References
- Aifantis, E. C. 1992. On the role of gradients in the localization of deformation and fracture. International Journal of Engineering Science 30 (10):1279–99. doi:https://doi.org/10.1016/0020-7225(92)90141-3.
- Alizade Hamidi, B., S. H. Hosseini, R. Hassannejad, and F. Khosravi. 2019. An exact solution on gold microbeam with thermoelastic damping via generalized green-Naghdi and modified couple stress theories. Journal of Thermal Stresses 43 (2):157–74. doi:https://doi.org/10.1080/01495739.2019.1666694.
- Apuzzo, A., R. Barretta, S. Faghidian, R. Luciano, and F. M. De Sciarra. 2018. Free vibrations of elastic beams by modified nonlocal strain gradient theory. International Journal of Engineering Science 133:99–108. doi:https://doi.org/10.1016/j.ijengsci.2018.09.002.
- Arefi, M., M. Kiani, and M. Zamani. 2018. Nonlocal strain gradient theory for the magneto-electro-elastic vibration response of a porous fg-core sandwich nanoplate with piezomagnetic face sheets resting on an elastic foundation. Journal of Sandwich Structures and Materials. doi:https://doi.org/10.1177/1099636218795378.
- Arefi, M., M. Pourjamshidian, and A. G. Arani. 2017. Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with fg-cntrcs face-sheets in electro-thermal environment. Applied Physics A 123 (5):323. doi:https://doi.org/10.1007/s00339-017-0922-5.
- Aydogdu, M. J. M. R. C. 2012. Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity. Mechanics Research Communications 43:34–40. doi:https://doi.org/10.1016/j.mechrescom.2012.02.001.
- Aydogdu, M. J. P. E. L.-D. S. 2009. Axial vibration of the nanorods with the nonlocal continuum rod model. Physica E: Low-dimensional Systems and Nanostructures 41 (5):861–4. doi:https://doi.org/10.1016/j.physe.2009.01.007.
- Ebrahimi, F., M. Nouraei, and A. Dabbagh. 2019. Modeling vibration behavior of embedded graphene-oxide powder-reinforced nanocomposite plates in thermal environment. Mechanics Based Design of Structures and Machines 48:1–24. doi:https://doi.org/10.1080/15397734.2019.1660185.
- Eringen, A. C. 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics 54 (9):4703–10. doi:https://doi.org/10.1063/1.332803.
- Farajpour, A., H. Farokhi, M. H. Ghayesh, and S. Hussain. 2018. Nonlinear mechanics of nanotubes conveying fluid. International Journal of Engineering Science 133:132–43. doi:https://doi.org/10.1016/j.ijengsci.2018.08.009.
- Farajpour, A., M. H. Yazdi, A. Rastgoo, and M. Mohammadi. 2016. A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mechanica 227 (7):1849–67. doi:https://doi.org/10.1007/s00707-016-1605-6.
- Fattahi, A. M., S. Sahmani, and N. A. Ahmed. 2019. Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations. Mechanics Based Design of Structures and Machines 1–30. doi:https://doi.org/10.1080/15397734.2019.1624176.
- Ghayesh, M. H., and A. Farajpour. 2019. A review on the mechanics of functionally graded nanoscale and microscale structures. International Journal of Engineering Science 137:8–36. doi:https://doi.org/10.1016/j.ijengsci.2018.12.001.
- Gholamzadeh Babaki, M. H., and M. Shakouri. 2019. Free and forced vibration of sandwich plates with electrorheological core and functionally graded face layers. Mechanics Based Design of Structures and Machines 1–18.
- Ghorbani, K., A. Rajabpour, and M. Ghadiri. 2019. Determination of carbon nanotubes size-dependent parameters: Molecular dynamics simulation and nonlocal strain gradient continuum shell model. Mechanics Based Design of Structures and Machines 1–18. doi:https://doi.org/10.1080/15397734.2019.1671863.
- Hayati, H., S. A. Hosseini, and O. Rahmani. 2017. Coupled twist–bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory. Microsystem Technologies 23 (7):2393–401. doi:https://doi.org/10.1007/s00542-016-2933-0.
- Hosseini, S., and O. Rahmani. 2018a. Bending and vibration analysis of curved fg nanobeams via nonlocal Timoshenko model. Smart Construction Research 2 (2):1–17. doi:https://doi.org/10.18063/scr.v0.401.
- Hosseini, S. A., and O. Rahmani. 2018b. Modeling the size effect on the mechanical behavior of functionally graded curved micro/nanobeam. Thermal Science and Engineering 1 (2).
- Humar, J. 1990. Dynamics of structures. Englewood Cliffs, NJ: Prentice-Hall.
- Jabbari Behrouz, S., O. Rahmani, and S. A. Hosseini. 2019. On nonlinear forced vibration of nano cantilever-based biosensor via couple stress theory. Mechanical Systems and Signal Processing 128:19–36. doi:https://doi.org/10.1016/j.ymssp.2019.03.020.
- Li, C. 2017. Nonlocal thermo-electro-mechanical coupling vibrations of axially moving piezoelectric nanobeams. Mechanics Based Design of Structures and Machines 45 (4):463–78. doi:https://doi.org/10.1080/15397734.2016.1242079.
- Li, L., Y. Hu, and X. Li. 2016. Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. International Journal of Mechanical Sciences 115:135–44. doi:https://doi.org/10.1016/j.ijmecsci.2016.06.011.
- Lim, C., G. Zhang, J. J. J. O. T. M. Reddy, and P. O. Solids. 2015. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids 78:298–313. doi:https://doi.org/10.1016/j.jmps.2015.02.001.
- Lu, L., X. Guo, and J. Zhao. 2019. A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Applied Mathematical Modelling 68:583–602. doi:https://doi.org/10.1016/j.apm.2018.11.023.
- Malekzadeh, P., and A. Farajpour. 2012. Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium. Acta Mechanica 223 (11):2311–30. doi:https://doi.org/10.1007/s00707-012-0706-0.
- Mindlin, R. D. 1964. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16 (1):51–78. doi:https://doi.org/10.1007/BF00248490.
- Nejad, M. Z., A. Hadi, and A. J. I. J. O. E. S. Rastgoo. 2016. Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory. International Journal of Engineering Science 103:1–10. doi:https://doi.org/10.1016/j.ijengsci.2016.03.001.
- Rahmani, O., S. Hosseini, I. Ghoytasi, and H. Golmohammadi. 2018. Free vibration of deep curved fg nano-beam based on modified couple stress theory. Steel and Composite Structures 26 (5):607–20.
- Rahmani, O., S. Hosseini, and H. Hayati. 2016. Frequency analysis of curved nano-sandwich structure based on a nonlocal model. Modern Physics Letters B 30 (10):1650136. doi:https://doi.org/10.1142/S0217984916501360.
- Sahmani, S., and D. M. Madyira. 2019. Nonlocal strain gradient nonlinear primary resonance of micro/nano-beams made of gpl reinforced fg porous nanocomposite materials. Mechanics Based Design of Structures and Machines 1–28.
- Shahgholian-Ghahfarokhi, D., M. Safarpour, and A. Rahimi. 2019. Torsional buckling analyses of functionally graded porous nanocomposite cylindrical shells reinforced with graphene platelets (GPLS). Mechanics Based Design of Structures and Machines 1–22. doi:https://doi.org/10.1080/15397734.2019.1666723.
- Tauchert, T. R. 1974. Energy principles in structural mechanics. New York: McGraw-Hill Companies.
- Zarepour, M., S. A. Hosseini, and M. Ghadiri. 2017. Free vibration investigation of nano mass sensor using differential transformation method. Applied Physics A 123 (3):181. doi:https://doi.org/10.1007/s00339-017-0796-6.
- Zarepour, M., S. H. Hosseini, and A. H. Akbarzadeh. 2019. Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen’s differential model. Applied Mathematical Modelling 69:563–82. doi:https://doi.org/10.1016/j.apm.2019.01.001.
- Zarezadeh, E., V. Hosseini, and A. Hadi. 2019. Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory. Mechanics Based Design of Structures and Machines 1–16. doi:https://doi.org/10.1080/15397734.2019.1642766.
- Zhu, X., and L. Li. 2017. Closed form solution for a nonlocal strain gradient rod in tension. International Journal of Engineering Science 119:16–28. doi:https://doi.org/10.1016/j.ijengsci.2017.06.019.
- Zorica, D., T. M. Atanacković, Z. Vrcelj, and B. J. J. O. E. M. Novaković. 2017. Dynamic stability of axially loaded nonlocal rod on generalized Pasternak foundation. Journal of Engineering Mechanics 143 (5):D4016003.