Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates

AIAA JOURNAL

Vol 40 No 7 July 2002

Exact Solution for Thermoelastic Deformationsof Functionally Graded Thick Rectangular Plates

Senthil S Velcurren

University of Maine Orono Maine 04469-5711and

R C Batradagger

Virginia Polytechnic Institute and State University Blacksburg Virginia 24061

An exact solution is obtained for three-dimensional deformations of a simply supported functionally gradedrectangular plate subjected to mechanical and thermal loads on its top andor bottom surfaces Suitable temper-ature and displacement functions that identically satisfy boundary conditions at the edges are used to reduce thepartial differential equations governing the thermomechanical deformations to a set of coupled ordinary differen-tial equations in the thickness coordinate which are then solved by employing the power series method The exactsolution is applicable to both thick and thin plates Results are presented for two-constituent metalndashceramic func-tionally graded rectangular plates that have a power law through-the-thickness variation of the volume fractionsof the constituents The effective material properties at a point are estimated by either the MorindashTanaka or theself-consistent schemes Exact displacements and stresses at several locations for mechanical and thermal loads areused to assess the accuracy of the classical plate theory the rst-order shear deformation theory and a third-ordershear deformation theory for functionally graded plates Results are also computed for a functionally graded platewith material properties derived by the MorindashTanaka method the self-consistent scheme and a combination ofthese two methods

I Introduction

A DVANCED composite materials offer numerous superiorproperties to metallic materials such as high speci c strength

and high speci c stiffness This has resulted in the extensive useof laminated composite materials in aircraft spacecraft and spacestructures For example a layer of a ceramic material when bondedto the surface of a metallic structure acts as a thermal barrier inhigh-temperature applications However the sudden change in ma-terial properties across the interface between discrete materials canresult in large interlaminar stresses leading to delamination Fur-thermore large plastic deformations at the interfaces may triggerthe initiation and propagation of cracks in the material One way toovercome these adverse effects is to use functionally graded materi-als in which material properties vary continuously This is achievedeither by gradually changing the volume fraction of the constituentmaterials usually in the thickness direction only or by changing thechemical structure of a thin polymer sheet to obtain a smooth vari-ation of in-plane material properties and an optimum response toexternal thermomechanical loads The former class of functionallygraded structures can be manufactured by high-speed centrifugalcasting12 in which layers are formed in the radial direction due todifferent mass densities of the constituents or by depositing layersof ceramic materials on a metallic substrate34 Lambros et al5 havedeveloped an ultraviolet irradiation process to obtain variations inYoungrsquos modulus in the plane of a sheet A directed oxidation tech-nique has been employed by Breval et al6 and Manor et al7 to obtaina ceramic layer on the outside surface

There are several three-dimensional solutions available for thethermoelastic analysis of inhomogeneous plates Most of thesestudies have been conducted for laminated plates that have

Received 26 April 2001 revision received 20 November 2001 acceptedfor publication 24 December 2001 Copyright cdeg 2002 by Senthil S Veland R C Batra Published by the American Institute of Aeronautics andAstronautics Inc with permission Copies of this paper may be made forpersonal or internal use on condition that the copier pay the $1000 per-copyfee to the Copyright Clearance Center Inc 222 Rosewood Drive DanversMA 01923 include the code 0001-1452 02 $1000 in correspondenc e withthe CCC

currenAssistant Professor Department of Mechanical EngineeringdaggerClifton C Garvin Professor Department of Engineering Science and

Mechanics MC 0219

piecewise constant material properties in the thickness directionThree-dimensional solutions for functionally graded plates are use-ful because they can be used as benchmarks to assess the ac-curacy of various two-dimensional plate theories Rogers et al8

have employed the method of asymptotic expansion to analyzethree-dimensional deformations of inhomogeneous plates How-ever the boundary conditions on the edges of the plate in their theoryare applied in an average sense like those in two-dimensional platetheories and the plate is assumed to be only moderately thick Tarnand Wang9 have also presented an asymptotic solution that may becarried out to any order but the manipulations become more andmore involved as one considers higher-order terms and numericalexamples are given only for laminated plates consisting of homo-geneous layers Cheng and Batra10 have also used the method ofasymptotic expansion to study the three-dimensional thermoelasticdeformations of a functionally graded elliptic plate

Tanaka et al1112 designed property pro les for functionallygraded materials to reduce the thermal stresses Reddy13 has pre-sented solutions for rectangular plates based on the third-order sheardeformation plate theory Reiter et al14 Reiter et al15 and Reiterand Dvorak16 performed detailed nite element studies of discretemodels containing simulated skeletal and particulate microstruc-tures and compared results with those computed from homogenizedmodels in which effective properties were derived by the MorindashTanaka17 and the self-consistent18 methods Cheng and Batra19 haverelated the de ections of a simply supported functionally gradedpolygonal plate given by the rst-order shear deformation theory(FSDT) and a third-order shear deformation theory (TSDT) to thatof an equivalent homogeneous Kirchhoff plate Lee and Yu20 andLee et al21 have expanded the mechanical displacements the elec-tric potential and the material moduli as a power series in thethickness coordinate derived plate equations of different orders forfunctionally graded piezoelectric materials and analyzed their freevibrations Batra22 has studied nite plane strain deformations ofa functionally graded incompressible hollow cylinder loaded by auniform pressure on the inner surface

The objective of this investigation is to present an exact solutionto the three-dimensional thermoelastic deformations of a simplysupported functionally graded rectangular thick plate We assumethat the plate is made of an isotropic material with material prop-erties varying smoothly in the thickness direction only By the use

1421

1422 VEL AND BATRA

of suitable temperature and displacement functions the governingpartial differential equations are reduced to a set of coupled ordinarydifferential equations in the thickness coordinate which are solvedby the power series method We consider a two-phase graded mate-rial with a power law variation of the volume fractions of the con-stituents through the thickness The effective material properties ata point are determined in terms of the local volume fractions and thematerial properties of the two phases either by the MorindashTanaka17

(also see Benveniste23 ) or by the self-consistent18 scheme Resultsare presented for an AlSiC graded rectangular plate Displacementsand stresses at critical locations for mechanical and thermal loadsare given for different length-to-thickness ratios exponents in thepower law through-the-thickness variation of the constituents anddifferent homogenization schemes We compare the exact resultswith those obtained from the classical plate theory2425 (CPT) theFSDT2627 and the TSDT2728 The results from the three plate theo-ries are quite different from the exact solution for thick functionallygraded plates

Results are also computed for a functionally graded plate forwhich the effective material properties in the ceramic rich and themetal rich regions are derived by the MorindashTanaka17 method and theeffective properties elsewhere are obtained by the self-consistent18

scheme A third-order transition function of Reiter and Dvorak16 isused in the transition regions to have a smooth variation of materialproperties between the regions For the thermal load this combinedmethod of homogenizing material properties gives plate deforma-tions and transverse normal and transverse shear stresses that differsigni cantly from those computed with either the MorindashTanaka orthe self-consistent scheme alone However through-the-thicknessvariation of the longitudinal stress given by the three homogeniza-tion methods is essentially the same except at points near the topsurface of the plate where the volume fraction of the ceramic is highFor the mechanical load the homogenization technique in uencesstrongly the transverse displacements but not the values of the stresscomponents

II Formulation of the ProblemWe use rectangular Cartesian coordinates xi i D 1 2 3 to de-

scribe the in nitesimal static thermoelastic deformations of anN -layer laminated plate occupying the region [0 L1] pound [0 L2] pound[iexclH=2 H=2] in the unstressed reference con guration Each layerof the laminated plate is made of an isotropic material with ma-terial properties varying smoothly in the x3 (thickness) direc-tion only The vertical positions of the bottom and the top sur-faces and the N iexcl 1 interfaces between the layers are denoted byH 1 D iexclH=2 H 2 H n H N and H N C 1 D H=2

The equations of mechanical and thermal equilibrium in theabsence of body forces and internal heat sources are (eg seeCarlson29)

frac34i j j D 0 (1a)

q j j D 0 (1b)

where frac34i j and q j are respectively the components of the Cauchystress tensor and the heat ux vector and where i j D 1 2 3 Acomma followed by index j denotes partial differentiation with re-spect to the position x j of a material particle and a repeated indeximplies summation over the range of the index

The constitutive equations for a linear isotropic thermoelasticmaterial are29

frac34i j D cedilkk plusmni j C 2sup1i j iexcl macrplusmni j T (2a)

q j D iexclmiddotT j (2b)

where cedil and sup1 are the Lame constants macr is the stress-temperaturemodulus middot is the thermal conductivity i j are components of thein nitesimal strain tensor T is the change in temperature of a mate-rial particle from that in the stress-free reference con guration andplusmni j is the Kronecker delta The material properties cedil sup1 macr and middot arefunctions of x3

The in nitesimal strain tensor is related to the mechanicaldisplacements u i by

i j D 12ui j C u ji (3)

The edges of the plate are assumed to be simply supported andmaintained at the reference temperature That is

frac3411 D 0 (4a)

u2 D u3 D 0 (4b)

T D 0 (4c)

at x1 D 0 and L1 and

frac3422 D 0 (4d)

u1 D u3 D 0 (4e)

T D 0 (4f)

at x2 D 0 and L2 The mechanicalboundary conditions prescribed on the top and the

bottom surfaces can be either a displacement component u j or thecorresponding traction component frac343 j However typically nonzeronormal and zero tangential tractions are prescribed on these twosurfaces Because the normal load can be expanded as a doubleFourier series in x1 and x2 it suf ces to consider loads of the form

frac3413 D frac3423 D 0 at x3 D sectH=2

frac3433x1 x2 sectH=2 D psect sin r x1 sin sx2 (5)

where pC and piexcl are known constants r D kfrac14=L1 s D mfrac14=L2 andk and m are positive integers The thermal boundary conditions onthe top and the bottom surfaces are speci ed as

sect T x1 x2 sectH=2 C raquosect q3x1 x2 sectH=2

D rsquosect sin rx1 sin sx2 (6)

When values of constants sect and raquosect are chosen appropriatelyvarious boundary conditions corresponding to either a prescribedtemperature a prescribed heat ux or an exposure to an ambienttemperature through a boundary conductance can be speci ed

The interfaces between adjoining layers are assumed to be per-fectly bonded together and in ideal thermal contact so that

[[u i ]] D 0 (7a)

[[frac34i3]] D 0 (7b)

[[T ]] D 0 (7c)

[[q3]] D 0 (7d)

on x3 D H 2 H 3 H N Here [[u i ]] is the jump in the value ofu i across an interface

The mechanical and the thermal problems are one-way coupledin the sense that the temperature eld is determined rst by solvingEqs (1b) and (2b) and the pertinent boundary conditions and thedisplacements are later obtained from Eqs (1a) and (2a) and therelevant boundary conditions

III Exact SolutionWe construct a local rectangular Cartesian coordinate system

x n

1 x n

2 and xn

3 with local axes parallel to the global axes andthe origin at the point where the global x3 axis intersects the mid-surface of the nth lamina In the local coordinate system the nth lam-ina occupies the region [0 L1] pound [0 L2] pound [iexclhn=2 hn=2] wherehn D H n C 1 iexcl H n We drop the superscript n for conveniencewith the understanding that all material constants and variablesbelong to this layer

We assume that within each layer the Lame constants cedil and sup1the stress-temperature modulus macr and the thermal conductivity middot

VEL AND BATRA 1423

are analytic functions of x3 and thus can be represented by a Taylorseries expansion about its midsurface as

[cedil sup1 macr middot] D1X

reg D 0

poundQ reg Qsup1reg Qreg Qmiddot reg

currenxreg

3 (8)

Note that cedil sup1 macr and middot are positive quantities for all x3 and there-fore Q 0 Qsup10 Q0 and Qmiddot 0 are positive

A Temperature FieldA solution for the change in temperature of points in the nth layer

is sought in the form

T D microx3 sin r x1 sin sx2 (9)

The assumed temperature function T identically satis es the bound-ary conditions (4c) and (4f) at the edges of the plate Substitutionfor T from Eq (9) into Eq (2b) and the result into Eq (1b) gives thefollowing second-order ordinary differential equation with variablecoef cients

middot [r 2 C s2micro iexcl micro 00] iexcl middot 0micro 0 D 0 (10)

where a prime denotes derivative with respect to x3 We assume asolution for micro in the form of a power series

microx3 D1X

deg D 0

Qmicro deg xdeg

3 (11)

The series (11) and the Taylor series for middot in Eq (8) are substitutedinto Eq (10) By multiplying the in nite series appropriately shift-ing the index of summation and equating each power of x3 to zerowe obtain the following recurrence relation

regX

deg D 0

Qmiddot deg pound

Qmicro reg iexcl deg C 2reg iexcl deg C 2reg iexcl deg C 1 iexcl Qmicro reg iexcl deg r 2 C s2curren

C Qmiddot deg C 1 Qmicro reg iexcl deg C 1deg C 1reg iexcl deg C 1 D 0 (12)

for reg D 0 1 2 Because Qmiddot 0 6D 0 corresponding to reg D 0 inEq (12) we obtain

Qmicro 2 D [r 2 C s2=2] Qmicro 0 iexclpound

Qmiddot 1macr

2 Qmiddot 0curren

Qmicro 1 (13)

Evaluation of the recursion formula (12) successively for reg D1 2 gives Qmicro reg C 2 in terms of arbitrary constants Qmicro 0 and Qmicro 1Substitution for Qmicro reg into Eq (11) and and the result into Eq (9)gives the following expression for the temperature change

T Dpound

Qmicro 0Atilde 0x3 C Qmicro 1Atilde 1x3curren

sin r x1 sin sx2 (14)

where Atilde 0x3 and Atilde 1x3 are known in nite series and Qmicro 0 andQmicro 1 are unknown constants There are two unknown constants foreach layer which result in a total of 2N unknowns for an N -layerplate The constants are determined by satisfying the thermal bound-ary conditions (6) on the top and the bottom surfaces of the plateand the interface continuity conditions (7c) and (7d) for the thermalquantities between adjoining layers This gives two conditions forthe top and the bottom surfaces and two conditions at each of theN iexcl 1 interfaces The resulting system of 2N linear algebraic equa-tions for the 2N unknowns is readily solved to yield the change intemperature and the heat ux vector for the entire plate

B Displacement FieldA solution for the displacement eld in the nth layer is sought in

the form

u1 D U1x3 cos rx1 sin sx2 u2 D U2x3 sin r x1 cos sx2

u3 D U3x3 sin rx1 sin sx2 (15)

which identically satis es the homogeneous boundary conditions(4andash4b) and (4dndash4e) at the simply supported edges Substitution for

u from Eq (15) into Eq (3) for and also for T from Eq (9) intoEq (2a) and then for frac34 into Eq (1a) gives the following coupledsystem of second-order ordinary differential equations

cedil C 2sup1U1r2 C cedilU2rs C sup1

iexclU1s

2 C U2rscent

iexcl cedilU 03r

iexcl sup10U 01 C U3r iexcl sup1U 00

1 C U 03r C macrmicror D 0

cedil C 2sup1U2s2 C cedilU1r s C sup1iexclU2s2 C U1rs

centiexcl cedilU 0

3s

iexcl sup10U 02 C U3s iexcl sup1U 00

2 C U 03s C macrmicros D 0

sup1U 01r C U 0

2s C sup1U3r 2 C s2 C cedil0U1r C U2s C cedilU 01r C U 0

2s

iexcl cedil0 C 2sup10U 03 iexcl cedil C 2sup1U 00

3 C macr 0micro C macrmicro 0 D 0 (16)

We assume a power series solution for the displacements as

Ui x3 D1X

deg D 0

QU deg

i xdeg

3 (17)

Inserting into the differential equations (16) the Taylor series forthe material properties cedil sup1 and macr from Eq (8) and the assumedpower series solution for the displacements and the temperaturechange from Eqs (17) and (11) we obtain the following recurrencerelations for every nonnegative integer reg

regX

deg D 0

iexclQ deg C 2 Qsup1deg

cent QU reg iexcl deg

1 r 2 C Q deg QU reg iexcl deg

2 r s

C Qsup1deg

plusmnQU reg iexcl deg

1 s2 C QU reg iexcl deg

2 rssup2

iexcl reg iexcl deg C 1 Q deg QU reg iexcl deg C 1

3 r

iexcl deg C 1 Qsup1deg C 1

plusmnreg iexcl deg C 1 QU reg iexcl deg C 1

1 C QU reg iexcl deg

3 rsup2

iexcl reg iexcl deg C 1 Qsup1deg


1 C QU reg iexcl deg C 1

3 rsup2

C Qdeg Qmicro reg iexcl deg r D 0

regX

deg D 0


centQU reg iexcl deg

2 s2 C Q deg QU reg iexcl deg

1 rs

C Qsup1deg


2 r 2 C QU reg iexcl deg

1 rssup2


3 s




3 ssup2




3 ssup2

C Qdeg Qmicro reg iexcl deg s D 0

regX

deg D 0

Qsup1deg reg iexcl deg C 1

plusmnQU reg iexcl deg C 1

1 r C QU reg iexcl deg C 1

2 ssup2

C Qsup1deg QU reg iexcl deg

3 r 2 C s2 C deg C 1 Q deg C 1


1 r C QU reg iexcl deg

2 ssup2

C reg iexcl deg C 1 Q deg



2 ssup2

iexcl deg C 1reg iexcl deg C 1iexclQ deg C 1 C 2 Qsup1deg C 1

cent QU reg iexcl deg C 1

3

iexcl reg iexcl deg C 2reg iexcl deg C 1iexclQ deg C 2 Qsup1deg


3

iexcl Qdeg C 1 Qmicro reg iexcl deg deg C 1 iexcl Qdeg Qmicro reg iexcl deg C 1reg iexcl deg C 1 D 0 (18)

The recurrence relations (18) are evaluated successively forreg D 0 1 to obtain QU reg C 2

1 QU reg C 2

2 and QU reg C 2

3 in terms of ar-bitrary constants QU 0

1 QU 1

1 QU 0

2 QU 1

2 QU 0

3 and QU 1

3 Thus thereare six unknown constants for each layer resulting in a total of 6Nunknowns for an N -layer plate The constants are determined by sat-isfying the mechanical boundary conditions (5) on the top and thebottom surfaces of the plate and the interface continuity conditions

1424 VEL AND BATRA

(7a) and (7b) for the mechanical quantities between adjoining lay-ers This yields six conditions for the top and the bottom surfacesand six conditions at each of the N iexcl 1 interfaces The resultingsystem of 6N linear algebraic equations for the 6N unknowns issolved to obtain displacements and stresses for the entire plate

The semi-inverse method of analyzing simply supported plateproblems as assumed in Eqs (9) and (15) is due toVlasov30 Srinivasand Rao31iexcl33 and Pagano34 and has been adopted by others to studypiezoelectric and thermoelastic plate problems Vel and Batra35iexcl37

have used the EshelbyndashStroh formalism to analyze deformations ofplates with arbitrary boundary conditions prescribed at the edges

IV Plate TheoriesThe displacement eld for the CPT2425 the FSDT2627 and the

TSDT132728 can be written as

udeg xi D u0deg iexcl x3u

03deg C gx3rsquodeg (19a)

u3xi D u03 (19b)

where u0deg u0

3 and rsquodeg are independent of x3 and the func-tion gx3 D 0 for the CPT gx3 D x3 for the FSDT andgx3 D x31 iexcl 4x2

3=3h2 for the TSDT In this section the Latinindices range from 1 to 3 and the Greek indices from 1 to 2 Func-tions u0 give displacements of a point on the midsurface of the plateand rsquo1 iexcl u0

31 and iexclrsquo2 C u032 are respectively the rotations of

the transverse normal to the midsurface about the x2 and x1 axesDisplacement elds (19) were proposed for studying isothermal de-formationsof a plate here we use them to analyze thermomechanicaldeformations in the presence of temperature gradients in the thick-ness direction Cheng and Batra10 among others employed thesedisplacement elds for analyzing thermomechanical deformationsof a functionally graded plate

For the bending of a linear functionally graded plate subjected toan arbitrary distributed normal load pxdeg on its surface the eldequations are

Ndeg acuteacute D 0 Mdeg acutedeg acute C p D 0 Pdeg acuteacute iexcl KRdeg D 0 (20)

where

[Ndeg acute Mdeg acute Pdeg acute] DZ H=2

iexclH=2

frac34deg acute[1 x3 g] dx3

Rdeg DZ H=2

iexclH=2

frac34deg 3g3 dx3

frac34deg acute D Hdeg acutefrac12 frac12 iexcl macrplusmndeg acuteT frac34deg 3 D 2Edeg 333 (21)

and constant K is the shear correction factor used only for the FSDTWe set K D 5

6 although this value was proposed by Reissner for a

homogeneous plate Because the temperature T is assumed to beknown in the CPT the FSDT and the TSDT we substitute theexact value for the temperature eld from Eq (9) into Eq (21)The transverse normal stress frac3433 is assumed to be negligible andneglected in all three plate theories this is inconsistent with theassumption of zero transverse normal strain implied by Eq (19b)For an isotropic material

Hdeg acutefrac12 D [ordmE=1 iexcl ordm2]plusmndeg acuteplusmnfrac12 C [E=21 C ordm]plusmndeg plusmnacutefrac12 C plusmndegfrac12plusmnacute

Edeg 33 D sup1plusmndeg (22)

where E ordm and sup1 are respectively Youngrsquos modulus Poissonrsquos ra-tio and the shear modulus For a functionally graded plate materialproperties are assumed to vary in the thickness direction only thatis E D Ex3 ordm D ordmx3 and sup1 D sup1x3

The boundary conditions for a simply supported plate are

N11 D 0 M11 D 0 P11 D 0 u03 D 0

u02 D 0 rsquo2 iexcl u0

32 D 0 at x1 D 0 L1

N22 D 0 M22 D 0 P22 D 0 u03 D 0


31 D 0 at x2 D 0 L2 (23)

A solution to the partial differential equations (20) can be obtainedby choosing the displacements and rotations as

poundu0

1 rsquo1

currenD [S1 S2] cos rx1 sin sx2

poundu0

2 rsquo2

currenD [S3 S4] sin r x1 cos sx2

u03 D S5 sin rx1 sin sx2 (24)

The assumed elds (24) satisfy boundary conditions (23) identicallySubstitution from Eq (24) into Eq (21) and the result into Eq (20)yields ve simultaneous linear equations for the ve unknowns Sk which can be readily solved The transverse shear stresses frac3413 and frac3423

and the transverse normal stress frac3433 are obtained by integrating thethree-dimensional equilibrium equations in the thickness direction

V Effective Moduli of Two-Phase CompositesConsider a functionally graded composite material that is fabri-

cated by mixing two distinct material phases for example a metaland a ceramic Often precise information about the size shapeand distribution of the particles may not be available and the ef-fective moduli of the graded composite must be evaluated basedonly on the volume fraction distributions and the approximate shapeof the dispersed phase Several micromechanics models have beendeveloped over the years to infer the effective properties of macro-scopically homogeneous composite materials We summarize twopopular methods for estimating the effective properties the MorindashTanaka and the self-consistent methods and use them to analyzefunctionally graded materials

A MorindashTanaka EstimateThe MorindashTanaka (see Refs 15 17 and 23) scheme for esti-

mating the effective moduli is applicable to regions of the gradedmicrostructure that have a well-de ned continuous matrix and adiscontinuous particulate phase as shown in Fig 1a It takes into

a)

b)

Fig 1 Two-phase material with a) particulate microstructure and b)skeletal microstructure

VEL AND BATRA 1425

account the interaction of the elastic elds between neighboringinclusions It is assumed that the matrix phase denoted by the sub-script 1 is reinforced by spherical particles of a particulate phasedenoted by the subscript 2 In this notation K1 sup11 middot1 and reg1 arethe bulk modulus the shear modulus the thermal conductivity andthe thermal expansion coef cient respectively and V1 is the vol-ume fraction of the matrix phase K2 sup12 middot2 reg2 and V2 are thecorresponding material properties and the volume fraction of theparticulate phase Note that V1 C V2 D 1 that the Lame constant cedil isrelated to the bulk and the shear moduli by cedil D K iexcl 2sup1=3 and thatthe stress-temperature modulus is related to the modulus of thermalexpansion by macr D 3cedil C 2sup1reg D 3K reg The following estimates forthe effective local bulk modulus K and shear modulus sup1 are use-ful for a random distribution of isotropic particles in an isotropicmatrix

K iexcl K1

K2 iexcl K1D V2

1 C 1 iexcl V2

K2 iexcl K1

K1 Ciexcl

43

centsup11

sup1 iexcl sup11

sup12 iexcl sup11D V2

iquestmicro1 C 1 iexcl V2

sup12 iexcl sup11

sup11 C f1

para(25)

where f1 D sup119K1 C 8sup11=6K1 C 2sup11 The effective thermalconductivity middot is given by38

middot iexcl middot1

middot2 iexcl middot1D

V2

1 C 1 iexcl V2middot2 iexcl middot1=3middot1

(26)

and the coef cient of thermal expansion reg is determined from thecorrespondence relation39

reg iexcl reg1

reg2 iexcl reg1D

1=K iexcl 1=K1

1=K2 iexcl 1=K1

(27)

B Self-Consistent EstimateThe self-consistent method1518 assumes that each reinforcement

inclusion is embedded in a continuum material whose effective prop-erties are those of the composite This method does not distinguishbetween matrix and reinforcement phases and the same overallmoduli is predicted in another composite in which the roles of thephases are interchanged This makes it particularly suitable for de-termining the effective moduli in those regions that have an inter-connected skeletal microstructure as shown in Fig 1b The locallyeffective moduli by the self-consistent method are

plusmn=K D V1=K iexcl K2 C V2=K iexcl K1 (28a)

acute=sup1 D V1=sup1 iexcl sup12 C V2=sup1 iexcl sup11 (28b)

where plusmn D 3 iexcl 5acute D K=K C 4sup1=3 These are implicit expressionsfor the unknowns K and sup1 Equation (28a) can be solved for K interms of sup1 to obtain

K D 1=[V1=K1 C 4sup1=3 C V2=K2 C 4sup1=3] iexcl 4sup1=3 (29)

and sup1 is obtained by solving the following quartic equation

V1K1

K1 C 4sup1=3C

V2 K2

K2 C 4sup1=3C 5

sup3V1sup12

sup1 iexcl sup12C

V2sup11

sup1 iexcl sup11

acuteC 2 D 0

(30)

The self-consistent estimate of the thermal conductivity coef cient40

is in the implicit form

V1middot1 iexcl middot

middot1 C 2middotC


middot2 C 2middotD 0 (31)

The self-consistent estimate of reg is obtained by substitution of theself-consistent estimate of the bulk modulus K from Eq (29) intothe correspondence relation (27) Because the quartic equation (30)and the quadratic equation (31) have to be solved to obtain the shearmodulus sup1 and the thermal conductivity middot it is easier to use theMorindashTanaka method than the self-consistent scheme

VI Results and DiscussionHere we present exact results for a representative simply sup-

ported square plate with its top surface subjected to either amechanical load or a thermal load

[frac3433x1 x2 H=2 T x1 x2 H=2]

D [pC T C] sinfrac14 x1=L1 sinfrac14 x2=L2 (32)

The bottom surface is traction free and held at the reference tem-perature that is frac34i3x1 x2 iexclH=2 D 0 and T x1 x2 iexclH=2 D 0

Because it is common in high-temperature applications to em-ploy a ceramic top layer as a thermal barrier to a metallic structurewe choose the constituent materials of the functionally graded plateto be Al and SiC with the following material properties11 For AlEm D 70 GPa ordmm D 03 regm D 234 pound 10iexcl6K and middotm D 233 WmKFor SiC Ec D 427 GPa ordmc D 017 regc D 43 pound 10iexcl6K middotc D65 WmK Recall that the bulk modulus and the shear modu-lus are related to Youngrsquos modulus E and Poissonrsquos ratio ordm byK D E=31 iexcl 2ordm and sup1 D E=21 C ordm We assume that the vol-ume fraction of the ceramic phase is given by the power-law-typefunction

Vc D V iexclc C

iexclV C

c iexcl V iexclc

centiexcl12

C x3=Hcentn

(33)

Here V Cc and V iexcl

c are the volume fractions of the ceramic phase onthe top and the bottom surfaces of the plate respectively and n is aparameter that dictates the volume fraction pro le through the thick-ness Cheng and Batra10 have also used a similar power law functionfor the volume fraction except that they have assumed V C

c D 1 andV iexcl

c D 0 We performed a convergence study for the temperature anddisplacements All results reported here have less than 0001 trun-cation error and are computed by retaining 200 terms in the seriesexpansion for the temperature and displacements in Eqs (11) and(17) respectively

The physical quantities are nondimensionalized by relations

[ Nu1 Nu2] D100Em H 2[u1 u2]

pC L31

Nu3 D100Em H 3u3

pC L41

[ Nfrac3411 Nfrac3422 Nfrac3412] D10H 2[frac3411 frac3422 frac3412]

pC L21

[ Nfrac3413 Nfrac3423] D10H [frac3413 frac3423]

pC L1 Nfrac3433 D

frac3433

pC(34)

for the applied mechanical load and by

OT DT

T C Oq3 D iexclq3 H

middotm T C [ Ou1 Ou2] D10[u1 u2]regm T C L1

Ou3 D100Hu3

regm T C L21

[ Ofrac3411 Ofrac3422 Ofrac3412] D10[frac3411 frac3422 frac3412]

Emregm T C

[ Ofrac3413 Ofrac3423] D100L1[frac3413 frac3423]

Em regm T C H Ofrac3433 D

100L21frac3433

Em regm T C H 2(35)

for the thermal load Nondimensional quantities for the mechanicalload are denoted by a superimposed bar and those for the thermalload by a superimposed caret

Consider a simply supported metalndashceramic square plateL2 D L1 with the metal (Al) taken as the matrix phase and theceramic (SiC) taken as the particulate phase That is P1 D Pm andP2 D Pc where P is either the volume fraction V or any materialproperty The exact solution for displacements and stresses at spe-ci c points in the plate is compared with the CPT the FSDT and theTSDT results in Fig 2 for length-to-thickness ratio L1=H rangingfrom 2 to 40 The effective material properties are obtained by theMorindashTanaka scheme The transverse de ection Nu3L1=2 L2=2 0of the plate centroid for the mechanical load predicted by the FSDTand the TSDT is in excellent agreement with the exact solutioneven for thick plates with L1=H lt 10 but the CPT solution exhibitssigni cant error for thick plates The error in the CPT value for the

1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)

Fig 2 Transverse de ection longitudinal stress and transverse shear stress vs length-to-thickness ratio for AlSiC functionally graded square platecomputed with MorindashTanaka homogenization scheme V iexcl

c = 0 V+c = 05 and n = 2 andashc) mechanical load and dndashf) thermal load

a)

b)

c)

d)

e)

f)

Fig 3 Transverse de ection longitudinal stress and transverse shear stress vs power law index n for AlSiC functionally graded square platecomputed with MorindashTanaka homogenization scheme V iexcl

c = 0 V+c = 10 and L1 H = 5 andashc) mechanical load and dndashf) thermal load

transverse de ection can be attributed to the large shear deformationthat occurs in thick plates The CPT and the FSDT give identicalvalues of the longitudinal stress Nfrac3411 and of the transverse shearstress Nfrac3413 which deviate from the exact solution as the length-to-thickness ratio decreases The TSDT gives accurate results for Nfrac3411

and Nfrac3413 even for thick plates When the plate is subjected to the ther-mal load both the longitudinal stress Ofrac3411 and the transverse shearstress Ofrac3413 given by each one of the three plate theories are inaccuratefor thick plates with L1=H lt 10 This could be due to the neglect

of transverse normal strains and the transverse normal stresses inthe plate theories Furthermore the displacement elds in the platetheories are constructed assuming isothermal conditions and needto be modi ed when thermal gradients are present as is the casein the present problem Note that the transverse shear and normalstresses have been computed by integrating the three-dimensionalelasticity equations in the thickness direction

Figure 3 shows plots of the nondimensional values of the trans-verse displacement u3 of the plate centroid the longitudinal stress

VEL AND BATRA 1427

a)

b)

Fig 4 Temperature at the center and heat ux on the bottom surface ofAlSiC functionally graded square plate vs volume fraction of ceramicon the top surface for power law index n = 1 2 and 5 computed withMorindashTanaka homogenization scheme V iexcl

c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)

Fig 5 Transverse de ection longitudinal stress and transverse shear stress vs ceramic volume fraction on the top surface of AlSiC functionallygraded square plate computed with MorindashTanaka homogenization scheme V iexcl

c = 0 n = 2 and L1H = 5 andashc) mechanical load and dndashf) thermal load

frac3411 and the transverse shear stress frac3413 vs the power law index nfor a thick square plate L1=H D 5 The effective properties areobtained by the MorindashTanaka scheme and the volume fraction Vc

of the ceramic phase is taken to be 0 and 1 on the bottom and thetop surfaces respectively It is seen that the differences betweenthe exact values and the CPT the FSDT and the TSDT values ofthese quantities for the thermal load do not change appreciably forincreasing values of the power law index n However for the me-chanical load applied on the top surface of the plate the errors in thetransverse de ection of the plate centroid and the stresses computedwith each one of the three plate theories increase with an increasein the value of n Whereas the CPT and the FSDT underpredict theTSDT overpredicts the transverse displacement at the plate centroidand the longitudinal stress at the centroid of the top surface For thethermal load all three plate theories underpredict this longitudinalstress For the mechanical load the transverse shear stress at theplate centroid computed from the TSDT solution is lower than thatobtained from the exact solution but the CPT and the FSDT givea value higher than that obtained from the analytical solution Forthe thermal load all three plate theories yield essentially the samevalue of the transverse shear stress at the point L1=2 L2=2 H=4and this value is lower than the analytical one

The change in temperature OT L1=2 L2=2 0 at the plate centroidvs the ceramic volume fraction V C

c on the top surface for the ther-mal load is shown in Fig 4a It is clear that the temperature changedecreases monotonically with an increase in the value of V C

c Thenondimensional transverse component of the heat ux on the bot-tom surface shown in Fig 4b decreases as the ceramic volumefraction on the top surface increases because the thermal conduc-tivity of the ceramic phase is much smaller than that of the metallicphase Figure 5 shows plots of the transverse displacements of theplate centroid and components of the stress at speci c points inthe plate vs the ceramic volume fraction on the top surface As isevident from Figs 5andash5f the percentage errors in the transversedisplacement of the plate centroid and stresses at the chosen pointsobtained from the plate theories do not change appreciably with thevalue of the ceramic volume fraction on the top surface For both the

1428 VEL AND BATRA

Table 1 Exact displacements stresses temperature and heat ux at speci c locations for the AlSiC functionally graded square plate whensubjected to mechanical and temperature loads MorindashTanaka scheme

V iexclc D 0 V C

c D 05 n D 2 V iexclc D 0 n D 2 L1=H D 5

Variable L1=H D 5 L1=H D 10 L1=H D 40 V Cc D 02 V C

c D 06 V Cc D 1

Nu10 L2=2 H=2 iexcl29129 iexcl28997 iexcl28984 iexcl36982 iexcl26708 iexcl17421Nu3L1=2 L2=2 0 25748 22266 21163 30254 24326 18699Nu3L1=2 L2=2 H=2 25559 22148 21155 29852 24196 18767Nfrac3411L1=2 L2=2 H=2 27562 26424 26093 23285 29359 41042Nfrac34120 0 H=2 iexcl15600 iexcl15529 iexcl15522 iexcl12163 iexcl17106 iexcl28534Nfrac34130 L2=2 0 23100 23239 23281 23516 22918 21805Nfrac3433L1=2 L2=2 H=4 08100 08123 08129 08300 08024 07623OT L1=2 L2=2 0 03938 04240 04343 04315 03800 03162Oq3L1=2 L2=2 iexclH=2 07316 08075 08335 08219 07001 05625Ou10 L2=2 H=2 iexcl12101 iexcl12124 iexcl12131 iexcl16989 iexcl10624 iexcl05326Ou3L1=2 L2=2 0 32497 33312 33567 50302 27110 07741Ou3L1=2 L2=2 H=2 44111 36337 33758 65843 37561 14115Ofrac3411L1=2 L2=2 H=2 iexcl41764 iexcl41555 iexcl41492 iexcl47912 iexcl38561 iexcl12077Ofrac34120 0 H=2 iexcl64804 iexcl64928 iexcl64966 iexcl55878 iexcl68047 iexcl87240Ofrac34130 L2=2 H=4 42264 44703 45501 17390 50227 79552Ofrac3433L1=2 L2=2 0 iexcl86829 iexcl91622 iexcl93191 iexcl35531 iexcl103675 iexcl173524

a)

b)

c)

d)

e)

f)

Fig 6 Through-the-thickness variation of the transverse de ection and stresses in AlSiC functionally graded square plate computed with MorindashTanaka homogenization scheme V iexcl

c = 0 V+c = 075 L1H = 5 and n = 2 andashc) mechanical load and dndashf) thermal load

mechanical and the thermal loads the transverse de ection of theplate centroid decreases nearly af nely with an increase in the valueof V C

c However the longitudinal stress at the centroid of the topsurface increases parabolically with a rise in the value of V C

c Withan increase in the value of V C

c the transverse shear stress decreasesparabolically for the mechanical load and it increases essentiallylinearly for the thermal load Note that the transverse shear stressesare evaluated at different points for the two loads The through-the-thickness variations of the displacements and longitudinal stresses atpoints on the centroidal axis and the shear stress variation at an edgeare shown in Fig 6 for a thick plate (L1=H D 5) When subjectedto the mechanical load the TSDT overestimates the transverse de- ection Nu3 at all points within the plate the CPT underestimatesit at all points and the FSDT value of Nu3 is close to the averagevalue given by the exact solution It is clear that the thickness of theplate changes and that the transverse normal strain is not uniformthrough the plate thickness The longitudinal stress Nfrac3411 does notvary as a linear function of the thickness coordinate x3 because the

plate is functionally graded and the material properties are functionsof x3 The transverse shear stress Nfrac3413 attains its maximum value atx3 D 011H Note that all three plate theories give very good val-ues of the longitudinal stress and the transverse shear stress Whenthe functionally graded thick plate with L1=H D 5 is subjected tothe temperature load the error exhibited by both the CPT and theFSDT values for the transverse de ection Ou3 on the top surface is26 The corresponding error in the TSDT value for Ou3 is 28The large errors are due to the assumption of inextensibility of thenormals to the midsurface inherent in all three plate theories consid-ered here The extensibility of transverse normals has been incor-porated in plate theories proposed by Vidoli and Batra41 Batra andVidoli42 Soldatos and Watson43 Kant44 Lee and Yu20 Lee et al21

and others Simmonds45 showed that accurate three-dimensionalstresses can be derived from the CPT by including the transverse ex-tensibility effects The transverse centroidal displacements and thestresses at the centroid of the top surface using the self-consistentscheme vs the length-to-thickness ratio are plotted in Fig 7 for

VEL AND BATRA 1429

Table 2 Exact displacements stresses temperature and heat ux at speci c locations for the AlSiC functionally graded square plate whensubjected to mechanical and temperature loads self-consistent scheme

V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)

Fig 7 Transverse de ection and longitudinal stress vs length-to-thickness ratio for AlSiC functionally graded square plate obtained with self-consistent homogenization scheme V iexcl


V iexclc D 02 V C

c D 08 and n D 2 The plots are qualitatively similarto those obtained by using the MorindashTanaka scheme and shown inFig 2

We have listed in Tables 1 and 2 values of displacements stressestemperature and heat ux at speci c points in a simply supportedAlSiC functionally graded plate for the two loads considered Val-ues listed in Table 1 are obtained by using the MorindashTanaka homog-enization scheme and those in Table 2 by the self-consistent methodThese should facilitate comparisons between the exact values andthose obtained from plate theories or other approximate methodssuch as the nite element method To capture well the through-the-thickness variation of material properties a very ne mesh in thethickness direction and hence throughout the domain will be re-quired Results presented herein should help decide the degree of neness of the mesh andor the choice of elements

In the examples studied so far we used a single homoge-nization scheme (either the MorindashTanaka method or the self-consistent method) to estimate the effective properties for the entireplate This approach is appropriate only for functionally gradedplates that have the same microstructure everywhere Reiter andDvorak16 performed detailed nite element studies of the responseof simulated discrete models containing both skeletal and par-ticulate microstructures and concluded that homogenized modelsof combined microstructures that employ only a single averag-ing method do not provide reliable agreements with the discretemodel predictions However close agreement with the discretemodel was shown by homogenized models that employ differ-ent effective property estimates for regions of the plate that havedifferent microstructures They state that in those parts of thegraded microstructure that have a well-de ned continuous matrix

1430 VEL AND BATRA

a)

b)

c)

d)

Fig 8 Effective properties for a) Youngrsquos modulus b) shear modulus c) coef cient of thermal expansion and d) thermal conductivity as a functionof the thickness coordinate x3 obtained by various homogenization schemes

a)

b)

c)

d)

Fig 9 Through-the-thickness variation of the transverse de ection and stresses in AlSiC functionally graded square plate obtained by varioushomogenization schemes mechanical load L1H = 5

VEL AND BATRA 1431

and discontinuous reinforcement the effective properties shouldbe approximated by the appropriate MorindashTanaka estimates andthat in skeletal microstructure that often form transition zonesbetween clearly de ned matrix and particulate phases the ef-fective properties should be approximated by the self-consistentmethod

We consider a functionally graded plate that has an af ne vari-ation of the ceramic volume fraction given by Vc D 1

2C x3=H

It is assumed to have a well-de ned continuous metallic ma-trix with discontinuous ceramic particles in the metal rich regioniexcl05H middot x3 middot iexcl02H adjacent to the bottom surface and a well-de ned continuous ceramic matrix with discontinuous metallic par-ticles in the ceramic rich region 02H middot x3 middot 05H adjacent to thetop surface The plate is assumed to have a skeletal microstruc-ture in the central region iexcl02H middot x3 middot 02H We use a combinedmodel wherein the effective properties in the metal rich regionadjacent to the bottom surface are obtained by the MorindashTanakascheme with a metallic matrix phase (MTM) the effective prop-erties in the ceramic rich region adjacent to the top surface areobtained by the MorindashTanaka scheme with a ceramic matrix phase(MTC) and the effective material properties in the central regionare obtained by the self-consistent scheme (SC) To accommodatethe discontinuities in homogenized material properties predicted atthe boundaries between the different regions we employ the third-order transition functions used by Reiter and Dvorak16 in transitionregions of width 005H centered at x3 D iexcl02H and 02H Thethrough-the-thickness variations of the effective Youngrsquos modulusshear modulus coef cient of thermal expansion and thermal con-ductivity obtained by the combined homogenization technique areplotted in Fig 8 In Fig 8 MTM MTC and SC signify effectiveproperties obtained by using a single averaging method through theentire thickness of the plate

In the combined model there are ve distinct regions in the thick-ness direction namely the three primary regions in which the ef-

a)

b)

c)

d)

Fig 10 Through-the-thickness variation of the transverse de ection and stresses in AlSiC functionally graded square plate obtained by varioushomogenization schemes thermal load L1H = 5

fective properties are obtained by MTM MTC and SC and thetwo transition regions at the boundaries between them Within eachregion the effective material properties are expanded as a Taylor se-ries and the solution to the thermal and the mechanical equilibriumequations are obtained in terms of 8 unknown constants resulting ina total of 40 unknowns The constants are determined by satisfyingthe mechanical and the thermal boundary conditions (5) and (6) onthe top and the bottom surfaces of the plate and the interface con-tinuity conditions (7) for the thermal and the mechanical quantitiesbetween adjoining layers This results in eight conditions for thetop and the bottom surfaces and eight conditions at each of the fourinterfaces between regions with distinct micromechanical modelsThe resulting system of 40 linear algebraic equations for the 40 un-knowns are solved to obtain the displacements and stresses for theentire plate

A comparison of the through-the-thickness variation of the de ec-tion and stresses obtained by the combined model and the two singleaveraging methods namely MTM and MTC are shown in Fig 9 fora thick plate L1=H D 5 subjected to the mechanical load It is clearfrom Fig 9a that there are signi cant differences in the values ofthe transverse de ection Nu3 given by the three homogenization tech-niques However the longitudinal stress Nfrac3411 the transverse shearstress Nfrac3413 and the transverse normal stress Nfrac3433 predicted by the threedifferent homogenization methods are essentially the same Thecorresponding displacements and stresses for the thermal load areshown in Fig 10 The combined model gives a smaller value for thetransverse de ection Ou3 than either the MTM or the MTC methodThe magnitude of the longitudinal stress Ofrac3411 from the MTM methodreaches a maximum value at a point situated slightly below the topsurface However the maximum value of Ofrac3411 is predicted to occur onthe top surface by the MTC and the combined model Unlike the re-sults for the mechanical load the MTM the MTC and the combinedmodel all give signi cantly different values for the transverse shearstress Ofrac3413 and the transverse normal stress Ofrac3433 for the thermal load

1432 VEL AND BATRA

VII ConclusionsWe have analyzed thermomechanical deformations of a simply

supported functionally graded AlSiC plate subjected to either a si-nusoidalpressure or a sinusoidal temperature eld on the top surfaceThe effective properties at points in the plate are obtained either bythe MorindashTanaka method or the self-consistent scheme or a com-bination of the two The volume fractions of the constituents andhence the effective material properties are assumed to vary in thethickness direction only The effective material properties the threecomponents of the displacement and the temperature eld are ex-panded in Taylor series in the thickness coordinate When in-planesinusoidal variations of the displacements and the temperature thatidentically satisfy boundary conditions at the edges are presumedordinary differential equations for functions giving their through-the-thickness variation are derived and solved analytically

The exact solutions of the problems studied here are comparedwith those obtained by the classical plate theory the FSDT andthe TSDT For thick functionally graded plates there are signi -cant differences between the exact solution and that obtained withany one of these three plate theories even when the transverse nor-mal and the transverse shear stresses are computed by integratingthe three-dimensional elasticity equations These differences couldbe due to displacement elds employed in plate theories that wereoriginally proposed for studying isothermal deformations of a plateIt is found that the displacements stresses and temperatures com-puted with either the MorindashTanaka scheme or the self-consistentmethod or their combination agree qualitatively but differ quanti-tatively For the thermal load the differences in the transverse dis-placements the transverse shear stresses and the transverse normalstresses are noticeable but those in the longitudinal stress are neg-ligible except near the top loaded surface For the mechanical loadonly the transverse displacements computed with the three methodsexhibit signi cant variations For each load the nonzero transversenormal strains vary through the plate thickness

The exact solutions presented here provide benchmark resultswhich can be used to assess the adequacy of different plate theoriesand also to compare results obtained by other approximate methodssuch as the nite element method

References1Berger R Kwon P and Dharan C K H ldquoHigh Speed Centrifu-

gal Casting of Metal Matrix Compositesrdquo 5th International Symposiumon Transport Phenomena and Dynamics of Rotating Machinery Maui HI1994

2Fukui Y ldquoFundamental Investigation of Functionally Gradient Mate-rials Manufacturing System Using Centrifugal Forcerdquo JSME InternationalJournal Series III Vol 34 No 1 1991 pp 144ndash148

3Choy K-L and Felix E ldquoFunctionally Graded Diamond-Like CarbonCoatings on Metallic Substratesrdquo Materials Science and Engineering AVol 278 No 1 2000 pp 162ndash169

4Khor K A and Gu Y W ldquoEffects of Residual Stress on the Per-formance of Plasma Sprayed Functionally Graded ZrO2NiCoCrAlY Coat-ingsrdquo Materials ScienceandEngineering A Vol 277 No 1ndash2 2000 pp 64ndash

765Lambros A Narayanaswamy A Santare M H and Anlas G ldquoMan-

ufacturing and Testing of a Functionally Graded Materialrdquo Journal of En-gineering Materials and Technology Vol 121 No 2 1999 pp 488ndash493

6Breval E Aghajanian K and Luszcz S J ldquoMicrostructure and Com-positionof AluminaAluminum Composites Made by the Directed Oxidationof Aluminumrdquo Journal of the American Ceramic Society Vol 73 No 91990 pp 2610ndash2614

7Manor E Ni H Levi C G and Mehrabian R ldquoMicrostructure Eval-uation of SiCAl203Al Alloy Composite Produced by Melt OxidationrdquoJournal of the American Ceramic Society Vol 26 No 7 1993 pp 1777ndash

17878Rogers T G Watson P and Spencer A J M ldquoExact Three-

Dimensional Elasticity Solutions for Bending of Moderately Thick Inho-mogeneous and Laminated Strips Under Normal Pressurerdquo InternationalJournal of Solids and Structures Vol 32 No 12 1995 pp 1659ndash1673

9Tarn J Q and Wang Y M ldquoAsymptotic Thermoelastic Analysis ofAnisotropic Inhomogeneou s and Laminated Platesrdquo Journal of ThermalStresses Vol 18 No 1 1995pp 35ndash58

10Cheng Z Q and Batra R C ldquoThree-Dimensiona l ThermoelasticDeformations of a Functionally Graded Elliptic Platerdquo Composites Part BVol 31 No 2 2000 pp 97ndash106

11Tanaka K Tanaka Y Enomoto K Poterasu V F and Sugano YldquoDesign of Thermoelastic Materials Using Direct Sensitivity and Optimiza-tion Methods Reduction of Thermal Stresses in Functionally Gradient Mate-rialsrdquo Computer Methods in Applied Mechanics and Engineering Vol 106No 1ndash2 1993 pp 271ndash284

12Tanaka K Tanaka Y Watanabe H Poterasu V F and SuganoY ldquoAn Improved Solution to Thermoelastic Material Design in Function-ally Gradient Materials Scheme to Reduce Thermal Stressesrdquo ComputerMethods in Applied Mechanics and Engineering Vol 109 No 3ndash4 1993pp 377ndash389

13Reddy J N ldquoAnalysis of Functionally Graded Platesrdquo InternationalJournal for Numerical Methods in Engineering Vol 47 No 1ndash3 2000 pp663ndash684

14Reiter T Dvorak G J and Tvergaard V ldquoMicromechanical Modelsfor Graded Composite Materialsrdquo Journal of the Mechanics and Physics ofSolids Vol 45 No 8 1997 pp 1281ndash1302

15Reiter T and Dvorak G J ldquoMicromechanical Modelling of Function-ally Graded Materialsrdquo IUTAM Symposium on Transformation Problems inComposite and Active Materials editedby Y Bahei-El-Din and G J DvorakKluwer Academic London 1997 pp 173ndash184

16Reiter T and Dvorak G J ldquoMicromechanical Models for GradedComposite Materials II Thermomechanica l Loadingrdquo Journal of theMechanics and Physics of Solids Vol 46 No 9 1998 pp 1655ndash1673

17Mori T and Tanaka K ldquoAverage Stress in Matrix and Average ElasticEnergy of Materials with Mis tting Inclusionsrdquo Acta Metallurgica Vol 211973 pp 571ndash574

18Hill R ldquoA Self-Consistent Mechanics of Composite Materialsrdquo Jour-nal of the Mechanics andPhysics of Solids Vol 13 No 2 1965 pp 213ndash222

19Cheng Z Q and Batra R C ldquoDe ection Relationships Between theHomogeneou s Kirchhoff Plate Theory and Different Functionally GradedPlate Theoriesrdquo Archives of Mechanics Vol 52 No 1 2000 pp 143ndash158

20Lee P C Y and Yu J-D ldquoGoverning Equations for a PiezoelectricPlate with Graded Properties Across the Thicknessrdquo IEEE Tansactions onUltrasonics Ferroelectrics and Frequency Control Vol 45 No 1 1998pp 236ndash250

21Lee P C Y Yu J-D and Shih W-H ldquoPiezoelectric Ceramic Diskswith Thickness Graded Material Propertiesrdquo IEEE Tansactions on Ultrason-ics Ferroelectrics and Frequency Control Vol 46 No 1 1999 pp 205ndash

21522Batra R C ldquoFinite Plane Strain Deformations of Rubberlike Materi-

alsrdquo International Journal for Numerical Methods in Engineering Vol 15No 1 1980 pp 145ndash160

23Benveniste Y ldquoA New Approach to the Application of MorindashTanakarsquosTheory of Composite Materialsrdquo Mechanics of Materials Vol 6 1987pp 147ndash157

24Jones R M Mechanics of Composite Materials Scripta WashingtonDC 1975

25Hyer M W Stress Analysis of Fiber-Reinforced Composite MaterialsMcGrawndashHill Higher Education New York 1998

26Whitney J M andPagano N J ldquoShearDeformation inHeterogeneousAnisotropic Platesrdquo Journal of Applied Mechanics Vol 37 No 4 1970pp 1031ndash1036

27Reddy J N Mechanics of Laminated Composite Plates Theory andAnalysis CRC Press Boca Raton FL 1997

28Reddy J N ldquoA Simple Higher-Order Theory for Laminated CompositePlatesrdquo Journal of Applied Mechanics Vol 51 No 4 1984 pp 745ndash752

29Carlson D E ldquoLinear Thermoelasticityrdquo Handbuch der Physik VIa2Springer Berlin 1972

30Vlasov B F ldquoOn One Case of Bending of Rectangular ThickPlatesrdquo Vestnik Moskovskogo Universiteta Serieiotildea Matematici MekhanikiAstronomii Fiziki Khimii Vol 2 No 2 1957 pp 25ndash34

31Srinivas S and Rao A K ldquoA Note on Flexure of Thick RectangularPlates and Laminates with Variation of Temperature Across ThicknessrdquoBulletin of the Polish Academy of Science Technical Sciences Vol 20 No 21972 pp 229ndash234

32Srinivas S and Rao A K ldquoBuckling of Thick Rectangular PlatesrdquoAIAA Journal Vol 7 No 8 1969 pp 1645 1646

33Srinivas S and Rao A K ldquoBending Vibration and Buckling of Sim-ply Supported Thick Orthotropic Rectangular Plates and Laminatesrdquo Inter-national Journal of Solids and Structures Vol 6 No 10 1970 pp 1463ndash

148134Pagano N J ldquoIn uence of Shear Coupling in Cylindrical Bending

of Anisotropic Laminatesrdquo Journal of Composite Materials Vol 4 No 31970 pp 330ndash343

35Vel S S and Batra R C ldquoAnalytical Solution for Rectangular ThickLaminated Plates Subjected to Arbitrary Boundary Conditionsrdquo AIAA Jour-nal Vol 37 No 11 1999 pp 1464ndash1473

36Vel S S and Batra R C ldquoThe Generalized Plane Strain Deformationsof Anisotropic Composite Laminated Thick Platesrdquo International Journalof Solids and Structures Vol 37 No 5 2000 pp 715ndash733

VEL AND BATRA 1433

37Vel S S and Batra R C ldquoGeneralized Plane Strain ThermoelasticDeformation of Laminated Anisotropic Thick Platesrdquo International Journalof Solids and Structures Vol 38 No 8 2001 pp 1395ndash1414

38Hatta H and Taya M ldquoEffective Thermal Conductivity of a Misori-ented Short Fiber Compositerdquo Journal of Applied Physics Vol 58 No 71985 pp 2478ndash2486

39Rosen B W and Hashin Z ldquoEffective Thermal Expansion Coef -cients and Speci c Heats of Composite Materialsrdquo International Journal ofEngineering Science Vol 8 1970 pp 157ndash173

40Hashin Z ldquoAssessment of the Self Consistent Scheme ApproximationConductivity of Particulate Compositesrdquo Journal of Composite MaterialsVol 2 1968 pp 284ndash300

41Vidoli S and Batra R C ldquoDerivation of Plate and Rod Equations for aPiezoelectric Body from a Mixed Three-Dimensiona l Variational PrinciplerdquoJournal of Elasticity Vol 59 Nos 13 2000 pp 23ndash50

42Batra R C and Vidoli S ldquoHigher-Order Piezoelectric Plate TheoryDerived from a Three-Dimensiona l Variational Principlerdquo AIAA JournalVol 40 No 1 2002 pp 91ndash104

43Soldatos K P and Watson P ldquoAccurate Stress Analysis of Lami-nated Plates Combining Two-Dimensional Theory with the Exact Three-Dimensional Solution for Simply-Supported Edgesrdquo Mathematics and Me-chanics of Solids Vol 2 No 4 1997 pp 459ndash489

44Kant T ldquoNumerical Analysis of Thick Platesrdquo Computational Meth-ods in Applied Mechanics and Engineering Vol 31 No 1 1982 pp 1ndash18

45Simmonds J G ldquoAn Improved Estimate for the Error in the ClassicalLinear Theory of Plate Bendingrdquo Quarterly of Applied Mathematics Vol 29No 3 1971 pp 439ndash447

A M WaasAssociate Editor

1422 VEL AND BATRA

of suitable temperature and displacement functions the governingpartial differential equations are reduced to a set of coupled ordinarydifferential equations in the thickness coordinate which are solvedby the power series method We consider a two-phase graded mate-rial with a power law variation of the volume fractions of the con-stituents through the thickness The effective material properties ata point are determined in terms of the local volume fractions and thematerial properties of the two phases either by the MorindashTanaka17

(also see Benveniste23 ) or by the self-consistent18 scheme Resultsare presented for an AlSiC graded rectangular plate Displacementsand stresses at critical locations for mechanical and thermal loadsare given for different length-to-thickness ratios exponents in thepower law through-the-thickness variation of the constituents anddifferent homogenization schemes We compare the exact resultswith those obtained from the classical plate theory2425 (CPT) theFSDT2627 and the TSDT2728 The results from the three plate theo-ries are quite different from the exact solution for thick functionallygraded plates

Results are also computed for a functionally graded plate forwhich the effective material properties in the ceramic rich and themetal rich regions are derived by the MorindashTanaka17 method and theeffective properties elsewhere are obtained by the self-consistent18

scheme A third-order transition function of Reiter and Dvorak16 isused in the transition regions to have a smooth variation of materialproperties between the regions For the thermal load this combinedmethod of homogenizing material properties gives plate deforma-tions and transverse normal and transverse shear stresses that differsigni cantly from those computed with either the MorindashTanaka orthe self-consistent scheme alone However through-the-thicknessvariation of the longitudinal stress given by the three homogeniza-tion methods is essentially the same except at points near the topsurface of the plate where the volume fraction of the ceramic is highFor the mechanical load the homogenization technique in uencesstrongly the transverse displacements but not the values of the stresscomponents

II Formulation of the ProblemWe use rectangular Cartesian coordinates xi i D 1 2 3 to de-

scribe the in nitesimal static thermoelastic deformations of anN -layer laminated plate occupying the region [0 L1] pound [0 L2] pound[iexclH=2 H=2] in the unstressed reference con guration Each layerof the laminated plate is made of an isotropic material with ma-terial properties varying smoothly in the x3 (thickness) direc-tion only The vertical positions of the bottom and the top sur-faces and the N iexcl 1 interfaces between the layers are denoted byH 1 D iexclH=2 H 2 H n H N and H N C 1 D H=2

The equations of mechanical and thermal equilibrium in theabsence of body forces and internal heat sources are (eg seeCarlson29)

frac34i j j D 0 (1a)

q j j D 0 (1b)

where frac34i j and q j are respectively the components of the Cauchystress tensor and the heat ux vector and where i j D 1 2 3 Acomma followed by index j denotes partial differentiation with re-spect to the position x j of a material particle and a repeated indeximplies summation over the range of the index

The constitutive equations for a linear isotropic thermoelasticmaterial are29

frac34i j D cedilkk plusmni j C 2sup1i j iexcl macrplusmni j T (2a)

q j D iexclmiddotT j (2b)

where cedil and sup1 are the Lame constants macr is the stress-temperaturemodulus middot is the thermal conductivity i j are components of thein nitesimal strain tensor T is the change in temperature of a mate-rial particle from that in the stress-free reference con guration andplusmni j is the Kronecker delta The material properties cedil sup1 macr and middot arefunctions of x3

The in nitesimal strain tensor is related to the mechanicaldisplacements u i by

i j D 12ui j C u ji (3)

The edges of the plate are assumed to be simply supported andmaintained at the reference temperature That is

frac3411 D 0 (4a)

u2 D u3 D 0 (4b)

T D 0 (4c)

at x1 D 0 and L1 and

frac3422 D 0 (4d)

u1 D u3 D 0 (4e)

T D 0 (4f)

at x2 D 0 and L2 The mechanicalboundary conditions prescribed on the top and the

bottom surfaces can be either a displacement component u j or thecorresponding traction component frac343 j However typically nonzeronormal and zero tangential tractions are prescribed on these twosurfaces Because the normal load can be expanded as a doubleFourier series in x1 and x2 it suf ces to consider loads of the form

frac3413 D frac3423 D 0 at x3 D sectH=2

frac3433x1 x2 sectH=2 D psect sin r x1 sin sx2 (5)

where pC and piexcl are known constants r D kfrac14=L1 s D mfrac14=L2 andk and m are positive integers The thermal boundary conditions onthe top and the bottom surfaces are speci ed as

sect T x1 x2 sectH=2 C raquosect q3x1 x2 sectH=2

D rsquosect sin rx1 sin sx2 (6)

When values of constants sect and raquosect are chosen appropriatelyvarious boundary conditions corresponding to either a prescribedtemperature a prescribed heat ux or an exposure to an ambienttemperature through a boundary conductance can be speci ed

The interfaces between adjoining layers are assumed to be per-fectly bonded together and in ideal thermal contact so that

[[u i ]] D 0 (7a)

[[frac34i3]] D 0 (7b)

[[T ]] D 0 (7c)

[[q3]] D 0 (7d)

on x3 D H 2 H 3 H N Here [[u i ]] is the jump in the value ofu i across an interface

The mechanical and the thermal problems are one-way coupledin the sense that the temperature eld is determined rst by solvingEqs (1b) and (2b) and the pertinent boundary conditions and thedisplacements are later obtained from Eqs (1a) and (2a) and therelevant boundary conditions

III Exact SolutionWe construct a local rectangular Cartesian coordinate system

x n

1 x n

2 and xn

3 with local axes parallel to the global axes andthe origin at the point where the global x3 axis intersects the mid-surface of the nth lamina In the local coordinate system the nth lam-ina occupies the region [0 L1] pound [0 L2] pound [iexclhn=2 hn=2] wherehn D H n C 1 iexcl H n We drop the superscript n for conveniencewith the understanding that all material constants and variablesbelong to this layer

We assume that within each layer the Lame constants cedil and sup1the stress-temperature modulus macr and the thermal conductivity middot

VEL AND BATRA 1423



reg D 0


currenxreg

3 (8)








microx3 D1X

deg D 0

Qmicro deg xdeg

3 (11)


regX

deg D 0

Qmiddot deg pound





Qmiddot 1macr

2 Qmiddot 0curren

Qmicro 1 (13)


T Dpound





the form






iexclU1s

2 C U2rscent

iexcl cedilU 03r




centiexcl cedilU 0

3s



sup1U 01r C U 0


2s




Ui x3 D1X

deg D 0

QU deg

i xdeg

3 (17)


regX

deg D 0




2 r s

C Qsup1deg



2 rssup2


3 r




3 rsup2




3 rsup2


regX

deg D 0




1 rs

C Qsup1deg



1 rssup2


3 s




3 ssup2




3 ssup2


regX

deg D 0




2 ssup2





2 ssup2




2 ssup2



3



3



1 QU reg C 2

2 and QU reg C 2


1 QU 1

1 QU 0

2 QU 1

2 QU 0

3 and QU 1


1424 VEL AND BATRA








u3xi D u03 (19b)

where u0deg u0







where


iexclH=2


Rdeg DZ H=2

iexclH=2

frac34deg 3g3 dx3









N11 D 0 M11 D 0 P11 D 0 u03 D 0


32 D 0 at x1 D 0 L1

N22 D 0 M22 D 0 P22 D 0 u03 D 0


31 D 0 at x2 D 0 L2 (23)


poundu0

1 rsquo1


poundu0

2 rsquo2









a)

b)


VEL AND BATRA 1425


K iexcl K1

K2 iexcl K1D V2

1 C 1 iexcl V2

K2 iexcl K1

K1 Ciexcl

43

centsup11

sup1 iexcl sup11



sup12 iexcl sup11

sup11 C f1

para(25)




V2


(26)


reg iexcl reg1

reg2 iexcl reg1D

1=K iexcl 1=K1

1=K2 iexcl 1=K1

(27)








V1K1

K1 C 4sup1=3C

V2 K2

K2 C 4sup1=3C 5

sup3V1sup12

sup1 iexcl sup12C

V2sup11

sup1 iexcl sup11

acuteC 2 D 0

(30)




middot1 C 2middotC






[frac3433x1 x2 H=2 T x1 x2 H=2]




Vc D V iexclc C

iexclV C

c iexcl V iexclc

centiexcl12

C x3=Hcentn

(33)



c D 1 andV iexcl



[ Nu1 Nu2] D100Em H 2[u1 u2]

pC L31

Nu3 D100Em H 3u3

pC L41


pC L21


pC L1 Nfrac3433 D

frac3433

pC(34)


OT DT

T C Oq3 D iexclq3 H


Ou3 D100Hu3

regm T C L21


Emregm T C



100L21frac3433

Em regm T C H 2(35)



1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)



a)

b)

c)

d)

e)

f)







VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1423



reg D 0


currenxreg

3 (8)








microx3 D1X

deg D 0

Qmicro deg xdeg

3 (11)


regX

deg D 0

Qmiddot deg pound





Qmiddot 1macr

2 Qmiddot 0curren

Qmicro 1 (13)


T Dpound





the form






iexclU1s

2 C U2rscent

iexcl cedilU 03r




centiexcl cedilU 0

3s



sup1U 01r C U 0


2s




Ui x3 D1X

deg D 0

QU deg

i xdeg

3 (17)


regX

deg D 0




2 r s

C Qsup1deg



2 rssup2


3 r




3 rsup2




3 rsup2


regX

deg D 0




1 rs

C Qsup1deg



1 rssup2


3 s




3 ssup2




3 ssup2


regX

deg D 0




2 ssup2





2 ssup2




2 ssup2



3



3



1 QU reg C 2

2 and QU reg C 2


1 QU 1

1 QU 0

2 QU 1

2 QU 0

3 and QU 1


1424 VEL AND BATRA








u3xi D u03 (19b)

where u0deg u0







where


iexclH=2


Rdeg DZ H=2

iexclH=2

frac34deg 3g3 dx3









N11 D 0 M11 D 0 P11 D 0 u03 D 0


32 D 0 at x1 D 0 L1

N22 D 0 M22 D 0 P22 D 0 u03 D 0


31 D 0 at x2 D 0 L2 (23)


poundu0

1 rsquo1


poundu0

2 rsquo2









a)

b)


VEL AND BATRA 1425


K iexcl K1

K2 iexcl K1D V2

1 C 1 iexcl V2

K2 iexcl K1

K1 Ciexcl

43

centsup11

sup1 iexcl sup11



sup12 iexcl sup11

sup11 C f1

para(25)




V2


(26)


reg iexcl reg1

reg2 iexcl reg1D

1=K iexcl 1=K1

1=K2 iexcl 1=K1

(27)








V1K1

K1 C 4sup1=3C

V2 K2

K2 C 4sup1=3C 5

sup3V1sup12

sup1 iexcl sup12C

V2sup11

sup1 iexcl sup11

acuteC 2 D 0

(30)




middot1 C 2middotC






[frac3433x1 x2 H=2 T x1 x2 H=2]




Vc D V iexclc C

iexclV C

c iexcl V iexclc

centiexcl12

C x3=Hcentn

(33)



c D 1 andV iexcl



[ Nu1 Nu2] D100Em H 2[u1 u2]

pC L31

Nu3 D100Em H 3u3

pC L41


pC L21


pC L1 Nfrac3433 D

frac3433

pC(34)


OT DT

T C Oq3 D iexclq3 H


Ou3 D100Hu3

regm T C L21


Emregm T C



100L21frac3433

Em regm T C H 2(35)



1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)



a)

b)

c)

d)

e)

f)







VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











1424 VEL AND BATRA








u3xi D u03 (19b)

where u0deg u0







where


iexclH=2


Rdeg DZ H=2

iexclH=2

frac34deg 3g3 dx3









N11 D 0 M11 D 0 P11 D 0 u03 D 0


32 D 0 at x1 D 0 L1

N22 D 0 M22 D 0 P22 D 0 u03 D 0


31 D 0 at x2 D 0 L2 (23)


poundu0

1 rsquo1


poundu0

2 rsquo2









a)

b)


VEL AND BATRA 1425


K iexcl K1

K2 iexcl K1D V2

1 C 1 iexcl V2

K2 iexcl K1

K1 Ciexcl

43

centsup11

sup1 iexcl sup11



sup12 iexcl sup11

sup11 C f1

para(25)




V2


(26)


reg iexcl reg1

reg2 iexcl reg1D

1=K iexcl 1=K1

1=K2 iexcl 1=K1

(27)








V1K1

K1 C 4sup1=3C

V2 K2

K2 C 4sup1=3C 5

sup3V1sup12

sup1 iexcl sup12C

V2sup11

sup1 iexcl sup11

acuteC 2 D 0

(30)




middot1 C 2middotC






[frac3433x1 x2 H=2 T x1 x2 H=2]




Vc D V iexclc C

iexclV C

c iexcl V iexclc

centiexcl12

C x3=Hcentn

(33)



c D 1 andV iexcl



[ Nu1 Nu2] D100Em H 2[u1 u2]

pC L31

Nu3 D100Em H 3u3

pC L41


pC L21


pC L1 Nfrac3433 D

frac3433

pC(34)


OT DT

T C Oq3 D iexclq3 H


Ou3 D100Hu3

regm T C L21


Emregm T C



100L21frac3433

Em regm T C H 2(35)



1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)



a)

b)

c)

d)

e)

f)







VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1425


K iexcl K1

K2 iexcl K1D V2

1 C 1 iexcl V2

K2 iexcl K1

K1 Ciexcl

43

centsup11

sup1 iexcl sup11



sup12 iexcl sup11

sup11 C f1

para(25)




V2


(26)


reg iexcl reg1

reg2 iexcl reg1D

1=K iexcl 1=K1

1=K2 iexcl 1=K1

(27)








V1K1

K1 C 4sup1=3C

V2 K2

K2 C 4sup1=3C 5

sup3V1sup12

sup1 iexcl sup12C

V2sup11

sup1 iexcl sup11

acuteC 2 D 0

(30)




middot1 C 2middotC






[frac3433x1 x2 H=2 T x1 x2 H=2]




Vc D V iexclc C

iexclV C

c iexcl V iexclc

centiexcl12

C x3=Hcentn

(33)



c D 1 andV iexcl



[ Nu1 Nu2] D100Em H 2[u1 u2]

pC L31

Nu3 D100Em H 3u3

pC L41


pC L21


pC L1 Nfrac3433 D

frac3433

pC(34)


OT DT

T C Oq3 D iexclq3 H


Ou3 D100Hu3

regm T C L21


Emregm T C



100L21frac3433

Em regm T C H 2(35)



1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)



a)

b)

c)

d)

e)

f)







VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











1426 VEL AND BATRA

a)

b)

c)

d)

e)

f)



a)

b)

c)

d)

e)

f)







VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1427

a)

b)


c = 0 and L1H = 5

a)

b)

c)

d)

e)

f)








1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











1428 VEL AND BATRA


V iexclc D 0 V C



c D 06 V Cc D 1


a)

b)

c)

d)

e)

f)









VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1429


V iexclc D 02 V C



c D 05 V Cc D 07


a)

b)

c)

d)



V iexclc D 02 V C




1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











1430 VEL AND BATRA

a)

b)

c)

d)


a)

b)

c)

d)


VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1431



2C x3=H



a)

b)

c)

d)




1432 VEL AND BATRA














































VEL AND BATRA 1433











1432 VEL AND BATRA














































VEL AND BATRA 1433











VEL AND BATRA 1433











Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates

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Transcript of Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates