Composites: Part B 45 (2013) 268–281

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Composites: Part B

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A novel higher-order shear deformation theory with stretching effectfor functionally graded plates

J.L. Mantari, C. Guedes Soares ⇑Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, Technical University of Lisbon, Portugal, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 January 2012Received in revised form 3 May 2012Accepted 7 May 2012Available online 8 June 2012

Keywords:A. PlatesB. StrengthC. Analytical modellingFunctionally graded materials

1359-8368/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compositesb.2012.05.036

⇑ Corresponding author. Tel.: +351 218 417607; faxE-mail address: guedess@mar.ist.utl.pt (C. Guedes

This paper presents an analytical solution to the static analysis of functionally graded plates (FGPs) byusing a new trigonometric higher-order theory in which the stretching effect is included. The governingequations and boundary conditions of FGPs are derived by employing the principle of virtual work.Navier-type solution is obtained for FGPs subjected to transverse bi-sinusoidal load for simply supportedboundary conditions. Benchmark results for the displacements and stresses of geometrically differentplates are obtained. The results are compared with 3D exact solution and with other higher-order sheardeformation theories, and the superiority of the present theory can be noticed.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are advanced compositematerials, in which the material properties are varied in a prede-termined manner. In the nature, FGM characteristics are found insea shells, bones, etc., and the understanding of the highly com-plexity of such kind of materials is contributing to the synthesizingof new kind of materials. In the industry, FGMs have been pro-posed, developed and successfully used since 1984 [1]. NowadaysFGMs are alternative materials widely used in aerospace, nuclearreactor, energy sources, biomechanical, optical, civil, automotive,electronic, chemical, mechanical, and shipbuilding industries.Functionally graded materials (FGMs) are both macroscopicallyand microscopically heterogeneous advanced composites whichare normally made from a mixture of ceramics and metals withcontinuous composition gradation from pure ceramic on one sur-face to full metal on the other. Such gradation leads to smoothchange in the material profile as well as the effective physicalproperties.

Classical composites structures suffer from discontinuity ofmaterial properties at the interface of the layers and constituentsof the composite. Therefore the stress fields in these regions createinterface problems and thermal stress concentrations under hightemperature environments. Furthermore, large plastic deformationof the interface may trigger the initiation and propagation of cracksin the material [2]. These problems can be decreased by gradually

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: +351 218 474015.Soares).

changing the volume fraction of constituent materials and tailoringthe material for the desired application. In fact, FGMs are materialswith spatial variation of the material properties. However, in mostof the applications available in the literature, as in the presentwork, the variation is through the thickness only. Therefore, theearly state development of improved production techniques, newapplications, introduction to effective micromechanical modelsand the development of theoretical methodologies for accuratestructural predictions, have encourage researchers and openedseveral research topics in this field.

Plates are often subjected to combinations of lateral pressureand thermal loading. However, plates and shells with FGM proper-ties are frequently under a thermal load to utilize perfect thermalresistance of FGMs. Therefore, it is interesting to analyze plates andshells under a general thermal load. Tauchert [3] gave a nice over-view of thermally induced flexure, buckling and vibration appliedto classical composite plates described by the Kirchhoff theory.With knowledge gained in classical composites, studies on ad-vanced composites are largely devoted to thermal stress analysis[4–6], and also to the fracture analysis of FG plates and shells[7,8]. Finot and Suresh [9] presented a closed form solution basedon the classical Kirchhoff’s theory of thin plates for the analysis ofmultilayered and functionally graded material plates, subjected tothermal loading. The dynamic thermoelastic response of the func-tionally graded cylinders and plates are obtained by Reddy andChin [10]. Praveen and Reddy [11] obtained the nonlinear transientthermoelastic response of the functionally graded ceramic metalplates using a plate finite element method, employing transverseshear strain, rotary inertia and von-Karman nonlinearity. Reddy

Fig. 1. Geometry of a functionally graded plate.

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 269

[12] presented Navier’s solutions, and finite element modelsincluding geometric non-linearity based on the third-order sheardeformation theory for the analysis of FGPs. Cheng and Batra[13] derived the field equations for a functionally graded plate byutilizing the first-order shear deformation theory and the third-or-der shear deformation theory and simplified them for a simplysupported polygonal plate. Literature survey shows that many pa-pers dealing with static and dynamic behaviour of functionallygraded materials (FGMs), have been published recently. An inter-esting literature review of abovementioned work may be foundin the paper of Birman and Byrd [14]. Therefore, for completeness,in the present article, the relevant work from 2007 up to now is de-scribed. For relevant papers perhaps not included in the abovementioned review paper, readers may consult Carrera et al. [15]and Mantari et al. [16].

Zenkour [17] investigated the static problem of transverse loadacting on exponentially graded (EG) rectangular plates using both2D trigonometric plate theory (TPT) and 3D elasticity solution. Sla-dek et al. [18] presented the static and dynamic analysis of func-tionally graded plates by the meshless local Petrov–Galerkinmethod. The Reissner–Mindlin plate bending theory was utilizedto describe the plate deformation. Numerical results were pre-sented for simply supported and clamped plates. Later the authorapplied the same method to solve problems plates and shells underthermal loading in Sladek et al. [19,20]. Abrate [21] deduced, byusing the classical plate theory, that no special tools are requiredto analyze functionally graded plates, because FGPs behave likehomogeneous plates.

Bo et al. [22] presented the elasticity solutions for the staticanalysis of functionally graded plates for different boundary condi-tions. Stress, free vibration and buckling analysis due to mechani-cal and thermal loads were given by Matsunaga [23–25] by using akind of generalized two-dimensional higher-order theory. Thisinteresting theory was obtained by using the method of power ser-ies expansion of continuous displacement components. Khabbazet al. [26] provided a nonlinear solution of FGPs using the firstand third-order shear deformation theories.

More recently, the thermo-bending problems of FGM sandwichplate, consisting of a homogeneous core-layer bounded with twoFGM face-sheet layers, were studied by Zenkour and Alghamdi[27], using a variety of equivalent single-layer theories (ESLTs), inwhich the material properties of the face-sheet layers were as-sumed to obey a power-law distribution of the volume fractionsof the constituents through the thickness coordinate.

Aghdam et al. [28] presented a static analysis of fully clampedfunctionally graded plates and doubly curved panels by using theextended Kantorovich method. Wu and Li [29] used a RMVT-basedthird-order shear deformation theory of multilayered FGPs undermechanical loads. The exponent-law distributions through thethickness and the power-law distributions of the volume fractionsof the constituents were used to obtain the effective properties.Talha and Singh [30] investigated the free vibration and static anal-ysis of functionally graded plates using the finite element methodby employing a quasi-3D higher-order shear deformation theory.Vaghefi et al. [31] presented a three-dimensional static solutionfor thick functionally graded plates by utilizing a meshless Pet-rov–Galerkin method. An exponential function was assumed forthe variation of Young’s modulus through the thickness of theplate, while the Poison’s ratio was assumed to be constant.RMVT-based meshless collocation and element-free Galerkinmethods for the quasi-3D analysis of multilayered composite andFGPs and circular hollow cylinders were presented by Wu et al.[32] and Wu and Yang [33]. Benachour [34] developed a novel fourvariable refined plate theory for free vibration analysis of platesmade of functionally graded materials with an arbitrary gradient.In that HSDT, the number of DOFs involved is only four. Thai and

Choi [35] studied the free vibration analysis of FG plates on elasticfoundation by using similar refined plate theory as the one pre-sented in [34]. Reddy and Kim [36] proposed a general nonlinearthird-order plate theory that accounts for geometric nonlinearity,microstructure-dependent size effects, and two-constituent mate-rial variation through the plate thickness using the principle of vir-tual displacements and Hamilton’s principle.

Carrera et al. [15] studied the effects of thickness stretching inFG plates and shells. The importance of the transverse normalstrain effects in mechanical prediction of stresses for FGPs waspointed out. In fact, this work is an extension of several FGM pa-pers published by using Carrera’s Unified Formulation (CUF), as de-scribed in [37–40]. Mantari et al. [16] presented bending results ofFGPs by using a new non-polynomial HSDT, different to the onepresented in here. Recently, Neves et al. [41] presented a quasi-3D hybrid polynomial and trigonometric shear deformation theoryfor the static and free vibration analysis of functionally gradedplates by using collocation with radial basis functions.

In the present paper, an analytical solution to the static analysisof functionally graded plates is developed using a new trigonomet-ric HSDT. In fact, an improved version of this HSDT [42], includingthe so-called ‘‘stretching effect’’, is presented. The theory accountsfor adequate distribution of the transverse shear strains throughthe plate thickness and tangential stress-free boundary conditionson the plate boundary surface, thus a shear correction factor is notrequired. The plate material is exponentially graded in the thick-ness direction. The plate’s governing equations and its boundaryconditions are derived by employing the principle of virtual work.Navier-type analytical solution is obtained for plates subjected totransverse bi-sinusoidal load for simply supported boundary con-ditions. Benchmark results for the displacement and stresses ofan exponential graded rectangular plate are obtained. The resultsare compared with 3D exact solution and with other higher-ordershear deformation theories, and the superiority of the present the-ory can be noticed.

2. Theoretical formulation

A rectangular plate of uniform thickness h made of a function-ally graded material is shown in Fig. 1, in which the rectangularCartesian coordinate system x, y, z, with the plane z = 0, coincidentwith the mid-surface of the plate, is shown. The material is inho-mogeneous and the material properties vary exponentiallythrough the thickness, as indicated in:

1 1.5 2 2.5 3 3.5 4 4.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

g( )

z

z

n=0.1n=0.3n=0.5n=0.7n=1.0n=1.5

Fig. 2. Exponential function g (�zÞ along the thickness of an EGM plate for differentvalues of the parameter ‘‘n’’.

270 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

PðzÞ ¼ gðzÞPb; gðzÞ ¼ en zhþ

12ð Þ;

Pt ¼ enPb;ð1a-bÞ

where P denotes the effective material property, Pt and Pb, denotethe property of the top and bottom faces of the plate, respectively,and n is the parameter that dictates the material variation profilethrough the thickness. The effective material properties of the plate,including Young’s modulus, E, and shear modulus, G, vary accordingto Eq. (1a), and Poisson ratio, m, is assumed to be constant. It isimportant to note that n is a parameter that dictates the materialvariation profile through the plate thickness and takes values great-er than zero.

A new trigonometric shear deformation theory for isotropic andcomposite laminated and sandwich plates, was recently developedby Mantari et al. [43]. This paper extends this theory to FG plates,including the stretching of the thickness, for the first time. The newdisplacement field is described in the following equations:

�uðx; y; zÞ ¼ uðx; yÞ þ z y�h1 þ y�@h3

@x� @w@x

� �þ tanðmzÞh1;

�vðx; y; zÞ ¼ vðx; yÞ þ z y�h2 þ y�@h3

@y� @w@y

� �þ tanðmzÞh2;

�wðx; y; zÞ ¼ wðx; yÞ þm sec2ðmzÞh3;

ð2a-cÞ

where u(x,y), v (x,y), w(x,y), h1(x,y), h2(x,y) and h3(x,y) are the sixunknown displacement functions of middle surface of the panel,whilst y� ¼ � sec2ðm h

2Þ and m ¼ 15h. In the case of the other well-

known trigonometric shear deformation theory, the shear strainshape function is f ðzÞ ¼ h

p sin ph z� �

and y⁄ = 0.In the derivation of the necessary equations, small strains are

assumed (i.e., displacements and rotations are small, and obeyHooke’s law). The linear strain expressions derived from the dis-placement model of Eqs. (2a-c), valid for thin, moderately thickand thick plate under consideration are as follows:

exx ¼ e0xx þ ze1

xx þ tanðmzÞe2xx;

eyy ¼ e0yy þ ze1

yy þ tanðmzÞe2yy;

ezz ¼ 2m2 sec2ðmzÞ tanðmzÞe4zz;

eyz ¼ e0yz þm sec2ðmzÞe3

yz;

exz ¼ e0xz þm sec2ðmzÞe3

xz;

exy ¼ e0xy þ ze1

xy þ tanðmzÞe2xy; ð3a-eÞ

where

e0xx ¼

@u@x; e1

xx ¼ y�@h1

@xþ @

2h3

@x2

!� @

2w@x2 ; e2

xx ¼@h1

@x;

e0yy ¼

@v@y

; e1yy ¼ y�

@h2

@yþ @

2h3

@y2

!� @

2w@y2 ; e2

yy ¼@h2

@y;

e4zz ¼ h3;

e0yz ¼ y� h2 þ

@h3

@y

� �; e3

yz ¼ h2 þ@h3

@y;

e0xz ¼ y� h1 þ

@h3

@x

� �; e3

xz ¼ h1 þ@h3

@x;

e0xy ¼

@v@xþ @u@y; e1

xy ¼ y�@h2

@xþ @h1

@yþ 2

@2h3

@x@y

!� 2

@2w@x@y

;

e2xy ¼

@h2

@xþ @h1

@y: ð4a-nÞ

The linear constitutive relations are given as:

rxx

ryy

rzz

syz

sxz

sxy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

Q 11 Q12 Q13 0 0 0Q 21 Q22 Q23 0 0 0Q 31 Q32 Q33 0 0 0

0 0 0 Q 44 0 00 0 0 0 Q 55

0 0 0 0 0 Q 66

2666666664

3777777775

exx

eyy

ezz

cyz

cxz

cxy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;: ð5Þ

in which, r = {rxx,ryy, rzz,syz, sxz, sxy}T and e ={exx,eyy,ezz,cyz,cxz,cxy}T are the stresses and the strain vectors withrespect to the plate coordinate system. The Qij expressions in termsof engineering constants are:

Q11 ¼ Q22 ¼ Q33 ¼EðzÞð1� vÞð1� 2mÞð1þ mÞ ;

Q12 ¼ Q13 ¼ Q23 ¼EðzÞv

ð1� 2mÞð1þ mÞ ;

Q44 ¼ Q55 ¼ Q66 ¼EðzÞ

2ð1þ mÞ : ð6a-cÞ

The modulus E, G and the elastic coefficients Qij vary throughthe thickness according to Eqs. (1a, b).

Considering the static version of the principle of virtual work,the following expressions can be obtained:

0 ¼Z h=2

�h=2

ZX

rðkÞxx dexx þ rðkÞyy deyy þ rðkÞzz dezz þ rðkÞyz deyz þ rðkÞxz dexz

h�"

þ rðkÞxy dexy

idxdy

dz��Z

Xqd�wdxdy

� �; ð7Þ

0 ¼Z

XN1de0

xx þM1de1xx þ P1de2

xx þ N2de0yy þM2de1

yy þ P2de2yy

þ R3de4

zz þ Q4de0yz þ K4de3

yz þ Q5de0xz þ K5de3

xz þ N6de0xy þM6de1

xy

þ P6de2xy � qd�w

�dxdy: ð8Þ

Table 1Non-dimensionalized center deflection �wða=2; b=2; 0Þ for various EG rectangular plates, a/h = 2.

b/a Theory n = 0.1 n = 0.3 n = 0.5 n = 0.7 n = 1.0 n = 1.5

6 3-D [17] 1.63774 1.48846 1.35184 1.22688 1.05929 0.82606Present 1.63654 1.47953 1.33644 1.20618 1.03325 0.79387Mantari et al. [42] 1.73465 1.56884 1.41822 1.28145 1.10032 0.84996TPT [17] 1.62939 1.47309 1.33066 1.20101 1.02823 0.79056HPT [17] 1.54777 1.39964 1.26493 1.14249 0.97956 0.75560

5 3-D [17] 1.60646 1.46007 1.32607 1.20349 1.03907 0.81024Present 1.60532 1.45130 1.31094 1.18315 1.01352 0.77867Mantari et al. [42] 1.70246 1.53972 1.39188 1.25762 1.07981 0.83401TPT [17] 1.59825 1.44493 1.30522 1.17804 1.00856 0.77540HPT [17] 1.51991 1.37444 1.24214 1.12188 0.96184 0.74184

4 3-D [17] 1.55146 1.41013 1.28074 1.16235 1.00352 0.78241Present 1.55042 1.40166 1.26610 1.14267 0.97884 0.75195Mantari et al. [42] 1.64584 1.48849 1.34553 1.21569 1.04374 0.80596TPT [17] 1.54348 1.39541 1.26048 1.13764 0.97395 0.74874HPT [17] 1.47089 1.33009 1.20201 1.08559 0.93065 0.71762

3 3-D [17] 1.44295 1.31160 1.19129 1.08117 0.93337 0.72750Present 1.44210 1.30373 1.17761 1.06279 0.91041 0.69925Mantari et al. [42] 1.53405 1.38735 1.25402 1.13291 0.97254 0.75060TPT [17] 1.43542 1.29771 1.17221 1.05795 0.90567 0.69615HPT [17] 1.37394 1.24238 1.12269 1.01386 0.86898 0.66977

2 3-D[17] 1.19445 1.08593 0.98640 0.89520 0.77266 0.60174Present 1.19408 1.07949 0.97503 0.87990 0.75377 0.57862Mantari et al. [42] 1.27760 1.15533 1.04413 0.94307 0.80929 0.62377TPT [17] 1.18798 1.07399 0.97009 0.87548 0.74936 0.57578HPT [17] 1.15080 1.04052 0.94012 0.84878 0.72712 0.55975

1 3-D [17] 0.57693 0.52473 0.47664 0.43240 0.37269 0.28904Present 0.57789 0.52240 0.47179 0.42567 0.36485 0.27939Mantari et al. [42] 0.63625 0.575172 0.519477 0.468742 0.401782 0.307905TPT[17] 0.57308 0.51806 0.46788 0.42216 0.36117 0.27712HPT[17] 0.58586 0.52955 0.47814 0.43127 0.36871 0.28246

Table 2Non-dimensionalized center deflection �wða=2; b=2; 0Þ for various EG rectangular plates, a/h = 4.

b/a Theory n = 0.1 n = 0.3 n = 0.5 n = 0.7 n = 1.0 n = 1.5

6 3-D[17] 1.17140 1.06218 0.96331 0.87378 0.75501 0.59193Present 1.17033 1.05825 0.95628 0.86359 0.74032 0.57128Mantari et al. [42] 1.19202 1.07885 0.97667 0.88437 0.76228 0.59545TPT[17] 1.16681 1.05509 0.95345 0.86107 0.73821 0.56969HPT[17] 1.00649 0.91087 0.82448 0.74640 0.64306 0.50178

5 3-D[17] 1.14589 1.03906 0.94236 0.85478 0.73859 0.57904Present 1.14484 1.03520 0.93545 0.84478 0.72419 0.55882Mantari et al. [42] 1.16628 1.05555 .95557 0.86525 0.74578 0.58253TPT[17] 1.14140 1.03210 0.93268 0.84231 0.72212 0.55726HPT[17] 0.98508 0.89150 0.80694 0.73050 0.62935 0.49105

4 3-D[17] 1.10115 0.99852 0.90560 0.82145 0.70979 0.55643Present 1.10013 0.99477 0.89891 0.81178 0.69589 0.53696Mantari et al. [42] 1.12113 1.01469 0.91856 0.83172 0.71685 0.55987TPT[17] 1.09682 0.99180 0.89625 0.80941 0.69390 0.53546HPT[17] 0.94753 0.85750 0.77615 0.70262 0.60529 0.47222

3 3-D[17] 1.01338 0.91899 0.83350 0.75606 0.65329 0.51209Present 1.01243 0.91546 0.82724 0.74704 0.64037 0.49408Mantari et al. [42] 1.03254 0.93450 0.84594 0.76593 0.66008 0.51541TPT[17] 1.00938 0.91272 0.82479 0.74486 0.63854 0.49270HPT[17] 0.87379 0.79076 0.71571 0.64787 0.55806 0.43525

2 3-D[17] 0.81529 0.73946 0.67075 0.60846 0.52574 0.41200Present 0.81448 0.73647 0.66547 0.60093 0.51508 0.39732Mantari et al. [42] 0.83246 0.75338 0.68192 0.61734 0.53188 0.41503TPT[17] 0.81202 0.73425 0.66350 0.59917 0.51361 0.39620HPT[17] 0.70700 0.63979 0.57901 0.52405 0.45126 0.35169

1 3-D[17] 0.34900 0.31677 0.28747 0.26083 0.22534 0.18054Present 0.34860 0.31519 0.28477 0.25710 0.22028 0.16972Mantari et al. [42] 0.36017 0.32589 0.29485 0.26676 0.22952 0.17854TPT[17] 0.34749 0.31419 0.28388 0.25631 0.21961 0.16922HPT[17] 0.31111 0.28146 0.25461 0.23027 0.19800 0.15377

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 271

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.2 0.4 0.6 0.8 1

w

z

3DPresentZenkour

b/a=1 b/a=2 b/a=3 b/a=4

Fig. 3. Distribution of non-dimensionalized displacement, �w, through the thicknessof a thick (a/h = 4) EGM plate (n = 0.5).

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.36 -0.24 -0.12 0 0.12 0.24 0.36

u

z

3DPresentZenkour

b/a=1b/a=2b/a=4

Fig. 4. Distribution of non-dimensionalized displacement, �u, through the thicknessof a thick (a/h = 4) EGM plate (n = 0.5).

272 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

where e(k) or r(k) are the stresses and the strain vectors of the kth

layer, q is the distributed transverse load; and Ni, Mi, Pi, Qi and Ki

are the resultants of the following integrations:

Ni;Mi; Pið Þ ¼XN

k¼1

Z zðkÞ

zðk�1ÞrðkÞi ð1; z; tan mzÞdz; ði ¼ 1;2;6Þ

Ni ¼XN

k¼1

Z zðkÞ

zðk�1ÞrðkÞi dz; ði ¼ 4;5Þ

ðQi;KiÞ ¼XN

k¼1

Z zðkÞ

zðk�1ÞrðkÞi ð1;m sec2 mzÞdz; ði ¼ 4;5Þ

Ri ¼XN

k¼1

Z zðkÞ

zðk�1ÞrðkÞi 2m2 sec2ðmzÞ tanðmzÞdz: ði ¼ 3Þ ð9a-dÞ

The static version of the governing equations are derived fromEq. (7) by integrating the displacement gradients by parts and set-ting the coefficients of du, dv, dw, dh1, dh2 and dh3 to zero separately.The equations obtained are as follows:

Table 3Non-dimensionalized centre deflection �wða=2; b=2; 0Þ for various EG rectangular plates, a/

b/a Theory n = 0.1 n = 0.3 n = 0.5 n

6 Present 1.0354 0.9363 0.8462 0.Mantari et al. [42] 1.0388 0.9405 0.8520 0.Present (TPT) 1.0321 0.9333 0.8436 0.

5 Present 1.0115 0.9147 0.8267 0.Mantari et al. [42] 1.0149 0.9189 0.8324 0.Present (TPT) 1.0083 0.9118 0.8241 0.

4 Present 0.9696 0.8768 0.7925 0.Mantari et al. [42] 0.9730 0.8809 0.7980 0.Present (TPT) 0.9665 0.8741 0.7900 0.

3 Present 0.8877 0.8027 0.7255 0.Mantari et al. [42] 0.8909 0.8066 0.7307 0.Present (TPT) 0.8849 0.8002 0.7233 0.

2 Present 0.7037 0.6364 0.5752 0.Mantari et al. [42] 0.7066 0.6397 0.5795 0.Present (TPT) 0.7015 0.6344 0.5734 0.

1 Present 0.2799 0.2531 0.2287 0.Mantari et al. [42] 0.2816 0.2550 0.2309 0.Present (TPT) 0.2790 0.2523 0.2280 0.

du :@N1

@xþ@N6

@y¼ 0;

dv :@N2

@yþ @N6

@x¼ 0;

dw :@2M1

@x2þ @

2M2

@y2þ2

@2M6

@x@yþq¼ 0;

dh1 : y�@M1

@xþy�

@M6

@yþ @P1

@xþ @P6

@yþ y�N5þQ 5 ¼ 0;

dh2 : y�@M2

@yþy�

@M6

@xþ @P2

@yþ @P6

@xþ y�N4þQ 4 ¼ 0;

dh3 : y�@N4

@yþ y�

@N5

@xþ@Q 4

@yþ @Q 5

@xþ y�

@2M1

@x2

þy�@2M2

@y2�2y�

@2M6

@x@y�R3� y�q¼ 0: ð10a-fÞ

By substituting the stress–strain relations into the definitions offorce and moment resultants of the present theory given in Eq. (9a-d) the following constitutive equations are obtained:

h = 10.

= 0.7 n = 1.0 n = 1.5 n = 2.0 n = 2.5 n = 3.0

7644 0.6558 0.5069 0.3913 0.3018 0.23247723 0.6670 0.5236 0.4115 0.3235 0.25397621 0.6538 0.5054 0.3901 0.3006 0.23147468 0.6406 0.4952 0.3823 0.2948 0.22717545 0.6516 0.5115 0.4020 0.3160 0.24807445 0.6387 0.4938 0.3810 0.2937 0.22617159 0.6141 0.4747 0.3664 0.2826 0.21777233 0.6247 0.4903 0.3854 0.3029 0.23777137 0.6123 0.4733 0.3653 0.2815 0.21676554 0.5622 0.4346 0.3355 0.2587 0.19926622 0.5720 0.4489 0.3528 0.2773 0.21766534 0.5605 0.4333 0.3344 0.2577 0.19835196 0.4457 0.3445 0.2659 0.2050 0.15795252 0.4536 0.3560 0.2797 0.2198 0.17245180 0.4444 0.3435 0.2651 0.2043 0.15722066 0.1772 0.1370 0.1057 0.0814 0.06272093 0.1807 0.1417 0.1112 0.0873 0.06842060 0.1767 0.1366 0.1053 0.0811 0.0624

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.14 -0.1 -0.06 -0.02 0.02 0.06 0.1 0.14

v

z

3DPresentZenkour

b/a=3

b/a=2b/a=4

Fig. 5. Distribution of non-dimensionalized displacement, �v , through the thicknessof a thick (a/h = 4) EGM plate (n = 0.5).

Table 4Nondimensionalized normal stresses �ryyða=2; b=2; h=2Þ for EG rectangular plates, a/h = 2.

b/a Theory n=0.1 n=0.3 n=0.5 n=0.7 n=

6

3-D[17] 0.29429 0.31008 0.32699 0.34508 0.37Present 0.27628 0.29544 0.31592 0.3378 0.37Mantari et al. [42] 0.21871 0.23447 0.25122 0.26900 0.29TPT[17] 0.29119 0.31184 0.33385 0.35731 0.39

HPT[17] 0.31192 0.33462 0.35873 0.38433 0.42

5

3-D[17] 0.29674 0.31277 0.32993 0.34829 0.37Present 0.27892 0.29833 0.31905 0.34119 0.37Mantari et al. [42] 0.22185 0.23784 0.25484 0.27288 0.30TPT[17] 0.29353 0.31439 0.33662 0.36032 0.39

HPT[17] 0.31327 0.33607 0.36030 0.38604 0.42

4

3-D[17] 0.30084 0.31727 0.33486 0.35368 0.38Present 0.28335 0.30317 0.32431 0.3469 0.38Mantari et al. [42] 0.22715 0.24354 0.26096 0.27945 0.30TPT[17] 0.29743 0.31864 0.34124 0.36533 0.40

HPT[17] 0.31543 0.33842 0.36285 0.38878 0.43

3

3-D[17] 0.30808 0.32525 0.34362 0.36329 0.39Present 0.29122 0.31177 0.33369 0.35707 0.39Mantari et al. [42] 0.23675 0.25387 0.27206 0.29138 0.32TPT[17] 0.30421 0.32606 0.34933 0.37410 0.41

HPT[17] 0.31890 0.34220 0.36695 0.39323 0.43

2

3-D[17] 0.31998 0.33849 0.35833 0.37956 0.41Present 0.30422 0.32613 0.34945 0.37427 0.41Mantari et al. [42] 0.25385 0.27231 0.29193 0.31276 0.34TPT[17] 0.31463 0.33758 0.36200 0.38796 0.43

HPT[17] 0.32223 0.34592 0.37109 0.39782 0.44

1

3-D[17] 0.31032 0.32923 0.34953 0.37127 0.40Present 0.29244 0.31468 0.33826 0.36325 0.40Mantari et al. [42] 0.25215 0.27100 0.29102 0.31227 0.34TPT[17] 0.29554 0.31811 0.34208 0.36750 0.40

HPT[17] 0.28882 0.31072 0.33398 0.35866 0.39

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 273

Ni ¼ Aije0j þ Bije1

j þ Eije2j þ Gije3

j þ Lije4j ; ði ¼ 1;2;4;5;6Þ

Mi ¼ Bije0j þ Dije1

j þ Fije2j þ Iije3

j þ eNije4j ; ði ¼ 1;2;6Þ

Pi ¼ Eije0j þ Fije1

j þ Hije2j þ Jije3

j þ Sije4j ; ði ¼ 1;2;6Þ

Q i ¼ Gije0j þ Iije1

j þ Jije2j þ Kije3

j þ Tije4j ; ði ¼ 4;5Þ

Ri ¼ Lije0j þ eNije1

j þ Sije2j þ Tije3

j þ Vije4j ; ði ¼ 3Þ ð11a-eÞ

whereðAij;Bij;Dij;Eij;Gij;Lij;Fij; Iij;Nij;Hij; Jij;Sij;Kij;Tij;VijÞ ¼R h=2�h=2 Q ðkÞij ð1;z;z2; tanmz;msec2 mz;2m2 sec2 mz tanmz;ztanmz;mzsec2 mz;

2m2zsec2 mz tanmz; tan2 mz;msec2 mz tanmz;2m2 sec2 mz tan2 mz;

m2 sec4 mz;2m3 sec4 mz tanmz;4m4 sec4 mz tan2 mzÞdz;ð12Þ

In what follows, the problem under consideration is solved forthe following simply supported boundary conditions prescribedat all four edges:

N1 ¼ M1 ¼ P1 ¼ v ¼ w ¼ h2 ¼ h3 at x ¼ 0; a;N2 ¼ M2 ¼ P2 ¼ u ¼ w ¼ h1 ¼ h3 at y ¼ 0; b: ð13a-bÞ

3. Solution procedure

Exact solutions of the partial differential Eqs. (10a-f) on arbitrarydomain and for general boundary conditions are difficult. Although

1.0 n=1.5 456 0.43051 374 0.44163 804 0.34981 547 0.46786

573 0.50345 821 0.43500 755 0.44614 236 0.35485 884 0.47187

764 0.50573 435 0.44257 394 0.45373 968 0.36337 446 0.47857

072 0.50943 534 0.45619 537 0.46732 297 0.37881 432 0.49035

572 0.51545 417 0.47989 483 0.49052 690 0.40636 003 0.50925

102 0.52203 675 0.47405 405 0.47848 773 0.40347 851 0.48508

852 0.47305

Table 5Nondimensionalized normal stresses �ryyða=2; b=2; h=2Þ for EG rectangular plates, a/h = 4.

b/a Theory n=0.1 n=0.3 n=0.5 n=0.7 n=1.0 n=1.5

6

3-D[17] 0.21814 0.23211 0.24699 0.26284 0.28857 0.33725 Present 0.21265 0.22547 0.23934 0.2544 0.27953 0.32937 Mantari et al. [42] 0.20097 0.21493 0.22976 0.24553 0.27105 0.31917 TPT[17] 0.23686 0.25204 0.26830 0.28574 0.31441 0.36990

HPT[17] 0.28170 0.30133 0.32219 0.34435 0.38024 0.44786

5

3-D[17] 0.22060 0.23476 0.24984 0.26591 0.29199 0.34133 Present 0.21524 0.2283 0.24241 0.25772 0.28323 0.33373 Mantari et al. [42] 0.20366 0.21781 0.23285 0.24883 0.27470 0.32346 TPT[17] 0.23912 0.25450 0.27097 0.28863 0.31764 0.37371

HPT[17] 0.28261 0.30231 0.32323 0.34547 0.38148 0.44934

4

3-D[17] 0.22470 0.23918 0.25460 0.27103 0.29770 0.34816 Present 0.21957 0.23302 0.24754 0.26327 0.28943 0.34105 Mantari et al. [42] 0.20818 0.22264 0.23802 0.25435 0.28081 0.33066 TPT[17] 0.24286 0.25858 0.27539 0.29342 0.32299 0.38004

HPT[17] 0.28399 0.30379 0.32483 0.34719 0.38338 0.45159

3

3-D[17] 0.23188 0.24692 0.26295 0.28002 0.30775 0.36021 Present 0.22721 0.24137 0.25663 0.27312 0.30044 0.35404 Mantari et al. [42] 0.21619 0.23122 0.24720 0.26417 0.29166 0.34346 TPT[17] 0.24931 0.26563 0.28307 0.30174 0.33230 0.39106

HPT[17] 0.28588 0.30583 0.32702 0.34954 0.38601 0.45471

2

3-D[17] 0.24314 0.25913 0.27618 0.29434 0.32385 0.37968 Present 0.23953 0.25497 0.27154 0.28936 0.3187 0.37562 Mantari et al. [42] 0.22943 0.24542 0.26240 0.28045 0.30967 0.36473 TPT[17] 0.25878 0.27609 0.29456 0.31428 0.34644 0.40788

HPT[17] 0.28539 0.30534 0.32655 0.34908 0.38556 0.45428

1

3-D[17] 0.22472 0.23995 0.25621 0.27356 0.30177 0.35885 Present 0.22372 0.23907 0.25544 0.27291 0.30137 0.35555 Mantari et al. [42] 0.21636 0.23157 0.24774 0.26492 0.29273 0.34508 TPT[17] 0.23457 0.25098 0.26842 0.28698 0.31706 0.37386

HPT[17] 0.24080 0.25783 0.27593 0.29515 0.32627 0.38482

274 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

the Navier-type solutions can be used to validate the present theory,more general boundary conditions will require solution strategiesinvolving, e.g., boundary discontinuous double Fourier series ap-proach (see for example Oktem and Guedes Soares [44]).

Solution functions that completely satisfy the boundary condi-tions in Eqs. (14) are assumed as follows:

0.15

0.20

0.25

0.30

0.35

0.40

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

b / a

σyy

3DPresentTPTMantari and Guedes Soares

Fig. 6. Non-dimensionalized in-plane stress �ryyða=2; b=2;h=2Þ as a function of theaspect ratio of the plate (b/a).

uðx; yÞ ¼X1r¼1

X1s¼1

Urs cosðaxÞ sinðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14aÞ

vðx; yÞ ¼X1r¼1

X1s¼1

Vrs sinðaxÞ cosðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14bÞ

wðx; yÞ ¼X1r¼1

X1s¼1

Wrs sinðaxÞ sinðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14cÞ

h1ðx; yÞ ¼X1r¼1

X1s¼1

h1rs cosðaxÞ sinðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14dÞ

h2ðx; yÞ ¼X1r¼1

X1s¼1

h2rs sinðaxÞ cosðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14eÞ

h3ðx; yÞ ¼X1r¼1

X1s¼1

h3rs sinðaxÞ sinðbyÞ; 0 6 x 6 a; 0 6 y 6 b ð14fÞ

where

a ¼ rpa; b ¼ sp

b: ð15Þ

Substituting Eqs. (14a)–(14f) into Eqs. (10a-f), the followingequations are obtained,

Kijdj ¼ Fj ði; j ¼ 1; . . . ;6Þ and ðKij ¼ KjiÞ: ð16aÞ

Elements of Kij in Eq. (16a) are given in Appendix A.

Table 6Non-dimensionalized normal stresses �rxxða=2; b=2;h=2Þ for EG rectangular plates, a/h = 10.

b/a Theory n = 0.1 n = 0.3 n = 0.5 n = 0.7 n = 1.0 n = 1.5 n = 2.0 n = 2.5 n = 3.0

6 Present 0.6014 0.6426 0.6864 0.7329 0.8084 0.9510 1.1177 1.3124 1.5394Mantari et al. [42] 0.6029 0.6443 0.6882 0.7350 0.8107 0.9536 1.1204 1.3150 1.5415Present (TPT) 0.6271 0.6707 0.7170 0.7661 0.8452 0.9935 1.1651 1.3637 1.5935

5 Present 0.5895 0.6299 0.6727 0.7184 0.7923 0.9321 1.0955 1.2865 1.5091Mantari et al. [42] 0.5910 0.6315 0.6746 0.7205 0.7947 0.9347 1.0982 1.2890 1.5111Present (TPT) 0.6149 0.6577 0.7031 0.7512 0.8287 0.9741 1.1424 1.3372 1.5626

4 Present 0.5686 0.6075 0.6488 0.6928 0.7641 0.8989 1.0566 1.2410 1.4560Mantari et al. [42] 0.5700 0.6092 0.6508 0.6950 0.7666 0.9016 1.0594 1.2434 1.4576Present (TPT) 0.5935 0.6348 0.6785 0.7249 0.7998 0.9401 1.1025 1.2907 1.5084

3 Present 0.5275 0.5635 0.6018 0.6425 0.7085 0.8335 0.9800 1.1514 1.3514Mantari et al. [42] 0.5288 0.5651 0.6037 0.6447 0.7112 0.8365 0.9828 1.1536 1.3523Present (TPT) 0.5514 0.5896 0.6302 0.6733 0.7427 0.8730 1.0240 1.1990 1.4017

2 Present 0.4340 0.4634 0.4947 0.5280 0.5822 0.6849 0.8056 0.9473 1.1130Mantari et al. [42] 0.4350 0.4649 0.4966 0.5303 0.5850 0.6881 0.8085 0.9490 1.1125Present (TPT) 0.4552 0.4867 0.5200 0.5554 0.6126 0.7201 0.8449 0.9898 1.1580

1 Present 0.2063 0.2199 0.2344 0.2499 0.2753 0.3240 0.3819 0.4506 0.5317Mantari et al. [42] 0.2062 0.2204 0.2355 0.2515 0.2774 0.3264 0.3835 0.4502 0.5278Present (TPT) 0.2196 0.2345 0.2503 0.2671 0.2944 0.3460 0.4065 0.4775 0.5603

Table 7Non-dimensionalized normal stresses �ryyða=2; b=2;h=2Þ for EG rectangular plates, a/h = 10.

b/a Theory n = 0.1 n = 0.3 n = 0.5 n = 0.7 n = 1.0 n = 1.5 n = 2.0 n = 2.5 n = 3.0

6 Present 0.1954 0.2065 0.2185 0.2317 0.2540 0.2988 0.3552 0.4255 0.5115Mantari et al. [42] 0.1960 0.2094 0.2237 0.2389 0.2635 0.3100 0.3642 0.4275 0.5011Present (TPT) 0.2223 0.2360 0.2507 0.2665 0.2926 0.3435 0.4054 0.4805 0.5708

5 Present 0.1980 0.2093 0.2216 0.2350 0.2577 0.3031 0.3602 0.4312 0.5179Mantari et al. [42] 0.1985 0.2122 0.2267 0.2421 0.2670 0.3140 0.3690 0.4331 0.5077Present (TPT) 0.2245 0.2385 0.2534 0.2694 0.2958 0.3473 0.4098 0.4855 0.5764

4 Present 0.2023 0.2140 0.2267 0.2406 0.2638 0.3104 0.3686 0.4407 0.5285Mantari et al. [42] 0.2028 0.2168 0.2316 0.2473 0.2728 0.3208 0.3770 0.4424 0.5187Present (TPT) 0.2283 0.2425 0.2578 0.2742 0.3012 0.3535 0.4170 0.4937 0.5857

3 Present 0.2099 0.2224 0.2358 0.2504 0.2748 0.3233 0.3835 0.4575 0.5472Mantari et al. [42] 0.2104 0.2248 0.2402 0.2565 0.2829 0.3328 0.3910 0.4589 0.5380Present (TPT) 0.2347 0.2495 0.2654 0.2825 0.3104 0.3645 0.4296 0.5080 0.6016

2 Present 0.2223 0.2360 0.2507 0.2666 0.2930 0.3447 0.4079 0.4846 0.5768Mantari et al. [42] 0.2225 0.2378 0.2541 0.2713 0.2993 0.3521 0.4137 0.4855 0.5692Present (TPT) 0.2441 0.2599 0.2768 0.2949 0.3244 0.3810 0.4486 0.5291 0.6246

1 Present 0.2063 0.2199 0.2344 0.2499 0.2753 0.3240 0.3819 0.4506 0.5317Mantari et al. [42] 0.2062 0.2204 0.2355 0.2515 0.2774 0.3264 0.3835 0.4502 0.5278Present (TPT) 0.2196 0.2345 0.2503 0.2671 0.2944 0.3460 0.4065 0.4775 0.5603

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7

σ

z

3DPresentZenkourb/a=2

b/a=4

b/a=1

xx

Fig. 7. Distribution of non-dimensionalized normal stress, �rxx , through the thick-ness of a thick (a/h = 4) EGM plate (n = 0.5).

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-1.8 -1.2 -0.6 0 0.6 1.2

σ

z

3DPresentZenkour

b/a=2b/a=4

b/a=1

xy

Fig. 8. Distribution of non-dimensionalized shear stress, �rxy , through the thicknessof a thick (a/h = 4) EGM plate (n = 0.5).

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 275

Table 8Non-dimensionalized shear stresses �rxzð0; b=2;0Þ for EG rectangular plates, a/h = 10.

b/a Theory n = 0.1 n = 0.3 n = 0.5 n = 0.7 n = 1.0 n = 1.5 n = 2.0 n = 2.5 n = 3.0

6 Present 0.4634 0.4626 0.4610 0.4586 0.4536 0.4416 0.4253 0.4065 0.3845Mantari et al. [42] 0.4633 0.4625 0.4609 0.4585 0.4536 0.4415 0.4252 0.4064 0.3842Present (TPT) 0.4776 0.4769 0.4753 0.4730 0.4681 0.4564 0.4405 0.4209 0.3981

5 Present 0.4579 0.4571 0.4556 0.4532 0.4483 0.4364 0.4203 0.4017 0.3800Mantari et al. [42] 0.4579 0.4571 0.4555 0.4531 0.4482 0.4363 0.4202 0.4016 0.3797Present (TPT) 0.4720 0.4713 0.4697 0.4674 0.4626 0.4510 0.4353 0.4159 0.3935

4 Present 0.4482 0.4475 0.4459 0.4436 0.4388 0.4271 0.4114 0.3933 0.3720Mantari et al. [42] 0.4482 0.4474 0.4458 0.4435 0.4387 0.4271 0.4113 0.3931 0.3717Present (TPT) 0.4620 0.4613 0.4598 0.4575 0.4528 0.4415 0.4261 0.4071 0.3851

3 Present 0.4286 0.4279 0.4264 0.4242 0.4196 0.4084 0.3934 0.3761 0.3558Mantari et al. [42] 0.4285 0.4278 0.4263 0.4241 0.4195 0.4084 0.3933 0.3760 0.3555Present (TPT) 0.4418 0.4411 0.4396 0.4375 0.4330 0.4221 0.4074 0.3893 0.3683

2 Present 0.3810 0.3803 0.3790 0.3770 0.3730 0.3630 0.3497 0.3344 0.3165Mantari et al. [42] 0.3809 0.3803 0.3789 0.3770 0.3729 0.3630 0.3496 0.3343 0.3162Present (TPT) 0.3927 0.3921 0.3908 0.3889 0.3849 0.3752 0.3621 0.3460 0.3273

1 Present 0.2380 0.2376 0.2368 0.2356 0.2330 0.2268 0.2185 0.2094 0.1985Mantari et al. [42] 0.2380 0.2376 0.2368 0.2356 0.2330 0.2268 0.2184 0.2093 0.1983Present (TPT) 0.2454 0.2450 0.2442 0.2430 0.2405 0.2344 0.2263 0.2162 0.2045

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

σ

z

3DPresentMantari and Guedes Soares Zenkour

xz

b/a=2

b/a=4

b/a=1

Fig. 9. Distribution of non-dimensionalized shear stress, �rxz , through the thicknessof a thick (a/h = 4) EGM plate (n = 0.5).

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.05 0.1 0.15 0.2 0.25

σ

z

b/a=1

b/a=2b/a=3

b/a=4

yz

3DPresentMantari and Guedes SoaresZenkour

Fig. 10. Distribution of non-dimensionalized shear stress, �ryz , through the thick-ness of a thick (a/h = 4) EGM plate (n = 0.5).

276 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

fdjgT ¼ Urs Vrs Wrs h1rs h2

rs h3rs

� ; ð16bÞ

fFjgT ¼ 0 0 Q rs 0 0 � y�Q rsf g; ð16cÞ

where Qrs are the coefficients in the double Fourier expansion of thetransverse load,

qðx; yÞ ¼X1r¼1

X1s¼1

Q rs sinðaxÞ sinðbyÞ: ð17Þ

4. Numerical results and discussions

In the present paper, results of simply supported EGPs by usinga new trigonometric HSDT are presented. These results are com-pared with: (a) 3D exact solutions [17]; (b) the well-known trigo-nometric plate theory (TPT), which includes sinus functionoriginally developed by Levy [45], corroborated and improved byStein [46], extensively used by Touratier [47] and recently adaptedto FGM and exponentially graded material (EGM) by Zenkour[17,48]; (c) a HSDT results also provided by Zenkour [17]; and(d) A recently HSDT developed by Mantari et al. [43], which weresuccessfully adapted to FGM by Mantari and Guedes Soares [42](without including the stretching effect).

Non-dimensionalized displacements and stresses given here arepresented according to the following definitions:

�u ¼ u 0;b2; z

� �10Ebh3

q0a4 ; �v ¼ v a2;0; z

�10Ebh3

q0a4 ;

�w ¼ wa2;b2; z

� �10Ebh3

q0a4 ;

�rxx ¼ rxxa2;b2; z

� �h2

q0a2 ;�ryy ¼ ryy

a2;b2; z

� �h2

q0a2 ;

�rxy ¼ rxyð0;0; zÞ10h2

q0a2 ;

�sxz ¼ sxz 0;b2; z

� �h

q0a; �syz ¼ syz

a2;0; z

� hq0a

; �z ¼ zh: ð20a-fÞ

In what follows, numerical results are presented for various as-pect ratios (a/b) and different values of the parameter n. Poisson’sratio is assumed to be a constant, m = 0.3. Fig. 2 shows the exponen-tial function g ð�zÞ along the thickness of an EGP for different valuesof the parameter n.

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 277

Table 1 and 2 present results of centre plate deflection,�wða=2; b=2;0Þ, for very thick plates, a/h = {2,4}, respectively. Theresults are in excellent agreement with 3D exact results. Thepresent results slightly under-predict the 3D solutions in all thecases, see Tables 1 and 2. However, the opposite can be foundin Vel and Batra [49], in which their 3D elasticity solutions andseveral other plate theories were compared. Mantari and GuedesSoares [42] also provide over-predicted values of centre platedeflection by using a HSDT in which the stretching effect wasnot included. Therefore, the importance of including the trans-verse expansion of plate’s thickness for accuracy and tendencyis crucial. The centre deflection, �w, decreases as n increase, andalso when b/a decrease.

Fig. 3 shows the transverse displacement distribution,�wða=2; b=2; zÞ, of exponentially graded plates, b/a = {1,2,3,4}, forn = 0.5 and a/h = 4. Results for b/a = 1, from both HSDTs (TPTand the present one) are almost the same. Moreover, the capa-bility of the present HSDT are shown for b/a > 1 and �z P 0. Ingeneral, for, approximately, �z > �0:3 the transverse displace-ments, �w, are under-predicted. However, the opposite occursfor �z < �0:3. The readers may also look at the results obtainedby Mantari and Guedes Soares [42] and compare with thepresent HSDT. The conclusion that can be drawn after this com-parison analysis is that the stretching effect allows getting im-proved transverse displacement and transverse normal stressesresults.

Results for other side-to-thickness, a/h ratios, other than 2 and 4are not provided by Zenkour [17]. Therefore, it was necessary, forcomparison purposes, to implement another numerical code con-sidering the well-known trigonometric shear deformation theory(TPT). Results of centre plate deflections for plates with side-to-thickness, a/h = 10, for different values of the parameter, n, and as-pect ratio, b/a, are presented in Table 3.

Fig. 4 shows the in-plane distribution of the displacement,�uð0; b=2; zÞ, through the thickness of EGPs, b/a = {1,2,4}, forn = 0.5 and a/h = 4. This figure shows that there are almost nodifferences between the present HSDT and TPT results. However,a small difference between the 3D solutions and the HSDTs canbe noticed, mainly for b/a = 4. Fig. 5 shows the in-plane distribu-tion of the displacement, �vða=2;0; zÞ, through the thickness ofthe exponentially graded plates, b/a = {2,3,4}, for n = 0.5 and a/h = 4. As in Fig. 4, no difference can be noticed between bothHSDTs and only a slight difference exists with 3D solutionresults.

Results of the in-plane normal stress, �ryyða=2; b=2; h=2Þ, ofsquare and rectangular thick EGPs, a/h = {2,4}, are presented in Ta-bles 4 and 5, respectively. Similarly, the present results are com-pared with 3D elasticity solution and with the other HSDTsabove mentioned. The results for the normal stresses, �ryy, increasewith both the increase of the parameter, n, and the decrease of as-pect ratio, b/a. The superiority of the present HSDT against TPT fora/h = 2 can be notice in some of shadow results provided in Table 4.However, for a/h = 4, the capability of the present new trigonomet-ric HSDT are noticed in all the cases.

The in-plane stresses, �ryyða=2; b=2; h=2Þ, increases as the aspectratio b/a decrease, approaching its largest value at b/a = 2, and thendecreases again for a square plate, as shown in Fig. 6, in which TPTresults are also provided. As Fig. 6 shows, for a/h = 4, the presentHSDT, which include the stretching effect, present better resultsthan TPT. Also the present improved results are compared withMantari and Guedes Soares [42].

Tables 6 and 7 present results for in-plane normal stresses, �rxx

and �ryy at (a/2,b/2,h/2), for plates with side-to-thickness, a/h = 10,for different values of both parameter n and aspect ration b/a. FromTable 6, it can be noticed that, �rxx; increases with both the increaseof the parameter n and the increase of the aspect ratio b/a. How-ever, �ryy; increase with both the increase of the parameter n andthe decrease of aspect ratio b/a, see Table 7.

Fig. 7 shows the distribution of in-plane normal stresses,�rxxða=2; b=2; zÞ, through the thickness of EGPs, b/a={1,2,4}, forn = 0.5 and a/h = 4. In general, the present HSDT presents better re-sults than TPT. Similar conclusion can be inferred for the distribu-tion of the another in-plane normal stresses, �ryy, not shown in thispaper for simplicity.

Fig. 8 shows the distribution of in-plane shear stresses,�rxyð0;0; zÞ, through the thickness of EGPs, b/a={1,2,4}, forn = 0.5 and a/h = 4. Results from both HSDTs are almost thesame, which may be explained by the fact that in-plane shearstress, rxy ¼ @�u

@y þ @�v@x, depends only of the in-plane displacements

�u and �v , and they are well modelled by both HSDTs, even whenthey have different shear strain shape functions. Identical resultsare found if the displacement field is modelled without theinclusion of the stretching effect (see Mantari and Guedes Soares[42]).

Table 8 present results for shear stress, �rxz; at (0,b/2,0) for a/h = 10. From this Table it can be noticed that �rxz decrease with boththe increase of the parameter, n, and the decrease of aspect ratio b/a. The results provided in Tables 5–8 may be used as benchmarkresults in futures works.

Figs. 9 and 10 show the distribution of transverse shear stresses,�rxzð0; b=2; zÞ and �ryzða=2;0; zÞ, through the thickness of EGPs, b/a 2 [1,4], for n = 0.5 and a/h = 4, respectively. The results of thepresent HSDT are compared with solutions of: (a) 3D exact solu-tions [17]; (b) the well-known trigonometric plate theory (TPT)[17]; and (c) A recently HSDT developed by Mantari et al. [43],which were successfully adapted to FGM by Mantari and GuedesSoares [42].

As is well-known, exz ¼ @ �w@x þ @�u

@z. By using the present HSDT,this will be represented by exz ¼ h1 þ @h3

@x

� �y� þm sec2ðmzÞ� �

. InMantari and Guedes Soares [42], rxz = h1(y⁄ + m sec2(mz)). Con-sidering the fact that the transverse shear stresses,rxz ¼ EðzÞ

2ð1þmÞ exz, were calculated at position (0,b/2,z), and at thatposition @h3

@x ¼ 0, then, the results of transverse shear stressesdo not depend of the higher-order terms of the plate’s transverseexpansion, i.e. the transverse shears stresses, including or notthe stretching effect, are the same. Figs. 9 and 10 show this fact.In these figures the present results and the one that comes fromMantari and Guedes Soares [42] are exactly the same. FromFigs. 9 and 10 can be noticed the capability and superiority ofthe present HSDT against the well-known trigonometric sheardeformation theory (TPT).

In general aspects, the displacements and the stresses do notvary as linear functions of the thickness coordinate z because theplate is EG and the material properties are functions of z. The dis-placements ð�u; �v and �w) and stresses (�rxx; �rxy; �rxz and �ryz) distri-butions obtained by using the present HSDT are in good agreementwith the exact 3D solution.

5. Conclusions

The static response of exponentially graded plates is presentedby using a new trigonometric higher-order shear deformation the-

278 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

ory that includes stretching effect. The plate material is exponen-tially graded in the thickness direction. The governing equationsand boundary conditions are derived by employing the principleof virtual work. These equations are solved via a Navier-type,closed form solution, for FG plates subjected to transverse bi-sinu-soidal load for simply supported boundary conditions.

The importance of including the transverse expansion of plate’sthickness for the accuracy and the tendency of the results isstressed out. The results are compared with 3D exact solutionand with other higher-order shear deformation theories, and thesuperiority of the present theory can be noticed. In general, forthick exponentially graded plates, the present new trigonometrictheory predicts displacements and stresses more accuracy whencompared to the well-known trigonometric plate theory (TPT).Therefore, benchmark results for the displacement and stressesof exponential graded rectangular plates are obtained, which canbe used for the evaluation of other HSDTs and also to compare re-sults obtained by using numerical methods such as the finite ele-ment and meshless methods.

Acknowledgment

The first author has been financed by the Portuguese Founda-tion of Science and Technology under the Contract NumberSFRH/BD/66847/2009.

Appendix A. Definition of constants in Eq. (16a)

The following proposed simple technique to calculate the Kij

element matrices (which comes from the governing Eqs. (10a-f))is perhaps more convenient and simple than the others [17]. Theadvantage of the present technique is that many shear deformationtheories can be calculated using the same following matrices, only‘‘y⁄’’ should be changed. In the present case, y� ¼ � sec2ðm h

2Þ (seeEq. (2a)–(2c). In the case of the other well-known trigonometricshear deformation theory, y⁄ = 0.

Calculation of N, M and P:

Nc1;M

c1; P

c1;Q

c1;R

c1

� �Nc

2;Mc2; P

c2;Q

c2;R

c2

� �Nc

3;Mc3; P

c3;Q

c3;R

c3

� �Nc

4;Mc4; P

c4;Q

c4;R

c4

� �Nc

5;Mc5; P

c5;Q

c5;R

c5

� �Nc

6;Mc6; P

c6;Q

c6;R

c6

� �

2666666664

3777777775

¼ ðAij;Bij; Eij;Gij;LijÞ

�a 0 0 0 0 00 �b 0 0 0 00 0 0 0 0 00 0 0 0 y� y�b0 0 0 y� 0 y�ab a 0 0 0 0

2666666664

3777777775

þ ðBij;Dij; Fij;Iij;eNijÞ

0 0 a2 �y�a 0 �y�a2

0 0 b2 0 �y�b �y�b2

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 �2ab y�b y�a 2y�ab

2666664

3777775

þ ðEij; Fij;Hij;Jij; SijÞ

0 0 0 �a 0 00 0 0 0 �b 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 b a 0

2666664

3777775

þðGij; Iij; Jij;Kij;TijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 1 b0 0 0 1 0 a0 0 0 0 0 0

26666664

37777775

þðLij; eNij;Hij;Jij; SijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 10 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775:ðA1Þ

where i, j = 1,2,6.First derivative of N, M and P with respect to x:

@ Nc1 ;M

c1 ;P

c1 ;Q

c1 ;R

c1ð Þ

@x@ Nc

2 ;Mc2 ;P

c2 ;Q

c2 ;R

c2ð Þ

@x@ Nc

3 ;Mc3 ;P

c3 ;Q

c3 ;R

c3ð Þ

@x@ Nc

4 ;Mc4 ;P

c4 ;Q

c4 ;R

c4ð Þ

@x@ Nc

5 ;Mc5 ;P

c5 ;Q

c5 ;R

c5ð Þ

@x@ Nc

6 ;Mc6 ;P

c6 ;Q

c6 ;R

c6ð Þ

@x

26666666666664

37777777777775

¼ ðAij;Bij;Eij;Gij;LijÞ

�a2 0 0 0 0 00 �ab 0 0 0 00 0 0 0 0 00 0 0 0 y�a y�ab0 0 0 �y�a 0 �y�a2

�ab �a2 0 0 0 0

266666664

377777775

þ ðBij;Dij;Fij;Iij;eNijÞ

0 0 a3 �y�a2 0 �y�a3

0 0 ab2 0 �y�ab �y�ab2

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 �2a2b �y�ab �y�a2 �2y�a2b

266666664

377777775

þ ðEij;Fij;Hij;Jij;SijÞ

0 0 0 �a2 0 00 0 0 0 �ab 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 �ab �a2 0

266666664

377777775

þ ðGij; Iij; Jij;Kij;TijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 a ab0 0 0 �a 0 �a2

0 0 0 0 0 0

26666664

37777775

þ ðLij; eNij;Hij;Jij;SijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 a0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

26666664

37777775: ðA2Þ

J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281 279

First derivative of N, M and P with respect to y:

@ Nc1 ;M

c1 ;P

c1 ;Q

c1 ;R

c1ð Þ

@y

@ Nc2 ;M

c2 ;P

c2 ;Q

c2 ;R

c2ð Þ

@y

@ Nc3 ;M

c3 ;P

c3 ;Q

c3 ;R

c3ð Þ

@y

@ Nc4 ;M

c4 ;P

c4 ;Q

c4 ;R

c4ð Þ

@y

@ Nc5 ;M

c5 ;P

c5 ;Q

c5 ;R

c5ð Þ

@y

@ Nc6 ;M

c6 ;P

c6 ;Q

c6 ;R

c6ð Þ

@y

26666666666666666666664

37777777777777777777775

¼ ðAij;Bij; Eij;Gij;LijÞ

�ab 0 0 0 0 0

0 �b2 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �y�b �y�b2

0 0 0 y�b 0 y�ab

�b2 �ab 0 0 0 0

266666666666664

377777777777775

þ ðBij;Dij; Fij;Iij;eNijÞ

0 0 a2b �y�ab 0 �y�a2b

0 0 b3 0 �y�b2 �y�b3

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 2ab2 �y�b2 �y�ab �2y�ab2

2666666666666664

3777777777777775

þ ðEij; Fij;Hij;Jij; SijÞ

0 0 0 �ab 0 0

0 0 0 0 �b2 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 �b2 �ab 0

266666666666664

377777777777775

þ ðGij; Iij; Jij;Kij; TijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �b �b2

0 0 0 b 0 ab

0 0 0 0 0 0

266666666666664

377777777777775

þ ðLij; eNij;Hij;Jij; SijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 b

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

266666666666664

377777777777775: ðA3Þ

Second partial derivative of N, M and P with respect to x:

@2 Nc1 ;M

c1 ;P

c1 ;Q

c1 ;R

c1ð Þ

@x2

@2 Nc2 ;M

c2 ;P

c2 ;Q

c2 ;R

c2ð Þ

@x2

@2 Nc3 ;M

c3 ;P

c3 ;Q

c3 ;R

c3ð Þ

@x2

@2 Nc4 ;M

c4 ;P

c4 ;Q

c4 ;R

c4ð Þ

@x2

@2 Nc5 ;M

c5 ;P

c5 ;Q

c5 ;R

c5ð Þ

@x2

@2 Nc6 ;M

c6 ;P

c6 ;Q

c6 ;R

c6ð Þ

@x2

266666666666666666664

377777777777777777775

¼ ðAij;Bij; Eij;Gij;LijÞ

a3 0 0 0 0 0

0 a2b 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �y�a2 �y�a2b

0 0 0 �y�a2 0 �y�a3

�a2b �a3 0 0 0 0

266666666666664

377777777777775

þ ðBij;Dij; Fij;Iij;eNijÞ

0 0 �a4 y�a3 0 y�a4

0 0 �a2b2 0 y�a2b y�a2b2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 2a3b �y�a2b �y�a3 �2y�a3b

266666666666664

377777777777775

þ ðEij; Fij;Hij;Jij; SijÞ

0 0 0 a3 0 0

0 0 0 0 a2b 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 �a2b �a3 0

266666666666664

377777777777775

þ ðGij; Iij; Jij;Kij;TijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �a2 �a2b

0 0 0 �a2 0 �a3

0 0 0 0 0 0

266666666666664

377777777777775

þ ðLij; eNij;Hij;Jij; SijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 �a2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

266666666666664

377777777777775: ðA4Þ

280 J.L. Mantari, C. Guedes Soares / Composites: Part B 45 (2013) 268–281

Second partial derivative of N, M and P with respect to y:

@2 Nc1 ;M

c1 ;P

c1 ;Q

c1 ;R

c1ð Þ

@y2

@2 Nc2 ;M

c2 ;P

c2 ;Q

c2 ;R

c2ð Þ

@y2

@2 Nc3 ;M

c3 ;P

c3 ;Q

c3 ;R

c3ð Þ

@y2

@2 Nc4 ;M

c4 ;P

c4 ;Q

c4 ;R

c4ð Þ

@y2

@2 Nc5 ;M

c5 ;P

c5 ;Q

c5 ;R

c5ð Þ

@y2

@2 Nc6 ;M

c6 ;P

c6 ;Q

c6 ;R

c6ð Þ

@y2

26666666666666666666664

37777777777777777777775

¼ ðAij;Bij; Eij;Gij;LijÞ

ab2 0 0 0 0 0

0 b3 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �y�b2 �y�b3

0 0 0 �y�b2 0 �y�ab2

�b3 �ab2 0 0 0 0

2666666666666664

3777777777777775

þ ðBij;Dij; Fij;Iij;eNijÞ

0 0 �a2b2 y�ab2 0 y�a2b2

0 0 �b4 0 y�b3 y�b4

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 2ab3 �y�b3 �y�ab2 �2y�ab3

2666666666666664

3777777777777775

þ ðEij; Fij;Hij;Jij; SijÞ

0 0 0 ab2 0 0

0 0 0 0 b3 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 �b3 �ab2 0

2666666666666664

3777777777777775

þ ðGij; Iij; Jij;Kij;TijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 �b2 �b3

0 0 0 �b2 0 �ab2

0 0 0 0 0 0

266666666666664

377777777777775

þ ðLij; eN ij;Hij;Jij; SijÞ

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 �b2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

266666666666664

377777777777775: ðA5Þ

Second partial derivative of N, M and P with respect to x and y:

@2 Nc1 ;M

c1 ;P

c1 ;Q

c1 ;R

c1ð Þ

@x@y

@2 Nc2 ;M

c2 ;P

c2 ;Q

c2 ;R

c2ð Þ

@x@y

@2 Nc3 ;M

c3 ;P

c3 ;Q

c3 ;R

c3ð Þ

@x@y

@2 Nc4 ;M

c4 ;P

c4 ;Q

c4 ;R

c4ð Þ

@x@y

@2 Nc5 ;M

c5 ;P

c5 ;Q

c5 ;R

c5ð Þ

@x@y

@2 Nc6 ;M

c6 ;P

c6 ;Q

c6 ;R

c6ð Þ

@x@y

26666666666666664

37777777777777775

¼ ðAij;Bij;Eij;Gij;LijÞ

�a2b 0 0 0 0 00 �ab2 0 0 0 00 0 0 0 0 00 0 0 0 �y�ab �y�ab

0 0 0 �y�ab 0 �y�a2b

ab2 a2b 0 0 0 0

2666666664

3777777775

þ ðBij;Dij;Fij;Iij;eNijÞ

0 0 a3b �y�a2b 0 �y�a3b

0 0 ab3 0 �y�ab2 �y�ab3

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 �2a2b2 y�ab2 y�a2b 2y�a2b2

2666666664

3777777775

þ ðEij;Fij;Hij;Jij;SijÞ

0 0 0 �a2b 0 00 0 0 0 �ab2 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 ab2 a2b 0

2666666664

3777777775

þ ðGij; Iij; Jij;Kij;TijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 �ab �ab2

0 0 0 �ab 0 �a2b

0 0 0 0 0 0

2666666664

3777777775

þ ðLij; eNij;Hij;Jij;SijÞ

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 ab

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

2666666664

3777777775: ðA6Þ

Example to get K(1, j), in the governing Eq. (10a):From the Eqs. (A2) and (A3), @Nc

1@x and @Nc

6@y can be easily obtained

and substituted in Eq. (A7).

Kð1; jÞ ¼ @Nc1

@xþ @Nc

6

@y; where j ¼ 1;2; . . . ;5: ðA7Þ

Following the same technique the coefficients associated withthe rest of the governing equations can be obtained, and in thisway the system of equations, see Eq. (16a), can be solved.

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