References
1. Witvrouw, A., and Mehta, A. The use of functionally graded poly-SiGe layers for MEMS applications", In Materials Science Forum. Trans. Tech. Publ., 492, pp. 255-260 (2005).
2. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., and Tong, P. Experiments and theory in strain gradient elasticity", Journal of the Mechanics and Physics of Solids, 51, pp. 1477-1508 (2003).
3. Malekzadeh, P. and Shojaee, M. A two-variable rstorder
shear deformation theory coupled with surface
and nonlocal e ects for free vibration of nanoplates",
Journal of Vibration and Control, 21, pp. 2755-2772
(2015).
4. Setoodeh, A.R., Rezaei, M., and Zendehdel Shahri,
M.R. Linear and nonlinear torsional free vibration
of functionally graded micro/nano-tubes based on
modi ed couple stress theory", Applied Mathematics
and Mechanics (English Edition), 37, pp. 1-16 (2016).
5. Peng, X.W., Guo, X.M., Liu, L., and Wu, B.J. Scale
e ects on nonlocal buckling analysis of bilayer composite
plates under non-uniform uniaxial loads", Applied
Mathematics and Mechanics (English Edition), 36, pp.
1-10 (2015).
6. Akgoz, B. and Civalek, O. A size-dependent shear
deformation beam model based on the strain gradient
elasticity theory", International Journal of Engineering
Science, 70, pp. 1-14 (2013).
7. Ma, H.M., Gao, X.L., and Reddy, J.N. A non-classical
Mindlin plate model based on a modi ed couple stress
theory", Acta Mechanica, 220, pp. 217-235 (2011).
8. Gholami, R. and Ansari, R. A most general strain
gradient plate formulation for size-dependent geometrically
nonlinear free vibration analysis of functionally
graded shear deformable rectangular microplates",
Nonlinear Dynamics, 84, pp. 2403-2422 (2016).
9. Akgoz, B. and Civalek, O. A new trigonometric beam
model for buckling of strain gradient microbeams",
International Journal of Mechanical Sciences, 81, pp.
88-94 (2014).
10. Akgoz, B. and Civalek, O. Bending analysis of FG
microbeams resting on Winkler elastic foundation via
strain gradient elasticity", Composite Structures, 134,
pp. 294-301 (2015).
11. Akgoz, B. and Civalek, O. microstructure-dependent
sinusoidal plate model based on the strain gradient
elasticity theory", Acta Mechanica, 226, pp. 2277-2294
(2015).
12. Akgoz, B. and Civalek, O. A novel microstructuredependent
shear deformable beam model", International
Journal of Mechanical Sciences, 99, pp. 10-20
(2015).
13. Li, A., Zhou, S., Zhou, S., and Wang, B.A. Sizedependent
model for bi-layered Kirchho micro-plate
based on strain gradient elasticity theory", Composite
Structures, 113, pp. 272-280 (2014).
14. Toupin, R.A. Elastic materials with couple-stresses",
Archive for Rational Mechanics and Analysis, 11, pp.
385-414 (1962).
15. Mindlin, R.D. and Tiersten, H.F. E ects of couplestresses
in linear elasticity", Archive for Rational
Mechanics and Analysis, 11, pp. 415-448 (1962).
16. Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong, P.
Couple stress based strain gradient theory for elasticity",
International Journal of Solids and Structures,
39, pp. 2731-2743 (2002).
17. Jomehzadeh, E., Noori, H.R., and Saidi, A.R. The
size-dependent vibration analysis of micro-plates based
on a modi ed couple stress theory", Physica E: Low-
Dimensional Systems and Nanostructures, 43, pp. 877-
883 (2011).
18. Askari, A.R. and Tahani, M. Analytical determination
of size-dependent natural frequencies of fully
clamped rectangular microplates based on the modi ed
couple stress theory", Journal of Mechanical Science
and Technology, 29, pp. 2135-2145 (2015).
19. Yin, L., Qian, Q., Wang, L., and Xia, W. Vibration
analysis of microscale plates based on modi ed couple
stress theory", Acta Mechanica Solida Sinica, 23, pp.
386-393 (2010).
20. Asghari, M. Geometrically nonlinear micro-plate formulation
based on the modi ed couple stress theory",
International Journal of Engineering Science, 51, pp.
292-309 (2012).
21. Gao, X.L, Huang, J.X, and Reddy, J.N. A nonclassical
third-order shear deformation plate model
based on a modi ed couple stress theory", Acta Mechanica,
224, pp. 2699-2718 (2013).
22. Tsiatas, G.C. A new Kirchho plate model based on
a modi ed couple stress theory", International Journal
of Solids and Structures, 46, pp. 2757-2764 (2009).
23. Shenas, A.G., Malekzadeh, P., and Mohebpour, S.
Vibrational behavior of variable section functionally
graded microbeams carrying microparticles in thermal
environment", Thin-Walled Structures, 108, pp. 122-
137 (2016).
24. Shenas, A.G., Ziaee, S., and Malekzadeh, P. Vibrational
behavior of rotating pre-twisted functionally
graded microbeams in thermal environment", Composite
Structures, 157, pp. 222-235 (2016).
25. Setoodeh, A.R. and Rezaei, M. Large amplitude free
vibration analysis of functionally graded nano/micro
beams on nonlinear elastic foundation", Structural
Engineering and Mechanics, 61, pp. 209-220 (2017).
A.R. Setoodeh and M. Rezaei/Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 799{812 811
26. Ke, L.L. and Wang, Y.S. Size e ect on dynamic
stability of functionally graded microbeams based on a
modi ed couple stress theory", Composite Structures,
93, pp. 342-350 (2011).
27. Setoodeh, A.R. and Afrahim, S. Nonlinear dynamic
analysis of FG micro-pipes conveying
uid based on
strain gradient theory", Composite Structures, 116,
pp. 128-135 (2014).
28. Thai, H.T. and Kim, S.E. A size-dependent functionally
graded Reddy plate model based on a modi ed
couple stress theory", Composites Part B: Engineering,
45, pp. 1636-1645 (2013).
29. Thai, H.T. and Choi, D.H. Size-dependent functionally
graded Kirchho and Mindlin plate models
based on a modi ed couple stress theory", Composite
Structures, 95, pp. 142-153 (2012).
30. Thai, H.T. and Vo, T.P. A size-dependent functionally
graded sinusoidal plate model based on a modi ed
couple stress theory", Composite Structures, 96, pp.
376-383 (2013).
31. Ke, L.L., Yang, J., Kitipornchai, S., Bradford, M.A.,
and Wang, Y.S. Axisymmetric nonlinear free vibration
of size-dependent functionally graded annular
microplates", Composites Part B: Engineering, 53, pp.
207-217 (2013).
32. Ansari, R., Shojaei, M.F., Mohammadi, V., Gholami,
R., and Darabi, M.A. Nonlinear vibrations of
functionally graded Mindlin microplates based on the
modi ed couple stress theory", Composite Structures,
114, pp. 124-134 (2014).
33. Lou, J. and He, L. Closed-form solutions for nonlinear
bending and free vibration of functionally graded microplates
based on the modi ed couple stress theory",
Composite Structures, 131, pp. 810-820 (2015).
34. Mohammadimehr, M. and Mohandes, M. The e ect
of modi ed couple stress theory on buckling and vibration
analysis of functionally graded double-layer Boron
Nitride piezoelectric plate based on CPT", Journal of
Solid Mechanics, 7, pp. 281-298 (2015).
35. He, L., Lou, J., Zhang, E., Wang, Y., and Bai,
Y.A. Size-dependent four variable re ned plate model
for functionally graded microplates based on modi ed
couple stress theory", Composite Structures, 130, pp.
107-115 (2015).
36. Shenas, A.G. and Malekzadeh, P. Free vibration
of functionally graded quadrilateral microplates in
thermal environment", Thin-Walled Structures, 106,
pp. 294-315 (2016).
37. Roozbahani, M.M., Heydarzadeh Arani, N., Moghimi
Zand, M., and Mousavi Mashhadi, M. Analytical
solutions to nonlinear oscillations of a microbeam using
higher order beam theory", Scientia Iranica, 23(5), pp.
2179-2193 (2016).
38. Sedighi, H.M., Shirazi, K.H., and Zare, J. An analytic
solution of transversal oscillation of quintic non-linear
beam with homotopy analysis method", International
Journal of Non-Linear Mechanics, 47, pp. 777-784
(2012).
39. Alipour, A., Zand, M.M., and Daneshpajooh, H.
Analytical solution to nonlinear behavior of electrostatically
actuated nanobeams incorporating van der
Waals and Casimir forces", Scientia Iranica, 22(3), pp.
1322-1329 (2015).
40. Liao, S.J. Advances in the homotopy analysis
method", World Scienti c, New York (2013).
41. Rao, G.V., Raju, I.S., and Raju, K.K. A nite
element formulation for large amplitude
exural vibrations
of thin rectangular plates", Computers &
Structures, 6, pp. 163-167 (1976).
42. Ventsel, E. and Krauthammer, T. Thin plates and
shells: theory: analysis, and applications", CRC Press,
New York (2001).
43. Mindlin, R.D. In
uence of rotary inertia and shear on
exural motions of isotropic elastic plates", Journal of
Applied Mechanics, 18(1), pp. 31-38 (1951).
44. Christoforou, A.P. and Swanson, S.R. Analysis of
simply-supported orthotropic cylindrical shells subject
to lateral impact loads", Journal of Applied Mechanics,
57(2), pp. 376-382 (1990).
45. Rao, S.S., Vibration of Continuous Systems, John
Wiley & Sons, New Jersey (2007).
46. Liao, S.J., Beyond Perturbation: Introduction to Homotopy
Analysis Method, CRC press, Boca Raton
(2004).
47. Han, W. and Petyt, M. Geometrically nonlinear
vibration analysis of thin, rectangular plates using
the hierarchical nite element method 2014 I: the
fundamental mode of isotropic plates", Computers &
Structures, 63, pp. 295-308 (1997).
48. Sundararajan, N., Prakash, T., and Ganapathi, M.
Nonlinear free
exural vibrations of functionally
graded rectangular and skew plates under thermal environments",
Finite Elements in Analysis and Design,
42, pp. 152-168 (2005).
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