Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations

doi:10.1016/j.ijnonlinmec.2017.02.010

International Journal of Non-Linear Mechanics

Volume 91, May 2017, Pages 69-75

https://doi.org/10.1016/j.ijnonlinmec.2017.02.010 Get rights and content

Highlights

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Nonlinear vibration of thermally postbuckled CNTRC beams is solved.
•
The initial deflection caused by thermal postbuckling is taken into account.
•
The beam-foundation interaction and temperature-dependent material properties are both taken into account.
•
FG reinforcement has a significant effect on the nonlinear vibration characteristics of thermally postbuckled CNTRC beams.

Abstract

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled carbon nanotube-reinforced composite (CNTRC) beams resting on elastic foundations. For the CNTRC beams, uniformly distributed (UD) and functionally graded (FG) reinforcements are considered where the temperature-dependent material properties of CNRTC beams are assumed to be graded in the thickness direction and estimated through a micromechanical model. The motion equations are derived based on a higher order shear deformation beam theory with including the beam-foundation interaction. The initial deflection caused by thermal postbuckling is also included. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled CNTRC beams under uniform temperature field. The effects of carbon nanotube (CNT) volume fraction and distribution patterns as well as foundation stiffness on the vibration characteristics of CNTRC beams are examined in detail.

Introduction

Thermal postbuckling and the vibration of thermally postbuckled isotropic and laminated beams with or without elastic foundation are conventional topics in structural mechanics. Compares with the small- and large-amplitude vibrations of compressed postbuckled beams [1], [2], [3], [4], [5], relatively few works have been made on the small- and large-amplitude vibrations of beams in thermal prebuckling and postbuckling equilibrium states. Li et al. [6] studied the natural frequencies of thermally buckled isotropic beams with various boundary conditions based on the Euler-Bernoulli beam theory. Esfahani et al. [7] studied small amplitude vibrations of a functionally graded ceramic-metal beam under in-plane thermal loading in the prebuckling and postbuckling regions based on the Timoshenko beam theory. The numerical results in these studies show that the fundamental frequencies of beams decrease with temperature rise in the prebuckled states and increase after the beams go into thermal postbuckling states.

Utilizing the concept of functionally graded (FG) materials, a new class of emerging composite materials, that is the FG carbon nanotube-reinforced composite (CNTRC), has been proposed by making use of CNTs as the reinforcements in a functionally graded pattern. The major difference between the conventional carbon fiber-reinforced composites and the carbon nanotube-reinforced composites lies in that the former can contain very high percentage of the carbon fibers (usually over 60% by volume), while the latter only has a low percentage of CNTs (about 2–5% by weight) [8], [9], [10], [11]. This is due to the fact that a larger CNT volume fraction in CNTRCs can actually lead to the deterioration of the mechanical properties of the composites [12]. The functionally graded distribution of CNTs in CNTRCs has the advantage to arrange the CNTs at the locations with the most effective reinforcements. Functionally graded carbon nanotube reinforced composites (FG-CNTRCs) were first proposed by Shen [13] with CNT distributions within an isotropic matrix designed specifically to grade them with certain rules along the desired directions to improve mechanical properties of the structures. Ke et al. [14] firstly studied the nonlinear free vibration of FG-CNTRC Timoshenko beams by using Ritz method. This work was then extended to the cases of FG-CNTRC Timoshenko beams with initial geometric imperfections by Wu et al. [15]. Wattanasakulpong and Ungbhakorn [16] presented linear bending, buckling and vibration of CNTRC beams resting on an elastic foundation based on a higher order shear deformation beam theory. Shen and Xiang [17] presented large amplitude vibration, nonlinear bending and thermal postbuckling of FG-CNTRC beams resting on Pasternak elastic foundations based on a higher order shear deformation beam theory. Lin and Xiang [18], [19] studied linear and nonlinear free vibrations of FG-CNTRC beams based on the Euler-Bernoulli and Timoshenko beam theories, respectively. To the best of the authors’ knowledge, there is no literature covering nonlinear vibration behavior of thermally postbuckled CNTRC beams, in particular for the case of the beam resting on an elastic foundation.

In the present work, we focus our attention on the linear and nonlinear free vibration of thermally postbuckled CNTRC beams resting on an elastic foundation in uniform temperature field. Distribution of the CNTs is assumed to be uniform (UD) or functionally graded (FG) through thickness. The motion equations are based on a higher order shear deformation beam theory and von Kármán-type nonlinear strain–displacement relationships. The beam-foundation interaction and the initial deflection caused by thermal postbuckling are also included. The material properties of CNTRCs are assumed to be temperature-dependent. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction and are estimated through a micromechanical model. Two ends of the beam are assumed to be simply supported and in-plane boundary conditions are assumed to be immovable.

Section snippets

Large amplitude vibration of thermally postbuckled beams

Consider a uniform beam of length L, width b, and thickness h, with two pinned ends and resting on an elastic foundation of Pasternak-type. The origin of the coordinate in located at the corner of the middle plane for the beam, where X-axis is longitudinal and Z-axis is perpendicular to the mid-plane. Let $\bar{U}$ be the displacement in the longitudinal direction, and $\bar{W}$ be the deflection of the beam. $Ψ_{x}$ is the mid-plane rotation of the normal about the Y axis. The foundation is assumed to be a

Solution procedure

A two-step perturbation technique [26] is adopted to solve the motion equations. Before carrying out the solution process, it is convenient to first define the following dimensionless quantities for such beams $\begin{array}{l} x = π \frac{X}{L}, (W, W^{*}) = \frac{(\bar{W}, {\bar{W}}^{*})}{L}, Ψ_{x} = \frac{1}{π} {\bar{Ψ}}_{x}, t = \frac{π \bar{t}}{L} \sqrt{\frac{E_{0}}{ρ_{0}}}, \\ ω_{L} = Ω_{L} \frac{L}{π} \sqrt{\frac{ρ_{0}}{E_{0}}}, (γ_{11}, γ_{12}, γ_{21}, γ_{22}) = \frac{1}{{\bar{D}}_{11}} (- S_{11}, S_{12}, - S_{21}, S_{22}), \\ γ_{23} = \frac{L^{2}}{π^{2} {\bar{D}}_{11}} S_{23}, γ_{13} = \frac{L^{2} {\bar{A}}_{11}}{π^{2} {\bar{D}}_{11}}, (γ_{14}, γ_{15}) = \frac{L}{π^{2} {\bar{D}}_{11}} ({\bar{B}}_{11} - \frac{4}{3 h^{2}} {\bar{E}}_{11}, \frac{4}{3 h^{2}} {\bar{E}}_{11}), \\ (γ_{16}, γ_{26}) = \frac{1}{{\bar{A}}_{11} L} ({\bar{B}}_{11}, {\bar{B}}_{11} - \frac{4}{3 h^{2}} {\bar{E}}_{11}), γ_{17} = - \frac{b I_{1} E_{0} L^{2}}{π^{2} ρ_{0} {\bar{D}}_{11}}, \\ (γ_{18}, γ_{19}, γ_{28}, γ_{29}) = - ({\hat{I}}_{5}, - \frac{4}{3 h^{2}} {\hat{I}}_{7}, {\tilde{I}}_{3}, - \frac{4}{3 h^{2}} {\tilde{I}}_{5}) \frac{b E_{0}}{ρ_{0} {\bar{D}}_{11}}, γ_{T 1} = \end{array}$

Numerical results and discussions

Numerical results are presented in this section for linear and nonlinear vibration characteristics of CNTRC beams resting on elastic foundations in the prebuckling and postbuckling equilibrium states. Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be $ρ^{m}$ =1150 kg/m³, $ν^{m}$ =0.34, $α^{m}$ =45(1+0.0005 $Δ$ T)×10⁻⁶/K and $E^{m}$ =(3.52-0.0034 T) GPa, in which T= T₀ + $Δ$ T and T₀ =300 K (room temperature). In such a way, $α^{m}$ =45.0×10⁻⁶/K and $E^{m}$

Concluding remarks

Small- and large-amplitude vibration analyses for thermally postbuckled CNTRC beams have been presented by a multi-scale approach. Micromechanical model is used to estimate the temperature dependent material properties of CNTRCs. The major difference between present model and previous one is that the present solution includes the initial deflection caused by thermal postbuckling. This initial deflection is not a constant and will be developed in the whole thermal postbuckling region. Solution

Acknowledgments

The support for this work, provided by the National Natural Science Foundation of China under Grant 51279103, is gratefully acknowledged.

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Vibration of thermally postbuckled carbon nanotube-reinforced composite beams resting on elastic foundations

Highlights

Abstract

Introduction

Section snippets

Large amplitude vibration of thermally postbuckled beams

Solution procedure

Numerical results and discussions

Concluding remarks

Acknowledgments

Int. J. Solids Struct.

Comput. Methods Appl. Mech. Eng.

Polymer

Comput. Mater. Sci.

Mater. Des.

Compos. Struct.

Compos. Struct.

Compos. Part B

Comput. Mater. Sci.

Eng. Struct.

Appl. Math. Model.

J. Sound Vib.

Compos. Struct.