Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation

Abstract

This paper presents an investigation on free vibration of functionally graded (FG) porous nanocomposite plates reinforced with a small amount of graphene platelets (GPLs) and supported by the two-parameter elastic foundations with different boundary conditions. Four different porosity distributions of the parent metal were reinforced by four various GPLs dispersion patterns (evenly or unevenly) along the thickness direction. To obtain graded distribution in both porosity and GPLs, the Young’s modulus, shear modulus and density of nanocomposites are assumed to vary through the thickness direction due to variation of internal pores and the volume fraction of GPLs. By employing Halpin-Tsai micromechanics model, effective elastic modulus of the nanocomposites is obtained according to the assumption of closed-cell cellular solids under Gaussian Random Field scheme. The governing equations of FG GPLs-reinforced metal foams resting on elastic foundations are derived from Hamilton’s principle by the means of classic plate theory with the consideration of von Kármán strain-displacement relation. Applying the differential quadrature method, the dimensionless natural frequencies of porous nanocomposite plates with different boundary conditions are obtained and present method is thoroughly validated with the results in open literature. The comprehensive parametric studies on different porosity coefficients, GPL dispersion patterns, weight fraction of GPL, aspect ratios, thickness ratios and parameters of elastic foundation on free vibration of FG GPLs-reinforced porous plates with various boundary conditions. An interesting finding is that the increase of porosity coefficients would lead to the linearly decrease of both mass density and stiffness of the plates, but the increase of porosity coefficients does not always induce the decrease of natural frequencies. Moreover, both porosity distribution and GPL dispersion pattern have a distinct effect on dynamic characteristics, the former plays a more important role than the latter in analysing mechanical properties of FG GPLs-reinforced plates.

Introduction

Compared with traditional composite materials, nanocomposite materials possess the extremely high surface to volume ratio of the reinforcing phase and/or high aspect ratio, in which one or more phases with nanoscale dimensions are embedded in a metal, ceramic, or polymer matrix. In general, the carbonaceous nanofillers, such as Carbon nanotubes (CNTs) in the form of 1D and Graphene platelets (GPLs) in the form of 2D phases, are introduced in the polymeric matrix as additives to improve their mechanical properties and maintain their functional applications as well. Due to the rapidly advanced production techniques and falling costs of material manufacture, nanocomposite materials are increasingly used in flexible batteries, replacement bones, structural components, lightweight sensors and so on in both academia and industry. As such, nanocomposite materials are identified as one of the most promising materials with a great growth potential in composite structures’ areas such as electronics, aerospace, biomedical, automotive and civil engineering. Thus, it is of a great importance to investigate mechanical characteristics of nano-reinforced structures since the application of such materials is becoming extensive and in-depth.

Numerous investigations have shown that the addition of small amounts of CNTs in polymer matrix can significantly strengthen the mechanical, thermal or physical properties of polymeric composites in both experimental analysis and theoretical methods. Material scientists found that only 0.1 wt% content of CNT can significantly improve strength, Young’s modulus and strain to failure of the composites in the laboratory [1]. Wuite and Adali [2] investigated the deflection and stress behaviour of CNTs reinforced polymer composite beams by using three different micromechanics models. By introducing the concept of FGMs, Shen [3] firstly studied the bending behaviours of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) plates by considering thermal effect. Following this idea, Ke et al. [4] applied Timoshenko beam theory and von Kármán geometric nonlinearity to investigate the nonlinear free vibration of FG-CNTRCs beams with different boundary conditions based on Ritz method. They [5] also analysed the dynamic stability characteristics of FG-CNTRC beams by employing Bolotin’s method and the influence of nanotube volume fraction, slenderness ratio and boundary conditions were discussed. FG-CNTRCs are also widely applied in intelligent structures, which bonded with piezoelectric layers on the surface of sensors or actuators. Rafiee et al. [6] conducted an investigation of actuator voltage on thermal bifurcation buckling load of piezoelectric FG-CNTRCs beams. The free vibration of FG-CNTRCs with surface-bonded piezoelectric layers with the consideration of thermal and voltage effect were also proposed by them [6]. Along with the thorough study, researchers found that the mechanical properties of GPLs-based nanocomposites out-perform that of CNTs reinforced materials and the reasons may be related to their high specific surface area, enhanced nanofiller – matrix adhesion/interlocking arising from their wrinkled (rough) surface, as well as the two-dimensional (planar) geometry of graphene platelets [7]. As a result, the mechanical behaviour of nanocomposites reinforced by graphene platelets nanofillers has become an emerging research field currently. Yang, as one of the main researchers on FG multilayer GPL-reinforced composites (GPLRC), has thoroughly investigated such material used in different structures under various conditions. For instance, Feng, Kitipornchai and Yang [8], [9] presented nonlinear free vibration and bending characteristics of multilayer nanocomposite beam reinforced by GPLs by using Timoshenko beam theory. Then, Yang, Wu and Kitipornchai also investigated the buckling and postbuckling [10], dynamic buckling [11] of GPLs-based composite beams and concluded that distributing more GPLs near the surface areas can significantly increase the stiffness of whole structures, and further magnify the natural frequency and increase the buckling load. Recently, the free and forced vibration of FG multilayer GPLRC plates within the framework of first-order shear deformation plate theory were proposed by Song, Kitipornchai and Yang [12]. The bending and vibration behaviours of trapezoidal plates made of nanocomposites were analysed based on FEM by Zhao et al. [13]. The thermal buckling and postbuckling [14], parametric instability [15] of the same target model were also discussed by Wu, Yang and Kitipornchai.

Most recently, as a new class of closed-cell metal reinforced foams, namely nanocomposite metal foams, were successfully produced owing to the rapidly developed nanotechnology [16], [17], [18], [19]. Such materials synergistically combine the remarkable properties of both metal foams (low density, light weight, excellent energy absorption capacity, etc.) and nanocomposites (high Young’s modulus, superior fracture strength, extreme thermal conductivity, etc.). Kitipornchai and his colleagues [20] first presented the micromechanics model of FG porous nanocomposites reinforced by GPLs where both internal pores and GPLs are layer-wise distributed in the matrix either evenly or non-evenly on the account of three specific different patterns. Inspired by this idea, a few latest studies were released. The nonlinear free vibration and postbuckling of shear deformable FG porous nanocomposite beams made of closed-cell metal reinforced by GPLs was investigated by Chen et al. [21] and they found that a small amount of GPLs additive can significantly strengthen the stiffness of the beam. Gao et al. [22] proposed nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load. Akbaş [23] studied the free vibration and static bending of a simply supported functionally graded (FG) plate with various porosity coefficients. The postbuckling analysis of imperfect GPLs reinforced beam with porosities was presented by Barati and Zenkour based on the refined beam theory [24]. Sahmani et al. [25] expanded this concept to micro/nano-scale and by employing nonlocal strain gradient theory, they investigated the size dependency effect of FG porous micro/nano-beams reinforced with GPLs.

As discussed above, most of existing studies about FG porous nanocomposites reinforced by GPLs focused on beam structures, few attempts have been made on application of GPLs-based porous foams to plate structures. However, in engineering practices, the plate/sheet structures are widely used all kinds of fields in order to match the desired functionality and optimize the structures [22], [26], [27], [28]. Furthermore, nanocomposites plate structures are found to be rested on or embedded in elastic foundation/medium [29], [30]. Under such circumstance, dynamic characteristics may be different from the conventional ones. Jamali et al. [31] studied the buckling analysis of nanoplate and nanocomposite plate with a central square hole embedded in the Winkler foundation. Garcia et al. [32] compared the bending and free vibration behaviours of FG GPLs and CNTs reinforced composites plates for both fully aligned and randomly oriented filler configurations. By applying Chebyshev polynomial function in the Lagrangian multipliers method, Kiani [33] studied the free vibration characteristics of CNTs-based composite thick plates, which are resting on point supports. Zhang et al. [29] analysed FG-CNTRC composites plates resting on elastic foundations involving the shear deformation effect and the influence of elastic foundation on vibration frequency of plates were examined. Based on first-order shear deformation plate theory (FSDT) and Hamilton principle, the static response and free vibration of FG-CNTRC plates resting on Winkler-Pasternak elastic foundations was solved by Navier solution [30]. An exact solution for vibration of thick rectangular plates made of metal porous was carried out by Rezaei and Saidi [34] and the effect of fluid saturated in porous medium were considered. As can be seen, the understanding of free vibration and nonlinear vibration plays an important role in design and analysis of FG porous nanocomposite plates reinforced by GPLs. However, from authors’ knowledge and above reviewed literature, no previous work has been done for dynamic behaviour of FG GPLs-reinforced porous plates by considering the effect of elastic medium, especially for different porosity distributions with various GPLs dispersion patterns inside. Thus, it is necessary to develop a reliable and computationally efficient method to analyse the free vibration and nonlinear vibration behaviour of FG GPLs-reinforced porous plates resting on elastic foundation. The proper understanding and development of dynamic characteristics of such structures can help researchers and engineers design and analysis of nano/micro-sized devices and systems.

The present work attempts to develop a reliable and computationally efficient model to investigate the vibration behaviour of rectangular plates made of FG GPLs-reinforced metal foams resting on the two-parameter elastic foundations with different boundary conditions. Four different GPLs dispersion patterns (evenly or unevenly) are layer-wise distributed in four various porosity distributions of metal matrix along the thickness direction. Equations of motion are derived from Hamilton’s principle by the means of classic plate theory with the consideration of von-Kármán strain-displacement relation. The dimensionless natural frequencies are computed by using differential quadrature scheme and the proposed method was validated with some reliable results known from literature. Then effects of different parameters, such as different porosity coefficients, GPL dispersion patterns, weight fraction of GPL, aspect ratios, thickness ratios and parameters of elastic foundation on free vibration of FG GPLs-reinforced porous plates with various boundary conditions are discussed in detail.