A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates

Abstract

The paper presents a novel nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates. To observe the length scale and size-dependency effects of nanostructures, the nonlocal strain gradient theory (NSGT) is considered. The present model is efficient to capture both nonlocal effects and strain gradient effects in nanoplate structures. In addition, the material properties of the FG-CNTRC are assumed to be graded in the plate thickness direction. Based on the higher order shear deformation theory (HSDT), the weak form of the governing equations of motion of the nanoplates is presented using the principle of virtual work. Afterward, the natural frequency and deflection of the nanoplates are made out of isogeometric analysis (IGA). Thanks to higher order derivatives and continuity of NURBS basic function, IGA is suitable for the weak form of NSGT which requires at least the third-order derivatives in approximate formulations. Effects of nonlocal parameter, strain gradient parameter, carbon nanotube (CNT) volume fraction, distributions of CNTs and length-to-thickness ratios on deflection and natural frequency of the nanoplates are examined and discussed in detail. Numerical results are developed to show the phenomenon of stiffness-softening and stiffness-hardening mechanisms of the present model.

1 Introduction

Due to superior features, carbon nanotubes (CNTs) are considered as “new materials for the twenty-first century” [1] and a potential constituent of reinforcement for nanocomposites, and the strongest and the most prevalent and resilient material. Therefore, CNTs have many attached attentions from scientists and open a new research direction in nanotechnology.

With the advantageous of functionally graded/composites materials, a lot of numerical methods has been devised to study these materials such as smoothed finite element methods [2,3,4], isogeometric analysis [5, 6]. To study mechanical behaviors of CNTs, two popular models including continuum mechanics (CM) and molecular dynamics (MD) are often used. The MD can predict micro/nanostructures with the high precision, but there are some limitations to computationally expensive costs and calculating on too short time scale. Therefore, it must currently be left to CM for simulating of longer times and large systems. To overcome these shortcomings, several CM theories have been developed and used such as the first- and second-order strain gradient theory of Mindlin [7, 8], the nonlocal elasticity theory [9], the first-order strain gradient theory [10, 11], the second-order strain gradient theory [12,13,14], couple stress theory [15], modified couple stress theory [16] and modified strain gradient elasticity theory [17], etc. In those models, effect of nonlocal elasticity or strain gradient is only considered. Hence, Lim et al. [18] combined nonlocal elasticity and strain gradient effects to propose the nonlocal strain gradient theory (NSGT). Xu et al. [19] studied analytical solutions of bending and buckling analyses of Euler–Bernoulli beams using NSGT. Size-dependent analysis based NSGT and finite element method for bending, buckling, and free vibration responses of nanobeams was developed by Rajasekaran and Khaniki [20]. Li and Hu [21] used NSGT to investigate analytical solutions of nonlinear bending deflections and free vibration frequency of functionally graded nanobeams. Bending and buckling analyses of nanobeams using analytical methods based on NSGT [22] were reported. In their model, both Euler–Bernoulli and Timoshenko nanobeams were considered. Free vibration analysis of isotropic nanoplates [23] and buckling analysis of orthotropic nanoplates in thermal environment [24] using NSGT and analytical solutions was also developed. Arefi et al. [25] studied bending analysis of a sandwich porous nanoplate integrated with piezoelectric. Transient responses of porous functionally graded nanoplates under various pulse loads were reported in Ref. [26]. Geometrically nonlinear vibration analysis of sandwich nanoplates was performed by Nematollahi and Mohammadi [27]. Jalaei and Thai [28] studied the dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field. For nanoshell structures, buckling and post-buckling behaviors [29], wave dispersion analysis [30], and free vibration analysis [31] have also examined.

In sum, available studies have only focused on using analytical methods based on NSGT to calculate and simulate nanostructures. As we have known, analytical solutions are suitable for the problems with simple geometries and boundary conditions. It is very difficult to apply real structures in practices. So, numerical methods have been developed and considered as a potential candidate to perform the real structures. To the best of author’s knowledge, a combination of NSGT and numerical methods has not been studied and published in the literature so far. Hence, this topic is very potential and interesting for researchers to explore. Due to that lack of the researches, the authors intend to fill this gap in the literature by developing a size-dependent numerical model using NSGT theory for nanoplate structures and plate structures. In nanoplates and plate structures, it always requires higher order continuity of basic functions for higher order derivative of approximate variables. IGA with NURBS basis functions proposed by Hughes and coworkers [32] can be considered as a very strong candidate to study plate structures. Nguyen et al. [33] reported an overview and computer implementation aspects of IGA. To express local refinement, an approach to generalize the geometry independent field approximation approach [34,35,36,37] was also proposed. This provides greater flexibility in the choice of the discretisation for the geometry and the field variables, hence allow accounting for localisation and local refinement without refining the patch in all coordinate directions and shows good convergence of IGA compared to alternatives [5, 6, 38]. The IGA has been successfully developed and applied to investigate size-dependent plate problems such as nonlocal theory [39,40,41,42,43], modified strain gradient elasticity theory [44, 45], modified couple stress theory [46, 47], structures with cutout [38, 48,49,50], locking free in plate/shell structures [38, 51,52,53,54] etc. Therefore, in this study, a novel size-dependent nonlocal strain gradient isogeometric model for static and free vibration analyses of functionally graded carbon nanotube-reinforced composites is developed.

2 Theoretical formulation

2.1 Functionally graded carbon nanotube-reinforced composite materials

Four configurations of CNTs including UD, FG-V, FG-O, and FG-X can be formulated as

$$ V_{{{\text{CNT}}}} = \left\{ \begin{gathered} V_{{{\text{CNT}}}}^{*} \hfill \\ \left(1 + \tfrac{2z}{h}\right)V_{{{\text{CNT}}}}^{*} \hfill \\ 2\left(1 - \tfrac{2\left| z \right|}{h}\right)V_{{{\text{CNT}}}}^{*} \hfill \\ 2\left(\tfrac{2\left| z \right|}{h}\right)V_{{{\text{CNT}}}}^{*} \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\,\begin{array}{*{20}c} {\text{(UD)}} \\ {\text{(FG-V)}} \\ {\text{(FG-O)}} \\ {\text{(FG-X)}} \\ \end{array} , $$

(1)

where

$$ V_{{{\text{CNT}}}}^{*} = \frac{{w_{{{\text{CNT}}}} }}{{w_{{{\text{CNT}}}} + (\rho_{{{\text{CNT}}}} /\rho_{{\text{m}}} ) - (\rho_{{{\text{CNT}}}} /\rho_{{\text{m}}} )w_{{{\text{CNT}}}} }}, $$

(2)

in which $w_{{{\text{CNT}}}}$ is the mass fraction of CNTs.

Based on the rule of mixtures, the effective material properties are expressed as

$$ E_{11} = \eta_{1} V_{{{\text{CNT}}}} E_{{{11}}}^{{{\text{CNT}}}} + V_{{\text{m}}} E^{{\text{m}}} , $$

(3)

$$ \frac{{\eta_{2} }}{{E_{22} }} = \frac{{V_{{{\text{CNT}}}} }}{{E_{{{22}}}^{{{\text{CNT}}}} }} + \frac{{V_{{\text{m}}} }}{{E^{{\text{m}}} }}, $$

(4)

$$ \frac{{\eta_{3} }}{{G_{12} }} = \frac{{V_{{{\text{CNT}}}} }}{{G_{{{12}}}^{{{\text{CNT}}}} }} + \frac{{V_{{\text{m}}} }}{{G^{{\text{m}}} }}, $$

(5)

$$ \nu_{12} = V_{{{\text{CNT}}}}^{*} \nu_{{{12}}}^{{{\text{CNT}}}} + V_{{\text{m}}} \nu^{{\text{m}}} , $$

(6)

$$ \rho = V_{{{\text{CNT}}}}^{{}} \rho_{{}}^{{{\text{CNT}}}} + V_{{\text{m}}} \rho^{{\text{m}}} , $$

(7)

where $\nu_{{{12}}}^{{{\text{CNT}}}} ,\,\,\rho_{{}}^{{{\text{CNT}}}}$ and $\nu^{{\text{m}}} ,\,\,\rho^{{\text{m}}}$ are Poisson’s ratio and density of CNTs and matrix, respectively; $E^{{\text{m}}}$ and $G^{{\text{m}}}$ are the Young’s modulus and shear modulus of the isotropic matrix; $E_{{{11}}}^{{{\text{CNT}}}}$, $E_{{{22}}}^{{{\text{CNT}}}}$ and $G_{{{12}}}^{{{\text{CNT}}}}$ are the Young’s and shear modulus of CNTs; $\eta_{1} ,\,\,\eta_{2}$ and $\eta_{3}$ are CNT efficiency parameters listed in Table 1; $V_{{{\text{CNT}}}}$ and $V_{{\text{m}}}$ are the CNTs and matrix volume fractions, in which $V_{{{\text{CNT}}}} + V_{{\text{m}}} = 1$.

Table 1 The CNTs’ efficiency parameters

2.2 Nonlocal strain gradient theory (NSGT)

Equations of motion of nonlocal elastic solids are given as follows

$$ t_{ij,j} + f_{i} = \rho \ddot{u}_{i} \;\;{\text{in}}\;V, $$

(8)

$$ t_{ij} n_{i} = g_{i} {\text{ on }}S, $$

(9)

where $t_{ij}$ is the total nonlocal stress components; f_i and $g_{i} \,$ are the body force and traction components, respectively; $\rho$ is density mass; $\ddot{u}_{i}$ is the acceleration field; V is the volume and S is the Neumann boundary.

Based on the NSGT [18], the nonlocal stress tensor is expressed by

$$ t_{ij} = t_{ij}^{0} - \nabla t_{ij}^{1} , $$

(10)

where

$$ t_{ij}^{0} \left( {\mathbf{x}} \right) = \int\limits_{V} {\alpha \left( {\left| {{\mathbf{x^{\prime}}} - {\mathbf{x}}} \right|} \right)} \sigma_{ij} \left( {{\mathbf{x^{\prime}}}} \right){\text{d}}V^{\prime}\left( {{\mathbf{x^{\prime}}}} \right), \ldots t_{ij}^{1} \left( {\mathbf{x}} \right) = l^{2} \int\limits_{V} {\alpha \left( {\left| {{\mathbf{x^{\prime}}} - {\mathbf{x}}} \right|} \right)\nabla } \sigma_{ij} \left( {{\mathbf{x^{\prime}}}} \right)\rm{d}V^{\prime}\left( {{\mathbf{x^{\prime}}}} \right), $$

(11)

in which $t_{ij}^{0}$ and $t_{ij}^{1}$ are the nonlocal stress and higher order stress tensors; $\alpha \left( {\left| {{\mathbf{x^{\prime}}} - {\mathbf{x}}} \right|} \right)$ is the nonlocal kernel function; ${\mathbf{x}}$ is a reference point in the body; l is the material length scale parameter; $\nabla = \frac{\partial }{\partial x} + \frac{\partial }{\partial y}$ is the differential operator; $\sigma_{ij}$ is the local Cauchy stress tensors of classical elasticity theory and satisfies

$$ \sigma_{ij} = C_{ijkl} \varepsilon_{kl} ,\;\;\varepsilon_{kl} = \frac{1}{2}\left( {u_{k,l} + u_{l,k} } \right), $$

(12)

where $\varepsilon_{kl}$ is strain components; $C_{ijkl}$ is elastic modulus coefficients and $u_{k}$ is displacements.

Using the special Helmholtz averaging kernel, the nonlocal constitutive in Eq. (11) can be rewritten as [55]

$$ Lt_{ij}^{0} = \sigma_{ij} ,\;Lt_{ij}^{1} = \lambda \nabla \sigma_{ij} ,\;Lt_{ij,j}^{0} = \sigma_{ij,j} ,\;Lt_{ij,j}^{1} = \lambda \nabla \sigma_{ij,j} , $$

(13)

where $L = \left( {1 - \mu \nabla^{2} } \right)$ defines the linear differential operator, in which $\nabla^{2} = \frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}$ and $\lambda = l^{2}$.

Similarly, Eq. (8) can be represented by

$$ Lt_{ij} = Lt_{ij}^{0} - L\nabla t_{ij}^{1} ,\;Lt_{ij,j} = Lt_{ij,j}^{0} - L\nabla t_{ij,j}^{1} , $$

(14)

Substituting Eq. (13) into Eq. (14), the equilibrium equation is performed as

$$ \sigma_{ij,j} - \lambda \nabla^{2} \sigma_{ij,j} + \left( {1 - \mu \nabla^{2} } \right)f_{i} = \left( {1 - \mu \nabla^{2} } \right)\rho \ddot{u}_{i} , $$

(15)

Applying the principle of virtual displacement, Eq. (15) can be presented

$$ \int\limits_{V} {\sigma_{ij,j} \delta u_{i} \rm{d}} V - \lambda \int\limits_{V} {\nabla^{2} \sigma_{ij,j} \delta u_{i} \rm{d}} V + \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)f_{i} \delta u_{i} \rm{d}} V = \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)\rho \ddot{u}_{i} \delta u_{i} \rm{d}} V, $$

(16)

where $\delta u_{i}$ is the virtual displacement.

The first term in Eq. (16) can be given by applying the integration by parts and divergence theorem

$$ \int_{V} {\sigma_{ij,j} } \delta u_{i} {\text{d}}V = - \int_{V} {\sigma_{ij} } \delta u_{i,j} {\text{d}}V + \int_{S} {\sigma_{ij} } n_{i} \delta u_{i} {\text{d}}S. $$

(17)

Similarly, the second term in Eq. (16) can be rewritten as

$$ \int_{V} {\nabla^{2} \sigma_{ij,j} } \delta u_{i} {\text{d}}V = - \int_{V} {\nabla^{2} \sigma_{ij} } \delta u_{i,j} {\text{d}}V + \int_{S} {\nabla^{2} \sigma_{ij} } n_{i} \delta u_{i} {\text{d}}S. $$

(18)

Substituting Eqs. (17) and (18) into Eq. (16), the final equation is expressed as

$$ \begin{gathered} - \int\limits_{V} {\sigma_{ij} \delta u_{i,j} \rm{d}} V + \int\limits_{S} {\sigma_{ij} n_{i} \delta u_{i} \rm{d}} S + \lambda \int\limits_{V} {\nabla^{2} \sigma_{ij} \delta u_{i,j} \rm{d}} V - \lambda \int\limits_{S} {\nabla^{2} \sigma_{ij} n_{i} \delta u_{i} \rm{d}} S + \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)f_{i} \delta u_{i} \rm{d}} V \hfill \\ = \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)\rho \ddot{u}_{i} \delta u_{i} \rm{d}} V. \hfill \\ \end{gathered} $$

(19)

The traction on the Neumann boundary in this study is ignored, Eq. (19) is rewritten as

$$ \int\limits_{V} {\sigma_{ij} \delta u_{i,j} \rm{d}} V - \lambda \int\limits_{V} {\nabla^{2} \sigma_{ij} \delta u_{i,j} \rm{d}} V + \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)\rho \ddot{u}_{i} \delta u_{i} \rm{d}} V = \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)f_{i} \delta u_{i} \rm{d}} V. $$

(20)

Using the symmetric condition, the virtual displacement vector can be defined as

$$ \delta u_{i,j} = \frac{1}{2}\left( {\delta u_{i,j} + \delta u_{j,i} } \right) = \delta \varepsilon_{ij} . $$

(21)

Substituting Eq. (21) into Eq. (20), the final form is described as

$$ \int\limits_{V} {\sigma_{ij} \delta \varepsilon_{ij} \rm{d}} V - \lambda \int\limits_{V} {\nabla^{2} \sigma_{ij} \delta \varepsilon_{ij} \rm{d}} V + \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)\rho \ddot{u}_{i} \delta u_{i} \rm{d}} V = \int\limits_{V} {\left( {1 - \mu \nabla^{2} } \right)f_{i} \delta u_{i} \rm{d}} V. $$

(22)

3 Displacement field

Based on HSDT [56], the displacement components can be formulated as

$$ {\overline{\mathbf{u}}}\left( {x,y,z} \right)\,\,{\mathbf{ = u}}^{1} \left( {x,y} \right) + z{\mathbf{u}}^{2} \left( {x,y} \right) + f(z){\mathbf{u}}^{3} \left( {x,y} \right), $$

(23)

where

$$ {\overline{\mathbf{u}}} = \left\{ {\begin{array}{*{20}c} {\overline{u}} \\ {\overline{v}} \\ {\overline{w}} \\ \end{array} } \right\}, \, {\mathbf{u}}^{1} = \left\{ {\begin{array}{*{20}c} u \\ v \\ w \\ \end{array} } \right\},\, \, \,{\mathbf{u}}^{2} = - \left\{ {\begin{array}{*{20}c} {\beta_{x} } \\ {\beta_{y} } \\ 0 \\ \end{array} } \right\}, \, {\mathbf{u}}^{3} = \left\{ {\begin{array}{*{20}c} {\theta_{x} } \\ {\theta_{y} } \\ 0 \\ \end{array} } \right\}, $$

(24)

in which u, v and w are the in-plane and transverse displacement components, respectively; $\theta_{x}$ and $\, \theta_{y}$ are two rotations; $\beta _{x} = w_{{,x}} {\text{ and }}\beta _{y} = w_{{,y}}$; $f\left( z \right) = z - 4z^{3} /3h^{2}$ [57].

The bending and shear strain components are defined as

$$ {{\mathbf{\varepsilon}}} = \left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } & {\varepsilon_{yy} } &{\gamma_{xy} } \\ \end{array} } \right\}^{{\text{T}}} ={\mathbf{ \varepsilon }}^{1} + z{{{\mathbf{ \varepsilon }}}}^{2} + f(z){{{\mathbf{\varepsilon}}}}^{3} ,\;{{{\mathbf{\gamma}}}} = \left\{ {\begin{array}{*{20}c} {\gamma_{xz} } & {\gamma_{yz} } \\ \end{array} } \right\}^{{\text{T}}} ={\mathbf{ \varepsilon }}^{s1} + f^{\prime}(z){\mathbf{ \varepsilon }}^{s2} , $$

(25)

where

$$ \begin{gathered} {{\varvec{\upvarepsilon}}}^{1} = \left\{ {\begin{array}{*{20}l} {u_{,x} } \\ {v_{,y} } \\ {u_{,y} + v_{,x} } \\ \end{array} } \right\},\;{{\varvec{\upvarepsilon}}}^{2} = - \left\{ {\begin{array}{*{20}c} {\beta_{x,x} } \\ {\beta_{y,y} } \\ {\beta_{x,y} + \beta_{y,x} } \\ \end{array} } \right\},\;{{\varvec{\upvarepsilon}}}^{3} = \left\{ {\begin{array}{*{20}c} {\theta_{x,x} } \\ {\theta_{y,y} } \\ {\theta_{x,y} + \theta_{y,x} } \\ \end{array} } \right\}, \hfill \\ {{\varvec{\upvarepsilon}}}^{s1} = \left\{ {\begin{array}{*{20}c} {w_{,x} - \beta_{x} } \\ {w_{,y} - \beta_{y} } \\ \end{array} } \right\},\;{{\varvec{\upvarepsilon}}}^{s2} = \left\{ {\begin{array}{*{20}c} {\theta_{x} } \\ {\theta_{y} } \\ \end{array} } \right\} \hfill \\ \end{gathered} $$

(26)

in which $f^{\prime}(z) = 1 - 4z^{2} /h^{2}$.

The constitutive relations can be defined as

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\tau_{xy} } \\ {\tau_{xz} } \\ {\tau_{yz} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {C_{11} } & {C_{12} } & 0 & 0 & 0 \\ {C_{21} } & {C_{22} } & 0 & 0 & 0 \\ 0 & 0 & {C_{66} } & 0 & 0 \\ 0 & 0 & 0 & {C_{55} } & 0 \\ 0 & 0 & 0 & 0 & {C_{44} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\gamma_{xy} } \\ {\gamma_{xz} } \\ {\gamma_{yz} } \\ \end{array} } \right\}, $$

(27)

where

$$ \begin{gathered} C_{11}^{{}} = \frac{{E_{11} }}{{1 - \nu_{12}^{2} }},\;C_{22}^{{}} = \frac{{E_{22} }}{{1 - \nu_{12}^{2} }},\;C_{12}^{{}} = C_{21}^{{}} = \frac{{\nu_{12} E_{11} }}{{1 - \nu_{12}^{2} }}, \hfill \\ C_{66}^{{}} = C_{55}^{{}} = C_{44}^{{}} = G_{12} . \hfill \\ \end{gathered} $$

(28)

Substituting Eqs. (25) and (27) into Eq. (22), the discrete equations for bending analysis of the nanoplates subjected to a transverse load $f_{0}$ can be described as [58]:

$$ \int_{\Omega } {\delta {\overline{\mathbf{\varepsilon }}}^{T} {\mathbf{C}}^{b} {\overline{\mathbf{\varepsilon }}}{\text{d}}\Omega } - \lambda \int_{\Omega } {\delta \left( {\nabla^{2} {\overline{\mathbf{\varepsilon }}}^{T} } \right){\mathbf{C}}^{b} {\overline{\mathbf{\varepsilon }}}{\text{d}}\Omega } + \int_{\Omega } {\delta {\overline{\mathbf{\gamma }}}^{T} {\mathbf{C}}^{s} {\overline{\mathbf{\gamma }}}{\text{d}}\Omega } - \lambda \int_{\Omega } {\delta \left( {\nabla^{2} {\overline{\mathbf{\gamma }}}^{T} } \right){\mathbf{C}}^{s} {\overline{\mathbf{\gamma }}}^{T} {\text{d}}\Omega } = \int_{\Omega } {\left( {1 - \mu \nabla^{2} } \right)f_{0} \delta w{\text{d}}\Omega } , $$

(29)

where

$$ \begin{gathered} {\overline{\mathbf{\varepsilon }}} = \left\{ {\begin{array}{*{20}c} {{{\mathbf{\varepsilon}}}^{1} } \\ {{{\mathbf{\varepsilon}}}^{2} } \\ {{{\mathbf{\varepsilon}}}^{3} } \\ \end{array} } \right\},\;{\overline{\mathbf{\gamma }}} = \left\{ {\begin{array}{*{20}c} {{{\mathbf{\varepsilon}}}^{s1} } \\ {{{\mathbf{\varepsilon}}}^{s2} } \\ \end{array} } \right\},\;{\mathbf{C}}^{b} = \left[ {\begin{array}{*{20}c} {\mathbf{A}} & {\mathbf{B}} & {\mathbf{E}} \\ {\mathbf{B}} & {\mathbf{D}} & {\mathbf{F}} \\ {\mathbf{E}} & {\mathbf{F}} & {\mathbf{H}} \\ \end{array} } \right],\;{\mathbf{C}}^{s} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{s} } & {{\mathbf{B}}^{s} } \\ {{\mathbf{B}}^{s} } & {{\mathbf{D}}^{s} } \\ \end{array} } \right] \hfill \\ \left( {A_{ij}^{{}} ,B_{ij}^{{}} ,D_{ij}^{{}} ,E_{ij} ,F_{ij} ,H_{ij} } \right) = \int_{ - h/2}^{h/2} {\left( {1,z,z^{2} ,f(z),zf(z),f^{2} (z)} \right)C_{ij}^{{}} {\text{d}}z} {\text{ where (}}i{,}j{\,= 1,2,6)} \hfill \\ \left( {A_{ij}^{s} ,B_{ij}^{s} ,D_{ij}^{s} } \right) = \int_{ - h/2}^{h/2} {\left( {1,f^{\prime}(z),f^{{\prime}{2}} (z)} \right)C_{ij}^{{}} {\text{d}}z} {\text{ where (}}i{,}j{\,= 4,5)}{\text{.}} \hfill \\ \end{gathered} $$

(30)

Similarly, the discrete equations for free vibration analysis are expressed as

$$ \int_{\Omega } {\delta {\overline{\mathbf{\varepsilon }}}^{T} {\mathbf{C}}^{b} {\overline{\mathbf{\varepsilon }}}{\text{d}}\Omega } - \lambda \int_{\Omega } {\delta \left( {\nabla^{2} {\overline{\mathbf{\varepsilon }}}^{T} } \right){\mathbf{C}}^{b} {\overline{\mathbf{\varepsilon }}}{\text{d}}\Omega } + \int_{\Omega } {\delta {\overline{\mathbf{\gamma }}}^{T} {\mathbf{C}}^{s} {\overline{\mathbf{\gamma }}}{\text{d}}\Omega } - \lambda \int_{\Omega } {\delta \left( {\nabla^{2} {\overline{\mathbf{\gamma }}}^{T} } \right){\mathbf{C}}^{s} {\overline{\mathbf{\gamma }}}^{T} {\text{d}}\Omega } + \int_{\Omega } {\left( {1 - \mu \nabla^{2} } \right)\delta {\overline{\mathbf{u}}\mathbf{I}}_{m} {\mathbf{\ddot{\overline{u}}}}{\text{d}}\Omega } = {\mathbf{0,}} $$

(31)

where

$$ \begin{gathered} {\overline{\mathbf{u}}} = \left\{ {\begin{array}{*{20}c} {{\mathbf{u}}^{1} } \\ {{\mathbf{u}}^{2} } \\ {{\mathbf{u}}^{3} } \\ \end{array} } \right\},{\mathbf{I}}_{m} {\mathbf{ = }}\left[ {\begin{array}{*{20}c} {{\mathbf{I}}_{1} } & {{\mathbf{I}}_{2} } & {{\mathbf{I}}_{4} } \\ {{\mathbf{I}}_{2} } & {{\mathbf{I}}_{3} } & {{\mathbf{I}}_{5} } \\ {{\mathbf{I}}_{4} } & {{\mathbf{I}}_{5} } & {{\mathbf{I}}_{6} } \\ \end{array} } \right], \hfill \\ \left( {{\mathbf{I}}_{1} ,{\mathbf{I}}_{2} ,{\mathbf{I}}_{3} ,{\mathbf{I}}_{4} ,{\mathbf{I}}_{5} ,{\mathbf{I}}_{6} } \right) = \int\limits_{ - h/2}^{h/2} {\rho_{e} \left( {1,z,z^{{2}} ,f(z),zf(z),f^{2} (z)} \right){\mathbf{I}}_{3x3} } {\text{d}}z, \hfill \\ \end{gathered} $$

(32)

where ${\mathbf{I}}_{3x3}$ is the identity matrix of size 3 × 3.

3.1 The NSGT formulation using NURBS basis function

The displacements using NURBS basis functions can be expressed as

$$ \begin{aligned} {\mathbf{u}}^{{h}} \left( {{x,y}} \right) & = \sum\limits_{{I = 1}}^{{mxn}} {\left[ {\begin{array}{*{20}c} {N_{I} \left( {x,y} \right)} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {N_{I} \left( {x,y} \right)} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {N_{I} \left( {x,y} \right)} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {N_{I} \left( {x,y} \right)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {N_{I} \left( {x,y} \right)} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {N_{I} \left( {x,y} \right)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 &{N_{I} \left( {x,y} \right)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {u_{I} } \\ {v_{I} } \\ {w_{I}^{{}} } \\ {\theta _{{xI}} } \\ {\theta _{{yI}} } \\ {\beta _{{xI}} } \\ {\beta _{{yI}} } \\ \end{array} } \right\}} , \\ & = \sum\limits_{{I = 1}}^{{mxn}} {{\mathbf{N}}_{I} \left( {{x,y}} \right)} {\mathbf{q}}_{{I}} , \\ \end{aligned} $$

(33)

where ${\mathbf{q}}_{I}$ is a vector that contains degrees of freedom combined with the control point I; N_I is the shape function defined as in Ref. [32].

Substituting Eq. (33) into Eq. (26), Eq. (26) can be rewritten as:

$$ {\overline{\mathbf{\varepsilon }}} = \left\{ {\begin{array}{*{20}c} {{{\mathbf{\varepsilon }}}^{1} } & {{{\mathbf{\varepsilon }}}^{2} } & {{{\mathbf{\varepsilon }}}^{3} } \\ \end{array} } \right\}^{T} = \sum\limits_{I = 1}^{{m\rm{x}n}} {\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}_{I}^{1} } &{{\mathbf{B}}_{I}^{2} } & {{\mathbf{B}}_{I}^{3} } \\ \end{array} } \right\}^{T} {\mathbf{q}}_{I} } = \sum\limits_{I = 1}^{{m\rm{x}n}} {{\overline{\mathbf{B}}}_{I}^{b} {\mathbf{q}}_{I} } ,{\overline{\mathbf{\gamma }}} = \left\{ {\begin{array}{*{20}c} {{{\mathbf{\varepsilon }}}^{s1} } & {{{\mathbf{\varepsilon }}}^{s2} } \\ \end{array} } \right\}^{T} = \sum\limits_{I = 1}^{{m\rm{x}n}} {\left\{ {\begin{array}{*{20}c} {{\mathbf{B}}_{I}^{s1} } & {{\mathbf{B}}_{I}^{s2} } \\ \end{array} } \right\}^{T} {\mathbf{q}}_{I} } = \sum\limits_{I = 1}^{{m\rm{x}n}} {{\overline{\mathbf{B}}}_{I}^{s} {\mathbf{q}}_{I} } , $$

(34)

where

$$ \begin{gathered} {\mathbf{B}}_{I}^{1} = \left[ {\begin{array}{*{20}l} {N_{I,x} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {N_{I,y} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {N_{I,y} } \hfill & {N_{I,x} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right],\;{\mathbf{B}}_{I}^{2} = - \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,x} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,y} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,y} } \hfill & {N_{I,x} } \hfill \\ \end{array} } \right], \hfill \\ {\mathbf{B}}_{I}^{3} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,x} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,y} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {N_{I,y} } \hfill & {N_{I,x} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right],\;{\mathbf{B}}_{I}^{s1} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & {N_{I,x} } \hfill & 0 \hfill & 0 \hfill & { - N_{I} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {N_{I,y} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - N_{I} } \hfill \\ \end{array} } \right], \hfill \\ {\mathbf{B}}_{I}^{s2} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]. \hfill \\ \end{gathered} $$

(35)

Inserting Eq. (33) into Eq. (24), Eq. (24) can be performed

$$ {\overline{\mathbf{u}}} = \left\{ {\begin{array}{*{20}c} {{\mathbf{u}}^{1} } & {{\mathbf{u}}^{2} } & {{\mathbf{u}}^{3} } \\ \end{array} } \right\}^{T} = \sum\limits_{I = 1}^{{m\rm{x}n}} {\left\{ {\begin{array}{*{20}c} {{\mathbf{N}}_{I}^{1} } & {{\mathbf{N}}_{I}^{2} } & {{\mathbf{N}}_{I}^{3} } \\ \end{array} } \right\}^{T} {\mathbf{q}}_{I} } = \sum\limits_{I = 1}^{{m\rm{x}n}} {{\overline{\mathbf{N}}}_{I}^{{}} {\mathbf{q}}_{I} } , $$

(36)

where

$$ \begin{gathered} {\mathbf{N}}_{I}^{{0}} = \left[ {\begin{array}{*{20}l} {N_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right],{\mathbf{N}}_{I}^{1} = - \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right], \hfill \\ {\mathbf{N}}_{I}^{2} = \left[ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & {N_{I} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right]. \hfill \\ \end{gathered} $$

(37)

Inserting Eqs. (34) and (36) into Eqs. (29) and (31), respectively, the final compact forms are expressed as follows:

$$ {\mathbf{Kq}}\rm{ = }{\mathbf{f,}} $$

(38)

$$ \left( {{\mathbf{K - }}\omega^{{2}} {\mathbf{M}}} \right){\overline{\mathbf{q}}}\rm{ = }{\mathbf{0,}} $$

(39)

where ${\mathbf{K}}$, ${\mathbf{M}}$ and ${\mathbf{f}}$ are the global stiffness matrix, mass matrix and force vector defined as

$$ \begin{gathered} {\mathbf{K}} = \int_{\Omega } {\left( {{\overline{\mathbf{B}}}^{b} } \right)^{T} {\mathbf{C}}^{b} {\overline{\mathbf{B}}}^{b} \text{d}\Omega } + \int_{\Omega } {\left( {{\overline{\mathbf{B}}}^{s} } \right)^{T} {\mathbf{C}}^{s} {\overline{\mathbf{B}}}^{s} {\text{d}}\Omega } - \lambda \int_{\Omega } {\left( {\nabla^{2} {\overline{\mathbf{B}}}^{b} } \right)^{T} {\mathbf{C}}^{b} {\overline{\mathbf{B}}}^{b} \text{d}\Omega } - \lambda \int_{\Omega } {\left( {\nabla^{2} {\overline{\mathbf{B}}}^{s} } \right)^{T} {\mathbf{C}}^{s} {\overline{\mathbf{B}}}^{s} {\text{d}}\Omega } , \hfill \\ {\mathbf{M}} = \int_{\Omega } {\left( {1 - \mu \nabla^{2} } \right){\overline{\mathbf{N}}}^{T} {\mathbf{I}}_{m} {\overline{\mathbf{N}}}{\text{d}}\Omega } ,\;{\mathbf{f}} = \int_{\Omega } {f_{0} \left( {1 - \mu \nabla^{2} } \right)\left\{ {\begin{array}{*{20}l} 0 \hfill & 0 \hfill & {w_{I} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right\}^{T} {\text{d}}\Omega } ,{\mathbf{q = \overline{q}}}e^{i\omega t} , \hfill \\ \end{gathered} $$

(40)

in which $\omega$ and ${\overline{\mathbf{q}}}$ are the natural frequency and modes shape, respectively.

4 Numerical examples

4.1 Verification

4.1.1 Strain gradient effect

We now consider an isotropic square plate (length a = 10, thickness h). Material properties can be given as Young’s modulus E = 30 × 10⁶, Poisson’s ratio v = 0.3 and density mass $\rho = 2300$ kg/m³. Without nonlocal effects, the present model is Aifantis’s pure strain gradient theory [10]. Analytical and finite element solutions based on Kirchhoff theory were investigated by Babu and Patel [59]. Effects of length scale parameter on convergence studies of the normalized first natural frequency and central deflection of the isotropic plate are listed in Tables 2 and 3, respectively. According to Ref. [59], FEM-C and FEM-NC were conforming and nonconforming finite elements. We see that the convergence of the proposed method is really good and the present results match very well with reference solutions. Besides, an increase of length scale parameter leads to a rise of the natural frequency and a decrease of the central deflection of the plate. Hence, it can be found that using strain gradient theory makes the stiffness of the plate rise.

Table 2 Non-dimensional natural frequency ($\overline{\omega } = \omega a^{2} \sqrt {{\raise0.7ex\hbox{${\rho h}$} \!\mathord{\left/ {\vphantom {{\rho h} D}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$D$}}}$, where $D = \frac{{Eh^{3} }}{{12(1 - \nu^{2} )}}$) of the simply supported isotropic square plate (a/h = 100)

Table 3 The central non-dimensional deflection $\left( {\overline{w} = \frac{1000wD}{{q_{0} a^{4} }}} \right)$ of the simply supported isotropic square plate (a/h = 100)

4.1.2 Nonlocal effect

In this section, nonlocal effect on FG-CNTRC nanoplates is studied. The matrix is made of PmPV with $E^{m} = 2.1\,\,{\text{GPa,}}$ $\nu^{m} = 0.34,$ $\rho^{m} = 1.16\,\,{\text{g/cm}}^{{2}}$, and the reinforcements are (10,10) SWCNTs with $E_{{{11}}}^{{{\text{CNT}}}} \, = 5.6466\,{\text{(TPa),}}\,\,E_{{{22}}}^{{{\text{CNT}}}} \, = 7.08\,{\text{(TPa),}}\,\,G_{{{12}}}^{{{\text{CNT}}}} \, = 1.9445\,{\text{(TPa),}}\,\,\alpha_{{{11}}}^{{{\text{CNT}}}} = 3.4584{,}\,\,\alpha_{{{22}}}^{{{\text{CNT}}}} = 5.1682$, and G₂₃ = G₁₃ = G₁₂.

First, a simply supported (SSSS) square nanoplate (length a = 1, µ = 1.5) is considered. Without strain gradient effects, the proposed model becomes Erigen’s nonlocal elasticity theory model. Table 4 shows the first three non-dimensional frequencies of the FG-CNTRC nanoplate. We observe that the present results match very well with reference solutions [43] for all cases of a/h and $V_{{{\text{CNT}}}}^{*}$.

Table 4 The three lowest non-dimensional frequencies $\overline{\omega } = (\omega a^{2} /h)\sqrt {\rho^{{\text{m}}} /E^{{\text{m}}} }$ of the SSSS FG-CNTRC nanoplates

Next, the central deflection of SSSS FG-CNTRC square nanoplate (length a = 1, a/h = 50) under a uniform load f₀ = 0.1 × 10⁴ is investigated. Effect of nonlocal parameter on defection of the nanoplate is indicated in Table 5. Again, we can see that an excellent agreement between the present results and reference ones [43] is obtained. Besides, the deflection of the nanoplate increases with a rise of the nonlocal parameter. So, we also find that the stiffness of the plate decreases once using the Erigen’s nonlocal elasticity theory.

Table 5 Non-dimensional deflection $\overline{w} = w/h$ of the SSSS FG-CNTRC square nanoplates

4.2 Square plate

To consider both strain gradient and nonlocal effects in nanoplates, a FG-CNTRC square nanoplate (length a = 1) is exampled.

First, effect of length scale and nonlocal parameters on free vibration analysis of the nanoplate is investigated. Table 6 shows the first five frequencies of the fully clamped (CCCC) nanoplate. It can be seen that the natural frequencies decrease with a rise of the nonlocal parameter and increase with a rise of both the length scale parameter and length-to-thickness ratio. Table 7 lists the first natural frequency of the SSSS FG-CNTRC square nanoplate with a/h = 50. Again, we can see that when the length scale parameter increases the natural frequency increases as well, while the natural frequency reduces when the nonlocal parameter rises.

Table 6 Effect of length scale and nonlocal parameters on the first five non-dimensional frequencies $\overline{\omega } = (\omega a^{2} /h)\sqrt {\rho^{{\text{m}}} /E^{{\text{m}}} }$ of the CCCC FG-CNTRC square nanoplate with $V_{{{\text{CNT}}}}^{*} = 0.11$

Table 7 Effect of length scale and nonlocal parameters on the first normalized natural frequency $\overline{\omega } = (\omega a^{2} /h)\sqrt {\rho^{{\text{m}}} /E^{{\text{m}}} }$ of the SSSS FG-CNTRC square nanoplate with a/h = 50

Next, effect of length scale and nonlocal parameters on deflection of the SSSS square nanoplate (length a = 1, a/h = 50) subjected to a uniform load f₀ = 0.1 × 10⁴ is examined. Deflections of the SSSS FG-CNTRC square nanoplate are indicated in Table 8. We can see that the deflection increases with a rise of the nonlocal parameter and decreases with an increase of the length scale parameter. Besides, we can see the smallest deflection in FG-X case and the largest deflection in FG-O case. So, the FG-X pattern can be considered the best reinforcement.

Table 8 Effect of length scale and nonlocal parameters on deflection $\overline{w} = w/h$ of the SSSS FG-CNTRC square nanoplate

As obtained in results of static and free vibration analyses of the plates, it is observed that mechanical characteristics of the nanoplates including stiffness-softening and stiffness-hardening mechanisms are significantly influenced using the NSGT.

4.3 Circular plate

A fully clamped FG-CNTRC circular nanoplate with radius R is now considered. Effects of length scale and nonlocal parameters on the first natural frequency of the nanoplate with R/h = 10 and 20 are listed in Tables 9 and 10, respectively. Again, the non-dimensional natural frequency of the nanoplate decreases with an increase of the nonlocal parameter and increases with a rise of the length scale parameter. Therefore, it can be observed that the phenomenon of stiffness-softening and stiffness-hardening mechanisms is found using the NSGT.

Table 9 Effect of length scale and nonlocal parameters on the first natural frequency ($\overline{\omega } = \omega R^{2} \sqrt {{\raise0.7ex\hbox{${\rho h}$} \!\mathord{\left/ {\vphantom {{\rho h} D}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$D$}}}$, where $D = \frac{{Eh^{3} }}{{12(1 - \nu^{2} )}}$) of the fully clamped circular nanoplate with R/h = 10

Table 10 Effect of length scale and nonlocal parameters on the first natural frequency ($\overline{\omega } = \omega R^{2} \sqrt {{\raise0.7ex\hbox{${\rho h}$} \!\mathord{\left/ {\vphantom {{\rho h} D}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$D$}}}$, where $D = \frac{{Eh^{3} }}{{12(1 - \nu^{2} )}}$) of the fully clamped circular nanoplate with R/h = 20

5 Conclusions

In this paper, a novel nonlocal strain gradient isogeometric model for FG-CNTRC nanoplates was proposed. The material properties of the FG-CNTRC are assumed to be graded in the thickness direction. To consider the length scale and size-dependency effects of nanostructures, the nonlocal strain gradient theory (NSGT) was considered. Through the numerical results, it can be withdrawn some interesting points:

• 1. The proposed model is capable of capturing both nonlocal effects and strain gradient effects in nanoplate structures.

• 2. The governing equation is approximated using isogeometric analysis (IGA) which easily satisfy at least the third-order derivatives in weak form of nanoplates.

• 3. Phenomenon of stiffness-softening and stiffness-hardening mechanisms of the proposed model was fully observed.

• 4. The distribution of CNTs has a lot of influence the stiffness of the plate.