Non-classical Wave Dynamics of Ultrathin Structures

Non-classical Wave Dynamics of Ultrathin Structures

S. Narendar

Defenc e Research and Development Laboratory, Kanchanbagh, Hyderabad-500 058

Abstract:

In this paper, the nonlocal elasticity theory has been incorporated into classical 1D-rod model to capture unique features of the rod like structures at Nanoscale, which are considered as ultra-thin structures, under the umbrella of continuum mechanics theory. The strong effect of the nanoscale has been obtained which leads to substantially different wave behaviors of nanoscale-rods from those of macroscopic rods. Nonlocal bar model is developed for nanorods. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The studies also show that the nonlocal scale parameter introduces certain band gap region in axial wave mode where no wave propagation occurs. This is manifested in the spectrum cures as the region where the wavenumber tends to infinite (or wave speed tends to zero). These results are also compared with the Born-Karman model and also with the second and fourth order strain gradient models. The results can provide useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Keywords: Nonlocal stress gradient model, Nonlocal strain gradient model, Lateral inertia, Wavenumber, Phase speed, Dispersion, Group speed, Nanorod, Born-Karman Model.

1. Introduction

A ultra-thin structure is defined as a material system or object where at least one of the dimensions lies below 100 nm. Nanostructures can be classified into three different categories: zero-dimensional; one-dimensional; two-dimensional. Zero-dimensional nanostructures are materials in which all three dimensions are at the nanoscale. A good example of these materials are buckminster fullerenes [1] and quantum dots [2]. One-dimensional nanostructures are materials that have two physical dimensions in the nanometer range while the third dimension can be large, such as in the carbon nanotube [3]. 2D nanostructures, or thin films, only have one dimension in the nanometer range and are used readily in the processing of complimentary metal-oxide semiconductor transistors [4] and micro-electro-mechanical systems (MEMS) [5]. Since the focus of this work is on one-dimensional nanostructures, all others from this point forward will cease to be discussed. One-dimensional nanostructures (here nanorods) have stimulated a great deal of interest due to their importance in fundamental scientific research and potential technological applications in nano-electronic, nano-optoelectronic and nano-electro-mechanical systems.

Classical continuum theories assume that the stresses in a material point depend only on the first-order derivative of the displacements, i.e. on the strains, and not on higher-order displacement derivatives. As a consequence of this limitation on the kinematic field, a classical continuum is not always capable of adequately describing heterogeneous phenomena. For instance, unrealistic singularities in the stress and/or strain field may occur nearby imperfections. Furthermore, severe problems in the simulation of localization phenomena with classical continua have been encountered, such as loss of well-posedness in the mathematical description. To avoid these types of deficiencies, it has been proposed to include higher-order strain gradients into the constitutive equations, so that the defects of the classical continuum may be successfully overcome [6-9]. The second-order strain gradients that are normally used for these purposes introduce accessory material parameters that reflect the microstructural properties of the material.

The higher-order gradients can improve the performance of the classical continuum in the sense that the dispersive behavior of the discrete model is reproduced with a higher accuracy, [10-12]. This is a direct consequence of the procedure that is commonly used to enhance the classical continuum with higher-order gradients: homogenization of the discrete medium may lead to higher-order gradients in a direct and straightforward manner. If regularization of singularities or discontinuities is required, higher-order gradients are used for smoothing the non-uniformity or singularities in the strain field. On the other hand, if a more accurate representation of the discrete microstructure is desired, the higher-order gradients are used to introduce a non-uniformity in the strain field.

Actually, the length scales associated with nanostructures like nanorods are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size. This makes a physically consistent classical continuum model formulation very challenging. The Eringen's nonlocal elasticity theory [13-15] is useful tool in treating phenomena whose origins lie in the regimes smaller than the classical continuum models. In this theory, the internal size or scale could be represented in the constitutive equations simply as material parameters. Such a nonlocal continuum mechanics has been widely accepted and has been applied to many problems including wave propagation, dislocation, crack problems, etc [16]. Recently, there has been great interest in the application of nonlocal continuum mechanics for modeling and analysis of nanostructures [17-27].

The study of wave propagation in nanostructures has attracted intensive attention in research because many crucial physical properties such as electrical conductance, optical transition and some dynamic behavior of carbon nanotubes (CNTs) are very sensitive to the presence of wave The present wave propagation studies using nonlocal continuum model has shown that the wave behavior in a nanorod is drastically different compared to the behavior of local or classical model. Hence, the main objective of this paper is to bring out the main effects that the nonlocal scale parameter to the wave propagation in nanorods.In this paper, a nonlocal Euler-Bernoulli bar model is developed for analyzing the ultrasonic wave propagation in nanorods. The effect of nonlocal scaling parameter on the wave propagation in nanorods and also the variation of the escape frequency with e0a is studied in detail. Here e0a = 0.5 nm and 1.0 nm are used, where a = 0.142 nm (C-C bond length). Also Eringen's nonlocal elasticity theory is used to model the dispersion characteristics of ultrasonic waves in the nanorod. This nonlocal elasticity theory assumes that the stress at a reference point to be a functional of the strain field at every point in the body. It allows one to account for the small scale effect that becomes significant when dealing with micro and nanostructures. This theory is extended to obtain the nonlocal second and fourth order strain gradient models. At the end, the unstable second order strain gradient model is made stable by considering the lateral inertia effects along with the nonlocal stress gradient model in the nonlocal rod formulation.

2 Mathematical Formulation

2.1 A review on theory of nonlocal elasticity

According to the theory of nonlocal elasticity [13-15], the stress at a reference point x is considered to be a functional of the strain field at every point in the body. In the limit when the effects of strains at points other than x are neglected, one obtains local or classical theory of elasticity. The basic equations for linear, homogeneous, isotropic, nonlocal elastic solid with zero body force are given by:

[illustration not visible in this excerpt],[illustration not visible in this excerpt],[illustration not visible in this excerpt], where [illustration not visible in this excerpt] is the elastic modulus tensor of classical isotropic elasticity, [illustration not visible in this excerpt] and [illustration not visible in this excerpt] are stress and strain tensors respectively, and [illustration not visible in this excerpt] is the displacement vector. [illustration not visible in this excerpt] is the nonlocal modulus or attenuation function incorporating the nonlocal effects into the constitutive equations. This nonlocal modulus is found by matching the curves of plane waves with those due to atomic lattice dynamics. Various different forms of [illustration not visible in this excerpt] have been reported in [13]. [illustration not visible in this excerpt] is the Euclidean distance, and [illustration not visible in this excerpt], where [illustration not visible in this excerpt] is an internal characteristic length, e.g., length of [illustration not visible in this excerpt] bond (0.142 nm) in CNT, granular distance etc., and [illustration not visible in this excerpt] is an external characteristic length e.g., wavelength, crack length, size of the sample etc. [illustration not visible in this excerpt] is a nonlocal scaling parameter, which has been assumed as a constant appropriate to each material in published literature and [illustration not visible in this excerpt] is the region occupied by the body. On the other hand, parameter [illustration not visible in this excerpt] was given as 0.39 by Eringen [14].

As the constitutive equation of nonlocal elasticity involves spatial integrals which represent weighted averages of the contributions of the strain tensors of all points in the body to the stress tensor at the given point, it is difficult mathematically to get the solution of nonlocal elasticity problems. However, it is pointed out by Eringen [14] that the integral constitutive equation can be converted exactly into an equivalent differential form for some kernels. This, of course, provides a great deal of simplicity and convenience for the application of the theory of nonlocal elasticity. In what follows, the following form for the kernel function:

[illustration not visible in this excerpt] will be adopted, which was suggested by Eringen [14]. In this equation, [illustration not visible in this excerpt] is the modified Bessel function. Hooke’s law for a uni-axial stress state by the nonlocal elasticity can be determined by [18] and given as[illustration not visible in this excerpt], where [illustration not visible in this excerpt] is the Young’s modulus of the material, [illustration not visible in this excerpt] is the coordinate with the origin at the left end of one-dimensional structure. Eq. (5) is the uniaxial stress version of Eq. (3.19) of ref. [14].

2.2 Nonlocal strain gradient models

Expansion of the general integral constitutive equation of nonlocal elasticity (ref. [14], Eq. (3.17)) for [illustration not visible in this excerpt], retention of only the first two terms, and simplification to the case of uniaxial stress produces [illustration not visible in this excerpt]. This is a second order strain gradient model with nonlocal scale effects.In the same way, retention of only the first three terms, and simplification to the case of uniaxial stress produces

Abbildung in dieser Leseprobe nicht enthalten.

This is a fourth order strain gradient model with nonlocal scale effects. Next, we will derive the governing equation of motion for nanorod based on the above mentioned two constitutive relations asmentione above.

2.3 Nonlocal governing partial differential equation for nanorods basedon nonlocal stress gradient model

Fig. 1 schematically describes a nanorod under discussion and serves to introduce the axial coordinate [illustration not visible in this excerpt], the axial displacement [illustration not visible in this excerpt], the length [illustration not visible in this excerpt], the Young's modulus [illustration not visible in this excerpt], and the density [illustration not visible in this excerpt]. The displacement field and strain for this nanorod are given by [illustration not visible in this excerpt][illustration not visible in this excerpt]. For thin rods nonlocal constitutive relation one dimensional form is[illustration not visible in this excerpt],where [illustration not visible in this excerpt] is the modulus of elasticity, [illustration not visible in this excerpt] and [illustration not visible in this excerpt] are the local stress and strain components in the [illustration not visible in this excerpt] direction, respectively. The equation of motion for an axial rod can be obtained as [illustration not visible in this excerpt] where [illustration not visible in this excerpt] is the axial force per unit length for local or classical elasticity and is defined by[illustration not visible in this excerpt]. Using these relations we have [illustration not visible in this excerpt]Substitution of the first derivative of [illustration not visible in this excerpt]in the above relation we obtain [illustration not visible in this excerpt]. Substituting [illustration not visible in this excerpt] into the equation of motion, we obtain

Abbildung in dieser Leseprobe nicht enthalten (1)

Eq. (1) is the consistent fundamental governing equation of motion for nonlocal rod model. When [illustration not visible in this excerpt], it is reduced to the equation of classical rod model.

3 Ultrasonic Wave Characteristics Nanorods

3.1 Computation of wavenumbers

For analyzing the ultrasonic wave dispersion characteristics in nanorods, we assume that a harmonic type of wave solution for the displacement field [illustration not visible in this excerpt] and it can be expressed in complex form as [28, 29],

Abbildung in dieser Leseprobe nicht enthalten (2)

where, [illustration not visible in this excerpt] and [illustration not visible in this excerpt] are the number of time sampling points and number spatial sampling points respectively. [illustration not visible in this excerpt] is the circular frequency at the [illustration not visible in this excerpt] time sample. Similarly, [illustration not visible in this excerpt] is the axial wavenumber at the [illustration not visible in this excerpt] spatial sample point. Substituting Eq. (2) into the governing partial differential equation (Eq. (1)), we get the dispersion relation as follows. Hereafter the subscript [illustration not visible in this excerpt] and [illustration not visible in this excerpt] in Eq. (2) are dropped for simplified notations. Here [illustration not visible in this excerpt].

The dispersion realtion is: [illustration not visible in this excerpt],where [illustration not visible in this excerpt].

This dispersion relation is solved for the wavenumbers as [illustration not visible in this excerpt].

The wave frequency is a function of wavenumber [illustration not visible in this excerpt], the nonlocal scaling parameter [illustration not visible in this excerpt] and the material properties ([illustration not visible in this excerpt]&[illustration not visible in this excerpt]) of the nanorod. If [illustration not visible in this excerpt], the wavenumber is directly proportional to wave frequency, which will give a non-dispersive wave behavior (for more details refer [29]). The cut-off frequency of this nanorod is obtained by setting [illustration not visible in this excerpt] in the dispersion relation. For the present case the cut-off frequency is zero, i.e., the axial wave starts propagating from zero frequency.

3.2 Escape frequency

Fig. 2 shows the spectrum relation plot as a function of nonlocal scale parameter [illustration not visible in this excerpt]. From the figure, we see that at certain frequencies, the wavenumber is tending to infinity and this frequency value decreases with increase in the scale parameter. Its value can be analytically determined by looking at the wavenumber expression (Eq. (16)) and setting [illustration not visible in this excerpt]. Which is given as [illustration not visible in this excerpt] The escape frequency is inversely proportional to the nonlocal scaling parameter and is independent of the diameter of the nanorod. The variation of the escape frequency with nonlocal scaling parameter is shown in Fig. 3.

3.3 Computation of wave speeds

For the present analysis, we are considering both wave speeds (i.e., phase and group speeds) of the wave and are defined as

[illustration not visible in this excerpt] and [illustration not visible in this excerpt].

These wave speeds are also depend on the nonlocal scaling parameter. When [illustration not visible in this excerpt], both the wave speeds are equal (i.e.,[illustration not visible in this excerpt]), which is already proved for local or classical bars/rods [29]. The phase speed and group speed dispersion curves with wave frequency are shown in Fig. 4 and Fig. 5, respectively.

4 Governing Partial Differential Equations of Nanorod

4.1 Based on second order strain gradient model

Fig. 1 schematically describes a nanobeam under discussion and serves to introduce the axial coordinate [illustration not visible in this excerpt], the axial displacement [illustration not visible in this excerpt], the rod length [illustration not visible in this excerpt], the rod Young's modulus [illustration not visible in this excerpt], and the density [illustration not visible in this excerpt]. The displacement field and strain for this nanorod are given by [illustration not visible in this excerpt][illustration not visible in this excerpt]. First we derive the governing differential equations for the nanorod using second order strain gradient model. The potential ([illustration not visible in this excerpt]) and kinetic ([illustration not visible in this excerpt]) energies of the nanorod are [illustration not visible in this excerpt]and[illustration not visible in this excerpt]

Here superscript [illustration not visible in this excerpt] represents for the second order strain gradient model. Assuming a uniform nanorod, these two relations can be rewritten as [illustration not visible in this excerpt]and [illustration not visible in this excerpt].Applying Hamilton's principle [illustration not visible in this excerpt]. Expanding this, and integrating by parts, we obtain the nonlocal governing partial differential equation for the nanorod obtained from the second order strain gradient model as

illustration not visible in this excerpt

This is a fourth order partial differential equation. One can substitute [illustration not visible in this excerpt] in Eq. (3), to recover the local or classical rod equation, which is a second order differential equation.

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