Abstract
In this paper, surface and thermal effects on the vibration characteristics of viscoelastic functionally graded (FG) nanobeams embedded in a viscoelastic foundation are investigated based on nonlocal strain gradient elasticity theory and Euler–Bernoulli beam model. The present model contains two parameters to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Gurtin–Murdoch elasticity theory is incorporated into the nonlocal strain gradient theory to develop a nonclassical beam model including the surface effects. The viscoelastic foundation consists of a Winkler–Pasternak layer together with a viscous layer of infinite parallel dashpots. A power-law model is adopted to describe a continuous variation of material properties of the FG nanobeam. The governing equations of nonlocal viscoelastic FG nanobeams are obtained using Hamilton’s principle and solved implementing an analytical solution for simply supported and clamped–clamped boundary conditions. The results are validated with those available in the literature. The effects of linear, shear and viscous layers of foundation, structural damping coefficient, surface elasticity, nonlocal parameter, length scale parameter, temperature change, power-law exponent, and slenderness ratio on the frequencies of viscoelastic FG nanobeams are examined.
Similar content being viewed by others
References
Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Yang, F.A.C.M., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)
Li, L., Hu, Y., Ling, L.: Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos. Struct. 133, 1079–1092 (2015)
Ebrahimi, F., Barati, M.R., Dabbagh, A.: A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int. J. Eng. Sci. 107, 169–182 (2016)
Aydogdu, M.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E 41, 1651–1655 (2009)
Beni, Y.T.: Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling. Mech. Res. Commun. 75, 67–80 (2016)
Thai, H.T.: A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int. J. Eng. Sci. 52, 56–64 (2012)
Ebrahimi, F., Nasirzadeh, P.: A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method. J. Theor. Appl. Mech. 53, 1041–1052 (2015)
Barati, M.R., Zenkour, A.M., Shahverdi, H.: Thermo-mechanical buckling analysis of embedded nanosize FG plates in thermal environments via an inverse cotangential theory. Compos. Struct. 141, 203–212 (2016)
Ebrahimi, F., Barati, M.R.: An exact solution for buckling analysis of embedded piezoelectro-magnetically actuated nanoscale beams. Adv. Nano Res. 4, 65–84 (2016)
Ebrahimi, F., Barati, M.R.: Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. J. Braz. Soc. Mech. Sci. Eng. 1–16 (2016). doi:10.1007/s40430-016-0551-5
Ebrahimi, F., Barati, M.R.: Electromechanical buckling behavior of smart piezoelectrically actuated higher-order size-dependent graded nanoscale beams in thermal environment. Int. J. Smart Nano Mater. 1–22 (2016)
Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Free vibration analysis of functionally graded size-dependent nanobeams. Appl. Math. Comput. 218, 7406–7420 (2012)
Eltaher, M.A., Emam, S.A., Mahmoud, F.F.: Static and stability analysis of nonlocal functionally graded nanobeams. Compos. Struct. 96, 82–88 (2013)
Şimşek, M., Yurtcu, H.H.: Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97, 378–386 (2013)
Ebrahimi, F., Barati, M.R.: Vibration analysis of nonlocal beams made of functionally graded material in thermal environment. Eur. Phys. J. Plus 131, 279 (2016)
Ebrahimi, F., Barati, M.R.: A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. Arab. J. Sci. Eng. 41, 1679–1690 (2016)
Zeighampour, H., Beni, Y.T.: Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Appl. Math. Model. 39, 5354–5369 (2015)
Shojaeian, M., Beni, Y.T.: Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens. Actuators A: Phys. 232, 49–62 (2015)
Shojaeian, M., Beni, Y.T., Ataei, H.: Electromechanical buckling of functionally graded electrostatic nanobridges using strain gradient theory. Acta Astronaut. 118, 62–71 (2016)
Hosseini-Hashemi, S., Nahas, I., Fakher, M., Nazemnezhad, R.: Surface effects on free vibration of piezoelectric functionally graded nanobeams using nonlocal elasticity. Acta Mech. 225, 1555–1564 (2014)
Beni, Y.T.: Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams. J. Intell. Mater. Syst. Struct. 27, 2199–2215 (2016)
Mehralian, F., Beni, Y.T.: Size-dependent torsional buckling analysis of functionally graded cylindrical shell. Compos. Part B: Eng. 94, 11–25 (2016)
Mehralian, F., Beni, Y.T., Ansari, R.: Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell. Compos. Struct. 152, 45–61 (2016)
Ebrahimi, F., Barati, M.R.: Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field. Appl. Phys. A 122, 1–18 (2016)
Ebrahimi, F., Barati, M.R.: Magnetic field effects on buckling behavior of smart size-dependent graded nanoscale beams. Eur. Phys. J. Plus 131, 1–14 (2016)
Ebrahimi, F., Barati, M.R.: Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment. J. Vib. Control (2016). doi:10.1177/1077546316646239
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003)
Lim, C.W., Zhang, G., Reddy, J.N.: A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015)
Li, L., Hu, Y.: Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int. J. Eng. Sci. 97, 84–94 (2015)
Li, L., Li, X., Hu, Y.: Free vibration analysis of nonlocal strain gradient beams made of functionally graded material. Int. J. Eng. Sci. 102, 77–92 (2016)
Ebrahimi, F., Barati, M.R.: Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory. Appl. Phys. A 122, 843 (2016)
Li, L., Hu, Y., Li, X.: Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory. Int. J. Mech. Sci. 115, 135–144 (2016)
Lei, Y., Adhikari, S., Friswell, M.I.: Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams. Int. J. Eng. Sci. 66, 1–13 (2013)
Pouresmaeeli, S., Ghavanloo, E., Fazelzadeh, S.A.: Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos. Struct. 96, 405–410 (2013)
Hashemi, S.H., Mehrabani, H., Ahmadi-Savadkoohi, A.: Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium. Compos. Part B: Eng. 78, 377–383 (2015)
Hosseini, M., Jamalpoor, A.: Analytical solution for thermomechanical vibration of double-viscoelastic nanoplate-systems made of functionally graded materials. J. Therm. Stresses 38, 1428–1456 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ebrahimi, F., Barati, M.R. Vibration analysis of viscoelastic inhomogeneous nanobeams resting on a viscoelastic foundation based on nonlocal strain gradient theory incorporating surface and thermal effects. Acta Mech 228, 1197–1210 (2017). https://doi.org/10.1007/s00707-016-1755-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-016-1755-6