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Macro and micro scale modeling of thermal residual stresses in metal

matrix composite surface layers by thehomogenization method

D. Golanski, K. Terada, N. Kikuchi

Abstract The modeling of thermal residual stresses gen-erated in TaC/stellite and TiC/stellite composite surfacelayers produced by the oscillating electron beam remeltingon low alloys steel is presented. The homogenizationmethod is applied to analyze the real composite micro-structures by utilizing the digital image-based (DIB) geo-metric modeling technique. Two scales of elastic stressanalysis are studied: macroscopic one referring to theglobal structure of composite layer produced over thesubstrate of low alloy steel and microscopic, comprisingthe selected unit cell of composite microstructure.

The results of the analysis show the microscopic stress to

be few times higher than the macroscopic one with stresslevel much above the elastic limit of matrix material, whichimplies the development of plastic field around the in-clusions. The ceramic inclusions within the unit cell arefound to be under high compressive stresses. Also, thecomposite surface layer stays in compression, mainly bythe influence of the stress component parallel to the layer/substrate interface. The effect of hardphase volume frac-tion is examined and it is found that for a small volumefractions the macro and micro stress does not differ sub-stantially between composites with TaC and TiC hard-phases despite their mismatch in thermophysicalproperties. Also, the stress modeling is presented for the

composite containing other inclusions and the problem ofthe selection between 2D and 3D model for the stressanalysis is discussed.

1IntroductionThere are several industry fields like aircraft, electrical,nuclear or building where metal matrix composites

(MMC) are preferably used for engineering applicationsreducing the use of traditional materials. High strength ofMMC together with high stiffness and high thermal sta-bility at elevated temperatures makes them very attractivefor engineering structures.

One of the fields where application of MMC can bebeneficial are composite surface layers produced over asubstrate of conventional material. The role of a surfacelayer is to secure the surface of material against the actionof external factors like corrosion as well as mechanicalfactors (intensive wear, cycling loading) resulting in theexcessive wear of material during service life (Rohatgi et al.

(1994)). An example of application of composite surfacelayers is for many machine parts and tools (e.g. drillingand boring tools) which are often subject to extremeworking conditions. Although the commonly used mate-rials for such applications are characterized by highstrength, wear and corrosion resistance, yet there is stillneed to improve the service condition for this kind oflayers because of the increase of working temperatures andloads. The metal matrix composite may be suitable for thispurpose as their unique properties can sustain these de-manding service conditions. By embedding hard ceramicsin the form of fibers, whiskers or particulates into themetal matrix, we can expect the hard inclusions to carry

on the main load resulting from the intensive surface wearduring the service life of machine parts. In such a system,elastic ceramic hardphases will allow to absorb the dy-namic load much better than the metal matrix, which canbe damaged through the accumulation of loads (see e.g.,Broutman and Krock (1974)).

The proper selection of materials for a composite layerover a given substrate material is very important as thisprocess should take into account many factors, of whichthe most important is chemical, metallurgical and physicalcompatibility between composite layer and the substratematerial and also between the composite componentsthemselves. To attain metallurgical compatibility we maycompose a composite with matrix material taken as the

metal alloy traditionally used for such layers. In thismeaning the surface material is enriched with hard in-clusions creating the composite with unique properties.

On the other hand, chemical compatibility requires thematrix and reinforcement to be at the equilibrium state inthermodynamically stable phases. This can be achieved bythe proper selection of the reinforcing phase for a givenmetal matrix. Usually ceramic materials like SiC, TiC, TiB2,ZrO2, Al2O3 are used as a hardphase while the metal ma-trix may be selected from titanium, aluminium, magne-sium or nickel alloys. Such combination of constituents

Computational Mechanics 19 (1997) 188 202 Springer-Verlag 1997

8

Communicated by S. N. Atluri, 14 August 1996

D. GolanskiVisiting Scholar, The Warsaw University of Technology,Welding Department, 85 Narbutta St., 02-524 Warsaw, Poland

K. Terada, N. KikuchiThe University of Michigan,Department of Mechanical Engineering,Computational Mechanics Laboratory, 2350 Hayward,Ann Arbor, MI 48109-2125, USA

Correspondence to: N. Kikuchi

The authors acknowledge the support of NSF MSS-93-01807,US Army TACOM, DAAE07-93-C-R125 and US Navy ONR,N00014-94-1-0022 and AFOSR-URI program, DoD-G-F49620-93-0289 and DEKABAN fund at The University of Michigan.

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will not assure the physical compatibility between com-posite components and also between the composite layerand the substrate material. The most important is theproblem of mismatch in thermal expansion coefficientsbetween thereinforcement andmatrix components. When aductile matrix has higher thermal expansion coefficientthan the reinforcement, then during formation of thecomposite the hard phase contains compressive residualstresses which seems to be preferable for brittle ceramicmaterials although the subsequent external loads may re-

verse thesign of these stresses. For thematrix materials withhigh yield stress it is advised to select reinforcing materialswith thermal expansion coefficient close to the matrix, be-cause the limited plasticity can lead to high internal stresses.However, the presence of residual stresses in compositescan be beneficial or detrimental depending upon the se-lection of constituent materials for the matrix and hard-phase. An example of positive effect of residual stresses isaluminum matrix composite reinforced with SiC particles.In this system, residual stresses induce the formation andfurther movement of dislocation in the plastic aluminummatrix, which leads to the strengthening of Al and sub-sequent residual stress relaxation (Shi and Arsenault

(1994)). The negative effect can be seen for composites withdecreased matrix plasticity and brittle reinforcements. Inthis case the composite can be damaged by the formationand growth of microcracks both through the matrix and thereinforcement. It was observed by MacKay (1990) for TiAlcomposites reinforced with SiC particles4.

The stress state is one of the key factors affecting thereliability and durability of the composite structure. Wecan distinguish two-scale levels of stress in the system ofcomposite surface layers. The first one refers to the mi-croscopic level of stress state within the composite and theother refers to the global structure, where the compositelayer is treated as a homogeneous body with uniformproperties. Both scales of stress are important parts of a

stress analysis in this system. The knowledge of stressstate can be especially desired in the process of controllingthe properties of manufactured composites (compositelayers) and their thermomechanical response under ap-plied loads.

However, the investigation of stress state in compositesis difficult because of the existence of different phases inone material. Thus analytical and numerical methods grewup to determine the stresses. There are several, less ormore precise, analytical models used to predict the overallcomposite thermomechanical response. These can includethe rule of mixture, shear lag model or Eshelbysequivalent inclusion model(see e.g., Taya and Arsenault

(1989)). The numerical methods are primary based on thefinite element method. The finite element model usuallyrepresents a small scale region with matrix and re-inforcement repeatedly distributed in a whole structureand the computed stresses are related to the microscopicstage of the problem. When a composite has randomlydistributed reinforcements, which is usually the case forthe particulate reinforced composites, the problem of de-termining micro and macroscopic stress is more compli-cated with the mentioned methods. To estimate the stressstate in real randomly distributed particles in the matrix

more sophisticated methods have been utilized. One of thepromising and advanced ones is the homogenizationmethod as it allows to describe both the micro and macrothermomechanical behavior of composite materials usingrigorous mathematical theory (Guedes and Kikuchi (1990)and Cheng (1992)). It should be noted that the theory candescribe both micro- (local) boundary value problemswith the help of the two-scale asymptotic expansionmethod if the periodic boundary condition is introducedon the representative volume element (RVE) (see Sanchez-

Palencia (1980)). Therefore, we assume that the selectedcomposite layers are statistically homogeneous and peri-odic so that the asymptotic homogenization method canbe applied if the RVE size is sufficiently large.

On the other hand, the continuum-based formulation ofthe homogenization method requires some kind of nu-merical methods such as finite element method. To carryout the computation effectively, modeling of appropriatemicrostructural geometry is essential since most of metalmatrix composites have more or less random nature intheir microstructural morphology. In this context, Hollis-ter and Kikuchi (1994) addressed the modeling issue in FEgeometric modeling of bone microstructure. They ex-

tensively utilized the digital images and their processingtechniques to construct the digitized 3D-FE model of abone microstructure by identifying each voxel as an finiteelement. Along with the asymptotic homogenizationmethod, their digital image-based (DIB) modeling tech-nique enabled the quantitative study of the macro- andmicromechanical characteristics of bones porous skeletonin the framework of linear elasticity. In order to success-fully take into account the effect of microstructural geo-metry, we shall utilize this novel method in ourhomogenization analyses. This method has been used inour work and the fromulae in linear elasticity will bepresented later in the paper.

The aim of this paper is to present some aspects on

modeling the stress state in metal matrix composite sur-face layers through the utilization of DIB geometricmodeling. This technique allows to analyze the real com-posite microstructures and to convert them into virtualrepresentation of 3D geometry as shown by Terada et al.(1996). For the stress analysis, we have selected nickel-basealloy as the matrix and four carbide ceramic materials: SiC,TiC, TaC and HfC for the hardphase. The homogenizationmethod is used to evaluate the effective material propertiessuch as elastic and thermal expansion coefficient matricesby taking a unit cell as a representative volume element.Also, other simplified models used to estimate the overallcomposite properties are presented for comparison. We

have conducted the analysis for linear elasticity case withthermal loading equal to the temperature difference be-tween composite formation and room temperature,therefore we analyzed the thermal residual stresses gen-erated upon cooling of composite layer. For high tem-perature drops, the thermal residual stresses maysignificantly influence the overall stress state in the com-posite layer as well as in the whole structure as they shallsuperimpose with the external service loads. The applica-tion of the homogenization method allows us to estimatethe stress state in the composites unit cell at the micro-

1

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scopic level by localizing the deformation of the globalstructure, which has been obtained in the stress analysis.

2Materials and processingFor the purpose of the analysis we based our work on thecomposite with stellite (nickel-based alloy) as a matrix andcarbide ceramics as inclusions. Stellites are widely used forsurface layers produced by plasma hardfacing methodsince they have high wear and corrosion resistance. The

chemical composition of the stellite used in the analysis isequivalent to Deloro Stellite 40. The following propertiesof the stellite were assumed in the analysis: density 8330 kg = m3,Youngs modulus E = 183 GPa, Poissonsratio 0: 3 and coefficient of thermal expansion 9 : 7 1 10 6 1 = K according to technical data of Deloro

Stellite (1970).The selection of reinforcing phase depends on many

factors affecting the properties of metal matrix composites.We have assumed four carbide particles: TaC, TiC, SiC andHfC as the reinforcement. Such selection was dictated bythe fact that the contribution of carbide metallic bondsshould effectively improve the wetting of ceramic particles

by the stellite matrix. Also, because the composite surfacelayers were obtained in the process of electron beam re-melting the high formation entalphy of these ceramicsprevents their decomposition during remelting.

The properties of ceramics were taken from the litera-ture (ASM Reference Book and Morrell (1989)) and arepresented in Table 1. As it is seen from the table the se-lected ceramics differ in their properties. Specifically, HfCand TaC have high density which is important when wetake into account the stiffness/density ratio as a commonindicator of composite weight with a given stiffness. Thethermal expansion coefficient () of the hardphases, whichhas the strong influence on the stress state generatedwithin a composite varies between ceramics in a wide

range from 4 to 7 : 4 1 10

6 1/K, while the Youngs modulushas similar values for all except TaC. All consideredceramics have high melting temperature and high hard-ness which makes them attractive for applications workingat elevated temperatures.

One of the advanced methods of producing this kind ofcomposite layers is a technique of remelting by a highenergy concentrated beam (electron, laser or plasma) thesurface of substrate material previously covered with amixture of metal and ceramic powders. The process of

coverage the surface with the powders can be donemanually, by spreading the powders over the surface ormechanically using e.g. the plasma spraying method (seee.g., Matejka (1989)). In fact, the former technique may beused separately for a direct deposition of composite layers,but because of the layer porosity and adhesive bonding tothe substrate it still needs a subsequent remelting in orderto reduce the porosity and create strong bonds at the layer/substrate interface.

In our analysis we have utilized the microstructures of

carbide ceramics/stellite composites produced by the os-cillating electron beam remelting technique over the sub-strate of low alloy steel (equivalent to ASTM A618 grade).Particularly, we based our work on TaC/stellite and TiC/stellite composites as the initial source for the furtheranalysis. The microstructures of these composite layers arepresented in Fig. 1. According to the DIB geometricalmodeling described in the next section we have specifiedthe composites unit cell by selecting them from the imagesof composite micrographs, which is marked and shown inFig. 1.

3

The homogenization method and microstructural modelingby digital images

3.1IntroductionIn this section, we shall present two rigorous tools toanalyze the mechanical behavior of the MMC whose ma-terial properties and configurations are described in theprevious section. One of the tools is the mathematicaltheory of homogenization and the other is the DigitalImage-Based (DIB) geometric modeling technique.

The so-called asymptotic homogenization method con-cerns composite media whose microstructure occupies a

fixed region with characteristic length

of its hetero-geneity. The theory asserts that if the selected re-presentative volume element (RVE) is infinitesimallysmall, the actual displacement, u, tends to the homo-genized displacement field, u0, which is the global solutionof the governing equations whose coefficients have beenhomogenized. While the effective properties can be de-rived from the micromechanical characteristics, the mi-cromechanical behaviors can be obtained by localizing theoverall structural response to the local one; these processesare called, respectively, homogenization and localization.The global-local approach was successfully applied to theengineering problems in both linear elasticity and elasto-plasticity with the help of Finite Element Method (Guedes

and Kikuchi (1990), Devries and Lene (1987), Duvaut andNuc (1983)).

After deriving the homogenization formulae for linearelasticity with temperature change, the procedure of con-struction of a unit cell by the DIB modeling is briefly de-scribed.

3.2The homogenization formulaeWe recall that the theory concerns statistically homo-geneous or periodic composite media of domain and

Table 1. The properties of ceramic reinforcements used for themetal matrix composites

Properties Ceramics

SiC HfC TaC TiC

Melting temperature (K) 2500 4203 4152 3340Density (g/cm3) 3.10 12.67 14.50 4.92Coefficient of thermal

expansion (10)6 1/K) 4.02 6.60 6.3 7.4Youngs modulus E (Gpa) 402 470 722 447Poissons ratio 0.142 0.180 0.240 0.190Shear modulus G (Gpa) 178 193 227 186Bend strength (Mpa) 459 234241 215310 282667Microhardness (Gpa) 28 22.630.5 1624 2024

0

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the representative volume element (RVE) occupying a

microscopic region V with characteristic length . Identi-fying the size of the RVE with , we introduce two differentscales: one of these is a macroscopic scale denoted by x, inthe domain at which the heterogeneities are invisibleand the other one is an microscopic one denoted byy x = which enlarges the RVE region by such thatV Y. Thus, the superscripts introduced in variablesindicate their orders as well as the dependency on bothx and= or y: Let the structure be subjected to a surfacetraction t and a prescribed homogenous displacementboundary conditions on t and u, respectively, withtemperature change T. According to the principle ofminimum total potential energy for equilibrium, the dis-placement u is the solution of the variational problemdefined in the domain :

Z

v : D x : u dx

Z

T v : D : b : v dx

Z

t

t^ x : v

dx; 8 v 1

with the constitutive relation

D x :

u

T x1

D x : u

T x 2

Here v is the virtual displacement, b x the body force,D x the elasticity tensor, x the coefficient of thermalexpansion (CTE) with D : :

With the help of the method of two-scale asymptoticexpansion (see, e.g., Sanchez-Palencia (1980)), the theoryasserts that if the selected RVE is periodic and in-

finitesimally small, the actual displacement, u

, tends tothe homogenized one, u0, which is the solution of thefollowing macroscopic equations whose coefficients havebeen homogenized.

Z

x v : DH : x u

0 dx

Z

Tx v : DH : Hdx

Z

bH: vdx

Z

t

t: vd 8 v

3

and that the Y-periodic characteristic deformations kh

and can be obtained by solving the following micro-scopic equations, respectively:

Z

Y

y w : D : y kh

dy

Z

Y

D : y w dy ; 8 w ; Y periodic 4

Z

Y

y w : D : y dy

Z

Y

: y w dy ; 8 w ; Y periodic 5

Here, the strains are denoted by 1= 2 vr r v x or y and the homogenized quantities in Eqn. (3)are calculated by averaging over the unit cell:

DH 1

j Yj

Z

Y

D : sdy ; H 1

j Yj

Z

y

D : y dy

and bH 1

j Yj

Z

Y

bdy 5

Also, the tensor of localization have been defined asskhij I

khij y ; ij

kh where Ikhij indicates the fourth order

identity tensor. Once the macroscopic displacement uo

and

T are obtained in the macroscopic region, thesevalues are localized to give the micromechanical responseof the unit cell. Therefore, the microscopic stress is de-fined by

o y

D y : s y

: x uo

T y y 6

On the other hand, the macroscopic quantities are ob-tained by taking the average over the RVE domain Y. Inorder that the macroscopic strain x u

o would be the

volume average of the microscopic one eo, it seems ap-

Fig. 1a,b. The microstructure of TaC/stellite (a) and TiC/stellite(b) composite surface layers produced on low alloy steel by theelectron beam remelting method

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propriate that the RVE domain and the microscopic dis-placement w1 would be periodic and therefore so are kh

and . In this way, we may call the RVE the unit cell.If we postulate that the displacement were Y-periodic,

the substitution of the variables obtained in the aboveprovides the following microscopic governing equations:Z

Y

wr y : D : r ykhdy

Z

YD

: r ywdy; 8

w:

Y

periodic;

7

and

Z

Y

wr y : D : r ydy

Z

Y

: r ywdy 8

Using these solutions kh and , the homogenized elasti-city tensor and the homogenized CTE are constructed bythe following formulae:

DH 1

j Yj

Z

Y

D :sdy ; 9

H

DH 1 :

1

j Yj

Z

Y

D : y 1

dy

; 10

and

bH 1

j Yj

Z

Y

bdy : 11

where j Yj indicates the volume of the unit cell.Thus, the effective properties can be derived from the

micromechanical characteristics and the micromechanicalbehaviors can be obtained by localizing the overall struc-tural response to the local one; these processes are called

the homogenization and the localization, respectively.We have skipped several steps in the formulation and

omitted some explanations, since they are not relevant toour present interest. One can refer to literature, for ex-ample, by Guedes and Kikuchi (1990) Devries and Lene(1987) or Duvaut and Nuc (1983) for the detailed deriva-tion of the homogenization formulae.

3.3Digital image-basedFE-geometric modelingfor theunit cellDigital image-based (DIB) technique is used to catch andmanipulate the image of composite microstructures (unitcells) so that they could be analyzed by the homogenizationmethod. Also, this technique is very helpful for preparingthe images of composite unit cells to some other additionalprocessing like generation of 3D structures or changingvolume fraction of inclusions. The main procedure ofpreparation the unit cell of a composite microstructure canbe divided into the following major four parts (see Fig. 2):

1. Capturing and Sampling: prior to digitalization, animage of a composite microstructure must be capturedby an optical sensor which is chosen upon the desiredformation modality. This process is assumed to be doneby a high resolution scanner unit.

2. Selecting and Thresholding: which are probably themost important operations. This process determines theunit cell size, its FE model size and the microstructuralconfigurations. By giving the thresholding pixel value,the actual morphology such as inclusion shape, volumefractions, etc., is determined.

3. Exporting (and Adjusting, if necessary): the binary datastored in the computer are exported into an ASCII fileand transferred to a UNIX platform so that our com-puter program can read and recognize the data. This setof data is actually the prototype of our FE model, which

is either two or three dimensional. Depending on ourneeds, the microstructural configuration is adjusted,e.g., the volume fraction is modified by changing thenumber of pixels.

4. Stacking: prior to or during the finite element analysis(FEA), the process is made using the exported data toconstruct the three-dimensional (3D) structure.

While the first three stages correspond to the pre-proces-sing of FEA, the last process includes both the main part andpost-processing of the FEA of the homogenization method.In order to construct 3D FE model in the DIB modeling, twodimensional digital images have to be combined. The forthprocess, namely the stacking, corresponds to such data

operation. In 3D FE modeling, each pixel in a 2D image isrecognized as a voxel which is identified as a finite elementin FEA. Then, the FE model obtained in this process is thedirect interpretation of the scanned image using two di-mensionally presented micrographs of real composite ma-terials along with image-processing software. Therefore, thehomogenization analyses can reflect the effects of the ori-ginal geometric configuration.

This method involves image processing which fully uti-lizes both hardware and software capability available. Forour particular purpose, the process of changing the vo-lume fraction of a constituent can be easily done by op-erating the voxel values of the digitized unit cell model.The more detailed description of the DIB modeling pro-

cedure, the related image processing and some applica-tions to the homogenization analyses are found in Teradaet al. (1996).

4The procedure of calculations and analysis

4.1Calculation schemeWe have implemented the following scheme for calcula-tion of macro and microscopic residual stresses in the

Fig. 2. The diagram showing DIB modeling sequence2

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MMC surface layers shown in Fig. 3. First, the homo-genization method is used to obtain the effective compo-site properties dependent on the properties of itsconstituents and their volume fraction. This is done for theselected unit cell taken from the microstructure of com-

posite. Next, the global structure is defined to compose thesubstrate material covered with a composite surface layerhaving uniform homogenized properties. Then, the stan-dard finite element method is applied to solve for thedisplacement and stress field in the macroscopic scale(global structure). The last step utilizes calculated globaldisplacement field taken for the selected unit cell from thecomposites global structure to computer the microscale(local) stress distribution for composite microstructurerepresented by the selected unit cell.

4.2Changing volume fractionThe initial hardphase volume fraction of TaC/stellite and

TiC/stellite equaled approximately 30% based on theweight method. In the process of digital image-based (DIB)modeling we have selected the unit cell as a representativevolume element from TaC/stellite and TiC/stellite com-posites (Fig. 1). After thresholding and adjusting we haveobtained digitized image of the selected composites mi-crostructures as seen in Fig. 4. These images contain 30450

pixels for TaC/stellite and 29225 pixels for TiC/stellitecomposites. The calculated volume fractions for these unitcells equaled 21 and 24%, respectively.

To see the effect of volume fraction on the thermo-mechanical response of selected composites we have gen-erated other volume fractions upon those digitized unitcells. As the typical hardphase volume fraction in MMCranges from 5 to 40% we have generated four more unitcells with volume fractions of 5, 13, 30 and 40% coveringthe range of application. Fig. 5 shows the unit cells with 5%

and 40% hardphase volume fraction for TaC/stellitecomposite as an example. The procedure we applied forthe volume fraction change randomly removes/addshardphase material (pixels) from/to the inclusion-matrixboundary. As it is seen from these pictures, rather tornshape boundary of inclusions is generated, but for thepurpose of the analysis this would not be a significantfactor.

5Results of calculations

5.1

Effective composite propertiesBy applying the homogenization method we have calcu-lated the engineering composites constants for the planestress case: Youngs modulus (E11 ; E22) Poissons ratio(v12), shear modulus (G12) and thermal expansion coeffi-cient (11 ; 22). There are many models found in the lit-erature elaborated to estimate the effective properties ofcomposites (see e.g., Vaidya and Chawla (1994) or Whit-ney and McCullough (1990)). For comparison with thehomogenization method we have included the calculatedeffective composite properties based on the classical ruleof mixture used primarily for the transverse direction offiber composites and two other often used models based

on approximated equations for particulate composites.These models are presented as follows:For the estimation of Youngs modulus we used the rule

of mixture model by Reuss (1929):

Ec 1

Vm = Em Vp = Ep 12

Fig. 3. The scheme of calculation procedure applied in thestress analysis

Fig. 4. The digitized unit cells of 21%TaC/stellite (left) and 24% TiC/stellitecomposites (right) after selection andthresholding

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and Halpin-Tsai (Halpin and Kardos (1967)) expression:

Ec Em 1 Vp

1 Vpwhere

Ep = Em 1

Ep = Em 13

where: E Youngs modulus, V volume fraction (c composite, m matrix, p particulate), is factorassumed to be 1.

For the estimation of thermal expansion coefficient weused two models:Kerners (1956) model for volumetric coefficient of ther-mal expansion 3 :

c m Vm pVp m p VmVp

2

1= Km 1 = Kp

Vm = Kp Vp = Km 3Gm = 4 14

where K is bulk modulus and G shear modulus.This model accounts for shear and isostatic stresses, but

the last part of this expression can be particularly negli-gible reducing the formula for to the classical rule ofmixture:

c mVm pVp 15

Turners model (1946), which accounts for the case ofhydrostatic stresses:

c mVmKm pVpKp

VmKm VpKp 16

The calculated homogenized composite properties werecombined and plotted in Fig. 6 with the presented sim-

plified models. These models are used for comparisonpurposes only. The measurement results are necessary tovalidate the analytical ones, although good agreement isobtained for Halpin-Tsai and Turner models. It can also beseen from the figure that the effective properties both of Eand are close each other for a small volume fraction. Theincrease of the hardphase volume fraction results in bi-linear increase of the homogenized properties and thehigher influence of ceramic properties on the effectiveproperties of the composite which is seen especially for theTaC ceramic having high Youngs modulus.

The rule of mixtures model gives a very rough esti-mation of composite properties except for the case of ef-fective density of composite which gives accurateapproximation. The density of a composite plays an im-portant role when we are taking into account stiffness/density (specific stiffness) ratio. Most ceramics are char-acterized by a higher Youngs modulus and lower density

than matrix material. This feature is used to producecomposites structures that are lighter than traditionalmaterials or have reduced cross section without the loss ofstiffness. Chawla (1987) discussed a simple model whichshows that for beams under compressive and flexuralloads the minimum weight of composite structure for agiven stiffness can be obtained while the term (E= 2)reaches maximum.

Fig. 7 shows the change of specific stiffness of TaC/stellite and TiC/stellite composites. The Youngs moduluswas obtained by the homogenization method while thedensity of composite was estimated by the rule of mix-ture:

h p 1 Vp m 1 Vm 17

Comparing to steel which E= 26 it is seen, that forTaC/stellite composite higher ratio is available when thehardphase volume fraction is greater than 30%. This is theresult of high density of TaC ceramic which is almost 2times the steel density. On the other hand, over 13%hardphase volume fraction is needed to assess for the samespecific density as steel in TiC/stellite composite.

5.2Finite element analysis of macroscopic stress

in the global structureThe finite element plane stress elastic analysis was con-ducted in order to obtain the stress and strain field in aglobal structure composed of a composite surface layerwith 1 mm thickness and 10 mm thick low alloy steel. Fig. 8shows the finite element mesh, which has 594 four nodeplain stress elements connected in 550 nodes.

The structure is assumed to be stress free and uniformtemperature drop of)1000 K is applied to simulate theformation of thermal residual stresses upon cooling fromcomposites fabrication (1293 K) to room temperature

Fig. 5. TaC/stellite digital imagewith generated hardphase volumefraction of 5% (left) and 40% (right)

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(293 K). For the linear analysis any level of temperatureloads could give the same quality results, however, theassumption of the real temperature drop would give us theview on the stress level comparing to the elastic limit ofapplied materials. It was also assumed in the calculations

that there is a perfect contact on the boundary of thecomposite layer and substrate material. The anisotropicproperties of composite are input in the analysis throughthe elastic and thermal expansion coefficients matrices andthe properties of substrate material are assumed to beisotropic and a typical one is for steel: 12 1 10 6 1 = K ; E 210 GPa ; v 0: 3 : The material

properties are assumed to be constant in the analyzedrange of temperature.

The finite element calculations show that the surfacelayer after cooling down to room temperature is in com-pression primarily by the influence ofxx stress compo-

nent parallel to the layer/substrate interface. The vonMises stress field shows the edges to be the regions of highstress concentration, but this singularity occurs in manysystems composed of materials with different propertiesand can be a separate field of research in this area. For ourpurpose we will narrow our study to regions laying far

Fig. 6. Comparison of calculated homogenized properties of TaC/stellite and TiC/stellite composites with respect to hardphasevolume fraction

Fig. 7. Variation of composite specific stiffness with hardphasevolume fraction for TaC/stellite and TiC/stellite composites

Fig. 8. The finite element model of the MMC surface layer producedover the low alloy steel

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from the edges. The xx stress and von Mises (M) stressdistribution is presented in Fig. 9 for one of TaC/stellitemodels with 21% hardphase volume fraction.

The overall stress distribution for both composite sys-tems and for each hardphase volume fraction is similarexpect for the stress level. The stress level primarily de-pends on the difference of strains in the x direction be-tween the layer and the substrate resulting from differentcontraction of composite and substrate during coolingprocess of composite structure. This is because of the

difference in thermal expansion coefficients between thecomposite and steel substrate. The increase of the hard-phase volume fraction alters the homogenized compositesproperties and as a result, the global stress level is alsochanged. As it was expected the residual stress in com-posite layer changes linearly with hardphase volumefraction because of the linearity of the problem. It is seenfrom the Fig. 10 that there is a symmetry across the zerostress line between xx and M stresses for both analyzedcomposites that confirms the main role ofxx stress in theeffective stress of the composite. It is also seen that for thesmall hardphase volume fractions there is a small differ-

ence in the stress level for TaC/stellite and TiC/stellitecomposites despite the differences in properties of TaCand TiC inclusions. Particularly, the ratio of TaC to TiCboth xx and M stress equals 1 for 5% inclusion volumefraction. This may result from a very close values of

homogenized properties of both composites with 5% vo-lume fraction, for which the effective composite propertiesdepend mainly on the matrix properties.

The stress level in TaC/stellite composite is increased byabout 100% from 5% hardphase volume fraction to 40%,while for the TiC/stellite this increase reached about 64%.Since the stress is compressive it is not seemed to bedangerous for the composite with hard ceramic inclusions.The variation of the von Mises stress and xx stresscomponent from the surface through the thickness of themodel shows the extreme of stress level just above thecomposite/substrate interface and the change for stresssign occurs in the substrate (Fig. 11). The xx stress in thesubstrate material may exceed the elastic limit near theboundary with composite layer, especially for the hard-phase volume fraction greater than 20%. This shall resultin the redistribution of stress at the composite/substrateboundary and would influence the microscopic stressfield.

Fig. 9a,b. Distribution ofxx macroscopic residual stress (a) andvon Mises (b) residual stress calculated for 21% TaC/stellitecomposite layer

Fig. 10. Variation of macroscopic residual stress with hardphase volume fraction change for 21% TaC/stellite and 24%TiC/stellite composites

Fig. 11. Thermal residual stress profile across the surface of 21%TaC/stellite and 24% TiC/stellite composite

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5.3Microscopic stress representation in the unit cellThe results of the microscale thermal residual stress dis-tribution are obtained for the composite unit cell takenfrom the boundary of composite/substrate interface (ele-ment 88 in Fig. 8), where the stress reaches the highestvalues according to the results of stress variation throughthe surface of the layer (see Fig. 11).

The local stress state in the micro scale is complex anddepends on the hardphase volume fraction. Generally,

ceramic inclusions contain most compressive residualstresses of magnitude few times higher than the macro-scopic stress in composite layer. The matrix stays underweak tension in regions laying far from the inclusions or incompression close to the hardphase boundary as well as inregions laying between inclusions located close each other.There is no stress component which could dominate theoverall composite stress response so the von Mises stressshall be the general indicator of the stress severity. Fig. 12shows von Mises M thermal residual stress distributionwithin the unit cell for 21% TaC/stellite and 24%TiC/stellite composites with a 3D plots of stress distribu-tion.

It can be seen from these pictures that stress con-centrates in regions laying between the particles and also

near the inclusion/matrix boundary. In order to estimatethe local stress in composite, von Mises stress was aver-aged and calculated separately for the hardphase and forthe inclusion for the given unit cell. The variation of thisaveraged stress with hardphase volume fraction is pre-sented in Fig. 13 for both analyzed composites. The in-crease of inclusion content results in considerable increaseof the average stress in inclusions. The increase of theaverage M residual stress from 5% to 40% hardphasevolume fraction is lower than for the global stresses, but

the magnitude of the maximal M stress is 3 to 4 times theaverage stress in inclusion. Such localized high stressescan cause the early cracking of ceramic even the stress iscompressive.

Also, high stresses may lead to the formation of a localplastic zone around the inclusions, which will result in theredistribution of this stress from the hardphase/inclusionboundary. Another reason for such a high stress con-centration, especially at the high volume fractions, mightbe the shape of the inclusion boundary, which is notsmooth because of randomly generated inclusion pixels inthe process of volume fraction change. Thus, the tornshape of the boundary works as the additional source of

geometric concentrator increasing the stress at the inclu-sion/matrix interface. The ceramic-metal interface is a

Fig. 12a,b. Distribution of local von Mises residual stress within the unit cell for 21% TaC/stellite ( a) and 24% TiC/stellite composites (b)

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region where high stress concentration occurs. So betterestimation of the stress severity would be obtained if theaveraging scheme could be narrowed to the stress valuestaken exactly from the boundary.

5.4Selection between 2D and 3D modelAll previous results were obtained for 2D plane stressmodels. We have repeated the applied calculation schemefor a 3D model of 21% TaC/stellite composite. The unit cellwas extruded in the z direction creating a 3D re-presentation of 2D plane structure. The global structure isalso extended in the z direction to form a cubic sub-strate with 1 mm composite layer as shown in Fig. 14.

The homogenized properties calculated for the 3D modeldiffer from those obtained in 2D analysis by 2 to 9% in thesame x-y plane. In out of x-y plane the differences raisedfrom 5 to 20%. The results of stress distribution in 3D

model, for x-y plane are similar to those obtained in the 2Danalysis except the stress level, which reaches more ex-treme values than for 2D case. From Fig. 15 presenting vonMises stress distribution we can see the extreme stressesexist also in the center of the model, which results fromboth xx and zz stress components as opposed to the edgeplanes, where one of the stress components reaches zero.

The difference between these two models is also seen inthe stress variation across the surface of the layer for thex-y plane as shown in Fig. 16. The M stress profile tendsto have sharp stress change at the composite/substrateboundary for the 3D model. The point of maximum Mstress is shifted from the near-boundary to the boundaryin comparison with the 2D case. The xx stress variationfor 3D case is characterized by lower than 2D stress level inthe composite layer with similar profile as for the 2Dmodel.

The microscopic stress results obtained in 3D analysisshows similar to 2D model stress distribution in the unitcell, which is seen in Fig. 17 for M stress. The averagelocal M stress is about 2 times higher in the hardphaseand about 1.5 times the average M stress in the matrix forthe 3D case anlysis.

The stress results calculated for 2D and 3D models showthat a good agreement is obtained for the stress field both

in macro and microscale. The selection between 2D or 3Danalysis case should be carefully considered according tothe complexity of the problem and computer efficiency.For a simple model analysis the 2D case could give sa-

Fig. 13. Variation of the average microscopic von Mises residualstress within the unit cell for 21% TaC/stellite and 24% TiC/stellitecomposites

Fig. 14. 3D model of global composite structure for calculation ofstress state.

Fig. 15. von Mises global stress distribution for 3D 21% TaC/stellitecomposite

Fig. 16. Comparison of thermal residual stress profiles across thesurface of 21% TaC/stellite composite for 2D and 3D case stressanalysis

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tisfactory results, especially for the purpose of comparisonwith other composite systems It is worthwhile to noticethat the calculation time for 3D case is much longer than

2D one, as a result of the total node numbers, which for 2Dcase equaled about 30000 while for 3D case 90000. Thisanalysis has been conducted on HP 9000/715 workstationand the total computing time was 7 to 8 times longer for3D model.

5.5Analysis extension for other hardphasesThe present method of homogenization combined with thedigital-image processing and finite element calculationsappears to be suitable to apply for other inclusion mate-rials even though we do not have their microstructures. Byusing the microstructure of one composite material we cansubstitute its properties with other, so that different in-clusions can be analyzed. This will give us comparableresults of the thermomechanical behavior of compositeswith different hardphases. Such modeling may simplify theprocess of initial selection of inclusion material for a givenmatrix or a matrix for a given hardphase.

As an example we present the results of such analysis,which are based on the TiC/stellite composite micro-structure with 24% of inclusion volume fraction. To ana-lyze the effect of inclusion material we substituted TiCproperties by TaC, SiC and HfC ceramics assuming iden-tical microstructure. Selected ceramics have differentthermal expansion coefficients and different Youngsmodulus, so the combined effect of these properties on the

stress level can be examined. We have studied the Youngsmodulus ratio and the CTE mismatch effect as they havethe main influence on the stress state in MMC.

In the macroscopic stress analysis, the effect of the dif-ference in the coefficients of thermal expansion (CTE)between the steel substrate and composite layer, and theeffect of the ratio of composite Youngs modulus Eh to theYoungs modulus of steel substrate Esub are presented inFig. 18a and Fig. 18b respectively. It is seen form thesepictures that the high Youngs modulus of TaC inclusioncontributes to the elevated stress level in the composite.

We may assume that the smallest ratio of Eh = Esub will notguarantee the lowest stresses, which is clearly seen for theSiC/stellite composite having the lowest CTE of SiC cera-mic among analyzed inclusions.

Therefore, the CTE mismatch between the compositeand substrate will play the main role in the generation ofstresses, although the high Youngs modulus of TaC in-clusion is responsible for higher stress level comparing toHfC and TiC inclusions which have higher CTE mismatch.The influence of high Youngs modulus of inclusion on the

stress increase is seen especially at higher volume fractionswhere the effect of inclusion properties on the effectiveproperties of composite become evident which is opposedto the low volume fractions. Similar trends are observedfor the average microscopic stress within the unit cell asseen in Fig. 19, where the von Mises stress is plotted se-parately for the matrix and inclusions. Herein, the CTEmismatch refers to the difference between the matrix andthe hardphase while the Youngs modulus ratio to thehardphase and matrix.

The results of the CTE mismatch show clearly that theYoungs modulus of TaC hardphase ceramic substantiallyincreased the stress level in the hardphase, although the

influence of the CTE mismatch seems to affect the stresslevel in a higher degree, which is shown for SiC inclusion.In both scales of stress analysis the CTE mismatch has the

Fig. 17. Von Mises local stress distribution within the x-y plane for3D 21% TaC/stellite composite

Fig. 18a,b. Influence of thermal expansion coefficient mismatchbetween the substrate and composite layer (a) and Young modulusratio of the composite to the substrate (b) on the von Mises stress inthe composite layer calculated for 24% TiC, HfC, TaC and SiC/stellitecomposites

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primarily effect on the stress level in the analyzed com-posites. The Youngs modulus effect becomes visible forhigh ratios of Ep = Em. This is seen in Fig. 19a for the HfCand TaC/stellite systems which have similar CTE mis-match, but the stress level sharply increased in TaC/stellitecomposite because of high Youngs modulus of TaCceramic. In the above analysis we assumed no influence oftemperature on the CTE and Youngs modulus (E) ofcomposite material. The CTE of most ceramics and metalincreases with temperature while the E decreases. There-fore, some authors use the product of (CTE 3 E) as aconstant value over the analyzed range of temperature andtreat it as a measure of joint effect of both the CTE and Eon the stress level in a structure. Such relation is presentedfor studied composites in Fig. 20 and shows that the Mstress reaches minimum for CTE 3 E 2: Although theCTE and E effect on the stress level is not equal we may use

this simple relation in the evaluation of stress for manycomposite systems.

6SummaryWe have presented some aspects on modeling macro andmicroscopic thermal residual stresses in the metal matrixcomposites by the homogenization method with applica-tion of digital image-base technique. The process of de-position advanced composite materials onto the substratematerial by using plasma spraying combined with electron

beam remelting method is very attractive for producingsurface layers with enhanced strength and resistance toexternal loads comparing to traditionally used materials.The embeding of ceramic inclusions in the metal matrixinevitably leads to the generation of internal stresses

during the formation and in service life of compositestructures. Upon the scale of mismatch in thermophysicalproperties of composites constituents, the high stress mayreduce substantially the service life of the structure. In theextreme case the composite may be damaged by crackpropagation through the matrix or inclusions. Thereforethe stress estimation in the MMC structures becomes im-portant.

In the present work the homogenization method waslimited to the linear elastic range. The application ofelastic-plastic analysis for that kind of microstructurescould be difficult to conduct nowadays as it requires muchlarger computer memory and calculation times. The as-sumption of linear elasticity is valid only for hard inclu-sions like advanced ceramics which we may treat as elasticbodies. The stress field in the global structure was calcu-lated using homogenized properties (E; G ; V; ) of studiedcomposites. It is shown that these properties strongly de-pend on the hardphase volume fraction. The results ofstress level indicate that the plastic field may be built up ina thin layer of substrate material just close to the boundarywith composite layer. As a result the global stress level inthe composite layer may change, which will affect themicroscopic stress within the composites unit cell.

Upon the results of microscopic stress distribution inthe unit cell of studied composite systems, we may say thatthe residual stress field is complex while inclusions are

generally under compression. There is a high stress con-centration at regions close to the inclusion/matrixboundary. The magnitude of the stress in the stellite ma-trix is much above the matrix elastic limit, which suggeststhat a local plastic field may be developed around ceramicinclusions and as a consequence the stresses shall be re-distributed from the region of stress concentration.

The change of hardphase volume fraction is a very at-tractive feature of the analysis. Without the need to pro-duce composite structures with specified hardphasevolume fractions we can simulate the thermomechanical

Fig. 19a,b. Influence of thermal expansion coefficient mismatchbetween the matrix and inclusion (a) and Young modulus ratio of theinclusion to the matrix (b) on the average von Mises stress within theunit cell calculated for 24% TiC, HfC, TaC and SiC/stellite composites

Fig. 20. The relation between the product of homogenized propertiesCTE3 E of stellite-based composite reinforced with different ceramicinclusions and von Mises stress calculated for global and local scale

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response of composite materials in the process of com-posite design. This is especially important for simulationof stresses in composite layers which are produced bydirect deposition onto the substrate material, because ofthe difficulty to control the real hardphase volume frac-tion. The calculations we have conducted for TaC/stelliteand TiC/stellite composites with hardphase volume frac-tion ranging from 5 to 40% reveals that the thermal re-sidual stresses both in the macroscale (global structure)and in the microscale (unit cell) of different composite

systems, do not vary substantially for the low (5%) hard-phase volume fractions. With the further increase ofhardphase content in the matrix the difference is ex-panding to reach the maximum at the highest volumefractions of inclusions.

In the analysis of thermal residual stresses the main roleplay the properties of constituent materials. The mostimportant is the mismatch in thermal expansion coeffi-cients between the hardphase and the matrix in the mi-croscopic stress analysis and between the composite layerand the substrate material in the macroscopic stressanalysis. Also, the stiffness of the local and global struc-tures characterized by Youngs modulus ratios affects the

stress generation in the ceramic/stellite composite sys-tems. Therefore, the selection of the hardphase materialfor a given matrix is of essential meaning. We have shownthe influence of hardphase material on the stress level forby selecting TaC, TiC, SiC and HfC ceramic inclusionswith 24% volume fraction in the stellite matrix and as-suming identical composite microstructure. It is evidentthat the increase of mismatch in the thermal expansioncoefficient between the matrix and the inclusion as well asthe increase of inclusion stiffness leads to the higherstresses in the composite, although the CTE mismatchseems to dominate the stress generation. On the otherhand, if we consider the composite specific stiffness, thenwe will see that HfC and TaC ceramics have very high

density, which can make them less attractive than con-ventional materials.

The selection between 2D and 3D models for the stressanalysis requires consideration of the efficiency of com-puter calculations versus obtained results. We have shownthat the 2D model applied in the stress analysis givescomparable results with the 3D one allowing for the sub-stantial reduction of computing time and memory.Therefore, the application of 3D models shall be carefullyconsidered according to the analysis we are going toconduct and expected results.

It is important to remember that the stress analysispresented in this paper refers to the thermal residual

stresses generated during the composite fabrication. In theservice life of the composite structure other stress fieldsmay develop resulting from the external service loads.These stresses will superimpose with the residual stresses,which can lead to a new, different stress state. If we expectthe composite structure to work under tensile loads thenthe compressive residual stresses are preferred in thecomposite as the overall stress level will be reduced. Al-though tensile stresses are generally more dangerous forthe brittle ceramic materials, the high compressive stresseswithin the composite layer are not preferred too, because

they can results cracking or delamination a thin compositelayer from the substrate.

The method presented in this paper can be an adequatetool to compare the macro (global) and microscopic(local) stress state in the metal matrix composites withhard inclusions, as well as to obtain the homogenizedcomposite properties. We shall emphasize, that this kindof analysis is unique as it allows the real microstructuresof composite materials to be analyzed. The presentedcalculation scheme may be applied to other kinds of

loading or the combined effect of multiple loads can beanalyzed.

Further development of this method would allow to ac-count for the plastic deformation in the elastic-plasticanalysis. At the present stage of computer, it is still adifficult task to deal with realistic complex micro-structures.

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