Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

doi:10.1016/j.compstruct.2018.05.116

Composite Structures

Volume 201, 1 October 2018, Pages 13-20

https://doi.org/10.1016/j.compstruct.2018.05.116 Get rights and content

Abstract

This study aims to investigate the postbuckling response of functionally graded (FG) nanoplates by using the nonlocal elasticity theory of Eringen to capture the size effect. In addition, Reddy’s third-order shear deformation theory is adopted to describe the kinematic relations, while von Kámán’s assumptions are used to account for the geometrical nonlinearity. In order to calculate the effective material properties, the Mori-Tanaka scheme is adopted. Governing equations are derived based on the principle of virtual work. Isogeometric analysis (IGA) is employed as a discretization tool, which is able to satisfy the $C^{1}$ -continuity demand efficiently. The Newton-Raphson iterative technique with imperfections is employed to trace the postbuckling paths. Various numerical studies are carried out to examine the influences of gradient index, nonlocal effect, ratio of compressive loads, boundary condition, thickness ratio and aspect ratio on the postbuckling behaviour of FG nanoplates.

Introduction

The use of nanostructures in modern technology is increasingly attracting the attention of many researchers recently owing to their advanced mechanical and electric characteristics [1]. Nanobeams and nanoplates are frequently adopted as fundamental components in biosensors, atomic force microscopes, micro-electro-mechanical systems, nano-electro-mechanical systems. Therefore, a comprehensive understanding of their structural behaviour is needed to be investigated. In fact, the mechanical responses of those small-scale structures are significantly size-dependent as experimentally verified. In order to investigate the behaviour of nanostructures, three common approaches have been used in the literature: atomistic model [2], hybrid atomistic-continuum mechanic model [3], and non-classical continuum mechanic model [4], [5]. In general, the two former approaches are less popular in practice due to their highly computational cost and complication. In contrast, the non-classical continuum mechanic model is widely employed owing to its simplicity. Based on this model, various theories have been proposed in the literature to capture the size effect in small-scale structures [6]. For nanostructures, the nonlocal elasticity theory [7], [8], [9], [10] is broadly used. According to this theory, the stress at a point of a continuum is assumed to be dependent not only on the strains at that point but also on the strains at other points in the body.

In addition to conventional analyses (e.g. static bending, free vibration, buckling), postbuckling analysis is important for proper design of nanostructures. However, it is seen that there has been only a small number of studies investigating this problem based on the nonlocal elasticity theory in the literature [6]. In a comprehensive study, Shen [11] investigated the nonlinear bending, vibration and postbuckling responses of a stiff thin film resting on an elastic foundation. The author also considered the thermal effect in this study. However, the solutions were only given for simply supported plates. The postbuckling analysis of graphene sheets in a polymer environment was addressed in the work of Naderi and Saidi [12]. The classical plate theory was employed to model the thin graphene sheets subjected to uniaxial and biaxial compressive loads, while the solutions were obtained by using the Galerkin method. By employing the first-order shear deformation theory, Ansari and Gholami [13] examined the buckling and postbuckling behaviour of magneto-electro-thermo nanoplates. In addition, the general differential quadrature method was adopted in their study to obtain the solutions. The buckling and postbuckling responses of peizoelectric nanoplates were also reported by Liu et al. [14]. The authors also adopted the first-order shear deformation theory to describe the kinematic relations, and the solutions were obtained based on the differential quadrature method. Gholami and Ansari [15] proposed a unified high-order shear deformation model to investigate the postbuckling behaviour of rectangular piezoelectric-piezomagnetic nanoplates. Based on the Isogeometric analysis (IGA) approach, Soleimani et al. [16] studied the postbuckling response of orthotropic single-layered graphene sheets under in-plane loadings. The nonlocal plate in this study was also modelled based on the assumptions of first-order shear deformation theory. Overall, it is seen that most of current studies on postbuckling analysis of nanoplates mainly focused on the size effect of graphene sheets, thin films or piezoelectic nanoplates.

In recent years, functionally graded materials (FGMs) appears to be an advanced composite material whose properties vary smoothly in prescribed directions. They have been increasingly used in small scale structures due to their favourable characteristics. Their application can be found in various components in high-tech devices such as Micro-electromechanical Systems (MEMS) and Nano-electromechanical Systems (NEMS). FG nanoplates in these devices could be manufactured based on the multilayer process which combines both chemical vapour deposition and high-growth rate plasma-enhanced chemical vapour deposition bulk layer [17]. Although the linear and nonlinear responses of nonlocal FG nanoplates were successfully studied by Natarajan et al. [18], Nguyen et al. [19] and Phung-Van et al. [20], the investigation on their postbuckling behaviour is still limited. Therefore, the main objective of this study is to study the postbuckling response of FG nanoplates based on the nonlocal elasticity theory.

In this study, the Mori-Taknaka scheme [21] is used to calculate the effective material properties of FGMs. The third-order shear deformation theory of Reddy [22] is employed to eliminate the use of shear correction factor. Furthermore, the geometrical nonlinearity is accounted based on the von Kámán’s assumptions. The nonlinear governing equations are derived based on the principle of virtual work. The system equation is obtained by employing the IGA approach [23] as a discretization tool, which is able to satisfy the $C^{1}$ -continuity requirement of interpolation functions naturally and efficiently [24]. The Newton-Raphson iterative scheme is adopted to trace the postbuckling paths. Geometrical imperfections are imposed to the initial geometry of the plate to obtain the bifurcation buckling paths. Several numerical examples are also presented to find out the influences of gradient index, nonlocal parameter, ratio of in-plane compressive loads, boundary condition, thickness ratio and aspect ratio on the postbucking response of FG nanoplates.

Section snippets

Material descriptions

As described in Fig. 1, a rectangular FGM plate consisting of two distinct materials with their properties varying continuously through the plate thickness is investigated in this study. The top $(z = h / 2)$ and bottom $(z = - h / 2)$ surfaces are prescribed ceramic and metal constituents, respectively. Volume fractions of ceramic $(V_{c})$ and metal $(V_{m})$ constituents at an arbitrary point in the plate’s volume are calculated by $V_{c} = {(\frac{z}{h} + \frac{1}{2})}^{κ}; V_{m} = 1 - V_{c}$ where $κ$ denotes the gradient index.

According to the Mori-Tanaka

Overview of nonlocal elasticity theory

According to the nonlocal elasticity theory of Eringen [7], [8], [9], [10], the stress at a point $x$ in an elastic continuum body is not only calculated based on the strain at its point but also based on the strains at all other points. The nonlocal stress tensor at point $x$ is given by $σ = \int_{V} ζ (|x^{'} - x|, τ) t (x^{'}) d x^{'}$ where $t$ is the classical macroscopic stress tensor at point $x$ . $ζ (|x^{'} - x|, τ)$ is the kernel function describing the nonlocal modulus. $τ = e_{0} a / ℓ$ is a nonlocal parameter describing the nonlocal effect.

IGA-based finite element formulations

According to the IGA approach, the middle domain $Ω$ of a rectangular plate can be expressed as a b-spline surface $Ω (x, y) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} R_{i, j}^{p . q} (ξ, η) B_{i, j}$ where $H_{i, j}^{p . q} (ξ, η)$ are the 2-dimensional b-spline basis functions and $B_{i, j}$ is the control net. The b-spline basis functions are constructed based on a tensor product of two univariate b-spline basis functions as $R_{i, j}^{p, q} (ξ, η) = \sum_{i = 1}^{n} \sum_{j = 1}^{m} N_{i, p} (ξ) M_{j, q} (η)$ in which, $N_{i, p} (ξ)$ and $M_{j, q} (η)$ are the basis functions in $ξ$ and $η$ directions. These two basis functions are defined based

Verification and convergence studies

To the best of the authors’ knowledge, there is no study on the postbuckling of FG nanoplates. Therefore, a postbuckling problem of an armchair graphene sheet addressed in the study of Naderi and Saidi [12] is revisited to verify the reliability and accuracy of the present approach. Material properties and geometrical information of the armchair graphene sheet are listed as follows: $E_{11} = 1949 GPa$ , $E_{22} = 1962 GPa$ , $ν_{12} = 0.201$ , $G_{12} = 846 GPa$ , $a \times b = 4.888 \times 4.855$  nm², and h = 0.156 nm, and the values of

Conclusions

In this paper, the postbuckling behaviour of FG nanoplates is investigated based on the IGA approach. The nonlocal elasticity theory is adopted to capture the size effect. The Mori-Tanaka scheme is used to evaluate the effective properties of FGMs. The kinematic relations are based on the third-order shear deformation theory, and von Kámán’s assumptions are used to account for the geometrical nonlinearity. The governing equations are derived based on the principle of virtual work. The

Acknowledgements

This research study was supported by La Trobe University under its Disciplinary Research Program (DRP) and Postgraduate Research Scholarship. This financial support is gratefully acknowledged

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Postbuckling analysis of functionally graded nanoplates based on nonlocal theory and isogeometric analysis

Abstract

Introduction

Section snippets

Material descriptions

Overview of nonlocal elasticity theory

IGA-based finite element formulations

Verification and convergence studies

Conclusions

Acknowledgements

Ultramicroscopy

J Sound Vib

Int J Eng Sci

Compos Struct

Int J Eng Sci

Int J Eng Sci

Compos Struct

Int J Eng Sci

Theor Appl Mech Lett

Compos Struct

Results Phys

Comput Mater Sci

Comput Methods Appl Mech Eng

Compos Part B: Eng