References
1. Ansari, R., Norouzzadeh, A., Shakouri, A., Bazdid-
Vahdati, M., and Rouhi, H. Finite element analysis
of vibrating microbeams/microplates using a
three-dimensional micropolar element", Thin-Walled
Struct., 124, pp. 489-500 (2018).
2. Tadi Beni, Y., Mehralian, F., and Zeighampour, H.
The modied couple stress functionally graded cylindrical
thin shell formulation", Mech. Adv. Mat. Struct.,
23, pp. 791-801 (2016).
3. Ansari, R., Norouzzadeh, A., Gholami, R., Faghih
Shojaei, M., and Darabi, M.A. Geometrically nonlinear
free vibration and instability of
uid-conveying
nanoscale pipes including surface stress eects", Micro
uidics Nano
uidics, 20, pp. 1-14 (2016).
4. Akgoz, B. and Civalek, O. Bending analysis of embedded
carbon nanotubes resting on an elastic foundation
using strain gradient theory", Acta Astronautica, 119,
pp. 1-12 (2016).
5. Mehralian, F. and Tadi Beni, Y. Size-dependent
torsional buckling analysis of functionally graded cylindrical
shell", Compos. Part B: Eng., 94, pp. 11-25
(2016).
6. Kroner, E. Elasticity theory of materials with long
range cohesive forces", Int. J. Solids Struct., 3, pp.
731-742 (1967).
7. Krumhansl, J. Some considerations of the relation
between solid state physics and generalized continuum
mechanics", In E. Kroner (Ed.), Mechanics of generalized
Continua, IUTAM symposia, Springer Berlin
Heidelberg, pp. 298-311 (1968).
8. Kunin, I.A. The theory of elastic media with microstructure
and the theory of dislocations", In E.
Kroner (Ed.), Mechanics of generalized Continua, IUTAM
symposia, Springer Berlin, Heidelberg, pp. 321-
329 (1968).
9. Eringen, A.C. Nonlocal polar elastic continua", Int.
J. Eng. Sci., 10, pp. 1-16 (1972).
10. Eringen, A.C. and Edelen, D.G.B. On nonlocal elasticity",
Int. J. Eng. Sci., 10, pp. 233-248 (1972).
11. Eringen, A.C. On dierential equations of nonlocal
elasticity and solutions of screw dislocation and surface
waves", J. Appl. Phys., 54, pp. 4703-4710 (1983).
12. Ansari, R., Shahabodini, A., and Rouhi, H. A nonlocal
plate model incorporating interatomic potentials
for vibrations of graphene with arbitrary edge conditions",
Curr. Appl. Phys., 15, pp. 1062-1069 (2015).
13. Shen, H.S., Shen, L., and Zhang, C.L. Nonlocal plate
model for nonlinear bending of single-layer graphene
sheets subjected to transverse loads in thermal environments",
Appl. Phys. A, 103, pp. 103-112 (2011).
14. Ansari, R., Faghih Shojaei, M., Shahabodini, A., and
Bazdid-Vahdati, M. Three-dimensional bending and
vibration analysis of functionally graded nanoplates
by a novel dierential quadrature-based approach",
Compos. Struct., 131, pp. 753-764 (2015).
15. Lu, P., Zhang, P.Q., Lee, H.P., Wang, C.M., and
Reddy, J.N. Non-local elastic plate theories", Proc.
Royal. Soc. A, 463 (2007).
DOI: 10.1098/rspa.2007.1903
16. Ansari, R. and Rouhi, H. Explicit analytical expressions
for the critical buckling stresses in a monolayer
graphene sheet based on nonlocal elasticity", Solid
State Commun., 152, pp. 56-59 (2012).
17. Ansari, R. and Norouzzadeh, A. Nonlocal and surface
eects on the buckling behavior of functionally graded
nanoplates: An isogeometric analysis", Physica E, 84,
pp. 84-97 (2016).
18. Nguyen, N.T., Hui, D., Lee, J., and Nguyen-Xuan,
H. An ecient computational approach for sizedependent
analysis of functionally graded nanoplates",
Comput. Meth. Appl. Mech. Eng., 297, pp. 191-218
(2015).
19. Norouzzadeh, A. and Ansari, R. Isogeometric vibration
analysis of functionally graded nanoplates with
the consideration of nonlocal and surface eects",
Thin-Walled Struct. (In Press).
20. Wang, C.M., Zhang, H., Challamel, N., and Duan,
W.H. On boundary conditions for buckling and vibration
of nonlocal beams", Eur. J. Mech. - A/Solids,
61, pp. 73-81 (2017).
21. Ansari, R., Gholami, R., and Rouhi, H. Sizedependent
nonlinear forced vibration analysis of magneto
-electro-thermo-elastic timoshenko nano- beams
based upon the nonlocal elasticity theory", Compos.
Struct., 126, pp. 216-226 (2015).
22. Yan, J.W., Tong, L.H., Li, C., Zhu, Y., and Wang,
Z.W. Exact solutions of bending de
ections for nanobeams
and nano-plates based on nonlocal elasticity
theory", Compos. Struct., 125, pp. 304-313 (2015).
23. Ansari, R., Ramezannezhad, H., and Gholami, R.
Nonlocal beam theory for nonlinear vibrations of
embedded multiwalled carbon nanotubes in thermal
environment", Nonlinear Dynam., 67, pp. 2241-2254
(2012).
24. Ansari, R., Gholami, R., Sahmani, S., Norouzzadeh,
A., and Bazdid-Vahdati, M. Dynamic stability analysis
of embedded multi-walled carbon nanotubes in
thermal environment", Acta Mechanica Solida Sinica,
28, pp. 659-667 (2015).
25. Sedighi, H.M., Keivani, M., and Abadyan, M. Modi
ed continuum model for stability analysis of asymmetric
FGM double-sided NEMS: corrections due to
nite conductivity, surface energy and nonlocal eect",
Compos. Part B: Eng., 83, pp. 117-133 (2015).
A. Norouzzadeh et al./Scientia Iranica, Transactions F: Nanotechnology 25 (2018) 1864{1878 1877
26. Civalek, O. and Demir, C. A simple mathematical
model of microtubules surrounded by an elastic matrix
by nonlocal nite element method", Appl. Math.
Comput., 289, pp. 335-352 (2016).
27. Rouhi, H. and Ansari, R. Nonlocal analytical Flugge
shell model for axial buckling of double-walled carbon
nanotubes with dierent end conditions", NANO, 7,
1250018 (2012).
28. Demir, C. and Civalek, O. Torsional and longitudinal
frequency and wave response of microtubules based on
the nonlocal continuum and nonlocal discrete models",
Appl. Math. Model., 37, pp. 9355-9367 (2013).
29. Ansari, R., Rouhi, H., and Mirnezhad, M. Stability
analysis of boron nitride nanotubes via a combined
continuum-atomistic model", Scientia Iranica, 20, pp.
2314-2322 (2013).
30. Natsuki, T., Matsuyama, N., and Ni, Q.Q. Vibration
analysis of carbon nanotube-based resonator using
nonlocal elasticity theory", Appl. Phys. A, 120, pp.
1309-1313 (2015).
31. Ansari, R., Shahabodini, A., and Rouhi, H. A
thickness-independent nonlocal shell model for describing
the stability behavior of carbon nanotubes
under compression", Compos. Struct., 100, pp. 323-
331 (2013).
32. Arani, A.G. and Hashemian, M. Surface stress eects
on dynamic stability of double-walled boron nitride
nanotubes conveying viscose
uid based on nonlocal
shell theory", Scientia Iranica, 20, p. 2356 (2013).
33. Polizzotto, C. Nonlocal elasticity and related variational
principles", Int. J. Solids Struct., 38, pp. 7359-
7380 (2001).
34. Challamel, N. and Wang, C. The small length scale
eect for a non-local cantilever beam: a paradox
solved", Nanotechnology, 19, p. 345703 (2008).
35. Challamel, N., Rakotomanana, L., and Le Marrec, L.
A dispersive wave equation using nonlocal elasticity",
Comptes Rendus Mocanique, 337, pp. 591-595 (2009).
36. Khodabakhshi, P. and Reddy, J.N. A unied integrodi
erential nonlocal model", Int. J. Eng. Sci., 95, pp.
60-75 (2015).
37. Fernandez-Saez, J., Zaera, R., Loya, J., and Reddy,
J.N. Bending of Euler-Bernoulli beams using Eringen's
integral formulation: A paradox resolved", Int.
J. Eng. Sci., 99, pp. 107-116 (2016).
38. Norouzzadeh, A. and Ansari, R. Finite element analysis
of nano-scale Timoshenko beams using the integral
model of nonlocal elasticity", Physica E, 88, pp. 194-
200 (2017).
39. Norouzzadeh, A., Ansari, R., and Rouhi, H. Prebuckling
responses of Timoshenko nanobeams based
on the integral and dierential models of nonlocal
elasticity: An isogeometric approach", Appl. Phys. A,
123, p. 330 (2017).
40. Mindlin, R.D. Micro-structure in linear elasticity",
Arch. Ration. Mech. Anal., 6, pp. 51-78 (1964).
41. Mindlin, R.D. Second gradient of strain and surface
tension in linear elasticity", Int. J. Solids Struct., 1,
pp. 417-438 (1965).
42. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., and
Tong, P. Experiments and theory in strain gradient
elasticity", J. Mech. Phys. Solids, 51, pp. 1477-1508
(2003).
43. Aifantis, E.C. On the role of gradients in the localization
of deformation and fracture", Int. J. Eng. Sci.,
30, pp. 1279-1299 (1992).
44. Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong,
P. Couple stress based strain gradient theory for
elasticity", Int. J. Solids Struct., 39, pp. 2731-2743
(2002).
45. Akgoz, B. and Civalek, O. Buckling analysis of linearly
tapered micro-columns based on strain gradient
elasticity", Struct. Eng. Mech., 48, pp. 195-205 (2013).
46. Nateghi, A. and Salamat-talab, M. Thermal eect
on size dependent behavior of functionally graded
microbeams based on modied couple stress theory",
Compos. Struct., 96, pp. 97-110 (2013).
47. Chen, W.J. and Li, X.P. Size-dependent free vibration
analysis of composite laminated Timoshenko beam
based on new modied couple stress theory", Arch.
Appl. Mech., 83, pp. 431-444 (2013).
48. Ansari, R., Norouzzadeh, A., Gholami, R., Faghih
Shojaei, M., and Hosseinzadeh, M. Size-dependent
nonlinear vibration and instability of embedded
uidconveying
SWBNNTs in thermal environment", Physica
E, 61, pp. 148-157 (2014).
49. Akgoz, B. and Civalek, O. Bending analysis of FG
microbeams resting on Winkler elastic foundation via
strain gradient elasticity", Compo. Struct., 134, pp.
294-301 (2015).
50. Zeighampour, H., Tadi Beni, Y., and Mehralian, F.
A shear deformable conical shell formulation in the
framework of couple stress theory", Acta Mechanica,
226, pp. 2607-2629 (2015).
51. Sadeghi, H., Baghani, M., and Naghdabadi, R. Strain
gradient thermoelasticity of functionally graded cylinders",
Scientia Iranica, 21, pp. 1415-1423 (2014).
52. Zeighampour, H. and Tadi Beni, Y. A shear deformable
cylindrical shell model based on couple stress
theory", Arch. Appl. Mech., 85, pp. 539-553 (2015).
53. Ansari, R., Gholami, R., Norouzzadeh, A., and Sahmani,
S. Size-dependent vibration and instability of
uid-conveying functionally graded microshells based
on the modied couple stress theory", Micro
uidics
Nano
uidics, 19, pp. 509-522 (2015).
54. Tadi Beni, Y. Size-dependent electromechanical
bending, buckling, and free vibration analysis of functionally
graded piezoelectric nanobeams", J. Intel.
Mat. Syst. Struct., 27, pp. 2199-2215 (2016).
55. Kanani, A., Koochi, A., Farahani, M., Rouhic, E.,
and Abadyan, M. Modeling the size dependent pullin
instability of cantilever nano-switch immersed in
1878 A. Norouzzadeh et al./Scientia Iranica, Transactions F: Nanotechnology 25 (2018) 1864{1878
ionic liquid electrolytes using strain gradient theory",
Scientia Iranica, 23, pp. 976-989 (2016).
56. Tadi Beni, Y. Size-dependent analysis of piezoelectric
nanobeams including electro-mechanical coupling",
Mech. Res. Commun., 75, pp. 67-80 (2016).
57. Ansari, R., Gholami, R., and Norouzzadeh, A. Sizedependent
thermo-mechanical vibration and instability
of conveying
uid functionally graded nanoshells
based on Mindlin's strain gradient theory", Thin-
Walled Struct., 105, pp. 172-184 (2016).
58. Hadjesfandiari, A.R. and Dargush, G.F. Couple stress
theory for solids", Int. J. Solids Struct., 48, pp. 2496-
2510 (2011).
59. Narendar, S. and Gopalakrishnan, S. Ultrasonic wave
characteristics of nanorods via nonlocal strain gradient
models", J. Appl. Phys., 107, 084312 (2010).
60. Polizzotto, C. A unifying variational framework for
stress gradient and strain gradient elasticity theories",
Eur. J. Mech. A/Solids, 49, pp. 430-440 (2015).
61. Li, L., Hu, Y., and Ling, L. Flexural wave propagation
in small-scaled functionally graded beams via a nonlocal
strain gradient theory", Compos. Struct., 133, pp.
1079-1092 (2015).
62. Farajpour, M.R., Rastgoo, A., Farajpour, A., and
Mohammadi, M. Vibration of piezoelectric nanolmbased
electromechanical sensors via higher-order nonlocal
strain gradient theory", IET Micro & Nano Lett.,
11, pp. 302-307 (2016).
63. Ebrahimi, F. and Barati, M.R. Nonlocal strain gradient
theory for damping vibration analysis of viscoelastic
inhomogeneous nano-scale beams embedded
in visco-Pasternak foundation", J. Vib. Control, 24,
pp. 2080-2095 (2018).
64. Mehralian, F., Tadi Beni, Y., and Karimi Zeverdejani,
M. Nonlocal strain gradient theory calibration using
molecular dynamics simulation based on small scale
vibration of nanotubes", Physica B, 514, pp. 61-69
(2017).
65. Ebrahimi, F. and Barati, M.R. Vibration analysis
of viscoelastic inhomogeneous nanobeams resting on
a viscoelastic foundation based on nonlocal strain
gradient theory incorporating surface and thermal
eects", Acta Mechanica, 228, pp. 1197-1210 (2017).
66. Hughes, T.J., Cottrell, J.A., and Bazilevs, Y. Isogeometric
analysis: CAD, nite elements, NURBS, exact
geometry and mesh renement", Comput. Meth. Appl.
Mech. Eng., 194, pp. 4135-4195 (2005).
67. Cottrell, J.A., Hughes, T.J., and Bazilevs, Y., Isogeometric
Analysis: Toward Integration of CAD and FEA,
John Wiley & Sons (2009).
68. Park, S. and Gao, X. Bernoulli-Euler beam model
based on a modied couple stress theory, journal
of micromechanics and microengineering", Journal of
Micromechanics and Microengineering, 16, pp. 2355-
2359 (2006).
69. Ansari, R., Faghih Shojaei, M., and Rouhi, H., Smallscale
Timoshenko beam element", Euro. J. Mech.
A/Solids, 53, pp. 19-33 (2015).
70. Zhang, Y., Wang, C., and Tan, V. Assessment of
Timoshenko beam models for vibrational behavior
of single-walled carbon nanotubes using molecular
dynamics", Adv. Appl. Math. Mech., 1, pp. 89-106
(2009).
71. Challamel, N., Zhang, Z., Wang, C.M., Reddy, J.N.,
Wang, Q., Michelitsch, T., and Collet, B. On nonconservativeness
of Eringen's nonlocal elasticity in
beam mechanics: correction from a discrete-based approach",
Arc. Appl. Mech., 84, pp. 1275-1292 (2014).