Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 4, pp. 1041-1052, Warsaw 2015 DOI: 10.15632/jtam-pl.53.4.1041 A NONLOCAL TIMOSHENKOBEAM THEORY FOR VIBRATION ANALYSIS OF THICK NANOBEAMS USING DIFFERENTIAL TRANSFORM METHOD Farzad Ebrahimi, Parisa Nasirzadeh Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran e-mail: febrahimy@eng.ikiu.ac.ir This article presents the solution for free vibration of nanobeams based onEringen nonlocal elasticity theory and Timoshenko beam theory. The small scale effect is considered in the first theory, and the transverse shear deformation effects as well as rotary inertia are taken into account in the latter one. Through variational formulation and the Hamilton principle, the governing differential equations of free vibration of the nonlocal Timoshenko beam and the boundary conditions are derived. The obtained equations are solved by the differential transformationmethod (DTM) for various frequencymodes of the beamswith different end conditions. In addition, the effects of slenderness and on vibration behavior are presented. It is revealed that the slenderness affects the vibration characteristics slightly whilst the small scale plays a significant role in the vibration behavior of the nanobeam. Keywords: free vibration, nanobeam, Eringen nonlocal elasticity theory 1. Introduction Nanostructures have significant mechanical, electrical and thermal performances that are supe- rior to conventional structuralmaterials. They have attractedmuch attention inmodern science and technology. For example, in micro/nano electromechanical systems (MEMS/NEMS); na- nostructures have been used in many areas including communications, machinery, information technology, biotechnology technologies, etc. So far, three main methods have been provided to study the mechanical behavior of nanostructures. These are the atomistic (Baughman et al., 2002), semi-continuum (Li and Chou, 2003) and continuummodels (Wang and Cai, 2006). Ho- wever, both atomistic and semi-continuum models are computationally expensive and are not suitable for analyzing large scale systems. Due to the inherent size effects, at nanoscale, themechanical characteristics of nanostructures are often significantly different fromtheirbehavior atmacroscopic scale. Sucheffects are essential for nanoscalematerials or structures and the influence on nano-instruments is great (Maranganti and Sharma, 2007). Generally, theoretical studies on size effects at nanoscale are by means of surface effects (Zhu et al., 2009), strain gradients in elasticity (Mindlin, 1964) and plasticity (Aifantis, 1984) aswell as nonlocal stress field theory (Eringen, 1983, 1972a), etc. Unfortunately, the classical continuum theories are deemed to fail for these nanostructures, because length dimensions at nano scale are often sufficiently small such that call the applicability of classical continuum theories into the question. Consequently, the classical continuum models need to be extended to consider the nanoscale effects. This can be achieved through the nonlocal elasticity theory proposed by Eringen (1972a) which considers the size-dependent effect. According to this theory, the stress state at a reference point is considered as a function of strain states of all points in the body. This nonlocal theory is proved to be in accordance with the atomic model of lattice dynamics and with experimental observations on phonon dispersion (Eringen, 1983). In nonlocal theory, nonlocal nanoscale in the constitutive equation could be considered simply 1042 F. Ebrahimi, P. Nasirzadeh as amaterial-dependent parameter. The ratio of internal characteristic scale (such as the lattice parameter, C-C bond length, granular distance, etc.) to external characteristic scale (such as crack length, wave length, etc.) is definedwithin a nonlocal nanoscale parameter. If the internal characteristic scale ismuch smaller than the external characteristic scale, the nonlocal nanoscale parameter approaches zero and the classical continuum theory is recovered. For analyzing these nanoscale beams,Euler-Bernoulli andTimoshenko beam theories appear to be inadequate, since they are scale free. For this problem, continuum mechanics is needed, and one of the efficient theories for nonlocal continuum mechanics is Eringen’s (Eringen, 1983; 1972a,b) theorywhich allows small scale effect by indicating that stress at one point is a function of strain at all points of the body. In the recent years, nanobeams and carbon nanotubes (CNTs) have held a wide variety of potential applications (Zhang et al., 2004; Wang, 2005; Wang and Varadan, 2006) such as sen- sors, actuators, transistors, probes, and resonators in nanoelectromechnical systems (NEMSs). Thus, establishing an accurate model of nanobeams is a key issue for successful NEMS design. As a result, nanotechnological research on free vibration properties of nanobeams is important because such components canbeused as design components in nano-sensors andnano-actuators. Furthermore,many researchersworked onbending, buckling andvibration of beam-like elements (Peddieson et al., 2003; Liew et al., 2008; Xu, 2006; Amara et al., 2010) and in some papers the Euler-Bernoulli theory has been applied for vibration of nanobeams (Lu et al., 2006; Zhang et al., 2005; Xu, 2006). But they used the Euler-Bernoulli theory which does not account trans- verse shear force and rotary inertia which are significant in stubby beams and high vibration frequencies. So in this paper, we used Timoshenko beam theory and the governing equations and boundary conditions for free vibration of a nonlocal Timoshenko beam have been derived via Hamilton’s principle. To the author’s best knowledge, there is no work reported on the ap- plication of DTM on vibration analysis of nonlocal Timoshenko beams with various boundary conditions. Furthermore, the solution procedure in this study is the differential transformation method (DTM)which is a semianalytical-numerical techniquedependingonTaylor series expan- sion. Thismethodwas first introduced by Zhou (1986) in his study about electrical circuits, and this method has the advantage of its simplicity in use as well as high accuracy. The results in this paper are provided by aMATLAB code with respect to DTM rules, for the first time. 2. Nonlocal Timoshenko beam equations and boundary conditions Consider a beamwith lengthL and cross sectional area ofA. Based onTimoshenkobeamtheory, strain-displacement and strain energy relations are as follows (Wang et al., 2000) εxx = z dφ dx γxz = φ+ dw dx (2.1) in which x is the longitudinal coordinate measured from the left end of the beam and z is the coordinatemeasured from themid-plane of the beam, w represents the transverse displacement and φ is rotation of the beam due to bending, εxx is the normal strain, γxz is the transverse shear strain, σxx is normal stress and σxz – transverse shear stress. The strain energy relation is as follows (Leissa and Qatu, 2011) U = 1 2 L∫ 0 ∫ A (σxxεxx +σxzγxz) dAdx (2.2) A nonlocal Timoshenko beam theory for vibration analysis... 1043 After substituting equations (2.1) into equation (2.2) andputting thebendingmoment and shear force into relation (2.2), the strain energy becomes U = 1 2 L∫ 0 ∫ A ( σxxz dφ dx +σxz ( φ+ dw dx )] dAdx = L∫ 0 1 2 [ M dφ dx +Q ( φ+ dw dx )] dx M = ∫ A σxxz dA Q = ∫ A σxz dA (2.3) where M is the bending moment and Q is the shear force. The kinetic energy T , by assuming free harmonic motion and rotary inertia effect, is T = 1 2 L∫ 0 (ρAω2w2+ρIω2φ2) dx (2.4) inwhich ω is the circular frequency of vibration and ρ and I are themass density and the second moment of area of the beam, respectively. ApplyingHamilton’s principle (Chow, 2013), requires δ(T −U)= 0= L∫ 0 [ −M dδφ dx −Q ( δφ+ dδw dx ) +ρAω2wδw+ρIω2φδφ ] dx (2.5) After performing integration by parts, we reach 0= L∫ 0 [(dM dx −Q+ρIω2φ ) δφ+ dQ dx +ρAω2w ) δw ] dx− [Mδφ]L0 − [Qδw]L0 (2.6) This results in the following equations dM dx = Q−ρIω2φ dQ dx =−ρAω2w (2.7) And the boundary conditions are in two forms of below relations Either w =0 or Q =0 Either φ =0 or M =0 (2.8) As can be seen, the equations appear to be the same as in local Timoshenko beam theory, but the shear force and bendingmoment expressions in nonlocal beam theorymust be different. The constitutive equation of classical elasticity is an algebraic relationship between stress and strain tensors while Eringen nonlocal elasticity includes spatial integrals which indicate the average effect of strain of all points of the body to the stress tensor at the given point (Eringen, 1972b; 1983). Since the spatial integrals in constitutive equations are mathematically difficult to solve, they can be converted into equal differential constitutive equations under certain conditions. The nonlocal constitutive stress-strain relation for an elastic material in the one dimensional case beam can be simplified as (Eringen, 1983) σxx − (e0a)2 d2σxx dx2 = Eεxx (2.9) in which E is the Young modulus, e0a is the scale coefficient that incorporates the small scale effect, a represents the internal characteristic length and e0 is a constant appropriate to each material which is measured experimentally. The local and nonlocal constitutive shear strain- 1044 F. Ebrahimi, P. Nasirzadeh -stress relations are the same, since form of the Eringen nonlocal constitutive model cannot be applied in the z direction σxz = Gγxz (2.10) in which G is the shear modulus. After multiplying the term (z dA) and integrating over the area A, equation (2.9) becomes M − (e0a)2 d2M dx2 = EI dφ dx (2.11) By integrating equation (2.10) over the area, we obtain Q = KsGA ( φ+ dw dx ) (2.12) inwhichKs is the shear correction factor in theTimoshenkobeamtheory in order to compensate for the error in assuming equal shear stress or strain in the whole beam thickness. Now by substituting equations (2.7) into equation (2.11), the moment can be reached as below M = EI dφ dx − (e0a)2 ( ρAω2w+ρIω2 dφ dx ) (2.13) And by utilizing equations (2.12) and (2.13) inTimoshenko beam equations (2.7), the governing equation for the vibration of nonlocal Timoshenko beammay be obtained as below EI d2φ dx2 −KsGA ( φ+ dw dx ) +ρIω2φ− (e0a)2 ( ρAω2 dw dx +ρIω2 d2φ dx2 ) =0 KsGA (dφ dx + d2w dx2 ) +ρAwω2 =0 (2.14) On thebasis of equation (2.8) anddue tovarious endings of thebeam, e.g. for a simply supported end, we have w =0 M = EI dφ dx − (e0a)2 ( ρAω2w+ρIω2 dφ dx ) =0 (2.15) and for a clamped end w =0 φ =0 (2.16) and for a free end M = EI dφ dx − (e0a)2 ( ρAω2w+ρIω2 dφ dx ) =0 Q = KsGA ( φ+ dw dx ) =0 (2.17) 3. Non-dimensional parameters Thenon-dimensionalparameters contributes to simplification of the equations and to themaking of comparisons in the studies possible. The non-dimensional parameters are introduced as the following terms x = x L w = w L λ2 = ω2 ρAL4 EI Ω = EI KsGAL 2 α = e0a L ε = L √ A√ I A nonlocal Timoshenko beam theory for vibration analysis... 1045 where λ2 is frequency parameter, Ω – shear deformation parameter, α – scaling effect arameter, ε – slenderness ratio. By applying the non-dimensional parameters to governing equations (2.14), the following relations are obtained Ω ( 1− α 2λ2 ε2 )d2φ dx2 + (Ωλ2 ε2 −1 ) φ− (α2λ2Ω +1)dw dx =0 dφ dx + d2w dx2 +λ2Ωw =0 (3.1) Also boundary conditions equations (2.15)- (2.17) appear for the simply supported end as w =0 M = ( Ω − Ωα2λ2 ε2 )dφ dx −Ωα2λ2w =0 (3.2) And for the clamped end as w =0 φ =0 (3.3) And for the free end as M = ( Ω − Ωα2λ2 ε2 )dφ dx −Ωα2λ2w =0 Q = KsGA ( φ+ dw dx =0 (3.4) 4. Differential transformation method The differential transformation method is one of the useful techniques to solve differential equ- ations with small calculation errors and capable of solving nonlinear equations with boundary condition value problems.Abdel-HalimHassan (2002) applied theDTMto eigenvalues and nor- malized eigenfunctions. Also Wang (2013) and Chen and Ju (2004) used the method in their studies. The DTM is a transformation technique based on the Taylor series expansion and is a useful tool to obtain analytical solutions to differential equations. The DTM is proved to be a good computational tool for various engineering problems. Using the differential transformation technique, ordinary and partial differential equations can be transformed into algebraic equ- ations from which a closed-form series solution can be obtained easily. In this method, certain transformation rules are applied to both the governing differential equations of motion and the boundary conditions of the system in order to transform them into a set of algebraic equations as presented in Table 1 and 2. Table 1.Basics of the differential transformmethod (Chen and Ju, 2004) Original function Transformed function f(x)= g(x±h(x) F(K)= G(K)±H(K) f(x)= λg(x) F(K)= λG(K) f(x)= g(x)h(x) F(K)= K∑ l=0 G(K − l)H(l) f(x)= dng(x) dxn F(K)= (k+n)! k! G(K +n) f(x)= xn F(K)= δ(k −n)=    0 if k 6= n 1 if k = n 1046 F. Ebrahimi, P. Nasirzadeh Table 2. Transformed boundary conditions (BC) based on DTM (Chen and Ju, 2004) X =0 X =1 Original Transformed Original Transformed BC BC BC BC f(0)= 0 F[0] = 0 f(1)= 0 ∞∑ k=0 F[k] = 0 df dx (0)= 0 F[1] = 0 df dx (1)= 0 ∞∑ k=0 kF[k] = 0 d2f dx2 (0)= 0 F[2] = 0 d2f dx2 (1)= 0 ∞∑ k=0 k(k−1)F[k] = 0 d3f dx3 (0)= 0 F[3] = 0 d3f dx3 (1)= 0 ∞∑ k=0 k(k−1)(k −2)F[k] = 0 The solution of these algebraic equations gives the desired results of the problem. It is different from the high-order Taylor series method because the Taylor series method requires symbolic computation of necessary derivatives of data functions and is expensive for large orders. The basic definitions and the application procedure of this method can be introduced as follows: The transformation equation of the function f(x) can be defined as (Chen and Ju, 2004) F[k] = 1 k! dkf(x) dxk ∣∣∣ x=x0 (4.1) where f(x) the original function and F[k] is the transformed function. The inverse transformation is defined as f(x)= ∞∑ k=0 (x−x0)kF[k] (4.2) Combining equations (4.1) and (4.2), one obtains f(x)= ∞∑ k=0 (x−x0)k k! dkf(x) dxk ∣∣∣ x=x0 (4.3) Considering equation (4.3), it is noticed that the concept of the differential transform is derived fromTaylor series expansion. In actual application, the function f(x) is expressed by a finite series, and equation (4.3) can be written as follows f(x)= n∑ k=0 (x−x0)k k! dkf(x) dxk ∣∣∣ x=x0 (4.4) which implies that the term in relation (4.5) is negligible f(x)= ∞∑ k=n+1 (x−x0)k k! dkf(x) dxk ∣∣∣ x=x0 (4.5) In this study, the natural frequencies determine the value of n. A nonlocal Timoshenko beam theory for vibration analysis... 1047 5. Solution with DTM According to the DTM rules given in Table 1, equations (3.1) will be transformed into the following equations Ω ( 1−α 2λ2 ε2 ) (k+1)(k+2)Φ(k +2)+ (Ωλ2 ε2 −1 ) Φ(k)− (α2λ2Ω +1)(k +1)W(k+1)=0 (5.1) (K +1)Φ(k+1)+(k+1)(k+2)W(k+2)+λ2ΩW(k)= 0 The rules of the DTM for defining boundary conditions are given in Table 2. W(k) and Φ(k) are transforms of w(x) and φ(x), respectively. By substituting values for k =0,1,2, . . ., α =0, ε =34.641 and Ω =0.2436 into equations (5.1), we can evaluate the amounts of W(2),W(3), . . . and Φ(2),Φ(3), . . . in terms of ω2 and some constants like c1, . . .. The values can be achieved with a computer program, and after substituting W(i) and Φ(i) into boundary conditions the following equation is obtained Nr1 (n)(ω)c1+Nr2 (n)(ω)c2 =0 r =1,2, . . . ,n (5.2) inwhichNs arepolynomials in termsofω correspondingto then-th term.When solving equation (5.2) in matrix form, the following eigenvalue equation may be obtained ∣∣∣∣∣ Nn11(ω) N n 12(ω) Nn21(ω) N n 22(ω) ∣∣∣∣∣ =0 (5.3) The solution to equation (5.3) gives ωnr which is the r-th estimated eigenvalue for the n-th repeat. The number of repeats can be obtained by equation (5.4) as |ωnr −ωn−1r | < δ (5.4) In the present study, δ = 0.0001 and this shows the accuracy of calculations. With respect to the differential transformation method and the algorithm above, a MATLAB code has been developed in order to determine vibration characteristics of the nonlocalTimoshenko nanobeam. 6. Results and discussion In the present study, the impact of the small scale coefficient as well as the effect of slenderness on the first, second and third frequencies of the nonlocal Timoshenko nanobeam are presented. Also, three types of boundary conditions are compared. In order to validate the computed results, a comparison between the present paper and the results obtained byWang et al. (2007) is performed. The mechanical properties of the nonlocal Timoshenko nanobeam are given in Table 3. Table 3.Mechanical properties of the nonlocal Timoshenko nanobeam (Wang et al., 2007) Property Unit E T·Pa 5.5 P g·cm−3 2.3 ν – 0.19 Also, the Timoshenko shear correction factor ks is taken 0.563. For calculating the exact difference between the results of the present paper and the available results in literature, relation (6.1) has been applied %difference=100 · |reference−present| present (6.1) 1048 F. Ebrahimi, P. Nasirzadeh As shown by the comparisons given in Tables 4 and 5, a close correlation between these results validates the proposedmethod of solution. Table 4.First three nondimensional frequencies √ λ of the nonlocal Timoshenko beam for both clamped ends and L/d =10 Mode 1 Mode 2 Mode 3 α pre- Wang et al. diff. pre- Wang et al. diff. pre- Wang et al. diff. sent (2007) [%] sent (2007) [%] sent (2007) [%] 0 4.53 4.45 1.766 7.19 6.95 3.33 9.6 9.2 4.1 0.1 4.4233 4.3471 1.72 6.67 6.4952 2.62 8.4 8.2 2.38 0.3 3.83 3.7895 1.057 5 4.9428 1.14 5.95 5.846 1.74 0.5 3.2657 3.242 0.725 4 3.994 0.15 4.75 4.6769 1.53 0.7 2.85 2.8383 0.41 3.45 3.4192 0.89 4.05 3.9961 1.33 Table 5.First nondimensional natural frequency √ λ of the nonlocal Timoshenko beam for two kinds of boundary conditions and L/d =10 Clamped-simple Simple-free α pre- Wang et al. diff. pre- Wang et al. diff. sent (2007) [%] sent (2007) [%] 0 3.82 3.7845 0.929 3.08 3.0929 0.418 0.1 3.73 3.6939 0.967 3.059 3.0243 1.13 0.3 3.23 3.2115 0.5727 2.91 2.6538 8.8 0.7 2.415 2.4059 0.37 2.4 2.0106 16 In addition, the convergence of the differential transformation method is perused. In Fig. 1, the convergence of the third frequency of the nonlocal Timoshenko beam with both clamped ends is presented. It illustrates that the third frequency converges at the 46th repeat, while the first and the second frequencies converged before, in this example at the 29th and 37th repeats. Fig. 1. Convergence of the third frequency, L/d=10, α =0 The variables in governing equations (3.1) are α, ε and Ω. α relates to the small scale effect, ε is in terms of the slenderness (L/d) and Ω relates to the mechanical properties and slenderness. So, it is possible to investigate the effects of slenderness and small scale on various frequencies and mode shapes of the nonlocal Timoshenko beam. Furthermore, determination of the magnitude of e0 is significant due to its prominent effect on the small scale coefficient. Some researchers worked on estimating themagnitude of e0a. For instance, Zhang et al. (2005) estimated the magnitude of the parameter for carbon nanotubes to be approximately 0.82. In A nonlocal Timoshenko beam theory for vibration analysis... 1049 this study, we adopt 0 ¬ α < 0.8 in our investigations as reported by Lu et al. (2006). As Figs. 2a,b,c show when the coefficient α equals zero, the frequency of the nonlocal Timoshenko beam equals its local counterpart. As the coefficient increases, the frequency ratio decreases, which means that the nonlocal beam frequency becomes smaller than the local counterparts. This reduction is especially noticeable in higher modes and cannot be neglected. In sum, the small scale effect makes the beammore flexible since in nonlocal theory elastic springs link the atoms together (Liew et al., 2008). Fig. 2. Effect of small scale on different frequencymodes, L/d =10: (a) clamped ends, (b) simply supported beam, (c) clamped-simply beam Figure 3 indicates that the small scale have significant effect on short beams and, as the beamgets longer, its impact becomes gradually negligible. So, the small scale will diminish for a very long and thin (slender) beam. Also, Fig. 4 illustrates that the nonlocal Timoshenko beam frequency approaches the local Timoshenko beam frequency as the slenderness increases. Fig. 3. Small scale effect on the frequency ratio with different values of L/d (both ends clamped) 1050 F. Ebrahimi, P. Nasirzadeh Table 6.First three frequencies √ λ of the nonlocal Timoshenko beamwith two kinds of boun- dary conditions e0a Mode L/d =10 L/d =20 L/d =30 Simply supported-simply supported beam 0 1 3.08057 3.12577 3.13451 0.1 3.08056 3.12577 3.13451 0.3 3.08056 3.12577 3.1345 0.5 3.08052 3.12576 3.1345 0.7 3.08047 3.12574 3.13449 0.9 3.08040 3.12572 3.13448 0 2 5.94588 6.18907 6.24037 0.1 5.94584 6.18906 6.24036 0.3 5.94558 6.18898 6.24033 0.5 5.94466 6.18882 6.24025 0.7 5.94425 6.18858 6.24015 0.9 5.94318 6.18826 6.24000 0 3 8.53236 9.15198 9.29787 0.1 8.53225 9.15194 9.29785 0.3 8.53139 9.15165 9.29771 0.5 8.52995 9.15107 9.29743 0.7 8.52936 9.15020 9.29702 0.9 8.52366 9.14904 9.29647 Clamped-simply supported beam 0 3.829744 3.901179 3.915187 0.1 3.829726 3.901175 3.915186 0.3 3.829653 3.901155 3.915176 0.5 1 3.829491 3.901111 3.915155 0.7 3.829248 3.901045 3.915136 0.9 3.828925 3.900957 3.915086 1 3.828732 3.900904 3.915063 0 6.644277 6.948166 7.01359 0.1 6.644219 6.948148 7.013581 0.3 6.642754 6.948008 7.013516 0.5 2 6.642824 6.947726 7.013386 0.7 6.641431 6.947305 7.013191 0.9 6.639576 6.946740 7.01293 1 6.638475 9.946408 7.012778 0 9.177342 9.888691 10.05988 0.1 9.177189 9.888635 10.05986 0.3 9.175962 9.888215 10.05966 0.5 3 9.173494 9.887381 10.05925 0.7 9.169836 9.886127 10.05866 0.9 9.164950 9.883445 10.05853 1 9.162052 9.883461 10.05806 A nonlocal Timoshenko beam theory for vibration analysis... 1051 Fig. 4. Effect of slenderness on the nonlocal beam frequency (α =0.7, both ends clamped) 7. Conclusion A semi-analytical method called the differential transformation method is generalized to ana- lyze vibration characteristics of a nanobeam. The formulation is based on the assumptions of Timoshenko beam theory and the nonlocal differential constitutive relations of Eringen. The transverse shear force and rotary inertia that become significant at short beams and higher fre- quencies are taken into account in the equations. Also, the effect of the small scale coefficient as well as the slenderness and boundary conditions in various frequency ratios are investigated. It is demonstrated that the DTMhas high precision and computational efficiency in the vibration analysis of nanobeams. References 1. Abdel-Halim Hassan I.H., 2002, On solving some eigenvalue problems by using a differential transformation,Applied Mathematics and Computation, 127, 1, 1-22 2. Aifantis E.C., 1984, On the microstructural origin of certain inelastic models, Journal of Engi- neering Materials and Technology, 106, 4, 326-330 3. Amara K., Tounsi A., Mechab I., Adda-Bedia E.A., 2010, Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field, Applied Mathematical Modeling, 34, 12, 3933-3942 4. BaughmanR.H., ZakhidovA.A., deHeerW.A., 2002,Carbon nanotubes – the route toward applications, Science, 297, 5582, 787-792 5. Chen C.O.K., Ju S.P., 2004, Application of differential transformation to transient advective- dispersive transport equation,Applied Mathematics and Computation, 155, 1, 25-38 6. Chow T.L., 2013,Classical Mechanics, CRCPress, Boca Raton, Florida, USA 7. Eringen A.C., 1972a, Linear theory of nonlocal elasticity and dispersion of plane waves, Interna- tional Journal of Engineering Science, 10, 5, 425-435 8. EringenA.C., 1972b,Nonlocalpolar elastic continua, International Journal of Engineering Scien- ce, 10, 1, 1-16 9. Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislo- cation and surface waves, Journal of Applied Physics, 54, 9, 4703-4710 10. Eringen A.C., Edelen D.G.B., 1972, On nonlocal elasticity, International Journal of Engine- ering Science, 10, 3, 233-248 11. Lee Y.Y., Wang C.M., Kitipornchai S., 2003, Vibration of Timoshenko beams with internal hinge, Journal of Engineering Mechanics, 129, 3, 293-301 12. Leissa A.W., Qatu M.S., 2011,Vibration of Continuous Systems, McGrawHill Professional 1052 F. Ebrahimi, P. Nasirzadeh 13. Li C., Chou T.W., 2003, A structural mechanics approach for the analysis of carbon nanotubes, International Journal of Solids and Structures, 40, 10, 2487-2499 14. LiewK.M.,HuY.,HeX.Q., 2008,Flexuralwavepropagation in single-walledcarbonnanotubes, Journal of Computational and Theoretical Nanoscience, 5, 4, 581-586 15. Lu P., Lee H.P., Lu C., Zhang P.Q., 2006, Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics, 99, 7, 073510 16. Maranganti R., Sharma P., 2007, Length scales at which classical elasticity breaks down for variousmaterials,Physical Review Letters, 98, 19, 195504 17. Mindlin R.D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, 16, 51-78 18. Peddieson J., BuchananG.R.,McNittR.P., 2003,Application of nonlocal continuummodels to nanotechnology, International Journal of Engineering Science, 41, 3, 305-312 19. Wang C.M., Reddy J.N., Lee K.H., Eds., 2000, Shear Deformable Beams and Plates: Rela- tionships with Classical Solutions, Elsevier 20. Wang C.M., Zhang Y.Y., HeX.Q., 2007,Vibration of nonlocal Timoshenko beams,Nanotech- nology, 18, 10, 105401 21. WangQ., 2005,Wavepropagation incarbonnanotubesvianonlocal continuummechanics,Journal of Applied Physics, 98, 12, 124301 22. WangQ.,VaradanV.K., 2006,Vibrationof carbonnanotubes studiedusingnonlocal continuum mechanics, Smart Materials and Structures, 15, 2, 659 23. Wang X., Cai H., 2006, Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes,Acta Materialia, 54, 8, 2067-2074 24. Wang Z.G., 2013, Axial vibration analysis of stepped bar by differential transformationmethod, Applied Mechanics and Materials, 419, 273-279 25. Xu M., 2006, Free transverse vibrations of nano-to-micron scale beams, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 462, 2074, 2977-2995 26. Zhang Y.Q., Liu G.R., Wang J.S., 2004, Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression,Physical Review B, 70, 20, 205430 27. Zhang Y.Q., Liu G.R., Xie X.Y., 2005, Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity,Physical Review B, 71, 19, 195404 28. Zhou J.K., 1986, Differential Transformation and its Applications for Electrical Circuits, Hu- azhong University Press,Wuhan, China 29. Zhu H., Wang J., Karihaloo B., 2009, Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms, Journal of Mechanics of Materials and Structures, 4, 3, 589-604 Manuscript received April 3, 2014; accepted for print July 1, 2015