The effect of the strand diameter on the damping characteristicsof fiber reinforced polymer matrix composites:Theoretical and experimental study

Nagasankar P. a,b,n, Balasivanandha Prabu S. a,n, Velmurugan R. c

a Department of Mechanical Engineering, CEG, Anna University, Chennai, Indiab Department of Mechanical Engineering, S.A. Engineering College, Chennai, Indiac Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai, India

a r t i c l e i n f o

Article history:Received 16 February 2014Received in revised form27 June 2014Accepted 8 September 2014Available online 22 September 2014

Keywords:Glass fibersEpoxyPolymer-matrix compositesDampingVibrationInterface damping

a b s t r a c t

The damping property of glass fiber reinforced polymer matrix composites with two different strand/fiber diameters, their different orientations and layups are investigated. It is found that the damping canbe improved at the negligible expense of stiffness, by generating more number of interfaces, i.e.,reducing the fiber diameter from 27.2 mm to 18.3 mm without compromising the dimensions of thecomposite specimen and the volume fraction of the fiber in the specimens. The natural frequencies andloss factors have been evaluated from experimental results, using the impulse technique. The sameproperties have also been evaluated theoretically by performing modal analysis, using Blevins' “Formulasfor Natural Frequency and Mode Shape”, and the three phase damping analysis using Ni and Adams's“the Specific Damping Capacity (SDC) model” and Gu et al. 's “the interfacial adhesion model” in theenergy dissipation relationship. A good agreement exists between the experimental and theoreticalvalues.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, all the weighty metallic structures in certainapplications, like the aerospace and automotive industries, aregetting replaced by light weight polymer matrix composites (PMC)[1]. These materials possess superior properties, such as highspecific strength and high energy dissipation, i.e. a high dampingvalue at the negligible loss of stiffness [2]. The metallic structurespossess inferior damping property. But, the use of a polymermatrix composite in structures improves the damping propertyconsiderably in several ways, and this has been reported in therecent past [3,4]. There are many types of damping/energydissipation like material damping, aerodynamic damping, viscousdamping [5] etc. This paper is confined only to the materialdamping. The energy can be dissipated from the materials bydifferent ways such as: the fiber orientations, visco-elastic beha-vior of the polymer composites, temperatures, damages, inter-faces, and flexible bonding at fiber-matrix interface. The followingseveral reports have explained these possible ways of improvingthe damping properties of polymer matrix composites.

Adams and Bacon [6] have predicted the effect of different fiberorientations and laminate geometry, on the flexural and torsionaldamping and modulus of fiber reinforced composites. Theircriterion was later used by Adams and Maheri [7] together withthe basic plane stress relations, to predict the moduli and thespecific damping capacity of the anisotropic beams with respect tofiber orientation. The results have shown that the longitudinalcomponent (Ψx) of the SDC was the sole contributor in the 01orientation and the major contributor in the 151 orientation offibers in beams, while the shear component (Ψxy) of the SDC wasthe major contributor in the 451 orientations, and the transversecomponent (Ψy) of the SDC was the major contributor in the60–751 orientations and the sole contributor in the 901 orienta-tions of the fibers. They also showed that the damping valuesincreased at a faster rate from 01 to 601, and increased slightlyfrom the 601 to 901 orientation of fibers. The effect of the beamaspect ratio was also considered in their predictions; i.e., the testson different widths of beams showed no significant effect on theSDC or the modulus.

The dynamic properties of hybrid (carbon-glass fiber lami-nated) composites were estimated by Ni et al. [8]. They used anenergy method and a finite element (FE) technique to demonstratethat the addition of a small amount of CFRP to the surface ofthe GFRP composite would improve the flexural modulus. Kishiet al. [9] studied the damping characteristics of fiber-reinforced

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International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2014.09.0030020-7403/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author at: Department of Mechanical Engineering, CEG, AnnaUniversity, Chennai, India. Tel.: þ91 44 22357747; fax: þ91 44 22357744.

E-mail addresses: karupsshankar@yahoo.in (N. P.),sivanandha@annauniv.edu (B.P. S.).

International Journal of Mechanical Sciences 89 (2014) 279–288

interleaved epoxy composites with different arrangements of thereinforcing carbon fiber, but they used several types ofthermoplastic-elastomer films as the interleaving materials. Huiand Ling [10] also investigated the damping behavior of laminatedcomposites with integral visco-elastic layers, under the effect ofthe ply angle of the complaint layers and the location of the visco-elastic layers.

The modal strain energy method and FE technique were usedby Mohan et al. [11] to study the modal parameters (resonancefrequencies and modal loss factors) of the multi-damping layer ofthe anisotropic laminated composite beam under the effect ofvarious temperatures. They found that the modal loss factorincreased with an increase in temperatures, with little reductionin the stiffness and strength. Youssef and Berthelot [12] alsopredicted the damping behavior of the composite under the effectof temperatures and found that the material became highly softand damped when close to the glass transition temperature of thepolymer. Vijayakumar and Sundareswaran [13] have studied thedynamic properties of polymer (epoxy) matrix composite underdifferent temperatures. They have found that the loss factor andthe natural frequency of epoxy/glass fiber composite can beincreased from 10% to 40% at 150 1C when the epoxy is modifiedwith cyanate content. The vibrational behavior of fiber glass/epoxyspecimen is also influenced by the temperature and moisture, i.e.,the natural frequency of the specimen is reduced with the increasein the temperature and moisture content [14].

In this study, improvement of the loss factor by the creation ofa larger number of interface regions is demonstrated; i.e., the fiberdiameter is reduced without change in the size of the laminate.Fibers with diameters of 18.3 μm (at the Standard Deviation (S.D.)of 0.64) and 27.2 μm (at the S.D. of 0.51) are considered for thisstudy. If the fiber diameter decreases, the number of fiber-matrixinterfaces increases, which leads to higher energy dissipationwhen the specimen is excited. The loss factor values have beenfound for two different diameters of the fiber in the two differentstrands with the same fiber-matrix proportions of the laminates.The frequencies obtained from the dynamic test have beencompared with those obtained from the modal analysis performedby using Blevins' [15] “Formulas for Natural Frequency and ModeShape”. Similarly, the loss factors obtained from the dynamic testhave also been compared with those obtained theoretically. Theloss factors obtained from the theoretical approach are in three

phases, in which the loss factors of the first two phases or thesystem (fiber and matrix) have been found by Ni and Adam's [16]“Specific Damping Capacity (SDC) model” and the loss factor of thethird phase or the damping between the interfaces by Gu [17] andGu et al. 's [18], “interfacial adhesion model”.

2. Materials and properties

Low temperature curing epoxy resin, Rotex EP-207S with aspecific gravity of 1.14 at 25 1C, a solvent based high temperaturecuring hardener, Rotex EH-210S, and the accelerator, Tertiaryamine which were supplied by ROTO Polymers, Chennai, havebeen used for the fabrication of the composite. The unidirectionalglass fiber of a density of 2.50 g/cm3

, supplied by SUNTECH Fibers,India, has been taken as reinforcement for the preparation of thecomposite laminate. The test specimens are made up of the uni-directional fiber mat, resin, accelerator and catalyst, using thesimple hand lay-up technique followed by pressing at roomtemperature. The two different test specimens of the dimension300�25�4 mm3 with a stacking of 12 layers for the smaller and4 layers for the larger strand diameter of the fiber category, werethus prepared from the laminate plates for the free vibration test.The test specimens, as mentioned above, have different stackingsequences, such as unidirectional and angle ply with 50% volumefraction of fiber at the S.D. of 1.3 for small diameter fiber and 1.37for large diameter fiber.

3. Experimental work

3.1. Dynamic mechanical analysis (DMA)

Dynamic mechanical analysis (DMA) has been performed in thethree point bending testing mode to check the effect of the visco-elastic properties like the loss factor, storage and loss modulusunder the influence of different temperatures as shown inFigs. 1 and 2. These visco-elastic properties have been recordedfrom the specimens of the pure epoxy and FRP materials ofdimension 35 mm�12.5 mm�3.3 mm using dual cantilever geo-metry. A constant strain amplitude of 1% and a frequency of 1 Hzhave been imposed on to the test specimen. The viscoelasticproperties have been monitored from 20 1C to 160 1C at a heating

30.40°C2737MPa 82.85°C

429.0MPa

92.91°C0.9447

0.2

0.4

0.6

0.8

Tan

Del

ta

0

100

200

300

400

500

0

500

1000

1500

2000

2500

3000

Sto

rage

Mod

ulus

(MP

a)

20 40 60 80 100 120 140 160

Temperature (°C)

Loss

Mod

ulus

(MP

a)

Fig. 1. DMA scan of the pure epoxy specimen showing the effect of storage modulus, loss modulus and tangent delta curves under the influence of various temperatures.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288280

rate of 5 1C/min. The temperatures at the maximum values of thestorage modulus, loss modulus and tangent delta are obtainedfrom their respective visco-elastic properties curves, and the glasstransition temperature (Tg) is also noted for these two types ofspecimens. It is observed that the glass transition temperature (Tg)is increased after the pure epoxy specimen is reinforced with theglass fiber.

From the analysis of the specimens, it is understood that thematrix/FRP remains in its glassy state during the entire experi-ment (impulse technique), performed at room temperature.Hence, in this study, the damping has been mainly improved fromthe dissipation of energy at the interfaces and the differentorientations of the fibers in the thermosetting polymers (Epoxy).

3.2. Impulse technique

Impulse technique, used elsewhere [19] has been performed tofind the vibration characteristics of the specimen, i.e., the values ofthe natural frequencies and loss factors. The schematic diagram ofthe impulse testing is shown in Fig. 3. One end of the laminatedspecimen is rigidly clamped in a firm support; the other end,which is free to vibrate, like a cantilever beam, is properlypositioned with an accelerometer. The input load is given by theinstrumented impacts hammer and the output (response) iscaptured by the accelerometer, and read by the National instru-ments data acquisition card. It is understood that the improperpositioning of the accelerometer and the clamping of the lami-nated specimen would influence the dynamic properties, whichmay deviate from their corresponding theoretical values. Using thehalf-power bandwidth method, the natural frequencies and theloss factors (η) have been determined under different fiberorientations of the first two modes as shown in Fig. 4. Theexpression for the loss factor (η) is given by the followingequation:

η¼ f 1� f 22f n

ð1Þ

where, f 1 and f 2¼bandwidth at the half-power points of resonantpeak for nth mode f n¼natural frequency

4. Theoretical calculation of the damping value for the FRPcomposites

As this study mainly focuses on improving the damping due tothe interfaces, it needs to be executed with a three phase (fiber,

93.44°C1458MPa

98.03°C0.450630.58°C

11793MPa

0.1

0.2

0.3

0.4

Tan

Del

ta

0

500

1000

1500

2000

Loss

Mod

ulus

(MP

a)

0

2000

4000

6000

8000

10000

12000

Sto

rage

Mod

ulus

(MP

a)

20 40 60 80 100 120 140 160

Temperature (°C)

Fig. 2. DMA scan of the FRP specimen showing the effect of storage modulus, loss modulus and tangent delta curves under the influence of various temperatures.

PC

Data Acquisition Card

Impact Hammer

FRP Specimen Accelerometer

Fixed end

Fig. 3. Schematic representation of impulse testing.

Frequency

Amplitude maximum, Amax

f1 f2fn

Amplitude factor =

Amax/√2

Fig. 4. Half-power bandwidth method.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288 281

matrix and interface) damping model. Related to this case, a threephase relation has been given in Eq. (2). It contains the two-phase(fiber and matrix) loss factors and the interfacial loss factor of thecomposite materials, in which the two-phase loss factors aretermed as system loss factor. Hence, the three phase relation hasbeen used here to calculate the loss factor of the composite inwhich the system and the interfacial loss factors are calculatedseparately.

ηc ¼ ηf þηmþηi ¼ ηsþηi ð2Þ

where

ηc¼the loss factor of the composite,ηi¼the loss factor attributable to the interface,ηs¼the loss factor attributable to the system (fiberþmatrix),

ηs ¼ηf Ef Vf þηmEmVm

Ef VFþEmVmð3Þ

where ηf and ηm¼the loss factors attributable to the fiber and thematrix respectively.

Since the system loss factor of the unidirectional compositeprovided in Eq. (3) could not effectively be applied to all theorientations of the fiber in the composites, the specific dampingcapacity (SDC) model has been used here to calculate the sameunder the different orientations and directions of the fiber in thecomposites.

As the loss factors and the modulii of the system are evaluatedonly from the properties of the fiber and matrix, the Sarvanos andChamis model [20] can be effectively used to calculate the aboveproperties (ESðLÞ ; ESðTÞ ; ESðLTÞ ;ηSðLÞ ;ηSðTÞ ;ηSðLTÞ ;νSðLTÞ etc.) of the system(the fiber and matrix) in the longitudinal, transverse and thelongitudinal–transverse directions as given in the followingsection.

Based on the rule of mixture, the longitudinal modulus and lossfactors are calculated as given in the following equations:

ESðLÞ ¼ Vf Ef ðLÞ þVmEm ð4Þ

ηSðLÞ ¼ηf ðLÞEf ðLÞVf þηmðLÞEmVm

Ef ðLÞVf þEmVmð5Þ

From the stored and the dissipated energy equations, thetransverse modulus and loss factors are calculated as given inthe following equations:

ESðTÞ ¼ 1�ffiffiffiffiffiffiVf

q� �Emþ

ffiffiffiffiffiffiVf

pEm

1� ffiffiffiffiffiffiVf

p1�ðEm=Ef ðTÞ Þ

� � ð6Þ

ηSðTÞ ¼ ηf ðTÞffiffiffiffiffiffiVf

q ESðTÞEf ðTÞ

þηmðTÞ 1�ffiffiffiffiffiffiVf

q� �ESðTÞEm

ð7Þ

where the subscripts ‘s’, ‘f ’ and ‘m’ are the system, fiber andmatrix, the subscripts of the subscripts ‘L’, ‘T’ and ‘LT’ are theproperties of the modulii and the loss factors in the longitudinal,transverse, longitudinal–transverse directions and ‘Vf ’ and ‘Vm’ arethe volume fractions of the fiber and matrix.

The in-plane modulus, loss factors and the Poisson's ratio of thesystem in the following equations are also calculated as in the caseof the transverse modulus and damping:

ηSðLTÞ ¼ ηf ðLTÞffiffiffiffiffiffiVf

q GSðLTÞ

Gf ðLTÞ

þηmðLTÞ 1�ffiffiffiffiffiffiVf

q� �GSðLTÞ

Gmð8Þ

GSðLTÞ ¼ 1�ffiffiffiffiffiffiVf

q� �Gmþ

ffiffiffiffiffiffiVf

pGm

1� ffiffiffiffiffiffiVf

p1�ðGm=Gf ðLTÞ Þ

� � ð9Þ

νSðLTÞ ¼ Vf νf ðLTÞ þVmνmðLTÞ ð10Þ

where, ‘νSðLTÞ ’, ‘νf ðLTÞ ’ and ‘νmðLTÞ ’ represent the Poisson's ratio of thesystem, fiber and matrix respectively, in the longitudinal-transverse directions.

It is known that the term SDC (Ψ) defines the ratio between thetotal damping energy and the maximum strain energy per cycle ofvibration. The SDC model adopted here was developed by Ni andAdams [16], who assumed that the stress-independent dampingcoefficients are applicable at low and normal amplitudes, and asymmetric layup of the laminate. These assumptions bring aboutno mid-plane strains under the classical laminate plate theory. Inaddition to neglecting σy and τxy, they also discussed that thetransverse strain, εy in each lamina would be much lesser than thelongitudinal and shear strains, and hence could be neglected.Later, the same SDC model was used with the loss factors by Yimet al. [21] in their theoretical approach. They are as follows:

ηs ¼Ψ s

2π¼ ΔW2πW

¼ΔWxþΔWyþΔWxy

Wbð11Þ

The dissipated energy in X-direction is evaluated as follows:

ΔWx ¼Z l

02Z h=2

0πηSðLÞσxεxdzdx¼ 2πηSðLÞ

Z l

0

Z h=2

0σxεxdzdx: ð12Þ

¼ 2πIn2

Z l

0M2

1dx ηSðLÞ

Z h=2

0m2 m2Cn

11þmnCn

16

� �Cn

11Q11þCn

12Q12þCn

16Q16� �

z2dz

" #

ð13Þwhere In is the normalizing factor and equals h3/12, h is thethickness of the specimen, l is the length of the beam, ηSðLÞ is thelongitudinal loss factor of the system, [Cijn] the normalized flexuralcompliance, [Qij] the stiffness of the lamina, M1¼Px, the bendingmoment under point loading or M1¼1/2 wx2, the bendingmoment under distributed loading, and m¼cosθ; n¼sinθ.

Similarly, the same energies in Y and XY-directions are eval-uated as follows:

ΔWy ¼ 2πIn2

Z l

0M2

1dx ηSðTÞ

Z h=2

0n2 n2Cn

11�mnCn

16

� �"

� Cn

11Q11þCn

12Q12þCn

16Q16� �

z2dz

#ð14Þ

ΔWxy ¼2πIn2

Z l

0M2

1dx ηSðLTÞ

Z h=2

0mn 2mnCn

11� m2�n2� �Cn

16

� �"

� Cn

11Q11þCn

12Q12þCn

16Q16� �

z2dz

#ð15Þ

where ηSðTÞ and ηSðLTÞ are the transverse and longitudinal–transverseshear loss factors of the system.

The bending strain energy of the FRP beam is calculated fromthe following equation

Wb ¼Z l

0M1k1dx¼

Cn

11

In

Z l

0M2

1dx ð16Þ

Stiffness in the lamina co-ordinate system, [Qij] is calculatedfrom the stiffness in the fiber co-ordinate system given in Eq. (18),[Qxy] using the following transformation relations:

Q11 ¼m4Qxxþn4Qyyþ2m2n2 Qxyþ2Qss� �

Q12 ¼m2n2 QxxþQyy�4Qss� �þ m4þn4� �

Qxy

Q16 ¼mn m2Qxx�n2Qyy� m2�n2� �Qxyþ2Qss

� �h ið17Þ

Qxx ¼E2SðLÞ

ESðLÞ �ν2SðLTÞ ESðTÞ

h i Qyy ¼ESðLÞ ESðTÞ

ESðLÞ �ν2SðLTÞ ESðTÞ

h i

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288282

Qxy ¼νSðLTÞESðLÞESðTÞ

ESðLÞ �ν2SðLTÞESðTÞh i Qss ¼ GSðLTÞ ð18Þ

where subscripts L and T refer to the longitudinal and transversedirections of the lay of fiber respectively, and can be taken as x andy directions of the fiber co-ordinate system.

The various properties of the fiber, matrix and the FRP speci-mens, with respect to the direction of the fiber were measuredfrom the mechanical and dynamic tests, and tabulated inTables 1 and 2, where the values of loss factors and Poisson's ratioof the fiber alone were taken from reference [20]. The twounidirectional strain gauges with 120 Ω resistance, both in thevertical and horizontal directions, were placed at the center of thespecimen, which was then held and loaded in the universal testingmachine (Manufacturer: Blue hill, Model: UTE 40T) to measure thePoisson's ratio. Then, the output data were captured through adata acquisition (FIE through an extensometer) system, by using asoftware ‘system 5000’.

The values of the flexural compliance of the laminate [Cijn] andthe stiffness of the lamina [Qij] are substituted in Eqs. (11–16), andthe loss factor values of the system with small and large stranddiameters of the fiber are calculated for all the fiber orientations (0to 901). As the same fiber and matrix category are used for thesmall and large diameters of the fiber in the system dampingmodel, no variation is observed from the calculated values of thesystem loss factor of these diameters. But, after including theinterfacial damping with the system damping, the damping valuesof the laminated plates with small fiber diameters are higher thanthose of the larger ones. It is because a larger number of interfacesthat exist in the small fiber diameter category release a largeramount of the absorbed energies, which results in the increase inthe loss factors.

Since this study focuses mainly on the interfacial damping, byincreasing the number of the interfaces at the reduction of thefiber diameters from 27.2 mm to 18.3 mm, a theoretical prediction isrequired for the same with the contribution of the fiber diameters.Related to this problem, a successfully developed Gu [17] interfacedamping model has been utilized to evaluate the interfacialdamping between the fiber and matrix of a single fiber polymercomposite. This model was carried out based on the unidirectionalcantilever beam vibration using the diameter of the fiber, the

interfacial shear strength between the fiber and matrix, Young'smodulus and the dimensions of this single wired fiber composite(Fig. 5).

Depending on the maximum force that could cause the fiberpullout or debond, the interfacial shearing force equation wasderived from the single fiber element, given in Fig. 6. Themaximum force (Ff¼τ π df dx) that could stretch the fiber in thematrix was substituted in the stored and dissipated energyequations obtained from the cantilever type impulse method.Then, the interfacial loss factor given in Eq. (19) was derived fromthose stored and dissipated energy equations. (Other details andsimplification of those energy equations can be referred from Gu[17] and Gu et al. ‘s [18] work).

ηi ¼50Exbh

5

64π2τdf l5 ð19Þ

where, ηi is the interfacial loss factor, Ex is the longitudinalmodulus of the system (fiber and matrix), b, h and l are the width,depth and length of the single fiber specimen, τ is the interfaceshear strength, df is the fiber diameter.

As the current study mainly concentrates on the effect of thevariation of the two different diameters of the fiber, the diametershave been accurately measured, using the SEM images capturedfrom these two categories of specimens (Figs. 7 and 8) in whichthe sizes are measured around 18.3 and 27.2 μm for the small andlarge fiber diameters. The Young's modulus values of the systemare calculated by the micro mechanical theory, in which themodulus values of the fiber and matrix have been measured fromthe mechanical tests. Gu [17] and Gu et al. 's [18] have alsoobtained the same modulus values from the specimens of theirdamping model. The dimensions of the same single wired com-posite specimen (2.835�0.857�30 mm3) with the measuredinterfacial shear strength (τ¼0.536 MPa) have been adopted fromtheir model. Since the two categories of the specimens used in thecurrent study have the same parameters of breadth, thickness andlength, the same dimensions have been taken for both thosespecimens. As the diameters of fibers used in the present studyare about 5 to 6 times smaller than the glass fiber used in theirmodel, those fiber diameters are multiplied by five and measuredfrom the SEM micrograph. As this experiment has been carried outfor the two different diameters of the fiber, the same equation can

Table 1Material properties of the FRP laminates with the small and large fiber diameters.

Volume fraction for small andbig strand dia. of fiber %

Young's modulus�103 (MPa) Poisson's ratio Shear modulus�103 (MPa)

Smaller dia. of thefiber (d¼17 mm)

Larger dia. of thefiber (d¼27 mm )

Smaller dia. of thefiber (d¼17 mm )

Larger dia. of thefiber (d¼27 mm )

Smaller dia. of thefiber ( d¼17 mm )

Larger dia.of thefiber (d¼27 mm )

50 Ex¼29.92 Ex¼31.29 νxy¼0.29 νxy¼0.285 Gxy¼2.29 Gxy¼2.474Ey¼5.91 Ey¼6.36

Table 2Material and damping properties of the fiber and the matrix for the small and large fiber diameters.

Materials Longi-tudinal modulus (GPa) Transversemodulus (GPa)

Shear modulus(GPa)

Longi-tudinalloss factor %

Transverseloss factor %

Longi-tudinal transverseloss factor %

Poisson'sratio

FiberSmall dia. d¼17 mm 60.54 60.54 25.22 0.1749a 0.1749a 0.09533a 0.2a

Large dia. d¼27 mm 61.07 61.07 25.44 0.1749a 0.1749a 0.09533a 0.2a

Matrix 2.737 2.737 1.015 1.3989 1.3989 1.5788 0.374

a Values were taken from the reference [20]

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288 283

be written separately with diameters df Sand df L as in the followingrelations:

For a small fiber diameter,

ηiS ¼50ExSbh

5

64π2τdf S L5 ð20Þ

For a large fiber diameter,

ηiL ¼50ExLbh

5

64π2τdf L L5 ð21Þ

where, ηiSand ηiLare the interfacial loss factors of the small andlarge fibers respectively, ExS and ExL are the longitudinal modulusof the system (fiber and matrix) with the small and large fibersrespectively, df S and df L are the diameters of the small and largefibers respectively.

Since the interface damping equations provided in Eqs. (19–21)are applicable only for the 01 orientation of the fiber in thelongitudinal direction, the same has been extended to the differentorientations (301, 451, 601 and 901) of the fiber as well. As thelongitudinal modulus values are decreased with the orientationsof the fiber, those modulus values are calculated based on thetransformation relations provided in the Eq. (24) of Section 5. Theinterfacial loss factors with respect to the different orientations ofthe small and large fibers have been calculated using the followingequations:

ηiSxðθ ¼ 0to90Þ¼50ExSðθ ¼ 0to90Þ

bh5

64π2τdf S L5 ð22Þ

ηiLxðθ ¼ 0to90Þ¼50ExLðθ ¼ 0to90Þ

bh5

64π2τdf L L5 ð23Þ

where, ηiLxðθ ¼ 0to90Þand ηiLxðθ ¼ 0to90Þ

are the interfacial loss factors of the

small and large fibers in the longitudinal direction and thedifferent orientations of fibers.

After the calculations of the system and interfacial loss factors(ηs and ηi) of the small and large fiber diameters, the theoreticalloss factors (ηc) are calculated from Eq. (2).

dx

df

fF+ ff

dFF dx

dx

fd dxτπ

Fig. 6. Forces acting on the single fiber element with its cross-section.

Fig. 7. Microscopic (300 SE) image showing dimensions of the small diameter offibers in the small strands.

Fig. 8. Microscopic (300 SE) image showing dimensions of large diameter of fibersin the large strands.

l

b

hdf

Matrix fiber

Fig. 5. Single fiber or wired composite model.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288284

5. Modal analysis

In the modal analysis, the natural frequencies have beeneffectively determined from the input values of the modulus(Ex, Ey, Gxy), Poisson's ratio (νxy), mass per unit area of the plate(γ) by using Blevins' [15] “Formulas for Natural Frequency andMode Shape” (1979). It was originally developed for calculatingthe frequency of 01 oriented FRP/isotropic plate in which themodulus values are to be directly substituted into the Blevinsformula. But, for the other oriented fibers, it is necessary tocalculate the modulus values of the laminate coordinate systembefore using the same in the formula. In order to obtain this, thetransformation relation has been used to convert the propertiesfrom the fiber coordinate system to the laminate coordinatesystem. The transformation relations of Ex, Ey and Gxy for thedifferent orientations of the fiber are shown in the followingequations:

1Ex

¼m2

ELm2�n2νLT� �þn2

ETn2�m2νTL� �þm2n2

GLTð24Þ

1Ey

¼ n2

ELn2�m2νLT� �þm2

ETm2�n2νTL� �þm2n2

GLTð25Þ

1Gxy

¼ 4m2n2

EL1þνLTð Þþ4m2n2

ET1þνTLð Þþ m2�n2

� �2GLT

ð26Þ

νxyEx

¼ νyxEy

¼m2

ELm2νLT �n2� �þn2

ETn2νTL�m2� �þm2n2

GLTð27Þ

where, Ex, Ey and Gxy and EL, ET and GLT are the longitudinal,transverse Young's modulus and the Shear modulus of thelaminate and the fiber co-ordinate system respectively;νxy, νyx¼Poisson's ratio of the laminate co-ordinate system inthe XY directions and νLT ,νTL¼Poisson's ratio of the fiber co-ordinate system in the longitudinal–transverse directions.

νyxEx ¼ νxyEy ð28Þ

Gxy ¼Exy

2 1þνxy� � ð29Þ

In the plate analysis, the following four orthotropic constantsare defined as given below to calculate Eq. (31):

Dx ¼Exh

3

12 1�νxνy� �; Dy ¼

Eyh3

12 1�νxνy� �; Dk ¼

Gxyh3

12; Dxy ¼Dxνyþ2Dk

ð30ÞAs the orthotropic axes align with the plate axes of a uniform

rectangular plate, which spans from x¼0 to x¼a and from y¼0 to

y¼b, an approximate expression (Eq. 31) for the natural frequencies,in hertz, of the plate is given as follows:

f ij ¼π

2γ1=2G41Dx

a4þG4

2Dy

b4þ2H1H2Dxy

a2b2þ4Dk J1J2�H1H2

� �a2b2

" #1=2

;

i¼ 1;2;3; :::; j¼ 1;2;3; :::; ð31Þ

where G, H and J are the dimensionless parameters (they arefunctions of the indices i and j and the boundary conditions on theplate), h¼thickness of the composite plate, n0 ¼mode number, γ isthe mass/unit area of the plate.

Blevins [15] provides the values of the constants G, H and J forthe first two nodes as follows: for mode 1, G1¼0.597, H1¼-0.0870and J1¼0.47 and for mode 2, G1¼1.494, H1¼1.347 and J1¼3.284.

Since the boundary condition is applied only at the left end ofthe cantilever beam and the other ends are free, the aboveconstants are taken as zero in all the modes i.e., G2¼H2¼ J2¼0.For the multiple mode calculations, the dimensionless parametersG, H and J are calculated as per the terms given in the followingequation:

G¼ n0 �1

2; H¼ n

0 �12

� �2

1� 2n0 �ð1=2Þ� �

π

" #;

J ¼ n0 �1

2

� �2

1þ 2n0 �ð1=2Þ� �

π

" #ð32Þ

It is observed that the frequency values calculated from theBlevins formula very well agree with the measured naturalfrequency values.

6. Results and discussions

The first and second mode frequencies obtained by the Blevinsformula and the experimental method, for the small (18.3 mm) andlarge (27.2 mm) diameters of the fiber category, are comparedunder different orientations of fibers as shown in Figs. 9 and 10.It shows a good correlation between the natural frequenciesobtained from both the methods. It is clearly understood fromthe charts given in Figs. 9 and 10, that the composite with thesmall diameter of the fiber exhibits a slightly lesser naturalfrequency values than that of the larger one, under all theorientations and the modes 1 and 2; i.e., both the small and largefiber/strand diameter curves in the theoretical (Blevins [15]equations) and experimental results are almost merged at all thepoints of orientations. The natural frequencies/the stiffness valuesof both the fiber diameters have significantly dropped while

Fig. 9. (a) Experimental and (b) theoretical natural frequencies under different orientations of the small and large diameter of the fiber for the Mode shape-I.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288 285

computing from 01 to 451 fiber orientations, but, are almostconstant, and minimum at 601 and 901 fiber orientations.

Similarly, the experimental and theoretical damping resultsobtained from the small and large diameters of the fiber cate-gories, under different stacking sequences (lay-up) of fibers of thecomposite materials, are shown in Figs. 11 and 12. It is observedfrom these charts that the composite with the small diameter ofthe fiber yields better damping values than those of the larger oneunder all the orientations. The deviations between the theoreticaland experimental values, could be due to various factors such asvariations in the thickness of the laminate and small hairlinecracks, fine impurities, improper bonding, tiny blow holes, etc. atthe interface between the fiber and matrix or at the matrix itself inthe structure. However, these errors do not affect the dynamicbehavior of the material to a great extent.

The damping value can only be achieved at the expense of stiffness.Since a high damping value is desired to be obtained at a smallerexpense of the stiffness value, it is done by generating more number ofinterfaces (i.e. more number of the same can be created by reducingthe strand diameter/diameter of the fiber, without changing thevolume fraction of the fiber and the matrix) under different stackingsequences of the fiber, which leads to a larger energy dissipation. It isevident from Figs. 9–12 that the different strand diameters of the fiberunder different stacking sequences greatly influence its dynamicbehavior. It is also noted that the loss factor and the naturalfrequency/stiffness are high in the unidirectional ply of the fiber at901 and 01 orientations respectively.

The loss factor values (shown in Figs. 11 and 12) of the smallerand larger strand diameters of the fiber have been comparedtheoretically and experimentally for the different orientations ofthe fibers. It is noted that the loss factor values have considerablyincreased when the large diameter of the fiber is replaced by thesmaller one at all fiber orientations. This variation is moresignificant at the 01 orientation of the fibers. In case the compositewith 01 fiber orientation (unidirectional composite) is alone used,the small diameter can be chosen at 01 fiber orientation itself. It isalso noted that the natural frequency/stiffness is maximum at thesame orientation. Despite the fact that the small diameter ispreferred irrespective of the fiber orientations in the case ofunidirectional composite, the better damping values need to beselected at a smaller expense of the stiffness or frequency, whichdepends on the orientations of the fiber in the angle ply compo-site, i.e., better combination of damping and stiffness or frequencyneed to be selected.

As the 01 and 901 orientations of small diameter of fiber givethe maximum natural frequency with minimum loss factor and

the minimum natural frequency with maximum loss factorrespectively, it is not advisable to select these two orientationsof the fiber (shown in the Figs. 9–12). In the other three orienta-tions (301,451 and 601), though, the frequency of 301 orientationseems to be better than that of the 451 and 601 orientations, itsloss factor value needs to be improved further. Even though, theFRP with 601 orientations gives good damping value, its frequency

Fig. 10. (a) Experimental and (b) theoretical natural frequencies under different orientations of the small and large diameter of the fiber for the Mode shape-II.

Fig. 11. Theoretical loss factors with the orientations of the small and largediameters of the fiber.

Fig. 12. Experimental loss factors with the orientations of the small and largediameters of the fiber.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288286

value is minimum. Hence, the orientations of 301 and 601 couldnot be selected. But, the 451 orientation of the smaller fiber yieldsthe required theoretical and experimental loss factor values (i.e.η¼0.0095 and 0.00963) nearing the maximum loss factor thatobtained from the 901 orientation (shown in Figs. 11 and 12). Italso gives moderate frequency/stiffness values at this orientation.Hence, the 451 orientation can be selected as a better choice of theorientations in this small strand diameter of fiber angle plycategory.

Though the fiber-matrix interfaces seem to be perfect in the300 SE microscopic images of the laminate specimen, there aresome micro gaps existing between the fiber-matrix interfaces(thicknesses of interfaces would be two to three microns) whichcan clearly be visible in the next higher SE microscopic image ofthe laminate specimen as shown in Fig. 13. The elastic deforma-tions/slips/dislocations with the friction that occurs between thefiber-matrix interfaces when the specimen is excited, would causea larger energy dissipation. When the interfacial damping modelof the small and large diameter fibers are alone reviewed, moreinterfacial loss factor values are observed from the smaller one(shown in Fig. 14) at all their orientations, due to a greater numberof interfaces which resulted in a larger energy dissipation in thesame category. Hence, it is understood from the interfacial damp-ing method (Eq. 30) that the diameter of the fiber is a majorconcern in increasing the damping of the laminated composite.Even these variations in the fiber diameters ( dfL�dfS¼27 mm�18 mm¼9 mm) cause a major impact on the interface

damping. If the variation between the two different diameters isstill high, a huge damping effect can be expected. It is alsointerestingly noted from Fig. 14 that the interfacial loss factordecreases with an increase in the orientations of the fiber unlikethe increase in the system loss factor with the fiber orientations.This drop is mainly due to the smaller energy dissipation causedby the smaller Young's modulus values obtained from the higherfiber orientations (i.e., the modulus value is maximum at the 01orientation and getting decreased gradually at the remainingorientations), which is one of the major influential parameterspresent in the evaluation of the interfacial damping Eq. (30).

From the study of the damping characteristics of any FRPmaterials, one can observe a significant variation in the theoreticaland experimental loss factor values at 01 orientation of fiber due tothe higher interface damping effect obtained from this orientationexplained above. After including the interface with the system ortwo phase damping (shown in Figs. 15 and 16), a good agreementexists between the 3 phase theoretical and the experimental lossfactor values as shown in Figs. 17 and 18 i.e., the error between thetheoretical and experimental damping is minimized to a greatextent.

On the whole, between the small and the large strand dia-meters of the fiber for the different orientations (01, 301, 451,601and 901) of the angle ply composite, better results have beenevidenced from the category of small strand or the fiber diameterdue to the interface damping effect and, in which their 451 fiberorientation gives the compromised results of the loss factor and

Fig. 13. Measurements of fiber diameter (about 27 mm) and interface thickness(about 3 mm) from Microscopic (500 SE) images.

Fig. 14. Interfacial loss factors values under different orientations of the fiber basedon Gu (1998) model.

Fig. 15. Comparison of experimental and theoretical 2 phase loss factors for thesmall fibers.

Fig. 16. Comparison of experimental and theoretical 2 phase loss factors for thelarge fibers.

N. P. et al. / International Journal of Mechanical Sciences 89 (2014) 279–288 287

the natural frequency. In case, the unidirectional composite isalone used, the small fiber diameter is chosen at its 01 fiberorientation itself. The deviations between the theoretical and theexperimental values have been greatly reduced.

7. Conclusions

In order to investigate the dynamic behavior of GFRP compo-sites, aiming to improve the damping without compromisingmuch on the stiffness/frequency values, two different diametersof the fiber with 01, 301, 451, 601 and 901 orientations in thepolymer matrix were fabricated. After carrying out detailedstudies of the frequency and the damping experimentally andtheoretically, it is concluded that the laminate with the smalldiameter and 451 orientation of the fiber exhibits better results.The diameter of the fiber plays an important role in this study.Even this small reduction in diameter has caused approximatelyan average of 18% increase from the overall damping (3 phasedamping) model. In case the fiber diameter differences arehigh, the percentage increase in the damping values can also beraised. After considering the 3 phase damping, i.e., including the

interface damping, the variation between the theoretical and theexperimental damping values has been significantly reduced. Fromthe 2 phase theoretical and the experimental damping analysis ofthe small and large fibers (Figs. 15 and 16), an average of morethan 20% variation is observed. But, when the 3 phase theoreticaland the experimental damping (Figs. 17 and 18) are compared,only an average of about 1–2% variation is obtained. Similarly, lessthan 5% error is observed between the theoretical (Blevins) andthe experimental calculations of the frequency values.

Acknowledgments

The authors would like to acknowledge the support ofM/s ROTO Polymers Pvt. Ltd. of Chennai, Atalon (DEWETRON-India Branch office) of Sriperumbudur and CIPETs of Chennai andBhubaneswar. Special mentions are also given to Anna UniversityChennai, IIT Madras and S.A. Engineering College Research centreof Chennai for their guidance and support.

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Fig. 17. Comparison of experimental and theoretical 3 phase loss factors for thesmall fibers.

Fig. 18. Comparison of experimental and theoretical 3 phase loss factors for thelarge fibers.

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