Wave Propagation in Rectangular Nanoplates Based on a New Strain Gradient Elasticity Theory with Considering in-Plane Magnetic Field

Document Type: Research Paper


School of Mechanical Engineering, Shiraz University, Shiraz, Iran


In this paper, on the basis of a new strain gradient elasticity theory, wave propagation in rectangular nanoplates by considering in-plane magnetic field is studied. This strain gradient theory has two gradient parameters and has the ability to compare with the nonlocal elasticity theory. From the best knowledge of author, it is the first time that this theory is used for investigating wave propagation in nanoplates. It is also the first time that magnetic field is considered in modeling the wave propagation in rectangular nanoplates. In this article, an analytical method is adopted to achieve an exact solution for the governing equation. To verify the present methodology, our results are verified with the results published by present authors and other researchers. It is obtained that with the increase of static gradient parameter, the frequencies are increase. It is also shown that the phase velocities increase for the increase of magnetic field.


 [1] K. Kiani, Free vibration of conducting nanoplates exposed to unidirectional in-plane magnetic fields using nonlocal shear deformable plate theories, Physica E: Low-dimensional Systems and Nanostructures, 57(2014) 179–192.
[2] Y. Leng, J. Zheng, J. Qu and X. Li, Thermal stability and magnetic anisotropy of nickel nanoplates, Journal of Materials Science, 44, 17(2009) 4599-4603.
[3] J. Klinovaja, M. J. Schmidt, B. Braunecker and D. Loss (2011). Carbon nanotubes in electric and magnetic fields, Phys. Rev. B, 84(2011) 085452.
[4] J. Kono, R. J. Nicholas and S. Roche, High magnetic field phenomena in carbon nanotubes, Carbon Nanotubes Topics in Applied Physics, 111(2008) 393-421
[5] A. Ghorbanpour Arani, A. Jalilvand and R. Kolahchi, Wave propagation of magnetic nanofluid-conveying double-walled carbon nanotubes in the presence of longitudinal magnetic field, Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems (2013) 1740349913488575.
[6] K.L. Metlov and K.Y. Guslienko, Stability of magnetic vortex in soft magnetic nano-sized circular cylinder, Journal of Magnetism and Magnetic Materials, 242–245(2002) Part 2, 1015–1017
[7] S. Li, H. J. Xie and X. Wang, Dynamic characteristics of multi-walled carbon nanotubes under a transverse magnetic field, Bull. Mater. Sci., 34, 1(2011) 45–52.
[8] B.K. Jang, Y. Sakka and S.K. Woo, Alignment of carbon nanotubes by magnetic fields and aqueous dispersion, J. Phys.: Conf. Ser.( 2009) 156 012005.
[9] T. Murmu, M.A. McCarthy & S. Adhikari, In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach, Composite Structures, 96(2013) 57–63.
[10] D. Yi, T.C. Wang and S. Chen, New strain gradient theory and analysis, Acta Mechanica Solida Sinica, 22 (2009), 45-52.
[11] E.C. Aifantis and H. Askes, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Physical Review B, 80(2009). 195412.
[12] P. Beskou and D. E. Beskos, Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates, Arch Appl Mech, 78(2008): 625–635. DOI 10.1007/s00419-007-0166-5.
[13] T. Murmu, M.A. McCarthy and S. Adhikari, In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach. Composite Structures, 96(2013) 57–63.
[14] Y.Z. Wang, F.M. Li and K. Kishimoto, Scale effects on flexural wave propagation in nanoplate embedded in elastic matrix with initial stress, Appl Phys A, 99(2010) 907–911. DOI 10.1007/s00339-010-5666-4.
[15] M. R. Nami and M. Janghorban, Wave propagation in rectangular nanoplates based on strain gradient theory with one gradient parameter with considering initial stress. Modern Physics Letters B, (2014), DOI: 10.1142/S0217984914500213.
[16] M. R. Nami and M. Janghorban, Static analysis of rectangular nanoplates using Trigonometric shear deformation theory based on nonlocal elasticity theory. Beilstein Journal of Nanotechnology, 4(2013) 968-973.
[17] Nami, M. R. and M. Janghorban, Static analysis of rectangular nanoplates using exponential shear deformation theory based on strain gradient elasticity theory, Iranian Journal of Materials Forming, 1(2014) 1-13.
[18] M.R. Nami, M. Janghorban and M. Damadam, Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory, Aerospace Science and Technology, 41(2015) 7-15.