A Numerical Model to Predict the Anisotropy of Polymer ...ResearchArticle A Numerical Model to...

Research ArticleA Numerical Model to Predict the Anisotropy ofPolymer Composites Reinforced with High-Aspect-RatioShort Aramid Fibers

Jianhong Gao 12 Xiaoxiang Yang 12 and Lihong Huang 1

1College of Chemical Engineering Fuzhou University Fuzhou 350108 China2Quanzhou Normal University Quanzhou 362000 China

Correspondence should be addressed to Xiaoxiang Yang yangxxfzueducn

Received 14 March 2019 Revised 31 May 2019 Accepted 10 June 2019 Published 2 July 2019

Academic Editor Lu Shao

Copyright copy 2019 Jianhong Gao 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

Some fiber types have a high aspect ratio and it is very difficult to predict their composites using traditional finite element (FE)modeling In this study an FE model was developed to predict the anisotropy of composites reinforced by short aramid fibersThree fiber distribution types were studied as follows perfectly aligned normally distributed and randomly distributed fibersTheelastic constants were obtained and for different alignment angles and parameters in the fiber orientation distribution functiontheir numerical results were compared to those of the MorindashTanaka model Good agreement was obtained thus the employedFE model is an excellent and simple method to predict the isotropy and anisotropy of a composite with high-aspect-ratio fibersTherefore the FE model was employed to predict the orientation distribution of a composite fiber with a nonlinear matrix Thepredicted and experimental results agree well

1 Introduction

Short-fiber-reinforced polymer composites have been widelyapplied in industrial fields In addition to the low cost andeasy processing versatility is an important characteristicthat is determined by the performance of the compositersquosconstituents fiber volume fraction short-fiber aspect ratiofiber orientation and interface between the matrix and fiber[1] A composite with aligned fibers can be recognized asan anisotropic material however a composite with randomfibers is isotropic and has completely different mechanicalproperties Generally the fiber orientation varies because ofdifferent flowpatterns inside themold processing conditionsand the rheological properties of the material [2] Thereforeit is difficult for the fibers in a composite to be completelyor randomly aligned The mechanical properties of a realcomposite with short fibers are always intermediate betweenthose of completely and randomly aligned fibers

Many analyses and numerical methods have been usedto predict the elastic properties of short-fiber-reinforcedcomposites Among the analysis methods the Mori-Tanaka

model has been proven to produce the most accurate pre-dictions [3] To assess the effect of fiber orientation onthe elastic properties classical models such as the Eshelbyequivalent inclusion HalpinndashTsai equation self-consistentshear lag and bounding models and their extensions havebeen applied to various materials Mortazavian et al [4] usedexperimental methods along with an analytical approachto study the tensile strength and elastic modulus of short-fiber-reinforced polymer composites They proved that alaminate analogy and modified TsaindashHill criteria providesatisfactory predictions Tian et al [5] investigated the effectof fiber orientation on the effective elastic properties ofshort-fiber-reinforced metal matrix composites using two-step mean-field homogenization procedures including theD-IReuss M-TReuss D-IReuss M-TVoigt and D-MV-R models Jiang et al [1] discussed the effects of fiberorientation distribution on the overall elastic properties ofcomposites usingMorindashTanakarsquos method Ogierman et al [6]studied the influence of process-induced orientation of thereinforcement onmechanical properties usingMorindashTanakarsquosmicromechanicalmodel and a numericalmethod Laspalas et

HindawiAdvances in Polymer TechnologyVolume 2019 Article ID 5484675 12 pageshttpsdoiorg10115520195484675

2 Advances in Polymer Technology

al [2] investigated the mechanical behavior of short-fiber-reinforced plastic composites by applying the TandonndashWengmicromechanical model considering the local fiber orien-tation distribution Chen et al [7] determined the effectivemoduli of composites containing misoriented fibers basedon the EshelbyndashMorindashTanaka theory Huang [8] presentedan analytical approach to evaluate the orientation effects onthe elastic properties of a composite containing randomlyoriented fibers Fu et al [9] modeled fiber orientation distri-butions using a laminate analogy approach

It is convenient to use analytical approaches howevercompared to a numerical method the microstructure fieldof deformation in composites cannot be directly obtainedTherefore numerical methods which predict the anisotropyhave been developed for different materials Tian et al [5]employed finite element (FE) methods with solid elementsfor short fibers to predict the effects of fiber orientation onthe effective elastic properties of metal matrix compositesOgierman et al [6] investigated the effect of fiber orientationusing a macroscale FE model with different boundary con-ditions Laspalas et al [2] studied the elasticity and failureof short-fiber-reinforced composites using FE simulationincluding an orientation average Jansson et al [10] employedthe FE method to predict the anisotropic and nonlinearbehavior of plastics reinforced by glass fibers However inFE modeling with multiple inclusions ldquojammingrdquo of fibers[11] easily occurs in which the generated new fibers intersectthe existing fibers Extensive efforts have been devoted todevelopingmethods for generating the representative volumeelements (RVEs) of composites reinforced by inclusions ofdifferent shapes Yi et al [12] proposed modified randomsequential adsorption (RSA) algorithms to generate an RVEof a random chopped fiber-reinforced composite materialwith straight and curved fibers Harper et al [11] presented amethod to generate RVEswith randomdiscontinuous carbonfibers in the plane and no limitation to the volume fraction ofthe fiber Li et al [13 14] presented modified RSA algorithmsfor generating RVEs with complex microstructures suchas fibers and spatially randomly distributed pores basedon the microstructure information of carboncarbon (CC)composites

However most numerical models have only focused oncomposites with a low-aspect-ratio fiber The aspect ratiorefers to the ratio of the fiber length to the diameter Fora high aspect ratio particularly when the fiber radius is ata micron scale and the fiber length is at a millimeter scalesuch as an aramid fiber it is very difficult to establish anumerical FE model using a traditional method particularlywhen the volume fraction of the fibers is high Only a fewpapers have reported on this topic [15 16] In our previousstudy [17] a numerical model was proposed to investigate themechanical properties of an aramid-fiber-reinforced rubbercomposite with a high aspect ratio It was proved that the FEmodel could simulate large deformationmechanical behaviorwell However the applicability of this proposed FE model indescribing the anisotropy of a composite remains unknown

In this study the FE model was further developed topredict the anisotropy of short-fiber-reinforced compositesThis paper is structured as follows In Section 2 the analytical

methods for predicting the elastic modulus of the compositeswith perfectly aligned completely random and probabilityorientation distribution fibers are introduced In Section 3the FE model and the algorithms of the different fiberorientation distributions are presented In Section 4 theboundary conditions for the different mechanical behaviorsare introduced and a means of obtaining the effective elasticconstants is presented In Section 5 the numerical results ofthe elastic constants with a different orientation distributionare compared to those of the MorindashTanaka model and a highcoincidence is shown Finally based on the aforementionedstudy the FE model was applied to composites with anonlinear matrix to predict the fiber orientation distributionThe numerical results were compared to the experimentalresults and found to agree well This study verifies thatthe employed FE model is excellent to predict the elasticproperties of a short-fiber-reinforced composite

2 Analytical Models

21 MorindashTanaka Model In the Eshelby theory of equivalentinclusion by replacing the elasticmoduli of the inclusionwiththose of the matrix an inhomogeneity problem can be equiv-alent to a homogeneity problemMeanwhile an eigenstrain isintroduced into the inclusion domain Based on the Eshelbyequivalent inclusion theory [18] theMorindashTanakamodel [19]considers the mutual effect between the inclusions in thecomposite thereby improving the accuracy of the predictionresults It has been proven that the MorindashTanaka model is areliable and simple method for inclusion problems and hasbeen widely used to predict effective elastic moduli

It is supposed that the fiber axes coincide with theprincipal direction of the composite matrix The compositeis subjected to a uniform stress 1205900 on the far field boundarywhich is regarded as the average composite stress For ahomogeneous material the constitutive relation is 1205900 = 1198711205760where 119871 is the elastic constant tensor of the composite Theaverage strain of the composite is as follows [19]

1205760 = (1 minus 1198621) 120576(0) + 1198621120576

(1) = (119868 + 1198621119860)119871minus10 1205900 (1)

where 120576(0) and 120576(1) are the average strain of the matrix andinclusions respectively 1198710 is the elastic constant tensor ofthe matrix C1 is the volume fraction of the inclusions A isexpressed as follows

119860 = 1198710 + (1198711 minus 1198710) [1198621119868 + (1 minus 1198621) 119878]minus1 (1198710 minus 1198711) (2)

where 119868 is the unit tensor of four orders and S is the Eshelbytensor of four orders which is related to the elastic propertyof the matrix and the shape of the inclusions The analyticalexpressions of S for an ellipsoid inclusion problem can befound in many previous studies [20 21] Thus the effectivemodulus of the composite can be obtained as follows

119871 = 1198710 (119868 + 1198621119860)minus1 (3)

Hence the effective elastic modulus 119871 is an explicit functionthat is easy to obtain

Advances in Polymer Technology 3

22 AnalyticalModel withOrientation Effects In the previoussection the deduction was based on the assumption thatthe fibers are oriented along the principal matrix axesHowever in practice because of processing conditions it isdifficult to achieve a perfect alignment and there is alwaysa fiber orientation distribution in a composite Therefore todetermine the real effective modulus the effect of the fiberorientation distribution should be considered In this sectionwe discuss the effective moduli for three cases a perfectlyaligned fiber whose principal axes present an angle referringto the principal matrix axes a probability fiber orientationdistribution and completely random fibers

Generally the orientation vector of fiber 119901 can beexpressed as 119901 = (cos 120579 sin 120579 cos120593 sin 120579 sin120593) where 120579 and120593 are the Euler angles as shown in Figure 1 x1 x2 and x3 arethe three directions of the global coordinates while 11990910158401 11990910158402and 11990910158403 are the three directions of the local coordinates In thiscase the general framework in the previous section remainsvalid The only change was to determine the tensor of Eq(3) in the local coordinates referring to the global structuralgeometry We denoted 119906119894 and 1199061015840119895 as the unit vectors of theglobal coordinates (0 x1 x2 and x3) and local coordinates(0 11990910158401 11990910158402 and 11990910158403) respectively A relationship between themcan be constructed as follows [1]

119906119894 = 1205721198951198941199061015840119895 (4)

where the transformation matrix 120572119895119894 is as follows

120572119895119894 = [[[cos120593 sin 120579 cos120593 sin 120579 sin120593minussin 120579 cos 120579 cos120593 cos 120579 sin120593

0 minussin120593 cos120593]]]

(5)

Therefore for the case of perfectly aligned fibers we obtaineda relationship between the elastic tensor 119871ijkl of the globalcoordinates and 119871i1015840j1015840k1015840l1015840 of the local coordinates as follows

119871 ijkl = 1205721198941015840119894120572119895101584011989512057211989610158401198961205721198971015840119897119871 i1015840j1015840k1015840l1015840 (6)

where 1205721198941015840119894 1205721198951015840119895 1205721198961015840119896 119886119899119889 1205721198971015840119897 are the elements of the transfor-mation matrix 120572119895119894 119871 ijkl and 119871 i1015840j1015840k1015840l1015840 are the elements of thecorresponding elastic tensor

In the case of a misaligned fiber the orientation averageof the elastic tensor of the composite is provided by thefollowing integration [1]

119871 = int120587120579=0

int2120587120593=0119871 (120579 120593) 120588 (120579 120593) sin 120579119889120579119889120593 (7)

where 120588(120579 120593) is the probability density function of the fiberorientation For a planar problem Eq (7) can be simplifiedcombined with a discrete algorithm [5 22] of the orientationaverage as follows

119871 = int120587120579=0119871 (120579) 120588 (120579) sin 120579119889120579

= 119873120579sum119894=0

(119871 (120579119894) + 119871 (120579119894+1))2 120596 (120579119894)(8)

x2

x1

x3

x2

x3

x1

a1

a2

a3

p

Figure 1 Orientation vector 119901 of a spatially oriented short fiber

x

y

D

D

Short fiber

Matrix

Figure 2 FE model of a short-fiber-reinforced composite squareshape with size DtimesD

where 120579 = 120587119873120579 is the incremental angle 119873120579 is the numberof the angle incrementation 120579119894 = 119894120579 and i is the integerwithin (0 119873120579) 120596(120579119894) is the probability of the fiber orientationwithin the range of (120579119894 120579119894+1) 119871(120579119894) is the elastic constantwhen 120579 = 120579119894 Eq (8) is a simplified algorithm that avoidsthe difficulty of an integral particularly for the complexprobability density function 120588(120579) The precision of the resultof Eq (8) depends on119873120579 because an increase in119873120579will lead tohigh calculation precision Previous studies show that if119873120579 isappropriately selected high-precision calculation results forelastic constants can be achieved

In the case of randomly distributed fibers 120596(120579119894) in eachangle range (120579119894 120579119894+1) is recognized to be equal It can becalculated by 120596(120579119894) = 1119873120579 Substituting it into Eq (8) theelastic tensor is obtained

3 Numerical Model Development

31 Method In the traditional method of finite element anal-ysis (FEA) the solid element is always used to simulate themechanical behavior of the fiberHowever using thismethodit is very difficult to establish a composite model with a highfiber volume fraction [12] Therefore in [17] an FE modelwas introduced for a planar problem of the elastic propertiesof a short-fiber-reinforced rubber composite However thusfar we did not know the applicability of this model for theanalysis of the anisotropic properties of a composite

The FE model was established as a square whose sizeis 119863 times 119863 [17] as shown in Figure 2 Short fibers weregenerated and the linear hybrid beam element (BH21H) was


(a) Perfectly aligned fibers (120579=50∘) (b) Normally distributed fibers (c) Completely random distribution

Figure 3 FE models for three fiber distribution types D=20 mm 119897f=1 mm

employed in the ABAQUS FE software For the matrix planestress continuum elements (CPS8R) were used Using theembedded technique which has been verified to yield quitesimilar results with those for the traditional method [23] themesh of the matrix and the fibers are independent of eachother as shown in Figure 3 and are more regular leading tomore precise numerical results Another advantage is that amodel with high volume fraction and high-aspect-ratio fibersis realized

32 Algorithm for Fiber Generation In this study planarproblems with three fiber distribution types ie perfectlyaligned probability function distributed and completelyrandom fibers were discussedThe detailed flow chart can bereferred to in the literature [17] To avoid fiber intersectionthe RSA algorithm was employed using Python languagein the ABAQUS software Three random numbers x yand 120579 which represent the moving distance along the xdirection moving distance along the y direction and therotation angle of the fiber respectively were used To ensurehigher accuracy of the fiber volume fraction each fiberwas generated with an equal length in the model includingthe fibers around the model edge For this reason theranges of x and y were set as follows 119909 isin (minus1198632 +(1198971198912)(1 minus sin2 120579)minus12 1198632 minus (1198971198912)(1 minus sin2 120579)minus12) 119910 isin(minus1198632 + (1198971198912)(1 minus cos2 120579)minus12 1198632 minus (1198971198912)(1 minus cos2 120579)minus12)where 119897f is the fiber length For different fiber orientationdistributions the FE models are shown in Figure 3 and theranges of 120579 are interpreted as follows

(1) Perfectly aligned fibers 120579 is the orientation angle ofthe fibers which is invariable for all the fibers in themodel

(2) Completely random fibers 120579 is a random numberwithin (minus1205872 1205872)

(3) Misaligned fibers with a probability function distri-bution the incremental angle 120579 = 120587119873120579 is set where119873120579 is the number of the angle incrementation First

119899119891119894 is calculated according to a given probabilityfunction of the fiber orientation 120588(120579119894) 119899119891119894 and 120588(120579119894)are the fiber amount and orientation probabilityrespectively distributed within an angle range (120579times119894120579 times (119894 + 1)) Subsequently fibers are generated withrandom angles within the range (120579 times 119894 120579 times (119894 +1)) and the corresponding generated fiber amount is119899119891119894 i is an integer between 0 and 119873120579 Therefore thisalgorithm is a discretized method whose precisiondepends on 120579 Decreasing the 120579 enhances thecalculation precision

33 Probability Density Function of the Fiber DistributionFor most real composites reinforced with aramid fibers of afixed aspect ratio there are always different fiber orientationdistributions that can be described by a symmetric functionsuch as a normal function and a nonsymmetric functioneg a Gamma Weibull or exponential function To studythe effect of fiber orientation distribution on the elasticproperties of the composite a normal distribution functionwas considered in this study For the planar problem only onevariable the Euler angle 120579 is included such that the functioncan be expressed as follows [1]

120588 (120579) = 1120590radic2120587 exp[minus12 (120579 minus 120583120590 )2] (9)

where 120583 (minusinfin lt 120583 lt infin) is the mean and 1205902 (120590 gt0) is the variance of the distribution Figure 4 shows thefiber distribution when in Eq (9) 120579 = 0 120583 = 0 and1205902 =30 100 and 500 (degree measure was used for 1205902) Thediagram shows that as 1205902 increases the fiber distribution isincreasingly disordered There are two extremes when 1205902 =0 the fibers perfectly align and when 1205902 is sufficiently largethe fiber distribution is completely random Indeed when1205902 increases to a large value the range of 120579 (minus1205872 1205872) isunable to accommodate all the fibers because of the normaldistribution characteristicsHowever fortunately the effect ofthe orientation angle of the fiber on themechanical properties


(

)(

)

2 = 30

2 = 100

2 = 500

240minus4minus2

030

025

020

015

010

005

000

Figure 4 Effect of variance 1205902 of the normal probability density function on fiber distribution 120579 = 0 120583 = 0

of the composite is periodic which indicates that all the fibersof a 120579119894 plusmn119899120587 orientation angle exert the same effect Accordingto this 120588(120579119894) can be flexibly calculated for large 12059024 Numerical Implementation

41 Model Parameters In this study aramid fiber which iswidely applied in industrial products was employed for theinvestigation Typically aramid fiber has a high aspect ratiobecause its diameter is very small at approximately 0012mmFor chopped fibers the length is changed to meet productrequirements Because an aramid fiber is short and rigid itis straight at the beginning Therefore a straight line can beemployed to simulate its initial elastic property in the FEmodel According to the product manual a fiber of modulus119864f= 70 GPa was selected for this study and the Poissonratio was 03 For the polymer composite a matrix modulusof 1 GPa was employed The mechanical properties of thecomposites containing the aforementioned components weresimilar to those of a composite with a resin matrix which iswidely used in the industryThe volume fraction of a fiberwasdefined by Vf = 120587119903f 2119897f119899f(119863 times 119863 times 119905) t is the model thicknessset to be 01 and 119899f is the total fiber amount in the modelAccording to [17] with increasing 119863119897f the numerical resultswill converge 119863119897f=30 was used in this study

42 Boundary Conditions The FEmodel in this study can beregarded as at amesoscopic scale If themodel with aminimalsize has the same effective mechanical properties as the bulkmaterial it can be recognized as the RVE Generally for amicroscopic model a periodic boundary condition will beemployed [24] In the literature [11] Harper et al adoptedperiodic boundary conditions and provided effectivematerialproperties within an inner RVE region In [17] consideringthe Hill condition [25] four boundary conditions wereimposed ie uniform traction uniformdisplacementmixed

boundary condition and periodic boundary condition forthe model uniaxial tension It has been proven that withincreasing model size results with different boundary con-dition tend to be the same

Therefore we believed the uniform displacement bound-ary condition would provide the most reasonable and correctnumerical results for the elastic property of uniaxial tensionThe boundary conditions for the different elastic behaviorsare described in Table 1 The variable u is the displacementload of the node on the corresponding edge the superscriptrepresents the load direction The subscript i or j representsany node on the corresponding edge and thus the expression119906119910119894 = 119906119910119895 implies that the displacement of each node is equalFor simple shear deformation 119906119909119894119909=1198712 = 119906119909119894119909=minus1198712 implies thatthe displacement along the x direction of the node on the119909 = 1198712 edge is equal to that of the corresponding node on the119909 = minus1198712 edge Among all the variables listed in Table 1 only120575119909 and 120575119910 are provided which are fixed values for each elasticbehavior For different elastic behaviors the deformations areshown in Figure 5Their elastic constants can be calculated byextracting the results using Python as follows

(1) Elastic Moduli 119864x and Vx

120590119909 = 119877119865119863 times 119905 = sum119899119894=1 119891119894119909=1198632119863 times 119905 = minus sum119899119894=1 119891119894119909=minus1198632119863 times 119905 (10)

120576119909 = 119880119863 = sum119899119894=1 119906119894119909=1198632 minus sum119899119894=1 119906119894119909=minus1198632119899119863 (11)


where 120590119909 is the stress in the x direction of the modelimposed by uniaxial tension 120576119909 and 120576119910 are the strain inthe x and y directions respectively RF is the reaction forceof the model edge which is loaded on and n is the node


Table 1 Boundary conditions for the different elastic constants

Elastic behavior Elastic constant(s) Boundary conditions

Uniaxial tension along the x direction 119864x Vxx=D2 119906119909 = 120575119909 x=- D2 119906119909 = minus120575119909y= D2 119906119910119894 = 119906119910119895 y=- D2 119906119910119894 = 119906119910119895

Uniaxial tension along the y direction 119864y Vyy=D2 119906119910 = 120575119910 y=- D2 119906119910 = minus120575119910x= D2 119906119909119894 = 119906119909119895 x=- D2 119906119909119894 = 119906119909119895

Simple shear 119864xyy=D2 119906119909 = 120575119909 119906119910 = 0 y=- D2 119906119909 = 119906119910 = 0119906119909119894119909=1198712 = 119906119909119894119909=minus1198712 119906119910119894119909=1198712 = 119906119910

119894119909=minus1198712

(a) Uniaxial tensile deformation along the xdirection

(b) Uniaxial tensile deforma-tion along the y direction

(c) Shear deformation

Figure 5 Different elastic deformations of short-fiber-reinforced composite

amount of one edge which is the same for each edge of thesquare model 119891119894119909=1198632 and 119906119894119909=1198632 are the reaction force anddisplacement of the node on the 119909 = 1198632 edge respectivelyThe elastic modulus and Poisson ratio in the x direction canbe expressed respectively as follows 119864119909 = 120590119909120576119909 and V119909 =120576119910120576119909(2) Elastic Moduli 119864y and Vy Replacing x with y in Eqs (10)(11) and (12) 119864y and Vy are obtained

(3) Shear Modulus 119864xy

119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)

where 120591119909119910 and 120574119909119910 are themodel shear stress and strain Othervariables have been previously defined

5 Verification and Discussion

51 Perfectly Aligned Fibers To investigate the applicability ofthe proposed model for predicting the anisotropic propertiesof composites with perfectly aligned fibers models with adifferent fiber orientation angle 120579 relative to the x axis wereestablished The boundary conditions shown in Table 1 wereimposed for the different elastic deformations and the resultswere compared to those of the MorindashTanaka model Figure 6shows this comparison when 119897f=1 mm and Vf=001 It can beseen that all the elastic constants in the numerical results119864x 119864y 119864xy Vx and Vy agree well with the results of theMorindashTanaka model

For the elastic modulus 119864x assuming that the uniaxialtension along the x direction is imposed the minimumoccurs at approximately 120579 = 60∘ but not at 90∘ To understandthis the matrix and fiber stresses were investigated Byaveraging the stress of the matrix and fiber of the RVE thecontributions from the resin and fiber were both consideredas follows [11]

120590119886V119890119909 = 120590119891119894119887119890119903minus119886V119890119909 + 120590119898119886119905119903119894119909minus119886V119890119909 = 1119881 (intV120590119891119894119887119890119903119909 119889119881

+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)

where 120590119886V119890119909 120590119891119894119887119890119903minus119886V119890119909 and 120590119898119886119905119903119894119909minus119886V119890119909 are the average stressof the model beam and matrix respectively and V is theRVE volumeThe subscripts a and b are the number of beamsand matrix elements respectively and the subscript i is theirintegration point number SF is the section force of the beamwhich is multiplied by cos 120579 its component in the loadingdirection x can then be obtained 119904119898 is the stress componentof each integrated point of the matrix material in the loadingdirection 119860119887 is the cross-sectional area of the beam element119878119881119874119871 is the integrated section volume of each beam elementIVOL is the parameter of the integration point volume of thematrix element


Ex -Simulation Exy -Simulation Ey -MoriminusTanaka

Ey -SimulationEx -MoriminusTanakaExy -MoriminusTanaka

Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy

x -Simulationy -Simulationx -MoriminusTanakay -MoriminusTanaka

x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy

Figure 6 Comparison between the numerical and MorindashTanaka model results for different elastic constants of a composite with 119897f=1 mmand Vf=001

ＰＲ

ＧＮＬＣＲ-ＰＲ

fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)

Figure 7 Average stress of the total model matrix and fibers of acomposite with 119897f=1 mm and Vf=001 when the uniaxial tension ofthe x direction is imposed

For the composite with 119897f=1 mm and Vf=001 120590119886V119890119909 120590119891119894119887119890119903minus119886V119890119909 and 120590119898119886119905119903119894119909minus119886V119890119909 were obtained as shown in Figure 7It can be seen that when 120579 lt 60∘ as 120579 increases the fiberstresses decrease which weakens the fiber reinforcementeffect Until 120579 increases to approximately 60∘ the stress inthe fibers is compressive Meanwhile for the average matrixstress it decreases when 120579 lt 50∘ and then increasesfrom nearly 60∘ Thus the total stress minimum occurs at

approximately 120579 = 60∘ which explains the minimum 119864X of120579 = 60∘52 Different Fiber Orientation Distributions To express theanisotropy of the composites a parameter 120572119894119895 was introducedthat refers to the deviation degree from isotropyThe isotropicfactor can be defined as follows [26]

120572119894119895 = 119871119879119894119895 minus 1198711198941198952119871(9minus119894minus119895)(9minus119894minus119895) with 119894 119895 = 1 2 and 3 (15)

where 119871119894119895 is the corresponding element in the stiffnessmatrix and 119871119879119894119895 is expressed by 119871119879119894119895 = (119871119894119894 + 119871119895119895)2 For theplanar problem only the factor 12057212 was used to describe thedeviation degree from isotropy The closer 12057212 is to 10 themore isotropic the composite is

Models with different variances of the distribution 1205902(120590 gt 0) were established when 120579 in Eq (9) was 0∘ 1205902 = 0implies that the fibers are perfectly aligned If 1205902 increasesto a sufficiently large value the fibers are considered to berandomly distributed For one 1205902 in the same manner theelastic constants 119864x 119864y 119864xy Vx 119886119899119889 Vy are calculated viasimulation and an analytical model Then 12057212 is obtainedThe relationship between 12057212 and the five elastic constantsis plotted as shown in Figure 8 The comparisons betweenthe numerical and MorindashTanaka model results are shown toagree well As shown in Figure 8 the minimum longitudinalmodulus 119864x occurs when 12057212 is near 1 indicating a randomlydistributed fiber Meanwhile the transverse modulus 119864yand shear modulus 119864xy both show a maximum value Inaddition with decreasing 12057212 119864y seldom changes at first butthe effect of 12057212 on 119864x is greater than that on 119864y


Ex ndash MoriminusTanakaExy ndash MoriminusTanakaEy ndash Simulation

Ey ndash MoriminusTanakaEx ndash SimulationExy ndash Simulation

11 12 13 1410Isotropic factor 12

0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy

x ndash MoriminusTanakay ndash MoriminusTanakax ndash Simulationy ndash Simulation


000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy

Figure 8 Relationship between elastic constants and isotropic factor 12057212

Normal distributed fibers

500 1000 1500 200002

10

11

1212

13

14

Figure 9 Relationship between the isotropic factor 12057212 andvariances 1205902

In addition the relationship between the isotropic factor12057212 and the variances 1205902 is shown in Figure 9 It can beseen that when 1205902 increases to approximately 2000 12057212is approximately 10 and tends to remain constant whichindicates a randomly distributed fiber Figure 9 can serveas a reference to determine the degree of the anisotropyof composites and is applicable to all short-fiber-reinforcedcomposites of normal orientation distribution In additionother anisotropic models with different fiber microstructuralparameters were calculated and there was good agreementas shown in Figures 6 and 7 Therefore we believe that theemployed FE model predicts the anisotropy of a compositevery well

6 Application to Nonlinear Material

61 Effect of Orientation Distribution The hyperelastic prop-erty of rubber composites reinforced by aramid fibers hasbeen studied [17] By averaging the experimental stressndashstraincurves along the x-direction (along the flowing direction)and y-direction (vertical to the flowing direction) specimensthe averaged curve can be recognized as the mechanicalresponse of a composite with randomly distributed fibers Bycomparing to the experimental data the numerical modelof isotropy with random fibers predicts the nonlinear elasticresponse of stressndashstrain well For real specimens becausethe short fibers in the composite are always oriented alongthe compound flowing direction anisotropy always existshowever only by means of macromechanics it is difficultto determine the orientation degree As described in thissection the anisotropic FE model was extended to nonlinearmaterial

The material parameters of the two components rubberand short aramid fibers are the same as [17] The OgdenN3 hyperelastic constitutive model was employed to describethe rubber matrix elastic behavior Based on the tensileexperiment results the material parameters were obtainedas follows 1205831=-8116 1205721=1975 1205832=5725 1205722=2388 1205833=5467and 1205723=-4495 For the aramid fibers the elastic modulusis 132 GPa the fiber diameter is 0012 mm and the studiedfiber length is 15 mm The difference in this study wasthe composite fiber distribution which was established withan anisotropic morphology Still a perfect interface andstraight morphology for the aramid fibers at the initialdeformation were supposed For fiber length 119897f=15 mm andfiber volume fraction Vf=00049 using the anisotropic FEmodel the stressndashstrain response of the composites withdifferent perfectly aligned angles was obtained as shown in


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040

6080Aligned angle ( ∘)

00

0102

030

Nominal strain

(a) Stressndashstrain response when the composites were loaded along thex direction

00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain

Aligned angle ( ∘)

(b) Stressndashstrain response when the composites were loaded alongthe y direction

Figure 10 Response of stressndashstrain relationship of composites with an aligned angle 120579 of fibers 119897f=15 mm and Vf=00049

00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2

(a) Stressndashstrain response when the composites were loaded along the xdirection

00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400

(b) Stress-strain response when the composites were loaded along the ydirection

Figure 11 Response of the stressndashstrain of composites with different variances 1205902 from the numerical results 119897f=15 mm Vf=00049

Figure 10 The curve shapes with blue scatters shown inFigures 10(a) and 10(b) are similar to those of119864x and119864y of thelinearmaterial shown in Figure 6 respectivelyTheminimumlongitudinal modulus shown in Figure 11 also occurs when120579 = 60∘ The stressndashstrain responses of the composites withdifferent variances 1205902 from the numerical results are shownin Figure 11 which corresponds to Figure 8 for the linearmaterial

62 Determination of the Orientation Distribution For thehyperelastic material the constitutive behavior is always

expressed in the form of strain energy W The hyperelasticconstitutive model of an isotropic material has been com-prehensively studied [27 28] However for an anisotropicmaterial many terms remain to be explored [29ndash31] Todescribe the orientation degree of the fibers as the isotropicfactor 12057212 of the linear material a variable 120573 = 119882119909119882119910was introduced where 119882119909 and 119882119910 are the strain energyof the elastic response when the tensile load is the along xand y directions respectively According to the numericalresults for fiber length 119897f=15 mm and fiber volume fractionVf=00049 the relationships between 120573 and 119882119909 and 119882119910 are


WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)

Figure 12 Relationship between 119882x (119882y) and the isotropic factor 120573 for a rubber composite reinforced with aramid fibers 119897f=15 mmVf=00049

500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122

Figure 13 Relationship between the isotropic factor 120573 and variances 1205902 for a rubber composite reinforced with aramid fibers 119897f=15 mmVf=00049

shown in Figure 12 It can be seen that when 120573 is near 10 119864xand 119864y coincide

From the effect of1205902 on120573 as shown in Figure 13 accordingto the experimental result 120573 = 109 the corresponding1205902 of the fiber orientation distribution was determined 1205902which was approximately 1400 is the most reasonable valueSimilarly 1205902 for 119897f=15 mm and Vf=00146 was determined tobe approximately 3000The experimental results [12] and thecorresponding stressndashstrain curves of the determined 1205902 arecompared in Figure 14There are good agreements indicatingthat the proposed method predicting the fiber orientationdistribution is applicable

7 Conclusion

In this study an FE model with an embedded technique wasemployed to predict the anisotropy of polymer composites

reinforced with short aramid fibers Using this FE methodone can obtain the RVE with high aspect ratio fibers in thecomposites To obtain the elastic properties of the real mate-rial the RVEs of three types of fiber orientation distributionswere established using Python language perfectly alignednormally distributed and randomly distributed fibers Thefive elastic properties tensile elastic moduli 119864x and 119864yshear elastic modulus 119864xy and Poisson ratios Vx and Vywere obtained by different elastic deformations with corre-sponding boundary conditions For different fiber orientationdistributions the numerical results were compared to thoseof the MorindashTanaka model and found to agree well

Based on the aforementioned conclusion the FE modelwas applied to predict the fiber orientation distributionAs a nonlinear material an isotropic factor in the form ofstrain energy was proposed to present the deviation degreefrom isotropy According to the relationships between the


Specimen of x direction experiment Specimen of y direction experiment Average curve experiment Longitudinal performance simulation Transverse performance simulation Randomsimulation

00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049

Longitudinal performance simulation Transverse performance simulation Randomsimulation

Specimen of x direction experiment Specimen of y direction experiment Average curve experiment

2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146

Figure 14 Comparison between the experimental and numerical results of the model with the determined 1205902

isotropic factor and variance the most reasonable variancesused to describe the fiber orientation distribution in thereal composite were determined Comparison between theexperimental results and numerical results of the determinedvariance indicated good agreements The FE model used inthis study is a simple and convenient methodThe numericalprediction for a composite with a high-aspect-ratio fiber hasits advantages In addition it can be applied to simulate thelarge deformation of a hyperelastic material that is difficultto realize using the traditional method because of the easyoccurrence of numerical nonconvergence The proposedFE model aids in predicting the anisotropy of short-fiber-reinforced composites and has a shorter experimental periodand lower cost

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The Natural Science Foundation Projects of the FujianProvince of China (grant number 2018J01427) and theNationalNatural Science Foundation ofChina (grant number11372074) supported this study

References

[1] B Jiang C Liu C Zhang B Wang and Z Wang ldquoThe effectof non-symmetric distribution of fiber orientation and aspectratio on elastic properties of compositesrdquo Composites Part BEngineering vol 38 no 1 pp 24ndash34 2007

[2] M Laspalas C Crespo M A Jimenez B Garcıa and J LPelegay ldquoApplication of micromechanical models for elasticityand failure to short fibre reinforced composites Numericalimplementation and experimental validationrdquo Computers ampStructures vol 86 no 9 pp 977ndash987 2008

[3] C L Tucker III and E Liang ldquoStiffness predictions forunidirectional short-fiber composites review and evaluationrdquoComposites Science and Technology vol 59 no 5 pp 655ndash6711999

[4] S Mortazavian and A Fatemi ldquoEffects of fiber orientationand anisotropy on tensile strength and elastic modulus ofshort fiber reinforced polymer compositesrdquo Composites Part BEngineering vol 72 pp 116ndash129 2015

[5] W Tian L Qi C Su J Zhou and Z Jing ldquoNumerical simula-tion on elastic properties of short-fiber-reinforcedmetal matrixcomposites Effect of fiber orientationrdquo Composite Structuresvol 152 pp 408ndash417 2016

[6] W Ogierman and G Kokot ldquoA study on fiber orientationinfluence on themechanical response of a short fiber compositestructurerdquo Acta Mechanica vol 227 no 1 pp 173ndash183 2016

[7] C-H Chen and C-H Cheng ldquoEffective elastic moduli ofmisoriented short-fiber compositesrdquo International Journal ofSolids and Structures vol 33 no 17 pp 2519ndash2539 1996

[8] J H Huang ldquoSome closed-form solutions for effective moduliof composites containing randomly oriented short fibersrdquoMate-rials Science and Engineering A Structural Materials PropertiesMicrostructure and Processing vol 315 no 1-2 pp 11ndash20 2001


[9] S-Y Fu X Hu and C-Y Yue ldquoThe flexural modulus of mis-aligned short-fiber-reinforced polymersrdquo Composites Scienceand Technology vol 59 no 10 pp 1533ndash1542 1999

[10] J Jansson T Gustafsson K Salomonsson et al ldquoAn anisotropicnon-linear material model for glass fibre reinforced plasticsrdquoComposite Structures vol 195 pp 93ndash98 2018

[11] L T Harper C Qian T A Turner S Li and N A WarriorldquoRepresentative volume elements for discontinuous carbonfibre compositesmdashpart 1 boundary conditionsrdquo CompositesScience and Technology vol 72 no 2 pp 225ndash234 2012

[12] Y Pan L Iorga and A A Pelegri ldquoNumerical generationof a random chopped fiber composite RVE and its elasticpropertiesrdquo Composites Science and Technology vol 68 no 13pp 2792ndash2798 2008

[13] X Chao L Qi J Cheng W Tian S Zhang and H LildquoNumerical evaluation of the effect of pores on effective elasticproperties of carboncarbon compositesrdquo Composite Structuresvol 196 pp 108ndash116 2018

[14] L Qi X Chao W Tian W Ma and H Li ldquoNumericalstudy of the effects of irregular pores on transverse mechanicalproperties of unidirectional compositesrdquo Composites Scienceand Technology vol 159 pp 142ndash151 2018

[15] M Schneider ldquoThe sequential addition and migration methodto generate representative volume elements for the homog-enization of short fiber reinforced plasticsrdquo ComputationalMechanics vol 59 no 2 pp 247ndash263 2017

[16] E Ghossein and M Levesque ldquoRandom generation ofperiodic hard ellipsoids based on molecular dynamics acomputationally-efficient algorithmrdquo Journal of ComputationalPhysics vol 253 pp 471ndash490 2013

[17] J Gao X Yang and L Huang ldquoNumerical prediction ofmechanical properties of rubber composites reinforced byaramid fiber under large deformationrdquo Composite Structuresvol 201 pp 29ndash37 2018

[18] J D Eshelby ldquoThe determination of the elastic field of anellipsoidal inclusion and related problemsrdquo Proceedings of theRoyal Society LondonAMathematical Physical and EngineeringSciences vol 241 pp 376ndash396 1957

[19] T Mori and K Tanaka ldquoAverage stress in matrix and averageelastic energy of materials with misfitting inclusionsrdquo ActaMetallurgica et Materialia vol 21 no 5 pp 571ndash574 1973

[20] G P Tandon and G J Weng ldquoThe effect of aspect ratio ofinclusions on the elastic properties of unidirectionally alignedcompositesrdquoPolymerComposites vol 5 no 4 pp 327ndash333 1984

[21] O Pierard C Gonzalez J Segurado J LLorca and I DoghrildquoMicromechanics of elasto-plastic materials reinforced withellipsoidal inclusionsrdquo International Journal of Solids and Struc-tures vol 44 no 21 pp 6945ndash6962 2007

[22] I Doghri and L Tinel ldquoMicromechanics of inelastic compositeswith misaligned inclusions numerical treatment of orienta-tionrdquo Computer Methods Applied Mechanics and Engineeringvol 195 no 13-16 pp 1387ndash1406 2006

[23] S A Tabatabaei S V Lomov and I Verpoest ldquoAssessment ofembedded element technique in meso-FE modelling of fibrereinforced compositesrdquo Composite Structures vol 107 pp 436ndash446 2014

[24] L Qi W Tian and J Zhou ldquoNumerical evaluation of effec-tive elastic properties of composites reinforced by spatiallyrandomly distributed short fibers with certain aspect ratiordquoComposite Structures vol 131 pp 843ndash851 2015

[25] R Hill ldquoElastic properties of reinforced solids some theoreticalprinciplesrdquo Journal of theMechanics and Physics of Solids vol 11no 5 pp 357ndash372 1963

[26] T Kanit F NrsquoGuyen S Forest D Jeulin M Reed and SSingleton ldquoApparent and effective physical properties of hetero-geneous materials Representativity of samples of two materialsfrom food industryrdquo Computer Methods Applied Mechanics andEngineering vol 195 no 33-36 pp 3960ndash3982 2006

[27] M C Boyce and E M Arruda ldquoConstitutive models of rubberelasticity a reviewrdquo Rubber Chemistry and Technology vol 73no 3 pp 504ndash523 2000

[28] X F Li and X X Yang ldquoA review of elastic constitutive modelfor rubber materialsrdquo China Elastomerics vol 15 no 1 pp 50ndash58 2005

[29] B Fereidoonnezhad R Naghdabadi and J Arghavani ldquoAhyperelastic constitutive model for fiber-reinforced rubber-likematerialsrdquo International Journal of Engineering Science vol 71pp 36ndash44 2013

[30] X Q Peng Z Y Guo and B Moran ldquoAn anisotropic hypere-lastic constitutive model with fiber-matrix shear interaction forthe human annulus fibrosusrdquo Journal of Applied Mechanics vol73 no 5 pp 815ndash824 2006

[31] E Chebbi M Wali and F Dammak ldquoAn anisotropic hyper-elastic constitutive model for short glass fiber-reinforcedpolyamiderdquo International Journal of Engineering Science vol106 pp 262ndash272 2016

CorrosionInternational Journal of

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Journal ofEngineeringVolume 2018

Applied ChemistryJournal of


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High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in



ChemistryAdvances in


Advances inPhysical Chemistry


BioMed Research InternationalMaterials

Journal of


Na

nom

ate

ria

ls


Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom


al [2] investigated the mechanical behavior of short-fiber-reinforced plastic composites by applying the TandonndashWengmicromechanical model considering the local fiber orien-tation distribution Chen et al [7] determined the effectivemoduli of composites containing misoriented fibers basedon the EshelbyndashMorindashTanaka theory Huang [8] presentedan analytical approach to evaluate the orientation effects onthe elastic properties of a composite containing randomlyoriented fibers Fu et al [9] modeled fiber orientation distri-butions using a laminate analogy approach

It is convenient to use analytical approaches howevercompared to a numerical method the microstructure fieldof deformation in composites cannot be directly obtainedTherefore numerical methods which predict the anisotropyhave been developed for different materials Tian et al [5]employed finite element (FE) methods with solid elementsfor short fibers to predict the effects of fiber orientation onthe effective elastic properties of metal matrix compositesOgierman et al [6] investigated the effect of fiber orientationusing a macroscale FE model with different boundary con-ditions Laspalas et al [2] studied the elasticity and failureof short-fiber-reinforced composites using FE simulationincluding an orientation average Jansson et al [10] employedthe FE method to predict the anisotropic and nonlinearbehavior of plastics reinforced by glass fibers However inFE modeling with multiple inclusions ldquojammingrdquo of fibers[11] easily occurs in which the generated new fibers intersectthe existing fibers Extensive efforts have been devoted todevelopingmethods for generating the representative volumeelements (RVEs) of composites reinforced by inclusions ofdifferent shapes Yi et al [12] proposed modified randomsequential adsorption (RSA) algorithms to generate an RVEof a random chopped fiber-reinforced composite materialwith straight and curved fibers Harper et al [11] presented amethod to generate RVEswith randomdiscontinuous carbonfibers in the plane and no limitation to the volume fraction ofthe fiber Li et al [13 14] presented modified RSA algorithmsfor generating RVEs with complex microstructures suchas fibers and spatially randomly distributed pores basedon the microstructure information of carboncarbon (CC)composites

However most numerical models have only focused oncomposites with a low-aspect-ratio fiber The aspect ratiorefers to the ratio of the fiber length to the diameter Fora high aspect ratio particularly when the fiber radius is ata micron scale and the fiber length is at a millimeter scalesuch as an aramid fiber it is very difficult to establish anumerical FE model using a traditional method particularlywhen the volume fraction of the fibers is high Only a fewpapers have reported on this topic [15 16] In our previousstudy [17] a numerical model was proposed to investigate themechanical properties of an aramid-fiber-reinforced rubbercomposite with a high aspect ratio It was proved that the FEmodel could simulate large deformationmechanical behaviorwell However the applicability of this proposed FE model indescribing the anisotropy of a composite remains unknown

In this study the FE model was further developed topredict the anisotropy of short-fiber-reinforced compositesThis paper is structured as follows In Section 2 the analytical

methods for predicting the elastic modulus of the compositeswith perfectly aligned completely random and probabilityorientation distribution fibers are introduced In Section 3the FE model and the algorithms of the different fiberorientation distributions are presented In Section 4 theboundary conditions for the different mechanical behaviorsare introduced and a means of obtaining the effective elasticconstants is presented In Section 5 the numerical results ofthe elastic constants with a different orientation distributionare compared to those of the MorindashTanaka model and a highcoincidence is shown Finally based on the aforementionedstudy the FE model was applied to composites with anonlinear matrix to predict the fiber orientation distributionThe numerical results were compared to the experimentalresults and found to agree well This study verifies thatthe employed FE model is excellent to predict the elasticproperties of a short-fiber-reinforced composite

2 Analytical Models

21 MorindashTanaka Model In the Eshelby theory of equivalentinclusion by replacing the elasticmoduli of the inclusionwiththose of the matrix an inhomogeneity problem can be equiv-alent to a homogeneity problemMeanwhile an eigenstrain isintroduced into the inclusion domain Based on the Eshelbyequivalent inclusion theory [18] theMorindashTanakamodel [19]considers the mutual effect between the inclusions in thecomposite thereby improving the accuracy of the predictionresults It has been proven that the MorindashTanaka model is areliable and simple method for inclusion problems and hasbeen widely used to predict effective elastic moduli

It is supposed that the fiber axes coincide with theprincipal direction of the composite matrix The compositeis subjected to a uniform stress 1205900 on the far field boundarywhich is regarded as the average composite stress For ahomogeneous material the constitutive relation is 1205900 = 1198711205760where 119871 is the elastic constant tensor of the composite Theaverage strain of the composite is as follows [19]

1205760 = (1 minus 1198621) 120576(0) + 1198621120576

(1) = (119868 + 1198621119860)119871minus10 1205900 (1)

where 120576(0) and 120576(1) are the average strain of the matrix andinclusions respectively 1198710 is the elastic constant tensor ofthe matrix C1 is the volume fraction of the inclusions A isexpressed as follows

119860 = 1198710 + (1198711 minus 1198710) [1198621119868 + (1 minus 1198621) 119878]minus1 (1198710 minus 1198711) (2)

where 119868 is the unit tensor of four orders and S is the Eshelbytensor of four orders which is related to the elastic propertyof the matrix and the shape of the inclusions The analyticalexpressions of S for an ellipsoid inclusion problem can befound in many previous studies [20 21] Thus the effectivemodulus of the composite can be obtained as follows

119871 = 1198710 (119868 + 1198621119860)minus1 (3)

Hence the effective elastic modulus 119871 is an explicit functionthat is easy to obtain




119906119894 = 1205721198951198941199061015840119895 (4)



0 minussin120593 cos120593]]]

(5)


119871 ijkl = 1205721198941015840119894120572119895101584011989512057211989610158401198961205721198971015840119897119871 i1015840j1015840k1015840l1015840 (6)



119871 = int120587120579=0

int2120587120593=0119871 (120579 120593) 120588 (120579 120593) sin 120579119889120579119889120593 (7)


119871 = int120587120579=0119871 (120579) 120588 (120579) sin 120579119889120579

= 119873120579sum119894=0

(119871 (120579119894) + 119871 (120579119894+1))2 120596 (120579119894)(8)

x2

x1

x3

x2

x3

x1

a1

a2

a3

p


x

y

D

D

Short fiber

Matrix




















(

)(

)

2 = 30

2 = 100

2 = 500

240minus4minus2

030

025

020

015

010

005

000


















119894119909=minus1198712







119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)






+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)





Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







119906119894 = 1205721198951198941199061015840119895 (4)



0 minussin120593 cos120593]]]

(5)


119871 ijkl = 1205721198941015840119894120572119895101584011989512057211989610158401198961205721198971015840119897119871 i1015840j1015840k1015840l1015840 (6)



119871 = int120587120579=0

int2120587120593=0119871 (120579 120593) 120588 (120579 120593) sin 120579119889120579119889120593 (7)


119871 = int120587120579=0119871 (120579) 120588 (120579) sin 120579119889120579

= 119873120579sum119894=0

(119871 (120579119894) + 119871 (120579119894+1))2 120596 (120579119894)(8)

x2

x1

x3

x2

x3

x1

a1

a2

a3

p


x

y

D

D

Short fiber

Matrix




















(

)(

)

2 = 30

2 = 100

2 = 500

240minus4minus2

030

025

020

015

010

005

000


















119894119909=minus1198712







119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)






+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)





Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls

















(

)(

)

2 = 30

2 = 100

2 = 500

240minus4minus2

030

025

020

015

010

005

000


















119894119909=minus1198712







119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)






+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)





Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





(

)(

)

2 = 30

2 = 100

2 = 500

240minus4minus2

030

025

020

015

010

005

000


















119894119909=minus1198712







119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)






+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)





Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls










119894119909=minus1198712







119864119909119910 = 120591119909119910120574119909119910 = sum119899119894=1 119891119894119910=1198632arctan (120575119909119863) (13)






+ intV120590119898119886119905119903119894119909119909 119889119881)

= 11198632119905 [[sum119886

(sum119894

119878119865119887119894119860119887 119878119881119874119871119887119894 cos 120579)]

+ [sum119887

(sum119894

119904119898119888119894119868119881119874119871119888119894)]]

(14)





Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







Ex

Ey

Exy

20∘ 40∘ 60∘0∘ 80∘

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy


x

y

20∘ 40∘ 60∘0∘ 80∘

00

01

02

03

04

05

x

y

(b) Vx and Vy


ＰＲ


fi＜Ｌ-ＰＲ

15717 -023803

-030407

-007313

20∘ 40∘ 60∘0∘ 80∘

0

5

10

15

20

25

30

Aver

age s

tress

(MPa

)











0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








0

200

400

600

800

1000

1200

1400

1600E

xE

yE

xy(M

Pa)

(a) 119864x 119864y and 119864xy



000

005

010

015

020

025

030

035

040

x

y

(b) Vx and Vy



500 1000 1500 200002

10

11

1212

13

14







00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

2040


00

0102

030

Nominal strain


00

04

08

12

16

Nom

inal

stre

ss (M

Pa)

00

0102

030

2040

6080

Nominal strain




00

04

08

12

16

0000000000000000000000000

Nom

inal

stre

ss (M

Pa)

000005

010015

020025

030

Nominal strain

20001600

1200800

400Variance 2


00

04

08

12

16

0Nom

inal

stre

ss (M

Pa)

000005

010015

020025

0302000

Nominal strain

16001200

800

Variance 2400

0000

0 05010

015020

0250320

minal strain

16001200

800

VaVV riance 2400







WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





WxWy

105 110 115 120100

025

026

027

028

029

030

Wx

Wy

(Jm

Ｇ3times10

3)


500 1000 1500 20000Variance 2

104

106

108

110

112

114

116

118

120

122




7 Conclusion






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






00

02

04

06

08

10

12

14

16

Nom

inal

stre

ss (M

Pa)

2=1400

005 010 015 020 025 030000Nominal strain

(a) 119897f=15 mm Vf=00049



2=3000

005 010 015 020 025 030000Nominal strain

00

02

04

06

08

10

12

14

16

18

20

Nom

inal

stre

ss (M

Pa)

(b) 119897f=15 mm Vf=00146



Data Availability




Acknowledgments


References



































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






























Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




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