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Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates
Accepted Manuscript Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates M. Memar Ardestani, L.W. Zhang, K.M. Liew PII: DOI: Reference:
S0045-7825(16)30604-1 http://dx.doi.org/10.1016/j.cma.2016.12.009 CMA 11255
To appear in:
Comput. Methods Appl. Mech. Engrg.
Received date: 19 June 2016 Revised date: 7 December 2016 Accepted date: 8 December 2016 Please cite this article as: M.M. Ardestani, L.W. Zhang, K.M. Liew, Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10.1016/j.cma.2016.12.009 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.
Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates
M. Memar Ardestani 1, L.W. Zhang 2,*, K.M. Liew 1,3,* 1 2
3
Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, Guangdong, China
Abstract A first attempt to investigate bending and vibration behaviors of carbon nanotube (CNT) reinforced functionally graded composite skew plates using the isogeometric method is performed. The formulation of the plate is based on Reddy’s higher order shear deformation theory (HSDT). The effect of CNT orientation angle on the mechanical behavior of CNT-reinforced composite plates with varying skew angles is investigated. The optimum fiber orientations for CNT-reinforced composite skew plates with simply supported, clamped and cantilevered boundary conditions are found. It is shown that selecting the correct fiber orientation angle is of crucial importance in obtaining the desired static and dynamic responses of CNT-reinforced composite skew plates. Keywords: Carbon nanotube; Composite; Bending; Free vibration; Reddy’s third-order shear deformation theory; Isogeometric method
*Corresponding authors. E-mail addresses: [email protected] (L.W. Zhang), [email protected] (K.M. Liew)
1
1. Introduction In 2005, isogeometric analysis (IGA) was proposed by Hughes et al. [1] and was originally aimed to fill the gap between the design and analysis stages. In this method, which of course is better known as the more general form of Finite Element Method (FEM), basis functions used for interpolating the field variables within the elements are the same as those used for constructing the geometrical model. These basis functions are constructed by making use of Non-Uniform Rational B-Splines (NURBS) which are capable of constructing the exact geometrical model. Furthermore, as opposed to FEM, this method provides a very convenient manner in which to achieve a higher order of continuity of basis functions [2]. Since the introduction of IGA, a considerable amount of research has been published on this subject. Specifically, in the field of plate structures, it has gained much attention as a suitable numerical technique. Tran et al. [3] presented static, dynamic and buckling analyses of functionally graded rectangular as well as circular plates based on Reddy’s higher-order shear deformation theory (HSDT) using the isogeometric method. They concluded that IGA does provide an effective computational tool for practical problems. An isogeometric analysis of CNT-reinforced composite plates based on HSDT is carried out by Phung-Van et al. [4]. They considered 5 degrees of freedom associated to each control points. For some plate configurations, the presented results in [4] showed significant deviation from those previously published results. In the present study, we will further examine the HSDT formulation using 7 degrees of freedom in an attempt to achieve more reliable and accurate results using IGA. Moreover, Valizadeh et al. [5] utilized a NURBS-based isogeometric method to analyze functionally graded plates. In their study, the plate displacement field was based on FSDT and they made use of an artificial shear correction factor to overcome the shear locking phenomenon when low order NURBS basis functions were used for thin plates. Using IGA, and incorporating a simple four-unknown shear and normal deformation theory, Thai et al. [6] studied the bending and free vibration behavior of functionally graded isotropic and sandwich plates. The free vibration and buckling analyses of laminated composite plates using the isogeometric method were presented by Shojaee et al. [7]. In their study, which is based on classical plate theory (CPT), they provided results for the vibration of elliptical and square plates, each containing a hole of complex shape. They have stated that by using the capability of NURBS to exactly represent complex geometric shapes, the isogeometric method would be an efficient and accurate technique for practical problems of engineering. In addition, Shojaee et al. [8] studied the free vibration of thin circular, skew, L-shaped and square-shaped (with complex cutout) plates using the 2
isogeometric method. The body of research using the isogeometric method has become vast and remains ongoing [9]–[14] . For more detailed information considering the computer implementation of the isogeometric method, readers are referred to [15], [16]. Carbon nanotubes (CNTs), since reported by Iijima [17], have been widely used as reinforcement in structures such as beams, plates, and shells made of composite materials, usually referred to as CNT-reinforced composites, due not only to their prominent mechanical properties but also to their unique thermal and electrical features [18], [19]. In particular, CNT-reinforced composite plates with diverse geometrical shapes have gained much attention as high performance structures for various applications in the aerospace, aeronautics and marine industries as well as in bridge structures. In fact, many researchers have devoted their attention to investigating the superior behavior of CNTs, through the framework of various schemes [20], [21]. In 2009, Shen analyzed the nonlinear bending behavior of CNT-reinforced composite plates [22]. In that paper, he introduced the concept of the functionally graded scheme into CNTreinforced composite plates, in which the material properties are graded along the thickness direction of the plate. In order to obtain the material properties he further proposed the CNT efficiency parameters which are estimated by matching the elastic modulus of CNT-reinforced composites obtained from the molecular dynamic (MD) simulation with those numerical results obtained from the rule of mixture. The static and free vibration of square CNT-reinforced composite plates based on first order shear deformation theory (FSDT) are presented by Zhu et al. [23]. In their study, they used four types of CNT distribution through the thickness of the plate, namely uniform distribution, FG-V, FG-O and FG-X. They stated that desired plate stiffness can be achieved by adjusting the distribution of the CNTs along the thickness direction of the plate. Moreover, they concluded that CNT distributions that have more CNTs near the top and bottom surfaces of the plate are more efficient. A thermoelastic analysis of CNT-reinforced composite plates was carried out by Alibeigloo and Liew [24], in which they incorporated the three-dimensional theory of elasticity to analyze the thermos-elastic behavior of the CNT-reinforced composite plates under thermo-mechanical loads. Alibeigloo and Emtehani further published a paper [25] on the static and vibration behavior of simply supported CNT-reinforced composite plates. They obtained a closed form solution for simply supported CNT-reinforced composite plate using Fourier series expansion and the state space technique and for the case of the clamped boundary condition, they used a semi-analytical technique. They further performed parametric studies on CNT-reinforced composite plate to describe their vibration behavior. Wang and Shen [26] studied the large amplitude vibration of 3
nanocomposite plates resting on an elastic foundation. In their study, the equations of motion are based on higher-order shear deformation plate theory in which plate-foundation interaction is taken into account. Moreover, using the Ritz method, Abdollahzadeh Shahrbabaki and Alibeigloo [27] presented a three-dimensional model for the free vibrational analysis of CNT-reinforced composite plates. Zhang et al. [28] studied the free vibration characteristics of CNT-reinforced composite triangular plates based on FSDT and using the element-free IMLS-Ritz numerical method. They presented a detailed study considering the effects of various geometrical and material parameters on the vibration of CNT-reinforced composite triangular plates. Using a unified formulation of finite prism methods (FPMs) and three-dimensional model, free vibration behavior of CNT-reinforced composite plates with various boundary conditions is investigated by Wu and Li [29]. Based on Eshelby-Mori-Tanaka approach, Sobhani Aragh et al. [30] investigated the vibration characteristics of continuously graded carbon nanotube-reinforced cylindrical panels. They concluded that continuously graded oriented CNTs can be used to achieve the desired effect in relation to the vibration behavior of structures. Furthermore, in the context CNT-reinforced composite plates, Lei et al. [31] compared the effective material properties estimated by the Eshelby-Mori-Tanaka approach and extended rule of mixture. They concluded that results obtained for the frequency parameters of the CNT-reinforced composite plate using the extended rule of mixture were slightly higher than those obtained based on Eshelby-Mori-Tanaka approach. Semianalytical solutions to buckling and free vibration analysis of carbon nanotube-reinforced composite thin plates utilizing multi-term Kantorovich-Galerkin method are proposed by Wang et al. [32]. Furthermore, Selim et al. [33] performed a study on the vibration of CNT-reinforced composite plates in a thermal environment based on Reddy’s higher-order shear deformation theory (HSDT). In their comprehensive study, a comparison between the results obtained based on FSDT and HSDT was made and a maximum relative difference of three percent for the natural frequency parameter for the first mode of free vibration at room temperature was reported. Vibrational analysis of carbon nanotube-reinforced composite quadrilateral plates subjected using a weak formulation of elasticity considering the thermal effects is presented by Ansari et al. [34]. Moradi-Dastjerdi et al. [35] presented the static analysis of CNT-reinforced composite cylinders utilizing the element-free Galerkin method. They used an axisymmetric model in their approach and investigated the influence of CNT distribution and volume fraction on the value as well as the location of maximum hoop stress. A nonlinear bending analysis of CNT-reinforced composite plates resting on a Pasternak foundation is presented by Zhang et al. [36], in which the influence of foundation stiffness as well as CNT distribution and CNT volume fractures is investigated. 4
The analysis of skew plates has long been interesting to researchers in the field of structural mechanics. Due to the presence of singular behavior on the obtuse angles of skew plates, an appropriate numerical scheme is usually required. One of the earliest studies to take this stress concentration into account is that of Morley [37]. He utilized the simple form of eigenfunctions to consider the singularities in obtuse angles. The results published in his paper are known to be benchmark solutions and, as such, will be used in our study for verification purposes. Rajaiah and Rao [38] also derived an exact solution to this problem by applying Stevenson’s tentative approach with complex variables. Butalia et al. [39] performed a comprehensive bending analysis of parallelogram-shaped plates using a Mindlin nine-node Heterosis element. In their study, the efficiency and the limitations of that element were explored and numerical results for the transverse shear forces are presented. Furthermore, Sengupta [40] carried out a performance study of simple finite elements for the bending analysis of skew rhombic plates. He proposed a simple three-node element and compared its accuracy and reliability with a triangular Mindlin plate element of 35 degrees of freedom (DOF) and a triangular plate element previously proposed by Zienkiewicz and Lefebvre [41] which had independent interpolation for slope, displacement and shear forces. He concluded that the proposed simple three-node element fits suitably for the analysis of skew plates. We will also use the results obtained by him in the following sections for numerical verification. A bending analysis of simply supported skew plates based on FSDT was carried out by Liew and Han [42]. They presented the results for both thin and thick plates with skew angles from slight to extreme cases. Some other references regarding the bending analysis of skew plates are found at [43]–[47]. Moreover, extensive work has also been conducted on the vibration analysis of skew plates. An analysis of thick skew plates based on FSDT using the pb-2 Rayleigh-Ritz method was presented by Liew et al. [48]. They apply special treatments to impose the boundary conditions on oblique edges. McGee and Leissa published a three-dimensional vibration analysis of skew cantilevered plates [49]. In that paper, they calculated accurate natural frequencies for skew plates with arbitrary skew angles. Moreover, McGee et al. [50] presented a free vibration study on clamped and simply supported thick rhombic plates based on HSDT. Krishna Reddy and Palaninathan [51] presented a free vibration analysis of skew plates by extending a general high precision three-node bending finite element to the free vibration analysis of laminated skew plates. They derived the consistent mass matrix in an explicit form and transformed element matrices to impose boundary condition on oblique edges. Further studies on the vibration analysis of skew plates can be found at [52]–[57].
5
Recently, studies dedicated to the bending and vibration analysis of CNT-reinforced composite skew plates have been carried out. García-Macías et al. [58] analyzed the static and free vibration behavior of CNT-reinforced composite skew plates using the shell finite element based on FSDT which was formulated in oblique coordinates. Kiani [59] presented the free vibration analysis of CNT-reinforced composite skew plates. To construct an eigenvalue problem, he used Ritz method in which shape functions are obtained according to Gram-Schmidt process. Moreover, a large deflection analysis of CNT-reinforced composite skew plates resting on Pasternak elastic foundations is carried out by Zhang and Liew [60]. In addition, an investigation on the vibration characteristics of moderately thick skew CNT-reinforced composite plates is presented by Zhang et al. [61]. For a more comprehensive overview of these papers on CNT-reinforced composite structures, readers are encouraged to refer to the review paper presented by Liew et al. [62]. In this paper, the isogeometric approach is utilized to analyze the bending and vibration behaviors of CNT-reinforced composite skew plates. The displacement field of the plate is based on Reddy’s third order shear deformation theory (HSDT). Considering the skew CNT-reinforced composite plate as a one-ply composite plate, the influence of CNT orientation angle on the mechanical response of the plate accounting for different plate aspect ratios, width-to-thickness ratios and boundary conditions is investigated. As will be demonstrated in detail, the choice of CNT orientation angle has a crucial effect on the desired responses of the CNT-reinforced composite skew plates. 2. B-splines and NURBS 2.1. Knot vector A fundamental feature that controls the behavior of a B-spline curve to be constructed is the knot vector. A knot vector is a set of ordered, non-decreasing, real numbers, known as knots, associated with each B-spline objects i.e. if 1,, i ,, n is a knot vector associated with a B-spline curve C, then i and i1 i for i 1, , n 1 . Ξ is said to be open if its first and last members are repeated p+1 times, and p is the polynomial order. For an open knot vector, Ξ, the number of members, n, is calculated from n b p 1 where b is the number of basis functions used in the construction of the B-spline curve. A knot vector is known to be uniform if the knots are equally spaced from each other; otherwise it is non-uniform. Knots divide the B-spline curve into sections recognized as elements. Within an element (i.e. between knots), functions are C and between
6
elements (i.e. over knots), functions are C p k where k represents the multiplicity of the corresponding knot.
2.2. Basis function Associated with a knot vector , are B-spline basis functions denoted by Ni , p for i 1, , n defined according to the following well-known Cox-de Boor recursive formulas, For p 0
1 if i i 1 , Ni ,0 ( ) otherwise. 0
(1)
For p 1
Ni , p ( )
i Ni , p 1 ( ) i p 1 N ( ). i p i i p 1 i 1 i 1, p 1
(2)
For cases of p=1, p=2, p=3 and p=4 with their corresponding open knot vectors written respectively
as
0,0,1, 2, 2 ,
0, 0, 0,1, 2, 2, 2 ,
0, 0, 0, 0,1, 2, 2, 2, 2
and
0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 the basis functions are shown in Fig. 1. 2.3. B-spline objects 2.3.1. B-spline curves A B-spline curve is defined as b
C( ) Ni , p ( )Pi ,
(3)
i 1
where Pi , i 1, 2,, b are the control points and Ni , p ( ) are the basis functions of degree p
associated with the knot vector 1 , 2 , , b p 1 calculated by Eqs. (1) and (2). 2.3.2. B-spline surfaces B-spline surfaces are known as tensor product extensions of B-spline curves and are defined as b
b
S , H ip, j, q , Pi , j ,
(4)
i 1 j 1
p ,q where Hi , j ( , ) i 1, 2,, b , j 1, 2,, b are the bivariate B-spline basis functions defined as
7
H ip, j,q ( , ) Ni , p ( ) M j ,q ( ),
(5)
in which N i , p ( ) and M j , q ( ) are the basis functions with polynomial degrees p and q along the
and
parametric
directions
associated
1 , 2 , , b q 1 , respectively.
with
P
2
i, j
knot
vectors
1 , 2 , , b p 1
and
is the control net associated with the B-spline
surface. 2.4. NURBS Introducing an additional variable referred to as weight into a more general form of B-spline function (i.e. rational B-splines), it is now possible to exactly represent conical shapes such as circles, cones and so on, which was not achievable through the use of classical non-rational Bsplines. As a favorable feature of NURBS, any projective transformation applied to NURBS objects can be performed by directly applying it to the control polygon [63]. As will be seen in the following sections, we make use of this property for the special case of the affine transformation of geometrical shapes. wi is usually treated as an additional coordinate for the control points and is kept constant under the affine transformation of control points. 2.4.1. NURBS basis function In this way, NURBS basis functions are defined as
Rip ( )
N i , p ( ) wi b
N I 1
I,p
,
(6)
( ) wI
where Ni , p are the B-spline basis functions defined in Eqs. (1) and (2) and wi are the weights associated with each control point. 2.4.2. NURBS objects Replacing the univariate and bivariate B-spline basis functions used in Eqs. (3) and (4) with the corresponding NURBS basis functions, one can construct the NURBS objects in one and two dimensions. Therefore, in addition to the knot vectors, control points and basis functions associated with each B-spline object, in the case of NURBS a new set of weights is also added. The formulations are written in more detail in the following.
8
A. NURBS curves
A NURBS curve associated with a knot vector 1 , 2 , , b p 1 is defined as b
C( ) Ri , p ( )Pi ,
(7)
i 1
where R i , p , i 1,2,..., b are the univariate NURBS basis functions of degree p in the direction defined in Eq. (6) and Pi are their associated control points. B. NURBS surfaces Given a control net, Pi , j , their corresponding weights, wi , j , and a knot vector corresponding to each parametric direction, NURBS surfaces are defined to be a tensor product of univariate NURBS. In other words b
b
S , Rip, j,q , Pi , j ,
(8)
i 1 j 1
p ,q in which Ri , j is the bivariate NURBS basis function and defined as
Rip, j, q ( , )
N i , p ( ) M j , q ( ) wi , j b
b
N I 1 J 1
I,p
,
(9)
( ) M J , q ( ) wI , J
where N i , p ( ) and M j , q ( ) are the univariate B-spline basis functions of degree p and q along the and parametric directions.
In order to increase the order of the NURBS basis functions, the k-refinement technique is used (i.e. the order of the basis function is elevated with the coarsest mesh before any internal knots are inserted). 3. Carbon nanotube-reinforced composites We consider a CNT-reinforced composite skew plate with skew angle , length a, width b and thickness h as shown in Fig. 2. The plate material is composed of an isotropic matrix and singlewalled carbon nanotubes (SWCNTs), aligned with orientation angle with respect to the positive x-direction as shown in Fig. 2, as reinforcement. The effective modulus of elasticity, employing the extended rule of mixture, is expressed as 9
E11 1VCNT E11CNT Vm E m ,
2 E22
3 G12
(10)
VCNT Vm m, CNT E22 E
(11)
VCNT Vm , G12CNT G m
(12)
CNT
CNT
CNT
where E11 , E22
and G12
are the Young’s moduli and shear modulus of the CNT, respectively.
E m and G m are the corresponding properties of the isotropic matrix. i (i 1,2,3) are the CNT efficiency parameters accounting for the scale-dependent material properties calculated by comparing the effective material properties obtained from MD simulations and that of numerical results [22]. VCNT and Vm are the CNT and matrix volume fractions related by
VCNT Vm 1.
(13)
The CNTs are arranged along the thickness of the plate according to four different distribution types: uniform distribution (UD), FG-V, FG-O and FG-X (see Fig. 3). Mathematically, these CNT distributions can be written as
VCNT
* VCNT 1 2 z V * CNT h 2 z * 2 1 h VCNT 2 z * 2 VCNT h
(UD CNTRC), (FG-V CNTRC), (14)
(FG-O CNTRC), (FG-X CNTRC),
* where VCNT is the volume fraction of CNTs and is defined as
* VCNT
wCNT (
wCNT CNT
m )(1 wCNT )
,
(15)
CNT m in which wCNT , and are the mass fraction of CNTs, densities of CNTs and matrix, * respectively. For all cases of CNT distribution, the value of VCNT is set to be constant. In addition,
the Poisson’s ratio, 12 , and the density, , of the CNT-reinforced composite plate are obtained from the following expressions. * 12 VCNT 12CNT Vm m ,
(16)
10
VCNT CNT Vm m .
(17)
It should be noted that following the assumptions made in [22] and [23], 12 is considered to be constant over the thickness of the plate. 4. Governing equations of CNT-reinforced composite skew plates 4.1. Displacement field and strain Employing the HSDT for a CNT-reinforced composite plate, the displacement field of the plate defined in the domain Ω can be expressed as
u( x, y, z) u0 ( x, y) zx ( x, y) cz3 x ( x, y),
(18)
v( x, y, z ) v0 ( x, y ) z y ( x, y ) cz 3 y ( x, y ),
(19)
w( x, y, z) w0 ( x, y),
(20)
where u , v and w are the displacement of a generic point ( x , y , z ) in the domain of the CNTreinforced composite plate along the x-, y- and z-directions, respectively. (u0 , v0 , w0 ) represent the displacements of a point at the mid-plane of the plate and x and y denote rotations of the cross section of the plate about the y- and negative x-axes, respectively. x and y are defined to be
( x w0 / x) and ( y w0 / y ) , respectively. The constant c is set to be 4 / 3h 2 . The strains of a point in the plate are given by
ε ε(0) zε(1) z3ε(3) ,
(21)
γ γ(0) z2γ(2) ,
(22)
where ε xx
ε (0)
yy xy , γ xz yz and T
T
x x u0 x x x y v0 (1) y (3) ,ε , ε c , y y y u0 v0 y y x x x x y x y y
11
(23)
and
γ (0)
w0 x x x , γ (2) 3c . y y w0 y
(24)
Then, the constitutive relations are expressed as
xx Q11 ( z ) Q12 ( z ) Q16 ( z ) 0 0 xx 0 0 yy yy Q12 ( z ) Q22 ( z ) Q26 ( z ) 0 0 xy , xy Q16 ( z ) Q26 ( z ) Q66 ( z ) 0 0 0 Q55 ( z ) Q45 ( z ) xz xz yz 0 0 0 Q45 ( z ) Q44 ( z ) yz
(25)
where Qij are the components of a transformed stiffness matrix in the global xy coordinate system and are related to Qij , the components of the stiffness matrix in the principal material coordinate system, through the following equation.
Q TQ T . T
(26)
In Eq. (26), T , the transformation matrix, is expressed as
m2 n2 2mn 0 0 2 2 m 2mn 0 0 n T mn mn (m2 n 2 ) 0 0 , m n 0 0 0 0 n m 0 0
(27)
where m and n denote cos( ) and sin( ) , respectively. is the angle between the principal material coordinate system and global xy coordinate system.
Q introduced in Eq. (26), together with its component definitions are given as 0 0 0 Q11 ( z ) Q12 ( z ) Q ( z ) Q ( z ) 0 0 0 22 12 Q 0 Q66 ( z ) 0 0 0 , Q55 ( z ) 0 0 0 0 0 Q44 ( z ) 0 0 0
(28)
12
Q11
E11
1 12 21
E22
, Q22
1 12 21
, Q12
21 E11 , Q44 G23 , Q55 G13 , Q66 G12 , 1 12 21
(29)
where E11 and E22 are the effective Young’s moduli of the CNT-reinforced composite plate in the principal material directions. G12 , G13 and G23 are the shear moduli and 12 and 21 are the Poisson’s ratios and are related to each other according to 21 (E22 E11 )12 . According to HSDT, the relationship between the stress resultants and strains can be written as
0 ε (0) 0 ε(1) 0 ε (3) , Ds γ (0) Fs γ (2)
N A B E 0 M B D F 0 P E F H 0 Qs 0 0 0 As s R 0 0 0 Ds
(30)
where the in-plane force resultants, moment resultants and transverse forces are expressed as N xx xx h/ 2 N N yy yy dz , h/ 2 N xy xy
(31)
M xx xx h/ 2 M M yy yy z dz , h /2 M xy xy
(32)
Pxx xx h /2 P Pyy yy z 2 dz , h /2 P xy xy
(33)
Q s h /2 xz Qs xs dz, Qy h /2 yz
(34)
s R h /2 xz Rs xs z 2 dz. Ry h /2 yz
(35)
The components used in Eq.(30) are given as
A , B , D , E , F , H Q 1, z, z , z , z , z dz ( A , D , F ) Q (1, z , z )dz for i, j 4and 5. h /2
ij
ij
ij
s ij
s ij
s ij
ij
ij
ij
h /2
h /2
h /2
2
2
3
4
6
ij
4
ij
13
for i, j 1, 2and 6,
(36) (37)
4.2. Stiffness matrices The total potential energy of the plate is expressed as Π U p W,
(38)
where U p denotes the strain energy of the plate and W is the work done by the external forces. The strain energy of the plate can be written as Up
1 ε T S ε d , 2
(39)
where ε (0) A B E (1) B D F ε ε ε (3) , S E F H γ (0) 0 0 0 (2) 0 0 0 γ
0 0 0 s
A Ds
0 0 0 . Ds F s
(40)
Considering the plate under a general lateral distributed load of q ( x , y ) , W can be written as
W q( x, y ) wd .
(41)
In the case of vibration analysis, W in Eq. (38) is replaced with plate kinetic energy, K, and expressed as
K
h /2 1 ( z )(u 2 v 2 w 2 )dzd . 2 h /2
(42)
Invoking the isoparametric concept, the field variables in the domain of the plate can be approximated as
u , h
b b
, d , i 1
i
(43)
i
h where {u } is the vector of field variables to be approximated and is represented as
u h u0
v0
w0 x y x y ; i , is the simpler notation for the NURBS shape T
p, q functions, Ri , , with designated degrees of continuity, p and q, for each parametric direction
14
and {di } is the vector of nodal degrees of freedom, d i u0 i
v0 i
w0i
xi yi xi yi , T
associated with the control point i in the control net. Substituting Eq. (43) into Eq. (38) and then taking variation, the following system of linear equation for the bending analysis of the plate is derived and expressed as
KU F.
(44)
It should be noted that U in Eq. (44) is composed of all instances of {d I } in the control net, that is U d1 d 2 d I
dn .
(45)
T
Furthermore, the eigenvalue equation of the free vibration analysis of the plate is obtained as
(K 2M)U 0,
(46)
where is the natural frequency of the plate and {U} is the vector containing the corresponding mode shapes. [ K ] , [ M ] and {F } are the global stiffness matrix, the global mass matrix and the global force vector, respectively which are obtained by assembling their corresponding element counterparts denoted by superscript ‘e’, given by
[Ke ] [Kem ] [Kbe ] [Kce ] [Kes ],
(47)
[K em ]ij e (Bim )T AB mj d,
(48)
[K be ]ij e (Bbi 1 )T DBbj1d e (Bbi 2 )T HBbj 2d
(49)
e (B ) FB d e (Bbi 2 )T FBbj1 d,
b1 T i
b2 j
[K ] e (B ) B B d e (Bbi 1 )T B B mj d e c ij
b1 j
m T i
(50)
e (B ) EB d e (Bbi 2 )T EB mj d,
b2 j
m T i
[Kes ]ij e (Bis1 )T AsBsj1d e (Bis 2 )T FsBsj2d
(51)
e (B ) D B d e (B ) D B d,
s1 T i
s
s2 j
s2 T i
s
s1 j
ˆ T ρH ˆ d, [M e ]ij e H i j
(52)
{Fe }i e qNTi d ,
in which
15
i , x 0 B 0 i , y i , y i , x
0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0
m i
0 0 0 i , x B 0 0 0 0 0 0 0 i , y
0 0 0 0 , 0 0
0
i , y i , x
b1 i
0 0 0 0 0 i , x B c 0 0 0 0 0 0 0 0 0 0 0 i , y b2 i
0 0 i, x Bis1 0 0 i, y 0 0 0 Bis 2 3c 0 0 0
(53)
(54)
0 i , y , i , x
(55)
i
0 0 0 , 0 i 0 0 0 0 i 0 , 0 0 0 i
(56) (57)
i 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 i 0 0 0 0 ˆ 0 0 0 0 0 0, H i i 0 0 0 0 i 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 i
0 I0 0 I 0 0 0 ρ I1 0 0 I1 cI 3 0 0 cI 3
0 0 I0 0
I1 0 0 I2
0 I1 0 0
cI 3 0 0 cI 4
0 0 0
0 cI 4 0
I2 0 cI 4
0 c I6 0
Ni 0 0 i
2
(58)
, cI 4 0 c 2 I 6 0 cI 3 0 0
(59)
0 0 0 0 .
(60)
The components used in [ρ] are defined as
Ii
h /2
h /2
z dz (i 0,1, 2,3, 4,6). i
(61)
16
5. Essential boundary conditions 5.1. Conventions Here, the boundary conditions used in studying the mechanical behavior of CNT-reinforced composite plates are presented as follows. Simply supported (S):
For upper and lower edges: u w x x 0,
(62)
For obliqueedges: v w y ' y ' 0.
(63)
Clamped (C):
For upper and lower edges: u v w x y x y 0,
(64)
For obliqueedges: u v w x ' y ' x ' y ' 0.
(65)
For free edges, no kinematic constraints will be imposed. The support of each edge of the plate will be denoted by S, C or F. In order to fully define the boundary conditions of a plate, these symbols will be written in sequence started from the bottom edge and will be continued to other edges in a counter clock wise direction. It should be mentioned that, following the approach used in [42] for bending analysis of skew plates, corner control points are considered to belong to the oblique edges. 5.2. Skew boundaries For the case of skew plates, in order to impose the appropriate boundary conditions on the nodes laid on the oblique edges of the plate it is necessary to introduce the oblique local coordinate system ( x y ) shown in Fig. 2. The relationships between position vectors, rotations and partial derivatives
in the global xy coordinate and the local x y coordinate are written as [48] x x sin( ) y, y cos( ) y,
(66)
x cos( ) x , y sin( ) x y ,
(67)
() / x () / x, () / y () / x tan( ) () / y sec( ),
(68)
where variables with prime symbol, ()׳, denote those obtained in the local x y coordinate system. Using the relations presented in Eqs. (66) to (68), and carrying out some manipulation, the expressions for x and y in terms of variables in an oblique coordinate system can be derived as
x cos( ) 1 x x ,
(69)
17
y tan( ) sin( ) x 1 sec( ) y tan( ) x sec( ) y.
(70)
Furthermore, the relationship between control variables in the global xy coordinate and the local x y coordinate for a control point I located on the oblique edge is written as d (I xy ) A xxyy d (I x y ) ,
(71)
where the transformation matrix, [ A xyxy ] , can now be constructed using Eqs. (66) to (70) as follows
A xy xy
1 sin( ) 0 cos( ) 0 0 0 0 0 0 0 0 0 0
. 0 1 0 0 sin( ) 0 cos( ) 1 0 1 0 0 tan( ) sin( ) 1 sec( ) tan( ) sec( )
0 0 1 0
0 0 0 cos( )
0 0 0 0
0 0 0 0
0 0 0 0
(72)
During the assembly process, only for those elements that have control points laid on oblique edges, element stiffness matrix, [K e ] , element mass matrix, [M e ] and element force vector, {F e} are transformed according to Eqs. (73), (74) and (75), respectively [64]. K e ( A e )T K e A e , M e ( A e )T M e A e ,
(74)
F e ( A e )T F e ,
(75)
(73)
in which the element transformation matrix, [Ae ] , is a diagonal matrix represented in Eq. (76), formed by placing either [ A xyxy ] or [I ] on its diagonal, depending on whether the corresponding
{dI } is related to the control point on the oblique edges. [I ] is the identity matrix of the same size as [A xy xy ] in Eq. (72). A xy x y Ae
I
0 I
0
A xy xy
. I
(76)
18
6. Numerical studies In this section, the capabilities of the IGA, as a numerical tool to be used in problems of skew plates, are first examined. Then, the bending and free vibration responses of CNT-reinforced composite skew plates as one-ply composite plates, are fully investigated. Throughout this study, several non-dimensional parameters have been used and it is convenient to introduce them here for future reference. For isotropic plates, the non-dimensional deflection, non-dimensional moment and nondimensional vibration parameters are defined in Eqs. (77), (78) and (79), respectively.
Wiso* wc 1600 D / (qa 4 ),
(77)
* M iso ( M c ) 40 / (qa 4 ),
(78)
* iso ( b 2 / 2 ) h / D ,
(79)
where D is the flexural rigidity of the plate and is defined as D Eh3 /12(1 2 ) . Moreover, subscript ‘c’ is used for parameters evaluated at the center point of the plate. For CNT-reinforced composite plates, the non-dimensional deflection and non-dimensional vibration parameter are defined in Eqs. (80) and (81), respectively.
W * wc / h,
(80)
( b 2 / h ) m / E m .
(81)
6.1. Bending analysis 6.1.1. Convergence and validation study Here we validate the accuracy and convergence of our IGA approach for the bending analysis of skew plates. The obtained results will be used in the sections that follow. We consider a rhombic plate (a=b) with the skew angle of shown in Fig. 2. Convergence studies are performed for two * different skew angles of 0 and 60 degrees; the non-dimensional deflections, Wiso , at the center of the
isotropic plates using NURBS functions of different degrees are obtained and shown in Figs. 4 and 5. As can be seen in Fig. 4 while the results obtained are convergent, higher rates of convergence can be achieved using NURBS of a higher degree. Moreover, as shown in Fig. 5, it is inevitable that a higher degree of NURBS functions will be utilized as skew angles become larger [65], [66], [67].
19
Having performed the numerical experiment, we have proved that quantic NURBS functions (i.e. NURBS with a continuity of degree four) perform optimally for the analysis of skew plates and, as such, they will be used in this study. The initial knot vectors used for both and directions are given as {0, 0, 0, 0, 0,1,1,1,1,1} , {0, 0, 0, 0, 0,1,1,1,1,1} , respectively. For knot insertion, the * knots are inserted in equally spaced manner. Moreover, values for non-dimensional deflection, Wiso , * and non-dimensional moments, M iso , at the center of the skew isotropic plates with skew angles of
30˚, 45˚ and 60˚ subjected to uniformly distributed loads are presented in Table 1. For comparison the results available in the literature are also provided in Table 1. As can be seen, the obtained results agree well with those previously reported for both simply supported and fully clamped boundary conditions. Based on the results listed in Table 1, the plates discretized using 19×19 elements will be utilized in all the analysis to be carried out in the following sections. Numerical integrations have been performed using a 4×4 gauss scheme within each element. 6.1.2. CNT-reinforced composite plates A comprehensive numerical analysis to investigate the mechanical behavior of CNT-reinforced composite plates is conducted in this study. The chosen matrix material is Poly-co-vinylene referred to as PmPV. The reinforcement material is comprised of armchair (10,10) SWCNTs. The properties of these two mentioned materials at room temperature are listed in Table 2 as reported in [22]. Three values of CNT volume fraction are considered here: 0.11, 0.14 and 0.17. The CNT efficiency parameters, i , (i 1, 2,3) , corresponding to each CNT volume fraction are given in Table 3. It is assumed that 2 3 and G12 G13 G23 , following the assumptions made in [22]. The thickness of the plate is set to be h=2 mm for all examples presented. We consider a CNT-reinforced composite skew plate with equilateral sides (a/b=1) under a uniformly distributed load q0 . In order to explore the effect of skew angle, a bending analysis of plates with varying skew angles has been performed. Three values of CNT volume fraction (0.11, 0.14 and 0.17), three different side to thickness ratios (i.e. 10, 20 and 50) and four types of CNT distribution (UD, FG-V, FG-O and FG-X) are considered. The results for non-dimensional central deflection, W * , are listed in Table 4. For comparison, results from [23], obtained for square plates based on first order shear deformation theory, are also included. Once again, the formulation presented in section 4 is proved to provide highly accurate results. The small discrepancies between the obtained results from IGA and those of [23] are due mainly to the assumed third-order and firstorder displacement fields used in HSDT and FSDT, respectively. 20
6.1.3. Optimum CNT fiber orientation angle This section aims to determine the optimum CNT orientation angle for CNT-reinforced composite plates, and to consider whether the values for non-dimensional deflection presented in Table 4 are the best achievable values. Usually, the degree of skewness of the plate is specified according to design geometric constraints but the type of material as well as its orientation can be chosen based on analyses related to the ultimate requirement of the structure. In order to show the effect of CNT orientation angle on the bending behavior of skew and non-skew plates, numerical investigations have been carried out. We consider a rhombic plate composed of a single-ply CNTreinforced composite, as shown in Fig. 2. The angle between the principal material coordinates and the problem global coordinates, , is varied between -90 and +90 degrees. The obtained values for the deflection of the corresponding CNT-reinforced composite skew plate are obtained for four types of CNT distribution and are plotted in Fig. 6. It can be seen that in the case of the square plate (i.e. 0 ), the right and left halves of the obtained plot are perfectly symmetrical with respect to 0 . In this case, the CNT orientation angle values equal to either 0 or 90 degree rotations give the lowest deflection values for both simply supported and clamped boundary conditions while for 45 , the highest deflection values are obtained. As the skew angle is increased, the plate loses its symmetricity, and the left half of the graph (in the case of a rhombic plate) gradually vanishes. Moreover, the peak points of the plots tend to appear at the lower values of . Upon inspection, it can be deduced that, in the cases of both simply supported and fully clamped boundary conditions, for the CNT-reinforced composite rhombic plate of skew angle , the lowest values of deflection can be achieved with the CNT orientation angle of (i.e. the minimum values of deflection are achieved when CNTs are aligned perpendicular to the oblique side of the skew plate). Furthermore, the highest values of deflection occur when the CNTs are aligned with the main diagonal of the plate. To further clarify the effect of CNT orientation on the bending response of CNT-reinforced composite skew plates, the values listed in Table 4 are regenerated in Table 5 incorporating the CNT orientation angles . The results listed in Table 5 show that by optimally aligning the CNT fibers embedded in the CNT-reinforced composite lamina, the non-dimensional deflection parameters are reduced at least by the factors of 1.1, 1.4, 2 and 2.5 for the cases of 15˚, 30˚, 45˚ and 60˚ skew angles, respectively.
21
* 0.11 , b/h=50 Furthermore, an extreme case with a reduction factor of 7.3 (for the parameters VCNT
and FG-X) is also achievable simply by changing the direction of CNT orientation. In order to attain a better understanding of the effect of changing the CNT orientation angle, the deflection profiles of the simply supported UD CNT-reinforced composite rhombic skew plates under uniform loading of q=-0.1 MPa along the x- and y-directions of the local coordinate system placed at the center of the skew plates are depicted in Fig. 7. Two different CNT orientation scenarios are applied: first CNT-reinforced composite plates with CNTs oriented along the global x-axis ( 0 ), as shown in Figs. 7(a)-(b); and second, CNT-reinforced composite skew plates with CNT orientation angles , as shown in Figs. 7(c)-(d). As can be seen, the CNT-reinforced composite skew plates experience a dramatic change in the way they deform under the same loading and boundary conditions, in the two cases. Next, the effect of varying the aspect ratio of the plate, a/b, on the bending behavior of the plate is investigated. For this case study, we have considered the length of the oblique edges, b, to be fixed and the length of the other two parallel edges, a, will be changed. The uniform CNT distribution with volume fraction 0.14 is chosen. The plate width-to-thickness ratio, b/h, is set to be 50. The results for plates with skewness values of 0˚, 30˚, 45˚ and 60˚ are illustrated in Fig. 8. As can be seen in Fig. 8, for the case of rectangular plate (i.e. 0 ), while the graphs maintain their symmetricity with respect to the CNT orientation angle 0 for all aspect ratios, it is noticeable that as the aspect ratios are increased, in contrast to the case of a/b=1, the CNT orientation 0 , no longer results in the minimum deflection of the plate. In fact, it can be observed that for all plates regardless of their skewness angle and their aspect ratios (except a/b=0.5), CNT orientation angle 90 gives the minimum deflection. In other words, in order to obtain the lowest deflections for CNT-reinforced composite skew plates with aspect ratios larger than 1, CNT fibers should be oriented perpendicular to the larger edge of the plate. Moreover, for aspect ratios larger than 1.5, the maximum deflections of the plates occurs when the CNT orientation angle is 0 . In order to comprehend the effect of CNT orientation angle on the deflection of the CNTreinforced composite skew plates, the influence of width-to-thickness ratio, b/h, should also be explored. In this case, a rhombic plate with a skewness angle of 45 degrees, composed of CNT* 0.14 , is chosen. Width-toreinforced composites with uniform CNT distribution and VCNT
thickness ratios are selected to be 5, 10, 20, 50 and 100. The results for normalized non-dimensional 22
deflection, W ** , defined as non-dimensional central deflection divided by non-dimensional central deflection for the case with CNT orientation angle 90 are plotted against CNT orientation angle in Fig. 9. It can be clearly noted that the greater the width-to-thickness ratio of the plate the more pronounced the CNT orientation effect.
6.2. Vibration analysis In this section, the vibration behavior of skew plates, specifically those comprising CNTreinforced composites, is investigated. First, the performance of the NURBS-based isogeometric method is verified and then parameters affecting the natural vibration response of the CNTreinforced composite plates such as CNT orientation angle, plate width-to-thickness ratio, CNT volume fractions and distributions are thoroughly explored. 6.2.1. Convergence and validation study A convergence study on the vibration of thin and thick isotropic skew plates with simply supported boundary condition on all edges has been carried out. Obtained values for the first eight * non-dimensional frequency parameters, iso , are listed in Table 6. It can be seen that the results for
all skew angles and plate thickness-to-width ratios agree well with those available in the literature. It can be deduced that the results obtained with the use of 19×19 elements are fully converged and thus this number of elements will be used in the case studies to be followed. 6.2.2. CNT-reinforced composite plates Having validated our isogeometric approach for analyzing the vibration of skew plates, in this section we further study the dynamic behavior of CNT-reinforced composite skew plates. The material properties are the same as those presented in section 6.1.2, used in the study concerning the bending of CNT-reinforced composite skew plates. We begin by ascertaining the optimum CNT orientation angle for UD and CNT-reinforced composite plates of different skewness angles. We consider a CNT-reinforced composite plate with a skewness angle of . The non-dimensional natural frequency parameters, , versus the CNT orientation angles, , which vary between 90 and + 90 , are plotted for skew angles
0 , 15 , 30 , 45 and 60 for simply supported and clamped boundary conditions in Fig. 10. In
23
* 0.14 ) are each case, four types of CNT distribution with a fixed CNT volume fraction (i.e. VCNT
considered. The width-to-thickness ratio is set to 50. As can be seen in Fig. 10, in the case of square plates (i.e. 0 ), the plots for both boundary conditions consist of two symmetric valleys. The valleys are split by the point named dividing point. As the skew angle increases this point moves toward the negative CNT orientation angle. Furthermore, the valley on the right becomes deeper while the left side valley becomes shallower and eventually changes direction and turns into a hill. Generally, these developments are faster for simply supported cases than for clamped boundary conditions. In all cases, as long as we have two distinct valleys, the CNT orientation angle corresponding to the dividing point between valleys can be considered as the point with Highest Achievable Frequency (HAF). This point changes its location to the peak of the left hill for 15 and 45 with simply supported and fully clamped boundary conditions, respectively. Moreover, the CNT orientation angle corresponding to the lowest point in the right valley can be understood as the point with the Lowest Achievable Frequency (LAF). For future references, the CNT orientation angles corresponding to HAF and LAF for CNTreinforced composite plates of different skew angles are further tabulated in Table 7. As we will see later, the values presented in Table 7 for simply supported boundary conditions are valid only for plates with width-to-thickness ratios greater than 20. In addition, it is noticeable from Figs. 10 that in all cases each CNT distribution has made its own frequency level and higher natural frequencies can be achieved by choosing a CNT distribution from FG-O to FG-V, FG-U and FG-X, respectively. In other words, FG-O and FG-X CNT distributions form the lower and upper bounds for the natural frequencies of a specific CNT-reinforced composite skew plate, respectively. From this conclusion, and using the values presented in Table 7, the first six frequency parameters for LAF and HAF corresponding to FG-O and FG-X CNT distributions with different volume fractions for CNT-reinforced composite plates with skew angles 0 , 15 , 30 , 45 and 60˚ with varying width-to-thickness ratios are presented in Table 8-12, respectively. As can be seen, the greater the CNT volume fraction, the higher the stiffness of the plate structure and as the result the higher the natural vibration frequency parameters. To further explain the trends in Tables 8-12, we define a non-dimensional frequency ratio, RHL , as the ratio of HAF to LAF for a particular CNT distribution and boundary condition. By inspecting Tables 8-12, it can be realized that, for simply supported plates, as the skew angle or the width-to-
24
thickness ratio increases the frequency ratio, RHL , also increases. Moreover, although the frequency parameter values (and as a result the frequency ratio values) are amplified for the clamped boundary condition, they nevertheless follow the same fashion as that for the simply supported boundary condition. In addition, according to Tables 8-12, it can be understood that when a first frequency ratio is more than 1, this does not necessarily mean that its subsequent modes have the same frequency ratio. Moreover, subsequent frequency ratios will not follow the same trend as for the first frequency ratio when the CNT distribution or boundary conditions are changed. Furthermore, one can notice that, for the case of simply supported square FG-O CNT-reinforced * 0.11 , b/h=20, the HAF is slightly less than that of the corresponding composite plate with VCNT
LAF. We will explain this issue in more detail in the descriptions related to Fig. 14. As an interesting case, the effect of CNT orientation angle on the vibration response of cantilever CNT-reinforced composite rhombic plates is also explored here. The plate is clamped on its bottom edge (i.e. the boundary conditions are denoted as CFFF). The material properties are the same as those used for the case of the simply supported and clamped CNT-reinforced composite plates. The obtained results are shown in Fig. 11. In terms of CNT distribution type, the graphs show a similar trend to what has been exhibited by CNT-reinforced composite plates with the simply supported and clamped boundary conditions (i.e. the FG-O and FG-X CNT distributions form the lower and upper bounds for the frequency range of the CNT-reinforced composite plates, respectively). According to Fig. 11, the values corresponding to HAF and LAF for cantilever CNT-reinforced composite rhombic plates are extracted and listed in Table 7, and will be used hereafter to obtain the corresponding mode shapes. Next, we investigate the effect of the aspect ratio of a CNT-reinforced composite skew plate on obtaining the optimum CNT fiber orientation angle for its dynamic behavior. For this approach, UD CNT-reinforced composite skew plates of different skew angles (i.e. 0 , 30 , 45 and 60 with varying aspect ratios: a / b 1, 1.2, 1.4, 1.5, 1.6, 1.8 and 2) are examined. The width-to-thickness ratio and CNT volume fraction are set to 50 and 0.14, respectively. Plots for natural frequency parameters versus corresponding CNT orientation angles are shown in Fig. 12. As can be observed in Fig. 12, for all cases, the higher the aspect ratios, the lower the natural frequency parameters. Moreover, for all aspect ratios higher than 1 (i.e. non-rhombic skew plates), the highest natural frequency parameter corresponds to the CNT orientation angle of 90 . Moreover, the CNT orientation angle value for the LAF is somewhere between zero and the
25
corresponding value for the case of a/b=1 for each skew angle. As the aspect ratio increases this point moves closer to zero and in fact, for aspect ratios larger than a/b=1.6 the point coincides with the CNT orientation angle 0 . To further continue our study, we explore the effect of width-to-thickness ratio on the optimum CNT orientation angle for the vibration of CNT-reinforced composite skew plates. A UD CNTreinforced composite rhombic plate of skew angle 45 with the varying width-to-thickness ratios of b / h 5, 10, 20, 50 and 100 with simply supported, clamped and cantilever boundary conditions is considered. The obtained results for frequency parameters are normalized with respect to the corresponding highest frequency parameter (presented in Table 7 for each boundary condition) and plotted against the changing CNT orientation angles from 90 to + 90 in Figs. 13. From Fig. 13, the following points can be expressed. First, as the width-to-thickness ratio is increased, the necessity to select the correct CNT orientation becomes crucial. Second, while the CNT orientation angle value corresponding to HAF for all width-to-thickness ratios remains the same, this is not necessarily the case for the corresponding value for LAF. To be exact, it should be pointed out that according to Fig. 13(a) the values presented in Table 7 for the case of the simply supported boundary condition are valid for width-to-thickness ratios larger than 20. From Figs. 13(b)-(c), it is clear that this limitation exists for neither the clamped nor the cantilever boundary conditions. Here, we will clarify the issue introduced in Table 8. Graphs of the non-dimensional vibration parameter versus CNT orientation angle, normalized with values of the vibration parameter evaluated at 0 , for the square FG-O CNT-reinforced composite plate with two values of CNT * * 0.11 and VCNT 0.14 ), are plotted in Figs. 14. As can be seen in Figs. volume fraction (i.e. VCNT * 0.11 ) for a simply supported square FG14(a)-(b), in the case of low CNT volume fractions ( VCNT
O CNT-reinforced composite plate, the variation of normalized vibration parameters is less than 2 percent, which is of minor importance in our study. Moreover, in this case the CNT-reinforced composite plate does not have the same response as described previously. However, as the CNT volume fraction or skewness angle of the plate is increased (see Fig. 14 and Fig. 13), or more robust CNT distributions (UD, FG-V or FG-X) are used the response of the plate gradually becomes closer to that presented earlier. Furthermore, from Figs. 14(c)-(d), it is notable that, for the clamped boundary condition, the response of the CNT-reinforced composite plate to the change in CNT orientation angle is independent of the CNT volume fraction value.
26
In Fig. 15, the first eight free vibration modes of clamped rhombic UD CNT-reinforced composite plates with skew angles of 0 , 15 , 30 , 45 and 60 are presented. Geometrical parameters are set to a/b=1, b/h=50. The CNT volume fraction of 0.14 is selected and the CNT orientation angles correspond to HAF. As can be noted, the mode shapes of skew plates are not necessarily the same due to their geometrical and CNT orientation angle differences. Moreover, as a comparison to mode shapes obtained with 0 reported in [58], here the mode shapes are aligned along the orientation of the CNTs. The first eight mode shapes of cantilever CNT-reinforced composite skew plates are presented in Fig. 16. All the parameters used are the same as for Fig. 15. It should be mentioned that, because of the looser constraints of the cantilever boundary condition, the mode shapes are not necessarily aligned with their corresponding CNT orientation angles. 7. Conclusion Bending and free vibration analyses of CNT-reinforced composite skew plates based on Reddy’s higher order shear deformation theory using the isogeometric method have been presented. The effect of CNT orientation angle on the static and dynamic behavior of CNT-reinforced composite plates has been investigated. The optimum CNT orientation angle for plates with different geometrical properties and CNT distributions is presented for simply supported, clamped and cantilever boundary conditions. It has been shown that the CNT orientation angle has a noticeable influence on the mechanical behavior of CNT-reinforced composite skew plates.
Acknowledgements The work described in this paper was fully supported by grants from the National Natural Science Foundation of China (Grant No. 11402142 and Grant No. 51378448) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9042047, CityU 11208914).
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32
Tables
Table 1 Convergence and accuracy studies of the skew plates subjected to uniformly distributed loads (h/a=0.01). Simply Supported Skew References Angle ( ) 30˚ IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] [37] [38] DQM [42] DQM [45]
* iso
* iso max
Clamped * iso min
* iso
* ( M iso ) max
* ( M iso ) min
W
(M )
(M )
W
4.1086 4.1107 4.1121 4.1079 4.1123 4.2824 4.10 4.096 4.1054
1.7033 1.7037 1.7040 1.6909 1.7075 1.7455 1.70 1.701 1.7006
1.3355 1.3361 1.3366 1.3267 1.3391 1.3670 1.33 1.332 1.3355
1.2335 1.2335 1.2335 1.2419 1.2335 1.2275
0.7914 0.7914 0.7914 0.7838 0.7916 0.7808
0.6177 0.6176 0.6176 0.6120 0.6175 0.6023
1.2336
0.7917
0.6177
45˚
IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] DQM [42] [38]
2.1144 2.1199 2.1239 2.1285 2.1330 2.2028 2.1204 2.107
1.2924 1.2943 1.2957 1.2892 1.2995 1.3258 1.2941 1.290
0.8794 0.8818 0.8836 0.8787 0.8866 0.9008 0.8872 0.876
0.6051 0.6051 0.6051 0.6087 0.6051 0.6032
0.5777 0.5776 0.5776 0.5718 0.5776 0.5710
0.3903 0.3901 0.3901 0.3850 0.3900 0.3803
60˚
IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] [37] [38] DQM [42] DQM [45]
0.6509 0.6552 0.6583 0.6587 0.6690 0.6841 0.653 0.6523 0.6556
0.7605 0.7635 0.7658 0.7628 0.7734 0.7853 0.76 0.762 0.7667
0.4331 0.4365 0.4391 0.4340 0.4481 0.4483 0.43 0.434 0.4428
0.1742 0.1743 0.1743 0.1746 0.1743 0.1737
0.3214 0.3212 0.3211 0.3175 0.3209 0.3172
0.1819 0.1817 0.1816 0.1779 0.1821 0.1748
0.1743
0.3213
0.1817
* * * Wiso wc 1600 D / ( qa 4 ),( M iso ) max ( M c ) max 40 / ( qa 4 ),( M iso ) min ( M c ) min 40 / ( qa 4 )
31
Table 2 Elastic properties of the (10,10) SWCNT and the polymer matrix [22]. SWCNT
Polymer matrix
CNT E11 5.6466(TPa)
E m 2.1(GPa)
CNT E22 7.0800(TPa)
m 1150 (kg/m3 )
CNT G12 1.9447 (TPa)
m 0.34
CNT 1400 (kg/m3 ) CNT 12 0.175
Table 3 CNT efficiency parameters for different values of volume fractions [22]. * VCNT 2 1 3 0.11
0.149
0.934
0.934
0.14
0.150
0.941
0.941
0.17
0.149
1.381
1.381
32
Table 4 Non-dimensional central deflection, W * wc / h , for simply supported FG-CNTRC plates with skewness of under uniform load of q=-0.1 MPa and CNT orientation angle 0 (a/b=1). 0
15
30
45
60
U FG-V FG-O FG-X
0 Ref. [23] 3.739×10-3 4.466×10-3 5.230×10-3 3.177×10-3
3.718×10-3 4.452×10-3 5.442×10-3 3.140×10-3
3.625×10-3 4.315×10-3 5.250×10-3 3.070×10-3
3.272×10-3 3.823×10-3 4.580×10-3 2.797×10-3
2.486×10-3 2.805×10-3 3.272×10-3 2.167×10-3
1.286×10-3 1.381×10-3 1.552×10-3 1.159×10-3
20
U FG-V FG-O FG-X
3.628×10-2 4.879×10-2 6.155×10-2 2.701×10-2
3.630×10-2 4.886×10-2 6.258×10-2 2.695×10-2
3.577×10-2 4.775×10-2 6.082×10-2 2.670×10-2
3.342×10-2 4.336×10-2 5.422×10-2 2.539×10-2
2.684×10-2 3.293×10-2 3.979×10-2 2.118×10-2
1.437×10-2 1.628×10-2 1.880×10-2 1.209×10-2
50
U FG-V FG-O FG-X
1.155 1.653 2.157 7.900×10-1
1.157 1.656 2.167 7.912×10-1
1.144 1.623 2.111 7.875×10-1
1.083 1.486 1.896 7.618×10-1
8.883×10-1 1.142 1.402 6.567×10-1
4.799×10-1 5.620×10-1 6.569×10-1 3.852×10-1
10
U FG-V FG-O FG-X
3.306×10-3 3.894×10-3 4.525×10-3 2.844×10-3
3.273×10-3 3.870×10-3 4.694×10-3 2.824×10-3
3.198×10-3 3.763×10-3 4.548×10-3 2.764×10-3
2.908×10-3 3.367×10-3 4.021×10-3 2.526×10-3
2.242×10-3 2.516×10-3 2.940×10-3 1.969×10-3
1.184×10-3 1.269×10-3 1.436×10-3 1.066×10-3
20
U FG-V FG-O FG-X
3.001×10-2 4.025×10-2 5.070×10-2 2.256×10-2
2.992×10-2 4.017×10-2 5.142×10-2 2.255×10-2
2.959×10-2 3.943×10-2 5.024×10-2 2.238×10-2
2.797×10-2 3.633×10-2 4.560×10-2 2.144×10-2
2.303×10-2 2.838×10-2 3.458×10-2 1.819×10-2
1.283×10-2 1.459×10-2 1.704×10-2 1.072×10-2
50
U FG-V FG-O FG-X
9.175×10-1 1.326 1.738 6.271×10-1
9.159×10-1 1.323 1.740 6.263×10-1
9.097×10-1 1.303 1.706 6.251×10-1
8.729×10-1 1.213 1.562 6.108×10-1
7.387×10-1 9.632×10-1 1.199 5.394×10-1
4.194×10-1 4.960×10-1 5.888×10-1 3.307×10-1
10
U FG-V FG-O FG-X
2.394×10-3 2.864×10-3 3.378×10-3 2.012×10-3
2.381×10-3 2.863×10-3 3.476×10-3 2.027×10-3
2.321×10-3 2.773×10-3 3.355×10-3 1.978×10-3
2.092×10-3 2.450×10-3 2.931×10-3 1.793×10-3
1.586×10-3 1.789×10-3 2.098×10-3 1.376×10-3
8.169×10-4 8.748×10-4 9.946×10-4 7.271×10-4
20
U FG-V FG-O FG-X
2.348×10-2 3.174×10-2 4.020×10-2 1.737×10-2
2.350×10-2 3.182×10-2 4.071×10-2 1.750×10-2
2.315×10-2 3.106×10-2 3.957×10-2 1.731×10-2
2.158×10-2 2.811×10-2 3.528×10-2 1.638×10-2
1.726×10-2 2.119×10-2 2.588×10-2 1.351×10-2
9.175×10-3 1.038×10-2 1.221×10-2 7.582×10-3
50
U FG-V FG-O FG-X
7.515×10-1 1.082 1.416 5.132×10-1
7.534×10-1 1.085 1.422 5.152×10-1
7.449×10-1 1.062 1.385 5.122×10-1
7.032×10-1 9.684×10-1 1.243 4.929×10-1
5.740×10-1 7.380×10-1 9.186×10-1 4.200×10-1
3.076×10-1 3.593×10-1 4.295×10-1 2.414×10-1
* VCNT
b/h
0.11
10
0.14
0.17
33
Table 5 Non-dimensional central deflection, W * wc / h , for simply supported FG-CNTRC plates with skewness of under uniform load of q=-0.1 MPa and CNT orientation angle (a/b=1). * VCNT 0.11
0.14
0.17
U FG-V FG-O FG-X
0 3.718×10-3 4.452×10-3 5.442×10-3 3.140×10-3
R 1.0 1.0 1.0 1.0
15 3.257×10-3 3.866×10-3 4.714×10-3 2.771×10-3
R 1.1 1.1 1.1 1.1
30 2.229×10-3 2.584×10-3 3.130×10-3 1.933×10-3
R 1.5 1.5 1.5 1.4
45 1.211×10-3 1.354×10-3 1.624×10-3 1.084×10-3
R 2.1 2.1 2.0 2.0
60 4.872×10-4 5.198×10-4 6.136×10-4 4.534×10-4
R 2.6 2.7 2.5 2.6
20
U FG-V FG-O FG-X
3.630×10-2 4.886×10-2 6.258×10-2 2.695×10-2
1.0 1.0 1.0 1.0
3.143×10-2 4.184×10-2 5.342×10-2 2.358×10-2
1.1 1.1 1.1 1.1
2.058×10-2 2.674×10-2 3.394×10-2 1.580×10-2
1.6 1.6 1.6 1.6
9.916×10-3 1.244×10-2 1.571×10-2 7.890×10-3
2.7 2.6 2.5 2.7
3.227×10-3 3.812×10-3 4.752×10-3 2.730×10-3
4.5 4.3 4.0 4.4
50
U FG-V FG-O FG-X
1.157 1.656 2.167 7.912×10-1
1.0 1.0 1.0 1.0
9.924×10-1 1.406 1.835 6.851×10-1
1.2 1.2 1.2 1.1
6.282×10-1 8.749×10-1 1.138 4.397×10-1
1.7 1.7 1.7 1.7
2.761×10-1 3.778×10-1 4.938×10-1 1.962×10-1
3.2 3.0 2.8 3.3
7.186×10-2 9.556×10-2 1.260×10-1 5.256×10-2
6.7 5.9 5.2 7.3
10
U FG-V FG-O FG-X
3.273×10-3 3.870×10-3 4.694×10-3 2.824×10-3
1.0 1.0 1.0 1.0
2.880×10-3 3.375×10-3 4.081×10-3 2.500×10-3
1.1 1.1 1.1 1.1
1.992×10-3 2.280×10-3 2.735×10-3 1.760×10-3
1.5 1.5 1.5 1.4
1.100×10-3 1.215×10-3 1.440×10-3 1.001×10-3
2.0 2.1 2.0 2.0
4.508×10-4 4.766×10-4 5.548×10-4 4.249×10-4
2.6 2.7 2.6 2.5
20
U FG-V FG-O FG-X
2.992×10-2 4.017×10-2 5.142×10-2 2.255×10-2
1.0 1.0 1.0 1.0
2.606×10-2 3.457×10-2 4.408×10-2 1.984×10-2
1.1 1.1 1.1 1.1
1.728×10-2 2.233×10-2 2.826×10-2 1.347×10-2
1.6 1.6 1.6 1.6
8.478×10-3 1.055×10-2 1.322×10-2 6.873×10-3
2.7 2.7 2.6 2.6
2.845×10-3 3.321×10-3 4.084×10-3 2.458×10-3
4.5 4.4 4.2 4.4
50
U FG-V FG-O FG-X
9.159×10-1 1.323 1.740 6.263×10-1
1.0 1.0 1.0 1.0
7.901×10-1 1.129 1.479 5.451×10-1
1.2 1.2 1.2 1.1
5.040×10-1 7.073×10-1 9.234×10-1 3.528×10-1
1.7 1.7 1.7 1.7
2.233×10-1 3.071×10-1 4.017×10-1 1.591×10-1
3.3 3.1 3.0 3.4
5.888×10-2 7.837×10-2 1.028×10-1 4.350×10-2
7.1 6.3 5.7 7.6
10
U FG-V FG-O FG-X
2.381×10-3 2.863×10-3 3.476×10-3 2.027×10-3
1.0 1.0 1.0 1.0
2.085×10-3 2.484×10-3 3.008×10-3 1.786×10-3
1.1 1.1 1.1 1.1
1.424×10-3 1.656×10-3 1.991×10-3 1.243×10-3
1.5 1.5 1.5 1.4
7.716×10-4 8.643×10-4 1.025×10-3 6.955×10-4
2.1 2.1 2.0 2.0
3.093×10-4 3.299×10-4 3.829×10-4 2.904×10-4
2.6 2.7 2.6 2.5
20
U FG-V FG-O FG-X
2.350×10-2 3.182×10-2 4.071×10-2 1.750×10-2
1.0 1.0 1.0 1.0
2.033×10-2 2.722×10-2 3.473×10-2 1.528×10-2
1.1 1.1 1.1 1.1
1.328×10-2 1.735×10-2 2.202×10-2 1.021×10-2
1.6 1.6 1.6 1.6
6.382×10-3 8.046×10-3 1.014×10-2 5.090×10-3
2.7 2.6 2.6 2.7
2.066×10-3 2.450×10-3 3.032×10-3 1.759×10-3
4.4 4.2 4.0 4.3
50
U FG-V FG-O FG-X
7.534×10-1 1.085 1.422 5.152×10-1
1.0 1.0 1.0 1.0
6.459×10-1 9.200×10-1 1.203 4.452×10-1
1.2 1.2 1.2 1.2
4.083×10-1 5.717×10-1 7.463×10-1 2.851×10-1
1.7 1.7 1.7 1.7
1.793×10-1 2.466×10-1 3.233×10-1 1.270×10-1
3.2 3.0 2.8 3.3
4.656×10-2 6.225×10-2 8.220×10-2 3.401×10-2
6.6 5.8 5.2 7.1
b/h 10
*The values listed in columns denoted with R show ratios of non-dimensional deflections obtained with 0 , presented in Table 4, to those with cases, presented here in Table 5.
34
Table 6 * Convergence study for first eight frequency parameters, iso (b 2 / 2 ) h / D , of a simply supported isotropic skew plate. h/b 1 2 3 4 5 6 7 8 0.001
0.2
0˚
N=17 N=19 N=21 N=23 Ref. [48]
2.0000 2.0000 2.0000 2.0000 2.0000
5.0004 5.0001 5.0000 5.0000 5.0000
5.0004 5.0001 5.0000 5.0000 5.0000
8.0004 8.0000 7.9999 7.9999 7.9999
10.0154 10.0045 10.0014 10.0005 9.9999
10.0154 10.0045 10.0014 10.0005 9.9999
13.0119 13.0033 13.0010 13.0002 12.9998
13.0119 13.0033 13.0010 13.0002 12.9998
30˚
N=17 N=19 N=21 N=23 Ref. [48]
2.5313 2.5298 2.5287 2.5279 2.5294
5.3340 5.3335 5.3334 5.3333 5.3333
7.2879 7.2831 7.2802 7.2782 7.2821
8.5043 8.4989 8.4970 8.4962 8.4966
12.4804 12.4550 12.4479 12.4456 12.4442
12.4881 12.4572 12.4487 12.4459 12.4442
14.2888 14.2666 14.2576 14.2527 14.2850
17.3513 17.2082 17.1670 17.1540 17.1471
60˚
N=17 N=19 N=21 N=23 Ref. [48]
6.7289 6.6952 6.6686 6.6470 6.7179
10.6612 10.6438 10.6381 10.6360 10.6354
15.2277 15.1131 15.0734 15.0566 15.0708
20.5791 20.1281 19.9748 19.9222 19.9000
21.7233 21.6171 21.5507 21.5027 21.6610
27.3839 26.0758 25.5705 25.3847 25.3953
30.3713 29.9869 29.8367 29.7792 29.7937
36.2693 33.3034 32.0024 31.4821 31.6761
0˚
N=17 N=19 N=21 N=23 Ref. [48]
1.7683 1.7683 1.7683 1.7683 1.7661
3.8692 3.8692 3.8692 3.8692 3.8580
3.8692 3.8692 3.8692 3.8692 3.8580
5.5980 5.5980 5.5980 5.5980 5.5737
6.6168 6.6168 6.6168 6.6168 6.5820
6.6168 6.6168 6.6168 6.6168 6.5820
8.0011 8.0011 8.0011 8.0011 7.9485
8.0011 8.0011 8.0011 8.0011 7.9485
30˚
N=17 N=19 N=21 N=23 Ref. [48]
2.1749 2.1745 2.1742 2.1739 2.1719
4.0769 4.0768 4.0767 4.0767 4.0637
5.2055 5.2047 5.2042 5.2037 5.1849
5.8601 5.8598 5.8596 5.8594 5.8321
7.7580 7.7577 7.7575 7.7573 7.7066
7.7580 7.7577 7.7575 7.7573 7.7066
8.5362 8.5352 8.5344 8.5338 8.4742
9.7075 9.7071 9.7068 9.7065 9.6240
60˚
N=17 N=19 N=21 N=23 Ref. [48]
4.7680 4.7597 4.7531 4.7478 4.7647
6.9286 6.9278 6.9271 6.9266 6.8841
8.8567 8.8543 8.8525 8.8510 8.7851
10.7408 10.7393 10.7383 10.7374 10.6296
11.1717 11.1637 11.1574 11.1522 11.0666
12.5679 12.5656 12.5639 12.5626 12.4167
13.9702 13.9684 13.9670 13.9659 13.7750
14.3836 14.3812 14.3796 14.3785 14.2005
35
Table 7 CNT orientations angle, , corresponding to HAF and LAF for FG-CNTRC skew plates of different skew angle, . Simply Supported (SSSS)
(degree)
Clamped (CCCC)
Cantilever (CFFF)
(degree) corresponding to LAF
HAF
0
45
15
40
30 45 60
25 30 10
0 -15 for FG-X -50 for all other dist. -60 -65 -75
LAF HAF
LAF HAF
45
0
0
90
35
-15
0
75
30 25 15
-35 -70 -75
-5 -20 -35
60 45 30
36
Table 8 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 0 ). Simply Supported * b/h Modes VCNT 0.11 20 1 2 3 4 5 6
LAF 13.392 23.423 35.152 35.645 49.535 50.544
FG-O FG-O HAF 13.380 18.295 29.549 38.955 38.955 42.609
RHL 1.00 0.78 0.84 1.09 0.79 0.84
Clamped
LAF 18.616 29.667 42.859 46.613 55.090 57.420
FG-X FG-X HAF 19.912 23.683 33.814 38.955 38.955 50.311
RHL 1.07 0.80 0.79 0.84 0.71 0.88
LAF 20.577 31.878 42.449 44.247 58.074 58.118
FG-O FG-O HAF 23.922 29.203 40.744 51.810 55.044 57.979
RHL 1.16 0.92 0.96 1.17 0.95 1.00
LAF 26.271 37.896 50.531 51.956 64.525 67.709
FG-X FG-X HAF 30.733 35.185 45.821 62.591 62.951 65.547
1.17 0.93 0.91 1.20 0.98 0.97
RHL
50
1 2 3 4 5 6
14.136 25.047 38.882 40.559 55.128 59.460
14.231 19.222 30.955 49.215 52.110 54.897
1.01 0.77 0.80 1.21 0.95 0.92
20.802 33.536 49.651 60.920 67.991 82.871
22.906 26.640 36.951 54.744 79.254 82.322
1.10 0.79 0.74 0.90 1.17 0.99
24.115 38.325 54.338 54.931 72.590 76.842
29.902 34.923 46.590 65.379 75.239 78.277
1.24 0.91 0.86 1.19 1.04 1.02
35.152 51.787 69.926 79.119 90.025 103.833
45.985 49.661 59.243 76.353 100.826 110.000
1.31 0.96 0.85 0.97 1.12 1.06
100
1 2 3 4 5 6
14.279 25.323 39.462 41.645 56.171 61.272
14.368 19.372 31.181 49.674 54.089 56.924
1.01 0.76 0.79 1.19 0.96 0.93
21.270 34.279 51.052 64.632 70.296 88.216
23.460 27.194 37.532 55.540 80.605 89.725
1.10 0.79 0.74 0.86 1.15 1.02
24.857 39.786 56.825 58.127 76.452 82.051
31.240 36.223 47.890 66.924 82.409 85.469
1.26 0.91 0.84 1.15 1.08 1.04
37.574 55.995 76.405 89.247 99.315 118.593
50.924 54.462 63.719 80.629 105.417 132.922
1.36 0.97 0.83 0.90 1.06 1.12
0.14 20
1 2 3 4 5 6
14.408 24.674 37.156 37.701 51.267 53.333
14.646 19.279 30.305 39.726 39.726 46.004
1.02 0.78 0.82 1.05 0.77 0.86
20.019 31.395 44.906 49.226 56.181 59.762
21.601 25.257 35.350 39.726 39.726 52.054
1.08 0.80 0.79 0.81 0.71 0.87
21.890 33.351 44.884 45.873 59.836 60.701
25.600 30.634 41.935 54.876 57.924 59.109
1.17 0.92 0.93 1.20 0.97 0.97
27.585 39.388 52.240 54.076 66.494 70.078
32.148 36.622 47.400 64.478 65.295 67.894
1.17 0.93 0.91 1.19 0.98 0.97
50
1 2 3 4 5 6
15.328 26.577 40.841 44.335 57.489 63.861
15.748 20.430 31.867 50.040 57.816 60.348
1.03 0.77 0.78 1.13 1.01 0.94
22.800 36.131 52.990 66.637 72.048 89.485
25.503 29.074 39.218 57.125 82.078 90.415
1.12 0.80 0.74 0.86 1.14 1.01
26.178 40.900 57.388 59.716 76.072 82.316
32.979 37.643 48.807 67.233 82.539 85.313
1.26 0.92 0.85 1.13 1.09 1.04
38.239 55.548 74.300 85.368 94.973 110.801
50.223 53.766 63.162 80.232 104.951 118.140
1.31 0.97 0.85 0.94 1.11 1.07
100
1 2 3 4 5 6
15.509 26.906 41.514 45.722 58.675 66.084
15.928 20.619 32.122 50.517 60.411 62.990
1.03 0.77 0.77 1.10 1.03 0.95
23.415 37.078 54.738 71.518 74.869 96.338
26.262 29.826 39.973 58.074 83.586 100.231
1.12 0.80 0.73 0.81 1.12 1.04
27.114 42.688 60.364 63.753 80.613 88.725
34.731 39.338 50.442 69.036 91.734 94.516
1.28 0.92 0.84 1.08 1.14 1.07
41.390 60.881 82.339 98.287 106.302 129.199
56.714 60.081 69.034 85.718 110.578 143.107
1.37 0.99 0.84 0.87 1.04 1.11
0.17 20
1 2 3 4 5 6
16.502 28.898 43.578 44.044 61.284 62.729
16.486 22.534 36.411 48.962 48.962 53.019
1.00 0.78 0.84 1.11 0.80 0.85
23.178 37.175 53.876 57.874 69.243 72.327
24.613 29.586 42.754 48.962 48.962 63.952
1.06 0.80 0.79 0.85 0.71 0.88
25.531 39.605 52.943 55.021 72.249 72.518
29.794 36.263 50.465 64.974 68.909 71.768
1.17 0.92 0.95 1.18 0.95 0.99
32.687 47.396 63.406 64.596 81.158 84.523
38.093 43.950 57.758 78.042 79.286 81.474
1.17 0.93 0.91 1.21 0.98 0.96
50
1 2 3 4 5 6
17.360 30.775 47.806 49.863 67.818 73.145
17.460 23.602 38.048 60.533 64.102 67.525
1.01 0.77 0.80 1.21 0.95 0.92
25.883 42.037 62.463 75.530 85.760 103.296
28.283 33.216 46.683 69.685 101.195 101.547
1.09 0.79 0.75 0.92 1.18 0.98
29.662 47.189 66.967 67.710 89.531 94.811
36.811 42.988 57.351 80.495 93.004 96.734
1.24 0.91 0.86 1.19 1.04 1.02
43.655 64.656 87.604 98.098 113.072 129.278
56.790 61.663 74.259 96.530 128.136 135.851
1.30 0.95 0.85 0.98 1.13 1.05
100
1 2 3 4 5 6
17.524 31.091 48.472 51.106 69.019 75.218
17.615 23.772 38.308 61.069 66.359 69.839
1.01 0.76 0.79 1.19 0.96 0.93
26.458 42.972 64.238 80.098 88.701 109.953
28.961 33.895 47.408 70.712 102.980 110.615
1.09 0.79 0.74 0.88 1.16 1.01
30.511 48.865 69.832 71.382 93.994 100.816
38.339 44.475 58.842 82.279 101.266 105.026
1.26 0.91 0.84 1.15 1.08 1.04
46.649 69.895 95.713 110.573 124.744 147.562
62.830 67.526 79.725 101.807 133.928 163.925
1.35 0.97 0.83 0.92 1.07 1.11
37
Table 9 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 15 ). Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 12.832 24.412 32.023 38.944 48.845 55.579
FG-O FG-O HAF 15.464 24.776 35.885 41.854 48.204 56.318
50
1 2 3 4 5 6
13.443 25.993 36.225 42.396 56.577 62.148
16.573 26.780 39.535 50.039 54.086 68.274
1.23 1.03 1.09 1.18 0.96 1.10
19.167 33.737 52.545 53.612 74.674 77.303
24.955 30.180 42.055 60.071 81.645 87.906
1.30 0.89 0.80 1.12 1.09 1.14
22.198 37.829 49.010 56.471 71.880 78.151
31.996 37.556 49.820 68.001 80.244 83.272
1.44 0.99 1.02 1.20 1.12 1.07
31.629 49.674 70.058 70.635 94.360 95.835
48.948 53.230 63.826 81.440 104.043 116.308
1.55 1.07 0.91 1.15 1.10 1.21
100
1 2 3 4 5 6
13.551 26.264 37.026 43.015 58.112 63.351
16.807 27.126 40.197 51.819 55.189 70.836
1.24 1.03 1.09 1.20 0.95 1.12
19.500 34.407 53.906 56.264 77.098 81.614
25.612 30.929 43.005 61.510 84.287 96.299
1.31 0.90 0.80 1.09 1.09 1.18
22.781 39.082 51.392 58.795 75.981 81.983
33.527 39.089 51.483 70.295 88.328 90.282
1.47 1.00 1.00 1.20 1.16 1.10
33.472 53.128 76.370 77.590 103.051 107.275
54.530 58.720 69.195 87.188 111.509 138.804
1.63 1.11 0.91 1.12 1.08 1.29
0.14 20
1 2 3 4 5 6
13.729 25.572 34.301 40.402 51.412 57.486
16.713 26.230 37.580 44.850 50.095 58.509
1.22 1.03 1.10 1.11 0.97 1.02
18.758 31.904 45.026 47.949 58.654 62.914
23.286 28.165 39.194 44.174 50.787 54.885
1.24 0.88 0.87 0.92 0.87 0.87
20.549 33.588 41.456 48.513 58.783 65.403
27.054 32.507 43.907 57.266 59.425 61.464
1.32 0.97 1.06 1.18 1.01 0.94
25.845 39.343 50.221 54.749 67.855 72.058
33.718 38.672 49.730 65.014 68.452 71.458
1.30 0.98 0.99 1.19 1.01 0.99
50
1 2 3 4 5 6
14.471 27.390 39.427 44.247 60.454 64.457
18.100 28.614 41.804 54.870 56.754 73.578
1.25 1.04 1.06 1.24 0.94 1.14
20.894 36.141 55.761 58.577 78.729 83.264
27.738 32.904 44.840 63.243 85.719 96.381
1.33 0.91 0.80 1.08 1.09 1.16
23.936 40.060 53.089 59.210 76.643 81.359
35.262 40.484 52.354 70.495 87.874 89.618
1.47 1.01 0.99 1.19 1.15 1.10
34.338 53.102 74.773 75.673 99.184 102.249
53.376 57.542 68.029 85.750 108.859 124.677
1.55 1.08 0.91 1.13 1.10 1.22
100
1 2 3 4 5 6
14.605 27.706 40.439 44.949 62.312 65.794
18.397 29.032 42.587 57.144 58.043 75.652
1.26 1.05 1.05 1.27 0.93 1.15
21.331 36.984 57.432 62.062 81.646 88.757
28.631 33.899 46.051 64.982 88.697 107.462
1.34 0.92 0.80 1.05 1.09 1.21
24.664 41.574 56.086 61.948 81.652 85.789
37.263 42.467 54.421 73.128 95.647 98.226
1.51 1.02 0.97 1.18 1.17 1.14
36.728 57.454 81.833 85.320 109.682 116.527
60.687 64.721 74.987 92.979 117.777 146.572
1.65 1.13 0.92 1.09 1.07 1.26
0.17 20
1 2 3 4 5 6
15.808 30.115 39.651 48.120 60.567 68.985
19.072 30.585 44.363 51.999 59.664 70.017
1.21 1.02 1.12 1.08 0.99 1.01
21.819 37.988 52.877 57.822 72.158 75.113
26.605 33.010 47.130 54.307 61.859 66.713
1.22 0.87 0.89 0.94 0.86 0.89
24.021 40.011 48.844 58.342 70.227 78.158
31.551 38.516 52.779 68.071 71.690 73.298
1.31 0.96 1.08 1.17 1.02 0.94
30.645 47.465 59.958 66.638 81.914 88.171
40.016 46.450 60.518 79.365 82.118 85.925
1.31 0.98 1.01 1.19 1.00 0.97
50
1 2 3 4 5 6
16.511 31.945 44.524 52.139 69.591 76.475
20.353 32.901 48.604 61.554 66.528 84.025
1.23 1.03 1.09 1.18 0.96 1.10
23.919 42.410 66.292 66.584 94.425 96.558
30.828 37.628 52.978 76.079 103.333 108.463
1.29 0.89 0.80 1.14 1.09 1.12
27.298 46.568 60.368 69.583 88.623 96.373
39.400 46.242 61.346 83.770 99.232 102.913
1.44 0.99 1.02 1.20 1.12 1.07
39.362 62.188 86.978 88.745 118.848 119.557
60.457 66.101 79.944 102.705 131.499 143.676
1.54 1.06 0.92 1.16 1.11 1.20
100
1 2 3 4 5 6
16.635 32.256 45.439 52.852 71.349 77.863
20.623 33.296 49.363 63.596 67.796 86.959
1.24 1.03 1.09 1.20 0.95 1.12
24.331 43.257 68.028 69.858 97.538 101.947
31.634 38.552 54.169 77.927 106.883 118.747
1.30 0.89 0.80 1.12 1.10 1.16
27.965 48.008 63.104 72.265 93.351 100.811
41.150 47.996 63.257 86.415 108.559 110.987
1.47 1.00 1.00 1.20 1.16 1.10
41.646 66.504 95.951 96.268 129.810 133.764
67.285 72.812 86.529 109.858 141.072 174.333
1.62 1.09 0.90 1.14 1.09 1.30
* b/h VCNT 0.11 20
RHL 1.21 1.01 1.12 1.07 0.99 1.01
LAF 17.479 30.236 42.537 45.889 57.427 60.129
FG-X FG-X HAF 21.514 26.440 37.400 43.227 49.318 52.806
RHL 1.23 0.87 0.88 0.94 0.86 0.88
38
LAF 19.389 32.252 39.235 46.972 56.380 62.563
FG-O FG-O HAF 25.317 30.996 42.554 54.298 57.686 58.558
RHL 1.31 0.96 1.08 1.16 1.02 0.94
LAF 24.584 37.858 48.173 52.985 65.521 69.966
FG-X FG-X HAF 32.277 37.193 48.087 63.045 66.077 69.054
1.31 0.98 1.00 1.19 1.01 0.99
RHL
Table 10 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 30 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 13.613 27.798 32.271 46.800 49.344 57.085
FG-O FG-O HAF 19.946 29.188 40.059 51.667 53.969 62.129
50
1 2 3 4 5 6
14.237 29.722 36.169 51.367 57.381 68.909
21.991 32.220 45.128 59.163 68.360 75.022
1.54 1.08 1.25 1.15 1.19 1.09
19.701 37.548 52.416 61.835 77.543 91.304
33.902 45.528 60.989 76.982 95.041 102.701
1.72 1.21 1.16 1.24 1.23 1.12
22.741 42.411 47.370 65.945 75.956 79.802
39.713 48.398 63.280 80.368 97.487 98.398
1.75 1.14 1.34 1.22 1.28 1.23
31.071 53.475 66.035 79.911 98.185 109.960
59.475 67.433 82.655 101.347 121.068 137.649
1.91 1.26 1.25 1.27 1.23 1.25
100
1 2 3 4 5 6
14.344 30.055 36.903 52.191 58.972 71.541
22.460 32.756 46.089 60.612 71.897 77.203
1.57 1.09 1.25 1.16 1.22 1.08
19.997 38.283 54.746 63.447 81.671 94.429
35.444 47.157 63.663 80.680 100.274 114.512
1.77 1.23 1.16 1.27 1.23 1.21
23.273 43.719 49.333 68.658 79.838 84.734
42.045 51.072 66.884 85.566 105.597 109.246
1.81 1.17 1.36 1.25 1.32 1.29
32.653 56.844 72.024 85.966 108.509 120.360
67.571 75.974 92.692 114.308 137.688 163.146
2.07 1.34 1.29 1.33 1.27 1.36
0.14 20
1 2 3 4 5 6
14.480 29.004 34.404 48.376 51.934 60.265
21.593 31.008 42.113 53.890 57.590 63.529
1.49 1.07 1.22 1.11 1.11 1.05
19.371 35.462 44.800 56.255 63.425 68.381
29.830 39.982 51.787 63.529 64.007 65.611
1.54 1.13 1.16 1.13 1.01 0.96
21.249 37.934 41.128 56.595 63.205 64.760
32.237 39.703 51.366 63.584 67.474 74.736
1.52 1.05 1.25 1.12 1.07 1.15
26.228 43.554 49.593 63.281 72.043 75.506
39.217 46.400 58.136 70.813 78.373 80.854
1.50 1.07 1.17 1.12 1.09 1.07
50
1 2 3 4 5 6
15.224 31.156 39.134 53.295 61.232 74.483
24.152 34.647 48.054 62.481 75.027 78.789
1.59 1.11 1.23 1.17 1.23 1.06
21.355 39.991 57.123 65.230 83.294 95.776
37.344 49.440 65.661 82.279 101.034 111.652
1.75 1.24 1.15 1.26 1.21 1.17
24.259 44.470 50.932 68.695 80.314 85.747
43.610 52.124 67.132 84.668 103.199 106.252
1.80 1.17 1.32 1.23 1.28 1.24
33.539 56.816 71.187 84.129 104.384 115.781
64.496 72.444 87.885 107.025 127.222 146.625
1.92 1.28 1.23 1.27 1.22 1.27
100
1 2 3 4 5 6
15.354 31.536 40.052 54.204 63.124 77.808
24.749 35.307 49.212 64.190 79.560 81.324
1.61 1.12 1.23 1.18 1.26 1.05
21.739 40.899 60.168 67.163 88.493 99.441
39.360 51.553 69.068 86.911 107.503 126.918
1.81 1.26 1.15 1.29 1.21 1.28
24.911 46.020 53.371 71.796 84.982 91.922
46.628 55.499 71.520 90.838 111.580 121.154
1.87 1.21 1.34 1.27 1.31 1.32
35.569 61.010 78.835 91.467 117.193 127.253
74.968 83.365 100.441 122.894 147.260 173.680
2.11 1.37 1.27 1.34 1.26 1.36
0.17 20
1 2 3 4 5 6
16.771 34.308 39.932 57.870 61.208 70.928
24.644 36.080 49.590 64.024 67.270 78.048
1.47 1.05 1.24 1.11 1.10 1.10
22.664 42.468 52.697 68.194 75.771 83.902
34.597 47.154 61.790 77.000 78.048 80.816
1.53 1.11 1.17 1.13 1.03 0.96
25.006 45.454 48.557 68.118 75.787 77.250
37.852 47.232 61.686 76.648 80.569 89.716
1.51 1.04 1.27 1.13 1.06 1.16
31.246 52.818 59.306 77.236 87.264 90.819
46.771 55.896 70.583 86.296 94.099 99.667
1.50 1.06 1.19 1.12 1.08 1.10
50
1 2 3 4 5 6
17.490 36.539 44.454 63.194 70.595 84.812
27.015 39.587 55.485 72.779 84.198 92.339
1.54 1.08 1.25 1.15 1.19 1.09
24.653 47.327 65.209 78.224 97.005 115.706
41.984 56.774 76.291 96.570 119.440 127.030
1.70 1.20 1.17 1.23 1.23 1.10
27.968 52.213 58.324 81.272 93.621 98.407
48.949 59.658 78.042 99.195 120.729 121.541
1.75 1.14 1.34 1.22 1.29 1.24
38.804 67.185 82.193 100.716 122.852 136.707
73.511 83.802 103.271 126.973 151.950 170.181
1.89 1.25 1.26 1.26 1.24 1.24
100
1 2 3 4 5 6
17.612 36.923 45.293 64.145 72.419 87.829
27.560 40.199 56.587 74.442 88.268 94.848
1.56 1.09 1.25 1.16 1.22 1.08
25.023 48.265 68.099 80.298 102.196 119.754
43.872 58.805 79.640 101.235 126.063 141.521
1.75 1.22 1.17 1.26 1.23 1.18
28.577 53.719 60.580 84.412 98.103 104.095
51.621 62.730 82.194 105.200 129.882 134.346
1.81 1.17 1.36 1.25 1.32 1.29
40.771 71.412 89.601 108.364 135.719 152.081
83.426 94.303 115.721 143.167 172.786 205.074
2.05 1.32 1.29 1.32 1.27 1.35
* b/h VCNT 0.11 20
RHL 1.47 1.05 1.24 1.10 1.09 1.09
LAF 18.113 33.732 42.331 54.021 60.641 66.804
FG-X FG-X HAF 27.895 37.807 49.397 61.415 62.129 64.304
RHL 1.54 1.12 1.17 1.14 1.02 0.96
39
LAF 20.204 36.670 39.069 54.818 60.916 62.011
FG-O FG-O HAF 30.319 37.903 49.509 61.429 64.214 71.660
RHL 1.50 1.03 1.27 1.12 1.05 1.16
LAF 24.986 41.999 47.552 61.316 69.640 72.824
FG-X FG-X HAF 37.695 44.787 56.328 68.757 75.789 79.299
1.51 1.07 1.18 1.12 1.09 1.09
RHL
Table 11 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 45 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 16.316 32.855 39.685 52.372 61.640 71.151
FG-O FG-O HAF 29.320 39.013 50.487 62.169 67.916 74.766
50
1 2 3 4 5 6
17.129 36.019 43.984 61.539 69.867 84.786
34.114 44.897 59.437 74.199 90.905 106.834
1.99 1.25 1.35 1.21 1.30 1.26
22.904 45.701 59.874 79.087 86.325 114.539
52.801 64.581 82.044 98.780 117.835 137.128
2.31 1.41 1.37 1.25 1.37 1.20
27.470 51.952 56.733 81.393 90.146 97.281
58.873 75.712 92.436 109.972 128.169 137.114
2.14 1.46 1.63 1.35 1.42 1.41
35.396 65.239 72.196 103.939 108.220 121.639
84.227 102.873 121.157 139.922 159.117 179.110
2.38 1.58 1.68 1.35 1.47 1.47
100
1 2 3 4 5 6
17.272 36.576 44.834 63.345 71.484 87.975
35.301 46.021 61.296 76.712 94.469 113.088
2.04 1.26 1.37 1.21 1.32 1.29
23.272 46.795 62.552 81.966 90.915 124.483
56.646 68.127 87.449 105.551 126.895 148.444
2.43 1.46 1.40 1.29 1.40 1.19
28.079 53.777 58.647 85.942 93.977 102.156
63.855 82.898 102.041 122.335 143.639 159.118
2.27 1.54 1.74 1.42 1.53 1.56
37.045 69.115 78.023 112.029 118.626 135.733
99.634 123.091 146.324 170.380 195.210 221.258
2.69 1.78 1.88 1.52 1.65 1.63
0.14 20
1 2 3 4 5 6
17.214 34.464 41.527 55.063 63.625 74.011
31.645 41.470 53.155 64.983 69.436 77.259
1.84 1.20 1.28 1.18 1.09 1.04
22.334 42.043 51.460 66.344 72.920 84.518
42.125 52.537 64.631 69.436 76.742 77.259
1.89 1.25 1.26 1.05 1.05 0.91
25.600 45.761 50.127 66.854 76.225 81.428
44.019 55.142 66.209 77.790 87.550 88.656
1.72 1.21 1.32 1.16 1.15 1.09
30.465 53.415 56.549 77.239 84.287 89.352
51.487 62.627 73.857 85.603 87.550 97.803
1.69 1.17 1.31 1.11 1.04 1.09
50
1 2 3 4 5 6
18.166 37.959 46.691 65.414 72.591 90.134
37.587 48.552 63.697 78.873 96.089 113.972
2.07 1.28 1.36 1.21 1.32 1.26
24.682 48.516 64.807 83.432 92.046 123.219
57.992 70.176 88.439 105.787 125.527 145.404
2.35 1.45 1.36 1.27 1.36 1.18
28.902 54.816 59.268 86.519 93.397 101.629
64.145 81.502 98.690 116.622 135.169 147.954
2.22 1.49 1.67 1.35 1.45 1.46
37.845 68.877 77.171 108.980 114.693 129.078
90.326 109.386 128.056 147.170 166.697 187.033
2.39 1.59 1.66 1.35 1.45 1.45
100
1 2 3 4 5 6
18.337 38.582 47.749 67.456 74.455 94.107
39.093 49.953 65.972 81.888 100.308 119.479
2.13 1.29 1.38 1.21 1.35 1.27
25.150 49.830 68.289 86.678 97.897 135.999
62.989 74.786 95.341 114.313 136.794 159.310
2.50 1.50 1.40 1.32 1.40 1.17
29.629 56.907 61.608 91.909 97.630 107.570
70.451 90.399 110.392 131.480 153.542 175.071
2.38 1.59 1.79 1.43 1.57 1.63
39.927 73.565 84.570 118.236 127.657 146.790
109.692 134.332 158.666 183.751 209.556 236.557
2.75 1.83 1.88 1.55 1.64 1.61
0.17 20
1 2 3 4 5 6
20.108 40.619 49.065 65.021 76.356 88.286
36.352 48.354 62.662 77.220 85.319 92.950
1.81 1.19 1.28 1.19 1.12 1.05
26.327 50.424 61.176 79.580 88.557 102.903
49.293 62.194 77.250 85.319 92.338 95.072
1.87 1.23 1.26 1.07 1.04 0.92
30.439 54.448 60.430 79.784 92.042 98.738
52.299 66.126 79.839 94.178 106.541 107.924
1.72 1.21 1.32 1.18 1.16 1.09
36.628 64.536 68.702 93.249 102.960 109.138
61.865 75.753 89.718 104.325 107.924 119.487
1.69 1.17 1.31 1.12 1.05 1.09
50
1 2 3 4 5 6
21.047 44.290 54.083 75.731 86.004 104.380
41.947 55.193 73.122 91.325 111.952 131.923
1.99 1.25 1.35 1.21 1.30 1.26
28.734 57.651 74.740 99.828 108.471 142.779
65.290 80.339 102.337 123.543 147.627 172.082
2.27 1.39 1.37 1.24 1.36 1.21
33.794 63.967 69.867 100.367 111.107 119.930
72.722 93.612 114.382 136.174 158.802 170.259
2.15 1.46 1.64 1.36 1.43 1.42
44.421 82.213 90.340 131.030 135.884 152.284
104.279 127.781 150.848 174.555 198.829 224.111
2.35 1.55 1.67 1.33 1.46 1.47
100
1 2 3 4 5 6
21.212 44.931 55.058 77.804 87.874 108.047
43.327 56.476 75.256 94.209 116.053 138.963
2.04 1.26 1.37 1.21 1.32 1.29
29.197 59.060 78.060 103.667 114.159 155.053
70.000 84.743 109.067 132.032 159.017 186.361
2.40 1.43 1.40 1.27 1.39 1.20
34.494 66.071 72.077 105.622 115.551 125.572
78.461 101.916 125.512 150.541 176.831 195.931
2.27 1.54 1.74 1.43 1.53 1.56
46.484 87.137 97.555 141.673 148.647 169.668
123.217 152.747 182.021 212.377 243.736 276.654
2.65 1.75 1.87 1.50 1.64 1.63
* b/h VCNT 0.11 20
RHL 1.80 1.19 1.27 1.19 1.10 1.05
LAF 20.991 40.099 48.902 63.609 70.252 82.050
FG-X FG-X HAF 39.781 49.961 61.895 67.916 73.826 75.662
RHL 1.90 1.25 1.27 1.07 1.05 0.92
40
LAF 24.604 43.856 48.744 64.030 74.042 79.492
FG-O FG-O HAF 41.751 52.790 63.749 75.225 84.723 85.867
RHL 1.70 1.20 1.31 1.17 1.14 1.08
LAF 29.164 51.441 54.612 74.593 81.763 86.713
FG-X FG-X HAF 49.794 60.768 71.796 83.316 85.867 95.270
1.71 1.18 1.31 1.12 1.05 1.10
RHL
Table 12 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 60 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 24.438 43.809 63.653 64.839 86.480 91.711
FG-O FG-O HAF 51.756 62.521 75.738 78.184 88.280 96.802
50
1 2 3 4 5 6
25.916 49.573 70.869 78.569 109.852 112.353
67.474 78.840 98.295 115.163 135.403 154.764
2.60 1.59 1.39 1.47 1.23 1.38
32.754 65.206 83.948 104.710 132.945 149.951
101.677 113.218 136.190 154.705 177.515 195.461
3.10 1.74 1.62 1.48 1.34 1.30
43.119 72.938 96.086 106.569 140.973 144.293
108.862 130.268 150.354 170.798 191.372 212.689
2.52 1.79 1.56 1.60 1.36 1.47
51.330 90.959 108.409 135.449 164.430 183.942
146.085 168.392 189.343 210.411 231.464 253.181
2.85 1.85 1.75 1.55 1.41 1.38
100
1 2 3 4 5 6
26.193 50.694 72.321 81.608 113.931 118.912
71.999 82.558 104.128 121.926 144.494 165.761
2.75 1.63 1.44 1.49 1.27 1.39
33.360 67.926 86.994 112.341 141.197 166.376
115.665 124.841 153.227 173.235 201.276 225.587
3.47 1.84 1.76 1.54 1.43 1.36
44.136 75.765 99.609 112.869 148.801 156.124
126.310 152.779 177.837 203.659 230.036 257.830
2.86 2.02 1.79 1.80 1.55 1.65
53.569 97.697 114.880 150.748 179.997 198.194
192.234 223.633 253.133 283.042 313.298 345.008
3.59 2.29 2.20 1.88 1.74 1.74
0.14 20
1 2 3 4 5 6
25.467 45.868 65.345 67.866 90.361 94.394
55.260 66.130 79.499 79.851 92.162 98.768
2.17 1.44 1.22 1.18 1.02 1.05
31.382 55.692 74.381 80.441 96.877 104.905
68.567 79.851 80.031 93.369 98.768 106.200
2.18 1.43 1.08 1.16 1.02 1.01
39.399 62.841 82.602 85.913 109.056 113.754
69.578 82.278 94.500 107.026 112.008 119.746
1.77 1.31 1.14 1.25 1.03 1.05
44.444 71.332 88.761 96.731 121.465 122.674
77.864 90.607 103.032 112.008 115.741 128.645
1.75 1.27 1.16 1.16 0.95 1.05
50
1 2 3 4 5 6
27.137 52.516 73.135 83.675 114.313 119.959
74.211 85.520 105.824 123.062 143.959 163.703
2.73 1.63 1.45 1.47 1.26 1.36
34.926 69.732 88.521 111.911 139.942 159.836
110.410 122.344 146.089 165.166 188.602 199.628
3.16 1.75 1.65 1.48 1.35 1.25
44.614 76.391 98.409 112.380 145.677 152.673
117.501 139.333 159.808 180.582 201.445 223.036
2.63 1.82 1.62 1.61 1.38 1.46
54.064 96.380 112.895 143.523 171.959 190.585
154.091 176.676 197.925 219.285 240.635 262.671
2.85 1.83 1.75 1.53 1.40 1.38
100
1 2 3 4 5 6
27.454 53.844 74.747 87.310 118.939 127.832
79.921 90.181 113.043 131.281 154.899 176.771
2.91 1.67 1.51 1.50 1.30 1.38
35.672 73.041 92.269 121.265 149.566 176.283
128.218 137.250 167.557 188.312 217.947 243.248
3.59 1.88 1.82 1.55 1.46 1.38
45.764 79.666 102.228 119.778 154.372 166.658
139.098 166.710 192.813 219.620 246.959 275.754
3.04 2.09 1.89 1.83 1.60 1.65
56.777 104.636 120.421 162.173 190.290 206.733
209.436 242.002 272.562 303.459 334.659 367.319
3.69 2.31 2.26 1.87 1.76 1.78
0.17 20
1 2 3 4 5 6
30.143 54.247 78.731 80.605 107.888 113.864
64.589 77.906 94.485 98.239 110.150 121.657
2.14 1.44 1.20 1.22 1.02 1.07
37.471 66.629 90.422 96.655 119.054 126.408
81.430 95.607 98.239 112.268 121.657 128.232
2.17 1.43 1.09 1.16 1.02 1.01
47.461 75.446 100.534 103.280 131.471 138.238
83.840 99.616 114.730 130.203 138.067 145.893
1.77 1.32 1.14 1.26 1.05 1.06
54.131 86.491 109.120 117.290 147.412 150.221
94.252 110.104 125.531 138.067 141.315 157.339
1.74 1.27 1.15 1.18 0.96 1.05
50
1 2 3 4 5 6
31.863 60.976 87.221 96.755 135.322 138.566
83.185 97.091 121.176 142.000 167.077 191.040
2.61 1.59 1.39 1.47 1.23 1.38
41.276 81.822 106.161 131.020 167.720 187.233
125.628 140.543 169.368 192.864 221.590 245.597
3.04 1.72 1.60 1.47 1.32 1.31
53.080 89.854 118.422 131.469 173.916 178.291
135.188 161.897 186.971 212.499 238.197 264.826
2.55 1.80 1.58 1.62 1.37 1.49
64.878 114.310 137.536 169.647 207.558 230.202
180.909 209.039 235.474 262.084 288.695 316.161
2.79 1.83 1.71 1.54 1.39 1.37
1 2 3 4 5 6
32.184 62.264 88.906 100.249 140.043 146.123
88.439 101.350 127.902 149.784 177.576 203.763
2.75 1.63 1.44 1.49 1.27 1.39
42.049 85.294 110.019 140.683 178.377 207.959
142.794 154.925 190.433 215.947 251.202 282.047
3.40 1.82 1.73 1.53 1.41 1.36
54.256 93.117 122.521 138.753 183.014 192.009
155.533 188.232 219.216 251.168 283.840 318.305
2.87 2.02 1.79 1.81 1.55 1.66
67.724 122.803 145.855 188.840 227.385 252.042
237.621 277.121 314.239 351.886 389.950 429.801
3.51 2.26 2.15 1.86 1.71 1.71
* b/h VCNT 0.11 20
100
RHL 2.12 1.43 1.19 1.21 1.02 1.06
LAF 29.761 53.190 71.587 77.333 94.767 101.297
FG-X FG-X HAF 65.720 76.906 78.184 90.117 96.802 102.745
RHL 2.21 1.45 1.09 1.17 1.02 1.01
41
LAF 38.335 60.727 80.995 82.846 105.156 111.032
FG-O FG-O HAF 66.610 79.211 91.299 103.691 109.849 116.269
RHL 1.74 1.30 1.13 1.25 1.04 1.05
LAF 42.891 68.900 86.333 93.703 117.963 119.362
FG-X FG-X HAF 75.761 88.268 100.439 109.849 112.883 125.513
1.77 1.28 1.16 1.17 0.96 1.05
RHL
Captions of Figures Fig. 1
(a) Linear (p=1), (b) Quadratic (p=2), (c) Cubic (p=3) and (d) Quartic (p=4) NURBS basis functions corresponding to knot vectors 0, 0,1, 2, 2 , 0, 0, 0,1, 2, 2, 2 , 0, 0, 0, 0,1, 2, 2, 2, 2 ,
0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 , respectively. (The graphs are drawn using more sampling points than those shown in the figures.)
Fig. 2
Schematic of a skew plate with skew angle , depicting its global coordinate ( xy ), local coordinate ( x y ) on the oblique edge and CNT orientation angle ( ) .
Fig. 3
Four types of CNT distributions: (a) UD (b) FG-V (c) FG-O (d) FG-X.
Fig. 4
Convergence study for central point deflection of a simply supported square plate ( 0 ).
Fig. 5
Convergence study for central point deflection of a simply supported rhombic plate with 60 .
Fig. 6
The non-dimensional deflection, W * wc / h , of FG-CNTRC plates with skew angle versus CNT orientation angle, with boundary conditions: (a)-(f) Simply supported and (g)-(l) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14 and uniformly distributed load q=-0.1 MPa).
Fig. 7
Profiles of non-dimensional deflection, W * w / h , for simply supported skew UD-CNTRC plates under uniformly distributed load of q=-0.1 MPa along (a) x-axis ( 0 ) (b) y-axis ( 0 ) (c) x-axis ( ) (d) y-axis ( ) (For all cases: a/b=1, b/h=50, VCNT =0.14).
Fig. 8
The non-dimensional central deflection versus CNT orientation angle for the simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (For all cases b/h=50, VCNT =0.14, UD and q=-0.1 MPa).
Fig. 9
Fig. 10
The normalized non-dimensional central deflection, W ** , of simply supported skew CNTRC plates with varying width-to-thickness ratio versus CNT orientation angle (a/b=1, VCNT =0.14, UD, 45 and q=-0.1 MPa). Natural frequency parameters of CNTRC plates of skew angle versus CNT orientation angle with boundary conditions: (a)-(e) Simply supported and (f)-(j) Fully clamped (For all cases a/b=1, b/h=50, VCNT 0.14 ).
Fig. 11
Natural frequency parameters of cantilever FG-CNTRC plates with skew angle versus CNT orientation angle (For all cases a/b=1, b/h=50, VCNT =0.14).
Fig. 12
Natural frequency parameters versus CNT orientation angle for vibration of simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (b/h=50, VCNT 0.14 , UD).
Fig. 13
Normalized frequency parameters, ** , versus CNT orientation angle for skew CNTRC plates with varying width-to-thickness ratios and boundary conditions: (a) Simply supported (b) Clamped (c) Cantilever (a/b=1, VCNT 0.14 , UD, 45 ).
Fig. 14
Normalized frequency parameters, ** , versus CNT orientation angle for square CNTRC plates of varying width-to-thickness ratios with boundary conditions and CNT volume fractions: (a) Simply supported, V*=0.11 (b) Simply supported, VCNT =0.14 (c) Clamped, VCNT =0.11 (d) Clamped, VCNT =0.14 (a/b=1, FGO, 0 ).
Fig. 15
First eight mode shapes of clamped skew UD-CNTRC plates with skew angle and CNT orientation angle =0.14). corresponding to HAF (a/b=1, b/h=50, VCNT
Fig. 16
First eight mode shapes of cantilever skew UD-CNTRC plates with skew angle and CNT orientation angle =0.14). corresponding to HAF (a/b=1, b/h=50, VCNT
42
1.0
1.0
N1,1
p=1
N1,2
p=2
N2,2
N2,1 N3,1
N3,2
0.8
Basis function value
Basis function value
0.8
0.6
0.4
N4,2
0.6
0.4
0.2
0.2
0.0
0.0 0.0
1.0
0.4
0.8
1.2
1.6
0.0
2.0
0.4
0.8
1.2
(a)
(b) 1.0
N1,3
p=3
1.6
N1,4
p=4
N2,4
N2,3 N3,3
N3,4
0.8
N4,3
Basis function value
Basis function value
0.8
N5,3
0.6
0.4
2.0
N4,4 N5,4 N6,4
0.6
0.4
0.2
0.2
0.0
0.0 0.0
0.4
0.8
1.2
1.6
0.0
2.0
0.4
0.8
1.2
(c)
(d)
1.6
2.0
Fig. 1. (a) Linear (p=1), (b) Quadratic (p=2), (c) Cubic (p=3) and (d) Quartic (p=4) NURBS basis functions corresponding to knot vectors 0, 0,1, 2, 2 , 0, 0, 0,1, 2, 2, 2 , 0, 0, 0, 0,1, 2, 2, 2, 2 , 0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 , respectively. (The graphs are drawn using more sampling points than those shown in the figures.)
43
y
y'
α
x'
β
b
a
x
Fig. 2. Schematic of a skew plate with skew angle , depicting its global coordinate ( xy ), local coordinate ( x y ) on the oblique edge and CNT orientation angle ( ) .
h
y z
y z
(a)
h
y z
h
h
y z
(b)
(c)
(d)
Fig. 3. Cross section of the CNTRC plates of thickness h with four types of CNT distributions: (a) UD (b) FG-V (c) FG-O (d) FG-X.
44
* Non-dimensional deflection Wiso
6.6
6.5
6.4
6.3
Ref. [42] p=4 p=3 p=2
6.2
6.1 9
11
13
15
17
19
21
23
25
27
29
31
33
Number of elements along each direction
* Non-dimensional deflection Wiso
Fig. 4. Convergence study for central point deflection of a simply supported square plate ( 0 ).
0.65 0.60 0.55
0.50
Ref. [42] p=4 p=3 p=2
0.45 0.40 9
11
13
15
17
19
21
23
25
27
29
31
33
Number of elements along each direction
Fig. 5. Convergence study for central point deflection of a simply supported rhombic plate with 60 .
45
Simply Supported
Clamped 0.7 FG-O FG-V UD FG-X
2.0
Non-dimensional deflection W*
Non-dimensional deflection W*
1.5
1.0
0.5
0.5
0.3
0.1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
+15 +30 +45 +60 +75 +90
(g) FG-O FG-V UD FG-X
Non-dimensional deflection W*
Non-dimensional deflection W*
0
CNT orientation angle ()
(a) 2.1
FG-O FG-V UD FG-X
1.7
1.3
0.9
FG-O FG-V UD FG-X
0.7
0.5
0.3
0.5
0.1 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(b)
(h)
FG-O FG-V UD FG-X
1.8
Non-dimensional deflection W*
Non-dimensional deflection W*
0.9
1.4
1.0
0.6
FG-O FG-V UD FG-X
0.7
0.5
0.3
0.1
0.2 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
(i)
FG-O FG-V UD FG-X
1.3
Non-dimensional deflection W*
Non-dimensional deflection W*
0.6
0.9
0.6
0.3
FG-O FG-V UD FG-X
0.5 0.4 0.3 0.2 0.1 0.0
0.1 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(d)
(j)
46
0.6
FG-O FG-V UD FG-X
0.5
Non-dimensional deflection W*
Non-dimensional deflection W*
0.25
0.4 0.3 0.2 0.1
0.20
0.15
0.10
0.05
0.00
0.0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(e)
+15 +30 +45 +60 +75 +90
(k)
0.030
FG-O FG-V UD FG-X
0.08
Non-dimensional deflection W*
Non-dimensional deflection W*
0
CNT orientation angle ()
CNT orientation angle ()
0.10
FG-O FG-V UD FG-X
0.06 0.04 0.02
FG-O FG-V UD FG-X
0.025 0.020 0.015 0.010 0.005
0.00
0.000 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(f) (l) Fig. 6. The non-dimensional deflection, W * wc / h , of FG-CNTRC plates with skew angle versus CNT orientation angle, with boundary conditions: (a)-(f) Simply supported and (g)-(l) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14 and uniformly distributed load q=-0.1 MPa).
47
1.0
Non-dimensional deflection W*
O
β
x
b
0.8
a
0.6
0.4
0.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
y α
Non-dimensional deflection W*
y α
β
0.8
a
0.6
0.4
0.2
-1.0
1.0
x
O
-0.8
-0.6
-0.4
-0.2
(a) O
x
b
0.8
a
0.6
0.4
0.2
α
-0.8
-0.6
-0.4
-0.2
0.6
0.8
1.0
β
O
x
b
0.8
a
0.6
0.4
0.2
0.0
0.0 -1.0
0.4
1.0
y
Non-dimensional deflection W*
Non-dimensional deflection W*
β
0.2
(b)
1.0
y
0.0
2(x/a)
2(x/a)
α
b
0.0
0.2
0.4
0.6
0.8
-1.0
1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
2(x/a)
2(x/a)
(c) (d) Fig. 7. Profiles of non-dimensional deflection, W * w / h , for simply supported skew UD-CNTRC plates under uniformly distributed load of q=-0.1 MPa along (a) x-axis ( 0 ) (b) y-axis ( 0 ) (c) x-axis ( ) (d) y-axis ( =0.14). ) (For all cases: a/b=1, b/h=50, VCNT
48
12
9.0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
10 8 6
Non-dimensional deflection W*
Non-dimensional deflection W*
4 2 0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
7.5 6.0 4.5 3.0 1.5 0.0
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(a)
(b)
5
2.0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
4 3
Non-dimensional deflection W*
Non-dimensional deflection W*
6
2 1
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
1.6 1.2
0.8 0.4
0.0
0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) Fig. 8. The non-dimensional central deflection versus CNT orientation angle for the simply supported CNTRC plates
Normalized non-dimensional deflection, W**
with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (For all cases b/h=50, VCNT 0.14 , UD and q=-0.1 MPa).
5
4
=45o a/b=1
b/h=100 b/h=50 b/h=20 b/h=10 b/h=5
3
2
1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Fig. 9. The normalized non-dimensional central deflection, W ** , of simply supported skew CNTRC plates with varying width-to-thickness ratio versus CNT orientation angle (a/b=1, VCNT =0.14, UD, 45 and q=-0.1 MPa).
49
Simply Supported
Fully Clamped 50
Natural frequency parameter (*)
Natural frequency parameter (*)
26 24 22 20 18 16 14 12
FG-X UD FG-V FG-O
10
45 40 35 30 25
FG-X UD FG-V FG-O
20
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(a)
(f) 55
Natural frequency parameter (*)
Natural frequency parameter (*)
28 26 24 22 20 18 16 14
FG-X UD FG-V FG-O
50 45 40 35 30 25
FG-X UD FG-V FG-O
12 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
36 32 28 24
16
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(g) Natural frequency parameter (*)
Natural frequency parameter (*)
(b)
20
0
CNT orientation angle ()
0
60
50
40
30
+15 +30 +45 +60 +75 +90
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
(h)
60
Natural frequency parameter (*)
Natural frequency parameter (*)
90
50
40
30
20
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
0
80 70 60 50 40 30
+15 +30 +45 +60 +75 +90
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(d)
(i)
50
Natural frequency parameter (*)
Natural frequency parameter (*)
160
100
80
60
40
FG-X UD FG-V FG-O
20
140 120 100 80 60
FG-X UD FG-V FG-O
40
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(e) (j) Fig. 10. Natural frequency parameters of CNTRC plates of skew angle versus CNT orientation angle with boundary conditions: (a)-(e) Simply supported and (f)-(j) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14).
51
10
Natural frequency parameter (*)
Natural frequency parameter (*)
10
8
6
4
2
FG-X UD FG-V FG-O
8
FG-X UD FG-V FG-O
6
4
2
0
0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(a)
(b) Natural frequency parameter (*)
Natural frequency parameter (*)
FG-X UD FG-V FG-O
6
4
2
-90 -75 -60 -45 -30 -15
0
Natural frequency parameter (*)
6
4
2
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
8
8
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
10
+15 +30 +45 +60 +75 +90
10
10
8
0
CNT orientation angle ()
CNT orientation angle ()
(d)
FG-X UD FG-V FG-O
6
4
2 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(e) Fig. 11. Natural frequency parameters of cantilever FG-CNTRC plates with skew angle versus CNT orientation angle (For all cases a/b=1, b/h=50, VCNT =0.14).
52
20
Natural frequency parameter (*)
Natural frequency parameter (*)
22
18 16 14 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
12 10 8
6
30
25
20 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
15
10
5
-105 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90 +105
50
40
30 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
20
-105 -90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(b) Natural frequency parameter (*)
Natural frequency parameter (*)
(a)
10
0
CNT orientation angle ()
CNT orientation angle ()
0
90 75 60 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
45 30 15
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90 +105
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) Fig. 12. Natural frequency parameters versus CNT orientation angle for vibration of simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (b/h=50, VCNT =0.14, UD).
53
Normalized frequency parameter (**)
1.0 0.9 0.8 0.7 0.6 0.5 0.4
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
=45o a/b=1 SSSS
-90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Normalized frequency parameter (**)
(a) 1.0 0.9 0.8 0.7 0.6 0.5 0.4
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
=45o a/b=1 CCCC
0.3 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Normalized frequency parameter (**)
(b) 1.0 0.9 0.8
=45o a/b=1 CFFF
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
0.7 0.6 0.5 0.4 0.3 0.2 0.1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c) Fig. 13. Normalized frequency parameters, ** , versus CNT orientation angle for skew CNTRC plates with varying width-to-thickness ratios and boundary conditions: (a) Simply supported (b) Clamped (c) Cantilever (a/b=1, VCNT 0.14 , UD, 45 ).
54
Normalized frequency parameter (**)
Normalized frequency parameter (**)
1.000
0.995
0.990
0.985
V*CNT=0.11
b/h=20 b/h=50 b/h=100
SSSS
0.980 -90 -75 -60 -45 -30 -15
0
1.00
0.99
0.98
0.97
0.96
+15 +30 +45 +60 +75 +90
(b) Normalized frequency parameter (**)
Normalized frequency parameter (**)
0
CNT orientation angle ()
(a) 1.00 0.96 0.92 0.88
0.80
SSSS
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0.84
V*CNT=0.14
b/h=20 b/h=50 b/h=100
b/h=20 b/h=50 b/h=100
* CNT
V
=0.11
CCCC
1.00 0.96 0.92 0.88 0.84 0.80
b/h=20 b/h=50 b/h=100
V*CNT=0.14 CCCC
0.76 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) ** Fig. 14. Normalized frequency parameters, , versus CNT orientation angle for square CNTRC plates of varying width to-thickness ratios with boundary conditions and CNT volume fractions: (a) Simply supported, VCNT =0.11 (b) Simply supported, VCNT =0.14 (c) Clamped, VCNT =0.11 (d) Clamped, VCNT =0.14 (a/b=1, FG-O, 0 ).
55
0
1* 43.426
2* 47.223
3* 56.997
4* 74.248
5* 98.726
6* 105.358
7* 107.521
8* 112.957
1* 56.487
2* 64.515
3* 79.708
4* 98.266
5* 117.863
6* 132.749
7* 139.345
8* 140.039
1* 80.742
2* 99.307
3* 117.543
4* 136.309
5* 155.539
6* 175.575
7* 179.209
8* 196.420
1* 142.204
2* 164.687
3* 185.783
4* 207.017
5* 228.243
6* 250.143
7* 272.522
8* 295.780
30
45
60
Fig. 15. First eight mode shapes of clamped skew UD-CNTRC plates with skew angle and CNT orientation angle corresponding to HAF (a/b=1, b/h=50, VCNT =0.14).
56
0
1* 7.479
2* 7.908
3* 11.235
4* 21.087
5* 37.646
6* 43.674
7* 44.126
8* 46.099
1* 7.517
2* 8.198
3* 13.107
4* 25.917
5* 43.769
6* 43.946
7* 45.736
8* 48.956
1* 7.608
2* 8.780
3* 16.665
4* 33.218
5* 44.570
6* 46.660
7* 54.144
8* 55.525
1* 7.943
2* 10.395
3* 24.653
4* 43.812
5* 48.095
6* 55.096
7* 56.414
8* 73.452
30
45
60
Fig. 16. First eight mode shapes of cantilever skew UD-CNTRC plates with skew angle and CNT orientation angle corresponding to HAF (a/b=1, b/h=50, VCNT =0.14).
57
Highlights
An isogeometric approach based on Reddy’s higher order shear deformation theory is developed for skew FG-CNTRC plates.
Boundary conditions along the oblique edges are specially treated by introducing transformation matrices for 7 degrees of freedom using geometrical considerations and relations in Reddy’s higher order shear deformation theory.
The influence of CNT fiber orientation angles are investigated for optimal bending and vibration behavior of FG-CNTRC skew plates.
58
S0045-7825(16)30604-1 http://dx.doi.org/10.1016/j.cma.2016.12.009 CMA 11255
To appear in:
Comput. Methods Appl. Mech. Engrg.
Received date: 19 June 2016 Revised date: 7 December 2016 Accepted date: 8 December 2016 Please cite this article as: M.M. Ardestani, L.W. Zhang, K.M. Liew, Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10.1016/j.cma.2016.12.009 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.
Isogeometric analysis of the effect of CNT orientation on the static and vibration behaviors of CNT-reinforced skew composite plates
M. Memar Ardestani 1, L.W. Zhang 2,*, K.M. Liew 1,3,* 1 2
3
Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
City University of Hong Kong Shenzhen Research Institute Building, Shenzhen Hi-Tech Industrial Park, Nanshan District, Shenzhen, Guangdong, China
Abstract A first attempt to investigate bending and vibration behaviors of carbon nanotube (CNT) reinforced functionally graded composite skew plates using the isogeometric method is performed. The formulation of the plate is based on Reddy’s higher order shear deformation theory (HSDT). The effect of CNT orientation angle on the mechanical behavior of CNT-reinforced composite plates with varying skew angles is investigated. The optimum fiber orientations for CNT-reinforced composite skew plates with simply supported, clamped and cantilevered boundary conditions are found. It is shown that selecting the correct fiber orientation angle is of crucial importance in obtaining the desired static and dynamic responses of CNT-reinforced composite skew plates. Keywords: Carbon nanotube; Composite; Bending; Free vibration; Reddy’s third-order shear deformation theory; Isogeometric method
*Corresponding authors. E-mail addresses: [email protected] (L.W. Zhang), [email protected] (K.M. Liew)
1
1. Introduction In 2005, isogeometric analysis (IGA) was proposed by Hughes et al. [1] and was originally aimed to fill the gap between the design and analysis stages. In this method, which of course is better known as the more general form of Finite Element Method (FEM), basis functions used for interpolating the field variables within the elements are the same as those used for constructing the geometrical model. These basis functions are constructed by making use of Non-Uniform Rational B-Splines (NURBS) which are capable of constructing the exact geometrical model. Furthermore, as opposed to FEM, this method provides a very convenient manner in which to achieve a higher order of continuity of basis functions [2]. Since the introduction of IGA, a considerable amount of research has been published on this subject. Specifically, in the field of plate structures, it has gained much attention as a suitable numerical technique. Tran et al. [3] presented static, dynamic and buckling analyses of functionally graded rectangular as well as circular plates based on Reddy’s higher-order shear deformation theory (HSDT) using the isogeometric method. They concluded that IGA does provide an effective computational tool for practical problems. An isogeometric analysis of CNT-reinforced composite plates based on HSDT is carried out by Phung-Van et al. [4]. They considered 5 degrees of freedom associated to each control points. For some plate configurations, the presented results in [4] showed significant deviation from those previously published results. In the present study, we will further examine the HSDT formulation using 7 degrees of freedom in an attempt to achieve more reliable and accurate results using IGA. Moreover, Valizadeh et al. [5] utilized a NURBS-based isogeometric method to analyze functionally graded plates. In their study, the plate displacement field was based on FSDT and they made use of an artificial shear correction factor to overcome the shear locking phenomenon when low order NURBS basis functions were used for thin plates. Using IGA, and incorporating a simple four-unknown shear and normal deformation theory, Thai et al. [6] studied the bending and free vibration behavior of functionally graded isotropic and sandwich plates. The free vibration and buckling analyses of laminated composite plates using the isogeometric method were presented by Shojaee et al. [7]. In their study, which is based on classical plate theory (CPT), they provided results for the vibration of elliptical and square plates, each containing a hole of complex shape. They have stated that by using the capability of NURBS to exactly represent complex geometric shapes, the isogeometric method would be an efficient and accurate technique for practical problems of engineering. In addition, Shojaee et al. [8] studied the free vibration of thin circular, skew, L-shaped and square-shaped (with complex cutout) plates using the 2
isogeometric method. The body of research using the isogeometric method has become vast and remains ongoing [9]–[14] . For more detailed information considering the computer implementation of the isogeometric method, readers are referred to [15], [16]. Carbon nanotubes (CNTs), since reported by Iijima [17], have been widely used as reinforcement in structures such as beams, plates, and shells made of composite materials, usually referred to as CNT-reinforced composites, due not only to their prominent mechanical properties but also to their unique thermal and electrical features [18], [19]. In particular, CNT-reinforced composite plates with diverse geometrical shapes have gained much attention as high performance structures for various applications in the aerospace, aeronautics and marine industries as well as in bridge structures. In fact, many researchers have devoted their attention to investigating the superior behavior of CNTs, through the framework of various schemes [20], [21]. In 2009, Shen analyzed the nonlinear bending behavior of CNT-reinforced composite plates [22]. In that paper, he introduced the concept of the functionally graded scheme into CNTreinforced composite plates, in which the material properties are graded along the thickness direction of the plate. In order to obtain the material properties he further proposed the CNT efficiency parameters which are estimated by matching the elastic modulus of CNT-reinforced composites obtained from the molecular dynamic (MD) simulation with those numerical results obtained from the rule of mixture. The static and free vibration of square CNT-reinforced composite plates based on first order shear deformation theory (FSDT) are presented by Zhu et al. [23]. In their study, they used four types of CNT distribution through the thickness of the plate, namely uniform distribution, FG-V, FG-O and FG-X. They stated that desired plate stiffness can be achieved by adjusting the distribution of the CNTs along the thickness direction of the plate. Moreover, they concluded that CNT distributions that have more CNTs near the top and bottom surfaces of the plate are more efficient. A thermoelastic analysis of CNT-reinforced composite plates was carried out by Alibeigloo and Liew [24], in which they incorporated the three-dimensional theory of elasticity to analyze the thermos-elastic behavior of the CNT-reinforced composite plates under thermo-mechanical loads. Alibeigloo and Emtehani further published a paper [25] on the static and vibration behavior of simply supported CNT-reinforced composite plates. They obtained a closed form solution for simply supported CNT-reinforced composite plate using Fourier series expansion and the state space technique and for the case of the clamped boundary condition, they used a semi-analytical technique. They further performed parametric studies on CNT-reinforced composite plate to describe their vibration behavior. Wang and Shen [26] studied the large amplitude vibration of 3
nanocomposite plates resting on an elastic foundation. In their study, the equations of motion are based on higher-order shear deformation plate theory in which plate-foundation interaction is taken into account. Moreover, using the Ritz method, Abdollahzadeh Shahrbabaki and Alibeigloo [27] presented a three-dimensional model for the free vibrational analysis of CNT-reinforced composite plates. Zhang et al. [28] studied the free vibration characteristics of CNT-reinforced composite triangular plates based on FSDT and using the element-free IMLS-Ritz numerical method. They presented a detailed study considering the effects of various geometrical and material parameters on the vibration of CNT-reinforced composite triangular plates. Using a unified formulation of finite prism methods (FPMs) and three-dimensional model, free vibration behavior of CNT-reinforced composite plates with various boundary conditions is investigated by Wu and Li [29]. Based on Eshelby-Mori-Tanaka approach, Sobhani Aragh et al. [30] investigated the vibration characteristics of continuously graded carbon nanotube-reinforced cylindrical panels. They concluded that continuously graded oriented CNTs can be used to achieve the desired effect in relation to the vibration behavior of structures. Furthermore, in the context CNT-reinforced composite plates, Lei et al. [31] compared the effective material properties estimated by the Eshelby-Mori-Tanaka approach and extended rule of mixture. They concluded that results obtained for the frequency parameters of the CNT-reinforced composite plate using the extended rule of mixture were slightly higher than those obtained based on Eshelby-Mori-Tanaka approach. Semianalytical solutions to buckling and free vibration analysis of carbon nanotube-reinforced composite thin plates utilizing multi-term Kantorovich-Galerkin method are proposed by Wang et al. [32]. Furthermore, Selim et al. [33] performed a study on the vibration of CNT-reinforced composite plates in a thermal environment based on Reddy’s higher-order shear deformation theory (HSDT). In their comprehensive study, a comparison between the results obtained based on FSDT and HSDT was made and a maximum relative difference of three percent for the natural frequency parameter for the first mode of free vibration at room temperature was reported. Vibrational analysis of carbon nanotube-reinforced composite quadrilateral plates subjected using a weak formulation of elasticity considering the thermal effects is presented by Ansari et al. [34]. Moradi-Dastjerdi et al. [35] presented the static analysis of CNT-reinforced composite cylinders utilizing the element-free Galerkin method. They used an axisymmetric model in their approach and investigated the influence of CNT distribution and volume fraction on the value as well as the location of maximum hoop stress. A nonlinear bending analysis of CNT-reinforced composite plates resting on a Pasternak foundation is presented by Zhang et al. [36], in which the influence of foundation stiffness as well as CNT distribution and CNT volume fractures is investigated. 4
The analysis of skew plates has long been interesting to researchers in the field of structural mechanics. Due to the presence of singular behavior on the obtuse angles of skew plates, an appropriate numerical scheme is usually required. One of the earliest studies to take this stress concentration into account is that of Morley [37]. He utilized the simple form of eigenfunctions to consider the singularities in obtuse angles. The results published in his paper are known to be benchmark solutions and, as such, will be used in our study for verification purposes. Rajaiah and Rao [38] also derived an exact solution to this problem by applying Stevenson’s tentative approach with complex variables. Butalia et al. [39] performed a comprehensive bending analysis of parallelogram-shaped plates using a Mindlin nine-node Heterosis element. In their study, the efficiency and the limitations of that element were explored and numerical results for the transverse shear forces are presented. Furthermore, Sengupta [40] carried out a performance study of simple finite elements for the bending analysis of skew rhombic plates. He proposed a simple three-node element and compared its accuracy and reliability with a triangular Mindlin plate element of 35 degrees of freedom (DOF) and a triangular plate element previously proposed by Zienkiewicz and Lefebvre [41] which had independent interpolation for slope, displacement and shear forces. He concluded that the proposed simple three-node element fits suitably for the analysis of skew plates. We will also use the results obtained by him in the following sections for numerical verification. A bending analysis of simply supported skew plates based on FSDT was carried out by Liew and Han [42]. They presented the results for both thin and thick plates with skew angles from slight to extreme cases. Some other references regarding the bending analysis of skew plates are found at [43]–[47]. Moreover, extensive work has also been conducted on the vibration analysis of skew plates. An analysis of thick skew plates based on FSDT using the pb-2 Rayleigh-Ritz method was presented by Liew et al. [48]. They apply special treatments to impose the boundary conditions on oblique edges. McGee and Leissa published a three-dimensional vibration analysis of skew cantilevered plates [49]. In that paper, they calculated accurate natural frequencies for skew plates with arbitrary skew angles. Moreover, McGee et al. [50] presented a free vibration study on clamped and simply supported thick rhombic plates based on HSDT. Krishna Reddy and Palaninathan [51] presented a free vibration analysis of skew plates by extending a general high precision three-node bending finite element to the free vibration analysis of laminated skew plates. They derived the consistent mass matrix in an explicit form and transformed element matrices to impose boundary condition on oblique edges. Further studies on the vibration analysis of skew plates can be found at [52]–[57].
5
Recently, studies dedicated to the bending and vibration analysis of CNT-reinforced composite skew plates have been carried out. García-Macías et al. [58] analyzed the static and free vibration behavior of CNT-reinforced composite skew plates using the shell finite element based on FSDT which was formulated in oblique coordinates. Kiani [59] presented the free vibration analysis of CNT-reinforced composite skew plates. To construct an eigenvalue problem, he used Ritz method in which shape functions are obtained according to Gram-Schmidt process. Moreover, a large deflection analysis of CNT-reinforced composite skew plates resting on Pasternak elastic foundations is carried out by Zhang and Liew [60]. In addition, an investigation on the vibration characteristics of moderately thick skew CNT-reinforced composite plates is presented by Zhang et al. [61]. For a more comprehensive overview of these papers on CNT-reinforced composite structures, readers are encouraged to refer to the review paper presented by Liew et al. [62]. In this paper, the isogeometric approach is utilized to analyze the bending and vibration behaviors of CNT-reinforced composite skew plates. The displacement field of the plate is based on Reddy’s third order shear deformation theory (HSDT). Considering the skew CNT-reinforced composite plate as a one-ply composite plate, the influence of CNT orientation angle on the mechanical response of the plate accounting for different plate aspect ratios, width-to-thickness ratios and boundary conditions is investigated. As will be demonstrated in detail, the choice of CNT orientation angle has a crucial effect on the desired responses of the CNT-reinforced composite skew plates. 2. B-splines and NURBS 2.1. Knot vector A fundamental feature that controls the behavior of a B-spline curve to be constructed is the knot vector. A knot vector is a set of ordered, non-decreasing, real numbers, known as knots, associated with each B-spline objects i.e. if 1,, i ,, n is a knot vector associated with a B-spline curve C, then i and i1 i for i 1, , n 1 . Ξ is said to be open if its first and last members are repeated p+1 times, and p is the polynomial order. For an open knot vector, Ξ, the number of members, n, is calculated from n b p 1 where b is the number of basis functions used in the construction of the B-spline curve. A knot vector is known to be uniform if the knots are equally spaced from each other; otherwise it is non-uniform. Knots divide the B-spline curve into sections recognized as elements. Within an element (i.e. between knots), functions are C and between
6
elements (i.e. over knots), functions are C p k where k represents the multiplicity of the corresponding knot.
2.2. Basis function Associated with a knot vector , are B-spline basis functions denoted by Ni , p for i 1, , n defined according to the following well-known Cox-de Boor recursive formulas, For p 0
1 if i i 1 , Ni ,0 ( ) otherwise. 0
(1)
For p 1
Ni , p ( )
i Ni , p 1 ( ) i p 1 N ( ). i p i i p 1 i 1 i 1, p 1
(2)
For cases of p=1, p=2, p=3 and p=4 with their corresponding open knot vectors written respectively
as
0,0,1, 2, 2 ,
0, 0, 0,1, 2, 2, 2 ,
0, 0, 0, 0,1, 2, 2, 2, 2
and
0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 the basis functions are shown in Fig. 1. 2.3. B-spline objects 2.3.1. B-spline curves A B-spline curve is defined as b
C( ) Ni , p ( )Pi ,
(3)
i 1
where Pi , i 1, 2,, b are the control points and Ni , p ( ) are the basis functions of degree p
associated with the knot vector 1 , 2 , , b p 1 calculated by Eqs. (1) and (2). 2.3.2. B-spline surfaces B-spline surfaces are known as tensor product extensions of B-spline curves and are defined as b
b
S , H ip, j, q , Pi , j ,
(4)
i 1 j 1
p ,q where Hi , j ( , ) i 1, 2,, b , j 1, 2,, b are the bivariate B-spline basis functions defined as
7
H ip, j,q ( , ) Ni , p ( ) M j ,q ( ),
(5)
in which N i , p ( ) and M j , q ( ) are the basis functions with polynomial degrees p and q along the
and
parametric
directions
associated
1 , 2 , , b q 1 , respectively.
with
P
2
i, j
knot
vectors
1 , 2 , , b p 1
and
is the control net associated with the B-spline
surface. 2.4. NURBS Introducing an additional variable referred to as weight into a more general form of B-spline function (i.e. rational B-splines), it is now possible to exactly represent conical shapes such as circles, cones and so on, which was not achievable through the use of classical non-rational Bsplines. As a favorable feature of NURBS, any projective transformation applied to NURBS objects can be performed by directly applying it to the control polygon [63]. As will be seen in the following sections, we make use of this property for the special case of the affine transformation of geometrical shapes. wi is usually treated as an additional coordinate for the control points and is kept constant under the affine transformation of control points. 2.4.1. NURBS basis function In this way, NURBS basis functions are defined as
Rip ( )
N i , p ( ) wi b
N I 1
I,p
,
(6)
( ) wI
where Ni , p are the B-spline basis functions defined in Eqs. (1) and (2) and wi are the weights associated with each control point. 2.4.2. NURBS objects Replacing the univariate and bivariate B-spline basis functions used in Eqs. (3) and (4) with the corresponding NURBS basis functions, one can construct the NURBS objects in one and two dimensions. Therefore, in addition to the knot vectors, control points and basis functions associated with each B-spline object, in the case of NURBS a new set of weights is also added. The formulations are written in more detail in the following.
8
A. NURBS curves
A NURBS curve associated with a knot vector 1 , 2 , , b p 1 is defined as b
C( ) Ri , p ( )Pi ,
(7)
i 1
where R i , p , i 1,2,..., b are the univariate NURBS basis functions of degree p in the direction defined in Eq. (6) and Pi are their associated control points. B. NURBS surfaces Given a control net, Pi , j , their corresponding weights, wi , j , and a knot vector corresponding to each parametric direction, NURBS surfaces are defined to be a tensor product of univariate NURBS. In other words b
b
S , Rip, j,q , Pi , j ,
(8)
i 1 j 1
p ,q in which Ri , j is the bivariate NURBS basis function and defined as
Rip, j, q ( , )
N i , p ( ) M j , q ( ) wi , j b
b
N I 1 J 1
I,p
,
(9)
( ) M J , q ( ) wI , J
where N i , p ( ) and M j , q ( ) are the univariate B-spline basis functions of degree p and q along the and parametric directions.
In order to increase the order of the NURBS basis functions, the k-refinement technique is used (i.e. the order of the basis function is elevated with the coarsest mesh before any internal knots are inserted). 3. Carbon nanotube-reinforced composites We consider a CNT-reinforced composite skew plate with skew angle , length a, width b and thickness h as shown in Fig. 2. The plate material is composed of an isotropic matrix and singlewalled carbon nanotubes (SWCNTs), aligned with orientation angle with respect to the positive x-direction as shown in Fig. 2, as reinforcement. The effective modulus of elasticity, employing the extended rule of mixture, is expressed as 9
E11 1VCNT E11CNT Vm E m ,
2 E22
3 G12
(10)
VCNT Vm m, CNT E22 E
(11)
VCNT Vm , G12CNT G m
(12)
CNT
CNT
CNT
where E11 , E22
and G12
are the Young’s moduli and shear modulus of the CNT, respectively.
E m and G m are the corresponding properties of the isotropic matrix. i (i 1,2,3) are the CNT efficiency parameters accounting for the scale-dependent material properties calculated by comparing the effective material properties obtained from MD simulations and that of numerical results [22]. VCNT and Vm are the CNT and matrix volume fractions related by
VCNT Vm 1.
(13)
The CNTs are arranged along the thickness of the plate according to four different distribution types: uniform distribution (UD), FG-V, FG-O and FG-X (see Fig. 3). Mathematically, these CNT distributions can be written as
VCNT
* VCNT 1 2 z V * CNT h 2 z * 2 1 h VCNT 2 z * 2 VCNT h
(UD CNTRC), (FG-V CNTRC), (14)
(FG-O CNTRC), (FG-X CNTRC),
* where VCNT is the volume fraction of CNTs and is defined as
* VCNT
wCNT (
wCNT CNT
m )(1 wCNT )
,
(15)
CNT m in which wCNT , and are the mass fraction of CNTs, densities of CNTs and matrix, * respectively. For all cases of CNT distribution, the value of VCNT is set to be constant. In addition,
the Poisson’s ratio, 12 , and the density, , of the CNT-reinforced composite plate are obtained from the following expressions. * 12 VCNT 12CNT Vm m ,
(16)
10
VCNT CNT Vm m .
(17)
It should be noted that following the assumptions made in [22] and [23], 12 is considered to be constant over the thickness of the plate. 4. Governing equations of CNT-reinforced composite skew plates 4.1. Displacement field and strain Employing the HSDT for a CNT-reinforced composite plate, the displacement field of the plate defined in the domain Ω can be expressed as
u( x, y, z) u0 ( x, y) zx ( x, y) cz3 x ( x, y),
(18)
v( x, y, z ) v0 ( x, y ) z y ( x, y ) cz 3 y ( x, y ),
(19)
w( x, y, z) w0 ( x, y),
(20)
where u , v and w are the displacement of a generic point ( x , y , z ) in the domain of the CNTreinforced composite plate along the x-, y- and z-directions, respectively. (u0 , v0 , w0 ) represent the displacements of a point at the mid-plane of the plate and x and y denote rotations of the cross section of the plate about the y- and negative x-axes, respectively. x and y are defined to be
( x w0 / x) and ( y w0 / y ) , respectively. The constant c is set to be 4 / 3h 2 . The strains of a point in the plate are given by
ε ε(0) zε(1) z3ε(3) ,
(21)
γ γ(0) z2γ(2) ,
(22)
where ε xx
ε (0)
yy xy , γ xz yz and T
T
x x u0 x x x y v0 (1) y (3) ,ε , ε c , y y y u0 v0 y y x x x x y x y y
11
(23)
and
γ (0)
w0 x x x , γ (2) 3c . y y w0 y
(24)
Then, the constitutive relations are expressed as
xx Q11 ( z ) Q12 ( z ) Q16 ( z ) 0 0 xx 0 0 yy yy Q12 ( z ) Q22 ( z ) Q26 ( z ) 0 0 xy , xy Q16 ( z ) Q26 ( z ) Q66 ( z ) 0 0 0 Q55 ( z ) Q45 ( z ) xz xz yz 0 0 0 Q45 ( z ) Q44 ( z ) yz
(25)
where Qij are the components of a transformed stiffness matrix in the global xy coordinate system and are related to Qij , the components of the stiffness matrix in the principal material coordinate system, through the following equation.
Q TQ T . T
(26)
In Eq. (26), T , the transformation matrix, is expressed as
m2 n2 2mn 0 0 2 2 m 2mn 0 0 n T mn mn (m2 n 2 ) 0 0 , m n 0 0 0 0 n m 0 0
(27)
where m and n denote cos( ) and sin( ) , respectively. is the angle between the principal material coordinate system and global xy coordinate system.
Q introduced in Eq. (26), together with its component definitions are given as 0 0 0 Q11 ( z ) Q12 ( z ) Q ( z ) Q ( z ) 0 0 0 22 12 Q 0 Q66 ( z ) 0 0 0 , Q55 ( z ) 0 0 0 0 0 Q44 ( z ) 0 0 0
(28)
12
Q11
E11
1 12 21
E22
, Q22
1 12 21
, Q12
21 E11 , Q44 G23 , Q55 G13 , Q66 G12 , 1 12 21
(29)
where E11 and E22 are the effective Young’s moduli of the CNT-reinforced composite plate in the principal material directions. G12 , G13 and G23 are the shear moduli and 12 and 21 are the Poisson’s ratios and are related to each other according to 21 (E22 E11 )12 . According to HSDT, the relationship between the stress resultants and strains can be written as
0 ε (0) 0 ε(1) 0 ε (3) , Ds γ (0) Fs γ (2)
N A B E 0 M B D F 0 P E F H 0 Qs 0 0 0 As s R 0 0 0 Ds
(30)
where the in-plane force resultants, moment resultants and transverse forces are expressed as N xx xx h/ 2 N N yy yy dz , h/ 2 N xy xy
(31)
M xx xx h/ 2 M M yy yy z dz , h /2 M xy xy
(32)
Pxx xx h /2 P Pyy yy z 2 dz , h /2 P xy xy
(33)
Q s h /2 xz Qs xs dz, Qy h /2 yz
(34)
s R h /2 xz Rs xs z 2 dz. Ry h /2 yz
(35)
The components used in Eq.(30) are given as
A , B , D , E , F , H Q 1, z, z , z , z , z dz ( A , D , F ) Q (1, z , z )dz for i, j 4and 5. h /2
ij
ij
ij
s ij
s ij
s ij
ij
ij
ij
h /2
h /2
h /2
2
2
3
4
6
ij
4
ij
13
for i, j 1, 2and 6,
(36) (37)
4.2. Stiffness matrices The total potential energy of the plate is expressed as Π U p W,
(38)
where U p denotes the strain energy of the plate and W is the work done by the external forces. The strain energy of the plate can be written as Up
1 ε T S ε d , 2
(39)
where ε (0) A B E (1) B D F ε ε ε (3) , S E F H γ (0) 0 0 0 (2) 0 0 0 γ
0 0 0 s
A Ds
0 0 0 . Ds F s
(40)
Considering the plate under a general lateral distributed load of q ( x , y ) , W can be written as
W q( x, y ) wd .
(41)
In the case of vibration analysis, W in Eq. (38) is replaced with plate kinetic energy, K, and expressed as
K
h /2 1 ( z )(u 2 v 2 w 2 )dzd . 2 h /2
(42)
Invoking the isoparametric concept, the field variables in the domain of the plate can be approximated as
u , h
b b
, d , i 1
i
(43)
i
h where {u } is the vector of field variables to be approximated and is represented as
u h u0
v0
w0 x y x y ; i , is the simpler notation for the NURBS shape T
p, q functions, Ri , , with designated degrees of continuity, p and q, for each parametric direction
14
and {di } is the vector of nodal degrees of freedom, d i u0 i
v0 i
w0i
xi yi xi yi , T
associated with the control point i in the control net. Substituting Eq. (43) into Eq. (38) and then taking variation, the following system of linear equation for the bending analysis of the plate is derived and expressed as
KU F.
(44)
It should be noted that U in Eq. (44) is composed of all instances of {d I } in the control net, that is U d1 d 2 d I
dn .
(45)
T
Furthermore, the eigenvalue equation of the free vibration analysis of the plate is obtained as
(K 2M)U 0,
(46)
where is the natural frequency of the plate and {U} is the vector containing the corresponding mode shapes. [ K ] , [ M ] and {F } are the global stiffness matrix, the global mass matrix and the global force vector, respectively which are obtained by assembling their corresponding element counterparts denoted by superscript ‘e’, given by
[Ke ] [Kem ] [Kbe ] [Kce ] [Kes ],
(47)
[K em ]ij e (Bim )T AB mj d,
(48)
[K be ]ij e (Bbi 1 )T DBbj1d e (Bbi 2 )T HBbj 2d
(49)
e (B ) FB d e (Bbi 2 )T FBbj1 d,
b1 T i
b2 j
[K ] e (B ) B B d e (Bbi 1 )T B B mj d e c ij
b1 j
m T i
(50)
e (B ) EB d e (Bbi 2 )T EB mj d,
b2 j
m T i
[Kes ]ij e (Bis1 )T AsBsj1d e (Bis 2 )T FsBsj2d
(51)
e (B ) D B d e (B ) D B d,
s1 T i
s
s2 j
s2 T i
s
s1 j
ˆ T ρH ˆ d, [M e ]ij e H i j
(52)
{Fe }i e qNTi d ,
in which
15
i , x 0 B 0 i , y i , y i , x
0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0
m i
0 0 0 i , x B 0 0 0 0 0 0 0 i , y
0 0 0 0 , 0 0
0
i , y i , x
b1 i
0 0 0 0 0 i , x B c 0 0 0 0 0 0 0 0 0 0 0 i , y b2 i
0 0 i, x Bis1 0 0 i, y 0 0 0 Bis 2 3c 0 0 0
(53)
(54)
0 i , y , i , x
(55)
i
0 0 0 , 0 i 0 0 0 0 i 0 , 0 0 0 i
(56) (57)
i 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 i 0 0 0 0 ˆ 0 0 0 0 0 0, H i i 0 0 0 0 i 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 i
0 I0 0 I 0 0 0 ρ I1 0 0 I1 cI 3 0 0 cI 3
0 0 I0 0
I1 0 0 I2
0 I1 0 0
cI 3 0 0 cI 4
0 0 0
0 cI 4 0
I2 0 cI 4
0 c I6 0
Ni 0 0 i
2
(58)
, cI 4 0 c 2 I 6 0 cI 3 0 0
(59)
0 0 0 0 .
(60)
The components used in [ρ] are defined as
Ii
h /2
h /2
z dz (i 0,1, 2,3, 4,6). i
(61)
16
5. Essential boundary conditions 5.1. Conventions Here, the boundary conditions used in studying the mechanical behavior of CNT-reinforced composite plates are presented as follows. Simply supported (S):
For upper and lower edges: u w x x 0,
(62)
For obliqueedges: v w y ' y ' 0.
(63)
Clamped (C):
For upper and lower edges: u v w x y x y 0,
(64)
For obliqueedges: u v w x ' y ' x ' y ' 0.
(65)
For free edges, no kinematic constraints will be imposed. The support of each edge of the plate will be denoted by S, C or F. In order to fully define the boundary conditions of a plate, these symbols will be written in sequence started from the bottom edge and will be continued to other edges in a counter clock wise direction. It should be mentioned that, following the approach used in [42] for bending analysis of skew plates, corner control points are considered to belong to the oblique edges. 5.2. Skew boundaries For the case of skew plates, in order to impose the appropriate boundary conditions on the nodes laid on the oblique edges of the plate it is necessary to introduce the oblique local coordinate system ( x y ) shown in Fig. 2. The relationships between position vectors, rotations and partial derivatives
in the global xy coordinate and the local x y coordinate are written as [48] x x sin( ) y, y cos( ) y,
(66)
x cos( ) x , y sin( ) x y ,
(67)
() / x () / x, () / y () / x tan( ) () / y sec( ),
(68)
where variables with prime symbol, ()׳, denote those obtained in the local x y coordinate system. Using the relations presented in Eqs. (66) to (68), and carrying out some manipulation, the expressions for x and y in terms of variables in an oblique coordinate system can be derived as
x cos( ) 1 x x ,
(69)
17
y tan( ) sin( ) x 1 sec( ) y tan( ) x sec( ) y.
(70)
Furthermore, the relationship between control variables in the global xy coordinate and the local x y coordinate for a control point I located on the oblique edge is written as d (I xy ) A xxyy d (I x y ) ,
(71)
where the transformation matrix, [ A xyxy ] , can now be constructed using Eqs. (66) to (70) as follows
A xy xy
1 sin( ) 0 cos( ) 0 0 0 0 0 0 0 0 0 0
. 0 1 0 0 sin( ) 0 cos( ) 1 0 1 0 0 tan( ) sin( ) 1 sec( ) tan( ) sec( )
0 0 1 0
0 0 0 cos( )
0 0 0 0
0 0 0 0
0 0 0 0
(72)
During the assembly process, only for those elements that have control points laid on oblique edges, element stiffness matrix, [K e ] , element mass matrix, [M e ] and element force vector, {F e} are transformed according to Eqs. (73), (74) and (75), respectively [64]. K e ( A e )T K e A e , M e ( A e )T M e A e ,
(74)
F e ( A e )T F e ,
(75)
(73)
in which the element transformation matrix, [Ae ] , is a diagonal matrix represented in Eq. (76), formed by placing either [ A xyxy ] or [I ] on its diagonal, depending on whether the corresponding
{dI } is related to the control point on the oblique edges. [I ] is the identity matrix of the same size as [A xy xy ] in Eq. (72). A xy x y Ae
I
0 I
0
A xy xy
. I
(76)
18
6. Numerical studies In this section, the capabilities of the IGA, as a numerical tool to be used in problems of skew plates, are first examined. Then, the bending and free vibration responses of CNT-reinforced composite skew plates as one-ply composite plates, are fully investigated. Throughout this study, several non-dimensional parameters have been used and it is convenient to introduce them here for future reference. For isotropic plates, the non-dimensional deflection, non-dimensional moment and nondimensional vibration parameters are defined in Eqs. (77), (78) and (79), respectively.
Wiso* wc 1600 D / (qa 4 ),
(77)
* M iso ( M c ) 40 / (qa 4 ),
(78)
* iso ( b 2 / 2 ) h / D ,
(79)
where D is the flexural rigidity of the plate and is defined as D Eh3 /12(1 2 ) . Moreover, subscript ‘c’ is used for parameters evaluated at the center point of the plate. For CNT-reinforced composite plates, the non-dimensional deflection and non-dimensional vibration parameter are defined in Eqs. (80) and (81), respectively.
W * wc / h,
(80)
( b 2 / h ) m / E m .
(81)
6.1. Bending analysis 6.1.1. Convergence and validation study Here we validate the accuracy and convergence of our IGA approach for the bending analysis of skew plates. The obtained results will be used in the sections that follow. We consider a rhombic plate (a=b) with the skew angle of shown in Fig. 2. Convergence studies are performed for two * different skew angles of 0 and 60 degrees; the non-dimensional deflections, Wiso , at the center of the
isotropic plates using NURBS functions of different degrees are obtained and shown in Figs. 4 and 5. As can be seen in Fig. 4 while the results obtained are convergent, higher rates of convergence can be achieved using NURBS of a higher degree. Moreover, as shown in Fig. 5, it is inevitable that a higher degree of NURBS functions will be utilized as skew angles become larger [65], [66], [67].
19
Having performed the numerical experiment, we have proved that quantic NURBS functions (i.e. NURBS with a continuity of degree four) perform optimally for the analysis of skew plates and, as such, they will be used in this study. The initial knot vectors used for both and directions are given as {0, 0, 0, 0, 0,1,1,1,1,1} , {0, 0, 0, 0, 0,1,1,1,1,1} , respectively. For knot insertion, the * knots are inserted in equally spaced manner. Moreover, values for non-dimensional deflection, Wiso , * and non-dimensional moments, M iso , at the center of the skew isotropic plates with skew angles of
30˚, 45˚ and 60˚ subjected to uniformly distributed loads are presented in Table 1. For comparison the results available in the literature are also provided in Table 1. As can be seen, the obtained results agree well with those previously reported for both simply supported and fully clamped boundary conditions. Based on the results listed in Table 1, the plates discretized using 19×19 elements will be utilized in all the analysis to be carried out in the following sections. Numerical integrations have been performed using a 4×4 gauss scheme within each element. 6.1.2. CNT-reinforced composite plates A comprehensive numerical analysis to investigate the mechanical behavior of CNT-reinforced composite plates is conducted in this study. The chosen matrix material is Poly-co-vinylene referred to as PmPV. The reinforcement material is comprised of armchair (10,10) SWCNTs. The properties of these two mentioned materials at room temperature are listed in Table 2 as reported in [22]. Three values of CNT volume fraction are considered here: 0.11, 0.14 and 0.17. The CNT efficiency parameters, i , (i 1, 2,3) , corresponding to each CNT volume fraction are given in Table 3. It is assumed that 2 3 and G12 G13 G23 , following the assumptions made in [22]. The thickness of the plate is set to be h=2 mm for all examples presented. We consider a CNT-reinforced composite skew plate with equilateral sides (a/b=1) under a uniformly distributed load q0 . In order to explore the effect of skew angle, a bending analysis of plates with varying skew angles has been performed. Three values of CNT volume fraction (0.11, 0.14 and 0.17), three different side to thickness ratios (i.e. 10, 20 and 50) and four types of CNT distribution (UD, FG-V, FG-O and FG-X) are considered. The results for non-dimensional central deflection, W * , are listed in Table 4. For comparison, results from [23], obtained for square plates based on first order shear deformation theory, are also included. Once again, the formulation presented in section 4 is proved to provide highly accurate results. The small discrepancies between the obtained results from IGA and those of [23] are due mainly to the assumed third-order and firstorder displacement fields used in HSDT and FSDT, respectively. 20
6.1.3. Optimum CNT fiber orientation angle This section aims to determine the optimum CNT orientation angle for CNT-reinforced composite plates, and to consider whether the values for non-dimensional deflection presented in Table 4 are the best achievable values. Usually, the degree of skewness of the plate is specified according to design geometric constraints but the type of material as well as its orientation can be chosen based on analyses related to the ultimate requirement of the structure. In order to show the effect of CNT orientation angle on the bending behavior of skew and non-skew plates, numerical investigations have been carried out. We consider a rhombic plate composed of a single-ply CNTreinforced composite, as shown in Fig. 2. The angle between the principal material coordinates and the problem global coordinates, , is varied between -90 and +90 degrees. The obtained values for the deflection of the corresponding CNT-reinforced composite skew plate are obtained for four types of CNT distribution and are plotted in Fig. 6. It can be seen that in the case of the square plate (i.e. 0 ), the right and left halves of the obtained plot are perfectly symmetrical with respect to 0 . In this case, the CNT orientation angle values equal to either 0 or 90 degree rotations give the lowest deflection values for both simply supported and clamped boundary conditions while for 45 , the highest deflection values are obtained. As the skew angle is increased, the plate loses its symmetricity, and the left half of the graph (in the case of a rhombic plate) gradually vanishes. Moreover, the peak points of the plots tend to appear at the lower values of . Upon inspection, it can be deduced that, in the cases of both simply supported and fully clamped boundary conditions, for the CNT-reinforced composite rhombic plate of skew angle , the lowest values of deflection can be achieved with the CNT orientation angle of (i.e. the minimum values of deflection are achieved when CNTs are aligned perpendicular to the oblique side of the skew plate). Furthermore, the highest values of deflection occur when the CNTs are aligned with the main diagonal of the plate. To further clarify the effect of CNT orientation on the bending response of CNT-reinforced composite skew plates, the values listed in Table 4 are regenerated in Table 5 incorporating the CNT orientation angles . The results listed in Table 5 show that by optimally aligning the CNT fibers embedded in the CNT-reinforced composite lamina, the non-dimensional deflection parameters are reduced at least by the factors of 1.1, 1.4, 2 and 2.5 for the cases of 15˚, 30˚, 45˚ and 60˚ skew angles, respectively.
21
* 0.11 , b/h=50 Furthermore, an extreme case with a reduction factor of 7.3 (for the parameters VCNT
and FG-X) is also achievable simply by changing the direction of CNT orientation. In order to attain a better understanding of the effect of changing the CNT orientation angle, the deflection profiles of the simply supported UD CNT-reinforced composite rhombic skew plates under uniform loading of q=-0.1 MPa along the x- and y-directions of the local coordinate system placed at the center of the skew plates are depicted in Fig. 7. Two different CNT orientation scenarios are applied: first CNT-reinforced composite plates with CNTs oriented along the global x-axis ( 0 ), as shown in Figs. 7(a)-(b); and second, CNT-reinforced composite skew plates with CNT orientation angles , as shown in Figs. 7(c)-(d). As can be seen, the CNT-reinforced composite skew plates experience a dramatic change in the way they deform under the same loading and boundary conditions, in the two cases. Next, the effect of varying the aspect ratio of the plate, a/b, on the bending behavior of the plate is investigated. For this case study, we have considered the length of the oblique edges, b, to be fixed and the length of the other two parallel edges, a, will be changed. The uniform CNT distribution with volume fraction 0.14 is chosen. The plate width-to-thickness ratio, b/h, is set to be 50. The results for plates with skewness values of 0˚, 30˚, 45˚ and 60˚ are illustrated in Fig. 8. As can be seen in Fig. 8, for the case of rectangular plate (i.e. 0 ), while the graphs maintain their symmetricity with respect to the CNT orientation angle 0 for all aspect ratios, it is noticeable that as the aspect ratios are increased, in contrast to the case of a/b=1, the CNT orientation 0 , no longer results in the minimum deflection of the plate. In fact, it can be observed that for all plates regardless of their skewness angle and their aspect ratios (except a/b=0.5), CNT orientation angle 90 gives the minimum deflection. In other words, in order to obtain the lowest deflections for CNT-reinforced composite skew plates with aspect ratios larger than 1, CNT fibers should be oriented perpendicular to the larger edge of the plate. Moreover, for aspect ratios larger than 1.5, the maximum deflections of the plates occurs when the CNT orientation angle is 0 . In order to comprehend the effect of CNT orientation angle on the deflection of the CNTreinforced composite skew plates, the influence of width-to-thickness ratio, b/h, should also be explored. In this case, a rhombic plate with a skewness angle of 45 degrees, composed of CNT* 0.14 , is chosen. Width-toreinforced composites with uniform CNT distribution and VCNT
thickness ratios are selected to be 5, 10, 20, 50 and 100. The results for normalized non-dimensional 22
deflection, W ** , defined as non-dimensional central deflection divided by non-dimensional central deflection for the case with CNT orientation angle 90 are plotted against CNT orientation angle in Fig. 9. It can be clearly noted that the greater the width-to-thickness ratio of the plate the more pronounced the CNT orientation effect.
6.2. Vibration analysis In this section, the vibration behavior of skew plates, specifically those comprising CNTreinforced composites, is investigated. First, the performance of the NURBS-based isogeometric method is verified and then parameters affecting the natural vibration response of the CNTreinforced composite plates such as CNT orientation angle, plate width-to-thickness ratio, CNT volume fractions and distributions are thoroughly explored. 6.2.1. Convergence and validation study A convergence study on the vibration of thin and thick isotropic skew plates with simply supported boundary condition on all edges has been carried out. Obtained values for the first eight * non-dimensional frequency parameters, iso , are listed in Table 6. It can be seen that the results for
all skew angles and plate thickness-to-width ratios agree well with those available in the literature. It can be deduced that the results obtained with the use of 19×19 elements are fully converged and thus this number of elements will be used in the case studies to be followed. 6.2.2. CNT-reinforced composite plates Having validated our isogeometric approach for analyzing the vibration of skew plates, in this section we further study the dynamic behavior of CNT-reinforced composite skew plates. The material properties are the same as those presented in section 6.1.2, used in the study concerning the bending of CNT-reinforced composite skew plates. We begin by ascertaining the optimum CNT orientation angle for UD and CNT-reinforced composite plates of different skewness angles. We consider a CNT-reinforced composite plate with a skewness angle of . The non-dimensional natural frequency parameters, , versus the CNT orientation angles, , which vary between 90 and + 90 , are plotted for skew angles
0 , 15 , 30 , 45 and 60 for simply supported and clamped boundary conditions in Fig. 10. In
23
* 0.14 ) are each case, four types of CNT distribution with a fixed CNT volume fraction (i.e. VCNT
considered. The width-to-thickness ratio is set to 50. As can be seen in Fig. 10, in the case of square plates (i.e. 0 ), the plots for both boundary conditions consist of two symmetric valleys. The valleys are split by the point named dividing point. As the skew angle increases this point moves toward the negative CNT orientation angle. Furthermore, the valley on the right becomes deeper while the left side valley becomes shallower and eventually changes direction and turns into a hill. Generally, these developments are faster for simply supported cases than for clamped boundary conditions. In all cases, as long as we have two distinct valleys, the CNT orientation angle corresponding to the dividing point between valleys can be considered as the point with Highest Achievable Frequency (HAF). This point changes its location to the peak of the left hill for 15 and 45 with simply supported and fully clamped boundary conditions, respectively. Moreover, the CNT orientation angle corresponding to the lowest point in the right valley can be understood as the point with the Lowest Achievable Frequency (LAF). For future references, the CNT orientation angles corresponding to HAF and LAF for CNTreinforced composite plates of different skew angles are further tabulated in Table 7. As we will see later, the values presented in Table 7 for simply supported boundary conditions are valid only for plates with width-to-thickness ratios greater than 20. In addition, it is noticeable from Figs. 10 that in all cases each CNT distribution has made its own frequency level and higher natural frequencies can be achieved by choosing a CNT distribution from FG-O to FG-V, FG-U and FG-X, respectively. In other words, FG-O and FG-X CNT distributions form the lower and upper bounds for the natural frequencies of a specific CNT-reinforced composite skew plate, respectively. From this conclusion, and using the values presented in Table 7, the first six frequency parameters for LAF and HAF corresponding to FG-O and FG-X CNT distributions with different volume fractions for CNT-reinforced composite plates with skew angles 0 , 15 , 30 , 45 and 60˚ with varying width-to-thickness ratios are presented in Table 8-12, respectively. As can be seen, the greater the CNT volume fraction, the higher the stiffness of the plate structure and as the result the higher the natural vibration frequency parameters. To further explain the trends in Tables 8-12, we define a non-dimensional frequency ratio, RHL , as the ratio of HAF to LAF for a particular CNT distribution and boundary condition. By inspecting Tables 8-12, it can be realized that, for simply supported plates, as the skew angle or the width-to-
24
thickness ratio increases the frequency ratio, RHL , also increases. Moreover, although the frequency parameter values (and as a result the frequency ratio values) are amplified for the clamped boundary condition, they nevertheless follow the same fashion as that for the simply supported boundary condition. In addition, according to Tables 8-12, it can be understood that when a first frequency ratio is more than 1, this does not necessarily mean that its subsequent modes have the same frequency ratio. Moreover, subsequent frequency ratios will not follow the same trend as for the first frequency ratio when the CNT distribution or boundary conditions are changed. Furthermore, one can notice that, for the case of simply supported square FG-O CNT-reinforced * 0.11 , b/h=20, the HAF is slightly less than that of the corresponding composite plate with VCNT
LAF. We will explain this issue in more detail in the descriptions related to Fig. 14. As an interesting case, the effect of CNT orientation angle on the vibration response of cantilever CNT-reinforced composite rhombic plates is also explored here. The plate is clamped on its bottom edge (i.e. the boundary conditions are denoted as CFFF). The material properties are the same as those used for the case of the simply supported and clamped CNT-reinforced composite plates. The obtained results are shown in Fig. 11. In terms of CNT distribution type, the graphs show a similar trend to what has been exhibited by CNT-reinforced composite plates with the simply supported and clamped boundary conditions (i.e. the FG-O and FG-X CNT distributions form the lower and upper bounds for the frequency range of the CNT-reinforced composite plates, respectively). According to Fig. 11, the values corresponding to HAF and LAF for cantilever CNT-reinforced composite rhombic plates are extracted and listed in Table 7, and will be used hereafter to obtain the corresponding mode shapes. Next, we investigate the effect of the aspect ratio of a CNT-reinforced composite skew plate on obtaining the optimum CNT fiber orientation angle for its dynamic behavior. For this approach, UD CNT-reinforced composite skew plates of different skew angles (i.e. 0 , 30 , 45 and 60 with varying aspect ratios: a / b 1, 1.2, 1.4, 1.5, 1.6, 1.8 and 2) are examined. The width-to-thickness ratio and CNT volume fraction are set to 50 and 0.14, respectively. Plots for natural frequency parameters versus corresponding CNT orientation angles are shown in Fig. 12. As can be observed in Fig. 12, for all cases, the higher the aspect ratios, the lower the natural frequency parameters. Moreover, for all aspect ratios higher than 1 (i.e. non-rhombic skew plates), the highest natural frequency parameter corresponds to the CNT orientation angle of 90 . Moreover, the CNT orientation angle value for the LAF is somewhere between zero and the
25
corresponding value for the case of a/b=1 for each skew angle. As the aspect ratio increases this point moves closer to zero and in fact, for aspect ratios larger than a/b=1.6 the point coincides with the CNT orientation angle 0 . To further continue our study, we explore the effect of width-to-thickness ratio on the optimum CNT orientation angle for the vibration of CNT-reinforced composite skew plates. A UD CNTreinforced composite rhombic plate of skew angle 45 with the varying width-to-thickness ratios of b / h 5, 10, 20, 50 and 100 with simply supported, clamped and cantilever boundary conditions is considered. The obtained results for frequency parameters are normalized with respect to the corresponding highest frequency parameter (presented in Table 7 for each boundary condition) and plotted against the changing CNT orientation angles from 90 to + 90 in Figs. 13. From Fig. 13, the following points can be expressed. First, as the width-to-thickness ratio is increased, the necessity to select the correct CNT orientation becomes crucial. Second, while the CNT orientation angle value corresponding to HAF for all width-to-thickness ratios remains the same, this is not necessarily the case for the corresponding value for LAF. To be exact, it should be pointed out that according to Fig. 13(a) the values presented in Table 7 for the case of the simply supported boundary condition are valid for width-to-thickness ratios larger than 20. From Figs. 13(b)-(c), it is clear that this limitation exists for neither the clamped nor the cantilever boundary conditions. Here, we will clarify the issue introduced in Table 8. Graphs of the non-dimensional vibration parameter versus CNT orientation angle, normalized with values of the vibration parameter evaluated at 0 , for the square FG-O CNT-reinforced composite plate with two values of CNT * * 0.11 and VCNT 0.14 ), are plotted in Figs. 14. As can be seen in Figs. volume fraction (i.e. VCNT * 0.11 ) for a simply supported square FG14(a)-(b), in the case of low CNT volume fractions ( VCNT
O CNT-reinforced composite plate, the variation of normalized vibration parameters is less than 2 percent, which is of minor importance in our study. Moreover, in this case the CNT-reinforced composite plate does not have the same response as described previously. However, as the CNT volume fraction or skewness angle of the plate is increased (see Fig. 14 and Fig. 13), or more robust CNT distributions (UD, FG-V or FG-X) are used the response of the plate gradually becomes closer to that presented earlier. Furthermore, from Figs. 14(c)-(d), it is notable that, for the clamped boundary condition, the response of the CNT-reinforced composite plate to the change in CNT orientation angle is independent of the CNT volume fraction value.
26
In Fig. 15, the first eight free vibration modes of clamped rhombic UD CNT-reinforced composite plates with skew angles of 0 , 15 , 30 , 45 and 60 are presented. Geometrical parameters are set to a/b=1, b/h=50. The CNT volume fraction of 0.14 is selected and the CNT orientation angles correspond to HAF. As can be noted, the mode shapes of skew plates are not necessarily the same due to their geometrical and CNT orientation angle differences. Moreover, as a comparison to mode shapes obtained with 0 reported in [58], here the mode shapes are aligned along the orientation of the CNTs. The first eight mode shapes of cantilever CNT-reinforced composite skew plates are presented in Fig. 16. All the parameters used are the same as for Fig. 15. It should be mentioned that, because of the looser constraints of the cantilever boundary condition, the mode shapes are not necessarily aligned with their corresponding CNT orientation angles. 7. Conclusion Bending and free vibration analyses of CNT-reinforced composite skew plates based on Reddy’s higher order shear deformation theory using the isogeometric method have been presented. The effect of CNT orientation angle on the static and dynamic behavior of CNT-reinforced composite plates has been investigated. The optimum CNT orientation angle for plates with different geometrical properties and CNT distributions is presented for simply supported, clamped and cantilever boundary conditions. It has been shown that the CNT orientation angle has a noticeable influence on the mechanical behavior of CNT-reinforced composite skew plates.
Acknowledgements The work described in this paper was fully supported by grants from the National Natural Science Foundation of China (Grant No. 11402142 and Grant No. 51378448) and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9042047, CityU 11208914).
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32
Tables
Table 1 Convergence and accuracy studies of the skew plates subjected to uniformly distributed loads (h/a=0.01). Simply Supported Skew References Angle ( ) 30˚ IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] [37] [38] DQM [42] DQM [45]
* iso
* iso max
Clamped * iso min
* iso
* ( M iso ) max
* ( M iso ) min
W
(M )
(M )
W
4.1086 4.1107 4.1121 4.1079 4.1123 4.2824 4.10 4.096 4.1054
1.7033 1.7037 1.7040 1.6909 1.7075 1.7455 1.70 1.701 1.7006
1.3355 1.3361 1.3366 1.3267 1.3391 1.3670 1.33 1.332 1.3355
1.2335 1.2335 1.2335 1.2419 1.2335 1.2275
0.7914 0.7914 0.7914 0.7838 0.7916 0.7808
0.6177 0.6176 0.6176 0.6120 0.6175 0.6023
1.2336
0.7917
0.6177
45˚
IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] DQM [42] [38]
2.1144 2.1199 2.1239 2.1285 2.1330 2.2028 2.1204 2.107
1.2924 1.2943 1.2957 1.2892 1.2995 1.3258 1.2941 1.290
0.8794 0.8818 0.8836 0.8787 0.8866 0.9008 0.8872 0.876
0.6051 0.6051 0.6051 0.6087 0.6051 0.6032
0.5777 0.5776 0.5776 0.5718 0.5776 0.5710
0.3903 0.3901 0.3901 0.3850 0.3900 0.3803
60˚
IGA-15×15 Elements IGA-17×17 Elements IGA-19×19 Elements FEM [40] Mindlin Element A [40] Mindlin Element B [40] [37] [38] DQM [42] DQM [45]
0.6509 0.6552 0.6583 0.6587 0.6690 0.6841 0.653 0.6523 0.6556
0.7605 0.7635 0.7658 0.7628 0.7734 0.7853 0.76 0.762 0.7667
0.4331 0.4365 0.4391 0.4340 0.4481 0.4483 0.43 0.434 0.4428
0.1742 0.1743 0.1743 0.1746 0.1743 0.1737
0.3214 0.3212 0.3211 0.3175 0.3209 0.3172
0.1819 0.1817 0.1816 0.1779 0.1821 0.1748
0.1743
0.3213
0.1817
* * * Wiso wc 1600 D / ( qa 4 ),( M iso ) max ( M c ) max 40 / ( qa 4 ),( M iso ) min ( M c ) min 40 / ( qa 4 )
31
Table 2 Elastic properties of the (10,10) SWCNT and the polymer matrix [22]. SWCNT
Polymer matrix
CNT E11 5.6466(TPa)
E m 2.1(GPa)
CNT E22 7.0800(TPa)
m 1150 (kg/m3 )
CNT G12 1.9447 (TPa)
m 0.34
CNT 1400 (kg/m3 ) CNT 12 0.175
Table 3 CNT efficiency parameters for different values of volume fractions [22]. * VCNT 2 1 3 0.11
0.149
0.934
0.934
0.14
0.150
0.941
0.941
0.17
0.149
1.381
1.381
32
Table 4 Non-dimensional central deflection, W * wc / h , for simply supported FG-CNTRC plates with skewness of under uniform load of q=-0.1 MPa and CNT orientation angle 0 (a/b=1). 0
15
30
45
60
U FG-V FG-O FG-X
0 Ref. [23] 3.739×10-3 4.466×10-3 5.230×10-3 3.177×10-3
3.718×10-3 4.452×10-3 5.442×10-3 3.140×10-3
3.625×10-3 4.315×10-3 5.250×10-3 3.070×10-3
3.272×10-3 3.823×10-3 4.580×10-3 2.797×10-3
2.486×10-3 2.805×10-3 3.272×10-3 2.167×10-3
1.286×10-3 1.381×10-3 1.552×10-3 1.159×10-3
20
U FG-V FG-O FG-X
3.628×10-2 4.879×10-2 6.155×10-2 2.701×10-2
3.630×10-2 4.886×10-2 6.258×10-2 2.695×10-2
3.577×10-2 4.775×10-2 6.082×10-2 2.670×10-2
3.342×10-2 4.336×10-2 5.422×10-2 2.539×10-2
2.684×10-2 3.293×10-2 3.979×10-2 2.118×10-2
1.437×10-2 1.628×10-2 1.880×10-2 1.209×10-2
50
U FG-V FG-O FG-X
1.155 1.653 2.157 7.900×10-1
1.157 1.656 2.167 7.912×10-1
1.144 1.623 2.111 7.875×10-1
1.083 1.486 1.896 7.618×10-1
8.883×10-1 1.142 1.402 6.567×10-1
4.799×10-1 5.620×10-1 6.569×10-1 3.852×10-1
10
U FG-V FG-O FG-X
3.306×10-3 3.894×10-3 4.525×10-3 2.844×10-3
3.273×10-3 3.870×10-3 4.694×10-3 2.824×10-3
3.198×10-3 3.763×10-3 4.548×10-3 2.764×10-3
2.908×10-3 3.367×10-3 4.021×10-3 2.526×10-3
2.242×10-3 2.516×10-3 2.940×10-3 1.969×10-3
1.184×10-3 1.269×10-3 1.436×10-3 1.066×10-3
20
U FG-V FG-O FG-X
3.001×10-2 4.025×10-2 5.070×10-2 2.256×10-2
2.992×10-2 4.017×10-2 5.142×10-2 2.255×10-2
2.959×10-2 3.943×10-2 5.024×10-2 2.238×10-2
2.797×10-2 3.633×10-2 4.560×10-2 2.144×10-2
2.303×10-2 2.838×10-2 3.458×10-2 1.819×10-2
1.283×10-2 1.459×10-2 1.704×10-2 1.072×10-2
50
U FG-V FG-O FG-X
9.175×10-1 1.326 1.738 6.271×10-1
9.159×10-1 1.323 1.740 6.263×10-1
9.097×10-1 1.303 1.706 6.251×10-1
8.729×10-1 1.213 1.562 6.108×10-1
7.387×10-1 9.632×10-1 1.199 5.394×10-1
4.194×10-1 4.960×10-1 5.888×10-1 3.307×10-1
10
U FG-V FG-O FG-X
2.394×10-3 2.864×10-3 3.378×10-3 2.012×10-3
2.381×10-3 2.863×10-3 3.476×10-3 2.027×10-3
2.321×10-3 2.773×10-3 3.355×10-3 1.978×10-3
2.092×10-3 2.450×10-3 2.931×10-3 1.793×10-3
1.586×10-3 1.789×10-3 2.098×10-3 1.376×10-3
8.169×10-4 8.748×10-4 9.946×10-4 7.271×10-4
20
U FG-V FG-O FG-X
2.348×10-2 3.174×10-2 4.020×10-2 1.737×10-2
2.350×10-2 3.182×10-2 4.071×10-2 1.750×10-2
2.315×10-2 3.106×10-2 3.957×10-2 1.731×10-2
2.158×10-2 2.811×10-2 3.528×10-2 1.638×10-2
1.726×10-2 2.119×10-2 2.588×10-2 1.351×10-2
9.175×10-3 1.038×10-2 1.221×10-2 7.582×10-3
50
U FG-V FG-O FG-X
7.515×10-1 1.082 1.416 5.132×10-1
7.534×10-1 1.085 1.422 5.152×10-1
7.449×10-1 1.062 1.385 5.122×10-1
7.032×10-1 9.684×10-1 1.243 4.929×10-1
5.740×10-1 7.380×10-1 9.186×10-1 4.200×10-1
3.076×10-1 3.593×10-1 4.295×10-1 2.414×10-1
* VCNT
b/h
0.11
10
0.14
0.17
33
Table 5 Non-dimensional central deflection, W * wc / h , for simply supported FG-CNTRC plates with skewness of under uniform load of q=-0.1 MPa and CNT orientation angle (a/b=1). * VCNT 0.11
0.14
0.17
U FG-V FG-O FG-X
0 3.718×10-3 4.452×10-3 5.442×10-3 3.140×10-3
R 1.0 1.0 1.0 1.0
15 3.257×10-3 3.866×10-3 4.714×10-3 2.771×10-3
R 1.1 1.1 1.1 1.1
30 2.229×10-3 2.584×10-3 3.130×10-3 1.933×10-3
R 1.5 1.5 1.5 1.4
45 1.211×10-3 1.354×10-3 1.624×10-3 1.084×10-3
R 2.1 2.1 2.0 2.0
60 4.872×10-4 5.198×10-4 6.136×10-4 4.534×10-4
R 2.6 2.7 2.5 2.6
20
U FG-V FG-O FG-X
3.630×10-2 4.886×10-2 6.258×10-2 2.695×10-2
1.0 1.0 1.0 1.0
3.143×10-2 4.184×10-2 5.342×10-2 2.358×10-2
1.1 1.1 1.1 1.1
2.058×10-2 2.674×10-2 3.394×10-2 1.580×10-2
1.6 1.6 1.6 1.6
9.916×10-3 1.244×10-2 1.571×10-2 7.890×10-3
2.7 2.6 2.5 2.7
3.227×10-3 3.812×10-3 4.752×10-3 2.730×10-3
4.5 4.3 4.0 4.4
50
U FG-V FG-O FG-X
1.157 1.656 2.167 7.912×10-1
1.0 1.0 1.0 1.0
9.924×10-1 1.406 1.835 6.851×10-1
1.2 1.2 1.2 1.1
6.282×10-1 8.749×10-1 1.138 4.397×10-1
1.7 1.7 1.7 1.7
2.761×10-1 3.778×10-1 4.938×10-1 1.962×10-1
3.2 3.0 2.8 3.3
7.186×10-2 9.556×10-2 1.260×10-1 5.256×10-2
6.7 5.9 5.2 7.3
10
U FG-V FG-O FG-X
3.273×10-3 3.870×10-3 4.694×10-3 2.824×10-3
1.0 1.0 1.0 1.0
2.880×10-3 3.375×10-3 4.081×10-3 2.500×10-3
1.1 1.1 1.1 1.1
1.992×10-3 2.280×10-3 2.735×10-3 1.760×10-3
1.5 1.5 1.5 1.4
1.100×10-3 1.215×10-3 1.440×10-3 1.001×10-3
2.0 2.1 2.0 2.0
4.508×10-4 4.766×10-4 5.548×10-4 4.249×10-4
2.6 2.7 2.6 2.5
20
U FG-V FG-O FG-X
2.992×10-2 4.017×10-2 5.142×10-2 2.255×10-2
1.0 1.0 1.0 1.0
2.606×10-2 3.457×10-2 4.408×10-2 1.984×10-2
1.1 1.1 1.1 1.1
1.728×10-2 2.233×10-2 2.826×10-2 1.347×10-2
1.6 1.6 1.6 1.6
8.478×10-3 1.055×10-2 1.322×10-2 6.873×10-3
2.7 2.7 2.6 2.6
2.845×10-3 3.321×10-3 4.084×10-3 2.458×10-3
4.5 4.4 4.2 4.4
50
U FG-V FG-O FG-X
9.159×10-1 1.323 1.740 6.263×10-1
1.0 1.0 1.0 1.0
7.901×10-1 1.129 1.479 5.451×10-1
1.2 1.2 1.2 1.1
5.040×10-1 7.073×10-1 9.234×10-1 3.528×10-1
1.7 1.7 1.7 1.7
2.233×10-1 3.071×10-1 4.017×10-1 1.591×10-1
3.3 3.1 3.0 3.4
5.888×10-2 7.837×10-2 1.028×10-1 4.350×10-2
7.1 6.3 5.7 7.6
10
U FG-V FG-O FG-X
2.381×10-3 2.863×10-3 3.476×10-3 2.027×10-3
1.0 1.0 1.0 1.0
2.085×10-3 2.484×10-3 3.008×10-3 1.786×10-3
1.1 1.1 1.1 1.1
1.424×10-3 1.656×10-3 1.991×10-3 1.243×10-3
1.5 1.5 1.5 1.4
7.716×10-4 8.643×10-4 1.025×10-3 6.955×10-4
2.1 2.1 2.0 2.0
3.093×10-4 3.299×10-4 3.829×10-4 2.904×10-4
2.6 2.7 2.6 2.5
20
U FG-V FG-O FG-X
2.350×10-2 3.182×10-2 4.071×10-2 1.750×10-2
1.0 1.0 1.0 1.0
2.033×10-2 2.722×10-2 3.473×10-2 1.528×10-2
1.1 1.1 1.1 1.1
1.328×10-2 1.735×10-2 2.202×10-2 1.021×10-2
1.6 1.6 1.6 1.6
6.382×10-3 8.046×10-3 1.014×10-2 5.090×10-3
2.7 2.6 2.6 2.7
2.066×10-3 2.450×10-3 3.032×10-3 1.759×10-3
4.4 4.2 4.0 4.3
50
U FG-V FG-O FG-X
7.534×10-1 1.085 1.422 5.152×10-1
1.0 1.0 1.0 1.0
6.459×10-1 9.200×10-1 1.203 4.452×10-1
1.2 1.2 1.2 1.2
4.083×10-1 5.717×10-1 7.463×10-1 2.851×10-1
1.7 1.7 1.7 1.7
1.793×10-1 2.466×10-1 3.233×10-1 1.270×10-1
3.2 3.0 2.8 3.3
4.656×10-2 6.225×10-2 8.220×10-2 3.401×10-2
6.6 5.8 5.2 7.1
b/h 10
*The values listed in columns denoted with R show ratios of non-dimensional deflections obtained with 0 , presented in Table 4, to those with cases, presented here in Table 5.
34
Table 6 * Convergence study for first eight frequency parameters, iso (b 2 / 2 ) h / D , of a simply supported isotropic skew plate. h/b 1 2 3 4 5 6 7 8 0.001
0.2
0˚
N=17 N=19 N=21 N=23 Ref. [48]
2.0000 2.0000 2.0000 2.0000 2.0000
5.0004 5.0001 5.0000 5.0000 5.0000
5.0004 5.0001 5.0000 5.0000 5.0000
8.0004 8.0000 7.9999 7.9999 7.9999
10.0154 10.0045 10.0014 10.0005 9.9999
10.0154 10.0045 10.0014 10.0005 9.9999
13.0119 13.0033 13.0010 13.0002 12.9998
13.0119 13.0033 13.0010 13.0002 12.9998
30˚
N=17 N=19 N=21 N=23 Ref. [48]
2.5313 2.5298 2.5287 2.5279 2.5294
5.3340 5.3335 5.3334 5.3333 5.3333
7.2879 7.2831 7.2802 7.2782 7.2821
8.5043 8.4989 8.4970 8.4962 8.4966
12.4804 12.4550 12.4479 12.4456 12.4442
12.4881 12.4572 12.4487 12.4459 12.4442
14.2888 14.2666 14.2576 14.2527 14.2850
17.3513 17.2082 17.1670 17.1540 17.1471
60˚
N=17 N=19 N=21 N=23 Ref. [48]
6.7289 6.6952 6.6686 6.6470 6.7179
10.6612 10.6438 10.6381 10.6360 10.6354
15.2277 15.1131 15.0734 15.0566 15.0708
20.5791 20.1281 19.9748 19.9222 19.9000
21.7233 21.6171 21.5507 21.5027 21.6610
27.3839 26.0758 25.5705 25.3847 25.3953
30.3713 29.9869 29.8367 29.7792 29.7937
36.2693 33.3034 32.0024 31.4821 31.6761
0˚
N=17 N=19 N=21 N=23 Ref. [48]
1.7683 1.7683 1.7683 1.7683 1.7661
3.8692 3.8692 3.8692 3.8692 3.8580
3.8692 3.8692 3.8692 3.8692 3.8580
5.5980 5.5980 5.5980 5.5980 5.5737
6.6168 6.6168 6.6168 6.6168 6.5820
6.6168 6.6168 6.6168 6.6168 6.5820
8.0011 8.0011 8.0011 8.0011 7.9485
8.0011 8.0011 8.0011 8.0011 7.9485
30˚
N=17 N=19 N=21 N=23 Ref. [48]
2.1749 2.1745 2.1742 2.1739 2.1719
4.0769 4.0768 4.0767 4.0767 4.0637
5.2055 5.2047 5.2042 5.2037 5.1849
5.8601 5.8598 5.8596 5.8594 5.8321
7.7580 7.7577 7.7575 7.7573 7.7066
7.7580 7.7577 7.7575 7.7573 7.7066
8.5362 8.5352 8.5344 8.5338 8.4742
9.7075 9.7071 9.7068 9.7065 9.6240
60˚
N=17 N=19 N=21 N=23 Ref. [48]
4.7680 4.7597 4.7531 4.7478 4.7647
6.9286 6.9278 6.9271 6.9266 6.8841
8.8567 8.8543 8.8525 8.8510 8.7851
10.7408 10.7393 10.7383 10.7374 10.6296
11.1717 11.1637 11.1574 11.1522 11.0666
12.5679 12.5656 12.5639 12.5626 12.4167
13.9702 13.9684 13.9670 13.9659 13.7750
14.3836 14.3812 14.3796 14.3785 14.2005
35
Table 7 CNT orientations angle, , corresponding to HAF and LAF for FG-CNTRC skew plates of different skew angle, . Simply Supported (SSSS)
(degree)
Clamped (CCCC)
Cantilever (CFFF)
(degree) corresponding to LAF
HAF
0
45
15
40
30 45 60
25 30 10
0 -15 for FG-X -50 for all other dist. -60 -65 -75
LAF HAF
LAF HAF
45
0
0
90
35
-15
0
75
30 25 15
-35 -70 -75
-5 -20 -35
60 45 30
36
Table 8 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 0 ). Simply Supported * b/h Modes VCNT 0.11 20 1 2 3 4 5 6
LAF 13.392 23.423 35.152 35.645 49.535 50.544
FG-O FG-O HAF 13.380 18.295 29.549 38.955 38.955 42.609
RHL 1.00 0.78 0.84 1.09 0.79 0.84
Clamped
LAF 18.616 29.667 42.859 46.613 55.090 57.420
FG-X FG-X HAF 19.912 23.683 33.814 38.955 38.955 50.311
RHL 1.07 0.80 0.79 0.84 0.71 0.88
LAF 20.577 31.878 42.449 44.247 58.074 58.118
FG-O FG-O HAF 23.922 29.203 40.744 51.810 55.044 57.979
RHL 1.16 0.92 0.96 1.17 0.95 1.00
LAF 26.271 37.896 50.531 51.956 64.525 67.709
FG-X FG-X HAF 30.733 35.185 45.821 62.591 62.951 65.547
1.17 0.93 0.91 1.20 0.98 0.97
RHL
50
1 2 3 4 5 6
14.136 25.047 38.882 40.559 55.128 59.460
14.231 19.222 30.955 49.215 52.110 54.897
1.01 0.77 0.80 1.21 0.95 0.92
20.802 33.536 49.651 60.920 67.991 82.871
22.906 26.640 36.951 54.744 79.254 82.322
1.10 0.79 0.74 0.90 1.17 0.99
24.115 38.325 54.338 54.931 72.590 76.842
29.902 34.923 46.590 65.379 75.239 78.277
1.24 0.91 0.86 1.19 1.04 1.02
35.152 51.787 69.926 79.119 90.025 103.833
45.985 49.661 59.243 76.353 100.826 110.000
1.31 0.96 0.85 0.97 1.12 1.06
100
1 2 3 4 5 6
14.279 25.323 39.462 41.645 56.171 61.272
14.368 19.372 31.181 49.674 54.089 56.924
1.01 0.76 0.79 1.19 0.96 0.93
21.270 34.279 51.052 64.632 70.296 88.216
23.460 27.194 37.532 55.540 80.605 89.725
1.10 0.79 0.74 0.86 1.15 1.02
24.857 39.786 56.825 58.127 76.452 82.051
31.240 36.223 47.890 66.924 82.409 85.469
1.26 0.91 0.84 1.15 1.08 1.04
37.574 55.995 76.405 89.247 99.315 118.593
50.924 54.462 63.719 80.629 105.417 132.922
1.36 0.97 0.83 0.90 1.06 1.12
0.14 20
1 2 3 4 5 6
14.408 24.674 37.156 37.701 51.267 53.333
14.646 19.279 30.305 39.726 39.726 46.004
1.02 0.78 0.82 1.05 0.77 0.86
20.019 31.395 44.906 49.226 56.181 59.762
21.601 25.257 35.350 39.726 39.726 52.054
1.08 0.80 0.79 0.81 0.71 0.87
21.890 33.351 44.884 45.873 59.836 60.701
25.600 30.634 41.935 54.876 57.924 59.109
1.17 0.92 0.93 1.20 0.97 0.97
27.585 39.388 52.240 54.076 66.494 70.078
32.148 36.622 47.400 64.478 65.295 67.894
1.17 0.93 0.91 1.19 0.98 0.97
50
1 2 3 4 5 6
15.328 26.577 40.841 44.335 57.489 63.861
15.748 20.430 31.867 50.040 57.816 60.348
1.03 0.77 0.78 1.13 1.01 0.94
22.800 36.131 52.990 66.637 72.048 89.485
25.503 29.074 39.218 57.125 82.078 90.415
1.12 0.80 0.74 0.86 1.14 1.01
26.178 40.900 57.388 59.716 76.072 82.316
32.979 37.643 48.807 67.233 82.539 85.313
1.26 0.92 0.85 1.13 1.09 1.04
38.239 55.548 74.300 85.368 94.973 110.801
50.223 53.766 63.162 80.232 104.951 118.140
1.31 0.97 0.85 0.94 1.11 1.07
100
1 2 3 4 5 6
15.509 26.906 41.514 45.722 58.675 66.084
15.928 20.619 32.122 50.517 60.411 62.990
1.03 0.77 0.77 1.10 1.03 0.95
23.415 37.078 54.738 71.518 74.869 96.338
26.262 29.826 39.973 58.074 83.586 100.231
1.12 0.80 0.73 0.81 1.12 1.04
27.114 42.688 60.364 63.753 80.613 88.725
34.731 39.338 50.442 69.036 91.734 94.516
1.28 0.92 0.84 1.08 1.14 1.07
41.390 60.881 82.339 98.287 106.302 129.199
56.714 60.081 69.034 85.718 110.578 143.107
1.37 0.99 0.84 0.87 1.04 1.11
0.17 20
1 2 3 4 5 6
16.502 28.898 43.578 44.044 61.284 62.729
16.486 22.534 36.411 48.962 48.962 53.019
1.00 0.78 0.84 1.11 0.80 0.85
23.178 37.175 53.876 57.874 69.243 72.327
24.613 29.586 42.754 48.962 48.962 63.952
1.06 0.80 0.79 0.85 0.71 0.88
25.531 39.605 52.943 55.021 72.249 72.518
29.794 36.263 50.465 64.974 68.909 71.768
1.17 0.92 0.95 1.18 0.95 0.99
32.687 47.396 63.406 64.596 81.158 84.523
38.093 43.950 57.758 78.042 79.286 81.474
1.17 0.93 0.91 1.21 0.98 0.96
50
1 2 3 4 5 6
17.360 30.775 47.806 49.863 67.818 73.145
17.460 23.602 38.048 60.533 64.102 67.525
1.01 0.77 0.80 1.21 0.95 0.92
25.883 42.037 62.463 75.530 85.760 103.296
28.283 33.216 46.683 69.685 101.195 101.547
1.09 0.79 0.75 0.92 1.18 0.98
29.662 47.189 66.967 67.710 89.531 94.811
36.811 42.988 57.351 80.495 93.004 96.734
1.24 0.91 0.86 1.19 1.04 1.02
43.655 64.656 87.604 98.098 113.072 129.278
56.790 61.663 74.259 96.530 128.136 135.851
1.30 0.95 0.85 0.98 1.13 1.05
100
1 2 3 4 5 6
17.524 31.091 48.472 51.106 69.019 75.218
17.615 23.772 38.308 61.069 66.359 69.839
1.01 0.76 0.79 1.19 0.96 0.93
26.458 42.972 64.238 80.098 88.701 109.953
28.961 33.895 47.408 70.712 102.980 110.615
1.09 0.79 0.74 0.88 1.16 1.01
30.511 48.865 69.832 71.382 93.994 100.816
38.339 44.475 58.842 82.279 101.266 105.026
1.26 0.91 0.84 1.15 1.08 1.04
46.649 69.895 95.713 110.573 124.744 147.562
62.830 67.526 79.725 101.807 133.928 163.925
1.35 0.97 0.83 0.92 1.07 1.11
37
Table 9 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 15 ). Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 12.832 24.412 32.023 38.944 48.845 55.579
FG-O FG-O HAF 15.464 24.776 35.885 41.854 48.204 56.318
50
1 2 3 4 5 6
13.443 25.993 36.225 42.396 56.577 62.148
16.573 26.780 39.535 50.039 54.086 68.274
1.23 1.03 1.09 1.18 0.96 1.10
19.167 33.737 52.545 53.612 74.674 77.303
24.955 30.180 42.055 60.071 81.645 87.906
1.30 0.89 0.80 1.12 1.09 1.14
22.198 37.829 49.010 56.471 71.880 78.151
31.996 37.556 49.820 68.001 80.244 83.272
1.44 0.99 1.02 1.20 1.12 1.07
31.629 49.674 70.058 70.635 94.360 95.835
48.948 53.230 63.826 81.440 104.043 116.308
1.55 1.07 0.91 1.15 1.10 1.21
100
1 2 3 4 5 6
13.551 26.264 37.026 43.015 58.112 63.351
16.807 27.126 40.197 51.819 55.189 70.836
1.24 1.03 1.09 1.20 0.95 1.12
19.500 34.407 53.906 56.264 77.098 81.614
25.612 30.929 43.005 61.510 84.287 96.299
1.31 0.90 0.80 1.09 1.09 1.18
22.781 39.082 51.392 58.795 75.981 81.983
33.527 39.089 51.483 70.295 88.328 90.282
1.47 1.00 1.00 1.20 1.16 1.10
33.472 53.128 76.370 77.590 103.051 107.275
54.530 58.720 69.195 87.188 111.509 138.804
1.63 1.11 0.91 1.12 1.08 1.29
0.14 20
1 2 3 4 5 6
13.729 25.572 34.301 40.402 51.412 57.486
16.713 26.230 37.580 44.850 50.095 58.509
1.22 1.03 1.10 1.11 0.97 1.02
18.758 31.904 45.026 47.949 58.654 62.914
23.286 28.165 39.194 44.174 50.787 54.885
1.24 0.88 0.87 0.92 0.87 0.87
20.549 33.588 41.456 48.513 58.783 65.403
27.054 32.507 43.907 57.266 59.425 61.464
1.32 0.97 1.06 1.18 1.01 0.94
25.845 39.343 50.221 54.749 67.855 72.058
33.718 38.672 49.730 65.014 68.452 71.458
1.30 0.98 0.99 1.19 1.01 0.99
50
1 2 3 4 5 6
14.471 27.390 39.427 44.247 60.454 64.457
18.100 28.614 41.804 54.870 56.754 73.578
1.25 1.04 1.06 1.24 0.94 1.14
20.894 36.141 55.761 58.577 78.729 83.264
27.738 32.904 44.840 63.243 85.719 96.381
1.33 0.91 0.80 1.08 1.09 1.16
23.936 40.060 53.089 59.210 76.643 81.359
35.262 40.484 52.354 70.495 87.874 89.618
1.47 1.01 0.99 1.19 1.15 1.10
34.338 53.102 74.773 75.673 99.184 102.249
53.376 57.542 68.029 85.750 108.859 124.677
1.55 1.08 0.91 1.13 1.10 1.22
100
1 2 3 4 5 6
14.605 27.706 40.439 44.949 62.312 65.794
18.397 29.032 42.587 57.144 58.043 75.652
1.26 1.05 1.05 1.27 0.93 1.15
21.331 36.984 57.432 62.062 81.646 88.757
28.631 33.899 46.051 64.982 88.697 107.462
1.34 0.92 0.80 1.05 1.09 1.21
24.664 41.574 56.086 61.948 81.652 85.789
37.263 42.467 54.421 73.128 95.647 98.226
1.51 1.02 0.97 1.18 1.17 1.14
36.728 57.454 81.833 85.320 109.682 116.527
60.687 64.721 74.987 92.979 117.777 146.572
1.65 1.13 0.92 1.09 1.07 1.26
0.17 20
1 2 3 4 5 6
15.808 30.115 39.651 48.120 60.567 68.985
19.072 30.585 44.363 51.999 59.664 70.017
1.21 1.02 1.12 1.08 0.99 1.01
21.819 37.988 52.877 57.822 72.158 75.113
26.605 33.010 47.130 54.307 61.859 66.713
1.22 0.87 0.89 0.94 0.86 0.89
24.021 40.011 48.844 58.342 70.227 78.158
31.551 38.516 52.779 68.071 71.690 73.298
1.31 0.96 1.08 1.17 1.02 0.94
30.645 47.465 59.958 66.638 81.914 88.171
40.016 46.450 60.518 79.365 82.118 85.925
1.31 0.98 1.01 1.19 1.00 0.97
50
1 2 3 4 5 6
16.511 31.945 44.524 52.139 69.591 76.475
20.353 32.901 48.604 61.554 66.528 84.025
1.23 1.03 1.09 1.18 0.96 1.10
23.919 42.410 66.292 66.584 94.425 96.558
30.828 37.628 52.978 76.079 103.333 108.463
1.29 0.89 0.80 1.14 1.09 1.12
27.298 46.568 60.368 69.583 88.623 96.373
39.400 46.242 61.346 83.770 99.232 102.913
1.44 0.99 1.02 1.20 1.12 1.07
39.362 62.188 86.978 88.745 118.848 119.557
60.457 66.101 79.944 102.705 131.499 143.676
1.54 1.06 0.92 1.16 1.11 1.20
100
1 2 3 4 5 6
16.635 32.256 45.439 52.852 71.349 77.863
20.623 33.296 49.363 63.596 67.796 86.959
1.24 1.03 1.09 1.20 0.95 1.12
24.331 43.257 68.028 69.858 97.538 101.947
31.634 38.552 54.169 77.927 106.883 118.747
1.30 0.89 0.80 1.12 1.10 1.16
27.965 48.008 63.104 72.265 93.351 100.811
41.150 47.996 63.257 86.415 108.559 110.987
1.47 1.00 1.00 1.20 1.16 1.10
41.646 66.504 95.951 96.268 129.810 133.764
67.285 72.812 86.529 109.858 141.072 174.333
1.62 1.09 0.90 1.14 1.09 1.30
* b/h VCNT 0.11 20
RHL 1.21 1.01 1.12 1.07 0.99 1.01
LAF 17.479 30.236 42.537 45.889 57.427 60.129
FG-X FG-X HAF 21.514 26.440 37.400 43.227 49.318 52.806
RHL 1.23 0.87 0.88 0.94 0.86 0.88
38
LAF 19.389 32.252 39.235 46.972 56.380 62.563
FG-O FG-O HAF 25.317 30.996 42.554 54.298 57.686 58.558
RHL 1.31 0.96 1.08 1.16 1.02 0.94
LAF 24.584 37.858 48.173 52.985 65.521 69.966
FG-X FG-X HAF 32.277 37.193 48.087 63.045 66.077 69.054
1.31 0.98 1.00 1.19 1.01 0.99
RHL
Table 10 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 30 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 13.613 27.798 32.271 46.800 49.344 57.085
FG-O FG-O HAF 19.946 29.188 40.059 51.667 53.969 62.129
50
1 2 3 4 5 6
14.237 29.722 36.169 51.367 57.381 68.909
21.991 32.220 45.128 59.163 68.360 75.022
1.54 1.08 1.25 1.15 1.19 1.09
19.701 37.548 52.416 61.835 77.543 91.304
33.902 45.528 60.989 76.982 95.041 102.701
1.72 1.21 1.16 1.24 1.23 1.12
22.741 42.411 47.370 65.945 75.956 79.802
39.713 48.398 63.280 80.368 97.487 98.398
1.75 1.14 1.34 1.22 1.28 1.23
31.071 53.475 66.035 79.911 98.185 109.960
59.475 67.433 82.655 101.347 121.068 137.649
1.91 1.26 1.25 1.27 1.23 1.25
100
1 2 3 4 5 6
14.344 30.055 36.903 52.191 58.972 71.541
22.460 32.756 46.089 60.612 71.897 77.203
1.57 1.09 1.25 1.16 1.22 1.08
19.997 38.283 54.746 63.447 81.671 94.429
35.444 47.157 63.663 80.680 100.274 114.512
1.77 1.23 1.16 1.27 1.23 1.21
23.273 43.719 49.333 68.658 79.838 84.734
42.045 51.072 66.884 85.566 105.597 109.246
1.81 1.17 1.36 1.25 1.32 1.29
32.653 56.844 72.024 85.966 108.509 120.360
67.571 75.974 92.692 114.308 137.688 163.146
2.07 1.34 1.29 1.33 1.27 1.36
0.14 20
1 2 3 4 5 6
14.480 29.004 34.404 48.376 51.934 60.265
21.593 31.008 42.113 53.890 57.590 63.529
1.49 1.07 1.22 1.11 1.11 1.05
19.371 35.462 44.800 56.255 63.425 68.381
29.830 39.982 51.787 63.529 64.007 65.611
1.54 1.13 1.16 1.13 1.01 0.96
21.249 37.934 41.128 56.595 63.205 64.760
32.237 39.703 51.366 63.584 67.474 74.736
1.52 1.05 1.25 1.12 1.07 1.15
26.228 43.554 49.593 63.281 72.043 75.506
39.217 46.400 58.136 70.813 78.373 80.854
1.50 1.07 1.17 1.12 1.09 1.07
50
1 2 3 4 5 6
15.224 31.156 39.134 53.295 61.232 74.483
24.152 34.647 48.054 62.481 75.027 78.789
1.59 1.11 1.23 1.17 1.23 1.06
21.355 39.991 57.123 65.230 83.294 95.776
37.344 49.440 65.661 82.279 101.034 111.652
1.75 1.24 1.15 1.26 1.21 1.17
24.259 44.470 50.932 68.695 80.314 85.747
43.610 52.124 67.132 84.668 103.199 106.252
1.80 1.17 1.32 1.23 1.28 1.24
33.539 56.816 71.187 84.129 104.384 115.781
64.496 72.444 87.885 107.025 127.222 146.625
1.92 1.28 1.23 1.27 1.22 1.27
100
1 2 3 4 5 6
15.354 31.536 40.052 54.204 63.124 77.808
24.749 35.307 49.212 64.190 79.560 81.324
1.61 1.12 1.23 1.18 1.26 1.05
21.739 40.899 60.168 67.163 88.493 99.441
39.360 51.553 69.068 86.911 107.503 126.918
1.81 1.26 1.15 1.29 1.21 1.28
24.911 46.020 53.371 71.796 84.982 91.922
46.628 55.499 71.520 90.838 111.580 121.154
1.87 1.21 1.34 1.27 1.31 1.32
35.569 61.010 78.835 91.467 117.193 127.253
74.968 83.365 100.441 122.894 147.260 173.680
2.11 1.37 1.27 1.34 1.26 1.36
0.17 20
1 2 3 4 5 6
16.771 34.308 39.932 57.870 61.208 70.928
24.644 36.080 49.590 64.024 67.270 78.048
1.47 1.05 1.24 1.11 1.10 1.10
22.664 42.468 52.697 68.194 75.771 83.902
34.597 47.154 61.790 77.000 78.048 80.816
1.53 1.11 1.17 1.13 1.03 0.96
25.006 45.454 48.557 68.118 75.787 77.250
37.852 47.232 61.686 76.648 80.569 89.716
1.51 1.04 1.27 1.13 1.06 1.16
31.246 52.818 59.306 77.236 87.264 90.819
46.771 55.896 70.583 86.296 94.099 99.667
1.50 1.06 1.19 1.12 1.08 1.10
50
1 2 3 4 5 6
17.490 36.539 44.454 63.194 70.595 84.812
27.015 39.587 55.485 72.779 84.198 92.339
1.54 1.08 1.25 1.15 1.19 1.09
24.653 47.327 65.209 78.224 97.005 115.706
41.984 56.774 76.291 96.570 119.440 127.030
1.70 1.20 1.17 1.23 1.23 1.10
27.968 52.213 58.324 81.272 93.621 98.407
48.949 59.658 78.042 99.195 120.729 121.541
1.75 1.14 1.34 1.22 1.29 1.24
38.804 67.185 82.193 100.716 122.852 136.707
73.511 83.802 103.271 126.973 151.950 170.181
1.89 1.25 1.26 1.26 1.24 1.24
100
1 2 3 4 5 6
17.612 36.923 45.293 64.145 72.419 87.829
27.560 40.199 56.587 74.442 88.268 94.848
1.56 1.09 1.25 1.16 1.22 1.08
25.023 48.265 68.099 80.298 102.196 119.754
43.872 58.805 79.640 101.235 126.063 141.521
1.75 1.22 1.17 1.26 1.23 1.18
28.577 53.719 60.580 84.412 98.103 104.095
51.621 62.730 82.194 105.200 129.882 134.346
1.81 1.17 1.36 1.25 1.32 1.29
40.771 71.412 89.601 108.364 135.719 152.081
83.426 94.303 115.721 143.167 172.786 205.074
2.05 1.32 1.29 1.32 1.27 1.35
* b/h VCNT 0.11 20
RHL 1.47 1.05 1.24 1.10 1.09 1.09
LAF 18.113 33.732 42.331 54.021 60.641 66.804
FG-X FG-X HAF 27.895 37.807 49.397 61.415 62.129 64.304
RHL 1.54 1.12 1.17 1.14 1.02 0.96
39
LAF 20.204 36.670 39.069 54.818 60.916 62.011
FG-O FG-O HAF 30.319 37.903 49.509 61.429 64.214 71.660
RHL 1.50 1.03 1.27 1.12 1.05 1.16
LAF 24.986 41.999 47.552 61.316 69.640 72.824
FG-X FG-X HAF 37.695 44.787 56.328 68.757 75.789 79.299
1.51 1.07 1.18 1.12 1.09 1.09
RHL
Table 11 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 45 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 16.316 32.855 39.685 52.372 61.640 71.151
FG-O FG-O HAF 29.320 39.013 50.487 62.169 67.916 74.766
50
1 2 3 4 5 6
17.129 36.019 43.984 61.539 69.867 84.786
34.114 44.897 59.437 74.199 90.905 106.834
1.99 1.25 1.35 1.21 1.30 1.26
22.904 45.701 59.874 79.087 86.325 114.539
52.801 64.581 82.044 98.780 117.835 137.128
2.31 1.41 1.37 1.25 1.37 1.20
27.470 51.952 56.733 81.393 90.146 97.281
58.873 75.712 92.436 109.972 128.169 137.114
2.14 1.46 1.63 1.35 1.42 1.41
35.396 65.239 72.196 103.939 108.220 121.639
84.227 102.873 121.157 139.922 159.117 179.110
2.38 1.58 1.68 1.35 1.47 1.47
100
1 2 3 4 5 6
17.272 36.576 44.834 63.345 71.484 87.975
35.301 46.021 61.296 76.712 94.469 113.088
2.04 1.26 1.37 1.21 1.32 1.29
23.272 46.795 62.552 81.966 90.915 124.483
56.646 68.127 87.449 105.551 126.895 148.444
2.43 1.46 1.40 1.29 1.40 1.19
28.079 53.777 58.647 85.942 93.977 102.156
63.855 82.898 102.041 122.335 143.639 159.118
2.27 1.54 1.74 1.42 1.53 1.56
37.045 69.115 78.023 112.029 118.626 135.733
99.634 123.091 146.324 170.380 195.210 221.258
2.69 1.78 1.88 1.52 1.65 1.63
0.14 20
1 2 3 4 5 6
17.214 34.464 41.527 55.063 63.625 74.011
31.645 41.470 53.155 64.983 69.436 77.259
1.84 1.20 1.28 1.18 1.09 1.04
22.334 42.043 51.460 66.344 72.920 84.518
42.125 52.537 64.631 69.436 76.742 77.259
1.89 1.25 1.26 1.05 1.05 0.91
25.600 45.761 50.127 66.854 76.225 81.428
44.019 55.142 66.209 77.790 87.550 88.656
1.72 1.21 1.32 1.16 1.15 1.09
30.465 53.415 56.549 77.239 84.287 89.352
51.487 62.627 73.857 85.603 87.550 97.803
1.69 1.17 1.31 1.11 1.04 1.09
50
1 2 3 4 5 6
18.166 37.959 46.691 65.414 72.591 90.134
37.587 48.552 63.697 78.873 96.089 113.972
2.07 1.28 1.36 1.21 1.32 1.26
24.682 48.516 64.807 83.432 92.046 123.219
57.992 70.176 88.439 105.787 125.527 145.404
2.35 1.45 1.36 1.27 1.36 1.18
28.902 54.816 59.268 86.519 93.397 101.629
64.145 81.502 98.690 116.622 135.169 147.954
2.22 1.49 1.67 1.35 1.45 1.46
37.845 68.877 77.171 108.980 114.693 129.078
90.326 109.386 128.056 147.170 166.697 187.033
2.39 1.59 1.66 1.35 1.45 1.45
100
1 2 3 4 5 6
18.337 38.582 47.749 67.456 74.455 94.107
39.093 49.953 65.972 81.888 100.308 119.479
2.13 1.29 1.38 1.21 1.35 1.27
25.150 49.830 68.289 86.678 97.897 135.999
62.989 74.786 95.341 114.313 136.794 159.310
2.50 1.50 1.40 1.32 1.40 1.17
29.629 56.907 61.608 91.909 97.630 107.570
70.451 90.399 110.392 131.480 153.542 175.071
2.38 1.59 1.79 1.43 1.57 1.63
39.927 73.565 84.570 118.236 127.657 146.790
109.692 134.332 158.666 183.751 209.556 236.557
2.75 1.83 1.88 1.55 1.64 1.61
0.17 20
1 2 3 4 5 6
20.108 40.619 49.065 65.021 76.356 88.286
36.352 48.354 62.662 77.220 85.319 92.950
1.81 1.19 1.28 1.19 1.12 1.05
26.327 50.424 61.176 79.580 88.557 102.903
49.293 62.194 77.250 85.319 92.338 95.072
1.87 1.23 1.26 1.07 1.04 0.92
30.439 54.448 60.430 79.784 92.042 98.738
52.299 66.126 79.839 94.178 106.541 107.924
1.72 1.21 1.32 1.18 1.16 1.09
36.628 64.536 68.702 93.249 102.960 109.138
61.865 75.753 89.718 104.325 107.924 119.487
1.69 1.17 1.31 1.12 1.05 1.09
50
1 2 3 4 5 6
21.047 44.290 54.083 75.731 86.004 104.380
41.947 55.193 73.122 91.325 111.952 131.923
1.99 1.25 1.35 1.21 1.30 1.26
28.734 57.651 74.740 99.828 108.471 142.779
65.290 80.339 102.337 123.543 147.627 172.082
2.27 1.39 1.37 1.24 1.36 1.21
33.794 63.967 69.867 100.367 111.107 119.930
72.722 93.612 114.382 136.174 158.802 170.259
2.15 1.46 1.64 1.36 1.43 1.42
44.421 82.213 90.340 131.030 135.884 152.284
104.279 127.781 150.848 174.555 198.829 224.111
2.35 1.55 1.67 1.33 1.46 1.47
100
1 2 3 4 5 6
21.212 44.931 55.058 77.804 87.874 108.047
43.327 56.476 75.256 94.209 116.053 138.963
2.04 1.26 1.37 1.21 1.32 1.29
29.197 59.060 78.060 103.667 114.159 155.053
70.000 84.743 109.067 132.032 159.017 186.361
2.40 1.43 1.40 1.27 1.39 1.20
34.494 66.071 72.077 105.622 115.551 125.572
78.461 101.916 125.512 150.541 176.831 195.931
2.27 1.54 1.74 1.43 1.53 1.56
46.484 87.137 97.555 141.673 148.647 169.668
123.217 152.747 182.021 212.377 243.736 276.654
2.65 1.75 1.87 1.50 1.64 1.63
* b/h VCNT 0.11 20
RHL 1.80 1.19 1.27 1.19 1.10 1.05
LAF 20.991 40.099 48.902 63.609 70.252 82.050
FG-X FG-X HAF 39.781 49.961 61.895 67.916 73.826 75.662
RHL 1.90 1.25 1.27 1.07 1.05 0.92
40
LAF 24.604 43.856 48.744 64.030 74.042 79.492
FG-O FG-O HAF 41.751 52.790 63.749 75.225 84.723 85.867
RHL 1.70 1.20 1.31 1.17 1.14 1.08
LAF 29.164 51.441 54.612 74.593 81.763 86.713
FG-X FG-X HAF 49.794 60.768 71.796 83.316 85.867 95.270
1.71 1.18 1.31 1.12 1.05 1.10
RHL
Table 12 Frequency parameters corresponding to LAF and HAF for FG-CNTRC skew plates with varying CNT volume fractions and width-tothickness ratios. (a/b=1, 60 ) Simply Supported
Clamped
Modes 1 2 3 4 5 6
LAF 24.438 43.809 63.653 64.839 86.480 91.711
FG-O FG-O HAF 51.756 62.521 75.738 78.184 88.280 96.802
50
1 2 3 4 5 6
25.916 49.573 70.869 78.569 109.852 112.353
67.474 78.840 98.295 115.163 135.403 154.764
2.60 1.59 1.39 1.47 1.23 1.38
32.754 65.206 83.948 104.710 132.945 149.951
101.677 113.218 136.190 154.705 177.515 195.461
3.10 1.74 1.62 1.48 1.34 1.30
43.119 72.938 96.086 106.569 140.973 144.293
108.862 130.268 150.354 170.798 191.372 212.689
2.52 1.79 1.56 1.60 1.36 1.47
51.330 90.959 108.409 135.449 164.430 183.942
146.085 168.392 189.343 210.411 231.464 253.181
2.85 1.85 1.75 1.55 1.41 1.38
100
1 2 3 4 5 6
26.193 50.694 72.321 81.608 113.931 118.912
71.999 82.558 104.128 121.926 144.494 165.761
2.75 1.63 1.44 1.49 1.27 1.39
33.360 67.926 86.994 112.341 141.197 166.376
115.665 124.841 153.227 173.235 201.276 225.587
3.47 1.84 1.76 1.54 1.43 1.36
44.136 75.765 99.609 112.869 148.801 156.124
126.310 152.779 177.837 203.659 230.036 257.830
2.86 2.02 1.79 1.80 1.55 1.65
53.569 97.697 114.880 150.748 179.997 198.194
192.234 223.633 253.133 283.042 313.298 345.008
3.59 2.29 2.20 1.88 1.74 1.74
0.14 20
1 2 3 4 5 6
25.467 45.868 65.345 67.866 90.361 94.394
55.260 66.130 79.499 79.851 92.162 98.768
2.17 1.44 1.22 1.18 1.02 1.05
31.382 55.692 74.381 80.441 96.877 104.905
68.567 79.851 80.031 93.369 98.768 106.200
2.18 1.43 1.08 1.16 1.02 1.01
39.399 62.841 82.602 85.913 109.056 113.754
69.578 82.278 94.500 107.026 112.008 119.746
1.77 1.31 1.14 1.25 1.03 1.05
44.444 71.332 88.761 96.731 121.465 122.674
77.864 90.607 103.032 112.008 115.741 128.645
1.75 1.27 1.16 1.16 0.95 1.05
50
1 2 3 4 5 6
27.137 52.516 73.135 83.675 114.313 119.959
74.211 85.520 105.824 123.062 143.959 163.703
2.73 1.63 1.45 1.47 1.26 1.36
34.926 69.732 88.521 111.911 139.942 159.836
110.410 122.344 146.089 165.166 188.602 199.628
3.16 1.75 1.65 1.48 1.35 1.25
44.614 76.391 98.409 112.380 145.677 152.673
117.501 139.333 159.808 180.582 201.445 223.036
2.63 1.82 1.62 1.61 1.38 1.46
54.064 96.380 112.895 143.523 171.959 190.585
154.091 176.676 197.925 219.285 240.635 262.671
2.85 1.83 1.75 1.53 1.40 1.38
100
1 2 3 4 5 6
27.454 53.844 74.747 87.310 118.939 127.832
79.921 90.181 113.043 131.281 154.899 176.771
2.91 1.67 1.51 1.50 1.30 1.38
35.672 73.041 92.269 121.265 149.566 176.283
128.218 137.250 167.557 188.312 217.947 243.248
3.59 1.88 1.82 1.55 1.46 1.38
45.764 79.666 102.228 119.778 154.372 166.658
139.098 166.710 192.813 219.620 246.959 275.754
3.04 2.09 1.89 1.83 1.60 1.65
56.777 104.636 120.421 162.173 190.290 206.733
209.436 242.002 272.562 303.459 334.659 367.319
3.69 2.31 2.26 1.87 1.76 1.78
0.17 20
1 2 3 4 5 6
30.143 54.247 78.731 80.605 107.888 113.864
64.589 77.906 94.485 98.239 110.150 121.657
2.14 1.44 1.20 1.22 1.02 1.07
37.471 66.629 90.422 96.655 119.054 126.408
81.430 95.607 98.239 112.268 121.657 128.232
2.17 1.43 1.09 1.16 1.02 1.01
47.461 75.446 100.534 103.280 131.471 138.238
83.840 99.616 114.730 130.203 138.067 145.893
1.77 1.32 1.14 1.26 1.05 1.06
54.131 86.491 109.120 117.290 147.412 150.221
94.252 110.104 125.531 138.067 141.315 157.339
1.74 1.27 1.15 1.18 0.96 1.05
50
1 2 3 4 5 6
31.863 60.976 87.221 96.755 135.322 138.566
83.185 97.091 121.176 142.000 167.077 191.040
2.61 1.59 1.39 1.47 1.23 1.38
41.276 81.822 106.161 131.020 167.720 187.233
125.628 140.543 169.368 192.864 221.590 245.597
3.04 1.72 1.60 1.47 1.32 1.31
53.080 89.854 118.422 131.469 173.916 178.291
135.188 161.897 186.971 212.499 238.197 264.826
2.55 1.80 1.58 1.62 1.37 1.49
64.878 114.310 137.536 169.647 207.558 230.202
180.909 209.039 235.474 262.084 288.695 316.161
2.79 1.83 1.71 1.54 1.39 1.37
1 2 3 4 5 6
32.184 62.264 88.906 100.249 140.043 146.123
88.439 101.350 127.902 149.784 177.576 203.763
2.75 1.63 1.44 1.49 1.27 1.39
42.049 85.294 110.019 140.683 178.377 207.959
142.794 154.925 190.433 215.947 251.202 282.047
3.40 1.82 1.73 1.53 1.41 1.36
54.256 93.117 122.521 138.753 183.014 192.009
155.533 188.232 219.216 251.168 283.840 318.305
2.87 2.02 1.79 1.81 1.55 1.66
67.724 122.803 145.855 188.840 227.385 252.042
237.621 277.121 314.239 351.886 389.950 429.801
3.51 2.26 2.15 1.86 1.71 1.71
* b/h VCNT 0.11 20
100
RHL 2.12 1.43 1.19 1.21 1.02 1.06
LAF 29.761 53.190 71.587 77.333 94.767 101.297
FG-X FG-X HAF 65.720 76.906 78.184 90.117 96.802 102.745
RHL 2.21 1.45 1.09 1.17 1.02 1.01
41
LAF 38.335 60.727 80.995 82.846 105.156 111.032
FG-O FG-O HAF 66.610 79.211 91.299 103.691 109.849 116.269
RHL 1.74 1.30 1.13 1.25 1.04 1.05
LAF 42.891 68.900 86.333 93.703 117.963 119.362
FG-X FG-X HAF 75.761 88.268 100.439 109.849 112.883 125.513
1.77 1.28 1.16 1.17 0.96 1.05
RHL
Captions of Figures Fig. 1
(a) Linear (p=1), (b) Quadratic (p=2), (c) Cubic (p=3) and (d) Quartic (p=4) NURBS basis functions corresponding to knot vectors 0, 0,1, 2, 2 , 0, 0, 0,1, 2, 2, 2 , 0, 0, 0, 0,1, 2, 2, 2, 2 ,
0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 , respectively. (The graphs are drawn using more sampling points than those shown in the figures.)
Fig. 2
Schematic of a skew plate with skew angle , depicting its global coordinate ( xy ), local coordinate ( x y ) on the oblique edge and CNT orientation angle ( ) .
Fig. 3
Four types of CNT distributions: (a) UD (b) FG-V (c) FG-O (d) FG-X.
Fig. 4
Convergence study for central point deflection of a simply supported square plate ( 0 ).
Fig. 5
Convergence study for central point deflection of a simply supported rhombic plate with 60 .
Fig. 6
The non-dimensional deflection, W * wc / h , of FG-CNTRC plates with skew angle versus CNT orientation angle, with boundary conditions: (a)-(f) Simply supported and (g)-(l) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14 and uniformly distributed load q=-0.1 MPa).
Fig. 7
Profiles of non-dimensional deflection, W * w / h , for simply supported skew UD-CNTRC plates under uniformly distributed load of q=-0.1 MPa along (a) x-axis ( 0 ) (b) y-axis ( 0 ) (c) x-axis ( ) (d) y-axis ( ) (For all cases: a/b=1, b/h=50, VCNT =0.14).
Fig. 8
The non-dimensional central deflection versus CNT orientation angle for the simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (For all cases b/h=50, VCNT =0.14, UD and q=-0.1 MPa).
Fig. 9
Fig. 10
The normalized non-dimensional central deflection, W ** , of simply supported skew CNTRC plates with varying width-to-thickness ratio versus CNT orientation angle (a/b=1, VCNT =0.14, UD, 45 and q=-0.1 MPa). Natural frequency parameters of CNTRC plates of skew angle versus CNT orientation angle with boundary conditions: (a)-(e) Simply supported and (f)-(j) Fully clamped (For all cases a/b=1, b/h=50, VCNT 0.14 ).
Fig. 11
Natural frequency parameters of cantilever FG-CNTRC plates with skew angle versus CNT orientation angle (For all cases a/b=1, b/h=50, VCNT =0.14).
Fig. 12
Natural frequency parameters versus CNT orientation angle for vibration of simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (b/h=50, VCNT 0.14 , UD).
Fig. 13
Normalized frequency parameters, ** , versus CNT orientation angle for skew CNTRC plates with varying width-to-thickness ratios and boundary conditions: (a) Simply supported (b) Clamped (c) Cantilever (a/b=1, VCNT 0.14 , UD, 45 ).
Fig. 14
Normalized frequency parameters, ** , versus CNT orientation angle for square CNTRC plates of varying width-to-thickness ratios with boundary conditions and CNT volume fractions: (a) Simply supported, V*=0.11 (b) Simply supported, VCNT =0.14 (c) Clamped, VCNT =0.11 (d) Clamped, VCNT =0.14 (a/b=1, FGO, 0 ).
Fig. 15
First eight mode shapes of clamped skew UD-CNTRC plates with skew angle and CNT orientation angle =0.14). corresponding to HAF (a/b=1, b/h=50, VCNT
Fig. 16
First eight mode shapes of cantilever skew UD-CNTRC plates with skew angle and CNT orientation angle =0.14). corresponding to HAF (a/b=1, b/h=50, VCNT
42
1.0
1.0
N1,1
p=1
N1,2
p=2
N2,2
N2,1 N3,1
N3,2
0.8
Basis function value
Basis function value
0.8
0.6
0.4
N4,2
0.6
0.4
0.2
0.2
0.0
0.0 0.0
1.0
0.4
0.8
1.2
1.6
0.0
2.0
0.4
0.8
1.2
(a)
(b) 1.0
N1,3
p=3
1.6
N1,4
p=4
N2,4
N2,3 N3,3
N3,4
0.8
N4,3
Basis function value
Basis function value
0.8
N5,3
0.6
0.4
2.0
N4,4 N5,4 N6,4
0.6
0.4
0.2
0.2
0.0
0.0 0.0
0.4
0.8
1.2
1.6
0.0
2.0
0.4
0.8
1.2
(c)
(d)
1.6
2.0
Fig. 1. (a) Linear (p=1), (b) Quadratic (p=2), (c) Cubic (p=3) and (d) Quartic (p=4) NURBS basis functions corresponding to knot vectors 0, 0,1, 2, 2 , 0, 0, 0,1, 2, 2, 2 , 0, 0, 0, 0,1, 2, 2, 2, 2 , 0, 0, 0, 0, 0,1, 2, 2, 2, 2, 2 , respectively. (The graphs are drawn using more sampling points than those shown in the figures.)
43
y
y'
α
x'
β
b
a
x
Fig. 2. Schematic of a skew plate with skew angle , depicting its global coordinate ( xy ), local coordinate ( x y ) on the oblique edge and CNT orientation angle ( ) .
h
y z
y z
(a)
h
y z
h
h
y z
(b)
(c)
(d)
Fig. 3. Cross section of the CNTRC plates of thickness h with four types of CNT distributions: (a) UD (b) FG-V (c) FG-O (d) FG-X.
44
* Non-dimensional deflection Wiso
6.6
6.5
6.4
6.3
Ref. [42] p=4 p=3 p=2
6.2
6.1 9
11
13
15
17
19
21
23
25
27
29
31
33
Number of elements along each direction
* Non-dimensional deflection Wiso
Fig. 4. Convergence study for central point deflection of a simply supported square plate ( 0 ).
0.65 0.60 0.55
0.50
Ref. [42] p=4 p=3 p=2
0.45 0.40 9
11
13
15
17
19
21
23
25
27
29
31
33
Number of elements along each direction
Fig. 5. Convergence study for central point deflection of a simply supported rhombic plate with 60 .
45
Simply Supported
Clamped 0.7 FG-O FG-V UD FG-X
2.0
Non-dimensional deflection W*
Non-dimensional deflection W*
1.5
1.0
0.5
0.5
0.3
0.1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
+15 +30 +45 +60 +75 +90
(g) FG-O FG-V UD FG-X
Non-dimensional deflection W*
Non-dimensional deflection W*
0
CNT orientation angle ()
(a) 2.1
FG-O FG-V UD FG-X
1.7
1.3
0.9
FG-O FG-V UD FG-X
0.7
0.5
0.3
0.5
0.1 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(b)
(h)
FG-O FG-V UD FG-X
1.8
Non-dimensional deflection W*
Non-dimensional deflection W*
0.9
1.4
1.0
0.6
FG-O FG-V UD FG-X
0.7
0.5
0.3
0.1
0.2 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
(i)
FG-O FG-V UD FG-X
1.3
Non-dimensional deflection W*
Non-dimensional deflection W*
0.6
0.9
0.6
0.3
FG-O FG-V UD FG-X
0.5 0.4 0.3 0.2 0.1 0.0
0.1 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(d)
(j)
46
0.6
FG-O FG-V UD FG-X
0.5
Non-dimensional deflection W*
Non-dimensional deflection W*
0.25
0.4 0.3 0.2 0.1
0.20
0.15
0.10
0.05
0.00
0.0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(e)
+15 +30 +45 +60 +75 +90
(k)
0.030
FG-O FG-V UD FG-X
0.08
Non-dimensional deflection W*
Non-dimensional deflection W*
0
CNT orientation angle ()
CNT orientation angle ()
0.10
FG-O FG-V UD FG-X
0.06 0.04 0.02
FG-O FG-V UD FG-X
0.025 0.020 0.015 0.010 0.005
0.00
0.000 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(f) (l) Fig. 6. The non-dimensional deflection, W * wc / h , of FG-CNTRC plates with skew angle versus CNT orientation angle, with boundary conditions: (a)-(f) Simply supported and (g)-(l) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14 and uniformly distributed load q=-0.1 MPa).
47
1.0
Non-dimensional deflection W*
O
β
x
b
0.8
a
0.6
0.4
0.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
y α
Non-dimensional deflection W*
y α
β
0.8
a
0.6
0.4
0.2
-1.0
1.0
x
O
-0.8
-0.6
-0.4
-0.2
(a) O
x
b
0.8
a
0.6
0.4
0.2
α
-0.8
-0.6
-0.4
-0.2
0.6
0.8
1.0
β
O
x
b
0.8
a
0.6
0.4
0.2
0.0
0.0 -1.0
0.4
1.0
y
Non-dimensional deflection W*
Non-dimensional deflection W*
β
0.2
(b)
1.0
y
0.0
2(x/a)
2(x/a)
α
b
0.0
0.2
0.4
0.6
0.8
-1.0
1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
2(x/a)
2(x/a)
(c) (d) Fig. 7. Profiles of non-dimensional deflection, W * w / h , for simply supported skew UD-CNTRC plates under uniformly distributed load of q=-0.1 MPa along (a) x-axis ( 0 ) (b) y-axis ( 0 ) (c) x-axis ( ) (d) y-axis ( =0.14). ) (For all cases: a/b=1, b/h=50, VCNT
48
12
9.0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
10 8 6
Non-dimensional deflection W*
Non-dimensional deflection W*
4 2 0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
7.5 6.0 4.5 3.0 1.5 0.0
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(a)
(b)
5
2.0
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
4 3
Non-dimensional deflection W*
Non-dimensional deflection W*
6
2 1
a/b=2 a/b=1.8 a/b=1.6 a/b=1.5 a/b=1.4 a/b=1.2 a/b=1 a/b=0.5
1.6 1.2
0.8 0.4
0.0
0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) Fig. 8. The non-dimensional central deflection versus CNT orientation angle for the simply supported CNTRC plates
Normalized non-dimensional deflection, W**
with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (For all cases b/h=50, VCNT 0.14 , UD and q=-0.1 MPa).
5
4
=45o a/b=1
b/h=100 b/h=50 b/h=20 b/h=10 b/h=5
3
2
1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Fig. 9. The normalized non-dimensional central deflection, W ** , of simply supported skew CNTRC plates with varying width-to-thickness ratio versus CNT orientation angle (a/b=1, VCNT =0.14, UD, 45 and q=-0.1 MPa).
49
Simply Supported
Fully Clamped 50
Natural frequency parameter (*)
Natural frequency parameter (*)
26 24 22 20 18 16 14 12
FG-X UD FG-V FG-O
10
45 40 35 30 25
FG-X UD FG-V FG-O
20
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(a)
(f) 55
Natural frequency parameter (*)
Natural frequency parameter (*)
28 26 24 22 20 18 16 14
FG-X UD FG-V FG-O
50 45 40 35 30 25
FG-X UD FG-V FG-O
12 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
36 32 28 24
16
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(g) Natural frequency parameter (*)
Natural frequency parameter (*)
(b)
20
0
CNT orientation angle ()
0
60
50
40
30
+15 +30 +45 +60 +75 +90
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
(h)
60
Natural frequency parameter (*)
Natural frequency parameter (*)
90
50
40
30
20
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
0
80 70 60 50 40 30
+15 +30 +45 +60 +75 +90
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(d)
(i)
50
Natural frequency parameter (*)
Natural frequency parameter (*)
160
100
80
60
40
FG-X UD FG-V FG-O
20
140 120 100 80 60
FG-X UD FG-V FG-O
40
-90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(e) (j) Fig. 10. Natural frequency parameters of CNTRC plates of skew angle versus CNT orientation angle with boundary conditions: (a)-(e) Simply supported and (f)-(j) Fully clamped (For all cases a/b=1, b/h=50, VCNT =0.14).
51
10
Natural frequency parameter (*)
Natural frequency parameter (*)
10
8
6
4
2
FG-X UD FG-V FG-O
8
FG-X UD FG-V FG-O
6
4
2
0
0 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(a)
(b) Natural frequency parameter (*)
Natural frequency parameter (*)
FG-X UD FG-V FG-O
6
4
2
-90 -75 -60 -45 -30 -15
0
Natural frequency parameter (*)
6
4
2
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c)
8
8
FG-X UD FG-V FG-O
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
10
+15 +30 +45 +60 +75 +90
10
10
8
0
CNT orientation angle ()
CNT orientation angle ()
(d)
FG-X UD FG-V FG-O
6
4
2 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(e) Fig. 11. Natural frequency parameters of cantilever FG-CNTRC plates with skew angle versus CNT orientation angle (For all cases a/b=1, b/h=50, VCNT =0.14).
52
20
Natural frequency parameter (*)
Natural frequency parameter (*)
22
18 16 14 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
12 10 8
6
30
25
20 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
15
10
5
-105 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90 +105
50
40
30 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
20
-105 -90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
(b) Natural frequency parameter (*)
Natural frequency parameter (*)
(a)
10
0
CNT orientation angle ()
CNT orientation angle ()
0
90 75 60 a/b=1 a/b=1.2 a/b=1.4 a/b=1.5 a/b=1.6 a/b=1.8 a/b=2
45 30 15
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90 +105
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) Fig. 12. Natural frequency parameters versus CNT orientation angle for vibration of simply supported CNTRC plates with varying aspect ratios and the skew angle (a) 0 (b) 30 (c) 45 (d) 60 (b/h=50, VCNT =0.14, UD).
53
Normalized frequency parameter (**)
1.0 0.9 0.8 0.7 0.6 0.5 0.4
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
=45o a/b=1 SSSS
-90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Normalized frequency parameter (**)
(a) 1.0 0.9 0.8 0.7 0.6 0.5 0.4
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
=45o a/b=1 CCCC
0.3 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
Normalized frequency parameter (**)
(b) 1.0 0.9 0.8
=45o a/b=1 CFFF
b/h=5 b/h=10 b/h=20 b/h=50 b/h=100
0.7 0.6 0.5 0.4 0.3 0.2 0.1 -90 -75 -60 -45 -30 -15
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
(c) Fig. 13. Normalized frequency parameters, ** , versus CNT orientation angle for skew CNTRC plates with varying width-to-thickness ratios and boundary conditions: (a) Simply supported (b) Clamped (c) Cantilever (a/b=1, VCNT 0.14 , UD, 45 ).
54
Normalized frequency parameter (**)
Normalized frequency parameter (**)
1.000
0.995
0.990
0.985
V*CNT=0.11
b/h=20 b/h=50 b/h=100
SSSS
0.980 -90 -75 -60 -45 -30 -15
0
1.00
0.99
0.98
0.97
0.96
+15 +30 +45 +60 +75 +90
(b) Normalized frequency parameter (**)
Normalized frequency parameter (**)
0
CNT orientation angle ()
(a) 1.00 0.96 0.92 0.88
0.80
SSSS
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
0.84
V*CNT=0.14
b/h=20 b/h=50 b/h=100
b/h=20 b/h=50 b/h=100
* CNT
V
=0.11
CCCC
1.00 0.96 0.92 0.88 0.84 0.80
b/h=20 b/h=50 b/h=100
V*CNT=0.14 CCCC
0.76 -90 -75 -60 -45 -30 -15
0
-90 -75 -60 -45 -30 -15
+15 +30 +45 +60 +75 +90
0
+15 +30 +45 +60 +75 +90
CNT orientation angle ()
CNT orientation angle ()
(c) (d) ** Fig. 14. Normalized frequency parameters, , versus CNT orientation angle for square CNTRC plates of varying width to-thickness ratios with boundary conditions and CNT volume fractions: (a) Simply supported, VCNT =0.11 (b) Simply supported, VCNT =0.14 (c) Clamped, VCNT =0.11 (d) Clamped, VCNT =0.14 (a/b=1, FG-O, 0 ).
55
0
1* 43.426
2* 47.223
3* 56.997
4* 74.248
5* 98.726
6* 105.358
7* 107.521
8* 112.957
1* 56.487
2* 64.515
3* 79.708
4* 98.266
5* 117.863
6* 132.749
7* 139.345
8* 140.039
1* 80.742
2* 99.307
3* 117.543
4* 136.309
5* 155.539
6* 175.575
7* 179.209
8* 196.420
1* 142.204
2* 164.687
3* 185.783
4* 207.017
5* 228.243
6* 250.143
7* 272.522
8* 295.780
30
45
60
Fig. 15. First eight mode shapes of clamped skew UD-CNTRC plates with skew angle and CNT orientation angle corresponding to HAF (a/b=1, b/h=50, VCNT =0.14).
56
0
1* 7.479
2* 7.908
3* 11.235
4* 21.087
5* 37.646
6* 43.674
7* 44.126
8* 46.099
1* 7.517
2* 8.198
3* 13.107
4* 25.917
5* 43.769
6* 43.946
7* 45.736
8* 48.956
1* 7.608
2* 8.780
3* 16.665
4* 33.218
5* 44.570
6* 46.660
7* 54.144
8* 55.525
1* 7.943
2* 10.395
3* 24.653
4* 43.812
5* 48.095
6* 55.096
7* 56.414
8* 73.452
30
45
60
Fig. 16. First eight mode shapes of cantilever skew UD-CNTRC plates with skew angle and CNT orientation angle corresponding to HAF (a/b=1, b/h=50, VCNT =0.14).
57
Highlights
An isogeometric approach based on Reddy’s higher order shear deformation theory is developed for skew FG-CNTRC plates.
Boundary conditions along the oblique edges are specially treated by introducing transformation matrices for 7 degrees of freedom using geometrical considerations and relations in Reddy’s higher order shear deformation theory.
The influence of CNT fiber orientation angles are investigated for optimal bending and vibration behavior of FG-CNTRC skew plates.
58