Exact solution for the vibrations of cylindrical nanoshells considering surface energy effect

Document Type : Research Paper


1 Department of Mechanical Engineering, University of Guilan

2 Department of mechanical engineering, University of Guilan


It has been revealed that the surface stress effect plays an important role in the mechanical behavior of
structures (such as bending, buckling and vibration) when their dimensions are on the order of
nanometer. In addition, recent advances in nanotechnology have proposed several applications for
nanoscale shells in different fields. Hence, in the present article, within the framework of surface
elasticity theory, the free vibration behavior of simply-supported cylindrical nanoshells with the
consideration of the aforementioned effect is studied using an exact solution method. To this end, first,
the governing equations of motion and boundary conditions are obtained by an energy-based
approach. The surface stress influence is incorporated into the formulation according to the Gurtin-
Murdoch theory. The nanoshell is modeled according to the first-order shear deformation shell theory.
After that, the free vibration problem is solved through an exact solution approach. To this end, the
dimensionless form of governing equations is derived and then solved under the simply-supported
boundary conditions using a Navier-type solution method. Selected numerical results are presented
about the effects of surface stress and surface material properties on the natural frequencies of
nanoshells with different radii and lengths. The results show that the surface energies significantly
affect the vibrational behavior of nanoshells with small magnitudes of thickness. Also, it is indicated
that the natural frequency of the nanoshell is dependent of the surface material properties.


[1]. Rafiee, M. A., Rafiee, J., Wang, Z., Song, H.,
Yu, Z. Z., and Koratkar, N., “Enhanced
Mechanical Properties of Nanocomposites at
Low Graphene Content,” ACS Nano, Vol. 3,
2009, pp. 3884-3890.
[2]. Xia, Y., and Xiao, H., “Au Nanoplate/Polypyrrole
Nanofiber Composite Film: Preparation,
Characterization and Application as SERS
Substrate,” J. Raman Spectrosc., Vol. 43, 2012,
pp. 469-473.
[3]. Abouzar, M. H., Poghossian, A., Pedraza, A.
M., Gandhi, D., Ingebrandt, S., Moritz, W., and
Schöning, M. J., “An Array of Field-Effect
Nanoplate SOI Capacitors for (Bio-) Chemical
Sensing,” Biosens. Bioelec., Vol. 26, 2011, pp.
[4]. Bi, L., Dong, J., Xie, W., Lu, W., Tong, W.,
Tao, L., and Qian, W., “Bimetallic Gold–Silver
Nanoplate Array as a Highly Active SERS
Substrate for Detection of Streptavidin/Biotin
Assemblies,” Anal. Chimica Acta, Vol. 805,
2013, pp. 95-100.
[5]. Khosroshahi, M. E., and Ghazanfari, L.,
“Synthesis of Three-Layered Magnetic Based
Nanostructure for Clinical Application,” Int. J.
Nanosci. Nanotechnol., Vol. 7, 2011, pp. 57-64.
[6]. Chang, S. T., and Hsieh, B. F., “TCAD Studies
of Novel Nanoplate Amorphous Silicon Alloy
Thin-Film Solar Cells,” Thin Solid Films, Vol.
520, 2011, pp. 1612-1616.
[7]. Odom, T. W., Huang, J. L., Kim, P., Lieber, C.
M., “Atomic Structure and Electronic
Properties of Single-Walled Carbon
Nanotubes,” Nature, Vol. 391, 1998, 62–64.
[8]. Ganesan, Y., Peng, C., Lu, Y., Ci, L.,
Srivastava, A., Ajayan, P. M., and Lou, J.,
“Effect of Nitrogen Doping on the Mechanical
Properties of Carbon Nanotubes,” ACS Nano,
Vol. 4, 2010, pp. 7637–7643. [9]. Suk, J. W., Piner, R. D., An, J., and Ruoff, R. S., “Mechanical Properties of Monolayer Graphene Oxide,” ACS Nano, Vol. 4, 2010, pp. 6557–6564.
[10]. Zakeri, M., and Shayanmehr, M., “On the Mechanical Properties of Chiral Carbon Nanotubes,” J. Ultrafine Grained Nanostruct. Mater., Vol. 46, 2013, pp. 01-09.
[11]. Kumar, C. S. S. R., “Nanomaterials for Biosensors,” Wiley-VCH, 2007, Wein-heim.
[12]. Zabow, G., Dodd, S. J., Moreland, J., and Koretsky, A. P., “The Fabrication of Uniform Cylindrical Nanoshells and Their Use as Spectrally Tunable MRI Contrast Agents,” Nanotechnology, 20, 2009, 385301.
[13]. Zhu, J., Li, J. -J., and Zhao, J. -W., “Obtain Quadruple Intense Plasmonic Resonances from Multilayered Gold Nanoshells by Silver Coating: Application in Multiplex Sensing,” Plasmonics, 8, 2013, 1493.
[14]. Yakobson, B. I., Brabec, C. J., and Bernholc, J., “Nanomechanics of Carbon Tubes: Instability Beyond Linear Response,” Phys. Rev. Lett., Vol. 76, 1996, pp. 2511–2514.
[15]. Pantano, A., Boyce, M. C., and Parks, D. M., “Mechanics of Axial Compression of Single and Multi-Wall Carbon Nanotubes,” ASME J. Eng. Mater. Technol., Vol. 126, 2004, pp. 279-284.
[16]. Yao, X., and Han, Q., “Buckling Analysis of Multiwalled Carbon Nanotubes Under Torsional Load Coupling With Temperature Change,” ASME J. Eng. Mater. Technol., Vol. 128, 2006, pp. 419-427.
[17]. Ansari, R., Hemmatnezhad, M., and Ramezannezhad, H., “Application of HPM to the Nonlinear Vibrations of Multiwalled Carbon Nanotubes,” Numer. Meth. Part. D. E. Vol. 26, 2009, pp. 490-500.
[18]. Mindlin, R. D., “Second Gradient of Strain and Surface Tension in Linear Elasticity,” Int. J. Solids Struct., Vol. 1, 1965, pp. 417-438.
[19]. 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, Vol. 51, 2003, pp. 1477-1508.
[20]. Ansari, R., Faghih Shojaei, M., Mohammadi, V., Gholami, R., and Rouhi, H., , “Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams Using the Most General Form of Strain Gradient Elasticity,” J. Mech., Vol. 30, 2014, pp. 161-172.
[21]. Eringen, A. C., Nonlocal Continuum Field Theories, Springer, 2002, New York.
[22]. Prasanna Kumar, T. J., Narendar, S., Gupta, B. L. V. S., and Gopalakrishnan, S., “Thermal vibration analysis of double-layer graphene embedded in elastic medium based on nonlocal continuum mechanics,” Int. J.Nano Dimens., Vol. 4, 2013, pp. 29-49.
[23]. Ansari, R., Faghih Shojaei, M., Shahabodini, A., and Bazdid-Vahdati, M., “Three-dimensional bending and vibration analysis of functionally graded nanoplates by a novel differential quadrature-based approach,” Compos. Struct., Vol. 131, 2015, pp. 753-764.
[24]. Mindlin, R. D., and Tiersten, H. F., “Effects of Couple-Stresses in Linear Elasticity,” Arch. Ration. Mech. Anal., Vol. 11, 1962, pp. 415-448.
[25]. Koiter, W. T., “Couple Stresses in the Theory of Elasticity,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen (B), Vol. 67, 1964, pp. 17-44.
[26]. Yang, F., Chong, A. C. M., Lam, D. C. C., and Tong, P., “Couple Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., Vol. 39, 2002, pp. 2731-2743.
[27]. Shaat, M., “Effects of Grain Size and Microstructure Rigid Rotations on the Bending Behavior of Nanocrystalline Material Beams,” Int. J. Mech. Sci., Vol. 94-95, 2015, pp. 27-35.
[28]. Shaat, M., and Abdelkefi, A., , “Modeling of Mechanical Resonators Used for Nanocrystalline Materials Characterization and Disease Diagnosis of HIVs,” Microsyst. Technol., 2015, DOI 10.1007/s00542-015-2421-y.
[29]. Shaat, M., and Abdelkefi, A., “Modeling the Material Structure and Couple Stress Effects of Nanocrystalline Silicon Beams for Pull-In and Bio-Mass Sensing Applications,” Int. J. Mech. Sci., Vol. 101-102, 2015, pp. 280-291.
[30]. Zhang, W. X., Wang, T. J., and Chen, X., “Effect of Surface/Interface Stress on the Plastic Deformation of Nanoporous Materials and Nanocomposites,” Int. J. Plast., Vol. 26, 2010, pp. 957–975.
[31]. Gurtin, M. E., and Murdoch, A. I., “A Continuum Theory of Elastic Material Surface,” Arch. Rat. Mech. Anal., Vol. 57, 1975, pp. 291-323.
[32]. Gurtin, M. E., and Murdoch, A. I., “Surface Stress in Solids,” Int. J. Solids Struct., Vol. 14, 1978, pp. 431-440.[33]. Chiu, M. S., and Chen, T., “Bending and Resonance Behavior of Nanowires Based on Timoshenko Beam Theory with High-Order Surface Stress Effects,” Physica E, Vol. 54, 2013, pp. 149-156.
[34]. Shaat, M., Mahmoud, F. F., Gao, X. L., and Faheem, A. F., “Size-Dependent Bending Analysis of Kirchhoff Nano-Plates Based on a Modified Couple-Stress Theory Including Surface Effects,” Int. J. Mech. Sci., Vol. 79, 2014, pp. 31-37.
[35]. Cheng, Ch. -H., and Chen, T., “Size-Dependent Resonance and Buckling Behavior of Nanoplates with High-Order Surface Stress Effects,” Physica E, Vol. 67, 2015, pp. 12-17.
[36]. Ghorbanpour Arani, A., and Roudbari, M. A., “Surface Stress, Initial Stress and Knudsen-Dependent Flow Velocity Effects on the Electro-Thermo Nonlocal Wave Propagation of SWBNNTs,” Physica B, Vol. 452, 2014, pp. 159-165.
[37]. Ansari, R., Ashrafi, M. A., Pourashraf, T., and Sahmani, S., “Vibration and Buckling Characteristics of Functionally Graded Nanoplates Subjected to Thermal Loading Based on Surface Elasticity Theory,” Acta Astronautica, Vol. 109, 2015, pp. 42–51.
[38]. Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., and Sahmani, S., “Surface Stress Effect on the Postbuckling and Free Vibrations of Axisymmetric Circular Mindlin Nanoplates Subject to Various Edge Supports,” Compos. Struct., Vol. 112, 2014, pp. 358-367.
[39]. Malekzadeh, P., and Shojaee, M., “Surface and Nonlocal Effects on the Nonlinear Free Vibration of Non-Uniform Nanobeams,” Compos. Part B, Vol. 52, 2013, pp. 84-92.
[40]. Sharabiani, P. A., and Haeri Yazdi, M. R., “Nonlinear Free Vibrations of Functionally Graded Nanobeams with Surface Effects,” Compos. Part B, Vol. 45, 2013, pp. 581-586.
[41]. Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., and Rouhi, H., “Nonlinear Vibration Analysis of Timoshenko Nanobeams Based on Surface Stress Elasticity Theory,” Eur. J. Mech. A/Solids, Vol. 45, 2014, pp. 143-152.
[42]. Ru, C. Q., “Simple Geometrical Explanation of Gurtin-Murdoch Model of Surface Elasticity with Clarification of Its Related Versions,” Sci. China Phys. Mech. Astron., Vol. 53, 2010, pp. 536–544.
[43]. Ru, C. Q., “A Strain-Consistent Elastic Plate Model with Surface Elasticity,” Continuum Mech. Thermodyn, 2015., DOI 10.1007/s00161-015-0422-9.
[44]. Shaat, M., Eltaher, M. A., Gad, A. I., and Mahmoud, F. F., “Nonlinear Size-Dependent Finite Element Analysis of Functionally Graded Elastic Tiny-Bodies,” Int. J. Mech. Sci., Vol. 77, 2013, pp. 356–364.
[45]. Lu, P., He, L. H., Lee, H. P., and Lu, C., “Thin plate theory including surface effects,” Int. J. Solids Struct., Vol. 43, 2006, pp. 4631–4647.
[46]. Du, C., Li, Y., and Jin, X., “Nonlinear Forced Vibration of Functionally Graded Cylindrical Thin Shells,” Thin-Walled Struct., Vol. 78, 2014, pp. 26–36.
[47]. Loy, C. T., Lam, K. Y., and Reddy, J. N., “Vibration of Functionally Graded Cylindrical Shells,” Int. J. Mech. Sci., Vol. 41, 1999, pp. 309–324.
[48]. Ansari, R., Gholami, R., Norouzzadeh, A., and Darabi, M. A., “Surface Stress Effect on the Vibration and Instability of Nanoscale Pipes Conveying Fluid Based on a Size-Dependent Timoshenko Beam Model,” Acta Mechanica Sinica, 31, 2015, pp. 708-719.
[49]. Zhu, R., Pan, E., Chung, P. W., Cai, X., Liew, K. M., and Buldum, A., “Atomistic Calculation of Elastic Moduli in Strained Silicon,” Semicond. Sci. Technol., Vol. 21, 2006, pp. 906-911.
[50]. Miller, R. E., and Shenoy, V. B., “Size-Dependent Elastic Properties of Nanosized Structural Elements,” Nanotechnology, Vol. 11, 2000, pp. 139-247.