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Composites: Part B 39 (2008) 1062–1068

Predictions of stiffness and strength of nylon 6/MMTnanocomposites with an improved staggered model

Ying Dai a,b,*, Yiu-Wing Mai a, Xing Ji a,b

a Center for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic Engineering J07,

University of Sydney, Sydney, NSW 2006, Australiab School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Received 3 February 2007; accepted 3 September 2007Available online 10 March 2008

Abstract

An improved staggered model has been developed for the predictions of stiffness and strength of nylon 6/MMT nanocomposites. Themodel is partitioned into four regions: A (for MMT platelet), B and D (for matrix: tensile deformation), and C (for matrix: tensile andshear deformations). From force equilibrium and deformation compatibility in these four regions, an approximate closed-form solutionfor the stiffness of nanocomposites is derived. Using the data for MMT platelets and nylon 6 from Ref. [Fornes TD, Paul DR. Modelingproperties of nylon 6/clay nanocomposites using composite theories. Polymer 2003;44:4993–5013], the stiffness and strength predictedgive fair to good agreement, respectively, with test results.� 2008 Elsevier Ltd. All rights reserved.

Keywords: A. Polymer-matrix composites (PMCs); B. Mechanical properties; C. Analytical modelling

1. Introduction

Nylon 6/clay nanocomposites are made up of nylon 6matrix filled with nano-sized montmorillonite platelets(MMT). Toyota researchers [1,2] were first to showremarkable improvements in some physico-mechanicalproperties of nylon 6 containing exfoliated organo-clay.The modulus is doubled, the strength increased by morethan 50%, and the heat distortion temperature enhancedby 80 �C compared to neat nylon 6. It is interesting thatthese nanocomposites offer exceptional stiffness andstrength at very low filler loading (e.g., 6.5 wt% clay),whereas three times this amount of glass fibers is neededto achieve the same improvements [3]. These advantageshave particular benefits to those industries, such as trans-

1359-8368/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesb.2007.09.005

* Corresponding author. Address: School of Aerospace Engineering andApplied Mechanics, Tongji University, Shanghai 200092, China. Tel.: +8621 65982383; fax: +86 21 65983267.

E-mail address: ydai@mail.tongji.edu.cn (Y. Dai).

portation and automotive, where lightweight structuresand components are critical to improve the fuel economy.

Although there have been numerous material syntheses,tests and characterizations of nylon 6/clay nanocompos-ites, the fundamental mechanisms for mechanical propertyenhancement have not been fully understood [4]. A betterunderstanding and prediction capability is important toaccelerate the development of these nanocomposites. Inthis regard, some qualitative and quantitative analyseshave been conducted with the aim of predicting the stiffen-ing effect of clay on nanocomposites [5]. Some researchersdirectly used those models which have been developed forpredicting properties of composite materials, e.g. rule-of-mixtures [6,7], parallel model [8], Halpin–Tsai equationand Mori–Tanaka method (see Refs. [3], [9]), etc. Modifiedmodels were further developed to predict the stiffness ofpolymer nanocomposites based on possible microstruc-tures around the clay platelets, e.g., combined parallel-ser-ies three-phase model [10], hierarchical model [11,12], etc.

Organo-clay has a platelet-like layered structure and hasa cationic exchange capacity. The layered platelets can be

Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068 1063

exfoliated and dispersed in a polymer-matrix under certainpre-treatment and processing conditions. These plateletshave a very high tensile modulus of �170 GPa, 1 nm thick-ness and aspect ratio in the range of 10–1000. This providesa large surface area, �750 m2/g of clay material [4]. Hence,the high filler modulus and large filler–matrix interface areawill significantly affect the mechanical properties of thenanocomposites. A staggered model, previously used byJager and Fratzl [13] to predict the elastic modulus andfracture stress of mineralized collagen fibrils, is introducedand modified in the present work for the predictions ofstiffness and strength of nylon 6/clay nanocomposites.

2. Theoretical models

2.1. Staggered model of Jager–Fratzl

Jager and Fratzl [13] developed a staggered model tostudy the mechanisms responsible for improvement of elas-

Fig. 1. Staggered arrangement of mineral particles in the fibril of boneand two adjacent elementary cells of the staggered model showing regionsof tensile (regions A and B) and shear (regions C and D) stresses, theamount of cell elongation, the dimensions of minerals and the distancesbetween them. After Ref. [12].

Table 2Comparison of stiffness between predictions and experimental results in Ref. [

Experimental results from Ref. [3] Jager–Fratzl model [12]

wt% V (%) Young’s modulus(GPa)

Young’s modulus(GPa)

Error(%)

1.6 0.65 3.49 5.05 44.83.2 1.31 3.92 6.98 78.24.6 1.91 4.59 8.90 93.97.2 3.03 5.7 13.0 128.4

tic modulus and fracture stress with mineral contentsdeposited within the collagen fibrils. Fig. 1 shows the stag-gered arrangement of mineral particles in the fibril. Twoelementary cells, one with regions (A and B) of tensilestress and another with regions (C and D) of shear stress,are also illustrated [13]. The deformation of the representa-tive volume element (RVE) is obtained by the superposi-tion of these two elementary cells. The introduction ofshear regions C and D allows the staggered arrangementto account for both enhanced modulus and fracture stress.However, in parallel arrays, when the tensile stress regions,A and B, are used, as in many current mechanical models,only the increase of modulus but not the fracture stress canbe obtained [13].

In the collagen fibrils, the mineral particles are around2–4 nm in thickness and may reach 100 nm in length. Thegeometrical shape of exfoliated MMT platelets are closeto that of mineral particles with a thickness �1 nm andlength �100 nm. It is therefore interesting to adopt thestaggered model to evaluate the stiffness and strength ofnylon 6/MMT nanocomposites. Table 1 lists the geometri-cal parameters of the MMT platelets, and elastic propertiesof MMT and nylon 6; and Table 2 shows the measuredstiffness of the nylon 6/MMT nanocomposites with differ-ent volume fraction of MMT. All these data are taken fromRef. [3]. It can be seen from Table 2 that the predicted stiff-ness values based on the Jager–Fratzl model are much lar-ger than the test results. Thus, the staggered model shouldbe modified.

In the Jager and Fratzl staggered model [13], the defor-mation of the representative volume element (RVE) isobtained by superposition of two elementary cells. How-ever, the superposition of two compatible strain fields willnot always yield a compatible strain field, so there is aproblem on deformation compatibility. In addition, theway the forces (normal forces on regions A and B and

Table 1Geometrical parameters of MMT platelets and elastic properties of MMTand nylon 6

Geometric parameters of MMTplatelets

Elastic properties

MMT MMT Nylon 6

Averagethickness d (nm)

0.94 Young’s modulus(GPa)

178 2.75

Averagelength l (nm)

91 Poisson’s ratio 0.2 0.35

3]

Improved staggered model FEA

Young’s modulus(GPa)

Error(%)

Young’s modulus(GPa)

Error(%)

3.81 9.17 3.65 4.514.74 20.9 4.24 8.315.57 21.3 4.85 5.857.14 25.3 6.14 7.73

a

Nylon

MMTNylon 6

MMT platelets

C FAC

FC

FACFB

A

FA

FAC

CD FF −

2CD FF −

D

FD

FC

2CD FF −

2CD FF −

FB2

CD FF −

B

FB

FB

1064 Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068

shear forces on regions C and D in Fig. 1) are added up togive the total force on the RVE is inadequate. Thus, toredress these inadequacies, the Jager–Fratzl staggeredmodel is revised as discussed in the following section.

2.2. Improved staggered model

Fig. 2a shows the idealized staggered arrangement of theMMT platelets in nylon 6 matrix and the representativevolume element (RVE) of the staggered model in Fig. 2bcontains all four regions of tensile and shear stresses, A,B, C and D. The model is assumed to be two-dimensional.The thickness and original length of the MMT platelet ared and l; the distances between the platelets in the longitudi-nal and transverse directions are a and b, respectively.Thus, the volume fraction of the MMT platelets can bederived as

V MMT ¼1

ð1þ b=dÞð1þ a=lÞ ð1Þ

b

B

FB

FB

FC2

FD

D

FC2CD FF −

2CD FF −

FA

AFAC

2CD FF −

Fig. 3. Sketch showing (a) deformations of MMT platelets and nylon 6matrix, and (b) forces acting and transferred in the four regions in theRVE of the staggered model.

2.3. Forces and deformations

With reference to the RVE in Fig. 2b, the MMT plateletis represented by region A under tensile deformation, andnylon 6 is partitioned into three regions, B, C and D withB and D under tensile deformation and C under both ten-sile and shear deformations.

The deformations and forces of these four regions areshown in Fig. 3. The MMT platelet in region A is elongatedlinearly by an amount DMMT by its resultant forces FA,F D�F C

2, FB and FAC. Nylon 6 in region B is loaded by force

FB and is extended by DNylon. Nylon 6 in region C has atensile elongation DMMT under force FCD and a sheardeformation DNylon under force FAC, respectively. There

Fig. 2. Schematic of (a) nanocomposite with ideally staggered arrange-ment of MMT platelets; (b) RVE of the staggered model, geometricalparameters of MMT platelets and nylon 6, and the partition of fourregions (tensile regions A, B and D, and shear and tensile region C) in themodel.

are two regions D which are triangular in shape. The tensiledeformations of the two triangles are added to DNylon, butthe shear deformations are neglected.

Hence, the sum of deformations in the longitudinaldirection on the symmetry plane [with area (b + d)h] arecoherent and equal to DTotal (=DMMT + DNylon). The forceon this plane is the sum of the forces acting in regions A, Band D, and is

F Total ¼ F A þ F B þ F D: ð2ÞThus, the Young’s modulus of the nanocomposites can begiven by

Ec ¼F Total

ðbþ dÞh

� DTotal

12ðaþ lÞ ¼

F A þ F B þ F D

ðbþ dÞhDMMT þ DNylon

12ðaþ lÞ

,,

That is,

Ec ¼ðaþ lÞF Total

2hðbþ dÞDTotal

¼ ðaþ lÞðF A þ F B þ F DÞ2hðbþ dÞðDMMT þ DNylonÞ

ð3Þ

The relationships of forces and deformations in Eq. (3) canbe obtained from each of the regions A–D as follows:

Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068 1065

Region A: Since the MMT platelet is very thin com-pared to its length, only tensile deformation in the longitu-dinal direction is considered and the MMT plateletelongates from a length l/2 to (l/2 + DMMT). In the interfaceof region of A and B and the symmetric plane, MMT notonly bear the forces in the platelet, FA, and region B, FB,but also the force from region D, F D�F C

2. The shear force

FAC acting at the interface between the MMT plateletand nylon 6 matrix, or regions A and C, can be integratedby assuming a roughly linear decrease of tensile strain fromthe symmetric plane to the extreme of region A. So, therelation of forces and deformations in region A is given by

DMMT ¼ KA � ðF A þ F B þ F D � F CÞ ð4Þ

where KA is the extensional compliance of region A, andhas the form

KA ¼l

2EMMT � d � h:

Region B: Nylon 6 in region B is only under tensiledeformation, and its length is changed from a/2 to (a/

2+DNylon) due to force FB. The relation of force and defor-mation is easily obtained from:

DNylon ¼ KB � F B ð5Þ

where KB is the extensional compliance of region B, and is

KB ¼a

ENylon � d � h:

Region C: Nylon 6 in region C is bonded to two MMTplatelets, and is extended as the elongation of MMT plate-lets by an amount DMMT. Meanwhile, shearing of the twoMMT platelets induces shear deformation in region Cequal to DNylon. Hence, the relations of forces and resultingdeformations are determined by

DMMT ¼ KCD � F C ð6ÞDNylon ¼ KC � F AC ¼ KC � ðF A � F BÞ ð7Þ

where KCD and KC are extensional and shear compliancesof region C given by

KCD ¼l

2ENylon � b� h; KC ¼

2bGNylon � l� h

Region D: Since the load taken up by MMT in region Dis much smaller than in region C, then force FD is largerthan FC. The two triangular region D can be joined to forma rectangle subject to force FD with an extension from a/2to (a/2 + DNylon) equal to their adjoining region B. Hence,the sum of tensile deformation of the rectangular region(D + D) is

DNylon ¼ KD � F D ð8Þ

where KD is extensional compliance of region D given by

KD ¼a

2ENylon � b� h

In Eqs. (4)–(8), there are totally six unknown variables. Wecan, therefore, express the four unknown forces (FA,FB, FC,and FD) and 1 unknown deformation DMMT in terms of theremaining unknown deformation DNylon. Thus, we have

DMMT ¼ CMNDNylon

F A ¼ 1KBþ 1

KC

� �DNylon

F B ¼ 1KB

DNylon

F C ¼ CMN

KCDDNylon

F D ¼ 1KD

DNylon

8>>>>>>>><>>>>>>>>:

ð9Þ

where, CMN ¼2

KBþ 1

KCþ 1

KD1

KAþ 1

KCD

.Substituting Eq. (9) into Eq. (3), a

closed-form solution of Young’s modulus or stiffness ofthe nanocomposites can be obtained from

Ec ¼ðaþ lÞ

2ðbþ dÞ � h�

2KBþ 1

KCþ 1

KD

1þ CMN

ð10Þ

By using the data of MMT platelets and nylon 6 inTable 1 and Eq. (10) we can predict the stiffness as a func-tion of MMT loading as shown in Table 2. It is clear thatthese predicted results of the improved staggered model arebetter than the original Jager–Fratzl model when com-pared to the test results. Errors of 20% and more for thehigh clay loading nanocomposites are largely due toassumptions of uniform strain distributions in regions ofthe RVE and omissions such as length and orientation dis-tributions of the platelets in the real materials.

2.4. Numerical results by FEM

Finite element analysis (FEA) is used to obtain numer-ical results for comparisons with those predicted by theJager–Fratzl model and modified staggered model.Fig. 4a shows the FE mesh of the RVE of the staggeredmodel and Fig. 4b and c give the tensile and shear straindistributions. In line with the modified staggered model,a shear deformation region between two MMT plateletsis clearly illustrated.

The Young’s modulus of the nanocomposites from FEanalysis is shown in Table 2 against the predictions fromthe staggered models. For easy comparisons, Fig. 5 plotsall the predicted results based on FEM, Jager–Fratzl modeland modified staggered model against MMT loading. Itcan be seen that FEM gives the closest results, followedby the modified staggered model, to the experimental datareported in [3]. Jager–Fratzl model gives unsatisfactorypredictions.

The errors due to the assumptions of uniform strain dis-tributions in the regions of the RVE in the improved stag-gered model can be estimated by comparison with theresults of FEM.

In the above FEA and staggered models, the distancesbetween the MMT platelets, a and b, are assumed equal,i.e., a/b = 1. Hence, it would be useful to assess the effectof distance ratio a/b on stiffness prediction. Fig. 6 shows

Fig. 4. (a) FE model, (b) tensile strain, and (c) shear strain distributions of the staggered model.

3

5

7

9

11

13

0 1 2 3

V %

You

ng's

Mod

ulus

(G

Pa)

Test results of Ref.[3]

FEM

Staggered model of Jager

Present staggered model

Fig. 5. Predicted and experimental (Ref. [3]) results for Young’s modulusof nylon 6/MMT nanocomposites.

a: b =

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0 1 2 3

V %

Rat

io o

f Y

oung

's M

odul

us

1:10

1:5

1:2

1:1

2:1

5:1

10:1

Fig. 6. Effects of distance ratio, a/b, on stiffness based on FE model.

1066 Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068

the ratio of Young’s modulus of nanocomposites to nylon6 for varying a/b ratio from 1:10 to 10:1 against MMTloading. It can be seen that the maximum error is �13%when a/b = 1:10 and volume fraction of MMT platelets is3.03%. Hence, the effect of the distance ratio is neglectedin our stiffness predictions.

3. Strength prediction

In the present staggered model, the stresses and strainsare homogenized. Hence, it is not able to accuratelydescribe the inhomogeneous distributions of stresses andstrains in some local regions in the nylon 6/MMT nano-composites, e.g., regions close to the ends of MMT plate-

lets. It is felt that this can be ignored in the strengthanalysis, since these stress concentration regions are verysmall and no larger than ten times the platelet thickness.Besides, the plastic strain of the matrix will smooth outthe stress concentration. For prediction of tensile strength,we need a valid failure criterion.

Here, we assume that the nanocomposite fails once thestress in nylon 6 reaches its limiting strength, whether intension or in shear. Thus, the matrix strength controls fail-ure, which has been observed to be valid for these nylon 6/MMT nanocomposites [3,14].

Table 3Comparison of strength between predictions and experimental results inRef. [13]

wt% Experimental results from Ref. [13] Improved staggered model

Yield strength(MPa)

Strengthimprovement

Strengthimprovement

Error(%)

0 69.7 1 1 –3.2 84.9 1.22 1.177 3.657.2 97.6 1.4 1.342 4.31

Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068 1067

From Eq. (9), the stresses in each region are given byRegion B:

rB ¼2

KB � d � hDNylon ¼

2ENylon

aDNylon ð11aÞ

Region C:

rC ¼CMN

KCD � b� hDNylon

sC ¼2

KC � l� hDNylon ¼

GNylon

bDNylon

ð11b; cÞ

andRegion D:

rD ¼1

KD � b� hDNylon ¼

2ENylon

aDNylon ð11dÞ

The tensile stress of the nanocomposites can be evalu-ated from Eqs. (2) and (9) and noting the elasticity relation-ship between E, G and m. Thus, we obtain

rcomp ¼l

4ð1þ mNylonÞbðbþ dÞ þ2

a

� �ENylon � DNylon ð12Þ

Since the tensile stresses in regions B and D are equal,the failure criterion of the nanocomposite becomes

MaxfrB; rCg ¼ rNylonc

or

sC ¼ sNylonc

8><>: ð13a; bÞ

where rNylonc and sNylon

c are tensile and shear strength of ny-lon 6, respectively.

Comparing the tensile stresses in regions B, C and thewhole nanocomposite,

rcomp

rB

¼ rcomp

rD

¼ ða=bÞ � ðl=dÞ8ð1þ mNylonÞð1þ b=dÞ þ 1 ð14aÞ

rcomp

rC

¼ 1

ð1þ b=dÞEMMT

ENylon

þ b=d� �

ð14bÞ

Eqs. (14a) and (14b) suggest several ways to improve thetensile strength of the nanocomposite. For example, themost effective ways are to increase the MMT platelet length(l) and/or decrease the distance (b) between the platelets.But b cannot be reduced by too much as it is limited bythe degree of exfoliation for a given MMT loading. Evenso, for small b, failure may occur by shear of nylon 6 gov-erned by Eq. (11c). The tensile strength of nylon 6 is 1.3–2.0 times its shear strength. This means shear failure willhappen first when the ratio a/b exceeds �2.5. Then, theimprovement of nanocomposite tensile strength can be cal-culated with respect to the tensile stress in the nylon 6 ma-trix, rcomp=rNylon

c , when sc ¼ sNylonc .

With the data of MMT platelets and nylon 6 in Table 1,assuming a/b = 1 and supposing failure of the nanocom-posites is given by MaxfrB; rCg ¼ rNylon

c , then the strengthratio of nanocomposites to nylon 6 matrix can be predicted

from Eq. (14a) and the results are shown in Table 3. Agree-ment with experimental data in Ref. [14] appears to begood.

4. Conclusions

An improved staggered model has been developed topredict the stiffness and strength of nylon 6/MMT nano-composites, and a closed-form solution for stiffness isobtained. A special feature of the improved model is that,besides the tensile deformation, shear deformation of thenylon 6 matrix between the MMT platelets are added asit also contributes to both stiffness and strength. Compar-isons with test data for nylon 6/clay nanocomposites showthat the improved staggered model gives better predictionsof stiffness than the original Jager–Fratzl model althoughmuch more accurate results can be obtained from FEAsolutions. Based on the matrix-dominated failure criterion,strength predictions calculated according to the improvedstaggered model are in good agreement with experimentalresults.

Finally, it is noted that continuum mechanics seems stillapplicable for good estimates of the mechanical propertiesof polymer nanocomposites. Although the improved stag-gered model does not give accurate stiffness predictionsfor nylon 6/clay nanocomposites, the major flaw is due tothe assumptions of uniform strain distributions in the dif-ferent regions of the RVE in the model. The computationalresults of FEA show that if more realistic deformationassumptions, such as those used in Hsueh’s modified two-dimensional stress transfer shear lag model for plateletreinforced composites [15,16], are adopted in the staggeredmodel the error can be reduced to a reasonable level.

Acknowledgements

Y.D. was Visiting Scholar and X.J. was Visiting Profes-sor at the CAMT, University of Sydney, supported by theAustralian Research Council (ARC) when parts of thiswork were completed. Y.W.M. would like to thank theARC for the award of an Australian Federation Fellow-ship. Financial supports from the Natural Science Founda-tion of China (No. 10372072), ‘‘Shu Guang Project” ofShanghai Education Committee and Shanghai LeadingAcademic Discipline Project (No. B302) are muchappreciated.

1068 Y. Dai et al. / Composites: Part B 39 (2008) 1062–1068

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