Acta MechDOI 10.1007/s00707-014-1297-8

S. I. Kundalwal · S. A. Meguid

Effect of carbon nanotube waviness on active dampingof laminated hybrid composite shells

Received: 25 September 2014 / Revised: 4 December 2014© Springer-Verlag Wien 2015

Abstract In this article, we investigate the effect of carbon nanotube (CNT) waviness on the active constrainedlayer damping (ACLD) of the laminated hybrid composite shells. In particular, the effect of CNT wavinesshas been studied for the case of a novel nano-tailored composite—continuous fuzzy fiber-reinforced composite(FFRC). The distinctive feature of the construction of the FFRC is that the uniformly spaced straight or wavyCNTs are radially grown on the circumferential surfaces of carbon fibers. The constraining layer of the ACLDtreatment is considered to be made of vertically or obliquely reinforced 1–3 piezoelectric composite material.A three-dimensional finite element model has been developed to study the damping characteristics of thelaminated FFRC shells integrated with the patches of ACLD treatment. Our results reveal that (i) the planarorientation of CNT waviness has a significant influence on the damping characteristics of the laminated FFRCshells, (ii) damping characteristics of the symmetric cross-ply, and antisymmetric angle-ply laminated FFRCshells are improved if CNT waviness is coplanar with the longitudinal plane of the carbon fiber, and (iii) for theantisymmetric cross-ply laminated FFRC shells, the performance of the ACLD patches becomes maximumfor attenuating the fundamental mode when CNT waviness is coplanar with the transverse plane of the carbonfiber.

1 Introduction

Owing to their remarkable elastic and scale-dependent physical properties, carbon nanotubes (CNTs) haveemerged as promising reinforcements to enhance the thermomechanical properties of CNT-based compositematerials [1–8]. Specifically, a hybrid fuzzy fiber composite system employing traditional advanced compos-ites and CNTs offer significant potential mechanical and multifunctional properties, such as laminate strength,toughness, electrical and thermal conductivities, damping and stress transfer characteristics [9–12]. In such“fuzzy fiber” composites, CNTs are radially grown on the circumferential surfaces of main reinforcements.The presence of radially grown CNTs on the circumferential surface of fiber enhances the fiber/matrix inter-facial bonding as well as the multifunctional properties of the resulting composite. For example, Veedu et al.[13] fabricated a 3D composite in which multi-walled CNTs are grown normal to the surface of micro-fiberfabric cloth layouts. The measured fracture toughness for crack initiation, interlaminar shear sliding fracturetoughness, in-plane strength, modulus, toughness, transverse thermal conductivity and damping characteristicsof this 3D composite exhibit 348, 54, 140, 5, 424, 51 and 514 % enhancements, respectively, over those of thebase composite. Their test results also revealed that the presence of CNTs in the transverse direction of this3D composite reduces the effective thermal expansion coefficient of the composite by 62 % as compared to

S. I. Kundalwal · S. A. Meguid (B)Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto,Toronto, ON M5S 3G8, CanadaE-mail: meguid@mie.utoronto.caTel.: +1 (416) 978 5741

S. I. Kundalwal, S. A. Meguid

that of the base composite. Yamamoto et al. [14] fabricated a hybrid composite composed of alumina fibers,in situ grown aligned CNTs and epoxy matrix. With a small volume fraction of CNTs, they reported that boththe fracture toughness for crack initiation and the steady-state fracture toughness exhibit an improvement of60 %, the electrical conductivity is enhanced by 6–8 orders of magnitude, and the thermal conductivity isincreased by ∼80 %. Recently, Kundalwal and Ray [15–17] determined the effective thermoelastic propertiesof the fuzzy fiber-reinforced composite (FFRC) by employing several micromechanics models. Their worksindicate that the effective thermoelastic properties of the FFRC are significantly improved over those of thecomposite without CNTs. Importantly, this novel multifunctional architecture needs to be extended for devel-oping laminated composite structural elements, such as beams, plates, shells and panels which find a widerange of engineering applications in aerospace and automobile sectors.

With the emergence of new piezoceramic materials, the concept of using a network of piezoelectric actu-ators and sensors forming a self-controlling and self-monitoring smart system in advanced structural designhas drawn considerable interest among researchers [18–30]. In view of its importance to the development ofCNT-based advanced structures, the subject of piezoelectric actuator/sensor has received attention from thescientific community. For example, Ramaratnam and Jalili [31] fabricated sensors using CNT-based piezo-electric polymers. The electromechanical response of these novel structures demonstrated an increase in thesensing capability of piezoelectric polymers by the addition of CNTs. Ray and Batra [32] investigated theperformance of a CNT-reinforced 1–3 piezoelectric composite (PZC) as a constraining layer in the activeconstrained layer damping (ACLD) of the laminated composite beam by using the finite element (FE) method.In their study, the computed controllability measure and frequency response reveal that a CNT-reinforced1–3 PZC layer performs better than the 1–3 PZC layer without CNTs. The ACLD treatment consists of con-straining layers made of piezoelectric and viscoelastic materials. The flexural vibration control by the ACLDtreatment is attributed to the dissipation of energy in the constrained viscoelastic layer due to its transverseshear deformations. Zhu et al. [33] presented bending and free vibration analysis of thin-to-moderately thickcomposite plates reinforced by single-walled CNTs using the first-order shear deformation theory (FSDT).In their study, it is found that both the CNT volume fraction and width-to-thickness ratio have pronouncedeffect on the natural frequencies and vibration mode shapes of a CNT-reinforced composite plate. Based ona three-dimensional theory of elasticity, Alibeigloo [34] investigated the bending behavior of a functionallygraded CNT-reinforced composite plate embedded in thin piezoelectric layers subjected to mechanical uniformload with simply supported boundary conditions. This study indicates that the effect of reinforced CNTs onthe transverse quantities, such as transverse normal and shear stresses in a functionally graded CNT-reinforcedcomposite plate, contrary to the in-plane quantities, is more significant. Recently, Kundalwal et al. [35] studiedthe damping characteristics of the smart laminated FFRC shells integrated with the patches of ACLD treat-ment. Their study revealed that radially grown straight CNTs on the circumferential surfaces of carbon fibersenhance the attenuation of vibrations and natural frequencies of the laminated FFRC shells over those of thelaminated base composite shells.

Carbon nanotube waviness is inherent to the fabrication process of CNT-reinforced polymer composites.Scanning electron microscopy (SEM) images shown in Fig. 1a, b [14,36] clearly demonstrate that CNTsremain highly curved when they are either embedded in the polymer matrix or grown on the circumferentialsurface of the fiber. Several studies reported that CNT waviness influences the effective elastic properties ofmicro- and nano-hybrid nanocomposites [16,17,37–41]. Based on a three-dimensional theory of elasticity,Jam et al. [42] investigated the effects of CNT aspect ratio and waviness on the vibrational behavior ofnanocomposite cylindrical panels, and their results indicate that the distribution pattern and volume fractionof CNTs have a significant effect on the natural frequencies of a nanocomposite cylindrical panel. A newconcept for the optimization of dynamic behavior of the laminated nanocomposite beam is introduced byRokni et al. [43] in which the fiber orientation factor in continuous fiber-reinforced composites is replaced bydifferent wt% of CNTs in each layer. This study revealed that the laminated nanocomposite beam with theoptimum distribution of CNTs in the polymer matrix causes a significant improvement in the effective Young’smodulus and damped natural frequencies over those of the beam with uniformly reinforced CNTs. Based ona mesh-free method, Moradi-Dastjerdi et al. [44] investigated the influence of orientation and aggregation ofCNTs on the axisymmetric natural frequencies of a functionally graded nanocomposite cylinder. They reportedthat the distribution pattern, aggregation, or evenly random orientations of CNTs have significant effect onthe effective stiffness and frequency parameter of a functionally graded nanocomposite cylinder. Since thedistribution pattern and CNT waviness influence the mechanical and vibrational behavior of CNT-reinforcedcomposite, the waviness of CNTs will also influence the damping characteristics of the laminated FFRC shells.Investigation of the effect of CNT waviness on the damping characteristics of the laminated FFRC shells is an

Effect of carbon nanotube waviness

Fig. 1 a SEM image of a compressed CNT-reinforced film (with permission from [36]). b SEM image of the alumina fiber onwhich radially aligned CNTs are grown (with permission from [14])

Fig. 2 a Fuzzy fiber coated with wavy CNTs being coplanar with the transverse (2–3) plane of the carbon fiber. b Fuzzy fibercoated with wavy CNTs being coplanar with the longitudinal (1–3) plane of the carbon fiber

important issue and has not yet been addressed. It is therefore the objective of this work to investigate the effectof CNT waviness on the vibrating thin laminated FFRC shells integrated with the ACLD patches. We accountfor the waviness of CNTs by considering the plane of wavy CNTs is coplanar with either the longitudinalplane or the transverse plane of the carbon fiber. Using the FSDT, a three-dimensional FE model is developedto study the damping characteristics of the laminated FFRC shells.

2 Elastic properties of the FFRC

To investigate the role of CNT waviness on the damping characteristics of the laminated FFRC shells, wavyCNTs are modeled as sinusoidal solid CNT fibers [37–40]. The orientation of the plane of wavy CNTs radiallygrown on the circumferential surface of a carbon fiber will influence the effective elastic properties of the FFRC[16,17,40,41]. Hence, two possible planar orientations of wavy CNTs are considered either in the transverse(2–3) plane or the longitudinal (1–3) plane of the carbon fiber, as shown in Fig. 2a, b, respectively. The radiallygrown wavy CNTs reinforce the polymer matrix surrounding the carbon fiber along the transverse directionto the fiber length. Figure 3 shows a schematic of the representative volume element of the polymer matrixnanocomposite containing a wavy CNT. This representative volume element is divided into infinitesimally thinslices of thickness dy. Averaging the elastic properties of these slices over the length (Ln) of the representativevolume element, the homogenized effective elastic properties of the polymer matrix nanocomposite can be

S. I. Kundalwal, S. A. Meguid

Fig. 3 The representative volume element of the polymer matrix nanocomposite containing a wavy CNT is coplanar with eitherthe longitudinal (1–3) plane or the transverse (2–3) plane of the carbon fiber

Fig. 4 Schematic of a FFRC lamina containing wavy CNTs (with permission from [16])

estimated. The CNT wave is defined by

x = A sin(ωy) or z = A sin(ωy); ω = nπ/Ln (1)

corresponding to the plane of CNT waviness being coplanar with the 1–3 plane and the 2–3 plane, respectively.Here, A and Ln denote the amplitude of CNT wave and the linear distance between CNT ends, respectively,and n represents the number of CNT waves. The running length (Lnr ) of the CNT wave is given by

Lnr =∫ Ln

0

√1 + A2ω2 cos2 (ωy) dy (2)

where the angle φ shown in Fig. 3 is given by

tan φ = dx/dy = Aω cos(ωy) or tan φ = dz/dy = Aω cos (ωy) (3)

corresponding to the plane of CNT waviness being coplanar with the 1–3 plane and the 2–3 plane, respectively.Note that for a particular value of ω, the value of φ varies with the amplitude of the CNT wave.

Figure 4 shows a schematic of the FFRC lamina in which the fuzzy fibers forming a hexagonal arrayare reinforced in the polymer matrix. The effective elastic properties of the FFRC containing wavy CNTsbeing coplanar with either the longitudinal plane or the transverse plane of the carbon fiber can be determinedby following the systematic modeling scheme developed in Ref. [16]. However, for the sake of brevity, thedevelopment of such modeling scheme can be found in that reference. In this paper, the focus is on thedevelopment of an FE model for investigating the effect of CNT waviness on the damping characteristics ofthe laminated FFRC shells.

Effect of carbon nanotube waviness

3 Basic equations

Figure 5 illustrates the longitudinal and transverse cross sections of the laminated FFRC shell integrated withthe two patches of ACLD treatment on its top circumferential surface. Such laminated FFRC shell is composedof N homogeneous and orthotropic layers, and the layers are assumed to be perfectly bonded together. Thelength, thickness and average radius of the shell are denoted by a, h and R, respectively. The thickness of thePZC layer is denoted by h p, and that of the viscoelastic layer of the ACLD treatment is represented by hv .The constraining layer of the patches of ACLD treatment is considered to be made of vertically or obliquelyreinforced 1–3 PZC material in which the piezoelectric fibers are coplanar with the xz plane as shown inFig. 6. The piezoelectric fiber orientation angle in the PZC layer with respect to the z axis is denoted by ψ .The mid-plane of the substrate FFRC shell is considered as the reference plane. The origin of the curvilinearlaminate coordinate system (x, y, z) is located on the mid-plane of the substrate FFRC shell such that thelines x = 0 and x = a represent the boundaries of the FFRC shell. The thickness coordinates (z) of thetop and bottom surfaces of any (kth) layer of the overall laminated FFRC shell are represented by hk+1 andhk (k = 1, 2, 3, . . . , N + 2), respectively.

The kinematics of deformations of the overall FFRC shell based on the FSDT has been demonstrated inFig. 7. As shown in this figure, u0 and v0 are the generalized translational displacements of a point (x, y) onthe reference plane (z = 0) of the substrate FFRC shell along the x and y axes, respectively; θx , φx and γx arethe generalized rotations of the normals to the middle planes of the substrate FFRC shell, viscoelastic layerand PZC layer, respectively, in the xz plane while θy , φy and γy represent their generalized rotations in theyz plane. Following the kinematics of deformations demonstrated in Fig. 7, the normal displacements at any

Fig. 5 Longitudinal and transverse cross sections of the laminated FFRC shell integrated with the two ACLD patches composedof 1–3 PZC layer

Fig. 6 Schematic of a lamina of the vertically reinforced 1– 3 PZC in which the piezoelectric fibers are coplanar with the verticalxz plane

S. I. Kundalwal, S. A. Meguid

Fig. 7 Deformation of any transverse cross section of the laminated FFRC shell integrated with the ACLD treatment which isparallel to a the xz plane and b the yz plane

point lying in any layer of the overall FFRC shell in the x and y directions represented by u and v, respectively,can be written as follows:

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) , (4)

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) (5)

in which a function within the bracket 〈 〉 represents the appropriate singularity function. Since the transversenormal strain is usually considered as negligible for thin structures, the radial displacement (w) is assumedto be linearly varying across the thickness of the substrate FFRC shell, viscoelastic layer and PZC layer.Consequently, the transverse displacement at any point in the overall FFRC shell can be written 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) (6)

in which w0 equals the transverse displacement of a point on the reference plane; θz , φz and γz are thegeneralized displacements representing the gradients of the transverse displacement in the substrate FFRCshell, viscoelastic layer and PZC layer, respectively, with respect to the thickness coordinate (z).

For convenience, the generalized displacement variables are separated into the 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. (7)

In order to implement the selective integration rule for avoiding the so-called shear locking problem in thethin laminated FFRC shell, the state of strains at any point in the overall FFRC shell is grouped into two strain

Effect of carbon nanotube waviness

vectors separating the transverse shear strains as follows:

{εb} = [εx εy εxy εz

]T and {εs} = [εxz εyz

]T (8)

where εx , εy and εz are the normal strains along the x, y and z directions, respectively; εxy is the in-plane shear strain; εxz and εyz are the transverse shear strains. Combining Eqs. (4)–(6) and (8), the vectors{εb}c , {εb}v and {εb}p defining the state of in-plane and transverse normal strains at any point in the substrateFFRC shell, viscoelastic layer and PZC layer, respectively, can be expressed as

{εb}c = {εbt } + [Z1] {εbr } , (9)

{εb}v = {εbt } + [Z2] {εbr } , (10)

{εb}p = {εbt } + [Z3] {εbr } , (11)

whereas the vectors {εs}c, {εs}v and {εs}p defining the state of transverse shear strains at any point in thesubstrate FFRC shell, viscoelastic layer and PZC layer, respectively, can be expressed as

{εs}c = {εst } + [Z4] {εsr } , (12)

{εs}v = {εst } + [Z5] {εsr } , (13)

{εs}p = {εst } + [Z6] {εsr } . (14)

The generalized strain vectors appearing in Eqs. (9)–(14) 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 θx θy γx γy

∂θz

∂x

∂θz

∂y

∂φz

∂x

∂φz

∂y

∂γz

∂x

∂γz

∂y

]T

.

Also, the various matrices [Z1], [Z2], [Z3], [Z4], [Z5] and [Z6] appearing in Eqs. (9)–(14) are given by

[Z1] =

⎡⎢⎢⎣

z 0 0 0 0 0 0 0 0 0 0 00 z 0 z/R 0 0 0 0 0 0 0 00 0 z 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0

⎤⎥⎥⎦ ,

[Z2] =

⎡⎢⎢⎣

h/2 0 0 0 z − h/2 0 0 0 0 0 0 00 h/2 0 h/2R 0 z − h/2 0 (z − h/2)/R 0 0 0 00 0 h/2 0 0 0 z − h/2 0 0 0 0 00 0 0 h/2 0 0 0 1 0 0 0 0

⎤⎥⎥⎦ and

[Z3] =

⎡⎢⎢⎣

h/2 0 0 0 hv 0 0 0 z − hN+2 0 0 00 h/2 0 h/2R 0 hv 0 hv/R 0 z − hN+2 0 (z − h/2)/R0 0 h/2 0 0 0 hv 0 0 0 z − hN+2 00 0 0 h/2 0 0 0 hv 0 0 0 1

⎤⎥⎥⎦ ,

[Z4] =[

1 0 0 0 0 0 z 0 0 0 0 00 z/R 0 0 0 0 0 z 0 0 0 0

],

[Z5] =[

h/2R 0 1 0 0 0 h/2 0 z − h/2 0 0 00 −h/2R 0 1 − (z − h/2)/R 0 0 0 h/2 0 z − h/2 0 0

]and

[Z6] =[

h/2R 0 hv/R 0 1 0 h/2 0 hv 0 z − hN+2 00 −h/2R 0 −hv/R 0 1 − (z − hN+2)/R 0 h/2 0 hv 0 z − hN+2

].

Corresponding to the description of the state of strains, the state of stresses at any point in the overall FFRCshell is described by the state of in-plane and out-of-plane stresses {σb}, and the state of transverse shearstresses {σs} as follows:

{σb} = [σx σy σxy σz

]T and {σs} = [σxz σyz

]T (15)

S. I. Kundalwal, S. A. Meguid

where σx , σy and σz are the normal stresses along the x, y and z directions, respectively; σxy is the in-planeshear stress; σxz and σyz are the transverse shear stresses.

The constitutive relations for the material of any orthotropic kth layer of the substrate FFRC shell are givenby {

σ kb

}=[Ck

b

] {εk

b

}and

{σ k

s

}=[Ck

s

] {εk

s

}; (k = 1, 2, 3, . . . , N ). (16)

The constraining PZC layer will be subjected to the applied electric field (Ez) acting along its thicknessdirection. Thus, the constitutive relations for the material of the PZC layer [27] can be expressed as{

σ kb

}=[Ck

b

] {εk

b

}+[Ck

bs

] {εk

s

}− {eb} Ez; k = N + 2, (17)

{σ k

s

}=[Ck

bs

]T {εk

b

}+[Ck

s

] {εk

s

}− {es} Ez, (18)

Dz = {eb}T{εk

b

}+ {es}T

{εk

s

}+ ε33 Ez (19)

in which Dz is the applied electric displacement along the z direction and ε33 is the transformed dielectricconstant. The transformed elastic coefficient matrices

[Ck

b

]and

[Ck

s

], and the transformed piezoelectric coeffi-

cient vectors {eb} and {es}, appearing in above equations, referred to the laminate coordinate system (x, y, z),are given by

[Ck

b

]=

⎡⎢⎢⎣

Ck11 Ck

12 Ck16 Ck

13Ck

12 Ck22 Ck

26 Ck23

Ck16 Ck

26 Ck66 Ck

36Ck

13 Ck23 Ck

36 Ck33

⎤⎥⎥⎦ ,

[Ck

s

]=[

Ck55 Ck

45Ck

45 Ck44

], {eb} =

⎧⎪⎨⎪⎩

e31e32e36e33

⎫⎪⎬⎪⎭ and {es} =

{e35e34

}. (20)

It may be noted that the transverse shear strains are coupled with the in-plane strains due to the orientationof the piezoelectric fibers in the vertical xz plane of the PZC layer, and the corresponding coupling elasticcoefficient matrix [C N+2

bs ] is given by

[C N+2

bs

]=[

C N+215 C N+2

25 0 C N+235

0 0 C N+246 0

]T

. (21)

The viscoelastic material is assumed to be linearly viscoelastic and isotropic, and is modeled by thecommonly used complex modulus approach, which is a frequency domain model. When the complex modulusapproach is employed, the constitutive relations for the viscoelastic material can also be represented by Eq.(16) with k = N + 1, while the complex shear modulus (G) and Young’s modulus (E) of the viscoelasticmaterial are given by [45]

G = G ′ (1 + iη) and E = 2G (1 + υ) (22)

where G ′ is the storage modulus,υ is Poisson’s ratio, andη is the loss factor at a particular operating temperatureand frequency. Using Eq. (22), the elastic coefficients of the viscoelastic material can be determined, and theresulting elastic coefficient matrix [C N+1

i j ] turns out to be a complex matrix.

4 Finite element (FE) model

In this section, we develop an FE model to study the frequency response functions (FRFs) of the transversedisplacement at the free end of the laminated FFRC shells. One end of a cantilevered laminated FFRC shellis constrained at x = 0, and a point load is applied to another end at x = a. The FRF for the transversedisplacement is evaluated at a point (a, 0, h/2) on the top surface of the FFRC shells. A time harmonic pointforce (1 N) is considered to act at the same point (a, 0, h/2) on the FFRC shell to excite the first few modesof the FFRC shells. The principle of virtual work is employed to derive the governing equations of the overallFFRC shell/ACLD system, such that

N+2∑k=1

∫Ω

(δ{εk

b

}T {σ k

b

}+ δ

{εk

s

}T {σ k

s

}− δEz ε33 Ez − δ {dt }T ρk {dt

})dΩ −

∫Aδ {dt }T { f } dA = 0,

(23)

Effect of carbon nanotube waviness

in which ρk is the mass density of the kth layer, { f } is the externally applied surface traction vector actingover the outer surface area (A) of the FFRC shell, and Ω represents the volume of the kth layer in concern.

The overall FFRC shell has been discretized by eight-noded quadrilateral isoparametric shell elements.Figure 8 demonstrates an FE mesh (12 × 8) of eight-nodded quadrilateral isoparametric elements. FollowingEq. (7), the generalized displacement vectors associated with the i th (i = 1, 2, 3, . . . , 8) node of an elementcan be represented as

{dti } = [u0i v0i w0i ]T and {dri } = [

θxi θyi θzi φxi φyi φzi γxi γyi γzi]T. (24)

Thus, the generalized displacement vectors {dt } and {dr } at any point within the element can be expressed interms of the nodal generalized displacement vectors

{de

t

}and

{de

r

}as follows:

{dt } = [Nt ]{de

t

}and {dr } = [Nt ]

{de

r

}(25)

in which

[Nt ] = [Nt1 Nt2 . . . Nt8]T , [Nr ] = [Nr1 Nr2 . . . Nr8]T , Nti = ni It , Nri = ni Ir ,{de

t

} =[{

det1

}T {de

t2

}T. . .

{de

t8

}T]T

and{de

r

} =[{

der1

}T {de

r2

}T. . .

{de

r8

}T]T, (26)

wherein It and Ir are the (3 × 3) and (9 × 9) identity matrices, respectively, and ni is the shape function of thenatural coordinates associated with the i th node. Using Eqs. (9)–(14) and (25), the strain vectors at any pointwithin the element can be expressed in terms of the nodal generalized displacement vectors as follows:

{εb}c = [Btb]{de

t

} + [Z1] [Brb]{de

r

}, {εb}v = [Btb]

{de

t

} + [Z2] [Brb]{de

r

},

{εb}p = [Btb]{de

t

} + [Z3] [Brb]{de

r

}, {εs}c = [Bts]

{de

t

} + [Z4] [Brs]{de

r

},

{εs}v = [Bts]{de

t

} + [Z5] [Brs]{de

r

}and {εs}p = [Bts]

{de

t

} + [Z6] [Brs]{de

r

}, (27)

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 . (28)

The elements of matrices [Btb], [Brb], [Bts] and [Brs] are presented in the Appendix. On substitution of Eqs.(17)–(19) and (27) into Eq. (23), and recognizing that Ez = V/h p with V being the applied voltage across

Fig. 8 FE mesh (12 × 8) of a lamina of composite

S. I. Kundalwal, S. A. Meguid

the thickness of the PZC layer, one can derive the following open-loop equations of motion of an elementintegrated with the ACLD treatment:

[Me] {de

t

} + [K e

tt

] {de

t

} + [K e

tr

] {de

r

} ={

Fetp

}V + {

Fe} , (29)[K e

r t

] {de

t

} + [K e

rr

] {de

r

} ={

Fer p

}V . (30)

The elemental mass matrix[Me

], the elemental stiffness matrices

[K e

t t

],[K e

tr

],[K e

r t

],[K e

rr

], the elemental

electro-elastic coupling vectors {Fetp}, {Fe

r p}, the elemental load vector {Fe}, and the mass parameter (m)appearing in Eqs. (29) and (30) are given by

[Me] =

∫ ae

0

∫ be

0m [Nt ]

T [Nt ] dxdy,[K e

t t

] = [K e

tb

] + [K e

ts

] + [K e

tbs

]pb + [

K etbs

]ps ,[

K etr

] = [K e

trb

] + [K e

trs

] + 1

2

([K e

trbs

]pb + [

K er tbs

]Tpb + [

K etrbs

]ps + [

K er tbs

]Tps

),

[K e

r t

] = [K e

tr

]T,

[K e

rr

] = [K e

rrb

] + [K e

rrs

] + [K e

rrbs

]pb + [

K errbs

]ps ,{

Fetp

}= {

Fetb

}p + {

Fets

}p ,

{Fe

r p

}= {

Ferb

}p + {

Fers

}p ,

{Fe} =

∫ ae

0

∫ be

0[Nt ]

T { f } dxdy and m =N+2∑k=1

ρk (hk+1 − hk) (31)

where ae and be are the length and circumferential width of an element under consideration.[K e

t t

],[K e

tr

]and[

K err

]are stiffness matrices of sizes (24 × 24), (24 × 72) and (72 × 72), respectively, and {Fe} is a vector of

size (24 × 1). The explicit forms of these matrices and vector are not shown here for the sake of clarity. Thevarious elemental stiffness matrices and the electro-elastic coupling vectors appearing in Eq. (31) are

[K e

tb

] =∫

A[Btb]T ([Dtb] + [Dtb]v + [Dtb]p

)[Btb] dxdy,

[K e

ts

] =∫

A[Bts]T ([Dts] + [Dts]v + [Dts]p

)[Bts] dxdy,

[K e

tbs

]pb =

∫A

[Btb]T [Dtbs]p [Bts] dxdy,[K e

tbs

]ps =

∫A

[Bts]T [Dtbs]p [Btb] dxdy,

[K e

trb

] =∫

A[Btrb]T ([Dtrb] + [Dtrb]v + [Dtrb]p

)[Brb] dxdy,

[K e

trbs

]pb =

∫A

[Btb]T [Dtrbs]p [Brs] dxdy,[K e

r tbs

]pb =

∫A

[Brb]T [Drtbs]p [Bts] dxdy,

[K e

trbs

]ps =

∫A

[Bts]T [Drtbs]Tp [Brb] dxdy,

[K e

r tbs

]ps =

∫A

[Brs]T [Dtrbs]Tp [Btb] dxdy,

[K e

trs

] =∫

A[Bts]T ([Dtrs] + [Dtrs]v + [Dtrs]p

)[Brs] dxdy,

[K e

rrb

] =∫

A[Brb]T ([Drrb] + [Drrb]v + [Drrb]p

)[Brb] dxdy,

[K e

rrs

] =∫

A[Brs]T ([Drrs] + [Drrs]v + [Drrs]p

)[Brs] dxdy,

[K e

rrbs

]pb =

∫A

[Brb]T [Drrbs]p [Brs] dxdy,[K e

rrbs

]ps =

∫A

[Brs]T [Drrbs]Tp [Brb] dxdy,

{Fe

tb

}p =

∫A

[Btb]T {Dtb}p dxdy,{

Ferb

}p =

∫A

[Brb]T {Drb}p dxdy,

{Fe

ts

}p =

∫A

[Bts]T {Dts}p dxdy and{

Fers

}p =

∫A

[Brs]T {Drs}p dxdy. (32)

Effect of carbon nanotube waviness

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

[Dtb] =N∑

k=1

∫ hk+1

hk

[Ck

b

]dz, [Dtrb] =

N∑k=1

∫ hk+1

hk

[Ck

b

][Z1] dz,

[Drrb] =N∑

k=1

∫ hk+1

hk

[Z1]T[Ck

b

][Z1] dz, [Dts] =

N∑k=1

∫ hk+1

hk

[Ck

s

]dz,

[Dtrs] =N∑

k=1

∫ hk+1

hk

[Ck

b

][Z4] dz, [Drrs] =

N∑k=1

∫ hk+1

hk

[Z4]T[Ck

s

][Z4] dz,

[Dtb]v = hv[C N+1

b

], [Dtrb]v =

∫ hN+2

hN+1

[C N+1

b

][Z2] dz,

[Drrb]v =∫ hN+2

hN+1

[Z2]T[C N+1

b

][Z2] dz, [Dts]v = hv

[C N+1

s

],

[Dtrs]v =∫ hN+2

hN+1

[C N+1

s

][Z5] dz, [Drrs]v =

∫ hN+2

hN+1

[Z5]T[C N+1

s

][Z5] dz,

[Dtb]p = h p

[C N+2

b

], [Dtrb]p =

∫ hN+3

hN+2

[C N+2

b

][Z3] dz,

[Drrb]p =∫ hN+3

hN+2

[Z3]T[C N+2

b

][Z3] dz, [Dts]p = h p

[C N+2

s

],

[Dtrs]p =∫ hN+3

hN+2

[C N+2

s

][Z6] dz, [Drrs]p =

∫ hN+3

hN+2

[Z6]T[C N+2

s

][Z6] dz,

[Dtbs]p =∫ hN+3

hN+2

[C N+2

bs

]dz, [Dtrbs]p =

∫ hN+3

hN+2

[C N+2

bs

][Z6] dz,

[Drtbs]p =∫ hN+3

hN+2

[Z3]T[C N+2

bs

]dz, [Drrbs]p =

∫ hN+3

hN+2

[Z3]T[C N+2

bs

][Z6] dz,

{Dtb}p =∫ hN+3

hN+2

− {eb} /h pdz, {Drb}p =∫ hN+3

hN+2

− [Z3]T {eb} /h pdz,

{Dts}p =∫ hN+3

hN+2

− {es} /h pdz and {Drs}p =∫ hN+3

hN+2

− [Z6]T {es} /h pdz. (33)

Finally, the elemental equations of motion are assembled to obtain the open-loop global equations of motionof the overall FFRC shell integrated with the ACLD patches as follows:

[M]{

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

q∑j=1

{F j

tp

}V j + {F} (34)

and

[Krt ] {X} + [Krr ] {Xr } =q∑

j=1

{F j

r p

}V j (35)

where [M] is the global mass matrix of size (864 × 864); [Ktt ], [Ktr ], [Krt ] and [Krr ] are the global stiffnessmatrices of sizes (864 × 864), (864 × 1,344), (1,344 × 864) and (1,344 × 1,344), respectively; {X} and{Xr } are the vectors of global nodal translational and rotational degrees of freedom of sizes (864 × 1) and(1,344×1), respectively;

{Ftp

}and

{Fr p

}are the global electro-elastic coupling vectors corresponding to the

j th ACLD patch of sizes (864 × 1) and (1,344 × 1), respectively; V j is the voltage applied to this patch; q is

S. I. Kundalwal, S. A. Meguid

the number of ACLD patches; and {F} is the global nodal force vector of size (864 × 1). The explicit formsof these matrices and vectors are not shown here for the sake of brevity.

In order to activate each patch of the ACLD treatment, a simple feedback control law has been implemented.The control voltage supplied to the constraining PZC layer of each patch is considered to be negativelyproportional to the velocity of a point on the outer circumferential surface of the FFRC shell. According tothis law, the control voltage supplied to the j th ACLD patch can be expressed in terms of the derivatives ofthe global nodal degrees of freedom as follows:

V j = −k jd w = −k j

d

[U j

t

] {X} − k j

d (h/2)[U j

r

] {Xr}

(36)

in which k jd is the control gain for the j th patch, and [U j

t ] and [U jr ] are the unit vectors for expressing

the transverse velocity of a point in terms of the derivative of the global nodal generalized displacements.Substituting Eq. (36) into Eqs. (34) and (35), the final equations of motion governing the closed-loop dynamicsof the overall FRC shell/ACLD system can be derived as follows:

[M]{

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

q∑j=1

k jd

{F j

tp

} [U j

t

] {X} +

q∑j=1

k jd (h/2)

{F j

tp

} [U j

r

] {X} = {F} ,

(37)

and

[Krt ] {X} + [Krr ] {Xr } +q∑

j=1

k jd

{F j

r p

} [U j

t

] {X} +

q∑j=1

k jd (h/2)

{F j

r p

} [U j

r

] {X} = 0. (38)

5 Numerical results and discussion

In this section, the numerical results are evaluated by using the FE model developed in the preceding section.Laminated symmetric/antisymmetric cross-ply and antisymmetric angle-ply thin cantilever cylindrical FFRCshells integrated with the two patches of ACLD treatment are considered for presenting the numerical results.The ACLD patches are placed on the outer circumferential surface of the laminated FFRC shell as demonstratedin Fig. 5 and are 180◦ apart from each other. The length of each patch is 67 % of the length of the shell, whilethe width of each patch is assumed to be one sixth of the length of the outer circumferential width of the shell.It should be noted that the locations of the ACLD patches correspond to an optimal placement of the ACLDtreatment such that the energy dissipation corresponding to the first two modes becomes maximum [46]. Toevaluate the results, the value of the carbon fiber volume fraction in the FFRC is kept constant to 0.7. Thedetermination of the CNT volume fraction (VCNT) in the FFRC is an important issue. It is obvious that theconstructional feature of the FFRC imposes a constraint on the maximum value of VCNT. Since in the polymermatrix nanocomposite the formation of the interphase is considered between a CNT and the surroundingpolymer matrix, the surface-to-surface distance between two adjacent CNTs at their roots is considered as 1.7nm [16]. The CNT volume fraction in the FFRC can be determined based on the surface-to-surface distance atthe roots of two adjacent CNTs (1.7 nm), CNT diameter (dn), running length of CNT (Lnr ), and carbon fibervolume fraction (v f ) as follows [16]:

VCNT = πd2n Lnr

d(dn + 1.7)2v f . (39)

The effect of CNT waviness on the damping characteristics of the laminated FFRC shells is investigated whenwavy CNTs are coplanar with either of the two mutually orthogonal planes. For such an investigation, thevalues of CNT wave frequency (ω) and maximum amplitude (A) for the zigzag (10, 0) CNT are consideredas 30π/Ln and 100dn , respectively. The material properties of the FFRC with respect to material coordinatesystem have been estimated following the systematic micromechanical scheme developed in Ref. [16]. Table 1summarizes the effective elastic properties of the FFRC containing either straight CNTs or wavy CNTs fordifferent values of VCNT and ω. From Table 1, it can be observed that the transverse effective elastic propertiesof the FFRC are significantly improved due to the radial growing of CNTs on the circumferential surfaces of

Effect of carbon nanotube waviness

Table 1 Material properties of the base composite and the FFRC (v f = 0.7)

Compositematerial

C11(GPa)

C12(GPa)

C22(GPa)

C23(GPa)

C33(GPa)

C44(GPa)

C55(GPa)

ρ (kg/m3)

Base composite(VCNT = 0)

167.7390 8.3014 15.9179 9.2379 15.9179 3.3400 4.3860 1,571

FFRC (VCNT =0.0150, ω = 0)

168.4045 9.2890 22.9629 11.5712 22.9629 5.6958 5.0660 1,574

FFRC (2–3plane, VCNT =0.1037, ω =30π/Ln)

160.5136 8.9645 24.9131 12.7065 24.9131 6.1033 59.2316 1,592

FFRC (1–3plane, VCNT =0.1037, ω =30π/Ln)

292.1637 11.5644 32.5068 26.0236 32.5068 3.2416 9.5223 1,592

Table 2 Comparisons of the fundamental natural frequency parameters (λ) of the clamped-free laminated composite shells

Shell type Source R/h a/R λ

0◦/90◦/0◦ Present FEM 20 5 0.4910Analytical [48] 20 5 0.4899

0◦/90◦ Present FEM 20 5 0.5620Analytical [48] 20 5 0.5581

carbon fibers over those of the composite without CNTs (VCNT = 0). The thicknesses of the laminated FFRCshells, constraining 1–3 PZC layer and viscoelastic layer, are considered as 0.003 m, 250 µm and 200 µm,respectively, while the value of R/h for the shells is taken as 50. The value of the length (a) of the laminatedFFRC shells is considered as 1 m. The materials of the piezoelectric fiber and the matrix of the PZC layerare considered as PZT5H and epoxy, respectively. Considering 60 % piezoelectric fiber volume fraction in thePZC layer, the following elastic and piezoelectric properties of the vertically reinforced 1–3 PZC layer withrespect to its material coordinate system are obtained from the existing micromechanics model [27] and areused for evaluating the numerical results:

C11 = 9.293 GPa, C12 = 6.182 GPa, C13 = 6.054 GPa, C33 = 35.44 GPa,

C23 = C13, C44 = 1.58 GPa, C66 = 1.544 GPa, C55 = C44, e31 = −0.1902 C/m2,

e32 = e31, e33 = 18.4107 C/m2, e24 = 0.004 C/m2, e15 = e24 and ρ = 5,090 kg/m3.

The viscoelastic material used by Chantalakhana and Stanway [47] is considered in this study to evaluatethe numerical results. The loss factor of this viscoelastic material remains invariant within a frequency range(0–600 Hz) of interest. The values of the complex shear modulus, Poisson’s ratio and density of the viscoelasticlayer are 20(1 + i) MPa, 0.49 and 1,140 kg m−3, respectively [47].

In order to verify the validity of the present FE model, the fundamental natural frequencies of the laminatedcomposite shells integrated with the inactivated patches of ACLD treatment of negligible thickness are firstcompared with the existing analytical results [48] of identical laminated composite shells without integratedwith the patches. Table 2 summarizes the outcome of this comparison. A nondimensional frequency parameter(λ) is used for comparing the fundamental natural frequencies of the laminated composite shells and is definedas

λ = ω(a2/h

)√ρ/ET (40)

in which ω represents the natural frequency of the overall shell; ρ and ET are the density and the transverseYoung’s modulus, respectively, of the orthotropic layers of the substrate shell. It may be observed from Table 2that the two sets of results predicted by the FE and analytical models are in excellent agreement, validatingthe FE model developed in this study.

Open- and closed-loop behavior of the laminated FFRC shells is studied by the FRF for the transversedisplacement at the free end of the laminated FFRC shells. The control gains (Kd) are selected as 1,000 and2,000 to control the first few modes of the laminated FFRC shells. For the sake of clarity in plots, a range offrequencies only around the first two or three modes of vibrations is presented. Figure 9a illustrates the FRF

S. I. Kundalwal, S. A. Meguid

100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10-4

Frequency (Hz)

w (a

, 0, h

/2) (

m)

Uncontrolled K

d = 1000

Kd = 2000

0.8

1

1.2

1.4

x 10-4

(a)

100 150 200 250 3000

20

40

60

80

100

120

Frequency (Hz)

Con

trol

Vol

tage

(Vol

t)

Kd = 1000

Kd = 2000

(b)

Fig. 9 a FRF for the transverse deflection (w) and b control voltages for the cantilever symmetric cross-ply (0◦/90◦/0◦) laminatedFFRC shells (ψ = 0◦, ω = 0)

100 150 200 250 3000

0.2

0.4

0.6

0.8

1x 10-4

Frequency (Hz)

w (a

, 0, h

/2) (

m)

VCNT

= 0

VCNT

= 0.0150, ω = 0V

CNT = 0.1037, ω = 30 π/L

n, 1-3 Plane

VCNT

= 0.1037, ω = 30 π/Ln, 2-3 Plane

(a)

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1x 10-4

Frequency (Hz)

w (a

, 0, h

/2) (

m)

VCNT

= 0

VCNT

= 0.0150, ω = 0V

CNT = 0.1037, ω = 30π /L

n, 1-3 Plane

VCNT

= 0.1037, ω = 30π /Ln, 2-3 Plane

(b)

Fig. 10 FRF for the transverse displacement (w) of the cantilever a symmetric cross-ply (0◦/90◦/0◦) and b antisymmetricangle-ply (−45◦/45◦/− 45◦/45◦) laminated shells (Kd = 2,000, ψ = 45◦)

of the cantilever symmetric cross-ply (0◦/90◦/0◦) laminated FFRC shells when the orientation angle of thepiezoelectric fibers (ψ) in the PZC layer is 0◦. This figure displays both uncontrolled and controlled responsesand clearly reveals that the active patches significantly improve the damping characteristics of the laminatedFFRC shell over the passive damping (uncontrolled). As the gain increases, the attenuation of amplitude ofvibrations also increases. Figure 9b demonstrates the control voltages required for the controlled responsesshown in Fig. 9a. It is observed from this figure that the maximum control voltage required to achieve theattenuation of amplitude of vibration is only 120 V when the value of Kd is 2,000. The subsequent results forinvestigating the influence of CNT waviness on the damping characteristics of the cantilever laminated FFRCshells are presented considering the value of Kd = 2,000.

The performance of the ACLD treatment controlling the thin laminated composite structural elements issignificantly affected by the piezoelectric fiber orientation angle in the two mutually orthogonal vertical planes(xz and yz) of the PZC layer [27–30,35,49]. Ray and coworkers [27,35] reported that the best performance ofthe constraining layer of the ACLD treatment is achieved for controlling the first two modes of the symmetriccross-ply, antisymmetric angle-ply and antisymmetric cross-ply laminated composite shells when the valuesof piezoelectric fiber orientation angle (ψ) are 45◦, 45◦ and 0◦ in the vertical xz plane of the PZC layer,respectively. Therefore, the FRF of the symmetric cross-ply, antisymmetric angle-ply and antisymmetric cross-ply laminated FFRC shells corresponding to these optimum values of ψ has been presented. Figure 10a, billustrates the effect of CNT waviness on the FRF of the cantilever symmetric cross-ply (0◦/90◦/0◦) and

Effect of carbon nanotube waviness

150 200 250 300 3500

1

2

x 10-4

Frequency (Hz)

w (a

, 0, h

/2) (

m)

VCNT

= 0

VCNT

= 0.0150, ω = 0V

CNT = 0.1037, ω = 30π /L

n, 1-3 Plane

VCNT

= 0.1037, ω = 30π /Ln, 2-3 Plane

Fig. 11 FRF for the transverse displacement (w) of the cantilever antisymmetric cross-ply (0◦/90◦/0◦/90◦) laminated shells(Kd = 2,000, ψ = 0◦)

antisymmetric angle-ply (−45◦/45◦/ − 45◦/45◦) laminated FFRC shells when the value of ψ is 45◦. Fromthese figures, the following observations can be made:

i. Compared with the FFRC containing straight CNTs (ω = 0): amplitudes of the first and second modes ofvibrations of the symmetric cross-ply laminated FFRC shells are attenuated by 15 and 22 %, respectively,whereas the first and second natural frequencies of the same are enhanced by 30 and 22 %, respectively,when the waviness of CNTs is coplanar with the 1–3 plane.

ii. Compared with the FFRC containing straight CNTs (ω = 0): amplitudes of the first and second modesof vibrations of the antisymmetric angle-ply laminated FFRC shells are attenuated by 19 and 23 %,respectively, whereas the first and second natural frequencies of the same are enhanced by 30 and 34 %,respectively, when the waviness of CNTs is coplanar with the 1–3 plane.

iii. Waviness of CNTs being coplanar with 2–3 plane enhances the first few natural frequencies and adverselyaffects the attenuation of amplitude of vibrations of the symmetric cross-ply and antisymmetric angle-plylaminated FFRC shells compared to those of the FFRC containing either straight CNTs or wavy CNTsbeing coplanar with the 1–3 plane.

iv. The attenuating capability of the ACLD patches becomes maximum for enhancing the damping char-acteristics of the symmetric cross-ply and antisymmetric angle-ply laminated FFRC shells when thewaviness of CNTs is coplanar with the 1–3 plane. This is attributed to the fact that the elastic propertiesof the FFRC are significantly improved when the waviness of CNTs is coplanar with the 1–3 plane (seeTable 1) which eventually enhances the attenuation of amplitude of vibrations and the natural frequenciesof the laminated FFRC shells.

Let us now demonstrate the effect of CNT waviness on the damping characteristics of the antisymmetric cross-ply laminated FFRC shells. Figure 11 illustrates the damping characteristics of the antisymmetric cross-ply(0◦/90◦/0◦/90◦) laminated FFRC shells when the value of ψ is 0◦. This figure illustrates that the amplitudeof the fundamental mode of vibration of the antisymmetric cross-ply laminated FFRC shells is significantlyattenuated when the waviness of CNTs is coplanar with the 2–3 plane over those of the other laminated shellswith and without CNTs. Compared with the FFRC containing straight CNTs (ω = 0), it is observed that theamplitude of the first mode is attenuated by 72 % if the waviness of CNTs is coplanar with the 2–3 plane, andthe corresponding value of VCNT is 0.113. It may be importantly observed from this figure that the naturalfrequencies of the antisymmetric cross-ply laminated FFRC shells are significantly improved if CNTs arepresent on the circumferential surfaces of carbon fibers.

6 Conclusions

In this article, a three-dimensional FE analysis is carried out to investigate the influence of CNT waviness onthe damping behavior of the cantilever laminated hybrid composite—fuzzy fiber-reinforced composite—shellsintegrated with the ACLD patches when wavy CNTs are coplanar with either of the two mutually orthogonalplanes of the carbon fiber. The following main inferences are drawn from our study:

S. I. Kundalwal, S. A. Meguid

i. A pronounced effect of CNT waviness on the damping characteristics of the laminated FFRC shells isobserved.

ii. The performance of the ACLD patches is maximized for controlling the first two modes of the symmetriccross-ply and antisymmetric angle-ply laminated FFRC shells when the waviness of CNTs is coplanarwith the longitudinal (1–3) plane of the carbon fiber. For the cantilever antisymmetric cross-ply lami-nated shells, attenuating capability of the ACLD patches for controlling the fundamental mode becomesmaximum when the waviness of CNTs is coplanar with the transverse (2–3) plane of the carbon fiber.

iii. Wavy CNTs can be utilized effectively to improve the damping behavior of the multifunctional nano-tailored composite structures exploiting their remarkable direction-oriented elastic properties.

Acknowledgments This work was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada andthe Discovery Accelerator Supplement by NSERC.

Appendix

The various sub-matrices Btbi , Btsi , Brbi and Brsi appearing in Eq. (28) read:

Btbi =

⎡⎢⎢⎢⎣

∂ni∂x 0 00 ∂ni

∂y 1/R∂ni∂y

∂ni∂x 0

0 0 0

⎤⎥⎥⎥⎦ , Btsi =

[0 0 ∂ni

∂x0 −1/R ∂ni

∂y

], Brbi =

⎡⎣ Brbi 0 0

0 Brbi 00 0 Brbi

⎤⎦ ,

Brbi =

⎡⎢⎢⎢⎣

∂ni∂x 0 00 ∂ni

∂y 0∂ni∂y

∂ni∂x 0

0 0 1

⎤⎥⎥⎥⎦ , Brsi =

⎡⎢⎢⎢⎢⎢⎢⎣

I 0 00 I 00 0 I

Brsi 0 00 Brsi 00 0 Brsi

⎤⎥⎥⎥⎥⎥⎥⎦, Brsi =

[0 0 ∂ni

∂x0 0 ∂ni

∂y

]

0 =⎡⎢⎣

0 0 00 0 00 0 00 0 0

⎤⎥⎦ , 0 =

[0 0 00 0 0

], and I =

[1 0 00 1 0

],

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