1 Introduction

According to application of nanobeams as nanosensors, actuators, nanogenerators, transistors, diodes, and resonators in nanoelectromechanical systems and in biotechnology ^[1-5], it is important to investigate the vibrational behavior of nanobeams. In addition, structures at the nanometer length scale are known to exhibit a size-dependent behavior ^[6-10]. Since the surface-to-bulk ratio is large in nanostructures, the surface energy effects cannot be ignored. Gurtin and Murdoch ^[11-12] presented a three-dimensional theory called the surface elasticity theory based on the continuum mechanics concept that takes into consideration the effects of surface energy. In their work, a surface was regarded as a mathematical deformable membrane of zero thickness fully adhered to the underlying bulk material. The equilibrium and constitutive equations for the bulk were the same as those in the classical theory of elasticity. In addition, a set of constitutive equations and the generalized Young-Laplace equation were applied to the surface.

Using the surface elasticity theory, various mechanical behaviors of nanobeams such as free and forced transverse vibration ^[13-14], buckling ^[15], postbuckling ^[16], bending ^[17], and wave prop-\linebreak agation ^[18] were analyzed, and in general, it was shown that the effects of the surface energy depend on the type of boundary condition. Another field that recently attracted significant attention of researchers is considering the surface energy effects on transverse vibration of cracked nanobeams ^[19-21]. The aim of research implemented the surface elasticity theory to model the problem is investigation of effects of four factors on mechanical behavior of nanostructures, i.e., (i) the effect of surface elasticity, (ii) the effect of surface density, (iii) the effect of surface stress, and (iv) the effect of satisfying the balance condition between the nanostructure bulk and its surfaces. For example, in the case of free transverse vibration of nanobeams, it was shown that the surface elasticity and stress may have both increasing and decreasing effects on natural frequencies depending on their signs ^[22]. In addition, it was reported that the surface density has a negligible effect at low mode numbers, while it is the other way round for its effect at higher mode numbers ^[23]. Finally, it was concluded that satisfying the balance condition causes a reduction on the transverse frequencies ^[19].

In several applications of nanobeams, such as drive shafts ^[24], torsion bar springs ^[25-26], linear nano servomotors and bearings ^[27], and torsional actuators ^[28], tensile and torsional loads are expected to occur. This implies that optimizing the design of new devices requires torsional behavior analysis of nanobeams besides analyses of other mechanical behaviors. Although there are several studies focusing on the torsional responses of these kinds of nanostructures, none of them has used the surface elasticity theory ^[29-33]. Gheshlaghi and Hasheminejad ^[29] studied the torsional vibration of carbon nanotubes using a modified couple stress theory ^[29]. Murmu et al. ^[30] analyzed the torsional vibration of carbon nanotube-buckyball systems using the nonlocal elasticity theory. Lim et al. ^[31] developed a new elastic nonlocal stress model for dynamic behaviors of circular nanorods/nanotubes. Using a nonlocal elasticity model, the effects of crack on free torsional vibration of nanorods ^[32] and the effects of elastic medium on torsional statics and dynamics of nanotubes ^[33] were investigated.

From the literature, it is understood that the surface elasticity theory has not been used for free torsional vibration analysis of nanobeams yet. The lack prompted the authors to model the free torsional vibration of nanobeams based on the surface elasticity theory and to investigate the surface energy effects on torsional frequencies.

The main purpose of the present work is to propose a comprehensive analytical model to study the surface energy effects on the free torsional vibration of nanoscale cracked beams. To this end, the governing equations of cracked nanobeams incorporating the surface energy effects are derived by Hamilton's principle and solved by the exact solution for various boundary conditions. Natural frequencies for cracked nanobeams with some typical boundary conditions for different crack positions, crack severities, values of the surface energy, mode numbers, and dimensions of nanobeam on the free torsional vibration of nanobeams are calculated.

2 Problem formulation 2.1 Surface elasticity theory

At the micro/nanoscale, the fraction of energy stored in the surfaces becomes comparable with that in the bulk, because of the relatively high ratio of the surface area to the volume of nanoscale structures. Therefore, the surface and induced surface forces cannot be ignored. Gurtin and Murdoch ^[11-12] presented a three-dimensional theory called the surface elasticity theory based on a continuum mechanics concept that takes into consideration the effects of surface energy. In their work, a surface was regarded as a mathematical deformable membrane of zero thickness fully adhered to the underlying bulk material. The equilibrium and constitutive equations for the bulk are the same as those in the classical theory of elasticity. In addition, a set of constitutive equations and the generalized Young-Laplace equation are applied to the surface. The constitutive relations of the surface layers, $S^+$ (upper surface) and $S^-$ (lower surface), given by Gurtin and Murdoch ^[11-12] are

(1)

(2)

in which $\tau_{0}$ is the residual surface tension under unconstrained conditions, $\lambda_{0}$ and $\mu_{0}$ are the surface Laméconstants, $\delta_{\alpha \beta}$ is the Kronecker delta, and $u_\alpha$ are the displacement components of the surfaces with $\alpha, \beta=x, \theta$. The signs + and $-$ denote the upper and lower surfaces, respectively.

2.2 Free torsional vibration analysis

In this section, the governing equation of motion of a cracked nanobeam in presence of the surface energy is derived by Hamilton's principle. To this end, we consider a nanobeam with the circular cross-section of constant area $A$ with the radius $R$ and the length $L$ $(0\leq x\leq L)$, as shown in Fig. 1.

The displacement components $(u_x, u_r, u_\theta)$ of the nanobeam (the bulk and the surface layers) parallel to the three coordinate axes $(x, r, \theta)$ are given by ^[34]

(3)

(4)

(5)

where $\phi(x, t)$ is the angle of twist of the cross-section along the $x$-coordinate.

Based on the displacement components, the strains in the nanobeam bulk are assumed to be

(6)

(7)

and the corresponding bulk stresses are given by

(8)

(9)

in which $G$ is the bulk shear modulus. In order to find the surface non-zero stress(es) corresponding to the non-zero strain(s), Eqs.(1) and (2) are implemented. Consider the same material properties for the surface layers of nanobeam results in the following surface stress-strain relation:

(10)

It is worth mentioning here that we know from continuum mechanics that the Laméconstant $\mu$ is identical to the shear modulus $G$. Therefore, the Laméconstant $\mu_0$ is shown as $G_0$.

After obtaining the bulk and surface stress-strain relations, it is possible to obtain the governing equation of motions of nanobeam with the aid of Hamilton's principle,

(11)

In Eq.(11), $U$ and $K$ are the strain and kinetic energies of the nanobeam, respectively, and are given by

(12)

(13)

where $U_{\rm b}$ and $K_{\rm b}$ are the bulk strain and kinetic energies, respectively, and $U_{\rm s}$ and $K_{\rm s}$ are the surface strain and kinetic energies, respectively.

Substituting Eqs.(12) and (13) into Eq.(11) and using Eqs.(6)-(10) yield the governing align of motion and boundary condition as follows:

(14)

(15)

where

(16)

It can be observed from Eqs.(14)-(16) that the surface energy parameters appear in the torsional governing equations of motion of nanobeam, but they will not have any effect on the boundary condition. In addition, the equation of torsional motion of the conventional beam ^[34] can be obtained from Eq.(14) by setting $\rho_0=G_0=0$.

The governing equation of motion, Eq.(14), can be solved using the separation-of-variables method as

(17)

where $\omega$ is the natural frequency of the torsional vibration.

Substitute Eq.(17) into Eqs.(14) and (15) and use the following dimensionless variables and constants given by

(18)

Then, we get the equation for the spatial function $\Phi(X)$ and the boundary condition as

(19)

(20)

Solving Eq.(19) gives

(21)

which is for an intact nanobeam.

Consider now that the nanobeam has a defect that can be modeled as a circumferential crack around the perimeter of a section located at a distance $L_{\rm c} (L^*=\frac{L_{\rm c}}{L} )$ from the left end. The effect of the crack is taken into account following the methodology first proposed by Freund and Herrmann ^[35]. Here, the cracked nanobeam has been considered as two intact nanobeams connected by a torsional linear elastic spring at the cracked section (see Fig. 1) to consider the additional strain energy due to the presence of the crack. Therefore, in the presence of the crack, the solution of Eq.(19) can be expressed as

(22)

Equation (22) with $i$ equal to 1 and 2 is the free torsional vibration solution of the segment one and two, respectively.

Since Eq.(22) has four unknown coefficients, obtaining the natural torsional frequencies requires two more conditions than those given in Eq.(20). The conditions are compatibility conditions at the crack section and given by

(i) Jump in twisting deflection,

(23)

(ii) Continuity of the twisting moment,

(24)

Applying the boundary and compatibility conditions, Eqs.(20), (23), and (24) to Eq.(22) yields a system of four homogeneous algebraic equations with $A_{11}, A_{12}, A_{21}$, and $A_{22}$ as unknowns. For a nontrivial solution of $A_{11}, A_{12}, A_{21}$, and $A_{22}$, the determinant of the $A_{11}, A_{12}, A_{21}$, and $A_{22}$ coefficients must be set to zero for each boundary type. The process gives the characteristic equation of the cracked nanobeam with fixed-fixed, fixed-free, and free-free end conditions as

(i) For the fixed-fixed cracked nanobeam,

(25)

(ii) For the fixed-free cracked nanobeam,

(26)

(iii) For the free-free cracked nanobeam,

(27)

The roots of the characteristic equations (25)-(27) are the torsional frequencies for the cracked nanobeam incorporating the surface energy effects.

3 Results and discussion 3.1 Comparison study

To confirm the reliability and accuracy of the present formulation and results, two comparison studies are conducted. Since there is not any work considering the surface energy effects on torsional frequencies of cracked nanobeams, the results of the present study are compared with those reported by Rao ^[34] and Loya et al. ^[32] for an intact macrobeam and cracked nanobeam, respectively, without considering the surface energy effects.

A comparative study for evaluation of the first five natural torsional frequencies $\omega$ between the present solution without considering the crack $(C= 0)$ and surface energy effects and the results given by Rao ^[34] is carried out in Table 1 for a beam with fixed-fixed, fixed-free, and free-free boundary conditions, and $L=1$ m, $G=30$ GPa, and $\rho=2~700$ kg$\cdot$m$^{-3}$. Table 1 confirms the reliability of the present formulation and results.

In Table 2, the first four dimensionless natural torsional frequencies $\lambda_{\rm c}=L\omega\sqrt{\frac{\rho}{G}}$ of a fixed-free cracked nanobeam without considering the surface effects are listed. The results given by Loya et al. ^[32] are also provided for direct comparison. It is assumed that the crack is located at the middle of nanobeam in Table 2. It is observed that the present results agree very well with those given by Loya et al. ^[32].

3.2 Benchmark results

To demonstrate the influence of the surface energy on the free torsional vibration of cracked nanobeams, variations of the frequency ratio, Eq.(28), versus the length and radius of nanobeam, mode number, crack severity, and position is illustrated in this section. A nanobeam with circular cross-section and three different boundary conditions, i.e., fixed-fixed, fixed-free, and free-free, is considered. The nanobeam made of aluminum and silicon with crystallographic direction of [100] and bulk and surface properties ^[36] given in Table 3 is also considered as illustrative examples.

(28)

where $r_{\rm f}$ is the frequency ratio, $f_{\rm c}$ is the natural frequency of cracked nanobeam with surface energy effects, and $f_{\rm n}$ is the natural frequency of nanobeam. In the following discussion and for the brevity, CrFr, CrSuFr, and ClFr stand for the natural frequency with only the crack effects, the natural frequency in presence of the both crack and surface energy effects, and the classical natural frequency, respectively.

Firstly, the effects of the crack and the surface energy on the torsional frequency ratios for various lengths of nanobeam are investigated. To this end, in Table 4, torsional frequency ratios (CrSuFr/ClFr, CrFr/ClFr, and SuFr/ClFr) for the aluminum and the silicon nanobeams with a crack severity of $C=2$ and radius of $R=1$~nm and two different crack positions are listed for three boundary conditions. From Table 4, the following interesting points can be addressed.

(i) Both the crack and the surface energy have a decreasing effect on the torsional frequency ratios. The decreasing effect of the crack is the result of the loss of rigidity of the structure, and the more flexible the structure is, the smaller its frequency ratio is. The decreasing effect of the surface energy can be explained in this way that the surface has a negative shear modulus, and its negative amount is increased by the surface residual stress resulting in the decrease in the torsional rigidity of nanobeam. Moreover, the surface density increases the kinetic energy of the system, and as we know, as the kinetic energy of the system increases, its natural frequency decreases. Consequently, the surface energy has a decreasing effect on the torsional frequency ratio. It should be noticed that the surface energy does not always have a decreasing effect on the nanostructures, because it has been reported that the surface energy has an increasing effect on transverse frequencies of the nanobeams and the nanoplates ^{[6, 13, 19, 37]}.

(ii) Although the effect of the crack depends on its position on nanobeam, it is observed that the effect of both the crack and the surface energy on the torsional frequency ratios is independent of the length of nanobeam. It is worth noting here that the effect of the surface energy on the transverse vibration of nanobeams depends on the length of the nanobeam, and the longer the nanobeam is, the more important the effect of the surface energy on the transverse frequencies is ^{[23, 38-39]}.

(iii) The effect of the crack on the torsional frequency ratios of nanobeams depends on the boundary conditions. However, the relation between the stiffness or the flexibility of the boundary condition and the effect of the crack on the torsional frequency ratios cannot be specifically determined. Moreover, it is observed that the effect of the surface energy is independent of the type of the boundary condition. However, the reported results of the effect of the surface energy on transverse frequencies state that the more the boundary condition is stiffened, the less important the effect of the surface energy is ^{[10, 19]}.

(iv) The effect of the crack on the torsional frequency ratios of the nanobeam is independent of the material of nanobeam. However, the surface energy depends on the material of nanobeam. Table 4 shows that the decreasing effect of the surface energy on the torsional frequency ratios of an aluminum nanobeam is more than a silicon one. The reason for this is the difference between the bulk and the surface mechanical properties of the two nanobeams (refer to linebreak Table 3).

Secondly, the effects of the crack and the surface energy on the torsional frequency ratios for various mode numbers are investigated. To this end, in Table 5, the torsional frequency ratios (CrSuFr/ClFr, CrFr/ClFr, and SuFr/ClFr) for aluminum and silicon nanobeams with a crack severity of $C=2$ and radius of $R=1$ nm and two different crack positions are listed for three boundary conditions. Table 5 not only reconfirms the results concluded from Table 4, but also shows that the decreasing effect of the surface energy is independent of the mode number, while the decreasing effect of the crack is dependent on the mode number and the crack position (see Figs. 2 and 3). The other result of Table 5 is that when the effects of the crack and the surface energy are considered simultaneously, their decreasing effect is less than that when they are considered separately. It is worth noting here that the references have reported that the effect of the surface energy on transverse frequencies of nanobeams depends on the mode number so that this effect first decreases and then increases by increasing the mode number ^[23].

Next, we turn our attention to the effects of the crack and the surface energy on the torsional frequency ratios for various radii of nanobeam. For this purpose, in Table 6, the torsional frequency ratios (CrSuFr/ClFr, CrFr/ClFr, and SuFr/ClFr) for aluminum and silicon nanobeams with various boundary conditions are listed for $L=10$ nm, $C=2$, and $L^*=0.5$. Table 6 again proves that the effect of the surface energy on the torsional frequency ratios is independent of the nanobeam material and the boundary condition. However, it is the other way round for the crack effects on the torsional frequency ratios. Also, from Table 6, it can be found that the effect of the crack and the surface energy on the torsional frequency ratios are independent of the nanobeam radius and dependent on the nanobeam radius. For a better comparison, the effect of dependence of the surface energy to the radius of nanobeam is shown in Fig. 4. Figure 4 shows that by increasing the radius of the nanobeam, the effect of the surface energy decreases. This condition is also true with the nanobeam in any boundary condition. Besides, it can be deduced that when the effect of the crack and the surface energy on frequency ratios are simultaneously applied by increasing the radius of the nanobeam, only the effect of the surface energy decreases and approaches their distinct horizontal asymptotes which are the frequency ratio curves with only considering the crack effects. In other words, for nanobeams with large radius, the effect of the surface energy on frequency ratios is inconsequential, and only the crack can have a decreasing effect on them.

Finally, the crack and surface energy effects on the torsional frequency ratios are investigated for various crack positions and crack severities. To this end, variations of first three torsional frequency ratios for an aluminum nanobeam versus the crack position are shown in Fig. 5, and the variations of first three torsional frequency ratios for an aluminum nanobeam versus the crack severity parameter $C$ are shown in Fig. 6 for different boundary conditions and for $L= 10$ nm, $R=1$ nm, and $C=1$. Figures 5 and 6 illustrate that the effect of the surface energy on the torsional frequency ratios is independent of the crack severity, and the crack position but the effect of the crack on torsional frequency ratios intensively depends on the crack position, the crack severity, and the mode number. As the number of the mode number increases, the effect of the crack on the torsional frequency ratios decreases. As the crack severity increases, the effect of the crack on the torsional frequency ratios increases (while the crack is effective on the frequency ratios). Besides, Figs. 5 and 6 show that the effect of the crack on the torsional frequency ratios in presence of the surface energy effects is less in comparison with the case that only the effect of the crack on torsional frequency ratios is considered.

4 Conclusions

In this study, the free torsional vibration of cracked nanobeams in the presence of the surface energy effects is investigated. The surface elasticity theory is used for considering the surface energy parameters in the governing equation of motion. Then, the governing equation of motion is derived by Hamilton's principle, and the torsional natural frequencies are analytically obtained for nanobeams with fixed-fixed, fixed-free, and free-free boundary conditions. The numerical results reveal that both the crack and the surface energy have a decreasing effect on the frequency ratios. The effect of the surface energy is independent of the length, the boundary condition, the mode number, the crack severity, and the crack position. However, it is observed that the effect of surface energy depends on the material and the radius of the nanobeam. By increasing the radius of the nanobeam, the effect of the surface energy decreases. For the nanobeams with a large radius, the effect of the surface energy can be neglected. Moreover, it is observed that the effect of the crack is independent of the nanobeam material but dependent on the boundary condition, the mode number, the crack severity, and the crack position. This study also shows that the effect of the crack with the presence of the surface effect on the torsional frequency ratios decreases in comparison with the case that only the effect of the crack on the torsional frequency ratios is considered. The results of this study are important because it is concluded that the effect of the surface energy on the free torsional vibration of nanobeams is different from that reported about the effect of the surface energy on free transverse vibration of nanobeams. Therefore, it is important to separately and specifically investigate the torsional behavior of nanobeams in the presence of the surface effects.