Vibrational analysis of sandwich sectorial plates with functionally graded sheets reinforced by aggregated carbon nanotube

Abstract

In the present work, by considering the agglomeration effect of single-walled carbon nanotubes, free vibration characteristics of functionally graded nanocomposite sandwich sectorial plates are presented. The volume fractions of randomly oriented agglomerated single-walled carbon nanotubes are assumed to be graded in the thickness direction. To determine the effect of carbon nanotube agglomeration on the elastic properties of carbon nanotube-reinforced composites, a two-parameter micromechanical model of agglomeration is employed. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is considered to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight carbon nanotubes. The two-dimensional generalized differential quadrature method as an efficient and accurate numerical tool is used to discretize the equations of motion and to implement the various boundary conditions. The proposed sectorial plates are simply supported at radial edges, while all possible combinations of free, simply supported, and clamped boundary conditions are applied to the other two circular edges. The benefit of using the considered power-law distribution is to illustrate and present useful results arising from symmetric and asymmetric profiles. The effects of agglomeration, geometrical, and material parameters together with the boundary conditions on the frequency parameters of the sandwich functionally graded nanocomposite plates are investigated. It is shown that the natural frequencies of structure are seriously affected by the influence of carbon nanotubes agglomeration. This study serves as a benchmark for assessing the validity of numerical methods or two-dimensional theories used to analyze the sandwich sectorial plates.

Keywords

Eshelby–Mori–Tanaka approach aggregated carbon nanotube sandwich structures two-dimensional generalized differential quadrature method sectorial plates vibration

Introduction

Nowadays, the use of carbon nanotubes (CNTs) in polymer/CNT composites has attracted wide attention [1]. A high aspect ratio, low weight of CNTs, and their extraordinary mechanical properties (strength and flexibility) provide the ultimate reinforcement for the next generation of extremely lightweight but highly elastic and very strong advanced composite materials. On the other hand, by using the polymer/CNT composites in advanced multilayered composite materials (sandwich structures), we can achieve structures with low weight, high strength, and high stiffness in many structures of civil, mechanical, and space engineering.

Functionally graded materials (FGMs) are advanced composite materials that are engineered to have a smooth spatial variation of material properties. This is achieved by fabricating the composite material to have a gradual spatial variation of the constituent materials’ relative volume fractions and microstructure [2]. Plates fabricated from FGMs have several engineering applications.

Owing to the superior properties against the conventional composite laminates, FGMs have found increasing applications in modern engineering designs, such as aircraft fuselage, rocking-motor casing, packaging materials in microelectronic industry, human implants, and so on. Functionally graded (FG) sectorial plates have extensive applications in different engineering branches. For mechanical engineering and aerospace engineering it can be used in different aircraft components such as turbine or fan blades and also vacuum filter segment with replaceable sector plates. In civil engineering, this kind of structure has many practical applications for curved bridge decks. Malekzadeh et al. [3] studied the free vibration analysis of FGM thin-to-moderately thick annular plates subjected to thermal environment and supported on two-parameter elastic foundation using first-order shear deformation theory (FSDT) as well as differential quadrature method (DQM). Chakraverty et al. [4] presented the effect of nonhomogeneity of the material properties on the vibration frequencies of circular and elliptic plates. They used boundary characteristic orthogonal polynomials as the basis function in the Rayleigh Ritz method to solve the problem. Free vibration and mode shape analysis of thick isotropic and FGM annular plates with variable thickness were studied by Efraim and Eisenberger [5] using exact element method. Hosseini-Hashemi et al. [6] performed the vibration of piezoelectric coupled thick annular FGPs subjected to different combinations of boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy’s third-order shear deformation theory (TSDT). Free and forced vibration of FGM annular sectorial plates with simply supported radial edges and arbitrary circular edges were investigated by Nie and Zhong [7]. The inhomogeneity of the plate was characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction, whereas Poisson’s ratio was assumed to remain constant. Yas and Tahouneh [8] investigated the free vibration analysis of thick FG annular plates on elastic foundations via DQM based on the three-dimensional (3D) elasticity theory. The same authors [9 –12] investigated the free vibration analysis of thick one- and two-directional FG annular and sectorial plates on Pasternak elastic foundations using DQ method. Tahouneh et al. [13] studied free vibration characteristics of annular continuous grading fiber-reinforced (CGFR) plates resting on elastic foundations using DQ method and recently, Tahouneh and Yas [14] used DQ method to investigate the influence of equivalent continuum model based on the Eshelby–Mori–Tanaka scheme on the vibrational response of nanocomposite annular plates. Tahouneh and Naei also [15] studied the effect of multi-directional nanocomposite materials on the vibrational response of thick shell panels. In their study, a sensitivity analysis was performed, and the natural frequencies were calculated for different sets of boundary conditions and different combinations of geometric, material, and foundation parameters. Tahouneh [16] used a semi-analytical approach composed of DQM and series solution to present a 3D elasticity solution for free vibration analysis of thick continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates. In this study, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach was employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight CNTs. Yas and Aragh [17] achieved the natural frequencies of rectangular CGFR plates resting on elastic foundations. The CGFR plate was simply supported at the edges and was assumed to have an arbitrary variation of fiber volume fraction in the thickness direction. The results obtained indicated the advantages of using CGFR plate with graded fiber volume fractions over traditional discretely laminated plates. Matsunaga [18] analyzed the natural frequencies and buckling stresses of FG plates using a higher order shear deformation theory which are based on the thickness series expansion of the displacement components. Zhou et al. [19] used the Chebyshev–Ritz method to study the free vibration of thick annular sector plates. McGee et al. [20] considered the effect of stress singularities on the vibration analysis of thick annular sector plates and presented the corner functions to improve the convergence of the numerical solutions. Hosseini-Hashemi et al. [21] employed the DQM to investigate free vibration of FGM circular and annular sectorial thin plates of variable thickness, resting on the Pasternak elastic foundation.

The discovery of CNTs by Iijima [22] has generated a great and sustained interest in carbon-based materials and nanotechnologies. CNTs have been shown to possess exceptional electrical, mechanical, and thermal properties, which are attractive for diverse potential applications ranging from nano-electronics to biomedical devices. A detailed summary of the mechanical properties of CNTs can be found in Ref. [23]. The exceptional mechanical properties of CNTs have shown great promise for a wide variety of applications, such as nanotransistors, nanofillers, semiconductors, hydrogen storage devices, structural materials, molecular sensors, field-emission-based displays, and fuel cells, to name just a few [24]. The addition of nano-sized fibers or nanofillers, such as CNTs, can further increase the merits of polymer composites [25]. These nanocomposites, easily processed due to the small diameter of the CNTs, exhibit unique properties [26, 27], such as enhanced modulus and tensile strength, high thermal stability, and good environmental resistance. This behavior, combined with their low density makes them suitable for a broad range of technological sectors such as telecommunications, electronics [28], and transport industries, especially for aeronautic and aerospace applications where the reduction of weight is crucial in order to reduce the fuel consumption. For example, Qian et al. [29] showed that the addition of 1 wt.% (i.e. 1% by weight) multiwall CNT to polystyrene resulted in 36–42% and 25% increases in the elastic modulus and the break stress of the nanocomposite properties, respectively. In addition, Yokozeki et al. [30] reported the retardation of the onset of matrix cracking in the composite laminates containing the cup-stacked CNTs compared to those without the cup-stacked CNTs. The properties of the CNT-reinforced composites (CNTRCs) depend on a variety of parameters including CNT geometry and the interphase between the matrix and CNT. Interfacial bonding in the inter-phase region between embedded CNT and its surrounding polymer is a crucial issue for the load transferring and reinforcement phenomena [31]. The traditional approach to fabricating nanocomposites implies that the nanotube is distributed either uniformly or randomly such that the resulting mechanical, thermal, or physical properties do not vary spatially at the macroscopic level. Experimental and numerical studies concerning CNTRCs have shown that distributing CNTs uniformly as the reinforcements in the matrix can achieve moderate improvement of the mechanical properties only [29, 32]. This is mainly due to the weak interface between the CNTs and the matrix where a significant material property mismatch exists. Barai and Weng [33] developed a two-scale micromechanical model to analyze the effect of CNT agglomeration and interface condition on the plastic strength of CNT/matrix inclusions, and the small scale addressed the property of the clustered inclusions. The concept of FGM can be utilized for the management of a material’s microstructure so that the vibrational behavior of a plate/shell structure reinforced by CNTs can be improved. According to a comprehensive survey of literature, the author found that there are few research studies on the mechanical behavior of FG CNTRC structures. For the first time, Shen [34] suggested that the nonlinear bending behavior can be considerably improved through the use of a FG distribution of CNTs in the matrix. He introduced the CNT efficiency parameter to account load transfer between the nanotube and polymeric phases. Compressive postbuckling and thermal buckling behavior of FG nanocomposite plates reinforced by aligned, straight single-walled carbon nanotube (SWCNTs) subjected to in-plane temperature variation were reported by Shen and Zhu [35] and Shen and Zhang [36]. They found that in some cases the CNTRC plate with intermediate CNT volume fraction does not have intermediate buckling temperature and initial thermal post buckling strength. Moreover, Ke et al. [37] investigated the nonlinear free vibration of FG CNTRC Timoshenko beams. They found that both linear and nonlinear frequencies of FG CNTRC beam with symmetrical distribution of CNTs are higher than those of beams with uniform or unsymmetrical distribution of CNTs. Kamarian et al. [38] studied vibration analysis of sandwich beams. The material properties of the FG nanocomposite sandwich beam are estimated using the Eshelby–Mori–Tanaka approach. Tornabene et al. [39] investigated the effect of CNT agglomeration on the free vibrations of laminated composite doubly curved shells and panels reinforced by CNTs. Fantuzzi et al. [40] studied free vibration of arbitrarily shaped FG CNT-reinforced plates using generalized differential quadrature method (GDQM). Some additional parametric studies were also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. Tornabene et al. [41] investigated the static response of composite plates and shells reinforced by agglomerated nanoparticles made of CNTs. A two-parameter agglomeration model was taken into account to describe the micromechanics of such particles, which showed the tendency to agglomerate into spherical regions when scattered in a polymer matrix. Marin [42] studied thermoelasticity of initially stressed bodies. They first wrote the mixed initial boundary value problem within the context of thermoelasticity of initially stressed bodies. Then they established some Lagrange type identities and also introduced the Cesaro means of various parts of the total energy associated to the solutions. Marin and Lupu [43] obtained a spatial estimate, similar to that of Saint-Venant type by using a measure of Toupin type associated with the corresponding steady-state vibration and assuming that the exciting frequency was lower to a certain critical frequency. Marin and Marinescu [44] extended the concept of domain of influence in order to cover the elasticity of microstretch materials.

Although there are research works reported on general sandwich structures, very little work has been done to consider even the vibration behavior of FG sandwich structures [45, 46]. Li et al. [47] studied free vibrations of FGSW rectangular plates with simply supported and clamped edges. Zenkour [48, 49] presented a two-dimensional (2D) solution to study the bending, buckling, and free vibration of simply supported FG ceramic–metal sandwich plates. Kamarian et al. [50] studied free vibration of FGSW rectangular plates with simply supported edges and rested on elastic foundations using DQM. Very recently, Wang and Shen [51] investigated the large amplitude vibration and the nonlinear bending of a sandwich plate with CNTRC face sheets resting on an elastic foundation on the basis of a micromechanical model and multi-scale approach. Tahouneh and Naei [52] investigated free vibration and vibrational displacements of thick laminated curved panels with finite via DQ method. The material properties varied continuously through the layers’ thickness according to a three-parameter power-law distribution. It was assumed that the inner surfaces of the FG sheets are metal rich, while the outer surfaces of the layers could be metal rich, ceramic rich, or made of a mixture of two constituents. More recently, Tahouneh and Naei [53] studied the vibrational response of thick nanocomposite laminated curved panels resting on two-parameter elastic foundation. An equivalent continuum model based on the Eshelby–Mori–Tanaka approach was employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight CNTs. In this study, both great and low amount of volume fraction of CNTs and their effects on vibrational response of sandwich curved panels were considered.

In all the studies mentioned above, the material properties of FG CNTRCs were assumed to be graded in the thickness direction and were estimated through the extended rule of mixture in which the CNT efficiency parameter was determined by matching the elastic modulus of CNTRCs observed from the molecular dynamics (MD) simulation results with the numerical results obtained from the extended rule of mixture. On the other hand, the extended rule of mixture is not applicable when CNTs are oriented randomly in the matrix. CNTs have low bending stiffness (due to small diameter and small elastic modulus in the radial direction) and high aspect ratio, which make CNTs easy to agglomerate in a polymer matrix [54, 55]. In order to achieve the desired properties of CNTRCs, it is critical to make CNTs uniformly dispersed in the matrix [56]. It has been observed in CNTRCs that a large amount of the nanotubes are concentrated in agglomerates [57]. Stephan et al. [58] observed that in the 7.5% concentration sample, a large amount of CNTs are concentrated in aggregates.

The specific objective of the present investigation is to provide a 3D elasticity solution for the analysis of the natural frequencies of FG nanocomposite sandwich sectorial plates. The volume fractions of randomly oriented agglomerated SWCNTs are assumed to be graded in the thickness direction of sheets. The direct application of the properties of CNTs in micromechanics models for predicting material properties of the nanotube/polymer composite is inappropriate without taking into account the effects associated with the significant size difference between a nanotube and a typical carbon fiber [59]. In other words, continuum micromechanics equations cannot capture the scale difference between the nano- and microlevels. In order to overcome this limitation, a virtual equivalent fiber consisting of nanotube and its interphase which is perfectly bonded to surrounding resin is applied. A two-parameter micromechanics model of agglomeration is used to determine the effect of CNT agglomeration on the elastic properties of randomly oriented CNTRCs. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented straight CNTs.

In general, the vibrational analysis of the plates is based on the classical thin plate theory and 2D approximate theories, that is, FSDT and higher order deformation theory. However, the results which are obtained based on the 2D plate theories have shortcomings. For the classical thin plate theory, it neglects the effect of the transverse shear deformation by the simplified assumption that the normal to the undeformed mid-plane remains normal after deformation. For case of the 2D approximate theories, the shear factor strongly relies on the boundary condition and they often need more hardware resources to obtain proper accuracy solution. Based on the above-mentioned issues, the three-dimensional (3D) elastic theory is presented in this research work to overcome the weakness of the 2D plate theories, which does not rely on any hypotheses or deserve any numerical precision and can be used to solve the vibration problem of thick plates.

Material properties of CNTRCs

Properties of the equivalent fiber

In this section, a virtual equivalent fiber consisting of nanotube and its interphase, which is perfectly bonded to surrounding resin, is introduced to obtain the mechanical properties of the CNT/polymer composite by using the results of multiscale finite element method (FEM). The equivalent fiber for SWCNT with chiral index of (10,10) is a solid cylinder with diameter of 1.424 nm. Rule of mixture is used inversely for calculating material properties of equivalent fiber [60]

\begin{array}{l} E_{L E F} = \frac{E_{L C}}{V_{E F}} - \frac{E_{M} V_{M}}{V_{E F}} \\ \frac{1}{E_{T E F}} = \frac{1}{E_{T C} V_{E F}} - \frac{V_{M}}{E_{M} V_{E F}} \\ \frac{1}{G_{E F}} = \frac{1}{G_{C} V_{E F}} - \frac{V_{M}}{G_{M} V_{E F}} \\ υ_{E F} = \frac{υ_{C}}{V_{E F}} - \frac{υ_{M} V_{M}}{V_{E F}} \end{array}

(1)

where E_LEF is the longitudinal modulus of equivalent fiber, E_TEF is the transverse modulus of equivalent fiber, G_EF is the shear modulus of equivalent fiber, υ_EF is the Poisson’s ratio of equivalent fiber, E_LC is the longitudinal modulus of composites, E_TC is the transverse modulus of composites, G_C is the shear modulus of composites, υ_C is the Poisson’s ratio of composites, E_M is the modulus of matrix, G_M is the shear modulus of matrix, υ_M is the Poisson’s ratio of matrix, V_EF is the volume fraction of the equivalent fiber, and V_M is the volume fraction of the matrix. E_LC, G_C, and E_TC are obtained from multiscale FEM or MD simulations, respectively. Mechanical properties of the developed equivalent fiber are listed in Table 1 [61]. It must be mentioned that in Ref. [61], material properties of the matrix are as

E^{m} = 2.1 GPa, ρ^{m} = 1150 {kg/m}^{3}, υ^{m} = 0.34

Table 1.

Material properties of equivalent fiber.

Mechanical properties	Equivalent fiber [61]
Longitudinal Young’s modulus (GPa)	649.12
Transverse Young’s modulus (GPa)	11.27
Longitudinal Shear modulus (GPa)	5.13
Poisson’s ratio	0.284
Density (kg/m³)	1400

Effect of CNT agglomeration on the properties of the composite

It has been found that in CNTRCs due to large aspect ratio (usually >1000), low bending rigidity of CNTs and van der Waals forces, CNTs have a tendency to bundle or cluster together. The effect of nanotube agglomeration on the elastic properties of randomly oriented CNTRC is presented in this section. Shi et al. [56] derived a two-parameter micromechanics model to determine the effect of nanotube agglomeration on the elastic properties of randomly oriented CNTRC (Figure 1). It is assumed that a number of CNTs are unidirectional (UD) throughout the matrix and that other CNTs appear in cluster form because of agglomeration, as shown in Figure 1. The total volume of the CNTs in the representative volume element (RVE), denote by V_r, can be divided into the following two parts

V_{r} = V_{r}^{c l u s t e r} + V_{r}^{m}

(2)

where

V_{r}^{c l u s t e r}

represents the volumes of CNTs inside a cluster and

V_{r}^{m}

is the volume of CNTs in the matrix and outside the clusters. The two parameters used to describe the agglomeration are defined as

μ = \frac{V_{c l u s t e r}}{V}, η = \frac{V_{r}^{c l u s t e r}}{V_{r}} 0 \leq η, μ \leq 1

(3)

where V is the volume of RVE,

V_{c l u s t e r}

is the volume of clusters in the RVE.

μ

is the volume fraction of clusters with respect to the total volume of the RVE, and

η

is the volume ratio of the CNTs inside the clusters over the total CNT inside the RVE.

μ = 1

denotes the case that all CNTs are uniformly dispersed in the matrix and with the decrease of

μ

, the agglomeration degree of CNTs is more severe. If

η = 1

, all the nanotubes are located in the clusters. The case

μ = η

means that the volume fraction of CNTs inside the clusters is the same as that of CNTs outside the clusters (fully dispersed). When

μ < η

, the bigger value of

η

denotes the more heterogeneous spatial distribution of CNTs. Thus, we consider the CNTRC as a system consisting of clusters of sphere shape embedded in a matrix. We may first estimate, respectively, the effective elastic stiffness of the clusters and the matrix, and then calculate the overall property of the whole composite system. The effective bulk modulus

K_{i n}

and shear modulus

G_{i n}

of the equivalent matrix inside the cluster and the effective bulk modulus

K_{o u t}

and shear modulus

G_{o u t}

of the equivalent matrix outside the cluster can be calculated by [38, 62]

K_{i n} = K_{m} + \frac{f_{r} η (δ_{r} - 3 K_{m} α_{r})}{3 (μ - f_{r} η + f_{r} η α_{r})}

(4)

K_{o u t} = K_{m} + \frac{f_{r} (1 - η) (δ_{r} - 3 K_{m} α_{r})}{3 (1 - μ - f_{r} (1 - η) + f_{r} (1 - η) α_{r})}

(5)

G_{i n} = G_{m} + \frac{f_{r} η (η_{r} - 2 G_{m} β_{r})}{2 (μ - f_{r} η + f_{r} η α_{r})}

(6)

G_{o u t} = G_{m} + \frac{f_{r} (1 - η) (η_{r} - 2 G_{m} β_{r})}{2 (1 - μ - f_{r} (1 - η) + f_{r} (1 - η) β_{r})}

(7)

where

α_{r} = \frac{3 (K_{m} + G_{m}) + k_{r} - l_{r}}{3 (G_{m} + k_{r})}

(8)

\begin{matrix} β_{r} = \frac{1}{5} [\frac{4 G_{m} + 2 k_{r} + l_{r}}{3 (G_{m} + k_{r})} + \frac{4 G_{m}}{G_{m} + P_{r}} \\ + \frac{2 [G_{m} (3 K_{m} + G_{m}) + G_{m} (3 K_{m} + 7 G_{m})]}{G_{m} (3 K_{m} + G_{m}) + m_{r} (3 K_{m} + 7 G_{m})}] \end{matrix}

(9)

δ_{r} = \frac{1}{3} [\frac{(2 k_{r} + l_{r}) (3 K_{m} + 2 G_{m} - l_{r})}{G_{m} + k_{r}} + n_{r} + 2 l_{r}]

(10)

\begin{array}{l} η_{r} = \frac{1}{5} [\frac{2}{3} (n_{r} - l_{r}) + \frac{8 G_{m} P_{r}}{G_{m} + P_{r}} + \frac{2 (k_{r} - l_{r}) (2 G_{m} + l_{r})}{3 (G_{m} + k_{r})} \\ + \frac{8 m_{r} G_{m} (3 K_{m} + 4 G_{m})}{3 K_{m} (m_{r} + G_{m}) + G_{m} (7 m_{r} + G_{m})}] \end{array}

(11)

The subscripts m and r stand for the quantities of the matrix and the reinforcing phase, K_m and G_m are the bulk and shear moduli of the matrix, respectively, and k_r, l_r, m_r, n_r, and p_r are the Hill’s elastic moduli for the reinforcing phase (CNTs), which can be found from the equality of the two following matrices [38, 62]

C_{r} = [\begin{array}{l} n_{r} l_{r} l_{r} 000 \\ l_{r} k_{r} + m_{r} k_{r} - m 000 \\ l_{r} k_{r} - m_{r} k_{r} + m_{r} 000 \\ 000 p_{r} 00 \\ 0000 m_{r} 0 \\ 00000 p_{r} \end{array}]

(12)

C_{r} = {[\begin{array}{l} \frac{1}{E_{L}} - \frac{υ_{T L}}{E_{T}} - \frac{υ_{Z L}}{E_{Z}} 000 \\ - \frac{υ_{L T}}{E_{L}} \frac{1}{E_{T}} - \frac{υ_{Z T}}{E_{Z}} 000 \\ - \frac{υ_{L Z}}{E_{L}} - \frac{υ_{T Z}}{E_{T}} \frac{1}{E_{Z}} 000 \\ 000 \frac{1}{G_{T Z}} 00 \\ 0000 \frac{1}{G_{Z L}} 0 \\ 00000 \frac{1}{G_{L T}} \end{array}]}^{- 1}

(13)

where E_L, E_T, E_Z, G_TZ, G_ZL, G_LT, ʋ_LT are material properties of the equivalent fiber, which can be determined from the inverse of the rule of mixture. It must be noticed that before the use of the rule of mixture, material properties of nanoscale RVE of nanocomposite must be obtained from multiscale FEM analysis or MD simulations. The effective bulk modulus K and the effective shear modulus G of the composite are derived from the Mori-Tanaka (MT) method as follows [38, 62]

K = K_{o u t} (1 + \frac{μ (\frac{K_{i n}}{K_{o u t}} - 1)}{1 + α (1 - μ) (\frac{K_{i n}}{K_{o u t}} - 1)})

(14)

G = G_{o u t} (1 + \frac{μ (\frac{G_{i n}}{G_{o u t}} - 1)}{1 + β (1 - μ) (\frac{G_{i n}}{G_{o u t}} - 1)})

(15)

In which

α = \frac{1 + S_{o u t}}{3 (1 - S_{o u t})}

(16)

β = \frac{2 (4 - 5 S_{o u t})}{15 (1 - S_{o u t})}

(17)

S_{o u t} = \frac{3 K_{o u t} - 2 G_{o u t}}{2 (3 K_{o u t} + G_{o u t})}

(18)

Figure 1.

Representative volume element (RVE) with Eshelby inclusion model of agglomeration of CNTs.

Finally, the effective Young’s modulus E and Poisson’s ratio υ of the composite are given by

E = \frac{9 K G}{3 K + G}

(19)

υ = \frac{3 K - 2 G}{6 K + 2 G}

(20)

Problem description

Consider a sandwich sectorial plate with total thickness h as depicted in Figure 2. A cylindrical coordinate system (r, θ, z) is used to label the material point of the FG sectorial plate. In the present work, V_cnt and V_m are considered as the CNT and matrix volume fraction, respectively. We assume that the volume fraction of CNTs varies through the thickness of FG-CNTR plate according to a generalized power-law distribution with four parameters as the following [62]

V_{c n t} = {\begin{array}{l} V^{*} (1 - {(1 - a (1 - \frac{z + 0.5}{h_{f}}) + b {(1 - \frac{z + 0.5}{h_{f}})}^{c})}^{p}), - 0.5 h \leq z \leq - 0.5 h + h_{f} \\ 0, - 0.5 h + h_{f} \leq z \leq 0.5 h - h_{f} \\ V^{*} (1 - {(1 - a (1 + \frac{z - 0.5}{h_{f}}) + b {(1 + \frac{z - 0.5}{h_{f}})}^{c})}^{p}), 0.5 h - h_{f} \leq z \leq 0.5 h \end{array}

(21)

where h and h_f are the thicknesses of plate and the face sheets, respectively, V* is the maximum possible amount of CNT volume fraction in the face sheets. Furthermore, volume fraction index p

(0 \leq p \leq \infty)

and the parameters a, b, and c indicate the CNT volume fraction profile through the thickness of structure. It should be noticed that the values of parameters a, b, and c must be chosen so that

(0 \leq V_{c n t} \leq V^{*})

. According to relation (21), the core of structure does not contain CNT, whereas the lower and upper face sheets are made of a mixture of the two constituents. Various material profiles through the thickness of face sheets can be illustrated by using the four-parameter power-law distribution. The through-thickness variations of volume fraction for some profiles are illustrated in Figures 3 to 5 for different amounts of parameter b, c, and p. The total volume fractions of CNTs (

V_{c n t}^{t o t a l}

) in the thickness direction for different types of material distribution profiles (Figures 3 to 5) is reported in Tables 2 to 4.

Figure 2.

Geometry of an FG-CNTR sandwich sectorial plate.

Figure 3.

Variation of the fiber volume fraction (V_cnt) through the thickness of the FG sandwich sectorial plate (a = 1, b = 0, c = 2).

Figure 4.

Variation of the fiber volume fraction (V_cnt) through the thickness of the FG sandwich sectorial plate (a = 1, b = 1, c = 2).

Figure 5.

Variation of the fiber volume fraction (V_cnt) through the thickness of the FG sandwich sectorial plate (a = 1, b = 1, c = 6).

Table 2.

The total volume fraction of CNTs ( $V_{c n t}^{t o t a l}$ ) in the thickness direction of the plate and with different amounts of parameters “P” for a = 1, b = 0, and c = 2.

P=0.1	P=0.2	P=0.5	P=1	P=2	P=5	P=10
1.09%	2.00%	4.00%	6.00%	8.00%	10.00%	10.90%

Table 3.

The total volume fraction of CNTs ( $V_{c n t}^{t o t a l}$ ) in the thickness direction of the plate and with different amounts of parameters “P” for a = 1, b = 1, and c = 2.

P = 0.1	P = 0.2	P = 0.5	P = 1	P = 2	P = 5	P = 10
0.22%	0.44%	1.05%	2.00%	3.60%	6.76%	9.19%

Table 4.

The total volume fraction of CNTs ( $V_{c n t}^{t o t a l}$ ) in the thickness direction of the plate and with different amounts of parameters “P” for a = 1, b = 1, and c = 6.

P = 0.1	P = 0.2	P = 0.5	P = 1	P = 2	P = 5	P = 10
0.56%	1.08%	2.47%	4.28%	6.65%	9.45%	10.66%

In order to make comparison between vibrational response of FG sandwich plates and single-layer FG plates, the variation of CNT distribution through the thickness of the plates is assumed as follows (Figure 6)

V_{c n t} = {\begin{array}{l} 2 * (\frac{2 | z |}{h}) * V_{C N T}^{*}, F G - X (T y p e 1) \\ V_{C N T}^{*}, U D (T y p e 2) \\ (1 + \frac{2 z}{h}) * V_{C N T}^{*}, F G - Λ (T y p e 3) \\ 2 * (1 - \frac{2 | z |}{h}) * V_{C N T}^{*}, F G - ⋄ (T y p e 4) \end{array}}

(22)

where

V_{C N T}^{*}

is the CNT volume fraction.

Figure 6.

Schematic configuration of a carbon nanotube-reinforced composite plate with four types of CNT distributions.

Governing equations

In the absence of body forces, the governing equations are as follows [64]

\begin{array}{l} \frac{\partial σ_{r}}{\partial r} + \frac{1}{r} \frac{\partial τ_{r θ}}{\partial θ} + \frac{\partial τ_{r z}}{\partial z} + \frac{σ_{r} - σ_{θ}}{r} = ρ \frac{\partial^{2} u_{r}}{\partial t^{2}} \\ \frac{\partial τ_{r θ}}{\partial r} + \frac{1}{r} \frac{\partial σ_{θ}}{\partial θ} + \frac{\partial τ_{θ z}}{\partial z} + \frac{2 τ_{r θ}}{r} = ρ \frac{\partial^{2} u_{θ}}{\partial t^{2}} \\ \frac{\partial τ_{r z}}{\partial r} + \frac{1}{r} \frac{\partial τ_{θ z}}{\partial θ} + \frac{\partial σ_{z}}{\partial z} + \frac{τ_{r z}}{r} = ρ \frac{\partial^{2} u_{z}}{\partial t^{2}} \end{array}

(23)

where

σ_{r}, σ_{θ}, σ_{z}

are the axial stress components,

τ_{r θ}, τ_{θ z}, τ_{r z}

are the shear stress components,

u_{r}, u_{θ}, u_{z}

are the displacement components,

ρ

denotes the material density and

t

is the time. The relations between the strain and the displacement are [64]

\begin{array}{l} ε_{r} = \frac{\partial u_{r}}{\partial r}, ε_{θ} = \frac{u_{r}}{r} + \frac{1}{r} \frac{\partial u_{θ}}{\partial θ}, ε_{z} = \frac{\partial u_{z}}{\partial z}, \\ γ_{θ z} = \frac{\partial u_{θ}}{\partial z} + \frac{1}{r} \frac{\partial u_{z}}{\partial θ}, γ_{r z} = \frac{\partial u_{r}}{\partial z} + \frac{\partial u_{z}}{\partial r}, \\ γ_{r θ} = \frac{1}{r} \frac{\partial u_{r}}{\partial θ} + \frac{\partial u_{θ}}{\partial r} - \frac{u_{θ}}{r} \end{array}

(24)

where

ε_{r}, ε_{θ}, ε_{z}, γ_{θ z}, γ_{r θ}, γ_{r z}

are the strain components. The constitutive equations for material are [65]

{\begin{array}{l} σ_{r} \\ σ_{θ} \\ \begin{array}{l} σ_{z} \\ τ_{z θ} \end{array} \\ \begin{array}{l} τ_{r z} \\ τ_{r θ} \end{array} \end{array}} = [\begin{array}{l} c_{11} c_{12} c_{13} 000 \\ c_{12} c_{22} c_{23} 000 \\ c_{13} c_{23} c_{33} 000 \\ 000 c_{44} 00 \\ 0000 c_{55} 0 \\ 00000 c_{66} \end{array}] {\begin{array}{l} ε_{r} \\ ε_{θ} \\ ε_{z} \\ γ_{z θ} \\ γ_{r z} \\ γ_{r θ} \end{array}}

(25)

where

c_{i j}

is the material elastic stiffness coefficient.

Using the 3D constitutive relations and the strain–displacement relations, the equations of motion in terms of displacement components for a linear elastic FG plate with infinitesimal deformations can be written as

\begin{array}{l} c_{11} \frac{\partial^{2} u_{r}}{\partial r^{2}} + c_{12} (- \frac{1}{r^{2}} \frac{\partial u_{θ}}{\partial θ} + \frac{1}{r} \frac{\partial^{2} u_{θ}}{\partial r \partial θ} + \frac{1}{r} \frac{\partial u_{r}}{\partial r} - \frac{1}{r^{2}} u_{r}) + c_{13} \frac{\partial^{2} u_{z}}{\partial r \partial z} \\ + \frac{c_{66}}{r} [\frac{\partial^{2} u_{θ}}{\partial θ \partial r} + \frac{1}{r} \frac{\partial^{2} u_{r}}{\partial θ^{2}} - \frac{1}{r} \frac{\partial u_{θ}}{\partial θ}] + c'_{55} (\frac{\partial u_{r}}{\partial z} + \frac{\partial u_{z}}{\partial r}) + c_{55} (\frac{\partial^{2} u_{r}}{\partial z^{2}} + \frac{\partial^{2} u_{z}}{\partial z \partial r}) \\ + \frac{1}{r} [c_{11} \frac{\partial u_{r}}{\partial r} + c_{12} (\frac{1}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{u r}{r}) + c_{13} \frac{\partial u_{z}}{\partial z} - c_{12} \frac{\partial u_{r}}{\partial r} \\ - c_{22} (\frac{1}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{u_{r}}{r}) - c_{23} \frac{\partial u_{z}}{\partial z}] = ρ \frac{\partial^{2} u_{r}}{\partial t^{2}} \end{array}

(26)

\begin{matrix} c_{66} (\frac{\partial^{2} u_{θ}}{\partial r^{2}} - \frac{1}{r^{2}} \frac{\partial u_{r}}{\partial θ} + \frac{1}{r} \frac{\partial^{2} u_{r}}{\partial r \partial θ} + \frac{1}{r^{2}} u_{θ} - \frac{1}{r} \frac{\partial u_{θ}}{\partial r}) + \frac{1}{r} [c_{12} \frac{\partial^{2} u_{r}}{\partial θ \partial r} + c_{22} (\frac{1}{r} \frac{\partial^{2} u_{θ}}{\partial θ^{2}} + \frac{1}{r} \frac{\partial u_{r}}{\partial θ}) + c_{23} \frac{\partial^{2} u_{z}}{\partial θ \partial z}] \\ + c'_{44} (\frac{1}{r} \frac{\partial u_{z}}{\partial θ} + \frac{\partial u_{θ}}{\partial z}) + c_{44} (\frac{1}{r} \frac{\partial^{2} u_{z}}{\partial z \partial θ} + \frac{\partial^{2} u_{θ}}{\partial z^{2}}) + \frac{2 c_{66}}{r} [\frac{\partial u_{θ}}{\partial r} + \frac{1}{r} \frac{\partial u_{r}}{\partial θ} - \frac{u_{θ}}{r}] = ρ \frac{\partial^{2} u_{θ}}{\partial t^{2}} \end{matrix}

(27)

\begin{matrix} c_{55} (\frac{\partial^{2} u_{r}}{\partial r \partial z} + \frac{\partial^{2} u_{z}}{\partial r^{2}}) + \frac{c_{44}}{r} (\frac{1}{r} \frac{\partial^{2} u_{z}}{\partial θ^{2}} + \frac{\partial^{2} u_{θ}}{\partial θ \partial z}) + c'_{13} \frac{\partial u_{r}}{\partial r} + c_{13} \frac{\partial^{2} u_{r}}{\partial z \partial r} + c'_{23} (\frac{1}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{u_{r}}{r}) \\ + c_{23} (\frac{1}{r} \frac{\partial^{2} u_{θ}}{\partial z \partial θ} + \frac{1}{r} \frac{\partial u_{r}}{\partial z}) + c'_{33} \frac{\partial u_{z}}{\partial z} + c_{33} \frac{\partial^{2} u_{z}}{\partial z^{2}} + \frac{c_{55}}{r} [\frac{\partial u_{r}}{\partial z} + \frac{\partial u_{z}}{\partial r}] = ρ \frac{\partial^{2} u_{z}}{\partial t^{2}} \end{matrix}

(28)

where

c'_{i j} = \frac{d c_{i j}}{d z}

Equations (26) and (27) represent the in-plane equations of motion along the r and $θ$ axes, respectively; and equation (28) is the transverse or out-of-plane equation of motion. The related boundary conditions are as follows

at z = −h/2 and h/2

τ_{z r} = 0, τ_{z θ} = 0, σ_{z} = 0

(29)

In this paper, three different kinds of boundary conditions are considered for circular edges including Clamped–Clamped (C–C), Simply supported–Clamped (S–C), and Free–Clamped (F–C).

The boundary conditions at edges are

Clamped (r = b) - Clamped (r = a) at r = a U_{r} = U_{θ} = U_{z} = 0 at r = b U_{r} = U_{θ} = U_{z} = 0

(30)

Simply supported (r = b) - Clamped (r = a) at r = b U_{θ} = U_{z} = σ_{r} = 0 at r = a U_{r} = U_{θ} = U_{z} = 0

(31)

Free (r = b) - Clamped (r = a) at r = a U_{r} = U_{θ} = U_{z} = 0 at r = b σ_{r} = τ_{r θ} = τ_{r z} = 0

(32)

Solution procedure

Using the geometrical periodicity of the plate, the displacement components for the free vibration analysis can be represented as

\begin{matrix} U_{r} (r, θ, z, t) = U_{r m} (r, z) \sin (\frac{m π θ}{α}) e^{i ω t} \\ U_{θ} (r, θ, z, t) = U_{θ m} (r, z) \cos (\frac{m π θ}{α}) e^{i ω t} \\ U_{z} (r, θ, z, t) = U_{z m} (r, z) \sin (\frac{m π θ}{α}) e^{i ω t} \end{matrix}

(33)

where m (=0,1,…,

\infty

) is the circumferential wavenumber,

ω

is the natural frequency, and

i (= \sqrt{- 1})

is the imaginary number. It is obvious that m = 0 means axisymmetric vibration. At this stage, the GDQ (a brief review of GDQ method is given in Appendix 1) rules are employed to discretize the free vibration equations and the related boundary conditions. Substituting the displacement components from equation (33) and then using the GDQ rules for the spatial derivatives, the discretized form of the equations of motion at each domain grid point

(r_{j}, z_{k})

with (j = 2,3,…,

N_{r} - 1

) and (k = 2,3,…,

N_{z} - 1

) can be obtained asEquation (26)

\begin{array}{l} {(c_{11})}_{k} \sum_{n = 1}^{N r} B_{j n}^{r} U_{r m n k} + {(c_{12})}_{k} (\frac{m π}{r_{j}^{2} α} U_{θ m j k} - \frac{m π}{r_{j} α} \sum_{n = 1}^{N r} A_{j n}^{r} U_{θ m n k} + \frac{1}{r_{j}} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} - \frac{1}{r_{j}^{2}} U_{r m j k}) \\ + {(c_{13})}_{k} \sum_{n = 1}^{N r} \sum_{r = 1}^{N z} A_{j n}^{r} A_{k r}^{z} U_{z m n r} + \frac{{(c_{66})}_{k}}{r_{j}} [- \frac{m π}{α} \sum_{n = 1}^{N r} A_{j n}^{r} U_{θ m n k} - \frac{m^{2} π^{2}}{r_{j} α^{2}} U_{r m j k} + \frac{m π}{r_{j} α} U_{θ m j k}] \\ + {(c'_{55})}_{k} (\sum_{n = 1}^{N z} A_{k n}^{z} U_{r m j n} + \sum_{n = 1}^{N r} A_{j n}^{r} U_{z m n k}) + {(c_{55})}_{k} (\sum_{n = 1}^{N z} B_{k n}^{z} U_{r m j n} + \sum_{n = 1}^{N r} \sum_{r = 1}^{N z} A_{j n}^{r} A_{k r}^{z} U_{z m n r}) \\ + \frac{1}{r_{j}} [{(c_{11})}_{k} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} + {(c_{12})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{1}{r_{j}} U_{r m j k}) + {(c_{13})}_{k} \sum_{n = 1}^{N z} A_{k n}^{z} U_{z m j n} \\ - {(c_{12})}_{k} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} - {(c_{22})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{1}{r_{j}} U_{r m j k}) - {(c_{23})}_{k} \sum_{n = 1}^{N z} A_{k n}^{z} U_{z m j n}] = - ρ_{k} ω_{2} U_{r m j k} \end{array}

(34)

Equation (27)

\begin{matrix} {(c_{66})}_{k} (\sum_{n = 1}^{N r} B_{j n}^{r} U_{θ m n k} - \frac{m π}{r_{j}^{2} α} U_{r m j k} + \frac{m π}{r_{j} α} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} + \frac{1}{r_{j}^{2}} U_{θ m j k} - \frac{1}{r_{j}} \sum_{n = 1}^{N r} A_{j n}^{r} U_{θ m n k}) \\ + \frac{1}{r_{j}} [{(c_{12})}_{k} \frac{m π}{α} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} + {(c_{22})}_{k} (- \frac{m^{2} π^{2}}{r_{j} α^{2}} U_{θ m j k} + \frac{m π}{r_{j} α} U_{r m j k}) + {(c_{23})}_{k} \frac{m π}{α} \sum_{n = 1}^{N z} A_{k n}^{ƶ} U_{z m j n}] + {(c'_{44})}_{k} (\frac{m π}{r_{j} α} U_{z m j k} + \sum_{n = 1}^{N z} A_{k n}^{ƶ} U_{θ m j n}) \\ + {(c_{44})}_{k} (\frac{m π}{r_{j} α} \sum_{n = 1}^{N z} A_{k n}^{ƶ} U_{z m j n} + \sum_{n = 1}^{N z} B_{k n}^{ƶ} U_{θ m j n}) \\ + \frac{2 {(c_{66})}_{k}}{r_{j}} [\sum_{n = 1}^{N r} A_{j n}^{r} U_{θ m n k} + \frac{m π}{r_{j} α} U_{r m j k} - \frac{U_{θ m j k}}{r_{j}}] = - ρ_{k}_{ω}^{2} U_{θ m j k} \end{matrix}

(35)

Equation (28)

\begin{array}{l} {(c_{55})}_{k} (\sum_{n = 1}^{N r} \sum_{r = 1}^{N ƶ} A_{k r}^{ƶ} A_{j n}^{r} U_{r m n r} + \sum_{n = 1}^{N r} B_{j n}^{r} U_{ƶ m n k}) \\ + \frac{{(c_{44})}_{k}}{r_{j}} (\frac{- m^{2} π^{2}}{r_{j} α^{2}} U_{ƶ m j k} - \frac{m π}{α} \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{θ m j n}) \\ + {(c'_{13})}_{k} \sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k} + {(c_{13})}_{k} \sum_{n = 1}^{N r} \sum_{n = 1}^{N ƶ}_{A}^{k r} A_{j n}^{r} U_{r m n r} + {(c'_{23})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{U_{r m j k}}{r_{j}}) \\ + {(c_{23})}_{k} (- \frac{m π}{r_{j} α} \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{θ m j n} + \frac{1}{r_{j}} \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{r m j n}) + {(c'_{33})}_{k} \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{ƶ m j n} \\ + {(c_{33})}_{k} \sum_{n = 1}^{N ƶ} B_{k n}^{ƶ} U_{ƶ m j n} + \frac{{(c_{55})}_{k}}{r_{j}} [\sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{r m j n} + \sum_{r = 1}^{N r} A_{j r}^{r} U_{ƶ m r k}] = - ρ_{k}_{ω}^{2} U_{ƶ m j k} \end{array}

(36)

where

A_{i j}^{r}

A_{i j}^{z}

and

B_{i j}^{r}

B_{i j}^{z}

are the first- and second-order GDQ weighting coefficients in the r and z directions, respectively.

In a similar manner, the boundary conditions can be discretized. For this purpose, using equation (33) and the GDQ discretization rules for spatial derivatives, the boundary conditions at z = −h/2 and h/2

Equation (29)

\begin{array}{l} \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{r m j n} + \sum_{n = 1}^{N r} A_{j n}^{r} U_{ƶ m n k} = 0, \frac{m π}{r_{j} α} U_{ƶ m j k} + \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{θ m j n} = 0, \\ {(c_{13})}_{k} (\sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k}) + {(c_{23})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{1}{r_{j}} U_{r m j k}) + {(c_{33})}_{k} (\sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{ƶ m j n}) = 0 \end{array}

(37)

where k = 1 at z = 0 and k =

N_{z}

at z = h, and j = 1,2,…,

N_{r}

. The boundary conditions at r = b and a stated in equations (30) to (32) become

Simply supported (S)

\begin{array}{l} U_{z m j k} = 0, U_{θ m j k} = 0, \\ {(c_{11})}_{k} (\sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k}) + {(c_{12})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{1}{r_{j}} U_{r m j k}) + {(c_{13})}_{k} (\sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{ƶ m j n}) = 0 \end{array}

(38.1)

Clamped (C)

\begin{array}{l} U_{r m j k} = 0, \\ U_{θ m j k} = 0, \\ U_{ƶ m j k} = 0 \end{array}

(38.2)

Free (F)

\begin{array}{l} {(c_{11})}_{k} (\sum_{n = 1}^{N r} A_{j n}^{r} U_{r m n k}) + {(c_{12})}_{k} (- \frac{m π}{r_{j} α} U_{θ m j k} + \frac{1}{r_{j}} U_{r m j k}) + {(c_{13})}_{k} (\sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{ƶ m j n}) = 0, \\ \sum_{n = 1}^{N r} A_{j n}^{r} U_{θ m n k} + \frac{m π}{α} U_{r m j k} - U_{θ m j k} = 0, \\ \sum_{n = 1}^{N ƶ} A_{k n}^{ƶ} U_{r m j n} + \sum_{n = 1}^{N r} A_{j n}^{r} U_{ƶ m n k} = 0 \end{array}

(38.3)

In the above equations, k = 2,…,

N_{z} - 1

; also j = 1 at r = b and j =

N_{r}

at r = a.

In order to carry out the eigenvalue analysis, the domain and boundary degrees of freedom are separated and in vector forms they are denoted as {U_d} and {U_b}, respectively. Based on this definition, the discretized form of the equations of motion and the related boundary conditions can be represented in the matrix form as

[\begin{array}{l} [A_{b b}] [A_{b d}] \\ [A_{d b}] [A_{d d}] \end{array}] {\begin{array}{l} {U_{b}} \\ {U_{d}} \end{array}} = {\begin{array}{l} {0} \\ - ρ ω^{2} {U_{d}} \end{array}}

(39)

where {U_d} and {U_b} are as follows

\begin{matrix} {U_{d}} = {{U_{r d}}, {U_{θ d}}, {U_{z d}}}^{T}, \\ {U_{b}} = {{U_{r b}}, {U_{θ b}}, {U_{z b}}}^{T} \end{matrix}

(40)

In relations (39) and (40), subscripts b and d correspond to the displacement vectors at boundaries and domain of the panel, respectively. Eliminating the boundary degrees of freedom, equation (39) becomes

([A] + ρ ω^{2} [I] {U_{d}}) = {0}

(41)

where

[A] = [A_{d d}] - [A_{d b}] {[A_{b b}]}^{- 1} [A_{b d}]

. The above eigenvalue system of equations can be solved to find the natural frequencies of the plates. The order of the above-mentioned matrices

[A_{d d}]

[A_{b b}]

[A_{b d}]

, and

[A_{d b}]

are (3N−6)* (3N−6), 6*6, 6* (3N−6), and (3N−6)*6, respectively.

Numerical results and discussion

In this section, the accuracy of the employed Mori–Tanaka model in estimating the effective mechanical properties of CNTRCs with the influence of CNTs agglomeration is investigated. As it is seen from Figure 7, the effective properties of randomly oriented clustered nanotube-reinforced composite are compared with the available experimental data obtained by Barai and Weng [33]. For this purpose, the matrix properties given by Odegard et al. [59] and the transversely isotropic properties of SWCNTs given by Shen and Li [66] are used to calculate the effective materials using Mori–Tanaka approach, which is provided in Table 5. The comparison shows that the experimental data for the agglomeration state of η = 1 and µ=0.4 are close to those estimated by Mori–Tanaka approach. Therefore, it can be concluded that the Mori–Tanaka model has an acceptable accuracy for prediction of the effective material properties of CNTRCs.

Figure 7.

Comparison of the Young’s moduli of CNTRCs at different degree of agglomerations with the experimental data [33].

Table 5.

Parameters used in the calculation of effective elastic properties of CNT-reinforced polymer composite for comparison of the employed Mori–Tanaka model and the experimental data.

Mechanical properties	Matrix phase [59]	CNT [69]
Isotropic Young’s modulus	0.85 GPa	–
Poisson’s ratio	0.4	–
Longitudinal Young’s modulus	–	1.06 TPa
Transverse bulk modulus	–	271 GPa
Transverse shear modulus	–	17 GPa
In-plane shear modulus	–	442 GPa
In-plane Poisson’s ratio	–	0.162

In this section, the convergence behavior and accuracy of the method in evaluating the non-dimensional natural frequencies of isotropic and FG sandwich annular sector plates with different sets of boundary conditions along the circular edges are investigated. McGee et al. [20] provided the exact results for sector plates with a re-entrant corner, based on the Mindlin plate theory. As a first example, the comparative studies of the fundamental frequency parameters are given in Table 6. It is seen from Table 6 that for thin plates $(h / a = 0.01)$ , there is an excellent agreement between the present 3D solutions and the classical solutions. For moderately thick plates $(h / a = 0.2)$ , the present 3D solutions also agree quite well with the Mindlin solutions. For very thick plates $(h / a = 0.4)$ , the discrepancies increase, particularly for C–C plates. The same problem has been analyzed by Zhou et al. [19]. It is obvious that the error of the Mindlin plate theory increases with the increase of the plate thickness, especially for very thick plates $(h / a \geq 0.4)$ . The error of Mindlin plate theory in Table 6 is about 2.7% in the worst case (α = 195, h/a = 0.4) for the first frequency. For higher frequencies, the Mindlin plate model is accurate. Although the 2D plate theories neglect transverse normal deformations, the results are acceptable and the effect of normal deformation is very negligible and no major influence on the results exists.

Table 6.

Comparison of fundamental frequency parameter $(Ω = ω a^{2} \sqrt{\frac{ρ h}{D}})$ for flexural vibration of annular sector plates with two straight edges simply supported for $\frac{b}{a} = 0.5$ .

α (deg)	h/a	Theories	C–C	F–C	F–S
195	0.01	McGee et al. [20]	90.0837	21.4263	10.8761
		Zhou et al. [19]	90.1125	21.4074	10.8522
		Present (N_r=N_z=9)	90.1102	21.4065	10.8513
		Present (N_r=N_z=13)	90.1124	21.4075	10.852
		Present (N_r=N_z=17)	90.1122	21.4076	10.8525
		Present (N_r=N_z=19)	90.1123	21.4076	10.8524
	0.2	McGee et al. [20]	70.809	19.9986	10.2268
		Zhou et al. [19]	71.9146	20.0967	10.2386
		Present (N_r=N_z=9)	71.9115	20.0954	10.2392
		Present (N_r=N_z=13)	71.9142	20.0964	10.238
		Present (N_r=N_z=17)	71.9143	20.0968	10.2385
		Present (N_r=N_z=19)	71.9143	20.0968	10.2384
	0.4	McGee et al. [20]	48.6618	17.5822	9.3661
		Zhou et al. [19]	50.0059	17.7636	9.3961
		Present (N_r=N_z=9)	50.0045	17.7653	9.3945
		Present (N_r=N_z=13)	50.0059	17.7641	9.3958
		Present (N_r=N_z=17)	50.0056	17.7638	9.3961
		Present (N_r=N_z=19)	50.0056	17.7638	9.3962
210	0.01	McGee et al. [20]	89.9678	20.9496	10.2631
		Zhou et al. [19]	90.0265	20.9368	10.2418
		Present (N_r=N_z=9)	90.0253	20.9347	10.2399
		Present (N_r=N_z=13)	90.026	20.9363	10.241
		Present (N_r=N_z=17)	90.0263	20.9369	10.2416
		Present (N_r=N_z=19)	90.0264	20.9369	10.2416
	0.2	McGee et al. [20]	70.7344	19.6097	9.6643
		Zhou et al. [19]	71.8406	19.7064	9.6751
		Present (N_r=N_z=9)	71.842	19.704	9.6733
		Present (N_r=N_z=13)	71.8401	19.7059	9.6745
		Present (N_r=N_z=17)	71.8407	19.7063	9.6751
		Present (N_r=N_z=19)	71.8406	19.7063	9.6752
	0.4	McGee et al. [20]	48.6117	17.2943	8.8769
		Zhou et al. [19]	49.9566	17.4733	8.9043
		Present (N_r=N_z=9)	49.9535	17.4714	8.9026
		Present (N_r=N_z=13)	49.9555	17.4725	8.9035
		Present (N_r=N_z=17)	49.9563	17.4736	8.9041
		Present (N_r=N_z=19)	49.9564	17.4735	8.9041
270	0.01	McGee et al. [20]	89.6828	19.7282	8.5788
		Zhou et al. [19]	89.7655	19.7258	8.5635
		Present (N_r=N_z=9)	89.7634	19.7219	8.5611
		Present (N_r=N_z=13)	89.7642	19.7245	8.5623
		Present (N_r=N_z=17)	89.7651	19.7257	8.563
		Present (N_r=N_z=19)	89.7653	19.7259	8.5633
	0.2	McGee et al. [20]	70.5516	18.6218	8.1304
		Zhou et al. [19]	71.6588	18.7149	8.1386
		Present (N_r=N_z=9)	71.6551	18.7117	8.1365
		Present (N_r=N_z=13)	71.6575	18.7139	8.1377
		Present (N_r=N_z=17)	71.6584	18.715	8.1386
		Present (N_r=N_z=19)	71.6586	18.715	8.1387
	0.4	McGee et al. [20]	48.4901	16.5657	7.5461
		Zhou et al. [19]	49.8361	16.7386	7.567
		Present (N_r=N_z=9)	49.8341	16.737	7.565
		Present (N_r=N_z=13)	49.8351	16.7382	7.5664
		Present (N_r=N_z=17)	49.8361	16.7387	7.5671
		Present (N_r=N_z=19)	49.836	16.7387	7.567
360	0.01	McGee et al. [20]	89.4931	18.8711	7.2502
		Zhou et al. [19]	89.6519	18.8831	7.2418
		Present (N_r=N_z=9)	89.6508	18.8801	7.2402
		Present (N_r=N_z=13)	89.6511	18.8816	7.2414
		Present (N_r=N_z=17)	89.6518	18.8827	7.2421
		Present (N_r=N_z=19)	89.652	18.8829	7.2421
	0.2	McGee et al. [20]	70.4307	17.9366	6.9363
		Zhou et al. [19]	71.5435	18.0283	6.9426
		Present (N_r=N_z=9)	71.5421	18.026	6.9403
		Present (N_r=N_z=13)	71.5439	18.0275	6.9415
		Present (N_r=N_z=17)	71.5433	18.028	6.9421
		Present (N_r=N_z=19)	71.5433	18.0285	6.9423
	0.4	McGee et al. [20]	48.4105	16.063	6.5171
		Zhou et al. [19]	49.7559	16.2316	6.5332
		Present (N_r=N_z=9)	49.7536	16.2295	6.5331
		Present (N_r=N_z=13)	49.755	16.2309	6.5321
		Present (N_r=N_z=17)	49.7563	16.2315	6.5328
		Present (N_r=N_z=19)	49.7561	16.2315	6.5331

As the second example, the convergence behavior and accuracy of the method for lowest non-dimensional frequency parameter $(ϖ = ω h \sqrt{ρ / C_{11}})$ of thick FG annular sector plates with two different sets of circular edge conditions including Clamped–Clamped and Clamped–Simply supported are studied in Tables 7 and 8. The results are compared with those of the 3D elasticity solutions of Nie and Zhong [7] which were obtained using the state space method (SSM). It is assumed that the material properties vary exponentially $(c_{i j} (z) = c_{i j}^{M} e^{(\frac{λ z}{h})}, ρ (z) = ρ^{M} e^{(\frac{λ z}{h})})$ through the thickness of the plate. Superscript M denotes the material property of the bottom surface of the plate, $λ$ is the material property graded index. One can see that an excellent agreement exists between the converged results of the presented approach and the other one.

Table 7.

The lowest non-dimensional frequency parameter $(ϖ = ω h \sqrt{\frac{ρ}{C_{11}}})$ for FGMs annular sector plates having clamped (r = b) and clamped (r = a) conditions.

α (deg)	$\frac{h}{a}$	$\frac{b}{a}$	m (circumferential wavenumber)		$λ$
α (deg)	$\frac{h}{a}$	$\frac{b}{a}$	m (circumferential wavenumber)		1	2	3	4	5
195	0.1	0.1	1	Nie and Zhong [7]	0.0663	0.0622	0.0566	0.0505	0.0446
				Present (N_r=N_z=9)	0.0651	0.0611	0.0553	0.0497	0.0432
				Present (N_r=N_z=13)	0.0661	0.062	0.0561	0.0502	0.044
				Present (N_r=N_z=17)	0.0664	0.0622	0.0564	0.0505	0.0444
				Present (N_r=N_z=19)	0.0664	0.0623	0.0564	0.0505	0.0445
			2	Nie and Zhong [7]	0.0795	0.0746	0.0677	0.0603	0.0531
				Present (N_r=N_z=9)	0.0781	0.0712	0.0666	0.0589	0.0519
				Present (N_r=N_z=13)	0.0791	0.0743	0.0677	0.0601	0.0528
				Present (N_r=N_z=17)	0.0793	0.0746	0.0679	0.0604	0.053
				Present (N_r=N_z=19)	0.0793	0.0747	0.0679	0.0603	0.053
		0.3	1	Nie and Zhong [7]	0.1041	0.098	0.0895	0.0801	0.071
				Present (N_r=N_z=9)	0.1049	0.0968	0.0888	0.0789	0.0721
				Present (N_r=N_z=13)	0.1041	0.0981	0.0896	0.0801	0.0712
				Present (N_r=N_z=17)	0.1039	0.0979	0.0898	0.0799	0.071
				Present (N_r=N_z=19)	0.1039	0.0979	0.0897	0.08	0.071
			2	Nie and Zhong [7]	0.1104	0.1039	0.0948	0.0849	0.0753
				Present (N_r=N_z=9)	0.1094	0.103	0.0933	0.0839	0.0741
				Present (N_r=N_z=13)	0.1103	0.1038	0.0946	0.0845	0.0755
				Present (N_r=N_z=17)	0.1106	0.104	0.095	0.085	0.0751
				Present (N_r=N_z=19)	0.1105	0.1039	0.095	0.085	0.0752
	0.3	0.1	1	Nie and Zhong [7]	0.404	0.3862	0.3611	0.3329	0.3046
				Present (N_r=N_z=9)	0.4026	0.3842	0.3593	0.3314	0.3035
				Present (N_r=N_z=13)	0.4038	0.3853	0.3604	0.3322	0.3045
				Present (N_r=N_z=17)	0.4041	0.3863	0.3609	0.3326	0.3047
				Present (N_r=N_z=19)	0.4041	0.3863	0.361	0.3327	0.3048
			2	Nie and Zhong [7]	0.5013	0.4781	0.4455	0.4091	0.373
				Present (N_r=N_z=9)	0.5001	0.4768	0.4438	0.4081	0.3719
				Present (N_r=N_z=13)	0.5008	0.4784	0.4449	0.409	0.3727
				Present (N_r=N_z=17)	0.5011	0.478	0.4453	0.4092	0.373
				Present (N_r=N_z=19)	0.5011	0.4779	0.4455	0.4092	0.3729
		0.3	1	Nie and Zhong [7]	0.5645	0.5436	0.5137	0.4796	0.445
				Present (N_r=N_z=9)	0.5633	0.5425	0.5119	0.4776	0.4466
				Present (N_r=N_z=13)	0.5641	0.544	0.513	0.479	0.4455
				Present (N_r=N_z=17)	0.5646	0.5436	0.5134	0.4794	0.4452
				Present (N_r=N_z=19)	0.5646	0.5435	0.5138	0.4796	0.4452
			2	Nie and Zhong [7]	0.6077	0.584	0.5504	0.5125	0.4744
				Present (N_r=N_z=9)	0.6061	0.5852	0.5494	0.512	0.4761
				Present (N_r=N_z=13)	0.6075	0.5846	0.55	0.5127	0.4754
				Present (N_r=N_z=17)	0.6079	0.5843	0.5505	0.5124	0.4748
				Present (N_r=N_z=19)	0.6079	0.5842	0.5505	0.5124	0.4746
210	0.1	0.1	1	Nie and Zhong [7]	0.0659	0.0619	0.0563	0.0502	0.0443
				Present (N_r=N_z=9)	0.0651	0.0603	0.055	0.0509	0.0451
				Present (N_r=N_z=13)	0.0665	0.0617	0.0555	0.0504	0.044
				Present (N_r=N_z=17)	0.0661	0.0621	0.056	0.0501	0.0445
				Present (N_r=N_z=19)	0.066	0.0621	0.0561	0.0501	0.0444
			2	Nie and Zhong [7]	0.0766	0.0719	0.0653	0.0581	0.0512
				Present (N_r=N_z=9)	0.0751	0.0705	0.0641	0.0573	0.05
				Present (N_r=N_z=13)	0.076	0.0717	0.065	0.0581	0.0508
				Present (N_r=N_z=17)	0.0765	0.072	0.0655	0.0583	0.0511
				Present (N_r=N_z=19)	0.0765	0.0721	0.0654	0.0583	0.051
		0.3	1	Nie and Zhong [7]	0.1039	0.0978	0.0892	0.0799	0.0708
				Present (N_r=N_z=9)	0.1025	0.0969	0.0883	0.0787	0.0681
				Present (N_r=N_z=13)	0.1033	0.0979	0.0892	0.0807	0.0693
				Present (N_r=N_z=17)	0.1038	0.0976	0.0895	0.0801	0.0701
				Present (N_r=N_z=19)	0.1037	0.0977	0.0895	0.08	0.0706
			2	Nie and Zhong [7]	0.109	0.1027	0.0937	0.0839	0.0744
				Present (N_r=N_z=9)	0.1099	0.1039	0.0944	0.0849	0.0757
				Present (N_r=N_z=13)	0.1095	0.1033	0.0939	0.0842	0.0749
				Present (N_r=N_z=17)	0.1091	0.1028	0.0936	0.0839	0.0744
				Present (N_r=N_z=19)	0.1092	0.1029	0.0935	0.0839	0.0745
	0.3	0.1	1	Nie and Zhong [7]	0.4002	0.3827	0.358	0.3302	0.3023
				Present (N_r=N_z=9)	0.4018	0.3815	0.3598	0.3318	0.3003
				Present (N_r=N_z=13)	0.4007	0.3824	0.3587	0.3308	0.3018
				Present (N_r=N_z=17)	0.4001	0.3829	0.3581	0.3303	0.3023
				Present (N_r=N_z=19)	0.4	0.3829	0.3582	0.3304	0.3023
			2	Nie and Zhong [7]	0.4832	0.4608	0.4294	0.3943	0.3594
				Present (N_r=N_z=9)	0.4813	0.4622	0.4277	0.3931	0.3577
				Present (N_r=N_z=13)	0.4826	0.4611	0.4288	0.394	0.3587
				Present (N_r=N_z=17)	0.4834	0.4605	0.4295	0.3944	0.3594
				Present (N_r=N_z=19)	0.4833	0.4606	0.4296	0.3944	0.3595
		0.3	1	Nie and Zhong [7]	0.563	0.5421	0.5123	0.4784	0.4439
				Present (N_r=N_z=9)	0.5618	0.5404	0.5105	0.4799	0.4424
				Present (N_r=N_z=13)	0.5629	0.5415	0.5117	0.479	0.4434
				Present (N_r=N_z=17)	0.5633	0.5422	0.5121	0.4785	0.4439
				Present (N_r=N_z=19)	0.5633	0.5421	0.5121	0.4784	0.444
			2	Nie and Zhong [7]	0.599	0.5756	0.5428	0.5056	0.4682
				Present (N_r=N_z=9)	0.5977	0.5771	0.5441	0.5041	0.4702
				Present (N_r=N_z=13)	0.5984	0.576	0.5433	0.505	0.469
				Present (N_r=N_z=17)	0.599	0.5756	0.5429	0.5055	0.4683
				Present (N_r=N_z=19)	0.5991	0.5755	0.5429	0.5057	0.4683

Table 8.

The lowest non-dimensional frequency parameter $(ϖ = ω h \sqrt{\frac{ρ}{C_{11}}})$ for FGMs annular sector plates having clamped (r = b) and simply supported (r = a) conditions.

$α (\deg)$	$\frac{h}{a}$	$\frac{b}{a}$	m (circumferential wavenumber)		$λ$
$α (\deg)$	$\frac{h}{a}$	$\frac{b}{a}$	m (circumferential wavenumber)		1	2	3	4	5
195	0.1	0.1	1	Nie and Zhong [7]	0.0442	0.0412	0.0372	0.0329	0.0289
				Present (N_r=N_z=9)	0.0431	0.0429	0.0361	0.0319	0.0271
				Present (N_r=N_z=13)	0.044	0.0417	0.037	0.0336	0.0282
				Present (N_r=N_z=17)	0.0444	0.0412	0.0376	0.0331	0.0287
				Present (N_r=N_z=19)	0.0444	0.0411	0.0374	0.0329	0.0287
			2	Nie and Zhong [7]	0.0582	0.0542	0.0488	0.0431	0.0377
				Present (N_r=N_z=9)	0.0594	0.0559	0.0477	0.0441	0.0362
				Present (N_r=N_z=13)	0.0588	0.0549	0.0483	0.0433	0.0372
				Present (N_r=N_z=17)	0.0584	0.0543	0.0487	0.043	0.0376
				Present (N_r=N_z=19)	0.0584	0.0544	0.0487	0.0429	0.0378
		0.3	1	Nie and Zhong [7]	0.0727	0.068	0.0617	0.0548	0.0484
				Present (N_r=N_z=9)	0.071	0.0689	0.0605	0.0538	0.0495
				Present (N_r=N_z=13)	0.0717	0.0685	0.0611	0.0544	0.0488
				Present (N_r=N_z=17)	0.0721	0.0682	0.0615	0.0548	0.0485
				Present (N_r=N_z=19)	0.0726	0.0682	0.0618	0.0548	0.0485
			2	Nie and Zhong [7]	0.0802	0.0751	0.0695	0.0604	0.0532
				Present (N_r=N_z=9)	0.079	0.0741	0.069	0.0619	0.0546
				Present (N_r=N_z=13)	0.0797	0.0754	0.0684	0.0613	0.0536
				Present (N_r=N_z=17)	0.08	0.075	0.068	0.0607	0.0533
				Present (N_r=N_z=19)	0.0803	0.075	0.068	0.0605	0.0531
	0.3	0.1	1	Nie and Zhong [7]	0.3152	0.2948	0.2687	0.2418	0.2166
				Present (N_r=N_z=9)	0.314	0.2959	0.2704	0.2431	0.2153
				Present (N_r=N_z=13)	0.3146	0.2951	0.2696	0.2425	0.2161
				Present (N_r=N_z=17)	0.315	0.2949	0.2691	0.242	0.2164
				Present (N_r=N_z=19)	0.3153	0.2949	0.2689	0.2416	0.2164
			2	Nie and Zhong [7]	0.4316	0.4039	0.3679	0.3304	0.2951
				Present (N_r=N_z=9)	0.4327	0.4029	0.3668	0.3317	0.2939
				Present (N_r=N_z=13)	0.4319	0.4037	0.3677	0.331	0.2945
				Present (N_r=N_z=17)	0.4314	0.4041	0.3682	0.3304	0.2952
				Present (N_r=N_z=19)	0.4314	0.4041	0.368	0.3304	0.295
		0.3	1	Nie and Zhong [7]	0.4565	0.4245	0.3922	0.36	0.329
				Present (N_r=N_z=9)	0.4551	0.4232	0.3939	0.36	0.3299
				Present (N_r=N_z=13)	0.456	0.4238	0.3926	0.3588	0.3293
				Present (N_r=N_z=17)	0.4563	0.4243	0.3921	0.3594	0.329
				Present (N_r=N_z=19)	0.4564	0.4243	0.392	0.36	0.329
			2	Nie and Zhong [7]	0.5198	0.4828	0.4442	0.4059	0.3693
				Present (N_r=N_z=9)	0.5209	0.4841	0.4457	0.4084	0.367
				Present (N_r=N_z=13)	0.5202	0.4833	0.4448	0.4071	0.3681
				Present (N_r=N_z=17)	0.5199	0.4828	0.4444	0.4064	0.3688
				Present (N_r=N_z=19)	0.5199	0.4826	0.4442	0.406	0.369
210	0.1	0.1	1	Nie and Zhong [7]	0.0438	0.0409	0.0369	0.0327	0.0287
				Present (N_r=N_z=9)	0.0422	0.042	0.0389	0.031	0.0299
				Present (N_r=N_z=13)	0.0433	0.0414	0.0378	0.032	0.029
				Present (N_r=N_z=17)	0.0437	0.0408	0.0371	0.0326	0.0287
				Present (N_r=N_z=19)	0.0437	0.0407	0.0371	0.0329	0.0287
			2	Nie and Zhong [7]	0.0552	0.0515	0.0463	0.0408	0.0357
				Present (N_r=N_z=9)	0.0571	0.0502	0.0488	0.0394	0.0331
				Present (N_r=N_z=13)	0.0559	0.0511	0.0475	0.0401	0.0345
				Present (N_r=N_z=17)	0.0552	0.0517	0.0468	0.0407	0.0351
				Present (N_r=N_z=19)	0.055	0.0517	0.0464	0.0408	0.0356
		0.3	1	Nie and Zhong [7]	0.0724	0.0678	0.0614	0.0546	0.0482
				Present (N_r=N_z=9)	0.0737	0.0661	0.0633	0.0567	0.0469
				Present (N_r=N_z=13)	0.0728	0.067	0.0623	0.0555	0.0477
				Present (N_r=N_z=17)	0.0722	0.0676	0.0618	0.0548	0.0481
				Present (N_r=N_z=19)	0.0722	0.0679	0.0615	0.0547	0.0481
			2	Nie and Zhong [7]	0.0787	0.0736	0.0667	0.0593	0.0522
				Present (N_r=N_z=9)	0.0807	0.0722	0.0683	0.0609	0.0503
				Present (N_r=N_z=13)	0.0797	0.073	0.0677	0.0602	0.0515
				Present (N_r=N_z=17)	0.079	0.0737	0.0671	0.0596	0.0521
				Present (N_r=N_z=19)	0.0786	0.0735	0.0669	0.0594	0.0523
	0.3	0.1	1	Nie and Zhong [7]	0.3103	0.2904	0.2648	0.2384	0.2137
				Present (N_r=N_z=9)	0.312	0.2918	0.2621	0.2403	0.2149
				Present (N_r=N_z=13)	0.3112	0.291	0.2636	0.2391	0.2141
				Present (N_r=N_z=17)	0.3105	0.2905	0.2644	0.2383	0.2131
				Present (N_r=N_z=19)	0.3101	0.2905	0.265	0.2384	0.2135
			2	Nie and Zhong [7]	0.4105	0.384	0.3495	0.3137	0.28
				Present (N_r=N_z=9)	0.4124	0.3854	0.3517	0.3155	0.2821
				Present (N_r=N_z=13)	0.4115	0.3846	0.3506	0.3147	0.2806
				Present (N_r=N_z=17)	0.4109	0.3842	0.3499	0.314	0.28
				Present (N_r=N_z=19)	0.4106	0.3842	0.3493	0.3138	0.2801
		0.3	1	Nie and Zhong [7]	0.4538	0.4221	0.3901	0.3582	0.3275
				Present (N_r=N_z=9)	0.4558	0.424	0.3922	0.3607	0.3299
				Present (N_r=N_z=13)	0.4549	0.4228	0.3911	0.3594	0.3287
				Present (N_r=N_z=17)	0.4544	0.4221	0.3904	0.3586	0.3279
				Present (N_r=N_z=19)	0.454	0.4221	0.39	0.3584	0.3277
			2	Nie and Zhong [7]	0.5077	0.4715	0.434	0.3968	0.3613
				Present (N_r=N_z=9)	0.5056	0.4701	0.4364	0.3962	0.3632
				Present (N_r=N_z=13)	0.5069	0.4711	0.4347	0.3978	0.3619
				Present (N_r=N_z=17)	0.5075	0.4716	0.4342	0.397	0.3612
				Present (N_r=N_z=19)	0.5076	0.4716	0.4342	0.3969	0.3612

According to the data presented in the above-mentioned tables, excellent solution agreements can be observed between the present method and those of the other methods. Fast rate of convergence of the method is quite evident, and it is found that only 19 grid points (N_r=N_z=19) along the z and r directions can yield accurate results. Based on the above studies, a numerical value of N_r=N_z=19 is used for the next studies.

After demonstrating the convergence and accuracy of the method, parametric studies for 3D vibration analysis of FG nanocomposite sandwich sectorial plates with different combinations of free, simply supported, and clamped boundary conditions at the circular edges, are computed. A comprehensive study is also carried out to investigate the effect of CNTs agglomeration on the vibrational response of sandwich structures.

Before analyzing the free vibration of FG nanocomposite sandwich plates, the effects of agglomeration degree ( $μ$ and $η$ ) on the effective longitudinal Young’s modulus and Poisson’s ratio of UD-CNTRC plate are investigated in Figures 8 and 9. Figure 8 represents the fact that the highest values of Young’s modulus are attained for the agglomeration state of $μ = η$ (fully dispersed), where the volume fraction of CNTs in the cluster and the matrix are equal. As it is observed, when $μ$ is less than $η$ ( $μ < η$ ), the effective Young’s modulus increases with increasing the value of $μ$ and has the maximum value when the CNTs are uniformly dispersed in the composite, i.e. $μ = η$ and for $μ > η$ , the effective stiffness decreases with the increase of $μ$ . The effect of agglomeration degree on the Poisson’s ratio for UD-plate is plotted in Figure 9. In contrast to Young’s modulus behavior, with the increase of $μ$ , Poisson’s ratio decreases for $μ < η$ and increases for $μ > η$ due to the fact that the Poisson’s ratio of the equivalent fiber described in properties of the equivalent fiber section is less than the Poisson’s ratio of matrix.

Figure 8.

Influence of CNT agglomeration parameters $μ$ and $η$ on the effective Young’s modulus of UD nanocomposite sandwich sectorial plate.

Figure 9.

Influence of CNT agglomeration parameters $μ$ and $η$ on the effective Poisson’s ratio of UD nanocomposite sandwich sectorial plate.

Using the relations presented in material properties of CNTRCs and problem description sections, it is also possible to observe the variations of the effective material properties through the thickness of the FGS-CNTR sectorial plate for different agglomeration parameters. For instance, by considering h_f/h = 0.3, a = 1, b = 0, p = 1, and V*=20%. The variations of CNT volume fraction through the thickness of FGS-CNTR sectorial plate can be found in Figure 10. The variations of Young’s modulus and Poisson’s ratio of FGS plates with respect to the different agglomeration parameters $μ$ and $η = 1$ are illustrated in Figures 11 and 12. As expected, at a constant value of z/h ratio, with the increase of parameter $μ$ , the effective Young’s modulus increases and on the other hand, the Poisson’s ratio decreases, because $μ < η = 1$ . It is also completely obvious that the agglomeration parameters have significant effects on the material properties. Therefore, it is concluded that the agglomeration of CNTs plays an important role in vibrational characteristics of FG-CNTR sectorial plates.

Figure 10.

Variations of CNT volume fraction through the thickness of FGS-CNTR sectorial plates.

Figure 11.

The variation of Young’s modulus along the thickness of the FG-CNTR sandwich sectorial plate with agglomeration effect.

Figure 12.

The variation of Poisson’s ratio along the thickness of the FG-CNTR sandwich sectorial plate with agglomeration effect.

Now free vibration characteristics of FG-CNTR sandwich sectorial plates is studied using Mori–Tanaka approach based on the equivalent fiber discussed in properties of the equivalent fiber section and using Table 1. Also, the material properties of the matrix are as

E^{m} = 2.1 GPa, ρ^{m} = 1150 {kg/m}^{3}, υ^{m} = 0.34

The non-dimensional natural frequency is as follows

Ω = ω \frac{a^{2}}{π^{2}} \sqrt{ρ_{m} h / D_{m}}, D_{m} = E_{m} h^{3} / 12 (1 - υ_{m}^{2})

(42)

The effect of agglomeration on vibrational response of FG sandwich sectorial plates by considering h_f/h = 0.3, a = 1, b = 0, p = 1, and V^*=20% (Figure 9) are depicted in Figures 13 to 15. It is clear that the lowest magnitude frequency parameter is obtained by using an FG plate, Type 4 and followed by Types 3, 2, 1 (Figure 6) and FG sandwich plate, respectively. It is also seen for great amount of $η$ , for instance $η = 1$ and for small amount of $μ$ , the FG sandwich sectorial plates have lower value of frequency than FG Type 1. It should be noted that for all the types of distributions of CNTs with the increase of $μ$ , the frequency parameters increase. It can be seen that the discrepancy between frequencies for the plates with Type 3 and Type 2 material distribution of CNTs remains almost unaltered with the increase of $μ$ .

Figure 13.

The variation of frequency parameters versus agglomeration parameter for different types of FG sectorial plates with F–C boundary condition (h/a=0.2, b/a=0.2, α=195°).

Figure 14.

The variation of frequency parameters versus agglomeration parameter for different types of FG-CNTR sectorial plates with S–C boundary condition (h/a=0.2, b/a=0.2, α=195°).

Figure 15.

The variation of frequency parameters versus agglomeration parameter for different types of FG-CNTR sectorial plates with C–C boundary condition (h/a=0.2, b/a=0.2, α=195°).

Figure 16 shows that for lower value of $η$ the frequency response of plates with different types of material distribution has changed. This figure also reveals that with the increase of $μ$ , the natural frequency increases, but for $μ$ approximately more than 0.5, the frequency parameters tend to decrease. It should be noted that this behavior has been seen in other two types of boundary condition but for brevity purpose they are not shown here.

Figure 16.

The variation of frequency parameters versus agglomeration parameter for different types of FG-CNTR sandwich sectorial plates with F–C boundary condition (η=0.5, h/a=0.2, b/a=0.2, α=195°).

In this section, the effect of material distributions and total volume percent of CNTs ( $V_{c n t}^{t o t a l}$ ) on the vibrational response of sandwich plates are investigated. It should be noted that various material profiles can be obtained by considering different amounts of parameters a, c, and p which are mentioned in Table 9. This table also shows the through-thickness variations of total volume percent of CNTs ( $V_{c n t}^{t o t a l}$ ) for various material profiles.

Table 9.

The through-thickness variations of total volume percent of CNTs ( $V_{c n t}^{t o t a l}$ ) for various material profiles and different amounts of parameters a, c, and p.

a	b	c	P
a	b	c	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0.25	0.2	2	9.70	11.55	11.91	11.98	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99
0.5	0.2	2	8.19	10.70	11.52	11.81	11.92	11.96	11.98	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99
1	0.2	2	5.19	7.503	8.69	9.40	9.86	10.18	10.42	10.60	10.75	10.86	10.96	11.04	11.11	11.17	11.23
1.25	0.2	2	3.69	5.15	5.69	5.77	5.80	5.80	5.80	5.99	5.99	5.99	5.99	5.99	5.99	5.99	5.99
0.2	0.2	2	9.99	11.66	11.94	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99
0.2	0.2	4	10.32	11.75	11.96	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99
0.2	0.2	8	10.53	11.80	11.97	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99
0.2	0.2	12	10.61	11.81	11.97	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99	11.99

The influence of parameters p and a on the fundamental frequency parameters of Free–Clamped, Supported–Clamped, and Clamped–Clamped FG-CNTR sandwich sectorial plates are shown in Figures 17 to 19. These figures show that with the increase of parameter p the non-dimensional natural frequency of sandwich plates increased. It should be taken into account that for small amount of parameter a, the frequency parameter steadily increased but for great amount of this parameter, there is a sharp increase of the frequency parameter. In order to have a better understanding, the material distribution profiles regarding Figures 17 to 19 are provided in Figures 20 to 22, respectively.

Figure 17.

Variation of the fundamental frequency parameter of FG-CNTR sandwich sectorial plates versus the power-law exponent P for various values of parameter a (η=0.5, h/a=0.2, b/a=0.2, α=195°).

Figure 18.

Variation of the fundamental frequency parameter of FG-CNTR sandwich sectorial plates versus the power-law exponent P for various values of parameter a (η=0.5, h/a=0.2, b/a=0.2, α=195°).

Figure 19.

Variation of the fundamental frequency parameter of FG-CNTR sandwich sectorial plates versus the power-law exponent P for various values of parameter a (η=0.5, h/a=0.2, b/a=0.2, α=195°).

Figure 20.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.25, b = 0.2, c = 2).

Figure 21.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.5, b = 0.2, c = 2).

Figure 22.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 1, b = 0.2, c = 2).

The influence of the parameter c on the free vibration of sandwich plates with FG-CNTR face sheets is investigated in Figure 23 when parameter c varies from 2 to 12. From Figure 23 it can be seen that with the increase of parameter c, the fundamental frequency increases. This is due to the fact that with the increase in the value of parameter c, the volume fraction of CNTs and therefore the frequency parameters increase. The material distribution profiles concerning Figure 23 are provided in Figures 24 to 27.

Figure 23.

Variation of the fundamental frequency parameter of FG-CNTR sandwich sectorial plates versus the power-law exponent P for various values of the parameter c (η=0.5, h/a=0.2, b/a=0.2, α=195°).

Figure 24.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.2, b = 0.2, c = 2).

Figure 25.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.2, b = 0.2, c = 4).

Figure 26.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.2, b = 0.2, c = 8).

Figure 27.

The material distribution profiles in the thickness direction of sandwich sectorial plates (a = 0.2, b = 0.2, c = 12).

The influences of the sector angle on the fundamental frequency parameter of FG sandwich sectorial plate with different circular edge conditions are shown in Figure 28. It is obvious that by increasing the sector angle, the frequency parameter decreases. Figure 28 also reveals that the Clamped–Clamped sandwich sectorial plate has the highest, whereas the Free–Clamped has the lowest frequency parameter values, implying that a sandwich sectorial plate with greater supporting rigidity will have higher vibrating frequencies.

Figure 28.

The influence of the sector angle on the non-dimensional natural frequency parameter of sandwich annular sector plates (b/a = h/a = 0.2, P = 1).

The variation of inner–outer radius ratio (b/a) with the frequency parameters of a Clamped–Clamped, Simply supported–Clamped, and Free–Clamped FG sandwich sectorial plates for different values of h/a ratios is shown in Figure 29. According to this figure, the general behavior of the frequency parameters of a sandwich sectorial plate for all b/a ratios is that the effects of the h/a ratios are more prominent at high inner-to-outer radius ratios. As it is observed, the frequency parameter decreases rapidly with the decrease of the b/a ratio and then remains almost unaltered for the b/a < 0.3.

Figure 29.

Variation of fundamental frequency parameters of sandwich annular sector plates versus b/a ratio for different boundary conditions at circular edges $(P = 1, μ = 0.2, η = 1, α = 195 °)$ .

Conclusion

In this research work, 2D GDQM is employed to obtain a highly accurate semi-analytical solution for free vibration of FG nanocomposite sandwich sectorial plates under various boundary conditions. The study is carried out based on the 3D, linear and small strain elasticity theory. The MT approach is implemented to estimate the effective material properties of the nanocomposite sandwich plate. The agglomeration effect of SWCNTs is considered in this study and it is shown that the natural frequencies of structure are seriously affected by the influence of agglomeration of the CNTs. Results presented the fact that mechanical properties and therefore free vibrations of FGS-CNTR plates are seriously affected by the agglomeration of CNTs. It was found that except some states, FGS types of structures improve the vibrational characteristics of CNTRCs. The effects of different boundary conditions, various geometrical parameters, different material profiles along the thickness of sandwich sectorial plates are also investigated in this study.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

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