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Isogeometric FE analysis of CNT-reinforced composite plates: free vibration

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Abstract

Free vibration behavior of carbon nanotube-reinforced composite plates is studied here, using isogeometric analysis (IGA) based on a higher-order shear deformation theory (HSDT). Carbon nanotubes are considered dispersed into matrix in four different types of volume fraction distributions varying across the thickness. The formulation developed is approximated according to the HSDT model and is incorporated into a computer program developed in-house. IGA uses the same basis function for exact geometric modeling and for analysis, i.e., non-uniform rational B-spline basis function which can easily model the complex geometries exactly, and it can also achieve any desired degree of continuity through the choice of interpolation order so that this method can easily fulfill the C1 continuity for the HSDT model. The effect of variation of width-to-thickness ratio, plate aspect ratio and other parameters, on natural frequencies, are evaluated from parametric studies after due validation with the given literature which is done in FORTRAN.

Introduction

After the advent of carbon nanotubes (CNTs), relevant research was initially focused on finding their thermal, mechanical, electrical and other properties through experimental and computational means. Once their basic properties became evident, including properties like lightweight, high elastic modulus, strength and resilience, their potential for use as a reinforcing element on polymer/metal matrix-based composites was recognized. Subsequently, evaluations of effective material properties of such CNTRCs caught the attention of some researchers, e.g., Liu and Chen [1]. Since then, CNTs are being considered as important as conventional carbon fibers, but are potentially superior to the latter due to the possibility of tailoring the properties of fiber–matrix composites at the nanoscale. Once the excellent mechanical properties of such CNTRCs became evident, they started to be researched for application as primary load-bearing structural components like plates. In this direction, Zhu et al. [2] investigated the static and vibration behavior of square CNTRC plates using the finite element method (FEM) based on Reissner–Mindlin theory. Some investigations on natural frequencies of skew CNTRC plates were presented very recently by Kiani [3] using FEM, by Ardestani et al. [4] using isogeometric analysis method and by Zhang [5] using mesh-free method. However, mostly square/rhombus plates having unit plate aspect ratio were considered therein. Using isogeometric analysis, static, dynamic and buckling behaviors of functionally graded rectangular and circular plate have been studied based on Reddy’s theory by Tran et al. [6].

Basically, the motivation behind the isogeometric analysis was the bottleneck that has existed since a long time in the creation of suitable model. The creation of suitable model involves many steps that take a lot of time, so it is necessary to introduce the concept that bridges the gap between the computer-aided design (CAD) and the IGA studied by Hughes et al. [7]. On this subject, a lot of research has been published. IGA has attained much attention in the numerical field, especially in structures. IGA uses the same basis function that was used by CAD for geometry exactness as well as the approximation of the solution for the analysis. One of the most important features of IGA is that they achieve a desired degree of smoothness by using the choice of interpolation order in comparison with FEM. That is why IGA easily fulfills the C1 continuity requirements of plate element using HSDT which is also used in the present study.

In this work, natural vibration behavior of CNTRC plates is investigated further using IGA based on the semi-refined HSDT (SRHSDT7) [8]. Considering plates of more general type of plan form become essential for use sometimes, non-rectangular plates of non-unit aspect ratio are considered herein. Four different types of volume fraction distributions of CNTs vary across the thickness, as per Van et al. [9]. The formulation developed is incorporated into a computer program developed in-house.

Methodology and problem description

As already mentioned, the present analysis is performed using IGA formulation based on HSDT. The volume fraction of CNT, considered for the four types of volume fraction distributions, varies with thickness direction z, such as uniformly distributed (UD) CNTRC, functionally graded V (FG-V)-type distribution, functionally graded X (FG-X)-type distribution and functionally graded O (FG-O)-type distribution of CNTRC plates as shown in Fig. 1 [9].

Fig. 1
figure1

Configuration of CNTRC plates: a UD CNTRC plate, b FG-V CNTRC plate, c FG-O CNTRC plate and d FG-X CNTRC plate

The distributions of CNTs along the thickness direction of the CNTRC plate are expressed by Eq. 1 [9]:

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

Isogeometric analysis

A set of coordinates in one dimension which is non-decreasing is knot vector:

$$\xi = \left\{ {\xi_{1} ,\xi_{2} , \ldots \xi_{n + p + 1} } \right\}$$
(2)

where n and p are the basis function number and the order of polynomial which is used to construct B-splines. Basis functions \(N_{i,p} \left( \xi \right)\) can be defined as:

$$N_{i,p} \left( \xi \right) = \frac{{\xi - \xi_{i} }}{{\xi_{i + p} - \xi_{i} }}N_{i,p - 1} \left( \xi \right) + \frac{{\xi_{i + p + 1} - \xi }}{{\xi_{i + p + 1} - \xi_{i + 1} }}N_{i + 1,p - 1}$$
(3)

When P = 0,

$$N_{i,p} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\xi_{i} \le \xi < \xi_{i + 1} } \hfill \\ 0 \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right\}$$
(4)

To understand the process, consider the open knot vectors in both directions as {1, 2,……8} and {1, 2……….8} = {0 0 0 1 2 3 3 3}.

Bivariate Basis Functions These are found by the tensor product of univariate basis functions:

$$N_{I}^{b} \left( {\xi ,\eta } \right) = N_{i,p} \left( \xi \right)M_{j,q} \left( \eta \right)$$
(5)

Considering the weight functions corresponding to every control point, NURBS basis function is written as:

$$N_{I} \left( {\xi ,\eta } \right) = \frac{{N_{I}^{b} w_{I} }}{{\mathop \sum \nolimits_{A}^{m \times n} N_{A}^{b} \left( {\xi ,\eta } \right)w_{A} }}$$
(6)

Displacement field

The spatial displacements of the plate in terms of reference plane displacements using the higher-order shear deformation theory are (Bhar et al. [8])

$$\left. \begin{aligned} U(x,y,z,t) & = u(x,y,t) + z\theta_{x} (x,y,t) + z^{3} \theta_{x}^{*} (x,y,t) \\ V(x,y,z,t) & = v(x,y,t) + z\theta_{y} (x,y,t) + z^{3} \theta_{y}^{*} (x,y,t) \\ W(x,y,z,t) & = w(x,y,t) \\ \end{aligned} \right\}.$$
(7)

u, v and w are the membrane displacements, \(\theta_{x}\) and \(\theta_{y }\) are the rotation terms, and \(\theta_{x}^{ *}\) and \(\theta_{y}^{ *}\) are the higher-order terms.

The laminate strain vector {ε̅} consists of the components:

$$\left\{ {\overline{\varepsilon } } \right\}^{T} = \left\{ { \, \left| {\left\{ {\overline{\varepsilon } } \right\}_{p}^{T} \, } \right|\left. { \, \left\{ {\overline{\varepsilon } } \right\}_{t}^{T} \, } \right|} \right\} = \left\{ { \, \left| {\left\{ {\overline{\varepsilon }_{m} } \right\}^{T} \, } \right|\left. { \, \left\{ {\overline{\varepsilon }_{k} } \right\}^{T} \, } \right|\left. { \, \left\{ {\overline{\varepsilon }_{k}^{*} } \right\}^{T} \, } \right|\left. { \, \left\{ {\overline{\varepsilon }_{\varphi } } \right\}^{T} } \right| \, \left. {\overline{\varepsilon }_{\varphi }^{*} } \right| \, } \right\}$$
(8)

where the subscripts m, k and φ represent collectively the membrane bending/curvature and transverse shear laminate strains, respectively. Now, the spatial constitutive for the kth orthotropic lamina is given by

$$\left\{ \sigma \right\}_{k} = \left\{ {\begin{array}{*{20}c} {\left\{ {\sigma_{p} } \right\}_{k} } \\ {\left\{ {\sigma_{t} } \right\}_{k} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\overline{Q}_{11} } & {\overline{Q}_{12} } & {\overline{{Q_{16} }} } & 0 & 0 \\ {\overline{Q}_{21} } & {\overline{Q}_{22} } & {\overline{Q}_{26} } & 0 & 0 \\ {\overline{Q}_{61} } & {\overline{Q}_{62} } & {\overline{Q}_{66} } & 0 & 0 \\ 0 & 0 & 0 & {\overline{Q}_{55} } & {\overline{Q}_{54} } \\ 0 & 0 & 0 & {\overline{Q}_{45} } & {\overline{Q}_{44} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{XX} } \\ {\varepsilon_{YY} } \\ {\gamma_{XY} } \\ {\gamma_{XZ} } \\ {\gamma_{YZ} } \\ \end{array} } \right\}$$
(9)

Governing equation of motion

Hamilton’s variational principle is used to derive the governing equation of motion:

$$\delta \mathop \int \limits_{{t_{1} }}^{{t_{2} }} \left( {T - U - W} \right){\text{d}}t = 0$$
(10)

After solving the above equation, we get the discretized system of equations:

$$\left( {K - \omega^{2} M} \right)d = 0$$
(11)

where

$$K = \int\limits_{\Omega } {\left( {\left\{ {\begin{array}{*{20}c} {B^{m} } \\ {B^{{b_{1} }} } \\ {B^{{b_{2} }} } \\ \end{array} } \right\}^{T} \left[ {\begin{array}{*{20}c} A & B & E \\ B & D & F \\ E & F & H \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {B^{m} } \\ {B^{{b_{1} }} } \\ {B^{{b_{2} }} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {B^{{s_{0} }} } \\ {B^{{s_{1} }} } \\ \end{array} } \right\}\left[ {\begin{array}{*{20}c} {A_{s} } & {B_{s} } \\ {B_{s} } & {D_{s} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {B^{{s_{0} }} } \\ {B^{{s_{1} }} } \\ \end{array} } \right\}} \right)} {\text{d}}\Omega$$
(12)

\(\omega\) is the natural frequency and M is the global mass matrix that can be calculated by

$$M = \int\limits_{\Omega } {N^{T} mN{\text{d}}\Omega }$$
(13)

where

$$N = \left\{ {\begin{array}{*{20}c} {N_{1} } \\ {N_{2} } \\ {N_{3} } \\ \end{array} } \right\}$$
(14)
$$N_{1} = \left[ {\begin{array}{*{20}c} {N_{I} } & 0 & 0 & 0 & 0 \\ 0 & {N_{I} } & 0 & 0 & 0 \\ 0 & 0 & {N_{I} } & 0 & 0 \\ \end{array} } \right],\quad N_{2} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {N_{I} } & 0 \\ 0 & 0 & 0 & 0 & {N_{I} } \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$
(15)
$$N_{3} = \left[ {\begin{array}{*{20}c} 0 & 0 & {N_{I,x} } & {N_{I} } & 0 \\ 0 & 0 & {N_{I,y} } & 0 & {N_{I} } \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$
(16)

Numerical results

In this section, analysis is being done by NURBS-based isogeometric analysis on bare plates to ensure the validity of the developed formulation into a computer program using FORTRAN. The material properties of the matrix are assumed as [9] Poisson’s ratio (νm) = 0.34 and elastic modulus (Em) = 2.1 GPa. The values of elastic modulus, shear modulus and Poisson’s ratio of CNT are: \(E_{11}^{\text{CNT}} = 5.6466 \times 10^{3 } \,{\text{GPa}}\), \(E_{22}^{\text{CNT}} = 7.08 \times 10^{3 } \,{\text{GPa}}\), \(G_{12}^{\text{CNT}} = 1.9445 \times 10^{3 } \,{\text{GPa}}\) and \(\nu_{12}^{\text{CNT}} = 0.175\). The CNT efficiency parameter of \(V^{*}_{\text{CNT}}\) = 0.11, η1 = 0.149 and η2 = 0.934, for \(V^{*}_{\text{CNT}}\) = 0.14, η1 = 0.150 and η2 = 0.941 and for \(V^{*}_{\text{CNT}}\) = 0.17, η1 = 0.149 and η2 = 1.381, where η1 and η2 are the CNTs’ efficiency parameters. In this paper, simply supported (SSSS) and clamped (CCCC) boundary conditions are used. The results are obtained in terms of non-dimensional frequency parameter \(\widetilde{\omega } = \left( {\omega a^{2} /h} \right)\sqrt {\rho^{\text{m}} /E^{\text{m}} }\) using NURBS-based isogeometric analysis for obtaining eigensolution. As evident from Table 1, the present results in terms of non-dimensional fundamental natural frequency are found to be in close conformity with those from the published literature, for rectangular CNTRC plates with simply supported boundary condition.

Table 1 Non-dimensional frequency parameters of CNTRC plate with simply supported boundary condition

First, we analyze the effect of aspect ratio and volume fraction of CNTs with simply supported (SSSS) boundary condition, and it shows that as the aspect ratio decreases, the non-dimensional natural frequency increases and it also increases with the increase of volume fraction of carbon nanotube as shown in Table 2.

Table 2 Non-dimensional natural frequency \(\overline{\omega } = \omega \left( {a^{2} /h} \right)\sqrt {\rho^{\text{m}} /E^{\text{m}} }\) for square plate with simply supported boundary condition

Next, we analyze the effect of width-to-thickness ratio with clamped (CCCC) boundary condition, and it can be seen that the non-dimensional natural frequency increases with the increase of b/h ratio and it also increases with the increase in volume fraction of CNTs as shown in Table 3.

Table 3 Non-dimensional natural frequency with clamped boundary condition under uniform load

From Fig. 2, convergence takes place at 8 × 8 elements with clamped boundary condition.

Fig. 2
figure2

Convergence of rectangular plate having clamped boundary conditions

Figure 3 shows the four types of CNTRC plates with different boundary conditions, and it can be seen that greater the width-to-thickness ratio greater will be the non-dimensional natural frequency. From all the four types of graphs, it can also be seen that the value of frequency in UD and FG-X type is greater than the FG-V and FG-O because the distribution of volume fraction of CNTs can affect the stiffness of the plate. In UD and FG-X, the reinforcement distribution is more in the top and bottom than the mid-plane. The stiffness is mainly influenced by the top and the bottom surface, i.e., if the volume fraction of CNTs is more in the top and bottom, the stiffness is more. So, the non-dimensional natural frequency is more in UD and FG-X distribution type than that in the other two types of distributions.

Fig. 3
figure3

Non-dimensional natural frequency of CNTRC plates with h/a = 0.1 a UD, b FG-V, c FG-O and d FG-X

Conclusions

This paper presents the plate formulation using NURBS-based isogeometric analysis with the semi-refined HSDT for free vibration of FG-CNTRC plate. The comparison and the convergence are carried out to assure the correctness of the given method. The following are the conclusions made with the present analysis: It can be seen that the non-dimensional frequency increases when the plate aspect ratio (b/a) decreases. As the volume fraction of CNTs increases, the non-dimensional natural frequency increases. As the b/h ratio increases, i.e., the thickness of the plate decreases, the non-dimensional frequency increases. It is concluded that the non-dimensional natural frequency in UD and FG-X type is more than that in the FG-V and FG-O type.

From the conclusion, it has been suggested that to reduce the non-dimensional natural frequency, the aspect ratio and volume fraction of CNT should be small and the thickness of the plate should be more.

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Correspondence to Ashish K. Singh.

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Singh, A.K., Bhar, A. Isogeometric FE analysis of CNT-reinforced composite plates: free vibration. SN Appl. Sci. 1, 1010 (2019). https://doi.org/10.1007/s42452-019-1027-x

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Keywords

  • Isogeometric analysis (IGA)
  • Carbon nanotube-reinforced composite (CNTRC)
  • Higher-order shear deformation theory (HSDT)
  • Non-uniform rational B-spline (NURBS)
  • FORTRAN