Vibration Analysis of an Axially Loaded Viscoelastic Nanobeam

Abstract

Vibration of an axially loaded viscoelastic nanobeam has been studied in this paper. Viscoelasticity of the nanobeam has been modeled as a Kelvin-Voigt material. Equation of motion and boundary conditions for an axially compressed nanobeam has been obtained with help of Eringen’s Nonlocal Elasticity Theory. Viscoelasticity effect on natural frequency and damping of nanobeam and critical buckling load have been investigated. Nonlocality effect on nanobeam structure in the view of viscoelasticity has been discussed.

Keywords

• viscoelastic nanobeam,nonlocal elasticity,vibration,axially loaded,buckling

References

1. Eringen A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, 54, 4703–10, 1983. doi:10.1063/1.332803
2. Eringen A.C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10, 1–16, 1972. doi:10.1016/0020-7225(72)90070-5
3. Lei Y., Murmu T., Adhikari S., Friswell M.I., Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beams, European Journal of Mechanics, A/Solids, 42, 125–36, 2013. doi:10.1016/j.euromechsol.2013.04.006
4. Lei Y., Adhikari S., Friswell M.I., Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams, International Journal of Engineering Science, 66–67, 1–13, 2013. doi:10.1016/j.ijengsci.2013.02.004
5. Chen C., Li S., Dai L., Qian C., Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces, Communications in Nonlinear Science and Numerical Simulation, 19, 1626–37, 2014. doi:10.1016/j.cnsns.2013.09.017
6. Pavlović I., Pavlović R., Ćirić I., Karličić D., Dynamic stability of nonlocal Voigt-Kelvin viscoelastic Rayleigh beams, Applied Mathematical Modelling, 39, 6941–50, 2015. doi:10.1016/j.apm.2015.02.044
7. Civalek Ö., Demir C., Buckling and bending analyses of cantilever carbon nanotubes using the Euler-Bernoulli beam theory based on non-local continuum model, Asian Journal of Civil Engineering, 12, 651–62, 2011
8. Akgöz B., Civalek Ö., Buckling Analysis of Cantilever Carbon Nanotubes Using the Strain Gradient Elasticity and Modified Couple Stress Theories, Journal of Computational and Theoretical Nanoscience, 8, 1821–7, 2011. doi:10.1166/jctn.2011.1888

9. Mercan K., Civalek Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Composite Structures, 143, 300–9, 2016. doi:10.1016/j.compstruct.2016.02.040
10. Mercan K., Civalek Ö., Buckling analysis of Silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ, Composites Part B: Engineering, 114, 34–45, 2017. doi:10.1016/j.compositesb.2017.01.067
11. Karličić D., Murmu T., Cajić M., Kozić P., Adhikari S., Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field, Journal of Applied Physics, 115, 234303, 2014. doi:10.1063/1.4883194
12. Ghorbanpour-Arani A.H., Rastgoo A., Sharafi M.M., Kolahchi R., Ghorbanpour Arani A., Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nanobeam systems, Meccanica, 51, 25–40, 2016. doi:10.1007/s11012-014-9991-0
13. Mohammadi M., Safarabadi M., Rastgoo A., Farajpour A., Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment, Acta Mechanica, 227, 2207–32, 2016. doi:10.1007/s00707-016-1623-4
14. Zhang Y., Pang M., Fan L., Analyses of transverse vibrations of axially pretensioned viscoelastic nanobeams with small size and surface effects, Physics Letters, Section A: General, Atomic and Solid State Physics, 380, 2294–9, 2016. doi:10.1016/j.physleta.2016.05.016
15. Ebrahimi F., Barati M.R., Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects, Applied Physics A: Materials Science and Processing, 123, 1–10, 2017. doi:10.1007/s00339-016-0511-z
16. Ebrahimi F., Barati M.R., Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory, Composite Structures, 159, 433–44, 2017. doi:10.1016/j.compstruct.2016.09.092
17. Ebrahimi F., Barati M.R., Effect of three-parameter viscoelastic medium on vibration behavior of temperature-dependent non-homogeneous viscoelastic nanobeams in a hygro-thermal environment, Mechanics of Advanced Materials and Structures, 25, 361–74, 2018. doi:10.1080/15376494.2016.1255831
18. 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 Mechanica, 228, 1197–210, 2017. doi:10.1007/s00707-016-1755-6
19. Ebrahimi F., Barati M.R., Damping Vibration Behavior of Viscoelastic Porous Nanocrystalline Nanobeams Incorporating Nonlocal–Couple Stress and Surface Energy Effects, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 2017. doi:10.1007/s40997-017-0127-8
20. Attia M.A., Mahmoud F.F., Analysis of viscoelastic Bernoulli–Euler nanobeams incorporating nonlocal and microstructure effects, International Journal of Mechanics and Materials in Design, 13, 385–406, 2017. doi:10.1007/s10999-016-9343-4
21. Attia M.A., Abdel Rahman A.A., On vibrations of functionally graded viscoelastic nanobeams with surface effects, International Journal of Engineering Science, 127, 1–32, 2018. doi:10.1016/j.ijengsci.2018.02.005
22. Oskouie M.F., Ansari R., Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects, Applied Mathematical Modelling, 43, 337–50, 2017. doi:10.1016/j.apm.2016.11.036
23. Oskouie M.F., Ansari R., Sadeghi F., Nonlinear vibration analysis of fractional viscoelastic Euler–Bernoulli nanobeams based on the surface stress theory, Acta Mechanica Solida Sinica, 30, 416–24, 2017. doi:10.1016/j.camss.2017.07.003
24. Ansari R., Faraji Oskouie M., Rouhi H., Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/nano-beams using the strain gradient theory, Nonlinear Dynamics, 87, 695–711, 2017. doi:10.1007/s11071-016-3069-6
25. Ansari R., Faraji Oskouie M., Gholami R., Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory, Physica E: Low-Dimensional Systems and Nanostructures, 75, 266–71, 2016. doi:10.1016/j.physe.2015.09.022
26. Ansari R., Faraji Oskouie M., Sadeghi F., Bazdid-Vahdati M., Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory, Physica E: Low-Dimensional Systems and Nanostructures, 74, 318–27, 2015. doi:10.1016/j.physe.2015.07.013
27. Cajic M., Karlicic D., Lazarevic M., Nonlocal vibration of a fractional order viscoelastic nanobeam with attached nanoparticle, Theoretical and Applied Mechanics, 42, 167–90, 2015. doi:10.2298/TAM1503167C
28. Marynowski K., Non-Linear Dynamic Analysis of an Axialy Moving Viscoelastic Beam, Journal of Theoretical and Applied Mechanics, 465–82, 2002
29. Eringen A.C., Nonlocal Continuum Field Theories. Springer New York, 2007
30. Civalek Ö., Demir Ç., Akgöz B., Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory, International Journal of Engineering and Applied Sciences, 1, 47–56, 2009
31. Akgöz B., Civalek Ö., Investigation of Size Effects on Static Response of Single-Walled Carbon Nanotubes Based on Strain Gradient Elasticity, International Journal of Computational Methods, 09, 1240032, 2012. doi:10.1142/S0219876212400324
32. Reddy J.N., Pang S.D., Nonlocal continuum theories of beams for the analysis of carbon nanotubes, Journal of Applied Physics, 103, 2008. doi:10.1063/1.2833431
33. Aydogdu M., A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration, Physica E: Low-Dimensional Systems and Nanostructures, 41, 1651–5, 2009. doi:10.1016/j.physe.2009.05.014
34. Arda M., Aydogdu M., Buckling of Eccentrically Loaded Carbon Nanotubes, Solid State Phenomena, 267, 151–6, 2017. doi:10.4028/www.scientific.net/SSP.267.151
35. Arda M., Aydogdu M., Nonlocal Gradient Approach on Torsional Vibration of CNTs, NOISE Theory and Practice, 3, 2–10, 2017
36. Lu P., Lee H.P., Lu C., Zhang P.Q., Dynamic properties of flexural beams using a nonlocal elasticity model, Journal of Applied Physics, 99, 073510, 2006. doi:10.1063/1.2189213
37. Eltaher M.A., Alshorbagy A.E., Mahmoud F.F., Vibration analysis of Euler-Bernoulli nanobeams by using finite element method, Applied Mathematical Modelling, 37, 4787–97, 2013. doi:10.1016/j.apm.2012.10.016
38. Romano G., Barretta R., Diaco M., Marotti de Sciarra F., Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams, International Journal of Mechanical Sciences, 121, 151–6, 2017. doi:10.1016/j.ijmecsci.2016.10.036
39. Li C., A nonlocal analytical approach for torsion of cylindrical nanostructures and the existence of higher-order stress and geometric boundaries, Composite Structures, 118, 607–21, 2014. doi:10.1016/j.compstruct.2014.08.008
40. Li C., Torsional vibration of carbon nanotubes: Comparison of two nonlocal models and a semi-continuum model, International Journal of Mechanical Sciences, 82, 25–31, 2014. doi:10.1016/j.ijmecsci.2014.02.023
41. Challamel N., Reddy J.N., Wang C.M., Eringen’s Stress Gradient Model for Bending of Nonlocal Beams, Journal of Engineering Mechanics, 142, 04016095, 2016. doi:10.1061/(ASCE)EM.1943-7889.0001161
42. Eptaimeros K.G., Koutsoumaris C.C., Tsamasphyros G.J., Nonlocal integral approach to the dynamical response of nanobeams, International Journal of Mechanical Sciences, 115–116, 68–80, 2016. doi:10.1016/j.ijmecsci.2016.06.013
43. Shaat M., Faroughi S., Abasiniyan L., Paradoxes of differential nonlocal cantilever beams: Reasons and a novel solution, 1–17, 2017