fig. 3.
Example 4. Considering the same material properties as Example 1, numerical results are presented for the forced vibrations of a clamped functionally graded circular plate (h/ a = 0.1 and k = 1) under the following boundary conditions: Case 1. rz ¼ srz ¼ shz ¼ 0 at z ¼ 0; rz ¼ 109 eixt ; srz ¼ 0; shz ¼ 0
at z ¼ h:
Case 2. rz ¼ srz ¼ shz ¼ 0 at z ¼ 0; rz ¼ 109 cosðhÞeixt ; srz ¼ 0;
4907
shz ¼ 0
at z ¼ h:
For Case 1, the through-the-thickness variation of displacements ur and uz, stresses rr, rh, rz and srz, at a chosen position (r/a = 0.5, h = 45), for a static load (x = 0) and for forcing frequencies x = 0.8x0, x = 0.95x0 and
It can be found from Fig. 3 that the displacement UR has a linear variation in the thickness direction while UZ is almost uniform either for a static load (x = 0) or for a forced vibration (x 5 0). The stresses rr, rh, rz, srz are nonlinear functions of the thickness coordinate. For Case 2, the through-the-thickness variation of displacements ur, uh and uz, stresses rr, rh, rz, shz, srz and srh, at a chosen position (r/a = 0.5, h = 45), for different forced frequencies x = 0, x = 0.8x0, x = 0.95x0 and x = 1.05x0, are shown in Fig. 4. From Fig. 4, it can be seen that the displacements and stresses have the same variation as Fig. 3. And UR, Uh vary linearly in the thickness direction while UZ is uniform either for a static load (x = 0) or for a forced vibration (x 5 0). The stresses rr, rh, rz, shz, srz, srh are nonlinear functions of the thickness coordinate. It is easy to observe from Figs. 3 and 4 that the displacements and stresses become larger as the forcing frequency
Fig. 3. Variation of physical quantities with coordinate z at a location (r/a = 0.5, h = 45) for the first frequency of a circular plate with h/a = 0.1 and k = 1 for Case 1: (a) displacement UR (m), (b) displacement UZ (m), (c) stress rr (Pa), (d) stress rh (Pa), (e) stress rz (Pa), (f) stress srz (Pa).
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Fig. 4. Variation of physical quantities with coordinate z at a location (r/a = 0.5, h = 45) for the first frequency of a circular plate with h/a = 0.1 and k = 1 for Case 2: (a) displacement UR (m), (b) displacement Uh (rad), (c) displacement UZ (m), (d) stress rr (Pa), (e) stress rh (Pa), (f) stress rz (Pa), (g) stress shz (Pa), (h) stress srz (Pa), (i) stress srh (Pa).
approaches the natural frequency of the circular plates. In the actual application, it should be avoided that the forcing
frequency is close to the natural frequency of the functionally graded circular plate.
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5. Conclusions
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c011 o2 2 o2 1 o 1 þ þ 2c044 or2 r2 oh2 r or r2
c012 o2 1 o 1 q0 o2 0 2 þ 2 þ 0 2; or r or r 2c44 c44 ot
0 k o kc13 o 1 kc0 o ; d 31 ¼ 0 þ ; d 32 ¼ 130 ; d 23 ¼ hr oh hc33 or r hrc33 oh
c044 o2 1 o2 1 o q0 o2 þ þ : d 33 ¼ 0 þ c33 or2 r2 oh2 r or c033 ot2 The matrix D3 is given as 2 3 d 14 d 15 d 16 6 7 ðA:3Þ D3 ¼ 4 d 24 d 25 d 26 5; d 22 ¼
The three-dimensional free and forced vibration analysis of FGMs circular plates with various boundary conditions are investigated in this paper by a semi-analytical approach (SSM-DQM). The analytical solution in the graded direction can be acquired using the state space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. The numerical results are compared with the existing numerical solutions and good agreement is displayed. Compared with finite element method, the semi-analytical method requires much less computational effort and it has great advantage in the computational efficiency. The lowest non-dimensional frequency of clamped circular plate increases with the increase of the thickness-to-width ratio of the plate and the circumferential wave number. And the lowest non-dimensional frequency decreases with the increase of the material property graded index. For the forced vibration problems, the displacements and stresses become larger as the forcing frequency approaches the natural frequency of the circular plates. In the actual application, it should be avoided that the forcing frequency is close to the natural frequency of the functionally graded circular plate. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 10432030) and the Natural Science Foundation of Shanghai (No. 05zr14122). Appendix A. Expression of the matrixes D1, D2, D3 The matrix 2 1 0 6 D1 ¼ 4 0 1 0 0
D1 is given as 3 0 7 0 5: 1
The matrix D2 2 d 11 d 12 6 D2 ¼ 4 d 21 d 22 d 31 d 32
is given as 3 d 13 7 d 23 5; d 33
ðA:1Þ
ðA:2Þ
where
c011 o2 1 o2 1 o 1 c0 o2 q0 o2 þ 2120 þ þ þ 0 2; 0 2 2 2 2 2 2r oh r or r c44 or 2r c44 oh c44 ot
2 2 0 0 c 1 o 3 o c 1 o 1 o ; d 12 ¼ 11 120 c044 2r oroh 2r2 oh 2c44 r2 oh r oroh k o ; d 13 ¼ h or
c0 1 o2 3 o c0 1 o2 1 o þ 2 2 120 ; d 21 ¼ 110 2c44 r oroh r oh 2c44 r oroh r oh d 11 ¼
d 34
d 35
d 36
where k o c0 o d 14 ¼ ; d 15 ¼ 0; d 16 ¼ 13 ; d 24 ¼ 0; h or c044 or k 1 o c0 o 130 ; d 25 ¼ ; d 26 ¼ h r oh rc44 oh
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