Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments

doi:10.1016/j.engstruct.2013.06.002

Engineering Structures

Volume 56, November 2013, Pages 698-708

https://doi.org/10.1016/j.engstruct.2013.06.002 Get rights and content

Highlights

•
We propose a multi-scale approach for nonlinear analysis of FG-CNTRC beams.
•
Beam with intermediate material properties does not necessarily have intermediate frequencies.
•
Beam with intermediate material properties does not necessarily have intermediate temperatures.
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Thermal postbuckling path of unsymmetric FG-CNTRC beams is no longer of the bifurcation type.

Abstract

This paper studies the behaviors of large amplitude vibration, nonlinear bending and thermal postbuckling of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation in thermal environments. Two types of carbon nanotube-reinforced composite (CNTRC) beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the beam thickness direction, and are estimated through a micromechanical model. The motion equations of a CNTRC beam on an elastic foundation are derived based on a higher order shear deformation beam theory. The thermal effects are also included in the motion equations and the material properties of CNTRCs are assumed to be temperature-dependent. Numerical studies are carried out for the nonlinear vibration, nonlinear bending and thermal postbuckling of CNTRC beams resting on Pasternak elastic foundations under different thermal environmental conditions. It is found that a CNTRC beam with intermediate CNT volume fraction does not necessarily have intermediate nonlinear frequencies, buckling temperatures and thermal postbuckling strengths. Thermal postbuckling path of unsymmetric FG-CNTRC beams is no longer the bifurcation type.

Introduction

Carbon nanotubes (CNT) possess exceptional mechanical, thermal and electrical properties and are used as significant reinforcement materials for high performance structural composites with substantial application potentials [1], [2], [3]. Carbon nanotube-reinforced composites (CNTRCs) have advanced mechanical properties such as high strength, high stiffness and light weight which can be applied as layers in advanced laminated structures. Recent researches on CNTRCs [4], [5], [6], [7] observed that only a low percentage of CNTs (2–5% by weight) can be added to the composites as more volume fraction in CNTRCs can actually cause the deterioration of their mechanical properties [8]. Shen [9] proposed to apply the functionally graded (FG) concept to CNTRCs in order to effectively make use of the low percentage of CNTs in the composite. He studied the nonlinear bending behavior of CNTRC plates with a linear distribution of CNTs along the thickness direction of the plates and observed that the load-bending moment curves of the plates can be considerably improved through the use of a functionally graded distribution of aligned CNTs in the matrix. Shen and his co-authors [10], [11], [12], [13], [14] further extended the study to the postbuckling and nonlinear vibration of CNTRC plates and shells and highlighted the influence of the FG-CNT distribution patterns on the mechanical behaviors of the CNTRC structures. The concept of functionally graded nanocomposites is strongly supported by a recent publication [15] in which a functionally graded CNT reinforced aluminum matrix composite was fabricated by a powder metallurgy route. Consequently, investigations on bending, buckling and vibration of CNTRC structures are recently emerged as an interesting field of study [16], [17], [18], [19], [20].

Several studies have been reported on the bending, buckling and vibration of CNTRC beams based on Euler–Bernoulli beam theory, Timoshenko beam theory and higher order shear deformation beam theory [21], [22], [23], [24], [25], [26]. Among those, Ke et al. [21] investigated the nonlinear free vibration of functionally graded CNTRC Timoshenko beams. They found that both linear and nonlinear frequencies of functionally graded CNTRC beams with symmetrical distribution of CNTs are higher than those of beams with uniform or unsymmetrical distribution of CNTs. This work was then extended to the cases of functionally graded CNTRC Euler–Bernoulli beams with piezoelectric layers by Rafiee et al. [22] and dynamic stability of functionally graded CNTRC Timoshenko beams by Ke et al. [23]. Yas and Heshmati [24] presented free vibration and linear buckling of functionally graded CNTRC Timoshenko beams on an elastic foundation by using the differential quadrature method. They [25] also presented a dynamic analysis of functionally graded CNTRC beams under the action of moving load by using the finite element method (FEM). Recently, Wattanasakulpong and Ungbhakorn [26] presented linear bending, buckling and vibration of CNTRC beams resting on an elastic foundation based on a higher order shear deformation beam theory. Like in the case of functionally graded ceramic–metal beams with simply supported boundary conditions, the bifurcation buckling of simply supported functionally graded CNTRC beams does not exist due to the stretching–bending coupling effect. Therefore, the bifurcation solutions for simply supported unsymmetric functionally graded CNTRC beams subjected to in-plane compression and temperature variation may be physically incorrect [24], unless the compressive load or the stress resultant caused by temperature rise is applied on the physical neutral surface of the CNTRC beam.

In the present work, we focus our attention on the nonlinear free vibration, nonlinear bending and thermal postbuckling of CNTRC beams resting on an elastic foundation in thermal environmental conditions. Two types of CNTRC beams, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. 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 thermal effects 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. The nonlinear vibration characteristics, nonlinear bending and thermal postbuckling behaviors of CNTRC beams resting on an elastic foundation under different sets of thermal environmental conditions are presented and discussed in details.

Section snippets

Effective material properties of functionally graded CNTRCs

We assume that the CNTRC layer is made from a mixture of aligned single-walled carbon nanotubes (SWCNTs) and matrix which is assumed to be isotropic. The SWCNT reinforcement is either uniformly distributed or functionally graded along the thickness direction of a CNTRC structure. At the nanoscale, the structure of the carbon nanotube strongly influences the overall properties of the composite. Several micromechanical models have been developed to predict the effective material properties of

Governing equations

Consider a uniform beam of length L, width b, and thickness h, with two pinned ends and resting on a two-parameter elastic foundation. The beam is exposed to elevated temperature and is subjected to a transverse static or dynamic load Q. Let $\bar{U}$ be the displacement in the longitudinal direction, and $\bar{W}$ be the deflection of the beam. $\bar{Ψ}$ _x is the mid-plane rotation of the normal about the Y axis. As is customary [36], [37], the foundation is assumed to be a compliant foundation, which means that no

Nonlinear vibration problem

For nonlinear vibration problem, we are to determine the relationship between the vibration amplitudes and the frequencies. We assume that the solutions of Eqs. (15), (16) can be expressed as $\begin{matrix} W (x, t) = W^{*} (x) + \tilde{W} (x, t), \\ Ψ_{x} (x, t) = Ψ_{x}^{*} (x) + {\tilde{Ψ}}_{x} (x, t), \end{matrix}$ where W^∗(x) is an initial deflection due to initial thermal bending moment, and $\tilde{W} (x, t)$ is an additional deflection. $Ψ_{x}^{*} (x)$ is the mid-plane rotation corresponding to W^∗(x). ${\tilde{Ψ}}_{x} (x, t)$ is defined analogously to $Ψ_{x}^{*} (x)$ , but is for $\tilde{W} (x, t)$ . Note that $W^{*} (x) = Ψ_{x}^{*} (x) = 0$

Nonlinear bending problem

For nonlinear bending problem, we are to determine the relationship between the applied pressure and the deflection of the beam. In this case, the static transverse load is assumed to be uniform, and Q(X, t) = q₀. Hence, W is independent of time. The governing equations can be written in the simple form as $γ_{11} \frac{\partial^{4} W}{\partial x^{4}} - γ_{12} \frac{\partial^{3} Ψ_{x}}{\partial x^{3}} - π \{\int_{0}^{π} [\frac{γ_{13}}{2} {(\frac{\partial W}{\partial x})}^{2} + γ_{14} \frac{\partial Ψ_{x}}{\partial x} - γ_{15} \frac{\partial^{2} W}{\partial x^{2}}] dx\} \frac{\partial^{2} W}{\partial x^{2}} + γ_{T 1} Δ T \frac{\partial^{2} W}{\partial x^{2}} - \frac{\partial^{2} M^{T}}{\partial x^{2}} = λ_{q} - (K_{1} W - K_{2} \frac{\partial^{2} W}{\partial x^{2}})$ $γ_{21} \frac{\partial^{3} W}{\partial x^{3}} - γ_{22} \frac{\partial^{2} Ψ_{x}}{\partial x^{2}} + γ_{23} (Ψ_{x} + \frac{\partial W}{\partial x}) - \frac{\partial S^{T}}{\partial x} = 0$ The solutions of Eqs. (31), (32) can also be determined by

Thermal postbuckling problem

As mentioned before, Eq. (7) is only valid in the case of the two ends of the beam are immovable, and therefore, only thermal postbuckling is discussed in this section. In such a case λ_q = 0, W is only a function of x. Unlike in Sections 4 Nonlinear vibration problem, 5 Nonlinear bending problem, ΔT is now an unknown. Let λ_T = ΔT, the governing equations can be written in the simple form as $γ_{11} \frac{\partial^{4} W}{\partial x^{4}} - γ_{12} \frac{\partial^{3} Ψ_{x}}{\partial x^{3}} - π \{\int_{0}^{π} [\frac{γ_{13}}{2} {(\frac{\partial W}{\partial x})}^{2} + γ_{14} \frac{\partial Ψ_{x}}{\partial x} - γ_{15} \frac{\partial^{2} W}{\partial x^{2}}] dx\} (\frac{\partial^{2} W}{\partial x^{2}} + \frac{\partial^{2} W^{*}}{\partial x^{2}}) - π \{\int_{0}^{π} [\frac{γ_{13}}{2} (2 \frac{\partial W}{\partial x} \frac{\partial W^{*}}{\partial x} + {(\frac{\partial W^{*}}{\partial x})}^{2}) + γ_{14} \frac{\partial Ψ_{x}^{*}}{\partial x} - γ_{15} \partial^{2}]\}$

Concluding remarks

The large amplitude vibration, the nonlinear bending and the thermal postbuckling analyses for CNTRC beams have been presented on the basis of a micromechanical model and multi-scale approach. A parametric study for UD- and FG-CNTRC beams with low CNT volume fractions has been carried out. The new findings are: the CNTRC beam with intermediate CNT volume fraction does not necessarily have intermediate nonlinear frequencies, buckling temperatures and thermal postbuckling strengths. Thermal

Acknowledgments

This study was supported by the National Natural Science Foundation of China under Grant 51279103, the Fund of State Key Laboratory of Ocean Engineering under Grant GKZD010059, and the IRIS Research Grant 20721-80872 of University of Western Sydney. The authors are grateful for the support.

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Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments

Highlights

Abstract

Introduction

Section snippets

Effective material properties of functionally graded CNTRCs

Governing equations

Nonlinear vibration problem

Nonlinear bending problem

Thermal postbuckling problem

Concluding remarks

Acknowledgments

Carbon

Mater Des

Compos Sci Technol

Comput Methods Appl Mech Eng

Polymer

Comput Mater Sci

Mater Des

Compos Struct

Mater Des

Comput Mater Sci

Compos B Eng

Comput Methods Appl Mech Eng

Compos Struct

Comput Methods Appl Mech Eng

Compos Struct

Mater Des

Mater Des

Compos Struct

Compos Struct

Int J Pres Ves Pip

Appl Math Model

Compos Mater Sci

Mech Mater