Free Vibration Analysis of Nanoplates using Differential Transformation Method

Subject Areas : Mechanical Engineering

Sayed Hassan  Nourbakhsh ¹ (Department of Mechanical Engineering, University of Shahrekord, Shahrekord, Iran *Corresponding author)
Mohsen  Botshekanan Dehkordi ² (Department of Mechanical Engineering, University of Shahrekord, Shahrekord, Iran)
Amir  Atrian ³ (Young Researchers and Elite Club, Najafabad Branch, Islamic Azad University, Najafabad, Iran)

Received: 2016-05-19 Accepted : 2016-09-10 Published : 2017-03-01

Keywords: Nano-plate, Semi-analytical, Free vibration, DTM,

Abstract :

In this paper, a free vibration of nano-plates is investigated considering the small scale parameter. The used rectangular nano plate is thin and under different boundary conditions. In order to obtain the natural frequencies of the nano-plates, classical plate theory on the basis of non-local theory is used. The governing equation is solved using a semi-analytical method DTM[1]. The results for free vibration of those plates are compared with the theoretical data published in the literature. Results show that DTM is a powerful, simple, accurate and fast method for solving equations in comparison with other methods. Non-local parameter is very effective in vibration of nano-plates and its influence is different in various boundary conditions. Influence of this parameter in simply supported-clamp boundary condition is higher than other boundary conditions.

References:

[1]     Iijima, S., “Helical microtubules of graphitic carbon”, Nature, Vol. 354, 1991, pp. 56- 58.

[2]     Bockrath, M., Cobden, D. H., Lu, J., Rinzler, A. G., Smalley, R. E., Balents, L., and McEuen, P. L., “Luttinger-liquid behavior in carbon nanotubes”, Nature, Vol. 397, 1997, pp. 598-607.

[3]     Bachtold, A., Hadley, P., Nakanishi, T., and Dekker C., “Logic Circuits with Carbon Nanotube Transistors”, Science, Vol. 294, 2001, pp. 294- 1317.

[4]     Kim, P., Lieber, C. M., “Nanotube Nanotweezers”, Science, Vol.  286, 1999, pp. 2148-2150.

[5]     Wang, C. M., Zhang, Y. Y., Ramesh, S. S., and Kitipornchai, S., “Buckling analysis of micro- and nano-rods/tubes based on non-local Timoshenko beam theory”, Journal of Physics D: Applied Physics, Vol. 39, 2006, pp. 3904–3909.

[6]     Lu, P., Lee, H. P., Lu, C., and Zhang, P. Q., “Dynamic properties of flexural beams using a non-local elasticity model”, Journal of Applied Physics, Vol. 99, 2006, 073510.

[7]     Duan, W. H., Wang, C. M., “Exact solutions for axisymmetric bending of micro/nanoscale circular plates based on non-local plate theory”, Nanotechnology, Vol. 18, 2007, 385704.

[8]     Wang, Q., Wang C. M., “The constitutive relation and small scale parameter of non-local continuum mechanics for modelling carbon nanotubes”, Nanotechnology, Vol. 18, 2007, 075702.

[9]     Gibson, R. F., Ayorinde, O. E., and Wen, Y. F., “Vibration of carbon nanotubes and their composites: a review”, Composites Science and Technology, Vol. 67, 2007, pp. 1–28.

[10]  Ball, P., “Roll up for the revolution”, Nature, Vol 414, 2001, pp. 142–144.

[11]  Baughman, R. H., Zakhidov, A. A., and de Heer, W. A., “Carbon nanotubes-the route toward applications”, Science, Vol. 297, 2002, pp. 787–792.

[12]  Bodily, B. H., Sun, C. T., “Structural and equivalent continuum properties of single-walled carbon nanotubes”, International Journal of Materials and Product Technology, Vol. 18, 2003, pp. 381–397.

[13]  Li, C., Chou, T.W., “A structural mechanics approach for the analysis of carbon nanotubes”, International Journal of Solids and Structures, Vol. 40, 2003, pp. 2487–2499.

[14]  Li, C., Chou, T.W., “Single-walled nanotubes as ultra-high frequency nano mechanical oscillators”, Physical Review B, Vol. 68, 2003, 073405.

[15]  Pradhan, S. C., Phadikar, J. K., “Nonlinear analysis of carbon nanotubes, Proceedings of Fifth International Conference on Smart Materials”, Structures and Systems, Indian Institute of Science, Bangalore, 24–26 July 2008, paperID19.

[16]  Phadikar, J. K., Pradhan, S. C., “Nonlinear finite element model of single wall carbon nanotubes”, Journal of the Institution of Engineers (India), Metallurgy and Materials Engineering Division, Vol. 89, 2008, pp. 3-8.

[17]  Toupin, R. A., “Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis”, Vol. 11, 1962, pp. 385-414.

[18]  Eringen, A. C., Suhubi, E. S., “Nonlinear theory of simple micro-elastic solids”, International Journal of Engineering Science, Vol. 2, 1964, pp. 189–203.

[19]  Fleck, N. A., Hutchinson, J. W., “Strain gradient plasticity,” Advances in Applied Mechanics, Vol. 33, 1997, pp. 295–361.

[20]  Eringen, A. C., “On differential equations of non-local elasticity and solutions of screw dislocation and surface waves”, Journal of Applied Physics, Vol. 54, 1983, 4703.

[21]   Chen, Y., Lee, J. D., and Eskandarian, A., “Atomistic viewpoint of the applicability of microcontinuum theories”, International Journal of Solids and Structures, Vol. 41, 2004, pp. 2085–2097.

[22]  Sun, C. T., Zhang, H. T., “Size-dependent elastic moduli of plate like nanomaterials”, Applied Physics, Vol. 93, 2003, pp. 1212–1218.

[23]  Zhu, R., Pan, E., and Roy, A.K., “Molecular dynamics study of the stress–strain behavior of carbon-nanotube reinforced Epon862 composites”, Materials Science and Engineering: A, Vol. 447, 2007, pp. 51–57.

[24]  Liang, Y. C., Dou, J. H., and Bai, Q. S., “Molecular dynamic simulation study of AFM single-wall carbon nanotube tip–surface interactions”, Science and Engineering: A, Vol. 339, 2007, pp. 206–210.

[25]  Eringen, A. C., “Non-local Continuum Field Theories”, Springer, New York 2002.

[26]  Nowinski, J. L., “On a non-local theory of longitudinal waves in an elastic circular bar”, Acta Mechanica, Vol. 52, 1984, pp. 189–200.

[27]  Zhou, Z. G., Wang, B., “Non-local theory solution of two collinear cracks in the functionally graded materials”, International Journal of Solids and Structures, Vol. 43, 2005, pp. 887–898.

[28]  Reddy, J. N., “Non-local theories for bending, buckling and vibration of beams”, International Journal of Engineering Science, Vol. 45, 2007, pp. 288–307.

[29]  Reddy, J. N., Pang, S. D., “Non-local continuum theories of beams for the analysis of carbon nanotubes”, Applied Physics, Vol. 103, 023511.

[30]  Peddieson. J., Buchanan, G. R., and McNitt, R. P., “Application of non-local continuum models to nanotechnology”, International Journal of Engineering Science, Vol. 41, 2003, pp. 305-312.

[31]  Wang, Q., Liew, K. M., “Application of non-local continuum mechanics to static analysis of micro- and nano-structures”, Physics Letters A, Vol. 363, 2007, pp. 236–42.

[32]  Shen, H., Zhang, C. L., “Torsional buckling and post buckling of double-walled carbon nanotubes by non-local shear deformable shell model”, Composite Structures, Vol. 92, 2010, pp. 1073–84.

[33]  Wang, C. M., Tan, V. B. C., and Zhang, Y. Y., “Timoshenko beam model for vibration analysis of multi-walled carbon nanotubes”, Journal of Sound and Vibration, Vol. 294, 2006, pp. 1060–1072.

[34]  Wang, Q., Varadan, V. K., “Wave characteristics of carbon nanotubes”, International Journal of Solids and Structures, Vol. 43, 2005, pp. 254–265.

[35]  Fu, Y. M., Hong, J. W., and Wang, X. Q., “Analysis of nonlinear vibration for embedded carbon nanotubes”, Journal of Sound and Vibration, Vol. 296, 2006, pp. 746–756.

[36]  Wang, Q., Zhou, G. Y., and Lin, K. C., “Scale effect on wave propagation of double walled carbon nanotubes”, International Journal of Solids and Structures, Vol. 43, 2006, pp. 6071–6084.

[37]  Wang, Q., Varadan, V. K., “Vibration of carbon nanotubes studied using non-local continuum mechanics”, Smart Materials and Structures, Vol. 15, 2006, pp. 659–666.

[38]  Lu, P., Lee, H. P., Lu, C., and Zhang, P. Q., “Application of non-local beam models for carbon nanotubes”, International Journal of Solids and Structures, Vol. 44, 2007, pp. 5289–5300.

[39]  He, X. Q., Kitipornchai, S., and Liew, K. M., “Resonance analysis of multi-layered grapheme sheets used as nanoscale resonators”, Nanotechnology, Vol. 16, 2005, pp. 2086–2091.

[40]  Kitipornchai, S., He, X. Q., and Liew, K. M., “Continuum model for the vibration of multi layered grapheme sheets”, Physical Review B, Vol. 72, 2005, pp. 754- 763.

[41]  Behfar, K., Naghdabadi, R., “Nanoscale vibrational analysis of a multi-layered grapheme sheet embedded in an elastic medium”, Composites Science and Technology, Vol. 7–8, 2005, pp. 1159–1164.

[42]  Reddy, J. N., “Mechanics of Laminated Composite Plates, Theory and Analysis”, Chemical Rubber Company, Boca Raton, FL, 1997.

[43]  Yoon, J., Ru, C. Q., and Mioduchowski, A., “Vibration of embedded multi wall carbon nanotubes”, Composites Science and Technology, Vol. 63, 2003, pp. 1533–1542.

[44]  Ru, C. Q., “Column buckling of multi walled carbon nanotubes with interlayer radial displacements”, Physical Review B, Vol. 62, 2000, pp. 16962–16967.

[45]  Li, C., Chou, T. W., “Elastic moduli of multi-walled carbon nanotubes and the effect of vander Waals forces”, Composites Science and Technology, Vol. 11, 2003, pp. 1517–1524.

[46]  Lu, B. P., Zhang, P. Q., Lee, H. P., Wang, C. M., and Reddy, J. N., “Non-local elastic plate theories”, Proceedings of the Royal Society A, Vol. 463, 2007, pp. 3225–3240.

[47]  Luo, X., Chung, D. D. L., “Vibration damping using flexible graphite”, Carbon, Vol. 38, 2000, pp. 1510–1512.

[48]  Zhang, L., Huang, H., “Young’s moduli of ZnO nanoplates: Abinitio determinations”, Applied Physics Letters, Vol. 89, 2006, id.183111 (3 pages).

[49]  Pradhan, S. C., Phadikar, J. K., “Non-local elasticity theory for vibration of nanoplates”, Sound and Vibration, Vol 325, 2009, pp. 206–223.

[50]  Aghababaei, R., Reddy, J. N., “Non-local third-order shear deformation plate theory with application to bending and vibration of plates”, Sound and Vibration, Vol. 326, 2009, pp. 277–289.

[51]  Pradhan, S. C., Phadikar, J. K., “Small scale effect on vibration of embedded multilayered graphene sheets based on non-local continuum models”, Physics Letters A, Vol. 373, 2009, pp. 1062-1069.

[52]  Pradhan, S. C., Phadikar, J. K., “Non-local elasticity theory for vibration of nanoplates”, Sound and Vibration, Vol. 325, 2009, pp. 206-223.

[53]  Pradhan, S. C., Phadikar, J. K., “Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and non-local elasticity theory”, Structural Engineering and Mechanics, Vol. 3,2009, pp. 193-213.

[54]  Pradhan, S. C., Murmu, T., “Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via non-local continuum mechanics”, Computational Materials Science, Vol. 47, 2009, pp. 268-274.

[55]  Murmu, T., Pradhan, S. C., “Vibration analysis of nanoplates under uniaxial pre stressed conditions via non-local elasticity,” Journal of Applied Physics, Vol. 106, 2009, doi:10.1063/1.3233914.

[56]  [56] Zhou, J. K., “Differential Transformation and Its Applications for Electrical Circuits”, Huazhong Univ. Press, Wuhan, China, 1986

[57]  Chen, C. K., Ho, S. H., “Solving partial differential equations by two-dimensional differential transform method”, Applied Mathematics and Computation, Vol. 106, 1999, pp.171-179.

[58]  Ayaz, F., “On the two-dimensional differential transform method”, Applied Mathematics and Computation, Vol. 143, 2003, pp. 361–374.

[59]  Ayaz, F., “Solutions of the system of differential equations by differential transform method”, Applied Mathematics and Computation, Vol. 147, 2004, pp. 547–567.

[60]  Arikoglu, A., Ozkol, I., “Solution of boundary value problems for integro-differential equations by using differential transform method”, Applied Mathematics and Computation, Vol. 168, 2005, pp. 1145–1158.

[61]  Catal, S., “Analysis of free vibration of beam on elastic soil using differential transform method”, Structural Engineering and Mechanics, Vol. 24, 2006, pp. 51-62.

[62]  Kaya, M. O., “Free vibration analysis of rotating timoshenko beam by differential transform method”, Aircraft Engineering and Aerospace Technology, Vol. 78, 2006, pp. 194-203.

[63]  Ozdemir, O., Kaya, M. O., “Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method”, Journal of Sound and Vibration, Vol. 289, 2006, pp. 413–420.

[64]  Ozdemir, O., Kaya, M. O., “Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method”, Meccanica, Vol. 41, 2006, pp. 661-670.