ARTICLE IN PRESS. Physica E

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1 Physica E 41 (2009) Contents lists available at ScienceDirect Physica E journal homepage: A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration Metin Aydogdu Department of Mechanical Engineering, Trakya University, Edirne, Turkey article info Article history: Received 2 March 2009 Received in revised form 21 April 2009 Accepted 22 May 2009 Available online 30 May 2009 PACS: Fg f abstract In the present study, a generalized nonlocal beam theory is proposed to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. After deriving governing equations, different beam theories including those of Euler Bernoulli, Timoshenko, Reddy, evinson and Aydogdu [Compos. Struct., 89 (2009) 94] are used as a special case in the present compact formulation without repeating derivation of governing equations each time. Effect of nonlocality and length of beams are investigated in detail for each considered problem. Present solutions can be used for the static and dynamic analyses of single-walled carbon nanotubes. & 2009 Elsevier B.V. All rights reserved. Keywords: Nonlocal elastic beam models Shear deformation Bending Buckling Vibration 1. Introduction Due to small length scale in micro- and nano-applications of beam, plate and shell-type structures nonlocal elasticity has been used in recent years. Basic difference between classical elasticity and nonlocal elasticity is definition of stress: stress at a point is function of strain at that point in local elasticity, whereas in local elasticity stress at a point is function of strains at all points in the continuum. In nonlocal elasticity, forces between atoms and internal length scale are considered in construction of constitutive equations [1 4]. Nonlocal elasticity has been used to study wave propagation in composites, elastic waves, dislocation mechanics, dynamic and static analysis of carbon nanotubes and nanorods [1 9]. Recently, molecular dynamic simulations and nonlocal continuum models are compared for wave propagation in single- and double-walled carbon nanotubes [10] and elastic buckling of single-layered graphene sheet [11]. Good agreement is observed between molecular dynamic simulations and nonlocal continuum modeling. Recently, Reddy [12] used different beam theories including those of Euler Bernoulli, Timoshenko, evinson [13] and Reddy Tel.: ; fa: address: metina@trakya.edu.tr [14] to analyze bending, buckling and vibration of nonlocal beams. In his study, different displacement functions are chosen in the first step and then all steps are repeated when deriving beam equilibrium and equations of motion. Also in his study lengthscale effect cannot be observed due to constant beam length, and free vibration frequencies are compared only for the fundamental frequency. In the present study, a compact generalized beam theory [15 19] is used to analyze bending, buckling and vibration of nanoscale beams using local and nonlocal elasticity. In the formulation of beam theories transverse shear deformation is modeled by help of a general function. After general formulation, each beam theory is found as a special case. Also a new shear deformation theory [20] proposed by Aydogdu in a previous study is used in the formulations. Free vibration results are given for the first three modes to see the effect of nonlocality in the higher modes. Also is changed in the analyses to see the length-scale effect in the investigated problems. Present formulation also can be used for nonlocal analyses of nanocomposite beams. 2. Beam theories In this section, derivation of the governing equations for the nanobeams is eplained. Governing equations are derived for nanocomposite beams and numerical results are given for an /$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi: /j.physe

2 1652 M. Aydogdu / Physica E 41 (2009) isotropic nanobeam. Consider a straight uniform composite beam having length with thickness h. The beam is assumed to be constructed of arbitrary number, N, of linearly elastic transversely isotropic layers. Therefore, the state of stress in each layer is given by [21] s ðkþ ¼ 11 ; t ðkþ z ¼ 55 g z, (1) where Q (k) are well-known reduced stiffnesses [21] and k is the number of layers. Assuming that the deformations of the beam take place in the z plane and upon denoting the displacement components along the -, y- and z-directions by U, V and W, respectively, the following displacement field can be written Uð; z; tþ ¼uð; tþ zw ; þ f ðzþu 1 ð; tþ, Vð; z; tþ ¼0, Wð; z; tþ ¼wð; tþ. (2) The displacement model (2) yields the following kinematic relations: ¼ u ; zw ; þ f ðzþu 1;, g z ¼ f 0 u 1, (3) where a prime denotes the derivative with respect to z and, represents partial derivative with respect to. Although different shape functions are applicable, only the ones which convert the present theory to the corresponding Euler Bernoulli beam theory (EBT), parabolic shear deformation beam theory (RBT) of Reddy [14], first order shear deformation beam theory (TBT) of Timoshenko [12] and general eponential shear deformation beam theory (ABT) of Aydogdu [20] are employed in the present study. This is achieved by choosing the shape functions as follows: EBT : f ðzþ ¼0; TBT : f ðzþ ¼z; RBT : f ðzþ ¼zð1 4z 2 =3h 2 Þ; ABT : f ðzþ ¼ðzÞð3Þ 2ðz=hÞ2 = ln 3 : The principle of virtual displacement 0 ¼ @dw þ 01 @ þ r 3 w r 2 dw @u þ r 1 02 N Mc dkc Ma dka Q du 1 f du þ qdw þ dt where the force and the moment resultants are defined in the following form: ðn c Þ¼ ðs Þdz; ðm c Þ¼ s zdz; ðm sd Þ¼ ðs Þf ðzþdz; ðq sd (4) Þ¼ ðt z Þf 0 ðzþdz. (6) r i ¼ R h=2 rzi dz; ði ¼ 0; 1; 2Þ; r jm ¼ R h=2 rzj f m j dz; ðj ¼ 0; 1; m ¼ 1; 2Þ The resultants denoted with a superscript c are the conventional ones of the classical beam theories, whereas the remaining ones with superscript sd are additional quantities incorporating the shear deformation effects. By substituting the stress strain relations into the definitions of the force and the moment resultants of the present theory, the following constitutive equations are obtained [15 20]: 2 3 N c 2 32 A 11 B 11 E 11 6 M c ¼ 6 D 11 F Sim H 11 M sd u ; w ; u 1; 3 h i 7 5 Q sd ¼ ½A 55 Š½u 1 Š. (7) The etensional, coupling, bending and transverse shear rigidities are defined as follows: A i11 ¼ E 11 ¼ H 11 ¼ 11 dz; A 55 ¼ f ðzþdz; D 11 ¼ 55 ðf 0 Þ 2 dz; B 11 ¼ z 2 dz; F 11 ¼ zdz; f ðzþzdz, ðf Þ 2 dz ðþ 0 ¼ dðþ=dz. (8) Upon employing principle of virtual displacement, the three variationally consistent governing equations of the beam are obtained as N c ¼ðr ; 0 u þ r 01 u 1 r 1 w ; Þ ;tt, M c ¼ðr ; 1 u ; þ r 11 u 1; þ r 0 w r 2 w ; Þ ;tt þ N e w ; þ q, M sd sd ; Q ¼ðr 01 u þ r 02 u 1 r 11 w ; Þ ;tt (9) where,tt denotes time derivatives. Moreover, the following sets of boundary conditions at the edges of the beam are obtained by the application of the virtual displacement principle at ¼ 0; either u or N c prescribed; either w or M c ; prescribed; either w or M c prescribed; either u 1 or M sd prescribed: 3. Nonlocal beam theories (10) Response of materials at the nanoscale is different from those of their bulk counterparts. Nonlocal elasticity is first considered by Eringen [1 4]. He assumed that the stress at a reference point is a functional of the strain field at every point of the continuum. Nonlocal stress tensor t at point 0 is defined by Z s ¼ Kðj 0 j; tþsðþd, (11) V where S( 0 ) is the classical, macroscopic stress tensor at point 0, K( 0, t) is the kernel function and t is a material constant that depends on internal and eternal characteristic length (such as the lattice spacing and wave length). Nonlocal constitutive relations for present nanobeams can be written as s 2 ¼ Q 11, s z s ¼ Q 55 2 z, (12) where m ¼ (e 0 a) 2 is nonlocal parameter, a an internal characteristic length and e 0 a constant. Choice of e 0 a (in dimension of length) is crucial to ensure the validity of nonlocal models. This parameter was determined by matching the dispersion curves based on the atomic models [1 4]. For a specific material, the corresponding nonlocal parameter can be estimated by fitting the results of atomic lattice dynamic and eperiment. A conservative estimate of the scale coefficient e 0 ao2.0 nm for a single-walled

3 M. Aydogdu / Physica E 41 (2009) carbon nanotube is proposed [22]. In this study, 0rmr4 is chosen to investigate nonlocality effects. Using Eqs. (7) and (8) following force-strain and moment strain relations are obtained: N c N 2 M c M 2 M sd M 2 ¼ A 11 þ B 11 k c þ E 11k sd, ¼ B 11 þ D 11 k c þ F 11k sd, ¼ E 11 þ D 11 k c þ H 11k sd. (13) Using Eqs. (7), (9), (12) and (13) following governing equations can be found in terms of displacements: A 11 u ; B 11 w ; þ E 11 u 1; ¼ 1 m d2 ðr d 2 0 u þ r 01 u 1 r 1 w ; Þ ;tt, B 11 u ; D 11 w ; þ F 11 u 1; ¼ 1 m d2 d 2 ½ðr 1 u ; þ r 11 u 1; þ r 0 w r 2 w ; Þ ;tt þ N e w ; þ qš, E 11 u ; F 11 w ; þ H 11 u 1; A 55 u 1 ¼ 1 m d2 ðr d 2 01 u þ r 02 u 1 r 11 w ; Þ ;tt (14) 4. Bending, buckling and vibration of simply supported nanobeams In this study, analytical solutions are given for simply supported isotropic nanobeams for bending, buckling and free vibration. The boundary conditions of simply supported beams are w ¼ M c ¼ Msd ¼ 0 at ¼ 0 and. (15) The following displacement field satisfies boundary conditions and governing equations. wð; tþ ¼ X1 u 1 ð; tþ ¼ X1 W m sin mp U 1m cos mp sin ot, sin ot. (16) For bending problem N e and all time derivatives set to zero the transverse load acting on nanobeam can be written as qðþ ¼ X1 Q m sin mp ; Q m ¼ 2 Z 0 qðþ sin mp d. (17) The Fourier coefficients Q m associated with uniform and point loads are given qðþ ¼q 0 ; Q m ¼ 4q 0 ; n ¼ 1; 3; 5... for uniform load, (18) mp qðþ ¼Q 0 dð 0 Þ; Q m ¼ 2Q 0 sin p 0, n ¼ 1; 2; 3... for point load; (19) where q 0 is the density of uniform load, d Dirac Delta function, Q 0 magnitude of point load and 0 (0r 0 r) application position of point load. For buckling, we set q and all time derivatives to zero, and for the free vibration we set N e and q to zero Bending For bending, we set N e and all time derivatives to zero. Inserting Eqs. (16) (19) in to Eq. (14) following static deflections can be obtained. Uniform load: 4½1 þðm½ð2m 1ÞpŠ 2 = 2 ÞŠq 0 4 d 11 ½ð2m 1ÞpŠ 5 EI EBT; (20) 4½1 þðm½ð2m 1ÞpŠ 2 = 2 ÞŠq 0 4 ½ d 11 ½ð2m 1ÞpŠ 5 þ f 2 11 ð½ð2m 1ÞpŠ7 =h 11 ½ð2m 1ÞpŠ 2 þ ka 55 2 ÞŠEI RBT; ABT and TBT, (21) where k ¼ 5 6 for TBT and k ¼ 1 for RBT and ABT. 4½1 þðm½ð2m 1ÞpŠ 2 = 2 ÞŠ½h 11 ½ð2m 1ÞpŠ 2 þ ka 55 2 Šq 0 4 ½ d 11 ½ð2m 1ÞpŠ 5 a 55 2 ŠEI Point load: 2½1 þðm½mpš 2 = 2 ÞŠQ 0 3 d 11 ½mpŠ 4 EI BT. (22) EBT; (23) 2½1 þðm½mpš 2 = 2 ÞŠQ 0 3 ½ d 11 ½mpŠ 4 þ f 2 11 ð½mpš6 =h 11 ½mpŠ 2 þ ka 55 2 ÞŠEI TBT; RBT and ABT; (24) 2½1 þðm½mpš 2 = 2 ÞŠ½h 11 ½mpŠ 2 þ ka 55 2 ŠQ 0 3 ½ d 11 ½mpŠ 4 a 55 2 ŠEI where (d 11, f 11, h 11, a 55 ) ¼ ((D 11, F 11, H 11, A 55 )/Do), Do ¼ D Buckling and vibration BT; (25) For buckling, q and all time derivatives are set to zero, and for vibration we set N e and q to zero in Eq. (14). Substituting Eq. (16) in Eq. (14) gives following eigen-value problem: f½kš l 2 ½MŠgfDg ¼0, (26) where K is the stiffness, M the inertia and load matrices for vibration and buckling, respectively, and D the column vector of unknown coefficients of Eq. (16). The eigen values (l) for which the determinant of coefficient matri of Eq. (26) is zero, leads to the free vibration frequencies and critical buckling load parameter in the case of buckling problem (it should be noted that in the case of buckling problem l 2 is critical buckling load parameter). 5. Numerical results In this section, numerical results are given for analytical solutions given in Section 2. Due to nondimensionalization, only the following material and geometrical properties are required in computations. n ¼ 0.3 and h ¼ 1 nm. Nondimensional terms are chosen in the following form: w ¼ 100w EI for point load; Q 0 3 w ¼ 100w EI for uniform load; q 0 4 qffiffiffi o ¼ o 2 r EI frequency parameter; N ¼ N e 2 EI critical buckling load parameter: The numerical results for bending under point load at the center and uniform load are given in Tables 1 and 2, respectively. Results

4 1654 M. Aydogdu / Physica E 41 (2009) Table 1 Comparison of dimensionless maimum center deflection under uniform load for simply supported beams. /h m EBT TBT RBT BT ABT Table 3 Dimensionless critical buckling load parameter for simply supported nanobeams. /h m EBT TBT RBT ABT Table 2 Comparison of dimensionless maimum center deflection under point load at the center for simply supported beams. /h m EBT TBT RBT BT ABT Table 4 Dimensionless frequency parameters for simply supported nanobeams (fundamental /h m EBT TBT RBT ABT are obtained using 100 terms in the series Eqs. (20) (25). According to these tables, length scale is more obvious for lower (see results for ¼ 10) and it decreases with increasing. This result can be seen from Eqs. (20) (25). Increasing nonlocal parameter m decreases stiffness of the beams and bending results increase. Nonlocality effects are more pronounced for point load. (see Table 2, ¼ 10). EBT predicts lower results for lower and /h ratios. With increasing results are converging to a certain value. Small differences are observed between different theories. Present results are compared with that of Reddy [12]. Good agreement is observed for uniform load but there are some deviations for point loads. This may be due to misprint in the Reddy s study. It should be noted that evinson beam theory is used only for bending problem in the present study. The nondimensional critical buckling loads are presented in Table 3. According to this table buckling loads decrease with increasing nonlocal parameter m. Critical buckling load parameters are insensitive to used theory. Similar to bending results, for lower, nonlocality is important and this effect is lost for higher. Present results are compared with results of Reddy [12] and good agreement is observed. Table 5 Dimensionless frequency parameters for simply supported nanobeams (second /h m EBT TBT RBT ABT

5 M. Aydogdu / Physica E 41 (2009) Table 6 Dimensionless frequency parameters for simply supported nanobeams (third /h m EBT TBT RBT ABT First three fleural nondimensional frequencies are presented in Tables 4 6 for different nonlocal parameter m,, /h ratios and for different theories. In Table 4, results are given for different theories. For given m, two frequencies sets are obtained for shear deformation theories, whereas only one set is found for E B theory. According to these tables, fundamental frequency is insensitive to used theory for ¼ 10 and results are approaching for increasing. Difference between theories increases with increasing mode number (Tables 5 and 6). This is due to small wavelength effect for higher modes. For second set important differences are observed between TBT and RBT, ABT. This may be due to second spectrum. 6. Conclusion A generalized nonlocal beam theory is used to study bending, buckling and free vibration of nanobeams. Nonlocal constitutive equations of Eringen are used in the formulations. Different beam theories including those of Euler Bernoulli, Timoshenko, Reddy, evinson and Aydogdu are used as a special case in the present formulation. Effect of nonlocality and length are investigated in detail for each considered problem. Present formulation can be etended to other classical boundary conditions. References [1] A.C. Eringen, Int. J. Eng. Sci. 10 (1972) 1. [2] A.C. Eringen, D.G.B. Edelen, Int. J. Eng. Sci. 10 (1972) 230. [3] A.C. Eringen, J. Appl. Phys. 54 (1983) [4] A.C. Eringen, Springer, New York, [5] J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41 (2003) 305. [6].J. Sudak, J. Appl. Phys. 94 (2003) [7] M.C. Ece, M. Aydogdu, Acta Mech. 190 (2007) 185. [8] P. u, H.P. ee, C. u, P.Q. Zhang, Int. J. Solids Struct. 44 (2007) [9] M. Aydogdu, Physica E 41 (2009) 861. [10] Y. Hu, K.M. iew, Q. Wang, X.Q. He, B.I. Yakobson, J. Mech. Phys. Solids 56 (2008) [11] A. Sakhaee-Pour, Comput. Mater. Sci. 45 (2009) 266. [12] J.N. Reddy, Int. J. Eng. Sci. 45 (2007) 288. [13] M. evinson, J. Sound Vib. 74 (1981) 81. [14] J.N. Reddy, J. Appl. Mech. 51 (1984) 745. [15] M. Aydogdu, Int. J. Mech. Sci. 11 (2005) [16] M. Aydogdu, Compos. Sci. Technol. 66 (2006) [17] M. Aydogdu, J. Reinf. Plast. Compos. 25 (2006) [18] M. Aydogdu, Int. J. Mech. Sci. 50 (2008) 837. [19] M. Aydogdu, Arch. Appl. Mech. 78 (2008) 711. [20] M. Aydogdu, Compos. Struct. 89 (2009) 94. [21] C.T. Herakovich, Wiley, [22] Q. Wang, J. Appl. Phys. 98 (2005)