Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments, Part I: Axially-loaded shells
Introduction
Recently, a new class of promising materials known as carbon nanotubes (CNTs) has drawn considerable attention. Experiments and simulations showed that CNTs have extraordinary mechanical properties over carbon fibers [1]. As the mechanical properties of composites depend directly upon the embedded fiber mechanical behavior, replacing conventional carbon fibers with CNTs can potentially improve composite properties, such as tensile strength and elastic modulus. Therefore, the introduction of carbon nanotubes into polymers may improve their applications in the fields of reinforcing composites [2], [3] electronic devices [4] and more.
Most studies on carbon nanotube-reinforced composites (CNTRCs) have focused on their material properties [5], [6], [7], [8], [9], [10]. Several investigations have shown that the addition of a small percentage of nanotubes (2–5% by weight) in a matrix may considerably increase the composite’s mechanical, electrical and thermal properties [6], [7], [8], [9], [10]. Wuite and Adali [11] found that the stiffness of CNTRC beams can be improved significantly by the homogeneous dispersion of a small percentage of CNTs. Vodenitcharova and Zhang [12] studied the pure bending and bending-induced local buckling of CNTRC beams. Formica et al. [13] presented the vibration behavior of CNTRC plates by employing an equivalent continuum model based on the Mori–Tanaka approach. They found that the improvement achieves a maximum when the carbon nanotubes are uniformly aligned with the loading direction.
The traditional approach to fabricating nanocomposites implies that the nanotube is distributed either uniformly or randomly such that the resulting mechanical, thermal, or physical properties do not vary spatially at the macroscopic level. Experimental and numerical studies concerning CNTRCs have shown that distributing CNTs uniformly as the reinforcements in the matrix can achieve moderate improvement of the mechanical properties only [14], [15]. This is mainly due to the weak interface between the CNTs and the matrix where a significant material property mismatch exists. Functionally graded materials (FGMs) are a new generation of composite materials in which the microstructural details are spatially varied through nonuniform distribution of the reinforcement phase. Two kinds of FGMs are designed to improve mechanical behavior of plate/shell structures. One is functionally graded unidirectional fibers reinforced composites [16], [17]. Another one, like functionally graded ceramic–metal materials, is functionally graded particles reinforced composites [18], [19], [20]. The concept of FGM can be utilized for the CNTRC by non-homogeneous distribution of CNTs into the composite plates with a specific gradient so that the buckling behavior of CNTRC plates can be improved. Shen [21] suggested that the nonlinear bending behavior can be considerably improved through the use of a functionally graded distribution of CNTs in the matrix. The effect of CNT volume fraction on the compressive postbuckling and thermal postbuckling behavior of functionally graded CNTRC plates was reported by Shen and Zhu [22] and Shen and Zhang [23]. They found that in some cases the CNTRC plate with intermediate CNT volume fraction does not have intermediate buckling temperature and initial thermal postbuckling strength. Moreover, Ke et al. [24] investigated the nonlinear free vibration of functionally graded CNTRC Timoshenko beams. They found that both linear and nonlinear frequencies of functionally graded CNTRC beam with symmetrical distribution of CNTs are higher than those of beams with uniform or unsymmetrical distribution of CNTs.
Motivated by these considerations, the present work focuses attention on the postbuckling analysis of CNTRC shells subject to axial compression in thermal environments. Unlike the carbon fiber-reinforced composites, the CNTRCs usually have a lower volume fraction of CNTs due to the fact their mechanical properties will deteriorate if the volume fraction increases certain limit [25]. A crucial problem is how to determine and increase the buckling load and postbuckling strength of CNTRC cylindrical shells under such a low nanotube volume fraction. The material properties of single-walled carbon nanotubes (SWCNTs) are assumed to be size-dependent and temperature-dependent and are obtained from molecular dynamics (MD) simulations. The material properties of functionally graded 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 a Kármán-type of kinematic nonlinearity and include thermal effects. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account but, for simplicity, the form of initial geometric imperfection is assumed to be the same as the initial buckling mode of the shell. The numerical illustrations show the full nonlinear postbuckling response of CNTRC cylindrical shells subjected to axial compression in environmental conditions.
Section snippets
Material properties of functionally graded CNTRCs
We assume that the CNTRC layer is made from a mixture of SWCNT and matrix which is assumed to be isotropic. The SWCNT reinforcement is either uniformly distributed (UD) or functionally graded (FG) in the thickness direction, as shown in Fig. 1. 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 CNTRCs, e.g. Mori–Tanaka scheme [26],
Multi-scale model for functionally graded CNTRC shells under axial compression
Consider an CNTRC cylindrical shell with mean radius R, length L and thickness h. The shell is referred to a coordinate system (X, Y, Z) in which X and Y are in the axial and circumferential directions of the shell and Z is in the direction of the inward normal to the middle surface. The corresponding displacements are designated by , and . and are the rotations of the normals to the middle surface with respect to the Y and X axes, respectively. The origin of the coordinate system
Analytical method and asymptotic solutions
Having developed the theory, we are now in a position to solve Eqs. (5), (6), (7), (8) with boundary conditions (12). Before proceeding, it is convenient first to define the following dimensionless quantities (with γijk in Eqs. (22), (23) below are defined as in [32])
Numerical results and discussion
Numerical results are presented in this section for perfect and imperfect, CNTRC cylindrical shells subjected to axial compression. We first need to determine the effective material properties of CNTRCs. Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix, and the material properties of which are assumed to be νm = 0.34, αm = 45(1 + 0.0005ΔT) × 10−6/K and Em = (3.52–0.0034T) GPa, in which T = T0 + ΔT and T0 = 300 K (room temperature). In such a way, αm = 45.0 × 10−6/K and Em = 2.5 GPa at T=300
Concluding remarks
Postbuckling behavior of functionally graded carbon nanotube-reinforced composite cylindrical shells subjected to axial compression in thermal environments has been presented on the basis of a micromechanical model and multi-scale approach. The scale effect of CNT reinforcements is considered by introducing the CNT efficiency parameter that is estimated by matching the elastic modulus of CNTRCs predicted by the MD simulations with the prediction of the extended rule of mixture. A parametric
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