Micromechanical modelling of functionally graded materials

Michael M. Gasik 1

Helsinki University of Technology, Vuorimiehentie 2 K, P.O. Box 6200, FIN-2015 Hut, Finland

Abstract

The problems in design of functionally graded materials (FGMs) are outlined and their modelling approaches are

reviewed. Due to the concentrational or structural gradients in FGMs, the ``normal'' approximations and models, used

for traditional composites, are not directly applicable to graded materials. The goal is to show the e�ciency of the

simplest models to provide the most accurate estimates of the properties and even to make simple elasto-plastic analysis

of FGM components without vast computations by FEMs or an array of empirical ®tting parameters. The development

of a micromechanical model for FGMs with an arbitrary non-linear 3D-distribution of phases and corresponding

properties is presented and the model application is discussed in comparison with other similar approaches. The model

allows the prediction of basic properties of a 3-D FGM, computations of thermal stresses, and, in some limits, it may be

used for pre-design evaluation of dynamic strain/stress distribution and inelastic behaviour. Since all equations of the

model are expressed in a simple analytical form, the model is rather ¯exible for computations and may be easily im-

plemented. As an example, results for W±Cu FGM are presented for application of upper divertor plates for the in-

ternational experimental thermonuclear reactor (ITER). Ó 1998 Elsevier Science B.V. All rights reserved.

PACS: 81.05.Mh; 81.40.Jj; 82.20.Wt; 62.20.Dc

Keywords: Graded materials; Micromechanical model; Stress analysis; Tungsten; Copper

1. Introduction

The trends in advanced materials R&D have formany years shown the struggle to obtain reliablehomogeneous materials such as metal alloys, ce-ramics, etc., striving that no signi®cant di�erencesin material properties will be revealed in the bulkvolume of ®nal components. New developments inthe hypersonic space plane (HySP) and hypersonicvelocity civil transport (HSCT), modern spaceprojects like new shuttle concept (HYFLEX), nextgeneration power systems etc., have created a lot

of speci®c problems in materials design and ap-plications. The development of the space planefaces a huge number of technical problems, inparticular in the ®eld of a superior thermally re-sistant material [1±5]. Usually ceramic composites,combining a ceramic matrix and a dispersionphase, are designed to use superior and sometimessynergetic characteristics of each of these constit-uent materials. They possess an evenly dispersedreinforcement phase, and the resulting propertiesare rather uniform. For the space environment itwill fail to withstand re-entry of the HSCT planedue to high thermal stresses generated by extremetemperature gradients.

For the purpose of attaining a superior stressrelaxation, materials possessing superior oxidation

Computational Materials Science 13 (1998) 42±55

1 Tel.: +358 9 451 2775; fax: +358 9 451 2799; e-mail:

mgasik@pop.hut.®

0927-0256/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 0 4 4 - 5

and thermal shock resistances, and other relatedcharacteristics, are highly desirable [3±5]. Thesegoals have become very common in many other®elds of modern engineering, such as electronics,communication systems, tool design, and otherspecial applications [5]. Such arti®cial materialswill be inhomogeneous and will be characterised(in the most common case) by a non-linear, graded3D-distribution of phases and correspondingproperties [1,3,5]. They are distinguished fromconventionally isotropic materials by gradients ofcomposition, phase distribution, porosity, texture,and related properties (hardness, density, resis-tance, thermal conductivity, Young's modulusetc.). The FGM is characterised not only by thepresence and appearance of compositional orother gradients but also by the sophisticated be-haviour of FGM component in comparison withconventional (macroscopically uniform) materials[1,7]. The initial idea of a graded material was tocombine the incompatible properties of heat re-sistance and toughness with low internal thermalstress, by producing a compositionally gradedstructure of distinct ceramic and metal phases,having in mind the development of a rocket enginewith long-term durability [3±5]. This concept wasbroadened to include a combination of dissimilarmaterials without having explicit boundaries forcreating of materials with new functions. The lat-est developments of theoretical principles ofFGMs, a variety of their processing techniquesand applications are summarised in several publi-cations [1±7], to which the reader should refer formore details. In this work, the main principles ofFGM structure modelling, justi®cation of severalmodel approaches, tungsten±copper FGM prop-erties evaluation with respect to their applicationin a fusion reactor are discussed.

2. Peculiarities of modelling for functional graded

materials

2.1. Some basic features of FGMs

In the simplest case, the structure of a materialis represented or replaced by the model-like systemof a matrix with embedded particles or grains. For

such composites the microstructural ®elds are as-sumed to be homogeneous, whereas for FGMsthey are heterogeneous. Due to the gradients inFGMs, the ``normal'' approximations and models,used for traditional composites, are not directlyapplicable to FGM. The situation becomes evenmore complicated, when an FGM has gradients onseveral levels, i.e. macro-, micro- and nano-scale,where defects such as vacancies and dislocationsstart to play an important role in the transferprocesses and mechanical behaviour of the speci-men [5].

The main methods actually used are based onthe ®nite element approach (FEM) and its varia-tions. Most numerical schemes use discrete distri-bution models, not all of them taking into accountpossible non-isotropic phase distribution. Whenthis distribution is not isotropic but complies witha certain law, the same observation of measure-ment process yields di�erent results depending onthe way the sample is placed. It is obvious that theresources necessary to de®ne, conduct, and in-terpret such an analysis, are prohibitive for com-plex structures [5,8]. Another opportunity wouldbe in a modelling of the FGM structure and in adeducing of ``structure-property'' relationships,e.g. as micromechanical model.

At the moment there is no general theory forFGMs, and only a few publications have beencontributed to this, except for fracture mechanicsand some related areas [9]. Recently a combinedtheoretical approach based on the equation ofmotion [10] and gauge transformation of Yang±Mills ®elds [11] has been developed for a descrip-tion of the mechanical (stress, strain) and tem-perature ®elds in a ``graded media''. On thecontrary to existing semi-empirical or solely nu-merical calculations, these approaches allow exactcomputation of crystalline solids with dislocations,phase grain boundaries etc., by means of a singletheoretical concept. The most essential feature ofthis approach is in exclusion of the singularities ofthe derivatives from consideration, since there areno singularities in the local domain by de®nition,and the integration removes singularities betweenthe domains [12]. The approach suggested could beextended to ®elds of any nature, such as mechan-ical, concentrational etc., whereas the form of

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 43

equations and the method of their solution remaininvariant for the kind of problem considered.

2.2. Review of modelling approaches

Despite the relative simplicity and smartness ofthe Yang±Mills ®elds application to FGMs, thereare yet not enough data for application of thistheory to engineering needs. That is why addi-tional models are used more extensively with FEMcalculations to reveal the peculiarities of FGMs.The majority of these models are employed for atwo-phase composite structure with certain parti-cle size distribution [5,13]. The experience of thesemodels' application has shown that most of themhave been employed mainly to estimate the e�ec-tive properties of the materials regardless of theunique FGM microstructure. Zuiker [13] has re-cently analysed the limitations of such FGMmodels and some of these models are consideredhere further in the same or similar form to ensurethe compatibility of results, discussed below.

In this work only these methods that use stan-dard micromechanical techniques are analysed.Since almost all of them are based on the repre-sentative volume cell or element de®nition, theconclusions drawn are limited to those cases,where this de®nition is valid [5,13]. Thus the goal isto show the e�ciency of the simplest models toprovide the most accurate estimates of the prop-erties and even to make simple elasto-plastic

analysis of FGM components without vast com-putations by FEMs or an array of the empirical®tting parameters.

The models, which have been developed forcomposites, are employed also for FGMs (Ta-ble 1), although with some modi®cations (a de-tailed survey of these models is presented in [13]).The rules of mixture (Voigt and Reuss) are in-cluded for reference, whereas the method of cells,the Tamura model [24] and the more complicatedCPA and SCM are left aside. These last twomethods have been shown to be almost identical,but they require more calculations than otherones. The Tamura model uses actually a linear ruleof mixtures, introducing one empirical ®tting pa-rameter known as ``stress-to-strain transfer'' [5,13],

q � r1 ÿ r2

e1 ÿ e2

: �1�

Estimates for q� 0 correspond to Reuss ruleand with q��1 to the Voigt rule, being invariantto the consideration of which phase is matrix andwhich is particulate. Certainly, Tamura method ishighly dependent on values of q and one has totake into account that FGMs usually should havea range of these values, varying through thethickness of the material [5]. Even when a homo-geneous (non-graded) composite is considered, thevalues of q change along with the degree of plasticdeformation. Dietrich e.a. [26] have shown that foran Ag±58%Ni composite this ratio varies from�1.7 to �0.4 for eeff � 0.05±0.4.

Table 1

Micromechanical models used for composites and FGMs properties evaluation

Model or method name Algorithm Fitting parameters Refs.

Reuss rule Explicit 0 [5,15]

Voigt rule Explicit 0 [5,15]

Sasaki±Kerner a Explicit 0 [13,17]

Mori±Tanaka a Explicit 0 [13,18]

Wakashima±Tsukamoto a Explicit 0 [13,18,19]

Hirano (fuzzy logic) a Explicit >1±3 [13,20]

Coherent potential approximation (CPA) Implicit 0 [21]

Self-consistent method (SCM) Semi-explicit 0 [13,22]

Method of cells Numerical 0 [23]

Tamura Explicit 1 [13,24]

Levin (for thermal expansion only) a Explicit 0 [14]

Gasik±Ueda (present work) a Explicit 0 [5,15]

a Methods that are analysed in this study.

44 M.M. Gasik / Computational Materials Science 13 (1998) 42±55

Levin's approach listed in Table 1 is used forevaluation of the thermal expansion coe�cientonly, providing that the bulk modulus was deter-mined by some other means [13,14],

a � a2 � �a1 ÿ a2�1K ÿ 1

K2

1K1ÿ 1

K2

; �2�

where K refers to the bulk modulus of a compositeof two phases 1 and 2. All other models were takenfor calculations of the properties of two materials,namely SiC±C composites and W±Cu pseudoal-loys. Calculations for SiC±C composites were doneby Zuiker [13] and here it was repeated to makesure that models parameters and equations pro-duce compatible results. Hence only results of W±Cu materials are presented in this work.

3. FGM micromechanical model

The micromechanical model, which was espe-cially developed for FGMs, relies on the mainassumption that a small representative volume el-ement is su�cient to predict the bulk properties ofFGMs [5,8,13,15]. Each sub-cell is assigned acorresponding function of volume concentrationof the second (particulate) phase v2(x, y, z). Thedistribution of the second phase for the structure isde®ned by any function, where the v2 changes from0 to 1. This function should satisfy the boundaryconditions S°(r)� v°2(x, y, z) for values {x; y; z} 2{0; 1}. The sub-cell dimensions should be chosento be small enough so that the structure inside itmay be considered as locally orthotropic. In thiscase the sub-cells became local representative vol-ume elements (LRVE). These cubic LRVEs areused to de®ne stress and strain components andmatch actual local properties [8,15]. The materialcharacteristics are stated by relating the stresscomponents on the LRVE's surfaces to their de-formation. However, due to the anisotropic natureof FGMs, combined with their limited availabilityin forms suitable for evaluation of mechanicalproperties and stresses along appropriate axes, thedetermination of these parameters experimentallyis very di�cult [5,8].

For FGMs an additional assumption was madeby Gasik and Lilius [15]. The second phase is

treated as ``inclusions'', and they are transformedinto a cube in each LRVE with the equivalentvolume, equal to the volume concentration of thisphase in this LRVE. In the 3-D space the positionof (i, j, k)th LRVE of the sample is described by aset of relative coordinates (x, y, z)ijk. Therefore,one can stipulate any real continuous concentra-tion function of these coordinates, resulting in aunique set of iso-concentration surfaces. Thecomplete set of the resulting equations and theirjusti®cation are presented elsewhere [5,15]. Therealso are published results of use of such model forestimation of properties in di�erent composites(FGM and non-FGM) and their analysis [16,27].The values of moduli, thermal conductivity etc.,change from one sub-cell to another according tothe volume concentration of the second phase v2.For instance, the original equation for the elasticmodulus derived from the geometry of the LRVEand the stress±strain relationship was shown as[5,15],

Eii � E1 1ÿ����m2

23

q1ÿ 1

1ÿ ����m2

3p �1ÿ E1=E2�

� �� �;

�3�where i represents one of the coordinate axes (X, Yor Z). This equation (as well as other modelequations) can be simpli®ed to,

Eii � E1 1� m2

FE ÿ ����m2

3p

� �; FE � 1

1ÿ E1=E2

�4�

in which form it was used in this work to reducecomputation time. From the last equation onemay clearly see that model equations have nosingularities over the whole range of v2 2 [0; 1].When two phases have equal moduli (E1�E2),FE ® 1 , but the module Eii ® E1 (�E2).

Anisotropy in the properties will appear aftergathering (assembling) the LRVEs into one blockto obtain the FGM specimen body [5]. The mic-romechanical model uses the transforming of thesub-cells to the LRVE, which could thereafter beapplied for computation and analysis. In addition,one can calculate thermal di�usion, speci®c heat,density, and other parameters in each LRVE. Thelast step is the accumulation of the LRVE in thewhole FGM-specimen, taking into account boun-dary conditions and continuity of the solid [5,15].

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 45

The comparison of the calculation results wasmade with the models mentioned above ®rst for a``composite'' of the tungsten±copper system withzero gradient in composition. Thus, the propertiespresented here are related to a homogeneous two-phase W±Cu composite. The initial parameters forW- and Cu-phases have been taken from di�erentsources [15] and assessed (Table 2) for room tem-perature (although calculations can be made atany temperature, as shown below). It was alreadyshown, that this model satis®es the Hashin±Shtrikman limits [5,15] as well as ``dilute com-posite'' approximations [13,16].

4. Model application example: W±Cu FGMs

The calculations of the properties of FGM andits elasto-plastic stress analysis have been made forthe tungsten±copper system, which is known to befree of mutual chemical interactions [15]. Besidesbasic properties of composites and graded speci-mens, the elasto-plastic analysis was made for aspeci®c application of this W±Cu FGM in an ex-perimental fusion reactor. In the InternationalThermonuclear Experimental Reactor (ITER),plasma facing components (PFCs) will be sub-jected to a high heat ¯ux. Therefore, the protectionof these components is a very important issue forthe design of ITER especially the divertor platesthat will be subject to high heat loads. The divertoris a ¯uid cooled component of a fusion reactor,and there are a number of factors which a�ect theheat transfer capabilities of the divertor: heat ¯uxdistribution, tube materials, orientation. The de-velopment of actively cooled components for the

divertor of a fusion reactor that can withstand thehigh heat ¯ux during transient and normal oper-ation of the reactor is one of the key issues for thedevelopment of the fusion reactor [16].

Although the divertor concept is mentioned themost here, the same approach is valid for ``®rstwall'' materials and assemblies, which are em-ployed in the fusion reactor. PFCs su�er frommany other factors such as erosion, runawayelectrons, neutron damage etc., but in the presentstudy mainly the behaviour of W±Cu divertorplates under thermal load and their optimisationby FGM concept are presented. Using the micro-mechanical model, the optimal elastic response ofa divertor plate was calculated. After that, inelasticbehaviour was introduced into the model, and theresulting stress relaxation maps were calculated[16].

4.1. Initial parameters and properties computation:comparison with other models

Basic properties were calculated in comparisonwith other models for a W±Cu composite, varyingthe volume fraction of tungsten from 0 to 1. Hereno gradient in concentration was applied [5]. Theinitial data for tungsten and copper were taken asshown in Table 2. Besides, temperature depen-dence of these properties was taken into account infurther stress analysis.

The examples of calculation of bulk modulus,CTE and shear modulus are presented in Figs. 1±3. The present model (Gasik±Ueda) is countedtwice: once for tungsten particulates embedded incopper matrix, and second for copper particu-lates in tungsten (marked as ``inverted''). For a

Table 2

Properties of tungsten and copper used for computation [5,15]

Tungsten Copper

Elastic modulus E [GPa] 411.4 ) 0.044 t 128 ) 0.0294 t

Shear modulus G [GPa] 159.5 ) 0.0184 t 47

Poisson's ratio l [)], at 20°C 0.285 0.364

Bulk modulus K [GPa], at 20°C 325 161

Thermal conductivity k [W mÿ1 Kÿ1] 162.29 ) 0.07 t + 0.0002 t2 401.42 ) 0.0625 t

Speci®c heat cp [J kgÿ1 Kÿ1], at 20°C 138 385

Linear CTE a á 106 [Kÿ1] 4.6 16.81 + 0.005t ) d4 á 10ÿ6 t

Density q [g cmÿ3], at 20°C 19.3 8.96

46 M.M. Gasik / Computational Materials Science 13 (1998) 42±55

comparison, the Sasaki±Kerner approach for theCTE (Fig. 3) was also calculated as ``inverted''one. Certainly, for small volume fraction of tung-sten ``inverted'' case is unlikely to take place, butan interesting feature is that the inverted Sasaki±

Kerner CTE values are lower for tungsten matrixthan for copper one.

The results show that the suggested microme-chanical model [15] gives values of bulk and shearmoduli higher than Sasaki±Kerner model and

Fig. 2. Shear modulus [GPa] of W±Cu composite vs. volume fraction of tungsten at room temperature.

Fig. 1. Bulk modulus [GPa] of W±Cu composite vs. volume fraction of tungsten at room temperature.

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 47

closer to Hirano's (fuzzy logic) approach, how-ever, without any arti®cial ®tting parameters. TheCTE values show a larger di�erence between thesemodels. ``Normal'' micromechanical calculationsare close to those of Voigt±Turner±Levin method[5,13±15], whereas ``inverted'' calculations givevalues varying between the Sasaki±Kerner [13,17]and Mori±Tanaka [18] values. The ``inverted''phase layout (copper grains in tungsten) giveshigher CTE values at lower tungsten volumefractions, coincides with Mori±Tanaka at m2� 0.5,and approaches the inverted Sasaki±Kerner valuesat m2 ® 1. It should be noted, that the presentmodel makes direct CTE calculations from themicromechanical principles, without relying on thevalues of the bulk modulus of a composite, as in allother methods. It is possible to show that errors,appearing during the bulk modulus computations,may accumulate or vice versa, cancel each otherwhen CTE is evaluated.

Therefore, the model [15] can be used for thecorrect prediction of properties of two-phase par-ticulate composites with the use of simple equa-tions on the contrary with other known models. Inprinciple, the model allows nesting of the phaseson di�erent levels. For example, it was applied for

a W±Cu±Diamond composite (a diamond tool forrock drilling applications), where the W±Cu com-posite (tungsten grain size 1±3 lm) forms a matrix[5]. For this ``composite matrix'', micromechanicalequations were applied ®rst to calculate propertieswith a small LRVE size. These results were furtheremployed on the second stage for the ``matrix-di-amonds'' composite (mean diamonds size 250±400lm) with coarser LRVE grid size [5,11,27].

4.2. FGM properties calculation

As was shown earlier [13,15], there is no othermodel, which could be directly applied to com-putation of FGM, using standard micromechani-cal approach. None of these models [13] in theiroriginal form is sensitive to the anisotropy ofmaterial. On the contrary, application of the de-veloped model [15] is possible to FGM with anarbitrary 3-D distribution function. This is clearlyseen, for instance, when di�erent functions of aconcentrational gradient (Table 3) are used forthermal conductivity calculations [15] (Fig. 4). Itwas also shown [5] that for phases with largerdi�erences in thermal conductivity (e.g. zirconia±nickel FGM [4,5,27]) these di�erences in X-, Y-, Z-

Fig. 3. Linear thermal expansion coe�cient [Kÿ1] of W±Cu composite vs. volume fraction of tungsten at room temperature.

48 M.M. Gasik / Computational Materials Science 13 (1998) 42±55

components of thermal conductivity may lead torather complicated heat ¯ow and resulting crack-ing due to thermal stresses. This e�ect can behardly calculated with other models (Table 1).

On the basis of the modelling equation like (4),a FGM plate of the W±Cu system with a 3-Dgradient of tungsten concentration was consideredfor modelling of properties and further elasticanalysis [16,27]. The initial conditions and the ge-ometry were taken as for a divertor plate, sub-jected to cyclic heat load of 15 MW mÿ2 [16](Fig. 5). In the case of a plasma disruption eventthe hot plasma is dumped along the magnetic ®eldlines to the divertor plates in a very short intervalof time, causing a sudden evaporation of a thinlayer of the divertor plate material. However, mostof the ablated material would be re-deposited backto the surface. These erosion and re-depositionprocesses during plasma±material interaction aremajor concerns in a design of PFCs [16]. The de-

position of heat by neutrons and impinging highenergy particles on the divertor is also non-uni-form. This uneven heat deposition will result in anuneven temperature distribution. This would leadto an exchange of radiation heat between thevarious parts of the reactor interior that couldhave signi®cant e�ect on the divertor plates[16,25]. In the present work, however, only a purethermal load in one direction was considered dueto a lack of experimental information about tem-perature distribution in divertors. Here and belowcalculated stresses are assumed to be stresses farfrom the edges, unless specially stated.

4.3. Elastic stress analysis

Before making elastic stress analysis of FGM,the one-dimensional Fourier equation has beensolved for a prescribed gradient function (Fig. 5),to get the quasi-static temperature distributionalong with the gradient of tungsten concentration.Obviously, the variation of power parameter p(Fig. 5) will result in di�erent maximal tempera-ture on the tungsten surface at ®xed temperature(20°C) on the ``cold'' side [16,25]. This distributionwas used to re-calculate the mechanical propertiesof tungsten and copper in LRVEs with the equa-tions in Table 2.

The elastic response of the W±Cu FGM for di-vertor plate conditions was modelled by applyingconventional plane strain equilibrium conditions

Fig. 4. Anisotropy of thermal conductivity [W mÿ1 Kÿ1] along

X-, Y- and Z-axes for four types of concentration functions [5],

Table 3.

Fig. 5. The model of a divertor plate (dimensions in mm) with

tungsten coating, W±Cu FGM layer, and copper substrate. VW

± tungsten concentration function [16], the whole concept by

Toshiba Corp. [25].

Table 3

Concentration distribution functions for W±Cu FGMs [5,15]

Name Function

Linear 1 ) X

Ellipse 1 ) (aX á X2 + aY á Y2 + aZ á Z2)1=2

Cylindrical 1 ) a á (X2 + Y2)1=2

3rd order 1 ) (aX á X3 + aYX á X á Y + aZY á Y2 á Z)

Values of the parameters (ai) are chosen in such a way that all

materials have a volume fraction of tungsten equal to 0.5.

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 49

as well as the continuity of the body [5,15,27]. Inevery LRVE the stress is assumed to consist ofcontributions from the ``internal'' and ``external''stresses. The internal stresses, related to LRVEstructure, are de®ned for matrix phase (A),

rintA �x; y; z� �

EA�x; y; z�1ÿ lA�x; y; z�

�ZT �x;y;z�

T0

�a�x; y; z� ÿ aA�x; y; z�� dT

0B@1CA;�5�

and ``complex'' phase (where both tungsten andcopper are present along the speci®c direction,[15]) A&B,

rintA&B�x; y; z� �

EA&B�x; y; z�1ÿ lA&B�x; y; z�

�ZT �x;y;z�

T0

�a�x; y; z� ÿ aA&B�x; y; z�� dT

0B@1CA:�6�

These stress components are orthotropic. Ex-ternal stresses in the LRVE appear from the ex-ternal loads �rext� and thermal strains mismatch:

rout;mA&B �x; y; z�

� Eml;n6�m�x; y; z�

1ÿ 1ÿ EA�x;y;z�EB�x;y;z�

� �� �1ÿ

����m2

23p� � ����

m23p

� rmext

EMR

� �emR ÿ em

l;n6�m�� �

; �7�

rout;mA �x; y; z� � rout;m

A&B �x; y; z�

� 1ÿ 1

�ÿEA�x; y; z�

EB�x; y; z��� �1ÿ ����

m23p �

� �;

�8�where superscripts l, m, n denote either coordinate(X, Y or Z), and subscript R the overall value (ofmodulus or strain) for the whole specimen in eitherdirection [27]. External stresses are anisotropic dueto the gradient of concentration m2.

The results of this elastic analysis are shown ascontour plots of thermal stresses �rext � 0� in thedivertor plate, where the anisotropy parameter ``p''varies from 0.2 to 5 (Figs. 6 and 7). It is necessaryto note that calculated high elastic stresses (manytimes more than the yield limit) may really appearin the material, however, for a short time. Theauthors [16,27,28] have shown that at the begin-ning of heat ¯ow through metal±ceramic FGMsafter about 0.1 s very high compressive stresses(1000±1800 MPa) are generated. They are ``relax-

Fig. 6. Elastic thermal stresses (MPa) in copper in a divertor plate (Fig. 5) along X- (left) and Y(Z)-axes (right) vs. plate thickness, mm,

and ``p'' value.

50 M.M. Gasik / Computational Materials Science 13 (1998) 42±55

ing'' after 100±300 s down to ``normal'' level ofabout 400 MPa due to plastic yielding.

4.4. Elasto-plastic stress analysis

The next step in the application of the modelwas in the elasto-plastic analysis of FGM plates.First of all, several sources were used in order toproperly assess the data of yield stress and theinelastic part of the stress±strain curve for puretungsten and copper [35±38]. It is necessary to saythat these data either have quite a large scatter (forCu) or just were not available at all (for W at lowtemperatures). Thus, a compromise approxima-tion of experimental data points and stress±straincurves was made by the author (Table 4). Theequations of Table 4 should be applied with care ±

they re¯ect more the tendency in properties andtheir di�erences rather than some ``true'' experi-mentally observed values. Furthermore, no creepe�ects were considered and the whole model re-mains quasi-static.

The yield onset point was de®ned for copperand tungsten. When the acting stress exceed theyield point, the plastic deformation equation isused, otherwise simple elastic analysis (5±8) is as-sumed to remain valid. There is little known aboutthe yield criteria for FGMs. Usually the von Misesequivalent stress criterion is used, although it isknown to be not directly applied to anisotropicmaterials. If the same stress is acting on severalconstituents, some of them may yield and somemay not. The general solution of the yielding cri-teria for a 3-D solid with an arbitrary anisotropy

Table 4

Plastic parameters for tungsten and copper [16]

Tungsten (20±1100°C) Copper

0.2% yield stress [MPa] 471.9 ) 56.225 ln(t) 447.17 ) 0.6857 t (<380°)

3770.1 exp()0.0096 t) (380±800°)

Yield onset point [MPa] 282.2 ) 37. 468 ln(t) 269.73 exp()0.0032 t)

Stress±strain relation:

ln (r[MPa])� ln [3251.9 ) 387.47 ln(t)] + 0.3195 ln(e) 7.934 ) 0.0042 t + [0.431 ) 0.0016 t + 4 á 10ÿ6 t2] ln(e) (<380°)

7.634 ) 0.0051 t + [0.816 ) 0.0024 t + 2 á 10ÿ6 t2] ln(e) (380±800°)

The copper recrystallisation temperature was taken as 380°C.

Fig. 7. Elastic thermal stresses (MPa) in tungsten in a divertor plate (Fig. 5) along X- (left) and Y(Z)-axes (right) vs. plate thickness,

mm, and ``p'' value.

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 51

usually cannot be analytically found. In the pres-ent study, the simple stress component was as-sumed to be the acting force for yielding of thespeci®c phase. If yielding occurs, the LRVE grid isre-calculated with the accumulation of resultinginelastic strains.

This procedure was ®rst applied to FGM di-vertor manufacturing process [25,29], where asintered tungsten skeleton is in®ltrated by moltencopper at 1100°C and after cooling is brazed to thecopper substrate. The ``zero-stress'' temperaturehence was taken as 1083°C for the FGM (copper

solidi®cation) and 20°C for copper substrate. Thetransient cool-down process was then calculatedand the resulting elastic and plastic strains andstresses were calculated as stated above. The sec-ond part of the analysis was in the application ofquasi-static heat ¯ow to a divertor (Fig. 5) andrepeating of the procedure once more with the newtemperature distribution and existing plastic de-formations and stresses.

The stress relaxation due to plastic deformationcan be observed in Figs. 8 and 9, where plasticstresses appear to be rather di�erent from those in

Fig. 8. Plastic thermal stresses (MPa) in Cu matrix (a, b) and Cu±W region (c, d) in a divertor plate (Fig. 5) along X- (a, c) and Y(Z)-

axes (b, d) vs. plate thickness (mm), and ``p'' value.

52 M.M. Gasik / Computational Materials Science 13 (1998) 42±55

elastic analysis (Figs. 6 and 7). One can see that anFGM with a low anisotropy coe�cient (0.1±1)provides similar thermal stresses over longerthickness and therefore is more advantageous incomparison with higher ``p'' values for this speci-men geometry [16,25]. It gives also about 100 MPalower stress values for the copper substrate.

5. Discussion

The micromechanical model, which was devel-oped for the prediction of the basic properties ofFGMs, seems to be useful for elastic and plasticstress analysis of FGMs even for a very simpleformulation. The model satis®es the Hashin±Shtrikman conditions and dilute approximationsconditions [13], unlike the majority of othermodels. The equations are simple, explicit, and canbe solved without additional assumptions or arti-®cial ®tting parameters. The computation processis easily implemented in any software (e.g. MapleV or MathCAD) in a symbolic form, so specialprogramming is not needed.

On the other hand, one has to consider any ofthe micromechanical model limitations, whenspeaking about elastic and even more plasticanalysis of such anisotropic materials. It is clear,that additional experimental measurements of re-sidual stresses and FEM analysis are necessary tojustify the model application in this case. Another

important limitation of the model lies in the par-ticular microstructure of materials (particles, em-bedded in a matrix). The model cannot be appliedfor ®bre reinforced materials or particulates withhigh aspect ratio. Certainly, for a given micro-structure it is always possible to re-formulate thebasic equations [15], taking into account the plyorientation, ®bre arrangement and phases' conti-guity.

A special problem for micromechanical model-ling of FGMs regardless of the kind of the modelapplied consists of yielding criteria for FGMs [30±32]. There are several attempts to compute yieldingon the free surface of FGMs, total yielding, plasticdeformation accumulation etc., but all the meth-ods utilise either pure FEM or some kind of nu-merical ®tting of FEM-results to certain geometryof the test specimens, i.e. joints [30±32]. The edgestresses also can be computed in a di�erent way.Usually various forms of the Airy stress functionare used [30],

rij�r; h� � KR

�r=R�x fij�h� � r0fij0�h�: �9�

Other studies suggest that Boussinesq stressfunction [34],

rii�r; h� � 2�1� l�p

r0 hÿ acoscos h

sin hÿ R=r

� �� ��10�

Fig. 9. Plastic thermal stresses (MPa) in tungsten in a divertor plate (Fig. 5) along X- (left) and Y(Z)-axes (right) vs. plate thickness

(mm), and ``p'' value.

M.M. Gasik / Computational Materials Science 13 (1998) 42±55 53

which is more simple, may be more advantageousover Airy function [30] for edge stress computationin multi-laminates and joints. The Boussinesqfunction may be constructed in di�erent formsdepending on the coordinate system used and theproblem symmetry [34].

The next important issue of FGM modelling isin crack propagation prediction and fractureanalysis [9]. The analytical solutions of stress in-tensity factors and local stress ®elds near the cracktip have been already established for di�erentcrack geometries although for uniform load con-ditions [33] or for ``graded±non-graded'' interfacesonly [9]. The model gives values of a stress ®eldtensor in every phase zone in each LRVE, and thisinformation may be applied when calculating theenergy release rate ahead of crack propagationpaths. This method, however, does require a sub-stantial assessment with respect to FGM condi-tions as for anisotropical bodies.

In the example of FGM divertors the modelwas shown to allow calculation of residual thermalstresses, when both elastic and plastic cases mighthappen at the plasma disruption conditions in athermonuclear reactor. This provides valuable in-formation for such components, where littleknown about their working conditions and evenless is known about the optimal gradient function.The general analysis con®rms that the usage ofFGM plates, like above, in divertors and ®rst wall-materials allows reduction of thermal stresses andimprovement of life-cycle and reliability. With themicromechanical model one may already pre-screen various graded designs before making FEMcalculations or testing specimens. Despite theseveral limitations, the FGM micromechanicalmodel is very useful for the evaluation of FGMproperties for components with simple geometry[15].

Acknowledgements

The author greatly acknowledges the contribu-tion of Prof. Sei Ueda (Tohoku University, Send-ai, Japan) for introduction of elastic stress analysisin to the model and valuable discussions about themodel development. The substantial part of this

work was supported by grants from the Ministryof Education of Japan and the Japanese Societyfor the Promotion of Science (JSPS).

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