Compo Part b....Transverse and Shear Properties

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Transverse and shear properties of fiber reinforced nanocomposites

Sumit Sharma , Rakesh Chandra

Department of Mechanical Engineering, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, India

a r t i c l e i n f o

Article history:

Received 18 February 2011

Received in revised form 24 August 2011

Accepted 30 November 2011

Available online 10 December 2011

Keywords:

A. Fibers

A. Carbon fiber

B. Elasticity

B. Mechanical properties

C. Finite element analysis (FEA)

a b s t r a c t

This paper compares the storage moduli and loss factors for three different types of short fiber reinforced

nanocomposites. Comparisons have been made for different fiber volume fractions (Vf) as well as for dif-

ferent fiber aspect ratios (l/d). Results show that staggered array of fibers give higher values of storagemodulus as compared to regular array. Aspect ratio has little effect onE022 and G

0

12, except at low aspect

ratios. The storage moduli show an increasing trend with increasing fiber volume fraction.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

Damping is the inherent material property that dissipates en-

ergy during cyclic loading. Finite element modeling (FEM) can be

used for numerical computation of bulk properties based on the

geometry, properties, and volume fraction of constituent phases.

FEM involves discretization of a material representative volume

element (RVE) into elements for which the elastic solution leads

to determination of stress and strain field. The coarseness of the

discretization determines the accuracy of the solution. Nano-scale

RVEs of different geometric shapes can be chosen for simulation of

mechanical properties. Finite element modeling is performed to

predict the response of a structure under prescribed boundary con-

ditions and time independent applied loads, when linear response

behavior can be assumed with reasonable accuracy. The desired re-

sponse quantities are generally, displacements, stresses, strains,

reactions, and energy. The basic equation for linear static analysis

may be written in the form;

Kfug fpg 1

where [K] is the linear stiffness matrix for the structure (known), {u}

is the nodal displacement vector (unknown) and {p} is the load vec-

tor (known).

Most of the work has been carried out for longitudinal storage

moduli and loss factors but little has been stated about transverse

and shear properties of discontinuous fiber reinforced nanocom-

posites. In this study, transverse and shear properties of short fiber

reinforced nanocomposites have been evaluated using Numerically

Integrated Method for System Analysis (NISA) software.

2. Literature review

In the approach, suggested by Mori and Tanaka[1] each inclu-

sion behaves as an isolated inclusion, subject to the averaged stress

fields acting on it from all the other inclusions. The stresses, acting

on an inclusion and caused by the presence of other inclusions, are

superimposed on the applied stress. The idea of Mori and Tanaka

was to combine the Eshelby approach and the effective field con-

cept. This is done by defining the strain concentration tensor,

which relates the strain in the inclusion to the unknown strain in

the matrix instead of the applied strain, as in the case of the dilute

distribution model.

Benveniste [2] expanded the relations suggested by Mori

Tanaka and provided a general method for determining the effec-

tive properties of fiber reinforced composites.

Chandra et al. [3] have provided comprehensive review of

damping involving macro-mechanical, micromechanical and vis-

co-elastic approaches. According to them, the various composite

damping mechanisms are: visco-elastic nature of matrix or fiber

materials, damping due to inter-phase, visco-plastic damping and

thermo-elastic damping. Chandra et al.[4]have predicted damping

coefficients of two-phase continuous fiber reinforced composites.

They studied the effect of shape of fiber cross section and fiber vol-

ume fraction on various damping coefficients using visco-elastic

correspondence principle. They also determined loss factors for

composites reinforced with cylindrical continuous fiber using 2D

micromechanical FEM/strain energy approach. They concluded

that longitudinal loss factor is independent of shape of fiber and

1359-8368/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.compositesb.2011.11.059

Corresponding author.

E-mail address:[email protected](S. Sharma).

Composites: Part B 43 (2012) 477487

Contents lists available atSciVerse ScienceDirect

Composites: Part B

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p o s i t e s b
http://dx.doi.org/10.1016/j.compositesb.2011.11.059mailto:[email protected]://dx.doi.org/10.1016/j.compositesb.2011.11.059http://www.sciencedirect.com/science/journal/13598368http://www.elsevier.com/locate/compositesbhttp://www.elsevier.com/locate/compositesbhttp://www.sciencedirect.com/science/journal/13598368http://dx.doi.org/10.1016/j.compositesb.2011.11.059mailto:[email protected]://dx.doi.org/10.1016/j.compositesb.2011.11.059


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transverse loss factor is slightly affected by aspect ratio of fiber

from low to moderate fiber volume fraction.

Hsueh[5]used Modified shear lag/Cox model to find the longi-

tudinal elastic moduli of unidirectional discontinuous fiber rein-

forced composites. He concluded that longitudinal storage

moduli increases with fiber volume fraction. However, nothing

has been stated about transverse and shear moduli.

Tucker and Liang[6]used HalpinTsai model to predict the lon-gitudinal elastic moduli and damping properties of short fiber rein-

forced composites. Theyconcluded that longitudinalmoduli of short

fiber reinforced composites increases with fiber volume fraction.

Wangetal. [7] studied the effect of fibervolumefractionand fiber

aspect ratio on the elastic properties of fiber reinforced nanocom-

posites in longitudinal direction. They concluded that longitudinal

modulus increases with fiber volume fraction and aspect ratio.

3. Representative volume elements (RVEs)

Several square RVE models for Vapour grown carbon fibers

(VGCF) in a matrix material are studied using the FEM in order

to evaluate material properties of the VGCF-based nanocompos-

ites. One of the arrangement of the fibers within the RVE is shown

in Fig. 1. Another type of arrangement of the fibers, where, two

centrally aligned short fibers are used for modeling on NISA

software, is shown inFig. 2. Similarly, the third type of arrange-

ment of fibers is shown inFig. 3.The models are drawn for differ-

ent fiber volume fractions and fiber aspect ratios. Each time, when

we consider a new fiber volume fraction or a new fiber aspect ratio,

a new model has to be drawn as we cannot use the same model for

different fiber volume fractions and aspect ratios. In this study the

Z-direction represents the fiber direction i.e. (1) direction and X

and Y-directions are transverse directions which refer to (3) and(2), respectively. With these two types of arrangements, modeling

is done on NISA. Different cases are considered in this modeling.

These are listed below:

(i) In the first case, length of the fibers is kept constant and

diameter is varied.

(ii) In the second case, diameter of the fibers is kept constant

and length is varied. Since, length of the fibers is varying

gap between the fibers is also varying.

Number of elements considered is 12,312. Number of elements

in the fiber is 4104 and the remaining are the elements in the ma-

trix material. Hexahedron elements are used for meshing. Strain

energy method is used to predict the properties of the VGCF/pp

nanocomposite. From 0.02 to 0.16 volume fractions, the models

are created on NISA software at different fiber aspect ratios. Results

of strain energy are taken from the NISA output file. The boundary

FibersMatrix

Fig. 1. Arrangement of fibers in RVE of Type 1.

Fibers

Matrix

Fig. 2. Arrangement of fibers in RVE of Type 2. The two fibers are touching the front and rear faces of RVE.

478 S. Sharma, R. Chandra/ Composites: Part B 43 (2012) 477487


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conditions for different types of loading are shown inTable 1. For

the case of transverse loading, the bottom of the RVE is fixed and

the top face is given a small displacement,Uy = 0.001, in theydirec-

tion, which is the direction perpendicular to the fibers.

The boundary conditions for the case of transverse loading are

as shown inFig. 4. In order to determine the storage moduli andthe corresponding loss factors for shear loading, such as, g31 (or

g12) the boundary conditions are as shown in theFig. 5. For finding

g21, the bottom of the RVE is constrained in all directions i.e., all the

nodes in the bottom face of the RVE have all degrees of freedom

equal to zero, and the nodes on the top face are given a small dis-

placement, Uz= 0.001, in the z direction, which is the fiber direction

Fibers

Matrix

Fig. 3. Arrangement of fibers in RVE of Type 3. The two fibers are not touching the front and rear faces of RVE, hence, fibers are shown as hidden.

Table 1

Boundary conditions for different types of loading.

Ux Uy Uz Tx Ty Tz

Transverse loading (22)

Constraint 0 0 0 0

Force/displacement 0.001 0 0 0

Longitudinal shear loading (12) (Bottom face fixed and top face given displacement in z direction)

Constraint 0 0 0 0 0 0

Force/displacement 0 0 0.001 0 0 0

Uy = 0.001

Fig. 4. Boundary conditions for transverse loading (22).

Uz = 0.001

Fig. 5. Boundary conditions for longitudinal shear loading (12).

85 85

200

(b)

10

97 97

200

(a)

2

Fig. 6. Different ways in which fibers can be arranged in the RVE. (All dimensions innm).

S. Sharma, R. Chandra/ Composites: Part B 43 (2012) 477487 479


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or 1 direction. Here, g31= g12, because of transverse isotropy. Fi-

bers in RVE ofFig. 3can be arranged in different ways. Two of these

are as shown inFig. 6. While making the models on NISA, several

arrangements are used.

These are explained in the following sections.

(i) Type 1: This is a staggered arrangement in which the RVE

consists of one full central fiber and the other fibers, onthe top and bottom faces of the RVE are half of the full fiber

i.e., they are semi-circles. On the top face, there are two

semi-circular fibers and similarly on the bottom face, there

are two other semi-circular fibers. This type of RVE is shown

inFig. 1.

(ii) Type 2:In this type of model, there are two central fibers sep-

arated by some inter-fiber spacing as shown inFig. 2.

(iii) Type 3:In this type, there are two central fibers separated by

some inter-fiber spacing as well as there is some distance

between the first fiber and the front face of the RVE and

between the second fiber and the back face of the RVE as

shown inFig. 3.

4. Theoritical formulation

The following theoretical formulation is used for predicting the

properties of Vapour Grown Carbon Fiber/Polypropylene (VGCF/

pp) nanocomposites:

Storage modulusE0 2 W

V e2 2

whereWis the total strain energy; Vis total volume of the RVE; e is

the strain (change in length per unit original length).

Loss factorg gfWfgmWm

W 3

where gfis the loss factor of fiber; gmis loss factor of matrix; Wfis

strain energy in the fiber; Wm is strain energy in the matrix; Wis

the total strain energy.

Loss modulusE00 g E0 4

Thus, the FEM results obtained through NISA software are used

in Eqs.(2)(4)for determining the moduli and corresponding loss

factors.

5. Modeling forE22

Boundary conditions for the case of transverse loading (22), are

shown inTable 1.Fig. 7shows the boundary conditions on NISA

software. Bottom face of the RVE is fixed and the top face is given

a small displacement of 0.001 inY-direction. InFig. 7, red1 colour

shows the fixed position and the yellow colour shows that a small

displacement is given to the top face.

Fig. 8shows the strain energy plot for transverse loading (22),

for a fiber volume fraction of 0.03 and an aspect ratio of 10, for

Type 1 RVE. Here, there are 12,312 elements in the model. Number

of elements in the fiber and matrix, respectively, is 4104 and 8208.

Displaced shape is shown in Fig. 9. Strain energy shared by fiber

and matrix is shown below:

Wf 7:51 108

Wm 9:79 105

Total volume of the RVE; V 1:48 105

Fig. 10 shows the strain energyplot for Type 2 RVE havinga fiber

volumefraction of 0.03 and an aspect ratio of 97, with an inter-fiber

spacing of 6 nm.Here,there are5832elementsin themodeland the

number of nodes is 6403. The RVE consists of 2160 elements in the

fiber and 3672 elements in the matrix. The strain energy shared byfiber and the matrix for transverse loading is shown below:

Wm 6:59 107

Wm 1:37 104

V 7:70 103

Fig. 11 shows another case of transverse loading where themod-

el belongs to the Type 3. Between the front face and the first fiber

there is a spacing of 2 nm. Then, there is a spacing of 2 nm between

the first and the second fibers followed by a further spacing of 2 nm

between the second fiber and the back face of the RVE. The effect of

increase in this inter-fiber spacing on the modulus and the loss fac-

torcan be estimated. RVE consists of 2160 elements in the fiber and

7560 elements in the matrix. The total number of elements in the

model is 9720 and the total number of nodes is 10,447. The strain

energy shared by fiber and the matrix are as shown below:

Wf 6:66 107

Wm 1:37 104


6. Modeling forG31

The boundary conditions for the case of longitudinal shear load-ing (31) are shown inTable 1.Fig. 12shows the boundary condi-

tions on NISA. Left face of the RVE is fixed and the right face is

given a small displacement (0.001) in the z-direction. Here, yellow

coloured dots represent fixed state and red coloured dots represent

a small displacement in the z-direction.

Fig. 13shows the strain energy plot for Type 1 RVE with a fiber

volume fraction of 0.03 and an aspect ratio of 10. Here, the RVE

consists of 12,312 elements and the number of elements in the fi-

ber and matrix, respectively, is 4104 and 8208. Displaced shape is

shown inFig. 14. Strain energy shared by fiber and the matrix are

as shown below:

Wf 4:42 108

Wm 1:65 107

Fig. 7. Boundary conditions for transverse loading (22).

1 For interpretation of color in Figs. 15 and 724, the reader is referred to the webversion of this article.



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Fig. 15shows the strain energy plot, for shear loading (31), for

Type 2 RVE, having an inter-fiber spacing of 6 nm, with fiber vol-

ume fraction of 0.03 and an aspect ratio of 97. The RVE consists

of 5832 elements and 6403 nodes. There are 2160 elements in

the fiber and 3672 elements in the matrix. Strain energy shared

by fiber and matrix for shear loading (31) is shown below:

Wf 5:80 109

Wm 4:92 105

V 7:69 103

Fig. 16 shows another case of transverse shear loading (31),

for Type 3 RVE having Vf= 0.03 and l/d= 97. Between the front

face and the first fiber there is a spacing of 2 nm. Then, there

is a spacing of 2 nm between the first and the second fibers

followed by a further spacing of 2 nm between the second fiber

and the back face of the RVE. The RVE consists of 2160 elements

in the fiber and 7560 elements in the matrix. The total number

of elements in the model is 9720 and the total number of nodes

is 10,447. Strain energy shared by fiber and the matrix is shownbelow:

After

Deformation

Before

Deformation

Fig. 9. Displaced shape for transverse loading for Type 1 RVE withVf= 0.03 and (l/d) = 10.

Fig. 8. Strain energy plot for transverse loading for Type 1 RVE with Vf= 0.03 and (l/d)=10.



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Wf 5:89 109

Wm 4:88 105


7. Comparison of results

Figs. 1724show the comparison of results obtained from FEM,

for three different types of RVEs. These results are explainedbelow:

Fig. 10. Strain energy plot for transverse loading for Type 2 RVE for Vf= 0.03 and (l/d) = 97 with inter-fiber spacing of 6 nm.

Fig. 11. Strain energy plot for transverse loading forVf= 0.03 and fiber aspect ratio (l/d) = 97 with inter-fiber spacing of Type 3.

Fig. 12. Boundary conditions for shear loading (31).



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7.1. Storage moduli

7.1.1. Variation of E022Fig. 17shows the variation ofE022withVfforl/d= 19. The results

again show that the RVE having staggered array of fibers giveshigher modulus as compared to the other two arrangements.

Staggered array of fibers gives higher values of storage modulus

as compared to regular array. The higher values probably occur

because this type of RVE tends to create long chains of nearly

touching fibers with a high degree of overlap.

The percentage increase inE022, fromVf= 0 to Vf= 0.06, for Type

1 RVE is 38.21%. Whereas, from Vf= 0.06 to 0.16, the percentage

increase inE022 is only 18.90%. At Vf= 0.06, the difference between

E022 values for Type 1 and Type 2 RVEs is 1.897%. This difference

increases to 3.17% at Vf = 0.16. Similarly, the difference between

E022 values for Type 1 and Type 3 RVEs at Vf= 0.06 is 5.6% and this

difference increases to 6.133% at Vf= 0.16.

Fig. 18 shows the variation of E022 with l/d for Vf= 0.03. The

results show that the transverse storage modulus decreases rapidlywithl/d tilll/d< 100. Thereafter, the decrease inE022 with increase

in l/d, for l/d> 100, is very slow and become insensitive to l/d at

large aspect ratios. Atl/d= 100, the percentage difference between

theE022values for FEM models of Type 1 and 2, is 0.00358%, while at

l/d= 1000, the difference reduces to 0.0000579%. Thus, it can be

seen that at largel/dratios,E0

22 is insensitive to the variation in l/d. Froml/d= 5 tol/d= 100, the percentage decrease in E022 for Type

1 RVE is 0.069% and from l/d= 100 to l/d= 1000, percentage de-

crease in E022 reduces to 0.0013%. A probable reason for rapid fall

of storage moduli till l/d= 100 is, increasing the fiber length de-

creases the number of fiber ends and thus the amount of matrix

between the fiber ends. Since, the amount of matrix is fixed, it

moves towards the space between the fibers and increases the in-

ter-fiber spacing and thus decreases the values ofE022.

7.1.2. Variation of G012Fig. 19shows the variation ofG012 withVffor l/d= 19. The trend

is similar to that ofFig. 17. Storage modulus (G012) increases 25.53%

for Type 1 RVE, for an increase in fiber volume fraction from Vf= 0

toVf= 0.06. Similarly, for an increase in fiber volume fraction fromVf= 0.06 to Vf= 0.16, G

0

12 increases 35.59%. At Vf = 0.06, the

Fig. 13. Strain energy plot for shear loading (31) for Type 1 RVE for Vf= 0.03 and (l/d) = 10.

Before

Deformation

After

Deformation

Fig. 14. Displaced shape for shear loading (31) for Vf= 0.03 and (l/d) = 10.



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difference betweenG012values for Type 1 and Type 2 RVEs is 3.62%.

This difference increases to 3.75% at Vf= 0.16. Similarly, the differ-

ence betweenG012 values for Type 1 and Type 3 RVEs at Vf= 0.06 is

6.20% and this difference increases to 6.25% at Vf= 0.16.

Fig. 20 shows the variation of G012 with l/d for Vf= 0.03. Here

also,G 012 decreases rapidly with l/dtilll/d< 100 and then becomes

almost constant. For Type 1 RVE, G012 decreases 0.609% for an in-

crease in fiber aspect ratio froml/d= 5 tol/d= 100. Whereas, from

l/d= 100 to l/d= 1000, the decrease in G 012 reduces to 0.001%. Atl/

d= 100, the difference between the G 012 values for FEM models of

Type 1 and 2, is 0.0377%. Whereas, at l/d= 1000, the difference re-

duces to 0.0328%, again showing thatl/dhas little effect on shear

storage modulus. Similarly, the difference between G0

12 values forType 1 and Type 3 RVEs atl/d= 100 is 0.09428% and this difference

decreases to 0.09423% at l/d= 1000. Here also, the staggered

arrangement i.e. Type 1 RVE, gives higher values of storage modu-

lus as compared to regular array i.e. Type 2 and Type 3 RVEs. The

higher values occur due to the fact that the Type 1 RVE creates long

chains of nearly touching fibers with a high degree of overlap. Also,

the values ofG012 are comparatively lower than that forE0

22. This is

due to the fact that in shear loading, the matrix properties are

dominating.

7.2. Loss factors

7.2.1. Variation ofg22

Fig. 21shows the variation ofg22withVfforl/d= 19 which indi-cates inverse relation vis-a-visE022. The percentage decrease in g22,

Fig. 15. Strain energy plot for shear loading (31) for Type 2 RVE havingVf= 0.03 and (l/d) = 97 with inter-fiber spacing of 6 nm.

Fig. 16. Strain energy plot for shear loading (31) for Vf= 0.03 and fiber aspect ratio (l/d) = 97 with inter-fiber spacing of Type 3.



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fromVf= 0 to 0.06, for Type 1 RVE is 0.0149% and fromVf= 0.06 to

0.16, the percentage decrease in g22 increases to 0.0298%. At

Vf= 0.06, the percentage difference between g22 values for Type 1

and Type 2 RVEs is 0.00116%. This difference increases to

0.00186% atVf= 0.16. Similarly, at Vf= 0.06, the percentage differ-ence between g22 values for Type 1 and Type 3 RVEs is 0.00217%.

This difference increases to 0.00353% at Vf= 0.16. Since, the lossfactor is inversely proportional to elastic modulus, hence, the trend

is opposite to that ofFig. 17.

Fig. 22shows the variation ofg22 with l/d for a constant fiber

volume fraction (Vf) of 0.03. The results show that the transverse

loss factor (g22) increases rapidly with l/dtilll/d< 100. Thereafter,

the increase in g22 with increase in l/d, for l/d> 100, is very slow

and become insensitive to l/d at large aspect ratios. At l/d= 100,

the percentage difference between the g22 values for FEM models

of Type 1 and 2, is 1.85 105 while at l/d= 1000, the difference

reduces to 1.83 105%. Similarly, atl/d= 100, the percentage dif-

ference between theg22 values for FEM models of Type 1 and 3 is

7.833 105% while at l/d= 1000, the difference reduces to

7.832 105%. Thus, it can be seen that at large l/d ratios, E022 is

insensitive to the variation inl/d. Froml/d= 5 tol/d= 100, the per-centage decrease inE022for Type 1 RVE is 0.069% and from l/d= 100

Fig. 17. Variation ofE022 with Vffor l/d= 19 for three different types of RVEs.

Fig. 18. Variation ofE022 with l/dfor Vf= 0.03 for three different types of RVEs.

Fig. 19. Variation ofG 012 with Vffor l/d= 19 for three different types of RVEs.

0.5295

0.5300

0.5305

0.5310

0.5315

0.5320

0.5325

0.5330

0.5335

0.5340

Storagemodulus(G12

'),GPa

Type 1

Type 2

Type 3

Fiber aspect ratio (l/d)

0 100 200 300 400 500 600 700 800 900 1000

Fig. 20. Variation ofG 012 with l/d for Vf= 0.03 for three different types of RVEs.



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to l/d = 1000, the percentage decrease in E022 reduces to 0.0013%.

Here also, the trend is opposite to that ofFig. 18as the loss factor

is inversely proportional to elastic modulus.

7.2.2. Variation ofg12Fig. 23shows the variation ofg12withVfforl/d= 19 which indi-

cates inverse relation vis-a-vis G012. Loss factor (g12) decreases

0.10% for Type 1 RVE, for an increase in fiber volume fraction from

Vf= 0 toVf= 0.06. Similarly, for an increase in fiber volume fraction

fromVf= 0.06 toVf= 0.16,g12decreases 0.1099% for Type 1 RVE. At

Vf= 0.06, the difference between g12 values for Type 1 and Type 2

RVEs is 0.00684%. This difference increases to 0.00902% atVf= 0.16.

Similarly, the difference betweeng12 values for Type 1 and Type 3

RVEs at Vf= 0.06 is 0.0138% and this difference increases to0.0139% atVf= 0.16.

Fig. 24shows the variation ofg12withl/dfor Vf= 0.03. For Type

1 RVE, g12 increases 0.006% for an increase in fiber aspect ratio

from l/d= 5 tol/d= 100. Whereas, from l/d= 100 to l/d= 1000, g12increases 0.00018%, for Type 1 RVE. At l/d= 100, the percentage dif-

ference between theg12 values for FEM models of Type 1 and 2, is

6.138 104% while at l/d= 1000, the difference reduces to

5.921 104%. Similarly, atl/d= 100, the percentage difference be-

tween theg12values for FEM models of Type 1 and 3 is 9.5 104%

while at l/d= 1000, the difference reduces to 9.25

10

4

%%. Theresults obtained show that the trend of variation of g12 with l/d

is same as that of variation ofg22 withl/d.

8. Conclusions

Staggered array of fibers gives higher values of storage modulus

as compared to regular array. Transverse storage modulus is

decreasing withl/drapidly, tilll/d= 100. After this limit, the mod-

uli become insensitive to variation inl/d. A probable reason for ra-

pid fall of storage moduli tilll/d= 100 is, increasing the fiber length

decreases the number of fiber ends and thus the amount of matrix

between the fiber ends. Since, the amount of matrix is fixed, it

moves towards the space between the fibers and increases the in-

ter-fiber spacing and thus decreases the values ofE0

22. Same reason-ing can be given for G012.

Fig. 21. Variation ofg22 with Vffor l/d= 19 for three different types of RVEs.

0.05999515

0.05999520

0.05999525

0.05999530

0.05999535

0.05999540

0.05999545

0.05999550

0.05999555

0.05999560

0.05999565

Lossfactor(22

)


Type 1

Type 2

Type 3

0 100 200 300 400 500 600 700 800 900 1000

Fig. 22. Variation ofg22 with l/d for Vf= 0.03 for three different types of RVEs.

0.059860

0.059880

0.059900

0.059920

0.059940

0.059960

0.059980

0.060000

0.060020

Lossfactor

(12

)

Fiber volume fraction (Vf)

Type 1

Type 2

Type 3

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

Fig. 23. Variation ofg12 with Vffor l/d= 19 for three different types of RVEs.

0.0599640

0.0599645

0.0599650

0.0599655

0.0599660

0.0599665

0.0599670

0.0599675

0.0599680

0.0599685

0.0599690

Lossfactor(12

)

Type 1

Type 2

Type 3


0 100 200 300 400 500 600 700 800 900 1000

Fig. 24. Variation ofg12 with l/d for Vf= 0.03 for three different types of RVEs.



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Following conclusions can be made from the above discussion:

(a) Storage moduli:

(i) Staggered array of fibers gives higher values of storage

modulus as compared to regular array.

(ii) All the storage moduli show an increasing trend with

increasing fiber volume fraction.

(iii) Storage moduli, E

0

22 and G

0

12 first decrease with l/d, tillthe fiber aspect ratio, l/d= 100, and thereafter become

constant.

(iv) Aspect ratio has little effect onE022and G0

12, except at low

aspect ratios.

(b) Loss factors:

(i) Loss factors g22 and g12 all decrease with Vf.

(ii) Loss factorsg22and g12all increase withl/d, tilll/d= 100

and then become constant.

(iii) Fibers having low aspect ratios give higher values of

damping as compared to the fibers having large aspect

ratios.

(iv) The highest loss factor is obtained at lowest fiber vol-

ume fractions. This coincides with the fact that the

damping in the composites is matrix dominated.

(v) Decrease in damping with increase in fiber volume frac-

tion is more for the fibers having large aspect ratios as

compared to the fibers having small aspect ratios.

References

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Compo Part b....Transverse and Shear Properties

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