Research ArticleStochastic Effect of Grain Elongation on NanocrystallineMaterials Strain and Strain Rate Produced by AccumulativeRoll-Bonding and Equal Channel Angular Pressing

P Baonhe Sob1 A Alfayo Alugongo1 and T Ba Bob Tengen2

1Department of Mechanical Engineering Faculty of Engineering and Technology Vaal University of TechnologyPrivate Bag X021 Vanderbijlpark 1900 South Africa2Department of Industrial Engineering and Operations Management Faculty of Engineering and TechnologyVaal University of Technology Private Bag X021 Vanderbijlpark 1900 South Africa

Correspondence should be addressed to P Baonhe Sob baonhe_sobrocketmailcom

Received 20 February 2017 Accepted 3 May 2017 Published 12 June 2017

Academic Editor Paulo M S T De Castro

Copyright copy 2017 P Baonhe Sob et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Severe plastic deformation techniques are acknowledged to produce elongated grains during fabrication of nanostructuredmaterials Previous models relating grain size to mechanical properties considered only equivalent radius thus ignoring otherapproaches of measuring grain sizes such as semiminor axis semimajor axis and major axis radii that determine true grain shapeIn this paper stochastic models of nanomaterials mechanical properties that include the ignored parameters have been proposed Theproposed models are tested with data from nanocrystalline aluminum samples The following facts were experimentally observedand also revealed by themodels Grain elongates to amaximum value and then decreases with further grain refinement due to grainbreakages Materials yield stress increases with elongation to a maximum and then decreases continuouslyThe varying approachesof measuring grain radius reveal a common trend of Hall-Petch and Reverse Hall-Petch Relationship but with different criticalgrain sizesMaterials with high curvature grains havemore enhanced yield stress Reducing strain rates leads tomaterials withmoreenhanced yield stress with critical strain rates values beyondwhich further reductions donot lead to yield stress enhancement It canbe concluded that by considering different approaches of measuring grain sizes reasons for different yield stress for nanomaterialsthat were observed but could not be explained have been dealt with

1 Introduction

Amongst the severe plastic deformation (SPD) techniquesdeveloped and tested the Equal Channel Angular Pressing(ECAP) and Accumulative Roll-Bonding (ARB) are uniquedue to their abilities to produce nanomaterials with ultrafinegrain structures and high shear strains [1 2] The ECAP pro-duces high shear strain without changing the sample shapeor dimension [1 2] Its main shortcoming is that only smallquantities of nanomaterials can be produced during experi-mentation which makes it not suitable for industrial appli-cation However the ARB solves this ldquosmall quantityrdquo prob-lem due to its ability to improve on productivity of bulknanostructure Both ECAP and ARB are unique and similarsince both methods involve large misorientation-angle grain

boundaries that lead to very high strains during the deforma-tion of nanostructures [3 4]

Materials are called nanomaterials because of theirrefined grain sizes and most nanomaterials properties havebeen characterized as a function of their grain sizes orsizes of the constituent structures The sizes of 3D graincan be described by the equivalent radius 119903 semiminoraxis radius 1199032 semimajor axis radius 1199031 and major axisradius 1199033 Hillert [5] initially established the model forevolution of grain size or grain growth assuming sphericalgrains while employing the Theory of Second Phase ParticleCoarsening Tengen [6] further states that the theoretical andexperimental investigations reveal that grain growth not onlyis brought about by Grain Boundary Migration (GBM) butalso is accompanied by Grain Rotation-Coalescence (GRC)

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 5418769 9 pageshttpsdoiorg10115520175418769

2 Advances in Materials Science and Engineering

mechanism Tengen [6] modified the Hillert Model [5] soas to consider GRC mechanism the random or stochasticnature of grain size by adding fluctuation terms that accountfor randomfluctuations due to GBMandGRC processes andvariable grain boundary energy

Despite these modifications of the models for grain sizeevolutions most nanomaterials mechanical properties aregiven as functions of spherical grains [7 8] although withrandom size distributionThe relationship between grain size(ie equivalent radius) and yield stress was first proposedby Hall and Petch [7 8] The Hall-Petch Relationship (HPR)states that as the average size of the grains in materialsgets finer the yield stress increases This relationship has itsshortcoming since infinite refinement does not lead to infiniteyield stress The HPRmodel was modified by Zhao and Jiang[9] to reveal the Reverse Hall-Petch Relationship (RHPR)which was later modified by Tengen et al [10] to consider thestochastic nature of grain size

It should be noted that all the above-mentioned modifiedmodels dealt only with the equivalent radius 119903 or made theassumptions that the grains in nanomaterials are spherical innature Experimentally this spherical shape of all the grains innanomaterials is not the case in fact majority of the grains innanomaterials are not nearly spherical natureThus themod-els that consider that the grains in nanomaterials are sphericalignore other important parameters such as semiminor axisradius 1199032 semimajor axis radius 1199031 and the major axis radius1199033 which can be used to closely define elongation and shapeIt must be remarked here that a grain undergoing elongationduring nanomaterial refinement might not change in equiv-alent radius (with equivalent radius defined as the radius ofan equivalent sphere or circle obtained by the displacementmethod of volumearea measurement) As such the impactsof grain elongation on nanomaterials mechanical propertiescannot be properly addressed with the use of equivalentradius onlyTherefore a stochastic model of grain elongation for3D grain undergoing severe plastic deformation that considersthe various approaches of measuring grain size is necessarywhich has been proposed in the current report The impactsof the different approaches of measuring grain sizes and theimpacts of grain elongations on nanomaterials are then dealtwith in this report The proposed models are tested with datafrom grain deformation in nanocrystalline aluminum

2 Methodology

21 Experimental Procedure of Accumulative Roll-Bonding(ARB) The experimental setup for ARB is shown schemat-ically in Figure 1(a) The ARB technology made use of con-ventional rolling facility In this report two shafts were placedhorizontally so that they were free to rotate by means of amechanically automated operationThree gauges (load gaugetemperature gauge and a time gauge or a stop watch) wereused during experimentation One measured the appliedforce which caused the deformation the second gaugemeasured the temperature at the time of experiment and thethird gauge measured the time of deformation All rotatingshafts were mounted on the support roll cage with roller

bearings to prevent friction during motion Extra supportshafts were mounted to ensure stability of the system duringoperation Power was transferred from the power generatorto the gearbox by a shaft Drive was also transferred fromthe gearbox to the rotating beam by a chain drive Duringexperimentation a plunger connected to another motor wasused to force or feed the AA6082-T6Aluminum (Al) throughthe rotating shafts The purpose of this second motor was tocontrol the feed rate or deformation rate or strain rate Therotating shafts gripped the AA6082-T6 Aluminum sampleand forced the sample through the rollers in the first pass

The deformed AA6082-T6 Aluminum was cut in twopieces and stacked together Before stacking the entiresurfaces of the strips were degreased with tetrachlorethyleneand wire-brushed (stainless steel brush) to achieve goodbonding The materials were joined together in the cornersusing aluminum wires and subsequently rolled The wholesequence of ldquorolling cutting face-brushing degreasing andstackingrdquo was repeated again for several ldquopassesrdquo until nano-materials with required characteristics were obtained

The samples with the deformed microstructures wereexamined using Transmission Electron Microscopy (TEM)The microstructure was observed to lengthen along thesemimajor axis 1199033 semimajor axis 1199031 and semiminor axis 1199032The observed nanostructures are presented in Figure 1(b)

22 Experimental Procedure of Equal Channel Angular Press-ing (ECAP) The experiments were carried out as demon-strated by the routes in Figures 2(a) and 2(b) using samplesof AA6082-T6 Aluminum The ECAP die tunnels weredesigned to intersect ldquointernallyrdquo at angle 120601 (120601 = 90∘) butsubtended by the arc or curved ldquoexternalrdquo surface Duringfabrication of nanomaterials a 50-ton hydraulic press wasused to press the aluminum sample through the ECAP dieshown in Figures 2(a)ndash2(c) Since it was very difficult toforce the AA6082-T6 Aluminum sample through the ECAPdie at the point where the two tunnels meet the AA6082-T6 Aluminum samples were heat-treated before each ECAPpass to enable the sample to pass through the ECAP dieas recommended by other researchers [1] It was observedthat heating the AA6082-T6 Aluminum sample led to graingrowth and subjecting the material through the ECAP dieled to grain refinement The samples with the deformedmicrostructures were examined using TEM to obtain valuesof semiminor axis radius semimajor axis radius and majoraxis radius

23 Justifications for Material Used for ARB and ECAPExperimentations In the current study AA6082-T6 was usedfor the experimentations Aluminum material is the mostused material in domestic and industrial applications Thereare several reasons why aluminum material is widely used insevere plastic deformation process Some of the reasons arethat it is easy to deform aluminum material when comparedwith othermaterials It is also easy to bend aluminum samplesto the desired shape during experimentations Aluminummaterial is lighter and cheaperwhen compared to othermate-rials However since most severe plastic deformations havebeen carried out on aluminum material it is therefore easy

Advances in Materials Science and Engineering 3

Stacking

Cutting Roll-bonging

Degreasingwire-brushing

(a)

Greater than 15 degreesLess than 15 degrees

1st pass 2nd pass 3rd pass

ND = r2

RD = r1

TD = r3

(b)

Figure 1 (a) Schematic of ARB experimental setup [11] and (b) microscopic observation

Plunger

Pressed sample

(a)

Plunger

Sample

Curvature region

90 degrees

Die

(b)

Plunger

Grain shear zone

Curvature regionTheoretical shear

zone

Deformed grain with different grain curvature

Y = r2

X = r1

Slip plane

Z = r3

(c)

Figure 2 Schematic of ECAP processing routes by Estrin and Vinogradov [1] (a) illustration of material deformation (b) schematic ofmaterial deformation and (c) schematic of grain refinement

to compare and validate experimental results and theoreticalresults which is critical aspect

24 Sample Preparation for TEM Examination StandardTEM thin foils that were 3mm in diameter were preparedby electrolytic twin jet polishing (at minus30∘C 30V) in StruersTenupol 2 filled with 6 solution of perchloric acid in

methanol The 3D observations were carried out at 200KVwith JEOL JEM 2000FX microscope equipped with an X-rayenergy dispersive spectrometer (XEDS) LINKAN 10000

25 Schematic of Experimental ObservationNeeded forModelsDerivations The material dealt with in this research hadgrains that were assumed to be initially closely spherical as

4 Advances in Materials Science and Engineering

23r113r1

r2r3r1

23r3

13r3

13r2

23r2

(a)

r2

r2 r2r1 r1

r3

r3

Cr2Cr1Cr

Cr3

(b)

Figure 3 (a) Initially spherical grains before deformation and (b) elongated grains due to deformation

shown in Figure 3(a) As the materials were subjected todeformation it was observed that the semimajor axis radius 1199031and major axis radius 1199033 initially increased (as shown in Fig-ure 3(b)) and then decreased since grain breakages took placein these directions As a result of this repeated lengtheningand grain breakage processes the effective lengths of 1199031 and1199033 decreased during grain refinement It was also observedduring grain refinement that the equivalent radius axis 119903 andsemiminor axis radius 1199032 decrease continuously It was furtherobserved that 1199032 and 1199033 evolved as proportions or fractions of119903 and 1199031 respectively26 Models Derivation Themodel for grain elongation in 2D[12] can be modified to be applicable to 3D grain from theprevious work of Sob et al [13] for grain elongation in 2Dto grain elongation in 3D grain by relating the cross-sectionalarea119860 of a 2D grain given in the previousmodel [12] from [13]to the volume in 3D grain of [13] since in 3D the density of amaterial is determined by it volumeThis relationship is givenas 119860 = 11988123 Thus the expression of elongation as defined byNash [12] is modified to

Elongation = 119875119903011988123119864 (1)

where 1199030 is initial length 119864 is Youngrsquos modulus and 119875 is theapplied force

The model for Youngrsquos modulus (119864) of nanomaterials asdefined by Sob et al [14] is given as

119864 = (119875119889)(1199033192119868) (2)

where it can be inferred that the moment of inertial is 119868 =1205871199034164 119889 = 119889119903 = (119903 minus 1199030) and 119903 is length or equivalent radiusof nanocrystalline grain

It is experimentally observed during grain size evolutionthat at any instant the radii can be related to the equivalentvolume of the grain through (43)1205871199033 = 119881 = (43)120587119903111990321199033This gives

1199033 = 119903111990321199033 (3)

By substituting expressions (2) and (3) into expression (1) thederived model of elongation for 3D grain is given as

Elongation = 3120587119903011990341119903211990331199033 (119903 minus 1199030) (4)

By employing the different experimental observations for 1199031199031 1199032 and 1199033 during grain refinement the following set ofgrain size variant evolutions (given in expressions (5)ndash(8))were established for 3D grain as defined by Sob et al [14] asfollows

Since 1199031 increased and instantaneously decreased afterbreakage the evolution of 1199031 during grain refinement can berepresented as

1198891199031 = 119872( 11199031198881 minus

11199031)119889119905 + 119903

12

1 119863119889119882(119905) minus 11988511990311198811119889 (119905) (5)

where 1199031198881 is local critical grain size 119885 and 119863 are constants119889119882(119905) is change of the Wiener process 1198811 = 120591111990321 definesrate of grain breakage 119872 = 119872119900(1 + 1198621198631199031) 119862119863 =4(119867119898)(ℎ119900)((119896)(119879)) 119879119898 = 119879ln(1198721199001119872) and 119872119874 =1198721198741 exp(inf)119879 as given by [6 10]

Since 1199033 evolves (ie decreases) as a fraction or pro-portion (Ratio1) of 1199031 during grain refinement 1199033 can berepresented by

1198891199033 = Ratio1 (1198891199031) (6)

Since equivalent radius 119903 decreases continuously during grainrefinement 119903 can be represented by

119889119903 = minus119874119903119889119905 + 119868119889119882 (119905) (7)

where 119874 and 119868 are constantsFor 1199032 that evolves or decreases as a fraction (Ratio2) of 119903

during grain refinement 1199032 can be represented by

1198891199032 = Ratio2 (119889119903) (8)

Moving now to other mechanical properties the model ofthe yield stress given by Reversed Hall-Petch Relationship(RHPR) as modified by Zhao and Jiang [9] is given by

120590 (119903) = 12059010158400 + 119860 (119903minus12) minus 119861 (119903minus1) minus 119862 (119903minus32) (9)

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

Size

(nm

)

Time (S)400200 600 800 10000

r2

r1

r3

r

(a)

200 400 600 800 10000Time (S)

0

00002

00004

00006

00008

0001

Elon

gatio

n

(b)

r2

r1

r3

r

0

00002

00004

00006

00008

0001

Elon

gatio

n

Size (nm)4020 60 80 1000

(c)

Figure 4 (a) Time evolution of size (b) time evolution of elongation and (c) evolution of elongation as function of size

where 12059010158400 = 1205900 + 119870119905 is bulk yield stress 119860 = 119870119889 isHPR proportionality constant 119861 = 119870119905[2ℎ119867119898119877119879119903] 119862 =119870119889[2ℎ119867119898119877119879119903] 119870119905 is a constant ℎ is atomic diameter in thecase of metal 119867119898 is the bulk melting enthalpy 119877 is idealgas constant and 119879119903 is the room temperature 119870119889⟩100119870119905 and1205900⟩10119870119905

The models of strain evolution for nanocrystalline mate-rial for the different approaches of measuring grain sizeevolution given in expressions (5)ndash(8) as defined by Sob etal [14] are as follows

1198891205761 = 119889 [11988911990311199031 ]

= 119889(119872( 11199031119888)(

11199031) minus

111990321 )119889119905 +

119862119863119889119882(119905)1199031

minus 119885119881111990321119889 (119905)1199031 1198891205763 = 119889 [11988911990331199033 ] = 119889(Ratio111988911990311199033 ) 119889120576119903 = 119889 [119889119903119903 ] = 119889(minus119874119903119889119905 + 119868119889119882 (119905)

119903 )

1198891205762 = 119889 [11988911990321199032 ] = 119889(Ratio21198891199031199032 ) (10)

Equations (1) to (10) are solved simultaneously using Engi-neering Equation Solver software (F-Chart Software Madi-son W153744 USA) while employing the lognormal distri-bution of grain size

3 Results and Discussion

To test the models proposed in this paper the data from(nanocrystalline) aluminum sample (some of which arefound in other papers [6]) are used which are 11987210158400 =001 nm2sminus1 119898 = 4 119862119862 = 12 119886 = 090 119863 = 10minus4ℎ0 = 025 nm 119879119898(infin) = 93347 K 1198621198810 = 03 119867119898(infin) =1071 KJMol1 12059010158400 = 167MPa 119870119905 = 13 1205900 = 1540MPa 119870119889 =130177MPa_nm12 119877 = 831 JKminus1molminus1 and 119879119903 = 300KTheadditional data obtained for thiswork are119874 = 00035 119868 = 111199031198881 = 195119903 1199030 = 100 nm 119885 = 04 Ratio1 = 081 Ratio2 =1071 and 1205911 = 0000008 The obtained results are presentedin Figures 4(a) 4(b) and 4(c)

6 Advances in Materials Science and Engineering

200 400 600 800 10000Time (S)

0

2

4

6St

rain

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

(a)

20 40 60 80 1000Size (nm)

0

2

4

6

Stra

in r

r2

r1

r3

r

(b)

20 40 60 80 1000Size (nm)

0

0001

0002

0003

0004

0005

r 1St

rain

r2

r1

r3

r

(c)

20 40 60 80 1000Size (nm)

0

02

04

06

08

1

12

Stra

in r 2

r2

r1

r3

r

(d)

Figure 5 (a) Time evolutions of strains and ((b)ndash(d)) strain evolutions as functions of grain size variants (nm)

It should be observed from Figure 4(a) that the naturesof decrease of the different approaches of measuring grainsizes are typical of experimental revelations of 3D grainsThedifferent approaches of measuring grain size such as 1199031 11990321199033 and 119903 have different characteristics and hence differentproperties (yield stress) due to different grain orientation andgrain curvatures as shown in Figures 3(a) and 3(b) Sincegrain breakage rotation grain migration and grain rotationcoalescence events take place during grain refinement byARB and ECAP whose rates depend on grain curvature orelongation differentmaterial yield stresses are being reportedas shown in Figures 3(a) and 3(b) It should be observed fromFigure 4(a) that 1199032 decrease as lower value when comparedto 119903 throughout the deformation process and similarly 1199031decrease as lower value when compared with 1199033 throughoutthe deformation range It was also observed that both 1199031 and1199033 initially started with breakage and as such smaller grainsizes which then lengthened and became larger than the sizesof 119903 and 1199032 The increase in grain elongation with time asshown in Figure 4(b) or with decreasing grain size as shownin Figure 4(c) can be explained by the fact that during grainrefinement new high angle grain boundaries are generatedIt is noted further from Figure 4(c) that depending on theapproach of measuring grain size or observing the grain size

the elongation values can become so large at larger grain sizeor the elongation values become larger at smaller grain size

It is shown from Figure 5(a) that at a time of forexample 600 sec different strains are obtained for the dif-ferent approaches of measuring grain size The nature of theevolution of strains depends on the direction along whichthe grain size is measured as shown in Figures 5(b)ndash5(d)The reasons for this nature of evolution of strain valueswith respect to the different directions of observation (ofthe grain size) are that it was noted experimentally thatalong the axial (1199031) and lateral (1199033) directions both change-in-grain size and grain size varied proportionately with botheither increasing or decreasing simultaneously while alongthe normal direction (1199032) or equivalent radius measured (119903)the increasing change-in-grain sizes were accompanied byreductions in the grain sizes making the strains approachinfinite values

From Figure 6 it is observed that yield stress evolved aspredicted by Hall-Petch to Reversed Hall-Petch Relationship(HPR-RHPR) when measured as functions of time differentapproaches of measuring grain size elongation strain andstrain rate The yield stress generally tended to increasewith decreasing gain size since it is reported [14 15] to bea result of dislocation motion from grain interiors to grain

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 3

Stacking

Cutting Roll-bonging

Degreasingwire-brushing

(a)

Greater than 15 degreesLess than 15 degrees

1st pass 2nd pass 3rd pass

ND = r2

RD = r1

TD = r3

(b)

Figure 1 (a) Schematic of ARB experimental setup [11] and (b) microscopic observation

Plunger

Pressed sample

(a)

Plunger

Sample

Curvature region

90 degrees

Die

(b)

Plunger

Grain shear zone

Curvature regionTheoretical shear

zone

Deformed grain with different grain curvature

Y = r2

X = r1

Slip plane

Z = r3

(c)

Figure 2 Schematic of ECAP processing routes by Estrin and Vinogradov [1] (a) illustration of material deformation (b) schematic ofmaterial deformation and (c) schematic of grain refinement

to compare and validate experimental results and theoreticalresults which is critical aspect

24 Sample Preparation for TEM Examination StandardTEM thin foils that were 3mm in diameter were preparedby electrolytic twin jet polishing (at minus30∘C 30V) in StruersTenupol 2 filled with 6 solution of perchloric acid in

methanol The 3D observations were carried out at 200KVwith JEOL JEM 2000FX microscope equipped with an X-rayenergy dispersive spectrometer (XEDS) LINKAN 10000

25 Schematic of Experimental ObservationNeeded forModelsDerivations The material dealt with in this research hadgrains that were assumed to be initially closely spherical as

4 Advances in Materials Science and Engineering

23r113r1

r2r3r1

23r3

13r3

13r2

23r2

(a)

r2

r2 r2r1 r1

r3

r3

Cr2Cr1Cr

Cr3

(b)

Figure 3 (a) Initially spherical grains before deformation and (b) elongated grains due to deformation

shown in Figure 3(a) As the materials were subjected todeformation it was observed that the semimajor axis radius 1199031and major axis radius 1199033 initially increased (as shown in Fig-ure 3(b)) and then decreased since grain breakages took placein these directions As a result of this repeated lengtheningand grain breakage processes the effective lengths of 1199031 and1199033 decreased during grain refinement It was also observedduring grain refinement that the equivalent radius axis 119903 andsemiminor axis radius 1199032 decrease continuously It was furtherobserved that 1199032 and 1199033 evolved as proportions or fractions of119903 and 1199031 respectively26 Models Derivation Themodel for grain elongation in 2D[12] can be modified to be applicable to 3D grain from theprevious work of Sob et al [13] for grain elongation in 2Dto grain elongation in 3D grain by relating the cross-sectionalarea119860 of a 2D grain given in the previousmodel [12] from [13]to the volume in 3D grain of [13] since in 3D the density of amaterial is determined by it volumeThis relationship is givenas 119860 = 11988123 Thus the expression of elongation as defined byNash [12] is modified to

Elongation = 119875119903011988123119864 (1)

where 1199030 is initial length 119864 is Youngrsquos modulus and 119875 is theapplied force

The model for Youngrsquos modulus (119864) of nanomaterials asdefined by Sob et al [14] is given as

119864 = (119875119889)(1199033192119868) (2)

where it can be inferred that the moment of inertial is 119868 =1205871199034164 119889 = 119889119903 = (119903 minus 1199030) and 119903 is length or equivalent radiusof nanocrystalline grain

It is experimentally observed during grain size evolutionthat at any instant the radii can be related to the equivalentvolume of the grain through (43)1205871199033 = 119881 = (43)120587119903111990321199033This gives

1199033 = 119903111990321199033 (3)

By substituting expressions (2) and (3) into expression (1) thederived model of elongation for 3D grain is given as

Elongation = 3120587119903011990341119903211990331199033 (119903 minus 1199030) (4)

By employing the different experimental observations for 1199031199031 1199032 and 1199033 during grain refinement the following set ofgrain size variant evolutions (given in expressions (5)ndash(8))were established for 3D grain as defined by Sob et al [14] asfollows

Since 1199031 increased and instantaneously decreased afterbreakage the evolution of 1199031 during grain refinement can berepresented as

1198891199031 = 119872( 11199031198881 minus

11199031)119889119905 + 119903

12

1 119863119889119882(119905) minus 11988511990311198811119889 (119905) (5)

where 1199031198881 is local critical grain size 119885 and 119863 are constants119889119882(119905) is change of the Wiener process 1198811 = 120591111990321 definesrate of grain breakage 119872 = 119872119900(1 + 1198621198631199031) 119862119863 =4(119867119898)(ℎ119900)((119896)(119879)) 119879119898 = 119879ln(1198721199001119872) and 119872119874 =1198721198741 exp(inf)119879 as given by [6 10]

Since 1199033 evolves (ie decreases) as a fraction or pro-portion (Ratio1) of 1199031 during grain refinement 1199033 can berepresented by

1198891199033 = Ratio1 (1198891199031) (6)

Since equivalent radius 119903 decreases continuously during grainrefinement 119903 can be represented by

119889119903 = minus119874119903119889119905 + 119868119889119882 (119905) (7)

where 119874 and 119868 are constantsFor 1199032 that evolves or decreases as a fraction (Ratio2) of 119903

during grain refinement 1199032 can be represented by

1198891199032 = Ratio2 (119889119903) (8)

Moving now to other mechanical properties the model ofthe yield stress given by Reversed Hall-Petch Relationship(RHPR) as modified by Zhao and Jiang [9] is given by

120590 (119903) = 12059010158400 + 119860 (119903minus12) minus 119861 (119903minus1) minus 119862 (119903minus32) (9)

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

Size

(nm

)

Time (S)400200 600 800 10000

r2

r1

r3

r

(a)

200 400 600 800 10000Time (S)

0

00002

00004

00006

00008

0001

Elon

gatio

n

(b)

r2

r1

r3

r

0

00002

00004

00006

00008

0001

Elon

gatio

n

Size (nm)4020 60 80 1000

(c)

Figure 4 (a) Time evolution of size (b) time evolution of elongation and (c) evolution of elongation as function of size

where 12059010158400 = 1205900 + 119870119905 is bulk yield stress 119860 = 119870119889 isHPR proportionality constant 119861 = 119870119905[2ℎ119867119898119877119879119903] 119862 =119870119889[2ℎ119867119898119877119879119903] 119870119905 is a constant ℎ is atomic diameter in thecase of metal 119867119898 is the bulk melting enthalpy 119877 is idealgas constant and 119879119903 is the room temperature 119870119889⟩100119870119905 and1205900⟩10119870119905

The models of strain evolution for nanocrystalline mate-rial for the different approaches of measuring grain sizeevolution given in expressions (5)ndash(8) as defined by Sob etal [14] are as follows

1198891205761 = 119889 [11988911990311199031 ]

= 119889(119872( 11199031119888)(

11199031) minus

111990321 )119889119905 +

119862119863119889119882(119905)1199031

minus 119885119881111990321119889 (119905)1199031 1198891205763 = 119889 [11988911990331199033 ] = 119889(Ratio111988911990311199033 ) 119889120576119903 = 119889 [119889119903119903 ] = 119889(minus119874119903119889119905 + 119868119889119882 (119905)

119903 )

1198891205762 = 119889 [11988911990321199032 ] = 119889(Ratio21198891199031199032 ) (10)

Equations (1) to (10) are solved simultaneously using Engi-neering Equation Solver software (F-Chart Software Madi-son W153744 USA) while employing the lognormal distri-bution of grain size

3 Results and Discussion

To test the models proposed in this paper the data from(nanocrystalline) aluminum sample (some of which arefound in other papers [6]) are used which are 11987210158400 =001 nm2sminus1 119898 = 4 119862119862 = 12 119886 = 090 119863 = 10minus4ℎ0 = 025 nm 119879119898(infin) = 93347 K 1198621198810 = 03 119867119898(infin) =1071 KJMol1 12059010158400 = 167MPa 119870119905 = 13 1205900 = 1540MPa 119870119889 =130177MPa_nm12 119877 = 831 JKminus1molminus1 and 119879119903 = 300KTheadditional data obtained for thiswork are119874 = 00035 119868 = 111199031198881 = 195119903 1199030 = 100 nm 119885 = 04 Ratio1 = 081 Ratio2 =1071 and 1205911 = 0000008 The obtained results are presentedin Figures 4(a) 4(b) and 4(c)

6 Advances in Materials Science and Engineering

200 400 600 800 10000Time (S)

0

2

4

6St

rain

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

(a)

20 40 60 80 1000Size (nm)

0

2

4

6

Stra

in r

r2

r1

r3

r

(b)

20 40 60 80 1000Size (nm)

0

0001

0002

0003

0004

0005

r 1St

rain

r2

r1

r3

r

(c)

20 40 60 80 1000Size (nm)

0

02

04

06

08

1

12

Stra

in r 2

r2

r1

r3

r

(d)

Figure 5 (a) Time evolutions of strains and ((b)ndash(d)) strain evolutions as functions of grain size variants (nm)

It should be observed from Figure 4(a) that the naturesof decrease of the different approaches of measuring grainsizes are typical of experimental revelations of 3D grainsThedifferent approaches of measuring grain size such as 1199031 11990321199033 and 119903 have different characteristics and hence differentproperties (yield stress) due to different grain orientation andgrain curvatures as shown in Figures 3(a) and 3(b) Sincegrain breakage rotation grain migration and grain rotationcoalescence events take place during grain refinement byARB and ECAP whose rates depend on grain curvature orelongation differentmaterial yield stresses are being reportedas shown in Figures 3(a) and 3(b) It should be observed fromFigure 4(a) that 1199032 decrease as lower value when comparedto 119903 throughout the deformation process and similarly 1199031decrease as lower value when compared with 1199033 throughoutthe deformation range It was also observed that both 1199031 and1199033 initially started with breakage and as such smaller grainsizes which then lengthened and became larger than the sizesof 119903 and 1199032 The increase in grain elongation with time asshown in Figure 4(b) or with decreasing grain size as shownin Figure 4(c) can be explained by the fact that during grainrefinement new high angle grain boundaries are generatedIt is noted further from Figure 4(c) that depending on theapproach of measuring grain size or observing the grain size

the elongation values can become so large at larger grain sizeor the elongation values become larger at smaller grain size

It is shown from Figure 5(a) that at a time of forexample 600 sec different strains are obtained for the dif-ferent approaches of measuring grain size The nature of theevolution of strains depends on the direction along whichthe grain size is measured as shown in Figures 5(b)ndash5(d)The reasons for this nature of evolution of strain valueswith respect to the different directions of observation (ofthe grain size) are that it was noted experimentally thatalong the axial (1199031) and lateral (1199033) directions both change-in-grain size and grain size varied proportionately with botheither increasing or decreasing simultaneously while alongthe normal direction (1199032) or equivalent radius measured (119903)the increasing change-in-grain sizes were accompanied byreductions in the grain sizes making the strains approachinfinite values

From Figure 6 it is observed that yield stress evolved aspredicted by Hall-Petch to Reversed Hall-Petch Relationship(HPR-RHPR) when measured as functions of time differentapproaches of measuring grain size elongation strain andstrain rate The yield stress generally tended to increasewith decreasing gain size since it is reported [14 15] to bea result of dislocation motion from grain interiors to grain

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Advances in Materials Science and Engineering

23r113r1

r2r3r1

23r3

13r3

13r2

23r2

(a)

r2

r2 r2r1 r1

r3

r3

Cr2Cr1Cr

Cr3

(b)

Figure 3 (a) Initially spherical grains before deformation and (b) elongated grains due to deformation

shown in Figure 3(a) As the materials were subjected todeformation it was observed that the semimajor axis radius 1199031and major axis radius 1199033 initially increased (as shown in Fig-ure 3(b)) and then decreased since grain breakages took placein these directions As a result of this repeated lengtheningand grain breakage processes the effective lengths of 1199031 and1199033 decreased during grain refinement It was also observedduring grain refinement that the equivalent radius axis 119903 andsemiminor axis radius 1199032 decrease continuously It was furtherobserved that 1199032 and 1199033 evolved as proportions or fractions of119903 and 1199031 respectively26 Models Derivation Themodel for grain elongation in 2D[12] can be modified to be applicable to 3D grain from theprevious work of Sob et al [13] for grain elongation in 2Dto grain elongation in 3D grain by relating the cross-sectionalarea119860 of a 2D grain given in the previousmodel [12] from [13]to the volume in 3D grain of [13] since in 3D the density of amaterial is determined by it volumeThis relationship is givenas 119860 = 11988123 Thus the expression of elongation as defined byNash [12] is modified to

Elongation = 119875119903011988123119864 (1)

where 1199030 is initial length 119864 is Youngrsquos modulus and 119875 is theapplied force

The model for Youngrsquos modulus (119864) of nanomaterials asdefined by Sob et al [14] is given as

119864 = (119875119889)(1199033192119868) (2)

where it can be inferred that the moment of inertial is 119868 =1205871199034164 119889 = 119889119903 = (119903 minus 1199030) and 119903 is length or equivalent radiusof nanocrystalline grain

It is experimentally observed during grain size evolutionthat at any instant the radii can be related to the equivalentvolume of the grain through (43)1205871199033 = 119881 = (43)120587119903111990321199033This gives

1199033 = 119903111990321199033 (3)

By substituting expressions (2) and (3) into expression (1) thederived model of elongation for 3D grain is given as

Elongation = 3120587119903011990341119903211990331199033 (119903 minus 1199030) (4)

By employing the different experimental observations for 1199031199031 1199032 and 1199033 during grain refinement the following set ofgrain size variant evolutions (given in expressions (5)ndash(8))were established for 3D grain as defined by Sob et al [14] asfollows

Since 1199031 increased and instantaneously decreased afterbreakage the evolution of 1199031 during grain refinement can berepresented as

1198891199031 = 119872( 11199031198881 minus

11199031)119889119905 + 119903

12

1 119863119889119882(119905) minus 11988511990311198811119889 (119905) (5)

where 1199031198881 is local critical grain size 119885 and 119863 are constants119889119882(119905) is change of the Wiener process 1198811 = 120591111990321 definesrate of grain breakage 119872 = 119872119900(1 + 1198621198631199031) 119862119863 =4(119867119898)(ℎ119900)((119896)(119879)) 119879119898 = 119879ln(1198721199001119872) and 119872119874 =1198721198741 exp(inf)119879 as given by [6 10]

Since 1199033 evolves (ie decreases) as a fraction or pro-portion (Ratio1) of 1199031 during grain refinement 1199033 can berepresented by

1198891199033 = Ratio1 (1198891199031) (6)

Since equivalent radius 119903 decreases continuously during grainrefinement 119903 can be represented by

119889119903 = minus119874119903119889119905 + 119868119889119882 (119905) (7)

where 119874 and 119868 are constantsFor 1199032 that evolves or decreases as a fraction (Ratio2) of 119903

during grain refinement 1199032 can be represented by

1198891199032 = Ratio2 (119889119903) (8)

Moving now to other mechanical properties the model ofthe yield stress given by Reversed Hall-Petch Relationship(RHPR) as modified by Zhao and Jiang [9] is given by

120590 (119903) = 12059010158400 + 119860 (119903minus12) minus 119861 (119903minus1) minus 119862 (119903minus32) (9)

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

Size

(nm

)

Time (S)400200 600 800 10000

r2

r1

r3

r

(a)

200 400 600 800 10000Time (S)

0

00002

00004

00006

00008

0001

Elon

gatio

n

(b)

r2

r1

r3

r

0

00002

00004

00006

00008

0001

Elon

gatio

n

Size (nm)4020 60 80 1000

(c)

Figure 4 (a) Time evolution of size (b) time evolution of elongation and (c) evolution of elongation as function of size

where 12059010158400 = 1205900 + 119870119905 is bulk yield stress 119860 = 119870119889 isHPR proportionality constant 119861 = 119870119905[2ℎ119867119898119877119879119903] 119862 =119870119889[2ℎ119867119898119877119879119903] 119870119905 is a constant ℎ is atomic diameter in thecase of metal 119867119898 is the bulk melting enthalpy 119877 is idealgas constant and 119879119903 is the room temperature 119870119889⟩100119870119905 and1205900⟩10119870119905

The models of strain evolution for nanocrystalline mate-rial for the different approaches of measuring grain sizeevolution given in expressions (5)ndash(8) as defined by Sob etal [14] are as follows

1198891205761 = 119889 [11988911990311199031 ]

= 119889(119872( 11199031119888)(

11199031) minus

111990321 )119889119905 +

119862119863119889119882(119905)1199031

minus 119885119881111990321119889 (119905)1199031 1198891205763 = 119889 [11988911990331199033 ] = 119889(Ratio111988911990311199033 ) 119889120576119903 = 119889 [119889119903119903 ] = 119889(minus119874119903119889119905 + 119868119889119882 (119905)

119903 )

1198891205762 = 119889 [11988911990321199032 ] = 119889(Ratio21198891199031199032 ) (10)

Equations (1) to (10) are solved simultaneously using Engi-neering Equation Solver software (F-Chart Software Madi-son W153744 USA) while employing the lognormal distri-bution of grain size

3 Results and Discussion

To test the models proposed in this paper the data from(nanocrystalline) aluminum sample (some of which arefound in other papers [6]) are used which are 11987210158400 =001 nm2sminus1 119898 = 4 119862119862 = 12 119886 = 090 119863 = 10minus4ℎ0 = 025 nm 119879119898(infin) = 93347 K 1198621198810 = 03 119867119898(infin) =1071 KJMol1 12059010158400 = 167MPa 119870119905 = 13 1205900 = 1540MPa 119870119889 =130177MPa_nm12 119877 = 831 JKminus1molminus1 and 119879119903 = 300KTheadditional data obtained for thiswork are119874 = 00035 119868 = 111199031198881 = 195119903 1199030 = 100 nm 119885 = 04 Ratio1 = 081 Ratio2 =1071 and 1205911 = 0000008 The obtained results are presentedin Figures 4(a) 4(b) and 4(c)

6 Advances in Materials Science and Engineering

200 400 600 800 10000Time (S)

0

2

4

6St

rain

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

(a)

20 40 60 80 1000Size (nm)

0

2

4

6

Stra

in r

r2

r1

r3

r

(b)

20 40 60 80 1000Size (nm)

0

0001

0002

0003

0004

0005

r 1St

rain

r2

r1

r3

r

(c)

20 40 60 80 1000Size (nm)

0

02

04

06

08

1

12

Stra

in r 2

r2

r1

r3

r

(d)

Figure 5 (a) Time evolutions of strains and ((b)ndash(d)) strain evolutions as functions of grain size variants (nm)

It should be observed from Figure 4(a) that the naturesof decrease of the different approaches of measuring grainsizes are typical of experimental revelations of 3D grainsThedifferent approaches of measuring grain size such as 1199031 11990321199033 and 119903 have different characteristics and hence differentproperties (yield stress) due to different grain orientation andgrain curvatures as shown in Figures 3(a) and 3(b) Sincegrain breakage rotation grain migration and grain rotationcoalescence events take place during grain refinement byARB and ECAP whose rates depend on grain curvature orelongation differentmaterial yield stresses are being reportedas shown in Figures 3(a) and 3(b) It should be observed fromFigure 4(a) that 1199032 decrease as lower value when comparedto 119903 throughout the deformation process and similarly 1199031decrease as lower value when compared with 1199033 throughoutthe deformation range It was also observed that both 1199031 and1199033 initially started with breakage and as such smaller grainsizes which then lengthened and became larger than the sizesof 119903 and 1199032 The increase in grain elongation with time asshown in Figure 4(b) or with decreasing grain size as shownin Figure 4(c) can be explained by the fact that during grainrefinement new high angle grain boundaries are generatedIt is noted further from Figure 4(c) that depending on theapproach of measuring grain size or observing the grain size

the elongation values can become so large at larger grain sizeor the elongation values become larger at smaller grain size

It is shown from Figure 5(a) that at a time of forexample 600 sec different strains are obtained for the dif-ferent approaches of measuring grain size The nature of theevolution of strains depends on the direction along whichthe grain size is measured as shown in Figures 5(b)ndash5(d)The reasons for this nature of evolution of strain valueswith respect to the different directions of observation (ofthe grain size) are that it was noted experimentally thatalong the axial (1199031) and lateral (1199033) directions both change-in-grain size and grain size varied proportionately with botheither increasing or decreasing simultaneously while alongthe normal direction (1199032) or equivalent radius measured (119903)the increasing change-in-grain sizes were accompanied byreductions in the grain sizes making the strains approachinfinite values

From Figure 6 it is observed that yield stress evolved aspredicted by Hall-Petch to Reversed Hall-Petch Relationship(HPR-RHPR) when measured as functions of time differentapproaches of measuring grain size elongation strain andstrain rate The yield stress generally tended to increasewith decreasing gain size since it is reported [14 15] to bea result of dislocation motion from grain interiors to grain

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 5

0

20

40

60

80

100

Size

(nm

)

Time (S)400200 600 800 10000

r2

r1

r3

r

(a)

200 400 600 800 10000Time (S)

0

00002

00004

00006

00008

0001

Elon

gatio

n

(b)

r2

r1

r3

r

0

00002

00004

00006

00008

0001

Elon

gatio

n

Size (nm)4020 60 80 1000

(c)

Figure 4 (a) Time evolution of size (b) time evolution of elongation and (c) evolution of elongation as function of size

where 12059010158400 = 1205900 + 119870119905 is bulk yield stress 119860 = 119870119889 isHPR proportionality constant 119861 = 119870119905[2ℎ119867119898119877119879119903] 119862 =119870119889[2ℎ119867119898119877119879119903] 119870119905 is a constant ℎ is atomic diameter in thecase of metal 119867119898 is the bulk melting enthalpy 119877 is idealgas constant and 119879119903 is the room temperature 119870119889⟩100119870119905 and1205900⟩10119870119905

The models of strain evolution for nanocrystalline mate-rial for the different approaches of measuring grain sizeevolution given in expressions (5)ndash(8) as defined by Sob etal [14] are as follows

1198891205761 = 119889 [11988911990311199031 ]

= 119889(119872( 11199031119888)(

11199031) minus

111990321 )119889119905 +

119862119863119889119882(119905)1199031

minus 119885119881111990321119889 (119905)1199031 1198891205763 = 119889 [11988911990331199033 ] = 119889(Ratio111988911990311199033 ) 119889120576119903 = 119889 [119889119903119903 ] = 119889(minus119874119903119889119905 + 119868119889119882 (119905)

119903 )

1198891205762 = 119889 [11988911990321199032 ] = 119889(Ratio21198891199031199032 ) (10)

Equations (1) to (10) are solved simultaneously using Engi-neering Equation Solver software (F-Chart Software Madi-son W153744 USA) while employing the lognormal distri-bution of grain size

3 Results and Discussion

To test the models proposed in this paper the data from(nanocrystalline) aluminum sample (some of which arefound in other papers [6]) are used which are 11987210158400 =001 nm2sminus1 119898 = 4 119862119862 = 12 119886 = 090 119863 = 10minus4ℎ0 = 025 nm 119879119898(infin) = 93347 K 1198621198810 = 03 119867119898(infin) =1071 KJMol1 12059010158400 = 167MPa 119870119905 = 13 1205900 = 1540MPa 119870119889 =130177MPa_nm12 119877 = 831 JKminus1molminus1 and 119879119903 = 300KTheadditional data obtained for thiswork are119874 = 00035 119868 = 111199031198881 = 195119903 1199030 = 100 nm 119885 = 04 Ratio1 = 081 Ratio2 =1071 and 1205911 = 0000008 The obtained results are presentedin Figures 4(a) 4(b) and 4(c)

6 Advances in Materials Science and Engineering

200 400 600 800 10000Time (S)

0

2

4

6St

rain

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

(a)

20 40 60 80 1000Size (nm)

0

2

4

6

Stra

in r

r2

r1

r3

r

(b)

20 40 60 80 1000Size (nm)

0

0001

0002

0003

0004

0005

r 1St

rain

r2

r1

r3

r

(c)

20 40 60 80 1000Size (nm)

0

02

04

06

08

1

12

Stra

in r 2

r2

r1

r3

r

(d)

Figure 5 (a) Time evolutions of strains and ((b)ndash(d)) strain evolutions as functions of grain size variants (nm)

It should be observed from Figure 4(a) that the naturesof decrease of the different approaches of measuring grainsizes are typical of experimental revelations of 3D grainsThedifferent approaches of measuring grain size such as 1199031 11990321199033 and 119903 have different characteristics and hence differentproperties (yield stress) due to different grain orientation andgrain curvatures as shown in Figures 3(a) and 3(b) Sincegrain breakage rotation grain migration and grain rotationcoalescence events take place during grain refinement byARB and ECAP whose rates depend on grain curvature orelongation differentmaterial yield stresses are being reportedas shown in Figures 3(a) and 3(b) It should be observed fromFigure 4(a) that 1199032 decrease as lower value when comparedto 119903 throughout the deformation process and similarly 1199031decrease as lower value when compared with 1199033 throughoutthe deformation range It was also observed that both 1199031 and1199033 initially started with breakage and as such smaller grainsizes which then lengthened and became larger than the sizesof 119903 and 1199032 The increase in grain elongation with time asshown in Figure 4(b) or with decreasing grain size as shownin Figure 4(c) can be explained by the fact that during grainrefinement new high angle grain boundaries are generatedIt is noted further from Figure 4(c) that depending on theapproach of measuring grain size or observing the grain size

the elongation values can become so large at larger grain sizeor the elongation values become larger at smaller grain size

It is shown from Figure 5(a) that at a time of forexample 600 sec different strains are obtained for the dif-ferent approaches of measuring grain size The nature of theevolution of strains depends on the direction along whichthe grain size is measured as shown in Figures 5(b)ndash5(d)The reasons for this nature of evolution of strain valueswith respect to the different directions of observation (ofthe grain size) are that it was noted experimentally thatalong the axial (1199031) and lateral (1199033) directions both change-in-grain size and grain size varied proportionately with botheither increasing or decreasing simultaneously while alongthe normal direction (1199032) or equivalent radius measured (119903)the increasing change-in-grain sizes were accompanied byreductions in the grain sizes making the strains approachinfinite values

From Figure 6 it is observed that yield stress evolved aspredicted by Hall-Petch to Reversed Hall-Petch Relationship(HPR-RHPR) when measured as functions of time differentapproaches of measuring grain size elongation strain andstrain rate The yield stress generally tended to increasewith decreasing gain size since it is reported [14 15] to bea result of dislocation motion from grain interiors to grain

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Materials Science and Engineering

200 400 600 800 10000Time (S)

0

2

4

6St

rain

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

(a)

20 40 60 80 1000Size (nm)

0

2

4

6

Stra

in r

r2

r1

r3

r

(b)

20 40 60 80 1000Size (nm)

0

0001

0002

0003

0004

0005

r 1St

rain

r2

r1

r3

r

(c)

20 40 60 80 1000Size (nm)

0

02

04

06

08

1

12

Stra

in r 2

r2

r1

r3

r

(d)

Figure 5 (a) Time evolutions of strains and ((b)ndash(d)) strain evolutions as functions of grain size variants (nm)

It should be observed from Figure 4(a) that the naturesof decrease of the different approaches of measuring grainsizes are typical of experimental revelations of 3D grainsThedifferent approaches of measuring grain size such as 1199031 11990321199033 and 119903 have different characteristics and hence differentproperties (yield stress) due to different grain orientation andgrain curvatures as shown in Figures 3(a) and 3(b) Sincegrain breakage rotation grain migration and grain rotationcoalescence events take place during grain refinement byARB and ECAP whose rates depend on grain curvature orelongation differentmaterial yield stresses are being reportedas shown in Figures 3(a) and 3(b) It should be observed fromFigure 4(a) that 1199032 decrease as lower value when comparedto 119903 throughout the deformation process and similarly 1199031decrease as lower value when compared with 1199033 throughoutthe deformation range It was also observed that both 1199031 and1199033 initially started with breakage and as such smaller grainsizes which then lengthened and became larger than the sizesof 119903 and 1199032 The increase in grain elongation with time asshown in Figure 4(b) or with decreasing grain size as shownin Figure 4(c) can be explained by the fact that during grainrefinement new high angle grain boundaries are generatedIt is noted further from Figure 4(c) that depending on theapproach of measuring grain size or observing the grain size

the elongation values can become so large at larger grain sizeor the elongation values become larger at smaller grain size

It is shown from Figure 5(a) that at a time of forexample 600 sec different strains are obtained for the dif-ferent approaches of measuring grain size The nature of theevolution of strains depends on the direction along whichthe grain size is measured as shown in Figures 5(b)ndash5(d)The reasons for this nature of evolution of strain valueswith respect to the different directions of observation (ofthe grain size) are that it was noted experimentally thatalong the axial (1199031) and lateral (1199033) directions both change-in-grain size and grain size varied proportionately with botheither increasing or decreasing simultaneously while alongthe normal direction (1199032) or equivalent radius measured (119903)the increasing change-in-grain sizes were accompanied byreductions in the grain sizes making the strains approachinfinite values

From Figure 6 it is observed that yield stress evolved aspredicted by Hall-Petch to Reversed Hall-Petch Relationship(HPR-RHPR) when measured as functions of time differentapproaches of measuring grain size elongation strain andstrain rate The yield stress generally tended to increasewith decreasing gain size since it is reported [14 15] to bea result of dislocation motion from grain interiors to grain

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 7

200 400 600 800 10000Time (S)

100

200

300

400Yi

eld

stres

s (M

Pa)

(a)

r2

r

r1

r3

20 40 60 80 1000Size (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(b)

00002 00004 00006 00008 00010Elongation (nm)

100

200

300

400

Yiel

d str

ess (

MPa

)

(c)

100

200

300

400

Yield

stre

ss (M

Pa)

2 4 60Strain

r2 rr1 r3

(d)

100

200

300

400

Yiel

d str

ess (

MPa

)

Strain rate r2 Strain rate r3

Strain rate r

1 15 2 25 3 35050Strain rate (1S)

(e)

Strain rate r1

0060020 004Strain rate (1S)

100

200

300

400

Yield

stre

ss (M

Pa)

(f)

Figure 6 Plots of yield stress as a function of (a) time (b) size (c) elongation (d) strain and ((e)-(f)) strain rate

boundaries and subsequent dislocation pile-ups at the grainboundaries It was observed that in the directions wherebythe sizes increased and subsequently decreased due to grainbreakage (ie 1199031 and 1199033) there were more material flow inthose directions which can without loss of generality beclaimed that there were more dislocation motions and pile-ups towards those directions and hencemore grain boundary

curvatures in those directions As a result the yield stresswhen measured as a function of grain size and strain withsize measured along the 1199031 and 1199033 directions show morerapid enhanced properties (see Figures 6(b) and 6(d)) Sinceelongation can be termed lengthening it can be concludedthat materials with elongated grains have more enhancedproperties which rapidly drop with continuous lengthening

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Materials Science and Engineering

Strain rate r2

Strain rate r3

Strain rate r1

Strain rate r

200 400 600 800 10000Time (S)

0

1

2

3

4St

rain

rate

(1S

)

(a)

0

1

2

3

4

20 40 60 80 1000Size (nm)

r3r2 rr1

1S

)St

rain

rate

(

(b)

20 40 60 80 1000Size (nm)

0

002

004

006

Stra

in ra

ter 1

(1S

)

r

r1

r3

r2

(c)

r3r1rr20

05

1

15

2

Stra

in ra

ter 2

(1S

)

20 40 60 80 1000Size (nm)

(d)

Figure 7 Plots of strain rate as a function of (a) time and (b)ndash(d) size (nm)

of grains as revealed in Figures 6(b) 6(c) and 6(d) and formost ARB processes at rolling facilities The reason for thesubsequent decrease in yield stress with further decrease ingrain size was that low plastic deformation occurred resultingin low materials hardness and properties Extreme plasticstraining led to distorted structures where the grain bound-aries and grain curvatures were in ldquononequilibriumrdquo states Atthese nonequilibrium states the grain boundaries and graincurvatures are characterized by more grain boundary energyenhanced free volume and the presence of long range elasticstresses [16]

Studying the effect of curvature on properties reference ismade to Figure 6(b) From the figure it is noted that at grainsize of 35 nm the material has the most enhanced property(ie yield stress) for radius measured along 119903 since the graincurvature of119862119903 is higher than the grain curvature of11986211990311198621199032and1198621199033 at that 35 nmWhen thematerial getsmore refined ata size of 30 nm the material has the most enhanced propertyfor the radius measured along 1199033 since grain curvature for1198621199033 is higher than the grain curvature of 1198621199031 1198621199032 and 119862119903Furthermore it has been revealed as shown in Figure 6(b)that different critical grain sizes exist for 1199031 1199032 1199033 and 119903 butgive the same yield stress value at those different critical grain

sizes It is observed from Figure 6(d) that the yield stresses forthe grain size variants increase with increasing strains due todifferentmisorientation angles and the grain curvatures (11986211990311198621199032 1198621199033 and 119862119903) during grain deformation

The different strain rates observed in Figures 6(e) and 6(f)have a general trend This implied that in order to increasethe material yield stress the rate of straining the materialhad to be reduced similar to observations made by otherresearchers [16ndash18]This is required so as to allow thematerialto relax and accommodate more plastic strain Furthermoreit is observed that indefinite reduction in strain rate or feedrate does not lead tomore property enhancement since it wasbecoming difficult for thematerial to be gripped by the rollersand get deformed It was also noted that the rate of strainingthe material depended on the direction of observation ormeasuring the grain size Figure 7 shows that in order to havemore grain refinements the strain rates have to be decreased

4 Conclusions

The current work was aimed at investigating the impactsof elongation or grain shapes on nanomaterials propertiesTo achieve that the model for elongation that has been

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Materials Science and Engineering 9

previously given for 2D was modified to be applicable to 3Dgrains Furthermore the stochastic natures of the grain sizevariants (ie different approaches of measuring or observinggrain sizes) were also taken into consideration

It was observed that the stochastic effect of grain elon-gation led to different mechanical properties when studiedas functions of different grain size variants due to differentgrain curvatures It was shown that the properties varied indifferent ways with equivalent radius semiminor axis radiussemimajor axis radius and major axis radius The presentanalysis revealed that the material has the most enhancedproperty for radiusmeasuredwith higher grain curvature andless enhanced property for radius with low grain curvature Ithas also been revealed that in order to increase the materialyield stress the rate of straining thematerial has to be reducedIt was also observed that materials with elongated grains havemore enhanced properties that rapidly drop with continuouslengthening of grainsThepresent analysis for 3D grain showsmore properties for nanomaterials that were not revealedfrom the models that dealt only with the equivalent radiusThese findings provide valuable insights into the possibilityof tailoring sample dimensions to elicit desired property

Disclosure

The abstract of this manuscript was approved at the inter-national conference of composites materials held in Istanbul2015 but the full paper was not submitted for journal consid-eration at the conference

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This paper is based on the work which is supported finan-cially by the National Research Foundation (NRF) and VaalUniversity of Technology (VUT)

References

[1] Y Estrin and A Vinogradov ldquoExtreme grain refinement bysevere plastic deformation a wealth of challenging sciencerdquoActa Materialia vol 61 no 3 pp 782ndash817 2013

[2] M Saravanan R M Pillai B C Pai M Brahmakumar and KR Ravi ldquoEqual channel angular pressing of pure aluminium-ananalysisrdquo Bulletin of Materials Science vol 29 no 7 pp 679ndash684 2006

[3] N Tsuji Y Saito S-H Lee and Y Minamino ldquoARB (accumu-lative roll-bonding) and other new techniques to produce bulkultrafine grained materialsrdquo Advanced Engineering Materialsvol 5 no 5 pp 338ndash344 2003

[4] A Rosochowski L Olejnik J Richert M Rosochowska andM Richert ldquoEqual channel angular pressing with convergingbillets-experimentrdquo Materials Science and Engineering A vol560 pp 358ndash364 2013

[5] M Hillert ldquoOn the theory of normal and abnormal graingrowthrdquo Acta Metallurgica vol 13 no 3 pp 227ndash238 1965

[6] T B Tengen Analysis of Characteristic of Random Microstruc-tures of Nanomaterials [PhD Thesis] Witwatersrand Johannes-burg 2008

[7] E O Hall ldquoThe deformation and ageing of mild steel IIIdiscussion of resultsrdquo Proceedings of the Physical Society SectionB vol 64 no 9 pp 747ndash753 1951

[8] N J Petch ldquoThe cleavage strength of polycrystaslrdquo Journal of theIron and Steel Institute vol 174 pp 25ndash28 1953

[9] M Zhao andQ Jiang ldquoReverse hall-petch relationship ofmetalsin nanometer sizerdquo in Proceedings of the 2006 IEEE ConferenceonEmerging TechnologiesmdashNanoelectronics pp 472ndash474 Singa-pore January 2006

[10] T B Tengen T Wejrzanowski R Iwankiewicz and K JKurzydlowski ldquoStochastic modelling in design of mechanicalproperties of nanometalsrdquoMaterials Science and Engineering Avol 527 no 16-17 pp 3764ndash3768 2010

[11] M Lewandowska ldquoMalgorzata lewandowska fabricationmeth-ods of nanomaterialsrdquo in Structure and Properties of Nanomate-rials pp 1ndash54 2006

[12] W A Nash Strength of Materials McGraw-Hill New York NYUSA 4th edition 1998

[13] P B Sob A A Alugongo and T B TengeN ldquoStochastic effectof grain elongation on nanocrystalline material yield stressproduced by accumulative roll bondingrdquo in Proceeding of the9th South African Conference of Computational and AppliedMechanics (SACAM rsquo14) Somerset West South Africa 2014

[14] P B Sob A A Alugongo and T B Tengen Modelling StrainRate Sensitive Nanomaterials Mechanical Properties The Effectsof Varying Definition MTECH [MS thesis] Vaal University ofTechnology Vanderbijlpark South Africa 2016

[15] G Wang and L Xiaodong ldquoPredicting the youngrsquos modulusof nanowire from first principle calculations on their surfaceand bulk materialsrdquo Journal of Applied Physics vol 104 no 11Article ID 113517 pp 1ndash33 2008

[16] K M Rahman V A Vorontsov and D Dye ldquoThe effect ofgrain size on the twin initiation stress in a TWIP steelrdquo ActaMaterialia vol 89 pp 247ndash257 2015

[17] L S Toth and C Gu ldquoUltrafine-grain metals by severe plasticdeformationrdquoMaterials Characterization vol 92 pp 1ndash14 2014

[18] I Sabirov M R Barnett Y Estrin and P D Hodgson ldquoTheeffect of strain rate on the deformation mechanisms and thestrain rate sensitivity of an ultra-fine-grained Al alloyrdquo ScriptaMaterialia vol 61 no 2 pp 181ndash184 2009

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014