NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells

Highlights

• NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells is performed in this paper.

• The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT).

• The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption.

• Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated.

• Some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.

Abstract

An investigation into the postbuckling and geometrically nonlinear behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells is carried out in this study. The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT). The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The incremental solutions are obtained by using a modified Riks method. In the present formulation, the rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated. Exact geometries of shells are modeled by using NURBS interpolation. Several verifications are given to show the high reliability of the proposed formulation. Especially, some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.

Introduction

The discovery of carbon nanotubes (CNTs) [1] has made a great stride for materials science. Nowadays, carbon nanotubes have been drawing a great interest by their notable electrical, thermal and mechanical characteristics [2], [3]. From the structural point of view, carbon nanotubes are impressive by their high strength, stiffness, aspect ratio and low density. Hence, CNTs can be considered as a good candidate for composite structures. In practice, shells are widely used to manufacture and construct the important and complicated components such as shells of ships, roofs of buildings and wings of airplanes, etc. For these reasons, behaviors of FG-CNTRC shells need to be studied particularly. Shells are categorized into membranes, thin or thick shells depending on their thicknesses. Using the Kirchhoff–Love assumption, the thin shell theory neglects transverse shear deformations. Therefore, it is only appropriate for thin shells. Thereafter, the first-order shear deformation shell theory (FSDT) was proposed to take into account the transverse shear deformations when shells become thicker. The FSDT is not only appropriate for both thin and moderate thick shells but also convenient in combinations with many numerical methods. This is due to its approximated displacement field only requests the $C^0$ continuity of basic functions. Higher-order shear deformation shell theories (HSDTs) have been developed to describe comprehensively the shear stresses and strains. Several HSDTs can be mentioned as third-order shear deformation theory [4], quasi-3D shear deformation theory [5], generalized shear deformation theories [6], [7], etc. In addition, the layerwise theories were proposed to fill a gap between equivalent single-layer model and 3D elastic model [8], [9]. A good review of shell theories is mentioned in [10], [11]. In this paper, postbuckling isogeometric analysis of FG-CNTRC shells based on the FSDT is focused.

To overcome some disadvantages of analytical approaches, many numerical methods have been investigated for analyses of shells such as finite element method (FEM) [12], [13], [14], [15], [16], [17], [18], [19], [20], smoothed FEM [21], [22], [23], meshless method [24], [25] and an isogeometric-meshless coupling approach [26], etc. In attempts to develop the advanced numerical methods, isogeometric analysis (IGA) or NURBS-based finite element analysis was proposed by Hughes et al. [27]. As an advantage of this method, NURBS basis functions possess high derivatives that allow convenient combinations with most of shell theories. In IGA, both exact geometry and approximated solutions are obtained from the same basis functions. Therefore, geometric data from computer-aided design (CAD) can be used directly for numerical simulation. It is asserted that the main procedures of IGA and FEM are same. Hence, isogeometric analysis can be considered as a unified approach that bridges the existing gap between CAD and FEM. This opens a new way for computational mechanics, especially for industrial problems with arbitrary geometries and required high accuracy. An overview and computer implementation aspects of IGA were presented in [28]. For linear analyses of shells, IGA has been successfully investigated for isotropic thin shells [29], [30], [31], [32], [33] and composite thick shells [34], [35]. In addition, it has been extensively studied for nonlinear problems [36], [37], [38], [39], optimization problems [40] and inverse problems [41] of isotropic shells. It should be emphasized that the application of IGA to nonlinear analyses of FG-CNTRC shells is limited. To the best of author’s knowledge, this is the first study that performs postbuckling analysis of FG-CNTRC shells using isogeometric analysis and the FSDT.

In this work, the formulation based on IGA and FSDT is developed for postbuckling analysis of FG-CNTRC shells. The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The significant effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are detected. Especially, several complex equilibrium paths of FG-CNTRC panels and cylinders are first provided. This could be useful for future references. The outline of this paper is organized as follows. Section 2 mentions the FSDT for FG-CNTRC shells. The FG-CNTRC shell formulation based on NURBS and FSDT is presented in Section 3. Numerical results are provided and discussed in Section 4. The paper is closed with some notable conclusions in Section 5.