Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

Abstract

This work aims at investigating the free torsional vibration of one-directional nanostructures with an elliptical shape, under different boundary conditions. The equation of motion is derived from Hamilton’s principle, where Eringen’s nonlocal theory is applied to analyze the small-scale effects. The analytical Galerkin method is employed to rewrite the equation of motion as an ordinary differential equation (ODE). After a preliminary validation check of the proposed formulation, a systematic study investigates the influence of the nonlocal parameters, boundary conditions, geometrical and mechanical parameters on the natural frequency of nanorods; the objective is to provide useful findings for design and optimization purposes of many nanotechnology applications, such as, nanodevices, actuators, sensors, rods, nanocables, and nanostructured aerospace systems.

1. Introduction

Small-sized structures such as nanorods, nanotubes, nanowires, nanobeams, and nanoshells are increasingly becoming key structural elements in nano- and micro-electromechanical systems (NEMS/MEMS), as well as in nanostructured coatings and materials for aerospace applications. In this context, several theories and experiments have been developed in the literature to study the mechanical, physical, electronic and thermal properties of atomic scaled structures, namely, boron nitride nanotubes [1,2], silica carbide nanotube/wires [3,4], graphene [5,6], and carbon nanotubes (CNTs) [7,8,9,10]. Among them, CNTs have been introduced in the late 1990s by Iijima [11,12], and have received an increased interest among the scientific community [13,14,15,16,17,18,19,20,21,22,23,24] due to their outstanding mechanical, thermal, and electrical properties [25,26,27,28]. CNTs can be partitioned in two parts, namely, the single-walled carbon nanotube (e.g., SWCNT) and the multi-walled carbon nanotube (e.g., MWCNT) [29]. It is well known that CNTs are hollow tubes rolled up by graphene sheets [11,12], which can be described as one-dimensional structures with lengths much higher than sectional sizes. At the same time, cross-sections are usually modeled with a circular shape, but the possible influence of noncircular shapes on the structural response would be of paramount importance in the design and analysis of many aircraft components and parts [30,31]. In this framework, the classical continuum mechanics has largely failed to verify findings at the nanoscale. This has increased the necessity of applying novel continuum mechanics theories and atomic/molecular dynamic simulations, along with the application of many experimental investigations and measurements in line with the high precision of minute sizes, see references [32,33,34,35,36,37], together with the review papers [38,39].

Starting with the nonlocal elasticity theory proposed for the first time by Eringen [40,41,42,43,44] to handle small-scale effects, many further works have merged nonlocal and gradient continuum formulations to study the size-dependent response of nanorods [45,46,47,48,49,50,51,52,53,54], nanobeams [55,56,57,58,59,60,61,62,63], and nanoplates/nanoshells [64,65,66,67,68,69,70,71]. More specifically, as far as the torsional vibration problem is concerned, different works in the literature have studied the size-dependent torsional behavior of nanorods and CNTs via theoretical and/or numerical nonlocal elasticity formulations. Ansari et al. [72] applied two approaches, namely the strain gradient theory (SGT), and molecular dynamics (MD), to determine the torsional vibration of CNTs, and its sensitivity to the nonlocal parameter. Fatahi-Vajari and Imam [73] investigated the torsional vibration of SWCNTs with zigzag and armchair structures by means of the MD, and surveyed the influence of the small-scale parameter, as well as the chiral effects on the structural response. In addition, a double CNT system connected by a torsional elastic medium was modeled by Arda and Aydogdu [74], for two various boundary conditions (BCs), to illustrate the effect of the elastic medium stiffness and the small-scale parameter on the torsional vibration of the nanostructure. Li [75] applied two reverse nonlocal models, including weakened and enhanced models to justify the nonlocal torsional natural frequencies of CNTs, and proposed a semi-continuum model with discrete atomic layers in the CNT cross section. In a more recent work, Aydogdu and Arda [76] analyzed the torsional frequencies of double carbon nanotubes, based on a nonlocal elasticity theory, while discussing the sensitivity of the response to the nonlocal parameter, the Van der Waals force interaction, and the nanotube length. Demir and Civalek [77] investigated both the torsional and axial vibration of microtubules based on a nonlocal continuous and discrete size-dependent rod model. In line with the previous work, Murmu et al. [78] performed a size-dependent parametric analysis of the torsional vibration of a single-walled CNT-buckyball system for nanoresonators, while tuning the mass inertia for design purposes. Suzuki et al. [79] proposed an ultra-precision sculpturing method in micro- and nano-structures for difficult-to-cut materials. More specifically, elliptical vibration cutting (EVC) was applied with a single crystal diamond as controlled cutting and machining tool for steel materials, while monitoring the structural vibration amplitudes. Similarly, Zhang et al. [80] applied the EVC technology to simplify the machining process of textured surfaces. Different EVC devices were employed, compared and verified in terms of their accuracy; the role of the elliptical vibrator was explained in detail for the manufacturing process of micro- and nano-structures. An elliptical vibration texturing was also adopted by Yang et al. [81] to colorize metallic surfaces with periodic micro- and nano-scale materials, and to optimize the metal surfaces properties.

Based on the above-mentioned practical problems, even more accurate methods have been adopted very recently to treat torsional vibrations, both analytically and/or numerically. In this framework, Mikeš and Jirásek [82] established a numerical method for free warping structures with an elliptical cross section, while adopting the finite element approach to handle more complicated cross sections. Barr [83] studied the torsional vibration of a uniform rod with a rectangular cross section to investigate the dispersion of torsional waves within rods, and its sensitivity to longitudinal stresses and inertia effects. In addition, Stephen [84] modeled beams with an elliptical cross section, and compared the dynamic torsional predictions by different theories. Francu et al. [85] employed the Airy functions to analyze the stress–strain analytical expressions of warping beams with different noncircular cross sections. Christides and Barr [86] modeled one or more pairs of symmetric cracks in uniform beams with a noncircular cross section, for the study of their torsional vibration, and the computation of their fundamental natural frequencies for different crack depths. In some further works by Loya et al. [87,88,89], different nonlocal cracked models were proposed to treat the torsional and bending vibration of CNTs, and their sensitivity to the nonlocal small-scale parameter, crack severity, and cracked section position, together with different boundary conditions.

According to the above literature review, however, there is a general lack of works focusing on the nonlocal dynamic torsion of nanostructures with noncircular cross sections. This topic is tackled here for elliptical nanorods in a nonlocal context, as a pioneering study of additional and more complicated shapes of practical interest. The nonlocal governing equation of the problem is obtained by means of Hamilton’s principle, and discretized via the Galerkin method. A systematic investigation aims at checking the sensitivity of the nonlocal dynamic twisting response to the nonlocal parameter, geometry parameters, and stiffness of the torsional spring, which could be of great interest for nanostructural design.

2. Nonlocal Elasticity Theory

According to the well-known MEMS/NEMS applications of micro/nano-roads under a torsional load (e.g., NEMS oscillators, torsional micromirrors, torsional miroscanners, etc.), nanostructures usually feature noncircular cross sections, primarily of elliptical shape, as visible in the TEM image of Figure 1, for a platinum nanowire on a MgO (110) substrate [90].

Let us consider a clamped-clamped (C-C), clamped-free (C-F), and clamped-torsional spring (C-T) nanorod, of length $l$, and radii $a$ and $b$ in the $y\text{-}$axis and $z\text{-}$axis, respectively, as depicted in Figure 2, Figure 3 and Figure 4. The torsional spring in Figure 4, is also denoted as $k_b$.

The displacement field for any arbitrary point under a torsional vibration, is defined by the following components

$$\begin{array}{l}u\left(x,t\right)=\psi \left(y,z\right)\frac{\partial \theta }{\partial x}\\ v\left(x,t\right)=-z\theta \left(x,t\right)\\ w\left(x,t\right)=y\theta \left(x,t\right)\end{array}$$

(1)

where the $x\text{-}$axis corresponds to the centerline of the CNT, $u$,$v$, and $w$ denote the displacement components in the $x$, $y$ and $z$ direction, respectively. Besides, $\theta$ is the angular twist, and $\psi (y,z)$ represents the warping function, defined as $\psi (y,z)=y\times z\left(\frac{b^2-a^2}{b^2+a^2}\right)$. The strain field is, thus, governed by the following kinematic relations

$$\begin{array}{l}{\epsilon }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=\left(\frac{\partial \psi }{\partial y}-z\right)\frac{\partial \theta }{\partial x}\\ {\epsilon }_{xz}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=\left(\frac{\partial \psi }{\partial z}+y\right)\frac{\partial \theta }{\partial x}\\ {\epsilon }_{yz}=\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}=0\\ {\epsilon }_{xx}={\epsilon }_{yy}={\epsilon }_{zz}=0\end{array}$$

(2)

which are, in turn, related to the stress components ${\sigma }_{xy}, {\sigma }_{xz}$ by means of the constitutive relations. The torsional couple acting on the twisted noncircular body is, thus, expressed as [91]

$$M={\displaystyle {\int }_A\left({\sigma }_{xy}\left(\frac{\partial \psi }{\partial y}-z\right)+{\sigma }_{xz}\left(\frac{\partial \psi }{\partial z}+y\right)\right)dA}$$

(3)

Hamilton’s Principle

Hamilton’s principle is then employed to derive the equation of motion in the following form

$${\displaystyle {\int }_0^t\delta \left(U-T+W_{ext}\right)dt}=0$$

(4)

where $U, T$ and $W_{ext}$ refers to the strain energy, the kinetic energy, and the external works, respectively. For noncircular nanorods, the strain energy is defined in a variational form as follows

$$\delta U={\displaystyle {\int }_V{\sigma }_{ij}\delta {\epsilon }_{ij}dV=}{\displaystyle {\int }_V({\sigma }_{xy}\delta {\epsilon }_{xy}+{\sigma }_{xz}\delta {\epsilon }_{xz})dV}$$

(5)

which is combined to Equation (2) to yield

$$\begin{array}{l}\delta U=\delta {\displaystyle \underset{V}{\int }\left({\sigma }_{xy}\left(\frac{\partial \psi }{\partial y}-z\right)\frac{\partial \theta }{\partial x}+{\sigma }_{xz}\left(\frac{\partial \psi }{\partial z}+y\right)\frac{\partial \theta }{\partial x}\right)dV}\\ =\delta {\displaystyle \underset{V}{\int }\left(\left(y{\sigma }_{xz}-z{\sigma }_{xy}\right)+\left({\sigma }_{xy}\frac{\partial \psi }{\partial y}+{\sigma }_{xz}\frac{\partial \psi }{\partial z}\right)\right)\frac{\partial \theta }{\partial x}dAdx}\end{array}$$

(6)

By a further combination of Equation (6) with Equation (3) we obtain the following variational expression for the strain energy

$$\begin{array}{l}\delta U={\displaystyle {\int }_0^l{\displaystyle {\int }_A\left(\left(y{\sigma }_{xz}-z{\sigma }_{xy}+{\sigma }_{xy}\frac{\partial \psi }{\partial y}+{\sigma }_{xz}\frac{\partial \psi }{\partial z}\right)\frac{\partial \delta \theta }{\partial x}\right)dxdA+}}\\ +{\displaystyle {\int }_0^l{\displaystyle {\int }_A\left({\sigma }_{xy}\frac{\partial \delta \psi }{\partial y}+{\sigma }_{xz}\frac{\partial \delta \psi }{\partial z}\right)\frac{\partial \theta }{\partial x}}})dxdA\end{array}$$

(7)

By integration of $\delta U$ within a certain lapse of time $\left[0,t\right]$ we obtain

$${\displaystyle {\int }_0^t\delta U}=-{\displaystyle {\int }_0^t\left({\displaystyle {\int }_0^l\frac{\partial M}{\partial x}\delta \theta dx}\right)}dt$$

(8)

The kinetic energy is defined as

$$T=\frac{1}{2}{\displaystyle \underset{V}{\int }\rho \left[{\left(\frac{\partial u}{\partial t}\right)}^2+{\left(\frac{\partial v}{\partial t}\right)}^2+{\left(\frac{\partial w}{\partial t}\right)}^2\right]dV}$$

(9)

By substituting the first derivative of Equation (1) with respect to the time into Equation (9), the kinetic energy can be rewritten as

$$T=\frac{1}{2}{\displaystyle \underset{V}{\int }\rho \left[{\psi }^2\left(y,z\right){\left(\frac{{\partial }^2\theta }{\partial x\partial t}\right)}^2+{\left(-z\frac{\partial \theta }{\partial t}\right)}^2+{\left(y\frac{\partial \theta }{\partial t}\right)}^2\right]dV}=I_1+I_2$$

(10)

where $I_1$ and $I_2$ include the axial and polar inertia terms, defined as

$$I_1=\frac{1}{2}\rho {\displaystyle {\int }_0^l{\displaystyle \underset{A}{\int }{\psi }^2\left(y,z\right)}}{\left(\frac{{\partial }^2\theta }{\partial x\partial t}\right)}^{2}dAdx$$

(11)

$$I_2=\frac{1}{2}\rho I_p{\displaystyle {\int }_0^l{\left(\frac{\partial \theta }{\partial t}\right)}^{2}dx}=\frac{1}{2}{I}_0{\displaystyle {\int }_0^l{\left(\frac{\partial \theta }{\partial t}\right)}^{2}dx}$$

(12)

$I_p$ and $I_0$ denote the polar moment of inertia, and mass inertia, respectively, expressed as

$$I_p={\displaystyle \underset{A}{\int }\left(y^2+z^2\right)}dA$$

(13)

$$I_0=\rho I_p$$

(14)

The first variation of $I_1$ can be determined as follows

$$\delta I_1={\displaystyle {\int }_0^l\rho I_{\psi }}\frac{{\partial }^2\theta }{\partial t\partial x}\frac{{\partial }^2\left(\delta \theta \right)}{\partial t\partial x}dx+{\displaystyle {\int }_0^l\rho I_{\theta }^2{\displaystyle \underset{A}{\int }\psi \delta \psi dA}}$$

(15)

where

$$I_{\psi }={\displaystyle \underset{A}{\int }{\psi }^{2}dA}$$

(16)

$$I_{\theta }={\displaystyle {\int }_0^l\rho \left(\frac{{\partial }^2\theta }{\partial t\partial x}\right)}dx$$

(17)

The first variation of $I_1$ in the Hamilton’s principle can be expressed as

$${\displaystyle {\int }_0^t\delta I_1}=\rho I_{\psi }{\displaystyle {\int }_0^t{\displaystyle {\int }_0^l\frac{{\partial }^4\theta }{\partial x^2\partial t^2}}\delta \theta dx}dt-\rho I_{\psi }{\displaystyle {\int }_0^t\frac{{\partial }^3\theta }{\partial x\partial t^2}}\delta \theta |\begin{array}{l}l\\ 0\end{array}dt$$

(18)

whereas, the first variation of $I_2$ is defined as

$$\delta I_2={\displaystyle {\int }_0^l\rho I_p\frac{\partial \theta }{\partial t}\frac{\partial \delta \theta }{\partial t}dx}=I_0{\displaystyle {\int }_0^l\frac{\partial \theta }{\partial t}\frac{\partial \delta \theta }{\partial t}dx}$$

(19)

By substitution of $I_2$ in Hamilton’s principle, we calculate

$${\displaystyle {\int }_0^t\delta I_2}=-I_0{\displaystyle {\int }_0^t{\displaystyle {\int }_0^l\frac{{\partial }^2\theta }{\partial t^2}dx}}$$

(20)

Finally, the variation of kinetic energy can be stated as

$$\begin{array}{l}\delta {\displaystyle {\int }_0^{t}T}=\rho I_{\psi }{\displaystyle {\int }_0^t{\displaystyle {\int }_0^l\frac{{\partial }^4\theta }{\partial x^2\partial t^2}}\delta \theta dx}dt-\rho I_{\psi }{\displaystyle {\int }_0^t\frac{{\partial }^3\theta }{\partial x\partial t^2}}\delta \theta |\begin{array}{l}l\\ 0\end{array}dt\\ +{\displaystyle {\int }_0^t{I}_{\theta }}{\displaystyle {\int }_A\psi \delta \psi dAdt}-I_0{\displaystyle {\int }_0^t{\displaystyle {\int }_0^l\frac{{\partial }^2\theta }{\partial t^2}dx}}\end{array}$$

(21)

In total absence of the external work, the combination of Equations (4), (8), and (21) yields the following equation of motion

$$\frac{\partial M}{\partial x}=I_0\frac{{\partial }^2\theta }{\partial t^2}-\rho I_{\psi }\frac{{\partial }^4\theta }{\partial x^2\partial t^2}$$

(22)

According to Eringen [41], the nonlocal constitutive relations of the nanostructure take the following form

$${\sigma }_{xy}-{\mu }^2\frac{{\partial }^2{\sigma }_{xy}}{\partial x^2}=G\left(\frac{\partial \psi }{\partial y}-z\right)\frac{\partial \theta }{\partial x}$$

(23)

$${\sigma }_{xz}-{\mu }^2\frac{{\partial }^2{\sigma }_{xz}}{\partial x^2}=G\left(\frac{\partial \psi }{\partial z}+y\right)\frac{\partial \theta }{\partial x}$$

(24)

$\mu$ being the nonlocal parameter, which accounts for the characteristic internal length of the covalent bonds of carbon. Thus, the nonlocal twisting moment is governed by the following relation

$$M-{\mu }^2\frac{{\partial }^{2}M}{\partial x^2}=G{\displaystyle \underset{A}{\int }\left({\left(\frac{\partial \psi }{\partial y}-z\right)}^2+{\left(\frac{\partial \psi }{\partial z}+y\right)}^2\right)}\frac{\partial \theta }{\partial x}dA=G{I}_{\Phi }\frac{\partial \theta }{\partial x}$$

(25)

with

$$I_{\Phi }={\displaystyle \underset{A}{\int }\left({\left(\frac{\partial \psi }{\partial y}-z\right)}^2+{\left(\frac{\partial \psi }{\partial z}+y\right)}^2\right)}dA$$

(26)

Substituting the first derivative of Equation (22) into Equation (25) yields

$$M-{\mu }^2\left(I_0\frac{{\partial }^3\theta }{\partial t^2\partial x}-\rho I_{\psi }\frac{{\partial }^5\theta }{\partial x^3\partial t^2}\right)=G{I}_{\Phi }\frac{\partial \theta }{\partial x}$$

(27)

Thus, the equation of motion for noncircular nanorods, takes the following final form

$$\rho I_{\psi }\frac{{\partial }^4\theta }{\partial x^2\partial t^2}-I_0\frac{{\partial }^2\theta }{\partial t^2}+{\mu }^2\left(I_0\frac{{\partial }^4\theta }{\partial t^2\partial x^2}-\rho I_{\psi }\frac{{\partial }^6\theta }{\partial x^4\partial t^2}\right)+G{I}_{\Phi }\frac{{\partial }^2\theta }{\partial x^2}=0$$

(28)

3. Analytical Solution

Based on the Galerkin method, we compute the theoretical solution of Equation (28), for nanorods with an elliptical cross-section, namely

$$\theta (x,t)={\displaystyle \sum _{n=1}^{\infty }{\Theta }_n(x)e^{i\omega t}}$$

(29)

${\Theta }_n$ being the $n\mathrm{th}$ mode shape, for a fixed boundary condition, and is expressed as

$${\Theta }_n=C_n\sin \left(Px\right)$$

(30)

where the parameter $P$ depends on the selected boundary condition. For a C-C, C-F, C-T, this parameter is defined as

$$P=n\pi /l$$

(31)

$$P=(2n-1)\pi /2l$$

(32)

$$P=\alpha /l$$

(33)

where $\alpha$ is determined by solving the following equation [91]

$$\tan \alpha =-\alpha \left(G{I}_{\Phi }/k_{b}l\right)$$

(34)

In the last relation, $k_b$ denotes the stiffness of the boundary spring. By substitution of Equation (29) into Equation (28) we obtain the following expression for the natural frequency

$${\omega }_n=\sqrt{\frac{G{I}_{\mathrm{\Phi }}{P}^2}{\rho I_{\psi }{P}^2+I_0+{\mu }^2\left(I_0{P}^2+\rho I_{\psi }{P}^4\right)}}$$

(35)

which can be redefined in dimensionless form as

$${\overline{\omega }}_n={\omega }_n\times l\times \sqrt{\frac{\rho }{G}}$$

(36)

4. Results and Discussion

This section is devoted to the numerical study of the vibration response for elliptical nanorods, with shear modulus $G=498\mathrm{GPa}$ and density $\rho =1330 {\mathrm{kg}/\mathrm{m}}^3$, in agreement with references [91,92,93,94]. A parametric investigation aims at checking the sensitivity of the response to some input parameters. The analysis starts with a comparative analysis of the first four dimensionless natural frequencies, for the reference circular case, under the assumptions $a=b=1\mathrm{nm}$, $\mu /l=0.1$ and $l=20\mathrm{nm}$, where we verify the accuracy of our results with respect to the available literature [95]. Then we consider the double effect of a varying geometrical ratio $b/a$ between the vertical and horizontal radius, and nonlocal parameter $\mu$, on a nanorod of length $l=$ 20 nm with C-C (

Table 1. Comparative evaluation of the first four dimensionless natural frequencies for a C-C circular nanorod with respect to the literature ($a=b=1\mathrm{nm}$, $\mu /l=0.1$, $l=20 \mathrm{nm}$ ).

	n = 1	n = 2	n = 3	n = 4
Present	2.9971	5.3201	6.8586	7.8247
Ref. [95]	2.9971	5.3201	6.8586	7.8247

) or C-F (

Table 2. Free torsional vibration of C-C elliptical nanorod with a varying $b/a$ ratio and nonlocal parameter $\mu$.

$\mathit{\mu }\times {10}^{-9}$	$\mathit{b}/\mathit{a}=0.1$	$\mathit{b}/\mathit{a}=0.2$	$\mathit{b}/\mathit{a}=1$	$\mathit{b}/\mathit{a}=5$	$\mathit{b}/\mathit{a}=10$
0	0.6220	1.2082	3.1415	1.2062	0.6208
1	0.6145	1.1935	3.1035	1.1916	0.6133
1.5	0.6055	1.1760	3.0578	1.1741	0.6043
2	0.5934	1.1526	2.9971	1.1508	0.5923

) boundary conditions. Based on the results from both tables, an increased nonlocal parameter yields a meaningless reduction of the natural frequency. At the same time, for a fixed nonlocal parameter, an increased dimensionless ratio $b/a$ gets a non monotonic variation of the natural frequencies, for both the selected boundary conditions. An increased frequency is noticed, first, for an increasing $b/a$ ratio from 0.1 up to 1 (i.e., when the elliptic shape reverts to the circular one). The contrary occurs for $b/a$ ratios higher than the unit value, with an overall decrease in the vibrational frequency of the nanostructure. In this last case, the elliptical shape becomes vertical, and the natural frequency reduces due to the decreased momentum of inertia compared to the horizontal elliptical shape, in agreement with Equation (26).

A further physically consistent justification for this behavior is the fact that a twisted horizontal elliptical cross section features a lower angular rotation at a certain lapse of time than a vertical cross section due its higher stiffness.

Moreover, for both boundary conditions, the frequency response is symmetric with respect to the circular limit case (i.e., when $b/a=1$), with a non linear variation for both boundary conditions. According to a comparative evaluation of results in Table 2 and

Table 3. Free torsional vibration of C-F elliptical nanorod with a varying $b/a$ ratio and nonlocal parameter $\mu$.

$\mathit{\mu }\times {10}^{-9}$	$\mathit{b}/\mathit{a}=0.1$	$\mathit{b}/\mathit{a}=0.2$	$\mathit{b}/\mathit{a}=1$	$\mathit{b}/\mathit{a}=5$	$\mathit{b}/\mathit{a}=10$
0	0.3110	1.2565	1.5708	1.2564	0.3108
1	0.3100	1.2527	1.5659	1.2525	0.3099
1.5	0.3089	1.2479	1.5600	1.2478	0.3087
2	0.3072	1.2413	1.5517	1.2412	0.3071

, C-C nanorods yield higher frequencies than C-F structures, for the same fixed geometry and nonlocal assumptions. A C-C structure, indeed, is expected to exhibit a stiffer behavior than a C-F one, because of the intrinsically different nature of the two boundary conditions.

All the previous results are plotted in Figure 5 and Figure 6, for a C-C and C-F boundary condition, respectively, where it is clearly visible that frequency computations based on a circular simplified assumption always overestimate the actual response of a more complicated actual geometrical cross section. All the plots in both figures, indeed, reach the peak value in the circular case (i.e., for $b/a=1$), which in turn, would be overestimated by a classical theory (i.e., for $\mu =0$), compared to a nonlocal theory. This means that possible nonlocalities within nanostructures must be properly estimated (also from an experimental standpoint) in order to provide accurate and physically consistent results.

A further systemic study considers the combined effect of the nonlocal parameter and stiffness of the boundary spring $\left(k_b\right)$ on the vibration response of a C-T elliptical nanorod with $a=0.4 \mathrm{nm}$ and $b=0.2 \mathrm{nm}$ (i.e., $b/a=0.5$), as summarized in

Table 4. Free torsional vibration of a C-T elliptical nanorod for a varying $\mu$ and $k_b$. $a=0.4 \mathrm{nm}$, $b=0.2 \mathrm{nm}$, and $l=20 \mathrm{nm}$.

$\begin{array}{l}{\mathit{k}}_{\mathit{b}}\times {10}^{-18}\\ \left(\mathbf{GPa}\times {\mathbf{nm}}^3\right)\end{array}$	$\mathit{\mu }\times {10}^{-9}$
	0	1	1.5	2
0.01	1.2815	1.2774	1.2723	1.2653
0.1	1.4690	1.4628	1.4552	1.4448
1	2.1225	2.1040	2.0817	2.0515
10	2.4638	2.4351	2.4006	2.3547

, and plotted in Figure 7. Based on the results, it is worth noticing that increased torsional stiffness of $k_b$ from $0.01\times {10}^{-18}$ to $10\times {10}^{-18} \mathrm{GPa}\times {\mathrm{nm}}^3$ enables higher frequencies for the same fixed nonlocality of the structure, which means a reduction in the structural deformability. The sensitivity of the response to $k_b$ is more pronounced for lower values of the nonlocal parameter, and reduces gradually for higher nonlocalities. Moreover, a clear decrease in the natural frequency is observed, once again, for an increased nonlocality of the nanostructure, while keeping fixed the torsional stiffness of the boundary spring. This reflects the great importance of a correct modeling of boundary conditions for design purposes.

5. Concluding Remarks

The free vibration of the torsional vibration of nanorods with an elliptical cross section is explored theoretically in this work, for three different boundary conditions, namely, a C-C, C-F, and C-T BCs. Hamilton’s principle is selected to derive the equation of motion. A Galerkin method is here employed to solve the governing equation of the problem, where a parametric study is performed to check for the sensitivity of the vibration response to different parameters, including the nonlocal parameter, the geometrical vertical-to-horizontal $b/a$ radii, along with the boundary conditions. Based on the systematic investigation, it seems that an increased $b/a$ ratio up to the unit value exhibits increasing values of the frequency, while reaching the peak value for $b/a=1$. It follows a decreasing branch for $b/a>1$, due to the reduction of the momentum of inertia compared to a horizontal state of the ellipse. Moreover, an increased nonlocal parameter reduces the natural frequency, thus verifying the inaccuracy of classical theories compared to the nonlocal formulations. The structural response of nanorods is also affected by the selected boundary condition, where C-C boundaries get higher frequencies, compared to a C-F boundary condition. In C-T nanorods, an increased stiffness of the spring provides higher natural frequencies, under the same fixed nonlocal assumptions. This sensitivity to the stiffness of the spring is more pronounced for lower values of the nonlocal parameter, and becomes meaningless for higher nonlocalities. A perfect circular shape always yields the highest natural frequencies, and could overestimate the actual twisting vibration response of a noncircular shape. These results are of great interest for design purposes, and could be extended to more complicated noncircular cross sections as a possible development of this work.