i so graded otho

7/28/2019 i So Graded Otho

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Jeong-Ho Kim

G. H. Paulino

1

Mem. ASME,

e-mail: [email protected]

Department of Civil and Environmental

Engineering,

University of Illinois at Urbana-Champaign,

Newmark Laboratory,

205 North Mathews Avenue,

Urbana, IL 61801

Isoparametric Graded FiniteElements for NonhomogeneousIsotropic and OrthotropicMaterialsGraded finite elements are presented within the framework of a generalized isoparametric

formulation. Such elements possess a spatially varying material property field, e.g.Youngs modulus E and Poissons ratio for isotropic materials; and principalYoungs moduliE11,E22, in-plane shear modulus G12, and Poissons ratio 12 fororthotropic materials. To investigate the influence of material property variation, bothexponentially and linearly graded materials are considered and compared. Severalboundary value problems involving continuously nonhomogeneous isotropic and ortho-tropic materials are solved, and the performance of graded elements is compared to thatof conventional homogeneous elements with reference to analytical solutions. Such solu-tions are obtained for an orthotropic plate of infinite length and finite width subjected tovarious loading conditions. The corresponding solutions for an isotropic plate are ob-tained from those for the orthotropic plate. In general, graded finite elements providemore accurate local stress than conventional homogeneous elements, however, such maynot be the case for four-node quadrilateral (Q4) elements. The framework described here

can serve as the basis for further investigations such as thermal and dynamic problems infunctionally graded materials. DOI: 10.1115/1.1467094

1 Introduction

Recent advances in material processing have allowed manufac-turing a wide diversity of functionally graded materials FGMs

1 3 . Such materials possess continuously graded propertieswith gradual change in microstructure 4,5 . The materials aremade to take advantage of desirable features of its constituentphases. For instance, in a thermal protection system, FGMs takeadvantage of heat and corrosion resistance typical of ceramics,and mechanical strength and toughness typical of metals.

FGMs are nonhomogeneous with regard to thermomechanicaland strength related properties. Depending on the processing tech-nique, they may exhibit either isotropic or anisotropic materialproperties. For instance, large bulk FGMs produced by sparkplasma sintering SPS technique may be modeled as isotropicmaterials 6 . On the other hand, materials processed usingplasma spray technique have generally a lamellar structure 7 ,while materials processed by electron beam physical vapor depo-sition PVD may have a columnar structure 8 . Thus, in study-ing the mechanics of the former class of materials fabricated bySPS , a nonhomogeneous isotropic model may be appropriate;and for the latter class of materials fabricated by plasma sprayingor PVD , a nonhomogeneous orthotropic model may suffice as afirst approximation. Thus, both types of material models, i.e., iso-tropic and orthotropic, are investigated here.

As the manufacturing of FGMs advances, new modeling tech-niques are also developed for such materials 3,9 . Here, wefocus on the finite element method for nonhomogeneous materialsusing a generalized isoparametric formulation. The graded ele-

ments obtained with this formulation are compared with conven-tional homogeneous elements, as illustrated by Fig. 1. Notice thatthe graded element incorporates the material property gradient atthe size scale of the element, while the homogeneous element

produces a stepwise constant approximation to a continuous ma-terial property field such as the one shown in Fig. 1.

This paper discusses and compares the behavior of graded ver-sus conventional homogeneous elements under various loading

conditions in both isotropic and orthotropic FGMs with respect toanalytical solutions which are either available in the literature or

derived in this work. The manuscript is organized as follows. Thenext subsection presents an example which serves as a motivation

to this work. In this example, the FGM leads to a stress redistri-bution with lower stress concentration factor SCF than the cor-responding problem with homogeneous material. Next, a brief lit-

erature survey and comments on previous related work are given.Section 2 presents some exact solutions for displacements andstresses in orthotropic FGMs. The exact solutions for isotropicFGMs are obtained as particular instances of those for orthotropic

FGMs. Section 3 reviews finite element formulations. Section 4addresses the generalized isoparametric graded finite element for-mulation. Sections 5 and 6 present finite element results forstresses in isotropic and orthotropic FGMs, respectively, whichare compared with analytical solutions. Finally, Section 7 providessome concluding remarks.

1.1 Motivation. Functionally graded composites, withsmooth variation of volume fractions, offer various advantagessuch as reduction of residual stress 10 and increased bondingstrength 11 . Moreover, if properly used, such materials mayalso lead to reduction of stress concentration or stress intensityfactors 12 . For example, Hasselman and Youngblood 13found that the maximum tensile thermal stresses in brittle ceram-ics can be reduced significantly by spatially varying thermal con-ductivity in a hollow circular cylinder subjected to radially inwardor outward steady-state heat flow, and Horgan and Chan 14investigated the effect of material nonhomogeneity on the re-sponse of linearly elastic isotropic hollow circular cylinders or

1To whom correspondence should be addressed.

Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS for publication in the JOURNAL OF APPLIED MECHAN-ICS. Manuscript received by the ASME Applied Mechanics Division, July 2, 2001;final revision Nov. 14, 2001. Associate Editor: M.-J. Pindera. Discussion on the paper

should be addressed to the Editor, Professor Lewis T. Wheeler, Department of Me-

chanical Engineering, University of Houston, Houston, TX 77204-4792, and

will be accepted until four months after final publication of the paper itself in

the THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS JOURNAL OF APPLIED

MECHANICS.

502 Vol. 69, JULY 2002 Copyright 2002 by ASME Transactions of the ASME


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disks under uniform internal or external pressure by varyingYoungs modulus with respect to the radial direction and foundthat the maximum hoop stress in a nonhomogeneous material doesnot, in general, occur on the inner surface in contrast with a ho-mogeneous material.

To further motivate the use of FGMs in engineering applica-tions, consider the isotropic FGM link bar of Fig. 2 a . The barhas unit thickness, it is subjected to unit axial tension load at theright end, and it is considered in a state of generalized planestress. The basic FGM constituents are titanium monoboride TiBand commercially pure titanium CP Ti as illustrated by Fig. 2 b .The elastic properties of the base materials are 15

ETi B375 GPa, Ti B0.14

ETi107 GPa, Ti0.34.

The graded region is incorporated with an exponential materialvariation. Thus Youngs modulus and Poissons ratio are functionsof the Cartesian coordinate y see Fig. 2 , i.e.,

E y ETi eEy, y Ti e

y, (1)

respectively, where 1/E and 1/ are the length scales of nonho-

mogeneity which are given by

E1

Wlog ETi B /ETi ,

1

Wlog Ti B /Ti , (2)

respectively, where W is the width of the symmetric model asshown in Fig. 2 b . Figure 3 a shows the finite element mesh forthe symmetric portion of the link bar with 1000 quadrilateral el-ements of eight nodes Q8 . These elements are graded finite el-ements as illustrated by Fig. 1 and explained subsequently in thispaper. Figure 3 b shows the xx stress contour for the homoge-neous link bar either TiB or Ti and Fig. 3 c shows the xx stress

contour for the FGM link bar TiB/Ti . The main stress values nodal average are summarized in Table 1. Notice that the maxi-mum stress location in the FGM bar is different from that in the

homogeneous barthe maximum stress occurs in A see Fig.

2 b for the homogeneous bar, while it occurs in B see Fig.2 b for the FGM bar. Moreover, the maximum stress is lower inthe FGM than in the homogeneous bar. Thus, the FGM leads tostress redistribution with a lower SCF as illustrated by Table 1 andFig. 3. In summary, this example shows, by means of elastic finiteelement analysis, that the stress response of (inhomogeneous)FGMs differ substantially from those of their homogeneouscounterparts.

1.2 Related Numerical Work. Several numerical modelshave been used to investigate FGMs, including integral equations

e.g., 1618 , the higher order model e.g., 19,20 , boundary

elements e.g., 21,22 , and finite elements e.g., 10,2330 .This work concentrates on the finite element method for FGMsusing the isoparametric concept for graded elements.

A few additional comments about the related work by Santareand Lambros 30 are in order. They have also published in this

journal a graded finite element model for nonhomogeneous mate-rials. However, their work differs from ours in the sense that theysample the material properties directly at the Gauss points of theelement, while we adopt a generalized isoparametric formulation.Although the two methods are different, they are equivalent forfine mesh discretizations 31 . They investigated the behavior of

Fig. 1 Homogeneous versus graded finite elements. a Prop-erty variation along one coordinate axis; b homogeneous el-ements; c graded elements. Notice that the property of the

homogeneous element corresponds to the property at the cen-troid of the graded element.

Fig. 2 FGM link bar units: N, mm: a geometry and boundary conditions; bsymmetric model

Journal of Applied Mechanics JULY 2002, Vol. 69 503


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four-node quadrilaterals Q4 for isotropic FGMs only. In additionto the bilinear element Q4 , we also investigate the behavior ofeight-node quadrilaterals Q8 for both isotropic and orthotropicFGMs. They investigated exponential material variation only.Here, we compare both exponential and linear material variations.Finally, we believe that the generalized isoparametric formulationis more natural to the finite element method than the Gauss pointsampling of material properties because the generalized formula-tion embraces the important isoparametric conceptthe sameshape functions are used to interpolate the unknown displace-ments, the geometry, and the material parameters.

2 Some Exact Solutions for Nonhomogeneous

Elasticity

Exact solutions for both isotropic and orthotropic functionallygraded materials FGMs will be used as reference solutions forthe numerical examples that follow. We consider an orthotropicfunctionally graded plate of infinite length and finite width sub-

jected to various loading conditions such as remote fixed grip,tension, and bending, as shown in Fig. 4. Both exponential andlinear material variations are considered. First, analytical solutionsfor stresses and displacements are developed for orthotropicFGMs and, afterwards, they are particularized e.g., in the limitfor isotropic FGMs. The analytical solutions for exponentiallygraded isotropic FGMs coincide with those of Erdogan and Wu

32 and Paulino and Kim 33 . The analytical solutions forlinearly graded isotropic and orthotropic FGMs are new solutionsderived in this work.

2.1 Exponential Material Variation. Consider a plate un-der generalized plane stress conditions see Fig. 4 made of anonhomogeneous orthotropic material. Assume the Poissons ratio

Table 1 Representative xx stress values Nmm2 for the link

bar of Figs. 2 and 3

xx

Location

A B

Homogeneous 2.908 2.137FGM 2.369 2.601

Fig. 4 An isotropic or orthotropic functionally graded plate:a geometry and material propertiesthe shaded portion indi-cates the symmetric region of the plate used in this analysis;b fixed grip loading with a schematic of the correspondingstresses at the end points of the plate; c tension loading; dbending loading

Fig. 3 FGM link bar under unit axial tension units: Nmm2: a mesh con-figuration with 1000 graded Q8 elements; b xx stress distributions for ho-mogeneous link bar both TiB and Ti; c xx stress distributions for FGM barTiBTi

504 Vol. 69, JULY 2002 Transactions of the ASME


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(12) constant, and the Youngs moduli and in-plane shear modu-lus with variations given by the following expressions:

E11 x E110

e11x

E22 x E220

e22x(3)

G12 x G120 e12x

12 x constant,

where E110E11(0), E22

0E22(0), and G12

0G12(0) are the mate-

rial properties at the x0 line see Fig. 4 a , and the coefficients

i j above are independent nonhomogeneity parameters character-ized by

111

Wlog

E11 W

E11 0

221

Wlog

E22 W

E22 0 (4)

121

Wlog

G12 W

G12 0 ,

where W is the width of the FGM plate as shown in Fig. 4. Notice

that in this case the i j parameters have units Length1.

For a corresponding nonhomogeneous isotropic material (EE11E22 , GG12 , 12), the Poissons ratio is assumed

constant and the Youngs modulus varies exponentially, i.e.,

E x E0ex(5)

x constant

where E0E(0). The nonhomogeneity parameter is given by

1

Wlog

E W

E 0 (6)

which has units Length 1.

2.1.1 Fixed Grip Loading. For fixed grip loading see Fig.4 b with y y(x,)0, the stress distribution becomes

y y x E220 0e

22x. (7)

Using strain-displacement relations and the boundary conditions

ux 0,0 0, uy x ,0 0, (8)

one obtains the displacements

ux x,y 120E22

0

E110

1

2211 e(2211)x1

(9)

uy x,y 0y .

Notice that for isotropic materials (EE11E22 , G12G, and12), the stress distribution 7 becomes 32

y y

x E00

ex (10)

and the displacements are obtained in the limit of Eq. 9 as(2211)0. Thus 33

ux x ,y 0x (11)

uy x,y 0y .

2.1.2 Tension and Bending. For tension and bending loads see Figs. 4 c and 4 d , respectively , the applied stresses aredefined by

NtW, MbW

2

6, (12)

where N is a membrane resultant along the xW/2 line see Fig.4 a , and M is the bending moment. For these two loading cases,

the compatibility condition2

yy / x2

0 gives y yAxB andthus

y y x E220

e22x AxB (13)

where the constants A with unit Length 1) and B dimension-less are determined from

0

W

y y x dxN,0

W

yy x xd xM (14)

by assuming

MNW/2 for tension(15)

N0 for bending.

Thus, for tension load, the stress distribution is given by Eq. 13

with

A22N

2E220 W22

2e22W222e

22WW222222

e22W222

W2e 222W2e22W1 ,

(16)

B22N

2E220 e

22W e22W W2222322W4 W

2222222W8 22W4

e22W1 e22W222

W2e222W2e22W1 .

For bending load, the stress distribution is also given by Eq. 13 ,however, the coefficients A and B for this case are

A22

2 M

E220 22 1e

22W

e22W222 W2e 222W2e22W1

,

(17)

B22

2M

E220 22We

22We22W1

e22W222

W2e 222W2e22W1,

respectively. For both tension and bending loads, using the strain-displacement relations and the boundary conditions 8 , one ob-tains the displacements

ux x ,y 12E22

0

E110

AxA

2211B

2211 e (2211)x

AB 2211

22112 A

2y 2

(18)

uy x,y AxB y .

The constants A and B refer to the appropriate loading case above,either tension Eq. 16 or bending Eq. 17 .



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For the isotropic case (EE11E22 , G12G, and 12), thestress distribution is obtained by Eqs. 16 and 17 for tensionand bending loads, respectively with 22 replaced by , whichagree with Erdogan and Wus 32 solution. The displacements areobtained in the limit of Eq. 18 as (2211)0. Thus 33

ux x ,y A

2x 2Bx

A

2y 2

(19)

uy x,y AxB y .

2.2 Linear Material Variation. Once again, consider a

plate under generalized plane stress conditions, as illustrated byFig. 4. Assume the Poissons ratio (12) is constant, and theYoungs moduli and in-plane shear modulus with variations givenby the following expressions cf. Eq. 3 :

E11 x E11011x

E22 x E22022x

(20)

G12 x G12012x

12 x constant,

where E110E11(0), E22

0E22(0), and G12

0G12(0) are the mate-

rial properties at the x0 line see Fig. 4 a and the coefficientsi j are independent nonhomogeneity parameters characterized by

11E11 W

E11 0W

22E22 W E22 0

W(21)

12G12 W G12 0

W.

Notice that in this case the i j parameters have units

Force / Length 3.For a corresponding nonhomogeneous isotropic material (E

E11E22 , GG12 , 12), the Poissons ratio is assumedconstant and the Youngs modulus varies linearly, i.e.,

E x E0x(22)

x constant

where E0E(0). The nonhomogeneity parameter is given by

E W E 0

W(23)

which has units Force / Length 3.

2.2.1 Fixed Grip Loading. For fixed grip loading see Fig.4 b with y y(x,)0, the stress distribution becomes

yy x 0 E22022x . (24)

Using strain-displacement relations and the boundary conditions.

ux 0,0 0, uy x ,0 0,

one obtains the displacements

ux x ,y 12 022

11x

E220 ln E11

022x

11

22E110 ln E11

011x

112

E220

11

22E110

112 ln E110

(25)

uy x,y 0y .

For isotropic materials (EE11E22 , G12G , and 12),the stress distribution 24 becomes

yy x 0 E0x , (26)

and the displacements are obtained from Eq. 25 as

ux x ,y 0x(27)

uy x,y 0y .

2.2.2 Tension and Bending. For tension and bending loads see Fig. 4 c and 4 d , respectively , the applied stresses are de-fined by Eq. 12 , i.e.,

NtW, MbW

2

6,

where N is a membrane resultant applied along the xW/2 line see Fig. 4 a , and M is the bending moment. For these two

loading cases, the compatibility condition 2yy / x20 gives

y yAxB and thus

yy x E22022x AxB , (28)

where the constants A with unit Length1

) and B dimension-less are determined from Eq. 14 , i.e.,

0

W

yy x dxN,0

W

y y x x dxM,

by assuming see Eq. 15

MNW/2 for tension(29)

N0 for bending.

Thus, for tension load, the stress distribution is given by Eq. 28with

A22N

16 22

2 W322E220 W2 E22

0 2W,

(30)

BN E22

022W

16 22

2 W322E220 W2 E22

0 2W.

For bending load, the stress distribution is also given by Eq. 28with

A36M 2E22

022W

222

W56E220 22W

46 E22

0 2W3,

(31)

B

36M 2E22022W

322W23E22

0W

222W6E220

222

W56E220 22W

46 E22

0 2W3.

For both tension and bending loads, using the strain-displacementrelations and the boundary conditions.

ux 0,0 0, uy x,0 0,

one obtains the displacements in closed form, which are given by



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ux x ,y 12 E11

0 22A

112

22B

11

E220

A

11 x 22A211x2

E220

B

11

E110 22B

112

E110

E220

A

112

E110 222A

113 ln 11xE110

E220

B112E11

0 1122BE110

E220 11A E11

0 222A

113 ln E110 A2 y 2,

(32)

uy x ,y AxB y .

For the isotropic case (EE11E22 , G12G, and 12), thestress distribution is obtained by Eqs. 30 and 31 for tension

and bending loads, respectively with 22 and E220

, replaced by

and E0, respectively. The displacements are obtained from Eq. 32 as

ux x ,y A

2x 2Bx

A

2y 2

(33)

uy x,y AxB y .

Notice that the form of the exact solutions for displacements inorthotropic FGMs differs significantly from that for isotropicFGMs because the former case depend on two principal Youngsmoduli, while in the latter case the explicit moduli dependence is

absent.

3 Basic Finite Element Formulation

Displacements for an isoparametric finite element can be writ-ten as

ue

i1

m

Niuie

(34)

where Ni are shape functions, ui is the nodal displacement corre-sponding to node i, and m is the number of nodal points in theelement. For example, for a Q4 element, the standard shape func-tions are

Ni 1i 1i /4, i1, . . . ,4 (35)

where (,) denote intrinsic coordinates in the interval 1,1and (i ,i) denote the local coordinates of node i. As usual,strains are obtained from displacements by differentiation as.

eB

eu

e (36)

where Be is the strain-displacement matrix of shape function de-

rivatives, and ue is the nodal displacement vector. Thus strain-stress relations are given by

eD

e x e (37)

where De(x) is the constitutive matrix, which is a function of

position for nonhomogeneous materials, i.e., De(x) De(x ,y ).The principle of virtual work

PVW

yields the following finite

element stiffness equations 34

keu

eF

e (38)

where Fe is the load vector and the element stiffness matrix is

ke

e

BeTD

e x Bede (39)

in which e is the domain of element e , and T denotes trans-pose. The reasoning above, at the element level, can be readilyextended to the whole domain, which leads to a system of alge-braic equations for the unknown displacements 34 .

4 Generalized Isoparametric Graded Finite Elements

For simplicity of notation, the superscript (e), denoting theelement, is dropped in this section. Material properties e.g., ateach Gaussian integration point can be interpolated from thenodal material properties of the element using isoparametric shapefunctions which are the same for spatial coordinates (x,y):

xi1

m

Nix i , yi1

m

Niy i (40)

and displacements (u ,v):

ui1

m

Niu i , vi1

m

Niv i . (41)

Thus, by generalization of the isoparametric concept, the Youngs

modulus EE(x) and Poissons ratio (x) are interpolated as

Ei1

m

NiEi , i1

m

Nii (42)

respectively, as illustrated by Fig. 5. Similar expansions can alsobe made to two-dimensional orthotropic materials where the fourindependent engineering elastic parameters are the principalYoungs moduli, E11E11(x), E22E22(x), in-plane shear modu-lus G12G12(x); and Poissons ratio 1212(x), i.e.,

E11i1

m

Ni E11 i , E22i1

m

Ni E22 i ,

(43)

G12

i

1

m

Ni G12 i , 12

i

1

m

Ni 12 i ,

as illustrated by Fig. 5.Some material models may be given in terms of the volume

fraction (V) of a material phase, p, e.g., the metal phase in aceramic/metal FGM 35 . In this case, the generalized isopara-

metric formulation consists of approximating Vp by the standardinterpolation

Vpi1

m

NiVip

(44)

where Vip

(i1,2, . . . ,m) are the values of Vp at the nodal points.

This approach offers a convenient framework to couple the finite

Fig. 5 Generalized isoparametric formulation for isotropic ororthotropic FGMs



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element method with micromechanics-based models, e.g., self-consistent scheme.

The above framework allows development of a fully isopara-metric formulation in the sense that the same shape functions areused to interpolate the unknown displacements, the geometry, andthe material properties. Thus, the actual variation of the materialproperties may be approximated by the element interpolationfunctions e.g., a certain degree of polynomial functions .

5 Numerical Examples

Although the finite element method offers a lot of flexibility interms of modeling material property variation, the actual choice ofproperties and boundary value problems in this section was dic-tated by the analytical solutions derived in Section 2 for the plateconfiguration of Fig. 4. Here the analytical solutions are comparedwith the numerical ones. The examples are divided into twogroups:

1. isotropic FGM plate2. orthotropic FGM plate

For each group, two material variations along the Cartesian direc-tion x are examined:

1. exponentially graded materials2. linearly graded materials

and also the following loading conditions are considered:

1. fixed grip2. tension loading3. bending loading

The relevant stress values obtained numerically by the finiteelement method are compared with the analytical results. Forfixed grip loading see Fig. 4 b , the stress yy is considered. Fortension applied parallel to the material gradation, the stress xx isthe quantity of interest, while for tension and bending loads ap-plied perpendicular to the material gradation see Figs. 4 c and4 d , respectively , the stress yy is the relevant quantity. More-over, for a few of the examples, the displacements computed nu-merically are also compared with the analytical results.

The finite element meshes consist of square elements Q4 or

Q8 with edges of unit length. For all the examples, 22 Gaussquadrature was employed. All the numerical stress values reportedhere are nodal values extrapolated directly from the Gauss pointsand without any averaging. The finite element program developedin this work was implemented by the authors in a simple codeusing MATLAB.

5.1 Isotropic Functionally Graded Plate. Figure 6 illus-trates an isotropic FGM plate with material variation in the Car-tesian direction x subjected to various loading conditions. Figure6 a shows the basic geometry, boundary conditions and proper-ties. The finite element mesh consists of 99 Q4 or Q8 elements

either graded or homogeneous as illustrated in Figs. 6 b to 6 d .The Youngs modulus varies from

E1E0E 0 to E2E W (45)

either exponentially as given by Eq. 5 or linearly as given by Eq. 22 with E11.0 and E28.0. The independent nonhomogeneityparameters are given by Eqs. 6 and 23 for the exponential andlinear material variations, respectively, with

ln 8/1 /9 and 7/9. (46)

Consistent units are employed here. The Poissons ratio is con-stant and it is selected as follows:

0.3 for tension and bending applied perpendicular to ma-terial gradation Figs. 6 b and 6 c , respectively

0.0 for tension load parallel to material gradation Fig.6 d .

The behavior of the elements homogeneous versus graded isas follows. Figure 7 shows the stress y y versus x for an expo-nentially graded isotropic plate subjected to a uniform displace-ment in the y direction with 0/H. According to Eq. 10 , thestress yy is uniform in the y-direction and thus the graph of Fig.7 is applicable to the entire range of y coordinates, i.e., 0yH see Fig. 6 a . In this case, the solution obtained with gradedQ4 elements matches the exact solution. This is expected becausethe exact displacement field is linear see Eq. 11 and Fig. 8 ,which is captured by linear isoparametric elements such as Q4.Moreover, because of the linearity of the analytical solution Eq.

11 , a single Q4 element could be used to predict the exactsolution. Figure 7 also shows that the stress obtained with homo-geneous Q4 elements is piecewise constant due to the fact thatthese elements have a single value for each material property,which leads to a piecewise constant material property approxima-tion as illustrated by Fig. 1. Therefore such homogeneous ele-ments predict the actual stress values only at their centroids where

Fig. 6 Isotropic FGM plate with material variation in thex-direction: a geometry, boundary conditions and materialproperties; b tension load perpendicular to material grada-tion; c bending load; d tension load parallel to material gra-dation. The finite element mesh 99 quads: either Q4 or Q8is illustrated in parts b through d with a representative Q4

element at the upper left hand corner

Fig. 7 Stress distribution yy using Q4 elements for fixedgrip 0H load applied perpendicular to the exponentialmaterial gradation



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the properties match the material gradation. Moreover, the ampli-tude of the nodal stress jumps for homogeneous Q4 elements in-creases with coordinate x in a nearly exponential fashion, as illus-trated by Fig. 7. These observations are consistent with those bySantare and Lambros 30 . Of course, the exact solution is alsorecovered with higher-order graded elements, e.g., Q8. The homo-geneous Q8 elements also lead to a piecewise constant nodal

stress profile with the stress at the midnode location along the xdirection matching the exact stress value, which occurs becausethe material properties at the mid-nodes match the actual materialproperties.

Figures 9 and 10 compare nodal stresses interpolated fromstresses at Gauss integration points using graded and homoge-neous Q4 and Q8 elements, respectively, which are subjected totension loading applied perpendicular to the material gradation.Figures 11 and 12 show such comparison considering linear ma-terial variation. On the left side of the domain in Figs. 912, theexact solution shows an increasing trend ofyy with x, while thehomogeneous elements either Q4 or Q8 give yy as a decreasingfunction of x in each individual element. Notice that this problemdoes not occur with the graded elements. In this case, the exactsolution for displacements is quadratic see Eqs. 19 and 33 forexponential and linear material variations, repectively , which co-incides with the order of interpolation for the Q8 element. More-over, the material variation for the linear case is captured by theelement shape functions. The stress results for the Q8 element

considering exponential and linear material variations are shownin Figs. 10 and 12, respectively. As expected, the homogeneousQ8 element shows piecewise variation while the graded Q8 ele-ment approaches the analytical solution quite well. The relativelysmall differences observed between the analytical and graded Q8solutions may be attributed to the finite plate length length/

Fig. 8 Displacements ux and uy using Q4 elements for fixedgripload applied perpendicular to the exponentialmaterial gra-dation in isotropic FGMs

Fig. 9 Stress distribution yy using Q4 elements for tensionload applied perpendicular to the exponential materialgradation

Fig. 10 Stress distribution yy using Q8 elements for ten-sion load applied perpendicular to the exponential materialgradation

Fig. 11 Stress distribution yy using Q4 elements for ten-sion loading applied perpendicular to the linear materialgradation

Fig. 12 Stress distribution yy using Q8 elements for ten-sionload applied perpendicular to the linearmaterial gradation



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width1 as shown in Fig. 6 utilized in the numericalcalculationthe analytical solution was derived for an infinitelylong plate of finite width.

A similar comparison is also made for a different loading caseconsisting of bending applied perpendicular to the material grada-tion. Figures 13 and 14 show the behavior of the Q4 and Q8elements, respectively, for the exponential variation. Figures 15and 16 show such comparison for the bending case consideringlinear material variation. The stress results for the Q8 elementconsidering exponential and linear material variations are shown

in Figs. 14 and 16, respectively. Similar comments to those madecomparing the Q8 homogeneous versus graded and analyticalsolutions for the tension load case also hold for the present bend-ing load case.

The above results lead to the following observations. The varia-tion of stress with position x is larger for linear than with expo-nential material variations cf. Figs. 9 and 11, 10 and 12, 13 and15, and 14 and 16 . In general, the amplitude of stress jumpsbetween Q4 elements is larger than between Q8 elements, espe-cially for conventional homogeneous elements cf. Figs. 9 and 10,

Fig. 13 Stress distribution yy using Q4 elements for bend-ing load applied perpendicular to the exponential materialgradation

Fig. 14 Stress distribution yy using Q8 elements for bend-ing load applied perpendicular to the exponential materialgradation

Fig. 15 Stress distribution yy using Q4 elements for bend-ing load applied perpendicular to the linearmaterial gradation

Fig. 16 Stress distribution yy using Q8 elements for bend-ing load applied perpendicular to the linearmaterial gradation

Fig. 17 Stress distribution xx using Q4 elements 99mesh for tension load applied parallel to the exponential orlinearmaterial gradation

Fig. 18 Stress distribution xx using Q8 elements 99mesh for tensionload applied parallel to the materialgradation



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11 and 12, 13 and 14, 15 and 16, and 17 and 18 . As expected, thegraded elements show superior performance to homogeneous el-

ements, i.e., the graded elements provide a better approximationto the exact solution in every element. Essentially, the gradedelements show good performance in terms of actual i.e., no av-eraging nodal stress (y y) along the y0 line and the homoge-neous elements behave well in terms of the averaged nodalstresses.

Figures 17 and 18 compare nodal stresses of graded versushomogeneous Q4 and Q8 elements (99 mesh , respectively,which are subjected to tension applied parallel to the materialgradation see Fig. 6 d . The exact solution is xx1.0. Differentfrom the observation above, it is interesting to observe in Fig. 17that the Q4 graded element shows poor performance when com-pared to Q4 homogeneous elements for both material variations

i.e., exponential and linear . Although mesh refinement for afixed material gradient increases the accuracy of the solution, thesame trend of Fig. 17 is observed for a finer mesh, e.g., 1818.Figure 17 shows that the Q4 graded elements provide piecewisecontinuous solutions to the nodal stresses (xx ), while the homo-geneous Q4 elements do recover the exact solution. This is thereverse of the effect seen in the previous load cases. However, ahigher order element such as Q8 either graded or homogeneousis able to capture the exact solution in this case, as shown in Fig.18.

A few additional remarks, regarding the behavior of Q4 ele-ments observed in Fig. 17, are in order. Both graded and homo-geneous elements lead to the same displacements at all nodes andthe same constant strains for each element. Notice that along the

y0 line, the nodal stress range has constant amplitude for theexponential material case, while it has decreasing amplitude forthe linear material case see Fig. 17 . The reason for this behavioris illustrated by Fig. 19 by investigating the strain distribution for

two mesh discretizations ( 99 and 1818 meshes . For instance,for the exponential material case, the nodal strains decrease expo-nentially while the Youngs modulus increase exponentially. Thusthe multiplication of these two factors cancel each other to give aconstant stress amplitude at the nodal points, as shown in Fig. 17.

5.2 Orthotropic Functionally Graded Plate. Figure 20shows orthotropic FGM plates, with material variation in the Car-tesian direction x, subjected to various loading conditions. Figure20 a shows the basic geometry, boundary conditions and materialproperty variation. The two principal Youngs moduli and in-planeshear modulus vary proportionally either with an exponentialfunction of x as given by Eq. 3 or with a linear function of x as

given by Eq. 20 . The independent nonhomogeneity parameters(i j and i j) are given by Eqs. 4 and 21 for the exponentialand linear material variations, respectively. The Poissons ratio isassumed constant.

For the examples in Fig. 20, the finite element mesh consists ofeither Q4 or Q8 graded or homogeneous elements under gener-alized plane stress. The mesh for the geometry of Figs. 20 b and20 c consists of 918 elements. For the sake of completeness,all the properties used in the numerical analyses are given asfollows. However, due to space limitations, not all the results areshown here, but they are reported elsewhere 36 . For the fixedgrip case and for tension and bending perpendicular to the mate-rial gradation, the following data were used for the finite elementanalysis:

E1101, E22

00.1, G12

00.5, 120.3

in which consistent units are employed. For tension parallel to thematerial gradation, the following data were used for the finiteelement analysis:

E1101, E22

00.1, G12

00.5, 120.0.

For the single case of fixed grip loading, only exponential ma-terial variation was considered. In this case, the i j parameters are

22 ln8 /9, 11/2, 12/3

so that the range of properties is the following

Fig. 19 Strain distribution xx using Q4 elements either 99 and 1818 for tensionload applied parallel to the materialgradation either exponential or linear

Fig. 20 Orthotropic FGM plate with material variation in thex-direction: a geometry, boundary conditions and materialproperties; b tension load perpendicular to material grada-tion; c bending load; d tension load parallel to the materialgradation. The finite element mesh Q4 or Q8 elements is illus-trated in parts b through d with a representative Q4 elementat the upper left hand corner



11/13

E11 1,2.828 , E22 0.1,0.8 , G12 0.5,1.0 .

For all other loading cases i.e., tension and bending perpendicularto the material gradation, and tension parallel to the material gra-dation , the i j parameters, characteristic of exponential materialvariation, are chosen so that the variations of E11 , E22 , and G12are proportional 16,17 , i.e.,

112212 ln8 /9, (47)

and the i j parameters, characteristic of linear material variation,are given by

117/9, 220.7/9, 123.5/9

so that the range of properties is the following:

E11 1.0,8.0 , E22 0.1,0.8 , G 12 0.5,4.0 .

Regarding the element behavior homogeneous versus graded ,several of the observations made for isotropic materials in theprevious section also hold for orthotropic materials. Thus ratherthan repeating those common observations, this section focuses onnew observations and insights. Moreover, the analytical solutions

of Section 2 show that, for exponential material gradation Section2.1 , the relevant stress quantity only depends on the nonhomoge-neous parameter 22 , and the displacements depend on both 11and 22 . For linear material gradation Section 2.2 , the relevantstress depends on 22 , and the displacements depend on both 11and 22 . This information will be helpful to understand the ex-amples reported below.

For proportional variation of material properties see Eq. 47 ,the change of ux with x is linear rather than the nonlinear func-tion of Eq. 9 , which is similar to the behavior of the isotropicplate under the same boundary conditions, i.e., fixed grip see Fig.8 . This behavior can be seen by the following limit:

lim(2211)0

ux lim(2211)0

12 0E22

0

E110

1

2211

e (2211)1120

E220

E110

x . (48)

Figures 21 and 22 compare nodal stresses interpolated fromstresses at Gauss points using graded and homogeneous Q4 andQ8 elements, respectively, which are subjected to tension loadapplied perpendicular to the exponential material gradation. Fig-ure 23 shows a comparison of the displacements (ux and uy)

computed numerically with those obtained by means of Eq. 18for all the element types investigated in the present loading case.The curves for uy indicate that the best elements in terms ofmatching the analytical solution Eq. 18 are Q8 graded, Q8homogeneous, Q4 graded and Q4 homogeneous, which is some-

how expected. Qualitatively, the nodal stress plots consideringlinear material variation are somewhat similar to those of Figs. 21and 22 and are not given here 36 .

A similar comparison is also made for a different loading caseconsisting of bending applied perpendicular to the material grada-tion. Figure 24 shows a comparison of the displacements (ux anduy) computed numerically with those obtained by Eq. 18 for allthe element types investigated in the present loading case. Asexpected, the Q8 elements capture the analytical solution Eq.

18 for uy better than the Q4 elements. For the sake of brevity,the nodal stress plots are not given here 36 .

Finally, a few comments regarding the case of tension loadingapplied parallel to the material gradation in orthotropic FGMs

Fig. 20 d are in order. Qualitatively, the counterintuitive behav-ior of homogeneous versus graded Q4 elements is similar to the

case involving isotropic nonhomogeneous materials illustrated byFigs. 17 and 18. Thus, for orthotropic case, the Q4 graded elementalso shows poor performance when compared to the Q4 homoge-neous elements for both material variations i.e., exponential orlinear . The reasons for such behavior are given in the last two

Fig. 21 Stress distribution yy using Q4 elements for ten-sion loading applied perpendicular to the exponentialmaterial

gradation in orthotropic FGMs E1101, E22

00.1, G12

00.5,

120.3

Fig. 22 Stress distribution yy using Q8 elements for ten-sion loading applied perpendicular to the exponentialmaterial

gradation in orthotropic FGMs E1101, E22

00.1, G12

00.5,

120.3

Fig. 23 Displacements ux and uy along y1 using Q4 andQ8 elements for tension load applied perpendicular to the ex-

ponential material gradation in orthotropic FGMs E1101,

E2200.1, G12

00.5, 120.3



12/13

paragraphs of the previous section and will not be repeated here.This is the reverse of the effect seen in the previous load cases forgraded orthotropic materials where the graded elements show su-

perior behavior to the corresponding homogeneous elements.Similarly to the isotropic case, a higher-order element such as Q8 either graded or homogeneous with 22 Gauss quadrature isable to capture the exact solution for this loading case.

5.3 Discussion. This study leads to the following remarks.The isotropic FGM plate see Fig. 6 has length over width ratioequal to 1 and the orthotropic FGM plate see Fig. 20 has ratio 2

for tension and bending loading cases . Because the analyticalsolution Section 2 was derived for an infinitely long plate, thehigher the aspect ratio within limits the better the numericalsolution with respect to the analytical one . For some load cases,e.g., tension and bending perpendicular to the material gradation,the homogeneous elements give y y as a decreasing function of xin each individual element on the left side of the domain, whilethe exact solution shows an increasing trend ofyy with x for this

portion of the domain. However, the graded elements show thesame trend as the exact solution in each element see, for ex-ample, Figs. 11 and 12 . The stress plots show that the graded Q8element gives a smoother stress profile than the graded Q4 ele-ment cf. Figs. 9 and 10 . For each loading case, the numericalvalues of the stress components other than the relevant normalstress quantity should approach zero. Thus the remainder of thisparagraph focus on the maximum magnitude of these stress values

obtained numerically which are theoretically zero. In general,these stress magnitudes are lower with Q8 than with Q4 elements.For tension parallel to the material gradation, the numerical valuesof the stress xy and yy are exactly zero for all cases investi-gated. For the fixed grip case, the largest magnitude of xy is

O(102) and occurs for the orthotropic plate with Q4 elements.

The largest magnitude of xx and xy is O(103) or less for all

other analyses for this loading case. For tension and bending per-pendicular to the exponential material gradation, the Q4 element

leads to spurious shear stresses ofO(100) for the orthotropic plate

and ofO(101) for the isotropic plate. Smaller magnitudes for themaximum shear stresses are obtained considering linear material

gradation. The stress xx is ofO(102) or less for all the analyses

involving these two loading cases.

6 Concluding Remarks

Graded finite elements, which incorporate the material propertygradient at the size scale of the element, have been presentedusing a generalized isoparametric formulation. Both linear Q4

and quadratic Q8 quadrilateral elements have been investigatedin detail. To address the influence of material property variation,both exponentially and linearly graded elements have been con-sidered and compared. Several plates with continuously nonhomo-geneous isotropic and orthotropic materials were considered underfixed grip, tension, and bending conditions. The performance ofgraded elements was compared to that of conventional elementswith respect to analytical solutions.

Higher-order graded elements e.g., quadratic and higher aresuperior to conventional homogeneous elements based on thesame shape functions. One should be careful when using graded

elements with linear shape functions e.g., Q4 as it may loseaccuracy in certain situations such as uniform traction parallel tothe material gradient direction. When using this element, we rec-ommend to average the nodal properties of the element, whichwould convert it to a regular homogeneous element. Thus thevalue of material properties at the integration points used to com-pute stresses depend on whether first-order or higher-order ele-ments are used. This simple procedure leads to a more robustelement. A similar procedure is used in the finite element codeABAQUS 37 for heat transfer analysis and also in theWARP3D code 38 .

Acknowledgments

We gratefully acknowledge the support from the National Sci-ence Foundation NSF under grant No. CMS-0115954 Mechan-

ics and Materials Program and from the NASA Ames ResearchCenter NAG 2-1424 to the University of Illinois at Urbana-Champaign.

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