286
7 Mathematical Models of Functionally Graded Beams in Temperature Field
5. Sankar, B.V.: An elasticity solution for functionally graded beams. Compos. Sci. Technol. 61,
689–696 (2001)
6. Fu, Y.Q., Du, H.J., Huang, W.M., Zhang, S., Hu, M.: TiNi-based thin films in MEMS applications: a review. Sens. Actuators A 112(2–3), 395–408 (2004)
7. Fu, Y.Q., Du, H.J., Zhang, S.: Functionally graded TiN/TiNi shape memory alloy films. Mater.
Lett. 57(20), 2995–2999 (2003)
8. Lee, Z., Ophus, C., Fischer, L.M., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S.: Metallic
NEMS components fabricated from nanocomposite Al-Mo films. Nanotechnology 17(12),
3063–3070 (2006)
9. Witvrouw, A., Mehta, A.: The use of functionally graded poly-SiGe layers for MEMS applications. Funct. Graded Mater. VIII 492–493, 255–260 (2005)
10. Fu, Y., Zhang, J.: Electromechanical dynamic buckling phenomenon in symmetric electric
fields actuated microbeams considering material damping. Acta Mech. 212, 29–42 (2010)
11. Moghimi, Z.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated
microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1046 (2010)
12. Jia, X.L., Yang, J., Kitipornchai, S.: Pull-in instability of geometrically nonlinear microswitches under electrostatic and Casimir forces. Acta Mech. 218, 161–174 (2011)
13. Chong, A.C.M., Lam, D.C.C.: Strain gradient plasticity effect in indentation hardness of
polymers. Mater. Res. 14(10), 4103–4110 (1999)
14. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
15. McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance
to microcantilever sensors. Micromech. Microeng. 15(5), 1060–1067 (2005)
16. Stolken, J.S., Evans, A.G.: Microbend test method for measuring the plasticity length scale.
Acta Mater. 46(14), 5109–5115 (1998)
17. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli–
Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)
18. Scheible, D.V., Erbe, A., Blick, R.H.: Evidence of a nanomechanical resonator being driven
into chaotic response via the Ruelle-Takens route. Appl. Phys. Lett. 81, 1884–1886 (2002)
19. Sanchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A., Ordejo’n, P.: Ab initio structural,
elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B 59, 12678 (1999)
20. Krishnan, A., Dujardin, E., Ebbesen, T., Yianilos, P., Treacy, M.: Young’s modulus of singlewalled nanotubes. Phys. Rev. B 58, 14013 (1998)
21. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of
carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)
22. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
23. Ansari, R., Shahabodini, A., Rouhi, H.: Prediction of the biaxial buckling and vibration
behavior of graphene via a nonlocal atomistic-based plate theory. Compos. Struct. 95, 88–94
(2013)
24. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling
and vibration. Phys. E 41, 1651–1655 (2009)
25. Lee, H.L., Chang, W.J.: Free transverse vibration of the fluid-conveying single-walled carbon
nanotube using nonlocal elastic theory. J. Appl. Phys. 103, 024302 (2008)
26. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to
nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
27. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci.
45, 288–307 (2007)
28. Thai, H.T., Vo, T.P.: A nonlocal sinusoidal shear deformation beam theory with application
to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54, 58–66 (2012)
29. Adali, S.: Variational principles for multi-walled carbon nanotubes undergoing buckling based
on nonlocal elasticity theory. Phys. Lett. A 372, 5701–5705 (2008)
30. Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P.: Exact nonlocal solution for postbuckling
of single-walled carbon nanotubes. Phys. E 43, 1730–1737 (2011)
7 Mathematical Models of Functionally Graded Beams in Temperature Field
5. Sankar, B.V.: An elasticity solution for functionally graded beams. Compos. Sci. Technol. 61,
689–696 (2001)
6. Fu, Y.Q., Du, H.J., Huang, W.M., Zhang, S., Hu, M.: TiNi-based thin films in MEMS applications: a review. Sens. Actuators A 112(2–3), 395–408 (2004)
7. Fu, Y.Q., Du, H.J., Zhang, S.: Functionally graded TiN/TiNi shape memory alloy films. Mater.
Lett. 57(20), 2995–2999 (2003)
8. Lee, Z., Ophus, C., Fischer, L.M., Nelson-Fitzpatrick, N., Westra, K.L., Evoy, S.: Metallic
NEMS components fabricated from nanocomposite Al-Mo films. Nanotechnology 17(12),
3063–3070 (2006)
9. Witvrouw, A., Mehta, A.: The use of functionally graded poly-SiGe layers for MEMS applications. Funct. Graded Mater. VIII 492–493, 255–260 (2005)
10. Fu, Y., Zhang, J.: Electromechanical dynamic buckling phenomenon in symmetric electric
fields actuated microbeams considering material damping. Acta Mech. 212, 29–42 (2010)
11. Moghimi, Z.M., Ahmadian, M.T.: Static pull-in analysis of electrostatically actuated
microbeams using homotopy perturbation method. Appl. Math. Model. 34, 1032–1046 (2010)
12. Jia, X.L., Yang, J., Kitipornchai, S.: Pull-in instability of geometrically nonlinear microswitches under electrostatic and Casimir forces. Acta Mech. 218, 161–174 (2011)
13. Chong, A.C.M., Lam, D.C.C.: Strain gradient plasticity effect in indentation hardness of
polymers. Mater. Res. 14(10), 4103–4110 (1999)
14. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory
and experiments. Acta Metall. Mater. 42, 475–487 (1994)
15. McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance
to microcantilever sensors. Micromech. Microeng. 15(5), 1060–1067 (2005)
16. Stolken, J.S., Evans, A.G.: Microbend test method for measuring the plasticity length scale.
Acta Mater. 46(14), 5109–5115 (1998)
17. Kong, S., Zhou, S., Nie, Z., Wang, K.: The size-dependent natural frequency of Bernoulli–
Euler micro-beams. Int. J. Eng. Sci. 46, 427–437 (2008)
18. Scheible, D.V., Erbe, A., Blick, R.H.: Evidence of a nanomechanical resonator being driven
into chaotic response via the Ruelle-Takens route. Appl. Phys. Lett. 81, 1884–1886 (2002)
19. Sanchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A., Ordejo’n, P.: Ab initio structural,
elastic, and vibrational properties of carbon nanotubes. Phys. Rev. B 59, 12678 (1999)
20. Krishnan, A., Dujardin, E., Ebbesen, T., Yianilos, P., Treacy, M.: Young’s modulus of singlewalled nanotubes. Phys. Rev. B 58, 14013 (1998)
21. Arash, B., Wang, Q.: A review on the application of nonlocal elastic models in modeling of
carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012)
22. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
23. Ansari, R., Shahabodini, A., Rouhi, H.: Prediction of the biaxial buckling and vibration
behavior of graphene via a nonlocal atomistic-based plate theory. Compos. Struct. 95, 88–94
(2013)
24. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling
and vibration. Phys. E 41, 1651–1655 (2009)
25. Lee, H.L., Chang, W.J.: Free transverse vibration of the fluid-conveying single-walled carbon
nanotube using nonlocal elastic theory. J. Appl. Phys. 103, 024302 (2008)
26. Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to
nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
27. Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci.
45, 288–307 (2007)
28. Thai, H.T., Vo, T.P.: A nonlocal sinusoidal shear deformation beam theory with application
to bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 54, 58–66 (2012)
29. Adali, S.: Variational principles for multi-walled carbon nanotubes undergoing buckling based
on nonlocal elasticity theory. Phys. Lett. A 372, 5701–5705 (2008)
30. Setoodeh, A.R., Khosrownejad, M., Malekzadeh, P.: Exact nonlocal solution for postbuckling
of single-walled carbon nanotubes. Phys. E 43, 1730–1737 (2011)