© The Institution of Engineering and Technology
The buckling analysis of a cantilever single-walled carbon nanotube embedded in an elastic medium with an attached spring is researched. The effects of axial compression load, attached spring, small size and surrounding elastic medium are taken into account at the same time. Theoretical formulation is carried out on the basis of the Bernoulli–Euler beam theory in conjunction with Eringen's non-local elasticity theory. Fourier sine series is selected for the simulation of single-walled carbon nanotube deflections. Winkler elastic foundation type is used to simulate the interaction between single-walled carbon nanotube and elastic medium. 2 × 2 coefficient matrix is derived with the aid of applying Stokes’ transformation to corresponding non-local boundary conditions. The critical buckling loads are calculated by using this coefficient matrix. Different validation studies are performed to endorse and corroborate the usefulness of the presented analytical method.
References
-
-
1)
-
35. Pradhan, S.C., Reddy, G.K.: ‘Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM’, Comput. Mater. Sci., 2011, 50, pp. 1052–1056 (doi: 10.1016/j.commatsci.2010.11.001).
-
2)
-
19. Kiani, K.: ‘A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect’, Int. J. Mech. Sci., 2010, 52, pp. 1343–1356 (doi: 10.1016/j.ijmecsci.2010.06.010).
-
3)
-
9. Fleck, N.A., Hutchinson, J.W.: ‘A reformulation of strain gradient plasticity’, J. Mech. Phys. Solids, 2001, 49, pp. 2245–2271 (doi: 10.1016/S0022-5096(01)00049-7).
-
4)
-
1. Poole, W.J., Ashby, M.F., Fleck, N.A.: ‘Micro-hardness of annealed and work-hardened copper polycrystals’, Scr. Mater., 1996, 34, pp. 559–564 (doi: 10.1016/1359-6462(95)00524-2).
-
5)
-
8. Fleck, N.A., Hutchinson, J.W.: ‘A phenomenological theory for strain gradient effects in plasticity’, J. Mech. Phys. Solids, 1993, 41, pp. 1825–1857 (doi: 10.1016/0022-5096(93)90072-N).
-
6)
-
10. Eringen, A.C.: ‘On differential equations of nonlocal elasticity and solutions of screw dislocation and surface-waves’, J. Appl. Phys., 1983, 54, pp. 4703–4710 (doi: 10.1063/1.332803).
-
7)
-
33. Yayli, M.O.: ‘A compact analytical method for vibration analysis of single-walled carbon nanotubes with restrained boundary conditions’, J. Vib. Control, 2016, 22, pp. 2542–2555 (doi: 10.1177/1077546314549203).
-
8)
-
16. Akgoz, B., Civalek, O.: ‘Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix’, Comput. Mater. Sci., 2013, 77, pp. 295–303 (doi: 10.1016/j.commatsci.2013.04.055).
-
9)
-
7. Toupin, R.A.: ‘Theory of elasticity with couple stresses’, Arch. Ration. Mech. Anal., 1964, 17, pp. 85–112 (doi: 10.1007/BF00253050).
-
10)
-
17. Demir, Ç., Civalek, O.: ‘Nonlocal finite element formulation for vibration’, Int. J. Eng. Appl. Sci. (IJEAS), 2016, 8, pp. 109–117.
-
11)
-
20. Narendar, S., Gopalakrishnan, S.: ‘Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics’, Phys. E, Low-Dimens. Syst. Nanostruct., 2011, 43, pp. 1185–1191 (doi: 10.1016/j.physe.2011.01.026).
-
12)
-
2. Lam, D.C.C., Yang, F., Chong, A.C.M., et al: ‘Experiments and theory in strain gradient elasticity’, J. Mech. Phys. Solids, 2003, 51, pp. 1477–1508 (doi: 10.1016/S0022-5096(03)00053-X).
-
13)
-
13. Peddieson, J., Buchanan, G.R., McNitt, R.P.: ‘Application of nonlocal continuum models to nanotechnology’, Int. J. Eng. Sci., 2003, 41, pp. 305–312 (doi: 10.1016/S0020-7225(02)00210-0).
-
14)
-
15. Wong, E.W., Sheehan, P.E., Lieber, C.M.: ‘Nanobeam mechanics: elasticity, strength and toughness of nanorods and nanotubes’, Science, 1997, 277, pp. 1971–1975 (doi: 10.1126/science.277.5334.1971).
-
15)
-
4. Eringen, A.C.: ‘Theory of micropolar plates’, Z. Angew. Math. Phys., 1967, 18, pp. 12–30 (doi: 10.1007/BF01593891).
-
16)
-
12. Pradhan, S.C., Murmu, T.: ‘Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory’, Phys. Lett. A, 2010, 373, pp. 4182–4188 (doi: 10.1016/j.physleta.2009.09.021).
-
17)
-
22. Yayli, M.O.: ‘Torsion of nonlocal bars with equilateral triangle cross sections’, J. Comput. Theor. Nanosci., 2013, 10, pp. 376–379 (doi: 10.1166/jctn.2013.2707).
-
18)
-
8. Koiter, W.T.: ‘Couple stresses in the theory of elasticity I and II’, Proc. K Ned Akad Wet (B), 1964, 67, pp. 17–44.
-
19)
-
20. Ece, M.C., Aydogdu, M.: ‘Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes’, Acta Mech., 2007, 190, pp. 185–195 (doi: 10.1007/s00707-006-0417-5).
-
20)
-
1. Fang, S.C., Chang, W.J., Wang, Y.H.: ‘Computation of chirality-and size-dependent surface Young's moduli for single-walled carbon nanotubes’, Phys. Lett. A, 2007, 371, (5), pp. 499–503 (doi: 10.1016/j.physleta.2007.06.076).
-
21)
-
3. Wang, L., Ni, Q.: ‘On vibration and instability of carbon nanotubes conveying fluid’, Comput. Mater. Sci., 2008, 43, (2), pp. 399–402 (doi: 10.1016/j.commatsci.2008.01.004).
-
22)
-
19. Civalek, O., Demir, C., Akgoz, B.: ‘Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen's nonlocal elasticity theory’, Int. J. Eng. Appl. Sci. (IJEAS), 2009, 2, pp. 47–56.
-
23)
-
2. Mir, M., Hosseini, A., Majzoobi, G.H.: ‘A numerical study of vibrational properties of single-walled carbon nanotubes’, Comput. Mater. Sci., 2008, 43, (3), pp. 540–548 (doi: 10.1016/j.commatsci.2007.12.024).
-
24)
-
34. Senthilkumar, V., Pradhan, S.C., Pratap, G.: ‘Small scale effect on buckling analysis of carbon nanotube with Timoshenko theory by using differential transform method’, Adv. Sci. Lett., 2010, 3, pp. 1–7 (doi: 10.1166/asl.2010.1080).
-
25)
-
21. Civalek, O., Demir, C.: ‘A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method’, Appl. Math. Comput., 2016, 289, pp. 335–352.
-
26)
-
12. Sudak, L.J.: ‘Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics’, J. Appl. Phys., 2003, 94, pp. 7281–7287 (doi: 10.1063/1.1625437).
-
27)
-
26. Reddy, J.N., Pang, S.D.: ‘Nonlocal continuum theories of beam for the analysis of carbon nanotubes’, J. Appl. Phys., 2008, 103, pp. 1–16.
-
28)
-
18. Pradhan, S.C.: ‘Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory’, Finite Elem. Anal. Des., 2012, 50, pp. 8–20 (doi: 10.1016/j.finel.2011.08.008).
-
29)
-
5. Mindlin, R.D., Tiersten, H.F.: ‘Effects of couple-stresses in linear elasticity’, Arch. Ration. Mech. Anal., 1962, 11, pp. 415–448 (doi: 10.1007/BF00253946).
-
30)
-
3. McFarland, A.W., Colton, J.S.: ‘Role of material microstructure in plate stiffness with relevance to microcantilever sensors.’, J. Micromech. Microeng., 2005, 15, pp. 1060–1067 (doi: 10.1088/0960-1317/15/5/024).
-
31)
-
24. Eringen, A.C., Edelen, D.G.B.: ‘On nonlocal elasticity’, Int. J. Eng. Sci., 1972, 10, pp. 233–248 (doi: 10.1016/0020-7225(72)90039-0).
-
32)
-
35. Bachtold, A., Hadley, P., Nakanishi, T., Dekker, C.: ‘Logic circuits with carbon nanotube transistors’, Science, 2001, 294, (5545), pp. 1317–1320 (doi: 10.1126/science.1065824).
-
33)
-
9. Arash, B., Wang, Q.: ‘A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes’, Comput. Mater. Sci., 2012, 51, pp. 303–313 (doi: 10.1016/j.commatsci.2011.07.040).
-
34)
-
15. Murmu, T., Pradhan, S.C.: ‘Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory’, Comput. Mater. Sci., 2009, 46, pp. 854–859 (doi: 10.1016/j.commatsci.2009.04.019).
-
35)
-
16. Yayli, M.O.: ‘On the axial vibration of carbon nanotubes with different boundary conditions’, Micro Nano Lett., 2014, 9, (11), pp. 807–811 (doi: 10.1049/mnl.2014.0398).
http://iet.metastore.ingenta.com/content/journals/10.1049/mnl.2016.0662
Related content
content/journals/10.1049/mnl.2016.0662
pub_keyword,iet_inspecKeyword,pub_concept
6
6