Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

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.


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 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.

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].
Vibration 2020, 3 FOR PEER REVIEW 3 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.

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 axis and -z axis, respectively, as depicted in  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-axis and z-axis, respectively, as depicted in Figures 2-4. 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     The displacement field for any arbitrary point under a torsional vibration, is defined by the following components where the x-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, θ is the angular twist, and ψ(y, z) represents the warping function, defined as ψ(y, z) = y × z b 2 −a 2 b 2 +a 2 . The strain field is, thus, governed by the following kinematic relations which are, in turn, related to the stress components σ xy , σ xz by means of the constitutive relations. The torsional couple acting on the twisted noncircular body is, thus, expressed as [91]

Hamilton's Principle
Hamilton's principle is then employed to derive the equation of motion in the following form 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 which is combined to Equation (2) to yield By a further combination of Equation (6) with Equation (3) we obtain the following variational expression for the strain energy By integration of δU within a certain lapse of time [0, t] we obtain The kinetic energy is defined as By substituting the first derivative of Equation (1) with respect to the time into Equation (9), the kinetic energy can be rewritten as where I 1 and I 2 include the axial and polar inertia terms, defined as I p and I 0 denote the polar moment of inertia, and mass inertia, respectively, expressed as The first variation of I 1 can be determined as follows where The first variation of I 1 in the Hamilton's principle can be expressed as whereas, the first variation of I 2 is defined as By substitution of I 2 in Hamilton's principle, we calculate Finally, the variation of kinetic energy can be stated as In total absence of the external work, the combination of Equations (4), (8), and (21) yields the following equation of motion According to Eringen [41], the nonlocal constitutive relations of the nanostructure take the following form µ 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 with Substituting the first derivative of Equation (22) into Equation (25) yields Thus, the equation of motion for noncircular nanorods, takes the following final form

Analytical Solution
Based on the Galerkin method, we compute the theoretical solution of Equation (28), for nanorods with an elliptical cross-section, namely Θ n being the nth mode shape, for a fixed boundary condition, and is expressed as where the parameter P depends on the selected boundary condition. For a C-C, C-F, C-T, this parameter is defined as P = nπ/l (31) where α is determined by solving the following equation [91] tan 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 which can be redefined in dimensionless form as

Results and Discussion
This section is devoted to the numerical study of the vibration response for elliptical nanorods, with shear modulus G = 498GPa and density ρ = 1330 kg/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 = 1nm, µ/l = 0.1 and l = 20nm, 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 µ, on a nanorod of length l = 20 nm with C-C (Table 1) or C-F (Table 2) 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). Table 1. Comparative evaluation of the first four dimensionless natural frequencies for a C-C circular nanorod with respect to the literature (a = b = 1nm, µ/l = 0.1, l = 20 nm ).  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 Tables 2 and 3, 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 Figures 5 and 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 µ = 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.   Table 4, and plotted in Figure 7. Based on the results, it is worth noticing that increased torsional stiffness of    Table 4, and plotted in Figure 7. Based on the results, it is worth noticing that increased torsional stiffness of A further systemic study considers the combined effect of the nonlocal parameter and stiffness of the boundary spring (k b ) on the vibration response of a C-T elliptical nanorod with a = 0.4 nm and b = 0.2 nm (i.e., b/a = 0.5), as summarized in Table 4, and plotted in Figure 7. Based on the results, it is worth noticing that increased torsional stiffness of k b from 0.01 × 10 −18 to 10 × 10 −18 GPa × 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.  which means a reduction in the structural deformability. The sensitivity of the response to b k 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.

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 / 1 b a = . It follows a decreasing branch for / 1 b a > , due to the reduction of the momentum of inertia compared to a

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.