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EFFECTIVE MATERIALS PROPERTIES: DETERMINATION AND

APPLICATION IN MECHANICAL DESIGN AND OPTIMIZATIONM. Grujicic, G. Cao and G. M. Fadel

Department of Mechanical Engineering

Program in Materials Science and Engineering

Clemson University, Clemson SC 29634

ABSTRACT

A finite element procedure for determination of effective mechanical, thermal and

thermo-mechanical properties of multi-phase materials is developed. To account for the

variation in phases-contiguity with the change in their volume fractions, a computer-based

procedure is proposed for generation of the materials microstructure. The procedure is

applied to the Co-WC two-phase material. The results show a significantly better

agreement with their experimental counterparts relative to that of the analytical methods

for predicting effective materials properties. Finally, the case of flywheel optimization is

used to demonstrate that accurate determination of effective materials properties can be

very critical in heterogeneous-materials design problems.

I. INTRODUCTION

Composite materials consisting of two or more constituent materials in which the

volume fractions of the constituent materials and their microstructures are spatially

distributed within a part (component) in such a way that certain performance of the part is

optimized are commonly referred to as Functionally Graded Materials (FGMs). As

shown by the schematic in Figure 1(a), the microstructure of a (two-phase) FGM can vary

in a complex fashion as the volume fractions of the materials is changed continuously from

one pure material to the other. In the limit of a small volume fractions of either material,

the microstructure can be treated as consisting of dispersed inclusions of the minor

material in a continuous matrix of the major material, Figures 1(b) and (d). In sharp

contrast, in the FGM region where the volume fractions of the two materials are

comparable, the microstructure consists of intertwined clusters of the two materials, Figure

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1(b). The microstructure in the remaining regions of the FGM can be the treated as a

mixture of the two basic microstructures described above.

Over the last three decades a large number of models have been developed to

predict the effective mechanical and thermo-mechanical properties of heterogeneous

(multi-phase) materials and their dependence on the materials microstructure. Some of

these models involve the use of the variational approach [e.g. 1], while others are based on

the probabilistic approach [e.g. 2]. There are also a number of self-consistent models such

as the ones proposed by Hill [3], Budiansky [4] and Hori and Nemat-Nasser [5]. All of these

models are based on the Eshelby's equivalent inclusion method [6], and are generally

successful in predicting the effective materials properties only for relatively simple

microstructures and for low volume fractions of the minor material [7]. To overcome these

limitations, several models are proposed [e.g. 8-10] within which the effective properties

are determined through the use of the finite element analysis of a number of carefully

discretized rectangular regions which contain the essential features of the materials

microstructure. Such regions are commonly referred to as the representative material

elements (RMEs). Optical metallography can be used to select RMEs [e.g. 11]. However,

discretization of the RMEs into finite elements is a tedious procedure. In the present work,

an efficient procedure is utilized for computer-based generation of the RMEs. This

procedure is then applied to determine the effective properties of the cobalt-tungsten

carbide (Co-WC) two-phase material.

The organization of the paper is as follows: A brief overview of the procedure for

computer-based generation of the materials microstructure is discussed in Section II.1.

Procedures for the finite-element based determination of the mechanical, thermal, and

thermo-mechanical properties are presented in Sections II.2, II.3, and II.4, respectively.

The results of application of the aforementioned procedures to the Co-WC two-phase

material are presented and discussed in Section III.1. A Co-WC flywheel optimization

problem is analyzed in Section III.2 in order to demonstrate the importance of accurate

determination of the effective materials properties. The main conclusions resulting from

the present work are summarized in Section IV.

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II. COMPUTATIONAL PROCEDURE

II.1 Microstructural Model

In the present work, the procedures for computer-based generation of RMEsinitially proposed by Reiter et al. [12] and Grujicic and Zhang [13] are extended to

anisotropic materials and used for determination of the effective mechanical, thermal and

thermo-mechanical properties. A brief description of this procedure as applied to a two-

phase material is given below:

(a) A two-dimensional rectangular region whose size corresponds to the size of the

RME (Determined by optical metallography) is first selected and divided into triangular

elements, Figure 2(a). The sides of the triangular elements are selected in such a way that

24 adjacent triangular elements form a perfect hexagonal region, Figure 2(a). For

materials with equiaxed grains, the size of the triangular elements is chosen so that the

width of the hexagonal regions is equal to the mean linear grain-intercept, as determined

through the use of quantitative metallography [e.g. 13]. For materials in which the grains

are elongated and aligned in a particular direction, which gives rise to materials

anisotropy, the width of the grains is set equal to the width of the hexagons, while the

length of the grains is used to determine the number of triangular elements which needs to

be combined to form the appropriate irregular hexagons, Figure 2(b). This procedure

ensures consistency between the characteristic length-scales of the computer-generated and

the experimentally-observed materials microstructures.

(b) Next, material 1 or material 2 are assigned to each hexagonal grain. This is done

using a statistical procedure within which the probability for a region to contain given

material is set prepositional to the volume fraction of that material in the two-phase

mixture. In addition, the procedure ensures that the periodic boundary conditions are

imposed on the resulting materials microstructure. In other words, the rectangular region

acts as a microstructural unit cell and an infinite two-phase crystal can be obtained by

repeating the cell in the x- and y-directions. Two examples of the microstructural unit cell

in a material containing equiaxed grains are shown in Figures 3(a)-(b).

(c) The boundaries of the rectangular domain are next subjected to the specific

mechanical or thermal loading and a finite element analysis carried out in order to

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determine the materials response. The response of the region is then used to determine the

effective materials properties. A more detailed description of this procedure for isotropic

two-phase materials with equiaxed grains is given below. The procedure can be readily

extended to anisotropic multi-phase materials with non-equiaxed grains.

II.1 Determination of the Effective Mechanical Properties

The finite-element procedure mentioned above is used in this section to determine

the effective Youngs module, E, the effective Poissons ratio, , and the effective yieldstress, y, in an isotropic two-phase material.

The (microstructural cell) rectangular region, such as the ones shown in Figures

3(a)-(b), is subjected to displacement-controlled plane-strain uniaxial extension in the x-

direction, while requiring that the domain remains rectangular during loading, Figure 4(a).

The normal strains 11 and 22 and the normal stress 11 are recorded during loading andthe effective properties computed as following:

)1( 2

11

11

=E (1)

11

22

11

22

1

=(2)

2

%2.0,111 +=

y(3)

where denotes an increment and 11, 0.2% is the 0.2% plastic-strain offset 11 stress.Equations (1) and (2) can be readily derived using the generalized isotopic Hookes law

with 22=0 and 33=0. Equation (3), on the other hand, arises from the Von Mises yieldcriterion. It should be noted that the constraint that the rectangular domain remainsrectangular during loading produces a non-zero 22 stress. However, 22 is found in thepresent work to be less than 0,1% of11 and its effect is deemed insignificant.

When a composite material consists of two metallic materials, in which yielding is

controlled by plastic deformation, determination of the effective yield stress (as well as of

the two elastic moduli) using Equation (3) is straightforward and requires specification of

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the yield stress of the two constituent materials within the finite-element based analysis

discussed above. However, when at least one of the materials is ceramic, a somewhat

different strategy has to be used since (tensile) stress in ceramic materials is generally

controlled by brittle fracture rather than plastic yielding. Depending on the defect content,

grain size and general quality of the ceramic material, brittle fracture can occur at

substantially different stress levels and can be intergranular, transgranular, or both. The

fracture phenomenon is frequently modeled using the cohesive-zone approach initially

proposed by Needleman [14]. This approach is utilized in the present work.

Within the cohesive-zone approach, fracture is modeled by prescribing the crack-

face traction vs. crack-face separation distance constitutive relation. In the present work

the following relation based on the exponential binding law [14] is used:

)exp()( maxn

n

n

nnn

UUeUT

= (4)

where Tn and Un are the normal crack traction and crack-face separation, respectively, and

max and n are material-dependent cohesive-zone parameters. They are respectively themaximum normal traction and the corresponding crack-face separation. The work of

fracture, n, is defined in terms ofmax and n as: nn max)1exp(= . And is related to theplane-strain mode-I critical stress intensity factor KIc as:

21

=

n

Ic

EK (5)

Thus, when KIc data is available for a given material, only one of the cohesive-zone

parameters (max or n) is independent and needs to be determined.The cohesive-zone parameters, max and n, can be determined by matching the

finite-element based and the experimental stress-strain curves for the (single-phase) brittle

material in question. In the present work this was done by inserting quadrilateral

interfacial elements along the edges of all triangular elements. The stiffness matrix of these

elements is a function ofmax and n. The procedure for deriving the stiffness matrix of theinterfacial elements is given in our previous work [15]. Since 24 adjacent triangular

elements form a grain, Figure 2(a), placing interfacial elements along the edges of all

triangular elements enables fracture to take place both transgranularly and

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intergranularly. Each interfacial element is then assigned max (and n) from a Gaussiandistribution with the mean value

max and with the standard deviation, 0.1

max ,

respectively. A simple optimization procedure withmax

as the sole design parameter is

next used to determine the optimal value of max which gives the best match between the

computed and the measured yield (failure) stress.

II.2 Determination of the Effective Thermal Properties

To determine the effective thermal conductivity of a two-phase material, the

(microstructural-cell) rectangular region is sandwiched between two single-phase

rectangular regions, Figure 4(b). The left and the right boundaries of the resulting three-

domain region are next subjected to the constant temperature boundary conditions, T 1 and

T2, respectively. The upper and the lower boundaries are insulated. The finite element

method is next used to determine the steady-state distribution of the temperature

throughout the three-domain region. The mean temperatures along the left and the right

boundaries of the middle (two-phase) domain, TL and TR, are next determined. The

effective thermal conductivity k is then computed using the constant heat-flux condition:

R

RR

ML

LL

d

TTk

d

TTk

d

TTk 2121

=

=

(6)

where the subscripts L, M, and R denote the left, middle, and right domains, and d is the

width of the particular domain in the x-direction.

Once the effective thermal conductivity is determined, the effective thermal

diffusivity, , can be computed from the relation:

pC

k= (7)

where the effective density, , and the effective specific heat, Cp, being microstructureinsensitive properties, are defined as the weighted average of the corresponding properties

of the two constituent materials.

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II.3 Determination of the Effective Thermo-mechanical Properties

The effective linear coefficient of thermal expansion, , can be determined throughthe use of the finite element method by subjecting the (microstructural cell) rectangular

region to a uniform heating while requiring that the shape of the region remains

rectangular. The effective linear thermal expansion coefficient then can be computed as:

dT

d thth )]1(2/[)(* 2211

++= (8)

where and , are the two normal thermal strains.th

11th

22

III. RESULTS AND DISCUSSION

III.1 Effective Properties of the Co-WC Two-phase Material

The procedure described above is utilized in this section to determine the effective

properties of cobalt-tungsten carbide (Co-WC) two-phase material. This material,

commonly referred to as a cermet, is frequently used as a cutting-tool material. In addition,

it has been recently suggested by Huang and Fadel [16] as a material for high-performance

flywheels.

The mechanical, thermal, and thermo-mechanical properties of the two constituent

materials (Co and WC) considered in the present work are summarized in Table 1. The

properties are obtained from the CES Materials Selector [17].

The computation of the effective materials properties of two-phase Co-WC material

as a function of the volume fractions of the two constituent materials is carried out using

the commercial finite element package Abaqus/Standard [18]. The procedure for

computation of the stiffness matrix of the interfacial elements is incorporated in the User

Element (UEL) subroutine of Abaqus/Standard. A detailed account of this procedure is

given in our previous work [15].

The finite element analysis of plane-strain tension of pure WC yielded the following

values of the cohesive zone parameters: max = 543 MPa and n =0 .26 m.The variations of the mechanical properties (the Youngs modulus, the Poissons

ration and the tensile yield stress), averaged over ten microstructural-cells with the volume

fraction of cobalt, fCo, are shown in Figures 5-6, respectively. For comparison, the results

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obtained using the self-consistent method [7] and the experimental results [18] are also

shown in Figures 5-6. The three sets of results are denoted as Finite Element, Self

Consistent and Experiment. A detailed account of the determination of the effective

mechanical properties using the self-consistent method and its application to a two-phase

metal-ceramic material is given in our previous work [14]. To account for the variation of

material microstructure with the volume fraction of cobalt, the self-consistent analysis is

carried out in such a way that for fCo0.15 and fCo0.85 the material is considered toconsist of isolated inclusions of the minor material in a finite matrix of the major material.

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with the respective experimental results than the self-consistent method. However, one

must recognize that the two computational approaches overall yield comparable results.

Hence, considering a relative computational simplicity of the self-consistent method, it is

not clear that the use of computationally more demanding finite-element formulation is

warranted. To help resolve this issue, the case of heterogeneous Co-WC flywheel

optimization recently carried out by Huang and Fadel [16] is redone in this section using

both the self-consistent method and finite-element method based effective mechanical

properties. One must recognize, never the less, that the knowledge of accurate mechanical

properties may be more critical in some applications than in others. The Co-WC flywheel

optimization problem analyzed by Huang and Fadel [16] involves determination of the

optimum flywheel profile and the optimum distribution of Co and WC in the radial

direction relative to two objectives: (a) a maximum stored kinetic energy; and (b) a

minimum value of the peak normal equivalent stress. The first objective arises from the

basic function of the flywheel while the second one concerns minimization of the

probability for failure of the flywheel. The two objectives are combined into a single

objective using the weighting method. The optimization is subject to the following

constraints: (a) the kinetic energy stored must exceeded 500kJ at the angular velocity of

630 rads/s; (b) the peak equivalent stress must not exceed 150 MPa; (c) the flywheel mass

must not exceed 100kg; (d) the flywheel thickness should be between 0.02 and 0.1m; and (e)

the flywheel inner and outer radii are fixed as 0.02 and 0.2m, respectively. The

optimization is carried out using the DOT optimization program [20].

The flywheel profile and material distribution results for the two (objective-

function) weighting factors equal 0.5 are shown in Figures 8 and 9, respectively. The results

obtained using the self-consistent method predicted effective properties are denoted as

Self Consistent while the ones obtained using the finite-element procedure are designated

as Finite Element. The overall profiles of the flywheel and the materials distributions

have common general features. That is, the flywheel is relatively thick and contains mostly

the strong material (Co) in the region next to the shaft where the operating stresses are

highest. In addition, the flywheel is relatively thick and contains mostly the heavy material

(WC) near its outer rim where the kinetic energy stored per unit volume is highest.

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However, details of the flywheel profiles and of the material distributions are significantly

different in the two cases.

The predicted optimum characteristics of the flywheel based on the self-consistent

and the finite-element method based effective mechanical prosperities are respectively: (a)

the maximum kinetic energy stored,503kJ and 534kJ; (b) the maximum equivalent stress,

137MPa and 119MPa; and (c) the flywheel mass, 91.2kg and 99.1 kg. These differences in

optimum characteristics of the flywheel in the two cases can be considered as significant.

Thus, based on all the results presented in this section, one may conclude that, at

least in the case of optimization of a multi-material flywheel, the choice of effective material

parameters is quilt important and that it can lead to significantly different design solutions.

IV. CONCLUSIONSBased on the results obtained in the present study the following main conclusions

can be drawn:

(a) The combination of a computer-based procedure for microstructure

generation and a finite element analysis of the mechanical and thermal responses can lead

to significant improvements in accurate prediction of the effective materials properties

over the predictions made by the analytical methods such as the self-consistent method.

(b) The improvements are particularly significant in materials in which the

volume fractions of the constituent phases are comparable;

(c) Relatively small differences in the effective materials properties may give rise

to significant differences in designs involving multi-phase materials.

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ACKNOWLEDGEMENTS

The material presented here is based on work supported by the National Science

Foundation, Grant Numbers DMR-9906268 and CMS-9531930, the U.S. Army Grant Number

DAAH04-96-1-0197, the Optomec Company, Albuquerque, and by he ALCOA Foundation.

The authors are indebted to Dr. Bruce A. MacDonald of NSF and Dr. David M. Stepp of ARO

for the continuing interest in the present work. The authors also acknowledge the support of

the Office of High Performance Computing Facilities at Clemson University.

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REFERENCES:

[1] Hashin, Z. and Strikman, S., A Variational Approach to the Theory of the Elastic

Behavior of Multiphase Systems,J. Mech. Phys. Solids, 11 (1963) 127.

[2] Chen, H.S. and Acrivos, A., The Effective Elastic Moduli of Composite Materials

Containing Spherical Inclusions at Non-Dilute Concentrations, Int. J. Solids Structures, 14

(1978) 349.

[3] Hill, R., Elastic Properties of Reinforced Solids: Some Theoretical Principles,J.

Mech. Phys. Solids, 13 (1965) 213.

[4] Budiansky, B., On the Elastic Moduli of Some Heterogeneous Materials, J. Mech.

Phys. Solids, 13 (1965) 223.

[5] Hori, M. and Nemat-Nasser, S., "Two Micromechanics Theories for Determining

Micro-Macro Relations in Heterogeneous Solids,Mech. Mater., 14 (1993) 189.

[6] Eshelby, J.D., The determination of the Elastic Field of an Ellipsoidal Inclusion, and

Related Problems,Proc. R. Soc. London, 241A (1958) 376.

[7] Nemat-Nasser, S. and Hori, M.,Micromechanics: Overall Properties of Heterogeneous

Materials, Elsevier Publishers, 1993.

[8] Bao, G., Hutchinson, J.W. and McMeeking, R.N., Particle Reinforcement of Ductile

Matrices Against Plastic Flow and Creep,Acta Metall. Mater., 39 (1991) 1871.

[9] Tvergaard, V., Analysis of Tensile Prosperities for a Whisker-Reinforced Metal-

Matrix Composite, Acta Metall. Mater., 38 (1990) 185.

[10] Christman, T., Needleman, A., and Suresh, S., An Experimental and Numerical

Study of Deformation in Metal-ceramic Composite,Acta Metall. Mater., 37 (1989) 3029.

[11] Grujicic, M., Erturk, T. and Owen, W.S., A Finite Element Analysis of the Effect of

the Accommodation Strain in the Ferrite Phase on the Work Hardening of a Dual-phase

Steel,Materials Science and Engineering, 82 (1986) 151.

[12] Reiter, T., Dvorak, G.J. and Tvergaard, V., Micromechanical Models for Graded

Composite Materials,J. Phys. Solids, 45 (1997) 1281.

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[13] Grujicic, M. and Zhang, Y., Determination of Effective Mechanical Properties of

Functionally Graded Materials,Mater. Sci. Engrg., A251 (1998) 64.

[14] Needleman, A., A Continuum Model for Void Nucleation by Inclusion Debonding,

J. Appl. Mech., 54 (1987) 525.

[15] Grujicic, M., and Zhao, H., Optimization of 316 Stainless Steal/Alumna Functionally

Graded Materials, Mater. Sci. Engrg., A252 (1998) 117.

[16] Huang, J., and Fadel, G.M., Heterogeneous Flywheel Modeling and Optimization,

Materials and Designs, 21 (2000) 111.

[17] Cambridge Engineering Selector, Version 3.1, Granta Design Ltd, Cambridge, UK,

2000.

[18] Abaqus Theory Manual, Version 5.8, Hibbitt, Karlsson and Sorensen, Inc.,

Providence, RI., 1998.

[19] Grujicic, M., Work in Progress, Clemson University, 2001.

[20] DOT Users Manual, Vanderplaats, Miura & Assoc. Inc., 1993.

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Table 1. Mechanical, thermal and thermo-mechanical properties of Co and WC.

Material E

(GPa) y

(MPa)

KIc

(MPa m )

k

(W/mK)

(kg/m3)

Cp

(J/kg K)

(m2/sec)

*(K-1)

Co 207 0.32 700 135 96 8,900 440 2.4510-8 12.810-6WC 645 0.21 488 3.5 58 15,500 204 1.5610

-8

5.810-6

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FIGURE CAPTIONS:

Figure 1. (a) A schematic representation of the variation in FGM microstructure:

(b) and (d) typical microstructure at small volume fractions of the minor material; (c)

typical microstructure at comparable volume fractions of the two materials.

Figure 2. Finite element meshing and grain assignment procedures proposed in the

present work: (a) for materials with equiaxed grains and; (b) for materials with elongated

grains.

Figure 3. Two examples of the microstructural cell used for determination of the

effective materials properties: (a) the minor-material volume fraction equals 0.1; (b) the

volume fraction of the two materials equal 0.5.

Figure 4. (a) The displacement-based plane-strain loading used for determination of

the effective mechanical properties; and (b) the thermal loading used for determination of

the effective thermal properties.

Figure 5. . Variation of the Youngs modulus and Poissons ratio with the volume

fraction of cobalt in Co-WC two-phase material. Error bars denote one standard deviation

over ten runs.

Figure 6. Variation of the yield stress and the thermal expansion coefficient with

the volume fraction of cobalt in Co-WC two-phase material. Error bars denote one

standard deviation over ten runs.

Figure 7. Variation of thermal conductivity and thermal diffusivity with the volume

fraction of cobalt in Co-WC two-phase material. Error bars denote one standard deviation

over ten runs.

Figure 8. Optimum flywheel profiles associated with the two sets of effective

mechanical properties.

Figure 9. Optimum distribution of cobalt in the radial direction of the flywheel

associated with the two sets of effective mechanical properties..

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Figure 1. (a) A schematic representation of the variation in FGM microstructure:

(b) and (d) typical microstructure at small volume fractions of the minor material; (c)

typical microstructure at comparable volume fractions of the two materials.

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(b)

(a)

Figure 2. Finite element meshing and grain assignment procedures proposed in the

present work: (a) for materials with equiaxed grains and; (b) for materials with elongated

grains.

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(a)

(b)

Figure 3. Two examples of the microstructural cell used for determination of the

effective materials properties: (a) the minor-material volume fraction equals 0.1; (b) the

volume fraction of the two materials equal 0.5.

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(a)

PrescribedDisplacements

MicrostructuralCell

RigidBody

Rigid Body

Thermally Insulated

(b)

Two-Phase

Material

Material

R

Material

L

TLT1

Thermally Insulated

T2TR

Figure 4. (a) The displacement-based plane-strain loading used for determination of

the effective mechanical properties; and (b) the thermal loading used for determination of

the effective thermal properties.

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Experiment [19]

FiniteElement

SelfConsistent [13]

Figure 5. . Variation of the Youngs modulus and Poissons ratio with the volume

fraction of cobalt in Co-WC two-phase material. . Error bars denote one standard

deviation over ten runs.

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Experiment [19]

SelfConsistent [13]

FiniteElement

Figure 6. Variation of the yield stress and the thermal expansion coefficient with

the volume fraction of cobalt in Co-WC two-phase material. Error bars denote one

standard deviation over ten runs.

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FiniteElement

Figure 7. Variation of thermal conductivity and thermal diffusivity with the volume

fraction of cobalt in Co-WC two-phase material. Error bars denote one standard deviation

over ten runs.

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Shaft

FiniteElement

SelfConsistent [13]

Figure 8. Optimum flywheel profiles associated with the two sets of effective

mechanical properties.

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Shaft

FiniteElement

SelfConsistent [13]

Figure 9. Optimum distribution of cobalt in the radial direction of the flywheel associated

with the two sets of effective mechanical properties.