Smart damping of laminated fuzzy fiber reinforced composite shells using 1–3 piezoelectric

composites

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2013 Smart Mater. Struct. 22 105001

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Smart Mater. Struct. 22 (2013) 105001 (16pp) doi:10.1088/0964-1726/22/10/105001

Smart damping of laminated fuzzy fiberreinforced composite shells using 1–3piezoelectric composites

S I Kundalwal, R Suresh Kumar and M C Ray

Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

E-mail: mcray@mech.iitkgp.ernet.in

Received 10 May 2013, in final form 1 July 2013Published 29 August 2013Online at stacks.iop.org/SMS/22/105001

AbstractThis paper deals with the investigation of active constrained layer damping (ACLD) of smartlaminated continuous fuzzy fiber reinforced composite (FFRC) shells. The distinctconstructional feature of a novel FFRC is that the uniformly spaced short carbon nanotubes(CNTs) are radially grown on the circumferential surfaces of the continuous carbon fiberreinforcements. The constraining layer of the ACLD treatment is considered to be made ofvertically/obliquely reinforced 1–3 piezoelectric composite materials. A finite element (FE)model is developed for the laminated FFRC shells integrated with the two patches of theACLD treatment to investigate the damping characteristics of the laminated FFRC shells. Theeffect of variation of the orientation angle of the piezoelectric fibers on the dampingcharacteristics of the laminated FFRC shells has been studied when the piezoelectric fibers arecoplanar with either of the two mutually orthogonal vertical planes of the piezoelectriccomposite layer. It is revealed that radial growth of CNTs on the circumferential surfaces ofthe carbon fibers enhances the attenuation of the amplitude of vibrations and the naturalfrequencies of the laminated FFRC shells over those of laminated base composite shellswithout CNTs.

(Some figures may appear in colour only in the online journal)

Nomenclature

a length of the substrate shell (m)Ck

ij elastic coefficients of the kth layer (GPa)

[Ckb], [C

ks ] elastic coefficient matrices for the kth layer

(GPa)D diameter of the composite fuzzy fiber (µm)Dx,Dy,Dz electric displacements in the x, y and z

directions, respectively (C m−2)

[Dtb], [Dtrb],

[Drrb], [Dts],

[Dtrs], [Drrs]

rigidity matrices for the kth layer

d diameter of the carbon fiber (µm){dt}, {dr} generalized translational and rotational

displacements{de

t }, {der } nodal generalized translational and rota-

tional displacements

E Young’s modulus of the viscoelastic mate-rial (MPa)

ET transverse Young’s modulus of the kth layer(MPa)

Ex,Ey,Ez applied electric field components in the x, yand z directions, respectively (C m−1 orV m−1)

{eb}, {es} piezoelectric stress coefficient matrices(C m−2)

eij transformed effective piezoelectric coeffi-cients of the PZC layer (C m−2)

{F} global nodal load vector (N){Ftp}, {Frp} global electro-elastic coupling vectors

(C m−1)

{Fe} nodal load vector (N)

{Fetp}, {F

erp} elemental electro-elastic coupling vectors

(C m−1)

10964-1726/13/105001+16$33.00 c© 2013 IOP Publishing Ltd Printed in the UK & the USA

Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

G complex shear modulus of the viscoelasticmaterial (MPa)

h1, h2 z-coordinates of the top surfaces of theviscoelastic and PZC layers (m)

h, hv, hp thicknesses of the substrate shell, theviscoelastic layer and the PZC layer,respectively (m)

Kd control gain[Ktt], [Ktr],

[Krt], [Krr]

global stiffness matrices (N m−1)

[Kett], [K

etr],

[Kert], [K

err]

elemental stiffness matrices (N m−1)

L half length of the RVE of the FFRC (µm)[M], [Me

] global and elemental mass matrices (kg)N total number of layers in the substrate shell[Nt], [Nr] shape function matricesni shape function of natural coordinates

associated with the ith nodeq number of patches of the ACLD treatmentR average radius of the substrate shell (m)u, v,w displacements along the x, y and z direc-

tions, respectively (m)u0, v0,w0 displacements of a point on the reference

mid-plane along the x, y and z directions,respectively (m)

(VCNT)max maximum CNT volume fraction in theFFRC

vf carbon fiber volume fraction in the FFRCx, y, z Cartesian coordinates{X}, {Xr} global nodal translational and rotational

displacement vectors (m)γx, γy, γz generalized rotations of the normal to the

middle plane of the PZC layer (rad){εb}, {εs} in-plane and transverse strain vectorsεx, εy, εz normal strains along the x, y and z

directions, respectivelyε33 dielectric constant in the 3-direction when

electric field is applied in the 3-direction(F m−1)

εxy, εyz, εxz shearing strainsη, ν loss factor and Poisson’s ratio of the

viscoelastic materialθx, θy, θz generalized rotations of the normal to the

middle plane of the substrate shell (rad)ρ mass density of the kth layer (g cm−3)

σx, σy, σz normal stresses along the x, y and zdirections, respectively

σxy, σyz, σxz shearing stressesφx, φy, φz generalized rotations of the normal to the

middle plane of the viscoelastic layer (rad)ψ piezoelectric fiber orientation in the vertical

plane of the PZC layer with respect to the zaxis (rad)

ω natural frequency (Hz)ω non-dimensional frequency parameterω1 non-dimensional frequency parameter

1. Introduction

Piezoelectric composites are now being extensively used asdistributed actuators and sensors for smart structures [1–6].Lin and Sodano [7] characterized a novel active piezoelectricstructural fiber that can be reinforced in composites to performsensing and actuation, in addition to providing additional loadbearing functionality. Piezoelectric composites are usuallycomposed of an epoxy matrix reinforced with monolithicpiezoelectric fibers providing a wide range of effectivematerial properties that are not offered by existing monolithicpiezoelectric materials. One of the commercially availablepiezoelectric composites is a lamina of vertically reinforced1–3 piezoelectric composite (PZC).1 In this 1–3 PZC lamina,the poling direction of the piezoelectric fibers is along thethickness of the lamina while the top and the bottom surfacesof the lamina are electroded. Micromechanical analyses[5, 8] reveal that the magnitude of the effective out-of-planepiezoelectric coefficient (e33) of this PZC is much larger thanthat of its effective in-plane piezoelectric coefficient (e31).Hence, the in-plane actuation of this PZC is negligible ascompared to its out-of-plane actuation. Ray and Pradhan [5, 9,10] extensively investigated the performance of the verticallyreinforced 1–3 PZC material as the constraining layer ofthe active constrained layer damping (ACLD) treatment foractive damping of smart composite structures. Saviz andMohammadpourfard [11] presented dynamic analysis for asimply supported laminated cylindrical shell integrated withthe piezoelectric layers subjected to local ring/pinch loads.In their study, attachment of the piezoelectric layers to thelaminated composite shell caused a reduction in the values ofthe natural frequencies.

Significant progress has been made in the area ofnanoscale science and technology in the past two decades. Asone of the most interesting nanomaterials, carbon nanotubes(CNTs) have received significant attention in terms oftheir potential applications. The discovery of CNTs [12]has created enormous interest among researchers to predicttheir exceptional elastic properties. It has been found thatCNTs have an axial Young’s modulus in the terapascalrange [13–16]. In order to exploit the excellent elasticproperties of CNTs, enormous effort is being exertedby researchers to develop CNT-reinforced polymer matrixcomposites [17, 18]. The high stiffness-to-weight ratiosestimated for CNT-reinforced composites may provide manyopportunities for aerospace structures. O’Donnell et al [19]investigated the potential applications of CNT-reinforcedcomposites for the airframes of commercial aircraft. Intheir study, it was reported that the structural mass ofa CNT-reinforced airframe is decreased by approximately14% when compared with that of an aluminum airframe,which eventually decreases the fuel consumption by anaverage of 9.8% and increases the flight range by anaverage of 13.2%. The stiffening effect of CNTs has beeninvestigated by Wuite and Adali [20] for use in structural

1 Piezocomposites, Materials Systems Inc., 543 Great Road, Littleton, MA01560, USA.

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

applications as CNT-reinforced composite beams. In theirstudy, aligned CNT-reinforced composite beams showedsignificant improvement in beam stiffness over traditionalcarbon fiber reinforced composite beam stiffness, and itwas also reported that the stacking sequence and the CNTdiameter are important parameters for reducing the maximumdeflection of the CNT-reinforced composite beam. Ray andBatra [21] studied the performance of CNT-reinforced 1–3PZC as a constrained layer in the ACLD treatment of alaminated composite beam by employing the finite element(FE) method. They found that the attenuating capability ofthe CNT-reinforced 1–3 PZC layer is significantly improvedover that of a PZC layer without CNTs. Zhu et al [22] carriedout bending and free vibration analyses of CNT-reinforcedcomposite plates by employing the FE model based on thefirst order shear deformation theory (FSDT). Their studyindicates that the CNT volume fraction, the CNT aspectratio and the boundary conditions of the plate significantlyinfluence the bending deflection of the CNT-reinforcedcomposite plate. Rokni et al [23] fabricated laminatedmulti-walled CNT-reinforced composite beams and measuredtheir effective stiffnesses, fundamental natural frequenciesand damping characteristics through static deflection and freevibration analyses. Their study reveals that the beam withthe optimum distribution of multi-walled carbon nanotubes(MWCNTs) in the polymer matrix achieves significantimprovement in the effective Young’s modulus and thedamped natural frequencies over those of the beam withuniformly distributed MWCNTs.

A review of the literature on two-phase CNT-reinforcedcomposites reveals that the exceptionally attractive propertiesof CNTs can be exploited to develop CNT-reinforcedcomposite structures where the CNTs are used as primaryreinforcements. For structural applications, the manufactureof two-phase unidirectional continuous CNT-reinforcedcomposites in large scale encounters some challengingdifficulties. Typical among these are agglomeration of CNTs,misalignment and the difficulty in manufacturing longCNTs [24]. In the case of three-phase hybrid CNT-reinforcedcomposite, the CNTs are grown on the circumferentialsurfaces of the advanced fiber reinforcements such as carbonfibers and alumina fibers. It seems that in comparison to themanufacture of long CNTs and the dispersion of long CNTsin the polymer matrix, direct growth of short CNTs on thecircumferential surfaces of the advanced fibers to achieveuniform distribution of CNTs throughout the composite ispractically more feasible and advantageous, and providesa means to tailor the multifunctional properties of existingadvanced fiber reinforced composites. Downs and Baker [25]reported initial research work on the growth of carbonnanofibers on the circumferential surfaces of commercialcarbon fibers. They modified the circumferential surface areaof the carbon fibers by coating with carbon nanofibers andfound significant enhancement in the adhesion properties atthe fiber/matrix interface. Bower et al [26] grew radiallyaligned CNTs on the circumferential surface of hair-thinoptical fiber glass using a high-frequency microwave plasmaenhanced chemical vapor deposition technique. They found

that the growth rate of CNTs is approximately 100 nm s−1

with total growth of 12 µm, and such CNT growth alwaysoccurs perpendicular to the substrate surface regardless ofthe substrate shape. The electrical self-bias imposed on thesubstrate surface was the primary mechanism responsible forthe alignment of the CNTs. Thostenson et al [27] synthesizedCNTs on the circumferential surfaces of carbon fibers usinga chemical vapor deposition (CVD) process and found thatthe presence of CNTs at the fiber/matrix interface improvesthe interfacial shear strength of the composites. In their work,the surfaces of the fibers after each step in the synthesizationprocess of growing CNTs were examined with scanningelectron microscopy and it was found that no pitting occurredon the fiber surfaces. De Riccardis et al [28] synthesizeda hybrid composite in which the carbon fiber was coatedwith a uniform distribution of CNTs on its circumferentialsurface. In their study, pseudo-mechanical tests performedon different samples (with and without CNT coated carbonfibers) revealed a good anchorage between the CNTs andthe carbon fiber. Veedu et al [29] fabricated a 3D compositein which MWCNTs were grown normal to the surfaceof micro-fiber fabric cloth layouts. The measured fracturetoughness for crack initiation, interlaminar shear slidingfracture toughness, in-plane strength, modulus, toughness,transverse thermal conductivity and damping characteristicsof this 3D composite exhibited 348%, 54%, 140%, 5%, 424%,51% and 514% enhancements, respectively, over those ofthe base composite. Their test results also revealed that thepresence of CNTs in the transverse (i.e., thickness) directionof this 3D composite reduces the effective thermal expansioncoefficient of the composite by 62% as compared to that of thebase composite. Kepple et al [30] reported that the growth ofCNTs on woven carbon fiber improved the fracture toughnessof the cured composite by of the order of 50%, while noloss in the structural stiffness of the final composite structurewas observed by them. Mathur et al [31] experimentallydemonstrated that the growth of CNTs on the circumferentialsurfaces of carbon fibers causes excellent improvement inthe mechanical properties of the carbon fiber reinforcedcomposite. They reported that the flexural strength isimproved by 20% for unidirectional, 75% for 2D and 66% for3D hybrid composites as compared to those of the compositewith neat reinforcements. Zhang et al [32] synthesizedMWCNTs on unsized carbon fibers and concluded that thegrowth of CNTs does not degrade the tensile properties of thecarbon fiber. Kulkarni et al [33] investigated a nanoreinforcedlaminated composite in which CNTs had been grown onthe circumferential surfaces of the carbon fibers by using aCVD process and nickel catalyst particles. In their study, thetransverse Young’s modulus predicted by the numerical modelwas about 1.4 times the experimental value. They attributedthis error to the assumptions adopted in the multiscale modeland the improper bonding of CNTs to the circumferentialsurfaces of the carbon fibers during the manufacture of thenanoreinforced laminated composite. Yamamoto et al [34]characterized the multifunctional properties of fuzzy fiberreinforced plastics in which dense and aligned CNTs weregrown on the circumferential surfaces of ceramic fibers

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 1. Schematic diagram of a lamina made of the FFRC.(Reprinted with permission from [38, p 151].)

by using a thermal CVD technique. They found thatthe small volume fraction of CNTs (1–3%) enhanced themultifunctional properties of the hybrid composite suchas interlaminar strength, mode I fracture toughness andtransverse thermal properties by 70%, 60% and 80%,respectively, compared to the composite without CNTs. Liuand Rao [35] developed a fuzzy finite element analysistechnique to model the uncertainties of the fuzzy compositematerial system within certain ranges. Sager et al [36]synthesized high density MWCNTs on T650 PAN-basedcarbon fiber. Four types of carbon fiber were studied: unsizedfiber, sized fiber, unsized with aligned MWCNTs and unsizedwith random MWCNTs. The interfacial shear strength ofCNT coated carbon fibers in epoxy was investigated using thesingle-fiber composite fragmentation test. The results of thefragmentation tests indicated that commercially sized fibershave the highest interfacial shear strengths while unsizedfibers have the lowest. Randomly oriented MWCNT andaligned MWCNT coated unsized fibers attained 71% and 11%increase in the interfacial shear strength over unsized fibers,respectively. Sager et al [36] attributed this enhancement inthe adhesion properties of the matrix to the fiber as due tothe presence of CNTs. Song et al [37] grafted radially alignedstraight CNTs on the circumferential surface of carbon fiberreinforcement and found that the radially grown straightCNTs significantly improved the out-of-plane and in-planecompressive strengths, and the interlaminar shear strength ofthe carbon fiber reinforced composite.

The literature reviewed on three-phase hybrid CNT-reinforced composites authenticates that the growth ofCNTs on advanced fibers results in enhancement of themultifunctional properties of the existing advanced fiberreinforced composites. Thus the current status of progress inresearch on CNT-reinforced composites brings to light thatthree-phase hybrid CNT-reinforced composite is a promisingcandidate material for achieving structural benefits from theexceptionally attractive properties of CNTs. The advancedfiber augmented with CNTs which are radially grown on itscircumferential surface is also called ‘fuzzy fiber’ [34] andthe resulting composite with such fuzzy fiber reinforcementsmay be called fuzzy fiber reinforced composite (FFRC) [34,38, 39]. Most recently, a novel FFRC reinforced with fuzzyfiber reinforcements was independently studied by Kundalwal

Figure 2. Fuzzy fiber coated with short CNTs radially grown on itscircumferential surface. (Reprinted with permission from[38, p 151].)

and Ray [38] and Chatzigeorgiou et al [39]. The distinctconstructional feature of this FFRC is that the uniformlyaligned CNTs are radially grown on the circumferentialsurfaces of the unidirectional carbon fiber reinforcements.Kundalwal and Ray [38] predicted that the transverse effectiveproperties of this novel FFRC will be significantly enhancedover their values without CNTs. Recently, the effectivethermomechanical properties of continuous FFRC and theload transfer of short FFRC have been extensively studiedby Kundalwal [40] using several analytical and numericalmicromechanics models. This study authenticates that thenovel FFRC with improved multifunctional properties canbe used for fabricating different FFRC laminated structureshaving a wide range of unexplored applications in numeroustechnological areas such as aerospace, automobile, medicine,defence, energy, etc. The laminated cylindrical FFRC shellis an important structural element and forms many criticalstructures in technological applications. To this end, however,use of the ACLD treatment to achieve active control of thelaminated FFRC shells has not yet been reported. For thisinvestigation, three-dimensional FE analysis of the laminatedFFRC shells integrated with the two patches of the ACLDtreatment has been carried out in this study. The effect ofvariation of the orientation angle of the piezoelectric fibers inthe two mutually orthogonal vertical planes of the PZC layeron the damping characteristics of the laminated FFRC shellsis also investigated.

2. Architecture of the novel fuzzy fiber reinforcedcomposite

Figure 1 shows a schematic sketch of a lamina of thecontinuous FFRC. The constructional feature of this novelFFRC is that short CNTs of equal length are uniformlyspaced and radially grown on the circumferential surfaces ofthe continuous carbon fiber reinforcements. The fuzzy fibercoated with radially aligned short CNTs on its circumferentialsurface is shown in figure 2. When this fuzzy fiber isembedded into the polymer material, the gaps between theCNTs are filled with the polymer material. Consequently,the radially aligned CNTs reinforce the polymer matrixsurrounding the carbon fiber along the direction transverseto the length of the carbon fiber. Thus the augmented fuzzyfiber can be viewed as a circular cylindrical composite

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 3. Transverse and longitudinal cross sections of the composite fuzzy fiber. (Reprinted with permission from [38, p 152].)

Figure 4. Schematic representation of the laminated FFRC shellintegrated with the two ACLD patches composed of thevertically/obliquely reinforced 1–3 PZC layer.

fuzzy fiber in which the carbon fiber is embedded in theCNT-reinforced polymer matrix nanocomposite and the radiusof the composite fuzzy fiber equals the sum of the radius ofthe carbon fiber and the length of a CNT. Cross sections ofthis composite fuzzy fiber are schematically demonstrated infigure 3. The novel FFRC being studied here is made of ahexagonal packing array of such composite fuzzy fibers andthe polymer matrix.

3. Theoretical formulation

In this section, an FE model is developed to analyze theACLD of laminated FFRC shells comprised of N laminae.Such a laminated FFRC shell integrated with the two patchesof the ACLD treatment on its top surface is illustrated infigure 4. The constructional feature of the patches of theACLD treatment can be such that the piezoelectric fibersare coplanar with either the xz plane or the yz plane of thePZC layer, as shown in figures 5(a) and (b), respectively. Theorientation angle of the piezoelectric fibers in the PZC layerwith respect to the z axis is denoted by ψ . The thickness ofthe PZC layer is hp and that of the viscoelastic layer of theACLD treatment is hv. The origin of the curvilinear laminatecoordinate system (xyz) is located on the mid-plane of thesubstrate FFRC shell such that the lines x = 0 and a representthe boundaries of the FFRC shells studied here. The thickness

Figure 5. Schematic diagram of a lamina of the verticallyreinforced 1–3 PZC in which the piezoelectric fibers are coplanarwith either (a) the vertical xz or (b) the vertical yz plane.

coordinates (z) of the top and the bottom surfaces of any(kth) layer of the overall shell are represented by hk+1 andhk (k = 1, 2, 3, . . . ,N + 2), respectively.

3.1. Displacement fields

Figure 6 illustrates the kinematics of the axial deformationsof the overall shell based on the FSDT. The length andthe average radius of the shell are denoted by a and R,respectively. As shown in this figure, u0 and v0 are thegeneralized translational displacements of a reference point

5

Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 6. Deformation of any transverse cross section of thelaminated FFRC shell integrated with the ACLD treatment which isparallel to (a) the xz plane and (b) the yz plane.

(x, y) on the mid-plane (z = 0) of the substrate compositeshell along the x- and y-axes, respectively; θx, φx and γx arethe generalized rotations of the normals to the middle planesof the substrate composite shell, the viscoelastic layer and thePZC layer, respectively, in the xz plane while θy, φy and γy

represent their generalized rotations in the yz plane. Followingthe FSDT, the axial displacements u and v at any point in theoverall shell integrated with the patches of ACLD treatmentcan be written as

u(x, y, z, t) = u0(x, y, t)+ (z− 〈z− h/2〉)θx(x, y, t)

+ (〈z− h/2〉 − 〈z− hN+2〉)φx(x, y, t)

+ (〈z− hN+2〉)γx(x, y, t) (1)

v(x, y, z, t) = v0(x, y, t)+ (z− 〈z− h/2〉)θy(x, y, t)

+ (〈z− h/2〉 − 〈z− hN+2〉)φy(x, y, t)

+ (〈z− hN+2〉)γy(x, y, t) (2)

in which u0 and v0 are the generalized displacements atany point on the reference mid-plane (z = 0), and thefunction within the bracket 〈〉 represents the appropriatesingularity functions. Since the transverse normal strainis usually considered as negligible for thin structures, theradial displacement (w) is assumed to be linearly varyingacross the thickness of the substrate composite shell, theviscoelastic layer and the PZC layer. Consequently, thetransverse displacement at any point in the overall shell canbe assumed as

w(x, y, z, t) = w0(x, y, t)+ (z− 〈z− h/2〉)θz(x, y, t)

+ (〈z− h/2〉 − 〈z− hN+2〉)φz(x, y, t)

+ (〈z− hN+2〉)γz(x, y, t) (3)

in which w0 equals the transverse displacement of a pointon the reference plane; θz, φz and γz are the generalizeddisplacements representing the gradients of the transversedisplacement in the substrate composite shell, the viscoelasticlayer and the PZC layer, respectively, with respect to thethickness coordinate (z).

For convenience, the generalized displacement variablesare separated into generalized translational {dt} and rotational{dr} displacements as follows:

{dt} = [u0 v0 w0]T and

{dr} = [θx θy θz φx φy φz γx γy γz]T.

(4)

3.2. Strain–displacement relations

The state of strain at any point in the overall shell continuumis grouped into the following two strain vectors:

{εb} = [εx εy εz εxy]T and {εs} = [εxz εyz]

T (5)

where εx, εy, εz are the normal strains along the x, y andz directions, respectively; εxy is the in-plane shear strain;εxz and εyz are the transverse shear strains. Using thelinear strain–displacement relations, the displacement fieldequations (1)–(3) and (5), the vectors {εb}c, {εb}v and {εb}pdefining the state of in-plane and transverse normal strainsat any point in the substrate composite shell, the viscoelasticlayer and the PZC layer, respectively, can be expressed as

{εb}c = {εbt} + [Z1]{εbr},

{εb}v = {εbt} + [Z2]{εbr} and

{εb}p = {εbt} + [Z3]{εbr}

(6)

whereas the vectors {εs}c, {εs}v and {εs}p defining the state oftransverse shear strains at any point in the substrate compositeshell, the viscoelastic layer and the PZC layer, respectively,can be expressed as

{εs}c = {εst} + [Z4]{εsr},

{εs}v = {εst} + [Z5]{εsr} and

{εs}p = {εst} + [Z6]{εsr}.

(7)

The various matrices appearing in equations (6) and (7) havebeen defined in the appendix while the generalized strainvectors are given by

{εbt} =

[∂u0

∂x

∂v0

∂y+

w

R

∂u0

∂y+∂v0

∂x0]T

{εst} =

[∂w0

∂x

∂w0

∂y−

v0

R

]T

{εbr} =

[∂θx

∂x

∂θy

∂y

∂θx

∂y+∂θy

∂xθz∂φx

∂x

∂φy

∂y

∂φx

∂y

+∂φy

∂xφz∂γx

∂x

∂γy

∂y

∂γx

∂y+∂γy

∂xγz

]T

and

{εsr} =

[θx θy φx φy γx γy

∂θz

∂x

∂θz

∂y

∂φz

∂x

∂φz

∂y

×∂γz

∂x

∂γz

∂y

]T

.

(8)

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

3.3. Constitutive relations

In coherence with the strain vectors, the states of the stressesat any point in the overall shell continuum are described bythe state of the in-plane and out-of-plane stresses {σb}, andthe state of the transverse shear stresses {σs} as follows:

{σb} =[σx σy σz σxy

]T and {σs} = [σxz σyz]T

(9)

where σx, σy and σz are the normal stresses along the x, yand z directions, respectively; σxy is the in-plane shear stress;σxz and σyz are the transverse shear stresses. The constitutiverelations for the material of any orthotropic layer of thesubstrate composite shell are given by

{σ kb } = [C

kb]{ε

kb} and {σ k

s } =

[Ck

s

]{εk

s }

(k = 1, 2, 3, . . . ,N)(10)

where

[Ckb] =

Ck

11 Ck12 Ck

13 Ck16

Ck12 Ck

22 Ck23 Ck

26

Ck13 Ck

23 Ck33 Ck

36

Ck16 Ck

26 Ck36 Ck

66

, [Cks ] =

[Ck

55 Ck45

Ck45 Ck

44

]

in which Ckij are the transformed elastic coefficients with

respect to the reference coordinate system appropriate for thethree-dimensional analysis.

The constraining PZC layer will be subjected to theapplied electric field acting along its thickness direction only.Thus the constitutive relations for the material of the PZClayer can be expressed as [5]

{σ kb } = [C

kb]{ε

kb} + [C

kbs]{ε

ks } − {eb}Ez,

{σ ks } =

[Ck

bs

]T{εk

b} + [Cks ]{ε

ks } − {es}Ez and

Dz = [eb]T{εk

b} + [es]T{εk

s } + ε33Ez; k = N + 2

(11)

in which Ez and Dz are the electric field and the electricdisplacement along the z-direction, respectively; ε33 isthe dielectric constant. It may be noted from the aboveconstitutive relations that the transverse shear strains arecoupled with the in-plane strains due to the orientation ofthe piezoelectric fibers in the vertical xz or yz plane of thePZC layer, and the corresponding coupling elastic coefficientsmatrices [CN+2

bs ] are given by

[CN+2bs ] =

[CN+2

15 CN+225 0 CN+2

35

0 0 CN+246 0

]T

or

[CN+2bs ] =

[0 0 CN+2

56 0

CN+214 CN+2

24 0 CN+234

]T (12)

according to whether the piezoelectric fibers are coplanarwith the vertical xz or yz plane, respectively, in the PZClayer. Also, the piezoelectric coefficient matrices {eb} and{es} appearing in equation (11) are referred to the reference

coordinate system (xyz) and are given by

{eb} = [e31 e32 e36 e33]T and

{es} = [e35 e34]T .(13)

The present analysis is concerned with the frequencyresponse of the shells integrated with the patches of the ACLDtreatment. Hence, the viscoelastic material is modeled byusing the complex modulus approach. The material of theviscoelastic layer is assumed to be linearly viscoelastic andisotropic, and the shear modulus (G) and Young’s modulus(E) of the viscoelastic material are given by [41]

G = G′(1+ iη) and E = 2G(1+ υ) (14)

where G′ is the storage modulus, υ is the Poisson’s ratioand η is the loss factor at a particular operating temperatureand frequency. Using equation (14), the elastic coefficients ofthe viscoelastic material can be determined and the resultingelastic coefficient matrix [CN+1

ij ] turns out to be a complexmatrix [41–43].

3.4. Finite element (FE) modeling

The principle of virtual work is employed to derive thegoverning equations of the overall shell/ACLD system andcan be expressed as [43]

N+2∑k=1

∫�

(δ{εkb}

T{σ k

b } + δ{εks }

T{σ k

s } − δEzε33Ez

− δ{dt}Tρk{dt}) d�−

∫Aδ{dt}

T{f } dA = 0 (15)

in which ρk is the mass density of the kth layer, {f } is theexternally applied surface traction acting over the surface area(A) and � represents the volume of the concerned layer.

The eight-noded isoparametric quadrilateral elementshave been used to discretize the overall shell. According toequation (4), the generalized displacement vectors associatedwith the ith (i = 1, 2, 3, . . . , 8) node of the element can bewritten as

{dti} = [u0i v0i w0i]T and

{dri} =[θxi θyi θzi φxi φyi φzi γxi γyi γzi

]T.

(16)

Thus the generalized displacement vectors {dt} and {dr} atany point within the element can be expressed in terms ofthe nodal generalized displacement vectors {de

t } and {der } as

follows:

{dt} = [Nt]{det } and {dr} = [Nt]{d

er } (17)

in which

[Nt] = [Nt1 Nt2 · · · Nt8]T ,

[Nr] = [Nr1 Nr2 · · · Nr8]T,

Nti = niIt, Nri = niIr

{det } = [{d

et1}

T{de

t2}T· · · {de

t8}T]T and

{der } = [{d

er1}

T{de

r2}T· · · {de

r8}T]T

(18)

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

wherein It and Ir are the (3× 3) and (9× 9) identity matrices,respectively, and ni is the shape function of the naturalcoordinates associated with the ith node. Using equations(6)–(8) and (17), the strain vectors at any point within theelement can be expressed in terms of the nodal generalizeddisplacement vectors as follows:

{εb}c = [Btb]{det } + [Z1][Brb]{d

er },

{εb}v = [Btb]{det } + [Z2][Brb]{d

er },

{εb}p = [Btb]{det } + [Z3][Brb]{d

er },

{εs}c = [Bts]{det } + [Z4][Brs]{d

er },

{εs}v = [Bts]{det } + [Z5][Brs]{d

er } and

{εs}p = [Bts]{det } + [Z6][Brs]{d

er }

(19)

while the nodal strain–displacement matrices [Btb], [Brb],[Bts] and [Brs] are given by

[Btb] = [Btb1 Btb2 · · · Btb8]T,

[Brb] = [Brb1 Brb2 · · · Brb8]T,

[Bts] = [Bts1 Bts2 · · · Bts8]T and

[Brs] = [Brs1 Brs2 · · · Brs8]T.

(20)

The elements of the matrices [Btb], [Brb], [Bts] and [Brs]

have been presented in the appendix. On substitution ofequations (11) and (19) into (15), and recognizing that Ez =

V/hp with V being the applied voltage across the thicknessof the PZC layer, one can derive the following open loopequations of motion of an element integrated with the ACLDtreatment:

[Me]{de

t } + [Kett]{d

et } + [K

etr]{d

er } = {F

etp}V + {F

e} (21)

[Kert]{d

et } + [K

err]{d

er } = {F

erp}V. (22)

The elemental mass matrix [Me], the elemental stiffness

matrices [Kett], [K

etr], [K

ert], [K

err], the elemental electro-elastic

coupling vectors {Fetp}, {F

erp}, the elemental load vector {Fe

}

and the mass parameter (m) appearing in equations (21) and(22) are given by

[Me] =

∫ ae

0

∫ be

0m[Nt]

T[Nt] dx dy

[Kett] = [K

etb] + [K

ets] + [K

etbs]pb + [K

etbs]ps,

[Ketr] = [K

etrb] + [K

etrs]

+12

([Ke

trbs]pb + [Kertbs]

Tpb + [K

etrbs]ps + [K

ertbs]

Tps

)[Ke

rt] = [Ketr]

T,

[Kerr] = [K

errb] + [K

errs] + [K

errbs]pb + [K

errbs]ps,

{Fetp} = {F

etb}p + {F

ets}p, {Fe

rp} = {Ferb}p + {F

ers}p,

{Fe} =

∫ ae

0

∫ be

0[Nt]

T{f } dx dy and

m =N+2∑k=1

ρk(hk+1 − hk).

(23)

The various elemental stiffness matrices and the electro-elastic coupling vectors appearing in equation (23) are

[Ketb] =

∫A[Btb]

T([Dtb] + [Dtb]v + [Dtb]p)[Btb] dx dy,

[Kets] =

∫A[Bts]

T([Dts] + [Dts]v + [Dts]p)[Bts] dx dy,

[Ketbs]pb =

∫A[Btb]

T[Dtbs]p[Bts] dx dy,

[Ketbs]ps =

∫A[Bts]

T[Dtbs]p[Btb] dx dy,

[Ketrb] =

∫A[Btrb]

T([Dtrb] + [Dtrb]v + [Dtrb]p)[Brb] dx dy,

[Ketrbs]pb =

∫A[Btb]

T[Dtrbs]p[Brs] dx dy,[

Kertbs

]pb =

∫A[Brb]

T[Drtbs]p[Bts] dx dy,

[Ketrbs]ps =

∫A[Bts]

T[Drtbs]

Tp [Brb] dx dy,

[Kertbs]ps =

∫A[Brs]

T[Dtrbs]

Tp [Btb] dx dy,

[Ketrs] =

∫A[Bts]

T([Dtrs] + [Dtrs]v + [Dtrs]p)[Brs] dx dy,

[Kerrb] =

∫A[Brb]

T([Drrb] + [Drrb]v + [Drrb]p)[Brb] dx dy,

[Kerrs] =

∫A[Brs]

T([Drrs] + [Drrs]v + [Drrs]p)[Brs] dx dy,

[Kerrbs]pb =

∫A[Brb]

T[Drrbs]p[Brs] dx dy,

[Kerrbs]ps =

∫A[Brs]

T[Drrbs]

Tp [Brb] dx dy,

{Fetb}p =

∫A[Btb]

T{Dtb}p dx dy,

{Ferb}p =

∫A[Brb]

T{Drb}p dx dy,

{Fets}p =

∫A[Bts]

T{Dts}p dx dy and

{Fers}p =

∫A[Brs]

T{Drs}p dx dy.

Also, the various rigidity matrices and the rigidity vectorsappearing in the above elemental matrices are given by

[Dtb] =

N∑k=1

∫ hk+1

hk

[Ckb] dz,

[Dtrb] =

N∑k=1

∫ hk+1

hk

[Ckb][Z1] dz,

[Drrb] =

N∑k=1

∫ hk+1

hk

[Z1]T[Ck

b][Z1] dz,

[Dts] =

N∑k=1

∫ hk+1

hk

[Cks ] dz,

[Dtrs] =

N∑k=1

∫ hk+1

hk

[Ckb][Z4] dz,

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

[Drrs] =

N∑k=1

∫ hk+1

hk

[Z4]T[Ck

s ][Z4] dz,

[Dtb]v = hv[CN+1b ],

[Dtrb]v =

∫ hN+2

hN+1

[CN+1b ][Z2] dz,

[Drrb]v =

∫ hN+2

hN+1

[Z2]T[CN+1

b ][Z2] dz,

[Dts]v = hv[CN+1s ],

[Dtrs]v =

∫ hN+2

hN+1

[CN+1s ][Z5] dz,

[Drrs]v =

∫ hN+2

hN+1

[Z5]T[CN+1

s ][Z5] dz

[Dtb]p = hp[CN+2b ],

[Dtrb]p =

∫ hN+3

hN+2

[CN+2b ][Z3] dz,

[Drrb]p =

∫ hN+3

hN+2

[Z3]T[CN+2

b ][Z3] dz,

[Dts]p = hp[CN+2s ],

[Dtrs]p =

∫ hN+3

hN+2

[CN+2s ][Z6] dz,

[Drrs]p =

∫ hN+3

hN+2

[Z6]T[CN+2

s ][Z6] dz

[Dtbs]p =

∫ hN+3

hN+2

[CN+2bs ] dz,

[Dtrbs]p =

∫ hN+3

hN+2

[CN+2bs ][Z6] dz,

[Drtbs]p =

∫ hN+3

hN+2

[Z3]T[CN+2

bs ] dz,

[Drrbs]p =

∫ hN+3

hN+2

[Z3]T[CN+2

bs ][Z6] dz,

{Dtb}p =

∫ hN+3

hN+2

−{eb}/hp dz,

{Drb}p =

∫ hN+3

hN+2

−[Z3]T{eb}/hp dz,

{Dts}p =

∫ hN+3

hN+2

−{es}/hp dz, and

{Drs}p =

∫ hN+3

hN+2

−[Z6]T{es}/hp dz.

It should be noted that as the elemental stiffness matricesassociated with the transverse shear strains are derivedseparately, one can employ the selective integration rule toavoid the so called shear locking problem in the case ofthin shells. Finally, the elemental equations of motion areassembled to obtain the open loop global equations of motionof the overall shell integrated with the ACLD patches as

follows:

[M]{X} + [Ktt]{X} + [Ktr]{Xr} =

q∑j=1

{Fjtp}V

j+ {F} (24)

and

[Krt]{X} + [Krr]{Xr} =

q∑j=1

{Fjrp}V

j (25)

where [M] is the global mass matrix; [Ktt], [Ktr], [Krt] and[Krr] are the global stiffness matrices; {X} and {Xr} are theglobal nodal translational and rotational degrees of freedom;{Ftp} and {Frp} are the global electro-elastic coupling vectorscorresponding to the jth patch; V j is the voltage applied to thispatch; q is the number of patches and {F} is the global nodalforce vector.

4. Closed loop model

In order to supply the control voltage to activate the patchesof the ACLD treatment, a simple velocity feedback controllaw has been used. According to this law, the control voltagesupplied to the jth patch can be expressed in terms of thederivatives of the global nodal degrees of freedom as follows:

V j= −kj

dw = −kjd[U

jt]{X} − kj

d(h/2)[Ujr]{Xr} (26)

where kjd is the control gain for the jth patch, [Uj

t] and [Ujr]

are the unit vectors for expressing the transverse velocity ofthe point concerned in terms of the derivative of the globalnodal generalized displacements. Substituting equation (26)into equations (24) and (25), the final equations of motiongoverning the closed loop dynamics of the overall shell/ACLDsystem can be derived as follows:

[M]{X} + [Ktt]{X} + [Ktr]{Xr} +

q∑j=1

kjd{F

jtp}[U

jt]{X}

+

q∑j=1

kjd(h/2){F

jtp}[U

jr]{Xr} = {F} (27)

and

[Krt]{X} + [Krr]{Xr} +

q∑j=1

kjd{F

jrp}[U

jt]{X}

+

q∑j=1

kjd(h/2){F

jrp}[U

jr]{Xr} = 0. (28)

5. Results and discussion

In this section, the numerical results obtained by the FEmodel developed in section 3.4 are presented. Laminatedsymmetric/antisymmetric cross-ply and antisymmetric angle-ply thin cantilever cylindrical FFRC shells integrated withthe two patches of the ACLD treatment are considered forthe presentation of the numerical results. The patches areplaced 180◦ apart from each other on the outer circumferential

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Table 1. Material properties of the FFRC and the constituent phases of the piezoelectric composite.

Material ReferenceCNT volumefraction C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C23 (GPa) C44 (GPa) C55 (GPa) ρ (g cm−2)

FFRC [38] VCNT = 0.034 77.31 7.65 7.65 25.60 13.19 6.20 2.19 1428vf = 0.3 VCNT = 0 76.90 6.71 6.71 11.03 7.10 1.96 2.09 1385

FFRC [38] VCNT = 0.021 145.36 8.87 8.87 23.74 12.04 5.85 3.62 1580vf = 0.6 VCNT = 0 144.98 7.80 7.80 14.31 8.62 2.84 3.44 1550

surface of the shell and one of the ends of the patchesis fixed at the clamped end of the shells. It should benoted that the locations of the ACLD patches correspondto an optimal placement of the ACLD treatment such thatthe energy dissipation corresponding to the first two modesbecomes maximum [44]. The length and width of the ACLDpatches are considered as 67% of the length of the shelland the one sixth of the outer circumference of the shell,respectively. The material properties of the FFRC with respectto the material coordinate system considered here are listed intable 1. It should be noted that the CNT volume fraction inthe FFRC is dependent on the carbon fiber volume fraction inthe FFRC [38]. Thus two different values of the carbon fibervolume fraction (0.3 and 0.6) in the FFRC are considered toinvestigate the effect of radially grown CNTs on the dampingcharacteristics of the laminated FFRC shells. From table 1, itmay be observed that the transverse effective elastic propertiesof the FFRC are significantly improved due to the radialgrowth of the CNTs on the circumferential surfaces of thecarbon fiber reinforcements over those of the base compositewithout CNTs (VCNT = 0). The elastic and piezoelectricproperties of the vertically reinforced 1–3 PZC layer with60% piezoelectric fiber volume fraction are taken from [5]and are given as: C11 = 9.29 GPa, C12 = 6.18 GPa, C13 =

6.05 GPa, C33 = 35.44 GPa, C23 = C13, C44 = 1.58 GPa,C66 = 1.54 GPa, C55 = C44, e31 = −0.19 C m−2, e32 = e31,e33 = 18.41 C m−2, e24 = 0.004 C m−2 and e15 = e24.

The thicknesses of the laminated FFRC shells, theconstraining 1–3 PZC layer and the viscoelastic layer areconsidered as 0.003 m, 250 µm and 200 µm, respectively,while the length and the value of R/h for the shells areconsidered as 1 m and 50, respectively. The viscoelasticmaterial used by Chantalakhana and Stanway [45] isconsidered in this study to evaluate the numerical results.The loss factor of this viscoelastic material remains invariantwithin the frequency range (0–600 Hz) of interest. The valuesof the complex shear modulus, the Poisson’s ratio and thedensity of the viscoelastic layer are 20(1 + i) MN m−2, 0.49and 1140 kg m−3, respectively [45].

To verify the validity of the present FE model, the naturalfrequencies of the laminated composite shells integrated withthe inactivated patches of the ACLD treatment with negligiblethickness are first computed by the present FE modeland, subsequently, compared with the existing analyticalresults presented in [46] for identical laminated compositeshells without the patches integrated. A non-dimensionalfrequency parameter (ω) is used to compare the fundamentalnatural frequencies of the laminated composite shells and is

Table 2. Comparison of the fundamental natural frequencyparameters (ω) of the clamped–free laminated composite shells.

Shelltype Source R/h a/R ω

0◦/90◦/0◦ Present FEM 20 5 0.4910Analytical [46] 20 5 0.4899

0◦/90◦ Present FEM 20 5 0.5620Analytical [46] 20 5 0.5581

defined as

ω = ω(a2/h)√ρ/ET (29)

in which ω represents the natural frequency of the overallshell; ρ and ET are the density and the transverseYoung’s modulus, respectively, of the orthotropic layers ofthe substrate shell. Table 2 illustrates this comparison ofthe fundamental natural frequencies of certain laminatedcomposite shells. It may be observed from this table that thetwo sets of results predicted by the present FE model and [46]are in excellent agreement, validating the present FE model.

Although the results predicted by the present FEmodel have been validated with the existing analyticalresults, an attempt has also been made to investigatethe convergence of the results predicted by the presentFE model. For this purpose, a convergence test forthe present FE model predicting the fundamental naturalfrequencies of the laminated composite shells integratedwith the inactivated ACLD patches has been carried out,and the results are compared with those of the laminatedcomposite shells without the ACLD patches integrated [47].A non-dimensional frequency parameter (ω1) is used tocompare the fundamental natural frequencies of the laminatedcomposite shells and is defined as

ω1 = ωR√ρ/ET. (30)

Table 3 illustrates the comparison of the convergence of thefundamental natural frequency parameters of the laminatedshells considering different mesh sizes. It is evident fromtable 3 that the value of the fundamental natural frequencyparameter remains almost the same beyond a mesh sizeof 6 × 12. A convergence test for a symmetric cross-ply(0◦/90◦/0◦) laminated FFRC shell has also been carried out inthis study, as demonstrated in table 4. Table 4 reveals that thefundamental natural frequencies of the symmetric cross-plylaminated FFRC shell become saturated beyond a mesh sizeof 6× 12. Therefore, for the subsequent results predicting the

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Table 3. Convergence of the fundamental natural frequency parameters (ω1) of the laminated composite shells with negligible thickness ofthe ACLD patches.

Shell type R/h a/RMeshsize

Present FEM(ω1)

Reference [47](ω1)

45◦/−45◦/−45◦/45◦ 100 4 4× 4 0.1543 0.17574× 8 0.1617 0.17576× 8 0.1682 0.17578× 8 0.1745 0.17576× 12 0.1753 0.17578× 12 0.1755 0.1757

Table 4. Convergence of the fundamental natural frequency (ω) ofthe laminated FFRC shells with negligible thickness of the ACLDpatches.

Shell type R/h a/RMeshsize

Present FEM(ω)

0◦/90◦/0◦ 100 5 4× 4 548.84× 8 629.66× 8 694.16× 12 742.18× 12 743.6

10× 12 742.6

frequency response functions (FRFs) of the laminated FFRCshells, the optimum mesh size of 6× 12 is considered.

The open and the closed loop behavior of the laminatedshells is studied by the FRFs evaluated at a point (a, 0, h/2)on the top surface of the shell. A time-harmonic point force(1 N) is considered to act at the same point (a, 0, h/2) on theshell to excite the first few modes of the shells. The controlgains (Kd) are selected as 600 and 1500 to control the firstfew modes of the laminated shells. For the sake of clarityin the plots, only the range of frequencies around the firstthree modes of vibration is presented in the results. Figure 7illustrates the FRFs of the cantilever symmetric cross-ply(0◦/90◦/0◦) laminated FFRC shells when the orientation angleof the piezoelectric fibers (ψ) in the PZC layer is 0◦. Thisfigure displays both uncontrolled and controlled responses,and clearly reveals that the constraining layer made of the 1–3PZC layer significantly enhances the damping characteristicsof the laminated FFRC shells over the passive damping(uncontrolled). The maximum control voltage required toachieve this attenuation of the amplitude of vibrations is only50 V when the value of Kd is 1500, as depicted in figure 8.

The effect of variation of the orientation angle of thepiezoelectric fibers in the two mutually orthogonal verticalplanes (xz and yz) of the PZC layer on the performance of theACLD patches has been studied by considering the maximumvalue of the piezoelectric fiber orientation angle as 45◦ inthe PZC layer (see footnote 1). It should be noted that thevalue of the piezoelectric fiber orientation angle is variedfrom 0◦ to 45◦ in the PZC layer to investigate the effectof the piezoelectric fiber orientation on the performance ofthe ACLD patches. However, for the sake of clarity in theplots, the FRFs corresponding to the four specific values ofthe piezoelectric fiber orientation angles (0◦, 15◦, 30◦ and45◦) have been presented in such a way that the optimum

Figure 7. FRFs for transverse deflection (w) of the cantileversymmetric cross-ply (0◦/90◦/0◦) laminated FFRC shells(ψ = 0◦, vf = 0.3,VCNT = 0.034).

Figure 8. Control voltages for the cantilever symmetric cross-ply(0◦/90◦/0◦) laminated FFRC shells shown in figure 7(ψ = 0◦, vf = 0.3,VCNT = 0.034).

performance of the ACLD patches can be demonstrated.Figures 9 and 10 illustrate the effect of the piezoelectricfiber orientation on the performance of the ACLD patches forimprovement of the damping characteristics of the cantileversymmetric cross-ply (0◦/90◦/0◦) laminated FFRC shells when

11

Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 9. Effect of different values of ψ in the xz plane of the PZClayer on the performance of the ACLD patches for control of thecantilever symmetric cross-ply (0◦/90◦/0◦) laminated FFRC shells(Kd = 1500, vf = 0.3,VCNT = 0.034).

Figure 10. Effect of different values of ψ in the yz plane of thePZC layer on the performance of the ACLD patches for control ofthe cantilever symmetric cross-ply (0◦/90◦/0◦) laminated FFRCshells (Kd = 1500, vf = 0.3,VCNT = 0.034).

the orientation angle of the piezoelectric fibers is variedin the vertical xz and yz planes, respectively. These figuresindicate that the maximum attenuation of vibration is obtainedfor the first mode of vibration when the piezoelectric fiberorientation angle is 45◦ in the vertical xz plane of thePZC layer. Figures 11 and 12 illustrate the effect of thepiezoelectric fiber orientation on the performance of theACLD patches for improving the damping characteristics ofthe cantilever antisymmetric angle-ply (−45◦/45◦/−45◦/45◦)laminated FFRC shells when the orientation angle of thepiezoelectric fibers is varied in the vertical xz and yz planes,respectively. In this case also the maximum attenuation ofvibration is achieved for the first two modes of vibration whenthe piezoelectric fiber orientation angle is 45◦ in the vertical xzplane of the PZC layer. Hence, the value of the piezoelectric

Figure 11. Effect of different values of ψ in the xz plane of thePZC layer on the performance of the ACLD patches for control ofthe cantilever antisymmetric angle-ply (−45◦/45◦/−45◦/45◦)laminated FFRC shells (Kd = 1500, vf = 0.3,VCNT = 0.034).

Figure 12. Effect of different values of ψ in the yz plane of thePZC layer on the performance of the ACLD patches for control ofthe cantilever antisymmetric angle-ply (−45◦/45◦/−45◦/45◦)laminated FFRC shells (Kd = 1500, vf = 0.3,VCNT = 0.034).

fiber orientation angle is considered as 45◦ in the verticalxz plane of the PZC layer for investigation of the effect ofradially grown CNTs on the attenuation of vibration of thesymmetric cross-ply and antisymmetric angle-ply laminatedFFRC shells. Figures 13 and 14 illustrate the FRFs of thesymmetric cross-ply and antisymmetric angle-ply laminatedshells, respectively, when the value of the piezoelectric fiberorientation angle is 45◦ in the vertical xz plane of the PZClayer. These figures clearly reveal that when the value ofVCNT 6= 0 the performance of the ACLD patches for controlof the first few modes of vibration becomes maximum andthe natural frequencies of the laminated FFRC shells aresignificantly improved over those of the laminated compositeshells without CNTs (VCNT = 0). This is attributed to the

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 13. FRFs for transverse displacement (w) of the cantileversymmetric cross-ply (0◦/90◦/0◦) laminated shells when thepiezoelectric fibers of the PZC layer are coplanar with the xz plane(Kd = 1500, ψ = 45◦).

Figure 14. FRFs for transverse displacement (w) of the cantileverantisymmetric angle-ply (−45◦/45◦/−45◦/45◦) laminated shellswhen the piezoelectric fibers of the PZC layer are coplanar with thexz plane (Kd = 1500, ψ = 45◦).

fact that the transverse elastic properties of the FFRC aresignificantly larger than those of the base composite withoutCNTs (VCNT = 0).

Figures 15 and 16 demonstrate the effect of thepiezoelectric fiber orientation on the performance ofthe ACLD patches for improvement of the dampingcharacteristics of the cantilever antisymmetric cross-ply(0◦/90◦/0◦/90◦) laminated FFRC shells when the orientationangle of the piezoelectric fibers is varied in the vertical xz andyz planes, respectively. Figure 15 depicts that the attenuatingcapability of the ACLD patches becomes maximum when thevalue of the piezoelectric fiber orientation angle is 45◦ and0◦ in the vertical xz plane of the PZC layer for controlof the first and second modes of vibration, respectively,

Figure 15. Effect of different values of ψ in the xz plane of thePZC layer on the performance of the ACLD patches for control ofthe cantilever antisymmetric cross-ply (0◦/90◦/0◦/90◦) laminatedFFRC shells (Kd = 1500, vf = 0.3,VCNT = 0.034).

Figure 16. Effect of different values of ψ in the yz plane of thePZC layer on the performance of the ACLD patches for control ofthe cantilever antisymmetric cross-ply (0◦/90◦/0◦/90◦) laminatedFFRC shells (Kd = 1500, vf = 0.3,VCNT = 0.034).

but the best performance of the ACLD patches is obtainedwhen the value of the piezoelectric fiber orientation angle is0◦ for control of the second mode of vibration. When thepiezoelectric fiber orientation angle is varied in the verticalyz plane of the PZC layer, the attenuation capability of theACLD patches is maximized if the value of the piezoelectricfiber orientation angle is 0◦ for control of the first three modesof vibration, as shown in figure 16. Therefore, the valueof the piezoelectric fiber orientation angle is considered as0◦ for investigation of the effect of radially grown CNTs onthe attenuation of vibrations of the cantilever antisymmetriccross-ply laminated FFRC shells. The significant effectof radially grown CNTs on the circumferential surfacesof the carbon fiber reinforcements has been observed forimprovement of the damping characteristics of the cantilever

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Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

Figure 17. FRFs for transverse displacement (w) of the cantileverantisymmetric cross-ply (0◦/90◦/0◦/90◦) laminated shells when thepiezoelectric fibers of the PZC layer are coplanar with the xz plane(Kd = 1500, ψ = 0◦).

antisymmetric cross-ply laminated FFRC shells, as shownin figure 17. It may also be noted that the performance ofthe ACLD patches for control of the attenuation of vibrationof the cantilever antisymmetric cross-ply laminated FFRCshells is significantly improved over the performance for thesymmetric cross-ply and antisymmetric angle-ply laminatedFFRC shells.

6. Conclusions

In this paper, the damping characteristics of smart laminatedfuzzy fiber reinforced composite (FFRC) shells have beeninvestigated. The distinct constructional feature of thenovel FFRC is that short CNTs are radially grown onthe circumferential surfaces of the continuous carbon fiberreinforcements. The following main conclusions are drawnfrom the investigations carried out in this paper.

(1) Since the radially grown CNTs on the circumferentialsurfaces of the carbon fibers eventually stiffen thepolymer matrix in the radial directions, the transverseeffective elastic coefficients of the resulting composite areimproved, which causes enhancement of the attenuationof the amplitude of vibration and the natural frequenciesof the laminated FFRC shells over those of the laminatedbase composite shells without CNTs.

(2) The FRFs of the symmetric cross-ply, the antisymmetriccross-ply and the angle-ply laminated FFRC shellsindicate that the active constraining layer of the ACLDtreatment made of the vertically/obliquely reinforced PZCsignificantly enhances the damping characteristics of thelaminated FFRC shells over passive damping.

(3) The performance of the ACLD patches is maximized forcontrol of the first two modes of the symmetric cross-plyand antisymmetric angle-ply laminated shells when the

value of the piezoelectric fiber orientation angle is 45◦ inthe vertical xz plane of the PZC layer. For the cantileverantisymmetric cross-ply laminated shells the maximumcontrol authority of the ACLD patches is achieved whenthe value of the piezoelectric fiber orientation angle is0◦ in the vertical plane of the PZC layer.

(4) The novel architecture of the FFRC allows one to modifythe static as well as the dynamic properties of thelaminated FFRC shells by changing the CNT volumefraction in the FFRC, keeping the value of the carbon fibervolume fraction constant.

Based on the above points, it may be concluded thatshort CNTs can be properly incorporated to constructnanocomposites with better elastic properties and dampingcharacteristics for the development of light weight smartcomposite structures.

Appendix

In equations (6) and (7), the matrices [Z1], [Z2], [Z3], [Z4],[Z5] and [Z6] are given by

[Z1] = [[Z1] o o], [Z2] = [(h/2)I [Z2] o],

[Z3] = [(h/2)I hvI [Z3]],

[Z4] = [ [Z4] o o zI o o],

[Z5] = [(h/2R)I1 [Z5] o (h/2)I (z− h/2)I o],

[Z6] = [(h/2R)I1 (hv/R)I1 [Z6] (h/2)I hvI (z− hN+2)I],

in which

[Z1] =

z 0 0 0

0 z 0 z/R

0 0 z 0

0 0 0 1

,

[Z2] =

(z− h/2) 0 0 0

0 (z− h/2) 0 (z− h/2)/R

0 0 (z− h/2) 0

0 0 0 1

,

[Z3] =

(z− hN+2) 0 0 0

0 (z− hN+2) 0 (z− hN+2)/R

0 0 (z− hN+2) 0

0 0 0 1

,

[Z4] =

[1 0

0 z/R

][Z5] =

[1 0

0 1− (z− h/2)/R

],

[Z6] =

[1 0

0 1− (z− hN+2)/R

]

[I] =

1 0 0 0

0 1 0 1/R

0 0 1 0

0 0 0 1

, [I1] =

[1 0

0 −1

],

14

Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

[I] =

[1 0

0 1

],

[o] =

[0 0

0 0

]and [o] =

[o o

o o

].

The various sub-matrices Btbi, Btsi,Brbi and Brsiappearing in equation (20) are given by

Btbi =

∂ni

∂x0 0

0∂ni

∂y1/R

∂ni

∂y

∂ni

∂x0

0 0 0

, Btsi =

0 0∂ni

∂x

0 −1/R∂ni

∂y

,

Brbi =

Brbi 0 0

0 Brbi 0

0 0 Brbi

, Brbi =

∂ni

∂x0 0

0∂ni

∂y0

∂ni

∂y

∂ni

∂x0

0 0 1

,

Brsi =

I 0 0

0 I 0

0 0 I

Brsi 0 0

0 Brsi 0

0 0 Brsi

, Brsi =

0 0∂ni

∂x

0 0∂ni

∂y

0 =

0 0 0

0 0 0

0 0 0

0 0 0

, 0 =

[0 0 0

0 0 0

]and

I =

[1 0 0

0 1 0

].

References

[1] Hwang W S and Park H C 1993 Finite element modeling ofpiezoelectric sensors and actuators AIAA J. 31 930–7

[2] Reddy J N 1999 On laminated composite plates withintegrated sensors and actuators Eng. Struct. 21 568–93

[3] Lam K Y and Ng T Y 1999 Active control of composite plateswith integrated piezoelectric sensors and actuators undervarious dynamic loading conditions Smart Mater. Struct.8 223–37

[4] Song G, Sethi V and Li H H 2006 Vibration control of civilstructures using piezoceramic smart materials: a reviewEng. Struct. 28 1513–24

[5] Ray M C and Pradhan A K 2007 On the use of verticallyreinforced 1–3 piezoelectric composites for hybrid dampingof laminated composite plates Mech. Adv. Mater. Struct.14 245–61

[6] Ray M C and Faye A 2009 Theoretical and experimentalinvestigations on the active structural-acoustic control of athin isotropic plate using a vertically reinforced 1–3piezoelectric composite Smart Mater. Struct. 18 015012

[7] Lin Y and Sodano H A 2008 Concept and model of apiezoelectric strauctural fiber for multifunctionalcomposites Compos. Sci. Technol. 68 1911–8

[8] Smith W A and Auld B A 1991 Modeling 1–3 compositepiezoelectrics: thickness-mode oscillations IEEE Trans.Ultrason. Ferroelectr. Freq. Control 38 40–7

[9] Ray M C and Pradhan A K 2008 Performance of verticallyand obliquely reinforced 1–3 piezoelectric composites foractive damping of laminated composite shells J. Sound Vib.315 816–35

[10] Ray M C and Pradhan A K 2009 Active damping of laminatedthin cylindrical composite panels using vertically/obliquelyreinforced 1–3 piezoelectric composites Acta Mech.209 201–18

[11] Saviz M R and Mohammadpourfard M 2010 Dynamic analysisof a laminated cylindrical shell with piezoelectric layersunder dynamic loads Finite Elem. Anal. Des. 46 770–81

[12] Iijima S 1991 Helical microtubules of graphitic carbon Nature354 56–8

[13] Treacy M M J, Ebbesen T W and Gibson J M 1996Exceptionally high Young’s modulus observed forindividual carbon nanotubes Nature 381 678–80

[14] Li C and Chou T W 2003 A Structural mechanics approach forthe analysis of carbon nanotubes Int. J. Solids Struct.40 2487–99

[15] Shen L and Li J 2004 Transversely isotropic elastic propertiesof single-walled carbon nanotubes Phys. Rev. B 69 045414

[16] Cho W S T 2006 Bending and shear moduli of single-walledcarbon nanotube Finite Elem. Anal. Des. 42 404–13

[17] Thostenson E T and Chow T W 2003 On the elastic propertiesof carbon nanotube-based composites: modelling andcharacterization J. Phys. D: Appl. Phys. 36 573–82

[18] Odegard G M, Gates T S, Wise K E, Park C and Siochi E J2003 Constitutive modeling of nanotube-reinforcedpolymer composites Compos. Sci. Technol. 63 1671–87

[19] O’Donnell S E, Sprong K R and Haltli B M 2004 Potentialimpact of carbon nanotube reinforced polymer compositeon commercial aircraft performance and economics ATIO:4th AIAA Aviat. Technol. Int. Oper. Conf. (Chicago)pp 1–10

[20] Wuite J and Adali S 2005 Deflection and stress behaviour ofnanocomposite reinforced beams using a multiscaleanalysis Compos. Struct. 71 388–96

[21] Ray M C and Batra R C 2007 A single-walled carbonnanotube reinforced 1–3 piezoelectric composite for activecontrol of smart structures Smart Mater. Struct. 16 1936–47

[22] Zhu P, Lei Z X and Liew K M 2012 Static and free vibrationanalyses of carbon nanotube-reinforced composite platesusing finite element method with first order sheardeformation plate theory Compos. Struct. 94 1450–60

[23] Rokni H, Milani A S, Seethaler R J and Stoeffler K 2012Improvement in dynamic properties of laminatedMWCNT–polystyrene composite beams via an integratednumerical–experimental approach Compos. Struct.94 2538–47

[24] Thostenson E T, Reng Z and Chou T W 2001 Advances in thescience and technology of carbon nanotubes and theircomposites Compos. Sci. Technol. 61 1899–912

[25] Downs W B and Baker R T K 1995 Modification of thesurface properties of carbon fibers via the catalytic growthof carbon nanofibers J. Mater. Res. 10 625–33

[26] Bower C, Zhu W, Jin S and Zhou O 2000 Plasma-inducedalignment of carbon nanotubes Appl. Phys. Lett. 77 830–2

[27] Thostenson E T, Li W Z, Wang D Z, Ren Z F and Chou T W2002 Carbon nanotube/carbon fiber hybrid multiscalecomposites J. Appl. Phys. 91 6034–7

[28] De Riccardis M F, Carbone D, Makris T D, Giorgi R, Lisi Nand Salernitano E 2006 Anchorage of carbon nanotubesgrown on carbon fibers Carbon 44 671–4

15

Smart Mater. Struct. 22 (2013) 105001 S I Kundalwal et al

[29] Veedu V P, Cao A, Li X, Ma K, Soldano C, Kar S,Ajayan P M and Ghasemi-Nejhad M N 2006Multifunctional composites using reinforced laminae withcarbon-nanotube forests Nature Mater. 5 457–62

[30] Kepple K L, Sanborn G P, Lacasse P A, Gruenberg K M andReady W J 2008 Improved fracture toughness of carbonfiber composite functionalized with multi walled nanotubesCarbon 46 2026–33

[31] Mathur R B, Chatterjee S and Singh B P 2008 Growth ofcarbon nanotubes on carbon fiber substrates to producehybrid/phenolic composites with improved mechanicalproperties Compos. Sci. Technol. 68 1608–15

[32] Zhang Q, Liu J, Sager R, Dai L and Baur J 2009 Hierarchicalcomposites of carbon nanotubes on carbon fiber: influenceof growth condition on fiber tensile properties Compos. Sci.Technol. 69 594–601

[33] Kulkarni M, Carnhan D, Kulkarni K, Qian D and Abot J L2010 Elastic response of a carbon nanotube fiber reinforcedpolymeric composite: a numerical and experimental studyComposites B 41 414–21

[34] Yamamoto N, Wicks S S, de Villoria R G, Ishiguro K,Steiner S A III, Garcia E J and Wardle B L 2009Mechanical, thermal, and electrical properties of wovenlaminated advanced composites containing aligned carbonnanotubes ICMM:17th Int. Conf. Compos. Mater.(Edinburgh) pp 27–31

[35] Liu Q and Rao S S 2005 Fuzzy finite element approach foranalysis of fiber-reinforced laminated composite beamsAIAA J. 43 651–61

[36] Sager R J, Klein P J, Lagoudas D C, Zhang Q, Liu J, Dai Land Baur J W 2009 Effect of carbon nanotubes on theinterfacial shear strength of T650 carbon fiber in an epoxymatrix Compos. Sci. Technol. 69 898–904

[37] Song Q, Li K Z, Li H L, Li H J and Ren C 2012 Graftingstraight carbon nanotubes radially onto carbon fibers andtheir effect on the mechanical properties of carbon/carboncomposites Carbon 50 3943–52

[38] Kundalwal S I and Ray M C 2011 Micromechanical analysisof fuzzy fiber reinforced composites Int. J. Mech. Mater.Des. 7 149–66

[39] Chatzigeorgiou G, Efendiev Y and Lagoudas D C 2011Homogenization of aligned fuzzy fiber composites Int. J.Solids Struct. 48 2668–80

[40] Kundalwal S I 2013 Micromechanical analysis of novelcontinuous and short fuzzy fiber reinforced compositesPhD Thesis Department of Mechanical Engineering, IndianInstitute of Technology Kharagpur, India

[41] Baz A and Ro J 1996 Vibration control of plates with activeconstrained layer damping Smart Mater. Struct. 5 272–80

[42] Ray M C and Baz A 1997 Optimization of energy dissipationof active constrained layer damping treatments of platesJ. Sound Vib. 208 391–406

[43] Jeung Y S and Shen I Y 2001 Development of isoparametric,degenerate constrained layer element for plate and shellstructures AIAA J. 39 1997–2005

[44] Ray M C and Reddy J N 2004 Optimal control of thin circularcylindrical shells using active constrained layer dampingtreatment Smart Mater. Struct. 13 64–72

[45] Chantalakhana C and Stanway R 2001 Active constrainedlayer damping of clamped–clamped plate vibrationsJ. Sound Vib. 241 755–77

[46] Reddy J N 1996 Mechanics of Laminated Composites PlatesTheory and Analysis (Boca Raton, FL: CRC Press)

[47] Timarci T and Soldatos K P 2000 Vibrations of angle-plylaminated circular cylindrical shells subjected to differentsets of edge boundary conditions J. Eng. Math. 37 211–30

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