Equivalent layered models for functionally graded plates

Computers and Structures xxx (2015) xxx–xxx

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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc

Equivalent layered models for functionally graded plates D. Kennedy ⇑, R.K.H. Cheng, S. Wei, F.J. Alcazar Arevalo Cardiff School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff CF24 3AA, United Kingdom

a r t i c l e

i n f o

Article history: Accepted 22 September 2015 Available online xxxx Keywords: Functionally graded Plates Vibration Dynamic stiffness Wittrick–Williams algorithm Transverse shear

a b s t r a c t Functionally graded plates whose material properties vary continuously through the thickness are modelled as exactly equivalent plates composed of up to four isotropic layers. Separate models are derived for analysis using classical plate theory, first-order and higher-order shear deformation theory. For cases where Poisson’s ratio varies through the thickness, the integrations required to obtain the membrane, coupling and out-of-plane stiffness matrices are performed accurately using a series solution. The model is verified by comparison with well converged solutions from approximate models in which the plate is divided into many isotropic layers. Critical buckling loads and undamped natural frequencies are found for a range of illustrative examples. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded (FG) materials can be defined as those which are formed by gradually mixing two or more different materials, with the main aim of adapting their physical properties to the external environment. The variation of properties is required to be as smooth as possible in order to avoid phenomena such as stress concentrations which could lead to the development or propagation of fractures. Nature provides examples of materials whose physical properties vary gradually, but the concept of synthetically manufactured FG materials was first developed in Japan in the early 1980s [1]. The simplest kind of FG material is made from gradually varying proportions of two constituent materials, usually with complementary properties. For example, in a FG material composed of metal and a ceramic reinforcement, the ceramic material contributes heat and oxidation resistance, while the metal provides toughness, strength and the bonding capability needed in order to minimize residual stresses. Furthermore some crucial properties, such as thermal insulation and impact resistance, can be conveyed to the material by varying the internal pore distribution. Pioneering manufacturing techniques include powder metallurgy, physical and chemical vapour deposition, plasma spraying, self-propagating high temperature synthesis and galvanoforming [1]. Property changes during FG material processing are commonly performed by functions of the chemical composition, microstructure or atomic order, which depend on the position within the ele⇑ Corresponding author.

ment [2]. Property variation through the thickness of a FG plate is achieved by bulk processing or stacking, layer processing by molecular or mechanical deposition, thermal and electrical preform processing or melt processing. It is also possible to vary properties in the same plane by means of technologies such as ultraviolet irradiation [3]. Jet solidification and laser cladding permit greater variation and are suitable for a wide range of layer thicknesses. Solid freeform fabrication is an advanced production technique which can be controlled by computers. Although the most important applications of FG materials have taken place in the aerospace industry, mechanical engineering, chemical plants and nuclear energy, they are now attracting attention in optics, sports goods, car components, and particularly in biomaterials by means of prostheses. Modern FG implants allow the bone tissues to penetrate between the metallic (often titanium) part and the bone by means of the hydroxyapatite (a transition porous material), forming a graded layup in which a suitable bonding is developed [1]. In engineering it is important to highlight the effects of FG materials in turbomachinery components such as rotating blades, since by varying the gradation it is possible to alter the natural frequencies in order to guarantee stability at particular spinning speeds. Finally it is worth mentioning smart applications, in which piezoelectric sensors and actuators are integrated into the FG material to control vibrations or static responses in structures [3]. Natural frequencies and critical buckling loads of FG plates have been tabulated by various authors [4–7], and it was shown by Abrate [8,9] that these results are proportional to those for homogeneous isotropic plates. Coupling between in-plane and out-ofplane behaviour can be accounted for by an appropriate choice of

E-mail address: [email protected] (D. Kennedy). http://dx.doi.org/10.1016/j.compstruc.2015.09.009 0045-7949/Ó 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Kennedy D et al. Equivalent layered models for functionally graded plates. Comput Struct (2015), http://dx.doi.org/ 10.1016/j.compstruc.2015.09.009

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

the neutral surface [10–12]. Thus the behaviour of FG plates can be predicted from that of similar homogeneous plates. These ideas were exploited to obtain an equivalent isotropic model for a FG plate [13] so that it can be analysed using existing methods based on classical plate theory (CPT) for homogeneous plates. This model was shown to give an exact equivalence when the two component materials have the same Poisson’s ratio, but otherwise a small approximation is introduced. The analysis of FG plates with varying Poisson’s ratio poses a greater challenge due to the complexity of the integrals which have to be evaluated in order to obtain the in-plane, coupling and flexural stiffness matrices, even when the analysis is restricted to CPT. Efraim proposed an alternative approach [14] in which these integrations are approximated, while the present authors performed the integrations accurately [15] using a series solution proposed by Dung and Hoa [16]. The present paper includes the previously derived CPT models and examples [13,15] and then makes important extensions to first-order (FSDT) and higher-order (HSDT) shear deformation theory so as to permit accurate solutions for thick functionally graded plates. Section 2 introduces an equivalent (single layer) isotropic plate model for use with CPT under the assumption that Poisson’s ratio does not vary through the thickness of the plate. This assumption is relaxed in Section 3, where an exactly equivalent plate composed of two isotropic layers is derived for CPT by solving an inverse problem to satisfy six independent stiffness requirements. Section 4 demonstrates that the extension of these models to FSDT is trivial, and then outlines extensions to HSDT giving equivalent plates with three and four layers, respectively. The numerical results in Section 5 verify the proposed models, and also demonstrate its accuracy in finding critical buckling loads and natural frequencies of FG plates, using the different plate theories. Section 6 summarises the conclusions and suggests further extensions to the method. 2. Equivalent isotropic plate for CPT with constant Poisson’s ratio Consider a FG plate of thickness h lying in the xy plane with the origin at mid-surface, having material properties which vary through the thickness (zÞ direction. Using standard notation, the plate constitutive relations of CPT are written as

N ¼ Ae0 þ Bj M ¼ Be0 þ Dj

ð1Þ

where the vectors N; M; e0 and j contain forces per unit length, perturbation moments per unit length, perturbation strains, and perturbation curvatures and membrane, coupling and out-of-plane given by

Z A¼

h=2

Z D¼

Z

h=2

EðzÞQ ðzÞdz B ¼

perturbation membrane bending and twisting mid-surface membrane twist, respectively. The stiffness matrices are

h=2

respectively, where

Q 11 ðzÞ Q 12 ðzÞ 6 Q ðzÞ ¼ 4 Q 12 ðzÞ Q 11 ðzÞ

0

0

Q 66 ðzÞ

0

VðzÞ ¼

 n 1 z þ 2 h

ð7Þ

The non-negative volume fraction index n controls the variation of the properties of the FG plate, as illustrated in Fig. 1. As n approaches zero the plate consists essentially of reinforcement material, while as n approaches infinity it consists essentially of matrix material. If both materials have the same Poisson’s ratio mm ¼ mr ¼ m0 , then

A ¼ AF Q 0

B ¼ BF Q 0

D ¼ DF Q 0

ð8Þ

where

2 1 1 6 m Q ðzÞ  Q 0 ¼ 4 0 1  m20 0

and

AF ¼ BF ¼ DF ¼

m0 1 0

  Ed EðzÞdz ¼ h Em þ nþ1

h=2

EðzÞzdz ¼ h2

R h=2

R h=2 h=2

2

EðzÞz dz ¼ 2

7 5

0 1 ð1 2

h=2

R h=2

3

0

9 > > > > > =

nEd ðnþ1Þðnþ2Þ

h3 12

ð9Þ

 m0 Þ

ð10Þ

 > > > 3ðn2 þnþ2ÞEd > Em þ ðnþ1Þðnþ2Þðnþ3Þ > ;

The presence of BF indicates coupling between the in-plane and outof-plane behaviour.  Now consider an isotropic plate of thickness h , Young’s modulus E , Poisson’s ratio m ¼ m0 , whose neutral surface is offset by d above the geometric mid-surface. The membrane, coupling and out-of-plane stiffness matrices are given by 

A ¼ E h Q 0



B ¼ E h d Q 0

D ¼ E

! 3 h  þ h d2 Q 0 12

ð11Þ

and is therefore equivalent to the FG plate of Eq. (8) if

n=00 n=0.2 n=0.5 n=1 n=2

7 5

Q 12 ðzÞ ¼ mðzÞQ 11 ðzÞ  Q 12 ðzÞÞ

ð6Þ

Here, subscripts m and r denote the properties of the metal and reinforcement components, respectively, and VðzÞ is a function representing the volume fraction of the reinforcement, which is assumed to follow the commonly encountered power law

ð3Þ

n=55 n

Em

-0.5

1 ðQ 11 ðzÞ 2

md ¼ mr  mm qd ¼ qr  qm

E(z)

3

with

Q 66 ðzÞ ¼

Ed ¼ Er  Em

ð2Þ

2

ð5Þ

where

Er

EðzÞQ ðzÞz dz

Q 11 ðzÞ ¼ 1m1ðzÞ2

mðzÞ ¼ mm þ VðzÞmd qðzÞ ¼ qm þ VðzÞqd

EðzÞQ ðzÞzdz

2

0

EðzÞ ¼ Em þ VðzÞEd

h=2

h=2

h=2

Young’s modulus EðzÞ, Poisson’s ratio mðzÞ and density qðzÞ are assumed to vary through the thickness according to the rule of mixtures

)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

z/h

ð4Þ

Fig. 1. Variation of Young’s modulus through the thickness of a FG plate with volume fraction index n.

Please cite this article in press as: Kennedy D et al. Equivalent layered models for functionally graded plates. Comput Struct (2015), http://dx.doi.org/ 10.1016/j.compstruc.2015.09.009

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u 2 u D B AF  F E ¼  h ¼ t12  F AF A2F h

BF d ¼ AF 

ð12Þ

For vibration analysis an equivalent density q is also defined,  so that the mass per unit area q h of the equivalent plate is equal to that of the FG plate, i.e.

q ¼

   h q qm þ d  nþ1 h

ð13Þ

Eqs. (12) and (13) give an exact isotropic equivalence for FG plates with constant Poisson’s ratio. For cases where Poisson’s ratio varies through the thickness of the plate, an approximate solution was proposed [13] in which the Poisson’s ratio for the equivalent isotropic plate took the mean value for the FG plate, i.e.



m ¼ mm þ



md

ð14Þ

nþ1

3. Equivalent two layer plate for CPT with varying Poisson’s ratio 3.1. Normalised stiffness matrices for FG plate

B11 ¼ D11 ¼

R h=2

EðzÞ h=2 1mðzÞ2

dz

A12 ¼

EðzÞ h=2 1mðzÞ2

zdz

B12 ¼

EðzÞ h=2 1mðzÞ2

z2 dz D12 ¼

R h=2

R h=2

R h=2

EðzÞmðzÞ h=2 1mðzÞ2

dz

EðzÞmðzÞ h=2 1mðzÞ2

zdz

R h=2

R h=2

1 1X cr 2 r¼0 k þ rn þ 1

with

"

cr ¼

1

ð1  m m Þ

þ rþ1

ð22Þ

9 > > > > =

> > > > EðzÞmðzÞ 2 ; z dz 2

h=2 1mðzÞ

ð23Þ

mrd

ð1 þ mm Þrþ1

3.2. Normalised stiffness matrices for two layer plate Consider the plate shown in Fig. 2, which is made from two layers of different isotropic materials. The figure shows the thicknesses ðh1 ; h2 Þ, Young’s moduli ðE1 ; E2 Þ and Poisson’s ratios ðm1 ; m2 Þ of the two layers. The thicknesses and Young’s moduli can be expressed in non-dimensional form as

h1 ¼ g1 h h2 ¼ g2 h E1 ¼ e1 Em

E2 ¼ e2 Em

ð24Þ

Application of Eq. (2) gives the independent elements of the stiffness matrices A; B and D in non-dimensional form as

A11 ¼ EAm11h ¼ K 1 þ K 2

A12 ¼ EAm12h ¼ m1 K 1 þ m2 K 2

B11 ¼ E2B11 ¼ L1 þ L2 h2

B12 ¼ E2B12 ¼ m1 L1 þ m2 L2 h2

D11

D12

¼

12D11 Em h3

m

¼ M1 þ M2

¼

12D12 Em h3

9 > > > =

> > > ¼ m1 M 1 þ m2 M2 ;

ð25Þ

where

e1 g1 1  m21

K1 ¼

L1 ¼ K 1 g2 ð15Þ

#

ð1Þr

m

When Poisson’s ratio varies through the thickness of the FG plate, Eq. (2) cannot be simplified to the form of Eq. (8), because the integrations become more complicated on account of the terms in mðzÞ appearing in the denominators of Eq. (4). The independent elements of A; B and D must be evaluated as

A11 ¼

Ik ¼

K2 ¼

e2 g2 1  m22

ð26Þ

L2 ¼ K 2 g1

ð27Þ





M1 ¼ K 1 g21 þ 3g22 M 2 ¼ K 2 3g21 þ g22

ð28Þ

3.3. Equivalence of FG plate and two layer plate

Note that as a result of Eq. (4),

1 1 1 ðA11  A12 Þ B66 ¼ ðB11  B12 Þ D66 ¼ ðD11  D12 Þ ð16Þ 2 2 2

A66 ¼

Writing

f¼

1 z þ 2 h

Ed ¼

Ed Em

ð17Þ

Eq. (15) can be written in non-dimensional form as

A11 ¼ EAm11h ¼ I0 þ Ed In

A12 ¼ EAm12h ¼ mm I0 þ md þ Ed mm In þ Ed md I2n



A11 A12 B11 B12 D11 D12

T

 T ¼ A11 A12 B11 B12 D11 D12 ð29Þ

i.e. if

)

9 B11 ¼ E2B11 > 2 ¼ ð2I 1  I 0 Þ þ Ed ð2I nþ1  I n Þ > mh =

2B12 B12 ¼ E h2 ¼ mm ð2I1  I0 Þ þ md þ Ed mm ð2Inþ1  In Þ > m > ; þEd md ð2I2nþ1  I2n Þ

Suppose that the non-dimensional stiffnesses of a FG plate and the two layer plate of Section 3.2 have been calculated by Eqs. (18)–(20) and (25), respectively. Then, for analysis using CPT, the two plates are exactly equivalent if

ð18Þ

L1 ¼ ð19Þ

a2 a1 K2 ¼ m1  m2 m1  m2

K1 ¼

b2

m1  m2

M1 ¼

9 11 D11 ¼ 12D ¼ ð12I2  12I1 þ 3I0 Þ þ Ed ð12Inþ2  12Inþ1 þ 3In Þ > > Em h3 > > > = 12 D12 ¼ 12D ¼ m ð 12I  12I þ 3I Þ m 2 1 0 3 Em h

> > þ md þ Ed mm ð12Inþ2  12Inþ1 þ 3In Þ > > > ; þEd md ð12I2nþ2  12I2nþ1 þ 3I2n Þ

d2

m1  m2

L2 ¼

ð30Þ

b1

ð31Þ

m1  m2

M2 ¼

d1

ð32Þ

m1  m2

where

a1 ¼ m1 A11  A12 a2 ¼ m2 A11  A12

ð33Þ

b1 ¼ m1 B11  B12

ð34Þ

b2 ¼ m2 B11  B12

ð20Þ where the integrals

Z Ik ¼ 0

1

f

h1

E1, ν1

h2

E2, ν2

k

1  ðmm þ md fn Þ

2

df

are evaluated by means of the series solution [16]

ð21Þ

Fig. 2. Equivalent two layer plate.

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

d1 ¼ m1 D11  D12

d2 ¼ m2 D11  D12

ð35Þ

E1 ¼

Thus from Eqs. (33)

m1 ¼

a1 þ A12

m2 ¼

A11

a2 þ A12

ð36Þ

A11

and in Eqs. (34) and (35)





a1 þ A12 B11

b1 ¼

A11



 B12



 a1 þ A12 D11

d1 ¼

A11

A11 

 D12

d2 ¼

 B12

ð37Þ



a2 þ A12 D11 A11

 D12

ð38Þ

L b g2 ¼  1 ¼  2 ¼ b1 þ b2 n2 K1 a2

ð39Þ

where

n1 ¼

1

n2 ¼

a1

1

b1 ¼

a2

B11

b2 ¼ B12 

A11

A12 B11

ð40Þ

A11

Also, from Eqs. (28), (30), (32) and (38),

M 1 d2 d2 ðb1 þ g2 Þ ¼ ¼ g21 þ 3g22 ¼ d1  b2 K 1 a2

ð41Þ

M 2 d1 d2 ðb1  g1 Þ ¼ ¼ 3g21 þ g22 ¼ d1  b2 K 2 a1

ð42Þ

where

d1 ¼

D11

d2 ¼ D12 

A11

A12 D11

ð43Þ

A11

Taking the difference between the simultaneous Eqs. (41) and (42),



d2 ðg1 þ g2 Þ 2 g22  g21 ¼  b2

ð44Þ

which, since the total thickness of the equivalent plate ðg1 þ g2 Þh cannot equal zero, implies that

d2 2b2

g2 ¼ g1 

ð45Þ

Substituting Eq. (45) into the sum of Eqs. (41) and (42) gives

8g21  4

d2 g þ b2 1

2 d2 2 2b2



2ðd1 b2  d2 b1 Þ b2

!

¼0

ð46Þ

Solving Eq. (46), using Eq. (45) and ignoring negative roots,



pffiffiffiffi

g1 ¼ u þ v

pffiffiffiffi



g2 ¼ u þ v

ð47Þ

where

u¼

d2 4b2

v¼

1 g1 þ g2

q ¼

ðd1 b2  d2 b1 Þ 4b2

ð48Þ

h1 ¼ g1 h h2 ¼ g2 h

ð49Þ

The Poisson’s ratios are obtained from Eqs. (36), (39) and (40) as

m1 ¼

A11



b2 þ A12 b1  g1



m2 ¼

1 A11



ð51Þ



qm þ



qd

ð52Þ

nþ1

4.1. First order shear deformation theory First order (FSDT) and higher order (HSDT) shear deformation plate theories require the satisfaction of additional constitutive relations beyond those of Eq. (1), and the use of additional stiffness coefficients beyond those listed in Eq. (2). In the case of FSDT, Eq. (1) is supplemented by the constitutive relation

T ¼ As c

b2 þ A12 b1 þ g2



ð50Þ

Finally, the Young’s moduli are found from Eqs. (24), (26) and (30) as

ð53Þ

between the transverse shear stresses T and transverse shear strains c. Because the two components of the FG plate are isotropic, the additional stiffness coefficients of As are obtained easily as

A44 ¼ A55 ¼ fA66

A45 ¼ 0

ð54Þ

where f is the shear correction factor. Thus no additional integrations are required beyond those already used to obtain A11 and A12 under the assumptions of CPT. 4.2. Higher order shear deformation theory In the case of HSDT, Eq. (1) is replaced by the constitutive relations [17,18]

2

2

3

32

3

#  0   " s T c A Ds 76 7 6 7 6 ¼ 4 M 5 ¼ 4 B D F 54 j 5  s s  T c D F E F H M j N

A

B

E

ð55Þ

where M ; j ; T and c denote higher order moments, curvatures, transverse shear stresses and transverse shear strains, respectively. Additional integrations are required to obtain the additional stiffness matrices E; F and H, while the stiffness coefficients of Ds and Fs corresponding to transverse shear effects are obtained as

D44 ¼ D55 ¼ D66

D45 ¼ 0

F 44 ¼ F 55 ¼ F 66

F 45 ¼ 0

ð56Þ

As a result there are more independent stiffness parameters than those listed in Eqs. (10) or (15). Therefore an equivalent plate has to comprise more than two layers, as outlined in the following discussion. To obtain the stiffness matrices of the FG plate, Eq. (2) is augmented by

E¼

R h=2 h=2

The thicknesses of the two layers are now obtained from Eq. (24) as

1



b2 1  m22 E g2 ðg1  b1 Þðm2  m1 Þ m

4. Extensions to shear deformation plate theories

Then, from Eqs. (27), (30), (31) and (37),

L b g1 ¼ 2 ¼ 1 ¼ b1  b2 n1 K 2 a1

E2 ¼

For vibration analysis an equivalent density q is defined, analogously to that of Eq. (13), such that the mass per unit area q ðh1 þ h2 Þ of the equivalent plate is equal to that of the FG plate, i.e.





a2 þ A12 B11

b2 ¼



b2 1  m21 E g1 ðg2 þ b1 Þðm2  m1 Þ m

9 R h=2 EðzÞQ ðzÞz3 dz F ¼ h=2 EðzÞQ ðzÞz4 dz = R h=2 ; H ¼ h=2 EðzÞQ ðzÞz6 dz

ð57Þ

If Poisson’s ratio remains constant through the thickness of the plate, then extension of the arguments of Section 2 leads to

E ¼ EF Q 0

F ¼ FF Q 0

H ¼ HF Q 0

ð58Þ

where

Z EF ¼

Z

h=2

h=2

EðzÞz3 dz F F ¼

Z

h=2

h=2

EðzÞz4 dz HF ¼

h=2

EðzÞz6 dz h=2

ð59Þ

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

E1

h1

E2

h2

h1

E1, ν1

h2

E2, ν2

h3

E3, ν3

h4 E4, ν4

h3 E3

(b)

(a)

Fig. 3. (a) Equivalent three layer plate; (b) equivalent four layer plate.

There are therefore six independent stiffness properties, AF ; BF ; DF ; EF ; F F and HF . An equivalent three layer plate, see Fig. 3 (a), can therefore be found by selecting six parameters, namely the thicknesses ðh1 ; h2 ; h3 Þ and Young’s moduli ðE1 ; E2 ; E3 Þ of the three layers, so that the six independent stiffness properties of the equivalent plate match AF ; BF ; DF ; EF ; F F and HF of the FG plate. Each layer is assumed to have the same Poisson’s ratio as the two components of the FG plate. If Poisson’s ratio varies through the thickness of the plate, extension of the arguments of Section 3 leads to twelve independent stiffness properties, namely those of Eq. (15) augmented by

E11 ¼ F 11 ¼ H11 ¼

R h=2

R h=2

EðzÞmðzÞ 3 z dz h=2 1mðzÞ2

EðzÞ h=2 1mðzÞ2

z3 dz

E12 ¼

EðzÞ h=2 1mðzÞ2

z4 dz

F 12 ¼

EðzÞ h=2 1mðzÞ2

z dz H12 ¼

R h=2

R h=2

6

R h=2

EðzÞmðzÞ h=2 1mðzÞ2

z4 dz

R h=2

9 > > > > = > > >

ð60Þ

> EðzÞmðzÞ 6 z dz ; h=2 1mðzÞ2

An equivalent four layer plate, see Fig. 3(b), can therefore be found by selecting twelve parameters, namely the thicknesses ðh1 ; h2 ; h3 ; h4 Þ, Young’s moduli ðE1 ; E2 ; E3 ; E4 Þ and Poisson’s ratios ðm1 ; m2 ; m3 ; m4 Þ of the four layers, so that the twelve independent stiffness properties of the equivalent plate match those of the FG plate given in Eqs. (15) and (60). The problems of selection of the layer parameters for the HSDT analyses of this section are not readily amenable to analytic solution. However, given suitable trial solutions, they can be solved numerically using the MATLAB software [19].

5. Numerical examples Table 1 summarises the forms of the equivalent plates for the various cases considered in Sections 2–4. Given any FG plate, the integrations given in these sections enable calculation of the independent stiffness parameters listed in the penultimate column of Table 1. Then solution of between 3 and 12 simultaneous nonlinear equations gives the layer parameters listed in the final column of the table, which define an equivalent plate, comprised of between one and four layers and having the same independent stiffness parameters. Hence structural analyses, such as the calculation of critical buckling loads or undamped natural frequencies,

can then be performed using existing forms of analysis for plates composed of isotropic layers. Alternatively, approximate solutions can be obtained by dividing the FG plate into a large number nl of isotropic layers, each of thickness h=nl , with the material properties for the ith layer given by Eq. (5) with

z¼

  h 2i  1 1 þ 2 nl

ð61Þ

and performing analysis for the resulting laminated plate. Under the assumption that the results from such analysis converge uniformly towards exact results as nl is increased, a set of results n o f nl =4 ; f nl =2 ; f nl from the layered model can be used [13] to give an extrapolated prediction

f ¼

 2 f nl =4 f nl  f nl =2

ð62Þ

f nl =4  2f nl =2 þ f nl

where f nl represents the result found for the approximate model with nl layers. Such predictions are taken as comparators for the equivalent isotropic and two layer models presented in this paper. Such analysis is available in the exact strip software VICONOPT [20], which covers free vibration, critical buckling and postbuckling analysis, and optimum design, of prismatic plate assemblies. In the simplest form of the analysis, which is described in detail in [21], the mode of vibration or buckling is assumed to vary sinusoidally in the prismatic (or longitudinal) direction, giving exact solutions for isotropic and orthotropic stiffened plates and panels in the absence of shear load. Analytical or numerical solution of the governing differential equations in the transverse direction avoids the usual discretisation required by the finite element and finite strip methods. In contrast to finite element analysis, this approach does not involve separate stiffness and mass matrices, but instead yields a transcendental dynamic stiffness matrix which cannot be handled by conventional linear eigensolvers. However the Wittrick–Williams algorithm [22] permits reliable, accurate and rapid convergence on any required eigenvalue, i.e. undamped natural frequency or critical buckling load, and the results which follow represent its first application to FG plates.

Table 1 Summary of equivalent plates for different cases. Plate theory

Poisson’s ratio

No of layers

No of independent stiffness parameters

Independent stiffness parameters

Layer parameters

CPT or FSDT CPT or FSDT HSDT HSDT

Constant Varying Constant Varying

1 2 3 4

3 6 6 12

AF ; B F ; D F A11 ; A12 ; B11 ; B12 ; D11 ; D12 AF ; BF ; DF ; EF ; F F ; HF A11 ; A12 ; B11 ; B12 ; D11 ; D12 ; E11 ; E12 ; F 11 ; F 12 ; H11 ; H12

h ; E ; d h1 ; h2 ; E1 ; E2 ; m1 ; m2 h1 ; h2 ; h3 ; E 1 ; E 2 ; E 3 h1 ; h2 ; h3 ; h4 ; E1 ; E2 ; E3 ; E4 ; m1 ; m2 ; m3 ; m4



Please cite this article in press as: Kennedy D et al. Equivalent layered models for functionally graded plates. Comput Struct (2015), http://dx.doi.org/ 10.1016/j.compstruc.2015.09.009

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

(a)

Table 2 Properties of equivalent isotropic plate, and convergence of stiffness matrices for an aluminium–ceramic FG plate.

140

E* (GPa) 120

n ¼ 0:2 Approximate, nl ¼ 128 240,010 A11 ðN m1 Þ

100

A12 B11 B12 D11 D12

80 60 0

2

4

6

8

10

12

14

16

18

20

n

(b)

h* (mm) 2.06

A12 B11 B12 D11 D12

2.04 2.02 2

1.96 4

6

8

10

12

14

16

18

20

n

(c)

A12 B11 B12 D11 D12

0.1

δ*

(mm) 0.08

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

n¼2

n¼5 167,470

221,870

203,730

185,600

55,468

50,933

46,400

41,866

8227.4 2056.8 78,641 19,660

14,501 3625.4 72,579 18,145

18,132 4533.1 67,911 16,978

18,132 4533.1 64,284 16,071

12,950 3237.6 60,138 15,034 167,480

221,860

203,730

185,600

60,002

55,467

50,933

46,400

41,866

8242.0 2060.4 78,626 19,656

14,506 3626.7 72,574 18,144

18,133 4533.4 67,911 16,978

18,133 4533.1 64,285 16,072

12,951 3238.0 60,141 15,034

105.678 1.9682 0.0654 221,870

96.655 1.9761 0.0890 203,730

86.549 2.0104 0.0977 185,600

76.268 2.0585 0.0773 167,470

Equivalent isotropic plate [13] 113.683 E ðGPaÞ  1.9792 h ðmmÞ 0.0343 d ðmmÞ 240,000 A11 ðN m1 Þ

1.98 2

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

n¼1

60,004

Extrapolated, Eq. (62) 240,000 A11 ðN m1 Þ

2.08

0

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

n ¼ 0:5

60,000

55,467

50,933

46,400

41,867

8242.4 2060.6 78,626 19,657

14,507 3626.7 72,574 18,143

18,133 4533.3 67,911 16,978

18,133 4533.3 64,284 16,071

12,952 3238.1 60,140 15,035

0.06

" # h ðEr  Em Þ d ¼ ffi pffiffiffiffiffiffiffiffiffi 2 2 pffiffiffiffi Er þ 2Em

0.04



0.02 0 0

2

4

6

8

10

12

14

16

18

20

n 

Fig. 4. Plots of (a) equivalent Young’s modulus E , (b) equivalent thickness h , and (c) neutral surface offset d against volume fraction index n, for an aluminium– ceramic FG plate of thickness h ¼ 2 mm.

ð63Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and occurs when n ¼ 2Er =Em . Table 2 demonstrates that the stiffness matrices obtained using VICONOPT by the approximate layered approach, with nl ¼ 128 and extrapolated by Eq. (62), converge exactly to those of the equivalent isotropic plate. Table 2 also lists the Young’s modulus  E , thickness h and neutral surface offset d of the equivalent isotropic plate.

5.1. Properties of FG plates with constant Poisson’s ratio 5.2. Critical buckling of FG plates with constant Poisson’s ratio The first example considered here is a FG plate of thickness h ¼ 2 mm, composed of an aluminium matrix reinforced by ceramic material. The aluminium and ceramic have Young’s modulus Em ¼ 70 GPa and Er ¼ 121 GPa, respectively, and both materials have the same Poisson’s ratio mm ¼ mr ¼ m0 ¼ 0:25. Fig. 4 shows  plots of the Young’s modulus E , thickness h and neutral surface offset d of the equivalent isotropic plate, obtained using Eqs. (10) and (12) for different values of the volume fraction index n. Fig. 4(a) demonstrates how the equivalent Young’s modulus varies from that of the reinforcement material when n ¼ 0 to that of the matrix material as n approaches infinity, in agreement with Fig. 1 which shows that these represent extreme cases where the FG plate is composed of a single isotopic material. For these extreme cases, the equivalent isotropic plate is  identical to the FG plate. Therefore it has thickness h ¼ h and its neutral surface coincides with the geometric mid-surface, i.e. d ¼ 0, as shown in Fig. 4(b) and (c), respectively. Fig. 4(b) shows that the thickness of the equivalent plate can be either larger or smaller than that of the FG plate, by up to about 3% for this example, depending on the value of n. Also, from Fig. 4(c), the neutral surface of the equivalent plate is always offset above its geometric mid-surface, by up to 5% of the true thickness h. It can be shown analytically from Eqs. (10) and (12) that the greatest offset is given by

A simply supported square aluminium–ceramic FG plate of length a ¼ 100 mm, thickness h ¼ 2 mm and the material properties given in Section 5.1 was loaded in uniform longitudinal compression. For seven values of the volume fraction index, ranging from n ¼ 0 (i.e. pure ceramic) to n ¼ 1 (i.e. pure metal), the software VICONOPT was used to find the critical buckling load, firstly

0 -1 -2

log10(ε)

-3

n=0.2 0.2 n=0.5 0.5 n=5 n=22 n=1

-4 -5 -6

1

2

4

8

16

32

64

128

log10(nl) Fig. 5. Relative errors in the critical buckling load of a square simply supported aluminium–ceramic FG plate with volume fraction index n, comparing a layered solution with nl layers and the analytical solution for an equivalent isotropic plate.

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7

D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

Table 3 Critical buckling load of a square simply supported aluminium–ceramic FG plate with volume fraction index n, comparing the analytical solution for an equivalent isotropic plate, the approximate layered solution with nl ¼ 128 layers, and extrapolations from the layered solutions obtained using Eq. (62). n

P cr (kN)

0 0.2 0.5 1 2 5 1

Relative error

Analytical

Approximate

Extrapolated

Approximate

Extrapolated

33.96899 30.92865 28.27660 26.17307 24.67906 23.34671 19.65148

33.96899 30.93497 28.27876 26.17314 24.67878 23.34599 19.65148

33.96899 30.92874 28.27660 26.17306 24.67906 23.34671 19.65148

0.00 2.04  104 7.64  105 2.94  106 1.14  105 3.10  105 0.00

0.00 2.73  106 3.18  107 5.53  108 9.00  109 4.71  109 0.00

Table 4 Properties of equivalent isotropic and two layer plates, and convergence of stiffness matrices for a Ti-AlOx FG plate, with mm ¼ 0:26 and mr ¼ 0:2884. n ¼ 0:2 Approximate, nl ¼ 128 348,630 A11 ðN m1 Þ

n ¼ 0:5

n¼1

n¼2

n¼5

429,440

509,850

590,000

669,960

98,624

119,020

138,660

157,770

176,500

36,816 9597.5 122,260 34,373

64,550 16,229 149,160 40,964

80,391 19,632 169,780 45,869

80,138 19,108 185,830 49,724

57,091 13,342 204,220 54,235

Extrapolated, Eq. (62) 348,690 A11 ðN m1 Þ

429,470

509,850

590,000

669,950

98,638

119,030

138,660

157,770

176,490

36,880 9612.1 122,310 34,389

64,572 16,233 149,180 40,964

80,396 19,633 169,780 45,869

80,142 19,109 185,830 49,723

57,100 13,344 204,210 54,232

200.884 1.9735 0.1527 429,890

245.712 1.9214 0.1603 510,490

290.726 1.8842 0.1381 590,660

330.279 1.8876 0.0867 670,420

A12 B11 B12 D11 D12

A12 B11 B12 D11 D12

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

Equivalent isotropic plate [13] 158.880 E ðGPaÞ  2.0190 h ðmmÞ 0.1072 d ðmmÞ 348,850 A11 ðN m1 Þ ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

98,958

119,910

139,980

159,160

177,480

37,402 10,610 122,520 34,754

65,638 18,308 149,550 41,714

81,815 22,434 170,160 46,659

81,588 21,985 186,010 50,123

58,118 15,386 204,100 54,032

Equivalent two layer plate E1 ðGPaÞ 225.542 m1 0.2770 0.5868 h1 ðmmÞ E2 ðGPaÞ 137.684 m2 0.2869 1.3814 h2 ðmmÞ 348,690 A11 ðN m1 Þ

303.602 0.2688 0.6297 156.770 0.2848 1.3093 429,460

346.969 0.2638 0.7617 177.391 0.2822 1.1716 509,850

358.940 0.2612 0.9951 199.647 0.2789 0.9547 590,000

354.721 0.2602 1.3724 225.007 0.2751 0.6072 669,960

A12 B11 B12 D11 D12

A12 B11 B12 D11 D12

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

Table 5 Properties of equivalent isotropic and two layer plates, and convergence of stiffness matrices for a Ti-AlOx FG plate, with mm ¼ 0:15 and mr ¼ 0:4.

98,639

119,030

138,660

157,770

176,500

36,882 9612.7 122,330 34,388

64,573 16,234 149,180 40,969

80,396 19,634 169,780 45,869

80,143 19,109 185,830 49,722

57,100 13,344 204,210 54,233

by using an analytical model of the equivalent isotropic plate defined in Section 2, and secondly by using an approximate layered model with up to nl ¼ 128 isotropic layers. The analytical results can be regarded as exact for this example because both materials have the same Poisson’s ratio. Fig. 5 shows that the results from the layered model converge towards the analytical results as nl is increased, giving 4–6 significant figures of accuracy when nl ¼ 128. Table 3 shows that these extrapolated predictions from Eq. (62) are even closer to the analytical results, demonstrating the correctness of the equivalent isotropic model.

n ¼ 0:2 Approximate, nl ¼ 128 366,820 A11 ðN m1 Þ

n ¼ 0:5

n¼1

n¼2

n¼5 646,550

438,360

508,380

577,660

129,300

132,910

131,050

125,520

117,370

33,887 4277.7 127,480 43,063

56,991 2328.7 150,910 42,637

70,005 1850.3 168,820 41,165

69,274 5530.5 182,900 40,346

49,162 6056.5 198,990 40,100

Extrapolated, Eq. (62) 366,870 A11 ðN m1 Þ

646,540

A12 B11 B12 D11 D12

A12 B11 B12 D11 D12

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

438,380

508,380

577,660

129,290

132,900

131,060

125,530

117,360

33,441 4266.1 127,550 43,055

57,011 2324.0 150,940 42,631

70,010 1850.3 168,820 41,164

69,278 5530.0 182,900 40,345

49,171 6055.1 198,980 40,099

200.884 1.9735 0.1527 440,630

245.712 1.9214 0.1603 510,730

290.726 1.8842 0.1381 579,310

330.279 1.8876 0.0867 647,210

Equivalent isotropic plate [13] 158.880 E ðGPaÞ  2.0190 h ðmmÞ 0.1072 d ðmmÞ 368,040 A11 ðN m1 Þ ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

131,880

139,530

140,450

135,170

124,050

39,459 14,140 129,260 46,317

67,277 21,304 153,280 48,540

81,854 22,510 170,240 46,817

80,020 18,671 182,440 42,568

56,106 10,754 197,040 37,765

Equivalent two layer plate E1 ðGPaÞ 219.628 m1 0.3007 0.6096 h1 ðmmÞ E2 ðGPaÞ 136.901 m2 0.3871 1.3643 h2 ðmmÞ 366,880 A11 ðN m1 Þ

296.554 0.2292 0.6621 154.867 0.3695 1.2886 438,370

339.572 0.1847 0.7957 174.719 0.3472 1.1509 508,380

354.349 0.1609 1.0240 196.802 0.3197 0.9360 577,660

353.286 0.1515 1.3899 222.547 0.2864 0.5938 646,540

A12 B11 B12 D11 D12

A12 B11 B12 D11 D12

ðN m1 Þ ðNÞ ðNÞ ðN mmÞ ðN mmÞ

129,290

132,900

131,050

125,520

117,370

33,443 4268.9 127,530 43,054

57,010 2324.1 150,920 42,632

70,009 1850.4 168,820 41,164

69,279 5529.9 182,890 40,346

49,170 6055.1 198,980 40,098

5.3. Properties of FG plates with varying Poisson’s ratio Abrate [8] studied the free vibration of FG plates composed of a titanium alloy (Ti) matrix reinforced by aluminium oxide (AlOx). The titanium had Young’s modulus Em ¼ 349:55 GPa, Poisson’s 3

ratio mm ¼ 0:26 and density qm ¼ 3750 kg m , while the aluminium oxide had Young’s modulus Er ¼ 122:56 GPa, Poisson’s 3

ratio mr ¼ 0:2884 and density qr ¼ 4429 kg m . Table 4 illustrates that the stiffness matrices for such a plate with thickness h ¼ 2 mm, obtained by VICONOPT using the approximate layered approach with nl ¼ 128 and extrapolated by Eq. (62), converge to

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8

D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx

0

Table 7 Natural frequencies xij of a square simply supported AlOx-Ti FG plate with volume fraction index n, normalised with respect to the corresponding fundamental natural frequency x11 .

-1

log10(ε)

-2

n=5 =5 n=0 =0.2 0.2 2 n=2 n=0 =0.5 5 n=11

-3 -4 -5

1

2

4

8

16

32

64

128

n

w11 /w11

w12 /w11

w22 /w11

w13 /w11

w23 /w11

w14 /w11

0 0.2 0.5 1 2 5 1

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2.5000 2.4987 2.4988 2.4989 2.4989 2.4989 2.5000

4.0000 3.9960 3.9962 3.9964 3.9965 3.9965 4.0000

5.0000 4.9933 4.9936 4.9939 4.9942 4.9942 5.0000

6.5000 6.4881 6.4886 6.4892 6.4896 6.4896 6.5000

8.5000 8.4787 8.4797 8.4807 8.4815 8.4814 8.5000

log10(nl) Fig. 6. Relative errors in the fundamental natural frequency of a square simply supported AlOx-Ti FG plate with mm ¼ 0:26; mr ¼ 0:2884 and volume fraction index n, comparing a layered solution with nl layers and the analytical solution for an equivalent isotropic plate.

Table 8 First three natural frequencies of equivalent isotropic and two layer plates, for a thick Ti-AlOx FG plate with mm ¼ 0:26 and mr ¼ 0:2884, found by CPT. n ¼ 0:5

n¼1

n¼2

n¼5

those of the equivalent two layer plate proposed in this paper, rather than to those of the previously proposed isotropic plate [13]. The reason for the superior accuracy of the two layer model is that it accounts exactly for the variation in Poisson’s ratio, in contrast to the approximation introduced by Eq. (14) for the isotropic plate. This accuracy is confirmed by Table 5, where the Poisson’s ratios of the component materials have been artificially adjusted to give a more extreme variation between them, namely mm ¼ 0:15 and mr ¼ 0:4.

Approximate, nl ¼ 128 x1 ðHzÞ 182.403 x2 ðHzÞ 450.361 x3 ðHzÞ 454.184

n ¼ 0:2

200.549 494.899 512.277

215.754 532.300 567.302

230.914 569.781 620.259

250.160 617.549 671.850

Extrapolated, Eq. (62) x1 ðHzÞ 182.445 x2 ðHzÞ 450.440 x3 ðHzÞ 454.221

200.562 494.926 512.221

215.752 532.298 567.302

230.910 569.771 620.257

250.152 617.531 671.847

Equivalent isotropic plate [13] x1 ðHzÞ 182.531 200.781 x2 ðHzÞ 450.788 496.121 x3 ðHzÞ 454.143 512.095

216.033 534.124 567.034

231.160 571.762 619.990

250.305 619.094 671.664

5.4. Natural frequencies of FG plates with varying Poisson’s ratio

Equivalent two x1 ðHzÞ x2 ðHzÞ x3 ðHzÞ

215.914 533.273 567.299

231.067 570.729 620.255

250.258 618.170 671.845

The software VICONOPT was used to find the fundamental natural frequencies of simply supported square FG plates of length a ¼ 100 mm with the composition described in Section 5.3, for seven values of the volume fraction index, ranging from n ¼ 0 (i.e. pure reinforcement) to n ¼ 1 (i.e. pure matrix). As in Section 5.2, results from an analytical model of the equivalent isotropic plate were compared with those from an approximate layered model with up to nl ¼ 128 isotropic layers, and with extrapolated predictions from the latter. Fig. 6 shows agreement to approximately 4 significant figures between the analytical and approximate results. However, in contrast to the example of Section 5.2, Table 6 shows that this accuracy cannot be significantly improved by extrapolation, indicating that as nl is increased the layered model converges to natural frequencies which differ slightly from those of the equivalent isotropic plate. The reason for this is the approximation introduced by Eq. (14) to obtain a constant value of Poisson’s ratio for the equivalent isotropic model. The natural frequencies xij of a simply supported square isotropic plate of length a and thickness h, with Young’s modulus E, Poisson’s ratio m and density q, are given by the expression [23]

xij ¼

layer plate 182.495 450.769 454.226

200.680 495.653 512.286

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#   E 2 2 i þj 2 2 4a 3qð1  m Þ

ph

ð64Þ

where i and j represent the number of half-waves in the longitudinal and transverse directions, respectively. Hence the natural frequencies can be written in the normalised form

xij 1  2 2  ¼ i þj x11 2

ð65Þ

Table 7 shows that this relationship is satisfied exactly for FG plates with extreme values of the volume fraction index n ¼ 0 and n ¼ 1. The remaining cases in Table 7 were obtained using the equivalent isotropic model, and show slight discrepancies from the analytical results due to the approximation in the representation of Poisson’s ratio by Eq. (14). However the discrepancies can be avoided by using the equivalent two layer model rather than the equivalent isotropic model.

Table 6 Fundamental natural frequency of a square simply supported AlOx-Ti FG plate with volume fraction index n, comparing the analytical solution for an equivalent isotropic plate, the approximate layered solution with nl ¼ 128 layers, and extrapolations from the layered solutions obtained using Eq. (62). n

0 0.2 0.5 1 2 5 1

w11

Relative error

Analytical

Approximate

Extrapolated

Approximate

Extrapolated

996.477 1163.675 1279.568 1376.225 1472.185 1594.158 1813.540

996.477 1163.006 1279.162 1376.351 1472.931 1595.245 1813.540

996.477 1163.274 1279.234 1376.343 1472.907 1595.199 1813.540

0.00 5.75  104 3.17  104 9.15  105 5.07  104 6.82  104 0.00

0.00 3.45  104 2.61  104 8.63  105 4.90  104 6.53  104 0.00

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D. Kennedy et al. / Computers and Structures xxx (2015) xxx–xxx Table 9 First three natural frequencies of equivalent isotropic and two layer plates, for a thick Ti-AlOx FG plate with mm ¼ 0:26 and mr ¼ 0:2884, found by FSDT. n ¼ 0:5

n¼1

n¼2

n¼5

Approximate, nl ¼ 128 x1 ðHzÞ 178.175 x2 ðHzÞ 426.587 x3 ðHzÞ 454.122

n ¼ 0:2

196.153 469.174 512.157

211.131 506.301 567.146

226.324 544.878 620.130

245.116 589.039 671.790

Extrapolated, Eq. (62) x1 ðHzÞ 178.220 x2 ðHzÞ 426.706 x3 ðHzÞ 454.169

196.167 470.217 512.171

211.130 506.299 567.146

226.320 543.869 620.129

245.110 589.023 671.786

Equivalent isotropic plate [13] x1 ðHzÞ 177.762 195.785 x2 ðHzÞ 424.072 468.006 x3 ðHzÞ 454.143 512.095

210.952 505.387 567.034

225.953 542.200 619.990

244.683 587.174 671.664

Equivalent two x1 ðHzÞ x2 ðHzÞ x3 ðHzÞ

211.133 506.289 567.182

226.173 543.049 620.160

244.979 588.299 671.804

layer plate 178.004 425.534 454.176

195.975 469.161 512.189

5.5. Vibration of thick FG plates with varying Poisson’s ratio VICONOPT was used to find the first three undamped natural frequencies of a thick Ti-AlOx FG plate with the same material properties as the plate of Table 4. The plate had length and width 100 mm and thickness h ¼ 10 mm. Tables 8 and 9 list the results obtained by the approximate layered approach with nl ¼ 128 and extrapolated by Eq. (62), and by the equivalent isotropic and two layer plates, using CPT and FSDT (with shear correction factor j ¼ 5=6Þ respectively. Both tables show that the approximate layered results converge to those of the two layer plate, making this an ideal representation of the FG plate whichever theory is chosen. The significant difference between most of the corresponding results in Tables 8 and 9 illustrates the necessity to account for transverse shear deformation in the vibration analysis of moderately thick plates. VICONOPT is currently unable to carry out buckling or vibration analysis based on HSDT. However the stiffness calculations of Section 4 have been coded and verified by the fourth author in an extensive parametric study [24] which will permit their future use in alternative HSDT software for plates composed of isotropic layers. 6. Conclusions and further work An equivalent two layer model has been developed for a functionally graded (FG) plate whose material properties, including Poisson’s ratio, vary continuously through the thickness. The model allows critical buckling loads and undamped natural frequencies of a FG plate to be obtained using existing methods based on classical plate theory (CPT) or first order shear deformation plate theory (FSDT) for plates composed of isotropic layers. Analytic expressions have been derived for the thickness, Young’s modulus, Poisson’s ratio and density of the component layers. The model gives an exact analogy with the FG plate, so extending and improving a previously proposed isotropic plate model which however remains exact if the matrix and reinforcement materials have the same Poisson’s ratio. The two layer plate model has been verified by using the software VICONOPT to obtain the stiffness properties for a simply supported FG plate. The correctness and accuracy of the model have been confirmed by comparing these results with well converged solutions from an approximate model in which the plate is divided into isotropic layers. Agreement has also been demonstrated in the calculation of undamped natural frequencies, using both CPT and FSDT, for a thick FG plate.

9

Because the equivalent model can be used directly with established software such as VICONOPT, the analysis is readily extended to FG plates with different loading and support conditions, and to prismatic panels containing such plates, e.g. aircraft wing and fuselage panels. Attention will also be given to FG plates whose volume fraction varies in different ways through the thickness of the plate. CPT and FSDT give satisfactory results for thin and moderately thick plates, respectively, but should be replaced by a more accurate higher order shear deformation theory when analysing thicker plates of the dimensions used in composite aircraft panels. The methods proposed in this paper have therefore been extended to obtain equivalent plates composed of three and four isotropic layers, which exhibit the additional material properties needed to fully represent the FG material when using higher order theories. Critical buckling and undamped vibration studies for such plates will be perfomed using alternative HSDT software for plates composed of isotropic layers. References [1] Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG. Functionally graded materials: design, processing and applications. Dordrecht (Netherlands): Kluwer Academic Publishers; 1999. [2] Kieback B, Neubrand A, Riedel H. Processing techniques for functionally graded materials. Mater Sci Eng: A 2003;362(1–2):81–105. [3] Byrd LW, Birman V. Modeling and analysis of functionally graded materials and structures. Appl Mech Rev 2007;60(1–6):195–216. [4] Liew KM, Yang J, Kitipornchai S. Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int J Solids Struct 2003;40 (15):3869–92. [5] Yang J, Shen H-S. Dynamic response of initially stressed functionally graded rectangular thin plates. Compos Struct 2001;54(4):497–508. [6] He XQ, Ng TY, Sivashanker S, Liew KM. Active control of FGM plates with integrated piezoelectric sensors and actuators. Int J Solids Struct 2001;38 (9):1641–55. [7] Yang J, Shen H-S. Vibration characteristics and transient response of sheardeformable functionally graded plates in thermal environments. J Sound Vib 2002;255(3):579–602. [8] Abrate S. Free vibration, buckling, and static deflections of functionally graded plates. Compos Sci Technol 2006;66(14):2382–94. [9] Abrate S. Functionally graded plates behave like homogeneous plates. Composites: Part B 2008;39(1):151–8. [10] Morimoto T, Tanigawa Y, Kawamura R. Thermal buckling of functionally graded rectangular plates subjected to partial heating. Int J Mech Sci 2006;48 (9):926–37. [11] Zhang DG, Zhou YH. A theoretical analysis of FGM thin plates based on physical neutral surface. Comput Mater Sci 2008;44(2):716–20. [12] Prakash T, Singha MK, Ganapathi M. Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates. Comput Mech 2009;43(3):341–50. [13] Kennedy D, Cheng RKH. An equivalent isotropic model for functionally graded plates. In: Topping BHV, editor. Proceedings of 11th international conference on computational structures technology. Stirlingshire (UK): Civil-Comp Press; 2012. http://dx.doi.org/10.4203/ccp.99.94 [Paper 94]. [14] Efraim E. Accurate formula for determination of natural frequencies of FGM plates basing on frequencies of isotropic plates. Proc Eng 2011;10:242–7. [15] Kennedy D, Wei S, Alcazar Arevalo FJ. Two layer model of functionally graded plate with varying Poisson’s ratio. In: Topping BHV, Iványi P, editors. Proceedings of 12th international conference on computational structures technology. Stirlingshire (UK): Civil-Comp Press; 2014. http://dx.doi.org/ 10.4203/ccp.106.34 [Paper 34]. [16] Dung DV, Hoa LK. Nonlinear analysis of buckling and postbuckling for axially compressed functionally graded cylindrical panels with the Poisson’s ratio varying smoothly along the thickness. Vietnam J Mech 2012;34(1):27–44. [17] Pradhan KK, Chakraverty S. Effects of different shear deformation theories on free vibration of functionally graded beams. Int J Mech Sci 2014;82:149–60. [18] Reddy JN. Analysis of functionally graded plates. Int J Numer Methods Eng 2000;47(1–3):663–84. [19] MATLAB version 8.4, release R2014b. Natick (MA, USA): The Mathworks, Inc.; 2014. [20] Kennedy D, Featherston CA. Exact strip analysis and optimum design of aerospace structures. Aeronaut J 2010;114(1158):505–12. [21] Wittrick WH, Williams FW. Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings. Int J Mech Sci 1974;16(4):209–39. [22] Wittrick WH, Williams FW. A general algorithm for computing natural frequencies of elastic structures. Quart J Mech Appl Math 1971;24(3):263–84. [23] Beards CF. Structural vibration: analysis and damping. Oxford: Elsevier; 1996. [24] Alcazar Arevalo FJ. Equivalent models of a functionally graded plate based on the classical plate theory and the third order shear deformation theory. MSc thesis. Cardiff University, UK; 2014.

Please cite this article in press as: Kennedy D et al. Equivalent layered models for functionally graded plates. Comput Struct (2015), http://dx.doi.org/ 10.1016/j.compstruc.2015.09.009