Yerel olmayan elastisite teorisi kullanılarak elastik ortam içerisindeki nano çubuğun eksenel titreşiminin incelenmesi

Bu çalışmada, yerel olmayan elastisite teorisi kullanılarak elastik ortamda bir nano çubuğun eksenel titreşimi ele alınmıştır. Probleme ait hareket denklemleri denge şartları vasıtasıyla elde edilmiş ve analitik olarak çözülmüştür. İki ucu ankastre ve bir ucu ankastre bir ucu serbest nano çubuğun serbest titreşim frekanslarını veren ifadeler yerel olmayan parametre ve elastik ortam parametrelerine bağlı olarak bulunmuştur. Ankastre-ankastre ve ankastre-serbest sınır koşulları için, titreşim frekansları ile elastik ortam parametresi ve yerel olmayan parametrenin ilişkileri incelenerek sonuçlar grafikler üzerinde gösterilmiştir. Sayısal sonuçlar için karbon nano çubuğa ait fiziksel ve malzeme özellikleri kullanılmıştır. Elde edilen sonuçlarla serbest titreşim frekanslarının boyuta önemli ölçüde bağlı olduğu ve boyut etkisinin yüksek modlarda daha etkili olduğu görülmüştür. Yerel teori ile elde edilen frekans değerleri, yerel olmayan elastisite teorisi kullanılarak elde edilenlerden çok farklıdır.

Anahtar Kelimeler:

Eksenel titreşim, Nano çubuk, Yerel olmayan elastisite teorisi

Investigation of axial vibration of nanorod in elastic media using nonlocal elasticity theory

In this study, axial vibration of a nano rod in an elastic media is discussed using the non-local elasticity theory. Equations of motion of the problem are obtained by means of equilibrium conditions and solved analytically. The expressions giving the free vibration frequencies of the fixed-fixed nanorod and fixed-free nanorod were found depending on the non-local parameter and the elastic medium parameters. For fixed-fixed and fixed-free boundary conditions, the relationships between vibration frequencies and elastic medium parameter and nonlocal parameter are examined and the results are shown on graphs. Physical and material properties of the carbon nanotube were used for numerical results. With the results, it was seen that free vibration frequencies are remarkably be subject to size and the size effect is more effective in high modes. The frequency values which obtained using the classical elasticity theory are very distinct than obtained using the non-local elasticity theory.

Keywords:

Axial vibration, Nanorod, Nonlocal elasticity theory,

PDF

___

Aifantis, E.C. (1999). Strain gradient interpretation of size effects, International Journal of Fructure, 95, 1-4. https://doi.org/10.1007/978-94-011-4659-3_16.
Arash, B., & Ansari, R. (2010). Evolution of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain. Physica E: Low-dimensional Systems and Nanostructures, 42, 2058-2064. https://doi.org/10.1016/j.physe.2010.03.028
Ansari, R., & Sahmani, S. (2011). Bending behavior and buckling of nano beams including surface stress effects corresponding to different beam theories. International Journal of Engineering Science, 49, 1244-1255. https://doi.org/10.1016/j.ijengsci.2011.01.007
Ansari, R., & Wang, Q. (2012). A review on the application of nonlocal elastic models in modelling of carbon nanotubes and graphenes. Computational Materials Science, 51, 303-313. https://doi.org/10.1016/j.commatsci.2011.07.040
Aydogdu, M. (2009a). Axial vibration of the nanorods with the nonlocal cotinuum rod model. Physica E: Low-dimensional Systems and Nanostructures, 41(5), 861-864. https://doi.org/10.1016/j.physe.2009.01.007
Aydogdu, M. (2009b). A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibrition. Physica E: Low-dimensional Systems and Nanostructures, 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014
Aydogdu, M. (2012). Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity, Mechanical Reseach and Communication, 43, 34-40. https://doi.org/10.1016/j.mechrescom.2012.02.001
Ece, M.C., & Aydogdu, M. (2007). Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes. Acta Mechanica, 190, 185-195. https://doi.org/10.1007/s00707-006-0417-5
Eringen, A.C. (1967). Theory of micropolar plates. Zeitschrift fur Angewandte Mathematik und Physik 18, 12-30.
Eringen, A.C. (1972). Nonlocal polar elastic continua. International Journal of Engineering Science, 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
Eringen, A.C. (1983). Interaction of dislocation with a crack. Journal of Applied Physics, 54, 6811-6817. https://doi.org/10.1063/1.332001
Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354, 56-58.
Peddieson, J., Buchanan, G.R., & McNitt, R.P. (2003). Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
Reddy, J.N.N. (2007). Nonlocal theories for bending, buckling and vibration of beams, International Journal of Engineering Science, 45(2-8), 288-307. https:// doi.org/10.1016/j.ijengsci.2007.04.004
Şimşek, M. (2016). Axial vibration analysis of a nanorod embedded in elastic medium using nonlocal strain gradient theory, Çukurova University Journal of the Faculty of Engineering and Architecture, 31(1), 213-221. https:// doi.org/10.21605/cukurovaummfd.317803
Peddieson, J., Buchanan, G.R., & McNitt, R.P. (2003). Application of nonlocal continuum models to nanotechnology, International Journal of Engineering Science, 41(3-5), 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0
Thai, H.T. (2012) A nonlocal beam theory for bending, buckling and vibration of nanobeams. International Journal of Engineering Science, 12, 56-64. https:// doi.org/10.1016/j.ijengsci.2011.11.011
Yang, F., Chong, A.C.M., Lam, D.C.C., & Tong, P. (2002). Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39, 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
Yang, J., Ke, L.L., & Kitipornchai, S. (2010). Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Physica E: Low-dimensional Systems and Nanostructures, 42:1727-1735. https:// doi.org/10.1016/j.physe.2010.01.035
Yaylı, M.O., Yanık, F., & Kandemir, S.Y. (2015), Longitudinal vibration of nanorods embedded in an elastic medium with elastic restraints at both ends. Micro and Nano Letters, 10, 641-644. https:// doi.org/10.1049/mnl.2014.0680