nang

D.

Un

rme 18

ate-strengthening eects involving (i) Orowan strengthening eect, (ii) enhanced dislocation density due to the residual plastic strain causedby the dierence in the coecients of thermal expansion between the matrix and particles, and (iii) load-bearing eect have been taken

than 100 nm have properties that are not seen in their

predict the bulk mechanical properties of MMNCs as afunction of the reinforcement, matrix, and processing con-

transformation approach. He et al. [11] and Holtz et al.

inating the mechanical behavior, especially the yieldstrength, were not fully considered. In the meantime,Ramakrishnan [15] proposed an analytical model for pre-dicting the yield strength of the microsized particulate-rein-forced metal matrix composite (MMCs), using a compositesphere model for the intra-granular type of MMCs and

* Corresponding author. Tel.: +1 416 979 5000x6487; fax: +1 416 9795265.

E-mail address: dchen@ryerson.ca (D.L. Chen).

Scripta Materialia 54 (2006) 13211microcrystalline counterparts [1,2]. However, nanostruc-tured materials generally suer from insucient ductilityand reduced toughness compared with the conventionalmicrocrystalline materials. On the other hand, metal matrixnanocomposites (MMNCs) are most promising in produc-ing balanced mechanical properties between nano- andmicro-structured materials, i.e., enhanced hardness,Youngs modulus, 0.2% yield strength, ultimate tensilestrength and ductility [39], due to the addition of nano-sized reinforcement particles into the matrix.

To facilitate the development of MMNCs, it is necessaryto develop constitutive relationships that can be used to

[12] qualitatively explained their results using Fan et al.smodel. Lurie et al. [13] developed a continuum mechanicsmodel by consideration of interactions between the nano-particles and the matrix. However, in order to use the con-tinuum mechanics approach the authors [1013] tried tomodify the interface between the matrix and reinforcementparticles. The diculty with the continuum approach isthat it ignores the inuence of particles on the microme-chanics of deformation and strengthening mechanisms,such as the location of particles, grain size, and dislocationdensity [14]. That is to say, the strengthening mechanismsor the types of MMNCs, which are the key factors in dom-into account in the model. The prediction is in good agreement with the experimental data reported in the literature. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Metal matrix nanocomposites; Yield strength; Orowan strengthening; Load-bearing eect; Enhanced dislocation density strengthening

1. Introduction

Nanocrystalline materials form an exciting area of mate-rials research because bulk materials with grain sizes of less

ditions. In the past few years, some modeling work [1013]has been done in this regard. Fan et al. [10] proposed ageneralized law of mixture by using a rigorous continuummechanics analysis and an equivalent microstructuralConsideration of Orowaparticulate-reinforced met

A model for predicti

Z. Zhang,

Department of Mechanical and Industrial Engineering, Ryerson

Received 28 October 2005; received in revised foAvailable onlin

Abstract

An analytical model for predicting the yield strength of particul1359-6462/$ - see front matter 2006 Acta Materialia Inc. Published by Elsedoi:10.1016/j.scriptamat.2005.12.017strengthening eect inl matrix nanocomposites:their yield strength

L. Chen *

iversity, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3

17 November 2005; accepted 8 December 2005January 2006

reinforced metal matrix nanocomposites has been developed. The

www.actamat-journals.com

326vier Ltd. All rights reserved.

Maincorporating two improvement parameters associatedwith the dislocation strengthening of the matrix and theload-bearing eect of the reinforcement. This model, repre-senting an incorporation of both continuum and microme-chanics approaches, has been used to predict the low-cyclefatigue life of discontinuous reinforced MMCs [16,17].However, Ramakrishnans model was applicable only forMMCs containing microsized particles.

The objective of this investigation was to model andpredict the yield strength of the intra-granular type ofMMNCs, which represents one of the most importantaspects of the nanocomposite strengthening mechanismsand eects. By considering the strengthening mechanismsof MMNCs, and incorporating Ramakrishnans modeland the Orowan strengthening eect, an analytical modelfor predicting the yield strength of particulate-reinforcedMMNCs has been proposed. The theoretical predictionsbased on this model were found to be in good agreementwith the experimental data reported in the literature.

2. Model development

Due to the excellent mechanical properties, MMNCshave attracted the interest of many researchers. A lot ofwork has been done involving dierent synthesis methods,structures, mechanical properties, and strengthening mech-anisms of MMNCs. Since the strengthening mechanisms ofMMNCs are fundamental to the development of the pres-ent model, they are rst summarized as follows.

2.1. Orowan strengthening mechanism

Orowan strengthening, caused by the resistance ofclosely spaced hard particles to the passing of dislocations,is important in aluminium alloys. It is widely acknowl-edged, however, that Orowan strengthening is not signi-cant in the microsized particulate-reinforced MMCs,because the reinforcement particles are coarse and theinterparticle spacing is large. Furthermore, since the rein-forcement is often found to lie on the grain boundaries ofthe matrix, it is unclear whether the Orowan mechanismcan operate at all under these circumstances [18]. For meltprocessed MMCs with the usually-used particles of 5 lm orlarger, Orowan strengthening has indeed been pointed outto be not a major factor [14]. In contrast, due to the pres-ence of highly-dispersed nanosized reinforcement particles(smaller than 100 nm) in a metal matrix, Orowanstrengthening becomes more favourable in MMNCs. Ithas been well established that the presence of a dispersionof ne (100 nm) insoluble particles in a metal can consid-erably raise the creep resistance, even for only a smallvolume fraction (

0:13Gmb dp

temperature [8,36] are used: rym = 97 MPa,Em = 42.8 GPa,m = 0.3, Gm = Em/[2(1 + m)] = 16.5 GPa, b = 0.32 nm,am = 28.4 106 (C)1, ap = 9.0 106 (C)1, Tprocess =300 C, Ttest = 20 C, and dp = 20, 30, 40, 50, 70 and100 nm. Two trends can be seen from Fig. 1: (i) a higher vol-ume fraction of nanoparticles leads to a higher yieldstrength; (ii) the nanoparticle size has a strong eect onthe yield strength. A small volume fraction of nanoparticu-lates of less than 0.06 can signicantly improve the yieldstrength of MMNCs.

3. Verication of the model and discussion

The yield strength predicted via the present model, i.e.,Eq. (9), in a nano-Al2O3 particulate-reinforced magnesiumnanocomposite as a function of nanoparticle size can beseen in Fig. 2. Clearly, the nanoparticle size has a signi-cant eect on the yield strength when the volume fractionis slightly higher, e.g., VpP 0.01. Another important pointis that the improvement in the yield strength of the

Y

Madislocation density in the matrix, caused by the thermalmismatch between the matrix and the reinforcementparticles.

As stated above, for MMNCs Orowan strengtheningmechanism should be taken into consideration. Whenseveral strengthening eects are simultaneously present,one way would be to use the rules of addition of thestrengthening contributions, e.g., by Lilholt [31]. In thisinvestigation Ramakrishnans approach [15] is considered,since it was shown that both additive and synergistic eectscould be taken into account. Thus, the yield strength ofparticulate-reinforced MMNCs, ryc, may be expressed as,

ryc rym1 fl1 fd1 fOrowan; 2where fOrowan is the improvement factor associated withOrowan strengthening of the nanoparticles. For particu-late-reinforced composites the general expression for fl is[15,16,32],

fl 0:5V p; 3where Vp is the volume fraction of the reinforcement nano-particles. fd has been expressed to be [33],

fd kGmb qp =rym; 4where Gm is the shear modulus of the matrix, b is the Bur-gers vector of the matrix, k is a constant, approximatelyequal to 1.25, q is the enhanced dislocation density whichis assumed to be entirely due to the residual plastic straindeveloped due to the dierence in the coecients ofthermal expansion (DCTE) between the reinforcementphase and the matrix during the post-fabrication cooling.For equiaxed particulates the following expression wasreported [34],

q 12 DaDTV pbdp1 V p ; 5

where dp is the particle size, Da is the dierence in thecoecients of the thermal expansion, DT is the dierencebetween the processing and test temperatures.

The improvement factor fOrowan related to the Orowanstrengthening of nanoparticles introduced in Eq. (2) canbe expressed as,

fOrowan DrOrowan=rym; 6where DrOrowan has been described by the OrowanAshbyequation [24],

DrOrowan 0:13Gmbk lnrb; 7

where r is the particle radius, r = dp/2, and k is the interpar-ticle spacing, expressed as [16,35],

k dp 12V p

13

1" #

. 8

Substituting Eqs. (3)(8) into Eq. (2) and considering

Z. Zhang, D.L. Chen / ScriptaDT = Tprocess Ttest, Da = am ap, one can derive the fol-lowing equation for the yield strength of MMNCs,B dp 12V p

13 1

ln2b

. 9b

Fig. 1 presents the analytical results of the eect of thevolume fraction (Vp) on the yield strength based onEq. (9) for dierent sizes of reinforcement nanoparticu-lates (dp). The data for the nano-Al2O3 particulate-reinforced magnesium nanocomposites tested at roomryc 1 0:5V p rym A B ABrym

; 9

A 1:25Gmb12T process T testam apV p

bdp1 V p

s; 9a

0

100

0 0.01 0.02 0.03 0.04 0.05 0.06Volume fraction of nanoparticles

Fig. 1. Yield strength as a function of volume fraction of nanoparticles fordierent particle sizes in nano-Al2O3 particulate-reinforced magnesiumnanocomposites tested at 20 C.200

300

400

500

600

ield

stre

ngth

, MPa

dp=20 nmdp=30 nmdp=40 nmdp=50 nmdp=70 nmdp=100 nm

terialia 54 (2006) 13211326 1323MMNCs becomes very strong when the nanoparticle sizeis smaller than about 100 nm. This is in agreement with

krishnans model [15], thus indicating that Orowanstrengthening eect should be taken into account inMMNCs. Since the tensile bar contained rod shapedAl2O3 nanoparticles [8], the strengthening eect of such arod shape should be higher than the spherical one [24]. Inour model all nanoparticles were assumed to be spherical.This is probably why our model slightly underestimatesthe rst two experimental data. On the other hand, withincreasing volume fraction of the reinforcement particles,the probability of forming the processing-induced voidsbecomes higher, leading to a degradation of the yieldstrength [38]. This would be the main reason why the thirdexperimental value was somewhat lower than our modelprediction, because in the present model no porosity wasconsidered within the nanocomposites.

To further verify our model, another comparisonbetween the present model prediction and the experimentaldata reported in Ref. [39] is shown in Fig. 4, where theeect of the particle shape related to Orowan strengtheningis also considered [24,39]. The following data for the Y2O3-reinforced titanium nanocomposites tested at room tem-perature are used: rym = 330 MPa [39]; Gm = 44.8 GPa,

6 1

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200Nanoparticle size, nm

Yiel

d st

reng

th, M

Pa

Vp=0.001Vp=0.005Vp=0.01Vp=0.02Vp=0.03Vp=0.04Vp=0.05

Fig. 2. Yield strength as a function of nanoparticle size for dierentvolume fractions in nano-Al2O3 particulate-reinforced magnesium nano-composites tested at 20 C.

1324 Z. Zhang, D.L. Chen / Scripta Materialia 54 (2006) 13211326the experimental results [8,36], and provides a theoreticalsupport to the terminology of nanotechnology, e.g.,dened by the US National Science Foundation [37], where. . . The novel and dierentiating properties and functionsare developed at a critical length scale of matter typicallyunder 100 nm. . . is specied. Most researchers [9,14] inthe area of nanocomposites have also done their researchby controlling the nanoparticle size below 100 nm. Thus,100 nm is the critical size for nanoparticulate-reinforcedMMNCs to produce excellent mechanical properties, com-pared to the counterpart of microparticulate-reinforcedMMCs.

Good agreement between the present model prediction,based on Eq. (9), and the experimental data is observedand shown in Fig. 3. It is seen that the present model can

be used to better predict the yield strength than Rama-

50

100

150

200

250

0.001 0.006 0.011 0.016Volume fraction of nanoparticles

Yiel

d st

reng

th, M

Pa

Present model, Eq. (9)Experimental data [8]Ramakrishnan's model [15]

Fig. 3. A comparison of the present model with Ramakrishnans model[15] and with the experimental data for nano-Al2O3 particulate-reinforcedmagnesium nanocomposites tested at 20 C [8].b = 0.29 nm [40]; am = 11.9 10 (C) [41], ap =9.3 106 (C)1 [42], Tprocess = 827 C for A1, B1, C1, D1and 900 C for A2, B2, C2, and dp = 2, 10, 9, 13, 40, 10and 30 nm [39]. On the basis of the values of the weightfraction given in Ref. [39], the following converted valuesof volume fraction Vp = 0.25, 0.38, 0.59, 0.59, 0.27, 0.41and 0.54% are utilized.

Again, good agreement between the model predictionfor the minimum sized reinforcement particles and theexperimental data is seen in Fig. 4, where a combined eect

400

500

600

700

800

900

400 500 600 700 800 900YS predicted by present model, MPa

Expe

rimen

tal Y

S, M

Pa

Fig. 4. A comparison of the prediction via the present model with theexperimental data for Y2O3 particulate-reinforced titanium nanocompos-

ites tested at room temperature, where the error bar was based on therange given in Ref. [39].

Maof the variation in the volume fraction of nanoparticles,thermomechanical treatment, and microstructure has beentaken into consideration.

The above comparison between the present modelprediction and the experimental data corroborates that itis necessary to consider Orowan strengthening in MMNCs.Fig. 5 shows an example of the comparison among thethree improvement factors (fl, fd, fOrowan) as a function ofthe volume fraction of nanoparticles with a size of 50 nmin nano-Al2O3 particulate-reinforced Mg nanocomposites.It is also seen that Orowan strengthening plays a signicant

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06Volume fraction of nanoparticles

Impr

ov

emen

t fac

tor

flfdfOrowan

Fig. 5. A comparison among the three improvement factors (fl, fd, fOrowan)as a function of the volume fraction of nanoparticles in nano-Al2O3particulate-reinforced Mg nanocomposites.

Z. Zhang, D.L. Chen / Scriptarole in MMNCs, while the load-bearing eect becomes verysmall.

4. Conclusions

(1) A model for predicting the yield strength of intra-granular type of metal matrix nanocomposites(MMNCs) is proposed on the basis of the strengthen-ing eects characterized by the modied shear lagmodel, enhanced dislocation density model, and theOrowan strengthening eect.

(2) It is shown that the yield strength of MMNCs isgoverned by the size and volume fraction of nanopar-ticles, the dierence in the coecients of thermalexpansion between the matrix and nanoparticles,and the temperature change after processing.

(3) The present model indicates that 100 nm is a criticalsize of nanoparticles to improve the yield strengthof MMNCs, below which the yield strength increasesremarkably with decreasing particle size.

(4) The proposed model shows excellent agreement withthe experimental data reported in the literature, indi-cating that it is necessary to consider Orowanstrengthening in MMNCs.

Nanostruct Mater 1997;9:2258.

[31] Lilholt H. In: in: Proc of the 4th Riso Int Symp on Metall Mater

Sci. Denmark, Roskilde: Riso National Laboratory; 1983. p.38192.

[32] Nardone VC, Prewo KM. Scripta Metall 1986;20:438.[33] Hansen N. Acta Metall 1977;25:8639.[34] Taya M, Arsenault RJ. Metal matrix compositesthermomechanical

behavior. New York (NY): Pergamon Press; 1989.[35] Meyers MA, Chawla KK. Mechanical behaviour of materials. Sad-Acknowledgements

The authors would like to thank the nancial supportprovided by the Natural Sciences and Engineering Re-search Council of Canada (NSERC), and the Premiers Re-search Excellence Award (PREA).

References

[1] Gleiter H. Mechanical properties and deformation behavior ofmaterials having ultra-ne microstructure. Dordrecht: Kluwer Aca-demic; 1993. p. 335.

[2] Mago MJ. Mechanical properties and deformation behavior ofmaterials having ultra-ne microstructure. Dordrecht: Kluwer Aca-demic; 1993. p. 36180.

[3] Sekino T, Niihara K. Nanostruct Mater 1995;6:6636.[4] Hwang S, Nishimura C, McCormick PG. Scripta Mater

2001;44:245762.[5] Audebert F, Prima F, Galano M, Tomut M, Warren PJ, Stone IC,

et al. Mater Trans 2002;43:201725.[6] Lu L, Lai M, Liang W. Compos Sci Technol 2004;64:200914.[7] Shao I, Vereecken PM, Chien CL, Searson PC, Cammarata RC. J

Mater Res 2002;17:14128.[8] Hassan SF, Gupta M. Mater Sci Technol 2004;20:13838.[9] Li XC, Yang Y, Cheng XD. J Mater Sci 2004;39:32112.[10] Fan Z, Tsakiropoulos P, Miodownik AP. J Mater Sci 1994;29:

14150.[11] He L, Allard LF, Ma E. Scripta Mater 2000;42:51723.[12] Holtz RL, Provenzano V. Nanostruct Mater 1997;8:289300.[13] Lurie S, Belov P, Volkov-Bogorodsky D, Tuchkova N. Comput

Mater Sci 2003;28:52939.[14] Lloyd DJ. Int Mater Rev 1994;39:2446.[15] Ramakrishnan N. Acta Mater 1996;44:6977.[16] Zhang Q, Chen DL. Scripta Mater 2004;51:8637.[17] Zhang Q, Chen DL. Int J Fatigue 2005;27:41727.[18] Clyne TW, Withers PJ. An introduction to metal matrix compos-

ites. Cambridge: Cambridge University Press; 1993.[19] Hazzledine PM. Scripta Metall Mater 1992;26:578.[20] Huang H, Bush MB, Fisher GV. Key Eng Mater 1997;127

31:11918.[21] Thilly L, Veron M, Ludwig O, Lecouturier F. Mater Sci Eng A

2001;309310:5103.[22] Thilly L, Veron M, Ludwig O, Lecouturier F, Peyrade JP, Askenazy

S. Philos Mag A 2002;82:92542.[23] Choi SM, Awaji H. Sci Tech Adv Mater 2005;6:210.[24] Dieter GE. Mechanical metallurgy. third ed. New York (NY): Mc-

Graw-Hill; 1986. p. 21220.[25] Vaidya RU, Chawla KK. Compos Sci Technol 1994;50:1322.[26] Dunand DC, Mortensen A. Mater Sci Eng A 1991;144:17988.[27] Arsenault RJ, Shi N. Mater Sci Eng 1986;81:17587.[28] Liu HZ, Wang AM, Wang LH, Ding BZ, Hu ZQ. J Mater Res

1997;12:118790.[29] Zhang HY, Maljkovic N, Mitchell BS. Mater Sci Eng A

2002;326:31723.[30] Liu HZ, Wang AM, Wang LH, Lou TP, Ding BZ, Hu ZQ.

terialia 54 (2006) 13211326 1325dle River (NJ): Prentice Hall; 1999. p. 4924.[36] Srikanth N, Hassan SF, Gupta M. J Compos Mater 2004;38:203747.

[37] Available from: www.nsf.gov/home/crssprgm/nano/omb_nifty50.htm.

[38] Tham LM, Su L, Cheng L, Gupta M. Mater Res Bull 1999;34:719.

[39] Castro V, Leguey T, Munoz A, Monge MA, Pareja R. Mater Sci EngA 2005;A4001:3458.

[40] Callister WD. Materials science and engineeringan introduction.sixth ed. New York (NY): John Wiley and Sons; 2003.

[41] Bever MB. Encyclopedia of materials science and engineering. NewYork (NY): Pergamon Press; 1986. p. 1059.

[42] Kingery WD. Introduction to ceramics. second ed. New York(NY): John Wiley and Sons; 1976. p. 595.

1326 Z. Zhang, D.L. Chen / Scripta Materialia 54 (2006) 13211326

Consideration of Orowan strengthening effect in particulate-reinforced metal matrix nanocomposites: A model for predicting their yield strengthIntroductionModel developmentOrowan strengthening mechanismEnhanced dislocation density strengtheningmechanismLoad-bearing effect of the reinforcementstrengthening mechanism

Verification of the model and discussionConclusionsAcknowledgementsReferences