Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations

doi:10.1016/j.physe.2011.08.020

Physica E: Low-dimensional Systems and Nanostructures

Volume 44, Issue 1, October 2011, Pages 229-248

https://doi.org/10.1016/j.physe.2011.08.020 Get rights and content

Abstract

The potential applications of nanoplates in energy storage, chemical and biological sensors, solar cells, field emission, and transporting of nanocars have been attracted the attentions of the nanotechnology community to them during recent years. Herein, the later application of nanoplates from nonlocal elastodynamic point of view is of interest. To this end, dynamic response of a nanoplate subjected to a moving nanoparticle is examined within the context of nonlocal continuum theory of Eringen. The fully simply supported nanoplate is modeled based on the nonlocal Kirchhoff, Mindlin, and higher-order plate theories. The non-dimensional equations of motion of the nonlocal plate models are established. The effects of moving nanoparticle's weight and existing friction between the surfaces of the moving nanoparticle and nanoplate on the in-plane and out-of-plane vibrations of the nanoplate are incorporated into the formulations of the proposed models. The eigen function expansion and the Laplace transform methods are employed for discretization of the governing equations in the spatial and the time domains, respectively. The analytical expressions of the dynamic deformation field associated with each nonlocal plate theory are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight path (an opened path) as well as an ellipse path (a closed path). The dynamic in-plane forces and moments of each nonlocal plate model are also derived. Furthermore, the critical velocity and the critical angular velocity of the moving nanoparticle for the proposed models are expressed analytically for the aforementioned paths. Part II of this work consists in a comprehensive parametric study where the effects of influential parameters on dynamic response of the proposed nonlocal plate models are scrutinized in some detail.

Graphical abstract

Dynamic response of a simply supported nanoplate under the mass weight and friction force of a moving nanoparticle is of concern using nonlocal continuum theory of Eringen. The explicit expressions of dynamic displacements and forces are obtained based on the Kirchhoff, Mindlin, and higher-order plate theories. The presented figure displays top view of the nanoplate subjected to a moving nanoparticle when it: (a) travels on an arbitrary straight line; (b) orbits on an arbitrary elliptical path which is wholly located on the top surface of the nanoplate.

Highlights

► Using nonlocal continuum theory, a nanoplate under a moving nanoparticle is studied. ► The nanoplate is modeled using Kirchhoff, Mindlin, and higher-order plate models. ► The mass weight of the moving nanoparticle and the frictional force are considered. ► Using Laplace method, dynamic deformations of the proposed models are obtained. ► The critical velocity of the moving nanoparticle is derived for each nonlocal model.

Introduction

Two-dimensional nano-scaled sheets or nanoplates are pivotal parts of the nanotechnology society, although having paid less attention than the nanobeams. The lack of effort on nanoplates is possibly related to the troubles in their productions [1]. Recent research on the nanoplates reveals a large body of potential applications for them, including energy storage and conversion nanodevices [2], chemical and biological sensors [3], [4], [5], solar cells [6], [7], [8], field emission nanodevices [9], [10], [11], photo-catalytic degradation of organic dye [12], and nanovehicle transporters. In this paper, the later application of the nanoplates from a nonlocal elastodynamic point of view is of concern.

The fabrication of nanovehicles was initiated by a research group from Rice University [13], [14]. The first nanovehicles were constructed to reply to the question of how fullerenes can move on the metal surfaces. Despite the chosen name, the first nanovehicles do not have a molecular motor. Thereafter, Morin et al. [15] developed novel nanovehicles with synthetic molecular motors powered by light. In fabrication of the new nanovehicles, the carborane wheels and a light powered helicene motor are utilized. As light strikes the motor, it turns around in one direction, resulting in the movement of the nanocar. The wheels of the aforementioned nanovehicles are generally made from spherical molecules of carbon, hydrogen, and boron. Early tests were performed in a bath of toluene solvent found that the motor turns around as planned. Nevertheless, further tests are being carried out to realize whether the motorized nanovehicle can travel across a flat surface. In the present work, the nanovehicle is mathematically modeled by a moving point load, including the friction between the surfaces of the moving nanovehicle and the nanoplate.

The aforementioned promising applications of nanoplates have brought challenges in dynamic simulation of them due to external loading. In general, plate-like nanostructures experience both in- and out-of-plane displacements due to externally applied loads. As a result, the dynamic response and controlling of them due to the applied loads would be generally more complex than the beam-like nanostructures. Additionally, due to the size dependency of the elastic field in the equivalent nanostructure of the nanoplates, the classical continuum mechanics cannot succeed in capturing their realistic dynamic response. In other words, the classical (local) continuum theory would be reliable for those structures in which their dominant sizes are negligible when compared to the wavelength of the propagated sound. As long as the two above-mentioned length scales are comparable, the results of the local continuum mechanics would be questionable since the deformation at a point can be influenced by the deformation of its neighboring points. Consequently, atomic-based models such as molecular dynamics, Mont Carlo method, density function theory, and tight binding molecular dynamics are commonly employed to examine the dynamic behavior of nanostructures. However, the capabilities of the atomistic models are limited because of their complexities and need of computational effort in analyzing of the problem.

Different classes of nonlocal elasticity have been developed to overcome the shortage of the classical (local) continuum mechanics in modeling the realistic behavior of micro- and nano-scale structures. In a pioneering work by Cosserat and Cosserat [16], a couple stress theory was proposed in which an independent rotation field is incorporated into the equations of motion of elastic materials. The couple stresses were defined as the work associated with the micro-curvatures (i.e., the spatial gradient of rotations). Toupin [17] and Mindlin [18] developed a more general couple stress theory which includes not only micro-curvatures, but also takes into account gradients of normal strain. In the proposed model by Mindlin [18], five length scale parameters of the material's crystals were incorporated into the equations of motion of linear elastic materials. Later, Eringen [19], [20] established a nonlocal continuum theory in which the nonlocal stress filed was written as a single volume integral of the classical stress filed multiplied by an appropriate kernel function. The kernel function has a certain influence domain controlled by the small length scales of the material. In order to determine the small-scale lengths, some justifications should be carried out between the dynamic results of the nonlocal continuum theory with those of a suitable atomic model. It was shown that the developed nonlocal continuum theory would be capable in predicting of removal of stress singularities from crack tips and dislocation lines. By revisiting the two later continuum models, Aifantis [21] proposed a simpler and robust continuum form of the above-mentioned theories. This higher gradient continuum form could also be gained from the proposed polar stress theory by Mindlin through a specific choice of parameters. Thereby, it would lead to a reduction of the number of length scale parameters from five to one [22]. Recently, the interest in nonlocal continuum mechanics has been substantially rising due to a large body of experimentally observed data on the size effect on the response of nanostructures. The interested readers are referred to see a recent paper by Gibson et al. [23] for a summary of the experimental evidence and theoretical works. The major advantages of the above-mentioned nonlocal and higher gradient continuum theories are in fairly accurate prediction of dynamic behavior of nanostrctures by taking much lower time and computational efforts with respect to the atomistic-based models. This issue would be more obvious for two-dimensional nanostructures with nanometer thickness and micro-scale length and width.

In a pioneering work on modeling of nanoplates, Sun and Zhang [24] and Zhang and Sun [25] developed a semi-continuum model to investigate the dispersion of flexural waves in nanoscale plate-like structures. The results showed that the suggested model could capture deflection according to the predicted results of the lattice dynamics. It was stated that the classical continuum theory of plates underestimates the real Young's modulus of nanomaterials. Moreover, the obtained results revealed that elastic properties of plate-like nanomaterials would be size dependent such that Young's modulus and Poisson's ratio approach to the respective bulk values as the number of atom layer increases in thickness of the nanoplate. Concerning dynamic behavior of nanoplates using nonlocal continuum mechanics, a few theoretical works have been conducted to explore free vibration of nanoplates. In this regard, Pradhan and Phadikar [26] formulated classical and first-order shear deformable plate theories using the nonlocal elasticity theory of Eringen. Effects of nonlocal parameter, length and thickness of the nanoplate, elastic modulus and stiffness of Winkler's foundation attached to the nanoplate on natural frequencies of the nanoplate are then investigated. In another work, Pradhan and Phadikar [27] studied free vibration of multilayered graphene sheets embedded in a polymer matrix employing nonlocal continuum mechanics. Using the principle of virtual work, equations of motion were derived. The effect of the small-scale parameter on the in-plane vibration of nanoplates was also explored by Murmu and Pradhan [28] using nonlocal continuum mechanics. The obtained results of the two later reference works indicated that nonlocal effect would be significant, and it is needed to be incorporated into the continuum model of nano plate-like structures. Wang et al. [29] studied the propagation characteristics of the longitudinal waves in nanoplates with small-scale effects. The dispersion relation was analyzed with different values of the wave number and scale coefficient. It was observed that the dispersion properties of the longitudinal wave are induced by the small-scale effects, which are vanished in local continuum models. Furthermore, some works regarding buckling analysis of nanoplates using nonlocal continuum mechanics could be found in the literature [30], [31], [32], [33].

Concerning vibration of nanotube structures acted upon by a moving nanoparticle, Kiani and Mehri [34] proposed nonlocal Euler-Bernoulli, Timoshenko, and higher-order beams to study the problem. The capabilities of various nonlocal beam theories in capturing the dynamic response of the equivalent continuum structure (ECS) associated with the nanotube structure were examined. The role of the small-scale effect parameter, the slenderness ratio of the ECS, and velocity of the moving nanoparticle on the time history and maximum dynamic deflection were then studied in some detail. In other works, Kiani [35], [36] investigated vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal double classical and shear deformable beams. The analytic expressions of dynamical deflections and nonlocal bending moments of the innermost and outermost tubes were obtained during excitation and free vibration phases. Moreover, the critical velocities of the moving nanoparticle associated with various nonlocal beam theories were expressed in terms of small-scale effect, geometry, and material properties of DWCNTs. Recently, Kiani [37] and Kiani and Wang [38] studied longitudinal and transverse vibrations of single-walled carbon nanotube (SWCNT) due to a moving nanoparticle in the context of nonlocal continuum theory of Eringen. In the proposed models, the inertial effects of the moving nanoparticle as well as the existing friction between the nanoparticle surface and the inner surface of the single-walled carbon nanotube (SWCNT) were considered. Furthermore, the possibility of moving nanoparticle separation from the inner surface of the SWCNT was investigated and the role of influential factors on this phenomenon was also addressed.

Regarding vibration of nanoplates under a moving nanoparticle, Kiani [39] examined the in-plane and transverse vibrations of a thin nanoplate subjected to a moving nanoparticle by considering small-scale effects. To this end, the equations of motion of the nanoplate were constructed based on the Kirchhoff plate theory when the moving nanoparticle traverses the nanoplate on an arbitrary straight line. The governing equations were analytically solved for nanoplates with simply supported edges. The effects of both the first and second small-scale parameters as well as the moving nanoparticle velocity on the vibration behavior of the thin nanoplate and dynamic amplitude factors of the displacements of the thin nanoplate were also explored.

A brief survey of the literature reveals that the dynamic behaviors of nonplates under moving nano-scale particles have not been covered thoroughly until now. To bridge this scientific gap, herein, vibration of a nanoplate under excitation of a moving nanoparticle is investigated using nonlocal continuum theory of Eringen. To this end, the nanoplate is simulated according to the nonlocal Kirchhoff, Mindlin, and higher-order plate theories in the case of simply supported boundary conditions. The non-dimensional equations of motion of the nonlocal plate models are established. The effects of moving nanoparticle's weight and existing friction between the surfaces of the moving nanoparticle and the nanoplate on the in- and out-of-plane vibrations of the nanoplate are incorporated into the formulations of the proposed models. Using eigen function expansion technique and the Laplace transform method, the analytical expressions of the deformation and stress fields associated with each model are provided in two cases: (i) the moving nanoparticle travels on an arbitrary straight path with a constant velocity; (ii) the moving nanoparticle orbits on an arbitrary ellipse path with a constant angular velocity. Furthermore, the critical values of velocity and angular velocity of the moving nanoparticle for the proposed models are expressed analytically. The obtained critical values would be valuable when we discuss on the effect of velocity of the moving nanoparticle on dynamic amplitude factors of the nanoplate's displacements.

In part II of the present work, the effects of influential parameters on time histories as well as dynamic amplitude factors of displacements of the proposed nonlocal plate models are determined and discussed in some details.

Section snippets

Definition and assumptions of the problem

Consider a rectangular isotropic nanoplate with uniform thickness t_p as illustrated in Fig. 1. The nanoplate is of width b, length a, density $ρ_{p}$ , Poisson's ratio $ν_{p}$ , Young's modulus E_p, and shear modulus G_p where $G_{p} = E_{p} / (2 (1 + ν_{p}))$ . The origin of the Cartesian coordinates system is located at a corner of the mid-plane of the nanoplate. The x, y, and z axes are taken along the length, width, and thickness of the nanoplate, respectively. The only out-of-plane applied load is due to the mass weight

Nonlocal continuum theory for plates

Based on the nonlocal continuum theory of Eringen [20], the stress field at a point $x$ depends on the stress field at all points of the body. This fact is attributed to the atomic theory of lattice dynamics and experimentally observed data of phonon dispersion. Lattice dynamic observations have shown that inter-atomic forces disappear with distance [20] (attenuation neighborhood hypothesis). Therefore, the nonlocal stress tensor $σ_{ij}$ at an arbitrary point $x$ , in elastic homogeneous isotropic

The nonlocal equations of motion

The displacement components of the nanoplate based on the local Kirchhoff plate theory are $u_{x}^{K} (x, y, z, t) = u_{0}^{K} (x, y, t) - {zw}_{0, x}^{K} (x, y, t),$ $u_{y}^{K} (x, y, z, t) = v_{0}^{K} (x, y, t) - {zw}_{0, y}^{K} (x, y, t),$ $u_{z}^{K} (x, y, z, t) = w_{0}^{K} (x, y, t),$ where $u_{0}^{K}$ , $v_{0}^{K}$ and $w_{0}^{K}$ are the displacements of the mid-plane of the nanoplate along the x, y and z axes, respectively. According to the linear elasticity, the strain components of the problem are expressed as $ε_{xx}^{K} = u_{0, x}^{K} - {zw}_{0, xx}^{K},$ $ε_{yy}^{K} = v_{0, y}^{K} - {zw}_{0, yy}^{K},$ $γ_{xy}^{K} = u_{0, y}^{K} + v_{0, x}^{K} - 2 {zw}_{0, xy}^{K},$ for an isotropic homogeneous

The nonlocal equations of motion

The local (or classical) Mindlin plate theory is based on the following displacement field: $u_{x}^{M} (x, y, z, t) = u_{0}^{M} (x, y, t) + z ψ_{x}^{M} (x, y, t),$ $u_{y}^{M} (x, y, z, t) = v_{0}^{M} (x, y, t) + z ψ_{y}^{M} (x, y, t),$ $u_{z}^{M} (x, y, z, t) = w_{0}^{M} (x, y, t),$ where $u_{0}^{M}$ , $v_{0}^{M}$ , and $w_{0}^{M}$ represent the displacement components of the mid-plane of the Mindlin nanoplate along x, y and z axes, correspondingly. $ψ_{x}^{M}$ and $ψ_{y}^{M}$ are, respectively, the rotation angle of the mid-plane point about the x and y axes. The small deformation-rotation strains are expressed as $ε_{xx}^{M} = u_{0, x}^{M} + z ψ_{x,}$

The nonlocal equations of motion

Based on the local higher-order plate theory of Reddy [40], the displacement components of the nanoplate are as $u_{x}^{H} (x, y, z, t) = u_{0}^{H} (x, y, t) + z ψ_{x}^{H} (x, y, t) - c_{1} z^{3} (ψ_{x}^{H} + w_{0, x}^{H}),$ $u_{y}^{H} (x, y, z, t) = v_{0}^{H} (x, y, t) + z ψ_{y}^{H} (x, y, t) - c_{1} z^{3} (ψ_{y}^{H} + w_{0, y}^{H}),$ $u_{z}^{H} (x, y, z, t) = w_{0}^{H} (x, y, t),$ where $u_{0}^{H}$ , $v_{0}^{H}$ , and $w_{0}^{H}$ are in order the displacements of the mid-plane of the nanoplate along the x, y, and z axes. The parameters $ψ_{x}^{H}$ and $ψ_{y}^{H}$ represent the angle of rotation of the higher-order nanoplate about the x and y axes. The nonzero strain components

The nonlocal in-plane forces and moments

In order to obtain the nonlocal internal forces, it is just necessary to construct the Green's function associated with Eq. (3) for the simply supported boundary conditions. To find the Green's function for this equation, we must solve the following PDE: $G - l_{1}^{2} \nabla^{2} G = - δ (ξ - ξ', η - η'); 0 < ξ, ξ' < 1, 0 < η, η' < 1,$ $G (0, η | ξ', η') = G (1, η | ξ', η') = G (ξ, 0 | ξ', η') = G (ξ, 1 | ξ', η') = 0,$ this yields the Green's function as follows: $G (ξ, η | ξ', η') = \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} \frac{4}{1 + π^{2} μ_{1}^{2} (i^{2} + k^{2} j^{2})} \times \sin (i π ξ') \sin (j π η') \sin (i π ξ) \sin (j π η),$ the nonlocal internal forces

A special case: $g_{i}^{[]} + g_{j}^{[]} = r_{m_{ij}}^{[]} (or n {\bar{ω}}^{[]} = r_{m_{ij}}^{[]}); [] = K or M or H; m = 1,2, 3; i, j \geq 1$ (The critical velocities (or the critical angular velocities))

In such a case, Eqs. (30), (76), (117) could not be utilized directly to determine the unknown coefficients required for assessing vibration of the nanoplate subjected to a moving nanoparticle since the denominators of these equations take zero values.

In the case of $g_{i}^{K} + g_{j}^{K} = r_{m_{ij}}^{K}$ , Eq. (29) is reexpressed as $L {Ω_{Iij}^{K}} = \frac{A_{Imij}^{K} + {sB}_{Imij}^{K}}{(s^{2} + (r_{m_{ij}}^{K})^{2})^{2}} + \sum_{J = 1, j \neq m}^{5} \frac{A_{IJij}^{K} + {sB}_{IJij}^{K}}{s^{2} + (r_{J_{ij}}^{K})^{2}},$ where $A_{Imij}^{K} = \lim_{s \to i r_{m_{ij}}^{K}} (s^{2} + (r_{m_{ij}}^{K})^{2})^{2} \frac{\sum_{n = 1}^{5} A_{I (2 n - 2) ij}^{K} s^{2 n - 2}}{\prod_{n = 1}^{5} (s^{2} + (r_{n_{ij}}^{K})^{2})},$ $B_{Imij}^{K} = \lim_{s \to i r_{m_{ij}}^{K}} (s^{2} + (r_{m_{ij}}^{K})^{2})^{2} \sum_{n = 1}^{5} A_{I (2}$

Conclusions

Vibration of a nanoplate under excitation of a moving nanoparticle is studied on the basis of the nonlocal continuum theory of Eringen. The nanoplate is modeled according to the nonlocal Kirchhoff, Mindlin, and higher-order plate theories in the case of fully simply supported boundary conditions. The moving nanoparticle is modeled by a moving point load involving the Coulomb friction between the surface of the nanoparticle and the top surface of the nanoplate. The nondimensional governing

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Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations

Abstract

Graphical abstract

Highlights

Introduction

Section snippets

Definition and assumptions of the problem

Nonlocal continuum theory for plates

The nonlocal equations of motion

The nonlocal equations of motion

The nonlocal equations of motion

The nonlocal in-plane forces and moments

A special case: gi[]+gj[]=rmij[](ornω¯[]=rmij[]);[]=KorMorH;m=1,2,3;i,j≥1 (The critical velocities (or the critical angular velocities))

Conclusions

Journal of Power Sources

Electrochemistry Communications

Applied Surface Science

International Journal of Engineering Science

Composites Science and Technology

Journal of Sound and Vibration

Physics Letters A

Physica E

Physica E

Physics Letters A

Computational Materials Science

Mechanics Research Communications

Physica E

Journal of Sound and Vibration

Physica E

Journal of Sound and Vibration

Advanced Materials

Journal of Cellular Biochemistry: Supplement

Journal of Materials Chemistry

Korean Journal of Chemical Engineering

A special case: $g_{i}^{[]} + g_{j}^{[]} = r_{m_{ij}}^{[]} (or n {\bar{ω}}^{[]} = r_{m_{ij}}^{[]}); [] = K or M or H; m = 1,2, 3; i, j \geq 1$ (The critical velocities (or the critical angular velocities))