Yerel olmayan elastisite teorisi kullanılarak nano ölçekli çubuklarda nonlineer dalga yayılımı

Abstract

Bu çalışmada, yerel olmayan elastisite teorisi kullanılarak nano ölçekli çıbuklarda zayıf nonlineer dalga yayılımı incelenmiştir. Mühendislik, fizik ve doğal bilimlerde birçok sistem nonlineerdir ve nonlineer denklemlerle modellenir. Lineer olmayan bilimin bir dalı olan dalga yayılımı son yıllarda yaygın olarak çalışılan konulardan biridir.  Yerel olmayan elastisite teorisi microelektromekanik ve nanoelektromekanik gibi sistemlerin analizinde gelişen popüler bir tekniktir. Formülasyonlarda Eringen’in yerel olmayan elastisite teorisine dayanan bünye denklemleri kullanılmıştır. Hareket denklemleri malzeme koordinatları cinsinden yazılmış ve nano ölçekli çubuğun doğrusal olmayan hareket denklemleri yerel olmayan elastisite teorisine göre elde edilmiştir. İndirgeyici pertürbasyon metodu kullanılarak zayıf nonlineer dalgaların hareketini yöneten evolüsyon denklemi olarak Korteweg de Vries (KdV) denklemi elde edilmiştir.  KdV denkleminde yerel olmayan etkiyi nümerik olarak gözlemleyebilmek için, karbon nanotüplerin fiziksel ve geometrik özellikleri göz önünde bulundurulmuştur.

Keywords

• Nano ölçekli çubuk,Yerel olayan elastisite teorisi,nonlineer dalgalar,indirgeyici pertürbasyon metodu

Propagation of weakly nonlinear waves in nanorods using nonlocal elasticity theory

Abstract

The present research examines the propagation of weakly solitary waves in nanorods by employing nonlocal elasticity theory. Many systems in physics, engineering, and natural sciences are nonlinear and modeled with nonlinear equations. Wave propagation, as a branch of nonlinear science, is one of the most widely studied subjects in recent years. Nonlocal elasticity theory represents a technique with increasing popularity for the purpose of conducting the mechanical analysis of microelectromechanical and nanoelectromechanical systems. The nonlinear equation of motion of nanorods is derived by utilizing nonlocal elasticity theory. The reductive perturbation technique is employed for the purpose of examining the propagation of weakly nonlinear waves in the longwave approximation, and the Korteweg-de Vries equation is acquired as the governing equation. The steady-state solitary-wave solution is known to be admitted by the KdV equation. To observe the nonlocal effects on the KdV equation numerically, the existence of solitary wave solution has been investigated using the physical and geometric properties of carbon nanotubes. 

Keywords

• Nanorod,nonlocal elasticity theory,Nonlinear waves,reductive perturbation technique

References

1. Eringen A.C. and Suhubi E.S., Nonlinear theory of simple micro-elastic solids-I, International Journal of Engineering Science, 2,189-203, (1964).
2. Eringen A.C., Simple microfluids, International Journal of Engineering Science, 2, 205-217, (1964).
3. Eringen A.C., Theory of micropolar elasticity in Fracture (Edited by H. Liebowitz), Vol. II Academic Press, New York, 621-729, (1968).
4. Kafadar C.B. and Eringen A.C., Micropolar media-I. The classical theory, International Journal of Engineering Science, 9, 271-305, (1971).
5. Eringen A.C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1-16, (1972).
6. Demiray H., A nonlocal continuum theory for diatomic elastic solids, International Journal of Engineering Science, 15, 623-644, (1977).
7. Eringen A.C., On differantial equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703-4710, (1983).
8. Toupin R.A., Elastic materials with coupled stresses, Archive for Rational Mechanics and Analysis, 11, 385, (1962).

9. Park S.K. and Gao X.L., Bernoulli-Euler beam model based on a modified coupled stress theory, Journal of Micromechanics and Microengineering, 16 (11), 2355-2359, (2006).
10. Ma H.M., Gao X.L. and Reddy J.N., A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids, 56(12), 3379-3391, (2008).
11. Murmu T. and Pradhan S.C., Small-scale effect on the vibration on the nonuniform nanocantiliver based on nonlocal elasticity theory, Physica E, 41, 1451-1456, (2009).
12. Senthilkumar V., Pradhan S.C. and Pratap G., Small-scale effect on buckling analysis of carbon nanotube with Timoshenko theory by using differential transform method, Advance Science Letters, 3, 1-7, (2010).
13. Rahmani O. and Pedram O., Analysis and modelling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory, International Journal of Engineering Science, 77, 55-70, (2014).
14. Eringen AC, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science, 10, 1-16, (1972).
15. Eringen A.C. and Edelen D.G.B., On nonlocal elasticity, International Journal of Engineering Science, 10, 233-248, (1972).
16. Demiray H, On the nonlocal theory of quazi-static dielectrics, International Journal of Engineering Science, 10, 285, (1972).
17. Thai H.T., A nonlocal beam theory for bending, buckling and vibration of nanobeams. International Journal of Engineering Science, 52, 56-64, (2012).
18. Aydogdu M., Axial vibration of the nonaroads with the nonlocal continuum rod model, Physica E: Low-dimensional Systems and Nanostructures, 41(5), 861-864, (2009).
19. Aydogdu M., Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal theory, Mechanics Research Communications, 43, 34-40, (2012).
20. Ansari R., Sahmani S, Bending behavior and buckling of nano beams including surface stress effects corresponding to different beam theories, International Journal of Engineering Science, 49,1244-1255, (2011).
21. Narendar S., Gopalakrishnan S, Nonlocal scale effects on ultrasonic wave characteristics of nanorods, Physica E: Low-dimensional Systems and Nanostructures, 42, 161-1604, (2010).
22. Filiz S., Aydogdu M., Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity, Computational Materials Science, 49, 616-627, (2010).
23. Narendar S., Gopalakrishnan S., Ultrasonic wave characteristics of nanorods via nonlocal strain gradient models, Journal of Applied Physics, 107(8), 084312, (2010).
24. Murmu T. and Adhikari S., Nonlocal effects in the longitudinal vibration of double-nanorod systems, Physica E, 43, 415-422, (2010).
25. Lim C.W. and Yang Y., Wave propagation in carbon nanotubes: nonlocal elasticity-induced stiffness and velocity enhancement effects, Journals of Mechanics of Materials and Structures, 5, 459-476, (2010).
26. Hu Y.G., Liew K.M., Wang Q., He X.Q., Yakobson B.I., Nonlocal shell model for elastic wave propagation single- and double-walled carbon nanotubes, Journal of the Mechanics and Physics of Solids, 56, 3475-3485, (2008).
27. Wang Q. and Varadan V.K., Wave characteristics of carbon nanotubes, International Journal of Solids and Structures, 43, 254-265, (2006).
28. Narendar S. and Gopalakrishnan S., Temperature effects on wave propagation in nanoplates. Composite Part B, 43, 1275-1281, (2012).
29. Narendar S. and Gopalakrishnan S., Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes, Computational Material Science, 47, 526-538 (2009).
30. Aydogdu M., Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, International Journal of Engineering Science, 56, 17-28, (2012).
31. Aydogdu M., Longitudinal wave propagation in multiwalled carbon nanotubes, Composite Structures, 107, 578-584, (2014).
32. Wu X.F., Dzenis Y.A., Wave propagation in nanofibers, Journal of Applied Physics, 100, 124318, (2006).
33. Challamel N., Rakotomanana L., Marrec L.L., A dispersive wave equation using nonlocal elasticity, Comptes Rendus Mecanique, 337, 591-595, (2009).
34. Narendar S. and Gopalakrishnan S., Nonlocal scale effects on ultrasonic wave characteristics of nanorods, Physica E: Low-dimensional Systems and Nanostructures, 42, 1601-1604, (2010).
35. Narendar S., Terahertz wave propagation in uniform nanorods: a nonlocal continuum mechanics formulation including the effect of lateral inertia, Physica E: Low-dimensional Systems and Nanostructures, 43, 1015-1020, (2011).
36. Cho H., Yu M.F., Vakakis A.F., Bergman L.A., McFarland D.M., Tunable, Broadband Nonlinear Nanomechanical Resonator, Nano Letters,. 10, 1793-1798, (2010).
37. Fu Y.M., Hong J.W., Wang X.Q., Analysis of nonlinear vibration for embedded carbon nanotubes, Journal of Sound and Vibration, 296, 746-756, (2006).
38. Yan Y., Wang W., Zhang I., Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes, Applied Mathematical Modelling, 35, 2279-2289, (2011).
39. Ansari R., Hemmatnezhad M., Rezapour J., The thermal effect on nonlinear oscillations of carbon nanotubes with arbitrary boundary conditions, Current Applied Physics, 11, 692-6, (2011).
40. Soltani P., Ganji D.D., Mehdipour I., Farshidianfar A., Nonlinear vibration and rippling instability for embedded carbon nanotubes, Journal of Mechanical Science and Technology, 26, 985-992, (2012).
41. Fang B., Zhen Y.X., Zhang C.P., Tang Y., Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory, Applied Mathematical Modelling, 37, 1096-1107, (2013).
42. Jeffrey A., Kawahara T., Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston (1982).
43. Demiray H., Propagation of weakly nonlinear waves in fluid-filled thin elastic tubes, Applied Mathematics and Computation, 133(1), 29-41, (2002).
44. Demiray H., Solitary waves in fluid filled elastic tubes: Weakly dispersive case, International Journal of Engineering Science, 39, 439-451, (2001).
45. Silling S.A., Solitary waves in a peridynamic elastic solid, Journal of the Mechanics and Physics of Solids, 96, 121-132, (2016).
46. Malvern L.E., Introduction to the Mechanics of a Continuum Medium, Prentice Hall, Englwood Cliffs, New Jersey, (1969).
47. Mousavi S.M., Fariborz SJ, Free vibration of a rod undergoing finite strain, Journal of Physics Conferance Series, 382(1), (2012).
48. Zhan-chun T., Single walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young’s moduli dependent on layer number, Physical Review B, 65, 233-237, (2002).
49. Wu X.F., Dzenis Y.A., Wave propagation in nanofibers, Journal of Applied Physics, 100, 124318, (2006).