Axisymmetric bending of two-directional functionally graded circular and annular plates

Acta Mechanica Solida Sinica, Vol. 20, No. 4, December, 2007 Published by AMSS Press, Wuhan, China. DOI: 10.1007/s10338-007-0734-9

ISSN 0894-9166

AXISYMMETRIC BENDING OF TWO-DIRECTIONAL FUNCTIONALLY GRADED CIRCULAR AND ANNULAR PLATES  Guojun Nie1

Zheng Zhong

(School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China)

Received 18 May 2007; revision received 6 December 2007.

ABSTRACT Assuming the material properties varying with an exponential law both in the thickness and radial directions, axisymmetric bending of two-directional functionally graded circular and annular plates is studied using the semi-analytical numerical method in this paper. The deflections and stresses of the plates are presented. Numerical results show the well accuracy and convergence of the method. Compared with the finite element method, the semi-analytical numerical method is with great advantage in the computational efficiency. Moreover, study on axisymmetric bending of two-directional functionally graded annular plate shows that such plates have better performance than those made of isotropic homogeneous materials or one-directional functionally graded materials. Two-directional functionally graded material is a potential alternative to the one-directional functionally graded material. And the integrated design of materials and structures can really be achieved in two-directional functionally graded materials.

KEY WORDS two-directional functionally graded materials, circular and annular plates, axisymmetric bending, semi-analytical numerical method, integrated design

I. INTRODUCTION Functionally graded materials (FGMs) are intentionally designed composite materials. The advantage of this kind of materials is that no internal boundaries exist and failures from interfacial stress concentrations can be avoided. In recent years, FGMs have gained considerable attention[1] . There are a lot of studies on the mechanics of FGMs at present. For example, Reddy et al.[2] have studied axisymmetric bending and stretching of functionally graded solid and annular circular plates using the first-order shear deformation Mindlin plate theory. Zhong and Shang[3] have gained an exact three-dimensional solution for a simply supported functionally gradient piezoelectric rectangular plate. Ma and Wang[4] have presented relationships between axisymmetric bending and buckling solutions of functionally graded circular plates based on third-order plate theory and classical plate theory. Zhu and Sankar[5] have studied a FGM beam whose Young’s modulus is given by a polynomial in the thickness coordinate. Shen and Noda[6] have performed the postbuckling analysis for a shear deformable functionally graded cylindrical shell of finite length subjected to combined axial and radial loads in thermal environment. Zhong and Yu[7] have obtained plane elasticity solutions for orthotropic functionally graded beams with arbitrary elastic moduli variations along the thickness direction under different end boundary conditions by means of the semi-inverse method. Shao and Wang[8] have achieved  

Corresponding author. E-mail: [email protected] Project supported by the National Natural Science Foundation of China (No.10432030).

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three-dimensional thermo-elastic analysis of a functionally graded cylindrical panel of finite length and subject to nonuniform mechanical and steady-state thermal loads. The plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate has been considered by Ding et al.[9] . At present, little has been done for two-directional functionally graded materials (2D-FGMs). However, the investigation on 2D-FGMs has shown that it is more capable of reducing thermal and residual stresses than one-directional FGMs[10] . In other words, if the FGMs have two-directional dependent material properties, more effective material can be obtained. The 2D-FGMs can be a potential alternative to the one-directional FGMs. Up to now, report on the static behavior of two-directional functionally graded circular and annular plates is not yet available in the literature. The aim of this study is to achieve an axisymmetric bending analysis of two-directional functionally graded circular and annular plates using a semi-analytical numerical method. The material properties are considered to be exponential functions in the thickness and radial direction. The governing equations are based on the three-dimensional elastic theory. A semi-analytical numerical method, which makes use of the state space method (SSM) and differential quadrature method (DQM), is employed in the static analysis of two-directional functionally graded circular and annular plates. The accuracy, convergence and superiority of the method presented are demonstrated through numerical examples. And the axisymmetric bending performance of two-directional functionally graded circular and annular plates is also studied.

II. BASIC FORMULATIONS 2.1. Governing Equations Consider a two-directional functionally graded circular and annular plate of total thickness h, inner radius b and outer radius a, as shown in Fig.1. The r-coordinate is taken radially outward from the center of the bottom plane, the z-coordinate along the thickness of the plate, and the θ-coordinate is taken along a circumference of the plate. Suppose that the grading of the material, applied loads, and boundary conditions are axisymmetric so that the circumferential displacement uθ is identically zero. The equilibrium equations of the axisymmetric Fig. 1 Two-directional functionally graded circular and annular plates. problem, in the absence of body forces, are ∂σr ∂τrz σr − σθ + + = 0, ∂r ∂z r where, σr , σθ , σz , τrz are components of the stress. The strains are related to the displacements by εr =

∂ur , ∂r

εθ =

ur , r

εz =

∂τrz ∂σz τrz + + =0 ∂r ∂z r

∂uz , ∂z

γrz =

∂uz ∂ur + ∂z ∂r

(1)

(2)

where ur , uz are components of the displacement; εr , εθ , εz , γrz are components of the strain. For a homogeneous, orthotropic material, the linear constitutive equations are σr = c˜11 εr + c˜12 εθ + c˜13 εz ,

σθ = c˜12 εr + c˜22 εθ + c˜23 εz

σz = c˜13 εr + c˜23 εθ + c˜33 εz ,

τrz = c˜55 γrz

(3)

where c˜ij denotes the elastic constants of the material and is assumed to be of the following form c˜ij = c0ij eλ1 (z/h)+λ2 (r/a)

(4)

where c0ij denotes the value at the center of the bottom plane of the plate, λ1 and λ2 denote the graded index along the thickness direction and radial direction, respectively.

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By considering Eq.(4), the equilibrium equations expressed by the displacements can be obtained from Eqs.(1)-(3) as follows:       2 ∂ 2 ur 1 ∂uz λ2 ∂uz 1 ∂ur λ2 ur ∂ uz λ2 ∂ur 0 0 c011 + + c + c + + + 12 13 r ∂r ∂r2 a ∂r a r ∂r∂z r ∂z a ∂z   2 2 ur 1 ∂uz λ1 ∂ur λ1 ∂uz ∂ ur ∂ uz + c055 + + =0 (5a) − c022 2 − c023 + r r ∂z ∂z 2 ∂r∂z h ∂z h ∂r     2   2 λ1 ∂ur λ1 ur ∂ ur 1 ∂ur ∂ uz λ1 ∂uz c013 + + c023 + + c033 + ∂r∂z h ∂r r ∂z h r ∂z 2 h ∂z   2 2 1 ∂uz ∂ ur ∂ uz λ2 ∂uz λ2 ∂ur 1 ∂ur + + + + =0 (5b) + c055 + r ∂z r ∂r ∂r∂z ∂r2 a ∂z a ∂r 2.2. Boundary Conditions Consider a solid circular plate (b = 0) with a clamped support at r = a. The boundary conditions are at r = a, ur = 0, uz = 0 (6) Consider a solid circular plate (b = 0) simply supported at r = a. The boundary conditions are at r = a,

σr = 0,

uz = 0

(7)

The regularity conditions of the solid circular plate on the central point are at r = 0,

ur = 0,

∂uz =0 ∂r

(8)

Consider an annular plate with a clamped support at the inner edge r = b and free at the outer edge r = a. The boundary conditions are at r = b,

ur = 0,

uz = 0;

at r = a,

σr = 0,

τrz = 0

(9)

Consider an annular plate with clamped inner edge r = b and outer edge r = a. The boundary conditions are at r = a, ur = 0, uz = 0 (10) at r = b, ur = 0, uz = 0; Consider an annular plate with clamped inner edge r = b and simply supported at the outer edge r = a. The boundary conditions are at r = b,

ur = 0,

uz = 0;

at r = a,

σr = 0,

uz = 0

(11)

The boundary conditions at the top and bottom surfaces of the plate are z = 0, z = h,

− (r), τrz = Krm

− σz = Kzm (r)

(12a)

τrz =

σz =

(12b)

+ Krm (r),

+ Kzm (r)

− − + + (r), Kzm (r), Krm (r), Kzm (r) are external loads. where, Krm

III. SOLUTION A semi-analytical numerical method, which makes use of SSM in the thickness direction and DQM along the radial direction, is employed to obtain the solutions of the differential equations (5a) and (5b). The SSM that originated from the modern control theory has been used to investigate the behavior of various plates[11, 12] . The analytical solution of first-order differential equations can be given using this method. DQM is a new numerical approach to solving differential equations. The basic idea of DQM is to approximate to 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[13] . The effective approximate solution of first-order differential equations can be obtained using this numerical method.

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3.1. Formulation for SSM Let the displacements ur , uz and their first derivatives ∂ur /∂z, ∂uz /∂z act as state variables and consider the following non-dimensional parameters Z=

z , h

R=

r , a

UR =

ur , h

UZ =

uz , h

c¯0ij =

c0ij c033

The governing equations, shown in Eqs.(5a) and (5b), can be written as      ∂ 0 D1 Π Π = Γ D 2 (R) D 3 (R) ∂Z Γ

(13)

(14)

where Π = {UR UZ }T , Γ = {∂UR /∂Z ∂UZ /∂Z}T , D 1 is the 2 × 2 identity matrix, D2 (R) and D 3 (R) are the functions of the variable R. In order to get the solution of Eq.(14), we need to apply the DQM approximation to the elements of matrixes D2 (R) and D 3 (R). 3.2. Application of DQM According to the differential quadrature rule, the partial derivatives with respect to R of the unknown functions UR , UZ at arbitrary point Ri can be expressed as:  N ∂UR  (1) = W URj , ∂R R=Ri j=1 ij  N ∂ 2 UR  (2) = W URj , ∂R2 R=Ri j=1 ij  N ∂ 2 UR  (1) ∂URj , = W ∂R∂Z R=Ri j=1 ij ∂Z

 N ∂UZ  (1) = W UZj ∂R R=Ri j=1 ij  N ∂ 2 UZ  (2) = W UZj ∂R2 R=Ri j=1 ij  N ∂ 2 UZ  (1) ∂UZj = W ∂R∂Z R=Ri j=1 ij ∂Z

(15)

where N denotes the total number of discrete points, URj , UZj , ∂URj /∂Z, ∂UZj /∂Z are the function (1) (2) values at the discrete point Rj , Wij and Wij are the weighting coefficients of the first derivative and second derivative, respectively. By substituting Eq.(15) into (14), the following state space equation at discrete points is then obtained      ∂ Πi Πi 0 D1 = (i = 1, 2, · · · , N ) (16) D 2 (Ri ) D3 (Ri ) Γi ∂Z Γ i where the elements of matrixes D 2 (Ri ) and D 3 (Ri ) are constants. The solution to Eq.(16) can be written as: F i (Z) = eAi ·Z F i (0)

(17)

where eAi ·Z is the matrix exponential function, F i (Z) and F i (0) are the values of the state variables at arbitrary plane Z and the bottom plane Z = 0, respectively, and   0 D1 Ai = D2 (Ri ) D3 (Ri ) 4N ×4N  T ∂URi (Z) ∂UZi (Z) F i (Z) = URi (Z) UZi (Z) (i = 1, 2, · · · , N ) (18) ∂Z ∂Z  T ∂URi (0) ∂UZi (0) F i (0) = URi (0) UZi (0) ∂Z ∂Z From Eq.(17), we get F i (1) = eAi F i (0) where F i (1) are the values of the state variables at the top plane Z = 1.

(19)

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For a solid circular plate, the boundary conditions and regularity conditions shown in Eqs.(6)-(8) can be expressed in discretized forms as follows: Clamped: Simply supported:

at R = 1, at R = 1,

Regularity conditions: at R = 0,

UZN = 0 UZN = 0

URN = 0, σRN = 0, UR1 = 0,

UZ1 = −

(20) (21) (1) N W1j j=2

(1)

W11

UZj

(22)

For an annular plate, the boundary conditions shown in Eqs.(9)-(11) can be expressed as follows: b , UR1 = 0, UZ1 = 0; a b Clamped-Clamped: at R = , UR1 = 0, UZ1 = 0; a b Clamped-Simply supported: at R = , UR1 = 0, UZ1 = 0; a at R =

Clamped-Free:

at R = 1, σRN = 0, τRZN = 0(23) at R = 1, URN = 0, UZN = 0 (24) at R = 1, σRN = 0, UZN = 0 (25)

The boundary conditions at the top and bottom surfaces of the plate (see Eqs.(12a) and (12b)) can be written in discretized forms At Z = 0, N

∂URi h (1) K − (Ri ) + Wij UZj = 0 rm ∂Z a j=1 c¯55 c033 eRi λ2 N ∂UZi h¯ c0 h¯ c0 (1) K − (Ri ) + 23 URi + 13 Wij URj = 0 zm ∂Z aRi a j=1 c¯33 c033 eRi λ2

(i = 1, 2, · · · , N )

(26a)

At Z = 1, N

∂URi h (1) K + (Ri ) + Wij UZj = 0 0 rm(λ +R λ ) ∂Z a j=1 c¯55 c33 e 1 i 2 N ∂UZi h¯ c0 h¯ c0 (1) K + (Ri ) + 23 URi + 13 Wij URj = 0 0 zm(λ +R λ ) ∂Z aRi a j=1 c¯33 c33 e 1 i 2

(i = 1, 2, · · · , N )

(26b)

Substituting the corresponding boundary conditions into Eq.(19), we can get the values of the state variables on the top and bottom surfaces. And then all the displacement and stress components of the plate can be obtained using Eqs.(17) and (3).

IV. NUMERICAL EXAMPLES 4.1. Verification of Results Example 1. Consider a solid circular plate (a = 1.0 m, h = 0.1a), which is clamped on the circumferential edge. The material properties are assumed as the exponent-law variation in the thickness and radial direction shown in Eq.(4). And the graded indexes (λ1 and λ2 ) are equal to one. Young’s modulus at the central point of the bottom plane is E = 380 GPa and the Poisson’s ratio is chosen as a constant, ν = 0.3. The boundary conditions on the top and bottom surfaces of the plate are at z = 0,

σz = τrz = 0;

at z = h,

σz = −1.0 GPa,

τrz = 0

(27)

A comparison of results between the present method and ANSYS is shown in Table 1. The equally spaced discrete points are nine while using the DQM. Three-dimensional eight-node solid element (Solid 5) is adopted while using the software ANSYS. The plate is divided into ten layers in thickness and

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Table 1. The deflection of two-directional functionally graded circular plate. (Unit: m)

0.0

Present ANSYS Error

0.000 −0.1523 −0.1513 0.0066

0.125 −0.1462 −0.1451 0.0076

0.250 −0.1297 −0.1289 0.0062

r/a 0.375 0.500 −0.1056 −0.0776 −0.1052 −0.0775 0.0038 0.0013

0.625 −0.0494 −0.0495 −0.0020

0.750 −0.0247 −0.0250 −0.0120

0.875 −0.0073 −0.0075 −0.0266

1.0

Present ANSYS Error

−0.1529 −0.1515 0.0092

−0.1468 −0.1456 0.0082

−0.1301 −0.1294 0.0054

−0.1059 −0.1055 0.0038

−0.0494 −0.0496 −0.0040

−0.0247 −0.0249 −0.0080

−0.0071 −0.0073 −0.0274

z/h

−0.0778 −0.0777 0.0013

the element size in the radial and circumferential directions is less than 0.1 m. And the total numbers of the elements and nodes are 14549 and 16745, respectively. From Table 1, it can be seen that the present results have a good agreement with the ANSYS results. It shows the accuracy of the presented solutions. At the same time, the superiority of the present method is also shown through this example. While using the semi-analytical numerical method, the effective approximate solution in the radial direction can be obtained using DQM by selecting fewer discrete points and simultaneously the analytical solution in the thickness direction can be given using SSM. 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. So compared with the finite element method, the semi-analytical numerical method requires much less computational effort and has great advantage in the computational efficiency. 4.2. Convergence Studies Example 2. To show the effect of the number of the selected discrete points, convergence studies are conducted for a two-directional functionally graded solid circular plate. The structural parameters of the circular plate are the same as those given in Example 1. The numerical results are shown in Fig.2. It can be seen from Fig.2 that the deflections of the plate approaches ANSYS solutions with an Fig. 2 Deflections of the plate for different discrete points. increase of discrete points. It demonstrates that the present method is of great convergence. 4.3. Axisymmetric Bending of 2D-FGMs Annular Plate Example 3. Considering the same top and bottom surface loads as given in Example 1, the axisymmetric bending of two-directional functionally graded annular plate simply supported at the inner radius (b = 0.1 m) and clamped at the outer (a = 1.0 m) is investigated. The material is isotropic and the material properties at an arbitrary point are E = E 0 eλ1 (z/h)+λ2 (r/a) ,

ν = ν 0 eλ1 (z/h)+λ2 (r/a)

(E 0 = 380 GPa,

ν 0 = 0.3)

(28)

The numerical results for different graded variations are shown in Figs.3-4.

Fig. 3. Variation of displacement uz (m) at the location (z/h = 0).

Fig. 4. Variation of stress σr (MPa) at the location (r/a = 0.55).

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It can be found from Fig.3 that the transverse displacement at the location (z/h = 0) for different graded variations is quite different. And the transverse displacement of two-directional functionally graded plate is least. It demonstrates the two-directional functionally graded plate is of great rigidity. Similarly, it is observed from Fig.4 that the radial normal stress at the location (r/a = 0.55) of twodirectional functionally graded plate is less than that of the isotropic homogenous and one-directional functionally graded plate. The maximum of the stress can be reduced and the distribution of the stress can approach uniformity by changing the graded indexes in the thickness and radial directions while using 2D-FGMs. This shows that 2D-FGMs have good carrying capacity. And it is a potential alternative to the one-directional FGM. And the integrated design of materials and structures can really be achieved in 2D-FGMs.

V. CONCLUSIONS Based on the three-dimensional elastic theory and assuming the mechanical properties having the exponential law variation in the thickness and radial direction, axisymmetric bending of two-directional graded circular and annular plates is studied in this paper using a semi-analytical numerical method. The effective approximate solution in the radial direction can be obtained using DQM by selecting fewer discrete points and the analytical solution in the thickness direction can be given using SSM. The accuracy, superiority and convergence of the method proposed are shown through the numerical examples. It is found that two-directional functionally graded circular and annular plates have better performance than homogeneous materials and one-directional FGMs. In order to improve the bearing load capacity of functionally graded structures and optimize the stress distribution, 2D-FGMs can be taken into consideration. And the integrated design of materials and structures can certainly be achieved in 2D-FGMs.

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