References
19
References
1. Guo, J.G., Zhao, Y.P.: The size-dependent bending elastic properties of nanobeams with
surface effects. Nanotechnology 18, 295701 (2007)
2. Lim, C.W., Wang, C.M.: Exact variational nonlocal stress modelling with asymptotic higherorder strain gradients for nanobeams. J. Appl. Phys. 101, 054312 (2007)
3. Wang, C.M., Zhang, Y.Y., Kitipornchai, S.: Vibration of initially stressed micro- and nanobeams. Int. J. Struct. Stab. Dyn. 7, 555–570 (2007)
4. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a
paradox solved. Nanotechnology 19, 345703 (2008)
5. Lim, C.W., Li, C., Yu, J.L.: The effects of stiffness strengthening nonlocal stress and axial
tension on free vibration of cantilever nanobeams. Interact. Multiscale Mech. 2(3), 223–233
(2009)
6. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling
and vibration. Physica E 41, 1651–1655 (2009)
7. Zhang, Y.Y., Wang, C.M., Challamel, N.: Bending and vibration of hybrid nonlocal beams.
J. Eng. Mech. ASCE J. Eng. Mech. 136(5), 562–574 (2010)
8. Murmu, T., Adhikari, S.: Nonlocal transverse vibration of double-nanobeam-systems. J. Appl.
Phys. 108, 083514 (2010)
9. Li, C., Lim, C.W., Yu, J.L., Zeng, Q.C.: Analytical solutions for vibration of simply supported
nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11(2), 257–271 (2011)
10. Hosseini-Hashemi, S., Fakher, M., Nazemnezhad, R.: Surface effects on free vibration analysis of nanobeams using nonlocal elasticity: a comparison between Euler-Bernoulli and Timoshenko. J. Solid Mech. 5(3), 290–304 (2013)
11. Simsek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain
gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)
12. Hashemi, S.H., Khaniki, H.B.: Vibration analysis of the Timoshenko non-uniform nanobeam
based on nonlocal theory: an analytical solution. Int. J. Nano Dimens. 8(1), 70–81 (2017)
13. Sahmani, S., Aghdam, M.M.: Small scale effects on the large amplitude nonlinear vibrations of
multilayer functionally graded composite nanobeams reinforced with graphene-nanoplatelets.
Int. J. Nanosci. Nanotechnol. 14(3), 207–227 (2018)
14. Jena, S.K., Chakraverty, S.: Free vibration analysis of Euler-Bernoulli nanobeam using differential transform method. Int. J. Comput. Mater. Sci. Eng. 7(3), 1850020 (2018)
15. Khaniki, H.B.: On vibrations of nanobeam systems. Int. J. Eng. Sci. 124, 85–103 (2018)
16. Jiang, H., Liu, R., Huang, Y., Hwang, K.C.: Thermal expansion of single wall carbon nanotubes. J. Eng. Mater. Technol. 126, 265–270 (2004)
17. Yao, X., Han, Q.: Buckling analysis of multiwalledcarbon nanotubes under torsional load
coupling with temperature change. J. Eng. Mater. Technol. 128, 419–427 (2006)
18. Lee, H.L., Chang, W.J.: A closed-form solution for critical buckling temperature of a singlewalled carbon nanotube. Physica E 41, 1492–1494 (2009)
19. Tounsi, A., Heireche, H., Beerabah, H.M., Benzair, A., Boumia, L.: Effect of small size on
wave propagation in double-walled carbon nanotubes under temperature field. J. Appl. Phys.
104, 104301 (2007)
20. Lim, C.W., Yang, Q.: Nonlocal thermal-elasticity for nanobeam deformation: exact solutions
with stiffness enhancement effects. J. Appl. Phys. 110, 013514 (2011)
21. Chen, Ch., Ma, M., Liu, J.Z., Zheng, Q., Xu, Z.: Viscous damping of nanobeam resonators:
humidity, thermal noise, and a paddling effect. J. Appl. Phys. 110, 034320 (2011)
22. Youssef, H.M., El-Bary, A.A., Elsibai, K.A.: Vibration of gold nano beam in context of twotemperature generalized thermoelasticity subjected to laser pulse. Lat. Am. J. Solids Struct.
11(13), 2460–2482 (2014)
23. Leijssen, R., Verhagen, E.: Strong optomechanical interactions in a sliced photonic crystal
nanobeam. Sci. Rep. 5, 15974 (2015)
24. Hoang, C.M.: Thermoelastic damping depending on vibration modes of nano beam resonator.
Commun. Phys. 25(4), 317–325 (2015)
19
References
1. Guo, J.G., Zhao, Y.P.: The size-dependent bending elastic properties of nanobeams with
surface effects. Nanotechnology 18, 295701 (2007)
2. Lim, C.W., Wang, C.M.: Exact variational nonlocal stress modelling with asymptotic higherorder strain gradients for nanobeams. J. Appl. Phys. 101, 054312 (2007)
3. Wang, C.M., Zhang, Y.Y., Kitipornchai, S.: Vibration of initially stressed micro- and nanobeams. Int. J. Struct. Stab. Dyn. 7, 555–570 (2007)
4. Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a
paradox solved. Nanotechnology 19, 345703 (2008)
5. Lim, C.W., Li, C., Yu, J.L.: The effects of stiffness strengthening nonlocal stress and axial
tension on free vibration of cantilever nanobeams. Interact. Multiscale Mech. 2(3), 223–233
(2009)
6. Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling
and vibration. Physica E 41, 1651–1655 (2009)
7. Zhang, Y.Y., Wang, C.M., Challamel, N.: Bending and vibration of hybrid nonlocal beams.
J. Eng. Mech. ASCE J. Eng. Mech. 136(5), 562–574 (2010)
8. Murmu, T., Adhikari, S.: Nonlocal transverse vibration of double-nanobeam-systems. J. Appl.
Phys. 108, 083514 (2010)
9. Li, C., Lim, C.W., Yu, J.L., Zeng, Q.C.: Analytical solutions for vibration of simply supported
nonlocal nanobeams with an axial force. Int. J. Struct. Stab. Dyn. 11(2), 257–271 (2011)
10. Hosseini-Hashemi, S., Fakher, M., Nazemnezhad, R.: Surface effects on free vibration analysis of nanobeams using nonlocal elasticity: a comparison between Euler-Bernoulli and Timoshenko. J. Solid Mech. 5(3), 290–304 (2013)
11. Simsek, M.: Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain
gradient theory and a novel Hamiltonian approach. Int. J. Eng. Sci. 105, 12–27 (2016)
12. Hashemi, S.H., Khaniki, H.B.: Vibration analysis of the Timoshenko non-uniform nanobeam
based on nonlocal theory: an analytical solution. Int. J. Nano Dimens. 8(1), 70–81 (2017)
13. Sahmani, S., Aghdam, M.M.: Small scale effects on the large amplitude nonlinear vibrations of
multilayer functionally graded composite nanobeams reinforced with graphene-nanoplatelets.
Int. J. Nanosci. Nanotechnol. 14(3), 207–227 (2018)
14. Jena, S.K., Chakraverty, S.: Free vibration analysis of Euler-Bernoulli nanobeam using differential transform method. Int. J. Comput. Mater. Sci. Eng. 7(3), 1850020 (2018)
15. Khaniki, H.B.: On vibrations of nanobeam systems. Int. J. Eng. Sci. 124, 85–103 (2018)
16. Jiang, H., Liu, R., Huang, Y., Hwang, K.C.: Thermal expansion of single wall carbon nanotubes. J. Eng. Mater. Technol. 126, 265–270 (2004)
17. Yao, X., Han, Q.: Buckling analysis of multiwalledcarbon nanotubes under torsional load
coupling with temperature change. J. Eng. Mater. Technol. 128, 419–427 (2006)
18. Lee, H.L., Chang, W.J.: A closed-form solution for critical buckling temperature of a singlewalled carbon nanotube. Physica E 41, 1492–1494 (2009)
19. Tounsi, A., Heireche, H., Beerabah, H.M., Benzair, A., Boumia, L.: Effect of small size on
wave propagation in double-walled carbon nanotubes under temperature field. J. Appl. Phys.
104, 104301 (2007)
20. Lim, C.W., Yang, Q.: Nonlocal thermal-elasticity for nanobeam deformation: exact solutions
with stiffness enhancement effects. J. Appl. Phys. 110, 013514 (2011)
21. Chen, Ch., Ma, M., Liu, J.Z., Zheng, Q., Xu, Z.: Viscous damping of nanobeam resonators:
humidity, thermal noise, and a paddling effect. J. Appl. Phys. 110, 034320 (2011)
22. Youssef, H.M., El-Bary, A.A., Elsibai, K.A.: Vibration of gold nano beam in context of twotemperature generalized thermoelasticity subjected to laser pulse. Lat. Am. J. Solids Struct.
11(13), 2460–2482 (2014)
23. Leijssen, R., Verhagen, E.: Strong optomechanical interactions in a sliced photonic crystal
nanobeam. Sci. Rep. 5, 15974 (2015)
24. Hoang, C.M.: Thermoelastic damping depending on vibration modes of nano beam resonator.
Commun. Phys. 25(4), 317–325 (2015)