Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories

Composite Structures 93 (2011) 973–982

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Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories M.E. Golmakani, M. Kadkhodayan * Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad 91775-1111, Iran

a r t i c l e

i n f o

Article history: Available online 1 July 2010 Keywords: Nonlinear bending FGM Annular plate DR method

a b s t r a c t This paper addresses the axisymmetric nonlinear bending analysis of an annular functionally graded plate under mechanical loading based on FSDT and TSDT. Using nonlinear von-Karman theory, the discretized equations are solved using the dynamic relaxation (DR) method combined with the finite difference technique. The effects of the material constant n, boundary conditions, thickness-to-radius ratio and shear deformation are studied. The results show that although, the difference between TSDT and FSDT becomes greater with an increasing thickness-to-external radius ratio, the effects of different types of boundary conditions is also of great importance. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Conventional laminated composite materials usually have an abrupt change in mechanical properties across the interface where two different materials are bonded together; this can result in cracking and large interlaminar stresses leading to delamination. One way to solve these problems is to employ functionally graded materials (FGMs) by gradually varying the volume fraction of the constituent materials, usually only in the thickness direction. FGMs were first introduced and designed by a group of Japanese scientists in 1984 as thermal barrier materials for aerospace structural applications and fusion reactors [1]. Typically, these materials are made from a mixture of ceramic and metal in which the ceramic constituent of the material provides high temperature resistance and protects the metal from corrosion and oxidation due to its low thermal conductivity. In addition, the FGM is toughened and strengthened by the metallic composition. Because of their functional gradation for optimized design, FGMs now have garnered significant attention as one of the most capable candidates for future intelligent composites in many engineering fields such as aerospace, high-speed computers, environmental sensors, energy conservation and power generation. In recent years, many studies have been done on the linear and nonlinear behavior of functionally graded plates [2–7]. In these studies, two conventional plate theories (classical plate theory (CPT) and first-order shear deformation plate theory (FSDT)) based on considering the effect of transverse shear strain through-the-thickness of the plate, have been utilized exten-

* Corresponding author. Tel.: +98 9153111869; fax: +98 5118763304. E-mail address: [email protected] (M. Kadkhodayan). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.06.024

sively. However, FGM plates under mechanical loading based on third-order shear deformation theory (TSDT) have received fewer investigations. In comparison with FSDT, which assumes that the transverse shear strain is constant along the thickness coordinate, TSDT employs a quadratic variation of transverse shear strain through-the-thickness. Using TSDT; leads to the satisfaction of the zero tangential traction boundary conditions on the surfaces of the plate; therefore, there is no need to employ shear correction factors with TSDT. Thus, because of the better approximation of TSDT, it is of central importance to use this theory for accurate and reliable structural analysis and design of FGMs. Ma and Wang [8] have employed TSDT and FSDT to solve the axisymmetric bending and buckling problems of functionally graded materials. Saidi et al. [9] have studied the axisymmetric bending and buckling problems of functionally graded circular plates using unconstrained third-order shear deformation plate theory. In these papers, the linear behavior of a circular FGM plate was investigated. For nonlinear cases, Reddy [10] has developed both theoretical and finite element formulations for thick rectangular FGM plates based on the TSDT, and the nonlinear dynamic responses of FGM plates subjected to suddenly uniform pressure were analyzed. Yang and Shen [11] have studied the nonlinear bending analysis of rectangular functionally graded plates subjected to thermo-mechanical loads under various boundary conditions based on TSDT. Sarfaraz Khabbaz et al. [12] have studied the nonlinear analysis of rectangular FGM plates under pressure loads based on FSDT and TSDT. However, to the best of our knowledge, no work has investigated the nonlinear bending of annular functionally graded plates; based on FSDT and TSDT. In the present paper, the axisymmetric nonlinear bending analysis of an annular functionally graded plate based on FSDT and TSDT is considered. The constitutive equations are obtained

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based on the nonlinear von-Karman plate theory for large deflections. The material properties of the functionally graded plates are assumed to vary continuously through-the-thickness of the plate. A higher-order determination technique, the Mori–Tanaka scheme [13], is used to determine the effective material properties of the FGM plate. To determine the advantage of this technique compared with the simple rule of mixtures, studies are conducted using the rule of mixtures and the Mori–Tanaka’s theory. Moreover, unlike most previous studies, the variation of Poisson’s ratio through-the-thickness is considered in this work. The plate is subjected to a uniform pressure loading and the boundary conditions are simply supported and clamped. The nonlinear equilibrium equations are solved using the dynamic relaxation (DR) numerical method combined with the finite difference discretization technique. The validation of the DR solutions is examined by comparing the results of the present study with those reported by Reddy et al. [6]. The parametric effects of the material gradient property n, boundary conditions, thickness-to-radius ratio and shear deformation on the nonlinear bending of functionally graded plates are discussed for both FSDT and TSDT in detail.

We investigate a functionally graded annular plate with thickness h, inner radius ri and outer radius ro. The FGM plate is subjected to axisymmetric transverse loading q. Axial symmetry in geometry and loading is assumed, and cylindrical coordinates (r, h, z) are considered. The annular plate is made from a mixture of ceramic and metal; the material properties of the composition are assumed to vary continuously and smoothly through-thethickness of the plate. There are many analytical and computational models that discuss the issue of achieving suitable functions for modeling the material properties of FGMs. Many previous studies have employed the simple rule of mixtures, which cannot accurately predict the elastic properties of actual FGMs for the widely varying Poisson’s ratios of the two constituents [14]. In the current study, however, the Mori–Tanaka scheme, which is a more realistic determination technique, is used for the modeling of material properties (E, m). To show the significant difference between the responses predicted by these two theories, the two homogenization techniques are compared in this work. According to the simple rule of mixtures, a material property through-the-thickness of the plate p can be expressed as [3]:

ð1Þ

where the subscripts m and c denote the metallic and ceramic constituents. The volume fractions of the metal Vm and ceramic Vc corresponding to the power law are assumed to be:

(

Vc ¼

1 2

n þ hz ;

ð2Þ

Vm ¼ 1  V c;

where z is the thickness coordinate (h/2 6 z 6 h/2), and n is a material constant. The effective bulk modulus K and the effective shear modulus G of the functionally graded material, according to the Mori–Tanaka [13] homogenization method, are as follows [14]:

8 KK c Vm > < K m K c ¼ 1þð1V m Þ K m K c

K c þð4=3ÞGc

GG V > : Gm Gcc ¼ 1þð1V mmÞGm Gc ;

;

ð3Þ

Gc þfc

where

Gc ð9K c þ 8Gc Þ fc ¼ : 6ðK c þ 2Gc Þ

(

9KG E ¼ 3KþG ;

According to these distributions, the bottom surface (z = h/2) of the functionally graded plate is pure metal, the top surface (z = h/2) is pure ceramic, and different volume fractions of metal can be obtained for different values of n. 3. Theoretical formulation 3.1. Governing equations based on TSDT The straightness of the transverse normal in FSDT can be eliminated by assuming that the transverse deflection changes through-the-thickness. As a result of this assumption, third-order shear deformation plate theory has been established [15,16]. In TSDT, a third-order polynomial is employed in the expansion of the displacement components through-the-thickness of the plate according to the following relations:

ur ðr; zÞ ¼ u0 ðrÞ þ z/ðrÞ  az3 ð/ðrÞ þ dw Þ; dr uz ðr; zÞ ¼ wðrÞ;

ð6Þ

where ur and uz are the displacements along the coordinates r and z; u0 and w are the radial and vertical displacements, respectively; / indicates the slope at z = 0 of the deformed line, which is straight in the non-deformed plate;, and a = 4/3h2. Substituting (6) into the nonlinear von-Karman strain–displacement relations gives:

  8   du0 d/ 1 dw 2 3 d/ d2 w > > < er ¼ dr þ 2 dr þ z dr  az dr þ dr2 ;   eh ¼ ur0 þ z /r  az3 /r þ 1r dw ; > dr >   : dw 2 dw crz ¼ / þ dr  3az / þ dr :

ð7Þ

For a plate, Hooke’s law is defined as:

8 rr ¼ ð1Em2 Þ ½er þ meh ; > > < rh ¼ ð1Em2 Þ ½eh þ mer ; > > :r ¼ E e ; rz

ð8Þ

2ð1þmÞ rz

where E = E(z) and m = m(z). Using the principle of virtual displacements, the following equilibrium equations for the Reddy plate can be derived:

8 dN 1 r þ r ðNr  Nh Þ ¼ 0; > > < dr dMr r þ 1r ðMr  Mh Þ  Q r  ar ðPr  Ph Þ  a dP þ 3aRr ¼ 0; dr dr > > : dQ r þ Q r  3a R  3a dRr þ a d2 Pr  a dPh þ N d2 w þ Nh dw þ q ¼ 0; r r dr r r dr r dr r dr dr2 dr 2 ð9Þ where Ni, Mi, Pi, i = r, h, are the stresses, moments and higher-order moment resultants, and Qr, Rr are the shear stress and higher-order shear stress resultants, respectively, expressed as:

8 R h=2 > ðN r ; M r ; Pr Þ ¼ h=2 rr ð1; z; z3 Þdz; > > < R h=2 ðN h ; M h ; Ph Þ ¼ h=2 rh ð1; z; z3 Þdz; > > > : ðQ ; R Þ ¼ R h=2 r ð1; z2 Þdz: r

ð4Þ

ð5Þ

3K2G m ¼ 2ð3KþGÞ :

(

2. FGM material properties

PðzÞ ¼ Pc V c þ Pm V m ;

Based on this method, the effective values of Young’s modulus, E, and Poisson’s ratio, m, are computed from:

r

h=2

ð10Þ

rz

Using (7), (8), and (10) gives the following constitutive relations:

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M.E. Golmakani, M. Kadkhodayan / Composite Structures 93 (2011) 973–982

8     2     > > þ A1 ur þ ðB  aEÞ d/ þ ðB1  aE1 Þ /r Nr ¼ A du þ 12 dw > dr dr dr > > 2  > dw > > > aE ddrw2  aE1 rdr ; > > > >     >       > 2 > > þ ðB  aEÞ /r þ ðB1  aE1 Þ d/ þ 12 dw Nh ¼ A ur þ A1 du > dr dr dr > > 2  > 1 dw > > d w > aE r dr  aE1 dr2 ; > > > >    >       > du 1 dw 2 > > þ B1 ur þ ðD  aFÞ d/ þ ðD1  aF 1 Þ /r > M r ¼ B dr þ 2 dr dr > > 2  > dw > > d w >  aF 1 rdr a F ;  > > dr 2 > >     > u     2 < þ 12 dw M h ¼ B r þ B1 du þ ðD  aFÞ /r þ ðD1  aF 1 Þ d/ dr dr dr 2  > 1 dw > > d w > aF r dr  aF 1 dr2 ; > > > >   > > > Q r ¼ ðC  3aGÞ / þ dw ; > dr >    > dw2     > > du 1 > þ E1 ur þ ðF  aHÞ d/ þ ðF 1  aH1 Þ /r > dr > Pr ¼ E dr þ 2 dr > > 2  > dw > > aH ddrw2  aH1 rdr ; > > > >     >       > 2 > > Ph ¼ E ur þ E1 du þ 12 dw þ ðF  aHÞ /r þ ðF 1  aH1 Þ d/ > dr dr dr > > > 2  > 1 dw > d w > aH r dr  aH1 dr2 ; > > > > > : R ¼ ðG  3aKÞ/ þ dw; r dr

and is not amenable to closed-form solution. Therefore, using numerical methods can be helpful. A variety of numerical techniques may be used to solve these types of boundary value problems. Here, the dynamic relaxation (DR) method [17,18] in conjunction with a finite difference discretization scheme has been selected to solve the nonlinear differential equations of the annular FGM plate. DR is an iterative method that generally aims to convert a static problem into a dynamic one to obtain a steady-state solution. The DR method is especially attractive for problems with highly nonlinear geometric and material behavior. Additionally, the explicit nature of the method makes it highly suitable for computers because all quantities may be treated as vectors, resulting in an easily programmable method with low storage requirements. Due to these benefits, many researchers use the DR method to solve both linear and nonlinear problems [19–25]. 4.1. The DR method The DR algorithm relies on transforming a boundary value problem into an equivalent time-stepping initial value problem. This transformation is achieved by adding inertia and damping terms to the right-hand sides of the plate equilibrium equations. The equilibrium equations become:

ð11Þ where

8 R h=2 E 2 3 4 6 > > > ðA; B; D; E; F; HÞ ¼ h=2 1m2 ð1; z; z ; z ; z ; z Þdz; < R h=2 Em ðA1 ; B1 ; D1 ; E1 ; F 1 ; H1 Þ ¼ h=2 1m2 ð1; z; z2 ; z3 ; z4 ; z6 Þdz; > > > : ðC; G; KÞ ¼ R h=2 E ð1; z2 ; z4 Þdz:

ð12Þ

h=2 2ð1þmÞ

In FSDT, the transverse shear strain is assumed to be constant with respect to the thickness coordinate. Thus, by setting a ¼ 0 in the governing equations of TSDT, the basic equations for FSDT can be obtained. 3.2. Boundary conditions The following cases of boundary conditions are used in this study:



where mu, m/, mw and cu, c/, cw are the elements of the diagonal fictitious mass and damping matrices M and C, respectively. The stability and convergence of the DR iterative procedure depends on the correct estimation of the mass matrix and nodal damping factor. According to Gershgörin’s theorem, the following inequality must be satisfied in determining mlii ½l : u; /; w: N   X  l kij ;

mlii P :25ðsn Þ2

At r ¼ r i ;

u ¼ 0;

w ¼ 0;

/ ¼ 0;

At r ¼ r o ;

u ¼ 0;

w ¼ 0;

/ ¼ 0:

where the superscript n indicates the nth iteration step and s is the increment of fictitious time. kij is the element of K that is calculated from:

ð13Þ

At r ¼ r i ;

u ¼ 0;

w ¼ 0;

/ ¼ 0;

At r ¼ r o ;

u ¼ 0;

w ¼ 0;

M r ¼ 0:

ð14Þ

(c) For an annular plate with simply supported inner and outer edges at r = ri and r = ro, respectively:



ð16Þ

(a) For an annular plate with clamped inner and outer edges at r = ri and r = ro:

(b) For an annular plate with clamped inner edge at r = ri and simply supported outer edge at r = ro:



8 dNr 1 d2 u du > > > dr þ r ðNr  Nh Þ ¼ mu dtt þ cu dt ; > > > d2 / dPr > 1 a r > dM > < dr þ r ðM r  Mh Þ  Q r  r ðPr  Ph Þ  a dr þ 3aRr ¼ m/ dt2 þc/ d/ ; dt > > > 2 2 2 > dQ r Q r 3a r h > þ r  r Rr  3a dR þ a ddrP2r  ar dP þ N r ddrw2 þ Nrh dw ¼ mw ddtw2 > dr dr dr dr > > > : þcw dw  q; dt

At r ¼ r i ;

u ¼ 0;

w ¼ 0;

M r ¼ 0;

At r ¼ r o ;

u ¼ 0;

w ¼ 0;

M r ¼ 0:

ð15Þ

4. Solution methodology of the nonlinear equations The system of expressed equations, which explains the nonlinear large deflection response of an FGM plate, is very complex

ð17Þ

j¼1

K¼

@P ; @X

ð18Þ

where X = u, /, w is the approximate solution vector, and P is the left-hand side of the equilibrium equations (16). The instant critical damping factor cni for node i at the nth iteration is given by Zhang et al. [26] as:

( cni ¼ 2

ðX ni ÞT Pni

ðX ni ÞT mnii X ni

)1=2 :

ð19Þ

To complete the transformation process, the velocity and acceleration terms must be replaced with the following equivalent central finite difference expressions [27]: 1 X_ n2 ¼ ðX n  X n1 Þ=sn ;   € n ¼ X_ nþ12  X_ n12 =sn : X

ð20Þ ð21Þ

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M.E. Golmakani, M. Kadkhodayan / Composite Structures 93 (2011) 973–982

Table 1 Comparison of the maximum dimensionless deflection obtained by the DR method with the results obtained by Reddy et al. [6] for thickness-to-radius ratio h/ro = 0.15. Material constant (n)

Reddy et al. [6]

0 2 4 8 10 50 100 1000 100,000

Present study

Clamped

Simply supported

Roller-supported

Clamped

Simply supported

Roller-supported

2.781 1.515 1.384 1.278 1.250 1.137 1.119 1.103 1.101

10.623 5.610 5.217 4.870 4.772 4.348 4.280 4.214 4.207

10.623 5.826 5.325 4.909 4.799 4.349 4.280 4.214 4.207

2.774 1.511 1.382 1.277 1.251 1.134 1.116 1.107 1.102

10.572 5.565 5.200 4.876 4.760 4.346 4.281 4.229 4.217

10.572 5.801 5.305 4.850 4.793 4.349 4.281 4.229 4.217

Table 2 Dimensionless parameters. Parameter

r

z

 u

 w

Definition

r/ro

z/h

ur/ro

w/h

rr rr h

2

qr2o

Nr

Mr

3 N r r 2o =Em h

Mr r 2o =Em h

4

Substituting relations (20) and (21) into the right-hand side of Eq. (16), equilibrium equations may be classified into initial value format as: 8  r 1 n 2sn cn n nþ1=2 > ¼ 2þ2ssn cn ðmnii Þ1 dN þ r ðN r  N h Þ i þ 2þsn cin u_ n1=2 ; > i dr > u_ i > i i > >  > nþ1=2 2sn n 1 dM r 1 a _ > ¼ 2þsn cn ðmii Þ þ r ðM r  M h Þ  Q r  r ðP r  P h Þ > /i > dr > i > < n 2sn cni n1=2 dP a dr þ 3aRr i þ 2þsn cn /_ i ; i >  > > 2 n > nþ1=2 1 dQ r Qr 2 s n 3 a r > _i ¼ 2þsn cn ðmii Þ þ r  r Rr  3a dR þ a ddrP2r w > dr dr > > i >  > n > 2 2sn cn > : _ n1=2 þ q þ 2þsn cni w :  ar dPdrh þN r ddrw2 þ Nrh dw i dr i

ð22Þ

i

To calculate the displacements, the velocities are integrated after each time step using the following equation:

unþ1 ¼ uni þ snþ1 u_ nþ1=2 : i i

ð23Þ

Similar equations may be set up to compute the other two displacement components, / and w. To apply the DR method to solve the system of equations, they should be discretized. To attain this discretization, the central finite difference technique is applied to replace the derivatives. Because of the axisymmetric nature of the loading and the plate geometry, only a radial line of the plate is used, and the governing plate equations are applied to the nodal points of this line. Therefore, Eqs. (20), (21) and (1)–(12), together with the appropriate boundary conditions in their finite difference forms, constitute the set of equations for the sequential DR method, which is very briefly outlined below: The DR flow chart:

Table 3 Comparison of the maximum dimensionless deflection of an SS annular FGM plate with effective elastic moduli obtained from the rule of mixtures and the Mori–Tanaka scheme. Material constant (n)

Maximum dimensionless deflection based on FSDT and TSDT with (a) h/ro = 0.15 and (b) h/ro = 0.3 Effective properties from rule of mixtures (a)

0 0.1 0.5 1 2 5 Metal

Effective properties from Mori– Tanaka scheme (a)

Effective properties from rule of mixtures (b)

Effective properties from Mori– Tanaka scheme (b)

FSDT

TSDT

FSDT

TSDT

FSDT

TSDT

FSDT

TSDT

2.46e3 2.95e3 3.66e3 4.32e3 5.16e3 6.50e3 1.58e2

2.49e3 2.99e3 3.70e3 4.39e3 5.34e3 6.88ee3 1.56e2

2.46e3 3.06e3 3.96e3 4.69e3 5.60e3 7.04e3 1.58e2

2.49e3 3.11e3 4.03e3 4.81e3 5.83e3 7.42e3 1.56e2

2.14e4 2.54e4 3.17e4 3.80e4 4.68e4 6.13e4 1.37e3

2.14e4 2.54e4 3.16e4 3.81e4 4.81e4 6.67e4 1.34e3

2.14e4 2.63e4 3.46e4 4.18e4 5.14e4 6.65e4 1.37e3

2.14e4 2.63e4 3.46e4 4.22e4 5.34e4 7.23e4 1.34e3

977

(a)

TSDT-Metal

0.5

FSDT-Metal

0.0006

TSDT-Ceramic

n=0.1

0.3

FSDT-Ceramic TSDT-n=1

n=1

0.1

FSDT-n=1

n=5

TSDT-n=20

-0.1

FSDT-n=20

0.0004

n=0.01

TSDT-n=0.5

w/h

Dimensionless thickness, z/h

M.E. Golmakani, M. Kadkhodayan / Composite Structures 93 (2011) 973–982

n=20

FSDT-n=0.5

-0.3 -0.5

0

1

2

3

0.0002

4

Dimensionless thickness, z/h

E/1011 Pa

(b)

0

0.5

0

0.2

0.4

n=0.1

0.3

0.6

0.8

1

r/ro

n=1

Fig. 3. Dimensionless vertical displacement of a CC annular FGM plate based on FSDT and TSDT with h/ro = 0.3.

n=5

0.1

n=0.01

-0.1

n=20

-0.3 TSDT-Metal

-0.5 0.14

0.17

0.2

0.23

0.26

FSDT-Metal TSDT-Ceramic

0.015

0.29

FSDT-Ceramic TSDT-n=1

Poisson's ratio, v

0.005

FSDT-n=1 TSDT-n=5 FSDT-n=5 TSDT-n=20 FSDT-n=20 TSDT-n=0.5 FSDT-n=0.5

0.010

w/h

Fig. 1. Through-the-thickness variations of the effective: (a) Young’s modulus and (b) Poisson’s ratio for different values of n from the rule of mixtures (dashed lines) and the Mori–Tanaka scheme.

0.005

TSDT-Metal FSDT-Metal TSDT-Ceramic

0.004

FSDT-Ceramic TSDT-n=1

0.000

FSDT-n=1

0

TSDT-n=20

0.003

w/h

FSDT-n=20 FSDT-n=0.5

0 0.2

0.6

0.8

1

Fig. 4. Dimensionless vertical displacement for a SS annular FGM plate based on FSDT and TSDT with h/ro = 0.15.

0.001

0

0.4

r/ro

TSDT-n=0.5

0.002

0.2

0.4

0.6

0.8

1

r/ro Fig. 2. Dimensionless vertical displacement of a CC annular FGM plate based on FSDT and TSDT with h/ro = 0.15.

4.2. Verification of DR analysis To show the efficiency and accuracy of the present numerical method, the results of the analysis for the axisymmetric bending of a functionally graded circular plate with clamped, roller-supported and simply supported boundary conditions are compared with those obtained by Reddy et al. [6]. Based on FSDT, the analysis of a uniform-thickness FGM plate is performed for one type of ceramic and metal combination. This combination contains Titanium and Zirconia with m = 0.288, Ec = 151.0 GPa and Em/Ec = 0.396. A comparison of the numerical results with the analytical results reported

by Reddy et al. [6] is shown in Table 1. Table 1 shows an acceptable agreement on the dimensionless maximum deflection of W max ¼ 3 64wDc =qo r 4o where Dc ¼ Ec h =12ð1  m2 Þ and the transverse load qo = 0.14 GPa and thus confirms the validity of the present numerical method.

5. Numerical results and discussion For the numerical investigation in this analysis, a uniformthickness FGM annular plate is considered under different boundary conditions including clamped–clamped (CC), clamped–simply (CS), simply–clamped (SC) and simply–simply (SS) supports at the inner and outer edges, respectively. The material properties are taken from Ferreira et al. [14] and are assumed to be Em = 70 GPa, Ec = 427 GPa, mm = 0.3 and mc = 0.17 for the metal, aluminum, and the ceramic, silicon carbide, respectively. In the following investigations, an annular FGM plate is considered with two ratios of thickness-to-external radius, h/ro = 0.15 and 0.3, under a mechanical pressure loading, q = 10 MPa, on the ceramic-rich top surface. The ratio of outer to inner radius is assumed to be ro/

M.E. Golmakani, M. Kadkhodayan / Composite Structures 93 (2011) 973–982

ri = 5, and ro = 50 mm for all cases. The dimensionless parameters are defined and presented in Table 2. In this table, the dimensionless radius, thickness, radial displacement, center deflection, stress, stress resultant and moment resultant are expressed with  ; w;  rr ; N r and M r , respectively. The maximum the symbols r ; z; u dimensionless deflections of an SS annular FGM plate with effective elastic moduli obtained from the rule of mixtures and the Mori–Tanaka scheme are compared to each other in Table 3. The results are presented based on FSDT and TSDT for different material constants with h/ro = 0.15 and 0.3. Fig. 1 depicts the throughthe-thickness variations of the effective Young’s modulus and the effective Poisson’s ratio as obtained by the rule of mixtures and the Mori–Tanaka scheme for different material constants n. It is apparent that the two homogenization schemes give results that are quite different. Except for n = 0.1 and 20, when the material properties tend toward those of homogeneous ceramic and metallic plates, respectively, there is a noticeable difference between the results obtained from these two methods. Hence, to determine the material properties more accurately and realistically, the Mori–Tanaka scheme is used in the following results. Figs. 2 and 3 show  of an annular FGM the dimensionless vertical displacement w plate along the radial direction based on FSDT and TSDT for two ratios of thickness to outer radius, h/ro = 0.15 and 0.3, with the CC boundary condition. It is observed that, with increasing thickness, the difference between TSDT and FSDT becomes greater; this is caused by a reduction in the accuracy of FSDT in thicker plates due to considering the shear strain as linear. In other words, a higher-order displacement field could achieve a higher-accuracy  vertical displacement. The dimensionless vertical displacement w along the radial direction for the SS boundary condition may also be studied (see Fig. 4). It can be seen that for thicker plate FSDT

Rotating angle

1.E-04

 in comparison with TSDT for the predicts larger values of w clamped boundary condition (see Fig. 3). In the simply supported case, however, TSDT predicts a larger deflection compared with FSDT, a result similar to that reported by Sarfaraz et al. [12] for rectangular FG plates. To explain these results, the displacement field of each theory (see Eq. (6)) may be studied in more detail. Figs. 5 and 6 present the values of the rotating angle / and the dimensionless radial dis along the radius, respectively, based on FSDT and placement u TSDT for a CC annular FG plate with n = 1 and h/ro = 0.3. It can be  values in both theories are seen that the predictions of the u approximately the same, while FSDT predicts higher values of /  , increasing than TSDT. Therefore, it is obvious that for constant u  (see Fig. 3). the rotating angle / increases w Figs. 7 and 8 demonstrate the dimensionless vertical displace based on FSDT and TSDT along the radial direction for ment w the CS and SC boundary conditions, respectively. The values of the vertical displacements for these two cases are between the values of the deflections predicted for the CC and SS boundary conditions in Figs. 2 and 4. However, the differences between the displacements computed by FSDT and TSDT for the SC support are more significant than those for the CS support. As mentioned

TSDT-Metal

0.009

FSDT-Metal TSDT-Ceramic FSDT-Ceramic TSDT-n=1 FSDT-n=1 TSDT-n=5

0.006

FSDT-n=5 TSDT-n=20

w/h

978

FSDT-n=20 TSDT-n=0.5 FSDT-n=0.5

0.003

TSDT FSDT

5.E-05 0.E+00

0.000 0

0.2

0.4

-5.E-05

0.6

0.8

1

r/ro

-1.E-04 0.2

0.4

0.6

0.8

1

Fig. 7. Dimensionless vertical displacement for a CS annular FGM plate based on FSDT and TSDT with h/ro = 0.15.

r/ro Fig. 5. Rotating angle / (rad) along the radius for a CC annular FGM plate based on FSDT and TSDT with h/ro = 0.3.

0.008

TSDT-Metal FSDT-Metal FSDT-Ceramic TSDT-n=1

0.006

FSDT-n=1 TSDT-n=5

4.E-06

FSDT-n=5

w/h

Dimensionless radial displacement

TSDT-Ceramic

2.E-06 0.E+00

TSDT-n=20

0.004

FSDT-n=20 TSDT-n=0.5 FSDT-n=0.5

0.002 TSDT

-2.E-06

FSDT

-4.E-06

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r/ro

 along the radius for a CC annular FGM Fig. 6. Dimensionless radial displacement u plate based on FSDT and TSDT with h/ro = 0.3.

0.000

0

0.2

0.4

0.6

0.8

1

r/ro Fig. 8. Dimensionless vertical displacement of an SC annular FGM plate based on FSDT and TSDT with h/ro = 0.15.

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Dimensionless thickness

Dimensionless thickness

0.5 TSDT-Metal or Ceramic FSDT-Metal or Ceramic

0.3 0.1 -0.1 -0.3 -0.5 -0.3

-0.1

0.1

0.3 0.1 -0.1 -0.3

TSDT-Metal or Ceramic FSDT-Metal or Ceramic

-0.5 -0.35

0.3

-0.15

Dimensionless radial stress 0.5

Dimensionless thickness

Dimensionless thickness

TSDT-n=5 FSDT-n=5

0.3 0.1 -0.1 -0.3

0.3 0.1 -0.1 -0.3

TSDT-n=5 FSDT-n=5

-0.5

-0.4

-0.2

0

-0.8

0.2

Dimensionless thickness

Dimensionless thickness

TSDT-n=2 FSDT-n=2

0.3

-0.2

0

0.2

0.1 -0.1 -0.3

0.3 0.1 -0.1 -0.3

TSDT-n=2 FSDT-n=2

-0.5

-0.4

-0.2

0

-0.7

0.2

-0.5

-0.3

-0.1

0.1

Dimensionless radial stress

Dimensionless radial stress 0.5

0.5

TSDT-n=0.1

Dimensionless thickness

Dimensionless thickness

-0.4

0.5

0.5

FSDT-n=0.1

0.3 0.1 -0.1 -0.3 -0.5 -0.35

-0.6

Dimensionless radial stress

Dimensionless radial stress

-0.5 -0.6

0.25

Dimensionless radial stress

0.5

-0.5 -0.6

0.05

-0.15

0.05

0.25

Dimensionless radial stress

0.3 0.1 -0.1 -0.3

TSDT-n=0.1 FSDT-n=0.1

-0.5 -0.35

-0.15

0.05

0.25

Dimensionless radial stress

Fig. 9. Dimensionless radial stress ðrr Þ through-the-thickness in a CC annular FGM plate with h/ro = 0.15 at the inner radius.

Fig. 10. Dimensionless radial stress (rr ) through-the-thickness in a CC annular FGM plate with h/ro = 0.3 at the inner radius.

above, this difference can be attributed to the displacement field applied by each theory. Figs. 9 and 10 show the variation of the dimensionless stresses at the inner radius for two thickness-to-radius ratios for the CC support. It can be seen that TSDT magnifies the stress on both

the top and bottom surfaces as the thickness increases. The differences between these two shear theories are small for thinner plates. However, there is a significant difference between TSDT and FSDT in predicting rr in thicker plates. Moreover, for FGM

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M.E. Golmakani, M. Kadkhodayan / Composite Structures 93 (2011) 973–982 0.5

Dimensionless thickness

Dimensionless thickness

0.5 0.3 0.1 -0.1 -0.3

TSDT-Metal or Ceramic FSDT-Metal or Ceramic

-0.5 -0.3

-0.1

0.1

0.3 0.1 -0.1 -0.3

TSDT-Metal or Ceramic FSDT-Metal or Ceramic

-0.5

0.3

-0.35

-0.15

Dimensionless radial stress

0.3 0.1 -0.1

TSDT-n=5 FSDT-n=5

-0.5 -0.65

-0.45

-0.25

-0.05

-0.1 -0.3

TSDT-n=5 FSDT-n=5 -0.55

-0.35

-0.15

0.05

Dimensionless radial stress 0.5

Dimensionless thickness

Dimensionless thickness

0.1

-0.75

0.5 0.3 0.1 -0.1 TSDT-n=2 FSDT-n=2

-0.5 -0.45

0.3

-0.5

Dimensionless radial stress

-0.3

0.25

0.5

Dimensionless thickness

Dimensionless thickness

0.5

-0.3

0.05

Dimensionless radial stress

-0.35

-0.25

-0.15

-0.05

0.05

0.3 0.1 -0.1 -0.3

FSDT-n=2 -0.5

-0.55

Dimensionless radial stress

TSDT-n=2

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

Dimensionless radial stress 0.5 0.3 0.1 -0.1 -0.3

TSDT-n=0.1 FSDT-n=0.1

-0.5 -0.3

-0.2

-0.1

0

0.1

Dimensionless radial stress Fig. 11. Dimensionless radial stress (rr ) through-the-thickness in an SS annular FGM plate with h/ro = 0.15 at the inner radius.

plates a larger material gradient constant, n, leads to greater compressive and tensile stresses on the top and bottom surfaces, respectively. Similarly, Figs. 11 and 12 illustrate the variation of the dimensionless stress through-the-thickness for the SS boundary condi-

Dimensionless thickness

Dimensionless thickness

0.5

0.3 0.1 -0.1 -0.3

TSDT-n=0.1 FSDT-n=0.1

-0.5

-0.35

-0.25

-0.15

-0.05

0.05

0.15

Dimensionless radial stress Fig. 12. Dimensionless radial stress (rr ) through-the-thickness in an SS annular FGM plate with h/ro = 0.3 at the inner radius.

tion with h/ro = 0.15 and 0.3. It is obvious that the differences between the FSDT and TSDT results increase in all cases for larger of thickness-to-radius ratios (see Figs. 9–12). However, it can be observed that increasing the plate thickness leads to smaller differ-

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Dimensionless radial moment resultant

0.03

0.008 n=1-a n=1-b n=0.5-a

0.006

n=0.5-b n=5-a

w/h

ences between FSDT and TSDT for the SS boundary conditions compared with the CC boundary condition. Fig. 13 displays the distribution of M r in the radial direction. The maximum absolute value of M r is obtained at the inner edge for the CS boundary condition. In Fig. 14, N r is plotted with respect to various values of the material gradient constant n for SS and CS plates based on FSDT and TSDT (The dashed lines are the results obtained from TSDT.). It can be seen that the two theories predict approximately the same values of N r for the SS support, but there is a significant difference between these values for the CS annular plate. Moreover, it can be observed that at n = 2, N r attains the maximum value. The area under the radial stress and thickness direction for this case is more than that for the other cases, which can be verified by Fig. 11. Finally, to demonstrate the effect of the variation of Poisson’s ratio through-the-thickness on the predicted results, the  of a CC annular FGM plate dimensionless vertical displacement w along the radius for both constant and variable ratios is illustrated in Fig. 15. It can be seen that using a constant Poisson’s ratio may cause considerable error in the calculated results for FGM analyses.

n=5-b

0.004

0.002

0.000 0

0.2

0.4

0.6

0.8

1

r/ro Fig. 15. Effect of variation of Poisson’s ratio (m) on the deflection of an SS annular FGM plate with h/ro = 0.15 for (a) m = 0.3 and (b) from the Mori–Tanaka scheme m = m(z).

6. Conclusions CC

0.02

CS

In this paper, the axisymmetric nonlinear bending analysis of an annular functionally graded plate with uniform thickness under mechanical loading is investigated based on FSDT and TSDT. Using nonlinear von-Karman theory, new nonlinear equilibrium equations are developed based on FSDT and TSDT, and then a dynamic relaxation method combined with the finite difference discretization technique is used to solve these equations. The effects of the material gradient constant n, boundary conditions and different thickness-to-radius ratios are studied in detail. Some general inferences are mentioned below:

SC SS

0.01

0

-0.01

-0.02

-0.03 0

0.2

0.4

0.6

0.81

1

1.2

r/ro Fig. 13. Dimensionless radial moment resultant (Mr ) along the radial direction of an annular FGM plate for different boundary conditions with h/ro = 0.15.

0.1

Maximum dimensionless radial stress resultant

SS CS

0.08

 With increasing thickness, the difference between TSDT and FSDT becomes greater. Therefore, to achieve higher accuracy, a higher-order displacement field is required.  First-order shear deformation plate theory in thicker plate pre compared with TSDT for the CC supdicts greater values of w port. The difference in the deflections predicted from TSDT and FSDT is largest for the SS boundary condition.  A greater material gradient constant n leads to greater compressive stress on the top surface. In addition a smaller n results in a reduction of tensile on the bottom surface.  The dimensionless stress resultant (N r ) has a maximum value at n = 2.  Noticeable error in the predicted results can result from assuming a constant Poisson’s ratio for FGMs with different of this ratio.

0.06

References 0.04

0.02

0 0

10

20

30

40

50

Material constant, n Fig. 14. Maximum dimensionless radial stress resultant (N r ) for various values of material gradient constant for SS and CS annular FGM plates with h/ro = 0.15 based on FSDT and TSDT (dashed lines).

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