Document Type : Research Paper
Author
Department of Mechanical Engineering, Islamic Azad University, Lamerd, 7434155388, Iran
Abstract
Keywords
Isogeometric Free and Forced Vibration Analyses of FGCNTs Plates based on a Logarithmic HigherOrder Shear Deformation Theory
Hassan Mohammadi^{ *}
Department of Mechanical Engineering, Islamic Azad University, Lamerd, 7434155388, Iran
KEYWORDS 

ABSTRACT 
Carbon nanotubes Free vibration Forced vibration Isogeometric analysis Logarithmic higherorder shear deformation theory 
This paper develops the new logarithmic higherorder shear deformation theory (LHSDT) incorporating isogeometric method for free and forced vibration analyses of functionally graded carbon nanotubes reinforced composite (FGCNTRC) plates. In this theory, a logarithmic function is employed to approximate the distribution of shear strains along the plate thickness which satisfies the condition of zero tractions on the top and bottom surfaces of the plate. The plate is assumed to be fabricated from a mixture of carbon nanotubes (CNTs) and a polymeric matrix. The CNTs are either uniformly distributed or functionally graded (FG) along the thickness direction of the plate. The modified rule of mixture scheme is applied to estimate the effective mechanical properties of FGCNTRC plates. The governing equations are derived from Hamilton’s principle. Furthermore, the Newmark approach is utilized to predict the temporal response of FGCNTRC plates under different transverse dynamical loadings. The applicability and efficiency of the present formulation in predicting vibrational characteristics of FGCNTRC plates are investigated through an extensive set of numerical examples considering different configurations of the plate. It is revealed that the computed results are in excellent agreement with other solution methods extracted by the 3D model and other plate theories. Eventually, a detailed parametric study is conducted to explore the influence of related parameters on the natural frequencies and temporal response of FGCNTRC plates. 
It has always been a demand in science and engineering to find novel materials with improved properties. CNTs have extraordinary mechanical, thermal, and electrical properties, and therefore, they are an appropriate candidate for use as reinforcement in a polymetric matrix [1,2]. In recent years, several research papers have been published to investigate different mechanical behaviors of carbon nanotubereinforced composite (CNTRC) structures [321]. In 2009, Shen [3] combined the concepts of FG distribution and CNTRCs and introduced a new class of materials known as FGCNTRCs.
There is a rich literature on the mechanical response of FGCNTRC plates in bending, vibration, and buckling. Especially, the vibration problem of FGCNTRC plate structures has been solved by many researchers during the last decade. In this regard, they have employed different analytical, semianalytical and numerical methods based on threedimensional (3D) elasticity theory or equivalentsinglelayer (ESL) theories. Depending on whether shear and normal deformation effects are taken into account, different ESL theories are derived from the 3D elasticity theory. In the simplest ESL theory, the classical plate theory (CPT), transverse shear deformation effects are ignored [22]. The CPT provides acceptable results only for thin plate structures. The next one is the firstorder shear deformation theory (FSDT) which considers constant transverse shear strains. In this theory, a shear correction factor is required in order to impose the condition of zerotractions at the inner and outer levels. The determination of this factor is difficult since it depends on the geometry and material of the plate [23]. The FSDT generates good results for thin and moderately thick plates; however, it suffers from shear locking phenomena [24]. Recently, some modifications have been devoted to this theory to eliminate the condition above [2426]. To overcome these obstacles, higherorder shear deformation theories (HSDTs) are developed. During past decades, numerous polynomial [2730] and nonpolynomial [3137] functions have been used by many researchers to predict the distribution of shear strains along the plate thickness. Recently, Zhu et al. [38] proposed a new logarithmic shear shape function and studied bending, free vibration, and buckling behaviors of FG plates via the isogeometric method. They showed that the presented LHSDT could predict accurate numerical results.
Isogeometric analysis (IGA) is a relatively novel numerical approach which Hughes and his coworkers proposed in 2005 [39]. IGA can be regarded as the extension of the traditional finite element analysis (FEA). In this method, the NonUniform Rational BSplines (NURBS) are employed for the description of geometry as well as unknown field variables. As a result, in this idea, geometric design and computational analysis are linked. The IGA possesses several advantages compared to the traditional FEA due to the exclusive characteristics of NURBS, including smoothness, highorder continuity, and reduction of total degreeoffreedom [4043].
Analytical methods provide solutions with high accuracy; however, their applications are restricted to certain types of plate problems. Abdollahzadeh Shahrbabaki and Alibeigloo [44] performed a 3D free vibration analysis of FGCNTRC rectangular plates using the Ritz method. They reported the computed results for several combinations of boundary conditions and different geometrical and material parameters. Zhang et al. [45] employed the statespace Levy method to determine the free vibration response of FGCNTRC plates. In their work, it is assumed that the plates are subjected to inplane loads. They computed the plates' natural frequencies and mode shapes based on the proposed approach. By using Navier’s method and in the context of the FSDT, Duc et al. [46] analyzed the static and free vibration behaviors of FGCNTRC plates resting on elastic foundations. They presented several numerical examples to verify the accuracy of the results compared to those obtained by previous approaches.
The semianalytical methods, which are neither analytical nor can be classified as numerical solutions, are also applied to investigate the vibrational behavior of FGCNTRC plates. Malekzadeh and Heydarpour [47] used a 3D semianalytical approach for static and free vibration analyses of laminated plates with FGCNTRC layers. They assumed that each layer is fabricated from a singlewalled carbon nanotubes (SWCNTs) mixture and an isotropic matrix. In their investigation, the layerwisedifferential quadrature method (LWDQM) is used to describe the displacement field in the thickness direction. Alibeigloo and Emtehani [48] obtained a closedform solution for static and free vibration responses of FGCNTRC plates. They used Fourier series expansion and the state space technique along the in plane and thickness directions, respectively. Wang et al. [49] employed the multiterm KantorovichGalerkin method to investigate free vibration and buckling behaviors of FGCNTRC plates based on the CPT. They solved the governing equations with the statespace approach.
In addition, several numerical techniques have already been employed to study the vibrational behavior of FGCNTRC plate structures. In this respect, Zhu et al. [5] applied the finite element method (FEM) in the context of the FSDT to evaluate natural frequencies and mode shapes for various patterns of FGCNTRC plates. Using the elementfree kpRitz method in the framework of the FSDT, Lei et al. [50] investigated free vibration analysis of laminated FGCNTRC plates in a thermal environment. They examined the influences of various parameters such as boundary condition, CNTs volume fraction, widthtothickness ratio, aspect ratio, and temperature change. Wu and Li [51] examine the 3D free vibration analysis of FGCNTRC rectangular plates. They developed a unified formulation of finite prism methods (FPMs) based on Reissner’s mixed variational theorem for FGCNTRC and fiberreinforced composite (FRC) plates. They assumed two opposite edges of the plates to be simply supported and the remaining edges to be clamped, simply supported or free. In the case of FGCNTRC plates, they verified the solution method with the FSDT based FEM and those obtained by ANSYS software. Malekzadeh and Zarei [52] performed free vibration analysis of quadrilateral laminated FGCNTRC plates. They discretized the governing equations according to the differential quadrature method (DQM). They investigated the effects of different related parameters. Zhang et al. applied the elementfree IMLSRitz method to study free vibration of various FGCNTRC plate structures with different configurations [5355]. Using the HSDT kinematic model and FEM, Mehar et al. [56] investigated the free vibration response of FGCNTRC plates subjected to elevated temperature. PhungVan et al. [57] applied the isogeometric method based on the thirdorder shear deformation theory (TSDT) of Reddy to study the static and dynamic behaviors of FGCNTRC plates. They compared the presented numerical values with those obtained by other numerical approaches. Based on the FSDT, GarciaMacias et al. [58] utilized a shell element formulated in the oblique coordinates to study the static and free vibration behaviors of FGCNTRC skew plates. They compared the computed numerical data with those obtained by ANSYS software. The Ritz method is used by Kiani [59] in order to evaluate natural frequencies of FGCNTRC moderately thick skew plates subjected to different boundary conditions. Ansari et al. [60] presented the generalized diﬀerential quadrature method (GDQM) based on the TSDT to analyze free vibration of FGCNTRC thick plates with arbitrary shapes. They showed the accuracy of the proposed model through a wide range of comparison studies. Using GDQM based on FSDT, Majidi et al. [61] performed vibration analysis of cantilever FG‑CNTRC trapezoidal plates. They presented the numerical results for a variety of included geometrical parameters. Mohammadi and Setoodeh [62] executed the free vibration behavior of FG‑CNTRC skew folded plates. They showed that the fundamental frequency ratio increases very considerably at too high skew angles.
Heretofore, numerous HSDTs have been proposed in the literature. To the best of the authors’ knowledge, the new LHSDT has not been employed to deal with different mechanical behaviors of FGCNTRC plates. Thus, this paper aims to investigate the accuracy and reliability of the proposed LHSDT when combined with the IGA in studying vibrational behaviors of FGCNTRC plates. The remainder of the paper is structured as follows. Section 2 provides the geometry and material description of FGCNTRC plates. Section 3 contains the kinematic and constitutive equations of the plate. In the following, the energy formulation of the plate is provided. Section 4 represents a brief review of the basic concepts of IGA. The isogeometric model of the FGCNTRC plate in free and forced vibration analyses is also addressed in this section. Then a comprehensive set of results is demonstrated in section 5 to show the capability and efficiency of the proposed formulation. Finally, some concluding remarks are drawn in section 6.
Figure 1(a) shows the geometry of an FGCNTRC plate with length a, width b, and thickness h. The inplane coordinates (x,y) are also displayed in this figure. As depicted in Fig. 1(b), it is assumed that the plate is fabricated from a polymetric matrix reinforced by SWCNTs with uniform distribution (UD) and three linear FG patterns. These patterns are denoted by FGX, FGO, and FGV. The volume fraction of CNTs in the cases above are specified by [5,53,54,57,59,62]
UD: 

(1a) 
FGX: 

(1b) 
FGO: 

(1c) 
FGV: 

(1d) 
In which z is the thickness coordinate variable. It is also notable that all considered patterns have the same total volume fraction of CNTs, namely, , which is given by [5,10,57,59,62]

(2) 
In Eq. (2), is the mass fraction of CNTs. Moreover, and denote mass densities of CNTs and matrix, respectively.
The modified rule of mixture is adopted in order to evaluate the apparent mechanical properties of the resulting nanocomposites as [5,10,57,59,62,63]

(3) 
where represent three efficiency parameters which are used to consider the sizedependent effects of CNTRC plates. These parameters for three different volume fractions of CNTs are given in Table 1 [5,10,57,62].


(a) 





(b) 

Fig. 1. (a) Geometrical model for an FGCNTRC plate; (b) Typical distribution patterns for CNTs along with the plate thickness. 
Table 1. Efficiency parameters for three different values of CNTs volume fractions [5,10,57,62]. 





0.11 
0.149 
0.934 
0.939 
0.14 
0.150 
0.941 
0.941 
0.17 
0.149 
1.381 
1.381 
Furthermore, and indicate elastic moduli of the CNTs and Young modulus of the matrix, respectively. Finally, the volume fraction of CNTs and matrix, which are denoted respectively by and , satisfy the following condition in analogy to the relation between volume fractions of matrix and reinforcing phase in FRCs

(4) 
The equivalent Poisson’s ratio and the mass density of the FGCNTRC plates through the thickness are obtained using the conventional rule of mixture [5,57,59,62]

(5) 

(6) 
where and represent Poisson’s ratio of CNT and matrix, respectively. As reported in [59], the equivalent Poisson’s ratio is weakly dependent on the thickness coordinate z and, consequently, the distribution of CNTs.
In a general five parameters HSDT, the displacement field of an arbitrary material point can be written as [64]

(7) 
in which and express displacement components and normal rotations of a material point in the midplane of the plate. Besides, is a kinematic function defines the distribution of transverse shear strains along the thickness of the plate. The function must satisfy the tangential stressfree boundary conditions at the top and bottom surfaces of the plate. According to this condition, various forms of transverse shear functions have been proposed by many researchers during past decades. These functions include polynomial functions [2730], trigonometric functions and their inverse [31,32], exponential function [33,34], hyperbolic functions [35,36] and combination functions [37]. Recently Zhu et al. [38] proposed a logarithmic type of function, which is used in this study

(8) 
For the considered distribution, one can easily check that

(9) 
Some proposed transverse shear functions are listed in Table 2 and are plotted in Fig. 2(a). Moreover, the derivative of these functions is displayed in Fig. 2(b), which confirms the condition of zero shear stress at the upper and lower surfaces of the plate.
Table 2. Some proposed transverse shear functions in the literature. 

Model 
Transverse shear function 
Reddy [27] 

Touratier [31] 

Soldatos [33] 

Karama et al. [35] 

Mantari et al. [37] 

Zhu et al. [38] 



(a) Shear shape functions 



(b) Derivatives of shear shape functions 

Fig. 2. (a) Some proposed shear shape functions; (b) Derivatives of the proposed shear shape functions. 
The strain tensor components generated by the above displacement field are given by [65]

(10) 
The nonzero strain components are usually gathered into a single vector and can be rewritten as [38,66,67]

(11) 
where

(12) 
and

(13) 
The generalized Hook’s law can be expressed as [57,6871]

(14) 
in which

(15) 
with

(16) 
The total potential energy of the plate can be expressed as [38]

(17) 
The strain energy of the plate can be written as

(18) 
where and denote the volume and midsurface area of the plate, respectively. With the aid of Eqs. (11) and (14), Eq. (18) can be rewritten as

(19) 
where C is the matrix of elastic constants, which is defined by

(20) 
By substituting of Eqs. (12) and (15), one can obtain

(21) 
with

(22) 
The kinetic energy is given by [38]

(23) 
In which is the global displacement vector defined by

(24) 
with

(25) 
By substituting Eq. (24) into Eq. (23), one can write

(26) 
Therefore, the kinetic energy of the plate can be expressed by

(27) 
where

(28) 
with

(29) 
in which I is the identity matrix of order .
Finally, the work generated by the external transverse load is calculated by

(30) 
In this section, the isogeometric model for studying the vibrational behavior of FGCNTRC plates is demonstrated.
In this subsection, some fundamental concepts of the IGA are reviewed. To have a detailed study, one can refer to [39,72]. In order to generate Bsplines and NURBS basis functions, a knot vector must be defined, which is a nondecreasing set of numbers, represented as . In this definition, is the ith knot, n denotes the number of basis functions, and p stands for the polynomial order. This study uses open and uniform knot vectors, which means that the knots are equally spaced. Moreover, the first and last knots are repeated p+1 times. The univariate Bspline basis functions are produced by inserting knot values into the wellknown Coxde Boor recursion formula, starting with the zerothorder (p=0) basis function as [73,74]

(31a) 
and for

(31b) 
It should be indicated that for p=0, 1, the generated polynomials are identical to those considered in the standard FEM. However, they are different for higherorder basis functions.
A BSpline curve of degree p is defined as follows

(32) 
where are coordinate positions of the ith control point.
A Bspline surface is easily obtained by the tensor product of two univariate basis functions of order p and q, respectively constructed on twoknot vectors of and . This definition is mathematically expressed by

(33) 
where and are two univariate Bspline basis functions in ξ and η directions, respectively. Also, is a net of control points.
Equation (33) is typically rewritten in the familiar notation which is used in finite element

(34) 
where is the basis function corresponding to the control point I.
To exactly describe the geometric model of various objects such as conic sections, NURBS are defined. In twodimensional space, a NURBS surface is defined as
with 
(35) 
in which is the weighting coefficient associated with the Ith control point. These numerical values control the flexibility of the surface at the control point location.
The field equations are discretized using NURBS basis functions as follows

(36) 
where is the number of control points for the whole plate and is the vector of nodal displacements associated with the control point I.
Substitution of Eq. (36) respectively into Eqs. (18), (23) and (30), then the resulting expressions into Eq. (17), and finally using Hamilton’s principle, the vibrational behavior is described by the following matrix form

(37) 
where is the stiffness matrix, is the mass matrix, and is the load vector. They are respectively expressed by

(38) 
with

(39) 
The timedependent part of the problem is solved using the Newmark scheme. In this approach, it is assumed that the initial state is known as . Knowing the predetermined initial state, the dynamic responses at the time , i.e., are obtained as [75]

(40) 
where

(41) 
and

(42) 
with

(43) 
The existing coefficients in Eq. (43) are assumed to be and .
In this section, the current numerical approach is verified. For this purpose, several numerical examples are presented to investigate the vibrational behavior of FGCNTRC plates. The results are separately presented in two subsections, respectively, for the free and forced vibration analyses. In all the examples, quartic order NURBS basis functions are used for the geometric description of the plate. Several combinations of simply supported (S), clamped (C), and free (F) boundaries are considered for the FGCNTRC plate. For simply supported and clamped edges, we have [38]

(44) 
It should be noted that the condition is implied by fixing the adjacent control points of the corresponding boundary.
In this study, the poly{(mphenylenevinylene)  co  [ (2,5dioctoxypphenylene) vinylene]} referred to PmPV is considered as the matrix and the (10,10) SWCNTs are chosen as the reinforcement of FGCNTRC plates. The material properties of these constituents at room temperature (T=300K) are listed in Table 3 [5,10,57,62]. Moreover, it is assumed that [5,57,62]. In addition, the nondimensional frequency parameter and central deflections are defined as

(45) 
where is the intensity of the applied transverse loading.
Firstly, it is necessary to investigate the convergence and stability of the proposed formulation by performing the convergence study and comparing the results with those available in the open literature. The fundamental frequency parameter for various types of supported FGCNTRC plates with CNT volume fraction are listed in Table 4. It can be observed that the convergence behavior is very excellent, and the converged values are obtained by considering only five elements in each direction of the parametric coordinate. Besides, the computed results for different withtothickness ratios are simultaneously compared with those extracted by other solution methods; analytical methods [46], semianalytical methods [48], and numerical techniques [5,50,51,57,62]. The present solution is very close to the semianalytical method conducted by Alibeigloo and Emtehani [48] and the 3Dbased FPM reported by Wu and Li [51].
Moreover, the computed results are in good agreement with those obtained by other approaches based on different plate models [5,46,50,57,62]. In another comparison study, Table 5 provides natural frequencies corresponding to the first three modes of vibration for UD, FGX, and FGO CNTRC plates having different types of boundary conditions. The parameters of the plates are taken to be . The computed data are compared with the results obtained by Wang et al. [49] via the KantorovichGalerkin method in the context of the CPT. Again, the accuracy and effectiveness of the method are evident by considering the fact that the present solution generates the lower bounds in all the considered modes.
Table 3. Material properties of (10,10) SWCNTs and PmPV matrix at the room temperature (300 K) [5,10,57,62]. 

(10, 10) SWCNTs 
(PmPV) matrix 










Table 4. Convergence and comparison study of fundamental frequency parameter for various types of simply supported FGCNTRC plates with different CNT volume fractions . 


Method 

Distribution pattern 

UD 

FGX 

FGO 

FGV 

10 
Present 
1 
13.5505 

14.6702 

11.3785 

12.4563 


3 
13.5487 

14.6687 

11.3766 

12.4544 


5 
13.5486 

14.6685 

11.3765 

12.4542 


7 
13.5486 

14.6685 

11.3765 

12.4542 


9 
13.5486 

14.6685 

11.3765 

12.4542 

Analytical [46] 

 

14.064 

10.779 

11.732 

Semianalytical [48] 

13.555 

14.668 

11.332 

12.263 

FEM (FSDT) [5] 

13.532 

14.616 

11.550 

12.452 

Elementfree kpRitz method (FSDT) [50] 

13.495 

14.578 

11.514 

12.416 

IGA (TSDT) [57] 

14.024 

15.254 

11.773 

12.755 
20 

1 
17.3158 

19.9069 

13.4256 

15.0749 


3 
17.3122 

19.9031 

13.4226 

15.0717 


5 
17.3120 

19.9029 

13.4224 

15.0714 


7 
17.3120 

19.9029 

13.4224 

15.0714 


9 
17.3120 

19.9029 

13.4224 

15.0714 

Analytical [46] 

 

18.571 

12.316 

13.855 

FEM (FSDT) [5] 

17.355 

19.939 

13.523 

15.110 

IGA (TSDT) [57] 

17.503 

20.241 

13.500 

15.127 
50 
Present 
1 
19.1597 

22.9026 

14.2548 

16.2047 


3 
19.1548 

22.8968 

14.2512 

16.2007 


5 
19.1545 

22.8964 

14.2510 

16.2003 


7 
19.1545 

22.8964 

14.2510 

16.2003 


9 
19.1545 

22.8964 

14.2510 

16.2003 

Analytical [46] 

 

20.959 

12.895 

14.716 

Semianalytical [48] 

19.168 

22.898 

14.280 

16.208 

IGA (CPT) [62] 

19.5813 

23.6446 

14.2484 

16.4471 

FEM (FSDT) [5] 

19.223 

22.984 

14.302 

16.252 

IGA (TSDT) [57] 

19.093 

22.880 

14.153 

16.093 

FPM (3D solution) [51] 

19.1547 

22.9020 

14.2370 

16.1758 
Table 5. Comparison study of first three frequency parameters for UD, FGX, and FGO CNTRC plates with different boundary conditions . 

Boundary condition 
Method 
Mode 
Distribution pattern 

UD 

FGX 

FGO 

FSFS 
Present 
1 
2.8839 

2.9813 

2.8123 


2 
4.8353 

4.9974 

4.7056 


3 
11.5308 

11.9190 

11.2471 

Wang et al. [49] 
1 
2.885 

2.982 

2.812 


2 
4.840 

5.003 

4.709 


3 
11.537 

11.926 

11.248 
SSFS 
Present 
1 
3.4781 

3.5946 

3.3897 


2 
12.1730 

12.5797 

11.8758 


3 
26.5833 

27.4691 

22.1269 

Wang et al. [49] 
1 
3.480 

3.597 

3.390 


2 
12.181 

12.589 

11.877 


3 
26.616 

27.507 

22.203 
CSFS 
Present 
1 
7.6637 

9.0278 

6.0065 


2 
14.1117 

15.2166 

13.0183 


3 
27.6422 

28.8974 

26.5876 

Wang et al. [49] 
1 
7.684 

9.061 

6.015 


2 
14.135 

15.251 

13.029 


3 
27.688 

28.955 

26.613 
FCFC 
Present 
1 
6.5380 

6.7567 

6.3803 


2 
7.8299 

8.0913 

7.6327 


3 
18.0045 

18.6062 

17.5700 

Wang et al. [49] 
1 
6.543 

6.762 

6.382 


2 
7.840 

8.103 

7.639 


3 
18.031 

18.636 

17.585 
CCCC 
Present 
1 
42.5275 

50.9762 

31.2851 


2 
46.4807 

54.5031 

36.2900 


3 
56.4779 

63.7455 

47.9855 

Wang et al. [49] 
1 
43.656 

52.969 

31.738 


2 
47.576 

56.451 

36.716 


3 
57.497 

65.564 

48.383 
CCSC 
Present 
1 
30.2226 

36.2246 

22.4526 


2 
35.5394 

41.0252 

28.9664 


3 
47.8413 

52.6696 

42.6551 

Wang et al. [49] 
1 
30.662 

37.011 

22.623 


2 
35.961 

41.786 

29.121 


3 
48.253 

53.389 

42.820 
After successively validating the proposed formulation, the effects of various geometrical and material parameters on the nondimensional frequency parameters of FGCNTRC plates are studied. Table 6 investigates the influences of CNTs distribution and their volume fraction on the fundamental frequency parameter of FGCNTRC plates. The results are prepared for square moderately thick plates subjected to different types of boundary conditions. It can be seen that by increasing the CNT volume fraction, the fundamental natural frequency changes significantly. This is possibly due to an increase in the stiffness of the FGCNTRC plate when more CNTs are dispersed into the background phase.
Moreover, one can observe that, for all cases under consideration, the FGX and FGO distributions of CNTs give the greatest and lowest natural frequencies, respectively. Accordingly, it can be deduced that when the regions near the top and bottom surfaces of the plate are enriched with more CNTs, the flexural rigidity and, consequently, the natural frequency of the plate increases. In addition, the effect of boundary conditions is examined in this table. The presented data reveals that the CFFF and CCCC plates possess the lowest and highest vibration frequency. Thus, it can be stated that when all other geometrical and material parameters are kept constant, plates with more constrained edges have higher natural frequencies. Finally, it can be concluded that, compared with the effects of CNTs distribution, CNTs volume fraction, and boundary condition, the CNTs distribution has a lower influence on the fundamental frequency parameter of the plate.
It will be substantial to consider the effect of the widthtothickness ratio on the frequency parameters of CNTRC plates having different FG patterns. In this study, five relative widthtothickness ratios are considered. The first six frequency parameters for SSSS and CCCC plates with CNT volume fraction are exhibited in Table 7. It can be observed that when we move from thin to thick plates, remarkable drops occur in the frequency parameters. It can also be seen that for , the frequency parameters are close to each other for all the CNTs distribution. Thus, it can be deduced that the effect of CNTs distribution is insignificant when the plate is comparably thick. These conclusions are true for two considered boundary conditions.
Table 6. The fundamental frequency parameter for various patterns of FGCNTRC plates with different CNT volume fractions subjected to different types of boundary conditions . 

Boundary condition 

Distribution pattern 

UD 

FGX 

FGO 

FGV 

SSSS 
0.11 
13.5486 

14.6685 

11.3765 

12.4542 

0.14 
14.3513 

15.3895 

12.1732 

13.2774 

0.17 
16.8297 

18.1733 

14.1343 

15.4474 
CCCC 
0.11 
18.6139 

19.3399 

16.7479 

18.0751 

0.14 
19.3053 

19.9684 

17.4754 

18.8295 

0.17 
23.2131 

24.0150 

21.0217 

22.5522 
CSCC 
0.11 
16.4770 

17.3325 

14.6043 

15.7524 

0.14 
17.1816 

17.9815 

15.3267 

16.5020 

0.17 
20.5269 

21.5398 

18.2500 

19.6260 
CSSS 
0.11 
13.9654 

15.0702 

11.8619 

12.9187 

0.14 
14.7570 

15.7918 

12.6352 

13.7273 

0.17 
17.3552 

18.7014 

14.7320 

16.0384 
CCFF 
0.11 
5.6700 

6.3689 

4.5126 

5.0379 

0.14 
6.1226 

6.8180 

4.9032 

5.4645 

0.17 
7.0225 

7.8801 

5.5778 

6.2246 
CFCF 
0.11 
6.1376 

6.3092 

6.0164 

6.1714 

0.14 
6.2466 

6.4839 

6.0783 

6.2879 

0.17 
7.6793 

8.0582 

7.4295 

7.7550 
CFFF 
0.11 
1.0197 

1.0534 

0.9952 

1.0256 

0.14 
1.0378 

1.0843 

1.0041 

1.0450 

0.17 
1.2758 

1.3501 

1.2256 

1.2889 
It is also worthwhile to investigate the variation of frequency parameters as the plate aspect ratio is varied. Table 8 contains the first six frequency parameters for different configurations of simply supported and fully clamped FGCNTRC plates with and having different aspect ratios . It can be seen that for all considered patterns of CNTs and certain values of widthtothickness ratio and volume fraction, the frequency parameters drop remarkably as the aspect ratio increases.
In this section, the transient response of FGCNTRC plates subjected to a distributed transverse load is demonstrated.
Firstly, the comparison study is performed for two examples. In the first one, we consider a homogeneous plate which is simply supported all around and subjected to a uniformly distributed step load of intensity . The parameters of the plate are given by
Figure 3 shows the variation of nondimensional central deflection versus time. The computed results are simultaneously compared with the FEM solution executed by Reddy [76]. An excellent agreement is revealed between both sets of results which demonstrates the capability of the proposed numerical approach to capture the temporal response of the plate.
Table 7. The first six nondimensional frequency parameters for various patterns of simply supported and fully clamped FGCNTRC plates have a different widthtothickness ratio . 


Mode 
SSSS 

CCCC 

UD 
FGX 
FGO 
FGV 
UD 
FGX 
FGO 
FGV 

5 
1 
10.9189 
11.2557 
10.0091 
10.6149 

14.2568 
14.6841 
13.1082 
14.1829 


2 
12.1538 
12.2406 
12.2406 
12.2382 

19.8667 
20.3033 
19.1084 
19.9120 


3 
12.1538 
12.2406 
12.2406 
12.2382 

23.3180 
23.5147 
23.5147 
23.4976 


4 
15.9014 
16.3453 
15.3050 
15.8541 

27.4858 
28.6501 
24.1766 
27.3102 


5 
23.2366 
23.4323 
21.5729 
23.2719 

28.6429 
29.0588 
28.1268 
28.7991 


6 
23.4920 
24.0826 
23.4323 
23.3866 

30.8184 
31.6067 
28.2331 
30.7553 

10 
1 
16.8297 
18.1733 
14.1343 
15.4474 

23.2131 
24.0150 
21.0217 
22.5522 


2 
22.0023 
23.3221 
19.9756 
21.2329 

29.6686 
30.6881 
27.8499 
29.3387 


3 
24.3076 
24.4812 
24.4812 
24.4800 

42.2209 
43.5286 
40.7311 
42.2372 


4 
24.3076 
24.4812 
24.4812 
24.4800 

45.7487 
47.0295 
41.3607 
44.7372 


5 
33.9269 
35.4487 
32.3595 
33.7498 

46.6360 
47.1994 
45.7015 
47.0122 


6 
41.0828 
42.4255 
36.9195 
39.5148 

49.6034 
51.1210 
47.0295 
48.8709 

20 
1 
21.3979 
24.6024 
16.5379 
18.5837 

35.9828 
38.7426 
30.1754 
33.0916 


2 
26.5680 
29.5479 
22.6224 
24.6132 

41.7247 
44.5474 
36.6482 
39.5135 


3 
39.6898 
42.7031 
36.5340 
38.7809 

55.1716 
58.3193 
50.8844 
53.9061 


4 
48.6152 
48.9623 
48.9623 
48.9617 

76.2045 
79.9479 
66.3736 
71.9371 


5 
48.6152 
48.9623 
48.9623 
48.9617 

76.2706 
80.2275 
70.3103 
75.7489 


6 
60.3951 
63.9225 
53.2979 
58.7331 

79.6238 
83.5865 
72.3239 
75.7675 

50 
1 
23.6068 
28.2655 
17.4918 
19.9179 

48.9376 
57.0414 
36.9501 
41.8914 


2 
28.8077 
33.1633 
23.6808 
26.0667 

54.0493 
61.8894 
43.1638 
47.9762 


3 
42.3600 
46.6015 
38.1759 
40.8859 

66.8718 
74.4535 
57.5791 
62.4560 


4 
64.7746 
69.6027 
60.7219 
64.2905 

88.9765 
96.7396 
80.8206 
86.2492 


5 
86.3863 
101.2261 
64.2266 
73.1467 

119.9988 
128.5968 
93.7105 
105.4651 


6 
89.0474 
101.5366 
67.7032 
76.4772 

121.0650 
137.2946 
97.4892 
109.0781 

100 
1 
23.9853 
28.9422 
17.6431 
20.1351 

52.4004 
62.8907 
38.3994 
43.9303 


2 
29.1946 
33.8388 
23.8495 
26.3049 

57.4093 
67.5611 
44.5783 
49.9683 


3 
42.8158 
47.3199 
38.4370 
41.2292 

70.0416 
79.7311 
59.0046 
64.4312 


4 
65.5165 
70.6188 
61.2604 
64.9589 

92.1967 
101.8170 
82.5360 
88.5229 


5 
91.6894 
102.9894 
66.4282 
76.2611 

123.8361 
134.1504 
101.5261 
116.1646 


6 
94.4276 
110.5898 
69.9674 
79.6720 

138.3135 
164.4419 
105.3397 
119.8165 

Table 8. The first six nondimensional frequency parameters for various patterns of simply supported and fully clamped FGCNTRC plates have a different aspect ratio . 


Mode 
SSSS 

CCCC 

UD 
FGX 
FGO 
FGV 
UD 
FGX 
FGO 
FGV 

1.0 
1 
16.8297 
18.1733 
14.1343 
15.4474 

23.2131 
24.0150 
21.0217 
22.5522 


2 
22.0023 
23.3221 
19.9756 
21.2329 

29.6686 
30.6881 
27.8499 
29.3387 


3 
24.3076 
24.4812 
24.4812 
24.4800 

42.2209 
43.5286 
40.7311 
42.2372 


4 
24.3076 
24.4812 
24.4812 
24.4800 

45.7487 
47.0295 
41.3607 
44.7372 


5 
33.9269 
35.4487 
32.3595 
33.7498 

46.6360 
47.1994 
45.7015 
47.0122 


6 
41.0828 
42.4255 
36.9195 
39.5148 

49.6034 
51.1210 
47.0295 
48.8709 

1.5 
1 
9.5878 
10.6265 
7.9529 
8.7255 

15.2301 
16.0076 
13.6379 
14.6173 


2 
16.2051 
16.3208 
15.6637 
16.3204 

23.8975 
24.8936 
22.6756 
23.7110 


3 
16.7919 
17.8521 
16.3208 
16.5258 

28.5707 
29.6788 
25.6880 
27.5779 


4 
24.3076 
24.4812 
21.3922 
23.1620 

34.2701 
35.4971 
31.8794 
33.6662 


5 
24.7563 
26.1551 
24.4812 
24.4800 

38.3499 
39.6338 
37.2924 
38.4841 


6 
28.6662 
30.0548 
25.9300 
27.5850 

42.9736 
43.3461 
40.3894 
43.3316 

2.0 
1 
6.6420 
7.3870 
5.6556 
6.1533 

11.7567 
12.4949 
10.5690 
11.2948 


2 
12.1538 
12.2406 
12.2406 
12.2404 

20.6276 
21.6457 
18.3472 
19.7098 


3 
15.0954 
15.9771 
14.1343 
15.0597 

21.8099 
22.7694 
20.8897 
21.7587 


4 
16.8298 
18.1734 
14.3465 
15.4474 

27.8821 
29.0346 
26.1123 
27.4354 


5 
22.0024 
23.3222 
19.9756 
21.2330 

31.8836 
33.1207 
28.5903 
30.6853 


6 
24.3076 
24.4812 
24.4812 
24.4800 

37.0588 
38.3190 
34.3944 
36.3522 

In the next example, a laminated plate with which is subjected to suddenly applied step load is considered. It is assumed that each lamina has the following material properties
Figure 4 plots nondimensional central deflection for CCFF and CFFF laminated square plates with the ply arrangement of against time. The transient solutions are compared with those obtained by Maleki et al. [77]. They employed the GDQM according to the FSDT to generate their results. Again, it can be observed that a very good agreement exists between the present solution and those reported in Ref. [77].
In the following, the dynamical load is defined as

(46) 

Fig. 3. Nondimensional central deflection of the simply supported homogeneous plate under step uniform load . 

Fig. 4. Nondimensional central deflection of CCFF and CFFF laminated square plates under step uniform load . 
where is a function of the time variable t as
Step loading: 

(47a) 
Triangular loading: 

(47b) 
Sinusoidal loading: 

(47c) 
Explosive blast loading: 

(47d) 
with .
It is worth noting that, for the step, triangular and sinusoidal loadings, it is assumed that the plates are subjected to the aforementioned dynamical loads in the interval of 0 to . After that, the load is eliminated, and the plates vibrate freely. However, in the case of explosive blast loading, it is assumed that the load is continuously applied to the plate.
Figures 58 investigate the influences of different FG patterns of CNTs on the nondimensional central deflection of the plates when they are subjected to dynamical step, triangular, sinusoidal, and explosive blast loadings. The plates are characterized by . It can be deduced that plates with FGX and FGO shapes have the upper and lower bounds for the central deflections and period of vibration, which is considered as a superior result in engineering design. As expected, the dynamic response of UD plates lies between the FGX and FGO ones. Thus, it can be concluded that the distribution of CNTs at the top and bottom surfaces of the plate is more beneficial than the dispersion of CNTs at the midsurface.
Another study assumed that plates with the same geometrical configuration and different CNTs volume fraction are subjected to loading the above types. Moreover, the distribution pattern of FGX is considered for the plates to account for the lower bound of deflection. Subsequently, the time histories of nondimensional central deflection of plates are plotted in Figs. 912. It can be observed that for a given pattern when CNTs volume fraction increases, the amplitude of vibration decreases noticeably. As a result, it can be expressed that the appropriate selection of FG patterns for CNTs and their volume fraction can improve the passive vibrational behavior of FGCNTRC plates.

Fig. 5. Nondimensional central deflection of the simply supported FGCNTRC plates with various patterns subjected to the step loading . 

Fig. 6. Nondimensional central deflection of the simply supported FGCNTRC plates with various patterns subjected to the triangular loading . 

Fig. 7. Nondimensional central deflection of the simply supported FGCNTRC plates with various patterns subjected to the sine loading . 

Fig. 8. Nondimensional central deflection of the simply supported FGCNTRC plates with various patterns subjected to the explosive blast loading . 

Fig. 9. Nondimensional central deflection of the simply supported FGXCNTRC plates with different CNTs volume fractions subjected to the step loading . 

Fig. 10. Nondimensional central deflection of the simply supported FGXCNTRC plates with different CNTs volume fractions subjected to the triangular loading . 

Fig. 11. Nondimensional central deflection of the simply supported FGXCNTRC plates with different CNTs volume fractions subjected to the sine loading . 

Fig. 12. Nondimensional central deflection of the simply supported FGXCNTRC plates with different CNTs volume fractions subjected to the explosive blast loading . 
In the present research, an efficient HSDT based isogeometric formulation is developed for free and forced vibration analyses of FGCNTRC plates. The transverse shear deformation along the plate thickness is estimated via a logarithmic function recently proposed by Zhu et al. [38]. It is shown that in the present solution, the convergence of the results is very fast. Besides, the computed data are in very close agreement with the semianalytical and 3D solutions. It can be concluded that the present LHSDT, when combined with the IGA, can predict very accurate natural frequencies and transient responses for FGCNTRC plates. In addition, a detailed parametric study is executed. It is demonstrated that
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