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287
31. Shen, H.S.: Buckling and postbuckling of radially loaded microtubules by nonlocal shear
deformable shell model. Theor. Biol. 264, 386–394 (2010)
32. Yang, Y., Lim, C.W.: A variational principle approach for buckling of carbon nanotubes based
on nonlocal Timoshenko beam models. Nano 6, 363–377 (2011)
33. Fu, Y., Du, H., Huang, W.M., Zhang, S.: A theoretical model for functionally graded shape
memory alloy cylinders subjected to internal pressure. Mater. Lett. 57, 2995–2999 (2003)
34. Praveen, G.N., Reddy, J.N.: Nonlinear transient thermoelastic analysis of functionally graded
ceramic-metal plates. Int. J. Solids Struct. 35(33), 4457–4476 (1998)
35. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414
(1962)
36. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration.
Mech. Anal. 11, 415–448 (1962)
37. Koiter, W.T.: Couple-stresses in the theory of elasticity, I and II. Proc. K. Ned. Akad. Wet. B
67, 17–44 (1964)
38. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent
behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)
39. Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1766 (2010)
40. Lazopoulos, K.A., Lazopoulos, A.K.: Bending and buckling of thin strain gradient elastic
beams. Eur. J. Mech. A-Solids 29, 837–843 (2010)
41. Ma, H.M., Gao, X.L., Reddy, J.N.: A non-classical Mindlin plate model based on a modified
couple stress theory. Acta Mech. 220, 217–235 (2011)
42. Ma, H.M., Gao, X.L., Reddy, J.N.: A nonclassical Reddy-Levinson beam model based on a
modified couple stress theory. Int. J. Multiscale Comput. Eng. 8, 167–180 (2010)
43. Park, S.K., Gao, X.L.: Bernoulli-Euler beam model based on a modified couple stress theory.
Micromech. Microeng. 16(11), 2355–2359 (2006)
44. Yang, F., Chong, M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for
elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
45. Batra, R.C., Porfiri, M., Spinello, D.: Vibrations of narrow microbeams predeformed by an
electric field. J. Sound Vib. 309, 600–612 (2008)
46. Jia, X.L., Yang, J., Kitipornchai, S., Lim, C.W.: Pull-in instability and free vibration of electrically actuated poly-SiGe graded micro-beams with a curved ground electrode. Appl. Math.
Model. 36, 1875–1884 (2012)
47. Lu, C., Lim, C., Chen, W.: Size-dependent elastic behavior of FGM ultra-thin films based on
generalized refined theory. Int. J. Solids Struct. 46, 1176–1185 (2009)
48. Mohammadi-Alasti, B., Rezazadeh, G., Borgheei, A.M., Minaei, S., Habibifar, R.: On the
mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and
nonlinear electrostatic pressure. Compos. Struct. 93, 1516–1525 (2011)
49. Zhang, J., Fu, Y.: Vibration of size-dependent functionally graded sandwich microbeams
with different boundary conditions based on the modified couple stress theory. Meccanica 47,
1649–1658 (2012)
50. Ansari, R., Gholami, R.: Size-dependent vibration of functionally graded curved microbeams
based on the modified strain gradient elasticity theory. Arch. Appl. Mech. 83, 1439–1449
(2013)
51. Ansari, R., Gholami, R., Sahmani, S.: Study of small scale effects on the nonlinear vibration
response of functionally graded Timoshenko microbeams based on the strain gradient theory.
J. Comput. Nonlinear Dyn. 7, 031009 (2012)
52. Asgharifard, S.P., Haeri Yazdi, M.R.: Nonlinear free vibrations of functionally graded
nanobeams with surface effects. Compos. B Eng. 45, 581–586 (2013)
53. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Free vibration analysis of functionally graded
size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012)
54. Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88 (2013)