Closed-form Solution of Dynamic Displacement for SLGS Under Moving the Nanoparticle on Visco-Pasternak Foundation

Document Type: Research Paper

Authors

1 Faculty of Mechanical Engineering, University of Kashan--- Institute of Nanoscience & Nanotechnology, University of Kashan

2 Faculty of Mechanical Engineering, University of Kashan

Abstract

In this paper, forced vibration analysis of a single-layered graphene sheet (SLGS) under moving a nanoparticle is carried out using the non-local elasticity theory of orthotropic plate. The SLGS under moving the nanoparticle is placed in the elastic and viscoelastic foundation which are simulated as a Pasternak and Visco-Pasternak medium, respectively. Movement of the nanoparticle is considered as a linear movement with constant velocity from an edge to another edge of graphene sheet. Using the non-linear Von Kármán strain-displacement relations and Hamilton’s principle, the governing differential equations of motion are derived. The differential equation of motion for all edges simply supported boundary condition is solved by an analytical method and therefore, the dynamic displacement of SLGS is presented as a closed-form solution of that. The influences of medium stiffness (Winkler, Pasternak and damper modulus parameter), nonlocal parameter, aspect ratio, mechanical properties of graphene sheet, time and velocity parameter on dimensionless displacement (dynamic displacement to static displacement of SLGS) are studied. The results indicate that, as the values of stiffness modulus parameter increase, the maximum dynamic displacement of SLGS decreases. Therefore, the results are in good agreement with the previous researches.

Keywords


[1] Sellers K., Mackay C., Bergeson L.L., Clough S.R., Hoyt M., Chen J., Henry K., Hamblen J., 2008, Nanotechnology and the Environment, CRC Press.

[2] Reddy J.N., 2003, Mechanics of laminated composite plates and shells, CRC press LLC, New York.

[3] Timoshenko S., 1959, Theory Of plates and shells, McGraw-Hill, Secound Edition.

[4] Vinson J.R., 2005, Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction, Springer, Netherlands.

[5] Eringen A.C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.

[6] Eringen A.C., 1983, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics 54: 4703–4710.

[7] Eringen A.C., 2002, Nonlocal Continuum Field Theories, Springer-Verlag, New York.

[8] Eringen A.C., Edelen D.G.B., On nonlocal elasticity, International Journa ofl Engineering Science 10: 233–248.

[9] Pradhan S.C., Phadikar J.K., 2009, Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration 325: 206-223.

[10] Murmu T., Adhikari S., 2011, Nonlocal vibration of bonded double-nanoplate-systems, Composites Part B: Engineering 42: 1901-1911.

[11] Murmu T., Adhikari S., 2011, Axial instability of double-nanobeam-systems, Physics Letters A 375: 601-608.

[12] Pradhan S.C., Kumar A., 2010, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity theory and differential quadrature method, Computational Materials Science 50: 239-245.

[13] Ghorbanpour Arani A., Shiravand A., Rahi M., Kolahchi R., 2012, Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation, Physica B: Condensed Matter 407: 4123-4131.

[14] Shu C., 1999, Diffrential Quadrature and its Application in Eengineering, springer.

[15] Zong Z., Zhang Y., 2009, Advanced Diffrential Quadrature Methods, CRC Press, New York.

[16] Kiani K., 2010, Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects, Physica E: Low-dimensional Systems and Nanostructures 42: 2391-2401.

[17] Kiani K., 2011, Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory, Journal of Sound and Vibration 330: 4896-4914.

[18] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations, Physica E: Low-dimensional Systems and Nanostructures 44: 229-248.

[19] Kiani K., 2011, Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part II: Parametric studies, Physica E: Low-dimensional Systems and Nanostructures 44: 249-269.

[20] Şimşek M., 2011, Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle, Computational Materials Science 50: 2112-2123.

[21] Ghorbanpour Arani A., Roudbari M.A., Amir S., 2012, Nonlocal vibration of SWBNNT embedded in bundle of CNTs under a moving nanoparticle, Physica B: Condensed Matter 407: 3646-3653.

[22] Pradhan S.C., Kumar A., 2011, Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method, Composite Structures 93: 774-779.