Vibration analysis of a Timoshenko non-uniform nanobeam based on nonlocal theory: An analytical solution

Document Type : Reasearch Paper

Authors

• Shahrokh Hosseini Hashemi ^{1, 2}
• Hossein Bakhshi Khaniki ¹

¹ School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16842-13114, Tehran, Iran

² Center of Excellence in Railway Transportation, Iran University of Science and Technology, Narmak, 16842-13114, Tehran, Iran

10.22034/ijnd.2017.24378

Abstract

In this article free vibration of a Timoshenko nanobeam with variable cross-section is investigated using nonlocal elasticity theory within the scope of continuum mechanics. Small scale effects are modelled after Eringen’s nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width through the beam length with constant thickness. Analytical solution is achieved for both Timoshenko beams and nanobeams with different boundary conditions including both ends being simply-supported (S-S), both ends being clamped (C-C) and one end clamped other free (C-F). It is shown that section variation accompanying small scale effects has a noticeable effect on natural frequencies of non-uniform Timoshenko beams at nano-scale. In order to illustrate these effects, Natural frequencies of single-layered graphene nano-ribbons (GNRs) with various boundary conditions are obtained for different nonlocal and non-uniform parameter which shows a great sensitivity to non-uniformity in different shape modes.

Keywords

• Free vibration
• Timoshenko beam
• analytical solution
• Nonlocal
• Nanobeam
• Variable cross-section

20.1001.1.20088868.2017.8.1.8.5

Main Subjects

• Nano Engineering
• Nano Mechanical Engineering

---

References

[1] Li Y., Cheng Y. F., (2016), Effect of surface finishing on early-stage corrosion of a carbon steel studied by electrochemical and atomic force microscope characterizations. Appl. Surf. Sci. 336: 95-103.

[2] Xie H, , Hussain D., Yang F., Sun L., (2015), Atomic force microscope caliper for critical dimension measurements of micro and nanostructures through sidewall scanning. Ultramicroscopy. 158: 8-16.

[3] Omori Y., Ikei H., Kugimiya Y., Masuda T., (2015), Nanometre-scale faulting in quartz under an atomic force microscope. J. Str. Geology. 79: 75-79.

[4] Taghipour Y., Baradaran G. H., (2016), A finite element modeling for large deflection analysis of uniform and tapered nanowires with good interpretation of experimental results. Int. J. Mech. Sci. 114: 111-119.

[5] Lee S. Y., Kang H. C., (2016), Characterization of individual ultra-long SnO₂ nanowires grown by vapor transport method. Mat. Lett. 176: 294-297.

[6] Pishkenari H. N., Afsharmanesh B., Tajaddodianfar F., (2016), Continuum models calibrated with atomistic simulations for the transverse vibrations of silicon nanowires. Int J. Eng. Sci. 100: 8-24.

[7] Lee W. C., Cho Y. H., (2007), Bio-inspired digital nanoactuators for photon and biomaterial manipulation. Curr. Appl. Phys. 7: 139-146.

[8] Hartbaum J., Jakobs P., Wohlgemuth J., Silvestre M., Franzreb M., Kohl M., (2012), Magnetic bead nanoactuator. Microelec. Eng. 98: 582-586.

[9] Zeng L., Pan Y., Zou R., Zhang J., Tian Y., Teng Z., Wang S., Ren W., Xiao X., Zhang J., Zhang L., Li A., Lu G., Wu A., (2016), 808 nm-excited upconversion nanoprobes with low heating effect for targeted magnetic resonance imaging and high-efficacy photodynamic therapy in HER₂-overexpressed breast cancer. Biomaterials. 103: 116-127.

[10] Liu M., Huang H., Wang K., Xu D., Wan Q., Tian J., Huang Q., Deng F., Zhang X., Wei Y., (2016), Fabrication and biological imaging application of AIE-Active luminescent starch based nanoprobes. Carbohyd. Polym. 142: 38-44.

[11] Yang W., Fu L. M., Wen X., Liu Y., Tian Y., Liu Y. C., Han R. C., Gao Z. Y., Wang T. E,. Sha Y. L., Jiang Y. Q., Wang Y., Zhang J. P., (2016), Nanoprobes for two-photon excitation time-resolved imaging of living animals: In situ analysis of tumor-targeting dynamics of nanocarriers. Biomaterials. 100: 152-161.

[12] Yan J. W., Tong L. H., Li C., Zhu Y., Wang Z. W., (2016), Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory. Compos. Struc. 125: 304-313.

[13] Sciarra F. M., Barretta R., (2014), A new nonlocal bending model for Euler–Bernoulli nanobeams. Mech. Rese. Communic. 65: 25-30.

[14] Şimşek M., Yurtcu H. H., (2012), Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos. Struct. 97: 378-386.

[15] Akgöz B., Civalek Ö. (2016), Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica. 119: 1-12.

[16] Zenkour A. M., Sobhy M., (2015), A simplified shear and normal deformations nonlocal theory for bending of nanobeams in thermal environment. Physica E: Low-Dimensional Systems and Nanostructures. 70: 121-128.

[17] Eltaher M. A., Khater M. E., Emam S. A., (2016), A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Mathem. Model. 40: 4109-4128.

[18] Jun Yu Y., Xue Z. N., Li C. L., Tian X. G., (2016), Buckling of nanobeams under nonuniform temperature based on nonlocal thermoelasticity. Compos. Struc. 146: 108-113.

[19] Ebrahimi F., Salari E., (2015), Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments. Compos. Struc. 128: 363-380.

[20] Tourki Samaei A., Hosseini Hashemi Sh., (2012), Buckling analysis of graphene nanosheets based on nonlocal elasticity theory. Int. J. Nano Dimens. 2: 227-232.

[21] Ansari R., Rouhi H., (2015), Nonlocal Flügge shell model for the axial buckling of single-walled Carbon nanotubes: An analytical approach. Int. J. Nano Dimens. 6: 453-462.

[22] Korayem A. H., Duan W. H., Zhao X. L., (2011), Investigation on buckling behavior of short MWCNT. Proced. Engin. 14: 250-255.

[23] Ansari R., Faraji Oskouie M., Sadeghi F., Bazdid-Vahdati M., (2015), Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory. Physica E: Low-Dimensional Systems and Nanostructures. 74: 318-327.

[24] Hosseini-Hashemi S., Nazemnezhad R., Rokni H., (2015), Nonlocal nonlinear free vibration of nanobeams with surface effects. Europ. J. Mech. A/Solids. 52: 44-53.

[25] Mohamed S. A., Shanab R. A., Seddek L. F., (2016), Vibration analysis of Euler–Bernoulli nanobeams embedded in an elastic medium by a sixth-order compact finite difference method. Appl. Mathemat. Modell. 40: 2396-2406.

[26] Hosseini Hashemi Sh., Bakhshi Khaniki H., (2016), Dynamic Behavior of Multi-Layered Viscoelastic Nanobeam System Embedded in a Viscoelastic Medium with a Moving Nanoparticle. J. Mech. In Press, 22 September.

[27] Prasanna Kumar T. J., Narendar S., Gupta B. L. V. S., Gopalakrishnan S., (2013), Thermal vibration analysis of double-layer graphene embedded in elastic medium based on nonlocal continuum mechanics. Int. J. Nano Dimens. 4: 29-49.

[28] Ebrahimi, F., Barati, M. R., (2017), Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Composite. Struct. 159: 433-444.

[29] Ansari R., Oskouie M. F., Rouhi H. (2016). Studying linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/nano-beams using the strain gradient theory. Nonlinear Dynam. 1-17.

[30] Pandeya A., Singhb J., (2015), A variational principle approach for vibration of non-uniform nanocantilever using nonlocal elasticity theory. Proced. Mater. Sci. 10: 497-506.

[31] Murmu T., Pradhan S. C., (2009), Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E. 41: 1451-1456.

[32] Malekzadeh P., Shojaee M., (2013), Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams. Composites: Part B. 52: 84-92.

[33] Hosseini Hashemi Sh., Bakhshi Khaniki H., (2016), Analytical Solution for Free Vibration of a Variable Cross-Cection Nanobeam. IJE Transact. B: Appl. 29(5):688-696.

[34] Hosseini Hashemi Sh., Bakhshi Khaniki H., (2016), Free vibration analysis of FGM non-uniform beams. IJE Transact.C: Aspects. 29: 1473-1479.

[35] Lee H. L., Chang W. J., (2010), Surface and small-scale effects on vibration analysis of a nonuniform nanocantilever beam. Physica E: Low-Dimensional Systems and Nanostructures. 43: 466-469.

[36] Behera L., Chakraverty S., (2014), Free vibration of non-uniform nanobeams using Rayleigh-Ritz method. Physica E: Low-dimensional Systems and Nanostructures. 67: 38-46.

 [37] Chang T. P., (2012), Small scale effect on axial vibration of non-uniform and non-homogeneous nanorods. Comp. Mater. Sci. 54: 23-27.

[38] Şimşek M., (2012), Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods. Comput. Mater. Sci.  61: 257-265.

[39] Ece M. C., Aydogdu M., Taskin V., (2007), Vibration of a variable cross-section beam. Mech. Res. Communic. 34: 78-84.

[40] Akgoz B., Civalek O., (2013), Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Struct. Engineer. Mech. 48: 195-205.

[41] Zeighampour H., Beni Y. T., (2015), Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory. Appl. Mathemat. Modell. 39: 5354-5369.

[42] Wang Q., Wang C. M., (2007), The constitutive relation and small scale parameter of nonlocal continuum mechanics for modeling carbon nanotubes. Nanotechnol. 18: 1-4.

[43] Wang C. M., Zhang Y. Y., He X. Q., (2007), Vibration of nonlocal Timoshenko beams. Nanotechnol. 10: 1-9.