Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates

Accepted Manuscript Free vibration analysis of arbitrarily shaped functionally graded Carbon Nanotubereinforced plates Nicholas Fantuzzi, Francesco Tornabene, Michele Bacciocchi, Rossana Dimitri PII:

S1359-8368(16)31901-1

DOI:

10.1016/j.compositesb.2016.09.021

Reference:

JCOMB 4513

To appear in:

Composites Part B

Received Date: 20 July 2016 Revised Date:

13 August 2016

Accepted Date: 8 September 2016

Please cite this article as: Fantuzzi N, Tornabene F, Bacciocchi M, Dimitri R, Free vibration analysis of arbitrarily shaped functionally graded Carbon Nanotube-reinforced plates, Composites Part B (2016), doi: 10.1016/j.compositesb.2016.09.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT FREE VIBRATION ANALYSIS OF ARBITRARILY SHAPED FUNCTIONALLY GRADED CARBON NANOTUBE-REINFORCED PLATES**

RI PT

Nicholas Fantuzzi1, Francesco Tornabene*1, Michele Bacciocchi1, Rossana Dimitri2

ABSTRACT. By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the

SC

reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix,

M AN U

CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some

TE D

additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different

EP

numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a

AC C

plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

KEYWORDS: A. Layered structures, B. Mechanical properties, C. Numerical analysis, C. Computational modelling, Functionally Graded Carbon Nanotubes.

1

DICAM - Department, School of Engineering and Architecture, University of Bologna, Italy. Department of Innovation Engineering, University of Salento, Lecce, Italy *Corresponding author: Francesco Tornabene, email: [email protected] web page: http://software.dicam.unibo.it/diqumaspab-project ** The results of this work have been presented at the 2016 International Workshop on Multiscale Innovative Materials and Structures (MIMS16), Cetara (Salerno), Italy, 28-30 October 2016. 2

1

ACCEPTED MANUSCRIPT 1.

INTRODUCTION

A laminated composite structure represents one of the most fascinating structural element due to its outstanding mechanical properties. Several lamination schemes made of different oriented plies are employed to design this kind of structure in order to reach specific static and dynamic responses. Similarly, various materials can be

RI PT

stacked for the same purposes. Recently, many papers have been published to investigate the structural behavior of such laminated composites [1-33]. With this aim, several authors have proposed different kinds of approaches in the last decades to deal with multilayered plates and shells. Among them, the work by Carrera [34] can be considered as the most famous higher-order approach for studying complex mechanical configurations. Later,

SC

the well-known Carrera Unified Formulation (CUF) has been generalized by Demasi [35], who introduced his Generalized Unified Formulation (GUF) as the next logical step of the previous approach. Starting from the

M AN U

work by Demasi [35], D’Ottavio [36] developed an optimized methodology termed Sublaminate Generalized Unified Formulation (S-GUF) to decrease the computation time by considering a particular structural theory for specific packages of layers. It should be noticed that the well-known Reissner-Mindlin theory or First-order Shear Deformation Theory (FSDT) can be obtained as a particular case of these more general higher-order approaches. This theory still represents a good compromise in terms of easy implementation, result accuracy and

TE D

computation time. Nevertheless, this aspect loses its validity if the mechanical configuration of a generic composite requires some particular expedients.

In general, most of the efforts of many scientists aims at developing and analyzing new class of composite

EP

materials. For instance, the well-known class of Functionally Graded Materials (FGMs) was introduced to reduce the inter-laminar stresses that cause the delamination phenomena [37-48]. As highlighted in the work by Ray and Reddy [49], the so-called smart composites raised to perform an active control and structural

AC C

monitoring. Nowadays, it is easy to find several examples in the literature which apply this class of composites [50-54]. Nevertheless, the most intriguing type of innovative composite medium is the class of nanocomposites, as illustrated in the review paper by Crainic and Marques [55]. Since the discoveries of CNTs by Iijima in the nineties [56, 57], these nanofibers made of Carbon have represented the most used reinforcing phase with a micrometric structure employing commonly in nanostructures. It should be noticed that in the literature different examples of nanofibers can be found. For this purpose, Mercan and Civalek in their work [58] proposed the use of Boron Nitride Nanotubes (BNNTs) as reinforcing fibers. Nevertheless, the nanoscale of these particles causes some difficulties related to the evaluation of the mechanical properties of the single nanoparticles. A huge

2

ACCEPTED MANUSCRIPT number of papers can be found in the literature to characterize the mechanical behavior of CNTs [59-67]. It should be noticed that sometimes the proposed values are considerably different. It is evident that this aspect affects also the overall mechanical properties of the CNT reinforced composites. The typical way for evaluating the global elastic properties of these composites consists in the theory of Mixture, as illustrated in the work by

RI PT

Shen [68]. The methodology in hand is employed in the works [69-82] to investigate the mechanical behavior of CNT reinforced composite structures. However, the paper of Shi et al. [83] represents a more valid alternative to evaluate the mechanical properties of these structures if compared to the previous approach. Indeed, Shi et al. [83] proposed a micromechanical model based on the agglomeration of CNTs when scattered in a polymeric

SC

matrix. To the best of the authors’ knowledge, this model can be used to represent more correctly the effective behavior of CNTs since it takes into account several aspects that are neglected in the classic theory of Mixture.

M AN U

According to this approach, the overall mechanical properties of the nanocomposite are evaluated by using the Mori-Tanaka scheme [84, 85]. The current method is followed in the works [86-88]. It is evident that from one side the structural response of a composite structure is affected by the medium which composes the stacking sequence. On the other hand, many studies are performed to design the shape of the structure to reach the maximum level of structural efficiency. For these purposes, discontinuities, holes and

TE D

cutouts, can influence the mechanical behavior of composite structures. In fact, their presence is unavoidable for practical requirements of many aerospace, automotive, mechanical and civil engineering applications. Classically, a solution to these problems is found numerically by means of the well-known Finite Element

EP

Method (FEM). Recently, these applications are studied through a strong-form based approach termed Strong Formulation Finite Element Method (SFEM) [89-98]. The main difference between SFEM and FEM is given by the fact that the former technique approximates the derivatives of the governing differential equations, whereas

AC C

in the latter the weak form of the same system is solved. In the present work, the SFEM method is employed to deal with the free vibration analysis of arbitrarily shaped plates reinforced by agglomerated CNTs. The Generalized Differential Quadrature (GDQ) method introduced by Shu [99] is the numerical tool used to find an approximated solution to the dynamic investigation at issue. A complete treatise concerning the GDQ method and some other correlated numerical techniques is shown in the review paper by Tornabene et al. [100]. In the recent years, the difficulties related to the representation of complex geometries have been overcome by using an isogeometric approach based on Computer-Aided Design (CAD). As illustrated in the work by Hughes et al. [101], CAD can be considered as the most efficient tool to define accurately arbitrary shapes. According to

3

ACCEPTED MANUSCRIPT the so-called Isogeometric Analysis (IGA) developed by Hughes et al. [101], the Non-Uniform Rational BSplines (NURBS) are employed to describe arbitrarily shaped domains due to their versatility and accuracy. On the other hand, the complete mathematical definition of these curves is illustrated in details in the books by Piegl and Tiller [102] and by Cottrell et al. [103]. This technique can be used to solve several kinds of structural

RI PT

problems, as it can be noticed from a brief list of works [104-121]. Recently, Fantuzzi and Tornabene [122] combine the accuracy and reliability of SFEM with the exact description of the geometry to investigate the free vibrations of laminated composite arbitrarily shaped plates. The acronym SFIGA, which stands for Strong Formulation Isogeometric Analysis, is introduced in the work [122] to specify this approach. The present work

SC

takes into account the same method in order to perform the dynamic analysis of CNT reinforced plates with

2.

M AN U

arbitrary domains and discontinuities.

MICROMECHANIC MODEL OF CARBON NANOTUBES

Following the theoretical framework presented by Shi et al. [83], it is possible to give an accurate esteem of CNT agglomeration and evaluate the effect of the CNT density on the dynamic response of composite plates enriched by Carbon nanofibers. The same approach has been employed by the authors in their previous work [88] for the

TE D

mechanical characterization of doubly-curved nanostructures. Thus, in the following only the main aspects of this procedure are presented for the sake of conciseness.

In general, CNTs are embedded in a polymeric matrix to enhance the structural behavior of several kinds of

EP

structures. Due to their extreme stiffness, coupled with their typical shape, these nanoparticles tend to gather together in some areas of the reinforced layer that can be assumed as spherical inclusions. As a consequence, the localization of CNTs within the polymeric matrix is not homogeneous, so that in some regions the concentration

AC C

of the nanoparticles is higher than in other areas, with respect to the average volume fraction of CNTs in the medium. Nevertheless, some particles can be still scattered in the matrix without concentrating in the spherical inclusions. As a consequence, CNTs are present in the matrix according two different manners: agglomerated in the inclusions or simply scattered in the reference domain. By hypotheses, CNTs are assumed to be randomly oriented in the reinforced lamina. The symbol Wrin is introduced now to denote CNTs concentrated in the inclusions, whereas Wrm is used to specify the reinforcing fibers sprinkled in the matrix. Consequently, the overall volume of CNTs is defined as follows

Wr = Wrin + Wrm

4

(1)

ACCEPTED MANUSCRIPT It is evident that the total volume of the medium W is given by the following summation

W = Wr + Wm

(2)

in which Wm represents the matrix volume. Once these quantities are defined, it is possible to introduce also the volume fraction of CNTs Vr and the volume fraction of the matrix Vm respectively as

Wr W , Vm = m W W

RI PT

Vr =

(3)

assuming that Vr + Vm = 1 . Following the theoretical approach given by Shi et al. [83], in order to correctly

SC

represent the agglomeration model for CNTs, the overall volume of the reinforcing particles Vr can be seen as the summation of two different terms, which are the volume of nanoparticles included in the inclusions Vrin and

M AN U

the volume of CNTs scattered in the polymeric matrix Vrm . Based on this crucial aspect, two parameters µ ,η can be introduced to control the agglomeration of CNTs. Their definition is shown below

µ=

Win W in , η= r W Wr

(4)

with η ≥ µ as limitation. In particular, µ defines the ratio between the volume of the inclusions Win with

TE D

respect to the total volume W of the considered ply, whereas η specifies the volume of CNTs in the inclusions

Wrin with respect to the overall volume Wr of the reinforcing phase. For the sake of clarity, some comments related to the values of µ ,η are reported. It should be noticed that for µ = 1 the tendency to agglomerate is

EP

absent and all the nanoparticles are scattered in the matrix. This can be assumed as a big spherical inclusion which corresponds to the whole ply. Vice versa, setting η = 1 the agglomeration is complete and each particle of

AC C

CNTs is inside the spherical inclusions. It is evident that a value of µ < 1 , with η ≥ µ , must be set to increase the heterogeneity of CNTs. These aspects are shown in graphical form in Figure 1a for the sake of clarity. By combining together definitions (3) and (4), the following relations are obtained

Wrin Vrη = Win µ

(5)

Vr (1 − η ) Wrm = W − Win 1− µ

(6)

By applying a gradual variation of the volume fraction of the reinforcing phase Vr , it is possible to vary the

5

ACCEPTED MANUSCRIPT mechanical properties along the plate thickness. If VC(

k)

(ζ )

denotes the through-the-thickness variation related

to the k -th ply, one gets

Vr (ζ ) = Vr*VC(

k)

(ζ )

(7)

 ρ  ρ Vr* =  r − r + 1  wr ρ m ρ m 

RI PT

where Vr* is defined as follows −1

(8)

nanoparticles wr assumes the following aspect

Mr Mr + Mm

M AN U

wr =

SC

ρ r and ρ m being the densities of CNTs and the matrix, respectively. On the other hand, the mass fraction of the

(9)

where M r and M m represent the CNTs and the matrix masses. The term Functionally Graded Carbon Nanotube (FG-CNT) is used to identify such nanostructures, highlighting the similarities with the well-known class of Functionally Graded Materials (FGMs). In fact, the same variation laws VC(

k)

(ζ )

that are commonly employed

for the definition of the volume fraction of the ceramic phase in FGMs can be used also for FG-CNT reinforced

TE D

structures. Alternatively, different through-the-thickness variations can be developed in a ad-hoc manner. In the present work, a four-parameter exponential law (4P) is introduced for this purpose. More details concerning this kind of variation can be found in the work [88]. In order to univocally identify the variation, the notation shown

EP

below can be used

top FG − CNTbottom(4P)( k k k k a( ) / b( ) / c ( ) / p( ) )

(10)

AC C

where (a ( k ) / b( k ) / c( k ) / p ( k ) ) denote the four parameters that specify the exponential law (4P) applied along the thickness hk of the k -th ply. In general, two different expressions of the (4P) law must be used. This choice is based on the material localized on the top and bottom surfaces of the reinforced layer, identified respectively by

ζ = hk 2 and ζ = − hk 2 . With reference to the expression (10), the words “ top ” and “ bottom ” are introduced for this purpose. For instance, if the top surface is completely made of the agglomerated nanoparticles and the bottom

one

shown

the

properties

of

the

polymer

matrix,

the

CNT FG − CNTPM(4P)( and the exponential law (4P) can be written as follows k k k k a ( ) / b ( ) / c( ) / p ( ) )

6

definition

(10)

becomes

ACCEPTED MANUSCRIPT (k ) c  (k ) ( k )  ζ k +1 − ζ  ( k )  ζ k +1 − ζ   VC (ζ ) = 1 − a  +b     hk   hk  

   

p

(k)

(11)

On the other hand, for the reverse situation one gets

 (k )  ζ − ζ k  +b     hk 

c

(k)

    

p

(k)

RI PT

 k  ζ −ζk k VC( ) (ζ ) =  1 − a ( )   hk  

(12)

PM and the corresponding nomenclature is FG − CNTCNT(4P)( . These two volume fraction distributions k k k k a ( ) / b( ) / c( ) / p( ) )

SC

are depicted in Figure 1b and Figure 1c respectively, for various values of the exponent p ( k ) = p . As it can be seen from Figure 1d, ζ k and ζ k +1 are the coordinates of the k -th layer measured along its thickness. It should

M AN U

be noticed that for a particular choice of the parameters (a ( k ) / b( k ) / c( k ) / p ( k ) ) , a mixture of the two constituents can be obtained on the external edges of the CNT reinforced ply.

At this point, it is possible to evaluate the overall mechanical properties of a FG-CNT plate. Due to the CNT agglomeration, it is necessary to esteem before the mechanical properties of the spherical inclusions and the hybrid matrix. For these purposes, the Eshelby-Mori-Tanaka scheme can be conveniently followed, as

TE D

highlighted in the work by Shi et al. [83]. According to this approach, the bulk modulus of the spherical inclusions K in = K in (ζ ) is defined as follows

Vrη (δ r − 3K mα r )

3 ( µ − Vrη + Vrηα r )

(13)

EP

K in (ζ ) = K m +

where K m denotes the bulk modulus of the polymer matrix. Analogously, the shear modulus of the inclusions

AC C

Gin = Gin (ζ ) can be evaluated as

Gin (ζ ) = Gm +

Vrη (η r − 2Gm β r )

2 ( µ − Vrη + Vrηβ r )

(14)

where Gm is the shear modulus of the matrix. The mechanical parameters K m and Gm can be evaluated also as a function of the Young’s modulus Em and the Poisson’s ratio ν m introduced for the matrix. For the sake of completeness, the definitions of K m and Gm are shown below

Km =

Em , 3 (1 − 2ν m )

Gm =

7

Em 2 (1 + ν m )

(15)

ACCEPTED MANUSCRIPT The same quantities must be computed for the hybrid matrix, in which the CNTs are scattered. As far as the bulk modulus of the hybrid matrix K out = K out (ζ ) is concerned, one gets

K out (ζ ) = K m +

Vr (1 − η )(δ r − 3K mα r )

3 (1 − µ − Vr (1 − η ) + Vr (1 − η ) α r )

(16)

Gout (ζ ) = Gm +

RI PT

Analogously, the definition of the shear modulus of the hybrid matrix Gout = Gout ( ζ ) is shown below

Vr (1 − η )(η r − 2Gm β r )

2 (1 − µ − Vr (1 − η ) + Vr (1 − η ) β r )

(17)

αr =

3 ( K m + Gm ) + k r + lr 3 ( Gm + k r )

SC

The mechanical parameters α r , β r , δ r and η r must be computed. They are defined as follows [87]

1

M AN U

2 ( Gm ( 3K m + Gm ) + Gm ( 3K m + 7Gm ) )  4Gm 1  4Gm + 2kr + lr  + + 5  3 ( Gm + k r ) Gm + pr Gm ( 3K m + Gm ) + mr ( 3K m + 7Gm )   

βr = 

δ r =  nr + 2lr + 3 

( 2kr + lr )( 3K m + Gm − lr )  Gm + kr

 

TE D

 2 ( k − l )( 2Gm + lr ) 8mr Gm ( 3K m + 4Gm ) 8G p 12 +  ( nr − lr ) + m r + r r 5 3 Gm + pr 3 ( Gm + k r ) 3K m ( mr + Gm ) + Gm ( 7 mr + Gm ) 

ηr =  

(18)

(19)

(20)

(21)

where the Hill’s elastic moduli of CNTs k r , lr , mr , nr , pr can be assumed equal to one of the sets of

EP

mechanical properties related to different kinds of nanoparticles shown in Table 1. In this table, various SingleWalled Carbon Nanotubes (SWCNTs) are taken into account for different values of chiral index. If the armchair

AC C

type is considered, the notation SWCNT ( n, n ) is used to specify the reinforcing phase. As highlighted in detail in the work [88], the mechanical properties of a CNT equivalent fiber are evaluated considering a constitutive model for a transversely isotropic medium. Thus, the Hill’s moduli listed in Table 1 are introduced following this hypothesis. More details about the constitutive relations based on this approach can be found in the book by Cristescu et al. [123] and in the work by Hill [124]. By means of the Eshelby-Mori-Tanaka approach, the global bulk modulus K = K (ζ ) of the composite layer in hand can be evaluated as follows

8

ACCEPTED MANUSCRIPT   K  µ  in − 1    K out  K (ζ ) = K out 1 +  K  1 + ν out  1 + (1 − µ )  in − 1  K  out  3 − 3ν out 

      

(22)

as a function of the Poisson’s ratio of the hybrid matrix ν out = ν out ( ζ ) , which is defined as

3K out − 2Gout 6 K out + 2Gout

RI PT

ν out (ζ ) =

(23)

Similarly, the effective shear modulus G = G (ζ ) of the composite ply can be assumed equal to

      

M AN U

SC

 G  µ  in − 1    Gout  G (ζ ) = Gout  1 +  Gin  8 − 10ν out  1 + (1 − µ )  − 1   Gout  15 − 15ν out 

(24)

Following this procedure, a generic FG-CNT reinforced layer turns out to be isotropic, as illustrated in the works [83, 88]. Thus, its Young’s modulus E = E (ζ ) is defined as follows

E (ζ ) =

9 KG 3K + G

(25)

TE D

whereas its Poisson’s ratio ν = ν (ζ ) assumes the following aspect

3K − 2G 6 K + 2G

ν (ζ ) =

(26)

EP

Finally, the theory of the Mixture can be used to compute the effective density of the composite

ρ (ζ ) = ( ρ r − ρ m )Vr + ρ m

(27)

AC C

It is important to underline that each quantity is a function of ζ , since a through-the-thickness variation is applied to the volume fraction of CNTs as shown in (7).

3.

FIRST-ORDER SHEAR DEFORMATION THEORY

In the current work, the mechanical behavior of arbitrarily shaped composite plates is analyzed by using the well-known FSDT. In fact, the micromechanical model and the homogenization technique just presented consider that the hybrid plies reinforced by agglomerated CNTs are isotropic media. Thus, the use of higherorder structural approaches can be avoided. According to the Reissner-Mindlin theory, the three-dimensional displacements U x , U x and W can be defined as follows

9

ACCEPTED MANUSCRIPT U x ( x, y , ζ , t ) = u x ( x, y , t ) + ζβ x ( x, y , t )

U y ( x, y , ζ , t ) = u y ( x, y , t ) + ζβ y ( x, y , t )

(28)

W ( x, y , ζ , t ) = w ( x, y , t )

where x, y are the in-plane coordinates of the reference system, whereas ζ is the coordinate that defines the

RI PT

plate thickness. On the other hand, the parameters u x , u y , w , β x and β y represent the degrees of freedom of the problem. It should be noticed that these five quantities are evaluated on the plate reference surface, which coincides with the middle plane. It is evident that this assumption is valid also if a laminated composite structure is considered. In this case, the total thickness of the plate is given by the following summation l

SC

h = ∑ hk k =1

(29)

M AN U

where hk represents the thickness of the k -th layer. For the sake of conciseness, the treatise is presented in compact matrix form. Thus, it is convenient to collect the degrees of freedom in the corresponding vector u . Since a dynamic problem is investigated, all the quantities are expressed as a function of the time variable t . The strain characteristics included in the vector ε can be defined as follows ε = Du

(30)

TE D

in which D represent the well-known kinematic operator defined in [93]. Afterwards, the stress resultant vector

S can be related to the strain characteristic vector introduced previously S = Aε

(31)

EP

where A is the stiffness matrix. As shown in the work [93], the following definitions can be employed to define

AC C

the terms collected in the matrix A . In particular, for i, j = 1, 2, 6 and τ = 0, 1, 2 , one gest l

Aij( ) = ∑ τ

ζ k +1

∫Q

k =1 ζ k

(k ) τ

ζ dζ

ij

(32)

whereas, the following definition l

Aij( ) = ∑ τ

ζ k +1

∫ κQ

k =1 ζ k

is used for i, j = 4, 5 and τ = 0 . The terms Qij(

k)

(k ) τ

ij

ζ dζ

(33)

are the plane stress-reduced elastic stiffnesses for an

orthotropic medium, which take into account also the orientation of the mechanical properties. It should be noticed that κ = 5 / 6 represents the shear correction factor. In general, the stiffness coefficients (32)-(33) can be

10

ACCEPTED MANUSCRIPT computed numerically. By means of the Hamilton principle, the five dynamic equations that govern the problem in hand can be easily obtained. In matrix compact form, they can be written as follows && D*S = Mu

(34)

where D* represents the equilibrium operator defined in [93]. On the other hand, M is the inertia matrix,

RI PT

&& is the vector that collects the second order derivatives of the five degrees of freedom with respect to whereas u the time variable t . By combining the kinematic equations (30), the constitutive relations (31) and the dynamic equations (34), it is possible to write the governing system as a function of the degrees of freedom. Thus, the

&& Lu = Mu

SC

fundamental system assumes the following aspect

(35)

The fundamental equations can be solved once the proper boundary conditions are introduced. As shown in

M AN U

Figure 2, each edge of an arbitrarily shaped domain is characterized by a generally oriented outward unit normal vector n . Its components nx , ny , representing the Cartesian projections, can be computed according the procedure illustrated in [122]. This vector is required to evaluate both the displacements and the stress resultants along the normal and tangential directions with respect to a general edge, denoted respectively by the subscripts n and t . In particular, the displacement components needed for the boundary conditions are shown below

TE D

u n = nx u x + n y u y ut = n x u y − n y u x wn = w

(36)

β n = nx β x + n y β y

EP

β t = nx β y − n y β x

As far as the stress resultants related to a generic edge are concerned, the following quantities must be computed

AC C

N n = N x nx2 + N y n y2 + 2 N xy nx n y

N ns = ( N y − N x ) nx n y + N xy ( nx2 − n y2 ) Tn = Tx nx + Ty n y

(37)

M n = M x nx2 + M y n y2 + 2 M xy nx n y M ns = ( M y − M x ) nx n y + M xy ( nx2 − n y2 )

If definitions (36) and (37) are properly combined, it is possible to obtain the boundary conditions for clamped (C), simply-supported (S), and free (F) edges.

11

ACCEPTED MANUSCRIPT 4.

MAPPING TECHNIQUE

The numerical technique considered in the present work represents a domain decomposition method. According to this approach, an arbitrarily shaped domain characterized by discontinuities, holes, cracks and curvilinear external edges can be divided into sub-domains, called finite elements (Figure 2). It is evident that the shape and

RI PT

the size of each sub-domain are related to the kind of problem under consideration. By means of a mapping technique that uses blending functions, each element can have a distorted shape. In general, the mapping procedure represents a coordinate transformation from a two-dimensional Cartesian space defined by the coordinates x, y into another regular domain, known as parent space, which is described by the natural

x = x (ξ1 , ξ 2 )

SC

coordinates ξ1 , ξ 2 . Analytically speaking, the mapping procedure is expressed as follows y = y (ξ1 , ξ 2 )

(38)

M AN U

As shown in the work [122], the parent space is limited according to the following relations

ξ1 ∈ [ −1,1] , ξ 2 ∈ [ −1,1]

(39)

If the mapping procedure is carried out by using the blending functions, the coordinate transformation assumes the following form

1 ( (1 − ξ 2 ) x1 (ξ1 ) + (1 + ξ1 ) x2 (ξ 2 ) + (1 + ξ 2 ) x3 (ξ1 ) + (1 − ξ1 ) x4 (ξ 2 ) ) + 2

TE D

x ( ξ1 , ξ 2 ) =

1 − ( (1 − ξ1 )(1 − ξ 2 ) x1 + (1 + ξ1 )(1 − ξ 2 ) x2 + (1 + ξ1 )(1 + ξ 2 ) x3 + (1 − ξ1 )(1 + ξ 2 ) x4 ) 4 y (ξ1 , ξ 2 ) =

1 ( (1 − ξ 2 ) y1 (ξ1 ) + (1 + ξ1 ) y2 (ξ 2 ) + (1 + ξ 2 ) y3 (ξ1 ) + (1 − ξ1 ) y4 (ξ 2 ) ) + 2

(41)

EP

1 − ( (1 − ξ1 )(1 − ξ 2 ) y1 + (1 + ξ1 )(1 − ξ 2 ) y2 + (1 + ξ1 )(1 + ξ 2 ) y3 + (1 − ξ1 )(1 + ξ 2 ) y4 ) 4

(40)

AC C

where the curvilinear edges of the distorted plate element are described by xi and yi , for i = 1, 2,3, 4 . On the other hand, xi and yi , for i = 1, 2,3, 4 , denotes the coordinates of the corners of the four-sided element. By means of the expressions (40), (41) it is possible to define an arbitrarily shaped domain. In the present work, NURBS curves are employed to define the external edges of each distorted element. More details about the NURBS interpolations can be found in the book by Piegl and Tiller [102]. For completeness, it should be recalled that a NURBS curve is completely defined once a set of three vectors is specified in terms of control points, their weights and knot vectors. These aspects are detailed in the work [122]. For the sake of conciseness, the mathematical description of these curves, as well as their properties, is omitted in the present paper. At this point, the variables of the problem and their derivatives with respect to the original domain coordinates x, y

12

ACCEPTED MANUSCRIPT must be transferred to the parent space defined by the natural coordinates ξ1 , ξ 2 . For this purpose, the first-order derivatives of a function with respect to the natural coordinates must be computed. By means of the chain rule of differentiation and using a matrix form, one gets ∂ ∂  ∂x  yξ1   ∂x    = J  yξ 2   ∂  ∂  ∂y   ∂y 

(42)

RI PT

 ∂   ∂ξ   x  1  =  ξ1  ∂   xξ 2  ∂ξ   2

where xξ1 = ∂x ∂ξ1 , xξ2 = ∂x ∂ξ 2 , yξ1 = ∂y ∂ξ1 , yξ2 = ∂y ∂ξ 2 , whereas J represents the Jacobian matrix

SC

related to the coordinate transformation in hand. Expression (42) denotes a one-to-one transformation. Thus, the Jacobian matrix can be inverted since its determinant does not assume a zero value, while the determinant of

J can be written as follows

M AN U

det J = xξ1 yξ 2 − yξ1 xξ2

(43)

By inverting the relation (42), the following equations are obtained

 ∂   ∂   ∂ξ  ξ 2, x   ∂ξ1    = J −1  1   ξ 2, y   ∂   ∂   ∂ξ   ∂ξ   2  2

TE D

∂  ∂x  ξ1, x  =  ∂  ξ1, y  ∂y 

(44)

where ξ1, x = ∂ξ1 ∂x , ξ1, y = ∂ξ1 ∂y , ξ 2, x = ∂ξ 2 ∂x , and ξ 2, y = ∂ξ 2 ∂y . Thus, the inverse of the Jacobian matrix J −1 can be written as

EP

J −1 =

1  yξ 2  det J  − xξ2

− yξ1   xξ1 

(45)

AC C

By comparing expressions (44) and (45), the following definitions can be obtained

ξ1, x = ξ 2, x

yξ 2

,

det J yξ =− 1 , det J

ξ1, y = − ξ 2, y =

xξ2 det J xξ1

(46)

det J

Similarly, starting from the definition (44), we determine the second-order derivatives with respect to the real coordinates x, y in the extended form, as in the following ∂2 ∂2 ∂2 ∂2 ∂ ∂ = ξ1,2x + ξ 2,2 x + 2ξ1, xξ 2, x + ξ1, xx + ξ 2, xx 2 2 2 ∂ξ1∂ξ 2 ∂ξ1 ∂ξ 2 ∂x ∂ξ1 ∂ξ 2

13

(47)

ACCEPTED MANUSCRIPT ∂2 ∂2 ∂2 ∂2 ∂ ∂ 2 2 = ξ + ξ + 2 ξ ξ + ξ1, yy + ξ 2, yy 1, y 2, y 1, y 2, y ∂y 2 ∂ξ12 ∂ξ 22 ∂ξ1∂ξ 2 ∂ξ1 ∂ξ 2

(48)

∂2 ∂2 ∂2 ∂2 ∂ ∂ = ξ1, x ξ1, y + ξ 2, x ξ 2, y + (ξ1, x ξ 2, y + ξ1, y ξ 2, x ) + ξ1, xy + ξ 2, xy 2 2 ∂x∂y ∂ξ1 ∂ξ 2 ∂ξ1∂ξ 2 ∂ξ1 ∂ξ 2

(49)

1 det J 2

 yξ2 yξ yξ  yξ 2 yξ1ξ2 − 2 det J ξ1 − yξ1 yξ 2ξ 2 + 1 2 det J ξ2  det J det J 

   

(50)

ξ1, yy =

1 det J 2

 xξ2 xξ xξ  xξ 2 xξ1ξ2 − 2 det J ξ1 − xξ1 xξ2ξ 2 + 1 2 det J ξ 2  det J det J 

   

(51)

SC

ξ1, xx =

1 det J 2

 yξ xξ yξ xξ  − yξ2 xξ1ξ 2 + 2 2 det J ξ1 + yξ1 xξ 2ξ 2 − 1 2 det J ξ2  det J det J 

ξ 2, xx =

1 det J 2

ξ 2, yy =

ξ 2, xy = xξ1ξ1 = ∂ 2 x ∂ξ12 ,

   

(52)

 yξ yξ yξ2  − yξ 2 yξ1ξ1 + 2 1 det J ξ1 + yξ1 yξ1ξ 2 − 1 det J ξ2  det J det J 

   

(53)

1 det J 2

 xξ xξ xξ2  − xξ 2 xξ1ξ1 + 2 1 det J ξ1 + xξ1 xξ1ξ 2 − 1 det J ξ2  det J det J 

   

(54)

1 det J 2

 xξ xξ yξ xξ  − yξ1 xξ1ξ 2 + 2 1 det J ξ1 + yξ2 xξ1ξ1 − 1 1 det J ξ2  J det det J 

   

(55)

M AN U

ξ1, xy =

TE D

where

RI PT

The following quantities are now introduced in Eqs. (47)-(49) for simplification purposes

xξ1ξ2 = ∂ 2 x ∂ξ1 ∂ξ 2 ,

xξ2ξ2 = ∂ 2 x ∂ξ 22 ,

yξ1ξ1 = ∂ 2 y ∂ξ12 ,

yξ1ξ2 = ∂ 2 y ∂ξ1 ∂ξ 2 , and

EP

yξ2ξ2 = ∂ 2 y ∂ξ 22 . On the other hand, det Jξ1 and det J ξ2 represent the first-order derivatives of the determinant of the Jacobian matrix with respect to coordinates ξ1 , ξ 2 . More specifically, Eq. (43) allows to write

AC C

det J ξ1 = xξ1 yξ1ξ2 − yξ1 xξ1ξ 2 + yξ2 xξ1ξ1 − xξ 2 yξ1ξ1 det J ξ2 = − xξ2 yξ1ξ 2 + yξ 2 xξ1ξ2 − yξ1 xξ2ξ 2 + xξ1 yξ2ξ2

(56)

Once each finite element is defined, the outward unit normal vectors can be computed along the external edges. In particular, the Cartesian components nx , n y of the normal vector along a generic edge denoted by ξ1 = ±1 are defined as follows  nx

ξ1

T

n y  =

xξ2 + yξ2 2

2

 yξ 2

− xξ 2 

T

(57)

On the other hand, when a general edge specified by ξ 2 = ±1 is considered, the Cartesian components nx , n y of the outward normal unit vector become

14

ACCEPTED MANUSCRIPT ξ2

T

n y  =

 nx

xξ1 + yξ1 2

2

 − yξ1

xξ1 

T

(58)

The expression of the normal vector is required also for applying the proper compatibility conditions between two faced elements. For the structural model at issue, they take the following aspect m)

N n( ) = N n(

ut( ) = ut(

m)

N nt( ) = N nt(

n n

n

m)

n

m)

n

wn( ) = wn(

Tn( ) = Tn( n

m)

m)

β n(n) = β n( m)

M n( n ) = M n( m )

βt (n) = βt ( m )

M nt( n ) = M nt( m )

RI PT

u n( ) = u n(

(59)

SC

where the superscripts ( n ) , ( m ) denote two adjacent general elements. Once the mapping technique is applied, the strong form of the governing equations is solved within each element. In particular, the GDQ method is taken

M AN U

into account to approximate the partial derivatives of the dynamic equations. This approach is named Strong Formulation Finite Element Method (SFEM), whose details are given in the works [92-98, 122].

5.

GENERAL DIFFERENTIAL QUADRATURE METHOD

In the present paper, the GDQ method is employed to solve the strong form of the governing equations since it

TE D

has shown great accuracy and reliability features [21-33, 88, 92-98]. The basic aspects of the technique are here illustrated for a one-dimensional domain [ x0 , x1 ] for shortness purposes. Nevertheless, the following definitions can be easily extended to a two-dimensional domain, as shown in the review paper by Tornabene et al. [100]. In

EP

general, the n -th order derivative of a smooth function f ( x ) can be evaluated at a generic point xi ∈ [ x0 , x1 ]

AC C

according to the following definition

d n f ( x) dx

n x = xi

IN

≅ ∑ ς ij( ) f ( x j ) for i = 1, 2,..., I N n

(60)

j =1

where I N denotes the total number of discrete points defined along the domain, and ς ij(

n)

refers to the weighting

coefficients of the summation, which are evaluated recursively as explained in the work [100]. It is clear that a proper grid distribution must be chosen to discretize the domain. For a two-dimensional domain, a discrete grid must be applied along the two principal directions. For this purpose, the symbols I N , I M are introduced to specify the total number of points within the reference domain. In the simplest case of a one-dimensional domain, the discrete node can be placed according to the following expression

15

ACCEPTED MANUSCRIPT   i − 1   ( x1 − x0 ) xi = 1 − cos  π   + x0 for i = 1, 2,..., I N .  2  I N −1   

(61)

This discrete grid (61) is known as Chebyshev-Gauss-Lobatto (Che-Gau-Lob). As shown in [21-33, 88, 92-98], this distribution is employed to solve different kinds of structural problems due to its accuracy. Nevertheless,

RI PT

different grid distributions can be chosen due to the general features of the present numerical approach. A list of the most common grid distributions employed in structural mechanics is shown in the review paper by Tornabene et al. [100]. It is important to specify that the points are located following the discrete grid in the parent space. Then, the mapping procedure shown above is used to move them in the distorted element.

SC

As mentioned in the previous section, a numerical approach must be used also to evaluate the stiffness coefficients defined in the Eqs. (32), (33). For this purpose, the Generalized Integral Quadrature (GIQ) method

M AN U

can be employed. This technique allows to approximate integrals by using the same fundamentals of the GDQ method. Let f ( x ) be a generic smooth function defined in a one-dimensional interval  xi , x j  . Its integral within the reference domain can be expressed as follows xj

T

∫ f ( x ) dx = ∑ w f ( x )

xi

k =1

ij k

k

(62)

TE D

where wkij denotes the weighting coefficients for the integral calculation. These coefficients can be evaluated starting from the definition of the weighting coefficients for the first-order derivatives as shown in the review paper by Tornabene et al. [100]. It should be noticed that the abovementioned grid distribution must be applied

FREE VIBRATION ANALYSIS

AC C

6.

EP

along the plate thickness.

By using the well-known variable separation method and by applying the GDQ method for the derivative approximations, the fundamental system of equations (35) can be written as follows Kδ = ω 2 Mδ

(63)

where K , M represent the discrete stiffness and mass matrices, respectively, δ is the modal shape vector, and

ω denotes the circular frequency. At this point, each matrix quantity can be subdivided in different terms related to the boundaries or linked to the nodes placed within the domain

16

ACCEPTED MANUSCRIPT  K bb K  db

K bd   δ b  0   δb  0 = ω2   δ  K dd   δ d  0 M dd     d 

(64)

where the subscript b , d specifies the boundary and internal nodes, respectively. After some easy manipulation, the Eq. (64) can be written as follows −1 dd

dd

)

−1 − K db K bb K bd ) − ω 2 I δ d = 0

(65)

RI PT

(M (K

where I represents the identity matrix. The solution of the generalized eigenvalue problem (65) allows to

7.

SC

compute the natural frequencies f = ω 2π and the corresponding modal shapes.

RESULTS AND DISCUSSIONS

M AN U

In the present section, the mechanical model just presented is employed to solve the free vibration analysis of several arbitrarily shaped plates reinforced by CNTs. In order to prove the validity of the current approach, some comparisons with the results available in the literature are performed. Thus, we test the capability of the proposed approach to deal with both arbitrary domains and CNT reinforced structures. All the numerical results are obtained by means of the GDQ method. In particular, the mapping procedure based on blending functions is used to describe the general shapes of each element of the considered plates. NURBS-based blending functions

TE D

are here used, as applied by the so-called isogeometric analysis (IGA). In what follows, all the results obtained with the proposed procedure are denoted by the acronym “SFIGA” (Strong Formulation Isogeometric Analysis)

analyses.

EP

for the sake of conciseness. A MATLAB code has been developed by the authors [125] for the numerical

The first application aims at comparing the present formulation with some results found in the literature for the

AC C

CNT orthotropic reinforced plates. For this purpose, the works by Zhang et al. [74] and Gracia-Macias et al. [75] are here taken as reference ones, when analyzing orthotropic skew. The reference geometry is a square plate of unit length of the edge a = b = 1 m . The horizontal edge (along the x -axis) of the square plate is identified as a and the vertical one (along the y -axis) as b . The skew angle is defined between the edge b and the horizontal axis. Two different studies are shown as a function of the skew angle. The first study is based on the variation of the volume fraction percentage of the CNTs according to a uniform distribution through the thickness. The second study analyses different aspect ratios. For all the applications in what follows, the plate is clamped on each edge, and therefore the effect of the natural boundary conditions is not present within results. The circular

17

ACCEPTED MANUSCRIPT

(

frequencies ω are presented in their dimensionless form according to the formula Ω = ωb 2 π 2

)

ρ m h Dm ,

where h is the plate thickness, ρ m and Dm are the density and the flexural stiffness of the matrix material, respectively. Please, note that the latter symbol is computed as Dm = Em

(12 (1 −ν )) , where 2 m

Em and ν m

RI PT

denote the Young’s modulus and the Poisson’s ratio of the matrix. For further details about the evaluation of the mechanical properties for the present case, the interested reader can refer to the work by Gracia-Macias et al. [75]. The cited references employed the theory of Mixture to deal with these composites. In the present computations very thin plates are analyzed with b = 1 m and the ratio h b = 0.001 . As far as the number of grid

SC

points is concerned, a grid of 25 × 25 points is assumed. The CNTs are uniformly distributed along the ζ direction in the present example. Thus, the material properties are not functionalized through the thickness.

M AN U

Tables 2 and 3 compare the numerical results given by our proposed method and those ones obtained by Refs. [74] and [75], where a very good agreement is observed. Moreover, it is worth noticing that Ref. [74] presented only the first 6 modes, whereas Ref. [75] showed the first 8 ones. The solution provided by the present formulation agrees well with both the solutions given by [74] and [75], and proposes the normalized frequencies up to the 10th order. Note also that the first part of Tables 2 and 3 reports the frequencies for a square plate, since

TE D

the skew angle is equal to 90 degrees.

The second example provided by Liu et al. [118] considers a straight lug depicted in Figure 3. The structure considers a rectangular shape connected to a semi-circular one. The present mesh has been taken exactly equal to

EP

the one of the reference work [118]. The geometry and their dimensions can be clearly seen in Figure 3 and the only boundary condition interests the left vertical edge which is clamped. The component is made of aluminum

AC C

with E = 70 GPa , ν = 0.3 , ρ = 2700 kg/m3 . In the reference paper, a FSDT has been implemented with a shear correction factor κ = π 2 12 , as assumed in the present computation. This test aims at verifying the validity of the domain decomposition technique, when approaching complex shapes. This is the main purpose of the following examples which consider distorted geometries. The main results are reported in Table 4. In the reference paper, two different solutions are provided, based on a weak formulation, i.e. the first one uses hierarchical finite elements with a differential quadrature integration, whereas the second one is a classical FEM solution as obtained with a commercial code. The hierarchical solution employs a 15 × 15 grid of points per element and the FEM solution took an average element edge side of 0.02 m. A further comparison has been also

18

ACCEPTED MANUSCRIPT provided by applying the NURBS-based isogeometric approach (IGA), as also done in [119-121]. The present strong form solution is listed in the last three columns of Table 4 for different number of points. It can be noted that a good approximation of the final solution can be obtained by including a few grid points per element. It should be remarked that the 6th frequency involves an axial mode that is not reported by the reference solution,

RI PT

since three equations of motions were implemented instead of a complete set of five equations as considered in our case. As visible from Table 4, a very good agreement between all the solutions is found.

The third application involves a comparison with a sectorial plate, already analyzed by the authors in a previous work [88], wherein the GDQ method was used in curvilinear coordinates without any geometrical mapping for

SC

solving doubly-curved shells reinforced by CNTs. The aim of the present application is to show the validity of the present procedure when single distorted domains are used within agglomerated CNTs functionalized through

M AN U

the thickness of the structure. The sectorial plate has an inner radius of 0.5 m, an outer radius of 1.5 m, an opening angle of 60°, and a thickness h = 0.1 m . The plate is clamped on three edges and free on the outer one (the circumference with the greater radius). The isotropic matrix has Em = 2.1 GPa , ν m = 0.34 and density

ρ m = 1150 kg/m3 . The CNT is characterized by kr = 271 GPa , mr = 17 GPa , pr = 442 GPa , nr = 1089 GPa , lr = 88 GPa , ρ r = 1400 kg/m3 , which represents the case of SWCNT (10,10) as shown in Table 1. The free

TE D

parameters for the following numerical investigations are wr , η , µ which correspond to the mass fraction of the CNTs, and the agglomeration parameters. The present finite element has a 25 × 25 grid of non-uniform points in all the computations. The natural frequencies are listed in Tables 5 and 6 according to the parametric

EP

study of the work [88]. Table 5 shows the natural frequency as a function of wr and µ by setting η = 1 .

AC C

Moreover, Table 6 reports the same quantities as a function of wr and η , by setting µ = 0.5 . It is worth noticing that the natural frequencies increase for higher values of wr , since the matrix is enriched with a larger quantity of CNTs which are stiffer than the matrix. At the same time, the frequencies increase enlarging the agglomeration parameter µ and decreasing η . For the sake of conciseness, it was not possible to compare frequency by frequency the present results with those ones presented in [88]. Therefore, the relative error of both solutions is plotted in Figure 4, where a small error below 1% can be noticed for all the cases presented in Table 5 (Figures 4a-d) and Table 6 (Figures 4e-h). The conclusion related to the previous case is that the present formulation can deal with single distorted domains. For this reason, the following application aims at considering a single solution as the reference one,

19

ACCEPTED MANUSCRIPT and studying the mesh distortion. The reference case here considered for our study on the free vibrations features

η = µ = 0.5 and wr = 0.4 . The mesh configurations are depicted in Figure 5. Figure 5a refers to the case of a single element, whereas Figure 5b considers the same geometry divided regularly in two elements. The mesh of Figure 5b will be considered as non-distorted mesh. The distortion here involves the horizontal edge division that

RI PT

rotates according to this rule. The left side of the aforementioned edge is rotated anti-clockwise by a fixed angle (10 or 20 degrees in the following) to enforce the distortion. The same procedure is performed on the right side, where the distortion is enforced in a clockwise direction. Thus, Figures 5c and 5d are obtained. A convergence analysis is performed and depicted in Figure 6 for the first 6 natural frequencies, where it is worth noticing that

SC

an increasing number of points per element decreases and stabilizes the relative percentage error. As expected, the mesh without any distortion reaches the minimum accuracy faster than in the other cases, but all the

M AN U

presented meshes provide a good solution. The same solution in terms of numerical frequencies is listed in Table 7 for the three meshes depicted in Figures 5b-d and for different number of grid points per element. The next example considers the annular plate and its distorted versions of Figure 7. The plate has an inner radius of 1 m and an outer radius of 5 m. The total plate thickness is h = 0.1 m . This geometry is distorted by moving the center of the inner circle by a quantity d with respect to the center of the figure (the origin of the Cartesian

TE D

system of axes). Please note that Figure 7a and 7b represent the same geometry. The former is described in curvilinear planar coordinates and the latter by Cartesian mapping of four elements through blending functions. The first geometry can be investigated by the GDQ method also because the local reference system is orthogonal

EP

and principal [21-33]. The second shape refers to a non-distorted mesh with d = 0 m . The other meshes of Figures 7c and 7d indicate the distorted meshes with d = 1 m and d = 2 m , respectively. For the present

AC C

applications, a lamination scheme of two plies is set as

FG − CNT PM

(

1 1 1 1 CNT (4 P ) a( ) =1/ b( ) = 0/ c( ) / p( )

)

/ FG − CNT CNT

(

2 2 2 2 PM (4 P ) a( ) =1/ b( ) = 0/ c( ) / p ( )

)

(66)

In other words, the plate is richer of CNTs on the top and bottom surfaces of the plate, whereas in the middle surface the material is fully polymeric. Between these three surfaces, a functionalization by the parameter p is defined. The same mechanical properties of the previous examples for the CNTs and the polymeric matrix are used with µ = 0.25, 0.5, 0.75 , η = 1 and wr = 0.2 . The results are presented in terms of natural frequencies for various values of µ and listed in Tables 8-10 for the meshes corresponding to d = 0 m, 1 m, 2 m . The same computations are depicted in Figures 8-11. Figures 8 and 9 show the variation of the first four frequencies as a

20

ACCEPTED MANUSCRIPT function of the exponent p ( ) = p ( ) = p . It can be noted that the GDQ solution coincides with the present one 1

2

when any distortion is applied. Different frequency values are observed, instead, for the distorted meshes ( d = 1 m, 2 m ). The curves are presented as a function of the agglomeration parameter µ = 0.25, 0.5, 0.75 . It can be noticed that the distortion leads to higher or lower values of frequency according to the modal shape that

RI PT

interacts with the circular hole. In fact, some curves for d = 2 m are lower than the ones with d = 0 m and viceversa. Figures 10 and 11 show the effect of the distortion on the first four natural frequencies as a function of the agglomeration parameter µ = 0.25, 0.5, 0.75 and for different power exponent values p = 0, 0.25, 0.5, 1, 2, 4 .

SC

For µ = 0.25 , it is worth noticing that the first and third frequencies increase with the distortion, while the second frequency decreases. The aforementioned effects are reduced if µ = 0.5, 0.75 . The fourth modal shape

M AN U

does not change for any value of µ and d . A comparison in terms of modal shapes is depicted in Figure 12 for the four structures shown in Figure 7. The solution obtained with a single domain in curvilinear coordinates is the same as the one obtained through a domain decomposition with four elements. Moreover, the modal shapes of the other two distorted meshes are shown below the previous ones, which graphically depict how the modal shapes change for varying positions of the hole.

TE D

The final numerical application refers to the plate represented in Figure 13, wherein a circular plate with two circular edges and a curved hole made of two parabolic edges is depicted. Due to its particular shape, this structure is named as “smile” plate. The outer circle has a radius of 5 m. The two small holes have a radius of 1

EP

m and their centers are placed at ( x, y ) = ( 2 m, 2 m ) and ( x, y ) = ( −2 m, 2 m ) , where ( x, y ) are the coordinates of the present Cartesian system. The two parabolas have the same ending points:

AC C

( 3 m, −1 m ) . The mid-point of the smaller parabola is ( 0 m, −2 m )

( −3 m, −1 m )

and

and the mid-point belonging to the larger

parabola is ( 0 m, −4 m ) . The outer edge is clamped, while maintaining free all the other edges. The present structure has the following lamination scheme PM FG − CNTCNT (4 P ) ( a =1/ b = 0/ c / p )

(67)

with the same mechanical properties of the previous numerical applications with η = µ = 0.5 and wr = 0.4 . In order to test the validity of the present results the solution is compared with a commercial FEM software. Due to difficulties in modelling functionalized materials through the plate thickness using commercial codes, the comparison is performed by introducing the two limit cases of isotropic materials for a full concentration of

21

ACCEPTED MANUSCRIPT CNTs for p = 0 and the only polymeric matrix p = ∞ . Table 11 reports the parametric investigation by changing the exponent p and the comparison between the present solution and that one obtained with a weak form-based finite element modeling. The FEM solution is obtained with a mesh of 405588 degrees of freedom with Quad8 elements. A further comparison has been also provided by applying the NURBS-based isogeometric

RI PT

approach (IGA), as also done in [119-121]. In the present application a grid of 15 × 15 points per element is employed. For the sake of completeness, the first nine mode shapes of the same plate are depicted in Figure 14,

SC

by setting p = 1 in equation (67), and a very good agreement is observed between the solutions.

CONCLUSION

The free vibration analysis has been performed for several arbitrarily shaped FG-CNT reinforced plates. The

M AN U

effect of the agglomeration of the reinforcing phase is investigated for varying values of the parameters that control both the density of CNTs within the spherical inclusions, and the quantities of nanofibers scattered in the polymeric matrix. The investigation is carried out for fixed values of the agglomeration parameter, and a varying exponent of the through-the-thickness volume fraction distribution. The results have been compared with the ones available in the literature. The free vibrations have been also evaluated for increasing mesh distortions.

TE D

Thus, the parametric study involves both the mechanical features and the geometric configuration. Indeed, it has been possible to deal with several geometries by means of the versatility of NURBS. Consequently, a mapping procedure that employs NURBS-based blending functions has been developed for this purpose. The strong form

EP

of the governing equations has been solved within each element that discretize the structure. The good agreement

AC C

between our results and some reference solutions has proven the validity of our proposed approach.

ACKNOWLEDGMENTS

The research topic is one of the subjects of the Centre of Study and Research for the Identification of Materials and Structures (CIMEST)-“M. Capurso” of the University of Bologna (Italy).

REFERENCES [1] [2] [3]

Wenzel C., D’Ottavio M., Polit O., Vidal P., Assessment of free-edge singularities in composite laminates using higher-order plate elements, Mech. Adv. Mat. Stuct., 2016, 23, 948-959. Vidal P., Polit O., D’Ottavio M., Valot E., Assessment of the refined sinus plate finite element: Free edge effect and Meyer-Piening sandwich test, Finite Elem. Anal. Des., 2014, 92, 60-71. Capozucca R., Magagnini E., Vibration of RC beams with NSM CFRP with unbonded/notched circular

22

ACCEPTED MANUSCRIPT

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

[20]

[21]

[22] [23] [24] [25]

[26]

[27]

[28]

RI PT

[9]

SC

[8]

M AN U

[7]

TE D

[6]

EP

[5]

rod damage, Compos. Struct., 2016, 144, 108-130. Caliri M.F., Ferreira A.J.M., Tita V., A review on plate and shell theories for laminated and sandwich structures highlighting the Finite Element Method, Compos. Struct., 2016. Doi: doi:10.1016/j.compstruct.2016.02.036. Sartorato M., de Medeiros R., Tita V., A finite element formulation for smart piezoelectric composite shells: Mathematical formulation, computational analysis and experimental evaluation, Compos. Struct., 2015, 127, 185-198. Pesic I., Lanc D., Turkalj G., Non-linear global stability analysis of thin-walled laminated beam-type structures, Comput. Struct., 2016, 173, 19-30. Biscani F., Giunta G., Belouettar S., Hu H., Carrera E., Mixed-dimensional modeling by means of solid and higher-order multi-layered plate finite elements, Mech. Adv. Mat. Struct., 2016, 23, 960-970. Nguyen H.X., Lee J., Vo T.P., Lanc D., Vibration and lateral buckling optimisation of thin-walled laminated composite channel-section beams, Compos. Struct. 2016, 143, 84-92. Brischetto S., An exact 3d solution for free vibrations of multilayered cross-ply composite and sandwich plates and shells, Int. J. Appl. Mechanics, 2014, 6, 1450076. Brischetto S., Convergence analysis of the exponential matrix method for the solution of 3D equilibrium equations for free vibration analysis of plates and shells, Compos. Part B-Eng., 2016, 98, 453-471. Dozio L., Alimonti L., Variable kinematic finite element models of multilayered composite plates coupled with acoustic fluid, Mech. Adv. Mater. Struct., 2016, 23, 981-996. Dozio L., A hierarchical formulation of the state-space Levy’s method for vibration analysis of thin and thick multilayered shells, Compos. Part B-Eng., 2016, 98, 97-107. Vescovini R., Dozio L., A variable-kinematic model for variable stiffness plates: Vibration and buckling analysis, Compos. Struct., 2016, 142, 15-26. Barbero E.J., Madeo A., Zagari G., Zinno R., Zucco G., Koiter asymptotic analysis of folded laminated composite plates, Compos. Part B-Eng., 2014, 61, 267-274. Zucco G., Groh R.M.J., Madeo A., Weaver P.M., Mixed shell element for static and buckling analysis of variable angle tow composite plates, Compos. Struct., 2016, 152, 324-338. Carpentieri G., Tornabene F., Ascione L., Fraternali F., An accurate one-dimensional theory for the dynamics of laminated composite curved beams, J. Sound Vib., 2015, 336, 96-105. Biswal M., Sahu S.K., Asha A.V., Vibration of composite cylindrical shallow shells subjected to hygrothermal loading-experimental and numerical results, Compos. Part B-Eng., 2016, 98, 108-119. Wang X., Wang Y., Static analysis of sandwich panels with non-homogeneous soft-cores by novel weak form quadrature element method, Compos. Struct., 2016, 146, 207-220. Kulikov G.M., Plotnikova S.V., Kulikov M.G., Monastyrev P.V., Three-dimensional vibration analysis of layered and functionally graded plates through sampling surfaces formulation, Compos. Struct., 2016, 152, 349-361. Kulikov G.M., Mamontov A.A., Plotnikova S.V., Mamontov S.A., Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells, Curved Layer. Struct. 2016, 3, 1-16. Tornabene F., Fantuzzi N., Bacciocchi M., Viola E., Accurate Inter-Laminar Recovery for Plates and Doubly-Curved Shells with Variable Radii of Curvature Using Layer-Wise Theories, Compos. Struct., 2015, 124, 368-393. Tornabene F., Fantuzzi N., Bacciocchi M., Dimitri R., Dynamic Analysis of Thick and Thin Elliptic Shell Structures Made of Laminated Composite Materials, Compos. Struct., 2015, 133, 278-299. Tornabene F., Brischetto S., Fantuzzi N., Viola E., Numerical and Exact Models for Free Vibration Analysis of Cylindrical and Spherical Shell Panels, Compos. Part B-Eng., 2015, 81, 231-250. Tornabene F., Fantuzzi N., Bacciocchi M., Dimitri R., Free Vibrations of Composite Oval and Elliptic Cylinders by the Generalized Differential Quadrature Method, Thin-Wall. Struct., 2015, 97, 114-129. Tornabene F., Fantuzzi N., Bacciocchi M., Viola E., A New Approach for Treating Concentrated Loads in Doubly-Curved Composite Deep Shells with Variable Radii of Curvature, Compos. Struct., 2015, 131, 433-452. Tornabene F., Fantuzzi N., Bacciocchi M., Viola E., Higher-Order Theories for the Free Vibration of Doubly-Curved Laminated Panels with Curvilinear Reinforcing Fibers by Means of a Local Version of the GDQ Method, Compos. Part B-Eng., 2015, 81, 196-230. Tornabene F., Fantuzzi N., Bacciocchi M., Higher-Order Structural Theories for the Static Analysis of Doubly-Curved Laminated Composite Panels Reinforced by Curvilinear Fibers, Thin-Wall. Struct., 2016, 102, 222-245. Tornabene F., Fantuzzi N., Bacciocchi M., The Local GDQ Method for the Natural Frequencies of

AC C

[4]

23

ACCEPTED MANUSCRIPT

[35] [36] [37] [38]

[39]

[40]

[41]

[42]

[43] [44] [45] [46] [47] [48]

[49] [50]

[51]

RI PT

[34]

SC

[33]

M AN U

[32]

TE D

[31]

EP

[30]

AC C

[29]

Doubly-Curved Shells with Variable Thickness: A General Formulation, Compos. Part B-Eng., 2016, 92, 265-289. Tornabene F., General Higher Order Layer-Wise Theory for Free Vibrations of Doubly-Curved Laminated Composite Shells and Panels, Mech. Adv. Mat. Struct, 2016, 23, 1046-1067. Tornabene F., Fantuzzi N., Viola E., Inter-Laminar Stress Recovery Procedure for Doubly-Curved, Singly-Curved, Revolution Shells with Variable Radii of Curvature and Plates Using Generalized HigherOrder Theories and the Local GDQ Method, Mech. Adv. Mat. Struct., 2016, 23, 1019-1045. Tornabene F., Fantuzzi N., Bacciocchi M., Neves A.M.A., Ferreira A.J.M., MLSDQ Based on RBFs for the Free Vibrations of Laminated Composite Doubly-Curved Shells, Compos. Part B-Eng., 2016, 99, 3047. Tornabene F., Dimitri R., Viola E., Transient Dinamic Responce of Genarally-Shaped Arches Based on a GDQ-Time-Stepping Method, Int. J. Mech. Sci., 2016, 114, 277-314. Bacciocchi M., Eisenberger M., Fantuzzi N., Tornabene F., Viola E., Vibration Analysis of Variable Thickness Plates and Shells by the Generalized Differential Quadrature Method, Compos. Struct., 2016. In Press. Doi: 10.1016/j.compstruct.2015.12.004. Carrera E., Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells, Arch. Comput. Methods Eng., 2002, 9, 87-140. Demasi L., ∞3 Hierarchy plate theories for thick and thin composite plates: The generalized unified formulation, Compos. Struct., 2008, 84, 256-270. D’Ottavio M., A Sublaminate Generalized Unified Formulation for the analysis of composite structures, Compos. Struct., 2016, 142, 187-199. Tornabene F., Viola E., Free Vibration Analysis of Functionally Graded Panels and Shells of Revolution, Meccanica, 2009, 44, 255-281. Viola E., Rossetti L., Fantuzzi N., Tornabene F., Static Analysis of Functionally Graded Conical Shells and Panels Using the Generalized Unconstrained Third Order Theory Coupled with the Stress Recovery, Compos. Struct., 2014, 112, 44-65. Tornabene F., Fantuzzi N., Bacciocchi M., Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories, Compos. Part B-Eng., 2014, 67, 490-509. Tornabene F., Fantuzzi N., Viola E., Batra R.C., Stress and strain recovery for functionally graded freeform and doubly-curved sandwich shells using higher-order equivalent single layer theory, Compos. Struct., 2015, 119, 67-89. Brischetto S., Tornabene F., Fantuzzi N., Viola E., 3D Exact and 2D Generalized Differential Quadrature Models for Free Vibration Analysis of Functionally Graded Plates and Cylinders, Meccanica, 2016. In Press. Viola E., Rossetti L., Fantuzzi N., Tornabene F., Generalized Stress-Strain Recovery Formulation Applied to Functionally Graded Spherical Shells and Panels Under Static Loading, Compos. Struct., 2016. In Press. Doi: doi:10.1016/j.compstruct.2015.12.060. Sofiyev A.H., Nonlinear free vibration of shear deformable orthotropic functionally graded cylindrical shells, Compos. Struct., 2016, 142, 35-44. Demir Ç., Mercan K., Civalek O., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel, Compos. Part B-Eng., 2016, 94-1-10. Fazzolari F.A., Natural frequencies and critical temperatures of functionally graded sandwich plates subjected to uniform and non-uniform temperature distributions, Compos. Struct., 2015, 121, 197-210. Fazzolari F.A., Reissner’s Mixed Variational Theorem and Variable Kinematics in the Modelling of Laminated Composite and FGM Doubly-Curved Shells, Compos. Part B-Eng., 2016, 89, 408-423. Lanc D., Turkalj G., Vo T.P., Brnić J., Nonlinear buckling behaviours of thin-walled functionally graded open section beams, Compos. Struct., 2016, 152, 829-839. Giunta G., Belouettar S., Ferreira A.J.M., A static analysis of three-dimensional functionally graded beams by hierarchical modelling and a collocation meshless solution method, Acta Mech., 2016, 227, 969-991. Ray M.C., Reddy J.N., Active control of laminated cylindrical shells using piezoelectric fiber reinforced composites, Compos. Sci. Technol., 2005, 65, 1226-1236. Raney J.R., Fraternali F., Amendola A., Daraio C., Modeling and in situ identification of material parameters for layered structures based on carbon nanotube arrays, Compos. Struct., 2011, 93, 30133018. Hadjiloizi D. A., Kalamkarov A.L., Metti Ch., Georgiades A. V., Analysis of Smart Piezo-MagnetoThermo-Elastic Composite and Reinforced Plates: Part I – Model Development, Curved Layer. Struct.,

24

ACCEPTED MANUSCRIPT

[60] [61] [62]

[63] [64] [65]

[66] [67] [68] [69] [70] [71]

[72]

[73] [74] [75]

[76]

[77]

RI PT

[59]

SC

[56] [57] [58]

M AN U

[55]

TE D

[54]

EP

[53]

2014, 1, 11-31. Hadjiloizi D. A., Kalamkarov A.L., Metti Ch., Georgiades A. V., Analysis of Smart Piezo-MagnetoThermo-Elastic Composite and Reinforced Plates: Part II – Applications, Curved Layer. Struct., 2014, 1, 32-58. H. Hariri, Y. Bernard, A. Razek, A two dimensions modeling of non-collocated piezoelectric patches bonded on thin structure, Curved Layer. Struct., 2015, 2, 15-27. Nasser H., Porn S., Koutsawa Y., Giunta G., Belouettar S., Optimal design of a multilayered piezoelectric transducer based on a special unit cell homogenization method, Acta Mech., 2016, 227, 1837-1847. Crainic N., Marques A.T., Nanocomposites: A state-of-the-art review, Key Eng. Mat., 2002, 230-232, 656-659. Iijima S., Helical Microtubles of Graphitic Carbon, Nature, 1991, 354, 56-58. Iijima S., Ichihashi T., Single-shell carbon nanotubes of 1-nm diameter, Nature, 1993, 363, 603-605. Mercan K., Civalek Ö, DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Compos. Struct., 2016, 143, 300-309. Popov V.N., Van Doren V.E., Balkanski M., Elastic properties of crystals of single-walled carbon nanotubes, Solid State Commun., 2000, 114, 395-399. Odegard G.M., Gates T.S., Wise K.E., Park C., Siochi E.J., Constitutive modeling of nanotube-reinforced polymer composites, Compos. Sci. Technol., 2003, 63, 1671-1687. Shen L., Li J., Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B, 2004, 69, 045414. Kalamkarov A.L., Veedu V.P., Mehrdad N., Ghasemi-Nejhad M.N., Mechanical properties modeling of carbon single-walled nanotubes: an asymptotic homogenization method, J. Comput. Theor. Nanos., 2005, 2, 124-131. Kalamkarov A.L., Georgiades A.V., Rokkam S.K., Veedu V.P., Ghasemi-Nejhad M.N., Analytical and numerical techniques to predict carbon nanotubes properties, Int. J. Solids Stuct., 2006, 43, 6832-6854. Ray M.C., Batra R.C., Effective Properties of Carbon Nanotube and Piezoelectric Fiber Reinforced Hybrid Smart Composites, J. App. Mech. - T. ASME, 2009, 76, 034503. Wang J.F., Liew K.M., On the study of elastic properties of CNT-reinforced composites based on element-free MLS method with nanoscale cylindrical representative volume element, Compos. Struct., 2015, 124, 1-9. Brischetto S., A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes, Compos. Part B-Eng., 2014, 61, 222-228. Brischetto S., A continuum shell model including van der Waals interaction for free vibrations of doublewalled carbon nanotubes, CMES Comp. Model Eng., 2015, 104, 305-327. Shen H.-S., Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments, Compos. Struct., 2009, 91, 9-19. Alibeigloo A., Liew K.M., Thermoelastic analysis of functionally graded carbon nanotube-reinforced composite plate using theory of elasticity, Compos. Struct., 2013, 106, 873-881. Liew K.M., Lei Z.X., Zhang L.W., Mechanical analysis of functionally graded carbon nanotube reinforced composites: a review, Compos. Struct., 2015, 120, 90-97. Alibeigloo A., Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity, Eur. J. Mech. A-Solid., 2014, 44, 104-115. Zhang L.W., Lei Z.X., Liew K.M., Computation of vibration solution for functionally graded carbon nanotube-reinforced composite thick plates resting on elastic foundations using the element-free IMLSRitz method, Appl. Math. Comput., 2015, 256, 488-504. Lei Z.X., Zhang L.W., Liew K.M., Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method, Compos. Struct., 2015, 127, 245-259. Zhang L.W., Lei Z.X., Liew K.M., Vibration characteristic of moderately thick functionally graded carbon nanotube reinforced composite skew plates, Compos. Struct., 2015, 122, 172-183. García-Macías E., Castro-Triguero R., Saavedra Flores E.I., Friswell M.I., Static and free vibration analysis of functionally graded carbon nanotube reinforced skew plates, Compos. Struct., 2016, 140, 473490. L.W., Lei Z.X., Liew K.M., Free vibration analysis of functionally graded carbon nanotube-reinforced composite triangular plates using the FSDT and element-free IMLS-Ritz method, Compos. Struct., 2015, 120, 189-199. Zhang L.W., Liew K.M, Large deflection analysis of FG-CNT reinforced composite skew plates resting on Pasternak foundations using an element-free approach, Compos. Struct., 2015, 132, 974-983.

AC C

[52]

25

ACCEPTED MANUSCRIPT

[84] [85]

[86] [87] [88]

[89] [90] [91] [92] [93] [94] [95] [96]

[97] [98] [99] [100] [101] [102] [103] [104]

RI PT

[83]

SC

[82]

M AN U

[81]

TE D

[80]

EP

[79]

Lei Z.X., Zhang L.W., Liew K.M., Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method, Compos. Part B-Eng., 2016, 84, 211-221. Alibeigloo A., Elasticity solution of functionally graded carbon nanotube-reinforced composite cylindrical panel subjected to thermo mechanical load, Compos. Part B-Eng., 2016, 87, 214-226. Lei Z.X., Liew K.M., Yu J.L., Buckling analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method, Compos. Struct., 2013, 98, 160-168. Zhang L.W., Lei Z.X., Liew K.M, Buckling analysis of FG-CNT reinforced composite thick skew plates using an element-free approach, Compos. Part B-Eng., 2015, 75, 36-46. Zhang L.W., Lei Z.X., Liew K.M, An element-free IMLS-Ritz framework for buckling analysis of FGCNT reinforced composite thick plates resting on Winkler foundations, Eng. Anal. Bound. Elem., 2015, 58, 7-17. Shi D.-L, Huang Y.Y., Hwang K.-C., Gao H., The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites, J. Eng. Mater. - T. ASME, 2004, 126, 250-257. Mori T., Tanaka K., Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions, Acta Metal., 1973, 21, 571-574. Sobhani Aragh B., Nasrollah Barati A.H., Hedayati H., Eshelby-Mori-Tanaka approach for vibrational behavior of continuously graded carbon nanotube-reinforced cylindrical panels, Compos. Part B-Eng., 2012, 43, 1943-1954. Formica G., Lacarbonara W., Alessi R., Vibrations of carbon nanotube-reinforced composites, J. Sound Vib., 2010, 329, 1875-1889. Kamarian S., Salim M., Dimitri R., Tornabene F., Free Vibration Analysis of Conical Shells Reinforced with Agglomerated Carbon Nanotubes, Int. J. Mech. Sci., 2016, 108-109, 157-165. Tornabene F., Fantuzzi N., Bacciocchi M., Viola E., Effect of Agglomeration on the Natural Frequencies of Functionally Graded Carbon Nanotube-Reinforced Laminated Composite Doubly-Curved Shells, Compos. Part B-Eng., 2016, 89, 187-218. Viola E., Tornabene F., Ferretti E., Fantuzzi N., On static analysis of plane state structures via GDQFEM and Cell method, CMES, 2013, 94, 421-458. Viola E., Tornabene F., Ferretti E., Fantuzzi N., GDQFEM numerical simulations of continuous media with cracks and discontinuities, CMES, 2013, 94, 331-368. Viola E., Tornabene F., Ferretti E., Fantuzzi N., Soft core plane state structures under static loads using GDQFEM and Cell method, CMES, 2013, 94, 301-329. Fantuzzi N., Tornabene F., Viola E., Generalized Differential Quadrature Finite Element Method for Vibration Analysis of Arbitrarily Shaped Membranes, Int. J. Mech. Sci., 2014, 79, 216-251. Viola E., Tornabene F., Fantuzzi N., Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape, Compos. Struct., 2013, 106, 815-834. Fantuzzi N., Tornabene F., Strong formulation finite element method for arbitrarily shaped laminated plates - I. Theoretical analysis, Adv. Aircraft Space. Sci., 2014, 1, 124-142. Fantuzzi N., Tornabene F., Strong formulation finite element method for arbitrarily shaped laminated plates - II. Numerical analysis, Adv. Aircraft Space. Sci., 2014, 1, 143-173. Fantuzzi N., Bacciocchi M., Tornabene F., Viola E., A.J.M. Ferreira, Radial Basis Functions Based on Differential Quadrature Method for the Free Vibration of Laminated Composite Arbitrary Shaped Plates, Compos. Part B-Eng., 2015, 78, 65-78. Fantuzzi N., Tornabene F., Viola E., Four-parameter functionally graded cracked plates of arbitrary shape: a GDQFEM solution for free vibrations, Mech. Adv. Mat. Struct., 2016, 23, 89-107. Fantuzzi N., Dimitri R., Tornabene F., A SFEM-Based Evaluation of Mode-I Stress Intensity Factor in Composite Structures, Compos. Struct., 2016, 145, 162-185. Shu C., Differential quadrature and its application in engineering, Springer, 2000. Tornabene F., Fantuzzi N., Ubertini F., Viola E., Strong formulation finite element method based on differential quadrature: a survey, Appl. Mech. Rev., 2015, 67, 020801 (55 pages). Hughes T.J.R., Cottrell J.A., Bazilevs Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 2005, 194, 4135-4195. Piegl L., Tiller W., The NURBS Book, Second Edition, Springer, 1997. Cottrell J.A., Hughes T.J.R., Bazilevs Y., Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009. Thai C.H., Nguyen-Xuan H., Nguyen-Thanh N., Le T.-H., Nguyen-Thoi T., Rabczuk T., Static, free vibration, and buckling analysis of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, Int. J. Numer. Methods Engrg., 2012, 91, 571-603.

AC C

[78]

26

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

SC

RI PT

[105] Thai C.H., Ferreira A.J.M., Carrera E., Nguyen-Xuan H., Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory, Compos. Struct., 2013, 104, 196-214. [106] Tran L.V., Ferreira A.J.M., Nguyen-Xuan H., Isogeometric analysis of functionally graded plates using higher-order shear deformation theory, Compos. Part B Eng., 2014, 51, 368-383. [107] Thai C.H., Ferreira A.J.M., Bordas S.P.A., Rabczuk T., Nguyen-Xuan H., Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory, Eur. J. Mech. A/Solids, 2014, 43, 89-108. [108] Natarajan S., Ferreira A.J.M., Nguyen-Xuan H., Analysis of cross-ply laminated plates using isogeometric analysis and unified formulation, Curved Layer. Struct., 2014, 1, 1-10. [109] Jari H., Atri H.R., Shojaee S., Nonlinear thermal analysis of functionally graded material plates using a NURBS based isogeometric approach, Compos. Struct., 2015, 119, 333-345. [110] Adam C., Hughes T.J.R., Bouabdallah S., Zarroug M., Maitournam H., Selective and reduced numerical integrations for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 2015, 284, 732-761. [111] Klinkel S., Chen L., Dornisch W., A NURBS based hybrid collocation-Galerkin method for the analysis of boundary represented solids, Comput. Methods Appl. Mech. Engrg., 2015, 284, 689-711. [112] Nguyen K.D., Nguyen-Xuan H., An isogeometric finite element approach for three-dimensional static and dynamic analysis of functionally graded material plate structures, Compos. Struct., 2015, 132, 423-439. [113] Phung-Van P., Abdel-Wahab M., Liew K.M., Bordas S.P.A., Nguyen-Xuan H., Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory, Compos. Struct., 2015, 123, 137-149. [114] Yin S., Yu T., Bui T.Q., Xia S., Hirose S., A cutout isogeometric analysis for thin laminated composite plates using level sets, Compos. Struct., 2015, 127, 152-164. [115] Ansari R., Norouzzadeh A., Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis, Physica E, 2016, 84, 84-97. [116] Thai C.H., Zenkour A.M., Wahab M.A., Nguyen-Xuan H., A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis, Compos. Struct., 2016, 139, 77-95. [117] Yu T., Yin S., Bui T.Q., Xia S., Tanaka S., Hirose S., NURBS-based isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method, Thin Wall. Struct., 2016, 101, 141-156. [118] Liu C., Liu B., Zhao L., Xing Y., Ma C., Li H., A differential quadrature hierarchical finite element method and its application to vibration and bending of Mindlin plates with curvilinear domains, Int. J. Numer. Meth. Eng., 2016. In press. Doi: 10.1002/nme.5277. [119] Dimitri R., De Lorenzis L., Wriggers P., Zavarise G. NURBS- and T-spline-based isogeometric cohesive zone modeling of interface debonding. Comput. Mech. 2014, 54, 369-388. [120] Dimitri R., De Lorenzis L., Scott M.A., Wriggers P., Taylor R.L., Zavarise G.Isogeometric large deformation frictionless contact using T-splines. Comput. Methods in Appl. Mech. Eng. 269, 394-414. [121] Dimitri R. Isogeometric treatment of large deformation contact and debonding problems with T-splines: a review. Curved Layer. Struct. 2015, 2, 59-90. [122] Fantuzzi N., Tornabene F., Strong Formulation Isogeometric Analysis (SFIGA) for Laminated Composite Arbitrarily Shaped Plates, Compos. Part B-Eng., 2016, 96, 173-203. [123] Cristescu N.D., Craciun E.-M., Soós E., Mechanics of elastic composites, CRC Press, 2004. [124] Hill R., A Self-Consistent Mechanics of Composite Materials, J. Mech., Phys. Solids, 1965, 13, 213-222. [125] Viola E., Tornabene F., Fantuzzi N., Bacciocchi M., DiQuMASPAB Software, DICAM Department, Alma Mater Studiorum - University of Bologna (http://software.dicam.unibo.it/diqumaspab-project).

27

RI PT

ACCEPTED MANUSCRIPT

M AN U

SC

b)

c)

EP

TE D

a)

d)

Figure 1. Schematic representation of a FG-CNT reinforced plate: a) agglomeration of CNTs varying the control parameters µ ,η ; b) effect

AC C

pf the variation of the exponent p ( k ) of the power law (4P) defined as FG − CNT PM p ( k ) of the power law (4P) defined as FG − CNT CNT

k k k k PM(4P)( a ( ) / b( ) / c( ) / p( ) )

k k k k CNT(4P)( a( ) / b( ) / c( ) / p( ) )

; c) effect pf the variation of the exponent

; d) identification of the k -th layer in the stacking sequence.

28

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

Figure 2. Arbitrarily shaped domain of a FG-CNT reinforced plate

Figure 3. Geometry of the straight lug taken from the work by Liu let al. [117].

29

RI PT

ACCEPTED MANUSCRIPT

b) wr = 0.1

TE D

c) wr = 0.2

M AN U

SC

a) wr = 0.05

f) wr = 0.1

AC C

EP

e) wr = 0.05

d) wr = 0.4

g) wr = 0.2

h) wr = 0.4

Figure 4. Relative errors between the solution obtained with the GDQ method without mapping [88] and the present solution with reference to the natural frequencies: a)-d) of Table 5; e)-h) Table 6.

30

a)

b)

c)

RI PT

ACCEPTED MANUSCRIPT

d)

M AN U

SC

Figure 5. Sectorial plate meshes used in the parametric investigation with and without distortion.

b) f 2

EP

TE D

a) f1

d) f 4

e) f 5

f) f 6

AC C

c) f 3

Figure 6. Relative errors between the solution obtained with the GDQ method without mapping [88] and the present solution with and without distortion.

31

RI PT

ACCEPTED MANUSCRIPT

b) d = 0 m

c) d = 1 m

M AN U

SC

a)

d) d = 2 m

AC C

EP

d =2 m .

TE D

Figure 7. Meshes used for the annular plate analysis: a) GDQ mesh using a single domain in curvilinear coordinates, b) SFIGA mesh with ne = 4 regular with d = 0 m , c) distorted annular mesh moving the hole of d = 1 m , d) distorted annular mesh moving the hole of

32

µ = 0.25

µ = 0.25

d)

M AN U

SC

a)

RI PT

ACCEPTED MANUSCRIPT

µ = 0.5

µ = 0.5

µ = 0.75

EP

c)

TE D

b)

e)

µ = 0.75 f)

Figure 8. First and second frequencies as a function of the exponent p for three values of the agglomeration parameter µ for η = 1 and

AC C

wr = 0.2 . Each curve represents a different level of distortion.

33

µ = 0.25

µ = 0.25

d)

M AN U

SC

a)

RI PT

ACCEPTED MANUSCRIPT

µ = 0.5

µ = 0.5

TE D

b)

µ = 0.75

EP

c)

e)

µ = 0.75 f)

Figure 9. Third and fourth frequencies as a function of the exponent p for three values of the agglomeration parameter µ for η = 1 and

AC C

wr = 0.2 . Each curve represents a different level of distortion.

34

ACCEPTED MANUSCRIPT

µ = 0.25

RI PT

µ = 0.25

d)

M AN U

SC

a)

µ = 0.5

µ = 0.5

e)

TE D

b)

µ = 0.75

EP

µ = 0.75

c)

f)

Figure 10. First and second frequencies as a function of the distortion d for three values of the agglomeration parameter µ for η = 1 and

AC C

wr = 0.2 . Each curve represents a different through the thickness power volume fraction.

35

ACCEPTED MANUSCRIPT

µ = 0.25

RI PT

µ = 0.25

d)

M AN U

SC

a)

µ = 0.5

µ = 0.5

TE D

b)

e)

µ = 0.75

µ = 0.75

EP

c)

f)

Figure 11. Third and fourth frequencies as a function of the distortion d for three values of the agglomeration parameter µ for η = 1 and

AC C

wr = 0.2 . Each curve represents a different through the thickness power volume fraction.

36

ACCEPTED MANUSCRIPT

No distortion d = 0 m

2nd mode

M AN U

1st mode

3rd mode

RI PT

2nd mode

SC

1st mode

3rd mode

1st mode

TE D

Distortion d = 1 m

2nd mode

3rd mode

AC C

EP

Distortion d = 2 m

1st mode

2nd mode

Figure 12. First three mode shapes of the annular plate with and without distortions.

37

3rd mode

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

EP

TE D

M AN U

Figure 13. Mesh configuration of the “smile” plate used in the computations.

38

2nd mode

3rd mode

M AN U

SC

1st mode

RI PT

ACCEPTED MANUSCRIPT

5th mode

6th mode

8th mode

AC C

7th mode

EP

TE D

4th mode

9th mode

Figure 14. First 9 mode shapes of the “smile” plate reinforced by CNTs with η = µ = 0.5 , wr = 0.4 and a linear through the thickness volume fraction with a = p = 1 .

39

ACCEPTED MANUSCRIPT Carbon Nanotubes SWCNT (5,5) SWCNT (6,6) SWCNT (10,10) SWCNT (15,15) SWCNT (20,20) SWCNT (50,50)

kr GPa  536 9.9 271 181 136 55

lr GPa  184 8.4 88 58 43 17

mr GPa  132 4.4 17 5 2 0.1

nr  GPa  2143 457.6 1089 726 545 218

pr  GPa  791 27 442 301 227 92

* VCNT = 0.12

45

Ref. [75]

Present

Ref. [74]

Ref. [75]

13.304 14.803 18.686 25.623 35.762 36.386 13.741 16.351 22.205 31.566 37.045 38.772 14.901 19.853 29.062 38.750 41.578 43.160 19.519 30.336 44.851 47.506 61.500 62.117 -

13.0538 14.3823 17.8530 23.1715 31.8996 35.7426 36.4624 38.2342 13.4471 15.6720 20.4977 28.0906 36.4444 37.3504 38.4512 41.4476 14.3946 18.6479 26.7126 37.4014 38.4475 41.6899 49.7066 50.5348 18.7081 28.5844 41.961 46.4368 57.7506 57.9898 75.3952 78.4185 -

16.0064 18.0246 23.0117 31.4757 43.2345 43.2650 44.4166 47.2101 52.5374 58.0172 16.6011 20.0677 27.4700 38.6984 44.1706 46.5355 51.9788 52.6353 62.1524 67.3868 18.1607 24.5955 35.9501 46.4996 50.6159 52.3945 64.5681 66.4571 83.9610 84.1155 24.2213 37.7056 54.8547 58.1132 74.3845 76.4338 96.3097 102.4212 108.7607 120.5811

16.027 18.087 23.327 32.514 43.722 44.862 16.625 20.164 27.931 40.080 44.638 47.029 18.194 24.770 36.695 47.034 52.318 53.022 24.306 38.184 56.232 58.847 76.983 78.307 -

15.7917 17.7137 22.5998 31.4317 41.8093 43.7683 44.3868 44.9723 16.2636 19.2775 25.6926 35.5932 43.9225 45.7816 48.2962 50.7851 17.5588 23.2105 33.6735 45.6978 48.1142 51.1101 62.0191 63.5476 23.3897 36.1798 53.2854 58.0262 73.1564 74.9533 95.7794 100.6950 -

Present

Ref. [74]

Ref. [75]

19.991 21.788 26.435 34.701 46.553 54.347 55.347 57.814 61.711 62.636 20.5220 23.7164 30.9181 42.4203 55.1277 56.7344 58.0569 62.0552 71.0444 74.6054 21.96339 28.21994 39.96521 55.66498 57.75273 62.49056 73.54721 74.46093 92.48527 94.36884 27.9023 42.2377 61.8943 67.8867 84.3153 85.7347 109.5348 114.4690 127.8517 137.0094

20.017 21.848 26.734 35.729 49.142 54.927 20.550 23.809 31.369 43.849 55.712 57.732 21.999 28.389 40.720 57.205 58.827 63.210 27.984 42.701 63.394 68.706 87.185 87.462 -

19.7446 21.4718 26.0166 34.6523 45.0572 47.7020 54.9867 56.0856 20.1599 22.9339 29.1320 39.1900 52.3513 54.9073 56.8337 60.9722 21.3205 26.8011 37.5648 53.0035 56.7587 61.3219 71.1036 71.3522 27.1443 40.6907 60.2382 68.4317 83.2442 84.8691 109.3371 112.4852 -

M AN U

SC

Ref. [74]

13.2866 14.7561 18.4508 24.8357 33.8112 36.0026 36.8365 38.8669 42.7724 45.1657 13.7209 16.2783 21.8563 30.4965 36.6560 38.3504 41.5331 42.3310 49.6783 53.3968 14.8744 19.7215 28.4914 38.2640 40.2445 42.6582 51.5687 52.8870 66.2924 66.9818 19.4548 29.9763 43.7667 46.9244 59.4481 60.7393 77.0875 81.4357 88.0453 96.4065

AC C

30

Present

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

TE D

60

* VCNT = 0.28

Ω

EP

skew angle 90

* VCNT = 0.17

RI PT

Table 1. Hill’s elastic moduli for different Single-Walled Carbon Nanotubes (SWCNT) by varying the chiral index (see Ref. [88]).

Table 2. First ten eigenfrequencies for different skew geometries and compared with the literature.

40

ACCEPTED MANUSCRIPT

4.0455 7.2247 9.3683 11.3363 12.9007 15.8261 17.8434 19.1525 20.7659 22.3821 4.6131 8.9422 10.0493 12.7363 17.0222 18.4040 18.4548 21.3956 24.3986 27.5511 5.8876 10.5356 13.5226 15.6708 19.9836 21.8356 25.1993 28.0335 28.9631 31.8008 10.1114 14.3033 21.3682 25.7780 29.2225 29.6513 37.8589 38.8719 47.4244 49.4847

a/b = 2 Ref. [74] 4.055 7.304 9.461 11.458 13.231 16.144 4.626 9.051 10.148 12.901 17.467 18.827 5.910 10.651 13.697 15.932 20.423 22.403 10.173 14.488 21.919 26.186 30.297 20.299 -

Ref. [75] 3.8633 6.6984 9.1930 10.9589 11.9110 15.0462 17.7122 18.9162 4.3790 8.2192 9.8701 12.1261 15.2323 17.2144 18.5912 20.9451 5.3706 10.1091 12.5785 14.8196 19.6324 20.9280 22.9664 26.5397 8.6907 13.2335 20.3526 22.3016 27.0132 27.8682 36.5696 36.9742 -

Present 3.1465 6.2899 6.7145 8.8258 11.6057 12.5820 13.3402 14.0017 18.8598 17.4331 3.7856 6.7031 8.7932 10.3729 12.2226 14.4335 16.6633 18.0801 19.4586 19.5956 5.1590 7.7736 12.3280 12.9099 15.0094 17.0114 21.0323 22.4642 24.8873 26.0649 9.5306 11.6027 15.9772 22.0041 25.4183 26.6444 29.0423 30.5650 36.2833 37.6710

a/b = 2.5 Ref. [74] 3.156 6.349 6.799 8.927 11.827 12.920 3.800 6.766 8.909 10.521 12.482 14.747 5.183 7.848 12.588 13.094 15.258 17.455 9.589 11.737 16.375 22.906 25.826 27.220 -

Ref. [75] 2.9586 6.2511 6.1689 8.4302 11.4818 11.6220 13.1281 13.2021 3.4908 6.5632 7.9273 9.7629 12.3587 13.9461 14.7044 16.7327 4.5718 7.3978 11.8484 11.8499 14.3152 16.2683 20.8109 21.6903 8.0034 10.3135 15.0589 21.0300 21.8432 23.3768 27.6982 28.0005 -

Present 2.7449 4.6994 6.5090 7.6878 8.2674 10.4308 12.4515 13.2299 13.2527 14.8534 3.4307 5.2008 8.4963 8.5782 9.7654 11.7668 13.8543 15.7006 16.5706 17.2682 4.8642 6.3716 9.4915 12.6991 13.6823 13.7129 16.3729 18.0424 21.7444 22.9089 9.3150 10.4174 13.0304 17.2104 22.4523 25.2266 26.0679 27.2095 29.1110 31.0965

Table 3. First ten eigenfrequencies for different skew geometries and compared with the literature.

f [Hz]

1 2 3 4 5 6 7 8 9 10

DQHFEM Ref. [118]

FEM Ref. [118] elem size 15 × 15 0.02 m 8.6190 8.6196 29.397 29.413 42.368 42.382 89.445 89.494 111.09 111.15 axial mode 151.99 152.05 181.43 181.56 212.53 212.69 -

IGA elem size 0.02 m 8.7110 30.688 42.315 90.178 112.582 129.122 157.064 190.907 221.537 287.793

Present 7×7

11× 11

15 × 15

8.7449 28.453 43.590 92.701 117.00 126.20 153.05 186.84 220.48 277.64

8.6235 29.303 42.352 89.580 111.13 127.72 152.02 181.54 212.73 275.45

8.6189 29.390 42.365 89.440 111.09 127.69 151.98 181.44 212.54 275.28

Table 4. Comparison with a straight lug component by several numerical approaches.

41

a/b = 3 Ref. [74] 2.755 4.740 6.596 7.781 8.449 10.624 3.445 5.245 8.670 8.697 9.905 12.015 4.888 6.426 9.688 12.890 13.893 14.161 9.370 10.523 13.319 17.920 23.805 25.614 -

RI PT

Present

SC

6.2983 8.7690 13.9038 16.2105 17.5708 20.9503 21.4734 26.9359 31.2498 31.4236 6.7792 10.5361 16.7219 17.0577 19.7185 23.4357 28.5616 31.2787 32.1194 34.0595 7.9366 13.9787 18.2743 20.9109 26.7441 28.8738 33.8285 37.8542 38.3509 42.6289 12.0141 20.4409 28.0910 29.5331 40.1923 41.0566 51.0584 52.9116 55.3366 64.6855

Ref. [75] 6.0874 8.2583 12.9399 15.9177 17.1348 20.0710 20.1964 25.7896 6.5715 9.7738 15.5079 16.6020 18.9644 21.8260 25.6026 28.7537 7.4977 13.0384 17.7808 19.8597 25.0125 27.6836 33.3166 36.7895 10.9893 19.2496 25.4377 27.9931 38.6406 39.3448 47.2396 50.9244 -

M AN U

30

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

a/b = 1.5 Ref. [74] 6.308 8.836 14.212 16.377 17.749 21.272 6.792 10.636 16.891 17.424 19.988 23.960 7.958 14.148 18.458 21.270 27.367 29.641 12.075 20.737 28.489 30.212 41.575 42.194 -

TE D

45

Present

EP

60

Ω

AC C

skew angle 90

Ref. [75] 2.5455 4.5680 5.9457 7.2574 8.2147 10.2374 11.4224 12.3322 3.1076 5.0592 7.6520 8.3166 9.3730 11.4054 14.3306 14.4855 4.2434 5.9532 9.2391 11.7179 12.8111 13.3469 15.9554 17.5885 7.5344 8.8580 11.8727 16.2833 21.4904 22.6422 23.5432 25.0394 -

ACCEPTED MANUSCRIPT µ 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

wr = 0.05

1 2 3 4 5 6 7 8 9 10

94.078 229.233 255.018 411.754 468.842 506.794 559.487 603.823 628.348 727.952

102.182 249.130 276.598 447.589 508.853 549.785 607.153 656.809 683.153 790.411

110.410 269.345 298.482 484.002 549.456 593.387 655.517 710.678 738.853 853.819

118.846 290.082 320.884 521.363 591.053 638.028 705.057 765.982 796.014 918.810

127.573 311.547 344.016 560.045 634.042 684.132 756.247 823.277 855.207 986.014

136.676 333.957 368.099 600.440 678.846 732.143 809.587 883.153 917.037 1056.098

146.254 357.555 393.380 642.987 725.932 782.552 865.630 946.271 982.180 1129.802

156.418 382.621 420.139 688.198 775.837 835.924 925.013 1013.397 1051.420 1207.984

167.302 409.492 448.710 736.679 829.203 892.925 988.492 1085.447 1125.694 1291.661

wr = 0.1

1 2 3 4 5 6 7 8 9 10

94.677 230.698 256.629 414.389 471.814 509.998 563.031 607.702 632.372 732.578

104.132 253.895 281.852 456.156 518.541 560.234 618.706 669.409 696.237 805.481

114.463 279.247 309.397 501.806 569.585 615.096 679.522 736.867 766.046 885.134

125.967 307.487 340.043 552.661 626.400 676.139 747.209 812.035 843.819 973.817

139.052 339.623 374.861 610.542 690.990 745.500 824.148 897.627 932.352 1074.672

154.316 377.134 415.408 678.116 766.271 826.289 913.807 997.612 1035.733 1192.282

172.685 422.318 464.092 759.537 856.767 923.313 1021.560 1118.186 1160.331 1333.764

195.715 479.034 524.923 861.783 970.034 1044.585 1156.379 1269.773 1316.860 1511.030

226.280 554.448 605.262 997.817 1120.012 1204.828 1334.792 1471.769 1525.235 1746.110

wr = 0.2

1 2 3 4 5 6 7 8 9 10

94.371 229.953 255.793 413.053 470.282 508.339 561.202 605.749 630.335 730.204

104.570 254.965 283.031 458.081 520.715 562.579 621.300 672.242 699.178 808.865

116.208 283.507 314.112 509.462 578.268 624.471 689.881 748.112 777.733 898.631

129.871 317.014 350.591 569.783 645.824 697.110 770.380 837.184 869.959 1004.006

146.484 357.766 394.925 643.150 727.950 785.396 868.236 945.539 982.140 1132.131

167.651 409.706 451.352 736.671 832.532 897.774 992.836 1083.707 1125.153 1295.340

196.476 480.489 528.054 864.153 974.824 1050.558 1162.331 1272.175 1320.142 1517.526

240.114 587.796 643.764 1057.500 1189.870 1281.133 1418.391 1558.381 1616.002 1853.683

320.903 787.039 856.316 1416.841 1586.498 1705.012 1890.240 2091.652 2166.374 2475.204

wr = 0.4

1 2 3 4 5 6 7 8 9 10

92.896 226.359 251.791 406.597 462.927 500.387 552.425 596.283 620.484 718.786

103.355 252.003 279.740 452.758 514.663 556.039 614.078 664.432 691.054 799.464

115.577 281.965 312.412 506.690 575.134 621.090 686.143 744.036 773.500 893.755

130.368 318.218 351.956 571.941 648.318 699.819 773.360 840.330 873.248 1007.862

149.102 364.135 402.043 654.585 741.015 799.535 883.832 962.285 999.583 1152.396

174.412 426.180 469.688 766.262 866.234 934.211 1033.055 1127.099 1170.307 1347.661

212.249 518.969 570.696 933.301 1053.319 1135.334 1255.977 1373.717 1425.701 1639.499

280.493 686.529 752.330 1235.061 1390.250 1497.114 1657.323 1819.745 1887.245 2165.580

487.036 1196.002 1295.327 2153.957 2403.967 2579.996 2863.102 3183.341 3294.750 3754.491

TE D

M AN U

SC

RI PT

f [Hz]

AC C

EP

Table 5. Comparison with a sectorial plate by using a single element ne = 1 with 25 × 25 grid and η = 1 .

42

ACCEPTED MANUSCRIPT η 0.5

0.6

0.7

0.8

0.9

1.0

wr = 0.05

1 2 3 4 5 6 7 8 9 10

167.302 409.492 448.710 736.679 829.203 892.926 988.492 1085.448 1125.695 1291.662

166.218 406.829 445.834 731.880 823.860 887.195 982.130 1078.347 1118.352 1283.312

162.478 397.632 435.923 715.308 805.436 867.444 960.193 1053.812 1092.992 1254.507

155.008 379.246 416.161 682.171 768.667 828.054 916.421 1004.716 1042.272 1196.989

141.878 346.897 381.502 623.846 704.110 758.949 839.585 918.212 952.974 1095.934

118.846 290.082 320.884 521.363 591.053 638.028 705.057 765.982 796.014 918.810

wr = 0.1

1 2 3 4 5 6 7 8 9 10

226.280 554.448 605.262 997.817 1120.013 1204.829 1334.792 1471.769 1525.236 1746.111

224.372 549.765 600.182 989.383 1110.590 1194.711 1323.568 1459.308 1512.337 1731.400

217.578 533.078 582.122 959.328 1077.066 1158.736 1283.642 1414.874 1466.361 1679.037

203.472 498.402 544.693 896.856 1007.523 1084.162 1200.833 1322.444 1370.774 1570.346

177.316 434.044 475.463 780.872 878.740 946.188 1047.521 1150.668 1193.258 1368.921

125.967 307.487 340.043 552.661 626.400 676.139 747.209 812.035 843.819 973.817

wr = 0.2

1 2 3 4 5 6 7 8 9 10

320.904 787.039 856.316 1416.842 1586.499 1705.013 1890.241 2091.653 2166.376 2475.205

317.751 779.304 847.910 1402.916 1570.918 1688.275 1871.679 2071.089 2145.080 2450.891

306.067 750.616 816.823 1351.252 1513.237 1626.357 1802.977 1994.741 2066.058 2360.816

280.972 688.953 750.180 1240.175 1389.462 1493.596 1655.584 1830.479 1896.120 2167.416

232.976 570.931 622.964 1027.525 1152.963 1240.112 1374.013 1515.779 1570.710 1797.668

129.871 317.014 350.591 569.783 645.824 697.110 770.380 837.184 869.959 1004.006

wr = 0.4

1 2 3 4 5 6 7 8 9 10

487.036 1196.003 1295.328 2153.958 2403.968 2579.997 2863.104 3183.343 3294.752 3754.493

481.956 1183.535 1281.793 2131.510 2378.872 2553.043 2833.208 3150.184 3260.422 3715.319

461.089 1132.238 1226.455 2039.092 2276.016 2442.791 2710.751 3013.479 3119.009 3554.535

414.119 1016.667 1102.186 1830.820 2044.758 2195.148 2435.500 2705.165 2800.231 3192.762

324.910 797.135 866.245 1435.177 1605.616 1724.941 1912.828 2119.366 2194.643 2505.719

130.368 318.218 351.956 571.941 648.318 699.819 773.360 840.330 873.248 1007.862

TE D

M AN U

SC

RI PT

f [Hz]

AC C

EP

Table 6. Comparison with a sectorial plate by using a single element ne = 1 with 25 × 25 grid and µ = 0.5 .

43

ACCEPTED MANUSCRIPT

491.444 1202.813 1285.750 2155.769 2353.397 2478.281 2864.295 3182.942 3342.007 3644.767

1 2 3 4 5 6 7 8 9 10

490.912 1213.409 1294.440 2213.155 2341.316 2457.625 2863.060 3036.672 3183.045 3676.090

13 × 13 486.914 1196.319 1295.466 2154.859 2404.249 2579.913 2863.415 3183.302 3295.759 3754.841

15 × 15 486.929 1196.046 1295.342 2154.236 2404.039 2579.956 2863.293 3183.323 3294.989 3754.606

487.035 1196.478 1295.264 2154.613 2404.159 2580.110 2863.337 3183.381 3295.673 3754.781

486.979 1196.127 1295.269 2154.079 2404.010 2580.028 2863.280 3183.377 3294.911 3754.567

487.212 1196.568 1295.056 2154.373 2404.217 2580.479 2863.222 3183.493 3296.601 3754.665

487.025 1196.165 1295.267 2153.887 2404.043 2580.065 2863.200 3183.476 3295.085 3754.498

RI PT

1 2 3 4 5 6 7 8 9 10

No distortion 9×9 11× 11 488.039 487.074 1200.489 1197.302 1295.434 1295.715 2161.746 2156.795 2409.955 2404.365 2600.546 2578.672 2864.474 2863.713 3183.134 3183.254 3306.263 3298.251 3764.894 3755.032 Distortion of 10 degrees 487.708 487.233 1200.685 1197.605 1296.983 1295.524 2164.015 2156.289 2411.323 2404.186 2600.295 2578.507 2863.261 2863.343 3183.354 3183.377 3308.864 3297.832 3767.398 3755.008 Distortion of 20 degrees 488.724 487.798 1200.601 1197.937 1297.268 1294.760 2169.633 2155.176 2412.902 2403.635 2607.368 2579.181 2862.840 2863.210 3183.570 3183.499 3332.852 3293.562 3782.710 3754.874

SC

7×7 493.901 1199.089 1278.111 2163.251 2355.168 2501.474 2866.002 3182.986 3294.191 3627.402

M AN U

f [Hz] 1 2 3 4 5 6 7 8 9 10

AC C

EP

TE D

Table 7. First ten frequencies of a sectorial plate reinforced by CNTs with η = µ = 0.5 and wr = 0.4 .

44

ACCEPTED MANUSCRIPT p

0.5

1

2

4

100

3.460

3.465

3.467

3.451

3.386

2.807

2

6.791

6.809

6.819

6.822

6.790

6.661

5.531

3

6.791

6.809

6.819

6.822

6.790

6.661

5.531

4

11.201

11.232

11.247

11.253

11.201

10.991

9.144

5

11.201

11.232

11.247

11.253

11.201

10.991

9.144

6

14.226

14.264

14.284

14.291

14.224

13.954

11.572

7

16.734

16.780

16.803

16.810

16.732

16.416

13.655

8

16.734

16.780

16.803

16.810

16.732

16.416

13.655

9

18.633

18.684

18.709

18.716

18.624

10

18.633

18.684

18.709

18.716

18.624

1

4.502

4.501

4.493

4.462

4.366

2

8.845

8.843

8.826

8.763

8.574

3

8.845

8.843

8.826

8.763

8.574

4

14.557

14.554

14.527

14.424

14.114

13.419

9.402

5

14.557

14.554

14.527

14.424

14.114

13.419

9.402

6

18.551

18.548

18.513

18.381

17.984

17.091

11.908

7

21.756

21.751

21.709

21.554

21.088

20.044

14.040

8

21.756

21.751

21.709

21.554

21.088

20.044

14.040

9

24.313

24.306

24.258

24.079

23.548

22.367

15.584

10

24.313

24.306

24.258

24.079

23.548

22.367

15.584

RI PT

0.25

3.451

18.264

15.146

18.264

15.146

4.151

2.888

8.149

5.689

8.149

5.689

1

6.252

6.204

6.145

6.008

5.720

5.228

3.007

2

12.263

12.168

12.049

11.778

11.210

10.243

5.918

3

12.263

12.168

12.049

11.778

11.210

10.243

5.918

4

20.135

19.978

19.784

19.340

18.410

16.829

9.772

5

20.135

19.978

19.784

19.340

18.410

16.829

9.772

6

25.754

25.554

25.307

24.737

23.543

21.510

12.391

7

30.104

29.868

29.577

28.908

27.511

25.143

14.594

8

30.104

29.868

29.577

28.908

27.511

25.143

14.594

9

33.771

33.502

33.170

32.409

30.824

28.143

16.217

10

33.771

33.502

33.170

32.409

30.824

28.143

16.217

TE D

µ = 0.75

0

1

SC

µ = 0.5

f [Hz]

M AN U

µ = 0.25

Table 8. First ten frequencies of a sectorial plate reinforced by CNTs by varying the agglomeration parameter µ and the functionally graded

AC C

EP

exponent p with η = 1 and wr = 0.2 for the mesh with no distortion d = 0 m .

45

ACCEPTED MANUSCRIPT p

1

2

4

100

1

3.443

3.452

3.457

3.459

3.443

3.378

2.802

2

6.814

6.832

6.842

6.845

6.813

6.685

5.552

3

6.998

7.018

7.027

7.031

6.998

6.866

5.699

4

11.173

11.203

11.219

11.224

11.172

10.961

9.111

5

11.207

11.238

11.253

11.258

11.206

10.994

9.137

6

13.094

13.130

13.149

13.154

13.092

12.843

10.657

7

16.585

16.630

16.653

16.660

16.582

16.268

13.529

8

16.600

16.645

16.668

16.675

16.597

16.284

13.542

9

18.989

19.041

19.067

19.074

18.982

10

21.364

21.422

21.452

21.460

21.358

1

4.488

4.488

4.479

4.448

4.353

2

8.872

8.870

8.853

8.790

8.601

3

9.118

9.116

9.099

9.034

8.840

4

14.536

14.533

14.505

14.401

14.091

13.394

5

14.583

14.580

14.552

14.448

14.136

13.436

9.397

6

17.067

17.064

17.031

16.909

16.543

15.720

10.965

7

21.568

21.563

21.522

21.367

20.905

19.870

13.912

8

21.586

21.582

21.540

21.385

20.923

19.887

13.925

9

24.741

24.735

24.686

24.506

23.970

22.774

15.902

10

27.826

27.819

27.765

27.564

26.964

25.623

17.897

RI PT

0.5

18.619

15.459

20.951

17.399

4.138

2.883

8.175

5.711

8.402

5.863

9.370

1

6.229

6.181

6.122

5.986

5.699

5.209

3.000

2

12.295

12.199

12.081

11.809

11.240

10.272

5.939

3

12.645

12.547

12.425

12.146

11.561

10.565

6.099

4

20.127

19.970

19.776

19.330

18.398

16.814

9.742

5

20.197

20.039

19.844

19.396

18.460

16.870

9.770

6

23.680

23.496

23.266

22.741

21.641

19.771

11.407

7

29.853

29.619

29.329

28.666

27.281

24.930

14.461

8

29.876

29.642

29.352

28.688

27.302

24.951

14.475

9

34.313

34.042

33.706

32.937

31.334

28.619

16.540

10

38.578

38.275

37.899

37.039

35.242

32.195

18.614

TE D

µ = 0.75

0.25

SC

µ = 0.5

0

M AN U

µ = 0.25

f [Hz]

Table 9. First ten frequencies of a sectorial plate reinforced by CNTs by varying the agglomeration parameter µ and the functionally graded

AC C

EP

exponent p with η = 1 and wr = 0.2 for the mesh with no distortion d = 1 m .

46

ACCEPTED MANUSCRIPT p

1

2

4

100

1

3.417

3.426

3.431

3.433

3.417

3.353

2.784

2

6.879

6.898

6.908

6.911

6.879

6.750

5.610

3

7.264

7.284

7.294

7.297

7.264

7.127

5.915

4

11.166

11.197

11.213

11.217

11.165

10.953

9.096

5

11.838

11.871

11.887

11.892

11.837

11.613

9.642

6

12.622

12.657

12.674

12.680

12.620

12.380

10.284

7

16.332

16.377

16.400

16.406

16.329

16.018

13.307

8

16.753

16.798

16.822

16.828

16.749

16.430

13.649

9

19.662

19.716

19.743

19.751

19.658

10

19.753

19.807

19.835

19.842

19.748

1

4.449

4.448

4.440

4.409

4.315

2

8.949

8.948

8.931

8.868

8.677

3

9.464

9.462

9.444

9.378

9.177

4

14.542

14.539

14.511

14.407

14.095

13.396

9.356

5

15.419

15.416

15.386

15.277

14.947

14.206

9.919

6

16.434

16.430

16.399

16.281

15.928

15.138

10.578

7

21.264

21.259

21.218

21.064

20.606

19.582

13.687

8

21.811

21.806

21.764

21.606

21.136

20.086

14.039

9

25.572

25.566

25.516

25.332

24.782

23.553

16.492

10

25.737

25.731

25.681

25.495

24.939

23.696

16.543

RI PT

0.5

19.285

16.038

19.370

16.082

4.103

2.864

8.249

5.770

8.724

6.085

1

6.166

6.119

6.060

5.925

5.642

5.157

2.979

2

12.391

12.295

12.176

11.903

11.331

10.357

5.999

3

13.124

13.023

12.898

12.610

12.005

10.972

6.331

4

20.158

20.001

19.806

19.358

18.422

16.833

9.731

5

21.377

21.210

21.004

20.531

19.540

17.856

10.317

6

22.774

22.596

22.375

21.869

20.811

19.015

11.001

7

29.468

29.236

28.949

28.292

26.921

24.595

14.233

8

30.226

29.988

29.694

29.020

27.613

25.227

14.599

9

35.397

35.118

34.774

33.985

32.339

29.550

17.143

10

35.697

35.415

35.067

34.270

32.605

29.783

17.207

TE D

µ = 0.75

0.25

SC

µ = 0.5

0

M AN U

µ = 0.25

f [Hz]

Table 10. First ten frequencies of a sectorial plate reinforced by CNTs by varying the agglomeration parameter µ and the functionally

AC C

EP

graded exponent p with η = 1 and wr = 0.2 for the mesh with no distortion d = 2 m .

f [Hz] 1 2 3 4 5 6 7 8 9 10

2D FEM 12.651 27.903 31.453 50.812 58.942 61.917 75.007 76.604 84.742 94.964

0 2D IGA 12.786 29.136 31.416 51.231 59.734 62.611 77.516 80.601 88.329 99.281

12.737 28.173 31.349 50.762 59.180 61.583 74.249 76.387 84.621 95.215

p 0.25

0.5

10.620 23.521 26.127 42.379 49.307 51.373 61.846 63.685 70.547 79.382

2.024 4.504 5.117 8.330 9.363 10.244 12.089 12.427 13.664 15.258

1

2 GDQ 7.292 5.352 16.200 11.934 17.870 13.065 29.145 21.448 33.780 24.724 35.183 25.755 42.237 30.782 43.617 31.902 48.349 35.401 54.403 39.808

4

100

4.014 9.015 9.845 16.237 18.548 19.460 23.083 24.005 26.624 29.895

2.676 5.965 6.714 10.891 12.405 13.341 15.770 16.273 17.929 20.107

2.028 4.552 5.113 8.327 9.404 10.204 11.980 12.406 13.642 15.291

∞ 2D FEM 2.024 4.504 5.117 8.330 9.363 10.244 12.089 12.427 13.664 15.258

2D IGA 2.046 4.703 5.111 8.399 9.489 10.359 12.493 13.075 14.242 15.952

Table 11. First ten frequencies of a “smile” plate reinforced by CNTs by varying the functionally graded exponent p with η = µ = 0.5 and wr = 0.4 . A 15 × 15 grid is employed for each element.

47