Composites Science and Technology 69 (2009) 1587–1591

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Composites Science and Technology

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Prediction of the variation of elastic modulus in ZrO2/NiCr functionallygraded materials

Xin Jin, Linzhi Wu*, Guo Licheng, Sun YuguoCenter for Composite Materials, Harbin Institute of Technology, Harbin 150080, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 December 2008Received in revised form 13 February 2009Accepted 22 February 2009Available online 6 March 2009

Keywords:A.Functionally graded materialsB.Elastic modulusB.InterfaceC.Mori–Tanaka methodC.Cohesive law

0266-3538/$ - see front matter Crown Copyright � 2doi:10.1016/j.compscitech.2009.02.032

* Corresponding author. Tel.: +86 451 86402376; faE-mail addresses: pzauzn@126.com (X. Jin), wlz@h

The variation of elastic modulus in the ZrO2/NiCr functionally graded materials (FGMs) is investigatedexperimentally and theoretically. To obtain the variation of the elastic modulus in the FGMs experimen-tally, 3-point bending tests are conducted on the homogeneous specimens with different volume frac-tions of NiCr. The experimental results show that the elastic moduli decrease obviously with theincrease of NiCr, which differ greatly from those predicted by the traditional Mori–Tanaka method. Fromthe fractographs of the homogeneous specimens, it is found that the particles/matrix interfaces arebonded weakly and easy to debond. Based on the microstructure and fracture mechanism of the FGMsand homogeneous specimens, the modified Mori–Tanaka method from Tan is applied to predict the elas-tic modulus. The debonding process of the particles/matrix interface is characterized by a cohesive law.The predicted results agree well with the experimental ones.

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1. Introduction

Functionally graded materials (FGMs) offer an advantageousmean of combining different materials and providing a spatial var-iation in composition and properties [1]. The principal motivationfor the FGMs development is that a spatial variation in compositionand properties at a joint between two different materials has thepotential to reduce the stress at the joint as compared to a bimate-rial interface. Additionally, there is an increased interfacialstrength at the joint and thus the probability of debonding isreduced.

For the optimal design and fabricating process of FGMs, theunderstanding of the relationship between the elastic propertiesand the material composition, as well as the microstructure, isindispensable. Micromechanical models provide a convenientmean to determine the distribution of the elastic properties withthe constituent material volume fraction and microstructure. Somemicromechanical models have been used successfully to predictthe variation of the elastic properties for many FGMs [2–7]. How-ever, for the FGMs with the varying microstructure, the applicabil-ity and accuracy of such models tend to be limited due to thegeometric and micromechanical assumptions on which they arebased [8]. Tohgo et al. [9] obtained the variation of elastic modulusfor the ZrO2/SUS304 FGMs experimentally, and found that the elas-tic moduli of the composites with intermediate composition areless essentially than those predicted by classical micromechanical

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x: +86 451 86402386.it.edu.cn (L. Wu).

models. This indicates that the elastic moduli can be influenced bysome microstructural factors which are not considered in the clas-sical micromechanical models. These microstructural factors canbe found and determined in the experimental investigation ofFGMs. In general, the property variation in FGMs can be deter-mined directly from the experiments by preparing and testingindividual homogeneous specimens with different material com-positions [9]. Hence, according to the microstructure and fracturemechanism of the FGMs and each homogeneous specimen, the ma-jor microstructural factors are determined. To exactly predict thevariation of elastic modulus in FGMs, these factors besides the vol-ume fraction of inclusion should be considered in the microme-chanical models.

In this paper, six-layered ZrO2/NiCr FGMs and six homogeneousspecimens with different volume fractions of NiCr are fabricated bypower metallurgy. The variation of elastic modulus in the FGMs isobtained from the mechanical tests of the homogeneous speci-mens. According to the microstructure and fracture mechanismof the ZrO2/NiCr FGMs and the homogeneous specimens, a modi-fied Mori–Tanaka method from Tan et al. [10] is applied to predictthe variation of elastic modulus.

2. Materials and experimental procedures

2.1. Materials

The ZrO2/NiCr FGMs used in the present investigation is fabri-cated by powder metallurgy. Commercially available ZrO2 andNi–20 wt.%Cr alloy (NiCr) powders are chosen as raw materials.

ights reserved.

S=32

Load

W =3

B=4

Fig. 1. Schematics of the 3-point bending test (dimensions in mm).

1588 X. Jin et al. / Composites Science and Technology 69 (2009) 1587–1591

The characteristics of raw materials are listed in Table 1. The ZrO2

and NiCr powders are mixed in volume ratios of 10–0, 9–1, 8–2, 7–3, 6–4 and 5–5, respectively, and each mixture is suspended inethanol, milled for 3 h by a ball mill and then dried. These mixedpowders are stacked sequentially so as to form a green compactwith graded composition in a steel die and compressed at�30 MPa. Then the green compacts are sintered under the hot-pressing condition of 1300 �C and 5 MPa for 1.5 h. With the sameprocess, six homogeneous specimens with different volume frac-tions of NiCr are also fabricated. Specimens for microstructureinspection and mechanical tests are cut by a diamond saw, andtheir surfaces are ground and polished carefully.

2.2. Mechanical testing

The variation of elastic modulus in the FGMs is obtained fromthe 3-point bending tests of the homogeneous specimens, asshown in Fig. 1. Elastic modulus E is calculated by the equation

E ¼ S3DP

4WB3Ddð1Þ

where S, W and B are the span length, specimen width and speci-men thickness, respectively; DP and Dd are the load incrementand deflection increment as measured from the linear part of theload–deflection curves. Bending tests are conducted on an Instron5569 universal testing machine at crosshead speed of 0.5 mm/min. And three test pieces are tested to get the average value of Efor each material composition.

2.3. Materials characterization

Optical microscopy (Olympus-SZX12) is used to characterizethe microstructure of the FGMs. The density of the sintered speci-mens is measured by Archimedes’ method. The fracture surfaces ofsome homogeneous specimens are examined by a scanning elec-tron microscope (SEM, Philips-XH3).

3.Results

3.1. Microstructure

The microstructure of the sintered ZrO2/NiCr FGMs is shown inFig. 2 in which the white and gray phases are NiCr and ZrO2,respectively. These micrographs show a good gradual variation inthe microstructure of the FGMs. The stepwise gradient in themicrostructure becomes comparatively unclear due to the sinter-ing process. Moreover, in the region from 0% NiCr to 50% NiCr,the microstructure is mainly characterized by the NiCr particlesdispersed in the ZrO2 matrix.

On the other hand, the pressure sintering such as hot-pressingis a more effective method to produce the dense specimens. Therelative densities of the FGMs and homogeneous specimens areshown in Fig. 3. The FGMs exhibit a low porosity level and a rela-tively high relative density 97.2%. The porosity levels of all homo-geneous specimens are also relatively low and their relativedensities are more than 94.5%. The relative density has a maximumvalue (98.4%) for the homogeneous specimen with 10% NiCr and aminimum value (94.6%) for the pure ZrO2.

Table 1Powder characteristics of raw materials.

Materials Particles size (lm) Purity (%)

NiCr <45 >98ZrO2 1.5 >99.9

3.2. Elastic modulus

Fig. 4 shows the resultant variation of the elastic modulus in theFGMs. The Young’s moduli decrease monotonously from 201.0 to119.5 GPa as the volume fractions of NiCr increase from 0% to 50%.

Fig. 5a–e shows the selected SEM micrographs of the fracturesurfaces of the homogeneous specimens with 50%, 40%, 30%, 20%and 10% NiCr, respectively. The fractographs of these specimensare mainly characterized by the debonded metal particles and theirtraces on the brittle fracture surface of the ceramic matrix. In addi-tion, no broken metal particles have been observed in Fig. 5a–e.These suggest that the interfacial strength between metal particlesand ceramic matrix is relatively low and the particles/matrix inter-faces are easy to debond.

4. Discussion

In the present material system, the Young’s moduli of NiCr alloyand ZrO2 are 194.0 [5] and 201.0 GPa, respectively. According tothe Mori–Tanaka method, the variation of the predicted elasticmodulus in the FGMs should not be obvious due to the elasticmodulus of the ceramic approaching to that of the metal. Fig. 6shows the variation of the elastic modulus predicted by the tradi-tional Mori–Tanaka method. The predicted values of elastic modu-lus are situated between 201.0 and 196.6 GPa. However, themeasured elastic modulus obviously decreases with the increaseof NiCr, as shown in Fig. 6. For the 50% NiCr specimen (the homo-geneous specimen with 50% NiCr), the measured elastic modulus(119.5 GPa) is 39.3% lower than the predicted one (196.6 GPa).The great difference between experimental results and theoreticalones can be attributed to the initial defects in the specimens or theweakly bonded particles/matrix interface.

In the powder metallurgy process, the pores can be treated asthe initiate defects. The pores have a deleterious effect on the elas-tic properties. When the pores are considered as the inclusion, theelastic modulus of the composites with the corresponding porositycan also be predicted by the traditional Mori–Tanaka method.However, it can be found from Fig. 3 that the porosity level is rel-atively low for all homogeneous specimens. The influence of thepores on the elastic modulus cannot be obvious. That is, the elasticmodulus of the 50% NiCr specimen with a relative density 96.9%should not decrease 39.3% as compared with that of the full denseone. Hence, the pores do not have prominent effect on the variationof elastic modulus in the ZrO2/NiCr FGMs.

The debonded NiCr particles and their traces in the matrix areobserved in the brittle fracture surface of the ZrO2, as shown inFig. 5. This indicates that the interfaces between the metal particlesand the ceramic matrix are bonded weakly and easy to debond. Tanet al. [10,11] have mentioned that the interfacial debonding cancause the reduction in the elastic modulus of composites as com-pared with those with the perfect bonded interface. They modifiedthe traditional Mori–Tanaka method to study the effect of interfacedebonding on the elastic modulus of the high explosive PBX 9501which consisted of 92.7 vol% energetic crystal (particle phase) and7.3 vol% polymeric binder (matrix phase). They considered the

Fig. 2. Microstructure of the ZrO2/NiCr FGMs.

0 10 2 0 30 4 0 5080

85

90

95

100

Rel

ativ

e de

nsity

(%

)

Volume fraction of NiCr (%)

non-FGM

FGMs

Fig. 3. Relative density of the FGMs and all homogeneous specimens.

0 1 0 2 0 3 0 4 0 5 080

120

160

200

240

E (G

Pa)

Volume fraction of NiCr (%)

Fig. 4. Relationship between elastic modulus and volume fraction of NiCr.

X. Jin et al. / Composites Science and Technology 69 (2009) 1587–1591 1589

spherical particles distributed in a statistically homogeneous man-ner in a continuous matrix. The interface debonding is character-ized by a three-stage (I, II, and III) cohesive law determined fromthe fracture test, as shown in Fig. 7. The piecewise linear cohesivelaw gives stress tractions in terms of displacement discontinuitiesacross the interface [12]. This interface cohesive law involves threeparameters, namely, the interface cohesive strength rmax, and thelinear modulus kr and softening modulus ~kr of the interface. Here,kr and ~kr are the slopes of the ascending and descending segments,respectively. The relation between the normal traction rint and theopening displacement [ur] at the interfaces is then given by

rint ¼ kr½ur �; ½ur� < rmax=kr; Stage-I;

rint ¼ ð1þ ~kr=krÞrmax � ~kr½ur �;rmax=kr < ½ur� < rmaxð1=kr þ 1=~krÞ; Stage-II;

rint ¼ 0; ½ur � > rmaxð1=kr þ 1=~krÞ; Stage-III: ð2Þ

When the particles/matrix interfaces are in Stage-I, the elastic prop-erties of composites are linear. However, once the interfacialstrength rmax is reached, the particles/matrix interfaces are moveto the Stage-II and then Stage-III, and hence the composites behavenonlinearly. More details regarding this cohesive law can be foundin Refs. [10–13] and will not be discussed here.

In the present paper, the main attention is paid to predict thevariation of elastic modulus in the FGMs with the weakly bondedparticles/matrix interfaces. Therefore, the linear modulus kr(Stage-I) is considered in the modified Mori–Tanaka method. Thelinear bulk modulus Kc of the composites with weakly bondedinterfaces is evaluated using the following equation:

Kc ¼1

3ð1�2mmÞEm

þ 9ð1�mmÞ2Em

f�fa1�fþfa

ð3Þ

where Em and mm are the Young’s modulus and Poisson’s ratio of thematrix, respectively; f is the volume fraction of the particles; a is theratio of the average stress in particles to the average stress in ma-trix. When the particles/matrix interfaces are in Stage-I, a is evalu-ated using the following equation [12]:

a ¼ 3ð1� mmÞ2Em

1rpkrþ ð1�2mpÞ

Epþ ð1þmmÞ

2Em

� � ð4Þ

where Ep and mp are the Young’s modulus and Poisson’s ratio of theparticles, respectively, and rp and kr are the particles radius and thelinear modulus of particles/matrix interfaces, respectively.

The modified Mori–Tanaka method is introduced into the cera-mic/metal FGMs to predict the variation of the elastic modulus. Forthe ZrO2/NiCr FGMs, the Young’s moduli of the ZrO2 matrix and theNiCr particles are Em = 201 GPa and Ep = 194 GPa, respectively, andthe Possion’s ratio of the ZrO2 matrix and the NiCr particles aremm = 0.30 and mp = 0.305 [5], respectively. Due to the small differ-ence between mm and mp, the Possion’s ratio of each homogeneousspecimens is assumed to be mc = 0.30. Therefore, the Young’s mod-ulus Ec of each homogeneous specimen can be calculated from thefollowing classical elastic relation:

Ec ¼ 3Kcð1� 2mcÞ: ð5Þ

The mean size of the NiCr particles is rp = 19 lm, which is measuredfrom the SEM micrographs of the fractured homogeneous speci-mens. Then the interfacial linear modulus kr is determined by

Fig. 5. SEM micrographs of fracture surface for the homogeneous specimens with (a) 50%, (b) 40%, (c) 30%, (d) 20%, and (e) 10% NiCr.

0 10 20 30 40 500

40

80

120

160

200

E (

GPa

)

Volume fraction of NiCr (%)

Experiment Mori-Tanaka Modified Mori-Tanaka

Fig. 6. The measured and predicted variation of the elastic modulus (E) in the ZrO2/NiCr FGMs.

int

[ur]

max

k

~k

Stage-I

Stage-II

Stage-III

Fig. 7. Interface cohesive model [11].

0 10 20 30 40 5050

100

150

200E

(G

Pa)

kσ /Kσ

rp=19 μm

f =50 %

Fig. 8. The variation of the elastic modulus E with the linear modulus kr normalizedby Kr = 18.4 GPa/lm.

1590 X. Jin et al. / Composites Science and Technology 69 (2009) 1587–1591

matching the calculated results of Eq. (5) with the above experi-mental data (Fig. 4). For the best fit to the data, the value of kr istaken as 18.4 GPa/lm. The variation of elastic modulus predictedby the modified Mori–Tanaka method is plotted in Fig. 6. The pre-dicted variation of the elastic modulus agrees reasonably with the

measured one. This method also reveals the influence of the particlevolume fraction on the elastic modulus for the composites with theweak particle/matrix interface. The increase of the particle volumefraction can cause obvious reduction in the elastic modulus.

In the modified Mori–Tanaka method, the elastic modulus E isalso influenced by the microstructural factor: linear modulus krof the weakly bonded interface. Fig. 8 shows the relationship be-tween E and kr for the 50% NiCr specimens as rp = 19 lm. It canbe found that the values of E increase with the increase of kr. Thismeans that the improvement of the interfacial performance canenhance the elastic modulus of the composites. As kr ?1, the va-lue of E calculated by the modified Mori–Tanaka method is identi-cal to that predicted by the traditional Mori–Tanaka method. It alsocan be found that, in the range of the normalized kr from 1/10 to10, the values of E increase substantially with the increase of kr.When the normalized kr increases from 1/5 to 1, the value of E in-creases from 79.8 to 127.6 GPa. However, when the normalized kris more than 20 or less than 1/20, the variation of E is insensitive tothe variation of kr. In addition, the predicted elastic modulus tendsto 55.5 GPa when kr ? 0.

X. Jin et al. / Composites Science and Technology 69 (2009) 1587–1591 1591

5. Conclusions

The variation of the elastic modulus in the ZrO2/NiCr FGMs isobtained experimentally from the homogeneous specimens withdifferent volume fractions of NiCr. It is found that the experimentalresults differ greatly from the predicted ones by the traditionalMori–Tanaka method. According to the Mori–Tanaka method, thepredicted elastic moduli should be situated between that of ZrO2

(201.0 GPa) and that of NiCr (194.0 GPa). However, the measuredelastic modulus reduces 40.7% with the increase of NiCr particlesfrom 0% to 50%. This is caused by the weakly bonded particles/ma-trix interfaces which are verified by the SEM observation of thefracture surface for the homogeneous specimens. The modifiedMori–Tanaka method by Tan et al. [10], which takes into accountof the interface debonding, is applied to predict the variation ofelastic modulus in the ceramic/metal FGMs. The predicted resultsagree well with the experimental ones. Hence, this study providesthat the modified Mori–Tanaka method is a reasonable microme-chanical method to estimate the variation of elastic modulus forthe ceramic/metal FGMs with weakly bonded particles/matrixinterfaces.

Acknowledgements

The authors are grateful for the financial support by NSFC(10432030, 10672049, 10872056 and 10602015), National ScienceFoundation for Excellent Young Investigators (10325208).

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