1 Introduction

The specific mechanical properties of carbon nanotubes (CNTs) lead to utilize it instead of common reinforcements in composite structures. Because of great mechanical properties of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) structures, many studies are conducted to investigate the responses of these structures, which some of them are reviewed here. Singh and Bhar [1] studied vibration characteristics of CNTRC plates by using the higher-order shear deformation theory (HSDT). Shen [2] investigated nonlinear static bending of FGCNT plate with simply supported boundary conditions under different transverse loads in thermal surroundings. Tornabene et al. [3] studied the linear static response of nanocomposite plates and shells reinforced by agglomerated CNTs based on several HSDTs. Zhu et al. [4] presented a numerical procedure based on finite element method for the natural frequency and static bending analyses of FGCNT plates for various boundary conditions by applying the first order shear deformation plate theory (FSDT). Lei et al. [5] studied the natural frequency and static bending responses of rectangular plates by applying element-free Ritz method and plate theories. Nonlinear vibration of CNT-RC plates based on a HSDT theory in thermal surroundings was studied by Wang and Shen [6]. They used perturbation technique to solve the obtained governing equations. Ansari et al. [7] performed natural frequency analysis of FG-CNTRC quadrilateral plates in thermal surroundings by using elasticity theory. The vibration characteristics of composite plates with FGCNT layers with arbitrary quadrilateral shape by using the FSDT and the differential quadrature method (DQM) were studied by Malekzadeh and Zarei [8]. Malekzadeh and Heydarpour [9] presented the natural frequency and static bending responses of simply supported composite plates with FGCNTRC layers by applying the Navier-layer wise and DQ methods. Natarajan et al. [10] used HSDT and normal deformable plate theory for static and natural frequency analysis of FGCNT plate and sandwich plates containing face sheets reinforced with CNTs. Alibeigloo and Emtehani [11] studied the natural frequency and static bending behavior of FGCNT rectangular plates under transverse uniform pressure for various boundary conditions by using DQM. Based on von-Karman assumptions and HSDT plate theory, nonlinear dynamic bending behavior of FGCNT plate supported on elastic medium in thermal surroundings were investigated by Wang and Shen [12]. Alibeigloo [13] by using an analytical solution based on theory of elasticity studied static bending response of FGCNT plate with piezoelectric layers. Phung-Van et al. [14] by using HSDT and isogeometric analysis methods studied static and dynamic bending response of the FGCNT plates with various essential boundary conditions.

Literature review shows the existence of a large number of researches on the response of FGCNTRC beams and rectangular plates. However, circular, sector, annular and annular sector plates are used in different engineering structures to sustain different static and dynamic loads. Hence, it is necessary to investigate response of these FGCNT structures under static and dynamic loads to satisfy the design requirements. In the literature, analyses correspond to FGCNT circular, annular and annular sector plates are limited in number. For example, Keleshteri et al. [15] and Mohammadzadeh-Keleshteri et al. [16] by using FSDT plate theory, von Karman starin-displacement assumptions, Hamilton principle and the generalized DQM studied the nonlinear vibration response of FGCNT annular sector plates with piezoelectric layers. Zhong et al. [17] based on FSDT and by using weak form approach and Ritz-variational energy method investigated vibration analysis of FGCNT annular and sector plates.

The literature review illustrates that a detailed study including; static, dynamic and free vibration analyses of FGCNT annular sector plate have not been investigate so far. Therefore, in this paper, static, dynamic and natural frequency behavior of composite FGCNT annular sector plates resting on viscoelastic foundation is investigated. In Sect. 2, geometry of plate resting on Kelvin-voight viscoelastic foundation and material properties of CNTRC are presented. The CNTs are considered to have uniform or functionally graded distributions in the plate thickness. The mechanical properties of composite annular sector plate composed of CNTs and a polymer matrix are evaluated by using the rule of mixtures. Then, the governing equations based on FSDT are derived and, finite element method is used to solve the governing equation of FGCNT plate. A 4-node two-dimensional element with 20 degree of freedom is applied to mesh the domain. In Sect. 3, first, static response of FGCNT annular sector plate have been verified by using data of a FGCNT square plate, and then, the effects of different factors such as distribution and volume fraction of CNTs, different boundary conditions, stiffness and damping coefficients of viscoelastic foundation and sector angles on natural frequency, static and transient dynamic responses of the plate have been studied.

2 Theoretical formulations

2.1 Description of the Geometry

An FGCNT annular sector plate resting on viscoelastic foundation with thickness h, angle of sector θ0, inner and outer radii R0 and R1 is considered (Fig. 1). The cylindrical coordinates r, θ and z at the mid-plane of the plate are used.

Fig. 1
figure 1

Description of geometry and coordinate system of FGCNTRC annular sector plate

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2.2 Material properties of CNTRC

In this study, the idea of FGMs is implemented to the nanocomposite structures reinforced by CNTs with low CNTs volume fractions. The carbon nanotubes are considered to have uniform or FG distributions in the plate thickness. UD-CNTRC denotes the uniform distribution of CNTs and FGX, FGO and FGV-CNTRC show the FG patterns of CNTs through the plate thickness (Fig. 2). The mechanical properties of CNTs are considered to be size-dependent and are estimated from molecular dynamics (MD) simulations. The mechanical properties of FG CNTRCs are obtained by using a micromechanical model in which the CNTs efficiency parameter is obtained from the MD simulation with the results estimated from the rule of mixture [2]. The effective material properties may be estimated by Mori–Tanaka scheme or the rule of mixture. The Mori–Tanaka scheme is applicable to nanoparticles and the rule of mixture is simple and convenient to apply for predicting the overall material properties and responses of the structures. Thus, in this study, the effective mechanical properties of plate, mixtures of isotropic polymeric matrix and CNTs are evaluated by using the rule of mixtures [18] as shown in Eq. (1). In this model, it is assumed that both CNTs and polymer are very well-bonded and equally strained [2].

$$\begin{aligned} \rho & = V_{CN} \rho^{CN} + V_{m} \rho^{m} \\ E_{11} & = \eta_{1} V_{CN} E_{11}^{CN} + V_{m} E^{m} \, \\ \frac{{\eta_{2} }}{{E_{22} }} & = \frac{{V_{CN} }}{{E_{22}^{CN} }} + \frac{{V_{m} }}{{E^{m} }},\frac{{\eta_{3} }}{{G_{12} }} = \frac{{V_{CN} }}{{G_{12}^{CN} }} + \frac{{V_{m} }}{{G^{m} }} \\ \upsilon_{12} & = V_{CN} \upsilon_{12}^{CN} + V_{m} \upsilon^{m} \\ \end{aligned}$$
(1)

where \({\text{E}}_{11}^{\text{CN}} ,{\text{E}}_{22}^{\text{CN}} ,{\text{G}}_{12}^{\text{CN}} ,\upsilon_{12}^{CN}\) are modulus of elasticity, shear modulus and Poisson’s ratio of carbon nanotubes, respectively. Also, \({\text{E}}^{\text{m}} ,{\text{G}}^{\text{m}}\) and \(\upsilon^{m}\) are the same properties of isotropic polymer matrix. \(\rho^{m}\) and \(\rho^{CN}\) are the mass density of matrix and CNTs, respectively. \({\text{V}}_{\text{CN }}\) is the volume fraction of CNTs, and \({\text{V}}_{\text{m}}\) is the volume fraction of polymer matrix (\({\text{V}}_{\text{CN }} + {\text{V}}_{\text{m}} = 1\)). \({\text{V}}_{\text{CN }}\) for different distribution of CNTs are shown in Table 1. CNTs efficiency parameters \(\eta_{i} ,\;i = 1,2,3\) are given in Table 2 [19, 20].

Fig. 2
figure 2

Different kinds of CNTs distribution

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Table 1 Mathematical representation of CNT distributions
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Table 2 CNTs efficiency parameters for different values of \({\text{V}}_{\text{CN}}^{ *}\)
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2.3 Governing equations

According to the FSDT plate theory, the displacement components of annular sector plate are considered as [21]:

$$\begin{aligned} u(r,\theta ,z) & = u_{0} (r,\theta ) + z\varphi_{r} (r,\theta ) \\ v(r,\theta ,z) & = v_{0} (r,\theta ) + z\varphi_{r} (r,\theta ) \\ w(r,\theta ,z) & = w_{0} (r,\theta ) \\ \end{aligned}$$
(2)

In Eq. (2), u, v, w, are displacement components along the r, θ and z directions, respectively, while \({\text{u}}_{0} ,{\text{v}}_{0} ,{\text{w}}_{0}\) are the same at the mid-plane. Also, \(\varphi_{r}\) and \(\varphi_{\theta }\) are respectively normal transverse rotations around r and θ. The corresponding strains related to the displacement field are defined as follows:

$$\begin{aligned} \varepsilon_{r} & = \varepsilon_{r}^{0} + zk_{r}^{k} ,\varepsilon_{\theta } = \varepsilon_{\theta }^{0} + zk_{\theta }^{k} \\ \gamma_{r\theta } & = \gamma_{r\theta }^{0} + zk_{r\theta } ,\gamma_{rz} = \gamma_{rz}^{0} ,\gamma_{\theta z} = \gamma_{\theta z}^{0} \\ \end{aligned}$$
(3)

where

$$\begin{aligned} \varepsilon_{r}^{0} & = \frac{{\partial u_{0} }}{\partial r},\varepsilon_{\theta }^{0} = \frac{{u_{0} }}{r} + \frac{1}{r}\frac{{\partial v_{0} }}{\partial \theta } \\ k_{r} & = \frac{{\partial \varphi_{r} }}{\partial r},k_{\theta } = \frac{1}{r}\frac{{\partial \varphi_{\theta } }}{\partial \theta } + \frac{{\varphi_{r} }}{r},k_{r\theta } = \frac{{\partial \varphi_{\theta } }}{\partial r} + \frac{1}{r}\frac{{\partial \varphi_{r} }}{\partial \theta } - \frac{{\varphi_{\theta } }}{r} \\ \gamma_{rz}^{0} & = \frac{\partial w}{\partial r} + \varphi_{r} ,\gamma_{\theta z}^{0} = \frac{1}{r}\frac{\partial w}{\partial \theta } + \varphi_{\theta } \frac{\partial w}{\partial \theta } + \varphi_{\theta } ,\gamma_{r\theta }^{0} = \frac{{\partial v_{0} }}{\partial r} + \frac{1}{r}\frac{{\partial u_{0} }}{\partial \theta } - \frac{{v_{0} }}{r} \\ \end{aligned}$$
(4)

The matrix form of the strain field is as follows

$$\begin{aligned} \left[ {\begin{array}{*{20}c} {\varepsilon_{r} } \\ {\varepsilon_{\theta } } \\ {\gamma_{r\theta } } \\ \end{array} } \right] = \left\{ {\begin{array}{*{20}c} {\varepsilon_{r}^{0} } \\ {\varepsilon_{\theta }^{0} } \\ {\gamma_{r\theta }^{0} } \\ \end{array} } \right\} + z\left\{ {\begin{array}{*{20}c} {k_{r} } \\ {k_{\theta } } \\ {k_{r\theta } } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial r}} & 0 & 0 & {z\frac{\partial }{\partial r}} & 0 \\ {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & {z\frac{1}{r}} & {\frac{z}{r}\frac{\partial }{\partial \theta }} \\ {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\frac{\partial }{\partial r} - \frac{1}{r}} & 0 & {\frac{z}{r}\frac{\partial }{\partial \theta }} & {z\left( {\frac{\partial }{\partial r} - \frac{1}{r}} \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{0} } \\ {v_{0} } \\ {w_{0} } \\ {\varphi_{r} } \\ {\varphi_{\theta } } \\ \end{array} } \right] = d_{1} Q \\ \left\{ {\begin{array}{*{20}c} {\gamma_{rz}^{0} } \\ {\gamma_{\theta z}^{0} } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & {\frac{\partial }{\partial r}} & 1 & 0 \\ 0 & 0 & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & 1 \\ \end{array} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{0} } \\ {v_{0} } \\ {w_{0} } \\ {\varphi_{r} } \\ {\varphi_{\theta } } \\ \end{array} } \right] = d_{2} Q \\ \end{aligned}$$

where

$$\begin{aligned} Q = \left[ {\begin{array}{*{20}c} {u_{0} } \\ {v_{0} } \\ {w_{0} } \\ {\varphi_{r} } \\ {\varphi_{\theta } } \\ \end{array} } \right], d_{1} & = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial r}} & 0 & 0 & {z\frac{\partial }{\partial r}} & 0 \\ {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & {z\frac{1}{r}} & {\frac{z}{r}\frac{\partial }{\partial \theta }} \\ {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\frac{\partial }{\partial r} - \frac{1}{r}} & 0 & {\frac{z}{r}\frac{\partial }{\partial \theta }} & {z\left( {\frac{\partial }{\partial r} - \frac{1}{r}} \right)} \\ \end{array} } \right], d_{2} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & {\frac{\partial }{\partial r}} & 1 & 0 \\ 0 & 0 & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & 1 \\ \end{array} } \\ \end{array} } \right], \\ \left\{ {\begin{array}{*{20}c} {\varepsilon_{r}^{0} } \\ {\varepsilon_{\theta }^{0} } \\ {\gamma_{r\theta }^{0} } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial r}} & 0 & 0 & 0 & 0 \\ {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & 0 & 0 \\ {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\frac{\partial }{\partial r} - \frac{1}{r}} & 0 & 0 & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{0} } \\ {v_{0} } \\ {w_{0} } \\ {\varphi_{r} } \\ {\varphi_{\theta } } \\ \end{array} } \right] = d_{3} Q \\ \left\{ {\begin{array}{*{20}c} {k_{r} } \\ {k_{\theta } } \\ {k_{r\theta } } \\ \end{array} } \right\} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {\frac{\partial }{\partial r}} & 0 \\ 0 & 0 & 0 & {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} \\ 0 & 0 & 0 & {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\left( {\frac{\partial }{\partial r} - \frac{1}{r}} \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {u_{0} } \\ {v_{0} } \\ {w_{0} } \\ {\varphi_{r} } \\ {\varphi_{\theta } } \\ \end{array} } \right] = d_{4} Q \\ \end{aligned}$$
(5)

where

$$d_{3} = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial r}} & 0 & 0 & 0 & 0 \\ {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} & 0 & 0 & 0 \\ {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\frac{\partial }{\partial r} - \frac{1}{r}} & 0 & 0 & 0 \\ \end{array} } \right],d_{4} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {\frac{\partial }{\partial r}} & 0 \\ 0 & 0 & 0 & {\frac{1}{r}} & {\frac{1}{r}\frac{\partial }{\partial \theta }} \\ 0 & 0 & 0 & {\frac{1}{r}\frac{\partial }{\partial \theta }} & {\left( {\frac{\partial }{\partial r} - \frac{1}{r}} \right)} \\ \end{array} } \right],$$

The relation between stress and strain in an orthotropic structure is as:

$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {\sigma_{r} } \\ {\sigma_{\theta } } \\ {\tau_{r\theta } } \\ \end{array} } \right\}^{k} = \left[ {\begin{array}{*{20}l} {C_{11} } \hfill & {C_{12} } \hfill & 0 \hfill \\ {C_{12} } \hfill & {C_{22} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {C_{66} } \hfill \\ \end{array} } \right]^{k} \left\{ {\begin{array}{*{20}c} {\varepsilon_{r} } \\ {\varepsilon_{\theta } } \\ {\gamma_{r\theta } } \\ \end{array} } \right\}^{k} \hfill \\ \left\{ {\begin{array}{*{20}c} {\tau_{rz} } \\ {\tau_{\theta z} } \\ \end{array} } \right\}^{k} = \left[ {\begin{array}{*{20}c} {C_{44} } & 0 \\ 0 & {C_{55} } \\ \end{array} } \right]^{k} \left\{ {\begin{array}{*{20}c} {\gamma_{rz} } \\ {\gamma_{\theta z} } \\ \end{array} } \right\}^{k} \hfill \\ C_{11} = \frac{{E_{11} }}{{1 - v_{12} v_{21} }},C_{12} = \frac{{v_{12} E_{22} }}{{1 - v_{12} v_{21} }} = \frac{{v_{21} E_{11} }}{{1 - v_{12} v_{21} }} \hfill \\ C_{22} = \frac{{E_{22} }}{{1 - v_{12} v_{21} }},C_{44} = G_{13} ,C_{55} = G_{23} ,C_{66} = G_{12} \hfill \\ \end{aligned}$$
(6)

The resultants of moment and force are given by integrating the stress components through the z direction:

$$\begin{aligned} \left\{ {\begin{array}{*{20}c} {N_{r} } \\ {N_{\theta } } \\ {N_{r\theta } } \\ \end{array} } \right\} = \mathop \int \limits_{{{\raise0.7ex\hbox{${ - h}$} \!\mathord{\left/ {\vphantom {{ - h} 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}^{{{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} {\sigma_{r} } \\ {\sigma_{\theta } } \\ {\tau_{r\theta } } \\ \end{array} } \right]} \\ \end{array} dz,\quad \left\{ {\begin{array}{*{20}c} {M_{r} } \\ {M_{\theta } } \\ {M_{r\theta } } \\ \end{array} } \right\} = \mathop \int \limits_{{{\raise0.7ex\hbox{${ - h}$} \!\mathord{\left/ {\vphantom {{ - h} 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}^{{{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \left[ {\begin{array}{*{20}c} {\sigma_{r} } \\ {\sigma_{\theta } } \\ {\tau_{r\theta } } \\ \end{array} } \right]zdz \hfill \\ \left\{ {\begin{array}{*{20}c} {Q_{r} } \\ {Q_{\theta } } \\ \end{array} } \right\} = {\text{K}}^{2} \mathop \int \limits_{{{\raise0.7ex\hbox{${ - h}$} \!\mathord{\left/ {\vphantom {{ - h} 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}^{{{\raise0.7ex\hbox{$h$} \!\mathord{\left/ {\vphantom {h 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \left[ {\begin{array}{*{20}c} {\tau_{rz} } \\ {\tau_{\theta z} } \\ \end{array} } \right]dz = K^{2} \mathop \sum \limits_{K = 1}^{n} \mathop \int \limits_{zk - 1}^{zk} \left\{ {\begin{array}{*{20}c} {\tau_{rz} } \\ {\tau_{\theta z} } \\ \end{array} } \right\}^{\left( k \right)} dz \hfill \\ \end{aligned}$$
(7)

where K = 5/6 is the shear correction factor.

By the integral from Eq. (7) along the thickness:

$$\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {N_{r} } \\ {N_{\theta } } \\ {N_{r\theta } } \\ {M_{r} } \\ {M_{\theta } } \\ \end{array} } \\ {M_{r\theta } } \\ {Q_{r} } \\ {Q_{\theta } } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {A_{11} } \hfill & {A_{12} } \hfill & 0 \hfill & {B_{11} } \hfill & {B_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {A_{12} } \hfill & {A_{22} } \hfill & 0 \hfill & {B_{12} } \hfill & {B_{22} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {A_{66} } \hfill & 0 \hfill & 0 \hfill & {B_{66} } \hfill & 0 \hfill & 0 \hfill \\ {B_{11} } \hfill & {B_{12} } \hfill & 0 \hfill & {D_{11} } \hfill & {D_{12} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {B_{12} } \hfill & {B_{22} } \hfill & 0 \hfill & {D_{12} } \hfill & {D_{22} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {B_{66} } \hfill & 0 \hfill & 0 \hfill & {D_{66} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {K^{2} A_{44} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {K^{2} A_{55} } \hfill \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{r}^{0} } \\ {\varepsilon_{\theta }^{0} } \\ {\gamma_{r\theta }^{0} } \\ {k_{r} } \\ {k_{\theta } } \\ {k_{r\theta } } \\ {\gamma_{rz}^{0} } \\ {\gamma_{\theta z}^{0} } \\ \end{array} } \right\}$$
(8)

where:

$$\left( {A_{ij} ,B_{ij} ,D_{ij} } \right) = \int\limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} {C_{ij} \left( {1,z,z^{2} } \right)dz}$$
(9)

The strain and kinetic energies for the annular sector plate is expressed as

$$\begin{aligned} \delta U = \delta U_{1} & = \frac{1}{2}\iiint {\varepsilon^{T} \sigma dV} = \iint {\left\{ {\begin{array}{*{20}c} {\varepsilon_{r}^{0} + N_{\theta } \varepsilon_{\theta }^{0} + N_{r\theta } \gamma_{r\theta }^{0} + M_{r} K_{r} + M_{\theta } K_{\theta } + } \\ {M_{r\theta } K_{r\theta } + Q_{r} \gamma_{rz} + Q_{\theta } \gamma_{\theta z} } \\ \end{array} } \right\}rdrd\theta } \\ & = \int {\left( {\left( {\left( {d_{3} Q} \right)^{T} A^{T} + \left( {d_{4} Q} \right)^{T} B^{T} } \right)(d_{3} \delta Q) + \left( {\left( {d_{3} Q} \right)^{T} B^{T} + \left( {d_{4} Q} \right)^{T} D^{T} } \right)(d_{4} \delta Q)} \right.} \\ \quad \left. { + \left( {d_{2} Q} \right)^{T} e^{T} \left( {d_{2} \delta Q} \right)} \right)rdrd\theta \\ \delta T & = \mathop \int \limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \rho \left( {\ddot{u}\delta u + \ddot{v}\delta v + \ddot{w}\delta w} \right)dV \\ \end{aligned}$$
(10)

where:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {\delta u = \delta u_{0} + z\delta \varphi_{r} } \hfill \\ { \delta v = \delta v_{0} + z\delta \varphi_{\theta } } \hfill \\ { \delta w = \delta w_{0} } \hfill \\ \end{array} } \right.\quad \quad \left\{ {\begin{array}{*{20}l} {\ddot{u} = \frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{r} }}{{\partial t^{2} }}} \hfill \\ {\ddot{v} = \frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{\theta } }}{{\partial t^{2} }}} \hfill \\ {\ddot{w} = \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}} \hfill \\ \end{array} } \right. \\ & T = \int {\int\limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} {\rho \left( {\left( {\frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{r} }}{{\partial t^{2} }}} \right)\left( {\delta u_{0} + z\delta \varphi_{r} } \right) + \left( {\frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{\theta } }}{{\partial t^{2} }}} \right)\left( {\delta v_{0} + z\delta \varphi_{\theta } } \right) + \frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}\delta w_{0} } \right)dzdA} } \\ \end{aligned}$$
(11)

By substituting Eqs. (10), (11) in Hamilton principle, we have:

$$\begin{aligned} & \int\limits_{{t_{1} }}^{{t_{2} }} {\left[ {\mathop {\iint }\limits_{{ - \frac{h}{2}}}^{{\frac{h}{2}}} \rho \left( {\left( {\frac{{\partial^{2} u_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{r} }}{{\partial t^{2} }}} \right)\left( {\delta u_{0} + z\delta \varphi_{r} } \right) + \left( {\frac{{\partial^{2} v_{0} }}{{\partial t^{2} }} + z\frac{{\partial^{2} \varphi_{\theta } }}{{\partial t^{2} }}} \right)\left( {\delta v_{0} + z\delta \varphi_{\theta } } \right)} \right.} \right.} \\ & \quad + \left. {\frac{{\partial^{2} w_{0} }}{{\partial t^{2} }}\delta w_{0} } \right)dzdA \\ & \quad + \int {\left( {\left( {\left( {d_{3} Q} \right)^{T} A^{T} + \left( {d_{4} Q} \right)^{T} B^{T} } \right)(d_{3} \delta Q) + \left( {\left( {d_{3} Q} \right)^{T} B^{T} + \left( {d_{4} Q} \right)^{T} D^{T} } \right)(d_{4} \delta Q)} \right.} \\ & \left. {\quad \left. { + \left( {d_{2} Q} \right)^{T} e^{T} \left( {d_{2} \delta Q} \right)} \right)rdrd\theta - \delta W} \right]dt = 0 \\ \end{aligned}$$
(12)

\(\delta W\) is virtual work of external forces, where the plate is subjected to a transverse pressure \(P_{z}\) is as following equation:

$$\delta W = \int P_{z} \delta w_{0} rdrd\theta$$
(13)

2.4 Finite element model of governing equations

Finite element method is used to solve the governing equation of FGCNT plate. A 4-node two-dimensional element with 20 degree of freedom is applied to mesh the solution domain (Fig. 3). Also, a local-coordinate system (\(\xi ,\eta\)) is used to express shape functions.

Fig. 3
figure 3

The schematic of the meshed annular sector plate and the natural coordinates

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The relationship between the natural and the global coordinates is as [22, 23]:

$$\xi = \frac{{2r - a^{\left( e \right)} - b^{\left( e \right)} }}{{\left( {b^{\left( e \right)} - a^{\left( e \right)} } \right)}},\quad \eta = \frac{{2\left( {\theta - \theta_{c} } \right)}}{{\beta^{\left( e \right)} }}$$
(14)

where \(- 1 \le \xi ,\,\,\eta \, \le 1\) are through the r and θ directions. \(a^{(e)} ,b^{(e)}\) and \(\beta^{(e)}\) are the inner and outer radii and angle of sector of element, respectively. Also, \(\theta_{c}\) is the circumferential coordinate of center of element.

The shape functions in the natural coordinates and the displacement vector of element are defined as follow

$$\left\{ {\begin{array}{*{20}c} {\varPsi_{i} } \\ {\varPsi_{j} } \\ {\varPsi_{k} } \\ {\varPsi_{m} } \\ \end{array} } \right\} = \frac{1}{4}\left\{ {\begin{array}{*{20}c} {\left( {1 + \xi } \right)\left( {1 - \eta } \right)} \\ {\left( {1 + \xi } \right)\left( {1 + \eta } \right)} \\ {\left( {1 - \xi } \right)\left( {1 + \eta } \right)} \\ {\left( {1 - \xi } \right)\left( {1 - \eta } \right)} \\ \end{array} } \right\}$$
(15)
$$Q^{\left( e \right)} = \left( {\left( {\begin{array}{*{20}c} {\varPsi_{1} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\varPsi_{1} } \\ \end{array} } \right) \ldots \left( {\begin{array}{*{20}c} {\varPsi_{4} } & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & {\varPsi_{4} } \\ \end{array} } \right)} \right)\left\{ {\begin{array}{*{20}c} {u_{01} } \\ {v_{01} } \\ {w_{01} } \\ {\varphi_{r1} } \\ {\varphi_{\theta 1} } \\ \vdots \\ {u_{04} } \\ {v_{04} } \\ {w_{04} } \\ {\varphi_{r4} } \\ {\varphi_{\theta 4} } \\ \end{array} } \right\} = \varPsi q(e)$$
(16)

where \(\varPsi_{n} , {\text{n}} = 1,2,3,4\) are the shape functions, and \(\varPsi\) is the matrix of shape functions. \(u_{0i} ,v_{0i} ,w_{0i} ,\varphi_{ri}\) and \(\varphi_{\theta i}\) are nodal degrees of freedom and are approximated as

$$\begin{aligned} u_{0} = \mathop \sum \limits_{i = 1}^{4} \varPsi_{i} U_{0i} ,v_{0} = \mathop \sum \limits_{i = 1}^{4} \varPsi_{i} V_{0i} ,w_{0} = \mathop \sum \limits_{i = 1}^{4} \varPsi_{i} W_{0i} \hfill \\ \varphi_{r} = \mathop \sum \limits_{i = 1}^{4} \varPsi_{i} \theta_{ri} ,\varphi_{\theta } = \mathop \sum \limits_{i = 1}^{4} \varPsi_{i} \theta_{\theta i} \hfill \\ \end{aligned}$$
(17)

Substituting Eq. (17) in Eq. (12) can be rewritten as

$$\begin{aligned} \int_{{\varOmega_{0}^{e} }} {\left[ {\left( {\left( {d_{3} \varPsi } \right)^{T} A^{T} d_{3} \varPsi + \left( {d_{4} \varPsi } \right)^{T} B^{T} d_{3} \varPsi + \left( {d_{3} \varPsi } \right)^{T} B^{T} d_{4} \varPsi + \left( {d_{4} \varPsi } \right)^{T} D^{T} d_{4} \varPsi + } \right.} \right.} \hfill \\ \left. {\left. {\left( {d_{2} \varPsi } \right)^{T} e^{T} d_{2} \varPsi } \right)d + \varPsi^{T} I\varPsi \ddot{q} - \varPsi^{T} P} \right]rdrd\theta = 0 \hfill \\ \end{aligned}$$
(18)

where \(P = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ {\begin{array}{*{20}c} {P_{z} } \\ 0 \\ 0 \\ \end{array} } \\ \end{array} } \right],d_{2} \varPsi = B_{2} , d_{3} \varPsi = B_{3} , d_{4} \varPsi = B_{4}\)

By sorting the above equation, the following equation is obtained for the each annular sector element:

$$\left( {k_{1} + k_{2} + k_{3} } \right)^{\left( e \right)} q^{\left( e \right)} + M^{e} \ddot{q}^{\left( e \right)} = F^{e}$$
(19)

Finally, after sum of the stiffness, mass and force element matrices, the finite element equations of the FGCNT plate is as

$$\left( {k_{1} + k_{2} + k_{3} } \right)q + M\ddot{q} = F$$
(20)

For the case that the plate is resting on the Kelvin-voight linear viscoelastic foundation, the relationship between force per unit area and deflection can be calculated according to the following equation [24]:

$$P_{v} = k_{w} w + c_{d} \frac{\partial w}{\partial t}$$
(21)

where \({\text{k}}_{\text{w}}\) is the elastic coefficient of the foundation in terms of (N/m3), and \({\text{C}}_{\text{d}}\) is the damping coefficient of the foundation in terms of (N s/m3). Therefore, in this case, by adding the effects of viscoelastic foundation on the plate, the governing Eq. (20) can be rearranged as:

$$\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right)q + M\ddot{q} + C\dot{q} = F$$
(22)

where \({\text{k}}_{4}\) is the stiffness matrix due to the elastic properties of the foundation, and C is the damping matrix due to damping property of foundation. The mass, stiffness, damping and force matrices are given in “Appendix”.

The considered essential boundary conditions of plate are as:

All edges clamped (CCCC):

$$u_{0} ,\,v_{0} ,\,w_{0} ,\,\varphi_{r} ,\,\varphi_{\theta } = 0\quad at\left( {r,0} \right),\left( {r,\theta_{0} } \right),\left( {a,\theta } \right),\left( {b,\theta } \right)$$
(23)

Clamped radial edges (FCFC):

$$u_{0} ,\,v_{0} ,\,w_{0} ,\,\varphi_{r} ,\,\varphi_{\theta } = 0\quad at\,\left( {r,0} \right),\left( {r,\theta_{0} } \right) = 0$$
(24)

Clamped circumferential edges (CFCF):

$$u_{0} ,\,v_{0} ,\,w_{0} ,\,\varphi_{r} ,\,\varphi_{\theta } = 0\quad at\;\left( {a,\theta } \right),\left( {b,\theta } \right)$$
(25)

Simply supported edges (SSSS):

$$\begin{aligned} u_{0} ,\,w_{0} = 0\quad at\;\left( {a,\theta } \right),\left( {b,\theta } \right) \hfill \\ v_{0} ,\,w_{0} = 0\quad at\;\left( {r,0} \right),\left( {r,\theta_{0} } \right) \hfill \\ \end{aligned}$$
(26)

Finally, Newmark integeration method [25] is applied to solve the Eq. (22) with respect to time. The natural frequency analysis of the plate converts to the solution of the eigen value problem as

$$\left( {\left( {k_{1} + k_{2} + k_{3} + k_{4} } \right) - M\omega^{2} } \right)q = 0$$
(27)

where \(\omega\) is the circular natural frequency and q is the vibration mode shapes.

3 Results and discussions

In this part, results of static, natural frequency and dynamic transient responses of FGCNT annular sector plate resting on viscoelastic foundation have been presented. The effect of various boundary conditions, stiffness and damping coefficients of viscoelastic foundation, CNT distributions, volume fraction of CNTs, slenderness ratio and sector angle have been studied. The mechanical and geometrical properties of FGCNT annular sector plate are presented in Table 3.

Table 3 Mechanical and geometrical properties of FGCNT annular sector plate
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3.1 Static analysis

3.1.1 verification

In this section, static response of FGCNT annular sector plate have been verified by using data of a FGCNT square plate [4]. Table 4 shows the effect of VCN and different distributions of nanotubes on the non-dimensional central deflection \(\tilde{W} = w_{0} /h\) of FGCNT square plates with CCCC edges. Therefore, the sector angle is assumed as a small value θ0 = 0.001 rad, and inner and outer radiuses of plate are chosen as large values: b = 200 m, a = 200.2 m and h = 0.02 m. These geometric dimensions lead to nearly a square plate with length-to-thickness ratio of 10:1. Also, the material properties are considered as previous section. Comparison between results in Table 4 shows excellent agreement between them.

Table 4 Effect of \(V_{CN}\) on the \(\tilde{W} = w_{0} /h\) of FGCNT square plate with all edges clamped (CCCC) subjected to an uniform pressure P = 0.1 MPa (a = 200 m, b = 200.2 m, h = 0.02 m, α = 0.001 rad)
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3.1.2 Static analysis of FGCNT annular sector plate

In this part, static response of FGCNT annular sector plate without foundation under a transverse pressure P = 1 MPa is investigated. The effects of \(V_{CN}\) on the non-dimensional central deflection \(\tilde{W} = w_{0} /h\) are shown in Table 5. Also, the effects of CNTs distribution and sector angle have been investigated. Table 5 denotes that \(V_{CN}\) has great influence on the deflection of plate. Increasing \(V_{CN}\) from 0.11 to 0.17 leads to more than 36% decrease in the deflection. This is because that by increasing \(V_{CN}\), stiffness of plate enhances. Also, results show that the maximum and minimum deflection belong to the FGO and FGX distributions, respectively. Therefore, it can be deduced that CNTs distribution close to upper and lower surface of plate are more appropriate than those distributions concentrated on near the mid-plane for enhancing the overall stiffness of plate. Also, results present that as the sector angle increases from 30° to 60°, central deflection of plate is also increases, and for 60° to 240°, it is almost identical. Table 6 shows the effect of different boundary conditions on the non-dimensional central deflection. In this case: θ0 = 120°, h = 0.05 m, VCN = 0.11. Results denote that the maximum and minimum non-dimensional central deflection is related to FCFC and CFCF boundary conditions, respectively. Also, comparisons between Tables 5 and 6 show that by decreasing the thickness of plate, deflection significantly is increased.

Table 5 Effects of \(V_{CN}\) on \(\tilde{W} = w_{0} /h\), P = 1 MPa, CCCC, θ0 = 30°, 60°,120°, 240° (a = 1 m, b = 2 m, h = 0.1 m)
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Table 6 Effects of different boundary conditions on \(\tilde{W} = w_{0} /h\), P = 1 MPa, θ0 = 120° (a = 1 m, b = 2 m, h = 0.05 m), VCN = 0.11
Full size table

3.2 Natural frequency analysis

3.2.1 Verification

In this section, natural frequency of FGCNT annular sector plate has been validated by using data of FGCNT square plate [4]. Table 7 shows the effects of VCN and distributions of carbon nanotubes on the non-dimensional natural frequency \(\tilde{\omega } = \omega (a^{2} /h)\sqrt {\rho^{m} /E^{m} }\) of CNTRC square plates with CCCC edges. Geometric dimensions and the mechanical properties are considered same as the previous sections. Comparison between results in Table 7 shows excellent agreement between them.

Table 7 Effects of VCN on \(\tilde{\omega } = \omega (a^{2} /h)\sqrt {\rho^{m} /E^{m} }\) of FGCNT square plate with all edges clamped (CCCC) (a = 200 m, b = 200.2 m, h = 0.02 m, θ0 = 0.001 rad)
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3.2.2 Natural frequency analysis of FGCNT annular sector plate

In this part, free vibration analysis of FGCNT annular sector plate without foundation is investigated. Table 8 shows the effects of VCN on the natural frequencies of CCCC plate. Also, the effects of CNTs distribution and sector angle have been investigated. Table 8 denotes that as VCN increases, natural frequencies are increased. It should be noted that by enhancing VCN, both of the stiffness and mass density of plate increases. However, the effect of CNTs on the stiffness of the structure is more considerable than the mass density. Therefore, the natural frequencies of plate are increased. Also, Table 8 denotes that minimum and maximum fundamental frequency belongs to the FGO and FGX distributions. Furthermore, results show that as the sector angle of plate increases, the fundamental frequency decreases. Table 9 shows the effects of different boundary conditions on natural frequencies of FGCNT annular sector plates for VCN = 0.11, θ0 = 120° and h = 0.05 m. Results show that the minimum natural frequencies are related to FCFC and fundamental frequencies of CCCC and CFCF boundary conditions and also fundamental frequencies of SSSS and SCSC boundary conditions are almost identical.

Table 8 Effects of VCN on natural frequency (HZ) of FGCNT annular sector plates with all edges clamped (CCCC), θ0 = 30°, 60°,120° (a = 1 m, b = 2 m, h = 0.1 m)
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Table 9 Effects of different boundary conditions on natural frequency (HZ) of FGCNT annular sector plates, \(V_{CN} = 0.11,\) α = 120° (a = 1 m, b = 2 m, h = 0.05 m)
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Figures 4, 5, 6 and 7 show the first six mode shapes of FGCNT annular sector plate for θ0 = 30°, 60°, 120° and 240°, respectively (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.1 m, CCCC). Figures 8, 9, 10 show the first six mode shapes of FGCNT annular sector plate for FCFC, SSSS and SCSC boundary conditions, respectively. (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.05 m, θ0 = 120°)

Fig. 4
figure 4

The first six mode shapes of FGCNT annular sector plate (FGV, \(V_{CN} = 0.11\), a = 1 m, b = 2 m, h = 0.1 m, θ0 = 30°, CCCC)

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Fig. 5
figure 5

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 60°, CCCC)

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Fig. 6
figure 6

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC)

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Fig. 7
figure 7

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 240°, CCCC)

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Fig. 8
figure 8

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.05 m, θ0 = 120°, FCFC)

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Fig. 9
figure 9

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.05 m, θ0 = 120°, SCSC)

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Fig. 10
figure 10

The first six mode shapes of FGCNT annular sector plate (FGV, VCN= 0.11, a = 1 m, b = 2 m, h = 0.05 m, θ0 = 120°, SSSS)

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3.3 Transient vibration analysis

3.3.1 Transient analysis of plate without foundation

Transient vibration analysis of FGCNT annular sector plate without foundation is conducted, and the effect of the different distribution of CNTs and VCN on time histories of centerpoint of plate are investigated (a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC). The plate is under an impulsive loading (Eq. 28), and the plate is unloaded in t = 0.005 (s).

$${\text{P}}_{\rm z} = \left\{ {\begin{array}{*{20}l} {400 {\text{t }}\left( {\frac{\text{MPa}}{\hbox{s}}} \right)} \hfill & {{\text{t}} \le 0.005} \hfill \\ 0 \hfill & {{\text{t}} \ge 0.005} \hfill \\ \end{array} } \right\}$$
(28)

Figure 11 shows the effects of VCN on the time history of centerpoint of plate for UD distribution. This figure show that by increasing VCN, the amplitude of transient vibration decreases and its frequency increases. Figure 12 shows the effect of different distributions of CNTs on time history of centerPoint of plate for VCN= 0.11. It can be seen that the minimum and maximum amplitude of vibration is related to FGX and FGO distributions. Figure 13 shows the transient vibration behavior of FGCNT annular sector plate.

Fig. 11
figure 11

The effect of VCN on time history of centerpoint of plate (UD, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC)

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Fig. 12
figure 12

The effect of different distribution of CNTs on time history of centerpoint of plate (\(V_{CN} = 0.11\), a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC)

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Fig. 13
figure 13

Dynamic behavior of FGCNT plate (FGO, VCN= 0.11, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC (1) t = 0.005 s, (2) t = 0.005725 s, (3) t = 0.00585 s, (4) t = 0.005975 s, (5) 0.0071 s, (6) t = 0.007225 s, (7) t = 0.00735 s, (8) t = 0.007475 s, (9) t = 0.008475 s)

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3.3.2 Transient analysis of plate resting on viscoelastic foundation

In this part, the effect of viscoelastic foundation on the transient behavior of plate is considered (a = 1 m, b = 2 m, h = 0.1 m, \(\theta_{0} = 1 20^\circ\), CCCC, VCN= 0.11, UD). The loading function is according to Eq. (28). Figure 14 show the effect of elastic coefficient of foundation on time history of centerpoint transverse displacement of plate. In this result, damping coefficient of the foundation is considered to be zero \(({\text{c}}_{\text{d}} = 0)\). Result illustrates that by increasing the elastic coefficient of the foundation, the stiffness of plate increases, and consequently, amplitude of transverse displacement decreases significantly, and also, frequency of transient vibration increases. Figure 15 shows the effect of damping coefficient of foundation on time history of centerpoint transverse displacement of plate. In this result, elastic coefficient of the foundation is considered to be zero \(({\text{k}}_{\text{w}} = 0)\). As it can be seen from this figure, by increasing damping of the foundation, amplitude of vibration diminishes and vibration of plate can be seen in three situations such as under-damped, critically-damped and over-damped.

Fig. 14
figure 14

The effect of different stiffness coefficient of viscoelastic foundation on time history of centerpoint of plate (\(V_{CN} = 0.11\), UD, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC)

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Fig. 15
figure 15

The effect of different damping coefficient of viscoelastic foundation on time history of centerpoint of plate (\(V_{CN} = 0.11\), UD, a = 1 m, b = 2 m, h = 0.1 m, θ0 = 120°, CCCC)

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4 Conclusions

A full comprehensive study about static, dynamic and natural frequency analyses of FGCNT annular sector plate has been investigated. A general solution based on FSDT in polar coordinate is presented that can be used for analyses of circular, annular and annular sector plates. Linear strain–displacement relationship is used to model the problem, and it is assumed that the plate is resting on simple linear Kelvin-voight viscoelastic foundation. Hamilton principle and finite element method have been used to derive the governing motion equations. A 4-node two-dimensional element with 20 degree of freedom is applied to mesh the domain. The influence of volume fraction of carbon and its distribution, different boundary conditions, damping and stiffness of viscoelastic foundation and sector angles on displacements and natural frequency of plate have been studied. Results show that volume fraction of CNTs has great effect on the deflection of plate, and increasing the \(V_{CN}\) from 0.11 to 0.17 leads to more than 36% decrease in the deflection. Also, the minimum and maximum displacements correspond to the FGX and FGO distributions, respectively. It means that CNTs distributions close to upper and lower surface of plate are more appropriate than those distribution concentrated on near the mid-plane for enhancing the overall stiffness of plate. Also, results denote that by increasing the \(V_{CN}\), natural frequencies increases, and minimum and maximum fundamental frequency corresponds to the FGO and FGX distributions. Furthermore, by increasing damping of the foundation, amplitude of vibration decreases and vibration of plate can be seen in three situations such as under-damped, critically-damped and over-damped. For the future studies, applying higher order theories for thicker plates, large deflection analysis, and also investigating the effects of nonlinear foundations can be considered.