0551

Thernlo- Elastoplastic Characteristics of Heat-ResistingFunctionally Graded Composites

.1. R. Cho* and D. Y. Hat

Schoo] of Mechanical Engineering, Pusan Nationa] University

Jangjeon-Dong, Kumjung-Ku, Pusan 609-7:35, KOREA

J. Tinsley Oden!

Texas Institute for Computational and Applied Mathematics

The University of Texas at Austin, Austin, TX 78712

January 23, 1999

Abstract

This paper is concerned with the study on thermo-elastoplast.ic characteristics of afunctionally graded material(FGM). Compared to the classical layered composites, itshows a wide range of thermo-elastoplastic characteristics according to the choice oftwo major parameters, the thickness-wise variation in material composition ofthe mid-dle graded-layer and its relative thickness ratio. Therefore, by selecting an appropriatecombination of the two parameters. one is expected to select the most suitable layeredcomposite for a given thermal circumstance. Here, we address the parametric investi-gation on its characteristics together with theoretical study on thermo-elastoplasticityand numerical techniques for its finite element approximations. Through the numeri-cal experiments, we examine the effects of the two parameters on thermo-elastoplasticcharacteristics.

1 Introduction

For the high temperature engineering applications, materials are required to have superiorthermo-mechanical properties, such as high temperature strength and creep resistance, ex-

•Assistant Professor.t Graduate Research Assistant .t Director of TICAM, and Cockrell Faculty Regents' Chair in Engineering #2.

1

cellent fracture toughness and thermal shock resistance. Since single-composed materialsare almost impossible to meet such inherently conflicting properties, laminated compositescomposed of appropriately selected metallic and ceramic materials were introduced and, tosome extent, have been successfully used for several decades [4].

However, owing to their methodology, there exists one inevitable disadvantage III

the classical composites, a discontinuity in material composition( and hence, one in thermo-mechanical properties) at the interface contacting layers. This material discontinuity leadsto sharp change in thermal stress distributions, and which may bring stress concentrationor cracking near the interfaces. In addition, those may be easily triggered for the initia-tion of plasticization or cracking, and eventually they have a potential reaching at a fatalunexpected failure [22, 23].

To resolve this problem, a notion of functionally graded materials(FGM) was intro-duced in late 1980's. In FGMs, a material composition varies continuously from one endto the other of the graded layer. This material technology was originally initiated for thedevelopment of heat-proof structures of space shuttles [21].

According to the development of advanced manufacturing techniques, such as chem-ical or physical vapor deposition, thermal spray, powder metallurgy, currently many ofextended active research are ongoing for diverse engineering applications to thermal partsof turboengines, high-speed tools, automobile engines, and so on. For a specific goal, anelaborate tailoring of the material composition variation and the geometry dimension isessential to satisfy the clesired functions. This is because that the success of finally designedFGM depends definitely on the suitability of combination of the volume fraction functionand the relative thickness of the graded layer.

In order to optimize the volume fraction function in material composition and theoverall geometry, one needs to explore the thermo-elastoplastic characteristics \vith respectto the two parameters. In other words, one needs to carry out the parametric analysisto examine the plastic zone and its location, distributions of equivalent plastic strain andthermal stress and so on for a variety of combinations of the two parameters.

First, in this paper, we address the parametric and material characteristics of FGMstogether with the fundamental theory on thermo-elastoplasticity problems subjected tothermal loadings. As a next step, we formulate heat diffusion problem and elastoplastic de-formation by means of the unconditionally stable Crank-Nicolson-Galerkin scheme and theincremental numerical technique. With the developed finite element approximation tech-niques, we conduct numerical experiments with a representative two-dimensional beam-likefunctionally graded structure. Through the numerical results, we investigate the effects ofthe variation in material composition and the relative thickness on the thermo-elastoplasticcharacteristics as well as the comparison with a classical layered composite( CLC).

2

2 Thermo-Elastoplastic FGMs

Figure 1 shows a three-dimensional symmetric functionally graded composite plate, where2d and 2da, respectively, indicate thicknesses of the plate and the middle graded layer. Asmentioned earlier, the material composition varies continuously through the thickness suchthat no discontinuity in material properties exist.

Throughout this paper, we assume that layers are perfectly bonded and initiallystress-free, and that the strains are small. Furthermore, the ceramic layer is linearly elastic,while the metallic layer obeys linearly elastic rigid-plastic behavior. Then, naturally thegraded layer becomes a transversely isotropic(with respect to the xy-plane) and linearlyelastic rigid-plastic material.

z

x

Figure 1: A symmetric heat-resisting functionally graded composite.

As discussed in our previous work [31: the thermo-mechanical behavior of FGMsdepends strongly on two parameters, the relative thickness ratio X of the graded layerdefined by

X = da / d, 0 ~ X ~ 1 (1)

and the volume fraction functions Vm(z) and Vc (z) of consti tuting metal and ceramic mate-rials(Here, subscripts m and c refer to metal and ceramic, respectively). In addition, FGMsshow the flexibility in thermo-elastoplastic characteristics along the two parameters, whichis the major predominance to the classical layered composites.

Since two volume fraction functions satisfy the following relation for every point inFGMs,

(2)

one needs to define either of the two. Here, we define the volume fraction function of the

3

(3)

metal such that

( )

NdG - Z

2dG

' I Z I~ dG

1, -d ~ Z ~ -dG

0, dG ~ Z ~ dwhere, the power index N is a positive real number. As a first step, we consider thecharacteristics of FGM as the two parameters approach their lower or upper limits. It isobvious that it approaches the classical bimaterial-type layered cornposite(CLC) as X tendsto zero while the full FGlvfs(f-FGMs) as X tends to unity.

0.8E:

I>c,g 0.6uJJ"E:::I-0 0.4>-a'ilE

0.2

o d Gvertical coordinate in the graded layer -

Figure 2: Plots of the metal volume fraction functions of the graded layer for different N.

On the other hand, the graded layer becomes to be dominated by metal or ceramiccompound according to the power index in Vm approaches 0 or +00, and which is 'welldepicted in Fig. 2. In this paper, we denote a FGM dominated by metal layer as themetal layer-extended CLC(m-CLC) while one dominated by ceramic layer as the ceramiclayer-extended CLC( c-CLC), respectively.

According to the two parameters, we construct two different families of FGMs. First,for a given power index N (0 < N < +(0), the X - family Fx is defined as a set of infinitelayered composites which are sequentially distinguished by the choice of relative thicknessratio such that

:F = {Mx : J\ltx = CLeo MX = f-FGM 0 < p < I}x po, 1 ,-- (4)

Similarly the other family, the N - family FN, is constructed by sequentially varying thepower index for a fixed relative thickness ratio X (0 < X < +(0):

FN = {J\It~ : lvf~ = m-CLC, M~ = c-CLC, 0 ~ q ~ +oo} (5)

4

The purpose of such FGM families is to examine sort of sequential variations inthermo-elastoplastic characteristics along the two defined parameters. In this study, ourconcerns focus on the size and location of plastic zones and the distributions of therma.lstresses and plastic strains caused by a cyclic heating. In our previous study [3], the para-metric analysis on thermal stress characteristics of thermo-elastic FGMs is represented.

3 Theory and Material Properties

3.1 Therlllo-Elastoplasticity Theory

Once after a state of stresses at a point in a functionally graded composite reaches at acritical level, a yielding phenomenon starts from that point. A condition for yielding isexpressed in the following generalized form describing a yield surface,

F(8, r) = f(8) - k2(r) = 0 }

k(r) = k(tp;T) = ko(tp) - k1(T)(6)

where, 8 and tp denote a deviatoric stress tensor of Cauchy stresses (T (Sij = O'ij - ()Oij, () =O'kkj3), and the equivalent plastic strain, respectively. In addition, r is a work-hardeningparameter and T indicates temperature.

For our study we aSsume that FGMs obey isotropic hardening rule and von-Misesyield criterion, then,

(7)

with O'y = O'y(lp; T) indicating a yield stress in uniaxial tension. Furthermore, we employPmndtl-Reuss equations for increments of plastic strains given by

deP = d)'" ~~, d)'" '" a scalar multiplier (8)

where, eP is a deviatoric strain tensor of plastic deformation. Assuming no creep, the totalincrement of deviatoric strain is composed of elastic and plastic strain increments,

de = dee + deP (9)

where, eij = Cij - ~Jijj3, ~ = ckk. Taking a total differentiation to the yield surfacefunction, we have the following relations

_ 8F . 8F. p 8F _dF - 88 . d8 + fJeP • de + aTdT - 0

5

(10)

Figure 3: Normality principle and Prandtl-Reuss relations.

1 (aF OF) of aFd>"= - A as: ds + aT dT , A = os : oeP

Furthermore, we obtainaF oJ2----sas - os -

a F = 2k dk. = 2 fj. (3 (3 = dk1 = _ dayoT dT YJ

2, dT V3dTand, in this paper, (3 is assumed to be temperature-interval-wise linear such that

(11)

(12)

(13)

{c, 6.T > 0

(3= ,0, 6.T ~ 0

Using, the results in Eqs.(10) and (11), we have

C rv constant (14 )

d>"

deP

- ~ (s : ds + 2(3 fh dT) }sd>"

(15)

Next, we establish the relation between equivalent stress a- and equivalent plastic strain Epstarting from the basic relations,

-0 = J3J2a = 2's: s

J3i2s: ds {£ ~ (16)da- = 3/2j'i:S = - s : d88 : 8 4J2

3ada- = -8: d82

6

0'

au •.... - ..... H • H • - • - ••

0'*

0' ._HHH.

Y

E

0'

Figure 4: Elastoplastic and plastic moduli for elastic work-hardening materials.

Then, plugging Eq.(16) into (15), we have

(17)

On the other hand, referring to Fig. 4 depicting the elastoplastic modulus ET and theplastic modulus H, the latter is defined by

_ aO' EETH(cp;T) = otp = E - E

T

Using the definition of k(r) in Eq.(6), we obtain

(18 )

(19)

thus~ we arrive at

dEp = v3d~o = ~ (dO' + v3 (3dT)

Then, equating Eqs.(17) and (20), the following relations are obtained:

(20)

(21)

(23)O'dO' 2

s : de = - + _0'2 d)'3G 3

38 10deP = 2HO'(dO' + v3(3dT) (22)

From the basic relation of de = d8/2G+deP (G rv the shear modulus) together with Eqs.(l6)and (22), we have

7

Substituting dii in Eq(21) into the above relation, we arrive at

8 : de = BdA - QdT (24)

with the definition of (Q = 0 for tlT :::;0)

(3iiQ=J3G (25)

Next, Prandtl-Reuss equations with Skk = 0 imply dtlP = ASkk = 0, so the plastic deforma-tion is incompressible. Hence, we can use the basic relation

da = (dtl - 3ndT) I< (26)

where, [«(= A + 2G/3) is the bulk modulus.

Now, we construct the relation of stress increments using the relations establishedso far,

(27)

dO' ds + Ida

2G( de - deP) + Ida

2G [de - 8dA - ~8dT] + Ida

2G [de - S(8 : de/B) - ~SdT] + (dtl- 3ndT)II{

where I denotes {I, 1, 1,0,0,0}T. It is worthy noting that we apply (~~ = 8) to the secondstep. From the relations of material moduli of linearly elastic materials,

2Gde + dtll]{ = Dede (28)

with De of elastic material matrix(6x6) containing Lame constants.

With the definition of two material matrices, the plastic material matrix Dp and theelastoplastic material matrix Dep given by,

(29)

we finally obtain the constitutive law for thermo-elastoplastic materials defined as [01-

lows(with s : de = s : de):

(30)QdO' = Depde - DeadT - 2G S sdT

Here, a vector a is defined by In. The second term in the RHS of Eq.(30) representsa change of normal stresses due to temperature variation, while the third term reflects

8

the effect of a contraction of yield surface in proportion to the temperature increase. Inparticular, Dp and the third term are identically zero before an initial yielding starts.

For the two-dimensional plane-stress functionally graded composites, we rewritethe above relation into the following form by splitting in-plane(referring to n) and trans-verse( referring to 1') parts,

(31)

(~33)

where in-plane and transverse stress and strain vectors are arranged as {ax, ay, aXy}T and{az, ayz, az:z:}T, respectively, and furthermore an = {I, 1, O}T and ar = {I, 0, O}T. Then,using the condition dur = 0 and the static condensation technique, we obtain the corre-sponding relation given by

dun = (Dnn _ Dnr Drr-1 Drn) den _ 2GQ (sn _ Drr-1 sr) dTep ep ep ep S ep (32)

{ (Dnn _ Dr7·-1 Drn) an + (Dnr _ Dr7·-1 Drr) ar} dT

e ep e e ep e

3.2 Elastoplastic Material Properties

Since the middle graded-layer is a two-phase composite material, its thermo-mechanicalproperties must be determined in terms of those of two constituents and their volume frac-tions. Many theoretical results on derivation for material properties of two-phase compositeshave been proposed by numerous investigators. The reader may refer to [12] for the detaileddiscussion on the approaches introduced so far. Here, basically we employ the rule of mix-tures, and we select A1203 as the ceramic layer and Ni for the metallic layer. Furthermore,we assume that all materials properties except for the coefficient of thermal expansion 0:,

the yield stress O"y, the ultimate stress O"u and strain eu, are temperature independent, andthat the Ni-Iayer and the middle graded-layer obey linear-elastic-isotropic hardening rulewhile the Al203 layer is linearly elastic.

In order to compute the temperature distribution, we use the linear rule of mixturesfor the density p, the specific heat c and the thermal conductivity K-

P = VmPm + (1 - Vm)p, }C = Vmcm + (1 - Vm)ccK- = VmK-m + (1 - Vm)K-c

For the composite material properties involved in the elastoplastic analysis, we use the samelinear rule for the Poisson's ratio v and the coefficient of thermal expansion 0:

v VmVm + (1 - Vm)vc }VmD:m + (1 - Vm)O:c

9

(34)

crGraded-Layer

Figure 5: A representation of the Young's modulus and the elastoplastic tangent modulusaccording to the modified rule of mixtures.

while the modified rule of mixtures is employed for the Young's modulus E, the yieldstress and the elastoplastic tangent modulus ET. The modified rule of mixtures has beenintroduced by Tamura et al. [17] and \.villiamson et al. [22].

According to the modified rule of mixtures, stresses and strains in the two-phasecomposites are assumed to be expressed by the linear rule,

(j : Vm(jm + (1 - Vm)(jc }C - Vmcm + (1 - Vm)cc

and the ratio of stress to strain transfer q is defined as

(35)

(36)

(38)

Substituting the relation (36) into Eq. (35), we have the expression for the Young's modulusof the graded-layer given by

E = [VmEm (:: ;~) + (1 - Vm)Ec] / [Vm (:: ;~) + (1 - V~)] (37)

Here, it has been reported that q of 4.5 CPa is suitable for arbitrary two-phase materials,from the experiment with dual-phase steel. A schematic representation of the expression(37) is shown in Fig. 5. Reminding the fact of linear elastic behavior of Al203 layer andusing Eq. (35), the yield stress of the graded layer is determined by

(jy = Vm(jym+(l-Vm)(jc

{ Ec (q + Em)}(jYm Vm + (1- Vm) Em q+ Ec

In addition, along the similar procedure, we have the elastoplastic tangent modulus givenby

(39)

10

time(sec)

Figure 6: A cyclic heating function applied to the top surface of the functionally gradedbeam.

It is worth to remind that, in accordance with the temperature-dependence of 0' y, 0' u andcu' ETm and ET do also vary a.long the temperature, and the former is determined onceEm' ay, au and Cu are given for the rigid-plastic materials.

4 Finite Element Approximations

The thermo-elastoplastic behavior of functionally graded materials is governed by two setsof field equations, heat diffusion equation and static equilibrium equation. Under the as-sumption of infinitesimal deformation, the two field equations are coupled weakly.

As is well known, heat diffusion equation is expressed by the following initial-boundary-value problem defined in the time interval t E (0, t*] and the spatial domain iV E n suchthat

V' . (K:V'T) + q = peT, -in (0, to] x nT = To, at t = 0

} (40)T = To, on z = -d (0 < t :s; t*)

T = f(t), on z = +d (0 < t :s; t*)vVhere, q denotes an internal heat source, and the temperature condition f(t) imposed onthe upper surface of FGMs is a cyclic function of the period tp, as illustrated in Fig. 6.For the temporal discretization, we make N uniform time partit.ions for a considering timeinterval, that is t1t = t* / N, tk+l = tk + t1t (k = 0, 1,· .. , N -1), and we employ the centeredmethod(i.e., the Crank-Nicolson scheme) for Tk+l/2 and Tk+l/2 such that

(y (Tk + Tk+1 )/2 = (t1Tk+1 + 2Tk)/2 }

(y (Tk+1 - Tk)/ t1t = t1Tk+1 / t1t

11

(41)

NextJor a variational formulation, we define the two scalar-function spaces V(O) and i/(O)for test and trial temperature fields, respectively

~(O) = {Q: Q E H1(O)I'DQ = o} }Vn(O) = V(O) + {wn}

(42)

where"D : H1(O) -t H1/2(OOD) denotes a trace operator, while {wn} are extended HI

functions satisfying Wn Iz=-d= To and Wn Iz=d= f(ln).

Assuming no heating source, we have the following semi-discrete initial-boundary-value problem for the temperature field Tk(x) = Tk-1 + /j.Tk ({/j.wd = {Wk -wk-d):

For a given Tk-I E Vk-1(O), find /j.Tk(x) E V(O) + {/j.wd

such that (k = 1,' .. , N) VQ E V(O)

in {pc/j.TkQ + ~t ",V(/j.Tk) . VQ} dO = -6t 1",VTk-1 . VQ dO

(43)

Now, we make a partition of 0 into a finite number of elements OK with boundaries on/.;', and we enforce this spatial partition obey regularity of time-space partitioning, in otherwords, /j.t/h = f1 with h of the mesh size and a finite positive constant g < +00. On theconstructed finite element mesh, we define finite element approximation spaces Vh(n) and{wn}h with the element-wise continuous basis functions {<Pi(X)};;'1 given by

N

Vh(O) :3 Qh = LQ7<Pi(X);=1N

{Wny :3 W~ = L fi(tn)¢>i(X)i=1

(44)

Substituting the finite element approximated forms into the previous variational form, wearrive at the well-known Crank-Nicolson- Galerkin scheme for successive matrix system ofsimultaneous linear equations

[/j.t] k k 1C + 2:K /j.T = - /j.t [j(]T - , k = 1,... , N (4.5 )

"Here, the two matrices (time-invariant for FGMs with temperature independent materialproperties) are

[Cjj] = in PC¢i¢j dO, [Kjj] = inK,V¢i . VcPj dD. (46)

For the stability and convergence analysis of this scheme, the reader may refer to [8].

12

Neglecting a body force of FGMs, the thermo-elastoplastic behavior is governed bythe following equilibrium equations

"ij(U),j = 0, in n )Hi = 0, on aOD (47)

t· - (T·n· on aONI - IJ J'

and the displacement-strain( Cauchy) relations and the constitutive law derived in the pre-vious section

2Cij = (lli,j + Hj,d (48)

du = Depde - DeadT - 2G~ sdT (49)

In Eq.(47), ti E U(anN) and nj denote components of applied external traction and out-ward unit normal, respectively. Defining a vector-valued function space V(O) of admissibledisplacement fields as

V(O) = {v(:v) E [H1(O)]31IDV = o}we construct a nonlinear variational formulation of the boundary value problem (47):

Find u E V(n) such that Vv E V(O)

r e(v): u(u)dO = r vTtdsIn JanN

(50)

(,j 1)

Since we consider FGMs subjected to a thermal loading (without application of externaltractions), let us express the constitutive law in the next form for the iterative numericalcomputation

IrQU = -DeaT - 2G-sT

S(,52)

Then, Eq.(51) ends up with

in e(v): [Depe(u)]dO = - in e(v): uI(u)dO (53)

Along the similar procedure for the heat diffusion problem, we construct finite elementapproximations vh of v such that

NVh(O) 3 vh, vI! = L Vk 0 ¢k(:V)

k=l

Vh(O) = v(n) n [CO(OW } (54)

Employing an expression in matrix form for 'Of and introducing a divergence-like operatordefining strain tensors by D, we have

(55)

13

(56)

In the above expression, ~ and v indicate a matrix containing finite element basis functionsand a nodal vector of vh, respectively. Substituting matrix forms into Eq.(53), we arrive atthe following system of simultaneous nonlinear equations:

(57)

I-Jere,

Kep = J BTDepB dfl }n (58)

f = - In BT(7'1dfl

For the iterative finite element analysis of the above nonlinear equation system, let llS

introduce a residual force vector :F defined by

:F = Kepu - f (59)

In accordance with the Crank-Nicolson-Galerkin scheme for successive increments of tem-perature field !:J.Tk

, we employ the linearized incremental theory for the increments !:J.uk asfollows:

Kk-l Auk = Afkep (60)

Here, since we assume that FGMs in our study are linearly isotropic work-hardening materi-als, we use D:;l, Qk-l, Sk-I and sk-l computed at the previous stage k-l for computationat the current stage k. With the computed increment !:J.uk, we calculate 6.ek, 6.(7'I(k) andupdate thermo-elastoplastic stress at the current stage k according to

!:J.(7'I(k) = -Dea!:J.Tk _ 2C (~s) k-I !:J.Tk

(7'k = (7'k-l + !:J.(7'k

(61)

(62)

(63)

(64)

Next, we compute D:p, Qk, Sk and sk for the next computation, and we compute 6.€"; andupdate €"; at the current stage

6.€"; = ~k (6(jk + V3j3k 6.Tk)

€"; = €";-l + 6.€"; (65)

where, Hk and 13k are determined from Eqs. (18) and (13), respectively. A schematic flowchart for the incremental scheme for thermo-elastoplastic approximation is presented in Fig.7, where 6.(7'k indicates D:p-16.ek.

Temp

14

k=k+ I Update

D k Qk Sk skep l , ,

FE Approximation for

Heat Diffusion

FE Approximation for ElastoplasticityKk-1.1U k= .1fkep

k kOutput .1U and Compute .1crTemp

Tk=Tk-l+ .1Tk. Uk=Uk-l+ .1Uk

Ok =crk-I +.1~Temp Temp -Temp

Ll£~ = (.1 cr k + 3J12l3k.1T k) / Hk

- k - k-l -Ep=Ep +.1Ek

p

No

Figure 7: Flow chart for the incremental analysis of thermo-elastoplastic FG'tvls.

15

zf(t) t

lLII)

.;!l Ex ~ 0

~'<-<11) .....>,1:~ :l... t

ToL=IOO mm ~ uniform ~

Figure 8: A plane-stress FGM beam and its finite element mesh.

5 Numerical Experiments

For the numerical simulation we took a simply supported symmetric plane-stress FGMbeam. As depicted in Fig. 8, its length L and the thickness 2d are 100mm and 10mm.,respectively, and furthermore its lower surface is kept to 7'0 of 2900 J{ while its upper surfa.ceis subjected to a cyclic temperature function f(t). Referring to Fig. 6, the cyclic period Tp is3.0sec and the peak temperature Tl is set to be 1190°K. The thermo-mechanical propertiesof two constituents are listed in Table 1, and the temperature-interval-wise variations of fourmaterial properties are contained in Tables 1 and 2. We note here that linear interpolationis applied to obtain their continuous temperature variations, and sixteen FGM cases (N of0.5, 1.0, 5.0 and 50 for each X Of 0.2,0.5, 0.7 and 1.0) together with a bimaterial are takenfor the comparative analysis.

Table 1: Material data of A.1203 and lVi.Properties Constituents

Ni Ah03Density (kg/m3) 8900.0 3970.0Young's modulus (Gpa) 199.5 393.0Poisson's ratio 0.3 0.25Specific heat (J/ kg . °[{) 444.0 775.0Thermal conductivity (W/m . °[{) 90.7 30.1Thermal expansion coefficient 7.44 ( 293 °K) 3.00 ( 2930 K)(0[{-I X 10-6) 9.89 (1110 °[{) 5.22 (11100[{)

In accordance with the purpose of this study, the parametric investigation of thermo-elastoplastic characteristics of FGMs, we focus on the distributions of corresponding behav-ior through the thickness, which implies that we exclude the edge effect owing to the finitedimension in the x-direction. To realize such a situation, we imposed no temperature gra-dient on both surfaces of x± at L/2 and we took the numerical results along the z-axis(i.e.,at x of 0).

16

According to the symmetry of thermo-elastoplasticity problem with respect to thez-axis, we took a half of the beam as a finite element approximation model. Basicallywe carried out the numerical simulation with ANSYS commercial software by utilizing theuser interface routine for simulation and material data, and we selected two-dimensional8-node quadrilateral finite elements(plane 77 for the temperature field and plane 82 for thedisplacement field [25]).

Table 2: Variations in material properties of Ni.

II Temperature (0 K) ~ ay(M Pa) I au(M Pa) I £u(%) I ETm(CPa) ~293 148 462 47 0.669400 153 459 46 0.666500 140 459 44 0.726600 138 462 46 0.705700 115 328 64 0.333800 100 245 68 0.213900 69 176 72 0.1491000 59 121 82 0.0761100 45 83 95 0.040

Along the x-direction we made 150 uniform partitions, while three layer-wise uni-form mesh partitions such that total 90 elements are almost quasi-uniformly constructedthrough the thickness regardless of the choice of two parameters X and N. In order toavoid the inherent oscillation problem of the Crank-Nicolson-Calerkin scheme, we carriedout a preliminary numerical experiment for the temperature approximation to determinethe appropriate time step. Since the analytic and numerical study on the time step andthe mesh size for this scheme has been reported many investigators, here we leave furtherreference [1,8, 24] to the reader.

Figure 9 presents the temperature-time histories at the position p in Fig. 8 obtainedusing three different uniform time steps 3.0,1.0 and O.lsec, respectively. For the preliminaryexperiment, we applied thecyclic( up to 9.0sec)-constant(Tl after that) heating function.The reason of such a observation position and a combination of X and N is because that themaximum eigen mode of Eq. (40) is proportional to (/\,/pc) and furthermore the considerabletemperature gradient for the FGM with N of 50 occurs at. the lower interface (i.e., at. theNi-gradient interface). From t.he plots in Fig. 9, we choose a uniform 6.t of O.lsec forentire numerical experiments in this study.

The dependence of temperature distribution on the volume fraction function is shownin Fig. 10 for which the FGM with X of 0.5 was taken. The temperature distribution showsnoticeable difference in its smoothness at two interfaces with respect to the value N. Theirvariations to the relative thickness ratio X for a fixed N are plotted in Fig. 11, where we

17

480

460

440

420".....,

~0 400Se::l 380(;j....~c.. 360EB

340

320

300

2800 5 10 15

time (sec)20

0.1 sec1.0 sec3.0 sec

25 30

Figure 9: Temperature-time histories at the point p for three uniform time steps(x = 0.2and N = 50).

observe the increase in the smoothness of curves in proportion to the extension of the gradedlayer.

In order to observe the edge effect in thermal stress and equivalent plastic straindistributions, we include three figures obtained with X of 0.2 and N of 1.0 showing thermalstress distributions at A in Fig. 12, at B in Fig. 13 and equivalent plastic strain distributionat E in Fig. 14. From the figures, we see the edge effect in the vicinity of the right end,however it disappears in the interior region distant from the right end. Accordingly it isobvious that the numerical results at ;1: of 0 are not influenced by the edge effect. For theelaborate analysis on the edge effect, it is widely known that the gradient meshing techniquetogether with a careful decision of element size are of great importance. The reader mayrefer to [2, 15] for the detailed discussion.

A series of next figures starting from Fig. 15 to Fig. 32 show thickness-wise distri-butions of equivalent plastic strain £p and corresponding thermal stress (Txx at time A. "-' Efor a bimaterial and X-FGM families.

For the bimaterial case(Fig. 15), initiation of plastic yielding starts from the bot-tom surface of the Ni-Iayer and it advanced into the mid-plane(approximately up tox = -3.0mm). According to the cyclic heating, additional successive equivalent plastics areobserved in the vicinity of the bottom surface. Corresponding thermal stress distributionsare shown in Fig. 16, where we see the effect associated with the plastic yielding near the

18

atB,D

...... ,,.;':' ./

,.:" I

.'~' ,-.-:"' ,-

..•• I

,'. I

.<:';"'; //':-' l

.,'. /'),

N=0.5 -+-1.0 -+--5.0 -+..50 .

800

400

600

1200

1000

200-5 -3 -1 1 3

coordinate in the thickness direction (mm)5

Figure 10: Temperature distributions through the thickness at time (A,C,E) and (B,D) forthe N-FGM family(x = 0.5).

1200

1100

1000

900,.-...~0 800<'"-'

~700Z

t':l..Il.lQ.. 600E!:l

500

400

300

200-5

bimaterialRT=0.2

0.50.71.0

-3 -1 1 3coordinate in the thickness direction (mm)

5

Figure 11: Temperature distributions through the thickness at time (A,C,E) and (B,O) forthe X-FGM family(N = 1.0).

19

FGM BEAMNOV 25 19961': 09: 34PLO'l' NO. 1THERMAL STRESS (AITIME=)'SX IAVG)RS'iS=ODHX =.8753-03SMN ::-. 890E-+09SMX =.535E.09c=J .690E+09c:::J -. 712E+09_ -.514E-+09_ -.4158 ..09_ -,257£+09_ -. 988E'~08_ .595E-OS_ . 21SE-09_ .376E·09

.535E-09

Figure 12: Thermal stress distribution at time A for X 0.2 and IV 1.0.

rGM B:=:AHNOV 2S 199614:11:47PLOT NO. 2THERMAL STRESS (B)TIME"'6SX (AVGIRSYS=ODMX =. 264E-04SMN :;:-.233£"09SMX =.188E+09c::J -. 233E+09c:;::J -,186£ ...09_ - 140E .. 09_ - 929E+08_ - .461E+09_ 706374_ .<75E.09_ . 943E'OB_ .141E+09

.ISSE-O'

Figure 13: Thermal stress distribution at time B for X

20

0.2 and N 1.0.

FGM BEAMNOV25 199614,13,14PLOT NO. 3PLASTIC STRAINTlME=15NLEPEQ IAVG IDH}( "'. B15E~03SHl{ =.005101= 0c:::l .567E-03_ .001134_ .0017_ .002267_ .OC2834_ .003401_ 003967_ .004534

.005101

Figure 14: Distribution of equivalent plastic strain at time E for X = 0.2 and N = 1.0.

lower surface of the Ni-Iayer. Despite of the existence of successive plastic yielding near thebottom surface, no noticeable time-variation in thermal stress distributions is found. Thisis because the plastic tangent modulus ET in the Ni-Iayer is too small when compared toEm and Ec, as listed in Table 2.

Figure 17 shows the results of the X-FGM family with N of 0.5. Regardless of therelative thickness ratio X, the plastic yielding occurs in the upper region of graded layerjust below the gradient-Al203 interface. It is worth to note the FGM with X of 0.2 hasthe plastic zone near the bottom surface, and furthermore the plastic behavior of the FGMwith X of 0.7 as well as one( for X of 0.2) near the bottom surface is not affected by cyclicheating. As the graded layer extends, the peak of equivalent plastic strain strictly decreases,but the f-FGM has exceptionally big value. The next two figures, Figs. 18 and 19 showthermal stress distributions at A,C,E and B,D, respectively. For the f-FGM, considerabletime-variation in thermal stress is observed according to the existence of large successiveplastic yielding. Even though it possess smoother and smaller thermal stress distribution,it has excessive plastic accumulation near the top surface. Hence, in this family, the case ofX = 0.7 is more attractive from the thermo-elastoplastic point of view.

Numerical results for the FGMs with N of 1.0 are presented in Figs. 20-22. Comparedto the case with N of 0.5, the plastic-affected region becomes wider, particularly for theFGMs with X of 0.5 and 0.7, and the magnitude for the f-FGM becomes considerablysmaller, which is owing to the relaxation of the steep gradient of the volume fraction Ileal'

21

the gradient-Al203 interface. The occurrence of plastic zone near the bottom surface up toX of 0.7 is another remarkable point. According to the extension of plastic zone showingsuccessive plastic yielding along the cyclic heating, the time-variations in thermal stress areshown in wider regions for X of 0.5,0.7 and 1.0, as shown in Figs. 21 and 22.

Distributions of equivalent plastic strain for the FGMs of N = 5.0 are shown in Figs.23-28 in the order of the relative thickness ratio. Referring to Fig. 2 associated with thevolume fraction variation, the plastic zones in the Ni-Iayer for this N-FGM family startsfrom the bottom surface and those at A show almost same against the relative thicknessratio. On the other hand, the plastic zone starting from the graded layer spreads out asthe graded layer extends, but they do not show remarkable additional effect by the cyclicheating. It is also noteworthy that there exists successive plastic yielding in the N i-layerand the affected region becomes wider a little bit as X increases. Corresponding thermalstress distributions are plotted in Figs. 27 and 28. Owing to the extremely small plast.ictangent modulus of Ni and its small increase by Al203 compound for the f-FGM case,considerable successive plastic yielding by the cyclic heating does not produce noticeabletime-variation in thermal stress distributions.

As depicted in Figs. 29 and 30 showing the plastic strain distributions of FGMs withN of 50, a negligible plastic zone existed in the graded layer for X of 0.2 completely disappearsfor the other relative thickness ratios. In addition, the plastic zone in the Ni-Iayer exhibitst.he trend of shrinkage towards the bottom surface as the relative thickness ratio increases.However, each case experiences remarkable successive plastic yielding in the region belowapproximately x of -O.4mm. Differing from the previous f-FGM with N of 5.0, from Figs.31 and 32, we see the slight variations in thermal stress distributions for the f-FGM nearthe bottom surface according to the considerable increase in ET due to the prevalence ofAl203 compound in that region.

By comparing the figures showing stress distributions, first vvesee the bimaterial dis-plays the steepest stress difference at the interface while the other FGrvls exhibit a wide rangeof variations according to the choice of two parameters. At the gradient-Al203 interface,considerable stress jumps occur except for the f-FGM when N is relatively small(N S 1.0),and which reflects the behavior of the m-CLCs. Along the increase in N, thermal stressdistributions seem to be smoother but they have noticeable stress changes again at theNi-gradient interface when N is 50, which follows the behavior of the c-CLCs. On theother hand, the f-FGM displays the remarkable stress change near the bottom surface onlywhen N is 50.

Figure 33 shows comparatively the total length of plastic zones in the graded layerand in the Ni.-Iayer, respectively. Here, we note that the f-FGM has a graded layer only.In the graded layer, it becomes larger as a whole as the relative thickness ratio increases,

22

3.5I

AB3.0 l- eD--- I E<""l

0 2.56l:.~

2.0<Il

U';:J<Il«lc.. 1.5EIII(;>

1.0'5C"III

0.5

0-5 -3 -I 1 3 5

verticallocalion (mm)

Figure 15: Distributions of equivalent plastic strain for the bimaterial.

A,e,EB,D

0.6

0.4

--- 0.2t<:Ip.,

~ "0

"'-' ">< o uumuuuuu>~', ____«llE.~

-0.2<Il

vi'<IlQ)

tl-0.4<Il

(;

§III

-0.6-5

-0.8

-1.0-5 -3 -1 I

verticallocalion (mm)3 5

Figure 16: Distributions of thermal stress for the bimaterial.

23

3.5I

AB

3.0 l- eD

---- I Ert'l0 2.5I

L.Ll'-"i::'ab 2.0til

U'..::1

~0.. 1.5CQ)

<is>1.0'3

C'"Q)

0.5 . RT=0.2(A-E)

I~0

-5 -3

RT=0.2

RT=0.5

-1 Ivertical location (mm)

3

RT=l.O

5

Figure 17: Distributions of equivalent plastic strain when N is 0.5.

0.6I

A0.41

eE

~i:l..

0.2c:>'-"~",I 0Sb/J';j;v; -0.2til

~til

<is-0.4

EQ)..s -0.6

-0.8

-1.0-5

RTinc.~

-3 -1 Ivertical location (mm)

3 5

Figure 18: Distributions of thermal stress at time A,C and E when N is 0.5.

24

0.6I

B0.4 I

D

,-.

~ 0.20'-'><~I 0EOJ:)

'Vien -0.2'"~'"<U -0.4

~;S -0.6

-0.8

-1.0-5 -3

({T=0.2~ RT=O.~

-1 Ivertical location (mm)

3 5

Figure 19: Distributions of thermal stress at time Band 0 when N is 0.5.

RT=0.2

+ RT=0.5(A-E)* RT=O.7(A-E)

5

RT=1.0

3

RT=0.5~

-1 1vertical location (mm)

-3

3.5I

A-B u ____

3.0 ~ C - ...D ..·.. ·

;:; t E --.--0 2.5~'-'c::'al:l 2.0'".!:l-'"(llc.. 1.5E<I.l

<U> 1.0'5g"

0.5

0-5

Figure 20: Distributions of equivalent plastic strain when N is 1.0.

25

0.6I

A

0.4 r BC

--- 0.20:1

"'"'Q>< 00:1'Eb.O.til

-0.2vi'ellII)

bell -0.4"aEII)

·0.6.s

-0.8

-1.0-5 -3 -1 1 3 5

vertical location (mm)

Figure 21: Distributions of thermal stress at time A,C and E when N is 1.0.

0.6I

B0.4 I

D

---0:1 0.2"'"'c:>'-"><0:1

1 0Eb.O.tilvi'

-0.2ellgell

"a-0.4

§II).s -0.6

-0.8

-1.0-5 -3 ·1 1

vertical location (mm)3 5

Figure 22: Distributions of thermal stress at time Band D when N is 1.0.

26

3.5

3.0

---f'l0 2.5~c

'03t: 2.0til

U....

15 t\1:.l0.i: "-; ~

\\0 , ':'~-a \\\> ].0 ···,i'::;g-

0.5

0-5 -3 -] I

vertical location (mm)3

ABCoE

5

Figure 23: Distributions of equivalent plastic strain "",hen N is 5.0 (X = 0.2).

3.5

3.0

---f'l0

25 ~'r..il ,'-' , i.c ': \

'@ \. \.t: 2.0 "itil'.\

.~til0:10. 1.5....c0-a>

1.0'3c:r0

0.5

0-5 -3 -I I

vertical location (mm)3

A-B ------C ....oE ----

5

Figure 24: Distributions of equivalent plastic strain when N is 5.0 (X = 0.5).

27

3.5

3.0

2.5 '.\

i"

"

2.0 hi;" \i.... \\

" \11.5 ~ ;;;

, ',:~.\"".\:~

\ ',\1,

1.0 " \.!\\'~\1

0.5

o-5 -3 - I 1

vertical location (mm)3

ABCDE

5

Figure 25: Distributions of equivalent plastic strain when IV is 5.0 (X = 0.7).

3.5

3.0

,.....,<")

2.5 1\0~'-'l::

'@

2.0.l:I'"u

'':::~

Q.. 1.5C0

«J:>

1.0'30'0

0.5

0-5 -3 -I 1

vertical location (mm)3

A-BCDE

5

Figure 26: Distributions of equivalent plastic strain when N is 5.0 (X = 1.0).

28

0.6I

RT=0.2(AIC,E)0.4 ~ 0.5(A,C,E)

0.7(A,C,E)1.0(A,C,E)

..-,0.2ro

0-Q>< o ~. ........ "-'_ .•~"" .... ".."I ....... ,..roEbll

'Vi -0.2cJ;en

~'" -0.4

t;J§(1)

-0.6-S

-0.8

-1.0-5 -3 -I ] 3 5

vertical location (mm)

Figure 27: Distributions of thermal stress at time A,C and E when N is 5.0.

RT=0.2(B,D)O.5(B,D)0.7(B,D)1.0(B,D)

0.6

0.4

'2 0.20-0......,

>< 0IroE.~

-0.2encJ;en(1)

!:len -0.4t;J§(1)

-0.6-S

-0.8

-1.0-5 -3

..' -' -::.-.-:..~.-:.~~?

-] Ivertical location (mm)

3 5

Figure 28: Distributions of thermal stress at time Band D when IV is 5.0.

29

Distributions of equivalent plastic strain when N is 50 (X = 0.2 and 0.5).

-I

A-B mno

C h

D-E --.--

(RT=O.5)

-3 ·2vertical location (mm)

-4o·1

3.5I

A-B __m

~=1 3.0

E ---- ;;;0 2.5~J::.~

(RT::O.2) 1 !:l 2.0:/I

.g'"<'lQ. l.SEIl<l)

5 1.03'

0.5

-3 -2vertical location (mm)

-4

3.5

3.0

""r<\0 2.5~~;II 2.0'".~~Q. 1.5C\l~)

5 1.00'Il

0.5

0-5

Figure 29:

3.5 3.5A- A-B m __

B m __

10 ~ C hh .. 3.0 C ..D-- D-

"" ; E --- "" E -..--r<\

25 "

r<\0 0 25~ .. ' ~ .,

'\ \ ,J:: J:: ..

'.. \ .,a \ ; 'a i

II 2.0 ...i !:l 2.0,

fI. .... \ (RT=O.7) '" \! (RT=1.0)u \, \\ .S! ':\... ~\

" :1 :;; .. ,~ " :.,'~ <'l ....,.r~'E.

• ":1 Q... ,

1.5 \, ....\\ 1.5 . ;

E E'.:.1.. '. \~ ~.}

Il '. ",",: ~ u .',-; \ '.\1, -; n .... , :i

\'.\1. \ "',l) \\\)

'5 1.0 '5 1.0 ..,\'..\ \~:l'\~

0-u u "1

0.5 r~ 1

0.5

0-4 -3 ·2 -I 0 ·5 -4 -3 -2 ·1 0

vertical location (mm) vertical location (mm)

Figure 30: Distributions of equivalent plastic strain when N is 50 (X = 0.7 and 1.0).

30

0.6I

RT=0.2(A,C,E) -0.4 ~ 0.5(A,C,E) __u_ •

0.7(A,C,E) ....m.

I.O(A).-...

0.2 ~ I.O(C)~~ I.O(E)'0.....,>< 0~Iebll.~

-0.2vi"en

~'" -0.4

t<lEI!)

-5 -0.6

-0.8

-1.0-5 -3 -I I

vertical location (mm)3 5

Figure 31: Distributions of thermal stress at time A,C and E when N is 50.

RTdec.

RT=0.2(B,D) -0.5(B,D) _m_0.7(B,D)I.O(B)I.O(D)

0.6

0.4

'2' 0.2~~>< 0~Ie.~

-0.2envi"'"I!).\::l

-0.4'"t<l§I!)

-0.6-5

-0.8

-1.0-5 -3 -I I

vertical location (mm)3 5

Figure 32: Distributions of thermal stress at time Band D when N is 50.

31

and for a given X it reaches the peak length when N is .5.0. On the other hand, in theNi-Iayer it shows the trend of increase as the power index N increases, while it for relativelybig power indices saturates as the graded layer extends. This figure implies that the widerregion in the graded layer is plasticized as N approaches 5.0 or X increases while thistendency in the N i-layer prevails as N increases.

The plastic occurrence region for each FGM case is plotted in Fig. 34(a), where thebimaterial case is included for which yielding occurs in the lower region of the Ni-Iayer.From the figure, the plastic zone for X of 0.2 is splitted into two regions. One in the gradedlayer is restricted within the region just below the gradient-Al203 interface when N is 0..5.but it moves down to the Ni-gradient interface as the power index N increases, while theother in the Ni-Iayer just extends to the Ni-gradient interface from the bottom with theincrease in N. This tendency is because, according to the increase in N, the positive peakthermal stress occurred at the gradient-Al203 interface moves to the N i-gradient interfaceand the thermal stress near the bottom surface increases negatively, which is well describedin our previous work [3].

According to the extension of the graded layer, the plastic zone within the region justbelow the gradient-Al203 interface(i.e., for N of 0.5) does not show considerable differenceeach other except for the change in its length and location, but it moves down to the otherinterface with the trend showing wider as N increases and even it disappears when N is 50.On the other hand, the plastic zones in the Ni-layer for N of 5.0 and 50 do not show anynoticeable change when compared to those of X with 0.2, however those of N ::; 1.0 shrinkto zero as X increases.

Variation in the locations of peak equivalent plastic strain in the graded layer is shownin Fig. 34(b), where (m) indicates its movement along the heating cycle(i.e., E f- A). Foreach relative thickness ratio, the peak position moves from the point just below the gradient-Al203 interface to the Ni -gradient interface according to the increase in N, and hence themovement width becomes larger as the relative thickness increases. It is worth to note thatthe FGMs with N of 1.0 experience considerable successive changes in the peak positionsalong the heating cycle.

Numerical values of peak equivalent plastic strain in the graded layer are recorded inTable 3, where (-) indicates that no noticeable change due to the heating cycle is observed.Regardless of the value N, the peak strain decreases to some extent but it increases as Xincreases. This is because that the two limiting cases of N -+ 0 or +00 produce steepchange in the material composition near the gradient-Al203 interface or the Ni-gradientinterface. Successive increase in the peak strains is remarkable when X is 1.0 or N is sma.llfor the other relative thickness ratios. Figure 35(a) illustrates this tendency.

Numerical values for the N i-layer are listed in Table 4, where those of the bimaterial

32

are included for a comparison purpose and (x) indicates no existence of the plastic zone. TheFGMs with N ~ 5.0 have larger peak values than the bimaterial case, and in addition, fora given relative thickness ratio the peak value and its successive plastic yielding uniformlyincrease as the power index N becomes larger. Since the Ni-region shrinks in proportion tothe increase in N and hence the temperature gradient at the Ni-gradient interface becomessteeper, which produces negatively bigger thermal stress in the N·i-Iayer. The numericalvalues at A and E are plotted in Fig. 35(b).

By comparing Figs. 3:3 and 35, we realize that the distribution of equivalent plasticstrain in the graded layer becomes more sharp as N approaches zero for every )(, andfurthermore this tendency in successive distributions according to the cyclic heating is moreremarkable when X is relatively small or large. In the Ni-Iayer, the variation in sharpnessat time A in proportion to the power index N is not noticeable, but which prevails as timegoes by.

Figure 36 presents the variations of I (1max - (1min I in the overall FGM beam at(A,C,E) and (B,D) respectively, for which we take the maximum value from the data fordifferent observation times for each case. When the top surface is heated to 1190° f{, theabsolute maximum differences are as a whole smaller than the bimaterial case and thosedecrease as the relative thickness ratio increases except for the case of N=50. It is worthto note that the absolute maximum difference becomes bigger as the power index decreaseswhen X is 0.2, but this tendency is totally reversed when X is unity. This is explained bythe fact that the absolute maximum difference for X of 0.2 is not affected by the plasticyielding but rather by the inherent stress distribution along the power index N, while itsinfluence on thermal stress by the plastic yielding near the top surface when X is unitybecomes considerable as N decreases, which is clear from the previous figures starting fromFig. 17 to Fig. 32. At B abd D, the absolute maximum difference slightly increases inproportion to the increase in N and all cases have bigger value than the bimaterial case.

6 Conclusion

In this paper we addressed the parametric investigation on thermo-elastoplastic characteris-tics of heat-proof functionally graded composites together with the theoretical formulationand the finite element approximations for the thermo-elastoplasticity problem. For the com-parative numerical analysis, we selected the power index of the volume fraction function andthe relative thickness ratio of the graded layer as considering parameters.

Through the numerical experiments with a plane-stress heat-resisting FGM subjectedto the heating cycle, we examined the distributions of equivalent plastic strain and thermalstress, and we analyzed the variations associated with the plastic zone, the peak equivalent

33

505power index N

505power index N

7

(graded layer) /~Il j (Ni-Iayer) RT~.2 -

H 0.5 .--6 0.7

5"" e§

4 E'" '"~ I)c c0 0N N,3 3 .2 3~ :;;:oj CIl0. 0.

Figure 33: Variations of total plastic-zone lengths in the graded layer and the Ni-Iayer.

N=SO

m, m

N::O.5 n _

l.0-5.0 - .... -SO .00\

\\\\\\\\\\\

\~.'.'-,'-. N-SO"'" -''''-,'

0.2

0.5

0.7

N::O.5 -l.0n ... n.

5.0 _,.'u.

50-

(RT~.5)

(RT~.2)

(RT'jO.7)

(RT::l.O)

(bimaterial)

'"ClCC~

,~..'""0.

....o'"cc';l~.r:.E'":a

-5 ·3 .J Ivenical coordinate (mm)

-5 -3 -I Ivertical coordinalC(mm)

Figure 34: Distributions of plastic zones(a) and locations of peak equivalent plasticstrains(b) in the graded layer.

34

Table 3: Peak equivalent plastic strains in the graded layer.

Parameters Peak value at each time (x 10 3)N X A B C D E

0.2 0.660 0.931 1.165 1.369 1.5460.5 0.5 0.596 0.774 0.919 1.037 1.135

0.7 0.252 - - - -1.0 0.714 1.397 1.947 2.386 2.7490.2 0.753 1.001 1.185 1.336 1.479

1.0 0.5 0.663 0.824 0.964 1.074 1.1640.7 0.443 - 0.446 - 0.4571.0 0.928 1.201 1.304 1.376 1.4210.2 0.497 0.516 0.534 0.548 0.558

5.0 0.5 0.297 - - - -0.7 0.348 - - - -1.0 0.928 1.353 1.757 2.157 2.5520.2 0.107 - - - -

50 0.5 0.021 - - - -

0.7 0.322 - - - -

1.0 0.736 1.237 1.668 2.079 2.471

Table 4: Peak equivalent plastic strains in the Ni-layer.

Parameters Peak value at each time (x 10-3)

N X A B C D Ebi- mat'l 0.583 0.723 0.856 0.988 1.120

0.2 0.187 - - - -

0.5 0.5 x x x x x0.7 x x x x x0.2 0.298 0.352 0.404 0.457 0.512

1.0 0.5 0.105 - - - -

0.7 0.065 - - - -0.2 0.756 1.046 1.329 1.609 1.888

5.0 0.5 1.040 1.485 1.919 2.350 2.7780.7 1.032 1.487 1.930 2.369 2.8060.2 1.158 1.552 1.936 2.318 2.698

50 0.5 1.321 1.874 2.413 2.948 3.4790.7 1.085 1.553 2.002 2.449 2.892

35

50

......(Ni·layer)

/,.-": -----.

/

5power index N

RT=O.2(A,E) -O.5(A,E) ..O.7(A,E) -

3.5

1.5

2.5

0.5

50

RT=O.2(A,E) -O.5(A.E) .---.O.7(A.E) __ .h

I.O(A,E) --

I.~i

/

5power index N

(graded layer)

RT=1.0(A)-------------------

./

/RT=I.O(E) /

o0.5

2.5

1.5

Figure 35: Variations in peak equivalent plastic strains at A and E in the graded layer( a)and the Ni-Iayer(b).

1.6 1.6N=O.5-

1.0 m __ ..

5.0 .-50 --

at (B,D)

.--...-0.4 ~h ~ ._<'::'::,:·.~-=-c --.'

. ._ .. .J)j[11aleti~L no. • _ •

0.6

0.8

1.4

1.2

.~.~.•._ .

birnaterial

al (A,C.E)

N=O.5-1.0 .......5.0 .. '.50 -0.4

0.8

0,6

1.2

1.4C:.,~ _

....' .....,...,...,'''\,

''\''"

~ ··~\\<;_'i.::'::"::h.".

0.20.2 0.5 0.7

relative thickness ratio

0.20.2 0.5 0,7

relative thickness ratio

Figure 36: Variations of I amax - amin I in the overall FGM beam.

36

plastic strain, the additional successive yielding and the absolute maximum difference inthermal stress distribution.

In the graded layer, the plastic zone occurs in the region near the gradient-Al203

interface when N is 0.5, but it moves down to the Ni-gradient interface together with thetrend becoming wider and smoother as N increases. In addition, the total length of plasticzone is maximum when N is 5.0 and it shows the trend of increase according to the extensionof the graded layer, but the peak equivalent strain and the sharpness in distributions showbigger values at lower power index(N ~ 1.0). Furthermore, the successive yielding due tothe cyclic heating prevails when X is 1.0 or N is small.

In the Ni-Iayer, plastic yielding occurs in the bottom region and the pla.stic zoneadvances towards the Ni-gradient interface(but, it shows saturation behavior) as N in-creases. In all cases, the peak equivalent plastic strain occurs at the bottom surface, a.ndit shows uniform increase in its magnitude and sharpness according to the increase in N astime goes.

Furthermore we observed that the thermal stress exhibits considerable stress jumpat the gradient-Al203 interface when N is relatively small except for the f-FGM while at theNi-gradient interface when N is 50. However, the stress concentration at the lVi-gradientinterface is not severe when compared to one at the gradient-Al203 interface.

From the numerical results, we observed that a wide range of thermo-elastoplasticcharacteristics is obtained by varying the combination of two parameters. And furthermore,we analyzed the effects of the two parameters on thermo-elastoplastic behavior of FGMs.Consequently this study leads one to the further study on the tailoring the most suitableheat-resisting FGM for a given purpose, for which one should be able to select the appro-priate combination of the volume fraction function and the relative thickness ratio. Thisultimate goal would be possible by the optimization technique together with the employ-ment of spline-like volume fraction functions.

Acknowledgements: The financial support given to the authors(JRC and DYH) by Pu-san National University in Korea under the Grant-in-Scientijic-Assistance (May, 1997-April,2001) and that given to the author(.JTO) by the Office of Naval Research under GTant No.N00014-95-1-0401 are gratefully acknowledged.

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