Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects
Highlights
►Nonlocal two variable refined plate theory is developed for nanoplates to study buckling analysis. ► Navier’s method is adopted for solving the governing equations. ► The closed-form solution for buckling load of a simply supported rectangular nanoplate is obtained. ► Effects of nonlocality, mode number and aspect ratio on buckling load are investigated. ► It can be concluded that the present theory does not require shear correction factor.
Introduction
Nanoplates such as graphene [1], the two-dimensional (2D) counterpart of three-dimensional graphite, has attracted vast interests in solid-state physics, materials science, and nanoelectronics since it was discovered in 2004 as the first free-standing 2D crystal. Graphene is considered as a promising electronic material in post silicon electronics. However, large-scale synthesis of high quality graphene represents a bottleneck for the next generation graphene devices. Existing routes for graphene synthesis include mechanical exfoliation of highly ordered pyrolytic graphite (HOPG) [2], eliminating Si from the surface of single crystal SiC [3], depositing graphene at the surface of single crystal [4] or polycrystalline metals [5], and various wet-chemistry based approaches [6], [7]. However, up to now no methods have delivered high quality graphene with large area required for application as a practical electronic material.
A great deal of research has been conducted to explore the promising properties of the single-layered graphene sheets (SLGSs) after appearance of the new method of graphene sheet preparation [8], [9], [10]. Furthermore, Katsnelson and Novoselov [11] have explored the unique electronic properties of the SLGSs. They have stated that the graphene sheet is an unexpected bridge between condensed matter physics and quantum electrodynamics. Moreover, Meyer et al. [12] have achieved the ability of distinguishing between single- and multi-layered graphene sheets by analyzing electron diffraction. On the other hand, Bunch et al. [13] have reported the experimental results of using electromechanical resonators made from suspended single- and multi-layered graphene sheets.
Because of the unique electrical, mechanical and thermal properties enable the nanostructures (such as graphene, carbon nanotube, nanorod, and nanofibre) to be used for the development of superconductive devices for micro-electromechanical system (MEMS) and nano-electromechanical system (NEMS) applications. Conducting experiments with nanoscale size specimens is both difficult and expensive. Hence, development of appropriate mathematical models for nanostructures is an important issue concerning the application of nanostructures. The modeling for the nanostructures is classified into three main categories. The approaches are atomistic [14], [15], continuum [14], [15] and hybrid atomistic-continuum mechanics [16], [17], [18]. The above atomic methods are limited to systems with a small number of molecules and atoms and therefore restricted to the study of small-scale modeling. In order to carry out analysis for a large-sized atomic system, other powerful and effective models for the analysis are needed. Continuum mechanics approach is less computationally expensive than the former two approaches. Further, it has been found that continuum mechanics results are in good agreement with those obtained from atomistic and hybrid approaches.
Nanotechnologies small scale makes the applicability of classical or local continuum models, such as beam, shell and plate models, questionable. Classical continuum models do not admit intrinsic size dependence in the elastic solutions of inclusions and inhomogeneities. At nanometer scales, however, size effects often become prominent, the cause of which needs to be explicitly addressed due to an increasing interest in the general area of nanotechnology [19]. Sun and Zhang [20] indicated the importance of a semi-continuum model in analyzing nanomaterials after pointing out the limitations of the applicability of classical continuum models to nanotechnology. In their semi-continuum model for nanostructured materials with plate like geometry, material properties were found completely dependent on the thickness of the plate structure contrary to classical continuum models. The modeling of such a size-dependent phenomenon has become an interesting research subject in this field [21], [22], [23]. It is thus concluded that the applicability of classical continuum models at very small scales is questionable, since the material microstructure, such as lattice spacing between individual atoms, becomes increasingly important at small size and the discrete structure of the material can no longer be homogeneities into a continuum. Therefore, continuum models need to be extended to consider the scale effect in nanomaterial studies. This can be accomplished through proposing nonlocal continuum mechanics models.
Nonlocal elasticity theory [24], [25], [26], [27], [28], [29] was proposed to account for the scale effect in elasticity by assuming the stress at a reference point to be a function of strain field at every point in the body. This way, the internal size scale could be simply considered in constitutive equations as a material parameter Only recently has the nonlocal elasticity theory been introduced to nanomaterial applications. As the length scales are reduced, the influences of long-range interatomic and intermolecular cohesive forces on the static and dynamic properties tend to be significant and cannot be neglected. The classical theory of elasticity being the long wave limit of the atomic theory excludes these effects. Thus the traditional classical continuum mechanics would fail to capture the small scale effects when dealing in nano structures. The small size analysis using local theory over predicts the results. Thus the consideration of small effects is necessary for correct prediction of micro/nano structures. Chen et al. [30] that the nonlocal continuum theory based models are physically reasonable from the atomistic viewpoint of lattice dynamics and molecular dynamics (MD) simulations. Peddieson et al. [31] applied nonlocal elasticity to formulate a nonlocal version of the Euler–Bernoulli beam model and concluded that nonlocal continuum mechanics could potentially play a useful role in nanotechnology applications.
In wave mechanics of nanostructures, one important outcome of the nonlocal elasticity is the realistic prediction of the dispersion curve i.e., frequency-wavenumber/wavevector relation. As shown in Eringen [24], the dispersion relationwhere e0a nonlocality parameter, closely matches with the BornKarman model dispersionwhen e0 = 0.39 is considered. However, among the two natural conditions at the mid-point and end of the first Brillouin zone:these relations satisfy only the first one. It was suggested that two-parameter approximation of the kernel function will give better results. This is reiterated by Lazar et al. [66] that one parameter (only e0a) nonlocal kernel will never be able to model the lattice dynamics relation and it is necessary to use the bi-Helmholtz type equation with two different coefficients of nonlocality to satisfy all the boundary conditions.
It is to be noted that the simple forms of the group and phase velocities that exist for isotropic materials permitted to tune the nonlocality parameters so that the lattice dispersion relation is matched. Further, by virtue of the Helmholtz decomposition, only one-dimensional Brillouin zone needs to be handled. Although the general form of the boundary conditions, i.e., group speed is equal to phase speed (at k = 0) or zero (at k = π/a), is still applicable, the expressions are difficult to handle. This is because, the Brillouin zone is really a two-dimensional region where four boundary conditions are involved.
Various size-dependent continuum theories which capture small scale parameter such as couple stress elasticity theory [32], strain gradient theory [33], modified couple stress theory [34] are reported. These modified continuum theories are being used for the analysis of small scale structures. However, the most reportedly used continuum theory for analyzing small scale structures is the nonlocal elasticity theory initiated by Eringen [24]. Using this nonlocal elasticity theory, some drawbacks of the classical continuum theory can be efficiently avoided and size-dependent phenomena can be satisfactorily explained. In nonlocal elasticity theory the small scale effects are captured by assuming the stress components at a point is dependant not only on the strain components at the same point but also on all other points in the domain [24], [35].
In the literature a great deal of attention has been focused on studying the buckling behavior of one-dimensional nanostructures using nonlocal elasticity theory. These nanostructures include nanobeams, nanorods and carbon nanotubes. On the contrary no work appears related to the buckling of biaxially compressed nanoplate based on two-variable refined plate theory. However some studies using nonlocal elasticity theory on mechanical behavior of isotropic nanoplates are recently reported by Murmu and Pradhan [36], [37], [38] and by Duan and Wang [39]. Recently Sakhaee-Pour [40] studied the buckling of graphene nanosheets via atomistic modeling. Understanding the importance of employing nonlocal elasticity theory in small scale structures, various research works have been reported on static, dynamic and stability analysis of 1D and 2D micro/nano structures [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64].
When compressive loads are applied onto the nanoplate, it tends to buckle. Understanding the buckling behavior is an important issue from design perspective. Consequently, numerous studies on elastic buckling of scale free plates can be found in literature. Nanoplates as an upcoming nano engineering structures can also be subjected to in-plane loads. These buckling in-plane loads would be influenced by small scale effects. For proper use of graphene sheets as MEMS or NEMS component, its stability response under in-plane load should be studied. Thus there is a strong encouragement for acquiring proper understanding and mathematical modeling of the buckling of nanoplates. Though buckling of nanoplates is an important factor for proper design of nanodevices, very few studies are reported.
In the present work attempt is made to study the buckling characteristics of nanoplates using two variable refined plate theory and nonlocal elasticity. Higher order plate theory represents the kinematics better, does not require shear correction factor and the present refined plate theory also does not require shear correction factor and yield more accurate interlaminar stress distributions [65]. In principle,it is possible to expand the displacement field in terms of the thickness coordinate up to any desired degree. However, due to the algebraic complexity and computational effort involved with higher order theories in return for gain in accuracy, theories higher than third order have not been attempted. Such type of complexity can be reduced in the refined plate theory. The two variable refined plate theory takes account of transverse shear effects and parabolic distribution of the transverse shear strains through the thickness of the plate, hence it is unnecessary to use shear correction factors. This avoids the need for shear correction coefficients used in FSDT.
Recently, Malekzadeh et al. [68] studied the small scale effect on the thermal buckling characteristic of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. They modeled the surrounding elastic medium as the two-parameter elastic foundation. The solution procedure was based on the transformation of the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. The fast rate of convergence of the method was shown and the results were compared against existing results in literature. Malekzadeh et al. [69] studied the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates using the nonlocal elasticity theory. The formulation was derived based on the first order shear deformation theory (FSDT). They investigated the effects of nonlocal parameter in combination with the geometrical shape parameters, thickness-to-length ratio and the boundary conditions on the frequency parameters of the nanoplates. Ansari et al. [70] developed the nonlocal elastic plate model accounting for the small scale effects to investigate the vibrational behavior of multi-layered graphene sheets under various boundary conditions. Their analysis provides the possibility of considering different combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit expressions for the nonlocal frequencies of a double-layered graphene sheet with all edges simply supported were also obtained. Narendar et al. [71] presented an ultrasonic type of wave propagation characteristics of monolayer graphene on silicon (Si) substrate. An atomistic model of a hybrid lattice involving a hexagonal lattice of graphene and surface atoms of diamond lattice of Si was developed to identify the carbon-silicon bond stiffness. Properties of that hybrid lattice model was mapped into a nonlocal continuum framework. Equivalent force constant due to Si substrate was obtained by minimizing the total potential energy of the system. For this equilibrium configuration, the nonlocal governing equations were derived to analyze the ultrasonic wave dispersion based on spectral analysis. Their analysis show that the silicon substrate affects only the flexural wave mode. The frequency band gap of flexural mode was also significantly affected by the Si substrate. The results also show that, the silicon substrate adds cushioning effect to the graphene and it makes the graphene more stable. The analysis also show that the frequency bang gap relations of in-plane (longitudinal and lateral) and out-of-plane (flexural) wave modes depends not only on the y-direction wavenumber but also on nonlocal scaling parameter. Hui-Shen [72] presented postbuckling, nonlinear bending and nonlinear vibration analyses for a simply supported stiff thin film resting on a two-parameter elastic foundation in thermal environments. The stiff thin film was modeled as a nonlocal orthotropic plate which contains small scale effects. The elastomeric substrate with finite depth was modeled as a two-parameter elastic foundation. The thermal effects were included and the material properties of the substrate were assumed to be temperature-dependent. The numerical results reveal that the small scale parameter reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies, but increases the nonlinear to linear frequency ratios of the thin film slightly. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of the thin film resting on an elastic foundation.
More recently, Narendar and Gopalakrishnan [73] estimated the critical buckling temperature of single-walled carbon nanotubes (SWCNTs), which were embedded in one-parameter elastic medium (Winkler foundation) under the umbrella of nonlocal continuum mechanics theory. An explicit expression for the non-dimensional critical buckling temperature was also derived in that work. The effect of the nonlocal small scale coefficient, the Winkler foundation parameter and the ratio of the length to the diameter on the critical buckling temperature was investigated in detail. Finally, Wang et al. [74] presented a study of the mechanisms of nonlocal effect on the transverse vibration of two-dimensional nanoplates, e.g., monolayer layer graphene and boron-nitride sheets. It was found that such a nonlocal effect stems from a distributed transverse force due to the curvature change in the nanoplates and the surface stress due to the nonlocal atom–atom interaction. A single equivalent vibration wavelength was defined to measure the nonlocal effect on the vibration of 2D nanoplates. The critical equivalent wavelength of order 0.55–2.23 nm was obtained for significant nonlocal effect on monolayer graphene.
This paper presents a formulation of nonlocal two variable refined plate theory for the buckling analysis of biaxially compressed nanoplate. The small scale effects are introduced using the nonlocal elasticity theory. Governing equations are derived from the principle of virtual displacements. Using the principle of virtual work, the nonlocal governing equations are derived for rectangular nanoplates and explicit relations for buckling loads are presented. The effects of the small scale on buckling loads considering various parameters are examined and discussed in detail. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories as applied to the stability analysis of nanoplates and nanoshells.
Section snippets
Theory of nonlocal elasticity: 2D-formulation
This theory assumes that the stress state at a reference point X in the body is regarded to be dependent not only on the strain state at X but also on the strain states at all other points X′ of the body. The most general form of the constitutive relation in the nonlocal elasticity type representation involves an integral over the entire region of interest. The integral contains a nonlocal kernel function, which describes the relative influences of the strains at various locations on the stress
Numerical results and discussion
To illustrate the effects of small scale, the ratio of the buckling loads obtained from both nonlocal and local theories are computed and plotted with respect to various nonlocal scaling parameter. Table 1 depicts the small scale effects on buckling load ratio (ξ) for biaxially compressed nanoplate. By small-size plates we mean plates with micro- or nano-dimensions. The plate is considered to be square a/b = 1 and is subjected to equal compressive in-plane loads in the x1 and x2 directions (i.e.,
Conclusion
In the present work, buckling analysis of a monolayer nanoplate under biaxial compression and at small scales is carried out with continuum models. Nonlocal elasticity theory is being employed into two variable refined plate theory to capture the buckling behavior of nanoplate. From the present work following conclusions are drawn:
- (a)
Buckling load ratio decreases with increase in nonlocal small scale parameter and this variation is more prominent in higher modes of buckling.
- (b)
As the size of the
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