Elsevier

Composite Structures

Volume 93, Issue 10, September 2011, Pages 2496-2503
Composite Structures

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part II: Pressure-loaded shells

https://doi.org/10.1016/j.compstruct.2011.04.005Get rights and content

Abstract

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.

Introduction

Buckling under external pressure is a basic problem for cylindrical shells. The early work [1] predicted that the postbuckling equilibrium path is unstable for the cylindrical shell subjected to external pressure, but the experimental results of themselves [2] and others [3], [4] did not support this conclusion. In fact, the buckling pressure was found to be greater than 20% of the theoretical prediction sometimes [5], [6]. It seems reasonable to conclude that the postbuckling equilibrium path of moderately long cylindrical shells is stable under external pressure. This conclusion is supported by experimental results of [3], [4] and is also recently supported by molecular dynamics (MD) simulations (no any hypothesis was made) for carbon nanotubes subjected to external pressure [7].

Shen and his coworkers [8], [9], [10] studied the buckling and postbuckling behavior of functionally graded ceramic-metal cylindrical shells subjected to lateral or hydrostatic pressure in thermal environments. In the above studies, the material properties were considered to be temperature-dependent and the effect of temperature rise and/or heat conduction on the postbuckling behavior was reported. It is concluded that the postbuckling behavior of moderately long FGM cylindrical shells subjected to external pressure is stable and the shell structure is virtually imperfection-insensitive. The same conclusion was recently reported in [11] for the FGM cylindrical shell without an elastic foundation subjected to internal pressure.

It has been shown in Part I [12] that the mid-plane symmetric functionally graded carbon nanotube (CNT) reinforcements can increase the buckling load as well as postbuckling strength of the shell subjected to axial compression. Now we need to know whether this reinforcement phase can also increase the buckling pressure as well as postbuckling strength of the carbon nanotube-reinforced composite (CNTRC) cylindrical shell under a low nanotube volume fraction when the shell is subjected to external pressure.

As in the case of axial compression, two kinds of CNTRC shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model in which the CNT efficiency parameter is estimated by matching the elastic modulus of CNTRCs observed from the MD simulation results with the numerical results obtained from the extended rule of mixture. The governing equations are based on a higher order shear deformation shell theory with von Kármán-type of kinematic nonlinearity and include thermal effects. A singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. It has been shown [13] the nonlinear prebuckling deformations are likely to play a great role in shells under external pressure. As a result, the nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. The numerical illustrations show the full nonlinear postbuckling response of CNTRC cylindrical shells subjected to lateral or hydrostatic pressure in environmental conditions.

Section snippets

Multi-scale model for functionally graded CNTRC shells under external pressure

Consider a cylindrical shell made of CNTRCs. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model, as expressed by Eqs. (1), (2), (3), (4) in Part I [12]. The shell is assumed to be geometrically imperfect, and is subjected to external pressure q in thermal environments. Based on Reddy’s higher order shear deformation theory [14] with a von Kármán-type of kinematic nonlinearity and include thermal effects,

Solution procedure

Introducing dimensionless quantities of (15a), (15b), (15c) in Part I [12], and letλq=q(3)3/4LR3/24πA11A221/8D11D223/8,δq=ΔxL(3)3/4LR1/24πD11D22A11A223/8The nonlinear Eqs. (1), (2), (3), (4) may then be written in dimensionless form asε2L11(W)-εL12(Ψx)-εL13(Ψy)+εγ14L14(F)-γ14F,xx=γ14β2L(W+W,F)+γ1443(3)1/4λqε3/2L21(F)+γ24L22(Ψx)+γ24L23(Ψy)-εγ24L24(W)+γ24W,xx=-12γ24β2L(W+2W,W)εL31(W)+L32(Ψx)-L33(Ψy)+γ14L34(F)=0εL41(W)-L42(Ψx)+L43(Ψy)+γ14L44(F)=0where all dimensionless linear operators L

Numerical results and discussion

Numerical results are presented in this section for perfect and imperfect, CNTRC cylindrical shells with a low nanotube volume fraction subjected to lateral or hydrostatic pressure in thermal environments. As in the case of axial compression, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the (10, 10) SWCNTs are selected as reinforcements. All the material properties are the same as those used in Part I [12]. The CNT efficiency parameters used in Eq. (1) in

Concluding remarks

Postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells subjected to external pressure in thermal environments has been presented on the basis of a micromechanical model and multi-scale approach. A parametric study for FG- and UD-CNTRC cylindrical shells with low nanotube volume fractions has been carried out. The buckling and postbuckling behaviors of CNTRC cylindrical shells under external pressure differ considerably from analogous behaviors for

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