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Axial Vibration Analysis of a Nanorod Embedded in Elastic Medium Using Nonlocal Strain Gradient Theory

Year 2016, Volume: 31 Issue: 1, 213 - 222, 15.06.2016
https://doi.org/10.21605/cukurovaummfd.317803

Abstract

Size-dependent axial vibration of a nanorod embedded in elastic medium is studied for the first time in this paper within the framework of the nonlocal strain gradient theory. The governing equation of motion of the problem is derived using the equilibrium condition and it is solved analytically to obtain the exact expression of vibration frequency for a fixed-fixed nanorod. The influences of the nonlocal parameter, the material length scale parameter and the elastic medium coefficient on the free vibration frequencies are investigated in detail. The results show that free vibration frequencies are significantly dependent on the size effects, and the size effects gain more importance at higher modes. Therefore, the classical continuum theory is inappropriate to investigate the mechanical behavior of nanostructures

References

  • 1. Eringen, A.C., 1972. Nonlocal Polar Elastic Continua; International Journal of Engineering Science, 10, pp. 1-16.
  • 2. Aifantis, E.C., 1999. Strain Gradient Interpretation of Size Effects; International Journal of Fracture, 95, pp. 1-4.
  • 3. 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, pp. 2731–2743.
  • 4. Eringen, A.C., 1967. Theory of micropolar plates; Zeitschrift fur Angewandte Mathematik und Physik, 18, pp. 12–30.
  • 5. Peddieson, J., Buchanan, G.R., McNitt R.P., 2003. Application of Nonlocal Continuum Models to Nanotechnology; International Journal of Engineering Science, 41, pp. 305-312.
  • 6. Reddy, J.N., 2007. Nonlocal Theories for Bending, Buckling and Vibration of Beams; International Journal of Engineering Science, 45, pp. 288-307.
  • 7. Adali S, 2008. Variational Principles for Multi-Walled Carbon Nanotubes Undergoing Buckling Based on Nonlocal Elasticity Theory; Physics Letters A, 372, pp. 5701-5705.
  • 8. Murmu, T., Pradhan, S.C., 2009. Buckling Analysis of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity and Timoshenko Beam Theory and Using DQM; Physica E, 41, pp. 1232-1239.
  • 9. Murmu, T., Adhikari, S., 2011. Axial Instability of Double-Nanobeam-Systems; Physics Letters A, 375 pp. 601-608.
  • 10. Pradhan, S.C., Reddy, G.K., 2011. Buckling Analysis of Single Walled Carbon Nanotube on Winkler Foundation Using Nonlocal Elasticity Theory and DTM; Computational Materials Science, 50, pp. 1052-1056.
  • 11. Ansari, R., Sahmani, S., Rouhi, H., 2011. Axial Buckling Analysis of Single-Walled Carbon Nanotubes in Thermal Environments Via the Rayleigh–Ritz Technique; Computational Materials Science, 50 pp. 3050–3055.
  • 12. Şimşek, M., Yurtcu, H.H., 2013. Analytical Solutions for Bending and Buckling of Functionally Graded Nanobeams Based on the Nonlocal Timoshenko Beam Theory; Composite Structures, 97, pp. 378-386.
  • 13. Aydogdu, M., 2009. A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration; Physica E, 41, pp. 1651-1655.
  • 14. Murmu, T., Pradhan, S.C., 2009. Thermo-Mechanical Vibration of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity Theory; Computational Materials Science, 46, pp. 854-859.
  • 15. Şimşek, M., 2010. Vibration Analysis of a Single-Walled Carbon Nanotube Under Action of a Moving Harmonic Load Based on Nonlocal Elasticity Theory; Physica E, 43, pp. 182-191.
  • 16. Şimşek, M., 2011. Nonlocal Effects in the Forced Vibration of an Elastically Connected Double-Carbon Nanotube System Under a Moving Nanoparticle; Computational Materials Science, 50, pp. 2112-2123,
  • 17. Şimşek, M., 2011. Forced Vibration of an Embedded Single-Walled Carbon Nanotube Traversed by a Moving Load Using Nonlocal Timoshenko Beam Theory; Steel Composite Structures, 11, pp. 59-76.
  • 18. Eltaher, M.A., Emam, S.A., Mahmoud, F.F., 2012. Free Vibration Analysis of Functionally Graded Size-Dependent Nanobeams; Applied Mathematics and Computation, 218, pp. 7406–7420.
  • 19. Thai, H.T., A., 2012.Nonlocal Beam Theory for Bending, Buckling, and Vibration of Nanobeams; International Journal of Engineering Science, 52, pp. 56–64.
  • 20. Şimşek M., 2012. Nonlocal Effects in the Free Longitudinal Vibration of Axially Functionally Graded Tapered Nanorods; Computational Materials Science, 61, pp. 257–265.
  • 21. Tsepoura, K.G., 2002. Papargyri-Beskou, S., Polyzos, D., Beskos, D.E., Static and Dynamic Analysis of a Gradient-Elastic Bar in Tension; Archive of Applied Mechanics, 72, pp. 483-497.
  • 22. Park, S.K., Gao, X.L, 2006. Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory; Journal of Micromechanics and Microengineering, 16, pp. 2355–2359.
  • 23. Ma, H.M., Gao, X.L., Reddy, J.N., 2008. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory; Journal of the Mechanics and Physics of Solids, 56, pp. 3379-3391.
  • 24. Şimşek M., 2010. Dynamic Analysis of an Embedded Microbeam Carrying A Moving Microparticle Based on the Modified Couple Stress Theory, International Journal of Engineering Science, 48, pp. 1721–1732.
  • 25. Reddy, J.N., 2011. Microstructure-Dependent Couple Stress Theories of Functionally Graded Beams; Jo.urnal of the Mechanics and Physics of Solids, 59, pp. 2382-2399.
  • 26. Şimşek, M., Kocatürk, T., Akbaş, Ş. D., 2013. Static Bending of a Functionally Graded Microscale Timoshenko Beam Based on the Modified Couple Stress Theory; Composite Structures, 95, pp. 740-747.
  • 27. Şimşek, M., Reddy, J.N, 2013. Bending and Vibration of Functionally Graded Microbeams Using a New Higher Order Beam Theory and the Modified Couple Stress Theory; International Journal of Engineering Science, 64, pp. 37–53.
  • 28. Şimşek, M., Reddy, J.N, 2013. A Unified Higher Order Beam Theory for Buckling of a Functionally Graded Microbeam Embedded in Elastic Medium Using Modified Couple Stress theory, Composite Structures, 101, pp. 47-58.
  • 29. Akgöz, B., Civalek, Ö., 2013. Longitudinal Vibration Analysis of Strain Gradient Bars Made of Functionally Graded Materials (FGM); Composites: Part B, 55, pp. 263-268.
  • 30. Şimşek, M., 2014. Nonlinear Static and Free Vibration Analysis of Microbeams Based on the Nonlinear Elastic Foundation Using Modified Couple Stress Theory and He’s Variational Method; Composite Structures, 112, pp. 264-272.
  • 31. Challamel N., 2013. Variational Formulation of Gradient or/and Nonlocal Higher-Order Shear Elasticity Beams, Composite Structures, 105, pp. 351–368.
  • 32. Lim C.W., Zhang G., Reddy J.N., 2015. A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and its Applications in Wave Propagation, Journal of the Mechanics and Physics of Solids, 78, pp. 298–313,
  • 33. Li L., Hu Y, 2015. Buckling Analysis of Size-Dependent Nonlinear Beams Based on a Nonlocal Strain Gradient Theory; International Journal of Engineering Science, 97, pp. 84–94.

Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi

Year 2016, Volume: 31 Issue: 1, 213 - 222, 15.06.2016
https://doi.org/10.21605/cukurovaummfd.317803

Abstract

Elastik zemine gömülü bir nano çubuğun boyut etkisine bağlı eksenel titreşimi yerel olmayan şekil değiştirme gradyanı teorisi çerçevesinde ilk olarak bu çalışmada incelenmiştir. Probleme ait yönetici hareket denklemi denge şartı kullanılarak çıkarılmış, iki ucu ankastre nano çubuğun serbest titreşim frekansına ait kesin ifadeyi elde etmek için yönetici denklem analitik olarak çözülmüştür. Yerel olmayan parametre, malzeme uzunluk ölçek parametresi ve elastik zemin parametresinin serbest titreşim frekansları üzerindeki etkisi detaylı olarak incelenmiştir. Elde edilen sayısal sonuçlar göstermiştir ki; serbest titreşim frekansları boyut etkisine önemli derecede bağlıdır ve boyut etkisi yüksek modlarda daha çok önem kazanmaktadır. Bu nedenlerden dolayı, klasik sürekli ortamlar mekaniği nano ölçekteki yapıların analizi için uygun değildir.

References

  • 1. Eringen, A.C., 1972. Nonlocal Polar Elastic Continua; International Journal of Engineering Science, 10, pp. 1-16.
  • 2. Aifantis, E.C., 1999. Strain Gradient Interpretation of Size Effects; International Journal of Fracture, 95, pp. 1-4.
  • 3. 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, pp. 2731–2743.
  • 4. Eringen, A.C., 1967. Theory of micropolar plates; Zeitschrift fur Angewandte Mathematik und Physik, 18, pp. 12–30.
  • 5. Peddieson, J., Buchanan, G.R., McNitt R.P., 2003. Application of Nonlocal Continuum Models to Nanotechnology; International Journal of Engineering Science, 41, pp. 305-312.
  • 6. Reddy, J.N., 2007. Nonlocal Theories for Bending, Buckling and Vibration of Beams; International Journal of Engineering Science, 45, pp. 288-307.
  • 7. Adali S, 2008. Variational Principles for Multi-Walled Carbon Nanotubes Undergoing Buckling Based on Nonlocal Elasticity Theory; Physics Letters A, 372, pp. 5701-5705.
  • 8. Murmu, T., Pradhan, S.C., 2009. Buckling Analysis of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity and Timoshenko Beam Theory and Using DQM; Physica E, 41, pp. 1232-1239.
  • 9. Murmu, T., Adhikari, S., 2011. Axial Instability of Double-Nanobeam-Systems; Physics Letters A, 375 pp. 601-608.
  • 10. Pradhan, S.C., Reddy, G.K., 2011. Buckling Analysis of Single Walled Carbon Nanotube on Winkler Foundation Using Nonlocal Elasticity Theory and DTM; Computational Materials Science, 50, pp. 1052-1056.
  • 11. Ansari, R., Sahmani, S., Rouhi, H., 2011. Axial Buckling Analysis of Single-Walled Carbon Nanotubes in Thermal Environments Via the Rayleigh–Ritz Technique; Computational Materials Science, 50 pp. 3050–3055.
  • 12. Şimşek, M., Yurtcu, H.H., 2013. Analytical Solutions for Bending and Buckling of Functionally Graded Nanobeams Based on the Nonlocal Timoshenko Beam Theory; Composite Structures, 97, pp. 378-386.
  • 13. Aydogdu, M., 2009. A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration; Physica E, 41, pp. 1651-1655.
  • 14. Murmu, T., Pradhan, S.C., 2009. Thermo-Mechanical Vibration of a Single-Walled Carbon Nanotube Embedded in an Elastic Medium Based on Nonlocal Elasticity Theory; Computational Materials Science, 46, pp. 854-859.
  • 15. Şimşek, M., 2010. Vibration Analysis of a Single-Walled Carbon Nanotube Under Action of a Moving Harmonic Load Based on Nonlocal Elasticity Theory; Physica E, 43, pp. 182-191.
  • 16. Şimşek, M., 2011. Nonlocal Effects in the Forced Vibration of an Elastically Connected Double-Carbon Nanotube System Under a Moving Nanoparticle; Computational Materials Science, 50, pp. 2112-2123,
  • 17. Şimşek, M., 2011. Forced Vibration of an Embedded Single-Walled Carbon Nanotube Traversed by a Moving Load Using Nonlocal Timoshenko Beam Theory; Steel Composite Structures, 11, pp. 59-76.
  • 18. Eltaher, M.A., Emam, S.A., Mahmoud, F.F., 2012. Free Vibration Analysis of Functionally Graded Size-Dependent Nanobeams; Applied Mathematics and Computation, 218, pp. 7406–7420.
  • 19. Thai, H.T., A., 2012.Nonlocal Beam Theory for Bending, Buckling, and Vibration of Nanobeams; International Journal of Engineering Science, 52, pp. 56–64.
  • 20. Şimşek M., 2012. Nonlocal Effects in the Free Longitudinal Vibration of Axially Functionally Graded Tapered Nanorods; Computational Materials Science, 61, pp. 257–265.
  • 21. Tsepoura, K.G., 2002. Papargyri-Beskou, S., Polyzos, D., Beskos, D.E., Static and Dynamic Analysis of a Gradient-Elastic Bar in Tension; Archive of Applied Mechanics, 72, pp. 483-497.
  • 22. Park, S.K., Gao, X.L, 2006. Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory; Journal of Micromechanics and Microengineering, 16, pp. 2355–2359.
  • 23. Ma, H.M., Gao, X.L., Reddy, J.N., 2008. A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory; Journal of the Mechanics and Physics of Solids, 56, pp. 3379-3391.
  • 24. Şimşek M., 2010. Dynamic Analysis of an Embedded Microbeam Carrying A Moving Microparticle Based on the Modified Couple Stress Theory, International Journal of Engineering Science, 48, pp. 1721–1732.
  • 25. Reddy, J.N., 2011. Microstructure-Dependent Couple Stress Theories of Functionally Graded Beams; Jo.urnal of the Mechanics and Physics of Solids, 59, pp. 2382-2399.
  • 26. Şimşek, M., Kocatürk, T., Akbaş, Ş. D., 2013. Static Bending of a Functionally Graded Microscale Timoshenko Beam Based on the Modified Couple Stress Theory; Composite Structures, 95, pp. 740-747.
  • 27. Şimşek, M., Reddy, J.N, 2013. Bending and Vibration of Functionally Graded Microbeams Using a New Higher Order Beam Theory and the Modified Couple Stress Theory; International Journal of Engineering Science, 64, pp. 37–53.
  • 28. Şimşek, M., Reddy, J.N, 2013. A Unified Higher Order Beam Theory for Buckling of a Functionally Graded Microbeam Embedded in Elastic Medium Using Modified Couple Stress theory, Composite Structures, 101, pp. 47-58.
  • 29. Akgöz, B., Civalek, Ö., 2013. Longitudinal Vibration Analysis of Strain Gradient Bars Made of Functionally Graded Materials (FGM); Composites: Part B, 55, pp. 263-268.
  • 30. Şimşek, M., 2014. Nonlinear Static and Free Vibration Analysis of Microbeams Based on the Nonlinear Elastic Foundation Using Modified Couple Stress Theory and He’s Variational Method; Composite Structures, 112, pp. 264-272.
  • 31. Challamel N., 2013. Variational Formulation of Gradient or/and Nonlocal Higher-Order Shear Elasticity Beams, Composite Structures, 105, pp. 351–368.
  • 32. Lim C.W., Zhang G., Reddy J.N., 2015. A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and its Applications in Wave Propagation, Journal of the Mechanics and Physics of Solids, 78, pp. 298–313,
  • 33. Li L., Hu Y, 2015. Buckling Analysis of Size-Dependent Nonlinear Beams Based on a Nonlocal Strain Gradient Theory; International Journal of Engineering Science, 97, pp. 84–94.
There are 33 citations in total.

Details

Journal Section Articles
Authors

Mesut Şimşek

Publication Date June 15, 2016
Published in Issue Year 2016 Volume: 31 Issue: 1

Cite

APA Şimşek, M. (2016). Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 31(1), 213-222. https://doi.org/10.21605/cukurovaummfd.317803
AMA Şimşek M. Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi. cukurovaummfd. June 2016;31(1):213-222. doi:10.21605/cukurovaummfd.317803
Chicago Şimşek, Mesut. “Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 31, no. 1 (June 2016): 213-22. https://doi.org/10.21605/cukurovaummfd.317803.
EndNote Şimşek M (June 1, 2016) Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 31 1 213–222.
IEEE M. Şimşek, “Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi”, cukurovaummfd, vol. 31, no. 1, pp. 213–222, 2016, doi: 10.21605/cukurovaummfd.317803.
ISNAD Şimşek, Mesut. “Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 31/1 (June 2016), 213-222. https://doi.org/10.21605/cukurovaummfd.317803.
JAMA Şimşek M. Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi. cukurovaummfd. 2016;31:213–222.
MLA Şimşek, Mesut. “Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, vol. 31, no. 1, 2016, pp. 213-22, doi:10.21605/cukurovaummfd.317803.
Vancouver Şimşek M. Yerel Olmayan Şekil Değiştirme Gradyanı Teorisi Kullanılarak Elastik Zemine Gömülü Nano Çubuğun Eksenel Titreşim Analizi. cukurovaummfd. 2016;31(1):213-22.