Photonic crystals in biology - NanoTR-VI

Poster Session, Thursday, June 17Theme F686 - N1123Free vibration analysis of carbon nanotubes based on nonlocal continuum and gradientelasticity theoriesÖmer Civalek 1 , Bekir Akgöz, Hakan Ersoy1 Akdeniz University, Civil Engineering Department, Division of MechanicsAntalya-TURKIYE, Tel: + 90- 242-310 6319, Fax: + 90-242-310 6306Abstract- Free vibration analysis of single walled carbon nanotubes (CNT) is presented based on the Euler-Bernoulli beamtheory. The size effect is taken into consideration using the Eringen’s non-local elasticity theory. Gradient elasticity theory isalso adopted for modeling. The governing differential equations for CNT vibration is being solved using the differentialquadrature (DQ) method. Numerical results are presented to show the effect of nonlocal behavior on frequencies of CNT.The concept of carbon nanotubes (CNTs) was firstintroduced in 1991 by Iijima [1] in Japan. Reviews on thedevelopment and application of such nano structures havebeen presented [2]. So, the studies of mechanical behaviorsof carbon nanotubes have being attracted more and moreattentions of scientists in the world and also have become anew research area of applied mechanics [3,4]. In thepresent work, the consistent governing equations for thebeam model for CNTs are derived for free vibrationanalysis. Nonlocal beam and couple stress beam theoriesare adopted for modeling. It is known that, the stress stateof any body at a point x is related to strain state at the samepoint x in the classical elasticity. But this theory is notconflict the atomic theory of lattice dynamics andexperimental observation of phonon dispersion. As statedby Eringen [5] the linear theory of nonlocal elasticity leadsto a set of integropartial differential equations for thedisplacements field for homogeneous, isotropic bodies.According to the nonlocal elasticity theory of Eringen’s,the stress at any reference point in the body depends notonly on the strains at this point but also on strains at allpoints of the body. This definition of the Eringen’snonlocal elasticity is based on the atomic theory of latticedynamics and some experimental observations on phonondispersion. In the present manuscript two differentapproaches are used for modeling of carbon nanotubes.Euler-Bernoulli beam-nonlocal model [5]42 W22 WEI A W ( e0a)A 0 (1)42xxEuler-Bernoulli beam-gradient elasticity theory [6] Wx Wx4422EI g EI A 0 (2)447Table 1. First three frequencies (10) of S-S carbon8nanotubes via gradient theory ( L 510m ,312 2 2300kg/ m , m , t 510 10E 10 N /m )Modeg/L (DQ results)0.005 0.015 0.1251 0.10388 0.10669 0.113742 0.41065 0.41103 0.423013 0.91863 0.92007 0.934857Table 2. First three frequencies (10) of S-S carbon8nanotubes via nonlocal theory( L 510m ,312 2 2300kg/ m , m , t 510 10E 10 N /m )Mode (e 0 a) 2 (DQ results)0 2 41 0.10273 0.10158 0.099622 0.40967 0.40863 0.405533 0.9172 0.90864 0.90637[1] S. Iijima, Nature, 354, 56 (2001).[2] D. Qian, G.J. Wagner, W.K. Liu, Appl. Mech. Rev.,55, 495(2002).[3] C.M. Wang, V.B.C. Tan, T.Y. Zhang, J. Sound Vib.294, 1060 (2006).[4] J.N. Reddy, S.D. Pang, J. Appl. Phys. 103, 023511(2008).[5] A.C. Eringen, J. Appl. Phys., 54, 4703 (1983).[6] S.P. Beskou, D. Polyzos, D.E. Beskos, Struct. Eng.Mech. 15, 705(2003).[7] Ö. Civalek, Engineering Structures, 26, 171(2004).The results obtained by differential quadrature (DQ)method [7] using two higher order elasticity theories arelisted in Tables 1-2. In table 1, first three frequencies ofsimple supported (S-S) carbon nanotubes are listed fordifferent gradient parameter. It is shown that, thefrequencies are increased gradually with the increasingvalue of g for all modes. Nonlocal parameter also affectedon frequencies (Table 2). When the nonlocal parametersare increased, the values of frequencies are decreased,significantly. It is possible to say that, the classical beamtheories can not to capture to size effect on mechanicalbehavior of nano sized structures. So, it is suitable to usesome higher order continuum theory such as nonlocalelasticity theory or gradient strain theory to investigate thesize effect on mechanical behaviour of nano/microstructures.6th Nanoscience and Nanotechnology Conference, zmir, 2010 705