Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels

doi:10.1016/j.cma.2014.01.024

Computer Methods in Applied Mechanics and Engineering

Volume 273, 1 May 2014, Pages 1-18

https://doi.org/10.1016/j.cma.2014.01.024 Get rights and content

Abstract

A large deflection geometrically nonlinear behavior of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels under uniform point transverse mechanical loading is studied. The analysis is carried out using the kp-Ritz method with kernel particle function is employed to construct the shape functions for the two-dimensional displacement approximations. Based on the first-order shear deformation shell theory, nonlinear governing equations are developed with geometric nonlinearity taking the form of von Kármán strains. It is assumed that carbon nanotubes are uniaxially aligned in the axial direction and are functionally graded in thickness direction of the cylindrical panels. The effective material properties of resulting CNTR-FG panels are estimated by employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach. A stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness and the membrane as well as shear terms are calculated by the direct nodal integration method to eliminate shear locking, for a very thin cylindrical panel. Several numerical example problems are examined to reveal the influences of volume fraction of carbon nanotubes, span angle, edge-to-radius ratio and thickness on nonlinear responses of the CNTR-FG panels. Moreover, effects of different boundary conditions and distribution type of carbon nanotubes are also investigated.

Introduction

Carbon nanotube reinforced composite materials have received wide attention from researchers since carbon nanotubes (CNTs) have become potential constituents of reinforcements because they have been demonstrated to have high strength and stiffness with high aspect ratio and low density. Applications of carbon nanotube reinforced composite materials may expand in areas such as reinforcing composites, electronic devices and many others. Therefore, theoretical and numerical analyzes play an important role in capturing and understanding the fine mechanical properties of these carbon nanotube reinforced composite materials.

Earlier investigations about carbon nanotube reinforced composite materials were mainly focused on their material properties and constitutive modeling. Since the performance of a composite material system is highly dependent on the interfacial characteristics of the reinforcement and the matrix material, Liao and Li [1] reported a study on interfacial characteristics of a carbon nanotube reinforced polystyrene composite system through molecular mechanics simulations and elasticity calculations. They found that interfacial shear stress of the CNT-polystyrene system is about 160 MPa, significantly higher than most carbon fiber reinforced polymer composite systems. Wong et al. [2] examined the mechanical properties of CNT/polystyrene and CNT/ epoxy thin film. The results showed that these polymers adhered well to CNT at the nanometer scale and CNT-polymer interfacial shear stress (at 0 K) is about 138 and 186 MPa for CNT/epoxy and CNT/polystyrene, respectively. With 1 wt% nanotubes addition dispersed homogeneously throughout polystyrene matrices, tensile tests on composite films revealed 36–42% increase in elastic modulus and 25% increase in breaking stress, implying significant load transfer across the nanotube-matrix interface [3]. Odegard et al. [4] proposed that the nanotube, the local polymer near the nanotube, and the nanotube/polymer interface can be modeled as an effective continuum fiber by using an equivalent-continuum modeling method at small length scales. By using a multiscale approach, Gao and Li [5] developed a shear-lag model for carbon nanotube-reinforced polymer composites. With nanotube modeled at the atomistic scale and matrix deformation analyzed by the continuum finite element method, Li and Chou [6] reported a multiscale modeling of the compressive behavior of carbon nanotube/polymer composites. Seidel and Lagoudas [7] modeled the effective elastic properties of carbon nanotube reinforced composites by using a composite cylinders micromechanics technique conjunction with the Mori–Tanaka and self-consistent techniques.

Although the above investigations are quite useful for understanding of constitutive and material properties of carbon nanotube reinforced composite materials, the ultimate purpose of the development of this advanced class of materials is usage in actual structural applications. Since carbon nanotube reinforced composites may be incorporated in the form of beam, plate or shell as structural components, there is a need to observe the global response of carbon nanotube reinforced beam, plate or shell. Wuite and Adali [8] presented a multiscale analysis of deflection and stress behavior of nanocomposite reinforced beams by using micromechanics relations to determine the elastic constants in terms of nanotube volume fraction. With effective material properties estimated by rule of mixture model, Yas and Samadi [9] carried out free vibration and buckling analyses of nanocomposite Timoshenko beams reinforced by single-walled carbon nanotubes resting on an elastic foundation. Rafiee et al. [10] studied large amplitude free vibration of functionally graded carbon nanotube reinforced composite beams with surface-bonded piezoelectric layers subjected to a temperature change and an applied voltage. For carbon nanotube reinforced plate, Zhu et al. [11] carried out static and dynamic analyses of functionally graded carbon nanotube reinforced composite plates using the finite element method. Formica et al. [12] studied vibration behaviors of CNTRC plates employing an equivalent continuum model, in accordance with the Eshelby–Mori–Tanaka approach. By using the mesh-free kp-Ritz, Lei et al. [13], [14], [15], [16], [17] reported some typical mechanical analysis of functionally graded carbon nanotube reinforced composite plates and panels. With nanocomposite plates reinforced by single-walled carbon nanotubes resting on an elastic foundation, a large amplitude vibration analysis in thermal environments is presented [18]. By employing an equivalent continuum model based on the Eshelby–Mori–Tanaka approach, Aragh et al. [19] studied natural frequency characteristics of a continuously graded CNT-reinforced cylindrical panel. Shen and Xiang [20] investigated the large amplitude vibration behavior of nanocomposite cylindrical shells in thermal environments. Shen [21] presented thermal buckling and postbuckling analysis of nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes subjected to a uniform temperature rise. Some further investigations about postbuckling analysis of nanocomposite cylindrical shells subjected to axial compression and lateral pressure were performed by Shen [22], [23].

In this paper, a first attempt to study large deflection geometrically nonlinear behavior of carbon nanotube-reinforced functionally graded (CNTR-FG) cylindrical panels is carried out using the mesh-free kp-Ritz method based on a modified version of Sander’s nonlinear shell theory to derive the discretized governing equations. With CNTs assumed uniaxially aligned in axial direction and functionally graded in thickness direction of the panels, the effective material properties of CNTR-FG cylindrical panels are estimated through a micromechanical model based on the Eshelby–Mori–Tanaka approach. To improve computational efficiency and eliminate shear and membrane locking, a stabilized conforming nodal integration scheme is employed to evaluate the system bending stiffness and the membrane; shear terms are calculated by the direct nodal integration method. A combination of the arc-length iterative algorithm and the modified Newton–Raphson method is adopted to solve the nonlinear system equations to track the full load–displacement path. Several computational simulation examples are presented to figure out the effects of volume fraction of CNTs, edge-to-radius ratio, span angle, thickness, boundary conditions and distribution types of CNTs on nonlinear responses of the CNTR-FG panels.

Section snippets

Carbon nanotube reinforced composite panels

A CNTR-FG cylindrical panel having the length L, radius R, span angle θ₀ and thickness h is shown in Fig. 1. A coordinate system (x, θ, z) is established on the middle surface of the panel. The panels are made of a mixture of single-walled carbon nanotubes (SWCNTs) and the matrix in which the CNTs are assumed to be uniaxially aligned in axial direction and functionally graded in thickness direction of the cylindrical panels (Fig. 2). According to distributions of CNTs in the thickness direction

Displacement filed and strains

According to the first-order shear deformation shell theory [32], the displacement field is expressed as $u (x, θ, z) = u_{0} (x, θ) + z ϕ_{x} (x, θ),$ $v (x, θ, z) = v_{0} (x, θ) + z ϕ_{θ} (x, θ),$ $w (x, θ, z) = w_{0} (x, θ),$ where u₀, v₀ and w₀ represent displacements of a point at the middle surface of the panel in x, θ and z directions; ϕ_x and ϕ_θ denote rotations of a transverse normal about positive θ and negative x axes, respectively.

In accordance with the above displacement field, the strain–displacement relations are given as $\{\begin{matrix} ε_{xx} \\ ε_{θ θ} \\ γ_{x θ} \end{matrix}\} = ε_{0} + z κ$

Numerical results and discussion

Numerical results are presented in this section for large deflection analysis of CNTR-FG cylindrical panels. Poly (methyl methacrylate), referred as PMMA, with material properties v_m = 0.34, α^m = 45(1 + 0.0005ΔT) × 10⁻⁶/K and E^m = (3.52 − 0.0034T) GPa, where T = T₀ + ΔT and T₀ = 300 K (room temperature) is selected as the matrix. Han and Elliott [39] obtained relatively low values of modulus for (10, 10) SWCNTs $(E_{11}^{CNT} = 600 GPa, E_{22}^{CNT} = 10 GPa, G_{12}^{CNT} = 17.2 GPa)$ since the effective thickness of CNTs was assumed as 0.34

Conclusions

In this paper, a first attempt to use the mesh-free kp-Ritz method for large deflection geometrically nonlinear analysis of CNTR-FG cylindrical panels subjected to mechanical loads is performed. CNTs are assumed to be graded in thickness direction of the cylindrical panel and effective material properties are estimated through a micromechanical model based on the Eshelby–Mori–Tanaka approach. The formulation is based on the first-order shear deformation shell theory. The two-dimensional